**Radio wave active Doppler imaging of space plasma structures:**

**Arrival angle, wave polarization, and Faraday rotation measurements
with the radio plasma imager**

Bodo W. Reinisch, Gary S. Sales, and D. Mark Haines

Center for Atmospheric Research, University of Massachusetts Lowell

Shing F. Fung

NASA Goddard Space Flight Center, Greenbelt, Maryland

William W. L. Taylor

Raytheon Information Technology and Scientific Services, Goddard Space
Flight Center, Greenbelt, Maryland

**Abstract.** Radio sounding in the magnetosphere by the radio plasma
imager on the IMAGE spacecraft will determine the dimensions and shape
of the cavity between the magnetopause and the plasmapause.
Omnidirectional transmission of pulsed radio signals results in echoes
arriving from many directions. Quadrature sampling and Doppler analysis
of the signals received on three orthogonal antennas will make it
possible to determine the angles of arrival of the echoes, their
polarization ellipses, and the Faraday rotation. Decomposition of the
echo signals into the two characteristic waves is used to identify the
O- and X-wave components.

# 1. Introduction

For the first time, an active radio plasma imager will operate in space
when NASA's satellite orbits the Earth. NASA's Magnetopause-to-Aurora
Global Exploration (IMAGE) mission is scheduled for launch on February
15, 2000, into a high-inclination elliptical orbit with an altitude of 7
*R~E~* at apogee. The IMAGE payload includes the radio plasma imager
(RPI), now being built at the University of Massachusetts Lowell, in
addition to far ultraviolet (FUV), extreme ultraviolet (EUV), and
neutral atom (NAI) imagers. The scientific objectives of RPI include the
detection of plasma influx into the magnetosphere during magnetic
substorms and storms and the assessment of the response of the
magnetopause and plasmasphere to variations of the solar wind \[*Green
et al*., 1998\]. Unlike the plasma wave instruments on Wind \[*Bougeret
et al*., 1995\] and Polar \[*Gurnett et al*., 1995\], RPI will use
active Doppler radar techniques for the remote sensing of plasma
structures. These techniques are similar to the ones used by the
Digisonde portable sounder (DPS), a modern ground-based ionosonde
\[*Reinisch et al*., 1997\]. In the frequency range from 3 kHz to 3 MHz,
RPI will omnidirectionally transmit 10-W radio wave pulses and receive
reflected echoes on three orthogonal antennas. Echo reflections occur at
plasma structures where the surface normals are parallel to the wave
normals of the incident waves and where the local plasma frequency
equals the wave frequency. The transmitted signals generally contain
both characteristic polarizations, the ionic or ordinary (O) mode, and
the electronic or extraordinary (X) mode, and there will be two echoes
from a given plasma structure. The O echo is reflected at the density
level *N* = 0.0124 *f* *^2^* (*f* is in hertz, and number density *N* is
per cubic meter), and the X echo is reflected at the density given by
*N~(x)~* = 0.0124 *f* (*f* - *f~H~*), with *f~H~* being the local
gyrofrequency at the reflection point \[*Fung and Green*, 1996;
*Davies*, 1990\].

