Estimation of linear wave polarization of the auroral kilometric radiation

Authors


Abstract

[1] There is a general view that the auroral kilometric radiation (AKR) polarization is fully circular with dominating right-hand extraordinary mode and some admixture of the left-hand ordinary mode. Most of up-to-date determinations of the wave modes of the AKR have been based on measurements of the circular component of the wave polarization, leaving the linear polarization parameters unknown. In space observations the polarimeter is usually not directed to the AKR source. However, having determined the polarization parameters in each of the three perpendicular planes, one is able to determine the polarization ellipse in the 3-D space. Results of statistical estimations of the AKR polarization parameters in the wave plane are presented. Data were provided by the Interball-2/Polrad triaxial polarimeter. It is the first time the dynamic spectra of all four Stokes parameters of the AKR are displayed in the wave plane. The AKR polarization is mostly circular with an estimated mean value of the semiaxes ratio of 0.99 ± 0.10 rms error, with a small contribution of the linear component at a level of 0.07 ± 0.09 rms error, and an unpolarized component at a level of 0.10 ± 0.08. The uncertainties of these estimations are too large to definitely conclude on presence or absence of some small contributions of the linear and upolarized components in the AKR. However, in the limits of the above uncertainties these estimations do not contradict expectations that the linear polarization is absent in the AKR. Some small contribution of the unpolarized component can be explained by a relatively high level of noise and board interefences.

1. Introduction

[2] The auroral kilometric radiation (AKR) is emitted at frequencies between 20 and 1000 kHz from sources located in the auroral region of the magnetosphere [Gurnett, 1974]. It is generated near the local gyro-frequency of electrons in the low-density source cavities [Benson and Calvert, 1979; Benson, 1985; Bahnsen et al., 1989; Ungstrup et al., 1990; Hilgers et al., 1991; Hilgers, 1992]. It propagates mostly in the right-handed extraordinary (R-X) mode. This was first inferred from an indirect estimate of its polarization obtained by Green et al. [1977] based on ray tracing calculations, and by Gurnett and Green [1978] based on observation of frequency cutoffs at low altitudes. First direct measurements of the AKR polarization were obtained by Kaiser et al. [1978] with planetary radio astronomy experiments on Voyager-1 and 2 during the early portions of each flight. They showed that the AKR signals were emitted primarily in the R-X mode assuming their source to be located in the Northern hemisphere. Unambiguous direct measurements of AKR polarization were made from the Dynamic Explorer-1 (DE-1) by Shawhan and Gurnett [1982], who confirmed that the AKR was generated predominantly in the R-X mode. Polarization measurements of the AKR made by Mellott et al. [1984] with DE-1 indicated presence of the R-X mode with a weaker contribution of the L-O mode by roughly a factor of 50. Presence of both the modes with the dominating R-X mode was confirmed by polarization observations from Akebono [Morioka et al., 1990] and Interball-2 satellites [Lefeuvre et al., 1998; Hanasz et al., 2000, 2003; Panchenko, 2004]. Observations of the R-X mode circular polarization of AKR led Wu and Lee [1979] to the now commonly accepted theory of AKR generation through the electron cyclotron maser instability. Akebono and Interball-2 used for the first time combinations of crossed electric and magnetic antennas to provide 3-D measurements of the Poynting flux and wave polarization of the AKR [Morioka et al., 1990; Lefeuvre et al., 1998]. For one event of the AKR the degree of polarization was estimated by Lefeuvre et al. [1998] to be about 80%. Polarimetry of the AKR with a triaxial electric antenna system allowed Panchenko [2004] to determine effective length vectors of the antennas.

[3] Since up-to-date determinations of the wave modes of the AKR have been based mostly on measurements of a circular component of the wave polarization projected on a single antenna plane the question of the presence or absence of its linear and unpolarized components is still open.

[4] Contrary to ground based radio polarimeters, space based polarimeters are usually not pointed to the observed radio source, especially in the case of the AKR, as the wave plane orientation is not known a priori. However, having determined the polarization parameters in each of the three perpendicular antenna planes one is able to determine the polarization ellipse in the 3-D space.

