Using the World Wide Lightning Location Network (WWLLN) to Study Very Low Frequency Transmission in the EarthIonosphere Waveguide: 1. Comparison With a FullWave Model
Abstract
We investigate a novel way to quantify Very Low Frequency transmission in the EarthIonosphere Waveguide, using data from the World Wide Lightning Location Network (WWLLN). The lightning signals from intense and longduration storm clusters are recorded at several stations. Any individual stroke amplitude is in principle unknown, so that the recorded electric field from that stroke varies semirandomly from the recorded field due to other strokes from that storm cluster. Thus, it is not possible to straightforwardly infer the channel characteristics from a stroke recorded at a single station. However, if two stations record the signal from the same stroke, then the interstation ratio of the recorded amplitude on the two fixed propagation paths is (in the absence of noise) independent of source power. We develop a procedure to provide information on timevariations in the waveguide transmission, using an approach based on ratios of amplitudes from pair of stations which record the same strokes. These amplituderatio data are then compared to an existing model of fullwave Very Low Frequency reflection from the underside of the ionosphere.
Key Points

Infer signal attenuation even without precise knowledge of the source strength

Model the nighttoday transition in very low frequency radio attenuation in Earth Ionosphere Waveguide

Model based on fullwave calculation of Very Low Frequency reflection from underside of ionospheric Dlayer
Plain Language Summary
Lightning strokes emit radio waves of varying strength. The global lightningmonitoring system known as the World Wide Lightning Location Network provides an opportunity to infer features of radio attenuation of these radio waves as they propagate from the lightning to the network sensors. Although we do not know the amplitude of individual lightning strokes, we can form ratio's of each stroke's signal between pairs of stations that detect and record the same stroke's signal. We investigate the use of such ratios to study the radio attenuation process in the waveguide formed between the Earth's conductive surface and the conductive ionosphere about 70–90 km above the surface. In particular, we are able to account for the difference between daytime and nighttime attenuation, using a physicsbased model of radio reflection from the ionosphere.
1 Introduction
We will describe a research effort to infer that part of the attenuation of broadband (5–20 kHz) radio signals in the Earthionosphere waveguide which is caused by reflections from the waveguide's upper boundary (ionospheric Dlayer). To attempt this, we exploit intense, pulsed broadband signals generated by lightning strokes.
Many previous measurements of this attenuation have exploited anthropogenic narrowband radio beacons (McRae & Thomson, 2000; Thomson, 1993; Thomson & Clilverd, 2001; Thomson & Rodger, 2005; Thomson et al., 2004, 2007). Those measurements relied on detailed and reliable knowledge of the source power of the beacons. Our approach differs from the beacon work, in that our sources’ power is taken to be completely unknown. Our source is also broadband.
A key innovation of inferring ionospheric effects using broadband lightning radio emissions (Cheng & Cummer, 2005; Cheng et al., 2006, 2007; Cummer et al., 1998) exploited the distortion of the spectrum of lightning, caused by reflections from the ionospheric boundary. This was a very good adaptation to the fact that the radio source amplitude is unknown. A similar approach was to use radio sensors close enough to the source that the “ground wave” as well as an ionospheric reflection could be jointly recorded, so that the “ground wave” served to calibrate the lightning source power (Jacobson et al., 2007; Shao & Jacobson, 2009). That approach is possible for shortrange observations, <800 km, which allow both the “ground wave” and the “sky wave” to be recorded.
The World Wide Lightning Location Network (WWLLN) detects, locates, and times lightning strokes globally, with ∼10km spatial accuracy and ∼10microsec temporal accuracy (Abreu et al., 2010; Holzworth et al., 2019; Hutchins et al., 2012, 2013; Rodger et al., 2004, 2005). Presently WWLLN comprises over 60 active stations, at each of which is a VLF (Very Low Frequency; 3–30 kHz) receiver. The location/timing is done by timeofgrouparrival (“TOGA”) of the stroke's VLF signal at the various stations (Dowden et al., 2002). The TOGA is determined by “dechirping” the recorded signal for the waveguide's dominant mode (first transverse magnetic, or first TM) for longrange propagation. For shorter ranges, for example, 1,000 km or shorter), the group time of arrival may be due to a mode mix involving higherorder modes than first TM (Dowden et al., 2002). This can make the shortrange path's timeofgroup arrival inconsistent with those from the longerrange paths, in which the first TM mode dominates. Therefore, when there is a mix of stations competing to control the solution for a stroke location/time, the shortrange stations are at a disadvantage, as their timing error will be higher. For this reason, the location/timing solutions tend to be provided by longerrange stations (severalthousand km).