# 2. Magnetospheric Sounding With RPI

A detailed description of the RPI instrument is given in a forthcoming
paper by *Reinisch et al*. \[1999a\], and the principles of the RPI
observations are discussed by *Green et al*. \[1998\] and *Benson et
al*. \[1998\] and in a feasibility paper by *Calvert et al*. \[1995\].
The present paper describes how RPI will determine the locations of the
different reflection points by measuring the time delay, angle of
arrival, and wave polarization of the received echoes. The RPI design
has three orthogonal antennas, two 500-m tip-to-tip dipoles in the spin
(*xy*) plane, and a 20-m tip-to-tip dipole along the spin (*z*) axis.
Each of the six antenna monopoles is connected to its own low-noise
receiver. The voltage gain of the *z* receiver is a factor of 500/20,
i.e., 28 dB, higher than the *x* and *y* receivers in order to
compensate for the shorter antenna length. All receivers have a 300-Hz
bandwidth, and the sensitivity is 8 nV/√Hz for the *z* receiver and 25
nV/√Hz for the *x* and *y* receivers. The *x* and *y* antennas are used
for transmitting the radio pulses and, after the end of the transmit
pulse, are connected to their respective receivers. The nominal pulse
width is 3.2 ms, giving an echo range resolution of 480 km, and the
receiver recovery time is 3.2 ms after the end of the transmit pulse,
corresponding to a dead range of about 1000 km. Most of the RPI
soundings will be taken near apogee, when the spacecraft is in the
northern magnetospheric cavity, extending from the plasmapause to the
magnetopause, where the plasma frequencies *f~p~* ≈ 9 √N are mostly less
than 10 kHz. For transmit frequencies *f* \> 20 kHz, this implies that
the echo delay time *t~e~* multiplied by half the free-space speed of
light *c* provides a fair estimate of the distances to the magnetopause
and the plasmapause. The true ranges can be calculated from the
measurements of *t~e~* as a function of frequency. For these conditions,
the magnetopause and plasmasphere echoes arriving at the spacecraft
location are essentially transverse waves with almost circular
polarization (except when the wave normal is perpendicular to the
geomagnetic field), and the imaging technique described in sections 3
and 4 can be applied. When operating at lower frequencies, *f* = 3-12
kHz, the assumption *f* \>\> *f~p~* is no longer valid and different
analysis techniques need to be developed.

The simulation in Figure 1 shows the expected delay times for a modeled
plasma frequency profile extending from 3 to 10 *R~E~* and a spacecraft
altitude of 6 *R~E~*. Rather than *t~e~*, the virtual range *R*′ = 0.5
*c t~e~* is shown along the vertical axis on the right side. We call
these *R′(f)* plots plasmagrams, and they serve as the basis for the
calculation of the true ranges. On the basis of the inversion program
for topside ionograms, *Huang and Reinisch* \[1982\] have developed a
program that calculates the plasma frequency profile from the plasmagram
echo traces. Profile inversion of the RPI plasmagrams has an important
advantage over the inversion of ground-based ionograms: RPI on IMAGE
will determine accurate local *N* values by measuring the
sounder-stimulated plasma resonances \[*Benson*, 1977\] and the thermal
noise resonances \[*Meyer-Vernet and Perche*, 1989\]. These measurements
will provide *N* values at the spacecraft location as the starting point
for the profile calculation, which is not available in the case of
ground-based sounding.

Considering the geometry of the magnetosphere and RPI's nearly
omnidirectional radiation pattern, the virtual range alone is
insufficient to determine the location of the reflectors. Clearly, the
angle of arrival of the echoes must also be measured. We have shown in a
feasibility paper \[*Calvert et al*., 1995\] that the expected angular
resolution of an instrument like the RPI is about 2^o^, assuming that
only a single echo of frequency *f* arrives at the spacecraft at a given
time. To satisfy this condition, the transmitted signal is pulsed with a
3.2-ms pulse width, thus limiting time-coincident echoes to targets
whose virtual ranges must be equal to within 480 km. Fourier analysis
then separates any time-coincident echoes by making use of the
direction-dependent Doppler shifts. It is very unlikely that echoes with
the same propagation delay from different directions have the same
Doppler shift *d* = **K ·** (**v-v***~S~*)*/*π, were **K =** (2π/λ)**k**
is the wave vector, **v** is the target velocity, **v***~S~* is the
spacecraft velocity, λ is the free-space wavelength, and **k** is the
wave normal. This echo source identification technique, using Doppler
analysis and direction finding \[*Reinisch*, 1996\], had been pioneered
for radio sounding from the ground by *Bibl and Reinisch* \[1978\].

# 3. The Wave Polarizations

Electromagnetic wave propagation in a magnetoionic medium is
anisotropic, and only waves with the so-called characteristic wave
polarizations are solutions of Maxwell's equations \[*Stix*, 1962;
*Budden*, 1985\]. The two characteristic polarizations are generally
right- and left-hand elliptical, and the two characteristic waves
propagate with different phase and group velocities that lead to Faraday
rotation effects. The two modes also have different plasma cutoff
frequencies, corresponding to different reflection levels and echo delay
times in plasma sounder observations. Since the tenuous magnetospheric
plasma is not homogeneous, the characteristic polarization slowly
changes along the ray path. Significant mode conversion does not occur
unless steep gradients are encountered \[*Budden*, 1985\]. For RPI
sounding, the characteristic polarizations at the spacecraft location
are defined, both for transmitted and echo signal, by the local plasma
frequency *f~pS~* and the magnetic field **B***~0S~*. Before describing
the characteristic polarizations, we discuss the polarization of the
waves transmitted by the RPI antennas.