[5] Suitable data for 3-D determinations of the AKR polarization were provided by the Interball-2/Polrad triaxial polarimeter, capable to measure the wave polarization parameters in each of the three nearly perpendicular planes composed by each pair of its three antennas [Hanasz et al., 1998]. Data from this instrument enabled us to retrieve the polarization ellipse of the AKR emission from its projections on each of the three antenna planes.

[6] The aim of this paper is to statistically examine the general view that the AKR polarization is fully circular and put some statistical constrains on its circularly and linearly polarized components. We show that: (1) the estimated mean value of the amount of linear polarization is small (0.07 ± 0.09 rms error) or even absent, and (2) the estimated mean value of the semiaxes ratio of the polarization ellipse is very close to 1 (0.99 ± 0.10 rms error). We also discuss estimation of some amount of the unpolarized component for the weak AKR and note that the degree of the total polarization becomes very close to 100% for strong emissions, which suggests that the AKR is fully polarized.

[7] The outline of this paper is as follows: in section 2 we describe polarization observations of the AKR with Interball-2. In section 3 we explain the nonlinear least squares method applied for estimations of the AKR polarization state and show the dynamic spectra of the Stokes parameters in the wave plane of AKR. In section 4 we statistically verify the dominance of circular polarization in the AKR emission and estimate the linear component of the AKR polarization to be small (0.07) and comparable to the rms error of ±0.09. The mean value of the unpolarized component of the AKR is estimated to be at a level of 0.10 ± 0.08. In section 5 we summarize and discuss our results and show that for the sufficiently strong AKR the degree of the total polarization is estimated to be practically 100%.

2. Observations

[8] Polrad was a triaxial spectro-polarimeter aboard the Interball-2 (Auroral Probe) spacecraft (apogee 19,140 km, perigee 780 km, and orbit inclination 62.8°) operated in the period from August 29, 1996 to May 30, 1998 [Hanasz et al., 1998]. AKR polarization was measured in the Northern hemisphere, at invariant latitudes between 60° and 82° and altitudes from 11,000 to 19,000 km. Its antenna system consisted in three quasi-orthogonal short antennas, two of them being dipoles (22 m tip-to-tip) in the plane nearly perpendicular to the spin axis pointed approximately toward the Sun, and one of them a monopole (11 m tip-to-center) deployed along the spin axis. The spacecraft was spinning with a period of 120 s, and was reoriented toward the Sun every few days in order to keep the solar batteries approximately perpendicular to the Sun's direction. The polarimeter consisted in a 9-channel step-frequency analyzer (SFA) operating in a frequency range from 4 to 1000 kHz with a 4 kHz frequency resolution, and a sweep period of 6 or 12 s and a time constant of 6 ms. Three polarimeter channels measured power densities of the waves captured by each of the three quasi-orthogonal antennas, three next channels measured power densities of each pair of the antenna signals combined in phase, and three other channels measured power densities of each pair of the antenna signals combined in quadrature. Four Stokes parameters (I, Q, U, V) [Krauss, 1966] were a posteriori computed on the ground for each of the three antenna planes. For more details, see Panchenko [2004].

3. Estimations of the AKR Polarization State

[9] The power density of a signal measured by each polarimeter channel can be considered as a sum of an isotropic and a nonisotropic components:

equation image

where i = 1, 9 denotes a number of the polarimeter channel, Ppi(…) is a nonisotropic component of the i-th channel output, Pui(…) is an isotropic component of the i-th channel output, equation image(θ, ϕ) - is a wave vector, θ, ϕ are direction angles of the wave incidence, tan(χ) = E2/E1 - where E2/E1 is a semiaxes ratio of the polarization ellipse, ξ- is an orientation angle (see Figure 1), Xeff, Yeff, Zeff - are effective length vectors of the antennas, ∣Xeff∣, ∣Yeff∣, ∣Zeff∣ - are effective lengths of the antennas. We assume that AKR is emitted from a point source therefore the nonisotropic component approximated as a plane wave.

Figure 1.

Geometry of a wave polarization ellipse in the spinning orthogonal coordinates of the spacecraft. ω, wave angular frequency; θ and ϕ, direction angles of the wave incidence; Ω = 2π/120 s, angular frequency of the spacecraft rotation; E1 and E2, semiaxes of the polarization ellipse; tanχ = E2/E1, ratio of the ellipse semiaxes; ξ, orientation of the polarization ellipse; Xs/c, Ys/c, Zs/c, orthogonal coordinate system of the spacecraft; Xeff, Yeff and Zeff, effective lengths vectors of the antennas (Xeff, Yeff and Zeff are not mutually orthogonal).

