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\begin{document}

\title{Using the World Wide Lightning Location Network (WWLLN) to study Very
Low Frequency transmission in the Earth-Ionosphere Waveguide: 2. Model
test by patterns of detection/non-detection}



\author[1]{Abram R Jacobson}%
\author[1]{Robert H. Holzworth}%
\author[2]{James B Brundell}%
\affil[1]{University of Washington}%
\affil[2]{University of Otago}%


\vspace{-1em}



  
  \date{June 05, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
This is the second half of a two-part study. In the first part, we had
used the World Wide Lightning Location Network's recorded signal
amplitudes to test a model of Very Low Frequency signal transmission
from the lightning to each sensor. The model predicts a dramatic
worsening of transmission at low magnetic latitudes, for nighttime
propagation (compared to daytime propagation) toward magnetic West.
However, we found that the use of amplitudes was ill-adapted for testing
the model under conditions of a deep outage of transmission. Since the
relative weakening of nighttime transmission is rather
counter-intuitive, we have now developed an alternative approach to
testing that model prediction. This alternative approach highlights the
patterns of detection/non-detection of several low-magnetic-latitude
WWLLN stations and compares those patterns with the appropriate patterns
of the model transmission.%
\end{abstract}%



\sloppy


\textbf{Using the World Wide Lightning Location Network (WWLLN) to study
Very Low Frequency transmission in the Earth-Ionosphere Waveguide: 2.
Model test by patterns of detection/non-detection}

\textbf{Abram R. Jacobson\textsuperscript{1} and Robert H. Holzworth}

Earth and Space Sciences Dept., University of Washington, Seattle U. S.

and

\textbf{James B. Brundell}

Physics Dept., University of Otago, Dunedin NZ

1: Corresponding author Abram R. Jacobson, abramj@uw.edu



\textbf{Abstract}

This is the second half of a two-part study. In the first part, we had
used the World Wide Lightning Location Network's recorded signal
amplitudes to test a model of Very Low Frequency signal transmission
from the lightning to each sensor. The model predicts a dramatic
worsening of transmission at low magnetic latitudes, for nighttime
propagation (compared to daytime propagation) toward magnetic West.
However, we found that the use of amplitudes was ill-adapted for testing
the model under conditions of a deep outage of transmission. Since the
relative weakening of nighttime transmission is rather
counter-intuitive, we have now developed an alternative approach to
testing that model prediction. This alternative approach highlights the
patterns of detection/non-detection of several low-magnetic-latitude
WWLLN stations and compares those patterns with the appropriate patterns
of the model transmission.



\textbf{Key points}

Prior knowledge: Low-magnetic-latitude Very Low Frequency radio
transmission in the Earth-Ionosphere Waveguide is suppressed for
propagation toward magnetic West, relative to magnetic East.

New finding: This magnetic-westward suppression is dramatically stronger
during night than during day.

New finding: The magnetic-westward suppression at night applies to
nearly the entire westward half-plane, from magnetic-South clockwise to
magnetic-North.

New finding: The magnetic-westward suppression at night is readily
observable for dip angles in the range -30 deg to +30 deg.

\textbf{Index terms:} 2487, 6934, 6964,

\textbf{1. Introduction}

This is not a new topic. East-west asymmetry in VLF (Very Low Frequency;
3-30 kHz) propagation in the Earth-Ionosphere Waveguide had been
inferred as early as the 1920's {[}see the historical review
by\emph{Crombie} , 1958{]}. The geomagnetic control over VLF propagation
is expected to depend strongly on two orientational parameters
{[}\emph{Budden} , 1985; \emph{Piggott et al.} , 1965; \emph{Pitteway} ,
1965; \emph{Wait and Spies} , 1960; \emph{Wait and Spies} ,
1964;\emph{Yabroff} , 1957{]}. One is the dip angle of the geomagnetic
field, and the other is the propagation magnetic azimuth of the VLF
wavefield. Along any long-range propagation Great Circle Path, both of
these orientational parameters can widely vary. Thus, e.g., it is only
very approximate to characterize the propagation magnetic azimuth ''of
the path'' by its value at a path endpoint. The research results we
review below can be broadly separated as to whether they just grossly
consider the whole path together, or whether they dissect the path into
small segments and use a different propagation azimuth and magnetic dip
within each segment. The former we will call ''bucket'' approaches, in
that they just characterize the entire path as being effectively at one
propagation magnetic azimuth. By contrast, the model comparisons which
consider the local nature of the two orientational parameters will be
called ''local'' approaches.

All studies of the azimuthal asymmetry of VLF propagation prior to the
present have been based on observing the effect of magnetic propagation
azimuth on received amplitudes. This is equally true when the signals
received were from narrow-band artificial beacons {[}e.g., \emph{Bickel
et al.} , 1970; \emph{Pappert and Hitney} , 1988{]} or when derived from
lightning strokes {[}e.g., \emph{Hutchins et al.} , 2013; \emph{Jacobson
et al.} , 2021; \emph{Taylor} , 1960{]}. In the already-published
{[}\emph{Jacobson et al.} , 2021{]} first half of the present study,
henceforth referred to as ''JHB1'', we followed the amplitude-comparison
approach. However, we reached a cul-de-sac with that approach, when we
tried to test a particular non-intuitive though impactful prediction of
our model {[}\emph{Jacobson et al.} , 2010; \emph{Jacobson et al.} ,
2009;\emph{Jacobson et al.} , 2012{]}. The prediction is that for
propagation at low dip angle, e.g. in the range -30 to +30 deg,
propagation toward the magnetic West is deeply attenuated during
dark-path conditions (night), \emph{relative to sunlit-path (day)
conditions} . Curiously, this counter-intuitive effect had not been
overtly remarked prior to JHB1, though the physics package of the
comprehensive, state-of-the-art path simulator, LWPC {[}\emph{Pappert
and Ferguson} , 1986{]}, certainly contains all the relevant physics.

We found in JHB1 that the sought-after dark-path conditions apparently
caused such a dearth of numerically sufficient lightning detections at
the pertinent stations, so that amplitudes could not be determined with
statistical accuracy. Thus JHB1 could not test its most interesting
model prediction. ''No detection, no amplitude''.

This shortcoming of JHB1 motivated the present study, which is the
second part of the study begun by JHB1. We completely change strategy in
this paper. Rather than comparing amplitudes, we examine observed
statistical patterns of detection/non-detection from ten selected
stations. We compare those patterns to predictions of the model. The
stations are chosen to represent all longitude sectors and all Universal
Times, and to include many diverse paths from the lightning locations to
the stations, in such a manner as to provide compelling statistical
evidence on the model's predictions. The number of paths included
exceeds 15-billion. Each of these paths is then dissected into 50 path
segments, for a grand total exceeding 750-billion path segments. At each
path segment, the geomagnetic dip angle and magnetic propagation azimuth
are calculated, along with the instantaneous solar zenith angle, are
combined to predict the modeled contribution of that path segment to the
overall integrated attenuation for that path.

\textbf{2. Background}

\textbf{2a. Prior observations of east-west asymmetry of VLF
propagation}

Here we review the carefully documented work that began in the 1950's.
Crombie described new measurements performed in New Zealand during 1957
{[}\emph{Crombie} , 1958{]}, one purpose of which was to investigate the
asymmetry earlier hinted at by the scattered results of the 1920's. The
16.6-kHz signal from a powerful transmitter (''GBR'') in Rugby, UK was
received by a magnetic loop and a vertical aerial at Wellington, NZ. The
receiver and the transmitter were nearly antipodal. By rotating the loop
antenna, Crombie was able to select separately the signal arriving on
either Great Circle Path, from respectively NNW or SSE (reckoned at
Wellington.) The signal strength on each orientation of the loop was
measured versus time during three multi-day periods. The results
confirmed not only that the signal arriving from the NNW had 10-15 dB
stronger amplitudes than the path arriving from the SSE, but also that
the diurnal variations were dissimilar between the two paths. Crombie
attributed this difference to the east-west components of each path,
although this was left notional. The detailed variation of the magnetic
azimuth along each path was not presented or addressed, so that
Crombie's work was in the ''bucket'' category. The diurnal variation was
not explained, but the gross difference between eastward and westward
propagation was noted.

