Single-satellite observations of lightning radio emissions normally do not independently provide useful thunderstorm location. The scientific value of these radio waveform recordings is greatly enhanced by knowing at least the approximate location of the source thunderstorm. Since the Very High Frequency radio emissions from lightning are always broadband and usually incoherent, radio interferometry is an obvious approach to direction-finding the source. However, radio-interferometry requires separated, deployed antennas, and for decametric radio waves, success with interferometric direction-finding on a satellite has not yet been reported. We describe a method for approximate location of source thunderstorms, using the statistical fading of the received electric field on two orthogonal, but co-located, antennas. The method is based on previous work geolocating polarized sources, and we show that it can be adapted efficiently to unpolarized radio fields, which constitute the majority of lightning emissions. We implement the method with dual-antenna radio recordings from the FORTE satellite, and demonstrate its capabilities using a wide variety of radio data.
 Radio interferometric direction finding relies on deployment of multiple separated antennas (at least three to cover solid angle in two dimensions). All other things being equal, the required baselines scale as the wavelength, so that an interferometer for f ∼30 MHz would be tenfold as large as one for f ∼300 MHz. We are concerned here with radio emissions in the low VHF, with decametric wavelengths. Ground-based VHF interferometers were developed for the study of lightning [see, e.g.,Rhodes et al., 1994; Shao et al., 1995, 1996; Shao and Krehbiel, 1996, and references therein] but have been largely superseded by differential-time-of-arrival lightning VHF mappers pioneered by P. Krehbiel and colleagues. A deployed interferometric antenna array is especially difficult to realize on a small, low-cost satellite, particularly in the lower VHF, due to the mechanical challenge of deploying from the satellite an antenna array that has stable and known shape and dimensions. As of early 2011,no successful space-based VHF interferometric direction-finder has been reported in the literature, for lightning mapping, radio astronomy, or any other research purpose.
 The FORTE satellite at 800–830 km altitude viewed a circular ground locus of great circle diameter 6 × 103 km, or almost the Earth radius RE. FORTE did not have separated antennas but did have a pair of co-located, mutually orthogonal antennas. VHF signals recorded by FORTE could,a priori, originate anywhere within this locus, or even slightly beyond it, due to ionospheric raybending [Jacobson and Shao, 2001; Roussel-Dupré et al., 2001]. Regarding lightning and other sources of radio-impulses, any geolocation improvement (relative to the limb circle of diameter 6 × 103 km) would be potentially valuable. In particular, it is possible to infer intracloud discharge height from the spacing between intracloud VHF pulse-pairs, if the latitude and longitude of the discharge is known even approximately [Jacobson et al., 1999]. This article describes a technique using the existing antenna arrangement on FORTE to realize, in some cases, such an improvement.
 What is the motivation to develop a VHF direction-finding capability, when the lightning-mapping product will be so rudimentary compared to optical imagers from space [e.g.,Boccippio et al., 1999, 2000a, 2000b, 2001; Boccippio, 2002; Christian et al., 1989; 2003; Goodman et al., 2000; Nesbitt et al., 2000; Nesbitt and Zipser, 2003; Petersen and Rutledge, 1998, 2001; Petersen et al., 2005; Toracinta and Zipser, 2001; Toracinta et al., 2002]? The optical imagers have 10-km (or better) transverse spatial resolution, millisecond-scale time resolution, and high detection efficiency. These imagers can supply virtually all information that meteorologists would need about the real-time, synoptic-scale patterns of lightning incidence versus horizontal coordinates. These facts prompt the question: Why even bother to develop a far less capable VHF lightning locater from space? The reason is simply that optical mappers, despite their overwhelming superiority for plan view mapping, give no information on the vertical location of the lightning discharge. Only VHF observing systems can provide that otherwise missing vertical location. From ground this is done principally with VHF lightning-mapping arrays, which provide 3-dimensional location of VHF emitters [Rison et al., 1999]. From space, a single low-Earth-orbit research satellite's recordings of VHF fields can be used to infer the emitter height provided that the emitter's horizontal location is known [Jacobson et al., 1999; Jacobson, 2003b; Smith et al., 2004]. For this purpose, the horizontal location need not be as accurate as is needed for mapping purposes. If the horizontal location is known even crudely (say, within 200 km), then that crude location allows a reasonable inference of the emission height based on the time delay between two VHF pulses. The first pulse is due to the direct path from the lightning to the satellite. The second pulse is due to a reflected path, involving a VHF specular reflection on the ground. This possibility of tracking the vertical dimension is the motivation for the present work in crude VHF direction-finding. This crude VHF direction finding is not meant to, and is not able to, supplant space-based optical lightning mappers. The sole added value of this VHF direction finding is to provide sufficient plan-view geolocation of VHF emitters that their pulse separations can be used to infer discharge height.
 There have been three related previous schemes to improve FORTE's inferred geolocation of lightning and other radio sources.