WWLLN in essentially its present form has operated globally and constantly for over a decade; WWLLN's operating costs are already borne by its lightninglocation function. It would be desirable and useful, as well as essentially costfree, to derive from WWLLN some collateral benefit in testing models of broadband VLF radio propagation in the lower ionosphere. Using lightning in such a model test exploits a powerful, naturallyoccurring, and widely distributed radio source. The GreatCircle Paths along which the lightning signals propagate from the lightning to the sensor collectively cover most of the planet. The rest of this paper describes one approach to developing a VLFpropagation model test using the WWLLN lightning data.
Since 2009, WWLLN has archived not only the stroke locations and times, but also estimates of the stroke VLF energy radiated into the Earthionosphere waveguide (Hutchins, 2014; Hutchins et al., 2012). Along with these estimates, WWLLN retains the signal packet's amplitude at each station participating in the solution, in analogtodigitalconverter units. Amplitude is defined as follows: The WWLLN station software examines each triggered record and “dechirps” the recorded signal as appropriate for the lowest Transverse Magnetic waveguide mode. The software then determines the peak amplitude of this dechirped wavetrain. It is stored in “analogtodigital converter units,” which is proportional to the wave electric field, except that we have not in the present exercise invoked a station calibration, which would be required to make that proportionality. A wealth of explanatory material on WWLLN is also available at the project's website (http://wwlln.net). The network provides this data for, typically, more than a halfmillion strokes per day.
For all but a handful of WWLLN stations, we have verified that the station's own system gain is stable over periods exceeding several days, so that repeated arrivals of the same electricfield amplitude from the same bearing will give essentially the same ADC amplitude. The routine verification of the stability of each station's system gain occurs as follows: One WWLLN station is the “master station” (Hutchins, 2014; Hutchins et al., 2012). This is Scott Base on the edge of Antarctica, in an exceptionally VLFquiet location. Unlike all the other stations, Scott Base's sensor is magnetic rather than electric. This further reduces the effect of local environmental noise. Every year, the Scott Base sensor is manually calibrated by being surrounded by a VLF electromagnet. We observe that ever since its initial calibration in 2009, the Scott Base sensitivity (wave magnetic field per analogtodigital increment) has remained essentially constant. Scott Base then spawns daily calibrations of all WWLLN stations by the “bootstrap” method of calibration transfer. This bootstrap daily calibration is the core procedure of WWLLN's energy estimates, which equally rely on the Long Wave Propagation Capability (LWPC) code (Pappert & Ferguson, 1986). From daytoday, we normally observe excellent daytoday stability of each station's bootstrapped calibration factor.
There is no automatic gain control in the receiver. The system gain is manual and is fixed in time, so that, absent any occasional equipment malfunction, the observed stability of a station's gain is what we expect. The stability of station gains over periods of days is necessary for the energy estimates.
The WWLLN estimate of stroke VLF energy radiated into the Earthionosphere waveguide relies on a periodic relative calibration of the network (Hutchins, 2014; Hutchins et al., 2012). The relative calibration relies on calculations of path attenuation in the Earthionosphere waveguide performed by the Long Wave Propagation Capability (LWPC) code, which was originally developed by the United States Navy (Pappert & Ferguson, 1986).
In this article we describe a research project which steps back from the LWPC support, but still relies on the station gain's temporal stability, at least over the course of a day. We investigate using WWLLN's recorded ADC amplitudes to infer the conditions of waveguide attenuation due to the lowerionosphere boundary at the top of the waveguide, in an independent manner that does not presuppose the LWPC.

