Polarization of Impulsive Signals at Venus
J. Geophys. Res., 96, 22, 741-22, 752, 1991
(Received: June 11, 1991; accepted: October 1, 1991)
Copyright 1991 by the American Geophysical Union.
Paper Number 91JE02506.
The impulsive plasma wave bursts detected by the Pioneer Venus Orbiter electric field detector in the nightside ionosphere of Venus have been attributed to atmospheric lightning. However, it has also been argued that the wave bursts are generated locally by plasma instabilities. The waves associated with local instabilities are most probably electrostatic in nature, while lightning-generated waves should be whistler mode waves, at least at the lowest frequencies. It may be possible to identify the wave modes through analysis of the wave polarization. We show that for typical ionospheric parameters the whistler mode wave electric field should be polarized predominantly perpendicular to the ambient magnetic field. However, the signals are often impulsive in nature, with durations much less than the spacecraft spin period, and an individual wave event is likely to be aliased since the event may occur at arbitrary spin phase and arbitrary intensity. Statistical analysis of the wave polarization using data acquired in the third nightside periapsis season shows that the 100-Hz data suffer from interference, which may be due to the interaction of the spacecraft with the ambient plasma. The interference is removed through visual inspection of the data, and we show that the 100-Hz waves are polarized perpendicular to the ambient magnetic field provided we restrict the data to those intervals in which the magnetic field is sufficiently far from horizontal to allow vertical propagation within the whistler mode resonance cone. The 100-Hz waves detected outside of the resonance cone are polarized parallel to the magnetic field, as are the waves at higher frequency. The waves consequently fall into two classes: whistler mode waves which are most likely due to atmospheric lightning, since the low phase speed of the whistler mode argues against an in situ instability, and a mode that is polarized parallel to the ambient field. This latter mode may be analogous to the anomalous parallel polarized wave fields detected in the terrestrial ionosphere above thunderstorms.
Impulsive plasma wave signals observed in the nightside ionosphere of Venus by the Pioneer Venus Orbiter electric field detector (OEFD) are often cited as evidence for lightning in the atmosphere of Venus, as discussed in the review by Russell . In his review, Russell points out that other evidence for lightning exists, but the question of lightning at Venus has not been clearly resolved. Searches for the optical signature of lightning using the Pioneer Venus star sensor [Borucki et al., 1991] place an upper limit of the lightning rate that is less than the terrestrial rate (100 flashes/s). In contrast, the Galileo plasma wave instrument has detected radio bursts in the several hundred kilohertz range that are consistent with lightning [Gurnett et al., 1991]. Lastly, recent studies of the OEFD data [Ho et al., 1991] have determined VLF burst rates that are comparable with the terrestrial lightning rate.
The hypothesis that the impulsive signals observed with the OEFD were due to lightning was first put forward by Taylor et al.  and expanded on further by Scarf et al. . In the latter paper, Scarf and colleagues pointed out that in general only those signals observed in the lowest-frequency channel of the wave instrument (100 Hz) could be whistler mode waves. As a consequence, signals occurring solely in the 100-Hz channel were counted as possible lightning events in subsequent studies [e.g., Scarf and Russell, 1983].
From their analysis, Scarf and Russell  concluded that the impulsive events were observed primarily over the highlands of Venus and postulated that active volcanism was responsible for the lightning assumed to produce the observed signals. However, as noted by Taylor et al. , the study of Scarf and Russell  suffered from orbital bias. Furthermore, Russell et al. [1988, 1989] have shown that the data are more clearly ordered by local time. It is important to note that the event definition used by Russell et al. is different than the definition used by Scarf; in the former any impulsive signal detected in any of the four frequency channels of the OEFD is considered as a possible lightning event. It is the high-frequency events which show strong local time clustering, with an occurrence rate maximum in the 2000 - 2200 local time range.
The OEFD data attributed to lightning hence fall into two classes. One consists of a relatively broadband signal observed mainly in the postdusk local time sector. The second consists of waves observed only in the 100-Hz channel at all local times in the nightside ionosphere, and these waves are thought to propagate in the whistler mode. If the Pioneer Venus Orbiter were equipped with either a search coil sensor or with an electric field detector capable of producing well resolved wave spectra, we could determine the mode of propagation with a reasonable degree of certainty directly from the wave measurements. Unfortunately, the OEFD only measures a single electric field component and furthermore has only four frequency filters at 100 Hz, 730 Hz, 5.4 kHz, and 30 kHz. Typically, only the 100-Hz channel is sensitive to whistler mode waves; the ambient magnetic field must be greater than 26 nT for fce > 730 Hz, where fce is the electron gyrofrequency. We must consequently determine the wave mode through less direct methods.
One possibility is to determine whether or not the signals are observed within the whistler mode resonance cone [Strangeway, 1991; Sonwalkar et al., 1991; C.-M. Ho et al., Control of VLF burst activity in the nightside ionosphere of Venus by the magnetic field orientation, submitted to J. Geophys. Res., 1991 (hereinafter referred to as C.-M. Ho et al., 1991)]. Since the refractive index is large in the ionosphere and the variation in refractive index is mainly due to changes in plasma density, refraction will cause the wave vector of whistler mode waves to align along the density gradient in the ionosphere as the waves penetrate the ionosphere from below. For a horizontally stratified ionosphere the wave vector will consequently point vertically upward. Hence a simple test uses the orientation of the magnetic field with respect to the vertical to determine if wave bursts correspond to vertically propagating whistler mode waves. This test was used by C.-M. Ho et al., 1991 who found that the burst rate inside the resonance cone was about 2 to 3 times that for bursts outside the resonance cone.