 (a) The first technique [Jacobson and Shao, 2001] was adapted for radio emitters that radiated linearly polarized waves. It was based on the fact that, to first order, the square E2of the electric field recorded using a linear-polarized antenna will undergo a distinctive amplitude modulation. The modulation is at frequency 2fcecos(β), where fceis the electron gyrofrequency at ionospheric height near the line-of-sight, and whereβ is the angle between the radio wave vector k and the geomagnetic field B, in the ionosphere. This amplitude beat is due to the dispersive Faraday rotation of the electric field going into, and out of, alignment with the linear antenna [Massey et al., 1998b]. Using this technique with FORTE, the radio wave vector k could be constrained to lie a cone of revolution around B, or more precisely, on a locus of constant fcecos(β). The cone's intersection with Earth provides the locus where the source could be located. The method requires only a linear-polarized antenna and otherwise depends only on the geomagnetic field. The method proved to be not particularly practical, in that the solution locus is neither compact nor geographically straightforward. Moreover, too little electron content, or too high a frequency, or unfortunate geometry (cos(β) too small) prevent the generation of a useful modulation beat.
 (b) The second technique [Shao and Jacobson, 2001, 2002] retrieves the polarization Stokes parameters for coherent, polarized (linear or elliptical) disturbances using FORTE's two, orthogonally polarized antennas at the same frequency. The Stokes parameters in the cross-nadir plane, in particular the ellipticity and the orientation of the major axis of the polarization ellipse, provide the two-dimensional direction of arrival. In principle this method can geolocate a source with only one recorded broadband pulse. The method requires not only simultaneous, but also synchronous (coherent) recording of the same passband on both antennas.
 (c) The third technique uses unpolarized signals, exactly the opposite of the second technique. The third technique requires the simultaneous recording of the same passband from both orthogonal antennas [Jacobson and Shao, 2002a]. This technique can roughly geolocate some repeating lightning emitters, but with ambiguity right/left of the satellite track. Moreover for the degenerate special case of a source along the satellite track, the method simply fails, as one antenna then receives essentially zero signal. Unfortunately, the technique is disadvantaged by its failure to exploit the synchronicity of recording between the two antennas. Rather, the third technique dealt only with the power on each antenna channel, not with the instantaneous electric field. This in effect served to lose half the data's information content.
 The rest of this article describes a fourth geolocation technique and demonstrates it with FORTE's orthogonal antennas, based on combining the second and third techniques described above. The new technique applies synchronous, coherent recording to incoherent, unpolarized (or, randomly polarized) signals, and retrieves significant new geolocation information. Like the second technique above, the new one uses the complex electric field, not just the power, on each antenna. Many lightning VHF signals, including the most intense, are randomly polarized, or more exactly, instantaneously linear but with the axis of polarization randomly or chaotically varying over all angles within the pulse duration [Jacobson et al., 2011]. By fully exploiting the synchronicity of data acquisition on the two antennas, this new technique achieves improved geolocation for randomly polarized signals undergoing random fading.
 We mention that an additional method was researched for geolocation using FORTE, based on fitting the variation of the retrieved total electron content (TEC) along the line-of-sight on a series of emissions from the same storm [Tierney et al., 2001]. This is not related to the new approach we introduce here, but has a great deal of utility in concert with good ionospheric-profile data from other sources. The slant TEC is the integrated electron density, integrated along the line-of-sight from the lightning source to the satellite. The slant TEC causes a characteristic dispersion in the VHF arrival time versus frequency, and this dispersion can be fitted automatically to retrieve the slant TEC parameter [Jacobson et al., 1999]. For a given ionospheric profile, the TEC tends to be greatest for the satellite near the horizon (grazing line-of-sight), and least when the satellite is at zenith (vertical line-of-sight), since the nadir angle controls the path length through the ionosphere. During daytime conditions there is more electron density present in the ionosphere, and typically the slant TEC during daytime is in the range
while in nighttime the TEC is severalfold less. The wide range in either case is due to (a) the variety of line-of-sight nadir angle and (b) to the ionospheric variability versus latitude, season, local time, solar conditions, etc [Huang and Roussel-Dupré, 2005, 2006].
2. FORTE Dual Antenna/Receiver System
 The data used here are from the broadband dual receivers on board FORTE [Jacobson et al., 1999]. Each receiver has an analog passband of about 22 MHz, tunable anywhere in the VHF. For this study both channels were fixed at the 26–48 MHz passband. The voltages from both receivers were synchronouslydigitized at 50 megasamples/s, with 12-bit digitization. The two receivers were fed, respectively, by the two orthogonal log-periodic antennas (LPAs) that were deployed on a 10-m boom directed to nadir. Details on the triggering and other aspects of the dual-receiver system have been published elsewhere [Jacobson et al., 1999, 2000; Jacobson and Shao, 2001, 2002a; Shao and Jacobson, 2001, 2002; Shao et al., 2005].
