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Journal ofGeophysicalResearch: Atmospheres
RESEARCH ARTICLE
10.1002/2014JD021624
Key Points:
• Thunder is the weighted sum of
acoustic signals from different parts
of a flash
•Thunder can be inverted to find
source energy of each channel in
the flash
•This method can locate current flow
in lightning
Correspondence to:
J. F. Anderson,
ajakef@gmail.com
Citation:
Anderson, J. F., J. B. Johnson,
R. O. Arechiga, and R. J. Thomas
(2014), Mapping thunder sources
by inverting acoustic and electro-
magnetic observations, J. Geophys.
Res. Atmos.,119, 13,287–13,304,
doi:10.1002/2014JD021624.
Received 7 FEB 2014
Accepted 13 NOV 2014
Accepted article online 19 NOV 2014
Published online 11 DEC 2014
Mapping thunder sources by inverting acoustic
and electromagnetic observations
J. F. Anderson1, J. B. Johnson2, R. O. Arechiga3, and R. J. Thomas3
1Department of Earth and Environmental Sciences, New Mexico Institute of Mining and Technology, Socorro, New Mexico,
USA, 2Department of Geosciences, Boise State University, Boise, Idaho, USA, 3Department of Electrical Engineering, New
Mexico Institute of Mining and Technology, Socorro, New Mexico, USA
Abstract We present a new method of locating current flow in lightning strikes by inversion of thunder
recordings constrained by Lightning Mapping Array observations. First, radio frequency (RF) pulses are
connected to reconstruct conductive channels created by leaders. Then, acoustic signals that would be
produced by current flow through each channel are forward modeled. The recorded thunder is considered
to consist of a weighted superposition of these acoustic signals. We calculate the posterior distribution
of acoustic source energy for each channel with a Markov Chain Monte Carlo inversion that fits power
envelopes of modeled and recorded thunder; these results show which parts of the flash carry current and
produce thunder. We examine the effects of RF pulse location imprecision and atmospheric winds on quality
of results and apply this method to several lightning flashes over the Magdalena Mountains in New Mexico,
USA. This method will enable more detailed study of lightning phenomena by allowing researchers to
map current flow in addition to leader propagation.
1. Introduction
Lightning strikes begin with the propagation of ionized channels called stepped leaders in areas with strong
electric fields. These leaders grow through series of ionization events (referred to as steps) that lengthen
the ionized channel in random directions beyond the original extent. These steps are separated by several
microseconds in time and tens of meters in space. In this manner, leaders form tortuous, dendritic structures
that are electrically conductive [Uman, 1987]. If one of these conductive channels reaches the ground, the
flash is referred to as cloud to ground (CG); otherwise, it is called intracloud (IC). Any channel may carry cur-
rent, but few typically do; only 35% of one study’s CG flashes were observed to contain more than a single
current-carrying channel [Valine and Krider, 2002].
Two distinct processes produce thunder, each affecting a different frequency band. Audible and
near-infrasonic thunder is produced mainly by rapid heating of conductive lightning channels in response
to current flow [Few, 1969]. During a typical lightning strike, a current pulse on the order of 3×104Atrav-
els along conductive channels. This current rapidly heats the surrounding air ( 3×104Kin 5×10−6s) and
raises the surrounding pressure to around 106Pa. The intense overpressure forms a shock wave that prop-
agates supersonically (up to 3300 m s−1)[Rakov and Uman, 2003]. While additional shock waves may be
generated by subsequent strokes, the time separating these strokes (on the order of 4×10−2s) is long
enough that shock waves from different strokes are spatially separated and do not interact with one another
[Few, 1974]. These shock waves quickly decay to acoustic waves. As a result, each stroke radiates acoustic
waves along the length of the current-carrying channel. Recordings of thunder from IC events feature lower
amplitudes and lower peak frequencies than recordings of thunder from CG strikes [Holmes et al., 1971;
Johnson, 2012].