## 3.1. Polarization of the Transmitted Waves

RPI will use two orthogonal 500-m thin-wire dipoles to transmit right-
or left-hand polarized waves by feeding two equal currents with phase
differences of ±90^0^ into the two antennas along the *x* and *y* axes
(Figure 2): *I~x~* = *I~0~e^i^*^ω*t*^ and *I~y~* = *±iI~x~*. Waves
propagating along the *z* axis are therefore circularly polarized; all
other waves have elliptical polarization degenerating to linear
polarization for directions perpendicular to *z*. In the *z′* direction,
defined by the angles θ and φ, the transmitted **E** field becomes

![](media/image1.pct)

Here *r* is the radial distance from the satellite, and **x**, **y,**
and **z** are unit vectors. The plus-or-minus sign determines the sense
of rotation; for the upper signs, **E** rotates from the +*x* to the
+*y* axis. *Ê~0~* is given by the power *P* radiated by each dipole. It
is the peak amplitude of the far field on the *z* axis evaluated for a
distance of 1 m. For a short dipole antenna with *L~a~* \<\< λ/2, *L~a~*
being the tip-to-tip dipole length, *Ê~0~* (V/m) ≈ 9.5*√P* (watts)
\[*Kraus*, 1988\].

![](media/image2.wmf) The polarization of electromagnetic waves is best
described in a coordinate system *x′y′z′* where the *z′* axis is
parallel to **k**. For convenience the orientation of the *x′* axis was
chosen so that **B**~0~ = *B~0~***b** lies in the *x′z′* plane (Figure
3a). The wave front is in the x′y′ plane (Figure 3b), and the field is
given by

The amplitudes of the vector components and the phase angle α*~0T~* can
be expressed in terms of *Ê~0~*, θ, and φ. The angle α*~0T~* describes
the orientation of **E***~T~* at time *t* = 0 (Figure 3b is valid for
transmission and reception). The root-mean-square (rms) amplitude of the
elliptically polarized transmitted wave, averaged over one cycle, is
given by

![](media/image3.wmf)

This result shows that the radiated power is not azimuth-dependent and
varies only by a factor of 2 for θ from 0^o^ to 90^o^.

**3.2. Characteristic Polarizations**

Each of the two characteristic polarization ellipses can be described in
terms of their semimajor axes, an axial ratio ρ = (minor axis/major
axis), and tilt angle τ. The names used for the two characteristic waves
vary according to the application. In ionospheric sounding they are
often called O and X waves; there is also a Z wave, which has the same
polarization as the O wave. These designations, however, specify not
only the senses of rotation but also the plasma densities at their
reflection levels. Plasma wave studies prefer the designations R-X wave
and L-O wave \[*Shawhan*, 1970; *Gurnett*, 1991\], meaning right- and
left-hand polarization with respect to the external magnetic field or
its component in the direction of propagation. In an external magnetic
field, the **E** vector of the R wave rotates with the same sense as an
electron that is accelerated by the **E** vector and is therefore said
to have "electronic polarization," and correspondingly, the L wave has
ionic polarization \[*Rawer and Suchy*, 1967\]. The latter nomenclature
is used in this paper, where the letters *e* and *i* indicate electronic
and ionic polarizations, respectively. In a collisonless plasma the
major axis of the *i*-wave polarization ellipse is in the magnetic
meridian, i.e., is parallel to the *x*′ axis in the coordinate system of
the wave front, and the major axis