Shortly after the work by Crombie, there was a systematic attempt to use
geolocated lightning to observe the zonal asymmetry of long-range
broadband VLF propagation {[}\emph{Taylor} , 1960{]}. Whereas Crombie
had relied on a discrete, narrow-band, man-made beacon, Taylor exploited
the powerful broadband emissions of lightning return strokes. Attention
was focused on daytime conditions, with most of the paths over seawater.
VLF receiver stations in the western United States and in Hawaii,
triggering off common lightning strokes, were used to crudely geolocate
the lightning, at least within \textless{}10\% of the path length, by
triangulating the direction found at each station. Each station measured
and recorded the vertical electric field with a vertical mast, and also
provided the direction of arrival from comparing signals on two
vertical-magnetic-loop antennas. As was necessary in that era, data were
recorded for off-line analysis using oscilloscopes and cameras. The
east-to-west attenuation from twenty lightning discharges was used to
determine a mean spectral attenuation (dB/1000km) for that direction of
propagation. The spectral attenuation was determined for the entire VLF
band. Similarly, the spectral attenuation for west-to-east attenuation
was determined using sixteen lightning discharges. All observations were
for entirely-daylit paths. It was found that attenuation east-to-west
exceeded attenuation west-to-east, by approximately 3 dB/1000km for f
\textless{} 8 kHz and by approximately 1 dB/1000km for f \textgreater{}
10 kHz.

Taylor's characterization of the paths as ''east to west'' versus ''west
to east'' {[}\emph{Taylor} , 1960{]} is in the bucket category. Moreover
Taylor did not consider the control by geomagnetic dip angle; rather,
all the paths were simply tagged as ''east to west'' or as ''west to
east'', regardless of magnetic dip, and then simply labeled with one
orientation. The lightning locations are not given {[}\emph{Taylor} ,
1960{]}, so it would not be possible to retrospectively model Taylor's
observations with a more local approach.

In 1969, the United States Naval Ocean Systems Center conducted airborne
measurements of VLF beacon signals on Great Circle Paths from the island
of Hawaii toward San Diego and from the island of Hawaii toward Wake.
The paths were, respectively, west-to-east and east-to-west paths,
entirely over seawater, and entirely nighttime. These data were later
presented and compared {[}\emph{Pappert and Hitney} , 1988{]} to
state-of-the-art, full-wave waveguide propagation calculations using the
LWPC {[}\emph{Pappert and Ferguson} , 1986{]}. The fixed frequencies of
the beacons at Hawaii were discretely between 10.9 kHz (the lowest) and
28.0 kHz (the highest). The airborne receiver recorded signal amplitudes
due to the multifrequency sounder for the first \textasciitilde{}4000 km
of each path. Thus the measurements were all done within 4000 km of
Hawaii. The VLF data were compared to a model that included detailed
tracking of the propagation azimuth and the magnetic dip angle locally
at all points along the propagation path. This was a local approach, and
was a critical advance over the bucket approach. It was found that the
eastbound (San Diego path) signal was very reproducible day-to-day, and
was essentially perfectly modeled by LWPC with a generic nighttime
profile {[}\emph{Pappert and Hitney} , 1988{]}. The westbound (Wake
path) signal, by contrast, was more variable day-to-day, and this
adversely affected the agreement with the model, although on average the
agreement was satisfactory. The variability for westbound propagation
was speculated to be related to sporadic electron-density features near
altitude 90 km. We note that the sampled paths did not delve lower than
about 30 deg in dip angle.

In addition to the airborne measurements using the multifrequency
beacon, the same aircraft was also deployed to measure the signal from
the unique 23.4-kHz signal ''NPM'' radiated from the area of Honolulu,
Hawaii with much higher power than the research multifrequency beacon.
The NPM signal was measured along Great Circle Paths from NPM toward
Seattle, Ontario (California), Samoa, and Wake Island. Results were
reported {[}\emph{Bickel et al.} , 1970{]}, similarly, out to
\textasciitilde{}4000 km range, and were entirely over seawater and at
night. The authors {[}\emph{Bickel et al.} , 1970{]} used an early
predecessor of LWPC to compare with waveguide theory, and found that the
model predictions of dependence on magnetic azimuth and magnetic dip
angle were robustly confirmed at 23.4 kHz by the airborne measurements.
Their model comparison was a local approach, exactly similar to that
used for the multifrequency beacon data {[}\emph{Pappert and Hitney} ,
1988{]}.

A more recent entry into the observation of propagation magnetic-azimuth
asymmetry was done with the World Wide Lightning Location Network, or
WWLLN {[}\emph{Hutchins et al.} , 2013{]}. It dealt with over-seawater
paths in the Pacific sector, using WWLLN stations at island locations
Suva, Tahiti, and Honolulu. This study is in the ''bucket'' category.
The study used lightning strokes jointly detected by all three of those
stations (along with other stations as well.) Each lightning stroke's
radiated VLF energy was determined with the WWLLN energy retrieval
described elsewhere {[}\emph{Hutchins et al.} , 2012{]}. The candidate
strokes were selected according to the following strict limiting
criteria:

(a) The WWLLN VLF energy determination for the stroke needed to have an
estimated error less than 10\% of the VLF energy.

(b) The stations participating in the location/energy determination
needed to be equally distributed east/west of the stroke location, to
within 25\%.

(c) The strokes were limited to those for which the three paths to Suva,
Tahiti, and Honolulu were \emph{all} either less than 5\%
daylit\emph{or} more than 95\% daylit.

The strokes were selected from those occurring from May 2009 to May
2013. With these criteria, only 0.2\% of the stroke population was
accepted, that is, only 2X10\textsuperscript{6} strokes were accepted.

The high-confidence energy retrievals for the
2X10\textsuperscript{6}accepted strokes allowed each of these stroke's
''normalized electric field'' to be derived for each stroke, so as to
use all the strokes despite their widely differing stroke VLF energies
{[}\emph{Hutchins et al.} , 2013{]}. The normalization was the rms
measured electric field (in units of \selectlanguage{greek}µ\selectlanguage{english}Vm\textsuperscript{-1}) divided by
the square root of the retrieved VLF energy (in units of J). This
normalization was tabulated for each of the strokes as the electric
field in dB above 1 \selectlanguage{greek}µ\selectlanguage{english}Vm\textsuperscript{-1}J\textsuperscript{-1/2}.

The 2X10\textsuperscript{6} accepted strokes were grouped into
azimuth/distance bins, with eight azimuth bins, each 45 deg wide, and
distance bins 500 km wide. The azimuth was the average magnetic azimuth
over the path, which is approximate, as the azimuth actually varies
along each path. Within each bin, the bin median was used to show
variations versus distance and azimuth. In order to highlight azimuthal
variation, each attenuation rate was normalized by an ''all-azimuth''
average. Thus the normalized-attenuation data vary azimuthally with a
mean of unity. The normalized-attenuation data were compared to the
standard theory of idealized sharp-boundary reflection from a magnetized
D-layer {[}\emph{Wait and Spies} , 1960{]}. The agreement between the
WWLLN results and the sharp-boundary model was rather good {[}see Figure
5 in \emph{Hutchins et al.} , 2013{]}. In part this agreement may be
fortuitous. The model uses simply a sharp-boundary ionosphere, which is
a problem. Moreover, the model did not explicitly treat ``day'' or
``night'', but rather tried two electron densities. However, the cited
model stuck with 2X10\textsuperscript{7} s\textsuperscript{-1} as the
fixed collision electron-neutral rate in the case of either of those
electron densities, so they really do not illuminate the difference
between night and day reflection conditions. Another cause for caution
at the good agreement between the model and the data is that the model
was for dip angle of 0 deg, whilst the range of dip-angle magnitude in
the paths in the WWLLN study was 0 deg to \textasciitilde{}45 deg.
Therefore it is not ruled-out that the good agreement of the Wait model
and the WWLLN data may have been partially fortuitous.