Below frequencies of about 2.5 Hz, thunder is dominated by an electrostatic relaxation of the cloud instead
of by rapid thermal expansion [Balachandran, 1979]. Before a strike can occur, a substantial charge must
accumulate in a cloud. Electrostatic forces cause these charged particles to repel one another. When cur-
rent flow in lightning depletes these charges, the electrostatic repulsion forces decrease, and charged
particles are drawn inward, producing a low-frequency acoustic rarefaction propagating as a planar wave
with vertical incidence below the cloud [Dessler, 1973]. However, subsequent researchers observed a
compression pulse before the rarefaction pulse, sometimes followed by additional compression pulses
[Balachandran, 1979, 1983; Bohannon et al., 1977]. This study attempts to locate current flow and model
ANDERSON ET AL. ©2014. American Geophysical Union. All Rights Reserved. 13,287
Journal of Geophysical Research: Atmospheres 10.1002/2014JD021624
thunder from thermal expansion; therefore, we do not address the charge relaxation mechanism in
this paper.
Previous studies have attempted to locate thunder sources by obtaining slowness vectors from array
recordings, finding lightning strike time from electromagnetic observations and backpropagating the thun-
der recordings accordingly [Few, 1970; Few and Teer, 1974; Teer and Few, 1974; MacGorman et al., 1981;
Arechiga et al., 2011; Johnson et al., 2011]. The general shapes and locations of thunder sources from the last
two studies were confirmed by comparison to maps of radio frequency (RF) pulses from leaders. However,
the raypaths (and backpropagated source locations) are sensitive to atmospheric temperature and wind
structure, which is typically poorly constrained. Consequently, acoustic data alone cannot unambiguously
reveal thunder source locations.
The New Mexico Tech Lightning Mapping Array (LMA) has proven a valuable tool for studying lightning
over the past decade. The LMA consists of networked antennas that detect and locate RF pulses produced
by ionization events in stepped leaders [Rison et al., 1999]. RF pulses above the network are located with
an uncertainty of 6–12 m root mean square (RMS) in the horizontal plane and 20–30 m RMS in the vertical
direction [Thomas et al., 2004]. These pulses are vertices in the stepped leader structure and can there-
fore be used to reconstruct the leaders. Knowledge of the stepped leader structure of the strike is useful
in modeling thunder, because audible and near-infrasonic thunder comes from current flow along some
subset of these channels. Therefore, LMA pulse locations are used to constrain thunder source locations in
this method.
One potential bias associated with the LMA is that RF pulses produced by positive electrical breakdown in a
negative charge region are less intense than RF pulses produced by negative breakdown in a positive charge
region. As a result, ionization events might not be detected equally in the two charge regions. However,
subsequent re-ionization events can have different polarity from the original breakdown so that detectable
ionization events occur along all parts of the strike. In practice, this issue was not problematic in our LMA
deployment: sufficient RF pulses were located in both regions for this method to accurately and completely
delineate the channels.
Acoustic modeling requires knowledge of the propagation medium’s structure. Unfortunately, thunder-
storm atmospheric structures can be complicated and difficult to measure. Storms require unstable air to
form and persist, so the temperature lapse rate in a thunderstorm must exceed the moist adiabatic lapse
rate (5–10◦Ckm
−1, depending on humidity). Because the intrinsic sound speed in air is proportional to the
square root of temperature, sound speed must also decrease with elevation, which causes acoustic waves
to be refracted upward. Consequently, thunder is rarely heard more than 25 km from a flash, because refrac-
tion prevents the thunder from reaching the ground beyond that point [Fleagle, 1949]. On the other hand,
low-level atmospheric structures below the storm, such as inversions, could amplify thunder generated at
low elevations.
Additionally, thunderstorms often include intense and sheared wind. Wind affects the speed at which sound
propagates and can have refractive effects of equal importance to those of the temperature lapse rate
[Fleagle, 1949]. Refraction from wind and temperature in these heterogeneous, anisotropic structures affects
arrival time and amplitude of acoustic waves. Because thunderstorm temperature and wind information
is not commonly available above the surface, predicting refractive effects on thunder signals is generally
not possible.