A more sophisticated version of the resonance cone test was employed by Sonwalkar et al. . They used detailed knowledge of the plasma density, magnetic field strength and orientation, and the orientation and spin phase of the OEFD antenna to predict the degree of spin modulation expected for a whistler mode wave propagating vertically in the ionosphere. This test not only uses the simple binary test of whether or not the waves are within the resonance cone but also tests the degree of polarization of the signal. In general, this depends on the direction of propagation and the plasma parameters. Sonwalkar et al. found that 6 of the 11 cases studied were consistent with whistler mode propagation.
In this paper we will only test for perpendicular or parallel polarization rather than the more complex analysis used by Sonwalkar et al. . We are motivated to do this for two reasons. The first is that Scarf and Russell  published two examples of wave polarization showing perpendicular orientation to argue that the 100-Hz waves are whistler mode waves. We wish to determine if the polarization found by Scarf and Russell is statistically significant. Second, our analysis should complement the more rigorous analysis used by Sonwalkar et al. in their case studies. Because our method is relatively simple, we can analyze many orbits of data. The polarization test is an important additional piece of information to be used for determining the source of the waves observed at Venus. The examples presented by Scarf and Russell  were used to counter a suggestion by Taylor and Cloutier  that the 100-Hz signals were ion acoustic waves, since these might be expected to be parallel polarized.
The outline of the paper is as follows. In the next section we discuss the polarization expected for whistler mode waves in the nightside ionosphere of Venus, and we give some examples of wave polarization. In the third section we present the results of our statistical analysis. First, we consider the effects of interference. We then discuss the results for waves that can propagate in the whistler mode, assuming a subionospheric source. Lastly, we present statistical results for nonwhistler mode waves. The last section summarizes the statistical results and presents some concluding remarks. The appendix describes the methods used to determine the statistical significance of our analysis.
As already discussed, it is often assumed that the wave electric field is polarized perpendicular to the ambient field for whistler mode waves. Before presenting specific examples, we should show under which conditions this is a reasonable assumption. Whistler mode waves propagating parallel to the ambient magnetic field are transverse electromagnetic waves, and in this case the wave field is polarized perpendicular to the ambient field. However, obliquely propagating whistler mode waves, especially those on the resonance cone, tend to be electrostatic. The wave field is polarized along the wave vector for electrostatic waves, and these waves will be polarized approximately parallel to the ambient field when the resonance cone angle is small. The degree of polarization of the wave field as a function of propagation direction and ambient plasma parameters has been taken into account explicitly by Sonwalkar et al. . We will show, however, that for most circumstances in the nightside ionosphere of Venus, whistler mode at 100 Hz will be polarized predominantly perpendicular to the ambient field.
2.1 Expected Whistler Mode Polarization
From cold-plasma theory [e.g., Boyd and Sanderson, 1969] it can be shown that the wave field polarization is given by
where Ez, is the electric field component parallel to the ambient magnetic field, Ex is the component in the plane containing the magnetic field and the wave vector, and Ey is orthogonal to the other two. The angle between the wave vector and the ambient field is given by , and is the refractive index. The variables S, D, and P are terms in the cold-plasma dielectric tensor, with
where the summation is carried over particle species, with ps being the species plasma frequency and s, being the species gyrofrequency (including the sign of the charge). In deriving (1) and (2) it is assumed that the wave fields vary as exp-i( t-k r).
Equation (1) gives the polarization in the plane perpendicular to the ambient field, while (2) gives the degree of polarization along the field. For example, the refractive index for parallel propagation is given by 2 = R or L, which on substitution into (1) gives iEx /Ey = 1 respectively, with, iEx /Ey = 1 corresponding to right-hand circular polarization.
Given the refractive index for a particular wave frequency and direction of propagation, (1) and (2) can be used to specify completely the electric field polarization. While we could use the complete cold-plasma dispersion relation to determine the refractive index, a simplified version is applicable for most parameter regimes in the Venus nightside ionosphere. First, since the lower hybrid frequency (fLHR) in a proton plasma = 0.65B Hz, where B is magnetic field magnitude in nT, fLHR would be less than 20 Hz, even if the magnetic field were as large as 30 nT. This is much less than 100 Hz, and the lower hybrid frequency would be even lower in an O+ plasma. We can consequently neglect the ions in the cold-plasma dispersion relation.
The cold-plasma dispersion relation can be further simplified using the quasi-parallel (or quasi-longitudinal) approximation provided sin2 < (2pe / 2e) cos , where pe and e are the electron plasma and gyrofrequencies (the latter does not include the sign of the electron charge). At Venus, nightside ionospheric densities are typically > 100 cm-3, while B < 30 nT. Consequently, 2pe / 2e 104, and the quasi-parallel approximation should be adequate, except for propagation near 90°. Since whistler mode waves can propagate perpendicular to the field only for f fLHR, we do not consider perpendicular propagation. Under the quasi-parallel approximation,
On substituting (3) into (1) and (2), assuming 2 1, we obtain the following:
Equations (4) and (5) depend only on the propagation direction and on the wave frequency with respect to the electron gyrofrequency, and so provided the quasi-parallel approximation is valid, these equations can be used to estimate the degree of polarization of whistler mode waves in the nightside ionosphere of Venus.