The legacy results cited above concern direct measurements of the VLF
amplitude. We now mention a different set of observations in which
east-west asymmetry, or ''non-reciprocity'', was revealed: The several
observations of VLF modal interference for narrow-band transmissions.
Here for brevity we mention only a few of these reports, because our
case (broadband emissions from lightning) has wide enough bandwidth to
essentially wash-out any mode-interference effects. WWLLN's passband of
useful VLF energy is roughly 5-20 kHz {[}see, e.g., Figure 2.2
in\emph{Hutchins} , 2014{]}. This mixes interferences of different
spatial scale, so that the net result is washed-out.

An early series of measurements on modal interference {[}\emph{Crombie}
, 1966{]} documented markedly different modal-interference wavelengths
for several discrete frequencies from 18 to 24 kHz. A more recent study
{[}\emph{Samanes et al.} , 2015{]} lacked westward paths and thus could
not address this non-reciprocity issue. An even more recent study
{[}\emph{Chand and Kumar} , 2017{]} did not yield unambiguous results on
non-reciprocity.

\textbf{2b. Prior modeling of east-west asymmetry of VLF propagation}

The reflection of radio waves from the underside of the ionosphere
became an active area of research during the 1950s {[}see the historical
references given by, e.g., \emph{Barber and Crombie} , 1959; \emph{Wait
and Spies} , 1960; \emph{Wait and Spies} , 1964; \emph{Yabroff} ,
1957{]}. The problem was nontrivial due to the anisotropy of the
dielectric, associated with the gyration of charged particles about the
geomagnetic field. This was especially true for VLF waves, whose height
of reflection occurs in the lowermost ionosphere, namely the D-layer.
The strong electron-neutral collision rate in the D-layer further
complicates models of VLF reflection. The models needed to address
practical challenges, e.g.:

(a) What is the VLF reflectivity?

(b) How does it depend on solar zenith angle?

(c) How does the reflectivity depend on angle-of-incidence?

(d) How does the reflectivity depend on local propagation magnetic
azimuth (reckoned clockwise from local magnetic North) and on local
magnetic declination (''dip angle'')?

(e) How does the reflectivity depend on electron-neutral collision rate?

Starting late in the 1950's, sharp-boundary treatments of the
collisional, anisotropic VLF reflection process were set up analytically
and solved numerically with newly available digital computers
{[}\emph{Barber and Crombie} , 1959; \emph{Wait and Spies} ,
1960;\emph{Wait and Spies} , 1964; \emph{Yabroff} , 1957{]}. The first
numerical model of an arbitrarily-layered (rather than just a sharp
boundary) D-layer {[}\emph{Piggott et al.} , 1965; \emph{Pitteway} ,
1965{]} followed quickly, although its physical implications appear to
have been only slowly appreciated. The Pitteway model for the
continuously varying D-layer solved the Maxwell Equations for the
altitude-dependent, anisotropic, and complex susceptibility tensor. All
of the sharp-boundary models, as well as the Pitteway model, dealt with
the elementary reflection of an incident plane wave.

Such plane-wave models are excellent for providing insights on
''process'' questions, such as those cited in the previous paragraph.
However, for long-range ''multi-hop'' propagation, it is more efficient,
though less heuristically instructive, to cast the problem in terms of
waveguide modes in the spherical-shell Earth-ionosphere waveguide
(EIWG). The modes are akin to cylindrical waves from a point source
within a parallel-plane waveguide, except that the waveguide elements
are (approximately) \emph{concentric spherical surfaces} {[}see the
illuminating tutorial by \emph{Cummer} , 2000{]}. A waveguide model
provides a point-to-point complete description of the VLF transmission
along any given Great Circle path. This includes all portions of the
path. The first portion consists of 3-dimensional expansion of the
wavefield into a hemisphere. The next portion takes account of the first
ionospheric reflection, which effectively is a transition to
spherical-shell EIWG propagation. This transition needs many
higher-order modes to describe the wavefield, because at such a short
range (e.g, \textless{} 1000 km) a broad range of plane-wave ''angles of
incidence'' are at play {[}\emph{Cummer} , 2000{]}. Ultimately, however,
at longer range the waveguide modes simplify. For a vertical-dipole
source near ground level, and a vertical-dipole receiver also near
ground level, the modes simplify at large distances to the fundamental
Transverse Magnetic (TM) mode. Thus the transmission characteristics
vary from 3-dimensional expansion into a hemisphere, to a single 2-D,
fundamental TM mode in the waveguide.

The waveguide approach was perfected in the Long Range Propagation
Capability, or LWPC {[}\emph{Pappert and Ferguson} , 1986{]} suite of
computer codes developed by the United States Navy. The LWPC includes an
atlas of Earth-surface conductivity. The user can select a D-layer
model, usually exponential profiles of electron density and of
electron-neutral collision rate. The LWPC contains ''everything'' in one
master code suite. LWPC uses just an approximation of the D-layer
electron-density profile, but that is justified by the impossibility of
knowing any better profile at any given instant.

One adverse side-effect of its end-to-end completeness is that the LWPC
blurs (to the LWPC user) the role of \emph{local} parameters, such as
solar magnetic propagation azimuth and local magnetic dip angle. These
vary along the path, but the LWPC's end-to-end approach path-integrates
over their local variations, and all the user sees is the result of the
path integration. Thus, despite its completeness, premiere accuracy, and
reliability, the LWPC is not pedagogically illuminating for exploring
individual local \emph{processes} in isolation.

\textbf{3. Recap of part 1 of the present study}

\textbf{3a. Plane-wave reflectivity and path transmission}

This article is part 2 of a two-part study; here, we briefly recap the
results of the first part, from JHB1. The work to follow entirely
depends on JHB1, and the reader should refer to that published article
for details beyond the brief recap here.

We rely on a numerical model of plane-wave reflection from a diffuse,
collisional, anisotropic D-layer {[}\emph{Jacobson et al.} ,
2010;\emph{Jacobson et al.} , 2009; \emph{Jacobson et al.} , 2012{]}.
Our model is a modernization of Pitteway's groundbreaking treatment
{[}\emph{Piggott et al.} , 1965; \emph{Pitteway} , 1965{]}. We represent
the electron-neutral collision rate by an exponentially declining
function of altitude as is common in this field. For the electron
density, we use an exponentially increasing function of altitude, also
common in the field {[}see, e.g., Eq. 3.23, Section 3.2.3,
in\emph{Volland} , 1995{]}. See Table 1 for details.

Figure 1 summarizes the prediction of our plane-wave reflection model.
The vertical axis is the amplitude reflection coefficient, \emph{R} ,
from the D-layer for a typical long-range-propagation angle of
incidence, in this case chosen as 85 deg. The reflection coefficient
shown has been averaged over all frequencies from 5 to 20 kHz. As shown
in JHB1, \emph{R} varies continuously with solar zenith angle, but we
show the pure-day and pure-night extreme cases only. On the left of
Figure 1 is shown (a) the day-profile D-layer result, while on the right
is shown (b) the night-profile D-layer result (refer to Table 1 for
profile parameters). The abscissa is the wave magnetic propagation
azimuth. A separate curve is shown for each abs(dip angle), from 5 deg
(blue) to 85 deg (red), in steps of 5 deg. The curves for dip = 30 deg
and 45 deg are labeled in the night profile. For both (a) and (b), the
curve for dip = 45 deg is dashed. The horizontal black line marks the
nadir of the night-profile reflectivity level for dip = 45 deg.

How do we employ the single-reflection reflectivity from a plane-wave
model, in the context of long-range (''multi-hop'') propagation of
quasi-cylindrical waves in a spherical-shell waveguide? The article on
the first half of this project, JHB1, shows how this is done
heuristically but with satisfactory agreement with observations: First,
we correct the wave amplitude for the varying cross-sectional area of a
ray-bundle on the spherical Earth (see Eq. 7 in JHB1). Second, we rely
in JHB1 on a free parameter ''\emph{r} '' , which is the effective
number of reflections per reference distance \selectlanguage{greek}ρ\selectlanguage{english}\textsubscript{0} = 1000
km (= 1 Mm). In JHB1 we demonstrated how comparison with observed
received electric-field amplitude resulted in a fit for \emph{r} in the
range 3 \textgreater{} \emph{r} \textgreater{} 2.

