© 2010 Marco Antonio Milla
STUDY OF COULOMB COLLISIONS AND MAGNETO-IONIC
PROPAGATION EFFECTS ON INCOHERENT SCATTER RADAR
MEASUREMENTS AT JICAMARCA
BY
MARCO ANTONIO MILLA
DISSERTATION
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Electrical and Computer Engineering
in the Graduate College of the
University of Illinois at Urbana-Champaign, 2010
Urbana, Illinois
Doctoral Committee:
Professor Erhan Kudeki, Chair
Professor Steven J. Franke
Professor Jianming Jin
Assistant Professor Jonathan J. Makela
ABSTRACT
In this dissertation, Coulomb collisions and magneto-ionic propagation effects on the
incoherent scatter radar measurements have been studied and analyzed in detail. The
present study aims at modeling radar observations of the equatorial ionosphere carried out
at the Jicamarca Radio Observatory (Lima, Peru) using antenna beams pointed perpen-
dicular to the Earth’s magnetic field B.
A Monte Carlo procedure based on the simulation of charged particle trajectories in
a magnetized plasma (with suppressed collective interactions) was developed to account
for the effects of Coulomb collisions on the shape of the incoherent scatter spectrum.
Statistics of simulated electron and ion trajectories, single-particle ACF’s, and associated
Gordeyev integrals are utilized in the general framework of incoherent scatter spectrum
models [e.g., Kudeki and Milla, 2006] to produce theoretical spectra for different plasma
configurations. Our simulation method effectively extends the procedure of Sulzer and
González [1999] into three dimensions and is valid for all magnetic aspect angles including
the direction perpendicular to B. The 3D trajectories, randomized by Coulomb collisions,
are described by a generalized Langevin equation with velocity-dependent friction and
diffusion coefficients taken from the standard Fokker-Planck collision model of Rosenbluth
et al. [1957]. A statistical analysis of the simulated trajectories shows that the ion motion is
well modeled as a Brownian-motion process with Gaussian displacement distributions (and
constant friction and diffusion coefficients), in which case, an analytical expression for the
single-ion ACF can be obtained [e.g., Woodman, 1967]. However, the simulated electron
motions do not fit a Brownian model because the electron displacement distributions in
the direction parallel to B are sharper than a Gaussian. To account for these effects on our
incoherent scatter spectrum model, a numerical library of electron statistics in an oxygen
ii
plasma (single-electron ACF’s) had to be developed. The library spans a set of densities,
temperatures, and magnetic fields as needed for Jicamarca F-region applications.
The antenna beams used in perpendicular-to-B radar observations at Jicamarca have
angular widths of the order of a degree. Within this range of small magnetic aspect angles,
different modes of magneto-ionic wave propagation are excited. These characteristic modes
vary from linearly polarized in the direction perpendicular to B (Cotton-Mouton regime)
to circularly polarized at aspect angles greater than 0.5◦ (Faraday rotation regime). In
order to model the magneto-ionic propagation effects on incoherent scatter radar measure-
ments, a computer algorithm based on the Appleton-Hartree equation for electromagnetic
wave propagation in a magnetized plasma was developed. Simulation studies show that
magneto-ionic propagation effectively modifies the shapes of the radar beams and does
have an impact on the incoherent scatter radar measurements because the polarization of
the incident and backscattered fields vary as they propagate through the ionosphere.
A soft-target radar equation, which incorporates our collisional incoherent scatter spec-
trum and magneto-ionic propagation models, is formulated to model the radar measure-
ments collected at Jicamarca. Voltages detected by the radar antenna are represented as
the beam-weighted sum of ionospheric backscattered signals corresponding to the range
of magnetic aspect angle directions illuminated by the antenna beam. This integration is
carried out numerically using a finite-element-like integration method that takes advantage
of the slow variation of physical parameters in the direction transverse to the geomagnetic
field. The resultant radar model is utilized in the inversion of ionospheric parameters
in a three-beam radar experiment conducted at Jicamarca. The experiment interleaves
radar observations with perpendicular-to-B and off-perpendicular antenna beams. The
data model matches very closely the different features of the measured data; for instance,
it predicts the enhancement of the measured power in the direction perpendicular-to-B at
ionospheric altitudes where the electron temperature is greater than the ion temperature.
F-region electron density and temperature ratio (Te/Ti) estimates were obtained using
a least-squares inversion algorithm. The inversion results show a good agreement with
ionosonde data, validating our model for incoherent scatter radar measurements.
iii
To Silvanita
iv
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to my adviser, Professor Erhan Kudeki,
for his guidance during the development of this dissertation. I greatly appreciate the
time he spent teaching me the “mysteries” of incoherent scatter. I very much enjoyed our
conversations and discussions on various topics of science; they were always insightful and
challenged my curiosity. I would also like to thank Professors Steven Franke, Jianming
Jin, and Jonathan Makela for agreeing to be members of my doctoral committee and for
their helpful suggestions and comments.
I am very thankful to the staff at the Jicamarca Radio Observatory, who always put
in great effort and show dedication and commitment to their work. Their professional
assistance was crucial during the experiments we conducted at the Observatory. I am very
much indebted to Dr. Jorge Chau for his support during my visits to Jicamarca. I would
also like to thank Dr. Ronald Woodman for our conversations on topics related to this
study.
I would like to thank my family for all their support throughout my years at the
University of Illinois. I have dedicated this dissertation to my wife, Silvanita; without her
love and patience, I would not have finished this work.
I acknowledge the use of the Turing Cluster maintained and operated by the Computa-
tional Science and Engineering Program at the University of Illinois. The Turing Cluster
is a 1536-processor Apple G5 X-serve cluster devoted to high-performance computing in
engineering and science. This work was funded by the NSF grant ATM-0215246 to the
University of Illinois.
v
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Perpendicular-to-B Incoherent Scatter Radar Measurements at Jicamarca . 1
1.2 Studying and Modeling Coulomb Collisions and Magneto-Ionic Propagation
Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
CHAPTER 2 INCOHERENT SCATTER SPECTRUM, LANGEVIN
EQUATION, AND FOKKER-PLANCK COLLISION MODEL . . . . . 10
2.1 Soft-Target Radar Equation and Ambiguity Function . . . . . . . . . . . . . 10
2.2 General Framework of IS Spectral Models . . . . . . . . . . . . . . . . . . . 12
2.3 Single-Particle ACF and Langevin Equation . . . . . . . . . . . . . . . . . . 14
2.4 The Link between Fokker-Planck and Langevin Equations . . . . . . . . . . 17
2.5 Spitzer Friction and Diffusion Coefficients . . . . . . . . . . . . . . . . . . . 18
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
CHAPTER 3 MONTE CARLO COMPUTATION OF PARTICLE
TRAJECTORIES AND SINGLE-PARTICLE ACF’S . . . . . . . . . . . 21
3.1 Computer Simulations of Plasma Particle Trajectories . . . . . . . . . . . . 21
3.2 Estimation of the Single-Particle ACF’s and Gordeyev Integrals . . . . . . . 26
3.3 Statistics of the Ion and Electron Trajectories . . . . . . . . . . . . . . . . . 31
3.3.1 Statistics of the Ion Displacements . . . . . . . . . . . . . . . . . . . 33
3.3.2 Statistics of the Electron Displacements . . . . . . . . . . . . . . . . 37
CHAPTER 4 COLLISIONAL INCOHERENT SCATTER SPECTRUM:
RESULTS AND COMPARISONS . . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Simulated Electron Gordeyev Integrals . . . . . . . . . . . . . . . . . . . . . 43
4.2 Simulated Collisional IS Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 45
CHAPTER 5 MAGNETO-IONIC PROPAGATION EFFECTS ON THE
INCOHERENT SCATTER RADAR MEASUREMENTS . . . . . . . . . 53
5.1 Propagation of Electromagnetic Waves in a Homogeneous Magnetoplasma . 53
5.2 Model for Radiowave Propagation in an Inhomogeneous Ionosphere . . . . . 62
5.3 Soft-Target Radar Equation and Magneto-Ionic Propagation . . . . . . . . . 71
vi
CHAPTER 6 MAGNETO-IONIC EFFECTS ON SIMULATED AND
MEASURED ISR DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.1 The Differential-Phase Radar Configuration . . . . . . . . . . . . . . . . . . 74
6.2 Magneto-Ionic Effects on the Radiation Patterns . . . . . . . . . . . . . . . 79
6.3 Simulated and Measured Power and Cross-Correlation Profiles . . . . . . . . 85
6.4 Magneto-Ionic Effects on Spectrum and Cross-Spectrum Measurements . . . 89
CHAPTER 7 THREE-BEAM EXPERIMENTS AT JICAMARCA . . . 93
7.1 Experiment Description and Antenna Configurations . . . . . . . . . . . . . 94
7.2 Radar Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3 Data Model and Inversion Technique . . . . . . . . . . . . . . . . . . . . . . 107
7.4 Results and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
CHAPTER 8 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . 123
APPENDIX A PARTICLE DYNAMICS DESCRIPTION OF “BGK
COLLISIONS” AS A POISSON PROCESS . . . . . . . . . . . . . . . . . . 127
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A.2 Derivation of the BGK Gordeyev Integral following a Particle Dynamics
Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
APPENDIX B FRICTION AND DIFFUSION COEFFICIENTS FOR
“TEST PARTICLES” IN NON-ISOTHERMAL PLASMAS . . . . . . . 137
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.2 Steady-State Solution of the Fokker-Planck Equation with Spitzer Collision
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.3 Modified Spitzer Friction Coefficient . . . . . . . . . . . . . . . . . . . . . . 145
B.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
AUTHOR’S BIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
vii
LIST OF TABLES
Table Page
3.1 Typical plasma parameters for an equatorial F-region ionosphere. . . . . . . 22
3.2 Representative physical parameters of a plasma. Definitions and sample
values are provided. The values were computed for an O+ plasma with
parameters given in Table 3.1. Note that the plasma and gyro frequency
formulas are given for angular frequencies, but the sample values are given
in Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Plasma parameters spanned by the library of single-electron ACF’s. . . . . 30
7.1 Radar operating parameters considered in the 3Ba and 3Bb experiments. . . 99
viii
LIST OF FIGURES
Figure Page
3.1 Sample trajectory of an electron moving in an O+ plasma with density Ne =
1012 m−3 and temperatures Te = Ti = 1000 K. The presence of a uniform
magnetic field B parallel to the z-axis was considered in the simulation. The
intensity of the field was assumed to be 25µT. On the top, the left panel
shows the particle trajectory in 3D space, while the panel on the right is the
projection of the trajectory on the xy-plane (i.e., the plane perpendicular
to B). On the bottom, the displacement of the electron in the direction
parallel to B is displayed as a function of time. The trajectory was sampled
every ∆t = 0.1µs. A total of 4×103 samples are displayed corresponding to
a time interval of 400µs. In all three panels, the red dots show the starting
location of the particle (the origin). In addition, the red curve depicts the
first 1.4µs of the simulated trajectory (about one gyroperiod). . . . . . . . . 25
3.2 Probability distributions of (a) ion and (b) electron velocities resulting from
the simulations. In each plot, the pdf’s of the velocity components (dis-
played in colors) are compared to a Gaussian distribution (black line). No-
tice that the velocity axes are normalized by the thermal speeds of the
corresponding particle species. The plasma parameters considered in the
simulations are given in Table 3.1. . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Probability distributions of the displacements of a simulated ion in the di-
rections perpendicular (top panels) and parallel (bottom panels) to the mag-
netic field. On the left, the displacement pdf’s are displayed as functions
of time delay τ . On the right, sample cuts of the pdf’s are compared to
a Gaussian distribution. Note that all distributions at all time delays are
normalized to unit variance. The displacement axis of each distribution at
every delay τ is scaled with the corresponding standard deviation of the
simulated displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Variances of the displacements of a simulated ion in the directions perpen-
dicular (top panels) and parallel (bottom panels) to the magnetic field. The
simulation results (color lines) are displayed for two time intervals: 5 ms
(left panels) and 500 ms (right panels). The dashed lines correspond to the
fits of the results to the theoretical expressions for 〈∆r2⊥〉 and 〈∆r2‖〉 of the
Brownian-motion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Simulated single-ion ACF’s at different magnetic aspect angles α for two
radar Bragg wavelengths: (a) λB = 3 m and (b) λB = 0.3 m. The simulation
results (color lines) are compared to theoretical ACF’s computed using ex-
pression (3.22) of the Brownian-motion approximation (dashed lines). Note
that there is effectively no dependence on aspect angle α. . . . . . . . . . . 36
ix
3.6 Same as Figure 3.3 but for the case of a simulated electron. All distributions
at all time delays are normalized to unit variance. Note that the distribu-
tions of the displacements parallel to B become narrower than a Gaussian
distribution in less than a millisecond. . . . . . . . . . . . . . . . . . . . . . 38
3.7 Same as Figure 3.4 but for the case of a simulated electron. In this case,
the results are displayed for time intervals of 0.02 ms (left panels) and 10 ms
(right panels). Note the different scales for the displacement variances. . . . 39
3.8 Simulated electron ACF’s for λB = 3 m at different magnetic aspect angles:
(a) α = 0◦, (b) α = 0.01◦, (c) α = 0.05◦, (d) α = 0.1◦, (e) α = 0.5◦, and (f)
α = 1◦. Note the different time scales used in each plot. . . . . . . . . . . . 40
4.1 Simulated electron Gordeyev integrals as functions of Doppler frequency
and magnetic aspect angle for two radar Bragg wavelengths: (a) λB = 3 m
and (b) λB = 0.3 m. An O+ plasma with electron density Ne = 1012 m−3,
temperatures Te = Ti = 1000 K, and magnetic field Bo = 25µT is considered. 44
4.2 Simulated incoherent scatter spectra as a function of magnetic aspect angle
and Doppler frequency for λB = 3 m (e.g., Jicamarca radar Bragg wave-
length). An O+ plasma with physical parameters given in Table 3.1 is
considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Sample cuts of the simulated spectra of Figure 4.2 taken at different mag-
netic aspect angles: (a) α = 0◦, (b) α = 0.01◦, (c) α = 0.05◦, (d) α = 0.1◦,
(e) α = 0.5◦, and (f) α = 1◦. Our simulation results (blue lines) are com-
pared to IS spectra computed using the Brownian-motion model (green
lines), the collisionless electron model (red lines), the simulation results of
Sulzer and González [1999] (light-blue lines), and the aspect angle depen-
dent collision frequency model of Woodman [2004] (magenta lines). Note
the different frequency scales used in each plot. . . . . . . . . . . . . . . . . 47
4.4 Electron density dependence of the simulated IS spectra for λB = 3 m at
aspect angles α = 0◦ (left panel), α = 0.1◦ (central panel), and α = 1◦
(right panel). An O+ plasma in thermal equilibrium with Te = Ti = 1000 K
and Bo = 25µT is considered. Note the different frequency scales used in
each plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 Te/Ti dependence of the simulated IS spectra for λ = 3 m at aspect angles
α = 0◦ (left panels), α = 0.02◦ (central panels), and α = 0.1◦ (right pan-
els). The same spectra are plotted on the top and bottom panels; however,
on the bottom, every spectrum has been normalized to its peak value. An
O+ plasma with Ne = 1012 m−3, Ti = 1000 K, and Bo = 25µT is consid-
ered. Each curve corresponds to different values of electron temperatures Te
(1000 K, 1500 K, 2000 K, 2500 K, and 3000 K). Note the different frequency
scales used in each column. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6 Electron scattering efficiency factor η(k) resulting from the frequency inte-
gration of the simulated incoherent scatter spectra as a function of electron-
to-ion temperature ratio Te/Ti and magnetic aspect angle α. An O+ plasma
with Ne = 1012 m−3, Ti = 1000 K, and Bo = 25µT is considered. . . . . . . 52
x
5.1 Coordinate system used for analyzing wave propagation in a magnetized
plasma. The magnetic field Bo is on the yz-plane at an angle θ from the
propagation vector k which is parallel to the zˆ-direction. The wave field
E has three mutually orthogonal components Ek, Eθ, and Eφ in directions
kˆ = zˆ, θˆ = −yˆ, and φˆ = xˆ, respectively. . . . . . . . . . . . . . . . . . . . 56
5.2 Geometry of wave propagation in an inhomogeneous magnetized ionosphere. 63
5.3 Electron density and Te/Ti profiles as functions of height. . . . . . . . . . . 66
5.4 Polarization coefficients for the mean square voltages detected by a pair of
orthogonal linearly polarized antennas placed at Jicamarca. The antennas
have very narrow beams and scan the ionosphere from north to south prob-
ing different magnetic aspect angle directions. Note that, for most pointing
directions, the polarization of the detected fields rotates (Faraday rotation
effect), except in the direction where the beam is pointed perpendicular to
B, in which case, the type of polarization changes from linear to circular
(Cotton-Mouton effect). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.5 Co-polarized (left panel), cross-polarized (middle panel), and total (right
panel) backscattered power detected by a pair of orthogonal linearly po-
larized antennas (see caption of Figure 5.4). Power levels are displayed in
units of electron density. In each plot, the dashed white lines indicate the
directions half a degree away from perpendicular to B, while the continuous
lines correspond to the directions one degree off. . . . . . . . . . . . . . . . 68
5.6 An example of the ALTAIR scan measurements collected during the 2004
EQUIS 2 campaign. In colors, levels of backscattered power calibrated to
match electron densities in the probed region are displayed. . . . . . . . . . 70
6.1 Antenna diagram of the differential-phase technique developed for the Ji-
camarca incoherent scatter radar system. In this mode of operation, the
Jicamarca antenna is phased to point perpendicular to B. In transmission,
the southeast polarized down antenna quarters are excited. In reception,
backscattered fields detected by the up and down polarizations of the north
and south quarters are combined to synthesize meridional and zonal polar-
ized field components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 Two-way antenna radiation pattern of the differential-phase radar configu-
ration (in free-space). In the left panel, the radiation pattern (in dB units
relative to its peak value) is displayed as a function of direction cosines with
respect to the x- and y-axes of the Jicamarca frame of reference. In the
right panel, the pattern is displayed as a contour plot in a rotated coordi-
nate system in which the x- and y-axes are now aligned with the east and
north directions (the Jicamarca frame of reference is rotated 51◦ positive
azimuth). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3 Meridional and zonal effective radiation patterns for r = 0, 100, 150, 200 km. 82
6.4 Same as Figure 6.3 but for r = 250, 300, 350, 400 km. . . . . . . . . . . . . 83
6.5 Same as Figure 6.3 but for r = 450, 500, 550, 600 km. . . . . . . . . . . . . 84
6.6 Simulated incoherent scatter signal power and coherence (magnitude and
phase) profiles for the differential-phase radar configuration of June 8, 2004. 87
xi
6.7 Measured incoherent scatter signal power and coherence (magnitude and
phase) profiles that correspond to the differential-phase experiment con-
ducted on June 8, 2004. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.8 Incoherent scatter Doppler spectra (left) and north-south baseline coher-
ence spectra (right) measured with the differential-phase radar technique at
Jicamarca on June 8, 2004. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.1 Antenna configurations of the 3Ba (left panel) and 3Bb (right panel) exper-
iments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2 Two-way radiation patterns of the west (top row) and east (bottom row)
beams used in the 3Ba (left) and 3Bb (right) experiments. The patterns are
displayed as functions of direction cosines in a coordinate system in which
x- and y-axes are aligned with the east and north directions at Jicamarca.
Note that in the 3Ba mode, the patterns of the west beams are identical,
but in the 3Bb mode, the west beams have different shapes. The same can
be observed in the case of the east beams. . . . . . . . . . . . . . . . . . . . 97
7.3 Two-way radiation patterns of the south beams used in the 3Ba (left) and
3Bb (right) experiments. Note that, in the north-south direction, the 3Ba
south beam is narrower than the 3Bb beam. . . . . . . . . . . . . . . . . . . 98
7.4 Sample self- and cross-spectra measured in the 3Ba experiment of June 19,
2008 (top rows), and in the 3Bb experiment of January 9, 2009 (bottom
rows). The sample data were obtained after five minutes of integration.
The self- and cross-spectra are plotted as functions of Doppler velocity and
radar range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.5 Range vs. time plots of the signal power and cross-correlation data measured
in the 3Ba experiment of June 19, 2008. On the left, the power data collected
by each of the radar beams are displayed in linear scale. On the right, the
magnitudes of the cross-correlation data are also plotted in linear scale,
while the phase data are plotted in degrees. . . . . . . . . . . . . . . . . . . 103
7.6 Same plots as described in Figure 7.5, but for the signal power and cross-
correlation data measured in the 3Bb experiment of January 9, 2009. . . . . 106
7.7 Sample plots of the results obtained from the inversion of the 3Ba and 3Bb
data (top and bottom panels, respectively). The estimated Ne and Te/Ti
profiles are plotted in the left column. In the other columns, the power
and cross-correlation measurements (dotted points) are compared with the
modeled profiles (continuous lines). . . . . . . . . . . . . . . . . . . . . . . . 117
7.8 Range vs. time plots of the electron density and Te/Ti estimates obtained
from the inversion of incoherent scatter signal power and cross-correlation
data collected in the 3Ba experiment of June 19, 2008 (left column), and
also in the 3Bb experiment of January 9, 2009 (right column). . . . . . . . . 118
7.9 Magnitudes and phases of the radar calibration constants obtained as part
of the inversion of the 3Ba and 3Bb radar data. . . . . . . . . . . . . . . . 120
7.10 Goodness-of-fit values resulting from the inversion of the 3Ba and 3Bb radar
data. The blue curves correspond to the values obtained by fitting only for
the electron density Ne and assuming that Te/Ti = 1 at all altitudes. The
green curves correspond to the values obtained by fitting for both parame-
ters, Ne and Te/Ti. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
xii
7.11 Comparison of the peak values of plasma frequencies (foF2) measured with
the 3Ba and 3Bb radar techniques (blue lines) and with an ionosonde system
also located at Jicamarca (red dots). . . . . . . . . . . . . . . . . . . . . . . 122
7.12 Comparison of the altitudes corresponding to the peak values of plasma
frequency presented in Figure 7.11. . . . . . . . . . . . . . . . . . . . . . . . 122
B.1 Relative difference between the effective electron temperature T effe computed
using (B.31) and the value of Te considered in the calculation of the Spitzer
coefficients for an oxygen plasma. The difference is displayed as a function
of Te/Ti. Each curve corresponds to different constant values of Te. . . . . . 144
B.2 Same as Figure B.1 but for the case of the effective ion temperature T effi . . . 145
xiii
CHAPTER 1
INTRODUCTION
1.1 Perpendicular-to-B Incoherent Scatter Radar
Measurements at Jicamarca
The incoherent scatter radars are the most powerful ground-based facilities built for
the study of the Earth’s ionosphere. Using these instruments, ionospheric plasma pa-
rameters such as densities and temperatures can be measured remotely as functions of
altitude. Only a small group of these radars is distributed around the World, and among
them, the Jicamarca radar is special by its location. Situated near Lima, Peru, the 50
MHz Jicamarca radar is the only incoherent scatter (IS) facility that is routinely used
for probing the equatorial ionosphere. Since it began operating in 1961, the Jicamarca
measurements have become the primary resource for scientists to conduct studies of the
equatorial electrodynamics.
One of the most important applications of the Jicamarca radar is the measurement
of ionospheric plasma drifts. For this purpose, the Jicamarca antenna array is phased to
point perpendicular to the Earth’s magnetic field B. These measurements are possible
on a pulse-to-pulse basis (as opposed to ACF measurements conducted with multi-pulse
techniques) because the bandwidth of the incoherent scatter radar (ISR) spectrum narrows
considerably when the radar beam is pointed perpendicular to B. The original technique of
Woodman and Hagfors [1969] has been recently improved by Kudeki et al. [1999]. Very ac-
curate plasma drifts determined from the Doppler shifts of the sharp peaks of the measured
spectra are obtained using the new technique. The procedure developed by Kudeki et al.
[1999] consists in fitting the measurements to a simplified ISR spectrum formula. This
1
equation results from the beam-weighted integration of theoretical spectra corresponding
to the range of magnetic aspect angles illuminated by the radar beam. The magnetic
aspect angle is the complement of the angle between the scattered field wavevector k and
the ambient magnetic field B. Kudeki et al. [1999] developed their model based on the
well-established collisionless IS theory [Fejer , 1961; Farley et al., 1961, and others] and
the collisional IS theory of Woodman [1967]. Analysis and comparisons of these theories
led Kudeki and his team to assume that electron Coulomb collisions could be neglected in
the formulation of the simplified model (though ion Coulomb collisions should still be con-
sidered). The simplified model predicted that the frequency bandwidth of the measured
spectrum would be proportional to the plasma temperature. Because of this, the initial
expectation of Kudeki and his team was to measure ionospheric temperatures (as well as
drifts) by applying their spectral estimation technique. To their surprise, however, the
temperature estimates they obtained were about half of what is expected for an F-region
ionosphere. Bhattacharyya [1998] carried out additional theoretical and experimental work
in an attempt to attribute the low temperature estimates to a modification of the spectrum
caused by magneto-ionic propagation effects. Despite these efforts, the temperatures ob-
tained were still underestimated. All this work gave Kudeki and his team enough evidence
to conclude that the actual measured ISR spectrum was in fact narrower than what the
theories available at that time were able to predict, and in consequence, a revision of the
IS theory for modes propagating perpendicular to B was needed.
The very same year, Sulzer and González [1999] revised the IS theory and showed that
electron Coulomb collisions do have a significant impact on determining the shape of the
IS spectrum at small magnetic aspect angles. In the literature, the effects of Coulomb
collisions on incoherent scattering have been studied before, but using simplified models.
Based on a qualitative analysis, Farley [1964] recognized that ion Coulomb collisions would
be responsible for the lack of O+ gyroresonance signatures on IS observations at F-region
heights. This analysis was later verified by the collisional IS theory of Woodman [1967],
theory that was based on the simplified Fokker-Planck collision model of Dougherty [1964].
These two studies concluded that electron Coulomb collisions would have very small effects
2
on IS observations at small magnetic aspect angles. However, Sulzer and González [1999]
showed that this is not the case. Based on the more complex Fokker-Planck Coulomb
collision model of Rosenbluth et al. [1957], Sulzer and González carried out intensive com-
putations to accurately model the effect of electron collision on the IS spectra for a set of
plausible ionospheric plasma parameters. Their results verified that the collisional spec-
trum is indeed narrower than what the collisionless theory predicts, a result that is in
better agreement with Jicamarca observations. Furthermore, they found that the effect
of electron collisions extends up to relatively “large” magnetic aspect angles (as large as
3 or 4 degrees). This important result was the key to solve another puzzle of Jicamarca
IS observations (using antenna beams pointed off-perpendicular to B), which was the sys-
tematic measurement of electron temperatures lower than ion temperatures at nighttime
when thermal equilibrium is expected [Aponte et al., 2001].
The collisional incoherent scatter spectrum calculations of Sulzer and González are
based on the statistics of simulated electron trajectories randomized by Coulomb colli-
sions. In order to simplify their computations, Sulzer and González assumed that the
geomagnetic field is sufficiently strong to keep the electrons moving along the same ge-
omagnetic field lines, such that diffusion across the field could be neglected. Under this
assumption, the problem can be reduced to the simulation of the electron motion in a
single dimension (i.e., in the direction parallel to B). This “strong” magnetic field approx-
imation is valid for Jicamarca IS observations at aspect angles larger than 0.1◦ (limit of
Sulzer and González simulations). However, for radar observations perpendicular to B, an
IS spectrum model valid for zero magnetic aspect angle is required. Given the small range
of magnetic aspect angles that remains to be modeled, one may think that extrapolation
of Sulzer and González spectrum would be just good enough; however, it is in this range
of aspect angles (between 0◦ and 0.1◦) that the spectrum becomes “super” sharp with its
bandwidth being reduced by a factor larger than ten. This is the cause of the sharp spec-
trum measured at Jicamarca with the antenna pointed perpendicular to B. Therefore, in
order to quantify all the features of the measured IS spectrum, an accurate model for all
magnetic aspect angles is needed.
3
More recently, Kudeki et al. [2003] developed a new incoherent scatter radar technique
for the measurement of ionospheric plasma densities using antenna beams pointed perpen-
dicular to B. An electromagnetic wave propagating through the ionosphere experiences
changes in its polarization caused by the presence of the Earth’s magnetic field. These
magneto-ionic propagation effects are significant for detection at 50 MHz [Farley , 1969].
The technique consists of transmitting radiowave pulses using a single linearly polarized
antenna. The waves excite both the ordinary and extraordinary modes of propagation in a
magneto-ionic medium. In reception, two orthogonal linearly polarized antennas are used
to collect the backscattered signals. The differential phase of the signals detected by both
orthogonal antennas depends on the difference between the refractive indices of the modes
of propagation. The model of these measurements was developed based on the Appleton-
Hartree equation for electromagnetic wave propagation in a magnetized plasma. Inversion
of the radar measurements yields electron density estimates as a function of height. Addi-
tional details of the experiment, the data model, and the estimation technique are given
in Feng et al. [2003] and Feng et al. [2004].
Depending on the direction of propagation, radiowaves traveling through the iono-
sphere experience different types of magneto-ionic effects [Yeh et al., 1999a, b]. If the
magnetic field is perpendicular to the propagation direction, changes in the polarization
are described by the Cotton-Mouton effect, and the characteristic modes of propagation
are linearly polarized. At relatively “large” magnetic aspect angles, the Faraday effect takes
place and the characteristic modes become circularly polarized. In the case of radiowave
propagation at 50 MHz, the transition between these two regimes happens at a critical
angle of approximately 0.5◦ measured with respect to the perpendicular-to-B direction. As
the Jicamarca antenna beam has a finite angular width, different modes of propagation for
small magnetic aspect angles are excited, modes that in general are elliptically polarized.
The resultant magneto-ionic effects on the differential-phase measurements were modeled
by Feng et al. [2003]. The model gets further complicated as the incoherent scatter radar
cross section (RCS) is also dependent on magnetic aspect angle. When the plasma is
not in thermal equilibrium, i.e., when the electron temperature is greater than the ion
4
temperature, the incoherent scatter RCS becomes larger at small magnetic aspect angles
[Farley , 1966]. This feature of the IS signal was not fully modeled by Feng et al. [2003].
In their approach, they considered a heuristic expression for the RCS since the IS theories
available at that time were not appropriate for this type of radar measurements. The
differential phase technique can be improved if an accurate incoherent scatter model for
modes propagating perpendicular to B is applied.
1.2 Studying and Modeling Coulomb Collisions and
Magneto-Ionic Propagation Effects
In this dissertation, Coulomb collisions and magneto-ionic propagation effects on the
incoherent scatter radar measurements have been studied and analyzed in detail. This
study aims at modeling radar observations of the equatorial ionosphere carried out at the
Jicamarca Radio Observatory (Lima, Peru) using antenna beams pointed perpendicular
to the Earth’s magnetic field B. The result of our studies is the development of a new
incoherent scatter spectrum model valid for all magnetic aspect angles (including the
direction perpendicular to B). This work is a continuation of the efforts of Bhattacharyya
[1998] and Feng [2002] to improve the capabilities of the incoherent scatter Jicamarca
radar.
Our investigations were divided in two stages:
1. The study of the effects of Coulomb collisions in determining the shape of the inco-
herent scatter spectrum.
2. The modeling of the incoherent scatter radar spectrum considering magneto-ionic
propagation effects.
The first stage of this research involved the development of a procedure to model the
incoherent scatter spectrum including the effects of Coulomb collisions. The proposed
method is based on the stochastic simulation of the trajectories of charged particles em-
bedded in a plasma. This procedure is effectively an extension of the method of Sulzer
and González [1999] into three dimensions. In the simulations, particle trajectories are
5
randomized by Coulomb interactions as described by the Fokker-Planck collision model of
Rosenbluth et al. [1957]. A statistical analysis of the simulated trajectories shows that the
electron statistics are poorly approximated by simplified collision models. The application
of this procedure in routine computations of collisional IS spectra is not practical because
of the large amount of computations that it requires. In order to circumvent this difficulty,
a numerical library of the statistics of the electron trajectories has been developed. The
library spans a set of densities, temperatures, and magnetic fields as needed for Jicamarca
F-region applications.
The second stage of this project corresponds to the modeling of the incoherent scatter
beam-weighted radar spectrum measured with antennas pointed perpendicular to B. At 50
MHz, the polarizations of radiowaves propagating through the ionosphere are modified due
to magneto-ionic effects. Therefore, the consideration of these effects in IS spectrum models
is necessary. Further complications arise because, within the range of small magnetic aspect
angles illuminated by the antenna beam, a variety of characteristic modes of propagation
are excited. These effects result in a distortion of the shape of the propagating wavefronts
and, thus, in a modification of the measured IS signals. On the other hand, the rapid
variation of the IS spectrum at small magnetic aspect angles imposes an additional level
of complexity to the model. Because of these features of the IS signal, the shape of the
radar beam has to be considered in the modeling, as it effectively determines the shape of
the measured spectrum. A model of the beam-weighted incoherent scatter radar spectrum
that takes into account magneto-ionic propagation and collisional effects was developed.
A comparison of measured and simulated incoherent scatter data validates to some extent
the accuracy of our model. The ultimate application of this model will be the estimation
of ionospheric physical parameters with Jicamarca antenna beams pointed perpendicular
to the geomagnetic field.
1.3 Dissertation Outline
The present dissertation is organized as follows. The general framework for incoherent
scatter theories is introduced in Chapter 2. The framework formulates the spectrum of
6
incoherently scattered radiowaves in terms of the so-called Gordeyev integrals [Kudeki and
Milla, 2010]. As discussed in Section 2.2, the Gordeyev integral is defined as the one-sided
Fourier transform of the characteristic function of the displacements of an electron or an
ion in the absence of collective interactions. As Coulomb collisions modify the particle
trajectories, collisions do have an impact on the shapes of these characteristic functions,
also termed single-particle ACF’s. It is through the modeling of the particle trajectories
that the effects of Coulomb collisions are introduced in our incoherent scatter spectrum
model.
The motion of a charged particle embedded in a plasma is modeled by a generalized
version of the Langevin equation. In this equation, the effect of Coulomb collisions is
described by the action of a friction force and random diffusion forces. The expected
values of these forces are given by the Fokker-Planck friction and diffusion coefficients
of Rosenbluth et al. [1957]. The general relationship between the Fokker-Planck and the
generalized Langevin equations is also discussed in Chapter 2.
A Monte Carlo procedure was developed to simulate the particle trajectories in a
magnetized plasma. The approach effectively extends the procedure of Sulzer and González
[1999] into three dimensions by letting the charged particles diffuse across the magnetic field
lines. Since the Langevin equation models the particle trajectories as Markov processes,
discrete time-update equations for the velocity and the position of a test particle can be
readily obtained. The computer program for the simulation of the particle trajectories
was developed based on these equations. The program is outlined in Chapter 3. Both
electron and ion trajectories were simulated. The use of these trajectories for the numerical
estimation of the single-particle ACF’s and the calculation of the corresponding Gordeyev
integrals are described in Section 3.2.
The statistics of our computed trajectories deserve a careful analysis. In Section 3.3,
velocity and displacement distributions resultant from the simulations are presented. We
have verified that the probabilistic distributions of the ion displacements in the directions
parallel and perpendicular to B can be very well approximated by independent Gaussian
distributions. In consequence, the ion trajectory can be described by a Brownian motion
7
process with constant friction and diffusion coefficients. This is convenient because, in this
particular case, an analytical expression for the Gordeyev integral can be derived [Wood-
man, 1967; Holod et al., 2005, and others]. On the other hand, however, electron statistics
show a different behavior. While in the direction perpendicular to B the displacement
distribution can still be approximated by a Gaussian function, in the parallel direction
the distribution is different. For this reason, the electron motion cannot be modeled as a
simple Brownian motion process. Moreover, no analytical expression for the corresponding
Gordeyev integral was found; therefore, a numerical library of electron Gordeyev integrals
for an oxygen plasma had to be developed. The library spans a set of densities, tempera-
tures, and magnetic fields as needed for Jicamarca F-region applications.
In Chapter 4, the computed electron and ion Gordeyev integrals are utilized in the
general framework of IS models to produce theoretical IS spectra for different magnetic
aspect angles, including the direction perpendicular to B. The analysis of our results shows
that the modeled spectrum saturates at very small magnetic aspect angles (< 0.01◦).
Additionally, we have also verified that the width of the perpendicular-to-B spectrum
is proportional to the electron Coulomb collision frequency. As the value of this collision
frequency is inversely proportional to the electron temperature, we found that the modeled
spectrum narrows at high temperatures and widens at low temperatures; this result is
discussed further in Section 4.2. Moreover, the dependence on temperature of our simulated
spectra for different magnetic aspect angles is analyzed at the end of this section.
The second stage of our investigations corresponds to the study of the magneto-ionic
propagation effects on the IS radar signals. As mentioned above, a radiowave propagat-
ing through the ionosphere experiences changes in its polarization due to the presence
of the Earth’s magnetic field. To account for these magneto-ionic propagation effects
on perpendicular-to-B radar observations at Jicamarca, a model for incoherent scatter
spectrum and cross-spectrum measurements is formulated in Chapter 5. In this model,
the voltages detected by the radar antennas are represented as beam-weighted sums of
the ionospheric backscattered signals arriving from the range of magnetic aspect angle
directions illuminated by the antenna beams. The model is effectively an extension of
8
the Appleton-Hartree solution for a homogeneous magneto-plasma to the case of an inho-
mogeneous ionosphere. Simulation studies based on our model show that magneto-ionic
propagation effectively modifies the shapes of the radar beams and does have an impact
on the IS radar measurements as described in further detail in Chapter 6.
In Chapter 7, the incoherent scatter data model of Chapter 5 is utilized in the inversion
of ionospheric physical parameters from radar data collected in three-beam radar exper-
iments conducted at Jicamarca. These experiments interleave radar observations with
perpendicular-to-B and off-perpendicular antenna beams. The data model matches very
closely the different features of the measured data, e.g., it predicts the enhancement of
the measured power in the direction perpendicular-to-B at ionospheric altitudes where the
electron temperature is greater than the ion temperature. F-region electron density and
temperature ratio (Te/Ti) estimates were obtained by applying a least-squares inversion
algorithm. The estimated results show a good agreement with ionosonde data, validating
our model for incoherent scatter radar measurements.
This dissertation is concluded in Chapter 8 with a discussion of the main results and
future extensions of our studies. The ultimate application of the incoherent scatter model
developed in this dissertation will be the estimation of ionospheric temperatures, simulta-
neously with electron densities and plasma drifts, using antenna beams pointed perpen-
dicular to the geomagnetic field at Jicamarca. This will be the fulfillment of objectives set
forth more than a decade ago, when spectral temperature estimations were first attempted
[Bhattacharyya, 1998].
9
CHAPTER 2
INCOHERENT SCATTER SPECTRUM,
LANGEVIN EQUATION, AND
FOKKER-PLANCK COLLISION MODEL
The problem of modeling and calculating the IS spectrum was first addressed by
Dougherty and Farley [1960], Fejer [1960], Salpeter [1960], Hagfors [1961], and others.
Different methods of derivation were followed but identical results were obtained in equiv-
alent physical settings. The fundamental principles that link these different approaches
constitute a framework for incoherent scatter spectral theories. A detailed description of
this general framework is presented by Kudeki and Milla [2010]. Aspects of the theory
relevant to our work are discussed in the first two sections of the present chapter.
The framework offers the alternative of formulating the IS spectrum problem in terms
of the statistics of single-particle trajectories. Then, to include the effects of Coulomb
collisions into the problem, a stochastic dynamical equation governing the motion of a
charged particle in a collisional plasma is introduced in Section 2.3. This equation, which
has the form of a generalized Langevin equation, is linked to the Fokker-Planck collision
model of Rosenbluth et al. [1957] in Section 2.4. The components of this collision model,
friction and diffusion coefficients, are then utilized to complete the formulation of the
Langevin equation. These coefficients are specified in Section 2.5 for a Maxwellian plasma.
2.1 Soft-Target Radar Equation and Ambiguity Function
In soft-target Bragg scattering, each infinitesimal volume dv = r2drdΩ of a radar
field-of-view behaves like a hard-target with a radar cross section dv
´
dω
2pi σv(k, ω), where
10
σv(k, ω), by definition, is the soft-target RCS per unit volume per unit Doppler frequency
ω/2pi, and k = −2korˆ denotes the relevant Bragg vector for a radar with a carrier wavenum-
ber ko and associated wavelength λ. As a consequence, the hard-target radar equation
generalizes for soft-target Bragg-scatter radars as [Milla and Kudeki , 2006]
Pr(t) =
ˆ
dr dΩ
dω
2pi
G(rˆ)A(rˆ)
(4pir)2
Pt(t− 2r
c
)σv(k, ω), (2.1)
where Pt(t) ∝ |f(t)|2 is the transmitted power carried by a pulse waveform f(t), G(rˆ)
and A(rˆ) are the gain and effective area of the radar antenna as a function of radial unit
vector rˆ, r is the radar range, and c the speed of light. Furthermore, equation (2.1), which
assumes an open radar bandwidth, can be recast for a match-filtered receiver output as
Pr(t)
EtKs
=
ˆ
dr dΩ
dω
2pi
G2(rˆ)A(rˆ)
4pi
|χ(t− 2rc , ω)|2
4pir2
σv(k, ω), (2.2)
where Et is the total energy of the transmitted radar pulse, Ks denotes a system calibration
constant including loss factors ignored in (2.1), and
χ(τ, ω) ≡ 1
T
ˆ
dt ejωt f(t)f∗(t− τ) (2.3)
has a magnitude known as the radar ambiguity function [e.g., Levanon, 1988]. In (2.3) the
normalization constant T denotes the duration of the pulse waveform f(t) and (2.3) itself
is effectively the normalized cross-correlation of the Doppler-shifted pulse f(t) ejωt with a
delayed pulse echo proportional to f(t − τ) that would be expected from a point target
located at a radar range R ≡ c τ/2. Hence, variable τ in (2.3) represents not only a time
delay but also a corresponding radar range R.
Let us denote by Sr(ω) the power spectrum of the signal detected by the radar receiver.
Thus, applying Parseval’s theorem, we have that Pr =
´
dω
2pi Sr(ω). Assuming that σ(k, ω)
varies slowly with the radar range r, and that the ambiguity function is almost flat within
the bandwidth of the RCS spectrum (which is a valid approximation in the case of short-
pulse radar applications), we can use equation (2.2) to obtain
11
Sr(ω)
EtKs
≈ δR
4piR2
ˆ
dΩ
G(rˆ)A(rˆ)
4pi
σv(k, ω), (2.4)
where R ≡ ct/2 is the measured radar range and
δR ≡
ˆ
dr |χ(2
c
(R− r), 0)|2 (2.5)
is the effective range depth of the radar scattering volume. An important system parameter
for soft-target radars is the backscatter aperture ABS , defined as
ABS ≡
ˆ
dΩ
G(rˆ)A(rˆ)
4pi
=
λ2
16pi2
ˆ
dΩG2(rˆ) (2.6)
as a function of the two-way antenna gain G2(rˆ).
The volumetric RCS spectrum of a quiescent ionosphere can be written as
σv(k, ω) = 4pir
2
e 〈|ne(k, ω)|2〉, (2.7)
where re is the classical electron radius and 〈|ne(k, ω)|2〉 denotes the space-time spectrum
of electron density fluctuations ne to be formulated in the next section.
2.2 General Framework of IS Spectral Models
In a plasma, electron and ion currents due to random thermal motions produce the
impressed electron and ion density fluctuations nte and nti, which can be regarded as
statistically independent random processes having the frequency spectra
〈|nte,i(k, ω)|2〉 = Ne,i
ˆ
dτe−jωτ 〈ejk·∆re,i〉 (2.8)
expressed in terms of ambient densities Ne,i and random particle displacements ∆re,i as
described in Kudeki and Milla [2010]. Electrostatic fields generated by the imbalance
between nte and nti in turn drive a conduction current so that the total current, including
the displacement current, vanishes in the electrostatic approximation. In the k-ω space,
12
the spectrum of total electron density fluctuations ne then exhibits a dependence on the
spectra of nte and nti that is given by [e.g., Kudeki and Milla, 2006, 2010]
〈|ne(k, ω)|2〉 = |jωo + σi(k, ω)|
2〈|nte(k, ω)|2〉+ |σe(k, ω)|2〈|nti(k, ω)|2〉
|jωo + σe(k, ω) + σi(k, ω)|2 , (2.9)
where σe(k, ω) and σi(k, ω) are the electron and ion conductivities, and o is the permit-
tivity of free space. This expression, valid for stationary single-ion plasmas, is derived in
Kudeki and Milla [2010] without making any assumption regarding the statistical distri-
butions of electrons and ions.
Expressions for the conductivity σs(k, ω) and the spectrum of thermal fluctuations
〈|nts(k, ω)|2〉 of each plasma species can be independently derived from plasma kinetic
equations. Note that macroscopic electric interactions between different particle species
have to be disregarded in the formulation of the kinetic equations as their effects have
already been considered in the derivation of 〈|ne(k, ω)|2〉. However, if each species is in
thermal equilibrium — i.e., if its velocity distribution is Maxwellian — it can be verified
that σs(k, ω) and 〈|nts(k, ω)|2〉 are related by the fluctuation-dissipation or generalized
Nyquist theorem [e.g., Callen and Welton, 1951; Kubo, 1966]
ω2q2s
k2
〈|nts(k, ω)|2〉 = 2KTsRe{σs(k, ω)}, (2.10)
where qs and Ts are the charge and temperature of particles of type s, and K is the
Boltzmann constant. Kudeki and Milla [2010] make use of this relation as follows: First,
the spectrum of thermal fluctuations is derived from a microscopic model of particle dy-
namics (where macroscopic “collective” interactions have been neglected), and then, the
fluctuation-dissipation theorem is used to determine the real part of the conductivity of
the corresponding particle species. Next, the imaginary part of σs(k, ω) is obtained by
applying Kramers-Kronig relations [e.g., Clemmow and Dougherty , 1969; Chew , 1990]. As
a result, it is found that
〈|nts(k, ω)|2〉
Ns
= 2Re{Js(ω)} (2.11)
13
and
σs(ω,k)
jωo
=
1− jωJs(ω)
k2h2s
, (2.12)
where Ns is the mean species density,
hs ≡
√
oKTs
Nsq2s
(2.13)
is the corresponding Debye length, and
Js(ω) ≡
ˆ ∞
0
dτe−jωτ 〈ejk·∆rs〉 (2.14)
is a Gordeyev integral, a one-sided Fourier transform of the characteristic function 〈ejk·∆rs〉
of random particle displacements ∆rs ≡ rs(t + τ) − rs(t) occurring over intervals τ in
the absence of collective interactions [see Kudeki and Milla, 2010]. This characteristic
function is also termed the single-particle ACF, since it can be interpreted as the normalized
correlation of the signal scattered by a single particle exposed to a radar pulse. In this way,
the problem of determining the incoherent scatter spectrum is reduced to the estimation
of single-particle ACF’s and the evaluation of the corresponding Gordeyev integrals (in
order to simplify the notation, subscript s is disregarded in the rest of this chapter).
2.3 Single-Particle ACF and Langevin Equation
The single-particle ACF 〈ejk·∆r〉 for a species in a given plasma can be calculated
directly from
〈ejk·∆r〉 =
ˆ
d(∆r) ejk·∆rf(∆r, τ), (2.15)
where f(∆r, τ) is the pdf of the particle displacement ∆r at a time delay τ . This is a
straightforward calculation as long as f(∆r, τ) is known either analytically or numerically.
The standard procedure for determining f(∆r, τ) starts with solving the Boltzmann ki-
netic equation for f(r, t) with a proper collision operator, e.g., the Fokker-Planck kinetic
equation of Rosenbluth et al. [1957] for Coulomb collisions, subject to an initial condition
14
[e.g., Holod et al., 2005]
f(r, 0) = δ(r). (2.16)
Although, analytical solutions of simplified versions of the Fokker-Planck kinetic equation
are available [e.g., Chandrasekhar , 1943; Dougherty , 1964], determining f(∆r, τ) would be
a daunting task when the full equation is considered.
An alternative approach to calculating 〈ejk·∆r〉 in the case of Coulomb collisions in-
volves producing, as the result of some simulation procedure, suitable sets of particle dis-
placement data ∆r for the same stochastic process described by the Fokker-Planck kinetic
equation. Assuming the process to be ergodic, a sufficiently large set of simulated samples
of particle trajectories r(t) can be used to compute 〈ejk·∆r〉 as well as any other statistical
function of displacements ∆r over time delays τ . Random trajectory simulations of course
require the availability of a stochastic equation describing how the particle velocities
v(t) ≡ dr
dt
(2.17)
may evolve under the influence of Coulomb collisions.
Restricting v(t) to be a Markovian random process constrains the stochastic evolution
equation of v(t) by very strict self-consistency conditions discussed by Gillespie [1996a, b]
to acquire the form of a Langevin equation
dv(t)
dt
= A(v, t) + C¯(v, t)W(t), (2.18)
where vector A(v, t) and matrix C¯(v, t) consist of arbitrary smooth functions of arguments
v and t. Above, W(t) is a random vector having statistically independent Gaussian white
noise components. A more natural way of expressing the Langevin equation is to cast it
in an update form, namely
v(t+ ∆t) = v(t) + A(v, t) ∆t+ C¯(v, t) ∆t1/2U(t), (2.19)
where ∆t is an infinitesimal update interval and U(t) is a vector composed of independent
15
zero-mean Gaussian random variables with unity variance, i.e.,
Ui(t) ≡ N (0, 1), (2.20)
where N (µ, σ2) denotes the normal random variable with mean µ and variance σ2. The
formal equivalency of (2.18) and (2.19) requires the i-th component of W(t) to be defined
as
Wi(t) = lim
∆t→0
(∆t)−1/2N (0, 1) = lim
∆t→0
N (0, 1/∆t), (2.21)
compatible with the requirement that 〈Wi(t+ τ)Wi(t)〉 = δ(τ).
Note that the Langevin equation (2.18) describing a Markovian process has the form
of Newton’s second law of motion, with the terms on the right representing forces per
unit mass that model the fluctuating Coulomb fields exerted on the particles in a plasma
and that will be regarded as the collision forces. The first term is a deterministic friction
force that causes particle deceleration, while the second term is a stochastic diffusion force
that is responsible for the random walk of the simulated particles in space. Considering
the Lorentz force on a charged particle in a magnetized plasma with a constant magnetic
field B, and not violating the strict format of (2.18), we can modify the equation by
adding a term qmv(t)×B to its right hand side. We would then have an additional term
q
mv(t)×B ∆t on the right-hand side of the update equation (2.19) as well.
Another relevant fact is that a special type of Markov process characterized by a linear
A(v, t) = −βv and a constant matrix C¯ = D1/2I¯, independent of v and t, is known
as Brownian motion process [e.g., Uhlenbeck and Ornstein, 1930; Chandrasekhar , 1943],
which is often invoked in simplified models of collisional plasmas [e.g., Dougherty , 1964;
Woodman, 1967; Holod et al., 2005]. In these models, friction and diffusion coefficients, β
and D, are constrained to be related by
D =
2KT
m
β (2.22)
for a plasma in thermal equilibrium.
16
2.4 The Link between Fokker-Planck and Langevin
Equations
In return for having restricted v(t) to the space of Markovian processes, we have gained
a stochastic evolution equation (2.18) with a plausible Newtonian interpretation and with
the potential of taking us beyond Brownian-motion-based collision models.
Letting v(t) to be Markovian turns out to have a second consequence of importance in
this work: namely, the evolution of probability density f(v, t) of a random variable v(t)
is known to be governed, when v(t) is Markovian, by the Fokker-Planck kinetic equation
having a “friction vector” and “diffusion tensor”
〈
∆v
∆t
〉
c
= A(v, t) (2.23)
and 〈
∆v∆vT
∆t
〉
c
= C¯(v, t)C¯T(v, t), (2.24)
respectively, specified in terms of the input functions of the Langevin equation. This
intimate link between the Langevin equation and the Fokker-Planck kinetic equation — in
describing Markovian processes from two different but mutually compatible perspectives —
was first pointed out by Chandrasekhar [1943] and more recently by Gillespie [1996b]. This
relationship is analogous to a similar relation between the BGK kinetic equation [Bhatnagar
et al., 1954] for charged particles colliding with neutrals and a particle dynamics formalism
in which this type of collisions is modeled as a discrete Poisson process, a model that is
discussed further by Milla and Kudeki [2009] (see Appendix A).
Since the Fokker-Planck friction vector and diffusion tensor for equilibrium plasmas
with Coulomb interactions have already been worked out by Rosenbluth et al. [1957] as
〈
∆v
∆t
〉
c
= −β(v)v (2.25)
and 〈
∆v∆vT
∆t
〉
c
=
D⊥(v)
2
I¯ +
(
D‖(v)−
D⊥(v)
2
)
vvT
v2
, (2.26)
17
in terms of scalar functions β(v), D‖(v), andD⊥(v) to be specified below, it follows that the
Langevin update equation to be used in Coulomb collision particle trajectory simulations
can be written as
v(t+ ∆t) = v(t) +
q
m
v(t)×B ∆t
− β(v)∆tv(t) +
√
D‖(v)∆t U1 vˆ‖ +
√
D⊥(v)
∆t
2
(U2 vˆ⊥1 + U3 vˆ⊥2) , (2.27)
including a DC magnetic field term, with vˆ‖(t), vˆ⊥1(t), and vˆ⊥2(t) denoting an orthogonal
set of unit vectors parallel and perpendicular to the particle trajectory, and Ui denoting
zero-mean Gaussian random variables with unit variance.
2.5 Spitzer Friction and Diffusion Coefficients
Expressions for the Fokker-Planck friction vector and diffusion tensor for a particle
moving in a magnetized plasma have been derived by different authors [e.g., Rostoker and
Rosenbluth, 1960; Ichimaru and Rosenbluth, 1970]. However, when the plasma is “weakly”
magnetized, i.e., if the particle Debye length is smaller than the mean gyroradius, magnetic
field effects in the friction vector and diffusion tensor can be neglected [Rostoker and
Rosenbluth, 1960] and the friction coefficient β(v) and velocity space diffusion coefficients
D‖(v) andD⊥(v) introduced above can be specified in (2.27) as formulated by the standard
collision model of Rosenbluth et al. [1957]. For a Maxwellian plasma, these coefficients
take the forms given by Spitzer [Spitzer , 1962]. Specifically, if a particle of type s colliding
against a background of particles of type s′ is considered, the Spitzer coefficients take the
forms [Callen, 2006]
β(v) =
∑
s′
(1 +
ms
ms′
)Ns′Γs/s′
1
C2s′
ψ(zs′)
v
, (2.28)
D||(v) =
∑
s′
2Ns′Γs/s′
ψ(zs′)
v
, (2.29)
D⊥(v) =
∑
s′
2Ns′Γs/s′
φ(zs′)− ψ(zs′)
v
, (2.30)
18
where Cs =
√
KTs/ms is the thermal speed of particles with mass ms, electric charge qs,
and equilibrium temperature Ts. Additionally,
Γs/s′ ≡
q2sq
2
s′
4pi2om
2
s
ln Λs/s′ (2.31)
in terms of “Coulomb logarithm” ln Λs/s′ (see below), while
zs ≡ v√
2Cs
(2.32)
in terms of particle speed v, and
ψ(z) ≡ φ(z)− zφ
′(z)
2z2
, (2.33)
where
φ(z) ≡ 2√
pi
ˆ z
0
e−x
2
dx (2.34)
is the well-known error function. Finally, defining a plasma Debye length
hD ≡
(∑
s
1
h2s
)−1/2
=
(∑
s
Nsq
2
s
oKTs
)−1/2
, (2.35)
and a minimum impact parameter
bmin ≡
qsqs′
12piomss′C
2
ss′
, (2.36)
where mss′ ≡ msmx′ms+ms′ is a reduced mass and C
2
ss′ ≡ C2s + C2s′ , we have
Λs/s′ ≡
hD
bmin
, (2.37)
which is known as the plasma parameter and is proportional to the number of particles of
type s inside a Debye cube. See Appendix B for a discussion regarding the application of
Spitzer friction and diffusion coefficients to the case of non-isothermal plasmas.
19
2.6 Summary
The general framework of the incoherent scatter theory formulates the electron den-
sity spectrum in terms of electron and ion Gordeyev integrals, functions that depend on
the statistics of individual particle trajectories. Since collisions in the plasma modify these
trajectories, they have an impact on the statistics of the particle displacements and also on
the corresponding Gordeyev integrals. It is through the estimation of these functions that
the effect of Coulomb collisions is considered in our incoherent scatter spectrum model.
Although the statistics of particle trajectories, i.e., velocity and displacement density dis-
tributions, can be obtained by numerically solving the Boltzmann kinetic equation with
the Fokker-Planck collision operator, we chose to utilize a different approach based on the
simulation of plasma particle trajectories using Langevin update equations. In the next
chapter, we describe our computer simulations which provide us with sample electron and
ion trajectories in magnetized and collisional F-region plasmas.
20
CHAPTER 3
MONTE CARLO COMPUTATION OF PARTICLE
TRAJECTORIES AND SINGLE-PARTICLE
ACF’S
The Monte Carlo procedure for the simulation of charged particle trajectories in a
plasma and the subsequent estimation of single-particle ACF’s is detailed in this chapter.
In the simulations, ion and electron trajectories are computed using a stochastic update
equation of the Langevin type where friction and diffusion forces account for the effects
of Coulomb collisions on the particle motion. The computer program developed for the
simulations is outlined in Section 3.1. The outcomes of the simulations, series of particle
positions, are then utilized in the computation of single-particle ACF’s and associated
Gordeyev integrals according to the algorithm described in Section 3.2. Velocity and
displacement distributions obtained from the simulations are analyzed in detail in Section
3.3, where samples of ion and electron ACF’s are displayed and discussed. The following
presentation follows closely the description of the procedure reported by Milla and Kudeki
[2010].
3.1 Computer Simulations of Plasma Particle Trajectories
Consider the motion of a particle governed by the Langevin update equation (2.27).
Denoting by v[n] the particle velocity vector at a discrete time n∆t, the updated velocity
21
Table 3.1 Typical plasma parameters for an equatorial F-region ionosphere.
Parameter Value
Plasma density Ne 1012 m−3
Magnetic field Bo 25µT
Electron temperature Te 1000 K
Ion temperature Ti 1000 K
Ion composition O+
after a sufficiently small time interval ∆t can be determined by using
v[n+ 1] = v[n] +
q
m
v[n]×B ∆t
− β(v)∆tv[n] +
√
D‖(v)∆t U1[n] vˆ‖ +
√
D⊥(v)
∆t
2
(U2[n] vˆ⊥1 + U3[n] vˆ⊥2) ,
(3.1)
where m and q denote the mass and the electric charge of the simulated particle (which
can be either an electron or an ion). The different terms in (3.1) on the right include
the changes in the velocity produced by the magnetic and collisional forces discussed in
Chapter 2. In addition, the particle position r[n] can be calculated from the velocities v[n]
using
r[n+ 1] = r[n] +
v[n+ 1] + v[n]
2
∆t, (3.2)
another update equation that can be obtained by approximating the integral of the velocity
vector over a time interval ∆t using the trapezoidal rule.
We have developed a computer program for the simulation of F-region particle tra-
jectories governed by (3.1) and (3.2). In the simulations, a homogeneous plasma in the
presence of a uniform magnetic field B ≡ Bo zˆ is considered. The plasma has electron
density Ne and electron and ion temperatures Te and Ti. The values of Bo, Ne, Te, and
Ti, which are typical of the equatorial F-region ionosphere, are listed in Table 3.1. For
completeness, a list of some representative plasma parameters is given in Table 3.2.
The F-region simulation runs as follows. First, we randomly pick some initial velocity
22
T
ab
le
3.
2
R
ep
re
se
nt
at
iv
e
ph
ys
ic
al
pa
ra
m
et
er
s
of
a
pl
as
m
a.
D
efi
ni
ti
on
s
an
d
sa
m
pl
e
va
lu
es
ar
e
pr
ov
id
ed
.
T
he
va
lu
es
w
er
e
co
m
pu
te
d
fo
r
an
O
+
pl
as
m
a
w
it
h
pa
ra
m
et
er
s
gi
ve
n
in
T
ab
le
3.
1.
N
ot
e
th
at
th
e
pl
as
m
a
an
d
gy
ro
fr
eq
ue
nc
y
fo
rm
ul
as
ar
e
gi
ve
n
fo
r
an
gu
la
r
fr
eq
ue
nc
ie
s,
bu
t
th
e
sa
m
pl
e
va
lu
es
ar
e
gi
ve
n
in
H
z.
P
ar
am
et
er
D
efi
ni
ti
on
E
le
ct
ro
n/
Io
n
D
efi
ni
ti
on
P
la
sm
a
T
he
rm
al
sp
ee
d
C
s
=
√ K
T
s
/m
s
12
3
.1
k
m
/
s
71
8
.3
m
/s
—
—
P
la
sm
a
fr
eq
ue
nc
y
ω
p
s
=
√ N s
q2 s
/
(
o
m
s
)
8
.9
8
M
H
z
52
.3
8
k
H
z
ω
p
=
( ∑ s
ω
2 p
s
) 1/2
8.
98
M
H
z
D
eb
ye
le
ng
th
h
s
=
C
s
/ω
p
s
2.
18
m
m
2.
18
m
m
h
D
=
( ∑ s
1/
h
2 s
) −1/2
1
.5
4
m
m
G
yr
of
re
qu
en
cy
Ω
s
=
q s
B
o
/m
s
69
9
.8
k
H
z
23
.8
2
H
z
Ω
p
=
( ∑ s
Ω
2 s
) 1/2
69
9.
8
k
H
z
M
ea
n
gy
ro
ra
di
us
ρ
s
=
√ 2
C
s
/Ω
s
39
.6
m
m
6
.7
9
m
—
—
23
v[0] and set the starting position of the particle to the origin (i.e., r[0] = 0). Next, the
values of friction and diffusion coefficients for the given particle speed are calculated. The
velocity increments resulting from the action of each of the simulated forces (magnetic,
friction, and diffusion) in a time interval ∆t are then computed. For the calculation
of the diffusive forces, we have to randomly pick three normal numbers. These values
are computed using the Ziggurat method for random generation of Gaussian variables
[Marsaglia and Tsang , 2000]. Once all the velocity increments are calculated, we proceed to
compute the new value of the velocity vector using (3.1), and subsequently, the position of
the particle is updated using (3.2). The algorithm is then repeated in a loop. The resulting
series of velocities and positions are stored in sequences of length M (typically equal to
217 samples). About 104 of these sequences are generated, which provides more than 109
simulated velocities and positions for a single particle in a given plasma configuration.
Note that the starting velocity and position of each new sequence is set equal to the last
velocity and position of the previous sequence. Only the initial velocity of the first sequence
is randomly chosen; the remaining samples of the series are calculated using the update
velocity equation (3.1).
A sample trajectory of an electron moving in an O+ plasma with densityNe = 1012 m−3
and temperatures Te = Ti = 1000 K is presented in Figure 3.1. The intensity of the
magnetic field was taken as 25µT. On the top, the left panel shows the particle trajectory
in 3D space, while the panel on the right is the projection of the trajectory on the xy-
plane (i.e., the plane perpendicular to B). On the bottom, the displacement of the electron
in the direction parallel to B is displayed as a function of time. As shown, the electron
moves approximately in a spiral path that is randomized by the action of collisions. As the
electron accelerates and decelerates at random rates, not only does the guiding center of its
trajectory randomly drift in space, but the diameter of the particle orbits also changes as a
function of time. Because of this, there are time intervals in which the electron orbits have
smaller or larger radii than the mean gyroradius ρe. For instance, the trajectory presented
in Figure 3.1 corresponds to a period when the orbit radius is smaller than ρe (which, in
this example, is approximately 40 mm). We can also see that, in the plane perpendicular
24
−0.1
0
0.1
−0.1
0
0.1
−5
0
5
10
15
x-axis [m]
Time Interval: 0-400 µs
y -axis [m]
z
-a
x
is
[m
]
−0.1 −0.05 0 0.05 0.1
−0.1
−0.05
0
0.05
0.1
x-axis [m]
y
-a
x
is
[m
]
Displacement ⊥ to B
0 50 100 150 200 250 300 350 400
−5
0
5
10
15
Time [µs]
z
-a
x
is
[m
]
Displacement ‖ to B
Figure 3.1 Sample trajectory of an electron moving in an O+ plasma with density Ne =
1012 m−3 and temperatures Te = Ti = 1000 K. The presence of a uniform magnetic field
B parallel to the z-axis was considered in the simulation. The intensity of the field was
assumed to be 25µT. On the top, the left panel shows the particle trajectory in 3D
space, while the panel on the right is the projection of the trajectory on the xy-plane (i.e.,
the plane perpendicular to B). On the bottom, the displacement of the electron in the
direction parallel to B is displayed as a function of time. The trajectory was sampled every
∆t = 0.1µs. A total of 4 × 103 samples are displayed corresponding to a time interval
of 400µs. In all three panels, the red dots show the starting location of the particle (the
origin). In addition, the red curve depicts the first 1.4µs of the simulated trajectory (about
one gyroperiod).
25
to B, the particle never returns to the same position after a gyroperiod. Additionally,
note that there is a large difference between the distances covered by the electron in the
directions parallel and perpendicular to B. These characteristics of the simulated electron
trajectories have implications for the shape of the computed single-particle ACF’s and
their corresponding Gordeyev integrals presented later in this chapter.
The simulation procedure just outlined was motivated by the earlier work of Sulzer
and González [1999]. Both simulation studies make use of Spitzer friction and diffusion
coefficients in order to include the effects of Coulomb collisions in IS spectral models.
However, the equations of motion employed by Sulzer and González [1999] differed from
our Langevin-based 3D update procedure. Sulzer and González considered the effect of
Coulomb collisions on particle displacements only in the direction of the ambient field
B, neglecting the diffusion and random walk effects across the field lines. That limits the
applicability of their results to magnetic aspect angles larger than about 0.1◦, in relation to
the 50 MHz Jicamarca radar observations. Our simulation results, by contrast, enable ACF
and Gordeyev integral calculations for all aspect angles, including the direction of exact
perpendicularity to the geomagnetic field B. Furthermore, our Langevin-based update
procedure does not suffer numerical instability issues when used with sufficiently small
update intervals ∆t.
3.2 Estimation of the Single-Particle ACF’s and Gordeyev
Integrals
In this section, we will describe our processing procedures of the particle trajectory
data for estimating the single-particle ACF’s and the corresponding Gordeyev integrals.
Consider a long sequence of N simulated particle positions uniformly sampled in time.
For a given radar Bragg vector k, the unbiased estimator for the single-particle ACF at a
discrete time delay m∆t is given by
R̂[m] ≡ ̂〈ejk·∆r[m]〉 = 1
N −mcrr[m], 0 ≤ m ≤ N − 1, (3.3)
26
where
crr[m] =
N−m−1∑
n=0
exp (jk · (r[n+m]− r[n])) =
N−1∑
n=m
exp (jk · (r[n]− r[n−m])) (3.4)
is the discrete correlation of a sequence of samples ejk·r[n]. In general, discrete correlations
can be computed efficiently using the FFT technique. In our case, however, we cannot
directly apply this procedure because the entire sequence of N particle positions is very
large and it cannot be fully allocated in the RAM of a computer (for instance, 109 sample
positions stored in double float format would require about 24 gigabytes of RAM). Since
we are just interested in the first M samples of this correlation (about 105 samples), let
us divide the full set of positions into L segments of length M , such that N = LM . We
can then re-express crr[m] as
crr[m] =
L−1∑
l=0
2M−1∑
n=0
gl[n]h
∗
l [(n−m)2M ], 0 ≤ m ≤M − 1, (3.5)
where (a)b denotes the modulo operation (amod b), and functions gl and hl are
gl[n] ≡