[10] The polarized and unpolarized parts of the AKR are considered as a nonisotropic radio emission (dependent on the angle between the wave vector and a direction of the antenna). The galactic background, the receiver noise as well as onboard random interferences are modeled as a uniform isotropic noise. The onboard interferences were mainly isotropic. Some of them were random at different frequencies whereas others appeared as long term (minutes or even hours) signals at constant frequencies. The latter were partly removed by means of data processing procedures.

[11] The Stokes parameters for each frequency step were determined from outputs of the polarimeter channels in the coordinate system of the spinning antennas [Hanasz et al., 1998, 2000]:

equation image

where indices k and j denote the names of the antennas in such a way that k = X when j = Y, k = Y when j = Z and k = Z when j = X, and the asterisk denotes a quarter wave shift of an input signal of the antenna k.

[12] Equations (1) and (2) were fitted to the channel outputs in order to find the modeled Stokes parameters in terms of the direction angles of the wave incidence (θ and ϕ), the parameters of the polarization ellipse (χ, ξ) and the effective length vectors of the antennas (Xeff, Yeff, Zeff). To derive the whole set of parameters the least squares fitting procedure was applied [Panchenko, 2004].

[13] With the retrieved direction of arrival of the AKR one can estimate the Stokes parameters in the wave plane. Similar determinations were made with the Cassini/RPWS experiment for the Saturnian Kilometric Radiation by Taubenschuss et al. [2006] and Cecconi et al. [2006]. An example of a set of radiospectrograms of the normalized Stokes parameters for the AKR event on Nov. 16, 1997 is shown in Figure 2. The left panels present them in the plane of the Y and Z antennas, deployed in the spacecraft spin plane [see also Hanasz et al., 2003], while the right panels present them after projection onto the wave plane of the AKR. Since observations with Interball-2/Polrad were limited only to the northern hemisphere (due to relatively low orbit the southern AKR sources could not be observed) the unambiguous determination of the wave modes was possible.

Figure 2.

Dynamic spectra of the Stokes parameters of AKR: intensity (I), degrees of linear (Q/I, and U/I) and circular (V/I) polarizations. (left) Dynamic spectra of the Stokes parameters determined with the pair of Y and Z antennas deployed in the spin plane (reproduced from Hanasz et al. [2003]). (right) Stokes parameters in the wave plane of AKR.

[14] Two features are typical for the set of dynamic spectra recorded with Y and Z antennas in the spin plane (Figure 2, left). First, the parameters of linear polarization Q/I and U/I are strongly modulated with the spin period, and second, the orbital motion of the spacecraft causes a slow change of the V parameter from positive to negative values (for the R-X mode of propagation), while the spacecraft moves from the “front” side to the “back” side of the polarimeter plane of Y and Z antennas.

[15] In the wave plane of the AKR (Figure 2, right) the negative values of the degree of circular polarization (V/I) fluctuate close to the value of −1 (dark blue) showing a very high degree of the R-X mode polarization. Three small patches of the L-O mode (8:40–8:45 UT, 9:07–9:13 UT and 9:28–9:37 UT), much less intense (about 2 orders of magnitude), are easily identified by a very high and positive degree of circular polarization V/I near the value of +1 (red).

[16] In both the modes variability of the degree of circular polarization is small and independent of variability of AKR intensity. Occurrence of the L-O mode before the R-X mode (8:40–8:48 UT) and at frequencies below the frequencies of the R-X mode (9:08–9:13 UT and 9:30–9:42 UT) indicates that the emission cone of the L-O mode was from time to time broader than that of the R-X mode.

[17] Note also that mode reversals from R-X to L-O were abrupt taking only one or two frequency steps of 4 kHz width (Figure 2, right bottom). This can happen when the edge of the emission cone of the R-X mode AKR is sufficiently sharp. With the dipole magnetic field model assumed the 4 kHz frequency step corresponds to a segment of the auroral magnetic field line about 100 km long, which when viewed from a distance of the satellite corresponds to an angle of a few degrees.

