Time domain waveform and azimuth variation of ionospherically reflected VLF/LF radio emissions from lightning

Authors


Abstract

[1] The Earth's geomagnetic field can, in principle, cause significant magnetic-azimuth variations of the lower ionosphere's reflection of 2–150 kHz radio waves emitted by lightning. There has been little published work on this azimuth variation, either modeled or observational. We use broadband emissions from negative cloud-to-ground lightning strokes to study the azimuthal variations systematically. The data are from the Los Alamos Sferic Array, operating in the United States' southern Great Plains during 2005. We compare the observations to a model of lower-ionosphere reflection of radio waves. The model recapitulates the basic features of the time domain reflection waveforms rather well, except at the lowest frequencies. The model transfer function describing the vertical electric field at the receiver is symmetric about 90° magnetic and about 270° magnetic. Two noteworthy features of the azimuth variation are both predicted by the model, and seen in the data: First, at the lowest frequencies (<30 kHz) there is enhanced reflection for eastward propagation, relative to westward propagation. Second, at the higher frequencies (>50 kHz) there is an opposite enhancement, of the reflection for westward propagation, relative to eastward propagation. The westward enhancement at >50 kHz depends sensitively on range and is most evident in nighttime conditions, while the eastward enhancement at <30 kHz occurs at all ranges studied. Range-dependent frequency modulations of the transfer function are the least for magnetic northward propagation (duplicated by magnetic southward).

1. Introduction

[2] Reflection from the ionospheric D region affects Very Low Frequency (VLF; 3–30 kHz) and Low Frequency (LF; 30–300 kHz) propagation between separated points on Earth's surface or within the Earth's atmosphere. At short ranges (<1000 km), the received signals comprise both ground wave and ionospheric reflection components, with the latter able to be modeled by discrete 1 and 2 hop contributions [Hu and Cummer, 2006; Jacobson et al., 2009, 2010; Nagano et al., 1975, 2003; Piggott et al., 1965; Pitteway, 1965]. For longer ranges the ground wave contribution becomes insignificant, while the propagation is most conveniently described by waveguide modes in the Earth-ionosphere waveguide [Cheng and Cummer, 2005; Cheng et al., 2006, 2007; Cummer, 1997; Cummer et al., 1998; McRae and Thomson, 2000; Thomson, 1993; Thomson and Clilverd, 2001; Thomson et al., 2004; Thomson and Rodger, 2005; Thomson et al., 2007]. All the cited waveguide applications exploit the Long Wave Propagation Code developed by the United States Navy [Morfitt and Shellman, 1976; Pappert and Ferguson, 1986; Pappert and Hitney, 1988].

[3] The D region, apart from lying below the “E region” which in turn lies below the “Fregion,” is uniquely the ionospheric region within which electron-neutral collisions profoundly affect the dielectric matrix. The collision rate is not so great as to remove magnetic anisotropy, but is large enough to create exceedingly complex and dissipative VLF wave propagation. During nighttime, there is only a very dilute remnant of theD region [Friedrich and Rapp, 2009], located higher in altitude than during daytime, and with lower density than during daytime [Cheng et al., 2006; McRae and Thomson, 2000; Thomson, 1993; Thomson et al., 2007].

[4] Remarkably few rigorous results have been published about the propagation magnetic azimuth's effect on VLF/LF propagation, either based on model predictions or data observation. Azimuth effects have both been briefly noted in passing, in the original discrete reflection model [see Piggott et al., 1965, Figure 4], and have been empirically observed, albeit with marginal statistical significance, in short-range lightning-generated VLF/LF data [Jacobson et al., 2007]. In order to explore azimuth controls over ionospheric reflection, the present work describes a systematic comparison of modeled reflection waveforms with VLF/LF recordings of signals from cloud-to-ground lightning strokes at short range (200 < 600 km), where the discrete reflection model provides insight and relative simplicity.

2. Scheme for Calculating D Layer Reflection

[5] In the model calculation, the Dregion electron-density profile is assumed to be exponential in the vertical coordinate z,

display math

with fixed reference density n0 = 3 × 108 m−3. Both the reference height z0and the logarithmic derivative q are varied. The electron-neutral collision rate is also exponential

display math

with its parameters fixed: ν0 = 5 × 106 s−1, h = 70 km, and p = 0.15 km−1, which are widely accepted constants [e.g., Thomson, 1993]. The notation [Volland, 1995] in equation (1a) is slightly simplified from that of some other authors [see, e.g., Thomson, 1993] who preserve the original format [Wait and Spies, 1964]. The geomagnetic field's dip angle is taken as 59°, and electron gyrofrequency fceas 1300 kHz, which are appropriate for the southeast/south-central part of the United States.