![](media/image4.wmf)of the *e*-ellipse is then aligned with the *y*′
axis \[*Budden*, 1985, chapter 4\]. The axial ratios for the two
polarization ellipses are the same, i.e., ρ*~e~* = ρ~*i*~ = ρ. The two
characteristic waves are given by

where *a~i~* and *a~e~* are the semimajor axes of the characteristic
polarizations and φ~0~ is the phase difference between the two
characteristic waves. The axial ratio ρ is uniquely determined by the
frequency *f*, the local plasma frequency *f~pS~*, the gyrofrequency
*f~HS~*, and the angle ψ between **z**′ and **b**~0~, ψ =
cos^-1^(**b**~0~·**z**′) \[*Kelso*, 1964\]:

![](media/image5.wmf)where

![](media/image6.wmf)

Relaxation sounding determines f*~pS~* and f*~HS~*, and models can be
used to estimate **b**~0~. Figure 4 illustrates how the axial ratio ρ
varies as function of ψ and *f* for a gyrofrequency of 1.5 kHz and a
local plasma frequency of 25 kHz. While circular polarization is a good
approximation for *X^2^* \<\< 1 and ψ \< 50^0^, elliptical polarization
must be considered in the general case. Since the variation of the axial
ratio with ψ is slow, there is no need to know **b**~0~ very accurately.

The polarization ellipse of the transmitted wave can be written as the
sum of the *e* and *i* ellipses. The *i* and *e* waves launched along
the *z*′ axis will be reflected at their respective reflection levels
and return to the spacecraft as *i* and *e* waves, i.e., O and X waves.
If they arrive at the same time, the resultant polarization ellipse can
also be decomposed into the two characteristic ellipses, as illustrated
in section 3.3.

**3.3. Decomposition Into Characteristic Polarizations**

Taking the characteristic waves as base functions, one can always
express each electric vector as

![](media/image7.wmf)With (3a) and (3b), the **E** vector becomes

![](media/image8.wmf)or

![](media/image9.wmf)

The vector components on the right side are known. For the transmitted
signal they are given by (1b), and for the received signal they can be
calculated from the receiver output voltages (section 4). Multiplying
(5c) by **x, y, z** gives three equations for *a~i~, a~e~*, and φ~0~:

![](media/image10.wmf)

The dot products **x⋅x**′, **x⋅y**′, etc., in terms of θ, φ are given in
Appendix A. Straightforward analysis shows that

![](media/image11.wmf)

where

![](media/image12.wmf)

The unknowns *a~i~, a~e~* and φ~0~ can be numerically obtained from (7a)
and (7b) in terms of the amplitudes *E*~x~, *E*~y,~ and *E*~z~. The echo
direction (θ,φ) must be calculated from the measured quadrature samples
of the three antenna signals as shown below.

**4. RPI Measurement of the Echo Wave Characteristics**

**4.1. The Echo Amplitude**

![](media/image13.wmf) In the *xyz* system, the rotating **E** vector of
the arriving echo signal can be expressed as

The same vector in the primed system becomes

![](media/image14.wmf)

The vector components and the phase angle  α*~oR~* (Figure 5) can be
obtained from (8a) by scalar multiplication with **x**′ and **y**′. Each
field component in (8a) produces its corresponding receiver output
voltage *V~m~*, where *m* = *x,y,z*. The final intermediate frequency
(IF) signal at each of the three receivers is digitized at 1.6-ms
intervals. For RPI, IF = 45 kHz, i.e., the IF is much larger than the
signal bandwidth of ±150 Hz. It is therefore possible to obtain the
amplitude and phase of each antenna signal from two quadrature samples
*I~m~* and *Q~m~* that are offset in time by a quarter IF cycle. When
the radio frequency (RF) signal is mixed with the local oscillator
signal, the RF phase α*~0R~* (Figure 5) is conserved. This is the same
technique that the ground-based Digisondes successfully used for many
years \[*Bibl and Reinisch*, 1978\] with IF = 225 kHz and a bandwidth of
15 kHz. The voltage vector

![](media/image15.wmf)