exp(jk · r[n+ lM ]), 0 ≤ n ≤M − 1,
0, M ≤ n ≤ 2M − 1,
(3.6)
and
hl[n] ≡

gl[n], 0 ≤ n ≤M − 1,
gl−1[n−M ], M ≤ n ≤ 2M − 1,
(3.7)
respectively (note that h0 = g0). Using equation (3.5), we can compute the correlation
crr[m] in an iterative way reducing significantly the amount of required computer mem-
ory. The inner summation in (3.5) is the circular cross-correlation of functions gl and hl,
calculation of which is performed using FFT’s.
Estimates Rˆ[m] of the single-particle ACF 〈ejk·∆r〉 are then obtained by dividing crr[m]
by LM −m. This means that the statistical variance of our estimates increases with m,
since for larger m fewer samples are used to estimate the ACF. Assuming that at every
27
M samples the simulated positions are independent random variables, we find that in the
worst-case scenario (i.e., when m = M − 1), the variance of our estimates will be at most
inversely proportional to L − 1. Thus, in order to secure small estimation errors, large
values of L have to be considered. In our calculations we used L = 104.
The procedure outlined above is used to compute 〈ejk·∆r〉 for different values of Bragg
vector k. In our calculations, we define
k =
2pi
λB
(cos(α) xˆ+ sin(α) zˆ) , (3.8)
where λB is the Bragg wavelength (e.g., λB = 3 m in the case of Jicamarca), and α is the
magnetic aspect angle that is the complement of the angle between k and the magnetic
field vector B = Bo zˆ. Notice that to estimate 〈ejk·∆r〉 for different aspect angles, we
do not need to generate a new set of particle positions. We took advantage of this and
performed many of these calculations in parallel on a computer. Our definition of vector
k is somewhat arbitrary. Notice that, for the direction perpendicular to B, we could have
chosen k to be 2piλB yˆ instead of
2pi
λB
xˆ. Therefore, for α = 0◦, there are two alternative
ways of computing the single-particle ACF that provide two sets of almost statistically
independent estimates of 〈ejk·∆r〉. Averaging these two sets, we could have reduced the
statistical errors of our estimates. In the future, we will take advantage of this in our
calculations.
Our next task is the computation of the Gordeyev integral J(ω) from discrete estimates
of 〈ejk·∆r〉. Since our samples are uniformly distributed in time, we could have simply taken
the FFT of the ACF estimates Rˆ[m] to perform the Gordeyev integral calculations, but
then samples of J(ω) would have been restricted to a discrete set of frequencies. We instead
used a chirp-Z transform algorithm described by Li et al. [1991] in our Gordeyev integral
computations, which allows the evaluation of the transform over any desirable range of
frequencies ω.
Given the set of frequencies ωk = ωo + k∆ω for 0 ≤ k ≤ K − 1, we can approximate
28
the Gordeyev integral (2.14) using the following summation
J(ωk) ≈ ∆t
M−1∑
m=0
w[m] Rˆ[m] e−jωkm∆t = ∆t
M−1∑
m=0
w[m] Rˆ[m] e−jωom∆t e−jkm∆ω∆t, (3.9)
where, in order to improve the accuracy of the approximation, ACF estimates Rˆ[m] are
weighted by composite quadrature coefficients w[m]. In our calculations, we use composite
Simpson’s rule coefficients which are given by
w[m] =