[6] The Dlayer radio reflection is calculated using a linear, frequency domain full-wave solution to Maxwell's Equations [Jacobson et al., 2009, 2010; Shao and Jacobson, 2009], with the anisotropic, collisional dielectric matrix appropriate to the lower ionosphere [Pitteway, 1965]. Our approach is linear and spectral, in contrast to the Finite difference Time domain approach [Hu and Cummer, 2006]. Our approach models only discrete ionospheric reflections. It does not (and cannot) describe waveguide modes in the Earth-ionosphere waveguide. Waveguide-mode calculations have been applied to longer-range propagation elsewhere [e.g.,Cheng and Cummer, 2005; Cheng et al., 2006, 2007; Cummer et al., 1998; McRae and Thomson, 2000; Thomson, 1993; Thomson and Clilverd, 2001; Thomson et al., 2004; Thomson and Rodger, 2005; Thomson et al., 2007] using the legacy code developed by the U. S. Navy [Morfitt and Shellman, 1976]. Our approach is for shorter ranges (up to several hundred km), for which the sky wave signal can be most straightforwardly synthesized by calculating discrete reflections from the ionosphere. The waveguide approach is not appropriate at these short ranges. Our linear, spectral reflection model has similar capabilities to the model of Nagano and coworkers [Nagano et al., 1975, 2003], but uses a different numerical approach modeled after Pitteway's [Piggott et al., 1965; Pitteway, 1965]. Our model makes approximations [Jacobson et al., 2009, 2010] which are invalid at the lowest frequency (2 kHz), and we treat our results with skepticism for the range f < 5 kHz. The approximations involve treating the wave headed for the ionospheric reflection as entirely separable from the “ground wave.” We approximate that the ionospheric reflection can be calculated as if the conducting ground were not present, and that the ground wave can be calculated as if the conducting ionosphere were not present. As shown elsewhere, this approximation loses validity at either the longest ranges or the lowest frequencies [see Jacobson et al., 2010, equation 2].

[7] The plane wave solutions for the vertical electric field are calculated on a grid stepped by 2 kHz in frequency and by 0.25° in angle-of-incidence [Jacobson et al., 2009]. The field's amplitude and phase are then smoothed, the phase is cleaned of “2π phase wraps,” and finally both smoothed phase and smoothed amplitude are interpolated onto a finer grid: 0.1 kHz in frequency and 0.025° in angle of incidence. These final plane wave gridded solutions are archived for a matrix of ionospheric parameters for use in equation (1a) (z0 = 67, 70, 73, 76, 79, 81, 83, 85, 87, 89, and 92 km; by q = 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50 km−1, and for eight cardinal magnetic azimuths: 0, 45, 90,135, 180, 225, 270, and 315°. The archive of plane wave solutions can be used for any choice of ground range and source height, and should suffice for any case where the D layer profile (exponential electron density and exponential collision rate) is applicable and where a discrete reflection suffices. These details are summarized in Table 1.

Table 1. Archive of Plane Wave Solutions
Parameter NameParameter Values
Geomagnetic field constantsfce = 1300 kHz, dip angle = 59°
Dregion electron-density formne = n0 exp(q(z − z0))
Dregion electron-density constant n03 × 108 m−3 (fixed)
Dregion electron-density parameter q0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50 km−1
Dregion electron-density parameter z067, 70, 73, 76, 79, 81, 83, 85, 87, 89, 92 km
Dregion electron-collision-rate formνen = ν0 exp(−p(z − h))
Dregion electron-collision-rate constantsν0 = 5 × 106 s−1, h = 70 km, p = 0.15 km−1
Frequency grid for calculations2 to 160 kHz, in 2 kHz steps
Angle-of-incidence grid for calculations1.0 to 89.0, in 0.25-° steps
Number of calculated plane waves∼2.5 million
Number of archived plane waves, after smoothing, phase-unwrapping∼497 million

[8] For the present work we have simplified our previous method of solution. That previous method summed over plane waves of all angle-of-incidence, to naturally home in on the the quasi-specular angle [Jacobson et al., 2009] and also provide normalization of the sky wave compared to the direct wave. The quasi-specular angle is the angle of stationary phase. This summation was performed to include the effects of diffraction due to a finite angular Fresnel zone. For the current work, we have simplified the method as follows: For each frequency on the interpolated grid from 2 kHz through 160 kHz in steps of 0.1 kHz, we calculate the phase at the receiver as a function of the plane wave interpolated angle-of-incidence. We tabulate the angle of stationary phase for each frequencyf, and use (for that frequency) only the plane wave corresponding to that angle-of-incidence in the solution for that frequency. When all frequencies have been done in this manner, we calculate the derivative of the phase at the receiver, with respect to frequency, and this gives the group delay t as a function of frequency.