The refractive index in (3) has a resonance when = cos-1 /e = r, the resonance cone angle. Inspection of (4) shows that the wave field is always right-hand polarized for all propagation angles < r, with the polarization being circular for parallel propagation and predominantly in the x direction for propagation near the resonance cone. The dominant electric field component in the plane perpendicular to the ambient field is consequently given by Ex. With regard to the parallel electric field,(5) shows that this is largest when the wave is propagating at the resonance cone angle, and in that case, Ez / Ex = / (2e - 2)1/2. For small /e, Ez / Ex /e . At Venus the OEFD detects whistler mode waves in the 100-Hz channel. The average magnetic field at low altitudes is about 18 nT [Ho et al., 1991], and so Ez / Ex < /e 0.20 on the average. In general, then, the assumption that the wave field is perpendicularly polarized for whistler mode waves is reasonable in the nightside ionosphere of Venus.
2.2 Examples of Observed Polarization
Having demonstrated that whistler mode waves will be predominantly perpendicularly polarized in the night ionosphere of Venus, we will present specific examples of the wave polarization. Figure 1 shows OEFD wave data acquired on orbit 526 by the Pioneer Venus Orbiter. The figure shows the log10 of the wave intensity (in V2/m2/Hz) as a function of time for the four frequency channels of the OEFD. For this orbit most of the wave activity is in the 100-Hz channel, and we give examples of the wave polarization for selected intervals as indicated by the horizontal bars in Figure 2 The first two intervals were chosen since they correspond to the intervals shown by Scarf and Russell . The last interval was chosen to show an interval which contains interference signals.
Fig. 1. Time series of the wave intensity as measured by the orbiter electric field detector. Six minutes of data acquired on orbit 526 are shown. Most of the impulsive signals occur in the 100-Hz channel only. The horizontal bars mark intervals for which polarization examples are given in Figure 2.
Figure 2 shows the log10 of the wave intensity as a function of spin phase measured over a 30-s interval, corresponding to roughly 2.5 spin periods (spin period 12 s). In the plot the x axis points toward the Sun. The data are plotted on a relative scale such that the minimum wave intensity in the interval corresponds to the origin of the polar plot, with the maximum giving the unit circle. The line labeled "B" is the direction of the average magnetic field in the spin plane, with the arrow head giving the fraction of the total field lying in the spin plane. The line labeled "E" is actually the line of maximum variance through the data, which we assume corresponds to the average wave electric field direction. To the left of each plot in Figure 2, we give the minimum and maximum wave intensities, together with the maximum/minimum variance ratio and the relative orientation of the "E" and "B" directions.
Fig. 2. Three polarization plots for orbit 526. (a) and (b) Polarization of naturally occurring signals. (c) Polarization of an interference signal. Each plot is drawn using a log10 relative scale, as discussed in the text. The naturally occurring signals are polarized perpendicular to the average magnetic field.
Figures 2a and Figure 2b show polarization diagrams for the first two marked intervals in Figure 1 These plots are not identical to those shown by Scarf and Russell  since the wave data are plotted on a relative scale and we use a 30-s interval when producing the plots, as opposed to a 15-s interval as used by Scarf and Russell. We selected 30-s intervals since this length was used by Russell et al. [1988, 1989] in their occurrence rate studies, and we will use their data base in conjunction with our study. These differences notwithstanding, the data in Figures 2a and 2b support the conclusion of Scarf and Russell  that the wave electric field is polarized perpendicular to the ambient field and hence whistler mode. The conclusion that these waves are whistler mode waves was also confirmed by Sonwalkar et al. , who in addition noted that the magnetic field orientation for the interval shown in Figure 1 was such that the waves were inside the whistler mode resonance cone.
Figure 2c shows data for an interval in which the data appear to be contaminated by an interference signal. The polarization plot shows an extremely clear alignment of the electric field. If this interval were to be included in a statistical study, then the resultant distribution of polarizations would be compromised. We will discuss this interference signal in more detail later.
As a counterexample to signals that show clear whistler mode polarization, we present data from orbit 501 in Figure 3. This figure shows qualitatively different wave signatures, which are characterized by impulsive signals at all frequencies. The interval shown in Figure 3 has also been studied by Sonwalkar et al. , who noted that the magnetic field was nearly horizontal at this time, and so vertically propagating whistler mode waves would lie outside of the resonance cone.
Fig. 3. Time series of wave intensity for orbit 501. Similar in format to Figure 1. On this orbit the impulsive signals are observed in all four channels.
The polarization plots for the intervals marked in Figure 3 are given in Figure 4. The format is similar to Figure 2, with Figure 4a and Figure 4b corresponding to the broadband signals in Figure 3, while Figure 4c gives another example of an interference signal. In Figure 4a the polarization is more parallel, while Figure 4b is more perpendicular. These events are probably not whistler mode waves.
Fig. 4. Three polarization plots for orbit 501. Similar in format to Figure 2. (a) The maximum variance direction is more parallel to the magnetic field than perpendicular, while (b) the wave field is perpendicular. (c) The interference signal is greater than 45° to the magnetic field and would bias the statistics to perpendicular polarization if included in the analysis.