 The characteristics of the antenna system on FORTE are key to the proposed method, so we review them here. Figure 1ais an artist's rendition of the FORTE satellite with its nadir-directed boom carrying two orthogonal LPAs. The primary frequency range of these LPAs is 25–90 MHz.Figure 1bshows the satellite coordinate system. z is directed out of the diagram, anti-nadirward (away from the center of the Earth). The axes x and y lie in the plane normal to nadir. The axis x points to the right flank (perpendicular to ram), and the axis y is the ram direction. The axes x, y, z form a right-hand coordinate system. The FORTE satellite had a nearly circular (altitude was 800–830 km), 70-degree-inclination orbit. The Earth limb was at approximately 63 deg nadir angle.
 Nadir angle and azimuth are natural parameters for direction-finding a radio source in the satellite frame. We define azimuth as rotation clockwise in the xy plane from the ram direction inFigure 1b. Figure 1b also shows the Earth's limb as a circle centered on the subsatellite point.
 The two LPAs (#1, along the x direction, and #2, along the y direction) were determined to have cross-polarization rejection of >60 dB (power) [Jacobson and Shao, 2001; Massey et al., 1998b; Shao and Jacobson, 2001]. This refers to the rejection of Ex by the y antenna, and of Ey by the x antenna. Because the two signals were synchronously digitized, their digital records can be linearly combined to construct the instantaneous electric field in any direction within the xy plane.
where ψ is the azimuth and ϕis the nadir angle (in satellite coordinates). The meaning of principal angle is as follows: Define the antenna “E plane” as that plane containing both the nadir direction and the antenna conductors. Project an arbitrary three-dimensional line-of-sight (from the satellite to the RF emitter) into that plane. This we shall call the E-plane-projected line-of-sight. The angle between the E-plane-projected line-of-sight, and the nadir direction, is the principal angle [seeShao and Jacobson, 2001, Figure 2]. For a dipole antenna in free space, the antenna lobe depends only on the principal angle, and the lobe is cylindrically symmetric around the antenna element (holding principal angle constant). A free-space dipole has a null atα = ±90 deg. The FORTE LPA system's lobe has been determined on orbit, by combining recordings from thousands of sources at known location [Jacobson and Shao, 2002a]. The lobe has been shown to have no significant sensitivity to the angle in the “H” plane, that is, in the plane containing the nadir and normal to the “E” plane. The best fit E-plane antenna lobef(α) for each antenna was determined [Jacobson and Shao, 2002a] to be
(where αis in radians.) This behavior of the LPA is slightly different from a short-dipole antenna [Shao and Jacobson, 2001].
 FORTE dual-channel VHF recordings with both channels covering the passband 26–48 MHz were programmed during (a) the 1-month period 199806 and (b) the 15-month period 199810 through 199912. These sixteen months produced >2 × 106dual-channel recordings, but most of those were of faint sources not amenable to this technique. We automatically select high signal-to-noise (SNR) recordings of strong sources, on the basis of the peak received intensity divided by the median received intensity during a record. This estimate of SNR is made without time editing but with both carrier suppression and dechirping [Jacobson et al., 1999]. To limit the amount of data processing we require SNR > 50 for this study. This is SNR threshold is arbitrary, and it is likely that much lower SNRs would also work well. We are left with ∼4 × 105dual-channel recordings in the compact data set for this study, and which we have automatically analyzed (see below).
3. The Direction-Finding Concept for Randomly Polarized Radio Noise Sources
 We note that “randomly-polarized” means that, in a practical sense, the radiated electric field's polarization varies randomly, so that even though it might be instantaneously linearly polarized, the random (or chaotic) variation of the axis of polarization makes the net signal “random.” It might be more accurate then to call this “jumbled” polarization, in that the signal consists of a progression of instantaneously linear-polarized components that undergo quasi-random jumbling of their orientation. This results in effectively “random” polarization after averaging for times longer than a correlation time. This occurs most of the time in the more powerful VHF radiation from lightning, because of the simultaneous contribution by multiple, quasi-independent radiators summing together to give a noisy but strong net signal. An example of quasi-independent radiators is a collection of streamers diverging from the tip of a leader [Gallimberti, 1979]. Another example is the multiplicity of leader channels prior to a negative cloud-to-ground stroke. In each of these examples, the individual radiator is certainly linearly polarized, but the net radiation in the far field from many randomly oriented radiators is effectively randomly polarized. That is, the far-fieldnet electric field's orientation in the plane normal to the propagation vector varies randomly versus time.
 A randomly polarized electric field vector E has the property that when decomposed into two orthogonal components in the plane normal to the propagation direction, those two components will be uncorrelated. The random distribution of E will be cylindrically symmetric around the propagation direction. Contours of constant probability amplitude will be circular in the plane normal to the propagation direction. This circular distribution is foreshortened into an oblong shape in the plane of the antennas [see Shao and Jacobson, 2001, Figure 2].