[9] Figure 1 shows the calculated ionospherically reflected vertical electric field at the receiver, versus angle of incidence of plane waves. The particular example is for ground range = 300 km and magnetic azimuth = 90°. For the parameters of equation (1a) we use z0 = 89 km and q = 0.50 km−1. Solutions at two frequencies are shown: Figure 1a shows 12 kHz, and Figure 1bshows 44 kHz. The real and imaginary parts are shown as solid and dashed curves, respectively. In addition, the phase of the electric field is shown as a faint parabola-like curve, with its constant offset adjusted to lie at the bottom of the ordinate scale. The bottom of each phase curve is the stationary point, corresponding to the specular angle-of-incidence.

Figure 1.

Vertical component of model ionospherically reflected electric field complex Fourier coefficient (solid: real part, dashed: imaginary part), versus angle of incidence, for (a) 12 kHz and (b) 44 kHz. The propagation azimuth is 90° magnetic. The model uses profile parameters (in equation (1a)) reference height z0 = 89 km and logarithmic derivative q = 0.50 km−1. The source-to-receiver range is 300 km, and the transmitter is at ground level. The thin parabolic curve is the phase versus angle of incidence, offset and rescaled to fit between −1 and +1. The total phase excursion (edge to bottom) is 43 rad for 12 kHz in Figure 1a and 152 rad for 44 kHz in Figure 1b.

[10] Figure 2ashows the quasi-specular angle-of-incidence, andFigure 2b shows the group range cτ, each as a function of frequency, for the same parameters as used in Figure 1. Both the quasi-specular angle of incidence and the group range are dispersive.

Figure 2.

(a) Model quasi-specular (i.e., stationary phase) angle of incidence versus frequency and (b) group range versus frequency for the same parameters asFigure 1 above.

[11] Spherical waves diverge, so that the amplitude varies as the inverse of path distance. We therefore adjust the amplitude of the sky wave by multiplying the ionospherically reflected amplitude at the receiver by a spherical correction factor which depends on frequency

display math

where c is the speed of light, and ground range is the Great circle distance from the lightning to the receiver. We express the result as the spectral complex transfer function of the vertical electric field [Jacobson et al., 2009], giving the complex ratio of the ionospherically reflected Ez(f) at the receiver to the direct wave's Ez(f) at the receiver.

3. Model Behavior Versus Propagation Path and D Layer Parameters

[12] The model calculation described above provides a complete toolkit for studying the variation of D layer VLF/LF radio reflection. The effect of range has been adequately described previously [Jacobson et al., 2007, 2008, 2009, 2010; Shao and Jacobson, 2009]. Here we emphasize the effect of the magnetic azimuth of propagation (from the lightning to the receiver) on the model solutions.

[13] To illustrate the azimuth effects, we study the model results for a nighttime case, z0 = 89 km and q = 0.50 km−1. Figure 3 shows the amplitude of the complex spectral transfer function giving Ez of the ionospheric reflection at the receiver, for (Figure 3a) ground range = 300 km, and (Figure 3b) ground range = 600 km. The transfer function is constructed using the plane wave archive as well as the choice of ground range, source height, and receiver height [Jacobson et al., 2009]. In our present study, the source and receiver heights are both zero. Note that Figure 3a is precisely the case treated in Figures 12. Magnetic azimuth in Figure 3 is coded by color.

Figure 3.

Magnitude of the complex Fourier transfer function for Ez versus frequency for (a) 300 km range and (b) 600 km range. The model uses profile parameters (in equation (1a)) reference height z0 = 89 km, and logarithmic derivative q = 0.50 km−1. Color identifies each cardinal magnetic azimuth. Since the transfer function is symmetric about 90° and is also symmetric about 270°, the 135° duplicates 45°, 180° duplicates 0°, and 315° duplicates 225°.

[14] There are three significant features of Figure 3: First, we note that magnetic azimuths 45 and 135° have the same transfer function, as do 0 and 180°, and as do 225 and 315°. Second, the transfer function for azimuth 0° (same as 180°) is relatively unmodulated versus frequency, while all the other cardinal azimuths show marked modulation versus frequency. The peaks and nulls of the modulation migrate in frequency as range is changed (comparing Figure 3a with Figure 3b). Third, the lowest-frequency (<20 kHz) amplitude is greatest for propagation toward the East (45, 90, and 135°) and lowest for propagation toward the West (225, 270, and 315°).