From the events shown in Figures 2 and 4 we see that not all events are consistent with whistler mode waves, and even those which do show perpendicular polarization are not clearly polarized. For example, it is the interference signal in Figure 1c which has the best defined polarization. Part of the reason for the relatively poor polarization data is due to the impulsive nature of the signals. The nightside signals often last for only a fraction of a spin period, unlike other plasma waves such as upstream plasma oscillations. Because the latter waves last for several spins the wave polarization can be determined quite readily [Crawford et al., 1990]. In addition to the aliasing caused by sampling a short-duration signal occurring at random spin phases, the interference phenomena may bias any statistical study of the polarization. How we address these problems will be discussed in more detail in the following section.
In order to carry out statistical analyses of the wave electric field polarization we reduce data such as those presented in Figures 2 and 4 to several parameters for inclusion in a data base. For each 30-s interval we determine the average direction of the magnetic field in the spin plane and the maximum variance direction for the wave power. These two angles allow us to estimate the average wave polarization in a 30-s interval. We can also calculate the maximum variance direction of the wave power using the instantaneous phase of the antenna with respect to the magnetic field direction in the spin plane. The latter method is less likely to be aliased because of fluctuations in the magnetic field. On the other hand, interference signals do not usually depend on magnetic field orientation but rather on spin phase. Since spin phase information is lost if we use the instantaneous magnetic field phase, we use both methods to determine the wave polarization.
In order to assess the reliability of the polarization determination with respect to the ambient magnetic field we also include in the data base the fraction of the magnetic field in the spin plane and the variability of the magnetic field in the averaging interval. In addition, the variance ratio (Smax /Smin in Figures 2 and 4) and the ratio of the maximum to minimum wave power are included as measures of the quality of the maximum variance determination. Lastly, in order to determine whether or not the waves lie within the whistler mode resonance cone, we include the spacecraft position.
We have elected to use data from the third season of nightside periapses to generate the data base in this initial statistical study. Most of the data acquired in the third season had 0.25-s resolution and periapsis altitude was usually below 170 km. Periapsis altitude was higher for later seasons, while data from earlier seasons tended to be of lower temporal resolution. However, we should be able to incorporate data from other seasons in subsequent studies using the insight gained from the analysis presented here. Only the data acquired in darkness on each orbit are included since the data suffer from interference due to photoelectron emission and solar cell interference when the spacecraft is in sunlight.
3.1 Effect on Interference
In the previous section we pointed out that the 100-Hz wave data often have interference signals present. Moreover, these signals are spin modulated and will consequently also result in a polarized signal as determined by the maximum variance method. As we shall show here, these interference signals will contaminate the polarization statistics. In Figure 5 the maximum variance phase angle in the spin plane is plotted as a function of orbit number for season III. The scatterplot shows a clear clustering of the maximum variance direction, which varies by about 1.4° per orbit, as indicated by the diagonal lines in the figure. These lines have been drawn by eye to bound the main cluster of points, using the parameters given at the bottom of the figure. In subsequent analyses we will refer to the interference reference angle = 18.5° - 1.4° (n - 522.5), where n is the orbit number. Maximum variance phase angles which fall in the range 20° are within the interference filter.
Fig. 5. Scatterplot of the maximum variance direction as a function of orbit number for all the 30-s intervals in the season III nightside periapsis passes. There is a noticeable clustering of the phase angle as a function of orbit. Using a visual fit to the data, we have defined a phase angle filter as shown by the diagonal lines in the plot. The filter parameters are given at the bottom of the figure.
We assume that the clustering of points in Figure 5 is due to interference since we would not expect any naturally occurring signal to depend in a systematic way on orbit number. However, it is not clear why an interference signal should show such a dependence. For example, impact ionization is known to generate wave signals due to the spacecraft motion through the dense ionospheric plasma at periapsis [Curtis et al., 1985], but the impact ionization signal shows no spin modulation. Moreover, photo-electron emission or solar cell interference are not probable causes since the data were acquired in darkness. At present we have not determined the cause of the interference signal, although the filter parameters may provide a clue. The interference signal precesses at 1.4° per orbit, while the Pioneer Venus Orbit precesses at 1.6°. This suggests that the interference signal is ordered roughly by the radius vector from the center of the planet to the spacecraft. Perhaps radial outflow of ionospheric plasma is responsible for the signal. However, this is highly speculative and further study is necessary to determine the cause of the interference.
In order to show that the interference signal biases the data we have plotted the polarization statistics for all of season III in Figure 6. Since there is a lot of information in the figure and other figures have a similar format, we will describe the figure in some detail here before discussing the results. At the left of the figure there are three panels. The largest of these shows a histogram of the percent occurrence of the relative phase between the maximum variance direction and the average magnetic field, plotted in 10° bins. Because of the 180° ambiguity in the maximum variance direction the phase angle is determined modulo 180, and the angles are restricted to the range 0°- 180°. The solid curve in the plot shows a sinusoidal fit to the histogram, with the solid circle giving the phase angle of the peak in the sinusoid. The error bar gives the 95% confidence limit, as discussed in th appendix.
Fig. 6. Phase angle histograms for all the season III data. The large histogram to the left shows the percent occurrence of different relative phases, while the two smaller histograms show the phase of the magnetic field and maximum variance directions. Auxiliary diagnostics are given at the right of the figure. The figure is described more fully in the text. The relative phase appears to be more perpendicular than parallel, although this is probably due to contamination by interference signals.
The upper panel near the center of the figure gives the percent occurrence for the orientation of the magnetic field with respect to the interference signal, where the interference reference angle is given by the filter parameters in Figure 5. The panel below the magnetic field phase shows the maximum variance phase with respect to the interference. Below this panel the phase angles and 95% confidence limits are given for the relative phase (R), magnetic field phase (B), and maximum variance phase (V). In addition, we include the probability (P) that the sinusoid fit is not random, as also discussed in the appendix.