Those two heuristics (correcting for the ray-bundle area, and invoking
an effective reflection-per-pathlength) were used in JHB1 to crudely
approximate long-range waveguide transmission in terms of the
single-hop, plane-wave reflectivity model. We define a ''logarithmic
reference transmission'', \emph{assuming perfect ground conductivity} ,
along the Great Circle Path segment \emph{L} \textsubscript{i,m} from
VLF emission point ''m'' to sensor point ''i'' :

\(ln(ref.\ transmission)\ =\ \ \frac{1}{\rho_{0}}\int_{0}^{L_{i,m}}{\ \ ln(R[Z_{i,m}}(t_{0}),\ \alpha_{i,m},\ I_{i,m}])ds_{i,m}\ +\ C(L_{i,m})\)Eq. (1)

where

\emph{L} \textsubscript{i,m} = arcdistance along Great Circle Path from
lightning location m to station i

\emph{Z} \textsubscript{i,m}(t) = time-dependent, location-dependent
solar zenith angle along path i,m

\selectlanguage{greek}α\selectlanguage{english}\textsubscript{i,m} = location-dependent magnetic propagation azimuth
along path i,m

\emph{I} \textsubscript{i,m} = location-dependent magnetic dip angle
along path i,m

\emph{R} (\emph{Z} \textsubscript{i,m}(t),
\selectlanguage{greek}α\selectlanguage{english}\textsubscript{i,m},\emph{I\textsubscript{i}} \textsubscript{,m}) local
instantaneous plane-wave reflectivity

\emph{ds} \textsubscript{i,m} = differential path element along Great
Circle Path i,m

\selectlanguage{greek}ρ\selectlanguage{english}\textsubscript{0} = 1000 km

The term \emph{C(L\textsubscript{i,m})} in Eq. (1) is the geometrical
correction due to the variation of ray-bundle cross-sectional area. We
tabulate the correction, relative to its value at the reference distance
1000 km:

\(C(L_{i,m})\ =ln\ \{\sqrt{\frac{sin(\rho_{0}/R_{E})}{sin(L_{i,m}/R_{E})}}\text{\ \ }\}\)Eq. (2)

where \emph{R\textsubscript{E}} is the Earth's radius.

The logarithmic reference transmission (Eq. 1) must be multiplied by the
fitted parameter \emph{r} to give an estimate of the actual logarithmic
path transmission \emph{assuming zero ground losses} (see Eq. 9b in
JHB1). This r parameter was fitted to lie in the range of 2 to 3.
Physically, it is the number of hops per 1000 km reference distance,
subject to our model's assumption of 85-deg angle-of-incidence.

Ignoring ground losses would be unacceptable if we were trying to
calculate absolute transmission in the waveguide. However, our
application involves examining the difference between day and night
conditions on the propagation anisotropy. The ground conductivity
effects are unchanged (on a given path) between day and night. Thus
modeling only D-layer losses is a satisfactory (though not perfect)
approach for our study of day-versus-night differences.

A further convenient simplification introduced in JHB1 is that we
actually solve for the log reflectivity ln(\emph{R)} only for the two
extreme cases of pure day and pure night. Any intermediate case is
approximated by a linear combination of pure-day and pure-night, using a
smooth function of solar zenith angle (see Eqs. 10-11 in
JHB1).\emph{This is done locally, at each point along the path integral
in Eq. (1), and for local solar zenith angle obtaining at the instant of
the lightning stroke.} There is a crucial difference between, on the one
hand, making the linear combination locally (which we do), versus, on
the other hand, evaluating the path integral along the entire path both
for an artifactual day and an artifactual night case, then taking a
linear combination of those two results based on the proportion of the
path that is daylit. The approach latter would be clearly incorrect.

\textbf{3b. Disfavoring of nighttime (relative to daytime)
magnetic-westward propagation}

Note that for abs(dip angle) \textless{} 45 deg, the night-profile
reflectivity (Figure 1b) for propagation toward magnetic west (270 deg)
is \emph{less than the day-profile reflectivity (Figure 1a)} . This
favoring of daytime over nighttime transmission for abs(dip angle)
\textless{} 45 deg actually applies over a broad azimuth sector centered
on magnetic west. Thus, for essentially half of all possible dip angles,
and for essentially half of all possible propagation magnetic azimuths,
the \emph{nighttime reflection is predicted by our
model}{[}\emph{Jacobson et al.} , 2010; \emph{Jacobson et al.} ,
2009{]}\emph{to be disfavored relative to the daytime reflection, and
for small dip angle deeply disfavored} . This surprising and
counter-intuitive feature is not remarked elsewhere in the VLF
literature, and thus the burden is on us to provide observational
support for this counter-intuitive claim. Intuition would suggest that
nighttime propagation should be \emph{less} lossy than daytime, because
the nighttime reference height (85 km in our model) has only 17\% as
much electron-neutral collision rate as does the daytime reference
height (73 km in our model).

The remainder of this article relies on the first part of the study
(JHB1) for a detailed development of the model theory. Readers should
consult JHB1, which was published as Open Access and hence is without
cost to the reader.

\textbf{3c. Summary of observational results from part 1 of the present
study}

We presented in JHB1 a method to study the behavior of the inter-station
ratio of VLF stroke amplitudes, for strokes that are simultaneously
recorded at multiple WWLLN stations. This approach combined numerous
recurrent strokes from long-duration lightning clusters to build a
time-series of the ratio for a major portion of the UT day. The time
variations of the sliding-averaged ratio are dominated by transient
excursions coinciding temporally with those periods when the solar
terminator is present along one or both of the paths. See, e.g., Figure
2 in JHB1. This strongly motivates a model incorporating significant
control by the solar zenith angle.

Our plane-wave model predicts that magnetic-westward propagation has
less waveguide transmission than does magnetic-eastward
propagation.\emph{Crucially, the anisotropy is extremely magnified for a
night ionosphere.} 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 solar-zenith-angle control on the waveguide transmission,
our model takes a weighted combination of pure-day and pure-night
solutions, \emph{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 plane-wave-reflection theory
successfully accounted for the gross features of the solar-terminator
transients; see, e.g, Figures 7-10 in JHB1.

Our model predicts, counter-intuitively, that the magnetic-westward
attenuation at low magnetic latitude will be much deeper during night
than during day conditions. Unfortunately, this suppression of
magnetic-westward propagation also largely eliminates the availability
of sufficiently numerous recurrent recordings of those signals at our
low-latitude stations. Thus the amplitude-ratio method pursued in JHB1
was inherently unable to check on the model's most intriguing and
counter-intuitive prediction.

Thus our method to follow, rather than using lightning detections that
exist, is designed to demonstrate the pattern of where and when
detections do \emph{not} exist.

\textbf{4. WWLLN evidence on day/night differences in anisotropic VLF
propagation}

\textbf{4a. WWLLN database}

The overall epoch for this study is 1 December 2009 through 31 May 2021,
Universal Time (UT). Within that overall epoch, numerous WWLLN recording
stations began operation, occasionally interrupted operation, and (in a
few cases) ceased operation. For much of the overall epoch, at any time
WWLLN had \textgreater{} 50 active stations worldwide. At present (2021)
the census is \textgreater{} 60 active stations.

Our methodology in the following is to develop statistics on
the\emph{patterns of detection and non-detection by selected stations.}
We use the entire WWLLN network product to define the overall population
of WWLLN-located lightning strokes. This population is defined in a
separate day file for each UT day. We then focus on ten selected
stations located at magnetic low and mid latitudes. We develop
statistical maps of the detection/non-detection of the overall WWLLN
stroke population, by each of these ten selected stations.