[15] We can take the spectral transfer function and model the time domain reflection, using a contrived input waveform, as shown in Figure 4, which treats the same cases as treated in Figure 3 above. The contrived waveform is just a convenience for producing a temporally narrow input, so that the ionospheric reflection can be clearly distinguished from the ground wave. The propagation magnetic azimuth is coded by color. The ionospheric reflection lags the ground wave more for short range (300 km) than for long range (600 km), due to the difference in angles of incidence. Figure 5shows expanded views of the reflected component, in time windows that are limited to exclude the ground wave. The “simplest” waveforms, that is, the least modulated, are for 0 (and 180)°. Those are, unsurprisingly, also the cardinal azimuths with the least frequency modulation of transfer-function amplitude (seeFigure 3above). The most time-modulated waveforms are for propagation toward the W (cardinal azimuths 225, 270, and 315°). The modulation is more dramatic at 300 km than at 600 km range.

Figure 4.

Time domain waveform for a contrived groundwave at (a) 300 km range and (b) 600 km range. The model uses profile parameters (in equation (1a)) reference height z0 = 89 km and logarithmic derivative q = 0.50 km−1. Color identifies each cardinal magnetic azimuth.

Figure 5.

Detail on the portion of Figure 4 containing the ionospheric reflection at (a) 300 km range and (b) 600 km range.

[16] We have explored the symmetry of the transfer function in detail, and have found that the entire complex transfer function for Ez is symmetric about 90°, and is also symmetric about 270°, for any given frequency. This is not true of horizontal components of E, but that is not of practical interest, because all that matters for ground-based stroke sources (e.g., cloud-to-ground strokes) and ground-based receivers is the vertical component, Ez [Jacobson et al., 2009]. We find that both the phase and amplitude of the transfer function for Ez, for any azimuth, can be accurately fitted with functions of azimuth of the form

display math

which has three fitting constants. These three constants must be fit separately for amplitude and phase, and separately at each frequency, using the archived solutions at the cardinal azimuths as input data for the fit.

[17] We perform the fit using equation (3) in the spectral domain. Figure 6 shows in color the interpolated amplitude of the transfer function at all azimuths, not just the cardinal azimuths, for (Figures 6a and 6b) range = 300 km, and (Figures 6c and 6d) range = 600 km. The horizontal axis is propagation azimuth, while the vertical axis is frequency. The left column (Figures 6a and 6c) is for nighttime conditions (coinciding with previous figures), while the right column (Figures 6b and 6d) is for midday conditions (z0 = 73 km, q = 0.30 km−1.) The low-frequency enhancement for eastward propagation is the red dome at the lower left of each quadrant ofFigure 6. The modulation versus frequency is much greater for nighttime conditions (Figures 6a and 6c) than for midday conditions (Figures 6b and 6d). The frequency modulation is strongest for westward propagation, and weakest for northward (and southward) propagation.

Figure 6.

Magnitude of complex transfer function versus magnetic azimuth (horizontal axis) and frequency (vertical axis). (a, c) Nighttime conditions (z0 = 89 km, q = 0.50 km−1). (b, d) Daytime conditions (z0 = 73 km, and q = 0.30 km−1. Figures 6a and 6b are for 300 km range, and Figures 6c and 6d are for 600 km range.

4. Model Versus Data Comparisons for Individual Strokes

[18] We are now ready to compare model against data for individual strokes. The data are from cloud-to-ground (CG) strokes gathered by the Los Alamos Sferic Array (LASA) during the period May–August 2005, in the Southern Great Plains of the United States [Shao et al., 2006; Smith et al., 2002; Wiens et al., 2008]. The array details are summarized in Table 2. The LASA Great Plains array [Shao et al., 2006; Wiens et al., 2008] was field calibrated by a portable flat-plate transfer-standard antenna, and then checked against concurrent National Lightning Detection Network [Cummins et al., 1998; Cummins and Murphy, 2009] inferrred electric field from NLDN's estimates of the stroke current. Since the current varies linearly as the radiated electric field in the model used by NLDN, the current reported by NLDN can be used to estimate the peak electric field referred to a reference distance. The LASA trigger in the 2005 May–August campaign occurred whenever the electric field amplitude exceeded a fixed level, regardless of algebraic sign of the electric field. In other words, the trigger was a rising-edge amplitude threshold.

Table 2. LASA Southern Great Plains Stations, 2005
SiteSymbolLatitude (°N)Longitude (°E)
Garden City, KSGCK37.971−100.850
Greely, COGRL40.447−104.638
Los Alamos, NMLAM35.871−106.327
Lubbock, TXLBB33.586−101.876
Norman, OKOUN35.211−97.440
Lincoln, NELNK40.820−96.704

[19] We limit the CGs to those lying within 600 km of the nominal array center (Garden City, Kansas), and having estimated peak-current amplitude >50 kA, using the standard approach based on the most commonly accepted radiation model for the stroke and used by, e.g., the National Lightning Detection Network [Cummins and Murphy, 2009]. In practice, we limit the study to negative CG strokes, due to their generally shorter duration than positive CGs. The shorter duration is convenient for separating the ionospheric reflection from at least the first half-cycle of the ground wave signal. For the data used here, LASA sampled the vertical electric field at 1 sample/microsec, with an antialiasing hardware 300 kHz low-pass filter. At the lowest frequencies, the preamplifier is AC coupled with an RC ∼ 1 millisec high-pass filter, corresponding to ∼150 Hz. The trigger is furnished by an amplitude threshold comparator [Smith et al., 2002].