[18] The linear components of wave polarization (Q/I and U/I) fluctuate around 0 (green) and similarly to V/I their variabilities are also small and independent on AKR intensity (Figure 2, right). This may suggest an absence of the linear component of polarization in the wave propagation of AKR. The above features are typical for other events of AKR.

4. Statistical Analysis of the AKR Polarization State

[19] For a statistical investigation we have selected 198 AKR events. A single event is defined as a continuous observation of the AKR in a time interval when the power density of the received signal was higher than the threshold value of 10−19 W m−2 Hz−1, established at a level 10 times the maximum galactic background emission in the frequency range below 600 kHz (∼10−20 Wm−2 Hz−1 [Dulk et al., 2001; Brown, 1973]), where most of the AKR emissions were observed. With this power threshold the background contribution was considered to be negligible. Figure 6 (top) shows an example of a spectral contour of the AKR event.

[20] Each event is divided into spectral windows 10 frequency steps wide (41 kHz) and 10 min long. For a sweep rate of the instrument of 1 sweep per 6 s such a window contains 1000 values (10 frequency steps times 100 sweeps) of a single normalized Stokes parameter obtained from measurements through the set of equations (2). The spectral window is sliding with a time step of 2 min in order to avoid effects of the spacecraft orbital motion. To reduce fluctuations, data averaging over the frequency bandwidth of the window is applied. In this way 9 time profiles (10 min long, 100 data points each) of the normalized Stokes parameters (Qxy/Ixy, Qyz/Iyz, Qzx/Izx, Uxy/Ixy, Uyz/Iyz, Uzx/Izx, Vxy/Ixy, Vyz/Iyz, Vzx/Izx) are obtained. A set of the model parameters (equation image(θ, ϕ), χ, ξ, Xeff, Yeff, Zeff) is retrieved for each of 2500 windows through fitting to the normalized Stokes parameters.

[21] Figure 3 shows the scatterplot of the retrieved values of the semiaxes ratio, E2/E1 = tanχ of the polarization ellipse, versus ξ, the orientation angle. One can easily notice that (1) their mean value is close to 1 over the whole range of orientation angles from 0° to 360°, which confirms that the AKR is most frequently circularly polarized, and (2) their scatter is strongly dependent on the orientation angle of the polarization ellipse. It maximizes up to ±0.6 around 0°, 90°, 180°, 270°, and minimizes around 45°, 135°, 225°, and 315°.

Figure 3.

Scatterplot of the semiaxis ratios (tanχ = E2/E1) of the polarization ellipse of AKR versus orientation angle ξ of the ellipse.

[22] This result points out the significance of numerical errors of the applied least squares procedure for the observed scatter of the retrieved parameters. It may suggest that for some input parameters of the model the set of linearized equations applied in the least squares procedure (see Appendix A) may be ill-posed (numerically very close to singularity), with no unique solution when the input signal decreases to the noise level or below.

[23] This is a well-known problem when the inversion is applied. An ill-posed system of linear equations describes situation when small fluctuations of the input data cause large changes in the output results. For example, the angular distance between an effective antenna direction and the vector of wave arrival is critical for determination of the radio source position and its polarization state [Ladreiter et al., 1995; Panchenko, 2003; Cecconi and Zarka, 2005]. In case when one of the effective antenna axes is pointed toward the radio source the determined direction of wave arrival will contain a significant error resulting from a signal decreasing below the noise level.

[24] The applied least squares procedure is examined with a Singular Value Decomposition (SVD) technique in order to decide for which input parameters of the model the system of linear equations becomes ill-posed [Connerney, 1981] (and references therein). The system of the least squares equations (see Appendix A) is analyzed in a similar way as in Ladreiter et al. [1995] and Panchenko [2004]. Using SVD technique we examined the dependencies between the semiaxes ratio E2/E1 = tanχ and the orientation angle ξ of the polarization ellipse in our model.