![](media/image16.wmf)is therefore proportional to the **E***~R~*
vector, i.e., **V** = β**E***~R~*. The proportionality factor β = *L\'G*
is the product of the effective antenna length *L\'* ≈ 0.5 *L~a~* and
the receiver voltage gain *G*. For RPI, *L\'~x,y~* ≈ 250 m , *L\'~z~* ≈
10 m, and the nominal receiver gains are *G~x~* = *G~y~* = 10^3^ and
*G~z~* = 25x10^3^, i.e., β = 2.5x10^5^. Careful calibration will be
required to correct the digital data for differences in β for the three
channels. The vector components in (8c) are

It follows that the digital samples taken at ω*t* = 0 and

ω*t* = π/2, i.e., the quadrature samples *I~m~* and *Q~m~*, are

![](media/image17.wmf)

The rms echo amplitude *E~R~* is then given by

![](media/image18.wmf)

**4.2. The Echo Angle of Arrival**

Since the quadrature vectors **I =** (*I~x~, I~y~, I~z~*) and **Q** =
(*Q~x~, Q~y~, Q~z~*) defined in (9b) are proportional to the field
vectors **E***~I~* and **E***~Q~* at ω*t* = 0 and π/2, respectively
(Figure 5), one can obtain the normal to the wave front directly from
the quadrature samples. It is standard technique to calculate the normal
to a plane by forming the cross product of two vectors within the plane
\[*Shawhan*, 1970\]. The wave normal of the arriving wave is therefore

![](media/image20.wmf)

If the **E** vector rotates clockwise when looking in the direction of
the wave normal, then **I**x**Q** points in the direction of **k***~R~*;
otherwise it points opposite to **k***~R~*. This ambiguity can be
resolved with the help of the signatures in a given plasmagram and the
use of models for the magnetopause and plasmapause. The angles θ and φ
for the direction of the arriving signal are simply given by

![](media/image21.wmf)

The sense of rotation of the **E** vector is determined by the direction
of the **I**x**Q** vector and is not dependent on **k***~R~*. The
identification of the echo in terms of O or X waves can easily be made
with the help of the reflection coefficients, as discussed in section
4.4.

![](media/image22.wmf) In a magnetoplasma, the **E** vector can have a
longitudinal component *E~z~*~′~, and therefore the above results are
only correct as long as *E~z′~* \<\< *E~x~*~′~ and *E~y′~*. This
is the case in the magnetospheric cavity where *X* \<\< 1. In principle,
the displacement vector **D** = **εE** could be used in the
calculations, where **ε** is the permittivity tensor at the location of
the spacecraft. Since the plasma frequency and Earth's magnetic field
**B**~0~ = *B~0~***b**~0~ are known from relaxation sounding and
magnetic field models, **ε** can be calculated \[*Budden*, 1985, chapter
3\]:

(12a)

where ε~0~ is the dielectric constant and

> ![](media/image23.wmf)

$\varepsilon_{2} = 1 - \frac{X}{(1 - Y)} = 1 - \frac{f_{N}^{2}}{\left\{ f\left( f - f_{H} \right) \right\}}$

![](media/image24.wmf) (12b)

> ![](media/image25.wmf)
> $G = \frac{- \text{XY}^{2}}{\left( 1 - Y^{2} \right)}$

This **D** vector, which rotates in the plane of the wave front, must be
used when *X* ≈ 1.

**4.3. The Echo Polarization Ellipse**

To completely describe the echo signal, it is necessary to determine its
polarization ellipse. The equation of an ellipse in the *x′y′* plane can
be written as

![](media/image26.wmf)

Expressing this ellipse as the sum of the two characteristic
polarization ellipses leads to the following relations for the
coefficients *A*, *B*, and *C* (see Appendix B):

![](media/image27.wmf)

Here *a~i~*, *a~e~*, and φ~0~ are known from (7a), and ρ is known from
(4). We are not using the subscript *R* (for received) in these
equations since they apply equally to received and the transmitted
signals. Using *A*, *B* and *C*, the axial ratio and tilt angle of the
polarization ellipse can be calculated. The tilt angle and the principal
axes of the ellipse are (Appendix B)

![](media/image30.wmf)

**4.4. Reflection Coefficients**

![](media/image32.wmf) Using the parameters of the characteristic
ellipses, one can express the rms amplitudes of the transmitted and
received fields in terms of these parameters. From (5b),

This relation is the same for the transmitted and the echo ellipse, and
the reflection coefficient γ, defined as the ratio of the received
amplitude *E~R~* to the range-normalized amplitude of the transmitted
wave *E~T~*/2*R*, becomes

![](media/image33.wmf)

![](media/image34.wmf)where *R* is the echo range. In general, the
reflection coefficients of the *i* and *e* waves are different, each
originating from a different reflection condition, and it is necessary
to define individual reflection coefficients for the two waves:

If γ*~e~* \<\< γ*~i~*, the echo is an ionic wave (O echo) and vice
versa. If γ~*e*~ ≈ γ*~i~*, the echo consists of equal O and X waves.