[20] Our early work on ionospheric reflection was with Narrow Bipolar Events [Jacobson et al., 2007, 2008, 2009, 2010] rather than with CG strokes. The Narrow Bipolar Event (NBE) was originally observed a third of a century ago [Le Vine, 1980], and was called a “short duration bipolar pulse.” Later researchers studied the same phenomenon but called it a “narrow bipolar pulse” [Willett et al., 1989]. More recently the name has evolved into “Narrow Bipolar Event” [Rison et al., 1999], although the same phenomenon is also called a “compact intracloud discharge” [Nag et al., 2010]. The Narrow Bipolar Event has the virtue of having a narrow source waveform, entirely occurring within a 50 microsec (or shorter) window, assuring (for ranges < 800 km) straightforward identification of the ionospheric reflection, which arrives entirely after the ground wave has passed the station. NBEs have the disadvantage, however, of being relatively few; their incidence has been estimated to be on the order of 1% of the overall stroke population, though that figure is highly variable [Smith et al., 1999, 2002; Suszcynsky and Heavner, 2003; Wiens et al., 2008]. Our choice in the current work to use negative CGs, rather than NBEs, is due to the CGs' being ∼100X more numerous. This high occurrence rate of CGs allows statistical averaging over multiple waveforms, as described in section 5 below. Such averaging over multiple waveforms would not usually be possible for NBEs within a statistically stationary duration of local time.

[21] We employ our model to predict the ionospheric reflections for both the first and second hops. (The second hop is implemented by additionally applying the transfer function twice, but with one half of the range.) The assumed input waveform is the first half-period of the recorded -CG pulse, followed by zeroes. The first half-cycle occurs before the ionospheric reflection's arrival, except for the longest ranges (>600 km). This simple model is serviceable here, losing realism only for the lower frequencies of our simulation, f < 5 kHz. The simulation does not extend below 2 kHz [Jacobson et al., 2009, 2010]. In order to use it as the source waveform, we compensate the first half-cycle for the Earth-curvature effects [Wait and Spies, 1964], using a frequency domain numerical implementation [Shao and Jacobson, 2009]. This compensation restores the source waveform that would propagate directly to the receiver on a straight path if there were no intervening curved-Earth conductor. It is not practical to straightforwardly estimate the source waveform from the ground wave's first half cycle for ranges >600 km, because of the encroachment of the reflected signal so that it partially overlaps even the first half cycle of the ground wave.

[22] The comparison between recorded data from LASA and predictions of our model will have significant limitations, based mainly on the difference between the recording bandwidth and the model bandwidth. The recording lower frequency limit is imposed by a ∼1 millisec RC filter prior to digitization, corresponding to a one pole high-pass filter at ∼0.15 kHz. The recording durations used here are typically ∼1.5 millisec or shorter. The “ground wave” first half-cycle is a negative pulse with width on the order of 0.05–0.2 millisec. This first half-cycle pulse, and it alone, constitutes our estimate of the “ground wave.” Our estimated ground wave does not include any coda extending beyond the first half-cycle. In the actual ground wave, there must be such a coda in order for the time integral of the groundwave to approach zero, as required since the receiver is in the radiation far field. The physical ground wave has opposite polarity from the initial half-cycle of the waveform. Our artificial truncation of the coda is necessitated by our inability to separate the actual ground wave's long coda from the ionospheric component in the recorded data. As long as one is not concerned about the accuracy of the model at the lowest frequencies, then it makes no practical difference whether or not the model input waveform contains the long compensating coda. We do not even calculate a model response below 2 kHz, so the consequences of omitting the long coda (in the model input) are not significant.

[23] The estimated ground wave, truncated in this way, is fed into the model to estimate a corresponding ionospheric reflection. The model of the reflection is increasingly unrealistic as frequencies lie below 5 kHz down to the calculation limit at 2 kHz [Jacobson et al., 2009, 2010].

[24] We illustrate the variation in azimuth behavior using simultaneous recordings on two stations, LBB and OUN, which have similar ranges (278 and 285 km) but have different propagation magnetic azimuths (westward and north/northeastward.) In Figure 7 we examine the westward reflection to station LBB in detail. In Figure 7a we show the recorded data as the heavy black curve, and the estimated source waveform in the thin black curve. The colors identify model curves for four nonduplicative cardinal azimuths, for nighttime conditions given by z0 = 92 km and q = 0.45 km−1. The azimuth closest to the actual azimuth is 270°, shown in red. Figure 7bshows the model at only two of the cardinal azimuths, 0° (blue) and 270° (red), as well as the data (in heavy black) and the estimated source waveform (in faint black). The northward (0°) model shows very little short-period ringing compared to the westward (270°) model in the times 350 to 400 microsec. The data's basic features are similar to those of the westward model.