To clarify ''detection'', we mean that a particular stroke is
''detected'' by a particular station \emph{if that station participates
in the solution for that particular stroke's location} . This is a
stringent definition of ''detection'', compared to a more permissive
definition that the station records a signal (whether or not that signal
has the requisite amplitude and signal-to-noise to allow participation
in the network solution.) The problem with the permissive definition is
that one can never be sure, or even reasonably sure, that a signal is
associated with a given stroke, if that signal cannot be used in the
position/time solution for that stroke.

Table 2 lists pertinent metadata about the ten selected stations. Three
of the stations (Atuona, Tahiti, and Honolulu) are in the Pacific ocean
and have dominantly over-seawater paths from abundant lightning in both
their Eastern and Western sectors. Another station (Costa Rica) is on a
relatively narrow land bridge between major oceans. Two stations (Peru
and Dakar) are on the western periphery of lightning-rich continents.

For the statistics on each of the ten selected stations, we define a
latitude band within which to include WWLLN strokes. This band is -40 to
+40 deg N for nine of the stations, but for Honolulu the band is
displaced to -30 to +50 deg N, in order to include the strokes in the
northern continental United States. The population of WWLLN strokes
within the latitude band is used to detect detection/non-detection by
the respective selected station.

The population of WWLLN strokes within the latitude band and available
for detection by the respective selected station varies from a maximum
of \textgreater{} 2X10\textsuperscript{9} strokes (for both Honolulu and
Tel Aviv) down to 3.2X10\textsuperscript{8} strokes (for Belem). This
disparity is driven mainly by the difference between the number of
operating days for Honolulu (3873) or for Tel Aviv (3987), versus for
Belem (566) within the overall epoch.

\textbf{4b. Qualitative demonstration of geomagnetic and zenith-angle
control over detection}

Before embarking on a systematic quantitative analysis, we show a
qualitatively clear example of the control over detection exerted by
solar zenith angle and by geomagnetic parameters. Figure 2 shows the
case of Atuona station, near the mid-Pacific. This example conveniently
illustrates the situation for low dip angle and nearly-zonal magnetic
propagation azimuth everywhere along the paths eastward and westward to
regions of abundant lightning. In Figure 2, the station is a black
rectangle symbol. In Figure 2(a), to the East of Atuona is shown a red
rectangular box in northwestern South America. We select all WWLLN
strokes within that box. For each stroke within that box, we calculate
the solar zenith angle (at D-layer height) for all points along the path
to Atuona from the stroke, and characterize each stroke by the
proportion of the path that is in daylight. Figure 2(b) shows the
daylit-fraction distribution of all strokes in the red box of Figure
2(a). The distribution is flat except for roughly equal peaks both at
pure-dark (daylit fraction = 0) and at pure-daylit (daylit fraction =
1). We now ask, what is the Atuona detection efficiency (DE) for these
strokes, versus the daylit fraction parameter? This is shown in Figure
2(c). The DE peaks toward maximum daylight, and is suppressed (by an
order of magnitude) for daylit fraction \textless{} 0.6.

Now let us define a ''control'' case, which is shown in the right column
of Figure 2. Figure 2(d) shows in red a ''West box'' over the
Australasia sector. It is slightly further from Atuona than the East
Box, but is roughly comparable in dip angle along the paths to Atuona.
Figure 2(e) shows the daylit-fraction distribution for all strokes in
the West box. Figure 2(f) shows the DE for Atuona detection of those
strokes, as a function of daylit fraction. Now the DE for eastward
propagation is relatively indifferent to daylit fraction, and the median
DE for the west box is two orders-of-magnitude higher than the median DE
for the East box (Figure 2c), and one order-of-magnitude greater than
the maximum DE for the East box (Figure 2c). We note that this is a case
where the paths to Atuona from either the West or the East box are
everywhere quasi-zonal in magnetic azimuth, and are everywhere at very
small dip angle (-20 deg to +20 deg). We chose this because of its
convenience for a qualitative exercise like Figure 2.

This example qualitatively demonstrates, within the context of low dip
angle, (a) the dramatic difference between propagation at eastward
magnetic azimuth versus westward magnetic azimuth, and (b) the extreme
favoring of daylit propagation over night propagation for westward
magnetic azimuth. This latter feature has not previously been remarked
in the VLF literature.

\textbf{4c. Geomagnetic context for the ten selected stations}

We now show the geomagnetic context of each of the ten selected stations
and of the propagation paths connecting them to lighting strokes in
their respective latitude band described in Table 2. The geomagnetic
model is the International Reference Geomagnetic Field, or IGRF
{[}\emph{V-MOD} , 2010{]}. Figure 3 contains a separate panel for each
of the ten stations. Color indicates the geomagnetic dip angle's
absolute value, in deg. Black is dip = 0 deg. For nine of the panels,
red is 74 deg, while for one (Figure 3i, Honolulu) the band is offset
and the maximum dip angle is 76 deg. In addition to color-coding,
discrete curves of \textbar{}dip\textbar{} = 30, 45 deg are overlaid on
the map. The station is marked with a rectangle symbol, either white or
black, to contrast with its immediate background color. The color
shading covers the latitude band in which strokes are considered for
detection/non-detection by the respective station.

Because of the extremely low transmission of VLF over Antarctic (or, to
a lesser extent, Arctic) ice, we wish to exclude strokes whose Great
Circle paths to the selected station reach further poleward than
geographic latitude +/- 55 deg. This excludes strokes roughly within a
cone centered on the respective station's antipode, which we blank-out
with white. Thus, for example, in Figure 3(a) the antipode of Atuona is
situated near the southern Red Sea. Each station has its own antipode,
and cone centered there, in which we do not gather statistics regarding
stroke detection by that station.

Recall that the model results (Figure 1) predict a nighttime increase of
attenuation for propagation toward magnetic West, relative to daytime,
for \textbar{}dip angle\textbar{} \textless{} 45 deg. This nighttime
increase in attenuation toward magnetic West becomes especially severe
for \textbar{}dip angle\textbar{} \textless{} 30 deg. Figure 3 shows
visually that this band of enhanced nighttime westward attenuation
occupies most of the important lightning prone areas {[}\emph{Christian
et al.} , 2003{]}, excepting only the continental United States. That
is, the nighttime disfavoring of magnetic-westward VLF propagation is
not going to be a mere academic curiosity confined to a small region,
but rather applies to most regions of relevance to global VLF lightning
location.

\textbf{4d. Spatial patterns of observed and predicted detectability}

\textbf{4d.1 Approach}

We now start a formal comparison of the observed and predicted detection
patterns of each of the ten selected stations. Separately for each
station, we define two cohorts of strokes within the accepted latitude
band. The first cohort contains the strokes whose Great Circle Paths to
the station are \textgreater{} 80\% sunlit at the instant of the stroke,
at D-layer height. This first cohort represents mostly-daytime
propagation. The second cohort contains the strokes whose Great Circle
Paths to the station are \textless{} 20\% sunlit, representing
mostly-nighttime propagation.

In addition to those statistics based on observation, we calculate the
logarithmic reference transmission (Eq. 1 above) for each Great Circle
Path under two artificial conditions: that the entire path be either in
daylight or in darkness. These yield ''day'' and ''night'' logarithmic
reference transmissions.

Finally, we calculate the instantaneous logarithmic reference
transmission, \emph{using the actual instantaneous solar zenith angle at
each point along each path} , for all strokes. The distribution of
logarithmic reference transmission shows the strokes that
are\emph{available} for the selected station to detect. The
sub-distribution of logarithmic reference transmission \emph{only for
the strokes that are detected by the selected station} shows the
relationship between detectability (by the selected station) and
logarithmic reference transmission (relative to the selected station).
We would expect that if the model has some correlation to observational
reality, then the strokes detected by the selected station would be
bunched at the high-transmission end of the distribution, and would be
sparse or absent in the low-transmission end of the distribution. On the
other hand, if the model were basically worthless, then there would be
no strong correlation between observed detectability and model-predicted
logarithmic reference transmission.

