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 RE 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 - fH), with fH 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 fp  9 N are mostly less than 10 kHz. For transmit frequencies f  > 20 kHz, this implies that the echo delay time te 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 te 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 >> fp 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 RE and a spacecraft altitude of 6 RE. Rather than te, the virtual range R = 0.5 c te 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 2o, 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-vS)/ were K = (2/)k is the wave vector, v is the target velocity, vS is the spacecraft velocity,  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 fpS and the magnetic field B0S. 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 ±900 into the two antennas along the x and y axes (Figure 2): Ix = I0eit and Iy = iIx. 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  and  the transmitted E field becomes

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. Ê0 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 La << /2, La being the tip-to-tip dipole length, Ê0 (V/m)  9.5P (watts) [Kraus, 1988]. 

   The polarization of electromagnetic waves is best described in a coordinate system xyz where the z axis is parallel to k. For convenience the orientation of the x axis was chosen so that B0 = B0b lies in the xz plane (Figure 3a). The wave front is in the xy plane (Figure 3b), and the field is given by

The amplitudes of the vector components and the phase angle 0T can be expressed in terms of Ê0, , and . The angle 0T describes the orientation of ET 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

This result shows that the radiated power is not azimuth-dependent and varies only by a factor of 2 for  from 0o to 90o.
3.2.  Characteristic Polarizations
  Each of the two characteristic polarization ellipses can be described in terms of their semimajor axes, an axial ratio  = (minor axis/major axis), and tilt angle . 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 

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., e = i = . The two characteristic waves are given by

where ai and ae are the semimajor axes of the characteristic polarizations and 0 is the phase difference between the two characteristic waves. The axial ratio is uniquely determined by the frequency f, the local plasma frequency fpS, the gyrofrequency fHS, and the angle  between z and b0, = cos-1(b0·z) [Kelso, 1964]:

where

Relaxation sounding determines fpS and fHS, and models can be used to estimate b0. Figure 4 illustrates how the axial ratio  varies as function of  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 X2 << 1 and elliptical polarization must be considered in the general case. Since the variation of the axial ratio with  is slow, there is no need to know b0 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

With (3a) and (3b), the E vector becomes

or

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 ai, ae, and 0:

The dot products xx, xy, etc., in terms of ,  are given in Appendix A. Straightforward analysis shows that

where

The unknowns ai, ae and 0 can be numerically obtained from (7a) and (7b) in terms of the amplitudes Ex, Ey, and Ez. The echo direction (,) 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

   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

The vector components and the phase angle 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 Vm, 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 Im and Qm 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 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 

is therefore proportional to the ER vector, i.e., V = ER. The proportionality factor L'G is the product of the effective antenna length L'  0.5 La and the receiver voltage gain G. For RPI, L'x,y  250 m , L'z  10 m, and the nominal receiver gains are Gx = Gy = 103 and Gz = 25x103, i.e.,  = 2.5x105. Careful calibration will be required to correct the digital data for differences in  for the three channels. The vector components in (8c) are

It follows that the digital samples taken at t = 0 and 
t = /2, i.e., the quadrature samples Im and Qm, are 

The rms echo amplitude ER is then given by

4.2.	The Echo Angle of Arrival
   Since the quadrature vectors I = (Ix, Iy, Iz) and Q = (Qx, Qy, Qz) defined in (9b) are proportional to the field vectors EI and EQ at t = 0 and /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

If the E vector rotates clockwise when looking in the direction of the wave normal, then IxQ points in the direction of kR; otherwise it points opposite to kR. 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  and  for the direction of the arriving signal are simply given by

The sense of rotation of the E vector is determined by the direction of the IxQ vector and is not dependent on kR. 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. 

   In a magnetoplasma, the E vector can have a longitudinal component Ez, and therefore the above results are only correct as long as Ez << Ex and Ey. This is the case in the magnetospheric cavity where X << 1. In principle, the displacement vector D = E could be used in the calculations, where  is the permittivity tensor at the location of the spacecraft. Since the plasma frequency and Earth’s magnetic field B0 = B0b0 are known from relaxation sounding and magnetic field models,  can be calculated [Budden, 1985, chapter 3]:
(12a)
where 0 is the dielectric constant and

        

                                          (12b)                                  

        

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 xy plane can be written as

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):

Here ai, ae, and 0 are known from (7a), and  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)

4.4.  Reflection Coefficients

   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 , defined as the ratio of the received amplitude ER to the range-normalized amplitude of the transmitted wave ET/2R, becomes

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  e << i, the echo is an ionic wave (O echo) and vice versa. If e  i, the echo consists of equal O and X waves.
5.  Faraday Rotation