To the right of Figure 6, we plot percent occurrence histograms for the diagnostic parameters. In descending order these are the maximum to minimum variance ratio, labeled "anisotropy," the log10 of the maximum over minimum wave power, the fraction of the average magnetic field in the spin plane, and the variability of the magnetic field, labelled "DeltaB/B", where DeltaB = (2x + 2y + 2z)1/2, with x, y, and z, being the standard deviations of the x, y and z components of magnetic field, respectively. These diagnostics show that about 50% of the data intervals are weakly anisotropic, < 2, but 25% are highly anisotropic, > 5. The weakly anisotropic signals are less likely to be due to real signals and may be the result of random fluctuations in the wave power, while the highly anisotropic signals may be interference. The second diagnostic shows that over 50% of the signals have a maximum power that is less than an order of magnitude more intense than the minimum, indicating that many of the signals included in the data base may not be significant in determining the polarization, which is also implied by the weak anisotropy.With regard to the magnetic field orientation, at least 60% of the intervals have over 90% of the average field in the spin plane, but only 50% of the intervals have the variation in the field less than the field magnitude.
At first sight the histogram of the relative phase in Figure 6 appears to support the interpretation that the average orientation of the wave electric fields tends to be more perpendicular than parallel to the magnetic field. The peak in the histogram is near 123°, and the sinusoidal fit is statistically significant. However, the peak is not centered on 90% which raises some doubt that the waves are perpendicularly polarized. Furthermore, the histogram of the maximum variance phase shows a large peak at 0° with respect to the interference signal phase angle determined in Figure 5. Obviously, since the same data are used in Figures 5 and 6, we would expect strong clustering of the maximum variance phase, but in addition, the magnetic field orientation also shows a peak around 74°. We might consequently expect a peak in the relative phase distribution around 107°, taking the difference of the peaks of the two distributions.
Since interference signals bias the determination of the wave polarization when we use the whole data set for season III, we have also considered a reduced data set. In this data set we only include the intervals for which a burst event occurs as defined by Russell, von Dornum, and Scarf [Russell et al., 1988] (hereinafter referred to as RvD&S). In this data set, each 30-s interval is classified as to whether or not any bursts occur in that interval. The RvD&S data set consists of 818 active intervals in season III, compared with the total of 3173 samples. With this data set we obtain a relative phase of 123° 23° (99.7% probability). This is very similar to the phase- obtained with the full season III data set and suggests that the polarization statistics are also contaminated by interference for the RvD&S data set. We must emphasize here that only the polarization data are compromised by the interference, the burst occurrence statistics published using the RvD&S data set are unaffected.
3.2 Removal of Interference
Given that the polarization data are probably contaminated by interference, we must derive some method for removing the interference signal from the data set. We could simply filter the data using the interference filter defined in Figure 5. However, not all the samples which fall within the interference filter are necessarily interference, and we find that removing those naturally occurring signals also biases the data. We consequently have cleaned the data primarily through visual inspection, using the interference filter parameters as a guide in the selection of the data. We have only applied this test to the RvD&S data set, rather than all of season III, since the former already excludes intervals of weak signal, and intervals containing only interference, such as the interval after 0930:30 in Figure 1, or after 1057:00 in Figure 3.
As a first pass through the data we rejected intervals that appeared to be strongly contaminated with interference, solely by scanning the time series. For example, the interval 1056:30 - 1057:00 in Figure 3 does contain a weak burst, but since the interference signal is likely to dominate the polarization determination, the interval is removed from the "cleaned" data set. Following this first pass through the data, 574/818 samples in the RvD&S date set were retained.
On the second pass through the data we determined if a datum fell within the interference filter, or if the signal was weak, with peak power less than an order of magnitude greater than the minimum wave power in each interval. Using this numerical filter as a guide, we again decided if a sample should be retained in the data set. For example, the interval from 0925:30 to 0926:00 in Figure 1 resulted in a maximum variance angle which fell within the filter range, but was retained in the data set. Similarly, the intervals from 1052:00 to 1053:30 in Figure 3 would be rejected by the phase angle filter but were retained in the cleaned data set. After the second pass, 99/202 samples which fell within the numerical filter were removed, while an additional 14/372 events which lay outside the filter were also removed. Consequently, 113 additional samples were deleted from the data base, giving a total of 461 samples in the cleaned data set.
The results of the phase angle analysis for the cleaned data set are shown in Table 1. The first row (average phase) shows the relative phase using the method employed in Figures 2, 4, and 6, where the electric field phase is determined from the difference of maximum variance phase and the average magnetic field direction in each 30-s interval. The second row (instantaneous phase) shows the relative phase using the instantaneous difference between the antenna direction and the magnetic field. The last two rows give the average magnetic field direction in the spin plane and the best fit maximum variance direction with respect to the interference reference angle. For each entry we give the best fit phase angle, 95% confidence limits, and the probability that the fit is not random. The low probability in the table shows that the "cleaned" data have no clear polarization of the 100-Hz signals, even though the magnetic field and maximum variance direction are both orientated near 70° to the interference reference angle. In the absence of any assumptions concerning the mode of propagation we would conclude that the 100-Hz data do not show clear parallel or perpendicular polarization.