The reader should keep in mind that stations do not all have the same
effectiveness in detecting lightning {[}\emph{Hutchins} , 2014{]}. We
will call this ''sensitivity'', but this does not mean something so
simple as system gain. Rather, the two most important factors are,
first, the level of background VLF noise affecting the selected station,
and, second, the abundance of nearby lightning {[}\emph{Hutchins} ,
2014{]}. The effect of abundant nearby lightning is to reduce the
ability of the station to participate in network detections of distant
lightning strokes. This is because each WWLLN station has a
software-adjusted trigger threshold for capturing a pulse to become a
candidate for participation in a network location of strokes. The
threshold is sluggishly (\textasciitilde{}2 minutes of inertia)
adjusted, so as to continuously keep the rate of station triggers not
greater than 3 per second. Abundant nearby lightning interacts with this
feedback to increase the trigger level and thus reduce the ability of
that station to trigger on distant lightning.

\textbf{4.d2 Ten case studies of patterns of detection/non-detection}

\textbf{Atuona case}

Figure 4(a) maps the density of the first cohort of strokes, having
mostly-sunlit paths to Atuona. The color scale is \emph{relative to the
maximum-density pixel in this plot} , with blue meaning 0.01Xmaximum,
and red meaning maximum. The resolution is 1-deg X 1-deg. The white
areas are \textless{} 0.01Xmaximum. The stroke density is displayed only
within the -40 to +40 deg N band. The curves of \textbar{}dip
angle\textbar{} = 30, 45 deg are shown in heavy black, while the
geomagnetic equator is shown as a thinner black curve. Neither the
\textbar{}dip angle\textbar{} curves, nor the stroke density, are shown
within the antipodal cone. Also, the -45 deg dip-angle curve is not
shown where (southmost South America) it is outside the latitude band.

Figure 4(b) is like Figure 4(a), \emph{except that the density is only
for the subset of the day-cohort strokes that are detected by Atuona} .
Thus comparison of Figure 4(a, b) gives a visual map of the pattern of
day-cohort detection/non-detection by Atuona. Note that the eastern
two-thirds of South America's day-cohort lightning is not detected by
Atuona, whilst the lightning in SE Asia and Indonesia, though no closer,
is largely detected.

Reminder: The color range in the second panel, Figure 4(b), is
determined \emph{only} by the densities in Figure 4(b), and
is\emph{unrelated} to the color range in the first panel, Figure 4(a).
Thus for example, the threshold for blue (0.01Xmaximum) is different
(and smaller) in Figure 4(b) than in Figure 4(a). This allows blue cells
to appear in the second panel, in principle, at a few locations that are
white (sub-blue) in the first panel.

Figure 4(c) maps the value of the day logarithmic reference
transmission, for all grid points within the selected latitude band,
regardless of the incidence of lightning there. The only exception is
that the transmission is whited-out in the antipodal cone. The displayed
value is lumped into just four ranges of logarithmic reference
transmission: \textgreater{} -2 (red), -2 to -3 (yellow), -3 to -4
(green), and \textless{} -4 (blue).

Figures (4d-e) are exactly like Figures 4(a-b) except for the second
cohort of strokes, having mostly dark paths to Atuona. Figure 4(f) is
like Figure 4(c), except for the \emph{night} logarithmic reference
transmission. In night conditions, the asymmetry becomes more dramatic.
Atuona detections in South America become insignificant. Comparing
Figures 4(c,f), we see that the model prediction is consistent with
observations.

Figure 4(g) shows histograms of the actual
\emph{instantaneous}logarithmic reference transmission, taking account
of the\emph{instantaneous solar zenith angle at each point along the
path} . This is not the contrived ''day'' or ''night'' prediction of
Figures 4 (c,f). The black curve in Figure 4(g) is for all
9.21X10\textsuperscript{8} strokes within the latitude band, while the
red curve is for only the 1.09X10\textsuperscript{8} strokes in that
band detected by Atuona. By comparing the two curves, it is apparent
that Atuona's detection rate falls off rapidly for logarithmic reference
transmission \textless{} -2, and is completely insignificant for
\textless{} -4. These are empirical facts based on the distribution of
lightning amplitudes, the proximity of the lightning to Atuona, the
performance of the network, and the performance of this particular
station. The empirical evidence of Figure 4(g) allows us to interpret
the model predictions for contrived pure-day (Figure 4c) and contrived
pure-night (Figure 4f). The red-shaded regions correspond to logarithmic
reference transmission \textgreater{} -2. We can thus interpret the red
regions in Figures 4(c,f) as having unimpeded detectability (at least as
far as D-layer effects are concerned.) The yellow-shaded regions are
predicted to have relatively lower detection success, though not zero.
Green is even lower, and there are predicted to be essentially no
detections in the blue-shaded regions, where logarithmic reference
transmission is \textless{} -4. With that as a guide, we can now
appreciate that the behavior of Figure 4(b), relative to Figure 4(a), is
roughly consistent with the contrived day model (Figure 4c) and the
empirical distribution (Figure 4g). Similarly, the behavior of Figure
4(e), relative to Figure 4(d), is roughly consistent with the contrived
night model (Figure 4f) and the empirical distribution (Figure 4g).
Notably, Atuona's complete non-detection of any lightning in South
America for mostly-dark paths (Figure 4e) is consistent with the
all-blue shading of South America (Figure 4f) in the night model.
Likewise, Atuona's strong detection in the Australasia sector for
mostly-dark paths (Figure 4e) is consistent with that region's being
shaded red (Figure 4f) in the night model.

The map-based displays (Figures 4a-f) are useful for illustrating the
geographic patterns of detection/non-detection by Atuona for two extreme
cases, as well as comparing those patterns with the respective model
predictions. However, these map-based displays are extremely complicated
to follow, and are patchy in their coverage due to the uneven
geographical and temporal occurrence of lightning {[}\emph{Christian et
al.} , 2003{]}.

Ultimately, the entire quantitative outcome of our data for Atuona can
be distilled into the far simpler and clearer Figure 4(g), which
uses\emph{all} the available strokes in the latitude band, without
parsing for contrived special cases (mostly-day, mostly-dark). The
parent distribution (black curve) shows that \textgreater{} 50\% of the
network-detected strokes have logarithmic reference transmission (to
Atuona) \textless{} -3, whilst by contrast, for the subset detected by
Atuona (red curve) there are very few detected strokes in that range.
Thus the predictive model is consistent with the pattern of Atuona's
detection/non-detection. (We take as axiomatic that large signals tend
to be easier to detect than are small signals.)

\textbf{Tahiti case}

Tahiti is close to Atuona though at somewhat larger \textbar{}dip
angle\textbar{} \textasciitilde{} 30 deg. Figure 5 is like Figure 4, but
for the Tahiti case. Because Tahiti has operated during almost the
entirety of the study epoch, it has more strokes both in the parent
distribution (1.94X10\textsuperscript{9}) and the Tahiti-detected
distribution (1.93X10\textsuperscript{8}) compared to Atuona. The Tahiti
detections are almost entirely confined to logarithmic reference
transmission (to Tahiti) \textgreater{} -2.5 (Figure 5g), consistent
with being a less sensitive station than Atuona. Tahiti almost totally
fails to detect lightning in the Americas in night conditions (Figure
5e), while Tahiti is highly successful with the Australasia sector.

Again, as commented earlier in the case of Atuona, all the quantitative
evidence is condensed into Figure 5(g); the geographical map
presentations are qualitative by comparison. And again, the sharp
cut-off of the red curve in Figure 5(g) shows that the model has
predictive value for Tahiti detection/non-detection.

\textbf{Peru case}

Figure 6 shows the Peru case. This is the first of our two cases on the
west margins of lightning-rich continents. Like Tahiti, Peru can detect
almost only for predicted logarithmic reference transmission (to Peru)
\textgreater{} -2.5 (Figure 6g). In the daytime case (Figures 6a-c),
Peru can detect well throughout South America. In the nighttime case
(Figures 6d-e), Peru cannot detect strokes in the eastern half of its
own continent, even as it detects in far more distant Micronesia. This
is well predicted by Figures 6(c,f). All the quantitative outcomes of
the data for Peru are summarized in Figure 6(g) ), and it indicates the
model is consistent with the Peru station's detection/non-detection. If
the model were fundamentally inconsistent with the observations, then
the red curve in Figure 6(g) would not be selective for the
high-transmission end of the abscissa.