 For lightning, it has been shown that there is an intimate connection between the “random polarization” thus defined, and the coherence of the signal defined as the time-bandwidth product [Jacobson and Shao, 2002b; Jacobson and Light, 2003]. The “coherent” signals thus defined include the attachment transient of a negative cloud-to-ground return stroke on seawater, whose VHF component is essentially a coherent, narrow pulse whose width seen by FORTE is limited by the inverse of the Nyquist bandwidth (∼50 nanosec). That pulse has a time-bandwidth product of ∼1. By comparison, the easiest pulse to record from space, in that it most easily competes with anthropogenic noise, is the VHF emission from CIDs [Holden et al., 1995; Jacobson, 2003a, 2003b; Jacobson and Light, 2003; Light and Jacobson, 2002; Massey and Holden, 1995; Massey et al., 1998a; Suszcynsky et al., 2000a]. These intense emissions have durations of ∼10 microsec but fading correlation times as short as the inverse bandwidth (50 nanosec) when observed with FORTE [Jacobson et al., 2011], for a time-bandwidth product of ∼200. Their polarization may be instantaneously linear but undergoes effectively random redirection at the fading rate. Therefore, for a scheme such as the present one, which averages over the full pulse duration, the CID VHF polarization is, in effect, robustly random.
Figure 2shows the direction-finding scheme for such randomly polarized electric fields. The xy plane is the nadir-normal plane discussed earlier. The random electric field's contours of constant amplitude, projected onto the xy plane, will be foreshortened into an oblate shape,with the minor axis of the oblate shape aligned with the azimuth of arrival (or that azimuth + 180 deg). This is precisely what was noted for the case of polarized wavefields [Shao and Jacobson, 2001], and it is equally true when the polarization varies randomly. In fact, with high-time-bandwidth-product signals, in which the signal duration is very large compared to the fade correlation time, the oblateness of the distributions is easier to study. Whereas the polarized-wavefield study dealt with deterministic E-fields having no randomness, the foreshortening idea applies also to the contours of a probability distribution when the field varies randomly or chaotically, provided that the averaging time is long enough to contain a large number of random amplitude variations. Although the signal vector is random, the two components in the xy plane must be synchronously recorded in order to reconstruct the instantaneous vector field and hence its statistical contours. The second method cited earlier [Jacobson and Shao, 2002a] failed to recognize this, using only the ratio of powerson the two antennas as algorithm inputs. As is the case with polarized non-random wavefields, the aspect ratio of those contours will depend only on the nadir angle. To the extent that a single recorded signal provides a statistically meaningful sample, we can estimate these parameters from a single recording. Of course, the azimuth is modulo 180 deg, so that even when a direction can be found, it is ambiguous between its value and its value +180 deg.
Figure 2 also motivates the practical procedure for determining the principal axes of the contours in the xy plane. We set up a regular grid of angles θ transforming the coordinates xy into new coordinates x′y′. The rotations are about z. This grid samples the range 0 to 180 deg counterclockwise, on a regular spacing. At each θ in this grid, we measure the width of a given contour along both x′ and y′ axes. In effect we close “calipers” around the distribution, with the closure axis determined by θ. The dashed lines in Figure 2 show the calipers touching a given contour of the amplitude distribution. These caliper fullwidths are 2Wx′ and 2Wy′ respectively. The widths are then graphed versus the rotation θ. The counterclockwise rotation θ giving the minimum width Wy′ equals minus the azimuth of arrival. (This same counterclockwise rotation θ gives the maximum width Wx′.) Once that the major axes and hence azimuth of arrival are determined in this manner, the ellipticity can provide an estimate the nadir angle. Together, the azimuth and nadir of arrival provide a geolocation, with twofold ambiguity of 180 deg in azimuth. The method depends completely on having a complex, jumbled-polarization lightning radio burst that randomly varies many times during the duration of the burst. We reiterate that this is a basic attribute of the strongest VHF emissions from thunderstorms [Jacobson and Light, 2003], namely CID emissions, so that the combination of jumbled polarization and long pulse duration required for this geolocation approach are quite well-adapted to space-based observations, in which the most intense radio emissions are favored.
 Our method of determining the arrival azimuth is not the only method; nor is it claimed to be either the most precise or the most computationally efficient. Rather, our method was adopted because it is based on a simple graphical concept of the foreshortening of the contours of constant amplitude of the electric field, in transforming from the plane normal to the line-of-sight, to the plane normal to the nadir.
4. Practical Signal Conditioning
 The ideal scheme outlined in Figure 2assumes that there are no competing signals likely to confuse the direction-finding. In practice, we need to take several steps to favor the principal lightning signal and to disfavor the competing signal. The competing signals are mainly due to anthropogenic quasi-continuous-wave narrowband carriers, such as found in communications channels. There is also a small but sometimes bothersome contribution from impulsive disturbances other than the lightning, including radars. We now show the practical steps in filtering-out the unwanted competing signals from a record containing a lightning signal.