Each stroke has an unknown and random radiated energy.

Each station is stable but has an unknown absolute calibration (V/m per ADC unit).

Each stroke's radiated VLF, departing from the stroke, is statistically isotropic versus departure azimuth.
2 Our Approach: Concept, and Preliminary Steps
2.1 Concept
Figure 1 shows this work's basic idea. A lightning stroke radiates VLF signals toward both stations #1 and #2. Although the singlestation signal amplitude is not useful by itself, because the lightning source strength is unknown, the ratio of the signals (#1 divided by #2) should, in principle, be independent of the unknown source strength. Due to noise, however, much averaging of the ratio (over temporally close strokes from approximately the same location), is required. Going into this project, it was unknown a priori if such averaging would yield a usefully “clean” timeseries of the twostation, slidingaveraged amplitude ratio. By “clean” we mean that the timevariations of the sliding averaged ratio must be primarily determined by ionospheric effects rather than by system drifts or statistical noise.
2.2 Identifying Stroke Spatial Clusters
To begin the project, we had to see if the WWLLNlocated strokes can be grouped into spatial clusters that contain enough members, distributed with enough temporal coverage of several hours to a day, to allow us to discern ionospheric effects. We use a simple search for strokeclusters in each day. Because we deal with a considerable number of days (over a decade's worth), we have automated the identification of stroke clusters. The automated procedure ingests all the data from a WWLLN day file, and outputs a somewhat smaller file containing only those strokes which fit into clusters whose population exceeds a threshold.
There is a tradeoff between choosing spatially compact clusters, or more extensive clusters. It is an example of the usual tradeoff of statistical strength versus data quality: A spatially compact cluster assures that the paths to a station remain essentially the same, from stroke to stroke during the time development of the cluster. However, the compact cluster contains fewer members, and so its usefulness to temporally track the timevarying amplitude ratio continually over a day with a running temporal average is impaired. A spatially larger cluster collects a larger stroke membership and provides stronger statistics and temporal coverage. However, the larger cluster risks mixing paths (to a given station) which differ sufficiently to compromise an understanding of ionospheric effects. We experimented with this tradespace and settled on a compromise choice of cluster size, explained as follows:
First, we take all strokes for a UT day (numbering typically 0.5–0.8 × 10^{6}), and take the nearestinteger versions of both their longitudes and latitudes. This places them on a spatial grid dimensioned 360 × 180, in longitude × latitude. We tally the population of each integer grid point. We then define larger (5° × 5°) boxes, overlapping 80% so they continue to be centered on each grid point of the 1° × 1° grid. These larger boxes must not be centered within 2° of either the longitude or the latitude boundaries, else the box's volume would be reduced. Hence we are slightly vignetting the spatial domain, by 2° at each of the four boundaries. This leads to a 4°wide band of “empty” longitudes at the International Date Line, within which we are prevented from identifying a 5° × 5° cluster. The large clusters are identified in the following order: First, the stroke membership of each 5° × 5° box is tallied. Second, the highestmembership box is identified. Third, the members of that highestmembership box are marked “already taken” and are removed from possible membership in subsequently identified clusters. We iterate these steps to identify other clusters, until the newest “winner” of the membership contest contains fewer than 8,000 strokes. Then we stop. The only accepted clusters have at least 8,000 members each, and any member stroke belongs exclusively to only one cluster.
Figure 2a shows the cluster result for one UT day, June 1, 2020. The membership is marked by color, from 8,000 (blue) to 78,725 (red). The four most populous clusters are all in or near the western Atlantic, from Central America up to alongside Florida. The overrepresentation of that region is quite typical of boreal summer. Partly, that is, due to that region's optimal WWLLN coverage, but partly also it is due to that region's longduration electrifiedstorm systems that last much of a day and that drift very slowly.
2.3 Temporal Running Averages of Amplitude Ratio
Here we face another tradeoff between statistical strength, and temporal resolution, analogous to the tradeoff seen earlier between statistical strength, and spatial resolution. After experimenting with different temporalaveraging windows, we settled on a sliding window of width 500 s, stepped by 100 s, so that there is a similar overlap factor (80%) as the spatial sampling earlier. We take the strokes from a given cluster and allocate them to these temporal windows. Now it is allowed for a stroke to belong to adjacent temporalaveraging windows. Within each 500swide window, we require the membership of that window to be at least 10. If smaller, that window does not contribute to the timeseries of averaged ratios. Within each accepted temporal window, the median (not average) amplitude ratio is tallied. We use the median to avoid corruption of the time series from occasional singlestroke saturations at a station's ADC.
Let us examine the mostpopulated cluster, shown as the red square in Figure 2a above. Figure 2b shows the time series (red square symbols) of the median amplituderatio of 500swide windows advanced by 100 s steps, for that cluster. The missing symbols are for occupancy <10 strokes. The smaller black square symbols indicate the estimated error from the internal scatter within each window. The two WWLLN stations in this example are Lisbon (Portugal), in the ratio's numerator, and St. Johns (Newfoundland, Canada) in the denominator. Although the cluster contains 78,725 strokes, only 14,931 of them are jointly located by Lisbon and St. Johns stations. The data curve in Figure 2b utilizes only the jointlylocated 14,931 strokes.