1
3 , m = 0,M,
4
3 , m = 1, 3, . . . ,M − 1,
2
3 , m = 2, 4, . . . ,M − 2,
(3.10)
where M is assumed to be an even number. To derive the integration algorithm, let us
use the identity
km =
1
2
(k2 +m2 − (k −m)2) (3.11)
to re-express equation (3.9) as
J(ωk) ≈ ∆t
M−1∑
m=0
w[m] Rˆ[m] e−jnωo∆tW k
2/2Wm
2/2W−(k−m)
2/2 (3.12)
where W = e−j∆ω∆t. Note that the previous equation is a convolution sum; then, if we
define
x[m] ≡

w[m] Rˆ[m] e−jmωo∆tWm2/2, 0 ≤ m ≤M − 1,
0, M ≤ m ≤M +K − 1,
(3.13)
and
y[m] ≡

W−m2/2, 0 ≤ m ≤ K − 1,
W−(M+K−m)2/2, K ≤ m ≤M +K − 1,
(3.14)
29
Table 3.3 Plasma parameters spanned by the library of single-electron ACF’s.
Plasma parameter Initial value Final value Step
log10Ne 11 13 0.5
Te 600 K 3000 K 200 K
Ti 600 K 2000 K 200 K
Bo 20µT 30µT 5µT
we can rewrite equation (3.12) in the following form
J(ωk) ≈ ∆tW k2/2
M+K−1∑
m=0
x[m] y[(k −m)M+K ], 0 ≤ k ≤ K − 1,
where the summation is the circular convolution of sequences x[m] and y[m], calculations
that can be efficiently performed using FFT’s. In addition, note that in order to avoid
aliasing artifacts, we need ωK∆t < pi.
In the above discussion, we made no distinction between electrons and ions since identi-
cal procedures are used to compute the Gordeyev integrals for each species. The estimated
electron and ion Gordeyev integrals are then used to compute theoretical incoherent scatter
spectra. As a result of our simulation studies, we found that the ion Gordeyev integrals can
be well represented by analytical means, as discussed in the next section of this chapter.
However, closed-formed expressions could not be found for the electron Gordeyev inte-
grals; therefore, a numerical electron ACF library had to be constructed for a wide range
of plasma parameters and magnetic aspect angles, as needed by Jicamarca applications.
The plasma parameters that this library spans are given in Table 3.3.
The construction of the library was computationally demanding and was carried out
using a cluster of computers. The Turing Cluster, maintained by the Computational
Science and Engineering Program at the University of Illinois, was used for this purpose.
The Turing system consists of 768 Apple Xserves, each with two 2 GHz G5 processors and
4 GB of RAM. In the cluster, hundreds of simulations run simultaneously, which allowed
us to build the library in less than three weeks. The same task would have taken many
30
months (probably more than two years) using a single desktop computer.
3.3 Statistics of the Ion and Electron Trajectories
If a plasma is in thermal equilibrium, i.e., if Te = Ti, one can show analytically that the
steady-state solution of the Fokker-Planck equation is given by the Maxwell-Boltzmann
velocity distribution [Montgomery and Tidman, 1964]
f(v) =
1
(2piC2)3/2
e−
1
2C2
(v2x+v2y+v2z), (3.15)
where vx, vy, and vz are the components of the particle velocity vector, and C =
√
KT/m
is the corresponding thermal speed. Notice that this pdf can be written as the product of
three independent Gaussians, one for each of the velocity components. Since the Fokker-
Planck and the Langevin equations are alternative representations of the same Markov
process, we expect the distributions of our simulated velocities to be Gaussians. Velocity
distributions resulting from independent ion and electron simulations in an O+ plasma are
presented in Figure 3.2. In the simulations, the plasma was considered to be in thermal
equilibrium with temperatures Te = Ti = 1000 K (the rest of the simulation plasma param-
eters are given in Table 3.1). The distributions were built using more than 109 sampled
velocities. In each panel, the expected Gaussian pdf is plotted on top of our simulation
results. As we can see, the computed distributions match exactly the Gaussian curves
in these two examples. We repeated the same test for different plasma parameters and
found that the agreement was excellent in all cases. Because of these results, we have the
confidence that our simulation procedure is working properly.
As we mentioned in Chapter 2, the single-particle ACF is a characteristic function;
therefore, its shape is determined by the time evolution of the pdf f(∆r, τ) of particle
displacements ∆r. If the plasma is magnetized, particles are forced to diffuse slowly
in the plane perpendicular to B, making the pdf f(∆r, τ) narrower in that plane than
in the parallel direction (for a given τ). In order to analyze the behavior of f(∆r, τ),
we have computed from the simulated trajectories the distributions of ion and electron
31
−3 −2 −1 0 1 2 3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Normalized velocity (v/Ci)
Io
n
v
el
o
ci
ty
d
is
tr
ib
u
ti
o
n
 
 
vx pdf
vy pdf
vz pdf
Gaussian
−3 −2 −1 0 1 2 3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Normalized velocity (v/Ce)
E
le
ct
ro
n
v
el
o
ci
ty
d
is
tr
ib
u
ti
o
n
 
 
vx pdf
vy pdf
vz pdf
Gaussian
(a) (b)
Figure 3.2 Probability distributions of (a) ion and (b) electron velocities resulting from
the simulations. In each plot, the pdf’s of the velocity components (displayed in colors) are
compared to a Gaussian distribution (black line). Notice that the velocity axes are normal-
ized by the thermal speeds of the corresponding particle species. The plasma parameters
considered in the simulations are given in Table 3.1.
displacements in the directions parallel and perpendicular to B. Since B = Bo zˆ, the
distributions of the parallel displacement correspond to f(∆z, τ). On the other hand, the
distributions of the perpendicular displacement were computed by averaging f(∆x, τ) and
f(∆y, τ). In addition, the variance and covariance of the components of the displacement
vector ∆r were computed from the simulated trajectories. These mean-square quantities
are defined as
〈∆ri ∆rj〉 = 〈(ri(t+ τ)− ri(t)) (rj(t+ τ)− rj(t))〉, (3.16)
where i and j denote the Cartesian coordinates. These quantities were estimated following
a procedure similar to the one used to compute 〈ejk·∆r〉.
Before presenting our statistics of the simulated ion and electron displacements, notice
that expression (2.15) for the ACF 〈ejk·∆r〉 is simply the Fourier transform of f(∆r, τ).
In principle, we could have computed 〈ejk·∆r〉 using the Fourier transforms of the dis-
placement distributions obtained in the simulations. This would have required, however,
a significant increase in the number of simulated particle positions in order to reduce the
statistical estimation errors to an acceptably small level. Since we are interested in eval-
uating 〈ejk·∆r〉 only for some discrete values of k, we consider the procedure outlined in
32
0 2
4 6
8 10−2 0
2
0
0.2
0.4
Time delay [ms]
Ion displacement distributions ⊥ to B
∆r⊥(τ )/σ⊥(τ )
∆
r
⊥
p
d
f
0 2
4 6
8 10−2 0
2
0
0.2
0.4
Time delay [ms]
Ion displacement distributions ‖ to B
∆r‖(τ )/σ‖(τ )
∆
r
‖
p
d
f
−3 −2 −1 0 1 2 3
0
0.1
0.2
0.3
0.4
∆r⊥(τ )/σ⊥(τ )
∆
r
⊥
p
d
f
Ion displacement distributions ⊥ to B
 
 
τ = 0.0ms
τ = 2.0ms
τ = 4.0ms
τ = 6.0ms
τ = 8.0ms
τ = 10.0ms
Gaussian
−3 −2 −1 0 1 2 3
0
0.1
0.2
0.3
0.4
∆r‖(τ )/σ‖(τ )
∆
r
‖
p
d
f
Ion displacement distributions ‖ to B
 
 
τ = 0.0ms
τ = 2.0ms
τ = 4.0ms
τ = 6.0ms
τ = 8.0ms
τ = 10.0ms
Gaussian
Figure 3.3 Probability distributions of the displacements of a simulated ion in the direc-
tions perpendicular (top panels) and parallel (bottom panels) to the magnetic field. On
the left, the displacement pdf’s are displayed as functions of time delay τ . On the right,
sample cuts of the pdf’s are compared to a Gaussian distribution. Note that all distribu-
tions at all time delays are normalized to unit variance. The displacement axis of each
distribution at every delay τ is scaled with the corresponding standard deviation of the
simulated displacements.
the previous two sections to be more accurate, involving fewer computations.
3.3.1 Statistics of the Ion Displacements
In Figure 3.3, we show the distributions of ion displacements in the directions perpen-
dicular and parallel to the magnetic field. Note that, at every delay τ , the distributions
have been normalized to unit variance by scaling the displacement axis of each distribu-
tion with the corresponding standard deviation of the simulated displacements.1 On the
1Note that in the limit τ → 0, we can write
lim
τ→0
∆ri(τ)
σi(τ)
= lim
τ→0
ri(t+ τ)− ri(t)
Cτ
= vi(t)/C.
Therefore, the distribution of ∆ri(τ)/σi(τ) at τ = 0 is equal to the distribution of the corresponding
velocity component.
33
left panels, the distributions are displayed as functions of τ , while, on the right panels,
sample cuts of these distributions are compared to a Gaussian pdf. For the time interval
considered in these plots, we can see that the shapes of the distributions do not change
with time delay and, also, that they match almost perfectly to the Gaussian curves. We
have observed that for τ up to and exceeding 10 ms, the distribution of the displacement
in the direction perpendicular to B remains Gaussian in shape. We also found that at
time delays of the order of hundreds of milliseconds, the parallel distribution eventually
becomes spikier than a Gaussian. However, by that time the single-ion ACF is negligibly
small — typical correlation times of the ion ACF for λB = 3 m are of the order of 1 ms
— and for that reason, ion displacements can be regarded as Gaussian random variables
for all practical purposes. Additionally, we verified that the components of the vector
displacement (i.e., ∆rx, ∆ry, and ∆rz) are mutually uncorrelated. The simulation results
presented here were computed for the plasma configuration of Table 3.1. The analysis was
repeated for other possible ionospheric plasmas and similar results were observed.
For the case of uncorrelated and jointly Gaussian ∆r components, it is known that
[e.g., Kudeki and Milla, 2010] the single-particle ACF takes the following form
〈ejk·∆r〉 = e− 12k2 sin2 α〈∆r2‖〉 × e− 12k2 cos2 α〈∆r2⊥〉, (3.17)
where, assuming a Brownian-motion process with distinct friction coefficients ν‖ and ν⊥
in the directions parallel and perpendicular to B, the mean square displacements will vary
as
〈∆r2‖〉 =
2C2
ν2‖
(
ν‖τ − 1 + e−ν‖τ
)
(3.18)
and
〈∆r2⊥〉 =
2C2
ν2⊥ + Ω2
(
cos(2γ) + ν⊥τ − e−ν⊥τ cos(Ωτ − 2γ)
)
(3.19)
in which γ ≡ tan−1(ν⊥/Ω), and C ≡
√
KT/m and Ω ≡ qB/m are, respectively, the
thermal speed and gyrofrequency of the simulated particles. In the literature, friction
coefficients ν‖ and ν⊥ are also referred to as “collision frequencies.” However, in the context
of Coulomb collisions, they should be interpreted not as the rate of collisions between
34
0 1 2 3 4 50
5
10
〈∆
r
2 ⊥
〉[
m
2
]
Variance of ion displacement ⊥ to B
 
 
〈|∆x|2〉
〈|∆y|2〉
Best fit
ν⊥ = 5.88 Hz
0 1 2 3 4 50
5
10
Time delay [ms]
〈∆
r
2 ‖〉
[m
2
]
Variance of ion displacement ‖ to B
 
 
〈|∆z |2〉
Best fit
ν‖ = 5.13 Hz
0 100 200 300 400 5000
50
100
150
〈∆
r
2 ⊥
〉[
m
2
]
Variance of ion displacement ⊥ to B
 
 
〈|∆x|2〉
〈|∆y|2〉
Best fit
ν⊥ = 5.88 Hz
0 100 200 300 400 5000
2
4
6
x 104
Time delay [ms]
〈∆
r
2 ‖〉
[m
2
]
Variance of ion displacement ‖ to B
 
 
〈|∆z |2〉
Best fit
ν‖ = 5.13 Hz
Figure 3.4 Variances of the displacements of a simulated ion in the directions perpen-
dicular (top panels) and parallel (bottom panels) to the magnetic field. The simulation
results (color lines) are displayed for two time intervals: 5 ms (left panels) and 500 ms
(right panels). The dashed lines correspond to the fits of the results to the theoretical
expressions for 〈∆r2⊥〉 and 〈∆r2‖〉 of the Brownian-motion model.
particles but instead as the rate of loss of momentum (deceleration) experienced by a
particle due to its interaction with the Coulomb fields generated by the swarm of electrons
and ions surrounding it.
To test the viability of a Brownian-motion model to represent the simulated ion data,
we fitted the expressions (3.18) and (3.19) to the variances of ion displacements obtained
in our simulations and compared the best-fit parameters ν‖ and ν⊥ to the Spitzer collision
frequency2 for ion-ion interactions, which for the case of a single-ion plasma is given by
[Callen, 2006]
νi/i =
Ne e
4 ln Λi
12pi3/2 2om
2
i C
3
i
, (3.20)
where Λi ≡ 12piNeh2ihD is the ion plasma parameter, and hi and hD are the ion and
plasma Debye lengths defined in Table 3.2.
An example of our fitting results is presented in Figure 3.4. In general, we found that
ν⊥ ≈ νi/i. For instance, for the case shown in Figure 3.4, the best-fit collision frequency is
ν⊥ = 5.88 Hz, while the Spitzer collision frequency is νi/i = 5.94 Hz. We also found that
2Spitzer collision frequency νs/s′ is the Maxwellian-averaged momentum relaxation rate of a particle of
type s due to its interaction with a background of particles of type s′. The inverse of νs/s′ can be interpreted
as the time interval over which the particle velocity vector rotates by about 90◦, an accumulated effect
due to many small Coulomb deflections known as Spitzer collisions.
35
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
Time [ms]
(a) Ion ACF (λB = 3 m)
α = 0.0◦
α = 0.5◦
α = 1.0◦
α = 90.0◦
Brownian
0 0.05 0.1 0.15 0.2 0.25 0.30
0.2
0.4
0.6
0.8
1
Time [ms]
(b) Ion ACF (λB = 0.3 m)
α = 0.0◦
α = 0.5◦
α = 1.0◦
α = 90.0◦
Brownian
Figure 3.5 Simulated single-ion ACF’s at different magnetic aspect angles α for two radar
Bragg wavelengths: (a) λB = 3 m and (b) λB = 0.3 m. The simulation results (color lines)
are compared to theoretical ACF’s computed using expression (3.22) of the Brownian-
motion approximation (dashed lines). Note that there is effectively no dependence on
aspect angle α.
the best-fit parallel collision frequency ν‖ is smaller than the Spitzer frequency by a factor
of ∼ 1.15, i.e., ν‖ < νi/i.
Finally, we note in the same figure that the variances of ∆r‖ and ∆r⊥ are very similar
for τ < 5 ms. This is expected because for τ in that interval ν‖τ  1 and ν⊥τ < Ωiτ  1,
in which case (3.18) and (3.19) indicate that
〈∆r2‖〉 ≈ 〈∆r2⊥〉 ≈ C2i τ2. (3.21)
Using this approximation in expression (3.17), we find that the single-ion ACF simplifies
to
〈ejk·∆ri〉 ≈ e− 12k2C2i τ2 , (3.22)
which is a well-known result for collisionless and non-magnetized plasmas that fits well the
simulated single-ion ACF’s for different magnetic aspect angles and Bragg wavelengths as
shown in Figure 3.5.
Evidently, in a magnetized F-region plasma with Coulomb collisions, the oxygen ions
diffuse along and across the magnetic field lines like Brownian-motion particles having
36
isotropic friction coefficients
ν⊥ ≈ ν‖ ≈ νi/i. (3.23)
But more significantly, the single-ion F-region ACF’s at 50 MHz are essentially the same
as in collisionless and non-magnetized plasmas because (a) the ions move by many Bragg
wavelengths λB = 2pi/k between successive Spitzer collisions, and (b) the ions are unable
to return to within λB/2pi of their starting positions after a gyroperiod as a consequence
of the ion-ion interactions (giving rise to Spitzer collisions). As a result, ion diffusion
is sufficient to eliminate the gyroresonance peaks from the single-ion ACF’s, as we have
shown above. As an upshot, we will be able to handle the ion terms analytically in the
general framework equations.
3.3.2 Statistics of the Electron Displacements
Next, we study the effects of Coulomb collisions on electron trajectories using pro-
cedures similar to those applied to ions. In Figure 3.6, the displacement distributions
resulting from the simulation of an electron moving in an O+ plasma are presented. The
plasma parameters considered in this simulation are also given in Table 3.1. The top and
bottom panels in Figure 3.6 correspond, respectively, to the distributions in the directions
perpendicular and parallel to B. On the left, the distributions are displayed as functions
of τ , while on the the right, sample cuts of the distributions are compared to a Gaus-
sian pdf. As in the ion case, we note that the normalized distributions for the direction
perpendicular to B do not vary with τ and match almost perfectly to Gaussian curves.
However, for displacements parallel to B, the normalized distributions do vary with τ , and
the shapes are distinctly non-Gaussian for intermediate values of τ . More specifically, at
very small time delays (τ . 1µs), the distributions are Gaussian, but then, in less than a
millisecond, the distribution curves become more “spiky” (positive kurtosis) than a Gaus-
sian. Although, at even longer delays τ the distributions once again relax to a Gaussian
shape, it is clear that the electron displacement in the direction parallel to B is not a
Gaussian random variable at all time delays τ .
Will a Brownian-motion model still prove useful to fit the simulation data for the elec-
37
0 2
4 6
8 10−2 0
2
0
0.2
0.4
Time delay [ms]
Electron displacement distributions ⊥ to B
∆r⊥(τ )/σ⊥(τ )
∆
r
⊥
p
d
f
0 2
4 6
8 10−2 0
2
0
0.2
0.4
Time delay [ms]
Electron displacement distributions ‖ to B
∆r‖(τ )/σ‖(τ )
∆
r
‖
p
d
f
−3 −2 −1 0 1 2 3
0
0.1
0.2
0.3
0.4
0.5
∆r⊥(τ )/σ⊥(τ )
∆
r
⊥
p
d
f
Electron displacement distributions ⊥ to B
 
 
τ = 0.0ms
τ = 2.0ms
τ = 4.0ms
τ = 6.0ms
τ = 8.0ms
τ = 10.0ms
Gaussian
−3 −2 −1 0 1 2 3
0
0.1
0.2
0.3
0.4
0.5
∆r‖(τ )/σ‖(τ )
∆
r
‖
p
d
f
Electron displacement distributions ‖ to B
 
 
τ = 0.0ms
τ = 2.0ms
τ = 4.0ms
τ = 6.0ms
τ = 8.0ms
τ = 10.0ms
Gaussian
Figure 3.6 Same as Figure 3.3 but for the case of a simulated electron. All distributions
at all time delays are normalized to unit variance. Note that the distributions of the
displacements parallel to B become narrower than a Gaussian distribution in less than a
millisecond.
trons even though the displacement process seems non-Gaussian? To answer this question
we first attempted to fit (3.18) and (3.19) to the simulated variances of the electron dis-
placements (as we did earlier for the ion displacements). An example of our fitting results
is presented in Figure 3.7. In general, we found that the simulated variance data are well
fitted by the Brownian-motion expressions with the best-fit results of
ν‖ ≈ νe/i (3.24)
and
ν⊥ ≈ νe/i + νe/e, (3.25)
where
νe/e =
Ne e
4 ln Λe
12pi3/2 2om
2
e C
3
e
(3.26)
38
0 0.005 0.01 0.015 0.020
1
2
3
x 10−3
〈∆
r
2 ⊥
〉[
m
2
]
Variance of electron displacement ⊥ to B
 
 
〈|∆x|2〉
〈|∆y|2〉
Best fit
ν⊥ = 2440.90 Hz
0 0.005 0.01 0.015 0.020
2
4
6
Time delay [ms]
〈∆
r
2 ‖〉
[m
2
]
Variance of electron displacement ‖ to B
 
 
〈|∆z |2〉
Best fit
ν‖ = 1468.74 Hz
0 2 4 6 8 100
0.02
0.04
〈∆
r
2 ⊥
〉[
m
2
]
Variance of electron displacement ⊥ to B
 
 
〈|∆x|2〉
〈|∆y|2〉
Best fit
ν⊥ = 2440.90 Hz
0 2 4 6 8 100
1
2
x 105
Time delay [ms]
〈∆
r
2 ‖〉
[m
2
]
Variance of electron displacement ‖ to B
 
 
〈|∆z |2〉
Best fit
ν‖ = 1468.74 Hz
Figure 3.7 Same as Figure 3.4 but for the case of a simulated electron. In this case, the
results are displayed for time intervals of 0.02 ms (left panels) and 10 ms (right panels).
Note the different scales for the displacement variances.
and
νe/i =
√
2νe/e =
√
2Ne e
4 ln Λe
12pi3/2 2om
2
e C
3
e
(3.27)
are the Spitzer electron-electron and electron-ion collision frequencies [Callen, 2006] with
Λe ≡ 12piNeh2ehD and electron Debye length he defined in Table 3.2. For instance, in
Figure 3.7 the best-fit frequencies were ν‖ = 1.469 kHz and ν⊥ = 2.441 kHz, while νe/i =
1.439 kHz and νe/i + νe/e = 2.457 kHz.
Next we examined whether the Brownian single-particle ACF model (3.17) can be used
to fit the simulated electron ACF’s using the best-fit friction coefficients identified above. In
Figure 3.8 we present the single-electron ACF’s that were computed for different magnetic
aspect angles and λB = 3 m. The blue curves correspond to the ACF’s obtained from
our simulations, while the green curves are the electron ACF’s calculated using expression
(3.17) together with our approximations for ν‖ and ν⊥. Additionally, the electron ACF’s
for a collisionless magnetized plasma are also plotted (red curves). We note that the
simulated and the Brownian ACF’s matched almost perfectly at α = 0◦, and also that
the agreement is still good at very small magnetic aspect angles (see panels a, b, and
c). However, substantial differences between the Brownian and simulated ACF’s become
evident as the magnetic aspect angle increases (see panels d, e, and f).
In summary, our study revealed that the single-electron ACF’s needed for ISR spectral
39
0
50
10
0
15
0
20
0
25
0
30
0
0
0.
2
0.
4
0.
6
0.
81
T
im
e
[m
s]
(a
)
E
le
c
tr
o
n
A
C
F
(α
=
0
◦ )
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
0
50
10
0
15
0
20
0
25
0
30
0
0
0.
2
0.
4
0.
6
0.
81
T
im
e
[m
s]
(b
)
E
le
c
tr
o
n
A
C
F
(α
=
0
.0
1
◦ )
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
N
o
-c
o
ll
is
io
n
s
0
10
20
30
40
50
60
0
0.
2
0.
4
0.
6
0.
81
T
im
e
[m
s]
(c
)
E
le
c
tr
o
n
A
C
F
(α
=
0
.0
5
◦ )
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
N
o
-c
o
ll
is
io
n
s
0
5
10
15
20
25
0
0.
2
0.
4
0.
6
0.
81
T
im
e
[m
s]
(d
)
E
le
c
tr
o
n
A
C
F
(α
=
0
.1
◦ )
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
N
o
-c
o
ll
is
io
n
s
0
0.
5
1
1.
5
2
2.
5
0
0.
2
0.
4
0.
6
0.
81
T
im
e
[m
s]
(e
)
E
le
c
tr
o
n
A
C
F
(α
=
0
.5
◦ )
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
N
o
-c
o
ll
is
io
n
s
0
0.
2
0.
4
0.
6
0.
8
1
0
0.
2
0.
4
0.
6
0.
81
T
im
e
[m
s]
(f
)
E
le
c
tr
o
n
A
C
F
(α
=
1
◦ )
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
N
o
-c
o
ll
is
io
n
s
F
ig
ur
e
3.
8
Si
m
ul
at
ed
el
ec
tr
on
A
C
F
’s
fo
r
λ
B
=
3
m
at
di
ffe
re
nt
m
ag
ne
ti
c
as
pe
ct
an
gl
es
:
(a
)
α
=
0◦
,
(b
)
α
=
0.
01
◦ ,
(c
)
α
=
0.
05
◦ ,
(d
)
α
=
0.
1
◦ ,
(e
)
α
=
0.
5
◦ ,
an
d
(f
)
α
=
1◦
.
N
ot
e
th
e
di
ffe
re
nt
ti
m
e
sc
al
es
us
ed
in
ea
ch
pl
ot
.
40
models cannot be accurately modeled as a Brownian-motion process over all aspect angles.
The fundamental reason for this is the deviation of the electron displacements parallel to
B from Gaussian statistics, despite the fact that displacement variances are well fitted by
the Brownian model. Certainly, a non-Gaussian process cannot be fully characterized by
a model that specifies its first and second moments only; this is particularly true for the
estimation of the characteristic function of the process 〈ejk·∆re〉 that depends on all the
moments of the process distribution.
What is the physical reason for the electron displacements to be non-Gaussian in the
direction parallel to B at intermediate time delays? We believe the answer is related to
the fact that at low and high electron speeds (in comparison with Ce), the Fokker-Planck
collision coefficients are dominated by different types of physical processes corresponding
to electron-ion and electron-electron interactions, respectively. While the electron-electron
collisions dominant at high speeds give rise to random changes in the electron speed,
the same quantity is conserved in electron-ion collisions (due to the large mass difference
of electrons and ions) dominating at low speeds. This asymmetry leads to the forma-
tion of a low-speed population of electrons (roughly after one electron-electron collision
time) staying close to their starting locations for longer periods of time (during which
many electron-ion collisions may take place) than would have been expected in a (speed
independent) Brownian-motion model for Coulomb collisions. As a result, the electron dis-
placements at intermediate delays τ have distributions that are sharper than a Gaussian,
a shape which is consistent with an enhanced population at small electron displacements.
This population eventually disperses, and the distributions relax back to a Gaussian form
at large time delays corresponding to many electron-electron collision times, as would be
expected in view of the central limit theorem. This description of what we suspect is going
on is also consistent with ν‖ fitting best νe/i (as opposed to νe/i + νe/e) at the interme-
diate time scales of importance to our fitting procedure. Note that despite our best-fit
result ν‖ ≈ νe/i, electron-electron Coulomb collisions still play a crucial role in shaping the
single-electron ACF by determining the time scale over which the displacements parallel
to B return back to normal (Gaussian) distribution.
41
As for the absence of similar effects in the direction perpendicular to B, this can be
explained by the fact that the main role the collisions play in the dynamics of the electrons
in the perpendicular plane is to knock them off their gyrocenters, a process that is not
sensitive to the distinctions between electron-electron and electron-ion collisions (and hence
ν⊥ ≈ νe/i + νe/e as we have found out).
Returning back to Figure 3.8, the fact that the correlation time of the simulated electron
signal is longer than what is predicted by the other models is a signature effect of Coulomb
collisions encountered at small (but non-zero) aspect angles. As discussed in the next
chapter, this feature of the electron ACF determines the width of the electron Gordeyev
integral; therefore, it has an impact on the shape of the incoherent scatter spectrum
averaged over small magnetic aspect angles. Since the Brownian-motion model cannot
be used for the calculation of electron ACF’s and no other simplified model was found, a
numerical library of electron ACF’s had to be built, as already described in the previous
section. The collisional IS spectra to be presented in the next chapter were produced using
the numerical library.
42
CHAPTER 4
COLLISIONAL INCOHERENT SCATTER
SPECTRUM: RESULTS AND COMPARISONS
In the present chapter, ion and electron Gordeyev integrals are utilized in the gen-
eral framework formulation of Chapter 2 to compute incoherent scatter spectra that take
into account Coulomb collision effects. The Gordeyev integrals are computed from their
corresponding single-particle ACF’s using the chirp-Z transform algorithm detailed in the
previous chapter. The electron Gordeyev integral, which resembles the shape of the IS
spectrum in the direction perpendicular to B, is analyzed first in Section 4.1. The char-
acteristics of the collisional spectra are discussed in Section 4.2, where our results are also
compared to other IS spectrum models.
4.1 Simulated Electron Gordeyev Integrals
Figure 4.1 shows the plots of Re{Je(ω)}, the real part of the electron Gordeyev integral
proportional to 〈|nte(k, ω)|2〉, computed using our new numerical library as a function of
Doppler frequency and magnetic aspect angle for two different Bragg wavelengths, λB =
3 m and λB = 0.3 m. Notice the difference between the frequency axes used in these plots.
In the first one, we are considering a range of ω/2pi from −1 kHz to 1 kHz, while in the
second plot, the frequency range is ten times wider. A few degrees away from perpendicular
to B, where the effect of collisions can be neglected, the bandwidth of the electron Gordeyev
integral Je(ω) is proportional to the product of Bragg wavenumber k and electron thermal
speed Ce. Thus, if the radar wavelength is reduced by a factor of ten, the bandwidth of
Je(ω) will increase by the same factor. If the same were true for viewing directions close
43
Frequency [kHz]
A
sp
ec
t
a
n
g
le
[d
eg
]
Electron Gordeyev integral (λB = 3m)
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
R
e
{J
e
}[
d
B
]
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
Frequency [kHz]
A
sp
ec
t
a
n
g
le
[d
eg
]
Electron Gordeyev integral (λB = 0.3m)
−10 −5 0 5 100
0.1
0.2
0.3
0.4
0.5
R
e
{J
e
}[
d
B
]
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
(a) (b)
Figure 4.1 Simulated electron Gordeyev integrals as functions of Doppler frequency and
magnetic aspect angle for two radar Bragg wavelengths: (a) λB = 3 m and (b) λB = 0.3 m.
An O+ plasma with electron density Ne = 1012 m−3, temperatures Te = Ti = 1000 K, and
magnetic field Bo = 25µT is considered.
to perpendicular to B, the plots displayed above would have looked identical. However,
that is not the case, and our results illustrate that the dependence of Je(ω) on k and Ce
deviates from kCe at small aspect angles.
We observe that both Gordeyev integrals become narrower as the magnetic aspect
angle decreases. However, in the case of λB = 0.3 m, Re{Je(ω)} stops shrinking at an
angle around 0.02◦, while in the case of λB = 3 m, the same happens at a smaller angle
(∼ 0.01◦). This is another effect of Coulomb collisions, namely the saturation of electron
Gordeyev integral width at very small α. Without having considered collisions, Re{Je(ω)}
for α = 0◦ would have approached a Dirac delta function at ω = 0 (accompanied by a
series of smaller deltas located at multiples of the electron gyrofrequency Ωe).
Trying out the Brownian-motion expression for the electron ACF as a guide, we noticed
that the bandwidth of Re{Je(ω)} in the limit α = 0◦ varies according to
k2C2e
ν⊥
ν2⊥ + Ω2e
. (4.1)
This expression is obtained by placing (3.19) into (3.17) and considering ν⊥τ  1, a limit
in which the electron ACF becomes an exponential function. Notice that the bandwidth
44
Figure 4.2 Simulated incoherent scatter spectra as a function of magnetic aspect angle
and Doppler frequency for λB = 3 m (e.g., Jicamarca radar Bragg wavelength). An O+
plasma with physical parameters given in Table 3.1 is considered.
dependence on kCe has changed from linear (at relatively large α) to quadratic (at very
small α), a change that is related to the transition between the collisionless and collisional
regimes of the electron diffusion. Furthermore, using ν⊥ ≈ νe/i+ νe/e from the last section
and taking Ωe  ν⊥, we can verify that bandwidth dependence (4.1) is proportional to
Ne√
Te
. (4.2)
Since at very small aspect angles the electron Gordeyev integral dominates the shape of
the overall incoherent scatter spectrum, we expect the IS spectral width to exhibit the
same dependence (4.2) on density and temperature. We will demonstrate this to be the
case by specific examples shown in the following section.
4.2 Simulated Collisional IS Spectra
Figure 4.2 shows a surface plot constructed from full IS spectrum calculations for
λB = 3 m (e.g., Jicamarca radar) as a function of aspect angle α and Doppler frequency
45
ω/2pi. In this self-normalized surface plot, we observe how the IS spectrum sharpens
significantly at small aspect angles. For instance, just in the range between 0.1◦ and 0◦,
the amplitude of the spectrum becomes ten times larger while its bandwidth is reduced by
the same factor.
Sample cuts of the surface plot taken at a number of magnetic aspect angles are dis-
played in Figure 4.3. In these plots, the simulated spectra (blue curves) are compared
to other spectral models. The green curves correspond to spectra computed using the
Brownian-motion model discussed in the previous section, while the red curves correspond
to the collisionless electron case (i.e., the case when Coulomb collisions for the electrons are
neglected but collisions for the ions are still considered). We note that the agreement with
the Brownian-motion spectra is excellent at very small aspect angles (α ≤ 0.01◦). However,
as α increases, our collisional spectra become narrower than the Brownian-motion spectra.
We also observe that the collisionless spectra are in general wider than our simulation
results. Additionally, note that at very small aspect angles, the collisionless spectrum has
a humped shape (similar to the one expected at large aspect angles but with a much nar-
rower spectral width). In this regime, electrons and ions exchange their roles in defining
the shape of the collisionless spectrum so that the spectral width becomes proportional to
the electron thermal speed Ce and sinα, as previously discussed by Kudeki et al. [1999].
In the limit of α → 0◦, the collisionless spectrum develops a delta function ω = 0
implying an infinite signal correlation time in the collisionless limit. The reason for this
is that in the absence of collisions, the magnetic field restricts the motion of the electrons
in the plane perpendicular to B, forcing them to gyrate always around the same magnetic
field lines. With collisions, the electrons manage to diffuse across the field lines, and
consequently the correlation time of the IS signal becomes finite and its spectrum broadens
out of its delta function limit. However, collisions do not cause spectral broadening at all
aspect angles. Notice that in Figure 4.3 the collisional spectrum for α > 0.01◦ is narrower
than the collisionless spectrum, implying that collisions in this range of aspect angles have
caused an increase of the IS signal correlation time. The explanation for this is that at
aspect angles larger than 0.01◦ the shape of the IS spectrum is dominated by electron
46
0
5
10
15
20
0246
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/N
e
(a
)
IS
-S
P
C
(α
=
0
◦ )
×
 
10
−
2
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
W
o
o
d
m
a
n
[2
0
0
4
]
0
5
10
15
20
0123456
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/N
e
(b
)
IS
-S
P
C
(α
=
0
.0
1
◦ )
×
 
10
−
2
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
N
o
-c
o
ll
is
io
n
s
W
o
o
d
m
a
n
[2
0
0
4
]
0
25
50
75
10
0
0
0.
51
1.
5
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/N
e
(c
)
IS
-S
P
C
(α
=
0
.0
5
◦ )
×
 
10
−
2
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
N
o
-c
o
ll
is
io
n
s
W
o
o
d
m
a
n
[2
0
0
4
]
0
50
10
0
15
0
20
0
25
0
0123456
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/N
e
(d
)
IS
-S
P
C
(α
=
0
.1
◦ )
×
 
10
−
3
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
N
o
-c
o
ll
is
io
n
s
W
o
o
d
m
a
n
[2
0
0
4
]
0
25
0
50
0
75
0
10
00
12
50
0
0.
2
0.
4
0.
6
0.
81
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/N
e
(e
)
IS
-S
P
C
(α
=
0
.5
◦ )
×
 
10
−
3
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
N
o
-c
o
ll
is
io
n
s
S
&
G
[1
9
9
9
]
0
30
0
60
0
90
0
12
00
15
00
0123456
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/N
e
(f
)
IS
-S
P
C
(α
=
1
◦ )
×
 