**5. Faraday Rotation**

![](media/image35.wmf) Using (14), one can specify the Faraday rotation
angle τ*~F~* that the wave undergoes along the entire propagation path:

![](media/image36.wmf)The integer factor *q* depends on the columnar
total electron content (TEC) along the path. For *X* \<\< 1 and *Y* \<\<
1, the Faraday rotation is \[*Davies*, 1990, chapter 8\]

This approximation does not account for the rotation effects at the
reflection point when *X* → 1, but it provides an order of magnitude
estimate. Assuming some typical values, *f* = 75 kHz, *R* = 2 *R~E~*,
*f~p~*~,ave~ = 25 kHz, and \[*f~H~* cosψ\]~ave~ = 1 kHz, one can obtain
estimates for the TEC and the total rotation:

![](media/image37.wmf)

This means that *q* in (19) is of order 19. The change of τ*~F~* with
frequency is

![](media/image38.wmf)

and the change of Faraday rotation for a 0.5% frequency step is δ*~F~* =
-0.01τ*~F~*, i.e., is smaller than 0.2 π for *f* \> 75 kHz. This
differential Faraday rotation measurement can therefore be used to
calculate the TEC, which in turn can be compared with the integrated
*N(s)* profile obtained from the plasmagram.

**6. Simulated Case Study**

To illustrate the results derived in the previous sections, it is
assumed that an echo is received from *R* = 2 *R~E~*, at θ = 100^o^ and
φ = 30^o^. No ray tracing was conducted for this case study, and the
Faraday rotation was assumed to be *q* π + 3π/4 (radians), and the RF
phase difference between the O and X echo was assumed to be φ~0*R*~ =
60^o^. The echo amplitudes are given for total reflection, no
focusing/defocusing, and zero absorption along the path. Table 1 gives
the details for this case study, along with the equation numbers used to
calculate their values.

The different polarization ellipses for this test case are shown in
Figure 6. On the left side is the transmitted polarization ellipse, and
on the right side is the received ellipse. The corresponding
characteristic polarization ellipses are shown below. Since ρ = 0.96 for
the given local conditions, they look like circles.

**7. Conclusions**

Radio wave active Doppler imaging in space can be done from a single
satellite with three orthogonal antennas. Signal modulation, quadrature
sampling, and Doppler analysis must be applied to determine the echo
range, angle of arrival, and wave polarization of the reflected signals.
As with any radio sounding which depends on total reflection, only
specific plasma structures can be detected: The plasma frequency must be
equal to the sounding frequency, and the density gradient must be
parallel to the wave normal. The wave polarization measurement
unambiguously identifies the O and X echoes. The techniques described
here for the exploration of Earth's magnetosphere with the radio plasma
imager on IMAGE can equally well be applied to topside ionospheric
sounding \[*Reinisch et al*., 1999b\] and to the probing of the
magnetospheres and ionospheres of other planets.

# Appendix A: Coordinate Transformation Between the *xyz* and *x′y′z′* Systems

We use a right-handed coordinate system aligned with the satellite
antennas such that the *x* and *y* axes lie along the orthogonal long
antennas and the *z* axis lies perpendicular to this plane along the
short antenna. The signal (both transmitted and received) is propagating
in the θ, φ direction with respect to the *xyz* system (Figure 1), where
θ is measured from the +*z* axis and φ is measured from the +*x* axis
(positive toward the +*y* axis). The polarization ellipse of the
propagating wave is described in the primed coordinates shown in Figure
2, where the *z′* axis is along the direction of propagation. The primed
system is then rotated about the z′-axis so that the local magnetic
field vector **B**~0~ lies in the *x′-z′* plane. The unit vectors **z**′
and **b**~0~ in the *xyz* system are