2.4 Relating AmplitudeRatio Time Series to Solar Zenith Angle
The data symbols in Figure 2b, following the amplitude ratio (Lisbon/St. Johns) during the course of a full day, indicate clear and systematic temporal variations that dwarf the statistical noise level. There are two dominant fast transients. The first occurs between ∼15,000 s (4.2 h) UT and ∼30,000 s (8.3 h) UT. The second occurs between ∼73,000 s (20.3 h) UT and the end of the UT day. These can be related to the controlling role of solar zenith angle. Since the paths from the lightning to either Lisbon or St. Johns are quite long, so that the solar zenith angle varies spatially along the path at any given instant of time, we show in Figure 2b the solar zenith angle (referred to ground altitude) at five different locations. The green curve is the solar zenith angle versus UT time at the center of the lightning cluster. At shortly before 60,000 s (16.7 h) UT, the sun culminates almost overhead the cluster. The solid black horizontal line marks 99.6°, the groundlevel solar zenith angle corresponding to the passage of the terminator at altitude 90 km. This altitude notionally is close to the nighttime ionospheric Dlayer altitude. The solid red curve in Figure 2b is the groundlevel solar zenith angle at St. Johns. The solid blue curve is for Lisbon. The dashed red and blue curves are for the midpoints of the lightningtoSt. Johns path and the lightningtoLisbon path, respectively. Each of these five colored curves intersects the horizontal black line for the 90kmaltitude terminator, and the times of those intersections are marked by corresponding colored vertical lines.
We see from Figure 2b that the “sunset” and “sunrise” periods are prolonged in time, due to the large scale of the paths from the lightning to the stations. From ∼1,000 s (0.3 h) UT to ∼15,000 s (4.2 h) UT, all portions of both paths at 90km altitude are in darkness. From ∼31,000 s (8.6 h) UT to ∼75,000 s (20.8 h) UT, all portions are in daylight. From ∼15,000 s (4.2 h) UT to ∼31,000 s (8.6 h) UT, the sunrise terminator is somewhere along at least one of the two paths, while from ∼75,000 s (20.8 h) UT thru the end of the UT day, the sunset terminator is somewhere along at least one of the two paths.
2.5 Sorting the SlidingAverage AmplitudeRatio Data
The slidingaverage amplituderatio data shown in Figure 2b is for just one pair of stations, for just one lightning cluster, and for just one single day. Typically we have a few acceptable clusters worldwide per day, and several station pairs per cluster. So, with the presence of more than a decade of days’ worth of data, we have available tensofthousands of time series to consider for further processing. Thus, the procedure for sorting the timeseries data is best automated.
In all data that we have studied, the solarterminator effects dominate the variations of the time series, compared to the residual variations during either “pure darkness” or “pure daylight” periods. To a firstorder approximation, the daytime ionization state of the Dregion is most controlled by the local solar zenith angle. But beyond that approximation, there are important complications due to the dependence of the photochemistry on the global atmospheric circulation pattern (Krivolutsky et al., 2015; Kulyamin & Dymnikov, 2015; Turco & Sechrist, 1972a, 1972b). This spatially and temporally modulates the availability of readily ionizable species, foremost of which is the molecule NO. According to both of the two modern numerical coupled Dregion/global circulation models (Krivolutsky et al., 2015; Kulyamin & Dymnikov, 2015), the departures from a simple solarzenithangle control are most apparent at low latitudes and at the latitude of the subsolar point.
We have no way in our simple model of VLF attenuation to model the irregularities introduced by global circulation, so instead we simply limit our project to those paths which exhibit dominant solarzenithangle control. This translates into limiting our project to those pairs of paths whose amplitude ratio is sufficiently constant during either the purenight phase of the record or the pureday phase of the record, compared to the terminatorpassage transitions. The data of Figure 2b above, for example, is quiet enough during the pureday or purenight portions, to justify testing whether the terminatorpassage transitions can be related to the rapid change of solar zenith angle during those transitions.
The automated sorting algorithm goes through all the time slidingaverage time series and slices each series into smaller pieces, so as to extract the periods of terminator passage. This slicing routine will be illustrated fully in Section 4. This will allow the model to focus on periods of rapid excursion, when there is the most reason to attribute control to the solarzenithangle changes occurring along the paths, but to avoid modeling the residual (and smaller) variations observed during pureday or purenight conditions. Logically, the latter are more likely related to complicated atmosphericcirculation effects on the photochemistry (Krivolutsky et al., 2015; Kulyamin & Dymnikov, 2015) than they are simply to rapid changes of the solar illumination.
3 The Approximate Model
3.1 Form of the Model
We seek to test whether the quantifiable changes in solar illumination during terminator passage can at least approximately account for the observed excursions of the slidingaverage amplitude ratio during terminator passage. Toward that end, we keep the model simple and heuristic, which suffices to compare with the noisy data on amplitude ratio. We want the model to stand independently of this field's standard tool (the LWPC code). We also want the model to be heuristic and transparent, something that the LWPC legacy code is not.
First we shall assume that the attenuation of a VLF signal in the Earthionosphere waveguide is an exponential function of distance. Because the local factors determining the attenuation vary along the path, as well as vary versus time, the exponential actually is of a path integration over local and instantaneous local variables. The path is the Great Circle Path from the lightning to the WWLLN station.