10
−
4
S
im
u
la
ti
o
n
B
ro
w
n
ia
n
N
o
-c
o
ll
is
io
n
s
S
&
G
[1
9
9
9
]
F
ig
ur
e
4.
3
Sa
m
pl
e
cu
ts
of
th
e
si
m
ul
at
ed
sp
ec
tr
a
of
F
ig
ur
e
4.
2
ta
ke
n
at
di
ffe
re
nt
m
ag
ne
ti
c
as
pe
ct
an
gl
es
:
(a
)
α
=
0◦
,
(b
)
α
=
0.
01
◦ ,
(c
)
α
=
0.
05
◦ ,
(d
)
α
=
0.
1
◦ ,
(e
)
α
=
0.
5◦
,
an
d
(f
)
α
=
1◦
.
O
ur
si
m
ul
at
io
n
re
su
lt
s
(b
lu
e
lin
es
)
ar
e
co
m
pa
re
d
to
IS
sp
ec
tr
a
co
m
pu
te
d
us
in
g
th
e
B
ro
w
ni
an
-m
ot
io
n
m
od
el
(g
re
en
lin
es
),
th
e
co
lli
si
on
le
ss
el
ec
tr
on
m
od
el
(r
ed
lin
es
),
th
e
si
m
ul
at
io
n
re
su
lt
s
of
Su
lz
er
an
d
G
on
zá
le
z
[1
99
9]
(l
ig
ht
-b
lu
e
lin
es
),
an
d
th
e
as
pe
ct
an
gl
e
de
pe
nd
en
t
co
lli
si
on
fr
eq
ue
nc
y
m
od
el
of
W
oo
dm
an
[2
00
4]
(m
ag
en
ta
lin
es
).
N
ot
e
th
e
di
ffe
re
nt
fr
eq
ue
nc
y
sc
al
es
us
ed
in
ea
ch
pl
ot
.
47
diffusion in the direction parallel to B. As collisions impede the free motion of particles,
electrons diffuse slower in a collisional plasma than in a collisionless one, which implies
that the electrons stay closer to their original locations for longer periods of time. As a
result, the correlation time of the signal scattered by the electrons also becomes longer,
causing the broadening of the IS signal ACF and the associated narrowing of the signal
spectrum in this aspect angle regime, as first explained by Sulzer and González [1999].
Also in Figure 4.3, we compare our simulated spectra to the results of Sulzer and
González [1999] for α ≥ 0.5◦ (see panels e and f). The comparison is good except for
a minor offset which was traced to a minor coding error in the simulation program used
by Sulzer and González [1999]. At smaller aspect angles, comparisons are made with the
spectrum model of Woodman [2004], which is effectively an ad hoc extrapolation of the
Sulzer and González [1999] model to very small magnetic aspect angles (including the
α→ 0◦ limit). Woodman [2004] fitted the Brownian-motion spectral model to the results
of Sulzer and González [1999] and developed an empirical formula for the electron collision
frequency that depends on the magnetic aspect angle. As can be observed in panels a, b,
c, and d, our simulated spectra are not quite as sharp as Woodman’s spectra (though the
differences are small). These differences can be expected since Woodman [2004] did not
really have the spectra to fit at very small aspect angles; therefore, his collision frequency
formula may require a little adjustment for α < 0.1◦.
Figure 4.4 shows the dependence of the simulated IS spectra on the mean electron
density Ne that controls the Coulomb collision rates. As discussed in the previous section,
we expect the spectrum at α = 0◦ to become wider as Ne grows (see relation (4.2)). This
behavior is clearly illustrated in the left panel of Figure 4.4. As Ne increases, collisions
become more frequent and consequently the electrons diffuse out to longer distances across
the magnetic field lines. Hence, the spectrum bandwidth increases with Ne, so long as
the collision frequency remains smaller than the gyrofrequency, otherwise the collisions
start impeding electron motion rather than just perturbing the gyromotions to facilitate
diffusion. In a highly collisional regime, characterized by collisional forces much larger
than magnetic forces (or by ν⊥  Ωe), electrons will find it very difficult to move in any
48
0 1 2 3 4 50
1
2
3
4
5
Frequency (ω/2pi)
〈|n
e
(k
,ω
)|2
〉/
N
e
IS-SPC (α = 0◦)
× 10−1
log10Ne = 11.0
log10Ne = 11.5
log10Ne = 12.0
log10Ne = 12.5
0 50 100 150 2000
2
4
6
Frequency (ω/2pi)
〈|n
e
(k
,ω
)|2
〉/
N
e
IS-SPC (α = 0.1◦)
× 10−3
No-collisions
log10Ne = 11.0
log10Ne = 11.5
log10Ne = 12.0
log10Ne = 12.5
0 500 1000 15000
2
4
6
Frequency (ω/2pi)
〈|n
e
(k
,ω
)|2
〉/
N
e
IS-SPC (α = 1◦)
× 10−4
No-collisions
log10Ne = 11.0
log10Ne = 11.5
log10Ne = 12.0
log10Ne = 12.5
Figure 4.4 Electron density dependence of the simulated IS spectra for λB = 3 m at
aspect angles α = 0◦ (left panel), α = 0.1◦ (central panel), and α = 1◦ (right panel). An
O+ plasma in thermal equilibrium with Te = Ti = 1000 K and Bo = 25µT is considered.
Note the different frequency scales used in each plot.
direction. As a result, the bandwidth of the IS spectrum at α = 0◦ will stop increasing
with Ne and eventually it will decrease. A similar effect can be observed as the aspect
angle increases. A few hundredths of a degree away from perpendicular to B, the shape
of the IS spectrum starts being controlled by diffusion along the magnetic field lines. In
this regime, the spectrum becomes narrower as Ne grows because the increasing number
of collisions impede the movement of the electrons (as already discussed above). This
behavior is observed in the central panel of Figure 4.4. Also note that the collisional
spectrum approaches the collisionless spectrum as Ne is reduced, which is consistent with
the dependence of collision frequencies on Ne. Finally, a comparison of the central and
rightmost panels of Figure 4.4 shows that collision effects become less significant at even
larger aspect angles where the spectrum is shaped by ion dynamics. In that regime, the
spectral shapes become independent of Ne as long as khe  1.
The temperature dependence of the simulated IS spectra at different aspect angles will
be examined with the help of Figure 4.5. Considering an O+ plasma with Ne = 1012 m−3
and Ti = 1000 K, we have computed the IS spectra for a Bragg wavelength of λB = 3 m
and for a set of electron temperatures Te (1000 K, 1500 K, 2000 K, 2500 K, and 3000 K).
Figure 4.5 displays the results for three different aspect angles, α = 0◦ (left panels),
α = 0.02◦ (center panels), and α = 0.1◦ (right panels). Notice that the same spectra are
plotted in the top and bottom panels, but in the bottom panels we have normalized each
spectrum by its peak value in order to emphasize the changes in spectral widths.
49
0
2
4
6
8
10
0
0.
51
1.
52
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/N
e
IS
-S
P
C
(α
=
0
◦ )
×
 
10
−
1
T
e
/
T
i
=
1
.0
T
e
/
T
i
=
1
.5
T
e
/
T
i
=
2
.0
T
e
/
T
i
=
2
.5
T
e
/
T
i
=
3
.0
0
10
20
30
40
50
0123
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/N
e
IS
-S
P
C
(α
=
0
.0
2
◦ )
×
 
10
−
2
T
e
/
T
i
=
1
.0
T
e
/
T
i
=
1
.5
T
e
/
T
i
=
2
.0
T
e
/
T
i
=
2
.5
T
e
/
T
i
=
3
.0
0
50
10
0
15
0
20
0
012345
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/N
e
IS
-S
P
C
(α
=
0
.1
◦ )
×
 
10
−
3
T
e
/
T
i
=
1
.0
T
e
/
T
i
=
1
.5
T
e
/
T
i
=
2
.0
T
e
/
T
i
=
2
.5
T
e
/
T
i
=
3
.0
0
2
4
6
8
10
0
0.
2
0.
4
0.
6
0.
81
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/〈|n
e
(k,0)|
2
〉
IS
-S
P
C
(α
=
0
◦ ) T
e
/
T
i
=
1
.0
T
e
/
T
i
=
1
.5
T
e
/
T
i
=
2
.0
T
e
/
T
i
=
2
.5
T
e
/
T
i
=
3
.0
0
10
20
30
40
50
0
0.
2
0.
4
0.
6
0.
81
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/〈|n
e
(k,0)|
2
〉
IS
-S
P
C
(α
=
0
.0
2
◦ )
T
e
/
T
i
=
1
.0
T
e
/
T
i
=
1
.5
T
e
/
T
i
=
2
.0
T
e
/
T
i
=
2
.5
T
e
/
T
i
=
3
.0
0
50
10
0
15
0
20
0
0
0.
2
0.
4
0.
6
0.
81
F
re
q
u
e
n
c
y
(ω
/
2
pi
)
〈|n
e
(k,ω)|
2
〉/〈|n
e
(k,0)|
2
〉
IS
-S
P
C
(α
=
0
.1
◦ ) T
e
/
T
i
=
1
.0
T
e
/
T
i
=
1
.5
T
e
/
T
i
=
2
.0
T
e
/
T
i
=
2
.5
T
e
/
T
i
=
3
.0
F
ig
ur
e
4.
5
T
e
/
T
i
de
pe
nd
en
ce
of
th
e
si
m
ul
at
ed
IS
sp
ec
tr
a
fo
r
λ
=
3
m
at
as
pe
ct
an
gl
es
α
=
0◦
(l
ef
t
pa
ne
ls
),
α
=
0.
02
◦
(c
en
tr
al
pa
ne
ls
),
an
d
α
=
0
.1
◦
(r
ig
ht
pa
ne
ls
).
T
he
sa
m
e
sp
ec
tr
a
ar
e
pl
ot
te
d
on
th
e
to
p
an
d
bo
tt
om
pa
ne
ls
;
ho
w
ev
er
,
on
th
e
bo
tt
om
,
ev
er
y
sp
ec
tr
um
ha
s
be
en
no
rm
al
iz
ed
to
it
s
pe
ak
va
lu
e.
A
n
O
+
pl
as
m
a
w
it
h
N
e
=
10
1
2
m
−3
,
T
i
=
10
00
K
,
an
d
B
o
=
25
µ
T
is
co
ns
id
er
ed
.
E
ac
h
cu
rv
e
co
rr
es
po
nd
s
to
di
ffe
re
nt
va
lu
es
of
el
ec
tr
on
te
m
pe
ra
tu
re
s
T
e
(1
00
0
K
,1
50
0
K
,2
00
0
K
,2
50
0
K
,a
nd
30
00
K
).
N
ot
e
th
e
di
ffe
re
nt
fr
eq
ue
nc
y
sc
al
es
us
ed
in
ea
ch
co
lu
m
n.
50
Figure 4.5 includes a wealth of information pertinent to the ultimate application of our
theory in estimating the ionospheric temperatures from Jicamarca radar data collected
with beams perpendicular to B. First, we note that the left panels of Figure 4.5 show
how the spectrum bandwidth at α = 0◦ is inversely proportional to the square root of
Te as discussed earlier. Second, we note that this dependence of spectral width on Te
changes at larger aspect angles as shown in the remaining panels of the figure — for aspect
angles larger than about α = 0.01◦, spectral width increases with increasing Te. Third,
the spectral amplitudes change rapidly with aspect angle. For instance, in the case of
Te = Ti = 1000 K, the peak of the spectrum at α = 0◦ is about ten times larger than the
one at α = 0.1◦.
Finally, the amplitude differences for different α are enhanced at higher values of Te
(in comparison with Ti), which is related to the aspect angle dependence of the volumetric
radar cross section (RCS) [e.g., Farley , 1966; Milla and Kudeki , 2006]
σv ≡ 4pir2eNeη(k) (4.3)
of incoherent scatter radar echoes. In (4.3), re is the classical electron radius and η(k) is
an electron scattering efficiency factor [see Milla and Kudeki , 2006] defined as
η(k) ≡
ˆ
dω
2pi
〈|ne(k, ω)|2〉
Ne
, (4.4)
which is primarily dependent on the temperature ratio Te/Ti and magnetic aspect angle α.
A plot of this factor obtained from our collisional simulation results is shown in Figure 4.6.
As we can observe, if the plasma is in thermal equilibrium (i.e., if Te = Ti), this factor is
1/2 at all angles α. We can also appreciate that η(k) at α = 0◦ is clearly increasing as
Te/Ti grows. On the other hand, at very large magnetic aspect angles, we can see that
the efficiency factor decreases as Te/Ti increases. In particular, note that our calculations
for α = 90◦ match the well-known formula (1 + Te/Ti)−1, as expected for moderate values
of Te/Ti and negligible Debye length [e.g., Farley , 1966]. Note that for α ≈ 1◦ the factor
is approximately independent of Te/Ti, but otherwise it increases and decreases with the
51
1 1.5 2 2.5 30.2
0.3
0.4
0.5
0.6
0.7
0.8
Te/Ti
η
(k
)
α = 0.
00
◦
0.10
◦
0.25
◦
0.50
◦
1.00◦
2.00◦
5.00◦
90.00◦
Figure 4.6 Electron scattering efficiency factor η(k) resulting from the frequency integra-
tion of the simulated incoherent scatter spectra as a function of electron-to-ion temperature
ratio Te/Ti and magnetic aspect angle α. An O+ plasma withNe = 1012 m−3, Ti = 1000 K,
and Bo = 25µT is considered.
temperature ratio at small and large aspect angles, respectively. The efficiency factor η(k)
plays an important role in Ne estimation from ISR power data, and its accurate determi-
nation in the collisional regime at small aspect angles is one of the main contributions of
this work.
The next stage of our studies, which is described in the following chapters, is the
modeling of incoherent scatter spectrum measurements carried out with Jicamarca antenna
beams pointed perpendicular to B. For this purpose, the measured spectrum has to be
carefully modeled because within the range of small magnetic aspect angles illuminated
by the antenna beams, the theoretical IS spectra vary quite rapidly. In addition, magneto-
ionic propagation effects are also considered into the model as these effects are significant at
50 MHz. This further complicates the description of Jicamarca measurements as different
modes of propagation are excited within the width of the antenna beams; these modes vary
from linearly polarized (at α = 0◦) to circularly polarized (at α > 1◦) with the transition
between these two regimes happening at an angle of approximately 0.5◦. The modeling of
the beam-weighted incoherent scatter radar spectrum including magneto-ionic propagation
and collisional effects is the subject of the next chapter of this dissertation.
52
CHAPTER 5
MAGNETO-IONIC PROPAGATION EFFECTS
ON THE INCOHERENT SCATTER RADAR
MEASUREMENTS
A radiowave propagating through the ionosphere experiences changes in its polarization
caused by the presence of the Earth’s magnetic field. In this chapter, a model for incoherent
scatter spectrum and cross-spectrum measurements that takes into account magneto-ionic
propagation effects is developed.
A mathematical description of radiowave propagation in a homogeneous magneto-
plasma is derived in the first section of this chapter. This basic formulation is extended to
the case of an inhomogeneous ionosphere in Section 5.2. The resultant wave propagation
model is used in Section 5.3 to reformulate the soft-target radar equation of Chapter 2 in
order to account for the magneto-ionic propagation effects on incoherent scatter spectrum
and cross-spectrum measurements.
5.1 Propagation of Electromagnetic Waves in a
Homogeneous Magnetoplasma
In this section, the characteristic modes of electromagnetic wave propagation in a ho-
mogeneous magnetoplasma are derived. We start by formulating Maxwell’s curl equations
53
in differential form
∇×E = −∂B
∂t
, (5.1)
∇×H = J + ∂D
∂t
. (5.2)
Considering wave field solutions proportional to ejωt−jk·r, where ω is the wave frequency
and k is the wave propagation vector, we can reformulate the previous equations in plane
wave form as
k×E = ωµoH, (5.3)
−jk×H = J + jωoE, (5.4)
where the differential operators ∇ and ∂/∂t have been replaced by −jk and jω, respec-
tively. Above, we have considered B = µoH and D = oE, which is a valid assumption as
long as current J includes components that are due to free and/or bound charged carri-
ers represented in terms of conductivity and/or susceptibility tensors. Eliminating H by
taking k× k×E, we have that
k× k×E = jωµo [J + jωoE] . (5.5)
Above, the propagation vector can be written as k = kon kˆ, where ko ≡ ω√µoo is the
wavenumber in free-space, n is the refractive index of the medium, and kˆ is the unit vector
parallel to the propagation direction. Now, we can rewrite (5.5) in the following form
n2 kˆ× kˆ×E︸ ︷︷ ︸
−E⊥
= − 1
jωo
J−E, (5.6)
which simplifies to
E− n2E⊥ = − 1
jωo
J, (5.7)
54
where E⊥ is the component of E that is perpendicular to kˆ. Without loss of generality, we
consider kˆ = zˆ; thus, breaking the previous equation into its components, we have that
(1− n2)Ex = − Jx
jωo
, (5.8)
(1− n2)Ey = − Jy
jωo
, (5.9)
Ez = − Jz
jωo
. (5.10)
Based on these equations, we can define the following polarization relations
R =
Ex
Ey
=
Jx
Jy
(5.11)
S =
Ez
Ex
=
Jz
Jx
(1− n2). (5.12)
In addition, the refractive index can be written as
n2 = 1 +
Jx/Ex
jωo
= 1 +
Jy/Ey
jωo
. (5.13)
In order to obtain the refractive index of the medium, we need to define the relations Jx/Ex
or Jy/Ey from an appropriate conductivity model. For instance, in an isotropic conductor,
the current and electric field vectors are related by J ≡ σE, where σ is the conductivity.
Thus, Jx/Ex = Jy/Ey = σ, which implies that n2 = 1 + σjωo , as is well known. Note that
J = σE also implies that there is no coupling between Ex and Ey. Therefore, R could
take any value, but S is equal to zero because there is no current along the propagation
direction.
In order to determine the conductivity of a magnetized plasma with a DC magnetic
field Bo, let us consider an electron moving in this medium under the influence of the
Lorentz force. Then, the equation of motion is given by
me
dv
dt
= −e(E + v ×Bo), (5.14)
where me and −e are the mass and charge of the electron. In phasor form, the motion
55
Bo = By yˆ + Bz zˆ
k θ
Ek
Eφ
Eθ
x
y
z
θˆ = −yˆ
φˆ = xˆ
Figure 5.1 Coordinate system used for analyzing wave propagation in a magnetized
plasma. The magnetic field Bo is on the yz-plane at an angle θ from the propagation
vector k which is parallel to the zˆ-direction. The wave field E has three mutually orthog-
onal components Ek, Eθ, and Eφ in directions kˆ = zˆ, θˆ = −yˆ, and φˆ = xˆ, respectively.
equation becomes
jωmev = −e(E + v ×Bo), (5.15)
which can be written as
− e
me
E = jωv +
e
me
v ×Bo. (5.16)
Next, multiplying both sides by −eNe and defining J ≡ −eNev, we have that
Nee
2
jωme
E = J− e
jωme
Bo × J. (5.17)
In the coordinate system of Figure 5.1, we have considered Bo = Byyˆ +Bzzˆ with compo-
nents By = Bo sin θ and Bz = Bo cos θ, where Bo is the magnitude of the field and θ is the
angle with respect to the z-axis. Now, we can recast the previous equation as
− jωoXE = J + jYT yˆ × J + jYLzˆ× J (5.18)
in which
X ≡ ω
2
p
ω2
, YL ≡ Ωe
ω
cos θ, and YT ≡ Ωe
ω
sin θ, (5.19)
where ωp ≡
√
Neq2e/ome is the plasma frequency and Ωe = eBo/me is the electron
56
gyrofrequency. Expanding equation (5.18) into its vector components, we have that
−jωoXEx = Jx + jYTJz − jYLJy, (5.20)
−jωoXEy = Jy + jYLJx, (5.21)
−jωoXEz = Jz − jYTJx. (5.22)
Early in this section, we found that Jz + jωoEz = 0. Then, using this result together
with (5.22), we can find that
Jz
Jx
= j
YT
1−X . (5.23)
Next, using the polarization relation R = Ex/Ey = Jx/Jy and the previous expression, we
can rewrite equations (5.20) and (5.21) in the following forms
−jωoXEx
Jx
= 1− Y
2
T
1−X − jYL
1
R
, (5.24)
−jωoXEy
Jy
= 1 + jYLR. (5.25)
Noting that Ex/Jx = Ey/Jy, we combine the equations above and find that
jYLR+
Y 2T
1−X + jYL
1
R
= 0, (5.26)
an equation that can be written as
jYL(1−X)R2 + Y 2TR+ jYL(1−X) = 0. (5.27)
Defining F ≡ −jYLR, the previous equation becomes
(1−X)F 2 − Y 2T F − (1−X)Y 2L = 0. (5.28)
Solutions of (5.28) are given by
F± =
Y 2T ±
√
Y 4T + 4Y
2
L (1−X)2
2 (1−X) , (5.29)
57
which are valid for any given ω and direction θ. In order to find the refractive indices, note
that
Ey
Jy
= −1 + jYLR
jωoX
= − 1− F
jωoX
. (5.30)
Thus, inserting this relation into (5.13), we obtain the following refractive indices
n2± = 1−
X
1− F± , (5.31)
which characterize the two possible modes of propagation in a magnetoplasma. This result
is known as the Appleton-Hartree equation [e.g., Budden, 1961]. Each of these refractive
indices, n+ and n−, is accompanied by an associated polarization vector. The relations
between the components of these vectors are given by the polarization relations R and S
defined before. In order to formulate these relations, we considered a coordinate system in
which kˆ = zˆ, θˆ = −yˆ, and φˆ = xˆ. In this system, the electric field components are given
by Ek, Eθ, and Eφ. Thus, after some math, we can find the following relations
Ek
Eθ
= −SR = F
2± − Y 2L − F±Y 2T
YLYT (1− F±) =
F±XYT
(1− F±) (1−X)YL , (5.32)
Ek
Eφ
= S = j
F 2± − Y 2L − F±Y 2T
YTF± (1− F±) = j
XYT
(1− F±) (1−X) , (5.33)
Eθ
Eφ
= − 1
R
= j
YL
F±
, (5.34)
where Y =
√
Y 2L + Y
2
T = Ωe/ω.
Let us discuss now some particular cases. In the limit of θ = 0◦ corresponding to
propagation parallel to the ambient magnetic field Bo (i.e., longitudinal propagation), we
have that YL = Y and YT = 0. Thus, F+ = Y and F− = −Y, which implies that refractive
indices (5.31) simplify to
n+ =
√
1− X
1− Y and n− =
√
1− X
1 + Y
. (5.35)
The polarization vectors associated with these modes are given by θˆ−jφˆ√
2
and θˆ+jφˆ√
2
, which
correspond to right- and left-circularly polarized waves. Note that the polarization vectors
58
are transverse to the propagation direction. Additionally, in the limit of θ = 90◦ corre-
sponding to propagation perpendicular to Bo (i.e., transverse propagation), we have that
YL = 0 and YT = Y. This implies that F+ = Y
2
1−X and F− = 0, which leads to the following
expressions for the refractive indices:
n+ =
√
1− X
1− F+ and n− =
√
1−X. (5.36)
While the polarization of the mode associated with n− is linear and parallel to the zˆ-
direction (i.e., parallel to Bo), the polarization vector of n+ has components in kˆ- and
φˆ-directions. If X  1 and Y  1, the kˆ-component can be considered negligible and the
polarization becomes linear and parallel to φˆ (a direction that is perpendicular to Bo).
Note that n− is equal to the refractive index of a plasma without magnetic field; because
of this, the modes associated with n− and n+ are known as the ordinary and extraordinary
modes (O- and X-modes), respectively.
In general, for propagation at intermediate angles θ, the propagation modes are ellip-
tically polarized. However, if X  1 and Y  1, e.g., the case of VHF propagation in the
ionosphere, the propagation modes can be considered quasi-longitudinal for a wide range
of angles θ. The approximation is valid as long as
Y 4T  4Y 2L (1−X)2 ⇒ Y 2T  2YL (1−X) , (5.37)
which implies the following condition:
tan θ sin θ  21−X
Y
≈ 2
Y
. (5.38)
Note that, since sin θ ≤ 1, a more stringent condition is tan θ  2/Y ; thus, we can define
a critical angle
θc ≡ tan−1
(
2
Y
)
(5.39)
that corresponds to the transition between longitudinal and transverse modes of propa-
gation. For instance in the case of Bo = 25µT and f = ω/2pi = 50 MHz, the electron
59
gyrofrequency is Ωe/2pi = 699.8 kHz and the critical angle is about 89.6◦. At Jicamarca,
antenna beams pointed perpendicular to Bo have a beam-width of the order of a degree;
thus, different modes of propagation ranging from longitudinal to transverse are excited
within the range of aspect angles illuminated by the beam.
As discussed above, in the presence of an ambient magnetic field, there are two possible
modes of electromagnetic wave propagation. Labeling the modes as ordinary and extraor-
dinary and denoting the refractive indices given in (5.31) as nO = n− and nX = n+, we
can express the transverse component of a propagating electric field as
Et = AO
(
θˆ − jφˆFO
YL
)
e−jkonOr +AX
(
θˆ − jφˆFX
YL
)
e−jkonXr, (5.40)
where AO and AX are the amplitudes of the O- and X-waves. Above, r is the distance
from the origin along the propagation direction kˆ and ko ≡ ω√µoo. In matrix notation,
we can express (5.40) as
 Eθ
Eφ
 =
 e−jkonOr e−jkonXr
−j FOYL e−jkonOr −j
FX
YL
e−jkonXr

 AO
AX
 , (5.41)
where Eθ and Eφ are the transverse field components in θˆ and φˆ directions. Defining Eθ,o
and Eφ,o as the field components at the origin, we have that
 Eθ,o
Eφ,o
 =
 1 1
−j FOYL −j
FX
YL

 AO
AX
 . (5.42)
Then, noting that FOFX
Y 2L
= −1, the propagating electric field (5.41) can be recast as
 Eθ
Eφ
 =

e−jkonOr + a2e−jkonXr
1 + a2
ja
(
e−jkonOr − e−jkonXr)
1 + a2
−ja (e−jkonOr − e−jkonXr)
1 + a2
a2e−jkonOr + e−jkonXr
1 + a2

 Eθ,o
Eφ,o
 ,
(5.43)
where the parameter a ≡ FOYL = −
YL
FX
determines the type of wave polarization and can
take values within the range 0 ≤ |a| ≤ 1. Note that the limits 0 and 1 correspond to the
60
cases of linearly and circularly polarized propagation modes. Defining n¯ ≡ nO+nX2 and
∆n ≡ nO−nX2 , we can rewrite (5.43) as Eθ
Eφ
 = e−jkon¯r
1 + a2
 e−jko∆nr + a2ejko∆nr 2a sin(ko∆nr)
−2a sin(ko∆nr) a2e−jko∆nr + ejko∆nr