[25] In the first step the input data were simulated using equations (1) and (2) with addition of the normally distributed random noise (ΔP/P = 1/equation image ≈ 0.2/equation image ≈ 0.06, where Δf = 4096 Hz is the receiver bandwidth, τ = 0.006 s is an integration time constant and 10 is a number of frequency steps). In the second step, the condition number of the least squares equations was derived by means of the SVD method. In the third step, the resolution matrix equation image, which describes the ability of the equations (see Appendix A) to obtain the linearly independent solution, was calculated for a selected critical level of the condition number (about 106). The nondiagonal elements of the resolution matrix equation image describe the degree of linear dependencies between the estimated parameters of the model.

[26] Figure 4 maps the calculated parameters equation image in the χ, ξ plane. It shows the ability of the angles χ and ξ to be linearly independent. In regions where this parameter is equal to 0.0 (black) the angles χ and ξ are linearly independent. In regions where it is greater than 0.0 (grey) the parameters χ and ξ are linearly dependent due to the ill-posed system of the least squares equations.

Figure 4.

Averaged parameter of linear dependence equation image between χ angle and ξ angle as a function of χ and ξ. Black color indicates absence of linear dependence between both parameters (equation image = 0). equation image has been calculated for the following input parameters of the model (equations (1) and (2)): θ = 10°–80°, ϕ = 0°–360°, Xeff = Yeff = Zeff = 1.

[27] The observed dependence of the angles χ and ξ comes out from numerical instabilities of applied mathematical routines. In other words, for the ranges of χ and ξ where equation image > 0.0 the values of the ratio E2/E1 could not be correctly estimated with our least squares fitting procedure. For further analysis we select the ranges of ξ ≈ 30°–60°, 120°–150°, 210°–240°, and 300°–330°, in which equation image > 0.1 for all values of χ. We assume that in these ranges χ and ξ are linearly independent. Only stable, linearly independent solutions obtained for the above ranges were selected for determinations of the mean values of the semiaxes ratio, and of the degree of the linear polarization. The results of estimation are:

equation image

[28] The mean value of the semiaxes ratio (3) is found to be very close to 1. This result suggests that the AKR is circularly polarized, but with an rms error of ±0.10 some small ellipticity can also be accepted. A small contribution of the estimated mean linear polarization, 0.07, is comparable to the rms error of ±0.09 and thus at this stage it is not possible to decide whether some small linear polarization is present or absent in the AKR.

[29] Figure 5 shows histograms of occurrence distributions of the estimated polarization parameters of the received radio signal: the semiaxes ratio E2/E1 (top), the degree of the linear polarization mL = equation image/I (middle) and the degree of the total polarization m = equation image/I (bottom). White columns contain all solutions, while grey ones comprise only linearly independent solutions. In both data sets of the top panel the distributions maximize at E2/E1 ∼ 1.0, which means that the circularly polarized AKR occurs most frequently. As it could be expected, the dispersions of the distributions of the semiaxes ratio E2/E1 and of the degree of the linear polarization mL (middle), were much smaller for the linearly independent solutions (grey), than those for all solutions (white).

Figure 5.

Occurrence distribution of semiaxes ratio of the AKR polarization ellipses (top), the degree of the linear polarization (middle) and the total degree of the AKR polarization (bottom). Whites columns, occurrence of all solutions; grey columns, occurrence of linearly independent solution. Columns are not stacked. Dispersions of the histograms are shown.

[30] Figure 5 (bottom) shows the occurrence distribution of the estimated values of the total polarization degree of the received radio signal. Its mean value was estimated to be 〈m〉 = 0.90 ± 0.08 (rms error). Comparison of the distributions for linearly independent solutions (grey) and for all solutions (white) shows that they have practically the same dispersions. It means that the dispersion of the total polarization degree distribution is not caused by instability of the solutions for the ill-posed system of least squares equations. It can be caused by other emissions like board interferences, instrumental noise or galactic background, which are present in the received radio signal. These non-AKR signals can introduce uncertainty in the AKR polarization measurements, which becomes significant when the AKR power density is small (less than 20 dB above the background signal).

[31] We checked this hypothesis for the event of a strong AKR emission on Nov. 16, 1997. Its contour shown in Figure 6 (top) marks a time-frequency range, where the power density of the observed radio emission was greater than 10 times the maximum galactic background. The dynamic spectrum of the AKR event was divided into windows each 10 min long (100 sweeps) and 10 frequency steps wide (41 kHz). The white windows in Figure 6 represent those parts of the dynamic spectra where more than 70% of single measurements were greater than 10 times the galactic background. Correspondingly, the grey windows represent those parts of the spectrum where more than 30% of single measurements were below 10 times the galactic background. In other words the grey windows contain measurements contaminated by the non-AKR emission (mainly local interferences, usually more intense than the galactic background), and the white windows contain measurements in which the non-AKR emission does not play role. The least squares procedure was applied to each data window in the same way as described earlier in this section.