C = geometrical (nonlossy) signaldilution factor for sphericalshell waveguide

P_{i,m} = path factor due to Earth's surface conductivity (constant in time), range is 0 < P_{i,m} < 1

E_{m} = source factor of lightning at location m

E_{i,m}(t) = signal at station i due to lightning at location m

G_{i} = gain (unknown) of station i

L_{i,m} = arcdistance along Great Circle Path from lightning location m to station i

Z_{i,m}(t) = timedependent, locationdependent solar zenith angle along path i,m

α_{i,m} = locationdependent magnetic propagation azimuth along path i,m

I_{i,m} = locationdependent magnetic dip angle along path i,m

g (Z_{i,m}(t), α_{i,m}, I_{i,m}) zenithangle and geomagnetic effect on local and instantaneous attenuation

ds_{i,m} = differential path element along Great Circle Path i,m
Notes:
The station gain G_{i} is taken to be unknown but stable in time.
The source factor E_{m} of lightning at location m is taken to be unknown but statistically omnidirectional.
The arguments of the zenith/geomagnetic factor g depend on position, and (via Z) on time.
The strategy is to form the ratio r_{i,k,m} (t) of the signals at two stations (i,k) observing the same lightning strokes (m) at the same times. The ratio is used in order to render the (unknown) lightning source strength irrelevant, as it is common to the signals on both stations i and k. When we say the “same strokes,” we mean that in detail: A stroke needs to be recorded at each of the two stations, for a “ratio” to be formed for that stroke and those two stations.
This timedifferenced form of the model equation enjoys some important simplifications: There is no dependence on the stations’ system gain. Nor is there any dependence on the ground losses (via the factors P_{i,m} etc).
However, this simplification comes at a cost: The model form of Equation 5 can be used only to fit that portion of the observed timeseries data that varies with time. Any constant baseline in that data is lost with the simplified Equation 5. In the comparisons with data below (Section 4), all we can fit is the data's temporal excursions, but Equation 5 gives no information on any constant baseline on which those excursions are superimposed.
In Equation 5 and the results to follow, the model needs accurate inputs of the magnetic propagation azimuth and magnetic dip angle. We calculate these inputs using the IGRF, or International Geomagnetic Reference Field (IAGA, 2003) multipole model of the vector geomagnetic field, as a function of latitude, longitude, and altitude, at low altitudes where solarwind and outermagnetosphere distortions are usually not relevant.
3.2 Choice of Physical Content of the Model
Equation 5 is the form of the model which we shall compare with data. As yet unspecified, however, is the actual physical content of the model, that is, the function g (Z_{i,m}(t), α_{i,m}, I_{i,m}) which contains all the effects of local, instantaneous solar zenith angle, as well as of both local propagation magnetic azimuth and local magnetic dip angle.
This exponential profile is characterized by three parameters: The reference electron density n_{e0}, the reference altitude z_{0}, and the logarithmic derivative q. Following the pioneering descriptions of the Dlayer (Volland, 1995; Wait & Spies, 1964), we use n_{e0} = 3 × 10^{8} m^{−3}. For daytime, we use z_{0} = 73 km and q = 0.3 km^{−1}. For nighttime, we use z_{0} = 85 km and q = 0.45 km^{−1}.
Figure 3 shows the predictions of the J2009 calculation for amplitude reflection during nominal (a) daytime and (b) nighttime conditions. Color indicates magnetic dip angle absolute value, from 0 (blue) through 85° (red) in 1° increments. The calculation assumes angle of incidence is 85°. Results shown here are an average over the passband 5–20 kHz, in 1kHz steps. The frequency average is appropriate to the spectrum of the broadband VLF signals captured by WWLLN stations (Hutchins, 2014). The choice of angle of incidence is not critical at this point; the effect of varying the angle from the chosen 85° to, for example, 80°, merely rescales the overall reflection coefficient deviation from unity. That scale factor will be adjusted anyway in our final fitting of the model to the data (below).
3.3 Preliminary Implications of the Model
The model for amplitude reflectivity (Figure 3) has a couple of key features that are perhaps not widely appreciated, although they should be latent in any correct fullwave calculation. The first key feature in Figure 3 is that the azimuthal asymmetry is profoundly modulated by the magnetic dip angle. The “east/west” asymmetry is most extreme at dip angle = 0° (Magnetic Equator) and gets smaller with increasing magnitude of dip angle, reaching zero at dip angle = ±90° (the Magnetic Poles). The latter should be trivially obvious by symmetry: For a vertical magnetic field, azimuth becomes an ignorable coordinate in the Maxwell's equations with the anisotropic susceptibility tensor (Pitteway, 1965), which are solved to get the electromagnetic wave field (Jacobson et al., 2009).