Figure 7.

Ez versus relative time, for a single stroke's signal recorded at station LBB, with 278.4° magnetic azimuth (approximately magnetic westward) and 278 km range. Heavy black curve: recorded data. Light black curve: corrected source waveform. (a) Data, corrected source waveform, and model (with azimuth identified by color) for four nonduplicative cardinal magnetic azimuths. (b) Zoom view of the ionospheric reflections only, showing the modeled reflections for magnetic northward (blue) and westward (red) propagation.

[25] In the main “ground wave” peak of Figure 7a, the model curves all extend down to −2.7 V/m, while the recorded data (black curve) attains only −2.3 V/m. The difference is due to the Earth-curvature correction [Shao and Jacobson, 2009; Wait and Spies, 1964]. Compensating for Earth curvature boosts the estimate of the radiated waveform relative to the “ground wave” recording. The compensation has been applied to the estimated source waveform (thin black curve).

[26] For the same stroke as in Figure 7, we now compare model versus data in Figure 8 for the N/NE propagating reflection, which was recorded at station OUN. (The format is the same as in Figure 7.) With N/NE propagation, the data lacks the higher-frequency ringing of westward propagation, and is best matched (from among the cardinal azimuths) by the model for northward propagation (blue curve). The westward propagating model (red curve) has pronounced higher-frequency ringing that is lacking in the data for N/NE propagation.

Figure 8.

Similar to Figure 7 but for the same stroke's signal recorded at station OUN, with 24.6° magnetic azimuth (approximately magnetic northward) and 285 km range.

[27] The success of the westward model in matching the data in Figure 7, and of the northward model in matching the data in Figure 8, is only partial. The main residuals between data and models occur in the lower-frequency (<5 kHz) end of the spectrum, where the model's simplifying assumptions are unrealistic [Jacobson et al., 2009, 2010].

[28] A word of explanation is needed about the choice of appropriate nighttime ionospheric parameters: Holding the reference electron density constant at 3 × 108 m−3, the exponential D region profile has only two independent parameters [Jacobson et al., 2010]. We try different combinations of the reference height (z0) and the logarithmic derivative steepness parameter (q) to get the best agreement between the data and the model prediction. Typically this provides a very sensitive and reliable determination of z0, but a less sensitive and less precise determination of q. This difference in the method's sensitivity to the two parameters was previously noted for wideband D region studies by the Duke University group [e.g., Hu and Cummer, 2006]. The reference height inferred in this manner for nighttime is always in the range 87–92 km, while the logarithmic derivative can vary from q 0.40 to 0.50 km−1. Our wideband method is not really able to infer q much more precisely than that. However, the reference height can be discriminated with ∼1 km precision.

5. Model Versus Data Comparisons for Multistroke Averages

[29] Individual CG sferics contain complex features that do not correlate well between different azimuths, for example due to channel branching and tortuosity [see, e.g., Willett et al., 2008, and references therein]. These features confuse the coordinated comparison of model versus data on multistation recordings on an individual stroke. Further confusion is due to electronic noise received or generated at each station, unrelated to, but competing with, the lightning signal. To avoid that confusion, we introduce multistroke averaging, in order to smooth out the complexities of the CG sferic, as well as the electronic noise, yet to retain a valuable testbed for comparison against the model, at least in the VLF spectral range. Averaging over many CG waveforms has been already been demonstrated with the same 2005 LASA data, in an empirical study of D region perturbations [Lay and Shao, 2011].

[30] We do not require the waveforms within an average to be of one type, namely “initial” or “subsequent” strokes, just so long as they are negative cloud-to-ground strokes. The point of the average is only to intentionally wash out the peculiar kinks and related details of the ground wave on any given stroke received at any given station, and concentrate instead only on the main feature, which is a reasonably compact (0.05–0.2 millisec) unipolar pulse. The point of the average isnot to realistically represent a particular stroke archetype, such as an initial stroke versus a subsequent stroke. In fact, for our purposes it is harmless to include both initial and subsequent strokes in the same average, as long as the individual stroke polarities are all negative. The model of the sky wave is a linear transformation of the input ground wave. The ionospheric reflection is completely indifferent to the pedigree of the stroke (e.g., “initial” versus “subsequent stroke”). Since the model of the ionospheric reflection is linear with regard to the ground wave “input,” the model of the sum of waveforms is the same as the sum of the models of the individual waveforms. In the present work we are not concerned with the properties of lightning strokes, but are using lightning strokes as powerful radio transmitters. Our model treats the complex Fourier spectrum of an input waveform (model input) and predicts the complex Fourier spectrum of the subsequent sky wave. For the purposes of this study, we are able to use both initial strokes and subsequent strokes in the same average.