︸ ︷︷ ︸
T¯
 Eθ,o
Eφ,o
 ,
(5.44)
where T¯ is a propagator matrix that maps the fields at the origin into the fields at a
distance r. Note that in the case of waves traveling in −kˆ direction, the same matrix T¯
can be used to propagate the fields from a distance r to the origin.
The propagator matrix can be eigenvalue-decomposed as T¯ = P¯ S¯ P¯H where
S¯ = e−jkon¯r
 e−jko∆nr 0
0 ejko∆nr
 (5.45)
is a diagonal matrix composed of the eigenvalues of T¯, and
P¯ =
1√
1 + a2
 1 −ja
−ja 1
 (5.46)
is an unitary matrix with columns equal to the eigenvectors of T¯ (corresponding to the
O- and X-modes, respectively). Returning to the xyz-coordinate system, we can express
the transverse electric field as Et = Eθ θˆ + Eφ φˆ. Then, using (5.44) and the eigenvalue
decomposition of T¯, we can show that
Et = e−jkon¯r
[
e−jko∆nrpOpHO + e
jko∆nrpXp
H
X
]
Eto, (5.47)
where Eto is the wave field at the origin, and
pO =
θˆ − ja φˆ√
1 + a2
and pX =
−ja θˆ + φˆ√
1 + a2
(5.48)
are the polarization vectors of the O- and X-waves expressed in cartesian coordinates.
61
This form of the transverse field is suitable for computational evaluation because it only
involves vector operations and there is no need to explicitly build the propagator matrix.
5.2 Model for Radiowave Propagation in an Inhomogeneous
Ionosphere
A radiowave propagating through the ionosphere will experience refraction and polar-
ization effects. Wave refraction is caused by the altitudinal variation of the refractive index
of the medium, which in this case is dependent on the amount of ionospheric electron den-
sity. Additionally, wave polarization variations are caused by the presence of the Earth’s
magnetic field, which is the controlling agent of the anisotropic nature of the ionospheric
plasma.
At VHF frequencies and for oblique propagation, refraction effects can be considered
negligible for most propagation directions because the plasma frequency ωp at ionospheric
altitudes is smaller than the wave frequency ω (i.e., X  1). It can be verified that this
assumption is valid as long as the propagation angle ψ (measured with respect to the
vertical axis) satisfies the condition
tanψ  2
X
. (5.49)
Thus, refraction might have an impact only at very low elevation angles (an angular regime
that is not of our interest). But for the same set of frequencies, polarization changes are
still significant despite the fact that the electron gyrofrequency Ωe is much smaller than
the wave frequency ω (i.e., Y  1). The reason for this is that the distances traveled by
the propagating fields are long enough (hundreds of kilometers) so that phase differences
between wave components propagating in distinct modes accumulate to detectable levels.
Taking these elements into consideration and noting that, at VHF frequencies, the longi-
tudinal components of the wave fields are negligibly small (as X  1 and Y  1), waves
propagating through the ionosphere can be represented as TEM (transverse electromag-
netic) waves.
62
zBi ∆r
θi
ri+1
ri
ri−1
r3
r2
r1 k
yx
kˆ
ψ
Figure 5.2 Geometry of wave propagation in an inhomogeneous magnetized ionosphere.
Consider plane wave propagation in an inhomogeneous magnetized ionosphere in an
arbitrary direction kˆ. To model the electric field of the propagating wave, we can divide
the ionosphere in slabs of equal width perpendicular to the propagation direction such
that within each slab the physical plasma parameters (as electron density, electron and
ion temperatures, and magnetic field) can be considered constants (see Figure 5.2).1 The
transverse component of the wave electric field propagates from the bottom to the top of
the i-th slab according to (5.47),
Ei = e
−jkon¯i∆r
[
e−jko∆ni∆rpOpHO + e
jko∆ni∆rpXp
H
X
]
︸ ︷︷ ︸
T¯i
Ei−1, (5.50)
which is the superposition of the O- and X-modes of magneto-ionic propagation detailed in
the previous section. Above, T¯i denotes the i-th propagator matrix (expressed in cartesian
coordinates), where ko ≡ 2pi/λo is the free-space wavenumber and ∆r is the width of the
slab, and where n¯i ≡ nO,i+nX,i2 and ∆ni ≡
nO,i−nX,i
2 are the mean and half difference
between the refractive indices of the propagation modes in the i-th layer. The polarization
1In the ionosphere, electron density and plasma temperatures can be considered to be functions of
altitude f(z). Thus, the values of these physical parameters at any position r from a radar placed at the
origin are given by f(r cosψ) where r is the radar range and ψ is the zenith angle.
63
vectors of the O- and X-modes are
pO =
θˆi − jai φˆi√
1 + a2i
and pX =
−jai θˆi + φˆi√
1 + a2i
, (5.51)
where ai ≡ FO,iYL,i = −
YL,i
FX,i
is the polarization parameter, and θˆi and φˆi are a pair of mu-
tually orthogonal unit vectors perpendicular to kˆ whose directions depend on the relative
orientation of the propagation vector k and the magnetic field Bi (see Figure 5.1). At
the interfaces between the slabs, the propagating fields are totally transmitted (i.e., wave
reflection is neglected), an assumption that is valid because ionospheric plasma parameters
change very slowly with altitude (having scale heights much larger than the wavelength of
the propagating fields). Therefore, the field components of an upgoing plane wave propa-
gating in the +kˆ direction (at a distance ri = i∆r from the origin) can be computed by
the successive application of the propagator matrices; that is,
Eui = T¯i · · · T¯2T¯1Euo , (5.52)
where Euo is the wave field at the origin (perpendicular to kˆ), and T¯1 . . . T¯i are the prop-
agator matrices from the bottom layer to the i-th layer. As the propagator matrices are
normal, it can be shown that
|Eui |2 = |Euo |2 , (5.53)
which is in agreement with the assumption of total transmission. Similarly, taking advan-
tage of the bidirectionality of the propagator matrices, the field components of a downgoing
plane wave propagating in the −kˆ direction (from the i-th layer to the ground) can be
written as
Edo = T¯1T¯2 · · · T¯iEdi , (5.54)
where Edi is the field at the top of the i-th layer.
In radar experiments, the transverse field component of the signal backscattered from
64
a radar range ri = i∆r can be modeled as
Ero ∝ κi T¯1T¯2 · · · T¯iT¯i · · · T¯2T¯1︸ ︷︷ ︸
Π¯i
Eto, (5.55)
where Eto and Ero are the fields transmitted and received by the radar antenna in the kˆ
direction. Above, Π¯i denotes a two-way propagator matrix that accounts for the polariza-
tion effects on the waves incident on and backscattered from the radar range ri (upgoing
and downgoing waves, respectively). In addition, κi is a scattering factor that depends on
the radar cross section (RCS) of the scatterers at the range ri (scatterers that in the case
of the ionosphere are randomly moving electrons).
As an example, consider the following incoherent scatter radar configuration. Two or-
thogonal linearly polarized antennas with very narrow beams (pencil beams) are placed at
Jicamarca. The antennas are steerable in the north-south plane such that different mag-
netic aspect angle directions can be probed by the radar beams. Considering a coordinate
system in which the x-, y-, and z-axes are parallel to the east, north, and zenith directions,
we can express the radar pointing direction as kˆ = cosψ zˆ + sinψ yˆ, where ψ is the zenith
angle. Furthermore, the radar antennas are oriented such that their polarization vectors
are given by
pˆ1 = xˆ and pˆ2 = − sinψ zˆ + cosψ yˆ, (5.56)
respectively. Note that pˆ1, pˆ2, and kˆ constitute a set of orthogonal unit vectors.
Operating at 50 MHz, the antenna beams scan the ionosphere from north to south to
build power maps of the backscattered signals. In every pointing direction, narrow pulses
are transmitted so that range filtering effects (due to the convolution of the pulse shape
with the response of the ionosphere) can be ignored. In transmission, only the first antenna
is excited, while, in reception, both antennas are used to collect the backscattered signals.
As a result, the co- and cross-polarized antenna voltages sampled at each range ri can be
expressed as
v1(kˆ) ∝ κi pˆT1 Π¯i pˆ1 and v2(kˆ) ∝ κi pˆT2 Π¯i pˆ1, (5.57)
65
0 2 4 6
0
100
200
300
400
500
600
700
800
900
N
e
 [1011 m−3]
H
ei
gh
t [k
m]
1 1.5 2
0
100
200
300
400
500
600
700
800
900
T
e
/Ti
H
ei
gh
t [k
m]
Figure 5.3 Electron density and Te/Ti profiles as functions of height.
where the two-way propagator matrix Π¯i (defined above) is dependent on the electron
density and magnetic field values along kˆ up to the range ri. As the scattering factor κi
is a random variable, statistics of the collected voltages have to be estimated in order to
analyze the characteristics of the backscattered signals. Thus, we can model the mean
square values of v1 and v2 as
〈|v1|2〉 ∝ σv Γ1 and 〈|v2|2〉 ∝ σv Γ2, (5.58)
where σv = 〈|κi|2〉 is the volumetric RCS of the medium, which is dependent on the electron
density, temperature ratio, and magnetic aspect angle at any given range. In addition, Γ1
and Γ2 are polarization coefficients defined as
Γ1 =
∣∣∣pˆT1 Π¯i pˆ1∣∣∣2 and Γ2 = ∣∣∣pˆT2 Π¯i pˆ1∣∣∣2 . (5.59)
Note that, given the orthogonality properties of the polarization vectors pˆ1 and pˆ2, it can
be verified that the polarization coefficients in (5.59) satisfy the relation Γ1 + Γ2 = 1.
To simulate the radar voltages, an ionosphere with the electron density and Te/Ti
profiles displayed in Figure 5.3 was considered. The density profile was obtained from
66
ionosonde measurements conducted at Jicamarca on June 8, 2004 (around local noon).
The Te/Ti profile is modeled as a layer of Gaussian shape centered at an altitude of 250 km
with a width of 120 km, the bottom and peak values of Te/Ti are 1 and 2, respectively.
In addition, the magnetic field above Jicamarca was computed using the International
Geomagnetic Reference Field (IGRF) model [e.g., Olsen et al., 2000] for the same date of
the density profile.
Let us first analyze magneto-ionic propagation effects on the radar voltages, disregard-
ing scattering effects. For this purpose, polarization coefficients Γ1 and Γ2 are displayed in
Figure 5.4 as functions of distance and altitude from Jicamarca (in the plots, the positive
horizontal axis is directed north). We can also think of these plots as the signal power
detected by the radar antennas in the case that the RCS of the medium was constant at
all heights. Note that, at low altitudes, where there is no ionosphere, signal returns will
be detected only by the co-polarized antenna (i.e., by the same antenna used on transmis-
sion). However, as the signal propagates farther through the ionosphere, magneto-ionic
effects start taking place. We can appreciate that, for most of the propagation directions,
the polarization vector of the detected field rotates such that signal from one polarization
goes to the other as the radar range increases (Faraday rotation effect). Note, however,
that there is a direction in which the wave polarization does not rotate much. In this
direction, the antenna beams are pointed perpendicular to the Earth’s magnetic field, and
it can be observed that the polarization of the detected fields varies progressively from
linear to circular as a function of height (Cotton–Mouton effect). Finally, note that at
higher altitudes, where the ionosphere vanishes, no more magneto-ionic effects take place,
and the polarization of the detected signal approaches a final state.
Next, scattering and propagation effects are considered in the simulation of the backscat-
tered power collected by the pair of orthogonal antennas described above. The incoherent
scatter volumetric RCS formulated in the previous chapter is used in the calculations.
In Figure 5.5, the simulated co-polarized (left panel) and cross-polarized (middle panel)
power data are displayed as functions of distance and altitude from Jicamarca. In addi-
tion, the right panel depicts the total power detected by both antennas. Note that power
67
Figure 5.4 Polarization coefficients for the mean square voltages detected by a pair of
orthogonal linearly polarized antennas placed at Jicamarca. The antennas have very nar-
row beams and scan the ionosphere from north to south probing different magnetic aspect
angle directions. Note that, for most pointing directions, the polarization of the detected
fields rotates (Faraday rotation effect), except in the direction where the beam is pointed
perpendicular to B, in which case, the type of polarization changes from linear to circular
(Cotton-Mouton effect).
Figure 5.5 Co-polarized (left panel), cross-polarized (middle panel), and total (right
panel) backscattered power detected by a pair of orthogonal linearly polarized antennas
(see caption of Figure 5.4). Power levels are displayed in units of electron density. In each
plot, the dashed white lines indicate the directions half a degree away from perpendicular
to B, while the continuous lines correspond to the directions one degree off.
68
levels are displayed as volumetric radar cross sections divided by 4pir2e (i.e., power levels
are in units of electron density). In each plot, the dashed white lines indicate the directions
half a degree away from perpendicular to B, while the continuous lines correspond to the
directions one degree off.
In the plots, we can observe that there is negligible backscattered power at low altitudes.
In real measurements, however, coherent echoes stronger than the incoherent scatter signal
can be detected at these altitudes. For instance, echoes from the electrojet at 100 km
altitude or from the so-called 150-km region can be measured daily at Jicamarca (note that
these echoes are only typical of the equatorial ionosphere). In our model, the presence of
this type of echo is ignored because (to our knowledge) there is no accurate physical model
for their radar cross sections. In the data analysis presented in the next chapter, we treat
these echoes as clutter. At higher altitudes between approximately 200 and 700 km (where
polarization effects are significant), co- and cross-polarized power maps exhibit features
that are similar to the ones observed in Figure 5.4. Note, however, that there is an
enhancement of the detected power in the direction where the antenna beams are pointed
perpendicular to B; this can be observed more clearly in the plot of the total power (right
panel of Figure 5.5). This feature is characteristic of the incoherent scatter process for
probing directions perpendicular to B and for heights where electron temperature exceeds
the ion temperature (i.e., Te > Ti) as described in the previous chapter. At even higher
altitudes, scattered signals become weaker and weaker as the ionospheric electron density
vanishes.
Power measurements similar to the simulation results presented above were conducted
for the first time a few years ago. Operating at a different frequency and located at a
different latitude, the UHF (422 MHz) ALTAIR radar was used to scan the low-latitude
ionosphere from north to south [e.g., Milla and Kudeki , 2006; Kudeki et al., 2006]. The
radar is located on Roi-Namur (an island of the Kwajalein Atoll in the Pacific sector),
and the measurements were carried out as part of the EQUIS 2 NASA rocket campaign
in September 2004 [e.g., Lehmacher et al., 2006; Friedrich et al., 2006]. The antenna
system at ALTAIR is dual-polarized; however, power signal was detected only by the
69
Figure 5.6 An example of the ALTAIR scan measurements collected during the 2004
EQUIS 2 campaign. In colors, levels of backscattered power calibrated to match electron
densities in the probed region are displayed.
co-polarized antenna because magneto-ionic propagation effects are negligible at UHF fre-
quencies (for the range of ionospheric altitudes probed in these experiments). An example
of the ALTAIR scan measurements is presented in Figure 5.6; note the similarity with our
simulation results (right panel of Figure 5.5). At Jicamarca, however, the antenna array
has limited steering capabilities, which makes scan measurements of the ionosphere not
possible in practice. But instead, the modularity of the array permits the simultaneous
observation of the ionosphere with different sections of the antenna phased to point into
different directions. Multi-beam radar configurations are currently under test and evalu-
ation at Jicamarca. The goal of these configurations is to make use of the aspect angle
dependence of the incoherent scatter signal in the data inversion of ionospheric densities,
temperatures, and drifts, all at the same time. The results obtained using one of these
configurations will be presented in the next chapter of this dissertation. The data model
is based on the propagation model presented in this section; however, an extra level of
complexity has to be considered because, within the range of aspect angles illuminated by
70
the antenna beams, propagation and scattering effects vary quite rapidly. For this rea-
son, the measured backscattered signals were carefully modeled taking into account the
shapes of the antenna beams. A model for the beam-weighted incoherent scatter spectrum
that considers magneto-ionic propagation and collisional effects is formulated in the next
section.
5.3 Soft-Target Radar Equation and Magneto-Ionic
Propagation
In this section, the soft-target radar equation derived in Chapter 2 is reformulated
using the wave propagation model described above. Consider a radar system composed of
a set of antenna arrays (located in the same area) with matched filter receivers connected
to the antennas used in reception. The mean square voltage at the output of the i-th
receiver can be expressed as
〈|vi(t)|2〉 = EtKi
ˆ
dr dΩ
dω
2pi
|T (rˆ)|2 |Ri(rˆ)|2
k2
Γi(r)
|χ(t− 2rc , ω)|2
4pir2
σv(k, ω), (5.60)
where t is the radar delay, Et is the total energy of the transmitted radar pulse, and Ki is
the i-th calibration constant (a proportionality factor that accounts for the gains and losses
along the i-th signal path). Integrals are taken over range r, solid angle Ω, and Doppler
frequency ω/2pi. In addition, k = −2korˆ denotes the relevant Bragg vector for a radar
with a carrier wavenumber ko and associated wavelength λ. Above, T (rˆ) and Ri(rˆ) are
the antenna factors of the arrays used in transmission and reception. Note that |T (rˆ)|2 and
|Ri(rˆ)|2 are antenna gain patterns and the product |T (rˆ)|2 |Ri(rˆ)|2 is the corresponding
two-way radiation pattern. The polarization coefficient Γi(r) is defined as
Γi(r) =
∣∣∣pˆTi Π¯(r) pˆt∣∣∣2 , (5.61)
where pˆt and pˆi are the polarization unit vectors of the transmitting and receiving anten-
nas, and Π¯(r) is the two-way propagator matrix for the wave field components propagating
71
along rˆ (incident on and backscattered from the range r). Note that pˆt and pˆi are normal
to rˆ, but they are not necessarily perpendicular to each other. In addition, χ(t, ω) is the
radar ambiguity function and σv(k, ω) is the volumetric RCS spectrum, functions that
have been defined in previous chapters. Similarly, the cross-correlation of the voltages at
the outputs of the i-th and j-th receivers can be expressed as
〈vi(t)v∗j (t)〉 = EtKi,j
ˆ
dr dΩ
dω
2pi
|T (rˆ)|2Ri(rˆ)R∗j (rˆ)
k2
Γi,j(r)
|χ(t− 2rc , ω)|2
4pir2
σv(k, ω),
(5.62)
where Ki,j is a cross-calibration constant (dependent on gains and losses along the i-th
and j-th signal paths), and Γi,j(r) is a cross-polarization coefficient defined as
Γi,j(r) =
(
pˆTi Π¯(r) pˆt
)(
pˆTj Π¯(r) pˆt
)∗
. (5.63)
Note that dispersion of the pulse shape due to wave propagation effects has been neglected
in our model.
Denoting by Si(ω) the self-spectrum of the signal at the output of the i-th receiver and
applying Parseval’s theorem, we have that
〈|vi(t)|2〉 =
ˆ
dω
2pi
Si(ω). (5.64)
Likewise, the cross-spectrum Si,j(ω) and the cross-correlation of the signals at the outputs
of the i-th and j-th receivers are related by
〈vi(t)v∗j (t)〉 =
ˆ
dω
2pi
Si,j(ω). (5.65)
Assuming that the ambiguity function is almost flat within the bandwidth of the RCS
spectrum σv(k, ω) (which is a valid approximation in the case of short-pulse radar appli-
cations), we can use equations (5.60) and (5.62) to obtain the following beam-weighted
spectrum and cross-spectrum models:
Si(ω) = EtKi
ˆ
dr
|χ(t− 2rc )|2
4pir2
ˆ
dΩ
|T (rˆ)|2 |Ri(rˆ)|2
k2
Γi(r)σv(k, ω) (5.66)
72
and
Si,j(ω) = EtKi,j
ˆ
dr
|χ(t− 2rc )|2
4pir2
ˆ
dΩ
|T (rˆ)|2Ri(rˆ)R∗j (rˆ)
k2
Γi,j(r)σv(k, ω), (5.67)
where
χ(t) =
1
T
f∗(−t) ∗ f(t) (5.68)
is the normalized auto-correlation of the pulse waveform f(t). In the radar equations (5.66)
and (5.67), the polarization coefficients Γi(r) and Γi,j(r) effectively modify the radiation
patterns; thus, the spectrum shapes are dependent not only on the scattering process but
also on the modes of propagation. This dependence further complicates the spectrum
analysis of radar data and the inversion of physical parameters.
73
CHAPTER 6
MAGNETO-IONIC EFFECTS ON SIMULATED
AND MEASURED ISR DATA
In this chapter, signatures of magneto-ionic propagation effects on incoherent scatter
radar measurements are analyzed and discussed in detail. This analysis is performed us-
ing simulated and measured data of the differential-phase technique used at Jicamarca
to measure F-region electron densities with antenna beams pointed perpendicular to the
geomagnetic field. First, the radar configuration of the differential-phase technique is de-
scribed. Next, magneto-ionic effects on the differential-phase antenna patterns are modeled
and simulated using the radar equations of Section 5.3. In the simulation, the ionospheric
plasma configuration specified in Section 5.2 is considered. The patterns are then utilized
to compute IS signal power and cross-correlation profiles. The different features of the
computed profiles are discussed and linked to the altitude variations of the simulated ra-
diation patterns. The computed profiles are then compared to real measurements carried
out at Jicamarca. The results of the comparisons show good agreement between mod-
eled and measured data. Finally, magneto-ionic effects on spectrum and cross-spectrum
measurements are identified and analyzed based on our simulation results.
6.1 The Differential-Phase Radar Configuration
The differential-phase technique developed for the Jicamarca incoherent scatter radar
system is used to measure high-precision drifts and ionospheric densities simultaneously
at F-region heights [e.g., Kudeki et al., 2003]. In this mode of operation, the Jicamarca
antenna is phased to point perpendicular to the Earth’s magnetic field. The Jicamarca
74
antenna consists of two superimposed square arrays of half-wave dipoles orthogonal to
each other with a diagonal oriented ∼ 6◦ east of the geographic north [Ochs, 1965]. The
arrays, labeled as “up” and “down,” are northeast and southeast polarized. As shown in
the antenna diagram of Figure 6.1, the up and down arrays are subdivided into quarters.
In transmission, all quadrants of the down (southeast polarized) array are excited. In
reception, backscattered fields detected by the up and down polarizations of the north and
south antenna quarters are combined using RF hybrids to synthesize meridional and zonal
polarized field components. After matched filter detection, the radar signals are digitalized
and stored in a computer. Further details of the differential-phase technique are given by
Feng et al. [2003, 2004].
The four antenna channels of the differential-phase radar configuration are:
• Channel 1: Meridional polarization of the north antenna quadrant.
• Channel 2: Zonal polarization of the north antenna quadrant.
• Channel 3: Meridional polarization of the south antenna quadrant.
• Channel 4: Zonal polarization of the south antenna quadrant.
Statistics of the signals detected by these channels, spectrum and cross-spectrum measure-
ments, can be modeled using the soft-target radar equations of Section 5.3. Assuming that
very short pulses are transmitted, the spectra measured by the i-th channel can take the
following form
Si(ω) =
EtKiδr
4pir2
ˆ
dΩWi(r)σv(k, ω), (6.1)
where δr is the effective radar pulse width defined as δr ≡ ´ dr |χ(t− 2rc )|2. Similarly, the
cross-spectra of the i-th and j-th channels can be modeled as
Si,j(ω) =
EtKi,jδr
4pir2
ˆ
dΩWi,j(r)σv(k, ω). (6.2)
In these equations, Wi(r) and Wi,j(r) denote effective two-way radiation patterns that
take into account the magneto-ionic propagation effects on the radar signals. Due to these
effects, the effective patterns vary smoothly as functions of the radar range. Note that
75
ND
H1
NU
SU
TX RX
DN
-p
ol
UP
-p
ol
M
ag
ne
tic
No
rth
SD
H2
Ch
1
Ch
2
Ch
3
Ch
4
I+ I-
O- O+
I+ I-
O- O+
F
ig
ur
e
6.
1
A
nt
en
na
di
ag
ra
m
of
th
e
di
ffe
re
nt
ia
l-p
ha
se
te
ch
ni
qu
e
de
ve
lo
pe
d
fo
r
th
e
Ji
ca
m
ar
ca
in
co
he
re
nt
sc
at
te
r
ra
da
r
sy
st
em
.
In
th
is
m
od
e
of
op
er
at
io
n,
th
e
Ji
ca
m
ar
ca
an
te
nn
a
is
ph
as
ed
to
po
in
t
pe
rp
en
di
cu
la
r
to
B
.
In
tr
an
sm
is
si
on
,t
he
so
ut
he
as
t
po
la
ri
ze
d
do
w
n
an
te
nn
a
qu
ar
te
rs
ar
e
ex
ci
te
d.
In
re
ce
pt
io
n,
ba
ck
sc
at
te
re
d
fie
ld
s
de
te
ct
ed
by
th
e
up
an
d
do
w
n
po
la
ri
za
ti
on
s
of
th
e
no
rt
h
an
d
so
ut
h
qu
ar
te
rs
ar
e
co
m
bi
ne
d
to
sy
nt
he
si
ze
m
er
id
io
na
la
nd
zo
na
lp
ol
ar
iz
ed
fie
ld
co
m
po
ne
nt
s.
76
the solid angle integrations in (6.1) and (6.2) need to be carefully evaluated because the
incoherent scatter RCS spectrum σv(k, ω) varies quite rapidly at small magnetic aspect
angles, the angular regime that is illuminated by the radar beams.
The effective radiation patterns to be considered in the self-spectrum model (6.1) are
given next. For the meridional polarized antenna configurations, channels 1 and 3, the
patterns are defined as
W1(r) = W3(r) =
1
k2
Gt(rˆ)Gr(rˆ) Γm(r), (6.3)
while, for the zonal polarized antennas, channels 2 and 4, we have that
W2(r) = W4(r) =
1
k2
Gt(rˆ)Gr(rˆ) Γz(r). (6.4)
In these expressions, k is the corresponding Bragg wavenumber (k = 4pi/λ) that normal-
izes the solid angle integration of the two-way patterns to units of backscatter aperture
(i.e., units of area). Above, Gt(rˆ) and Gr(rˆ) are the free-space radiation patterns of the
antenna arrays used in transmission and reception (full and quarter arrays, respectively).
In addition, the polarization coefficients of the meridional and zonal patterns, Γm(r) and
Γz(r), are defined as
Γm(r) =
∣∣∣pTm Π¯(r) pˆt∣∣∣2 and Γz(r) = ∣∣∣pTz Π¯(r) pˆt∣∣∣2 , (6.5)
where pˆt is the polarization unit vector of the field transmitted in the direction rˆ, and pm
and pz are the polarization vectors of the meridional and zonal components of the scattered
fields detected at the antenna level. In addition, Π¯(r) denotes the two-way propagator
matrix defined in Section 5.2.
In the Jicamarca frame of reference — a coordinate system in which the antenna is
parallel to the xy-plane and the dipoles of the down- and up-arrays are aligned with the x-
and y-axes — the polarization unit vectors of the down and up antennas can be expressed
77
as
pˆd =
rˆ× rˆ× xˆ
|ˆr× rˆ× xˆ| and pˆu =
rˆ× rˆ× yˆ
|ˆr× rˆ× yˆ| (6.6)
for fields propagating along rˆ. Since, in transmission, the down-array is excited, we have
that
pˆt = pˆd, (6.7)
while, in reception, the meridional and zonal polarization vectors are given by
pm =
1√
2
(pˆu − pˆd) and pz = 1√
2
(pˆu + pˆd) . (6.8)
To model the cross-spectrum measurements, expressions for the effective two-way ra-
diation patterns corresponding to each of the six pairs of antenna configurations are given
next. For the cross-spectra between cross-polarized channels of the same antenna quad-
rants, the effective patterns for the north pair (channels 1 and 2) and the south pair
(channels 3 and 4) are equal to
W1,2(r) = W3,4(r) =
1
k2
Gt(rˆ)Gr(rˆ) Γm,z(r), (6.9)
where the polarization coefficient Γm,z is defined as
Γm,z(r) =
(
pˆTm Π¯(r) pˆt
)(
pˆTz Π¯(r) pˆt
)∗
. (6.10)
In the case of the cross-spectra between co-polarized channels of different antenna quad-
rants, the pattern for the pair of meridional antenna channels takes the form
W1,3(r) =
ejkrˆ·D
k2
Gt(rˆ)Gr(rˆ) Γm(r), (6.11)
while, for the pair of zonal channels, the pattern is equal to
W2,4(rˆ) =
ejkrˆ·D
k2
Gt(rˆ)Gr(rˆ) Γz(r), (6.12)
78
where D is a distance vector from the center of the south quadrant to the center of the north
quadrant. Finally, we have the case of the cross-spectra between cross-polarized channels of
different antenna quadrants. The pattern for the pair composed of the meridional channel
of the north quadrant and the zonal channel of the south quadrant is given by
W1,4(r) =
ejkrˆ·D
k2
Gt(rˆ)Gr(rˆ) Γm,z(r), (6.13)
and the pattern for the pair composed of the zonal channel of the north quadrant and the
meridional channel of the south quadrant is equal to
W2,3(r) =
ejkrˆ·D
k2
Gt(rˆ)Gr(rˆ) Γ
∗
m,z(r). (6.14)
In Figure 6.2, we present the two-way radiation pattern Gt(rˆ)Gr(rˆ) of the differential-
phase radar configuration in free space.1 In the left panel, the pattern (in dB units relative
to its peak value) is displayed as a function of θx and θy (direction cosines with respect to
the x- and y-axes of the Jicamarca frame of reference). In the right panel, the pattern is
displayed as a contour plot in a rotated coordinate system in which the x- and y-axes are
now aligned with the east and north directions (the Jicamarca antenna frame of reference
is rotated 51◦ positive azimuth). In the plots, the continuous black lines trace the positions
where the propagation directions are perpendicular to the geomagnetic field at radar ranges
r = 100, 500, 1000 km.
6.2 Magneto-Ionic Effects on the Radiation Patterns
Simulation results showing the altitudinal variation of the radiation patterns of the
differential-phase radar configuration caused by the magneto-ionic propagation of the radar
signals are presented next. In the simulation, the electron density and Te/Ti profiles dis-
played in Figure 5.3 are considered. The Earth’s magnetic field components for June 2004
are calculated using the IGRF model. The simulated effective radiation patterns are dis-
1The Jicamarca antenna radiation patterns can be efficiently computed using a 2D FFT-based algorithm
that takes advantage of the spatial uniformity of the antenna array.
79
θ x
θ
y
D
2W
 =
 7
9.
4 
dB
 −
 A
BS
 =
 5
97
1.
2
 
 
−
0.
05
0
0.
05
−
0.
08
−
0.
06
−
0.
04
−
0.
020
0.
02
0.
04
0.
06
0.
08
−
40
−
35
−
30
−
25
−
20
−
15
−
10
−
50
10
0 
km
50
0 
km
10
00
 k
m
θ x
θ
y
D
2W
 =
 7
9.
4 
dB
 −
 A
BS
 =
 5
97
1.
2
 
 
−
0.
05
0
0.
05
−
0.
08
−
0.
06
−
0.
04
−
0.
020
0.
02
0.
04
0.
06
0.
08
−
40
−
35
−
30
−
25
−
20
−
15
−
10
−
50
10
0 
km
50
0 
km
10
00
 k
m
F
ig
ur
e
6.
2
T
w
o-
w
ay
an
te
nn
a
ra
di
at
io
n
pa
tt
er
n
of
th
e
di
ffe
re
nt
ia
l-p
ha
se
ra
da
r
co
nfi
gu
ra
ti
on
(i
n
fr
ee
-s
pa
ce
).
In
th
e
le
ft
pa
ne
l,
th
e
ra
di
at
io
n
pa
tt
er
n
(i
n
dB
un
it
s
re
la
ti
ve
to
it
s
pe
ak
va
lu
e)
is
di
sp
la
ye
d
as
a
fu
nc
ti
on
of
di
re
ct
io
n
co
si
ne
s
w
it
h
re
sp
ec
t
to
th
e
x
-
an
d
y
-a
xe
s
of
th
e
Ji
ca
m
ar
ca
fr
am
e
of
re
fe
re
nc
e.
In
th
e
ri
gh
t
pa
ne
l,
th
e
pa
tt
er
n
is
di
sp
la
ye
d
as
a
co
nt
ou
r
pl
ot
in
a
ro
ta
te
d
co
or
di
na
te
sy
st
em
in
w
hi
ch
th
e
x
-
an
d
y
-a
xe
s
ar
e
no
w
al
ig
ne
d
w
it
h
th
e
ea
st
an
d
no
rt
h
di
re
ct
io
ns
(t
he
Ji
ca
m
ar
ca
fr
am
e
of
re
fe
re
nc
e
is
ro
ta
te
d
51
◦
po
si
ti
ve
az
im
ut
h)
.
80
played in Figures 6.3, 6.4, and 6.5 in the rotated coordinate system specified above. Each
row of plots corresponds to different radar ranges. In the left and right panels, the merid-
ional and zonal polarized radiation patterns are displayed as functions of direction cosines.
In the middle panels, the polarization states of the wave field components corresponding
to different propagation directions are shown.
In Figure 6.3, we can observe that the initial polarization vector (at r = 0 km) is pointed
southeast; therefore, both meridional and zonal field components are equally excited and
the associated patterns look identical to each other. Note that the initial patterns are
actually the radiation patterns in free-space (disregarding near field effects). In the first
100 km, the radiation patterns vary very little, but as soon as the ionosphere appears,
magneto-ionic effects start taking place. At 200 km, Faraday rotation effects are noticeable
at aspect angle directions around one degree (∼ 0.02 dc) to the north and south of the
loci of perpendicularity (depicted as black lines). In the north, the polarization of the
wave fields rotates counter-clockwise, while, in the south, the contrary happens. Because
of this, we can observe that power from the northern part of the meridional pattern goes
to the zonal pattern, while power from the southern part of the zonal pattern goes to
the meridional one. Note, however, that in the close vicinity of the perpendicular-to-B
direction, rotation does not take place.
In Figure 6.4, beam shape differences between the meridional and zonal patterns be-
come more evident. Note that at 250 km and 300 km, the sidelobes of the patterns have
clearly changed their shapes. At higher altitudes around 350 km and 400 km (close to the
peak of the density profile), the polarization of the backscattered fields at one degree from
the north and south of the loci of perpendicularity have rotated about 90 degrees. Also
note that around the perpendicular direction, the polarization states have changed from
linear to elliptical. Moreover, we can see that the shapes of the mainlobes have started to
look different.
Finally, in Figure 6.5, we can observe that not only have the mainlobe shapes become
different but their centers have also been displaced. Note that the zonal beam pattern
is now pointing to the north of the locus of perpendicularity, while the meridional beam
81
θ
x
θ y
Meridional Beam
Height = 0.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 0.0 km
θ
x
Zonal Beam
Height = 0.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
θ
x
θ y
Meridional Beam
Height = 100.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 100.0 km
θ
x
Zonal Beam
Height = 100.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
θ
x
θ y
Meridional Beam
Height = 150.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 150.0 km
θ
x
Zonal Beam
Height = 150.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
θ
x
θ y
Meridional Beam
Height = 200.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 200.0 km
θ
x
Zonal Beam
Height = 200.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
Figure 6.3 Meridional and zonal effective radiation patterns for r = 0, 100, 150, 200 km.
82
θ
x
θ y
Meridional Beam
Height = 250.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 250.0 km
θ
x
Zonal Beam
Height = 250.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
θ
x
θ y
Meridional Beam
Height = 300.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 300.0 km
θ
x
Zonal Beam
Height = 300.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
θ
x
θ y
Meridional Beam
Height = 350.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 350.0 km
θ
x
Zonal Beam
Height = 350.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
θ
x
θ y
Meridional Beam
Height = 400.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 400.0 km
θ
x
Zonal Beam
Height = 400.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
Figure 6.4 Same as Figure 6.3 but for r = 250, 300, 350, 400 km.
83
θ
x
θ y
Meridional Beam
Height = 450.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 450.0 km
θ
x
Zonal Beam
Height = 450.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
θ
x
θ y
Meridional Beam
Height = 500.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 500.0 km
θ
x
Zonal Beam
Height = 500.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
θ
x
θ y
Meridional Beam
Height = 550.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 550.0 km
θ
x
Zonal Beam
Height = 550.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
θ
x
θ y
Meridional Beam
Height = 600.0 km
−0.02 0 0.02
0.01
0.02
0.03
0.04
0.05
0.06
0.07
−0.02 0 0.02
θ
x
Polarization Vector (RX)
Height = 600.0 km
θ
x
Zonal Beam
Height = 600.0 km
 
 
−0.02 0 0.02
−40
−30
−20
−10
0
Figure 6.5 Same as Figure 6.3 but for r = 450, 500, 550, 600 km.
84
pattern is still pointing, approximately, perpendicular to B. As the altitude increases
and the electron density vanishes, we can appreciate that magneto-ionic propagation ef-
fects are less severe, and the radiation patterns eventually relax toward their final shapes.
Note that the final polarization states have changed a lot compared to their initial linear
state. At 600 km, for instance, the polarization of the wave field components propagating
perpendicular to B have become more or less circular. We can also observe at the same
altitude, but one degree to the north or south of the perpendicular direction, that the
polarization states have remained linear but rotated many degrees. The behavior of the
radiation patterns described above has an impact on the measured signals, as we discuss
next.
6.3 Simulated and Measured Power and Cross-Correlation
Profiles
The effective radiation patterns simulated in the previous section are used to compute
signal power and cross-correlation profiles. Based on the soft-target radar equations (6.1)
and (6.2), we can model the power measured by the i-th channel as follows,
〈|vi|2〉 =
ˆ
dωSi(ω) =
EtKiδr
4pir2
ˆ
dΩWi(r)σv(k), (6.15)
while the cross-correlation between the i-th and j-th channels can be expressed as
〈viv∗j 〉 =
ˆ
dωSi,j(ω) =
EtKi,jδr
4pir2
ˆ
dΩWi,j(r)σv(k), (6.16)
where Wi(r) and Wi,j(r) are the effective two-way radiation patterns, and σv(k) is the
volumetric IS radar cross section (see Chapter 4).
As discussed in previous chapters, the incoherent scatter RCS is a function of the
angle between the wave propagation direction k and the ambient magnetic field B. At
the magnetic equator, it can be verified that σv(k) varies progressively in the north-south
plane (which is nearly parallel to the geomagnetic field lines). In fact, in the vicinity of the
85
direction perpendicular to B, the magnitude of σv(k) varies even more rapidly. However,
in the east-west plane, σv(k) changes very slowly. To take advantage of this behavior in
the numerical evaluation of the solid angle integrals in (6.15) and (6.16), we consider a
coordinate system in which the x- and y-axes are aligned with the east and north directions
of the Jicamarca location (see, for instance, the right panel of Figure 6.2). After recasting
the integrals in terms of θx and θy (direction cosines with respect to the x- and y-axes of
the coordinate system considered), we compute (6.15) and (6.16) applying a finite-element
integration method. At every radar range, values of σv(k) are finely sampled in the θy
direction but very coarsely in the θx direction. Then, the integrals are computed by adding
the samples of σv(k) weighted by numerical coefficients that account for the shapes of the
effective radiation patterns.
Signal power and cross-correlation profiles are simulated for the differential-phase radar
configuration of June 8, 2004, considering the set of ionospheric plasma parameters given
in the previous chapter. As mentioned before, four reception channels are used in the
differential-phase experiments. In the left panel of Figure 6.6, the simulated power profiles
of the four antenna channels are plotted in arbitrary units. Channels 1 and 2 (blue and
green curves) correspond to the meridional and zonal polarized signals detected by the
north quadrant of the Jicamarca array, while channels 3 and 4 (red and light-blue curves)
correspond to the signals detected by the south quadrant. Note that the meridional power
profiles (blue and red curves) overlap each other. In the case of the zonal profiles (green
and light-blue curves), the same happens. However, both the meridional and zonal profiles
are different from each other. This dissimilarity is an effect caused by the magneto-
ionic propagation of the radar signals in the ionosphere. According to the simulation
results presented in the previous section, the shapes of the effective radiation patterns
and, therefore, their corresponding radar volumes vary as functions of the radar range.
These variations, however, are different for different antenna polarizations. Therefore,
non-co-polarized antennas, such as the ones used by the differential-phase technique, detect
distinct signal power profiles. The simulated cross-correlations of the antenna signals are
used to compute coherence profiles for different pairs of antenna channels. Coherence is
86
0 2 4
0
100
200
300
400
500
600
700
800
Signal
R
an
ge
 [k
m]
 
 
[1] North M−pol
[2] North Z−pol
[3] South M−pol
[4] South Z−pol
0 0.25 0.5 0.75 1
0
100
200
300
400
500
600
700
800
Coherence
 
 
[1,2]
[3,4]
[1,3]
[2,4]
−180 −90 0 90 180
0
100
200
300
400
500
600
700
800
Phase [deg]
 
 
[1,2]
[3,4]
[1,3]
[2,4]
Figure 6.6 Simulated incoherent scatter signal power and coherence (magnitude and
phase) profiles for the differential-phase radar configuration of June 8, 2004.
defined as
ρij =
〈viv∗j 〉√〈|vi|2〉〈|vj |2〉 . (6.17)
The magnitude and phase of the computed coherence profiles are displayed in the middle
and right panels of Figure 6.6. The coherences between the cross-polarized signals of the
north (channels 1 and 2) and of the south (channels 3 and 4) antenna quadrants are plotted
as blue and green curves. As expected, these coherence profiles overlap each other. At low
altitudes, where magneto-ionic effects are negligible, the coherence is effectively one, since
the beam shapes of the meridional and zonal polarized antenna arrays are still the same.
However, it can be observed that at 200 km the coherence magnitudes start to decrease.
This happens because, as the ionospheric density increases, the meridional and zonal radi-
ation patterns change their shapes due to magneto-ionic propagation effects. Eventually,
the patterns become somewhat different from each other, which leads to a reduction of the
magnitude of the coherence. The coherences between the co-polarized signals of different
antenna quadrants, meridional signals (channels 1 and 3) and zonal signals (channels 2
87
and 4), are plotted as red and light-blue curves. Note that the magnitude and phase of
the coherence profiles vary as functions of height differently for distinct polarizations. We
can observe, once more, that at low altitudes the coherence profiles look similar. As alti-
tude increases, note in both cases that the magnitude of the coherence first rises, which
implies that the radiation patterns have become more compact. At higher altitudes, the
coherence of the meridional polarized signals decreases, while that of the zonal signals
increases. The reason for this is also related to the altitudinal variations of the effective
radiation patterns. While in the case of the meridional beams, the patterns become wider
(power is spread over aspect angle), in the case of the zonal beams, the patterns become
more compact (power is concentrated in some region). The phase of the coherence profiles
brings further information regarding the effective centers of the beam shapes. Note that,
when there is no ionosphere, the meridional and zonal beams are pointing to the same
direction (which is slightly to the north of the loci of perpendicularity and corresponds to
a coherence phase of about 90◦). In the case of the meridional coherence, we can observe
that the phase (red curve) is decreasing. This is in agreement with the fact that, as the
radar range increases, the centers of the meridional patterns move to the south of the loci
of perpendicularity. This effect can be observed in the simulated patterns displayed in the
previous section. In the case of the zonal coherence, the contrary happens, i.e., the phase
(light-blue curve) increases. This is also expected because, at ionospheric altitudes, the
zonal polarized patterns are pointing to the north of their initial pointing direction.
In addition to the simulation results, we present in Figure 6.7, measured power and
coherence profiles obtained after a five-minute integration of incoherent radar signals col-
lected during the differential-phase experiment conducted on June 8, 2004. We can observe
that the simulated profiles resemble very closely the actual measurements. This gives us
confidence that our incoherent scatter model is capable of reproducing the different features
of real measurements. In the following section, the characteristics of spectrum and cross-
spectrum measurements are analyzed based on the simulation results presented above.
88
0 100 200 300
0
100
200
300
400
500
600
700
800
Signal
R
an
ge
 [k
m]
 
 
0 0.25 0.5 0.75 1
0
100
200
300
400
500
600
700
800
Coherence
2004−06−08 14:30:00
 
 
−180 −90 0 90 180
0
100
200
300
400
500
600
700
800
Phase [deg]
 