The second (and more novel) phenomenon laid bare by Figure 3 is that the “eastwest asymmetry” is much more pronounced for night than for day conditions. This is somewhat counterintuitive, because the daytime reflection environment is lower in altitude—and hence more collisional—than is the nighttime reflection environment (Volland, 1995; Wait & Spies, 1964). Notwithstanding that intuition, the magneticwestward attenuation is predicted in Figure 3 to be in daytime than in nighttime.
As noted above, several studies have observed an “east/west asymmetry” in the VLF path attenuation (Crombie, 1958; Hutchins et al., 2013; Taylor, 1960). These previous observations of anisotropic VLF attenuation based on broadband lightning signals did not treat the variation of the magnetic propagation azimuth along the Great Circle Path (from the lightning to the receiver). Moreover, those prior observations’ presentations did not include that the entire anisotropy is controlled by the magnetic dip angle. A path segment at dip angle 0 is not equivalent to a path segment at dip angle 45°, even if the magnetic propagation azimuth were the same. For a given path, the anisotropic transmission needs to be treated locally at each place along the path, and these local contributions must be integrated over the path. For paths of thousands of km, it is not meaningful to assert that there is a “magnetic azimuth” of that path. Only a local path differential segment has a single magnetic azimuth or a dip angle; the entire path, in general, does not.
3.4 Possible Relationship With Previous Work on DLayer VLF Penetration
The model's prediction of deep attenuation of magneticwestward propagation at low dip angle and during nighttime might be consistent with an older observation involving lightning VLF signals recorded on the C/NOFS satellite (de La Beaujardiere, 2004) using the VEFI, or Vector Electric Field Instrument (Pfaff et al., 2010). The VLF signal arrives at the satellite only by penetrating the ionospheric Dlayer and then continuing as a magnetospheric oblique whistler wave. The C/NOFS orbital inclination (13°) ensured that most of the recorded VLF waveforms were recorded at low magnetic latitude (Jacobson et al., 2011, 2018). Moreover, the VLF recordings almost all occurred when the satellite was in darkness, so that the ionospheric transmission was for a nighttime Dlayer. The Dlayer penetration process was inferred to be more efficient for westwardpropagating than for eastwardpropagating VLF signals, based on the statistical behavior of the VEFI recorded signals from lightning strokes that had been located by WWLLN (see Figure 4 in Burkholder et al., 2013). This was later seen more quantitatively (see the striking anisotropy in Figure 15a of Jacobson et al., 2018).
In the present instance we observe that the magneticwestward propagation in the EarthIonosphere Waveguide is more attenuated than is eastward. Thus, we infer that the transmission in the waveguide, and the transmission upward from the waveguide into the ionosphere, have qualitatively opposite tendencies versus magnetic azimuth. This might be consistent. If a wave is attenuated due to Dlayer effects within the EarthIonosphere Waveguide, the causes of the attenuation logically include (a) lossy collisional processes, and (b) upward penetration of wave energy through the Dlayer. Either of these processes serves to reduce the forward transfer within the EarthIonosphere Waveguide. Hence the opposite tendencies versus magnetic propagation azimuth (see above) are compatible.
3.5 Adapting a PlaneWaveReflection Model for a SphericalShell Waveguide
This ratio is the “C” factor in Equation 1 above. However, in the transition from Equations 4 to 5, we lose the effect of this factor because its effect is not timedependent.
A more serious concern in using a planewave reflection model is that the planewave approach presents solutions for discrete, preordained anglesofincidence, whereas in the waveguide approach there are discrete waveguide modes rather than anglesofincidence. The connection between these approaches has been discussed elsewhere (see, e.g., the instructive Section 2.1 in Cummer, 1997, 2000). Here, in this very approximate model, our approach is as follows: For a given arcdistance L from the lightning to the receiver, suppose that the wave undergoes n successive reflections at the ionosphere (so that the path is said to be “nhop”). Then the path transmission factor for the plane wave will vary as R^{n}, where R is the coefficient of amplitude reflection for a single reflection. Take ρ_{0} = 1,000 km as a reference arcdistance. Then we shall leave as a free parameter the number of reflections in the reference distance ρ_{0} = 1,000 km. Call that free parameter “r.” Using the free parameter r, we will adapt our singlereflection model (Figure 3) to distributed reflection over arbitrary arcdistances. Since r is the number of reflections in the reference arcdistance ρ_{0}, the number of reflections in the arcdistance ρ is greater by the ratio ρ/ρ_{0}.