\textbf{Costa Rica case}

Figure 7 shows the Costa Rica case. Costa Rica can detect mostly for
predicted logarithmic reference transmission (to Costa Rica)
\textgreater{} -2.5 (Figure 7g). During night conditions (Figures 7d-e),
Costa Rica loses detection for strokes in much of eastern South America.
During those same night conditions, Peru gains detections in the far
more distant Australasia and Micronesia sector, where there are few
daytime detections (Figures 7a-b). All the quantitative outcomes of the
data for Costa Rica are summarized in Figure 7(g), and it indicates the
model is consistent with the Peru station's detection/non-detection.

\textbf{Belem case}

Figure 8 shows the Belem case. One of our least sensitive stations,
Belem is essentially unable to detect for strokes whose predicted
logarithmic reference transmission (to Belem) \textless{} -2.0 (Figure
8g). Some of western Africa lightning is detected by day, but nothing by
night. This is consistent with the day and night model predictions
(Figures 8c,f). All the quantitative outcomes of the data for Belem are
summarized in Figure 8(g) ), and it indicates the model is consistent
with the Belem station's detection/non-detection.

\textbf{Dakar case}

Figure 9 shows the Dakar case. Like Peru, Dakar lies on the western edge
of a lightning-prone continent. Figure 9(g) shows that Dakar cannot
detect lightning whose predicted logarithmic reference transmission (to
Dakar) \textless{} -2.0. During daytime Dakar can detect strokes in all
but the easternmost part of Africa (Figures 9a-b), but during the night,
Dakar loses all of Africa coverage save for the western bulge proximal
to the station itself (Figures 9d-e). All of these effects are
consistent with Figures 9(c,f,g). The model predictions are grossly
consistent with the Dakar pattern of detection/non-detection.

\textbf{Pune case}

Figure 10 shows the Pune case. Pune is between the Australasia's and
Africa's lightning-prone regions. As seen in Figure 10(g), Pune's
detections are almost all for strokes whose predicted logarithmic
reference transmission (to Pune) \textgreater{} -2.3. During nighttime
(Figure 10e) but not during daytime (Figure 10b), Pune can detect
significant lightning in South America. Similarly, during nighttime
(Figure 10e) Pune cannot detect lightning further east than Thailand,
while in daytime (Figure 10b), Pune's detection extends further eastward
over Borneo. These are predicted by the model in Figures 10(c,f,g). The
model predictions are consistent with the pattern of Pune
detections/non-detections.

\textbf{Singapore case}

Figure 11 shows the Singapore case. Similar to Pune, Singapore
detections are mostly confined to strokes whose predicted logarithmic
reference transmission (to Singapore) \textgreater{} -2.3 (Figure 11g).
The geographical patterns for the night cohort of strokes show extreme
asymmetry favoring strokes on the West of the station and disfavoring
strokes on the East of the station. The pattern of
detection/non-detection by Singapore is grossly consistent with the
model predictions (Figures 11 c,f,g).

\textbf{Honolulu case}

We complete this survey with two cases where the selected station is at
higher magnetic latitude than so far. Figure 12 shows the Honolulu case.
The bulk of Honolulu's detections are for strokes whose predicted
logarithmic reference transmission (to Honolulu) \textgreater{} -2.5
(Figure 12g), with a low tail out to -4, as had been the case with
Atuona. For the nighttime cohort of strokes (Figures 12d,e), Honolulu
detects very few strokes in South America, whilst its more distant
detections on the West go all the way across India. The Honolulu
patterns of detection/non-detection are grossly consistent with the
model predictions (Figures 12c,f,g).

\textbf{Tel Aviv case}

Finally, Figure 13 shows the Tel Aviv case, located at dip angle
\textgreater{} 45 deg. Virtually all of Tel Aviv's detections are for
strokes whose predicted logarithmic reference transmission (to Tel Aviv)
are \textgreater{} -2.5 (Figure 13g). Despite its relatively high
magnetic latitude, long range detection eastward or westward into the
low magnetic latitudes still displays the asymmetry favoring strokes
from South America over strokes in Australasia, particularly for the
night cohort (Figure 13e). The Tel Aviv pattern of
detection/non-detection is grossly consistent with the model predictions
(Figures 13c,f,g).

\textbf{4e. Closing the observational case that the model is consistent
with the data}

We have presented observations of the geographical patterns of
detection/non-detection for ten selected stations around the globe, at
low and low-middle latitudes. The geographical patterns are shown for
separately for mostly-day, and then mostly-night transmission paths. The
observed patterns of detection/non-detection are consistent with the
patterns of predicted logarithmic reference transmission, for the
respective day or night cases. More quantitatively, the distributions of
actual logarithmic reference transmission to each selected station, both
for the parent distribution and for the subset of strokes in whose
location the selected station is a participant (i.e. detected by the
selected station), show that the paths for \emph{detected} strokes are
clustered at the high-transmission end of the parent distribution. Thus
the model predicts which cases are more likely to be detected, and which
are not.

As mentioned earlier, this logic rests on an axiom: All other things
being equal, a strong pulse is more likely to be detected than is a weak
pulse. And we assume, all other things being equal, that paths involving
relatively weaker transmission will cause weaker detected pulses than
will paths involving relatively stronger transmission.

We now perform a ''sanity check'' on this key assumption. Figure 14
shows distributions of the detected, raw ADC amplitudes for pulses
detected by the Tel Aviv station. The ADC is 16-bits deep (0 to 65535),
but we show the distributions out to only ADC level 5000. The two panels
in Figure 14 are for two adjacent tranches of modeled logarithmic
reference transmission: (a) \textgreater{} -2, and (b) in the range -2
down to -2.5. The shoulder at about ADC level 100 - 200 corresponds to
the local-time servo adjustments of the station's software trigger
threshold. The higher-transmission distribution (a) contains
1.8X10\textsuperscript{8} detections, while the lower-transmission
distribution (b) contains only \textasciitilde{}12\% as many detections.
Moreover, in (a) the high-transmission distribution's tail, relative to
the distribution's peak, is much more relatively populated than in (b).
Finally, whereas in (a) the peak occurs at ADC level \textasciitilde{}
800, in (b) it has retracted to \textasciitilde{} 500.

Thus Figure 14 supports the picture that the high-transmission
population's extended tail (to higher values of detected ADC level)
becomes depleted at lower transmission, with those tail members being
swept to the left end of the distribution. Most of those then are swept
to sub-threhold ADC level, but some remain above the threshold and
constitute the peak in (b). Let us see if this picture makes
quantitative sense. The change in logarithmic reference transmission
between these two tranches is in the range 0.5 to 0.75, depending on
where, within a tranche, it is figured. The first part of this study
demonstrated that the ''r'' parameter, which multiplies the logarithmic
reference transmission to give the actual physical transmission, was
fitted by the data of JHB1 to lie in the range from 2 to 3 Nepers (see
Figures 6 and 9, and discussion thereof, in JHB1). Let us choose 2.5
Nepers. Then the change in logarithmic reference transmission between
these two tranches in the range 0.5 to 0.75 Nepers corresponds to a
change in \emph{physical} logarithmic transmission in the range 1.25 to
1.88 Nepers. In linear amplitudes, the range is a multiplicative factor
from \textasciitilde{}3.5 to \textasciitilde{}6.5. This implies that the
transition from Figure 14(a) to 14(b) can be understood as taking tail
members in (a) and moving them leftward (to lower ADC level) in (b) down
to ADC levels only 1/3.5 to 1/6.5 as big. Fortunately, we see that the
tail in (a) contains sufficient population to permit this simple occur.
Thus the relative distributions of detected raw ADC levels are
consistent with the predicted transmissions of adjacent tranches of the
transmission distribution.