 
[1] North M−pol
[2] North Z−pol
[3] South M−pol
[4] South Z−pol
[1,2]
[3,4]
[1,3]
[2,4]
[1,2]
[3,4]
[1,3]
[2,4]
Figure 6.7 Measured incoherent scatter signal power and coherence (magnitude and
phase) profiles that correspond to the differential-phase experiment conducted on June
8, 2004.
6.4 Magneto-Ionic Effects on Spectrum and
Cross-Spectrum Measurements
In this section, some of the features of incoherent scatter spectrum and cross-spectrum
measurements are analyzed. The sample spectra considered in this analysis were acquired
during the differential-phase experiment conducted at Jicamarca on June 8, 2004. In the
left panel of Figure 6.8, meridional and zonal polarized self-spectrum measurements are dis-
played as functions of Doppler velocity (or frequency) for different radar ranges. In general,
the incoherent scatter spectrum measured with perpendicular-to-B antennas is composed
of a narrow spectral component with a sharp peak and a wide spectral component that
is typically aliased in pulse-to-pulse spectrum observations. We can appreciate these fea-
tures of the measured spectra in the plots of Figure 6.8. Note that at low altitudes, where
the ionosphere and the magneto-ionic propagation effects are negligible, the four spectra
have the same shape. However, as altitude increases, we can observe that the meridional
89
IS
R
D
op
pl
er
Sp
ec
tr
a
−
20
04
−
06
−
08
14
:3
0:
00
IS
R
C
oh
er
en
ce
Sp
ec
tr
a
−
20
04
−
06
−
08
14
:3
0:
00
N
or
th
−
So
ut
h
ba
se
lin
e
in
te
rf
er
om
et
er
0
0.
01
0.
02
 
R
an
ge
 =
 5
28
 k
m
M
ag
ni
tu
de
 
0
0.
01
0.
02
 
R
an
ge
 =
 4
52
 k
m
 
0
0.
01
0.
02
 
R
an
ge
 =
 3
76
 k
m
 
0
0.
01
0.
02
 
R
an
ge
 =
 3
00
 k
m
 
−
15
0
−
10
0
−
50
0
50
10
0
15
0
0
0.
01
0.
02
 
R
an
ge
 =
 2
28
 k
m
Ve
lo
ci
ty
 [m
/s]
 
 
M
−p
ol
 (N
ort
h)
Z−
po
l (N
ort
h)
M
−p
ol
 (S
ou
th)
Z−
po
l (S
ou
th)
0
0.
2
0.
4
0.
6
0.
81
 
R
an
ge
 =
 5
28
 k
m
M
ag
ni
tu
de
 
−
18
0
−
9009018
0
Ph
as
e 
[de
g]
 
0
0.
2
0.
4
0.
6
0.
81
 
R
an
ge
 =
 4
52
 k
m
 
−
18
0
−
90090
 
0
0.
2
0.
4
0.
6
0.
81
 
R
an
ge
 =
 3
76
 k
m
 
−
18
0
−
90090
 
0
0.
2
0.
4
0.
6
0.
81
 
R
an
ge
 =
 3
00
 k
m
 
−
18
0
−
90090
 
−
10
0
0
10
0
0
0.
2
0.
4
0.
6
0.
81
 
R
an
ge
 =
 2
28
 k
m
Ve
lo
ci
ty
 [m
/s]
−
10
0
0
10
0
−
18
0
−
90090
Ve
lo
ci
ty
 [m
/s]
 
 
M
−p
ol
 (N
 
×
 
S*
)
Z−
po
l (N
 
×
 
S*
)
F
ig
ur
e
6.
8
In
co
he
re
nt
sc
at
te
r
D
op
pl
er
sp
ec
tr
a
(l
ef
t)
an
d
no
rt
h-
so
ut
h
ba
se
lin
e
co
he
re
nc
e
sp
ec
tr
a
(r
ig
ht
)
m
ea
su
re
d
w
it
h
th
e
di
ffe
re
nt
ia
l-
ph
as
e
ra
da
r
te
ch
ni
qu
e
at
Ji
ca
m
ar
ca
on
Ju
ne
8,
20
04
.
90
spectra become sharper than the zonal spectra. This behavior is related to the altitudi-
nal variation of the effective radiation patterns presented earlier in this chapter. Due to
magneto-ionic propagation effects, the centers of the radiation patterns are displaced at
F-region heights. While the meridional pattern points approximately perpendicular to the
magnetic field, the zonal pattern points to the north of the locus of perpendicularity. This
implies that incoherent scatter signals coming from off-perpendicular directions contribute
more to shaping the zonal spectra than the meridional spectra. As off-perpendicular signal
contributions have wide spectral bandwidths, the zonal polarized signals have wider spec-
tra than the meridional signals. Although the differences between the meridional and zonal
spectra are noticeable, they are not very large and can be masked by the statistical uncer-
tainties of the measurements. Because of this, a large number of collected spectra need to
be averaged in order to reduce the uncertainties to a level that allows the identification of
the spectrum differences.
In the right panel of Figure 6.8, the magnitude and phase of the cross-spectra between
co-polarized signals are displayed for different radar ranges. The blue curves correspond
to the cross-spectra of the meridional signals (channels 1 and 3), while the green curves
correspond to the cross-spectra of the zonal signals (channels 2 and 4). As expected, the
meridional and zonal cross-spectra look the same at low altitudes; however, as altitude
increases, the magnitudes and phases of both cross-spectra become different. Note, for
instance, that at altitudes above ∼ 400 km, the meridional cross-spectra are still peaky,
but the zonal spectra have become almost flat. This is related to the fact that, at these
altitudes, the zonal radiation patterns point away from the perpendicular-to-B direction.
Therefore, zonal signals measured by the radar are mostly composed of wide spectrum
(off-perpendicular) components. Also note that the phases of the measured cross-spectra
vary in altitude. In general, the zonal phase is greater than the meridional phase, which
is related to the fact that the effective zonal beam points to the north of the meridional
beam.
In this chapter, we have developed an incoherent scatter model that takes into account
magneto-ionic propagation. Simulated power and cross-correlation data have been com-
91
pared with real measurements. The results of the comparisons show excellent agreement
between simulated and measured data. The comparison of spectrum measurements with
simulated spectra obtained with our model will be the subject of future studies. Also
part of these investigations will be the analysis of the sensitivity of the spectrum and
cross-spectrum models to electron and ion temperatures. In the following chapter, our
incoherent scatter model is applied to the inversion of ionospheric densities and Te/Ti pro-
files using power and cross-correlation measurements obtained with a multi-beam radar
configuration.
92
CHAPTER 7
THREE-BEAM EXPERIMENTS AT
JICAMARCA
In the present chapter, the incoherent scatter data model described in this dissertation
is applied to the inversion of ionospheric plasma parameters using radar data collected in
recent multi-beam experiments at Jicamarca. In these experiments, different sections of
the Jicamarca antenna array are phased to point west, east, or south of the local zenith.
While the west and east beams are aimed perpendicular to the magnetic field, the south
beams are pointed in a direction ∼ 4◦ to the south of the loci of perpendicularity. The
details of the radar experiments and the antenna configurations are given in Section 7.1.
Statistics of the backscattered signals collected in these radar experiments are processed
to estimate state parameters of the equatorial ionosphere. For instance, vertical and zonal
plasma drifts are determined from the Doppler shifts of the frequency spectra measured
with the antenna beams pointed perpendicular to B (the west and east beams). The
spectral fitting technique developed to estimate the drifts was the subject of previous
research and was reported a decade ago by Kudeki et al. [1999]. In this chapter, we
describe a different inversion procedure developed for the estimation of electron densities
and electron-to-ion temperature ratios. Using the signal power and cross-correlation data
collected in these experiments (with all the antenna beams), we infer electron density
profiles corrected for the Te/Ti variations throughout the bottom-side of the F-region.
The estimation procedure exploits the magnetic aspect angle dependence of the incoherent
scatter RCS at heights where the electron temperature exceeds the ion temperature (i.e.,
when Te > Ti). The development of this technique was motivated by the work of Milla and
93
Kudeki [2006]. Samples of spectrum and cross-spectrum measurements, as well as power
and cross-correlation data are displayed in Section 7.2. The forward model of the radar
data and the parameter estimation technique are outlined in Section 7.3.
Electron density and Te/Ti maps resultant from the inversion of the incoherent scatter
radar data collected in these multi-beam experiments are presented in the last section
of this chapter. The goodness-of-fit criterion is used to analyze the quality of the fitting
results. In addition, our density estimates are compared to ionosonde measurements carried
out simultaneously at Jicamarca.
7.1 Experiment Description and Antenna Configurations
Two types of multi-beam incoherent scatter radar experiments with antenna beams
pointed into three different directions were carried out at Jicamarca in recent years. The
experiments were referred to as the DEWD 3Ba and 3Bb modes (names that stand for
Densities and East-West Drifts measured with three beams). In a first data acquisition
campaign, the radar was operated in the 3Ba mode from June 16 to 19, 2008. Radar
experiments in the 3Bb mode were conducted in two subsequent campaigns, from June 23
to 24, 2008, and also from January 9 to 12, 2009.
As mentioned before, the modularity of the Jicamarca array was exploited in order to
configure different sections of the antenna to point west, east, or south of the Jicamarca
zenith. A total of six antenna beams were configured, two for each pointing direction.
While the west and east beams are aimed perpendicular to the magnetic field, the south
beams are pointed in a direction ∼ 4◦ to the south of the loci of perpendicularity. The
antenna configurations used in these experiments are displayed in Figure 7.1. In the 3Ba
mode, the Jicamarca antenna is configured as follows.
• West Beam 1: West quadrant of the up-polarization.
• West Beam 2: East quadrant of the up-polarization.
• East Beam 1: West quadrant of the down-polarization.
94
TX RX
DN
-p
ol
UP
-p
ol
M
ag
ne
tic
No
rth
2
15
4
36
5
6
RX
1 
W
U:
 W
es
t 1
 (Q
ua
rte
r)
RX
2 
EU
: W
es
t 2
 (Q
ua
rte
r)
RX
5 
SU
-N
U:
 S
ou
th
 (C
o-
Po
l)
RX
3 
W
D:
 E
as
t 1
 (Q
ua
rte
r)
RX
4 
ED
: E
as
t 2
 (Q
ua
rte
r)
RX
6 
SD
-N
D:
 S
ou
th
 (X
-P
ol)
TX RX
DN
-p
ol
UP
-p
ol
M
ag
ne
tic
No
rth
1
12
3
34
5
6
RX
1 
EU
-W
U:
 W
es
t (
Bo
wt
ie)
RX
2 
NU
: W
es
t (
Qu
ar
te
r)
RX
5 
SU
: S
ou
th
 (C
o-
Po
l)
RX
3 
ED
-W
D:
 E
as
t (
Bo
wt
ie)
RX
4 
ND
: E
as
t (
Qu
ar
te
r)
RX
6 
SD
: S
ou
th
 (X
-P
ol)
F
ig
ur
e
7.
1
A
nt
en
na
co
nfi
gu
ra
ti
on
s
of
th
e
3B
a
(l
ef
t
pa
ne
l)
an
d
3B
b
(r
ig
ht
pa
ne
l)
ex
pe
ri
m
en
ts
.
95
• East Beam 2: East quadrant of the down-polarization.
• South Beam 1: South and north quadrants of the up-polarization.
• South Beam 2: South and north quadrants of the down-polarization.
In the 3Bb mode, the configuration is somewhat different.
• West Beam 1: East and west quadrants of the up-polarization (bowtie antenna).
• West Beam 2: North quadrant of the up-polarization.
• East Beam 1: East and west quadrants of the down-polarization (bowtie antenna).
• East Beam 2: North quadrant of the down-polarization.
• South Beam 1: South quadrant of the up-polarization.
• South Beam 2: South quadrant of the down-polarization.
Note that the west beams are co-polarized in the northeast direction (up-polarization).
The east beams are also co-polarized but in the southeast direction (down-polarization).
However, the south beams are cross-polarized: the first beam is polarized northeast, while
the second one is polarized southeast.
The two-way radiation patterns of the radar beams used in the 3Ba and 3Bb ex-
periments are presented in Figures 7.2 and 7.3. The patterns are displayed in colors as
functions of direction cosines, θx and θy, with respect to the x- and y-axes of a coordinate
system in which the antenna is located at the origin. In this coordinate system, the z-axis
is perpendicular to the plane of the Jicamarca antenna, and the x- and y- axes are aligned
with the local east and north directions (i.e., the coordinate system is rotated 51◦ positive
azimuth with respect to the direction of alignment of the “down” antenna dipoles). In the
plots, the black lines trace the loci of magnetic perpendicularity for June 2008 at different
altitudes (100, 500, and 1000 km). As indicated earlier in this section, the west-beam
antennas in the 3Ba mode are quarters of the Jicamarca array aligned in the east-west
direction. On the other hand, in the 3Bb mode, the west-beam antennas have different
shapes and are aligned in the north-south direction. One of them is still a quarter array,
96
3B
a
tw
o-
w
ay
an
te
nn
a
pa
tt
er
ns
3B
b
tw
o-
w
ay
an
te
nn
a
pa
tt
er
ns
 
θ
y
W
es
t B
ea
m
 1
 (U
p)
D
2W
 =
 7
3.
91
 d
B
AB
S 
= 
28
33
.6
 −
 C
O
H 
= 
0.
52
−
0.
1
−
0.
050
0.
050.
1
 
 
W
es
t B
ea
m
 2
 (U
p)
D
2W
 =
 7
3.
91
 d
B
AB
S 
= 
28
33
.6
 −
 C
O
H 
= 
0.
52
 
 
−
40
−
30
−
20
−
100
θ x
θ
y
Ea
st
 B
ea
m
 1
 (D
n)
D
2W
 =
 7
3.
84
 d
B
AB
S 
= 
28
30
.4
 −
 C
O
H 
= 
0.
52
−
0.
1
−
0.
05
0
0.
05
0.
1
−
0.
1
−
0.
050
0.
050.
1
θ x
 
Ea
st
 B
ea
m
 2
 (D
n)
D
2W
 =
 7
3.
84
 d
B
AB
S 
= 
28
30
.4
 −
 C
O
H 
= 
0.
52
 
 
−
0.
1
−
0.
05
0
0.
05
0.
1
−
40
−
30
−
20
−
100
 
θ
y
W
es
t B
ea
m
 1
 (U
p)
D
2W
 =
 7
8.
51
 d
B
AB
S 
= 
46
90
.5
 −
 C
O
H 
= 
0.
56
−
0.
1
−
0.
050
0.
050.
1
 
 
W
es
t B
ea
m
 2
 (U
p)
D
2W
 =
 7
5.
10
 d
B
AB
S 
= 
30
94
.3
 −
 C
O
H 
= 
0.
56
 
 
−
40
−
30
−
20
−
100
θ x
θ
y
Ea
st
 B
ea
m
 1
 (D
n)
D
2W
 =
 7
8.
09
 d
B
AB
S 
= 
43
84
.4
 −
 C
O
H 
= 
0.
54
−
0.
1
−
0.
05
0
0.
05
0.
1
−
0.
1
−
0.
050
0.
050.
1
θ x
 
Ea
st
 B
ea
m
 2
 (D
n)
D
2W
 =
 7
4.
59
 d
B
AB
S 
= 
29
02
.4
 −
 C
O
H 
= 
0.
54
 
 
−
0.
1
−
0.
05
0
0.
05
0.
1
−
40
−
30
−
20
−
100
F
ig
ur
e
7.
2
T
w
o-
w
ay
ra
di
at
io
n
pa
tt
er
ns
of
th
e
w
es
t
(t
op
ro
w
)
an
d
ea
st
(b
ot
to
m
ro
w
)
be
am
s
us
ed
in
th
e
3B
a
(l
ef
t)
an
d
3B
b
(r
ig
ht
)
ex
pe
ri
m
en
ts
.
T
he
pa
tt
er
ns
ar
e
di
sp
la
ye
d
as
fu
nc
ti
on
s
of
di
re
ct
io
n
co
si
ne
s
in
a
co
or
di
na
te
sy
st
em
in
w
hi
ch
x
-a
nd
y
-a
xe
s
ar
e
al
ig
ne
d
w
it
h
th
e
ea
st
an
d
no
rt
h
di
re
ct
io
ns
at
Ji
ca
m
ar
ca
.
N
ot
e
th
at
in
th
e
3B
a
m
od
e,
th
e
pa
tt
er
ns
of
th
e
w
es
t
be
am
s
ar
e
id
en
ti
ca
l,
bu
t
in
th
e
3B
b
m
od
e,
th
e
w
es
t
be
am
s
ha
ve
di
ffe
re
nt
sh
ap
es
.
T
he
sa
m
e
ca
n
be
ob
se
rv
ed
in
th
e
ca
se
of
th
e
ea
st
be
am
s.
97
3Ba mode 3Bb mode
θ
x
θ y
South Beam
D2W = 79.7 dB − ABS = 6942
 
 
−0.1 −0.05 0 0.05 0.1
−0.1
−0.05
0
0.05
0.1
−40
−35
−30
−25
−20
−15
−10
−5
0
θ
x
θ y
South Beam
D2W = 73.8 dB − ABS = 4353
 
 
−0.1 −0.05 0 0.05 0.1
−0.1
−0.05
0
0.05
0.1
−40
−35
−30
−25
−20
−15
−10
−5
0
Figure 7.3 Two-way radiation patterns of the south beams used in the 3Ba (left) and
3Bb (right) experiments. Note that, in the north-south direction, the 3Ba south beam is
narrower than the 3Bb beam.
but the other is a combination of two quarters forming a “bowtie” antenna configuration
(see Figure 7.1). Because of this, we can observe in the right panels of Figure 7.2 that
the 3Bb west-beam radiation patterns look different from each other. The same can be
observed in the case of the east-beam radiation patterns. As we can expect, the bowtie
antennas have narrower beams in the east-west direction. Moreover, their radiation pat-
terns have greater directivities and ABS factors, which implies that the data collected by
the bowtie antennas will have better SNR values. In each plot, the values of the expected
coherences between the co-polarized signals of the west-beam and east-beam antenna pairs
are given. In general, the expected coherence is ∼ 0.5 (a reference value that is calculated
disregarding magneto-ionic propagation effects).
In Figure 7.3, the two-way radiation patterns of the south beams used in the 3Ba and
3Bb experiments are displayed. Note that, in the north-south direction, the 3Ba south
beam is narrower than the 3Bb beam. This is because the south-beam antennas in the 3Ba
mode are the combination of two quadrants of the Jicamarca array (the north and south
quadrants), but in the 3Bb mode, the antennas are only single quadrants. In addition,
note that the ABS factors of the south beams are in general greater than those of the west
and east beams. The reason for this is that the south beams have lower sidelobes than the
98
Table 7.1 Radar operating parameters considered in the 3Ba and 3Bb experiments.
Radar parameter West-East South
Inter-pulse period 1000 km 1000 km
Pulse width 45 km 45 km
Pulse code 3-baud Barker 3-baud Barker
Number of pulses 128 32
Sampling rate 5 km 5 km
Number of samples 198 198
other beams.
Despite the different antenna configurations, the radar system was operated in the 3Ba
and 3Bb experiments with the same transmission and reception parameters (see Table 7.1).
In transmission, electromagnetic wave pulses were emitted in sequences that alternate
between perpendicular-to-B and off-perpendicular antenna beams. Each sequence started
with 128 pulses transmitted with the west and east beams (the four beams were excited
at the same time). These pulses were then followed by 32 pulses transmitted with the
up-polarization of the south beams. The separation between pulses (inter-pulse period
or IPP) was 6.667 ms, which is equivalent to a radar range of 1000 km. The length of
each transmitted pulse was 45 km. To improve the range resolution, the radar pulses were
coded with a 3-baud Barker sequence + +− that was flipped every other pulse (the baud
length was 15 km). In reception, the backscattered signals coming from the ionosphere were
sampled at the antenna outputs using digital receivers. The demodulation and the matched
filter detection of the signals are performed numerically in these receivers. Samples of the
in-phase and quadrature radar voltages were obtained every 5 km (i.e., the radar returns
were oversampled by a factor of three with respect to the baud length). These samples
were recorded and stored in a computer for later analysis using the data processing and
inversion techniques described in the following sections.
99
7.2 Radar Measurements
As mentioned before, observations of the upper atmosphere were conducted alternating
between perpendicular-to-B and off-perpendicular radar beams. The perpendicular-to-B
observations were carried out with the west and east beams, while the off-perpendicular
observations were performed with the south beams. The antenna voltages recorded during
each sequence of radar pulses constitute a block of data. In each block, 128 samples
per range gate per channel were collected with the west and east beams (4 channels). In
addition, 32 samples (also per range gate per channel) were acquired with the south beams
(2 channels). In both cases, a total of 198 range gates were sampled. Using the fast Fourier
transform (FFT), time series of the radar voltages (for every radar range) are converted to
frequency domain. As the time separation between these samples is equal to the interpulse
period, the maximum frequency of the spectrum is equal to 75 Hz (i.e., half the sampling
frequency). This is equivalent to a maximum Doppler velocity of 225 m/s.1 The resultant
self- and cross-spectrum estimates are then averaged with the results of the subsequent
data blocks. Approximately 280 spectra are averaged in 5 minutes of data integration.
This procedure reduces the statistical uncertainties of the estimated spectra.
In Figure 7.4, we present sample plots of the self- and cross-spectrum measurements
carried out at Jicamarca while operating in the 3Ba and 3Bb modes. The top panels
correspond to the data obtained in the 3Ba experiment of June 19, 2008, and the bottom
panels correspond to the 3Bb experiment of January 9, 2009. The data presented in these
plots were obtained after five minutes of incoherent integration. In each row, the first and
second plots are the self-spectra of the signals collected with one pair of antenna beams
(i.e., with either the west, east, or south beams). In addition, the third and fourth plots
are the magnitude and phase of the cross-spectrum between the same pair of signals. Note
that the magnitude of the cross-spectrum is normalized by the square root of the product
of the associated self-spectra (i.e., the plot is the magnitude of the coherence spectrum).
1The Doppler velocity vd is related to the frequency shift f by
vd =
λ
2
f,
where λ is the radar wavelength, e.g., λ = 6 m at Jicamarca.
100
R
an
ge
 [k
m]
W1−Up [dB]
 
 
 
−200 0 200
300
400
500
600
700
W2−Up [dB]
 
 
 
 
−200 0 200
W1 W2* [Mag]
 
 
 
 
−200 0 200
W1 W2* [Phase]
 
 
 
 
−200 0 200
R
an
ge
 [k
m]
E1−Dn [dB]
 
 
 
−200 0 200
300
400
500
600
700
E2−Dn [dB]
 
 
 
 
−200 0 200
E1 E2* [Mag]
 
 
 
 
−200 0 200
E1 E2* [Phase]
 
 
 
 
−200 0 200
Velocity [m/s]
R
an
ge
 [k
m]
S−Up [dB]
 
 
−200 0 200
300
400
500
600
700
Velocity [m/s]
S−Dn [dB]
 
 
 
−200 0 200
Velocity [m/s]
SU SD* [Mag]
 
 
 
−200 0 200
Velocity [m/s]
SU SD* [Phase]
 
 
 
−200 0 200
DEWD 3Ba − Date: 19−Jun−2008 13:00:00
50
52
54
56
50
52
54
56
0
0.2
0.4
0.6
0.8
1
−150
−100
−50
0
50
100
150
49
50
51
52
53
54
55
49
50
51
52
53
54
55
0
0.2
0.4
0.6
0.8
1
−150
−100
−50
0
50
100
150
44
45
46
47
48
49
50
44
45
46
47
48
49
50
0
0.2
0.4
0.6
0.8
1
−150
−100
−50
0
50
100
150
R
an
ge
 [k
m]
W1−Up [dB]
 
 
 
−200 0 200
300
400
500
600
700
W2−Up [dB]
 
 
 
 
−200 0 200
W1 W2* [Mag]
 
 
 
 
−200 0 200
W1 W2* [Phase]
 
 
 
 
−200 0 200
R
an
ge
 [k
m]
E1−Dn [dB]
 
 
 
−200 0 200
300
400
500
600
700
E2−Dn [dB]
 
 
 
 
−200 0 200
E1 E2* [Mag]
 
 
 
 
−200 0 200
E1 E2* [Phase]
 
 
 
 
−200 0 200
Velocity [m/s]
R
an
ge
 [k
m]
S−Up [dB]
 
 
−200 0 200
300
400
500
600
700
Velocity [m/s]
S−Dn [dB]
 
 
 
−200 0 200
Velocity [m/s]
SU SD* [Mag]
 
 
 
−200 0 200
Velocity [m/s]
SU SD* [Phase]
 
 
 
−200 0 200
DEWD 3Bb − Date: 09−Jan−2009 13:20:00
50
52
54
56
50
52
54
56
0
0.2
0.4
0.6
0.8
1
−150
−100
−50
0
50
100
150
49
50
51
52
53
54
55
49
50
51
52
53
54
55
0
0.2
0.4
0.6
0.8
1
−150
−100
−50
0
50
100
150
49
50
51
52
53
54
49
50
51
52
53
54
0
0.2
0.4
0.6
0.8
1
−150
−100
−50
0
50
100
150
Figure 7.4 Sample self- and cross-spectra measured in the 3Ba experiment of June 19,
2008 (top rows), and in the 3Bb experiment of January 9, 2009 (bottom rows). The sample
data were obtained after five minutes of integration. The self- and cross-spectra are plotted
as functions of Doppler velocity and radar range.
101
In the plots, we can appreciate that the spectra measured with the west and east beams
have narrow frequency bandwidths, as is expected for radar observations with antenna
beams pointed perpendicular to B. Note that the magnitude of the associated cross-
spectrum measurements also have narrow shapes. As mentioned before, the Doppler shifts
of the spectra measured with the west and east beams can be used to determine vertical
and zonal plasma drifts. The fact that the perpendicular-to-B spectra have very sharp
peaks makes it possible to estimate with high accuracy the Doppler shifts and the plasma
drifts. The signal processing technique developed for the estimation of these parameters
was the subject of previous research [Kudeki et al., 1999] and is not discussed in this
dissertation.
We can also see in Figure 7.4 that the spectra measured with the south beams, which
are pointed off-perpendicular to B, are very flat. The reason for this is that the bandwidth
of the off-perpendicular radar signal is greater than the pulse-to-pulse sampling frequency.
Therefore, the measured spectra are aliased. In addition, note that the signals collected
with the south beams experience Faraday rotation effects. As these beams are orthogonally
polarized, we can see how the polarization of the backscattered signal rotates, i.e., how the
signal goes from one polarization to the other as the radar range increases. These effects
will be discussed in more detail later in this section.
In view of Parseval’s theorem, the frequency components of the self- and cross-spectrum
measurements are summed to obtain signal power and cross-correlation profiles. In Figure
7.5, we present range vs. time plots of the power and correlation data measured in the
3Ba experiment of June 19, 2008. In the left column, the power data collected with all the
radar beams are displayed in linear scale. Additionally, the cross-correlation data of the
three pairs of beams are presented in the right column. The magnitudes of the correlation
are displayed in linear scale, and the phases are plotted in degrees.
First, note that there are marked differences between the power and correlation data
of the perpendicular-to-B (west and east) beams and the data of the off-perpendicular
(south) beams. As studied in the previous chapter, radiowaves propagating through the
ionosphere experience changes in their polarizations due to the presence of the Earth’s
102
 R
an
ge
 [k
m]
DEWD 3Ba − Signal Power Map (West beams) − Date: 19−Jun−2008
 
 
 Beam: W1−Up 
250
300
350
400
450
500
550
600
Si
gn
al
0
100
200
300
400
Time [Hours]
R
an
ge
 [k
m]
 
 
 Beam: W2−Up 
6 8 10 12 14 16 18 20 22 24
250
300
350
400
450
500
550
Si
gn
al
0
100
200
300
400
 
R
an
ge
 [k
m]
DEWD 3Ba − Cross−Correlation Map (West beams) − Date: 19−Jun−2008
 
 
 Pair: W1 W2* 
250
300
350
400
450
500
550
600
M
ag
ni
tu
de
0
50
100
150
200
Time [Hours]
R
an
ge
 [k
m]
 
 
 Pair: W1 W2* 
6 8 10 12 14 16 18 20 22 24
250
300
350
400
450
500
550
Ph
as
e 
[de
g]
−150
−100
−50
0
50
100
150
 
R
an
ge
 [k
m]
DEWD 3Ba − Signal Power Map (East beams) − Date: 19−Jun−2008
 
 
 Beam: E1−Dn 
250
300
350
400
450
500
550
600
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Figure 7.5 Range vs. time plots of the signal power and cross-correlation data measured
in the 3Ba experiment of June 19, 2008. On the left, the power data collected by each
of the radar beams are displayed in linear scale. On the right, the magnitudes of the
cross-correlation data are also plotted in linear scale, while the phase data are plotted in
degrees.
103
magnetic field. These polarization variations are dependent on the modes of propagation
excited by the radar beams. As the properties of these modes vary with magnetic aspect
angle, power and correlation profiles measured with antenna beams pointing in different
directions will have, to some extent, distinct shapes.
When the beam is pointed (at least one degree) away from perpendicular to B, the
modes excited by the beam are in the Faraday rotation regime, i.e., the propagating fields
rotate. This behavior can be observed in the power plots of the south beams. As these
beams are orthogonally polarized, we can see how power goes from one polarization to the
other as the radar range increases. A minimum of power in one polarization corresponds
to a maximum in the other polarization. Note that the transition between consecutive
minimum and maximum values in a power profile corresponds to a 90◦ rotation of the
polarization of the detected field. Faraday rotation effects can also be observed in the
cross-correlation plots. The transition between consecutive minimum values in a profile of
cross-correlation magnitude, minima that coincide with sudden changes in the correlation
phase, is related to a 180◦ rotation of the polarization vector.
As studied in the previous chapter, the modes of magneto-ionic propagation vary quite
rapidly around the perpendicular-to-B direction. The modes range from the Cotton-
Mouton regime (at exact perpendicularity) to the Faraday rotation regime (at aspect
angles around and greater than a degree). As the perpendicular-to-B (west and east)
beams in the 3Ba (and also in the 3Bb) experiments have widths of the order of a degree,
the signals collected with these beams are the result of the combination of different modes
of propagation. This complicates the description of the measurements collected with these
beams.
Although there are no apparent signatures of Faraday rotation in the power plots
of the west and east beams, some amount of power actually “rotates.” However, note
that the west beams as well as the east beams are co-polarized; therefore, the fraction
of power that goes to the orthogonal polarization is not measured in these experiments.
Additionally, note that these pairs of antennas are in practice radar interferometers aligned
in the east-west direction. As the characteristics of the propagating fields vary in the north-
104
south direction due to magneto-ionic propagation effects, the correlation profiles measured
with these interferometers are not dependent on the aspect angle variations of the radar
signals. Using the correlation and power information, we have computed coherence profiles
and noticed that the magnitudes and phases of the coherences are constant in height —
a result that is expected, as we discuss next. The coherence measured with a radar
interferometer (assuming a beam-filling scattering process) depends on the shapes of the
antenna beams along the direction of alignment of the antennas. The magnitude of the
coherence is determined by the widths of the radar beams, while the phase is dependent on
the pointing direction of the antennas. As mentioned before, magneto-ionic propagation
effects do not modify the shapes of beams in the east-west direction; therefore, as we found
in our measurements, the coherence values should be equal at all radar ranges. In addition,
we noted that the magnitudes of the coherences are approximately equal to the theoretical
values presented in the previous section (see Figure 7.2). This result gives us confidence
that the actual shapes of the radar beams are very close to the theoretical patterns.
Similar plots, but for the data measured in the 3Bb experiment of January 9, 2009, are
presented in Figure 7.6. In general, the features of the power and correlation measurements
presented in these plots are similar to the features of the 3Ba measurements displayed in
Figure 7.5. However, there are some differences that need to be pointed out. First,
note that the power measured with one of the west beams is stronger than the power
of the other beam. A similar difference can be observed between the signals collected
with the east beams. The reason for these differences is that the antennas of each pair
are of different sizes, one of them is twice as large as the other (see the description of
the antenna configurations in the previous section). In addition, note that the west-
and east-beam pairs are also radar interferometers, but the antennas are now aligned in
the north-south direction. Therefore, we expect that magneto-ionic propagation effects
will have an impact on the cross-correlation profiles measured with these interferometers.
However, we can observe that the phase of the correlation is almost constant as function
of height. Although altitudinal variations of the phase were expected, there are probably
two reasons to explain the lack of variations. First, note that, due to solar minimum
105
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Figure 7.6 Same plots as described in Figure 7.5, but for the signal power and cross-
correlation data measured in the 3Bb experiment of January 9, 2009.
106
conditions, the ionospheric densities in the last few years are smaller than typical values;
thus, magneto-ionic effects were relatively weak during these experiments. On top of that,
note that the separation between the interferometric antennas is not as large as in the
3Ba configuration (or in the DVD configuration of the previous chapter). Because the
phase variations are amplified by the separation between antennas, the 3Bb north-south
interferometers are not very sensitive to magneto-ionic effects.
In the following section, the different features of the power and cross-correlation data
presented above are modeled based on the incoherent scatter and magneto-ionic propaga-
tion theories developed in the previous chapters. The model is then used to invert physical
parameters of the equatorial ionosphere.
7.3 Data Model and Inversion Technique
In this section, we describe the forward model of the power and cross-correlation mea-
surements gathered in the 3Ba and 3Bb experiments. The data inversion technique used
for the estimation of electron densities and Te/Ti profiles is also described. Additionally,
some sample plots of the outcomes of the fitting results are presented.
In Chapter 5, a model for the propagation of radiowaves in a magnetized ionosphere
was developed. Based on this model, soft-target radar equations (5.60) and (5.62) were
formulated to represent the power and the cross-correlation of the incoherent scatter sig-
nals collected by the radar receivers. These equations take into account the aspect angle
variations of the radar signals due to incoherent scatter and magneto-ionic propagation
effects. In order to simplify these equations, it can be assumed that the bandwidth of the
incoherent scatter RCS spectrum σv(k, ω) is narrower than the frequency response of the
matched-filter receivers used for the detection of the antenna signals (a valid approxima-
tion in the case of short-pulse radar applications). After simplifying equations (5.60) and
(5.62), the power of the signal collected by the i-th antenna receiver can be expressed as
Pi = Ki
ˆ
dr
|χ(t− 2rc )|2
4pir2
ˆ
dΩWi(r)σv(k) +Ni, (7.1)
107
while the complex cross-correlation of the signals collected by the i-th and j-th antenna
receivers can be modeled as
Ci,j = Ki,j
ˆ
dr
|χ(t− 2rc )|2
4pir2
ˆ
dΩWi,j(r)σv(k) +Ni,j . (7.2)
Above, Ki and Ki,j are calibration constants that account for the gains and losses along
the paths of the radar signals. These constants can take the forms
Ki = |gi|2 and Ki,j = gig∗j , (7.3)
where gi and gj are complex proportionality factors for the voltages detected by the i-
and j-th radar receivers. In addition, χ(t) is the normalized auto-correlation of the pulse
waveform f(t) that can be expressed as
χ(t) =
1
T
f∗(−t) ∗ f(t). (7.4)
In equations (7.1) and (7.2), the volumetric incoherent scatter RCS σv(k) is weighted by
the effective two-way radiation patterns Wi(r) and Wi,j(r), which are smoothly varying
functions of the radar range that account for the magneto-ionic propagation effects on the
radar signals. In addition, Ni and Ni,j denote the power and cross-correlation of the noise
signals detected by the antenna receivers.
As mentioned before, in the 3Ba and 3Bb experiments, electromagnetic waves are
transmitted simultaneously with the antennas pointed in the west and east directions.
The west antennas are up-polarized, while the east ones are down-polarized. Despite the
orthogonality of the polarizations of these antennas, some amount of power may go from
the west beam to the east beam, or vice versa, due to magneto-ionic propagation effects.
In order to take into account this possible “cross-talk,” the expressions for the effective
radiation patterns in (7.1) and (7.2) become somewhat complicated. Denoting the pair of
west beams by w1 and w2 and the pair of east beams by e1 and e2, the effective two-way
108
radiation patterns for the power model (7.1) can take the following forms,
Ww1(r) =
|Rw1(rˆ)|2
k2
(
|Tw(rˆ)|2 Γuu(r) + |Te(rˆ)|2 Γdu(r) + 2Re
{
Tw(rˆ) T ∗e (rˆ) Γu,du (r)
})
,
(7.5)
Ww2(r) =
|Rw2(rˆ)|2
k2
(
|Tw(rˆ)|2 Γuu(r) + |Te(rˆ)|2 Γdu(r) + 2Re
{
Tw(rˆ) T ∗e (rˆ) Γu,du (r)
})
,
(7.6)
We1(r) =
|Re1(rˆ)|2
k2
(
|Te(rˆ)|2 Γdd(r) + |Tw(rˆ)|2 Γud(r) + 2Re
{
Te(rˆ) T ∗w (rˆ) Γd,ud (r)
})
, (7.7)
We2(r) =
|Re2(rˆ)|2
k2
(
|Te(rˆ)|2 Γdd(r) + |Tw(rˆ)|2 Γud(r) + 2Re
{
Te(rˆ) T ∗w (rˆ) Γd,ud (r)
})
, (7.8)
while the patterns for the cross-correlation model (7.2) can be expressed as
Ww1,w2(r) =
Rw1(rˆ)R∗w2(rˆ)
k2
(
|Tw(rˆ)|2 Γuu(r) + |Te(rˆ)|2 Γdu(r) + 2Re
{
Tw(rˆ) T ∗e (rˆ) Γu,du (r)
})
,
(7.9)
We1,e2(r) =
Re1(rˆ)R∗e2(rˆ)
k2
(
|Te(rˆ)|2 Γdd(r) + |Tw(rˆ)|2 Γud(r) + 2Re
{
Te(rˆ) T ∗w (rˆ) Γd,ud (r)
})
,
(7.10)
where k = 2pi/λB = 4pi/λ is the Bragg wavenumber. Above, Tw(rˆ) and Te(rˆ) are the
antenna-array factors of the west and east transmitting antennas. In addition, Rw1(rˆ),
Rw2(rˆ), Re1(rˆ), and Re2(rˆ) are the antenna-array factors of the receiving antennas. These
factors are properly normalized such that their squared magnitudes are antenna gain pat-
terns. Note that the products of the forms |Tw(rˆ)|2|Rw(rˆ)|2 and |Te(rˆ)|2|Re(rˆ)|2 correspond
to the two-way radiation patterns displayed in Figure 7.2. In the previous equations, the
coefficients multiplying the antenna-array factors can be expressed in the following form,
Γk,li,j (r) =
(
pˆTi Π¯(r) pˆk
)(
pˆTj Π¯(r) pˆl
)∗
, (7.11)
and represent polarization coefficients that account for the magneto-ionic propagation
effects on the polarizations of the returned fields. In this expression, Π¯(r) denotes the
two-way propagator matrix defined in Section 5.2. In addition, pˆi, pˆj , pˆk, and pˆl are
109
polarization unit vectors. The indices i and j denote the receiving antennas, while k and
l denote the transmitting antennas. In the equations for the effective radiation patterns,
these indices are either u for the up-polarization, or d for the down-polarization of the
Jicamarca antenna. Expressions for the polarization unit vectors pˆu and pˆd are given
in the previous chapter. Furthermore, in order to simplify the notation, we consider the
following equivalences Γki (r) ≡ Γk,ki,i (r), Γk,li (r) ≡ Γk,li,i (r), and Γki,j(r) ≡ Γk,ki,j (r).
Similarly, denoting the pair of south beams by s1 and s2, the effective two-way radiation
patterns for the corresponding power models take the following forms,
Ws1(r) =
|Rs1(rˆ)|2
k2
|Ts(rˆ)|2 Γuu(r), (7.12)
Ws2(r) =
|Rs2(rˆ)|2
k2
|Ts(rˆ)|2 Γud(r), (7.13)
while the effective pattern for the cross-correlation model can be expressed as
Ws1,s2(r) =
Rs1(rˆ)R∗s2(rˆ)
k2
|Ts(rˆ)|2 Γuu,d(r). (7.14)
Above, Ts(rˆ) is the antenna-array factor of the transmitting antenna that is up-polarized,
while Rs1(rˆ) and Rs2(rˆ) are the antenna-array factors of the receiving antennas that are
up- and down-polarized, respectively.
In the set of equations that model the power and correlation measurements, the vol-
umetric incoherent scatter RCS σv(kˆ) and the two-way propagator matrix Π¯(r) are the
only functions that depend on ionospheric parameters. The RCS σv(kˆ) depends mainly
on electron density Ne, temperature ratio Te/Ti, and magnetic aspect angle α, while the
matrix Π¯(r) only depends on the electron density and the magnetic field. Using the mag-
netic field values provided by the IGRF model [e.g., Olsen et al., 2000], we have applied
the power and cross-correlation models presented above to invert Ne and Te/Ti profiles
from the data collected in the 3Ba and 3Bb experiments.
For this purpose, we consider the ionosphere to be horizontally stratified. The electron
110
density profile Ne(z) is represented as a piecewise linear function
Ne(z) =
L∑
l=1
N le4
(
z − zl
∆z
)
(7.15)
for z ≥ z1, where 4(t) is the standard triangular function
4(t) =