[31] We use 5 min epochs for averaging. The 5 min period is short enough to allow us to assume stationary D region conditions, except during strong solar flares [Thomson and Clilverd, 2001; Thomson and Rodger, 2005]. During each 5 min epoch, we search for clusters of lightning, as follows: We place each lightning at the origin of the x(Eastward) versus y(Northward) plane, and graph the positions of the recording stations (of which there are up to six) for that stroke. Each cluster is associated with a storm cell. We define a cell as lying within a 20 km × 20 km pixel. This is a convenience based on searching for contiguous VLF emmitters, and is not meant to imply that all storms are contained within 20 km dimension. The most populous pixel is chosen, the waveforms within that pixel are averaged over all the strokes, and then this pixel's strokes are removed from treatment by the next iterations. We then iterate and find the most populous of the remaining pixels, etc etc. We terminate the iterations when there is no pixel remaining with at least 20 remaining strokes. Each accepted pixel's accumulated waveform is called an “average recording,” and is labeled by the recording station, the propagation range, the propagation azimuth, and the epoch time. Table 3 summarizes the average waveforms.

Table 3. Average Waveforms
Parameter NameParameter Values
Averaging Epoch5 min
CampaignMay–August 2005
Maximum range for waveform averaging600 km
Pixel size for waveform averaging20 km × 20 km
Minimum pixel census for average20 strokes
Number of 5 min epochs with lightning5280 epochs
Number of 5 min epochs in 9–15 CST (noon)843 epochs
Number of 5 min epochs in 22–02 CST (midnight)1033 epochs
Number of averages in 9–15 CST (noon)4270 averages
Number of averages in 22–02 CST (midnight)5530 averages

[32] Altogether we have 5280 5 min epochs containing CG strokes during the 4 month period May–August 2005. Let us focus on the periods of the day when ionospheric D layer conditions may be expected to be “stationary” with respect to a diurnal cycle [McRae and Thomson, 2000; Smith et al., 2002, 2004; Thomson, 1993; Thomson et al., 2007], namely local noon and local midnight. For noon we choose all archived 5 min epochs lying within ±3 h relative to zone local (CST) 12 h. This yields 843 5 min epochs. For midnight we choose ±2 h relative to zone local 0 h, yielding 1033 5 min epochs. Each epoch can contain several 20 km × 20 km averages, for two reasons: First, each storm is recorded by at least four stations. Second, there can be (and usually is) more than one storm present during any one 5 min epoch. This means that the epoch file can contain average recordings from different storms, possibly with systematically different waveforms. For example, station OUN can be involved with storms in two different locations, and the averaged waveforms for these two different locations can differ in even their first half-cycles.

[33] We have a total of 4270 average recordings for local noon, and 5530 average recordings for local midnight. The propagation vectors' terminii for these recordings are shown in Figure 9, in which the lightning location is at the origin, and the recording station's location relative to the stroke is shown as a black symbol. Each of the average recordings is an average over at least 20 stroke recordings. If there are fewer than 20 strokes in an epoch/pixel, we do not allow an average.

Figure 9.

Maps of recording stations relative to lightning location, for storms used for accumulating average waveforms. Each point is therefore the propagation vector from the lightning to the recording station. Horizontal axis: eastward component of propagation vector. Vertical axis: northward component. Shown are (a) 4270 averages during zone local midday ±3 h and (b) 5530 averages during zone local midnight ±2 h. Each average comprises at least 20 strokes within a 20 km × 20 km pixel.

[34] A nighttime example of averaged sferics from this procedure is shown in Figure 10. Four different range/azimuth clusters are found in this 5 min epoch beginning at 05:20 UT on 6 July 2005. Each cluster is labeled with it magnetic azimuth, the ground range, and the number of individual recordings being averaged. We stress that there is no requirement for the storm recorded at one station to be the same as the storm recorded at another station. The storm seen from LAM (Figure 10d) is not the same storm as seen from the other three stations. The red curve is the model solution for z0 = 89 km and q = 0.45 km−1. For each propagation path, the model solution is interpolated to that path's actual azimuth. Figure 10shows the data in heavy black curves, and the model in heavy red curves. The estimated source waves, estimated separately for each cluster, are shown as faint black curves. Each estimated source wave is clamped to zero after the first half-cycle of the waveform. The comparison of data to model inFigure 10 shows partial agreement on most (though not all) of the features. Clearly the averaging has simplified the comparison by suppressing complicated details of individual strokes. In general, the ionospheric reflection relative to the ground wave increases with increasing range, until the last panel (azmag = 271°), where the weak VLF reflectivity for westward propagation suppresses the ionospheric signal. The 61 record average in Figure 10dwashes out the details of any high-frequency response in the data, so the oscillations seen in the modeled ionospheric reflection are not fully matched by any feature in the data. InFigure 10c, for eastward propagation, there is a clear second-hop signature that is partially matched by the model.