Figure 3 shows raw spectrograms of the recorded electric field on the y antenna (Figure 3a) and the x antenna (Figure 3b). The voltages are sampled at 50 megasamples/sec. The FFT window is 128 samples long and is advanced by 8 samples per pixel step. The lightning signal is due to a CID geolocated by the National Lightning Detection Network (NLDN) [Jacobson et al., 2000]. The lightning source is near the limb of Earth (63 deg) seen from FORTE. Although the source is intracloud, the expected splitting between the direct and the ground-reflected pulse cannot be resolved in this display, due to the near-grazing incidence [Jacobson et al., 1999]. The overall dispersion is due to the electronic refractivity of the ionosphere, dominated by a 1/f2 term [Massey et al., 1998b; Roussel-Dupré et al., 2001]. The further broadening and splitting at low frequencies are due to birefringence in the geomagnetic field [Massey et al., 1998b].
 In addition to the lightning signal in Figure 3, there are strong carriers as well, for example near f = 27–28 MHz. These carriers serve to confuse the picture of geolocation presented in Figure 2. Carriers tend to be highly linear-polarized [see alsoShao and Jacobson, 2001, Plate 1; see Shao and Jacobson, 2002, Figure 2]. The present technique of direction-findingrequires unpolarized VHF radiation, so carriers are unsuitable even as test cases for validation of the method. To suppress these carriers, we perform a joint, dual-channel prewhitening as follows: The entire record (in this case, ∼410 microsec duration) is Fourier-transformed in a single coherent transform. This allows the carriers to be identified with sharp frequency resolution, in this case ∼2.5 kHz (the inverse of the record duration). Most of the carriers occupy only one or a few such frequencies, so excising them from the Fourier transform can be done with little loss of the broadband lightning energy. When viewed with this narrow frequency resolution (as opposed to the blunt resolution ofFigure 3), the carriers are orders-of-magnitude stronger than the lightning,but only within narrowly defined frequencies. We identify the top 10% of frequencies in either of the channels, in terms of spectral power. The complex Fourier components of these top-10% are then suppressed in amplitude (without changing their phase) to the 90th percentile (which is the bottom of the top-10%).This prewhitening is joint to the two channels, so that the manipulation of Fourier coefficients is identical between the two channels. This prewhitening can also be called “carrier suppression.” The choice of the top 10% of Fourier coefficients to be flattened, rather than some other percentage, was found empirically to capture most of the carriers while leaving most of the desired lightning power unaffected.
 Following the carrier suppression, we “dechirp” the signal by fitting the optimal Total Electron Content (TEC) to remove the leading term (varying as TEC/f2) in the dispersion [Jacobson et al., 1999]. This dechirping concentrates the VHF energy in a narrower portion of the record window. This temporal concentration of the desired lightning signal is next exploited to excise those portions of the record that are not related to the lightning VHF energy. We call this process “time editing.” Time editing is useful when there are impulsive man-made disturbances, such as pulsed radars, that are not addressed by the carrier suppression. After the time-editing, we restore the original gross dispersion.
Figure 4shows the progression of these data-editing steps. In each case the signal shown is the voltage time series, for the y antenna (Figure 4, left) and the x antenna (Figure 4, right). In the Figures 4a and 4b we show the raw, unedited, undechirped time series. In Figures 4c and 4d we show the time series after carrier suppression and optimal dechirping. The units in Figures 4c–4h have been rescaled to a relative scale, which retains the intercalibration of the two channels. Figures 4e and 4fshow the time-editing in which regions below a fractional threshold are zeroed. InFigures 4g and 4h we have restored the chirp (proportional to TEC/f2).
 These frequency- and time- data-editing steps, implemented automatically as the “front end” of the direction-finding algorithm, result in a modified pair of time series that are more dominated by the lightning wave train than is the raw data.Figure 5 is like Figure 3, but for the edited final time series. The carriers that had been obvious in Figure 3are now sensibly absent. The time editing, which was applied on the dechirped data, is now evident as the dispersed boundary between the green and the blue pixels. Barely visible are faint “ghost” pulses in the purple edited-out zones. These are artifacts of the Fourier manipulation in carrier suppression. Their power-density level is 5–6 orders of magnitude lower than the main signal; they have no sensible effect on the direction finding.
5. Illustration of Direction-Finding
 We are now able to subject the edited data of this example (Figures 4g, 4h, and 5) to the direction-finding “caliper” procedure motivated inFigure 2. Figure 6 shows scatter diagrams of the voltages time series for the y channel versus the x channel. One point accrues every 20 ns. The two panels show the raw data (Figure 6a, corresponding to Figures 4a and 4b) and the final edited, rechirped data (Figure 6b, corresponding to Figures 4g and 4h). In Figure 6bwe drill a hole at the origin, removing any data with voltage amplitude (in any direction) less than 5% of the maximum voltage. Removing the amplitude points below the 5% level has been found empirically to improve the solution in two related ways: First, the time editing creates transient voids in the data, and these contain no useful information; removing the amplitude points below the 5% level gets rid of these voids. Second, the electronic noise superimposed on the signal tends to have much lower amplitude than the lightning signal itself, and we can minimize the effect of that noise by removing the amplitude points below the 5% level. Our “caliper” approach does not require time-continuous data, so it is permissible to remove discrete samples according to the 5% criterion.