Figure 6.

(bottom) Scatterplot showing dependence of the estimated degree of the total polarization of AKR, m = equation image/I on the power density of the AKR emission, for the event on 16 November 1997. Each point is a result of estimation with the least squares procedure applied to each box. The open circles represent mean values of the degree of the total polarization estimated for the white boxes 10 min long and 41 kHz wide (10 frequency steps of SFA) shown in the top panel. The solid circles represent mean values of the degree of the total polarization estimated for the grey boxes of the same size where the background emission was more than 30% that of AKR. Note the grey boxes concentrate along the contour of the dynamic spectrum of this event (see also Figure 2).

[32] Figure 6 (bottom) shows the scatter plot of the values of the total polarization degree (m = equation image/I) of the received radio signal versus the mean values of the AKR power density for each data window. The open circles mark the mean total polarization degree for the white windows and, correspondingly, the solid circles represent the mean total polarization degree for the grey windows. One can easily see that the estimated values of the degree of the total polarization for a strong AKR (open circles) concentrate very closely to 100% independently of the radio signal power (above ∼10−18 W m−2 Hz−1), whereas for a weaker radio signal the estimated values of this parameter are scattered down to 20% and show tendency to depend on the radio signal power. The latter is therefore related to contribution of the non-AKR emissions in the received signal. For more reliable measurements of the strong AKR (above ∼10−18 W m−2 Hz−1) the estimated degree of the total polarization reaches 100%.

[33] This event suggests that the intrinsic total polarization degree of the AKR is higher than that determined for windows with a significant contribution of the non-AKR radio emission. For many other events, the AKR was weaker and the corresponding contribution of the non-AKR emissions was then sufficiently high to increase the estimated mean value of the unpolarized component to 〈m〉 = 0.10 ± 0.08.

5. Summary and Discussion

[34] The main results of this work are as follows:

[35] 1. Statistical estimations show that the degree of the linear polarization of the AKR is small at a level of 0.07 ± 0.09 rms error. This result does not contradict the earlier expectations on absence of the linear polarization, though its uncertainty is too large to definitely conclude on presence or absence of a small amount of the linear polarization in the AKR. The mean semiaxes ratio of the polarization ellipse is estimated to be 〈E2/E1〉 = 0.99 ± 0.10 (rms error). This result suggests that most of the AKR is circularly polarized.

[36] 2. The total polarization degree of the AKR is estimated at a high level of 〈m〉 = 0.90 ± 0.08 (rms error). This error is too large to definitely conclude on presence or absence of some small contribution of the unpolarized component of AKR at a level of 10%. However, estimations of the total polarization degree at a level of 100% for the event of a strong AKR allow us to suppose that some amount of the unpolarized component observed for the weaker AKR is caused by the nonnegligible contribution of the non-AKR background emissions like board interferences, instrumental noise and galactic background, which suggests that the AKR is fully polarized. This supposition needs verification with a more sensitive instrument.

[37] 3. The dynamic spectra of all Stokes parameters of the AKR are determined in the wave plane directly from measurements in the three quasi-perpendicular antenna planes. Their presentation in the wave plane allows for unambiguous identification of the AKR wave modes and simultaneous determination of the AKR power density distribution in each mode, which can be important for interpretation of AKR generation and propagation.

[38] This paper reports statistical estimations of the polarization parameters of the AKR. In the limits of uncertainty (≈9%) they do not contradict expectations that the linear polarization is absent in the AKR. Also, the mean semiaxes ratio estimated to be 0.99 ± 0.10 confirms that most of the AKR is circularly polarized. Some small contribution of the unpolarized component (0.10 ± 0.08) can be explained by the effect of a relatively high level of local interferences and noise.