   Using (14), one can specify the Faraday rotation angle F that the wave undergoes along the entire propagation path:

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 RE, fp,ave = 25 kHz, and [fH cos]ave = 1 kHz, one can obtain estimates for the TEC and the total rotation:

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

and the change of Faraday rotation for a 0.5% frequency step is F = -0.01F, i.e., is smaller than 0.2  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 RE, at  = 100o and  = 30o. No ray tracing was conducted for this case study, and the Faraday rotation was assumed to be q  + 3/4 (radians), and the RF phase difference between the O and X echo was assumed to be 0R = 60o. 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   = 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 xyz 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  direction with respect to the xyz system (Figure 1), where  is measured from the +z axis and  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 B0 lies in the x-z plane.  The unit vectors z and b0 in the xyz system are

From the dot product of these two vectors we get :

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

Since 

(see Figure 2), one gets

Finally, since x = yx z,
Appendix B:  Characteristics of the Polarization Ellipse
   Equation (5b) describes the electric field in the propagation plane:

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

where

In the xy plane, a rotation of the axes through an angle  aligns them with the major and minor axes (and  of the polarization ellipse. This rotation angle  is called the tilt angle of the ellipse.  The equation of the ellipse now takes the form

The rotation angle  that accomplishes this transformation, in terms of the coefficients A, B, and C, is

In this form the coefficients A and C are

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

Which of the above two expressions is the major axis and which is the minor axis of the polarization ellipse depends on : If 0    /2, then a is the major axis, and if /2    , then a 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.
References
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Gurnett, D.A., Auroral plasma waves, in Auroral Physics, edited by C.-I. Meng, M.J. Rycroft, and L.A. Frank, pp 241-254, Cambridge Univ. Press, New York, 1991.
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Reinisch, B.W., Modern ionosondes, in Modern Ionospheric Science, edited by H. Kohl, R. Rüster, and K. Schlegel, pp. 440-458, Eur. Geophys. Soc., Katlenburg-Lindau, ProduServ GmbH Verlagsservice, Berlin, Germany, 1996.
Reinisch, B.W., D.M. Haines, K.Bibl, I. Galkin, X. Huang, D.F. Kitrosser, G.S. Sales, and J.L. Scali, Ionospheric sounding support of OTH radar, Radio Sci., 32, 1681-1694, 1997.
Reinisch, B.W., et al., The Radio Plasma Imager: Remote sensing of magnetospheric plasmas, Space Sci. Rev., in press, 1999a.
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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 RE With  = 100o and  = 30o

List of Figure Captions

Figure 1.  Simulated plasmagram for the spacecraft at 6 RE. The N(r) profile in the radial direction is shown as a dashed line for altitudes from 3 to 10 RE. 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  < 90.

Figure 3.  The polarization ellipse in the xy plane. Letters a and b are the semimajor and seminminor axes. (a) The orientation of the xyz system. (b) The tilt angle of the polarization ellipse.

Figure 4.  The axial ratio of the characteristic polarization ellipses for f = 30, 100, and 300 kHz. The curves were calculated for fHS = 1.5 kHz and fpS = 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 RE. The N(r) profile in the radial direction is shown as a dashed line for altitudes from 3 to 10 RE. 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  < 90.

Figure 3.  The polarization ellipse in the xy plane. Letters a and b are the semimajor and seminminor axes. (a) The orientation of the xyz system. (b) The tilt angle of the polarization ellipse.

Figure 4.  The axial ratio of the characteristic polarization ellipses for f = 30, 100, and 300 kHz. The curves were calculated for fHS = 1.5 kHz and fpS = 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 RE With  = 100o and  = 30o 
Functions	Specifications	

Transmission	
f = 75 kHz, P/dipole = 1 W, Ê0 = 9.5 V/m 	
Reflection point 	R = 2 RE,  = 100o,  = 30o	
Local conditions	B0 = (0.50, 0, 0.87), fHS =1.5 kHz,  =74o,
FpS =25 kHz,   = 0.96	
Transmit signal evaluated at 1 m	ET = 6.8 V/m (equation (2b)), aT = 9.5 V/m, T = 0.17, T = 75o,
aiT = 5.6 V/m, aeT = 4.1 V/m,  0T =150o (equations (7a) and (7b))	
Echo signal	ÊRx = 0.19, ÊRy = 0.27, ÊRz = 0.19 V/m,    
ER = 0.27 V/m, aR = 0.37 V/m, R = 0.15 (equation (B7)), 
R = 30o,
aiR = 0.22 V/m, aeR = 0.16 V/m, 0R=60o	
Receiver output voltages	Vx = 48 mV, Vy = 68 mV, Vz = 48 mV (equation (9c))

	
Figure 1

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Figure 6