![](media/image39.wmf)

![](media/image40.wmf)

From the dot product of these two vectors we get ψ:

![](media/image41.wmf)

The direction cosines of **z**′ are easily found from (A1):

![](media/image42.wmf)

#### Since 

![](media/image43.wmf)

(see Figure 2), one gets

![](media/image44.wmf)

![](media/image45.wmf)Finally, since **x**′ = **y**′x **z**′,

**Appendix B: Characteristics of the Polarization Ellipse**

Equation (5b) describes the electric field in the propagation plane:

![](media/image46.wmf)

![](media/image47.wmf)

Eliminating the time dependence from these equations leads to the
general equation for an ellipse:

![](media/image48.wmf)

where

![](media/image49.wmf)

![](media/image51.wmf)

In the *x′y′* plane, a rotation of the axes through an angle τ aligns
them with the major and minor axes (ξ and η) of the polarization
ellipse. This rotation angle τ is called the tilt angle of the ellipse.
The equation of the ellipse now takes the form

![](media/image52.wmf)

![](media/image53.wmf)The rotation angle τ that accomplishes this
transformation, in terms of the coefficients *A*, *B*, and *C*, is

In this form the coefficients A′ and C′ are

![](media/image54.wmf)

![](media/image55.wmf)

Finally, the magnitudes of the principal axes of the polarization
ellipse are

![](media/image56.wmf)

Which of the above two expressions is the major axis and which is the
minor axis of the polarization ellipse depends on τ: If 0 ≤ τ ≤ π/2,
then *a~η~* is the major axis, and if π/2 ≤ τ ≤ π, then *a~ξ~* is the
major axis.

**Acknowledgments.** This research was supported by NASA subcontracts
83822 to UML and 83814 to Raytheon ITSS from SwRI, and by NASA contract
NASW-97002 to Raytheon ITSS.

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\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

S.F. Fung, NASA Goddard Space Flight Center, Code 630, Greenbelt, MD
20771. (shing.fung@gsfc.nasa.gov)

D.M. Haines, B.W. Reinisch, and G.S. Sales, Center for Atmospheric
Research, University of Massachusetts Lowell, 600 Suffolk Street,
Lowell, MA 01854. (Mark_Haines@uml.edu, Bodo_Reinisch@uml.edu, and
Gary_Sales@uml.edu)

W.W.L. Taylor, Raytheon Information Technology and Scientific Services,
Goddard Space Flight Center, Code 630, Greenbelt, MD 20771.
(taylor@nssdc.gsfc.nasa.gov)

(Received January 28, 1999; revised July 6, 1999;

accepted July 9, 1999.)

**\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_**

Copyright 1999 by the American Geophysical Union.

Paper number 1999RS900075.

0048-6604/99/1999RS900075\$11.00

**List of Table Captions**

**Table 1.** Case Study for Echo From *R* = 2 *R~E~* With θ = 100^o^ and
φ = 30^o^

**List of Figure Captions**

**Figure 1.** Simulated plasmagram for the spacecraft at 6 *R~E~*. The
*N(r)* profile in the radial direction is shown as a dashed line for
altitudes from 3 to 10 *R~E~*. A magnetospheric *N* distribution model
and three-dimensional ray tracing were used to calculate the echo
traces. The ray direction need not coincide with the radial direction.
The virtual range for the plasmagram is shown on the right. \[adapted
from *Benson et al*., 1998\]

**Figure 2**. Wave propagation along the *z′* axis in the antenna
coordinate system. The senses of the *i* and *e* waves are shown for ψ
\< 90°.

**Figure 3.** The polarization ellipse in the *x′y*′ plane. Letters a
and b are the semimajor and seminminor axes. (a) The orientation of the
*x′y′z′* system. (b) The tilt angle τ  of the polarization ellipse.

**Figure 4.** The axial ratio ρ(ψ) of the characteristic polarization
ellipses for *f* = 30, 100, and 300 kHz. The curves were calculated for
*f~HS~* = 1.5 kHz and *f~pS~* = 25 kHz.