Figure 6b reveals a clear oblong distribution of the data, elongated along an axis roughly between the yaxis and the negative-x axis (or equivalently, between the xaxis and the negative-y axis). Note also that the xy distribution's oblong shape is clearer in the filtered data (Figure 6b) than in the raw data (Figure 6a).
 We next evaluate the percentiles of the voltage amplitudes along the rotated axes x′, y′. This is done for each gridded rotation angle, from 0 through 175 deg, in 36 steps of 5°. Figure 7 shows 1/2 the caliper widths vs gridded rotation, for the 90th, 70th, and 50th percentiles of the distributions. The solid and dashed curves show Wx′ and Wy′ respectively. The distributions in Figure 7 pass three obvious “sanity checks”: First, the fractional modulation of the caliper width, as a function of angle, is roughly consistent between the three percentiles of the distribution. Second, the location of a maximum is roughly the same between differing percentiles. Third, the variation versus angle follows a clear and smooth shape, without much superimposed random variation.
 In the rest of this article, we will focus on the 90th-percentile curves as the standard product. For each data file containing synchronously sampled time domain data for both the x and y antennas at the same passband 26–48 MHz, we generate a summary file with a fit to the 90th-percentile curves such as those inFigure 7. The fitting function is a cosine of twice the rotation angle, with three adjustable parameters: An offset independent of angle (the “DC” term), an angular center, and a cosine amplitude.
 The data treatment and xy-plane analysis just described has been automatically implemented for all ∼4 × 105dual-channel recordings that are in the passband 26–48 MHz and that meet SNR requirements (seesection 2).
6. Ground-Truthing the Direction-Finding Method
 We wish to compare the azimuth and nadir from this technique with ground-truth knowledge of the location of lightning sources. To do this we use FORTE data records for which the lightning location has been determined using coincidence with other systems, including the National Lightning Detection Network, or NLDN [Cummins et al., 1998; Jacobson et al., 2000]. There were ∼3 × 105FORTE VHF records equipped with ground-truth geolocations [Jacobson, 2003a]. These include ∼46-thousand records whose VHF pulsewidth (after dechirping) exceeds 10 microsec. These “wide” pulses are overwhelmingly likely to be randomly polarized [Jacobson and Light, 2003]. The 10-microsec criterion serves to exclude both resolved leader steps and negative cloud-to-ground attachment transients, either of which can produce highly polarized VHF. Those polarized pulses have very narrow pulsewidths, down to 50 ns [Jacobson and Shao, 2002b; Jacobson and Light, 2003], so the pulsewidth is an excellent criterion for separating randomly polarized from cleanly polarized emissions.
 Taking the diffuse VHF records, we further require that both antennas' radio receivers be at the same 26–48 MHz passband. This leaves 24,344 acceptable records for which other systems have provided geolocations. These will constitute the groundtruth data in what follows. We expect that the cases with larger nadir angles will have better-behaved azimuth retrievals, as there will be sensible foreshortening of the amplitude distribution in the xy plane. Likewise, cases with incidence from near nadir will not be as successful.
Figure 8 shows the retrieved apparent azimuth (vertical axis) versus the azimuth from external geolocation (horizontal axis), for only those 8548 records for which the geolocated source nadir exceeds 40 deg. (The true source azimuth could lie in the entire −180 to +180 domain, but has been transferred to the range 0–180 deg by adding 180 deg when its value is negative.) The overall trend is for most of the azimuth retrievals to grossly lie along an oscillatory path relative to the equality line. Most of the scatter from this oscillation is estimated by eye to lie within ±8 deg, but there are many (∼10% of total) outliers at much wider scatter from the smooth oscillation.
 The smooth oscillation in Figure 8 was first noted during retrieval of the Stokes parameters for polarized VHF signals (see Figure 7 and discussion thereof in the work by Shao and Jacobson ). We calculate that oscillation based on the E-plane and H-plane antenna responses given byequations (1) and (2). The calculated oscillation is shown as the orange curve, based on equations (1) and (2). The calculation is the same as in earlier polarization studies [Shao and Jacobson, 2001, 2002] and will be easily compensated in azimuth retrievals (see below). The oscillation does not vary much with nadir angle over the practical range we will exploit (nadir angle >40 deg).