Note that R(Z, α, I) is the singlereflection amplitude reflection coefficient as a function of solar zenith angle, propagation magnetic azimuth, and dip angle. Figure 3 above presents two special cases of R(Z, α, I), for idealized daylight and idealized darkness ionospheric conditions.
3.6 Integrating the Solar Zenith Angle Into the Model
The first ln(R) on the rhs of Equation 11 is for Z = 0°, to designate the natural logarithm of the day model from Figure 3a, and the second is for Z = 180°, to designate the natural logarithm of the night model from Figure 3b. The only thing that changes as a function of time in this recipe is the dayweighting factor f(t), while the R(0,α,I) and R(180,α,I) are independent of time, varying only versus propagation magnetic azimuth and magnetic dip angle. Using this linear combination of timeinvariant models gives an efficienttoimplement transition between day and night conditions in the modeled logarithmic amplitude reflection. Hopefully this approach does no more violence to the physics than has already been done by use of an exponential profile of electron density. We emphasize that this is applied locally, to the local magnetic and local, instantaneous solarzenithangle conditions at each differential element along the path integrals.
4 Comparing the Model With AmplitudeRatio Data
4.1 SinglePair Parameter Fit
We now illustrate a parameter fit by comparing data and model for a single pair ratio. Figure 5a shows the location of the most prolific lightning cluster, in the western Atlantic (coinciding with the redcolored cluster in Figure 2a above) and the paths from that cluster to both Lisbon and St. Johns. Color indicates the absolute value of the dip angle, from 0° (blue) to 90° (red). Figure 5b shows the same paths, but with color indicating propagation magnetic azimuth as in the color wheel. This singlepair example samples only a very small portion of the dipangle/magneticazimuth space. Figures 6a and 6b shows the loci of the two paths (cluster to Lisbon, and cluster to St. Johns) superposed on the display of model reflectivity solutions for (a) pure day and (b) pure night. The curves are parametrized by dip angle, and this allows the loci of azimuth/dip to be transferred to this azimuth/reflectivity plane. It is obvious that the singlepair example has zero coverage of the magneticwestward half of each solution space. There is even incomplete coverage of the magneticeastward half. From Figures 6a and 6b alone, we can already see that the result of this parameter fit will not in any way sample the westward lobe in the model.
In Figure 6c, we vary the assumed reflectionsperMm parameter (r) along the abscissa. Recall that this parameter now multiplies the ln(R) function in Equations 9a9c. The ordinate shows the sum of the squares of residuals between the model (with the assumed r parameter) and the data. This is, in effect, a “penalty function.” The “bestfit” value of the parameter is where the penalty is minimum. That occurs at r ∼ 2.7 in this singlepair case.
A word of caution on the model: As stated earlier in the discussion of Equation 5, the model as it is fitted to the data is capable of fitting only the excursions in the data, but not any constant baseline. During the fit, the model curve is rigidly adjusted up or down to constrain its temporal average to match the temporal average of the data. The only feature of the data that is, meaningfully fitted is the data's temporal excursion. The baseline is arbitrary.
Figure 7 shows the data (natural logarithm of slidingmedian amplitude ratio) versus time (black squares). The bestfit model from Equations 9a9c (at r = 2.7) is shown as corresponding teal squares. The solar zenith angle at the lightning cluster, at the Lisbon station, and at the St. Johns station are show in green, blue, and red respectively. The two halves of Figure 7 are the two transition periods found by the sorting process described earlier in Section 2. The left half of Figure 7 is the sunrise transient, while the right half is the sunset transient. The data in these two halves are natural logarithms of two tranches of the amplituderatio data in Figure 2b above.
The data/model comparison shown in Figure 7 already indicates a limitation of our model. While the sign and approximate magnitude of the transients are fitted adequately by the model, there are complications in the data that are not captured by the model. Note the overshoots near the end of the sunrise transient; these details are beyond the skill of our model.