1− |t|, |t| < 1
0, otherwise.
(7.16)
The density values at the nodes zl are N le = Ne(zl). The nodes are spaced every ∆z =
15 km (which is the length of the baud of the Barker code used in transmission). We
have considered z1 = 210 km and zL = 600 km, thus L = 27. The density profile below
z1 is also modeled by linear functions characterized by a single parameter N0e at a node
z0 = 150 km. It is assumed that below z = 85 km the electron density vanishes. In addition,
the temperature ratio profile Te(z)/Ti(z) is modeled as a layer with a Gaussian shape
Te(z)/Ti(z) = Tm exp
(
−(z − zT )
2
2w2T
)
+ 1, (7.17)
where the constraint Te/Ti ≥ 1 is implemented by default. This profile is characterized by
the following parameters: Tm + 1 is the peak of the temperature ratio, zT is the altitude
where Te/Ti peaks, and wT is the width of the Gaussian profile. Modeling the temperature
ratio profile as a single layer can be considered a valid approximation for the range of
altitudes probed in these experiments. Typical measurements of equatorial ionospheric
temperatures (with other radar techniques at Jicamarca) show that, at daytime hours,
there is a layer in the bottom-side of the F-region where Te/Ti ≥ 1.
In the numerical implementation of the data model presented above, we have computed
samples of the power and cross-correlation data every ∆r = 15 km at radar ranges rn =
n∆r where n is an integer. For this purpose, equations (7.1) and (7.2) have been recast
111
as discrete convolutions taking the following forms:
Pi[n] = |gi|2 |χ[n]|2 ∗
(
1
4pir2n
ˆ
dΩWi(rn)σv(k)
)
+Ni (7.18)
and
Ci,j [n] = gi g
∗
j |χ[n]|2 ∗
(
1
4pir2n
ˆ
dΩWi,j(rn)σv(k)
)
+Ni,j , (7.19)
where
χ[n] =
1
M
c[−n] ∗ c[n] (7.20)
is the discrete auto-correlation of the code c[n] of length M . In our 3Ba and 3Bb experi-
ments, the radar pulses were modulated by a 3-baud Barker sequence + +− with a baud
length equal to 15 km; thus, c[n] = [1, 1,−1] and M = 3. In this case, it can be verified
that χ[n] is given by
χ[n] = −1
3
δ[n+ 2] + δ[n]− 1
3
δ[n− 2]. (7.21)
In order to numerically evaluate the solid angle integrals in (7.18) and (7.19), we reformu-
late the integrals in terms of θx and θy, which are direction cosines of a coordinate system
in which the x- and y-axes are aligned with the east and north directions (see Figures 7.2
and 7.3). Note that the differential solid angle dΩ becomes dθxdθy/
√
1− θ2x − θ2y . Using
this coordinate system, we can take advantage of the magnetic aspect angle dependence of
the functions Wi(rn), Wi,j(rn), and σv(k). As we know, these functions vary quite rapidly
in the north-south direction, but very slowly in the east-west direction. Therefore, values
of these functions are computed in a grid that is finely sampled in the θy-direction with a
separation between samples of 0.002 dc (∼ 0.11◦), and coarsely sampled in the θx-direction
with a separation of 0.113 dc (∼ 6.5◦). A range of approximately ±20◦ with respect to the
antenna on-axis direction is covered in both θx- and θy-directions. The grid is composed of
349 nodes in θy and 7 nodes in θx; thus, in order to calculate the integrals, a total of 2443
samples is computed for every function. A finite-element-like integration method based on
rectangular elements is used to evaluate the integrals. Note that the same operation has
112
to be repeated for every range gate.
A trust-region approach for minimizing non-linear least-squares problems subject to
simple inequality conditions [e.g., Coleman and Li , 1996] was used for the inversion of the
power and cross-correlation data. In comparison with the standard Levenberg-Marquardt
algorithm, the trust-region method is more robust for large-scale problems. Additionally,
the algorithm allows us to impose lower and upper bounds on the solution of the problem.
The following list of parameters are fitted using this minimization algorithm.
• Electron densities N le at the nodes zl for 0 ≤ l ≤ L.
• Temperature ratio parameters Tm, zT , and wT .
• Calibration factors gi for the six antenna channels.
• Noise power levels Ni for the six antenna channels.
• Noise cross-correlation values Ni,j for the three pairs of antenna channels.
Note that gi and Ni,j are complex numbers. The parameters listed above are estimated
by minimizing the following cost function
E2 = E2w1,w2 + E2e1,e2 + E2s1,s2, (7.22)
where
E2i,j =
∑
n
δmTi,j [n] M¯
−1
i,j [n] δmi,j [n] (7.23)
is the error cost function of the data collected with the i-th and j-th antenna channels. In
this expression, δmi,j [n] is the vector of the differences between the measured and modeled
data at the radar range rn that is defined as
δmi,j [n] =

δPi[n]
δPj [n]
δRi,j [n]
δQi,j [n]

=

pi[n]− Pi[n]
pj [n]− Pj [n]
ri,j [n]−Ri,j [n]
qi,j [n]−Qi,j [n]

, (7.24)
113
where pi[n] and pj [n] denote the power measurements, and ri,j [n] and qi,j [n] correspond
to the real and imaginary parts of the cross-correlation data. In addition, Pi[n], Pj [n],
Ri,j [n] = Re{Ci,j [n]}, and Qi,j [n] = Im{Ci,j [n]} are the modeled power (7.18) and cross-
correlation (7.19). In the cost function (7.23), M¯i,j [n] is the covariance matrix of the vector
of data measurement errors δmi,j [n], i.e., M¯i,j [n] =
〈
δmTi,j [n] δmi,j [n]
〉
. Assuming that
δmi,j [n] is a zero-mean Gaussian random vector, M¯i,j [n] can be written in the following
form [e.g., Feng et al., 2004]
M¯i,j =
1
I

P 2i R
2
i,j +Q
2
i,j PiRi,j PiQi,j
R2i,j +Q
2
i,j P
2
j PjRi,j PjQi,j
PiRi,j PjRi,j
1
2
(
R2i,j −Q2i,j + PiPj
)
Ri,jQi,j
PiQi,j PjQi,j Ri,jQi,j
1
2
(
Q2i,j −R2i,j + PiPj
)

,
(7.25)
where I is the number of samples used to compute the power and cross-correlation mea-
surements. In this expression, indices n were omitted in order to simplify the notation.
The assumption of a Gaussian vector δmi,j [n] can be justified because every element of
δmi,j [n] results from the average of a large number of nearly uncorrelated samples. Note
that the radar data is collected every 5 km; however, in the cost function (7.23), we have
only considered samples that are spaced every 15 km. The form of the cost function (7.23)
is valid if the samples at every range gate are not correlated. This is our case, as samples
spaced at least a distance equal to the resolution of the pulse shape (which is 15 km) can
be considered to be uncorrelated. However, in order to include the oversampled data, a
more complicated cost function has to be considered because the data corresponding to
neighboring gates spaced every 5 km are correlated. The implementation of the inversion
algorithm including the oversampled data will be the subject of future research. The sum
in (7.23) is taken over the set of data between 210 km and 600 km; thus 14 ≤ n ≤ 40,
which implies that 27 range gates were fitted.
In order to estimate the parameters listed above, we apply the following two-step
strategy.
114
First step
• Initial values for the electron density, noise, and calibration parameters are deter-
mined. In this step, it is assumed that Te/Ti = 1 at all heights.
• The following constraints are specified: electron densities N le and noise levels Ni
must be positive, and gw1, ge1, and gs1 must be real and positive. The rest of the
parameters are not constrained.
• The trust-region method is used to minimize the cost function (7.22) subject to the
constraints specified above. In this first step, we only fitted for N le, Ni, Ni,j , and gi.
Second step
• The estimated parameters obtained in the first step are used as an initial guess. In
addition, we define initial values for Tm, zT , and wT .
• The same constraints specified in the first step are considered in this one. Addition-
ally, the Te/Ti parameters are restricted to be positive.
• The trust-region method is used again to minimize the cost function (7.22) but
including in the minimization the Te/Ti parameters. In this second step, we solve
for all the parameters listed above, i.e., N le, Tm, zT , wT , Ni, Ni,j , and gi.
We apply the two-step strategy because we notice that the inversion of N le is more robust
than the inversion of the Te/Ti parameters. When fitting directly for the densities and the
Te/Ti parameters all together in a single step, the solutions for Te/Ti tend to be erratic,
which is probably related to the fact that the power and cross-correlation data is somewhat
less dependent on Te/Ti than on the densities. Estimating the densities in a first step and
subsequently the densities plus the Te/Ti parameters provides better results. Note that
we have not considered any explicit regularization term in the cost function (7.22). On
the other hand, modeling the Te/Ti profile as a Gaussian function (characterized with a
few parameters) regularizes, in some sense, the problem because a continuity condition on
115
the Te/Ti profile is imposed. The mathematical relationship between our approach and
standard regularization procedures will be the subject of future studies.
In Figure 7.7, sample plots of the results obtained from the inversion of the 3Ba and
3Bb data are displayed. In these plots, the estimated electron density and Te/Ti profiles
are presented in the left column. In the other columns, the power and cross-correlation
measurements (dotted points) are compared with the modeled profiles resultant from the
inversion (continuous lines). We can appreciate that the comparison is good, because
measured and modeled profiles match closely; however, there are some discrepancies (par-
ticularly in the cross-correlation data of the south-beams of the 3Ba experiment) that
might be related to a poor characterization of the bottom part of the density profile. Fur-
ther analysis and discussion regarding the characteristics of the results are presented in
the following section.
7.4 Results and Comparisons
In Figure 7.8, we present plots of the electron density and Te/Ti estimates obtained
from the inversion of incoherent scatter power and cross-correlation data collected in the
3Ba experiment of June 19, 2008 (left column), and also in the 3Bb experiment of Jan-
uary 9, 2009 (right column). To obtain these results, the data model and fitting technique
described in the previous sections were applied. Both the Ne and Te/Ti profiles are plotted
in linear scale. First, note that the estimated densities and temperature ratios are within
the ranges of values expected for an equatorial ionosphere. Electron densities of the order
of 1011 m−3 are typical of solar-minimum conditions. A layer of Te/Ti > 1 located at
the bottom of the F-region is characteristic of the daytime ionosphere. The Te/Ti layer
has peak values greater than two and relaxes toward thermal equilibrium (Te = Ti) as
time reaches the night hours. However, we can also observe that the Te/Ti values fluc-
tuate from one profile to the next one. The origin of these fluctuations is the subject of
current analysis. Apparently, the magnitudes of the fluctuations are of the order of the
statistical uncertainties expected for these estimates. Note that the radar data collected
in the 3Ba and 3Bb experiments can be considered (to some extent) to be less dependent
116
0 1 2 3 4
x 1011
250
300
350
400
450
500
550
N
e
 [m−3]
H
ei
gh
t [k
m]
1 1.5 2
250
300
350
400
450
500
550
T
e
/Ti
H
ei
gh
t [k
m]
400 600 800
250
300
350
400
450
500
550
Power (East−West)
R
an
ge
 [k
m]
 
 
W1−Up
W2−Up
E1−Dn
E2−Dn
150 200 250 300
250
300
350
400
450
500
550
Power (South)
R
an
ge
 [k
m]
 
 
S−Up
S−Dn
0 100 200
250
300
350
400
450
500
550
CCF (Real)
R
an
ge
 [k
m]
 
 
W1 W2*
E1 E2*
SU SD*
−5 0 5 10 15
250
300
350
400
450
500
550
CCF (Imag)
R
an
ge
 [k
m]
DEWD 3Ba − Date: 19−Jun−2008 13:00:00
0 2 4 6
x 1011
250
300
350
400
450
500
550
600
N
e
 [m−3]
H
ei
gh
t [k
m]
1 1.5 2
250
300
350
400
450
500
550
600
T
e
/Ti
H
ei
gh
t [k
m]
400 600 800 1000
250
300
350
400
450
500
550
600
Power (East−West)
R
an
ge
 [k
m]
 
 
W1−Up
W2−Up
E1−Dn
E2−Dn
400 600 800
250
300
350
400
450
500
550
600
Power (South)
R
an
ge
 [k
m]
 
 
S−Up
S−Dn
−200 −100 0 100 200
250
300
350
400
450
500
550
600
CCF (Real)
R
an
ge
 [k
m]
 
 
W1 W2*
E1 E2*
SU SD*
0 50 100 150
250
300
350
400
450
500
550
600
CCF (Imag)
R
an
ge
 [k
m]
DEWD 3Bb − Date: 09−Jan−2009 13:20:00
Figure 7.7 Sample plots of the results obtained from the inversion of the 3Ba and 3Bb data
(top and bottom panels, respectively). The estimated Ne and Te/Ti profiles are plotted
in the left column. In the other columns, the power and cross-correlation measurements
(dotted points) are compared with the modeled profiles (continuous lines).
117
Ti
m
e 
[H
ou
rs]
Height [km]
D
EW
D
 3
Ba
 −
 E
le
ct
ro
n 
D
en
si
ty
 (N
e) 
− D
ate
: 1
9−
Ju
n−
20
08
 
 
6
8
10
12
14
16
18
20
22
24
25
0
30
0
35
0
40
0
45
0
50
0
55
0
60
0
Ne [m
−3
]
00.
5
11.
5
22.
5
33.
5
44.
5
55.
5
x 
10
11
Ti
m
e 
[H
ou
rs]
Height [km]
D
EW
D
 3
Bb
 −
 E
le
ct
ro
n 
De
ns
ity
 (N
e) 
− D
ate
: 0
9−
Ja
n−
20
09
 
 
6
8
10
12
14
16
18
20
22
25
0
30
0
35
0
40
0
45
0
50
0
55
0
60
0
Ne [m
−3
]
01234567x 
10
11
Ti
m
e 
[H
ou
rs]
Height [km]
D
EW
D
 3
Ba
 −
 T
em
pe
ra
tu
re
 ra
tio
 (T
e/T
i) −
 D
ate
: 1
9−
Ju
n−
20
08
 
 
6
8
10
12
14
16
18
20
22
24
25
0
30
0
35
0
40
0
45
0
50
0
55
0
60
0
Te/Ti
11.
1
1.
2
1.
3
1.
4
1.
5
1.
6
1.
7
1.
8
1.
9
2
Ti
m
e 
[H
ou
rs]
Height [km]
D
EW
D
 3
Bb
 −
 T
em
pe
ra
tu
re
 ra
tio
 (T
e/T
i) −
 D
ate
: 0
9−
Ja
n−
20
09
 
 
6
8
10
12
14
16
18
20
22
25
0
30
0
35
0
40
0
45
0
50
0
55
0
60
0
Te/Ti
11.
1
1.
2
1.
3
1.
4
1.
5
1.
6
1.
7
1.
8
1.
9
2
F
ig
ur
e
7.
8
R
an
ge
vs
.
ti
m
e
pl
ot
s
of
th
e
el
ec
tr
on
de
ns
it
y
an
d
T
e
/T
i
es
ti
m
at
es
ob
ta
in
ed
fr
om
th
e
in
ve
rs
io
n
of
in
co
he
re
nt
sc
at
te
r
si
gn
al
po
w
er
an
d
cr
os
s-
co
rr
el
at
io
n
da
ta
co
lle
ct
ed
in
th
e
3B
a
ex
pe
ri
m
en
t
of
Ju
ne
19
,
20
08
(l
ef
t
co
lu
m
n)
,
an
d
al
so
in
th
e
3B
b
ex
pe
ri
m
en
t
of
Ja
nu
ar
y
9,
20
09
(r
ig
ht
co
lu
m
n)
.
118
on Te/Ti than on Ne. As we know, if Te/Ti > 1, the incoherent scatter RCS increases
as the magnetic aspect angle α decreases and reaches its maximum value at α = 0◦ (i.e.,
at the angle of perpendicular-to-B propagation). However, the power increments mea-
sured with perpendicular-to-B beams are somewhat smaller than the theoretical values
for α = 0◦ because signal contributions with smaller RCS are averaged by the antenna
beams. This reduces the sensitivity of the radar data to changes in Te/Ti. Additionally,
note that the fluctuations of the Te/Ti values are larger in the 3Ba case than in the 3Bb
case. This might be related to the fact that the 3Ba two-way radiation patterns are broader
in the north-south direction than the 3Bb patterns, because a wider range of signal con-
tributions with smaller RCS is collected by the 3Ba beams (see antenna description in
Section 7.1). In addition, note that the incoherent scatter RCS is directly proportional to
Ne (see Section 4.2). Moreover, the altitudinal variations of the measured signals due to
magneto-ionic propagation effects are dependent on the electron density values but not on
Te/Ti. Although it is likely that the Te/Ti fluctuations are due to the statistical errors of
the estimated parameters, further studies need to be carried out in order to fully identify
the origin of these fluctuations.
In the density plots, we can observe patches of enhanced densities resembling “bead”
shapes. Whether these beads were artifacts of the inversion procedure needed to be an-
alyzed. First, we examined whether these events are correlated with fluctuations on the
transmitted power. After analyzing records of the line voltage applied to the transmitters
used in these experiments, we did not find any correlation between the fluctuations of the
supplied voltage and the patches of enhanced electron density. We also noted that the
voltage fluctuations follow closely the variations of the radar calibration parameters pre-
sented in Figure 7.9. As there are no systematic variations that may be the cause for the
density increments, we consider the origin of the density beads to be geophysical. Whether
these variations are due to solar activity or due to perturbations of the neutral atmosphere
will be the subjects of future investigations.
It is also important to describe the characteristics of the radar calibration constants
resulting from the inversion of the radar data. Note in Figure 7.9 that the calibration
119
−30
−25
−20
M
ag
ni
tu
de
 [d
B]
DEWD 3Ba − Calibration parameters − Date: 19−Jun−2008
 
 
 
W1−Up
W2−Up
E1−Dn
E2−Dn
S−Up
S−Dn
6 8 10 12 14 16 18 20 22 24
−10
−5
0
5
Time [Hours]
Ph
as
e 
[de
g]
 
 
W1−Up
W2−Up
E1−Dn
E2−Dn
S−Up
S−Dn
−24
−23
−22
−21
−20
−19
M
ag
ni
tu
de
 [d
B]
DEWD 3Bb − Calibration parameters − Date: 09−Jan−2009
 
 
 
W1−Up
W2−Up
E1−Dn
E2−Dn
S−Up
S−Dn
6 8 10 12 14 16 18 20 22
−30
−20
−10
0
Time [Hours]
Ph
as
e 
[de
g]
 
 
W1−Up
W2−Up
E1−Dn
E2−Dn
S−Up
S−Dn
Figure 7.9 Magnitudes and phases of the radar calibration constants obtained as part of
the inversion of the 3Ba and 3Bb radar data.
constants of the west and east antenna channels exhibit approximately the same variations
in time. This is expected if we assume that the only system parameter that may fluctuate
during the operation of the radar is the amount of transmitted power. However, we can
observe that the features of the calibration parameters of the south channels do not follow
necessarily the time variations of the calibration parameters of the other channels. The fact
that two different transmitters were used in these experiments (one for the west and east
beams and the other for the south beams) might be the reason for the distinct behaviors of
the calibration parameters. However, we can also see that the behaviors of the calibration
constants of the south beams are somewhat different from each other (particularly in the
case of the 3Ba data). These differences become more evident at time intervals in which
the electron temperature is greater than the ion temperature (i.e., Te/Ti > 1). The cause
for these unexpected differences might be related to a poor description of the electron
density profile at altitudes below 200 km (a segment of the profile that is characterized
by a single parameter, as explained in Section 7.3). The shape of the lower part of the
electron density profile determines the amount of Faraday rotation that the south-beam
fields experience up to the first altitude of the data samples used in our inversions. If a
simplified description of the bottom part of the density profile is causing an inaccurate
estimation of these parameters, the use of ionosonde data to help on the inversion of the
lower part of the density profile might be needed to properly model the radar data and
improve the quality of the inversions. Further testing and analysis is needed in order to
identify the origin of the differences between the calibration parameters.
120
6 8 10 12 14 16 18 20 22 24
0
2
4
6
8
10
12
14
16
18
Time [Hours]
ε2
 
/ N
m
e
a
su
re
m
e
n
ts
DEWD 3Ba − Goodness of Fit − Date: 19−Jun−2008
 
 
Te/Ti = 1 (Fixed)
Te/Ti ≥ 1
6 8 10 12 14 16 18 20 22
0
2
4
6
8
10
12
Time [Hours]
ε2
 
/ N
m
e
a
su
re
m
e
n
ts
DEWD 3Bb − Goodness of Fit − Date: 09−Jan−2009
 
 
Te/Ti = 1 (Fixed)
Te/Ti ≥ 1
Figure 7.10 Goodness-of-fit values resulting from the inversion of the 3Ba and 3Bb radar
data. The blue curves correspond to the values obtained by fitting only for the electron
density Ne and assuming that Te/Ti = 1 at all altitudes. The green curves correspond to
the values obtained by fitting for both parameters, Ne and Te/Ti.
In Figure 7.10, goodness-of-fit values obtained from the inversion of the 3Ba and 3Bb
radar data are presented and can be used to analyze the quality of the inversion results. The
goodness-of-fit is defined as E2/Nmeasurements. In the plots, the blue curves correspond to
the values obtained by fitting only for the electron density Ne and assuming that Te/Ti = 1
at all altitudes. The green curves correspond to the values obtained by fitting for both
parameters, Ne and Te/Ti. The goodness-of-fit values are around five in the case of the 3Ba
results, while they are around three in the case of the 3Bb results. As expected, note that
the goodness-of-fit values are smaller when both ionospheric parameters, Ne and Te/Ti,
are fitted, implying that Te/Ti correction is needed. However, the goodness-of-fit values
obtained are, in some sense, somewhat large, because a “good” fitting result corresponds to
a goodness-of-fit value close to one. This indicates that the model has not totally captured
the features of the data; therefore, there is still space for improvement.
In addition, in order to validate our estimated densities, we have compared our results
with ionosonde measurements conducted simultaneously at Jicamarca. In Figure 7.11,
we are comparing the peak values of the plasma frequencies (foF2 = 12pi
√
Nee2/meo)
measured with the 3Ba and 3Bb radar techniques (blue points) and with the ionosonde
measurements (red points). In addition, the altitudes corresponding to the peak values
of the plasma frequencies are compared in Figure 7.12. Although the ionosonde data is
noisy in the 3Ba case, we can observe that the agreement is very good, particularly in the
121
6 8 10 12 14 16 18 20 22 24
0
1
2
3
4
5
6
7
8
Time [Hours]
fo
F2
 [M
Hz
]
DEWD 3Ba − F2 plasma frequency − Date: 19−Jun−2008
 
 
ISR
Digisonde
6 8 10 12 14 16 18 20 22
0
1
2
3
4
5
6
7
8
9
10
11
Time [Hours]
fo
F2
 [M
Hz
]
DEWD 3Bb − F2 plasma frequency − Date: 09−Jan−2009
 
 
ISR
Digisonde
Figure 7.11 Comparison of the peak values of plasma frequencies (foF2) measured with
the 3Ba and 3Bb radar techniques (blue lines) and with an ionosonde system also located
at Jicamarca (red dots).
6 8 10 12 14 16 18 20 22 24
0
50
100
150
200
250
300
350
400
Time [Hours]
H
ei
gh
t [k
m]
DEWD 3Ba − Height of foF2 − Date: 19−Jun−2008
 
 
ISR
Digisonde
6 8 10 12 14 16 18 20 22
0
50
100
150
200
250
300
350
400
Time [Hours]
H
ei
gh
t [k
m]
DEWD 3Bb − Height of foF2 − Date: 09−Jan−2009
 
 
ISR
Digisonde
Figure 7.12 Comparison of the altitudes corresponding to the peak values of plasma
frequency presented in Figure 7.11.
3Bb case. Note that the oscillations on the plasma frequency curves are correlated with
the patches of enhanced densities described before. The fact that the same behavior can
be observed in both the radar and the ionosonde measurements reinforces our conclusion
that the origin of these patches is geophysical.
122
CHAPTER 8
CONCLUSIONS AND FUTURE WORK
In this dissertation, we have studied in detail Coulomb collisions and magneto-ionic
propagation effects on incoherent scatter radar measurements carried out at Jicamarca.
Our studies show that both processes modify the shapes of the ISR spectra measured with
antenna beams pointed perpendicular to B at 50 MHz; thus, their effects should not be
neglected in the modeling of radar measurements. To perform this analysis, we developed
a new incoherent scatter spectrum model that takes into account both types of effects.
The model is valid for all magnetic aspect angles, including the direction perpendicular to
the ambient magnetic field.
In the first stage of our work, we focused on modeling the effects of Coulomb collisions.
The procedure to model the IS spectrum is based on the simulation of the trajectories
of charged particles moving in a magnetized plasma with suppressed collective interac-
tions. The effects of Coulomb collisions are considered using the Fokker-Planck model of
Rosenbluth et al. [1957]. The results of our studies show that, in contrast with the case
of the ions, the statistics of the electron displacements cannot be fully approximated by
simplified Coulomb collision models. Thus, in order to apply our collisional spectral model
in routine computations of the IS spectrum, a numerical library of single-electron ACF’s
had to be built. The library, however, was only developed for the case of oxygen plasmas.
This limits the applicability of our model to the region around and below the peak of the
F-region. In order to study the effects of Coulomb collisions on radar observations of the
topside ionosphere, we have to extend our model to the case of hydrogen and helium plas-
mas (because H+ and He+ are the dominant ion species at the top of the F-region). This
123
future work will require the simulation of a large number of particle trajectories for differ-
ent plasma settings, a very computationally demanding task that will require the use of a
cluster of computers, such as the Turing Cluster maintained by the Computational Science
and Engineering (CSE) Program at the University of Illinois. Basic questions regarding
the nature of Coulomb collisions will be addressed in these future studies. For instance,
whether the Coulomb collision process in ionospheric H+ and He+ plasmas is Gaussian
or not, it is an important question to answer. In the case of Gaussian particle trajec-
tories, a Brownian-motion model characterized by constant collision coefficients could be
used to simplify the description of the process. The collisional model resultant from this
investigation will provide the Jicamarca Radio Observatory with unique capabilities.
In the second stage of our studies, we have modeled the beam-weighted incoherent
scatter radar spectrum taking into account magneto-ionic propagation effects. The model
developed in Chapter 5 is effectively an extension of the Appleton-Hartree solution for a
homogeneous magneto-plasma to the case of an inhomogeneous ionosphere. Simulation
studies based on our model show that magneto-ionic propagation effectively modifies the
shapes of the antenna beams used in our radar experiments. The model was developed
based on the assumption that refraction effects can be neglected. Although refraction is
weak at 50 MHz, given the long distances traveled by the radar signals, some features of
our measurements might, to some extent, be the result of refraction effects, particularly
in the case of radar observations above the F-region peak. Note that refraction may cause
the bending of the propagating fields; therefore, the angular shifts of the antenna beams
that were observed in our radar experiments might be caused not only by magneto-ionic
polarization effects, but also by refraction effects. In order to analyze these possible effects,
we are planning to develop a more sophisticated propagation model based on the WKB
(Wentzel, Kramers, and Brillouin) approximation of Budden [1961] for the propagation of
radiowaves in an anisotropic inhomogeneous ionosphere.
In order to simplify the formulation of our spectrum model, we considered that the
shape of the radar ambiguity function in frequency domain is much wider than the inco-
herent scatter spectrum. Thus, the frequency dependence of the ambiguity function was
124
neglected by replacing this function with the ACF of the pulse waveform (see Chapter 5).
In the case of perpendicular-to-B radar measurements, this assumption is well justified
because the IS spectrum is very narrow; however, in the case of off-perpendicular radar
measurements, the IS spectrum is wider, and, therefore, its shape might be filtered by
the radar ambiguity function. Some preliminary calculations have shown that the power
measured with off-perpendicular radar beams around the F-region peak may be attenu-
ated by five percent due to ambiguity-function filtering effects. This attenuation, however,
depends on the shape and length of the radar pulse, as well as on the width of the IS
spectrum, which (at these aspect angles) is mainly controlled by the ionospheric temper-
atures. Further studies need to be conducted in order to quantify more carefully these
attenuation effects and verify if they will have an impact on the estimation of ionospheric
plasma parameters.
In Chapter 4, we studied the dependence of the collisional spectra on electron and ion
temperature variations and noticed that the bandwidth of the perpendicular-to-B spec-
trum is inversely proportional to Te/Ti; however, at larger aspect angles, the bandwidth
dependence is different, because the spectral width increases with increasing Te/Ti. As
the IS spectrum measured by the radar results from the combination of signal contribu-
tions arriving from different directions, the dependence of the measured ISR spectrum
might be more complicated, particularly because typical spectrum measurements with
perpendicular-to-B beams are aliased. The analysis of the sensitivity of the beam-weighted
IS spectrum model to electron and ion temperatures will be the subject of future studies.
In Chapter 7, we applied our incoherent scatter data model to the estimation of electron
densities and Te/Ti profiles from radar measurements carried out in multi-beam radar ex-
periments at Jicamarca. In these experiments, different sections of the Jicamarca antenna
array are phased to point west, east, or south of the local zenith. While the west and east
beams are aimed perpendicular to the magnetic field, the south beams are pointed in a
direction ∼ 4◦ to the south of the loci of perpendicularity. In addition to electron density
and Te/Ti estimates, radar calibration parameters were also obtained from the inversion
procedure. The quality of our inversion results was analyzed based on the goodness-
125
of-fit criterion. Moreover, comparisons between our density estimates and independent
ionosonde measurements carried out simultaneously at Jicamarca showed excellent agree-
ment. Although the quality of our inversion results can be considered good, a more careful
analysis of the statistical errors of our estimates is still needed. There are certain features
in our estimated results that require further investigation. For instance, in the inverted
density maps, we can observe patches of enhanced electron density that resemble “bead”
shapes, whose origin might be due to solar activity or perturbations of the neutral atmo-
sphere. In addition, we can also see that the estimated Te/Ti values fluctuate from profile
to profile. Although it is likely that the Te/Ti fluctuations are within the range of the
statistical errors for these parameters, further analysis is needed.
The ultimate application of our incoherent scatter model is the simultaneous estimation
of drifts, densities, and temperatures of the equatorial ionosphere (an objective not yet
achieved by the radar techniques available at Jicamarca). For this purpose, a composite
experiment that interleaves multi-pulse and pulse-to-pulse radar measurements has been
recently proposed for its application at Jicamarca. This experiment will be carried out
with antenna beams pointed perpendicular to B. As mentioned before, the pulse-to-
pulse incoherent scatter spectrum measured at Jicamarca is composed of narrow and wide
bandwidth spectral components. The plasma drifts are determined from the Doppler shift
of the narrow spectrum component. However, the wide spectrum component is typically
aliased in pulse-to-pulse radar observations. As the bandwidth of the wide component is
directly proportional to plasma temperatures, we will measure this spectrum component
using a multi-pulse radar technique in an attempt to invert temperature profiles from these
measurements. This will be the fulfillment of objectives set forth more than a decade ago,
when spectral Te estimations were first tried [Bhattacharyya, 1998] and the inadequacy of
the incoherent scatter theory close to perpendicular-to-B was first realized.
126
APPENDIX A
PARTICLE DYNAMICS DESCRIPTION OF
“BGK COLLISIONS” AS A POISSON PROCESS
The Gordeyev integral for plasma particles colliding with neutrals is obtained using
a particle dynamics formalism in which the collisions are modeled as a discrete Poisson
process. The result leads to an electron density fluctuation spectrum model for partially
ionized plasmas that is identical with the spectral model obtained from BGK plasma
kinetic equations. This isomorphism between the Poisson process and the BGK operator
is analogous to a similar relation between the Brownian-motion process and the Fokker-
Planck operator with constant coefficients. We take advantage of this analogy to derive
a collisional IS spectrum model that takes into account collisions with both neutrals and
charged species.
A.1 Introduction
The purpose of the following discussion is to show that the incoherent scatter spec-
trum model derived from the Boltzmann kinetic equation with the BGK collision opera-
tor [Dougherty and Farley , 1963] can also be obtained by using a particle dynamics ap-
proach where particle collisions are modeled as a Poisson process. This is analogous to the
case of the Fokker-Planck operator (with constant coefficients) and the Brownian-motion
(Ornstein-Uhlenbeck) collision process (described by the Langevin equation), which lead
to identical spectral models when utilized in the Boltzmann equation and in the particle
dynamics formalism, respectively [e.g., Chandrasekhar , 1943; Gillespie, 1996b; Holod et al.,
2005].
127
In general, incoherent scatter spectral theories pertinent to various types of ionospheric
plasmas (magnetized, collisional, etc.) in thermal equilibrium can be expressed in terms of
appropriately derived Gordeyev integrals utilized within the general framework described
in Chapter 2 [e.g., Kudeki and Milla, 2006, 2010]. In this framework, Gordeyev integral
Js(ω) ≡
ˆ ∞
0
dτ e−jωτ 〈ejk·∆rs〉 (A.1)
is defined as the one-sided Fourier transform of the characteristic function 〈ejk·∆rs〉 of
random particle displacement vector ∆rs for species s over time intervals τ (in a plasma
with suppressed collective interactions). Different types of plasmas are distinguished by
different types of ∆rs statistics. Such statistics depend on the physical processes govern-
ing individual particle motions and determine the species conductivities σs(k, ω) and the
corresponding spectra of thermally impressed density fluctuations 〈|nts(k, ω)|2〉, functions
that “collectively drive” the observed electron density spectrum 〈|ne(k, ω)|2〉. The char-
acteristic function 〈ejk·∆rs〉 is also termed the single-particle signal correlation (or ACF),
since signal returns from a single particle exposed to a radar pulse would be proportional
to ejk·rs , with rs = rs(t) denoting the particle trajectory.
In this appendix, we show that BGK-based incoherent scatter spectral models can also
be derived from a particle dynamics approach and that they constitute no exception to the
general framework presented in Chapter 2. In particular, the BGK-based Gordeyev integral
is obtained, in Section A.2, by assuming a Poisson collision process, and it is shown that
its inclusion in the general framework of Chapter 2 yields the usual BGK-model results
[e.g., Dougherty , 1963; Dougherty and Farley , 1963] for the electron density spectrum.
The appendix is concluded in Section A.3 with further discussions and implications of our
results; for instance, an extension of our approach to include collisions with both neutrals
and charged particles is proposed here.
128
A.2 Derivation of the BGK Gordeyev Integral following a
Particle Dynamics Approach
Clemmow and Dougherty [1969] state that the BGK model can be used to simulate the
effect of binary collisions, e.g., collisions between charged and neutral particles, and also
that these collisions could be imagined to be a Poisson process. However, this explanation
was given more to provide a possible interpretation of the BGK model than to establish a
direct relationship with the Poisson collision process. The mathematical proof that these
two collision models are directly linked is provided next.
Assume that in a time interval τ , a particle of type s (electron or ion) collides n times
with neutral particles constituting a medium. If collision events are independent from each
other and particles move in straight-line orbits in between collisions, we can then express
the particle displacement over interval τ as the random vector
∆rs =
n∑
l=0
vl(tl+1 − tl), (A.2)
where t0 ≡ 0, tn+1 ≡ τ, and tl is the time of l-th collision such that 0 < t1 < ... < tn < τ .
Let the number of collision events n invoked above be a Poisson random variable with
a collision frequency ν and pmf
p(n) = e−ντ
(ντ)n
n!
for n ≥ 0. (A.3)
Given that there are n collision events in an interval τ , the conditional pdf of collision
times tl (for 1 ≤ l ≤ n) is given by [e.g., Hajek , 2009]
f(t1, .., tn|n) =