 The data of Figure 8are selected to have ground-truth source nadir angle >40 deg. This tends to provide a well-foreshortened distribution function on the xy plane. The more foreshortening, the more robust is the azimuth determination. If the distribution is too circular, then determination of the major-axis orientation becomes non-robust.Figure 9shows the case with 12228 sources closer to the subsatellite point, with nadir angle in the range 20 to 40 deg. The core scatter is now doubled (by comparison with nadir angle >40 deg), and there are more outliers. This confirms what was already intuitively obvious, namely, that azimuth retrievals for sources closer to nadir are likely to be poor-quality compared to retrievals for sources closer to the Earth's limb. Of course, absent ground-truth information on the true nadir angle of the source, this judgment must be based instead on an observable, namely, the fractional modulation of the curves of Wx′ and Wy′ (see Figure 7).
 The groundtruthing of the nadir-angle retrieval has been quite encouraging, as evidenced byFigure 8. The groundtruthing of the fractional modulation is less encouraging, suggesting that nadir-angle retrievals will be far less trustworthy than azimuth retrievals using this method.Figure 10shows a scatterplot of groundtruth nadir angle to source (vertical axis) versus retrieved contrast ratio (horizontal axis), including all of the 24344 recording pairs in the ground-truth data set. The contrast ratio is defined as the ratio of the fitted sinusoidal amplitude to the fitted angle-independent (“DC”) term; seeFigures 2 and 7. We expect the contrast ratio ideally to be based on simple foreshortening of the plane normal to propagation after projection onto the xy plane:
 This simple model is shown as the smooth gray curve in Figure 10. At larger values of nadir angle, the data shows less contrast than predicted by model. An analogous discrepancy was noted earlier in the studies of polarized signals [Shao and Jacobson, 2001, 2002], where it was pointed out that for lightning sources near the Earth's limb, raybending in the ionosphere is expected to produce effectively steeper incidence of the wave vector onto the xy plane. We expect this effect to operate here, as well, inhibiting the contrast in the xy-plane for near-grazing incidence. It is not clear, however, that all the discrepancy evident ifFigure 10 can be ascribed to raybending.
 In view of this systematic discrepancy (whatever its cause), we conclude that practical geolocating of jumbled-polarization emitters would be best done by combining azimuth retrievals from several successive emissions, and then “triangulating” the source location. In this approach, we do not use the contrast ratio for geolocation, but only for selecting those events whose azimuth retrievals are more likely to be robust. We illustrate this in the next section.
 We caution that the triangulation concept based on intersections of Great circle segments from several points on the satellite pass can be done only for lightning sources in a subset of ranges from the satellite path. For sources too close to the satellite, the nadir angle is small, and the foreshortening is too slight to allow robust determination of the principal axes and hence azimuth of arrival. On the other hand, for a source at the limb of the Earth seen by the satellite, there is too short a period of visibility by the satellite to allow robust triangulation, even though the foreshortening is the maximum possible.
7. Examples of Triangulations With NLDN Storm Groundtruth
 As noted earlier, the locus of Earth visible to FORTE at any one instant has a ∼6-thousand km diameter. The speed of FORTE is ∼7.5 km/s, so that a source near the orbit track is visible to FORTE for ∼800 s, while a source off to the side is visible for a somewhat shorter duration. The FORTE on-board memory allowed recording and storage of >103successive dual-channel recordings each lasting 410 microsec. As the satellite moves along its orbit, a given radio source's azimuth in satellite frame migrates. For each suitable recording, we will attempt to determine the source azimuth (modulo 180 deg), then construct a Great circle arc along this azimuth, centered on the subsatellite point, and 6000 km long. It is a very rare thunderstorm that emits VHF only once; more typical is 10s to 100 s of successive emissions over tens of minutes [Jacobson et al., 1999]. Thus triangulation of thunderstorms can be attempted for most FORTE “passes” within view of a storm.
 During Summers of 1998 and 1999, we had detailed NLDN “stroke-level” data of a research grade [Jacobson et al., 2000], with extended coverage due to suspension of the 625-km maximum range in the usual NLDN product of that era [Cummins et al., 1998]. This allowed some detection (albeit with reduced detection efficiency) into areas normally not covered by NLDN, beyond the south and southeast borders and coastlines of the continental U.S. We now use this NLDN data to indicate the location of storms occurring during the FORTE pass within view of the storms. This is not a detailed event-to-event correlation as had been used inFigures 8–10, but rather just a storm geographic locator provided by contemporaneous NLDN stroke reports. The three examples to follow will show increasing levels of complexity.
Figure 11illustrates a simple triangulation for a satellite pass of several-hundred seconds starting at 11:59:15 UT on 1999 August 27.Figure 11a shows the retrieved source azimuth in the satellite frame for each event, versus the time of that event relative to the start of the display. There is a smooth evolution downward in azimuth. The black points indicate events for which the contrast ratio is <0.1 (see Figure 10), and red points have contrast >0.1. Figure 11b shows the slant Total Electron Content (TEC) for the same points, using an automated fit to the observed dispersion [Jacobson et al., 1999]. Figure 11c shows the Great circle arcs along the retrieved azimuths, plotted only for contrast ratio >0.1. Each arc extends from the satellite to the limb at ∼3000 km transverse from the satellite track. The satellite position at the instant of each event is shown as a red symbol. The green square marks the first satellite position at its first recording of this ascending pass. The Great circle arcs converge in a region west of the track.