[39] The weak point of these estimations is the relatively high uncertainty of measurements, which comes out from a high interference level onboard Interball-2 and from a small size of the antennas compared to the wavelengths of AKR, occurring most frequently below 0.6 MHz (L ≤ λ/10 [Ortega-Molina and Daigne, 1984], for waves longer than 500 meters). Future search for the linear polarization of the AKR would need a more sensitive receiver, lower level of board interferences (electromagnetically clean spacecraft) and a triaxial polarimeter equipped with antennas about twice as long as those on Interball-2. The latter would increase the gain of the antennas by about 6 dB. With their length of 44 meters they can still be considered as short for emissions at frequencies below 0.6 MHz.

[40] Fluctuation of the measured power density ΔP/P of the polarimeter channels is also an important parameter to achieve more accurate determination of the AKR polarization state. Its value can be estimated as ΔP/P = 1/equation image. For a measurement at a single frequency step of the Interball-2/Polrad Δf = 4096 Hz and integration time τ = 0.006 s the fluctuation level is ΔP/P ≈ 0.2. For reference, the corresponding fluctuation levels for Wind/WAVES/RAD experiment (Δf = 3000 Hz and τ = 0.154 s [Bougeret et al., 1995]) is ΔP/P ≈ 0.047, and for the Cassini/RPWS experiment working in a high frequency mode (HFR) (Δf = 25000 Hz and τ = 0.08 s [Gurnett et al., 2004]) is ΔP/P ≈ 0.02. Relatively large signal fluctuations in Polrad measurements are mainly due to the short time constant. Thus, the measured signal fluctuation level can be reduced at the expense of the time or frequency resolution.

Appendix A

[41] To fit the modeled Stokes parameters (equations (1) and (2)) to observations, the nonlinear least squares method has been used. The method is based on solving the following system of equations in each iteration step [Press et al., 1992]:

equation image

where ϒ2 = equation imageσj−2(SjobsSjmod)2, and N - number of data points; σi -uncertainty of measurements; Siobs and Sjmod - observed and modeled Stokes parameters; X = (equation image(θ, ϕ), χ, ξ, Xeff, Yeff, Zeff) - parameters of the model; n, m = (equation image(θ, ϕ), χ, ξ, Xeff, Yeff, Zeff) - components of the X and X0 initial guess of X.

[42] To evaluate the solutions of the equations (A1), the Singular Value Decomposition method has been applied. The solutions can be expressed in a matrix form:

equation image

where equation image and equation image are composed of the eigenvectors of equation imageTequation image and equation imageequation imageT, equation image = −equation imageXo, equation image = equation imageXo and diagonal elements of matrix equation image are the singular values (λj) of equation image. If the number of significant singular values (λi) is less than the number of unknowns, then the system of equations (A1) is ill-conditioned and has no unique solution.

[43] The ratio of the largest singular value to the smallest one C = λmax/λmin is the condition number. If C is very large, the system of equations (A1) is ill-posed and has no unique solution when noise on the data is considered [Connerney, 1981; Ladreiter et al., 1995]. Zeroing the columns of eigenvalues in matrix equation image corresponding to very small λ < λmax/Ccr, the resolution matrix can be obtained as equation image = equation imageequation imageT. The matrix equation image describes the ability of the equations (A1) to obtain the linearly independent solution. The nondiagonal elements of the matrix equation image describe the linear dependencies between each input parameters of the model. The level of the critical value of the condition number Ccr depends merely on noise in the observation data and input parameters of the model.

Acknowledgments

[44] The authors are grateful to the referees for their careful review and critical remarks which significantly improved the paper. The Polish space experiment Polrad was led by the Space Research Center P.A.S. Thanks are due to Z. Krawczyk from the Aviation Institute (Warsaw) for development of the polarimeter Polrad, and M. M. Mogilevsky and T. V. Romantsova from the Space Research Institute R.A.S. (Moscow) for coordination and management of the wave experiments on board Interball-2. The Interball mission was managed by the Space Research Institute R.A.S., Lavochkin Space Association and Babakin Center (Russia). This work was financed by the Space Research Institute and the OELZELT commission of the Austrian Academy of Sciences and supported by the Ministry of Science in Poland through the grant 4T12E 006 30. J.H. is grateful to the Austrian Academy of Sciences cooperation program for covering the cost of his visit to the Space Research Institute of A.A.S. in Graz.

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