3.3 Polarization Inside the Resonance Cone
Recent work [Sonwalkar et al., 1991; C.-M. Ho et al., 1991] has shown that the 100-Hz emissions in the nightside ionosphere of Venus fall into two classes. These classes are separated by whether or not the 100-Hz waves can propagate in the whistler mode. Since the refractive index for whistler mode waves in the Venus nightside ionosphere is large [Strangeway, 1991; Sonwalkar et al.,1991], the wave vector of any signal propagating through the ionosphere from below must be aligned along the plasma density gradient, which we assume is vertical. If the magnetic field is sufficiently far from vertical then the wave vector will lie outside of the resonance cone, and the waves cannot propagate as whistler mode waves, assuming a subionospheric source. Under the assumption of vertical propagation the requirement that the waves can propagate as whistler mode waves is Br cos-1f/fce where Br is the angle between the magnetic field and the radius vector in the range 0° - 90°. Using the average direction of the magnetic field in each 30-s interval, we calculate the resonance cone angle for 100-Hz waves and the orientation of the field with respect to the radius vector. The data can then be filtered for propergation inside and outside of the resonance cone.
The results for the cleaned data set are shown in Figure 7. On comparison with the diagnostic parameters shown in Figure 6, most of the extremely high anisotropy signals have been removed, and the peak wave power is usually at least an order of magnitude above background. The magnetic field is predominantly in the spin plane and well ordered. Both the histogram and sinusoidal fit show that the magnetic field phase is parallel to the interference reference angle, while the maximum variance phase is perpendicular. The histogram and fit show that the relative phase is perpendicular. The data in Figure 7 indicate that the wave electric field is polarized mainly perpendicular to the ambient magnetic field for waves inside the whistler mode resonance cone. This result also holds if we use the instantaneous phase, where we obtain a best fit relative phase angle of 82° 44° (87.8% probability).
Fig. 7. Phase statistics for the "cleaned" subset of the RvD&S data. The data are further restricted to those intervals for which the average magnetic field is sufficiently vertical to allow whistler mode propagation inside the resonance cone, assuming a subionospheric source. The 100-Hz waves that fall into this category are on the average polarized perpendicular to the ambient field as expected for whistler mode waves.
However, as already pointed out when discussing the results for the data not selected using the resonance cone condition, some care must be given to the effect of filtering out the interference signals. Figure 7 shows that the magnetic field is mainly aligned parallel to the interference reference angle. This is a consequence of the resonance cone criterion, which selects field directions that are mainly parallel to the radius vector. The maximum variance direction is near 90° for the cleaned data set. We might consequently expect the cleaned to be perpendicularly polarized, but it is not clear if this is solely a consequence of the filtering applied to the data.
To demonstrate that the cleaned data set is not biased by the filtering method employed, Table 2 shows the best fit phase angles for those data which have maximum variance phase angles within the interference filter. Only 35 of the 218 samples in the cleaned data set for which vertical whistler mode propagation is allowed fall into this category. Since the interference filter restricts the maximum variance phase angle, while the resonance cone condition might be expected to bias the magnetic field orientation toward radial, we would expect the relative phase to be biased toward parallel polarization. However, Table 2 shows this is not the case, and the magnetic field is orientated perpendicular to the interference reference angle, although with low probability. As a consequence, the relative phase is also weakly perpendicular with the probability being somewhat higher using the instantaneous phase. From Table 2 we conclude that the phase determination shown in Figure 7 is not biased by the filtering, method and those 100-Hz emissions which can propagate vertically within the whistler mode resonance cone are on the average polarized perpendicular to the ambient magnetic field.
Table 2. Best fit phase angles for cleaned data inside the resonance cone and inside the interference filter.
3.4 Polarization Outside the Resonance Cone
At Venus the electron gyrofrequency is usually only a few hundred hertz, and the waves observed at higher frequencies by the OEFD are presumably not whistler mode waves. It has been argued [Russell, 1991; C.-M. Ho et al., 1991] that the waves which are not whistler mode waves are analogous to the anomalous signals observed in the terrestrial ionosphere [Kelley et al., 1985] in association with lightning. In the terrestrial case the signals are field aligned. The polarization for the higher frequencies are given in Table 3. For completeness we also include the 100-Hz data, selected for inside and outside the resonance cone. Since the 100-Hz data are contaminated by interference signals, we have used the cleaned data set. We use the RvD&S data set for the higher frequencies.
In general, the nonwhistler mode waves are polarized parallel to the ambient field. Interestingly, the probability is low when the polarization is determined using the average phase but is quite high using the instantaneous phase. Phase angle histograms for the 30-kHz channel are shown in Figure 8, using the instantaneous phase. We assume the impulsiveness of the signals results in the poor phase determination using the average phase. The mode of propagation for the nonwhistler mode waves is not known, but the data in Table 3 do suggest that the waves are polarized along the magnetic field. This is consistent with the terrestrial observations of Kelley et al. . However, it should be noted that parallel polarization might also be expected for in situ plasma instabilities.Fig. 8. Phase statistics for the 30-kHz burst intervals in the RvD&S data set. In this case the relative phase is calculated using the instantaneous angle between the OEFD and antenna and the magnetic field direction in the spacecraft spin plane. Although there are relatively few intervals and the statistics are poor, the waves are mainly polarized parallel to the average field.