Figure 11calso shows the contemporaneous NLDN stroke locations as blue squares. These strokes occur anytime during the actual duration of the FORTE pass, temporally bounded between the first and last FORTE records shown. This case is relatively simple, in that most of the Great circle arcs converge onto the same intense and very compact concentration of NLDN strokes. The other NLDN strokes at the extreme left are beyond the limb, while the strokes in the upper portion (above the convergence) are within the limb but are not accompanied by any FORTE-detected VHF records.
Figure 12 shares the same format as Figure 11but presents a more complicated storm triangulation. In this case the various storm clusters (marked with a blue square for each stroke) are greatly elongated approximately E-W. The Great circle arcs of most of the recordings in this pass converge onto a portion of this elongation east of the track. In this case, the storm cluster at the convergence is itself elongated, unlike the compact target storm structure atFigure 11's convergence. The double traces of the TEC in Figure 12b, between the times 100 and 300 s into the pass, suggest that the convergence splits into a closer and a further subconvergence, with (respectively) lower and higher line-of-sight TEC [Tierney et al., 2001].
Figure 13illustrates a common occurrence of there being more than one storm center to which inferred Great circle arcs converge. The retrieved satellite-frame source azimuth (Figure 13a) indicates that there are two major trends, with the second trend (starting after 200 s into the pass) eventually containing the majority of the records. The first-occurring trend, always at greater azimuth, has records occurring over the entire pass. The TEC associated with recordings from a single storm is normally expected to minimize at the point on the orbit of closest approach to the storm [Jacobson et al., 1999; Tierney et al., 2001]. With that reasoning, the slant TEC (Figure 13b) shows closest approach to the first-occurring storm center at ∼100 to 200 s into the pass, and to the second at ∼400 to 450 s. The map (Figure 13c) shows that the first-occurring convergence is onto an intense and compact cluster of NLDN strokes. South of about 20 deg N, however, the coverage by research-grade NLDN (as of 1999) attenuated sharply [Jacobson et al., 2000], and for this example there are no groundtruthing strokes from NLDN for the second convergence, which is found at ∼10 deg N.
8. Ill-Conditioned Case: Storm on the Satellite Track
 If a storm lies on or near the satellite track, then the retrieved azimuths will tend to lie along the ram or anti-ram (0 or 180 deg azimuth in satellite frame.) Moreover, the contrast ratio (seeequation (3)) will become unusably small when the satellite passes nearly over the storm. In this case triangulation (on its own) fails, so that it is not practical to geolocate the source on the basis of retrieved azimuth alone. Figure 14illustrates this situation. Many of the retrieved-azimuth points (Figure 14a) lie near 180 deg. The TEC feature associated with this along-track source is marked “A” inFigure 14b. The closest approach to the storm occurs near ∼650–700 s, when TEC is a minimum. Near this closest approach, the contrast ratio dips below 0.1 (as inferred for the black points in Figures 14a and 14b), exactly as we would expect as the nadir angle approaches zero. This source's azimuth retrievals in fact become noisy near this closest approach. The blue square on the map (Figure 14c) shows the location of a single ground-truth storm located by FORTE's own on-board optical imager, the LLS [Davis et al., 2002; Suszcynsky et al., 2000b, 2001]. This located storm coincides with the region where the retrieval contrast ratio dips below 0.1 In cases like this, of course, the combination of the TEC and the retrieved azimuth can allow the source location to be inferred, as being closest (along the track) to the minimum in the TEC-versus-time series of the pertinent group of records.
 To avoid this ill-conditioned case, we must require that the contrast ratio exceeds ∼0.1 at closest approach. Fromequation (3), this is equivalent to requiring that the minimum nadir angle exceed ∼35 deg.
 Adapting the satellite xy-plane analysis [Shao and Jacobson, 2001, 2002] to unpolarized, incoherent radio noise from lightning can approximately geolocate thunderstorms when certain conditions are met:
 (a) The satellite must record successive radio signals from the same thunderstorm over a period in which the satellite-frame azimuth to the source changes significantly (>30 deg).
 (b) The nadir angle of the storm at closest approach must exceed ∼35 deg to avoid an ill-conditioned solution.
 (c) The recorded radio signals must be randomly polarized. In a practical sense this requires that the recorded radio signals must undergo many random fades during the typical pulse duration.
 (d) The radio signals must be synchronously digitized on two mutually orthogonal, co-located antennas.
 ARJ and RHH at the University of Washington have been supported in this work by the Defense Advanced Research Agency's NIMBUS program, led by Matthew Goodman, and by the National Science Foundation through proposal 0947130, Dark Lightning, administered by Chungu Lu. X-MS at the Los Alamos National Laboratory has been supported by the United States Department of Energy. None of this work would have been possible without the diligent support of the FORTE flight-operations team, led by Diane Roussel-Dupré and Phillip Klingner.