**Figure 5**. The wave normal in terms of the quadrature vectors.

**Figure 6.** Polarization ellipses for the case study. In this example,
the *i* ellipse is larger than the *e* ellipse.

**Figure 1.** Simulated plasmagram for the spacecraft at 6 *R~E~*. The
*N(r)* profile in the radial direction is shown as a dashed line for
altitudes from 3 to 10 *R~E~*. A magnetospheric *N* distribution model
and three-dimensional ray tracing were used to calculate the echo
traces. The ray direction need not coincide with the radial direction.
The virtual range for the plasmagram is shown on the right. \[adapted
from *Benson et al*., 1998\]

**Figure 2**. Wave propagation along the *z′* axis in the antenna
coordinate system. The senses of the *i* and *e* waves are shown for ψ
\< 90°.

**Figure 3.** The polarization ellipse in the *x′y*′ plane. Letters a
and b are the semimajor and seminminor axes. (a) The orientation of the
*x′y′z′* system. (b) The tilt angle τ  of the polarization ellipse.

**Figure 4.** The axial ratio ρ(ψ) of the characteristic polarization
ellipses for *f* = 30, 100, and 300 kHz. The curves were calculated for
*f~HS~* = 1.5 kHz and *f~pS~* = 25 kHz.

**Figure 5**. The wave normal in terms of the quadrature vectors.

**Figure 6.** Polarization ellipses for the case study. In this example,
the *i* ellipse is larger than the *e* ellipse.

**Table 1.** Case Study for Echo From *R* = 2 *R~E~* With θ = 100^o^ and
φ = 30^o^

+------------------------------+---------------------------------------+
| ## Functions                 | ## Specifications                     |
+------------------------------+---------------------------------------+
| Transmission                 | *f* = 75 kHz, *P*/dipole = 1 W,       |
|                              | *Ê~0~* = 9.5 V/m                      |
+------------------------------+---------------------------------------+
| Reflection point             | *R* = 2 *R~E~*, θ = 100^o^, φ = 30^o^ |
+------------------------------+---------------------------------------+
| Local conditions             | **B**~0~ = (0.50, 0, 0.87), *f~HS~*   |
|                              | =1.5 kHz, ψ =74^o^,                   |
|                              |                                       |
|                              | *F~pS~* =25 kHz, ρ = 0.96             |
+------------------------------+---------------------------------------+
| Transmit signal evaluated at | *E~T~* = 6.8 V/m (equation (2b)),     |
| 1 m                          | *a~T~* = 9.5 V/m, ρ*~T~* = 0.17,      |
|                              | τ*~T~* = 75^o^,                       |
|                              |                                       |
|                              | *a~iT~* = 5.6  V/m, *a~eT~* = 4.1     |
|                              | V/m, φ*~0T~* =150^o^ (equations (7a)  |
|                              | and (7b))                             |
+------------------------------+---------------------------------------+
| Echo signal                  | *Ê~Rx~* = 0.19, *Ê~Ry~* = 0.27,       |
|                              | *Ê~Rz~* = 0.19 μV/m,                  |
|                              |                                       |
|                              | *E~R~* = 0.27 μV/m, *a~R~* = 0.37     |
|                              | μV/m, ρ*~R~* = 0.15 (equation (B7)),  |
|                              |                                       |
|                              | *τ~R~* = 30^o^,                       |
|                              |                                       |
|                              | *a~iR~* = 0.22 μV/m, *a~eR~* = 0.16   |
|                              | μV/m, φ*~0R~*=60^o^                   |
+------------------------------+---------------------------------------+
| Receiver output voltages     | *V~x~* = 48 mV, *V~y~* = 68 mV,       |
|                              | *V*~z~ = 48 mV (equation (9c))        |
+------------------------------+---------------------------------------+

#####  

![](media/image57.jpeg){width="5.194444444444445in"
height="4.700694444444444in"}

> Figure 1

![](media/image60.png){width="2.8in" height="2.642361111111111in"}
Figure 2 Figure 3

Figure 4

![](media/image61.pct)

Figure 5

##### ![](media/image62.pct)Figure 6