Table 3 also shows that for the cleaned data set roughly half of the 100-Hz intervals are associated with horizontal magnetic fields. The fraction is slightly higher for the full RvD&S data set, 510/818 (62%) of the intervals in the RvD&S data set have 100-Hz events outside of the resonance cone. In terms of percent occurrence the RvD&S data set is not affected by interference and the distribution of events for this data set should be compared with the distribution of magnetic field orientation. We find that 2386/3173 (75%) of the intervals in the full season III data set have magnetic field orientation which does not allow vertical whistler mode propagation. Hence the percent occurrence rate is much higher for 100-Hz events inside the resonance cone (39%) than outside (21%). A similar result was found by C.-M. Ho et al., 1991 using burst rate statistics.
4. Summary and Conclusions
Given the restrictions of the Pioneer Venus OEFD, wave mode identification is often ambiguous. In particular, since the OEFD only detects one component of the wave electric field, and only at four discrete frequencies, it is difficult to identify whistler mode waves. One method for identifying the wave mode is to determine the wave polarization, taking advantage of the spacecraft spin. This has been shown to be quite successful when studying electrostatic plasma waves upstream of the bow shock [Crawford et al., 1990]. We have pointed out that for typical nightside ionosphere plasma densities and magnetic field strengths, the whistler mode wave electric field should be mainly perpendicular to the magnetic field. Hence we might expect to be able to determine the wave mode of the impulsive bursts in the nightside ionosphere from the polarization.
Specific examples [Scarf and Russell, 1988] have been published to show that the waves detected in the nightside ionosphere are whistler mode. However, the signals are usually rather impulsive in nature, and often only last for some fraction of the spin period. It is reasonable to ask if the examples published by Scarf and Russell are typical or just fortuitous. In order to address this question we have presented a statistical study of the wave polarization using data from the third nightside periapsis season.
In performing the statistical analysis of the wave polarization we assume that the direction of maximum variance of the wave power gives the orientation of the wave electric field. However, the maximum variance method does not discriminate between real and artificial signals, and the method may be aliased because of the impulsive properties of the signals. Indeed, we find that much of the 100-Hz data are contaminated by interference signals which are probably associated with the spacecraft interaction with the ionosphere. Even when we restrict our analysis to burst intervals in the RvD&S data set, we find that the polarization statistics are biased by interference. It should be noted that the interference only affects the polarization study, previous studies which determined percent occurrence rates are unaffected by the interference.
The spin phase at which the interference signal is maximum appears to be roughly ordered by the radius vector from the spacecraft to the center of the planet. Since the magnetic field is mainly horizontal in the nightside ionosphere, we obtain an apparent perpendicular polarization that is due to the interference signals. To counteract this, we have derived a "cleaned" data set in which those intervals which appeared to be contaminated by interference are deleted from the data base by visual inspection. This data set was found to display no preferred polarization direction for 100-Hz waves.
Recently, Strangeway , Sonwalkar et al. , and C.-M. Ho et al., 1991 have pointed out that not all 100-Hz waves are necessarily whistler mode waves. If the source of the whistler mode waves is subionospheric, then the large increase in refractive index encountered by the waves on entering the ionosphere will cause the wave vector to be aligned along the density gradient, which we assume is vertical. Since whistler mode waves can only propagate within the resonance cone, horizontal ambient magnetic fields will preclude whistler mode waves. Consequently, we have further subdivided the data into intervals for which whistler mode propagation is or is not allowed. The cleaned data set showed little bias due to interference, and the wave fields were found to be polarized perpendicular to the ambient magnetic field for propagation inside the resonance cone.
The cleaned data set also showed that 100-Hz waves detected outside the resonance cone tend to be polarized parallel to the ambient field. A similar result was found for the higher frequencies, which did not suffer from interference. These higher-frequency signals are mainly observed in the postdusk local time sector [Russell et al., 1989], and as discussed by Russell , may be analogous to the signals detected above terrestrial thunderstorms [Kelley et al., 1985]. In the terrestrial case the waves are also polarized along the magnetic field.
The polarization statistics are consistent with the results of Sonwalkar et al.  and C.-M. Ho et al., 1991 that the impulsive signals detected in the nightside ionosphere of Venus fall into two classes. Prior to these studies the classification was usually based on the wave frequency alone, with the 100-Hz bursts assumed to be whistler mode waves, and the higher frequencies being due to some anomalous wave propagation mechanism. It now appears that the 100-Hz waves also fall into these two, classes. We find that roughly half of the intervals containing 100-Hz bursts are associated with vertical fields and are consistent with whistler mode waves propagating from below the ionosphere. However, since the magnetic field is mainly horizontal in the nightside ionosphere, the normalized rate of occurrence is higher for whistler mode waves. A similar result was found in the burst rate study of C.-M. Ho et al., 1991.
In conclusion, the results presented here support the hypothesis that atmospheric lightning is responsible for the waves observed in the nightside ionosphere of Venus. The local time dependence of the higher-frequency burst rate [Russell et al., 1989; Ho et al., 1991] is the most telling argument for a lightning source for the nonwhistler mode waves, but the additional information that these waves are parallel polarized suggests that the signals may be similar to those reported by Kelley et al. . It should be emphasized, however, that a local electrostatic instability might generate parallel polarized waves, and the polarization data alone are not sufficient to discriminate between sources at the higher frequencies. With regard to the 100-Hz waves the whistler mode identification argues against a local source. First, the 100-Hz waves were found to be polarized perpendicular to the ambient magnetic field when whistler mode waves could propagate vertically from below. It is not clear why any local instability would depend on the ambient magnetic field orientation. Second, Maeda and Grebowsky  suggested that whistler mode waves could be generated in situ, but Strangeway  pointed out that the high refractive index of whistler mode waves means that the thermal electron Landau damping is important and any instability must have a large growth rate to overcome this damping. Since it appears unlikely that whistler mode waves are generated locally in the nightside ionosphere of Venus, atmospheric lightning is a probable source.