Choose a pair of stations and a lightning cluster to analyze.

Calculate the runningaverage time series of observed binmedian amplitude ratio.

Calculate the solar zenith angle versus time at the lightning cluster, at the two stations, and at a grid of points along each lightningtostation GreatCircle Path.

Check to see if the timeseries of amplitude ratio (from step b above) shows clearly identifiable transient excursions during times of day/night transitioning along the paths.

If there are clear amplituderatio excursions concurrent with the day/night transitioning, then snip the time series (from step b above) into pieces. Each retained piece will encompass one complete day/night transition.

These retained snippets, concurrent with day/night transitions, are now compared with the theory. The adjustable parameter is “r,” the number of reflections per 1,000 km of path. During the fit of a snippet of data to the model, the timeaveraged difference (between the snippet and the model of the snippet) is removed, so that all we are fitting is the temporal transient of the amplitude ratio, but not the offset of the amplitude ratio. For example, in Figure 7 above, there are two snippets, for each of the two day/night transitions. Each snippet is freely floated up or down to minimize the mean residual versus the model. The only effect contributing to the fit of the parameter “r” is the transient excursion within each snippet.
4.2 MultiplePair Parameter Fit
The same day as shown in Figures 57 above can provide a fit of the free parameter using all stationpairs that include Lisbon. Figure 8a indicates the pathpairs for Lisbon and the six other stations with which Lisbon has pairratio data satisfying the selection criteria described above in Section 2. In this case, there are three lightning clusters through which Lisbon has pairs amongst the other stations. These three clusters include the two in the western Atlantic, and one near the Aegean Sea. Color indicates the absolute value of the magnetic dip angle, from 0° (blue) to 90° (red). Figure 8b shows the propagation magnetic azimuth in color as shown in the color wheel. Now the sampling of dip angle/magnetic azimuth is much richer (compared to the singlepair example above). Figures 9a and 9b shows the multipair sampling, in the same format as Figure 6 above. The multipair sampling of dip angle/magnetic azimuth provides much better coverage of diverse magnetic propagation azimuths than had been the case with the singlepair case above. Regarding coverage of dip angle, the multipair coverage is still poor. It is true that the path to the Belem station crosses the Magnetic Equator and hence contains dip angle magnitudes down to zero, but this is done at a nearly magnetic southward azimuth and hence does not sample very deep into the lowestreflectivity solutions, for example, R < 0.65. Those lowestreflectivity conditions occur only in night and where the azimuth is westerly and where the dip angle is low.
Figure 9c shows the penalty function in fitting the r parameter for the multipair comparison of model with data. Compared to the singlepair penalty function (Figure 6c above), the contrast (between the heights at the wings and at the minimum) is degraded, though there is still a clear bestfit determination (r ∼ 2.8). One of the data/model comparisons is shown in Figure 10, for Lisbon paired with Forks (Washington State, U.S.). The transition shown here is a mixed transition, due to the great separation between Forks and Lisbon in local time. In fact, the time of sunrise at Lisbon (blue zenith curve) almost exactly coincides with the time of sunset at Forks (red zenith curve). The gross features of the transient in the data (black squares) are approximated by the model (teal squares), but the data's “overshoot” near the end is not.
The parameter fits shown in Figures 6 and 9 are typical of what we see: fitted values of r in the range 2 < r < 3.5, although some pathpairs, due to their collapsed geometry, yield no convergent solution. It is important to caution that this parameter fit does not necessarily prove the underlying correctness of the model. Rather, we have merely shown that, if we apply the model, the gross features of the day/night transitions can be predicted with parameter fits in the range 2 < r < 3.5.
4.3 A Fundamental Limitation of This Method
The “holy grail” of this section would have been to observationally test, by the amplituderatio method, the model's prediction of the broad and deep westward, nighttime transmission minimum.
Our method requires identifying copiouslyemitting clusters of lightning. Within a cluster, the strokes must occur often enough in time to provide good statistics in each 500swide sliding average. The strokes within the cluster must occur quasicontinuously over enough of the day to permit us to track ionospheric changes associated with the passage of the solar terminator. That the cluster meeting these requirements exists, is not by itself enough. Additionally, the strokes in the cluster must be simultaneously detected in farflung station pairs whose coverage in the solution plane (see Figure 3) rigorously tests the predictions of the model. In order to probe the effects of magneticwestward propagation at low dip angle, we must detect copious strokes at a station which, according to the model, will receive only a highly attenuated signal. If the model is correct, that requirement is impossible to meet. Highly attenuated signals at this station will either (a) be lost in the competition against other stations to locate the stroke, or (b) not even trigger the station at all.
And there's the rub. Despite inspection of hundreds of cases of clusters that are well detected by station pairs, no combinations of clusters and stations that we have found are able to test the model for lower reflectivity levels than about 0.6 in the solution space. That is, the cluster/stations combinations that are found do not provide magneticwestward propagation at low dip angle. We note from Figure 3 that the dramatic prediction of the model occurs for nighttime conditions in the magneticwestward sector at low magnetic latitudes. The predicted suppression of the local reflectivity there is to a level as low 0.05. However, we have found no station pairs, for any copiousemitting clusters, with propagation paths dwelling for a significant fraction of their length anywhere below a local reflectivity of 0.6.
Thus the stationpairratio method is not able, so far, to provide a test of the predicted deep nighttime suppression of magneticwestward transmission at lowtomoderate dip angles. Nor have we found observations contradicting the model in those conditions. Rather, there is just no data of sufficient quantity for addressing that part of the solutions space: R < 0.6.
This is not an accident or bad luck. Rather, it is inevitable that a method requiring reception of copious strokes will fail along paths where the transmission is deeply suppressed. We cannot make ratios of amplitudes from two stations if these are of strokes that are not even recorded at one of those stations.
5 Discussion and Conclusions
We have presented a method to study the behavior of the interstation ratio of VLF stroke amplitudes, for strokes that are simultaneously recorded at multiple WWLLN stations. This approach combines numerous recurrent strokes from longduration lightning clusters to build a timeseries of the ratio for a major portion of the UT day. The time variations of the slidingaveraged ratio are dominated by transient excursions coinciding temporally with those periods when the solar terminator is present along one or both of the paths. This strongly motivates a model incorporating significant control by the solar zenith angle.
We have adapted an existing fullwave theory of reflection from the Dlayer to model transmission in the EarthIonosphere Waveguide at the frequencies (5–20 kHz) contributing to WWLLN signals. The model predicts that magneticwestward propagation has less waveguide transmission than does magneticeastward propagation. This anisotropy is modulated by magnetic dip angle: The anisotropy is strongest at low dip angle, and weakest at large dip angle.
To account for solarzenithangle control on the waveguide transmission, our model takes a weighted combination of pureday and purenight solutions, determined locally for every path element along the Great Circle Path from the lightning to the WWLLN station, and for the exact Universal Time of the stroke.
The model solution based on the planewavereflection theory successfully accounts for the gross features of the solarterminator transients. However, small, residual time variations not obviously related to the solarzenithangle control are also observed, but are outside the scope of our model.
Our model predicts, counterintuitively, that the magneticwestward attenuation at low magnetic latitude will be much deeper during night than during day conditions. Unfortunately, this suppression of magneticwestward propagation also largely eliminates the availability of recurrent recordings of those signals at our lowlatitude stations. Thus the amplituderatio method is inherently unable to check on one of the model's most intriguing predictions.
The present report is preliminary. A later paper will develop an alternative approach to testing the model prediction of deep attenuation in conditions of low dip angle and magneticwestward propagation.
Acknowledgments
The authors are using data from the World Wide Lightning Location Network, a collaborative consortium of over 70 worldwide collaborators, managed at the University of Washington. The data would not exist but for the cooperative efforts of all of WWLLN's participants.
Open Research
Data Availability Statement
For sale to researchers who are not WWLLN participants, WWLLN data are available, at a nominal price to cover overhead costs of running the network, and archiving/distributing the data. To find out about such data access, see http://wwlln.net/