Figure 10.

Waveforms for four averages during the 5 min epoch beginning at 05:20 UT on 6 July 2005. The recordings at (d) station LAM come from a different spatial pixel from that recorded by (a) LBB, (b) GCK, and (c) OUN. Heavy black curve: recorded data. Light black curve: corrected source waveform. Red curve: model interpolated to exact azimuth of propagation vector, for nighttime conditions (z0 = 89 km, q = 0.45 km−1). Each panel has its own time offset.

[35] The fall off of model accuracy below 5 kHz, and the complete lack of model response below 2 kHz, lead to the model's inability to reproduce the single half-cycle character of the reflected wave at shortest range, e.g., range = 200 km inFigure 10a. In this case the estimated ground wave is a unipolar pulse, which would require a model all the way down to “DC” to reproduce. The model's lack of “DC” response leads to the reflection being more bipolar in the model than in the data.

[36] Figure 11 shows another example of data/model comparisons with a set of four contemporaneous nighttime averages, from the five minutes beginning at 05:40 UT on 6 July 2005. As in Figure 10, the reflectivity tends to increase with increasing range, except for the longest range, where the VLF reflectivity is suppressed due to the westward propagation azimuth. The most evident residuals between data and model are related to the model's skill at the lowest frequencies (<5 kHz). This is most apparent in Figure 11a, where the recorded ionospheric reflection is basically a unipolar pulse, while the modeled reflection is necessarily bipolar, due to the lack of any “DC” response in the model.

Figure 11.

Similar to Figure 10 but for 20 min later.

[37] Figure 12 shows a pair of simultaneous averages, with ranges differing by only 2%. This example shows (Figure 12a) almost eastward and (Figure 12b) almost westward propagation. The twofold lower VLF reflectivity in Figure 12b is due in part to westward propagation. The two panels are for different storms occurring contemporaneously during the same epoch.

Figure 12.

Comparison between averages for (a) eastward and (b) westward propagation, on similar ranges, during the 5 min epoch beginning at 05:30 UT on 7 July 2005. The two averages' signals come from different spatial pixels.

[38] Another example is shown, for a similar pair of averages, in Figure 13. As in Figure 12, the two panels in Figure 13 are for different storms. This example also shows (Figure 13a) almost eastward and (Figure 13b) almost westward propagation, with similar ranges. We see the high-frequency enhancement in all single-stroke (unaveraged) examples we've examined of westward propagation. However, the effect washes out upon multistroke averaging, due to the ability of even small propagation differences to change the group delay by a radian period 1/w, thus scrambling the phase between successive strokes.

Figure 13.

Similar to Figure 12 but for the 5 min epoch beginning at 06:10 UT on 31 May 2005. The two averages' signals come from different spatial pixels.

6. Conclusions and Discussion

[39] The discrete reflection model, using an exponential background Dregion profile, partially matches the basic features of recorded VLF/LF waveforms from -CG lightning strokes at short range (<600 km). This comparison is done assuming that the first half-cycle of the recorded waveform is primarily due to the ground wave.

[40] The model's most obvious shortcoming is its poor agreement with data at the lowest frequencies (<5 kHz). This is not altogether surprising, as the lowest frequencies are where our model approximations lose validity [Jacobson et al., 2009, 2010].

[41] The model predicts a strong asymmetry in the reflection for frequencies <30 kHz, with stronger D region reflectivity for eastward, and weaker reflectivity for westward propagation. This is true for all ranges.

[42] The model also predicts, however, an enhanced reflectivity for westward propagation in certain higher-frequency bands (f > 50 kHz, with the precise frequency depending on range), leading to a quasi-oscillatory ringing waveform.

[43] Both of these azimuth-dependent model features are confirmed by comparison with real data. The lower-frequency eastward enhancement can be seen both in waveforms of individual strokes, and in storm-averaged waveforms. The higher-frequency westward enhancement can be seen only in individual waveforms, because the short time scales of the enhanced features tend to misalign between different strokes in the average.

[44] The transfer function amplitude experiences the least frequency modulation at 0° magnetic azimuth (duplicated by 180°).

Acknowledgments

[45] One author (A.J.) was partly supported by a grant from the National Science Foundation (“Using Powerful, Low-Frequency Radio Waves from Lightning to Diagnose theD region Ionosphere”) and by a subcontract from the Los Alamos National Laboratory.