Appendix: Statistical Error Analysis
In order to readily compare phase angle distributions such as those plotted in Figures 6 - 8 we have elected to fit the histograms with a sinusoid, using a least squares analysis. This allows us to reduce the information in each histogram to a single phase angle. However, the histograms also convey information about the degree of confidence in the determination of the phase angle. For example, the relative phase histogram in Figure 6 is much smoother than that shown in Figure 8. This implies that in a statistical sense we should have more confidence in the phase angle as determined in Figure 6, in the absence of any consideration of other factors, such as contamination by interference. To this end, we carry out a probability and error analysis using methods for curvilinear regression, such as discussed by Pollard .
Each histogram is divided into 10° bins covering a range of 180°. We shall denote the bin angles by xi and the corresponding number of samples in a bin by yi. We fit a sinusoid to these data of the form
where is the mean of the yi, and for convenience we use x1 as a reference angle. Equation (Al) can also be written as
with y0 = (ys2 + yc2 )1/2 and x0 = 1/2 tan-1ys/ yc. The form given by (A1) is used to perform the regression analysis, while (A2) defines the best fit phase angle (x0).
The coefficients in (A1) are determined through minimization of the residual sum of squares
In order to determine the significance of the coefficients given by (A3a and A3b) we compare the ratio of the variance of the data due to regression over the residual variance to that expected for two chi-square random variables [Pollard, 1977]. The variance ratio of the chi-square distributions follows the F distribution. With the particular functional form we have chosen the test statistic can be calculated using the relationship
where the left-hand side of (A4) gives the residual sum of squares, which we denote by Sr2, and the second term on the right-hand side is the regression sum of squares, which we denote by Sf2.
There are 18 degrees of freedom in the data, and we calculate three regression parameters, including the mean. Since we are only testing for the coefficients ys, and yc, being significantly nonzero, there are only 2 degrees of freedom for the variance of the regression, and 15 for the residual. Dividing the regression and residual sum of squares by their respective degrees of freedom gives their variance, sf2 = Sf2 /2, and sf2 = sf2 /15. The variance ratio, F = sf2 / sr2, is compared to the F distribution for two chi-square random variables with 2 and 15 degrees of freedom, usually denoted F2,15.
Rather than simply compute the test statistic F, and compare that with a specific value of the F2,15 probability distribution (e.g., the upper 5%), we calculate the probability that F > F2,15 If we denote the probability that F > F2,15 by P and Q = 1 - P, then
See, for example, (26.6.4) of Abramowitz and Stegun . In the figures and tables of the main text we give P as a percentage.
The residual variance can also be used to give confidence limits on the coefficients ys,c denoted by ys,c Following Pollard , the confidence limits are calculated using the Student's t distribution, giving ys,c = t15 sr/91/2. The value of t15 used depends on the particular confidence limit required. For example, to obtain the 95% confidence limits, we use the value of t15 corresponding to the upper and lower 2.5% of the t distribution.
Since we use (A2) to specify the fit, we wish to express the error as an angular measure. Noting that ys,c does not depend on the magnitude of the individual coefficients ys,c, and that Sf2 = 9y02, we define
and from this we calculate an angular error given by
The functional form used to define the angular error in (A7) is somewhat arbitrary but has the advantage of only depending on the test statistic F, which we use to calculate the probability. The actual error depends on the confidence limit required, as shown in Figure A1. With the form given by (A7), the limiting error is 67.5° when F = 0, independent of the confidence limit. The figure shows that the error is reduced by roughly a factor of 2 if we use a 70% confidence limit, rather than the 95% limit actually employed in the main text. This is to be expected since these confidence limits correspond to roughly one and two standard deviations respectively for a normal probability distribution.Figure A1. Probability and angular error as a function of the test statistic. The probability is given by the single curve that approaches 100% for high values of the test statistic. The angular error depends on the degree of confidence desired, as indicated by the percentage labels. The horizontal dashed lines give the angular error for different confidence limits, assuming a test statistic that is 80% probable. In this paper we use 95% confidence limits when determining the error on the fit.
It is not clear that (A7) is the best form to be used for assigning an error to the fit. For example, Figure A1 shows that the 80% confidence limit is around 34° when the fit is just significant at the 80% level. An error of 45° seems more appropriate. Consequently, we might consider a form such as x0 = 0.5 sin-1 (F2,15 /F)1/2, which does give a 45° error when the fit is just significant at the particular confidence level chosen. There is little difference between the two forms when the fit is moderately or highly significant. However, the main purpose of the error analysis is to allow us to compare the best fit phase angles as determined for different subsets of the data. The error is defined even when > 1 in (A7), while the alternative form is not defined for F < F2,15. As long as the larger errors (> 35°) are mainly used for comparative purposes rather than as absolute error estimates, the derivation given here is probably adequate.
Acknowledements. This work was motivated by the previous work of the late F. L. Scarf, to whom I am greatly indebted. C. T. Russell kindly provided both the magnetometer data and the Russell, von Domum, and Scarf data set. I also wish to thank him for much useful discussion and encouragement. This work was supported by NASA grant NAG2-485.
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