n!
τn if 0 < t1 < ... < tn < τ,
0 else.
(A.4)
This is in effect a uniform distribution over all ordered sets of collision times 0 < t1 < ... <
129
tn < τ , which can also be written as
f(t1, .., tn|n) = n!
τn
n∏
l=0
u(tl+1 − tl) (A.5)
in terms of unit-step function u(t) and satisfies
´
dtn · · ·
´
dt1f(t1, .., tn|n) = 1 as all pdf’s
do. Additionally, assume that the particle velocities vl between collision events, for l ∈
[0, n], constitute a set of independent and identically distributed random variables. Taking
the distribution of vector velocities vl as Maxwellian, we have
f(vl) =
1
(2piC2)3/2
e−
1
2C2
v2l , (A.6)
where C ≡
√
KT
m is the thermal speed of the particles.
Let us next compute the single-particle ACF 〈ejk·∆rs〉, the characteristic function of
displacements ∆rs, required in the general framework outlined in Chapter 2, where the
expected value will be computed over random variables v0, · · · ,vn, t1, · · · , tn, and n using
the probability distributions defined above. We note that
〈ejk·∆rs〉 =
〈
ejk·
∑n
l=0 vl(tl+1−tl)
〉
v,t,n
=
〈
n∏
l=0
ejk·vl(tl+1−tl)
〉
v,t,n
, (A.7)
where the subscripts on the right indicate successive expected value operations to be per-
formed. Starting with the expectations over independent Gaussian random variables vl,
we have
〈ejk·∆rs〉 =
〈
n∏
l=0
e−
1
2
k2C2(tl+1−tl)2
〉
t,n
. (A.8)
Expectations over t1, · · · , tn next yield〈
n!
τn
ˆ
dtn · · ·
ˆ
dt1
n∏
l=0
e−
1
2
k2C2(tl+1−tl)2u(tl+1 − tl)
〉
n
, (A.9)
130
where integration limits run from −∞ to ∞. Finally, using p(n), we find that the ACF is
∞∑
n=0
νne−ντ
ˆ
dtn · · ·
ˆ
dt1
n∏
l=0
e−
1
2
k2C2(tl+1−tl)2u(tl+1 − tl). (A.10)
Now we define
g(t) ≡ e−νt− 12k2C2t2 u(t), (A.11)
to rearrange (A.10) as
〈ejk·∆rs〉 =
∞∑
n=0
νn
ˆ
dtn · · ·
ˆ
dt1
n∏
l=0
g(tl+1 − tl)
=
∞∑
n=0
νn
ˆ
dtng(τ − tn) · · ·
ˆ
dt1g(t2 − t1)g(t1). (A.12)
Clearly, the integral chain above is n successive convolutions of n + 1 realizations of g(t)
evaluated at t = τ . Thus, the ACF of particles of type s with a Maxwellian velocity
distribution undergoing a Poisson collision process reduces to
〈ejk·∆rs〉 =
∞∑
n=0
νn g(τ) ∗ · · · ∗ g(τ)︸ ︷︷ ︸
n+1
. (A.13)
Since 〈ejk·∆rs〉 is the characteristic function of ∆rs, i.e., the Fourier transform of the pdf
of ∆rs, our particle displacement model with Poisson collisions is uniquely described by
(A.13).
The corresponding Gordeyev integral (A.1), i.e., a one-sided Fourier transform of the
single-particle ACF (A.13), is then (using the properties of the convolution and the Fourier
transform)
Js(ω) = G(ω)
∞∑
n=0
νnGn(ω), (A.14)
where
G(ω) ≡
∞ˆ
0
dτe−jωτe−ντ−
1
2
k2C2τ2 (A.15)
131
is the Fourier transform of g(t). The convergence of the series in (A.14) is guaranteed as
∞∑
n=0
νnGn(ω) =
1
1− νG(ω) (A.16)
since |νG(ω)| < 1 for any ν ≥ 0. Hence, Gordeyev integral (A.14) reduces to
Js(ω) =
G(ω)
1− νG(ω) , (A.17)
a well-known result previously derived by Dougherty [1963] using the BGK collision oper-
ator and Boltzmann kinetic equation for unmagnetized plasmas.
A.3 Discussion
We can argue, based on our result in Section A.2, that from a particle dynamics per-
spective, the BGK collision process is a Poisson process. As a consequence, any of the
expected quantities that can be obtained using the BGK kinetic equation can also be de-
rived using our stochastic model for particle dynamics, for example, species conductivities
σs(k, ω) and the spectra of thermally impressed density fluctuations 〈|nts(k, ω)|2〉.
As we have shown above, the general framework for incoherent scatter spectrum models
presented in Chapter 2 can be developed independent of plasma kinetic equations on the
basis of only the following fundamental relations: the fluctuation-dissipation or Nyquist
theorem [e.g., Callen and Welton, 1951], the Kramers-Kronig relations [e.g., Clemmow
and Dougherty , 1969; Chew , 1990], and the Maxwell equations. Therefore, plasmas can
also be studied based on these principles; whether we use kinetic equations or the particle
dynamics approach in solving plasma problems concerning particles in thermal equilibrium
(including the phenomenon of Landau damping) is a matter of choice.
Let us now calculate, as an example, the covariance matrix of particle displacements
in a Poisson collision process. The covariance matrix is defined as
〈∆rs∆rTs 〉 =
〈
n∑
l=0
n∑
l′=0
vlv
T
l′ (tl+1 − tl)(tl′+1 − tl′)
〉
v,t,n
. (A.18)
132
Since vl are independent random variables, we have
〈∆rs∆rTs 〉 =
〈
n∑
l=0
vlv
T
l (tl+1 − tl)2
〉
v,t,n
. (A.19)
Furthermore, provided that the components of vl are Gaussian and independent, the co-
variance matrix of vl is given by
〈vlvTl 〉 = C2 I, (A.20)
where I is the identity matrix. Thus,
〈∆rs∆rTs 〉 = C2 I
〈
n∑
l=0
(tl+1 − tl)2
〉
t,n
. (A.21)
133
After some math, it can be verified that1
〈
n∑
l=0
(tl+1 − tl)2
〉
t
=
2τ2
n+ 2
. (A.22)
Taking the expected value of (A.22) with respect to n and substituting in (A.21) gives
〈∆rs∆rTs 〉 = C2 I
∞∑
n=0
e−ντ
(ντ)n
n!
2τ2
n+ 2
=
2C2
ν2
I
∞∑
n=0
e−ντ
(n+ 1)(ντ)n+2
(n+ 2)!
, (A.23)
which simplifies as
〈∆rs∆rTs 〉 =
2C2
ν2
I
(
ντ − 1 + e−ντ) . (A.24)
This is a very interesting result because the same mathematical expression can be
1Let sn(τ) ≡
〈∑n
l=0(tl+1 − tl)2
〉
t
for n ≥ 0. Taking expectations and expanding terms, we find that
sn(τ) =
n!
τn
τˆ
0
dtn · · ·
t2ˆ
0
dt1
n∑
l=0
(tl+1 − tl)2
=
n!
τn
τˆ
0
dtn · · ·
t2ˆ
0
dt1(τ − tn)2 + n!
τn
τˆ
0
dtn · · ·
t2ˆ
0
dt1
n−1∑
l=0
(tl+1 − tl)2
=
n
τn
τˆ
0
dtn(τ − tn)2tn−1n + n
τn
τˆ
0
dtnt
n−1
n
(n− 1)!
tn−1n
tnˆ
0
dtn−1 · · ·
t2ˆ
0
dt1
n−1∑
l=0
(tl+1 − tl)2
︸ ︷︷ ︸
sn−1(tn)
=
2τ2
(n+ 1) (n+ 2)
+
n
τn
τˆ
0
dt tn−1sn−1(t).
Using this recursive equation, we can compute
s0(τ) = τ
2,
s1(τ) =
τ2
3
+
1
τ
τˆ
0
dt τ2 =
2
3
τ2,
s2(τ) =
τ2
6
+
2
τ2
τˆ
0
dt
2
3
τ3 =
1
2
τ2,
s3(τ) =
τ2
10
+
3
τ3
τˆ
0
dt
1
2
τ4 =
2
5
τ2,
and so forth. Then, by induction, we obtain
sn(τ) =
2τ2
n+ 2
.
134
derived in the context of a Brownian-motion collision model [e.g., Chandrasekhar , 1943],
an approach that it is often used to describe Coulomb collisions [e.g., Zagorodny and Holod ,
2000]. In the Brownian-motion formalism, the effects of collisions on particle motion are
considered to be caused by the action of a friction force and random diffusion forces.
A parameter analogous to ν is also defined, but it is regarded as a friction coefficient.
Although the expressions for 〈∆rs∆rTs 〉 are the same for the Poisson and the Brownian-
motion models, the corresponding expressions for the single-particle ACF’s are not equal
and lead to different incoherent scatter spectral shapes [e.g., Hagfors and Brockelman,
1971]. The differences, however, are only noticeable for intermediate values of ν since
both spectra converge to the same asymptotic expressions in the collisionless/frictionless
(ν → 0) as well as high collision/friction (ν → ∞) limits. The difference at intermediate
ν can be attributed to ∆rs(τ) being a Gaussian random variable at each τ in case of a
Brownian-motion process, but only so in τ → 0 and ∞ limits for a Poisson process. Note
that when ∆rs(τ) is strictly Gaussian, and only then, the ACF 〈ejk·∆rs〉 can be shown to
reduce to e−
1
2
k2〈∆r2s〉, where 〈∆r2s〉 is a diagonal element of 〈∆rs∆rTs 〉.
A generalization of the results presented in Section A.2 can be easily performed. Notice
that the assumption that between collisions the particles move in straight-line orbits was
not a necessary condition and it was only considered in order to recover the classical results
of the BGK collision model. In between neutral collision events, we could have considered
the particles moving in helical orbits due the action of an external magnetic force or even
move randomly because of Coulomb interactions with other charged particles constituting
a plasma. Either of these assumptions would have led to different definitions of the function
G(ω), but it can be shown that the form of the Gordeyev integral Js(ω), i.e., equation
(A.14), would have remained the same. In general, it is found that
G(ω) =
∞ˆ
0
dτe−jωτe−ντ 〈ejk·∆ris〉, (A.25)
where 〈ejk·∆ris〉 is the single-particle ACF of the process that takes place between Poisson
collision events. For instance, let us consider an unmagnetized plasma in which both
135
neutral and Coulomb collisions are relevant. Modeling the neutral collisions as a Poisson
process of frequency ν, and Coulomb collisions as a Brownian-motion process with friction
coefficient β, function G(ω) for a particle undergoing both types of collisions takes the
following form
G(ω) =
∞ˆ
0
dτe−jωτe−ντe−
k2C2
β2
(βτ−1+e−βτ)
. (A.26)
This expression together with (A.14) provides us with a model for ionospheric incoherent
scatter spectrum measurements detected from regions in which both neutral and Coulomb
collisions are expected to be important, e.g., the equatorial 150-km region. More general
extensions of the Poisson collision model can be pursued; for instance, a velocity dependent
collision frequency ν(v) could be considered into the theory. Further generalizations of this
type will be the subject of future studies.
136
APPENDIX B
FRICTION AND DIFFUSION COEFFICIENTS
FOR “TEST PARTICLES” IN
NON-ISOTHERMAL PLASMAS
In this appendix, it is shown that the steady-state solution of the Fokker-Planck equa-
tion with the standard Spitzer coefficients for a plasma do not relax toward a Maxwellian
distribution in the case of unequal electron and ion temperatures. However, if typical
F-region plasma configurations are considered, deviations of the steady-state distributions
from exact Maxwellian are very small. For instance, we have verified that the second order
moments of the distributions (i.e., effective temperature values) have negligible dependence
on the difference between the values of Te and Ti considered for the plasma. In addition, a
modified version of the Spitzer friction coefficient is formulated in order to obtain steady-
state Maxwellian distributions even in the case of unequal electron and ion temperatures
(Te 6= Ti).
B.1 Introduction
Some of the problems in plasma physics can be reduced to the statistical analysis of
the dynamics of one of the particles constituting the medium — a “test” particle problem
[e.g., Rostoker and Rosenbluth, 1960]. For this analysis, a probability density function fs
of a particle of type s is defined such that fs(r,v, t)drdv is the probability of finding the
particle at a position r with velocity v at a time t. The time evolution of the probability
137
function fs is governed by the Boltzmann kinetic equation
∂fs
∂t
+ v · ∇rfs + F
m
· ∇vfs =
(
δfs
δt
)
c
, (B.1)
where
(
δfs
δt
)
c
accounts for the changes in the distribution due to particle collisions. Note
that fs must be properly normalized such that
´
fs(r,v, t)drdv = 1. If at a time to,
position ro and velocity vo are set, the density function fs becomes a conditional pdf
fs(r,v, t|ro,vo, to) (also referred as a transition probability function). This constitutes an
initial value problem, since equation (B.1) is solved subject to the condition
fs(r,v, to|ro,vo, to) = δ(r− ro)δ(v − vo). (B.2)
The solution of this problem is of principal interest in plasma physics, and it can be used,
for instance, to calculate the spectrum of electron density fluctuations [e.g., Rosenbluth
and Rostoker , 1962; Woodman, 1967].
If Coulomb interactions are the dominant collision process, the right-hand side of (B.1)
can take the form of a Fokker-Planck collision operator
(
δfs
δt
)
c
= −∇v ·
[〈
∆v
∆t
〉
fs
]
+
1
2
∇v∇v :
[〈
∆v∆vT
∆t
〉
fs
]
, (B.3)
where ∇v is the gradient operator in velocity space, ∇v = xˆ (∂/∂vx) + yˆ (∂/∂vy) +
zˆ (∂/∂vz) .
1 Above,
〈
∆v
∆t
〉
and
〈
∆v∆vT
∆t
〉
are the dynamical friction vector and velocity
diffusion tensor, functions of particle velocity that can be formulated in terms of the
Rosenbluth potentials [Rosenbluth et al., 1957]. A typical exercise in plasma books is the
evaluation of
〈
∆v
∆t
〉
and
〈
∆v∆vT
∆t
〉
for the case of a multiple-component plasma in thermal
equilibrium [e.g., Montgomery and Tidman, 1964; Gurnett and Bhattacharjee, 2005]. The
resultant friction and diffusion coefficients are known as the Spitzer coefficients [Spitzer ,
1The differential operations in equation (B.3) are defined as
∇v ·A(v) =
∑
i
∂
∂vi
Ai(v) and ∇v∇v : B¯(v) =
∑
i
∑
j
∂
∂vi
∂
∂vj
Bi,j(v),
where A(v) is a vector function and B¯(v) is a tensor.
138
1962]. Explicit expressions for these functions are provided in Section 2.5.
In the literature, the Fokker-Planck collision model with the Spitzer coefficients is
used to study the effects of Coulomb collisions on particle distributions in non-isothermal
plasmas, i.e., in plasmas in which electron and ion temperatures are not equal [e.g., Milla
and Kudeki , 2010]. Although this can be considered a valid extension of the model, we
found that there is a discrepancy between the assumptions made to derive the coefficients
of the Fokker-Planck equation and the solutions obtained from it. As we discuss in Section
B.2, steady-state solutions of the Fokker-Planck equation (B.3) with the standard Spitzer
coefficients do not relax towards Maxwellian distributions if electron and ion temperatures
are not equal. However, to derive the Spitzer coefficients, it is assumed that the particle
distributions for the different species in a non-isothermal plasma are Maxwellian. Despite
the fact that deviations of the steady-state solutions of (B.3) from Maxwellian distributions
are very small, this issue brings up the question of the validity of the model.
In Section B.3, a modified version of the Spitzer friction coefficient is derived. This
modified expression makes the steady-state solutions of the Fokker-Planck equation be
exactly Maxwellian (even if electron and ion temperatures are different). Whether such a
correction can be justified based on more fundamental physical principles than the ones
exposed here will be the subject of a future investigation. A short discussion of our results
presented in Section B.4 concludes this appendix.
B.2 Steady-State Solution of the Fokker-Planck Equation
with Spitzer Collision Coefficients
In the case of a homogeneous stationary plasma, the friction vector and diffusion tensor
of the Fokker-Planck equation (B.3) can be expressed in the forms
〈
∆v
∆t
〉
= −β(v)v (B.4)
and 〈
∆v∆vT
∆t
〉
=
D⊥(v)
2
I +
(
D‖(v)−
D⊥(v)
2
)
vvT
v2
, (B.5)
139
where friction coefficient β(v) and diffusion coefficients D‖(v) and D⊥(v) are only depen-
dent on particle speed v. Note that particle diffusion is characterized by two coefficients,
D‖(v) and D⊥(v), one for the diffusion in the direction parallel to the particle trajectory
and the other for the diffusion in the perpendicular plane. Plugging expressions (B.4) and
(B.5) into the Fokker-Planck equation (B.3), we get
(
δfs
δt
)
c
= ∇v · [βv fs] + 1
2
∇v∇v :
[(
D⊥
2
I +
(
D‖ −
D⊥
2
)
vvT
v2
)
fs
]
. (B.6)
The second term in the right-hand side of this equation can be expanded using the following
tensor identities
∇v∇v :
[
gI
]
= ∇2vg (B.7)
and
∇v∇v :
[
g
vvT
v2
]
=
2
v2
g +
4
v2
v · ∇vg + vv
T
v2
: ∇v∇vg, (B.8)
where g(v) is a scalar function. As a result, (B.6) becomes
(
δfs
δt
)
c
= ∇v · [βv fs] +∇2v
[
D⊥
4
fs
]
+
1
v2
[(
D‖ −
D⊥
2
)
fs
]
+
2v
v2
· ∇v
[(
D‖ −
D⊥
2
)
fs
]
+
1
2
vvT
v2
: ∇v∇v
[(
D‖ −
D⊥
2
)
fs
]
. (B.9)
In steady state, the density function fs is time independent; therefore,
(
δfs
δt
)
c
= 0. (B.10)
In addition, since β(v), D‖(v), and D⊥(v) are isotropic functions of particle speed v, the
steady-state solution of (B.9) should also be an isotropic function, i.e., fs = fs(v). Then,
velocity derivatives in (B.9) can be written as
∇vg = v
v
∂g
∂v
, ∇v · g = 1
v2
∂
∂v
(
v2g · vˆ) , ∇2vg = 1v2 ∂∂v
(
v2
∂g
∂v
)
, (B.11)
140
and
∇v∇vg =
(
I− vv
T
v2
)
1
v
∂g
∂v
+
vvT
v2
∂2g
∂v2
, (B.12)
leading to
(
δfs
δt
)
c
=
1
v2
∂
∂v
(
v3βfs
)
+
1
v2
∂
∂v
(
v2
∂
∂v
(
D⊥
4
fs
))
+
1
v2
((
D‖ −
D⊥
2
)
fs
)
+
2
v
∂
∂v
((
D‖ −
D⊥
2
)
fs
)
+
1
2
∂2
∂v2
((
D‖ −
D⊥
2
)
fs
)
= 0, (B.13)
which, after simplification, becomes
(
δfs
δt
)
c
=
1
2v2
∂
∂v
(
∂
∂v
(
v2D‖fs
)
+
(
2v3β − vD⊥
)
fs
)
= 0. (B.14)
Thus, the problem of computing the steady-state distribution fs reduces to solving
∂
∂v
(
v2D‖fs
)
+
(
2v3β − vD⊥
)
fs = C, (B.15)
where C is a constant of particle speed v.
Among the different possible values that C can take, we choose C = 0 because it can
be shown that, in this case, the solution of (B.15) does not have a singularity at the origin
(i.e., at v = 0). Then, equation (B.15) becomes
∂
∂v
(
v2D‖fs
)
+
(
2v3β − vD⊥
)
fs = 0, (B.16)
or, equivalently,
∂fs
∂v
+
1
v2D‖
(
∂
∂v
(
v2D‖
)
+ 2v3β − vD⊥
)
fs = 0. (B.17)
141
The solution of this equation is simply2
fs(v) =
1
Ws
exp
− vˆ
0
p(v)dv
 , (B.18)
where
p(v) =
1
v2D‖
(
∂
∂v
(
v2D‖
)
+ 2v3β − vD⊥
)
(B.19)
and Ws is a scalar such that
´
fs(v)dv
3 = 1.
For the Spitzer coefficients defined in Section 2.5, namely
β(v) =
∑
s′
(1 +
ms
ms′
)Ns′Γs/s′
1
C2s′
ψ(zs′)
v
, (B.20)
D||(v) =
∑
s′
2Ns′Γs/s′
ψ(zs′)
v
, (B.21)
D⊥(v) =
∑
s′
2Ns′Γs/s′
φ(zs′)− ψ(zs′)
v
, (B.22)
we can show, after some algebra, that
p(v) =
v
C2s
(∑
s′
Ts
Ts′
Ns′q
2
s′ ln Λss′ψ(zs′)∑
s′ Ns′q
2
s′ ln Λss′ψ(zs′)
)
, (B.23)
which can also be written in the following form
p(v) =
v
C2s
1 + ∑s′ 6=sNs′e2s′ ln Λss′
(
Ts
Ts′
− 1
)
ψ(zs′)∑
s′ Ns′e
2
s′ ln Λss′ψ(zs′)
 . (B.24)
Then, placing this expression in (B.18), we can readily find that the steady-state density
2A first order differential equation of the form
∂f(x)
∂x
+ p(x)f(x) = q(x)
has the following solution
f(x) = e−g(x)
xˆ
0
q(x)eg(x)dx+ f(0)e−g(x),
where g(x) =
´ x
0
p(x)dx.
142
function fs is equal to
fs(v) =
1
Ws
exp
− 1
C2s
v2
2
+
vˆ
0
v
∑
s′ 6=sNs′e
2
s′ ln Λss′
(
Ts
Ts′
− 1
)
ψ(zs′)∑
s′ Ns′e
2
s′ ln Λss′ψ(zs′)
dv
 . (B.25)
Note that, if the different species constituting a plasma are in thermal equilibrium (i.e.,
if Ts = Ts′), the density function (B.25) simplifies to
fs(v) =
(
1
2piC2s
)3/2
exp
(
− v
2
2C2s
)
, (B.26)
thus, the steady-state distributions for all plasma species are exactly Maxwellian (which is
in agreement with the assumption made to derive the Spitzer coefficients). On the other
hand, however, if the plasma is not isothermal, the steady-state distributions deviate from
the Maxwellian form as described by (B.25). For instance, let consider a single-ion plasma
where Te > Ti. In this case, the electron and ion steady-state distributions are given by
fe(v) =
1
We
exp
(
− 1
C2e
(
v2
2
+
(
Te
Ti
− 1
)
he(v)
))
(B.27)
and
fi(v) =
1
Wi
exp
(
− 1
C2i
(
v2
2
+
(
Ti
Te
− 1
)
hi(v)
))
, (B.28)
respectively, where functions he and hi are defined as
he(v) ≡
vˆ
0
v
ln Λeiψ(zi)
ln Λeeψ(ze) + ln Λeiψ(zi)
dv (B.29)
hi(v) ≡
vˆ
0
v
ln Λieψ(ze)
ln Λieψ(ze) + ln Λiiψ(zi)
dv. (B.30)
An analysis of these functions shows that the deviations of fe and fi from Maxwellian
distributions are very small. To quantify the deviations, we have used the steady-state
density functions (B.27) and (B.28) to calculate effective electron and ion temperatures
and, then, compared these results to the values of Te and Ti considered to specify the
Spitzer coefficients. The effective particle temperatures have been computed using the
143
     
      
      
      
      
      
      
Te/Ti
T
e
f
f
e
−
T
e
T
e
[%
]
1 2 3 4 5
−5
−4
−3
−2
−1
0
×10−3
Te = 500 K
Te = 1000 K
Te = 1500 K
Te = 2000 K
Te = 2500 K
Te = 3000 K
Te = 3500 K
Figure B.1 Relative difference between the effective electron temperature T effe computed
using (B.31) and the value of Te considered in the calculation of the Spitzer coefficients
for an oxygen plasma. The difference is displayed as a function of Te/Ti. Each curve
corresponds to different constant values of Te.
following expression
T effs ≡
ms
3κ
ˆ
v2fs(v)dv
3 =
4pims
3κ
∞ˆ
0
v4fs(v)dv, (B.31)
wherems is the mass of particles of type s (either electrons or ions) and κ is the Boltzmann
constant. In Figure B.1, the relative difference between the effective electron temperature
T effe computed using (B.31) and the temperature Te considered in the calculation of the
Spitzer coefficients for an oxygen plasma is displayed as a function of the ratio Te/Ti.
Note that the different curves in Figure B.1, corresponding to different constant values of
Te, overlap each other. As we can see in this plot, T effe is lower than Te for any Te > Ti
and the magnitude of the relative difference between these values increases with Te/Ti;
however, the difference is very small (of the order of 10−3 percent). In the case of the
ion temperatures (see Figure B.2), T effi is greater than Ti for any Te > Ti. Although the
differences between T effi and Ti are larger than the ones found in the case of the electron
temperatures, such differences are still small (of the order of one percent). Note that the
144
     
      
      
      
      
      
      
Te/Ti
T
e
f
f
i
−
T
i
T
i
[%
]
1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Ti = 500 K
Ti = 1000 K
Ti = 1500 K
Ti = 2000 K
Ti = 2500 K
Ti = 3000 K
Ti = 3500 K
Figure B.2 Same as Figure B.1 but for the case of the effective ion temperature T effi .
ion temperature difference reaches its maximum value around Te/Ti = 3. Finally, for larger
values of Te/Ti, the effective temperature T effi becomes closer to Ti.
B.3 Modified Spitzer Friction Coefficient
In the previous section, we found that the steady-state solution of the Fokker-Planck
equation (B.6) is given by
fs(v) =
1
Ws
exp
− vˆ
0
p(v)dv
 , (B.32)
where p(v) has been defined in (B.19).
If we want the distribution fs to be Maxwellian, function p(v) is constrained to be
equal to v/C2s , where Cs is the thermal speed of particle species s. Then, placing this
condition in (B.19), we have that
v
C2s
=
1
v2D‖(v)
(
∂
∂v
(
v2D‖(v)
)
+ 2v3β − vD⊥(v)
)
. (B.33)
145
Solving for the friction coefficient β(v), we obtain
β(v) =
1
2v3
(
v3
C2s
D‖(v) + vD⊥(v)−
∂
∂v
(
v2D‖(v)
))
. (B.34)
Using Spitzer’s definitions for the diffusion coefficients D‖(v) and D⊥(v) in (B.34), we
can find, after some manipulation, the following expression for the friction coefficient
β(v) =
∑
s′
Ns′Γs/s′
(
1 +
Ts′
Ts
ms
ms′
)
1
C2s′
ψ(zs′)
v
, (B.35)
which can also be written in the following form:
β(v) =
∑
s′
Ns′Γs/s′
(
1 +
ms
ms′
)
1
C2s′
ψ(zs′)
v
+
∑
s′
Ns′Γs/s′
(
1− Ts
Ts′
)
1
C2s
ψ(zs′)
v
. (B.36)
Above, the first sum is equal to the standard Spitzer friction coefficient (defined in Section
B.2), while the second term is the “correction” needed to make the distribution exactly
Maxwellian.
B.4 Discussion
Concerns regarding the applicability of the Fokker-Planck equation with the Spitzer
coefficients to study the effects of Coulomb collisions on the shape of the incoherent scatter
spectrum were raised during our studies. First, we were concerned about the compatibility
of this collision model with the general framework of incoherent scatter spectrum theories
presented in Chapter 2. As we have discussed earlier in this dissertation, one of the funda-
mentals principles on which our general framework is based is the fluctuation-dissipation
or generalized Nyquist theorem. The theorem relates the spectrum of thermally driven
density fluctuations and the conductivity function of each plasma species. However, the
simple form of the theorem given in Chapter 2 (see expression (2.10)) is only valid if the
velocity distribution of the particle species is Maxwellian. Although a more general formu-
lation of the theorem is available [Sitenko, 1999], its application would further complicate
the spectrum model. As we have shown in Section B.2, the steady-state solution of the
146
Fokker-Planck equation with the Spitzer coefficients is exactly Maxwellian only in the case
of strict thermal equilibrium (i.e., when Te = Ti). However, for typical F-region plasma
configurations, it was verified that, if the electron and ion temperatures are not equal, the
deviation of the steady-state distribution from a Maxwellian form is very small and can
be neglected for any practical purpose. Therefore, the fluctuation-dissipation theorem, as
stated in equation (2.10), can be considered a good approximation for the Fokker-Planck
Spitzer collision model used in our studies.
A second concern that drew our attention was whether ionization and recombination
processes should be considered in our collision model. As we have already mentioned,
Spitzer coefficients come from a more general formulation of the Fokker-Planck equation
in terms of the so-called Rosenbluth potentials [Rosenbluth et al., 1957]. In deriving these
potentials, it is assumed that the plasma is fully ionized; thus, particles in the medium
are only allowed to experience Coulomb collisions. Since ionization and recombination
are disregarded, Rosenbluth’s model predicts the relaxation of the plasma to strict ther-
mal equilibrium. Then, the reader may question the use of Spitzer coefficients to study
non-isothermal plasmas, because this state is the result of a balance between ionization,
recombination, and transport phenomena, which are processes that have been neglected in
our model. Since production and recombination are relatively slow processes, their effects
on incoherent scatter radar signals are typically ignored. This can be considered a valid
approach for most radar viewing directions except, maybe, for observations perpendicular
to the geomagnetic field, in which case the correlation time of the incoherent scatter signal
becomes of the order of one second. This issue will be investigated in future studies.
In an attempt to develop a more appropriate collision model for non-isothermal plas-
mas, we derived a modified expression for the Spitzer friction coefficient based on the
assumption that the particle distributions in a stationary plasma are Maxwellian even in
the case of unequal electron and ion temperatures. The derivation of the modified coef-
ficient is given in Section B.3. As we have already mentioned, the first term in (B.36) is
the standard expression for the Spitzer friction coefficient, while the second term is the
“correction” needed to make the distribution exactly Maxwellian. If we associate the cor-
147
rection term with ionization and recombination processes, we can interpret this term as
the heating (cooling) rate required by the electrons (ions) to sustain the Te > Ti station-
ary state. Whether the modified Spitzer friction coefficient can be derived from a physical
model for ionization and recombination processes will be the subject of future research.
148
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AUTHOR’S BIOGRAPHY
Marco Antonio Milla was born to Graciela Bravo and Felipe Milla on April 16, 1975,
in Lima, Perú. He finished his bachelor studies in electrical engineering at the Pontificia
Universidad Católica del Perú in 1997. He joined the Jicamarca Radio Observatory as
a signal-processing engineer in 1998, and worked there for about five years in different
research and engineering projects. In 2004, he started his graduate studies joining the
Remote Sensing and Space Sciences group of the Department of Electrical and Computer
Engineering at the University of Illinois at Urbana-Champaign. He received his M.S. degree
in 2006 and is currently pursuing the Ph.D. degree.
His doctoral research involves the development of incoherent scatter radar techniques
for the estimation of ionospheric physical parameters. In particular, he has studied
Coulomb collisions and magneto-ionic propagation effects on the incoherent scatter spec-
trum measured with antenna beams pointed perpendicular to the Earth’s magnetic field.
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