\textbf{5. Conclusions}

This is the second part of a two-part study of broadband VLF propagation
from lighting strokes to WWLLN stations. The first part of the study
(JHB1) had developed a model for the effects of the ionospheric D-layer
on long-range VLF transmission in the Earth-Ionosphere Waveguide. The
model makes the counter-intuitive prediction that, for dip angles in the
range -30 to +30 deg, propagation toward the west half of magnetic
azimuths will be dramatically worse during conditions of darkness (along
the propagation path) than during conditions of daylight. This feature
had never been remarked before in the literature, although it is in fact
also embedded in the standard LWPC code. We surmise that the reason the
feature had never been remarked is that the LWPC is an end-to-end
treatment that tends to obscure, to the code's user, the details of
differential transmission at any one point on the propagation path.

Our model had been applied in JHB1 to explaining the inter-station
ratios of signal amplitudes from the same stroke at different stations.
That approach, in common with all virtually all other approaches done by
prior workers, was based on the measurement of VLF amplitudes. However,
we found that the amplitude-based method was inadequate to the test the
model's counter-intuitive prediction regarding the day/night control of
westerly propagation. That is because the amplitude-based approaches
require detections in order for amplitudes to be determined. ''No
detections, no amplitudes''.

This second part of the study circumvents that problem by adopting an
opposite approach. Rather than use received signal amplitudes as the raw
data, we now use the observed statistical patterns of
detection/non-detection. We compare those patterns to our model's
predictions of the D-layer contributions to path transmission. By
focusing on the variations between daylit and dark conditions, we also
avoid the confounding effect of ground losses, as the latter are
invariant between daylit and dark conditions.

We highlight the geographical patterns of detection/non-detection from
each of ten selected stations arranged around diverse longitudes. For
each of these stations, we identify strokes whose paths are either
mostly daylit or mostly dark. The patterns of detection/non-detection in
these two special cases are then compared with the predictions of our
transmission model, for either all-lit paths or all-dark paths
respectively. The spatial agreement between observation and model is
good. We then use \emph{all} the strokes, not just those whose paths are
mostly lit or mostly dark, and calculate the modeled logarithmic
reference transmission along each stroke's path to the selected station,
taking account of the instantaneous solar zenith angle at each point
along the path. We tally the distribution of logarithmic reference
transmission, both for the parent population of strokes, and for the
subset of strokes that are detected by the selected station. We find
consistently, for all of our ten selected stations, that the detected
subset's distribution of logarithmic reference transmission is entirely
crowded to the high-transmission end. This suggests that the model's
predictions of transmission are pertinent.

Finally, and most importantly, our ten case studies robustly demonstrate
that for dip angles in the range -30 to +30 deg, during conditions of
darkness there is dramatically worse transmission from magnetic East to
West then from magnetic West to East, whereas for daylit conditions,
this is much less pronounced. These findings are operationally
significant for long-range lightning detection. For example, WWLLN's
Pacific stations Atuona, Tahiti, and Honolulu are not able in dark-path
conditions to contribute significantly to locating lightning in South
America, though they are extremely useful over comparable distances with
lightning in Australasia. Similarly, under dark-path conditions, Peru
basically misses the eastern half of its own continent, and Dakar sees
even less of its own continent. For the same reason, during dark-path
conditions, Pune is very good for detecting lightning in Africa but
misses almost all lightning at similar distances in Australasia. These
effects are not subtle, when viewed geographically in terms of areas of
detection and non-detection.

\textbf{Acknowledgements}

The authors are using data from the World Wide Lightning Location
Network, a collaborative consortium of over seventy worldwide
collaborators, managed at the University of Washington. The data would
not exist but for the cooperative efforts of all of WWLLN's
participants. 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/

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\textbf{Figure captions}

Figure 1: Model amplitude reflectivity (vertical axis) vs propagation
magnetic azimuth (horizontal axis). Color marks magnetic dip angle, from
5 deg (blue) to 85 deg (red). Curve for dip angle = 45 deg is dashed.
Model assumes angle-of-incidence is 85 deg, and is averaged over the
band 5 - 20 kHz. (a) Day case. (b) Night case, with labels on curves for
dip angle = 30, 45 deg. The horizontal black line on both panels is
through the minimum of the curve for dip angle = 45. deg in (b). See
Table 1 for parameter values.

Figure 2: Comparison of Atuona detection efficiency against strokes in
two roughly equidistant geographic boxes. (a) Showing the East box, over
northwestern South America. (b) For all strokes in East box, showing the
distribution of fraction of instantaneous path (to Atuona) that is
daylit. This includes all strokes in the East box, not just those
detected by Atuona. Horizontal resolution is 0.01. (c) Detection
efficiency (DE) of Atuona against strokes in East box, as a function of
the instantaneous daylit fraction of the path. (d) Showing the West box.
(e) For all strokes in West box, showing the distribution of fraction of
instantaneous path (to Atuona) that is daylit. (f) Detection efficiency
(DE) of Atuona against strokes in West box, as a function of the
instantaneous daylit fraction of the path. Note the order-of-magnitude
difference in DE scales between (c) East and (f) West boxes.

Figure 3: Geomagnetic setting of each of the ten selected stations. Each
station is shown as a rectangle, either white or black so as to contrast
with the background. The color shading of the background is magnitude of
dip angle, from 0 (black) to 74 deg (red). Only (i) has a different
color scale: 0 (black) to 76 deg (red). The colors are shown within
latitude bands -40 to +40 deg, except -30 to +50 deg for (i) Honolulu.
Also, discrete curves are drawn, where dip angle = -30, -45, +30, and
+45 deg. The white cone at each station's antipode is excluded from the
analyses, because the Great Circle Paths from strokes within those cones
extend poleward of +/- 55 deg latitude.

Figure 4: Patterns of detection/non-detection for Atuona. (a) Spatial
density of WWLLN-located strokes within the latitude band for which the
Great Circle Path is \textgreater{} 80\% sunlit. The maximum for this
density is red, while blue is 1\% of the maximum. White areas outside of
the antipodal cone correspond to stroke density less than 1\% of the
maximum. The discrete lines are at dip angle = 0, +/- 30, and +/- 45
deg. Neither the stroke density, nor the curves, are shown within the
antipodal cone or outside of the latitude band. (b) Spatial density of
the subset of strokes in whose location Atuona participates, for which
the Great Circle Path is \textgreater{} 80\% sunlit. The new maximum is
shown as red, and 1\% of this new maximum is shown as blue. (c) Day
model logarithmic reference transmission (see text) versus position, in
four ranges: \textgreater{} -2 (red), from -2 to -3 (yellow), from -3 to
-4 (green), and \textless{} -4 (blue). Model is not shown within the
antipodal cone or outside of the latitude band. (d) Similar to (a) but
for Great Circle Paths \textless{} 20\% sunlit. (e) Similar to (b) but
for Great Circle Paths \textless{} 20\% sunlit. (f) Similar to (c) but
for night model. (g) Histogram of logarithmic reference transmission.
Black curve: all strokes in latitude band excluding antipodal cone. Red
curve: only those strokes in whose location Atuona participates.

Figure 5: Similar to Figure 4, but for Tahiti station.

Figure 6: Similar to Figure 4, but for Peru station.

Figure 7: Similar to Figure 4, but for Costa Rica station.

Figure 8: Similar to Figure 4, but for Belem station.

Figure 9: Similar to Figure 4, but for Dakar station.

Figure 10: Similar to Figure 4, but for Pune station.

Figure 11: Similar to Figure 4, but for Singapore station.

Figure 12: Similar to Figure 4, but for Honolulu station. Note that
latitude band is -30 to +50 deg, for this station only.

Figure 13: Similar to Figure 4, but for Tel Aviv station.

Figure 14: Distribution of raw amplitude (ADC level) for WWLLN-located
strokes detected by Tel Aviv station. (a) For logarithmic reference
transmission \textgreater{} -2.0. (b) For logarithmic reference
transmission from -2.0 to -2.5.

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