Other propagation effects further alter thunder signals. In high-amplitude waves, such as thunder close to
the source, the linear acoustic wave equation becomes invalid, and nonlinear propagation effects (mainly
advection of momentum and energy by wave crests) cause compressions to propagate faster than the nor-
mal speed of sound. Because thunder waves begin with compressions, the front travels faster than the rest
of the wave, making the wave lengthen as it propagates. This nonlinear effect can be significant up to 1 km
away from thunder sources [Otterman, 1959]. Turbulence is another common complication of thunderstorm
atmospheres, although its consequences to sound wave propagation are difficult to quantify. Multipathing
by topographic scattering can further complicate thunder; however, due to the computational expense of
calculating reflected raypaths, and the lower amplitudes of scattered waves compared to direct arrivals, it is
not considered here. Another factor (mainly affecting high frequencies) is intrinsic attenuation, which dis-
sipates energy in acoustic waves. Attenuation is roughly equal to an elevation-dependent coefficient times
ANDERSON ET AL. ©2014. American Geophysical Union. All Rights Reserved. 13,288
Journal of Geophysical Research: Atmospheres 10.1002/2014JD021624
Figure 1. Schematic outline of the inversion method using synthetic data. (a) Each RF pulse is connected to its near-
est neighbor that occurs earlier in time. These connections can be traced backward to reconstruct conductive channels.
(b) Each channel (here channel 3) is discretized as a set of finely spaced interpolated acoustic sources. Travel times
and amplitudes are calculated for waves propagating from each source to the receiver. These arrivals are represented
as time-shifted, amplitude-scaled delta functions and are superposed to form an arrival function. (c) Arrival functions
are filtered, and the power envelope is calculated for the filtered recorded data. (d) A nonlinear inversion is performed
to match the power envelope of a weighted superposition of filtered arrival functions to the power envelope of the
recorded data.
frequency squared, and it never exceeds 7.35 ×10−2dB/km over the low elevations and frequencies stud-
ied here (less than 24 Hz and less than 12 km) [Sutherland and Bass, 2004; de Groot-Hedlin, 2008]. The flashes
studied here occur within 12 km of the microphones, so no interesting part of our signals is ever attenu-
ated by more than 2 dB, and the bulk of the signals is attenuated much less than that. Because the effect of
attenuation on our signals is weak, it is not considered here.
2. Description of Method
This inversion method locates and quantifies thunder generation in lightning flashes (Figure 1). It requires RF
pulse locations (computed by the LMA) and acoustic recordings of thunder. We identify conductive channels
ANDERSON ET AL. ©2014. American Geophysical Union. All Rights Reserved. 13,289
Journal of Geophysical Research: Atmospheres 10.1002/2014JD021624
Figure 2. The number of channels in a flash can be reduced by discarding those with short independent length. This
reduces the computational expense of forward modeling thunder signals, as well as reducing the number of free
parameters in the inverse problem. In this flash, the number of channels to consider was reduced from 431 to 7.
by connecting RF pulses (section 2.1), and, assuming a reasonable atmosphere, forward model the acoustic
signal that would be recorded for each conductive channel (section 2.2). Treating the thunder as a weighted
superposition of the channels’ acoustic signatures, we then invert to find the acoustic amplitude of each
channel that optimizes the fit of modeled thunder power envelopes to recorded thunder power envelopes
(section 2.3). Because the true structure of the atmosphere around the strike is unknown, we repeat this
process many times, over many reasonable atmospheres, until an optimal fit is found.
2.1. Conductive Channel Identification
The LMA provides a catalog of RF pulses occurring during an event but does not show the conductive chan-
nels connecting them. In order to reconstruct conductive channels, we use the principle that leaders may
branch as they propagate but do not merge. Consequently, each vertex in a leader can connect to any num-
ber of later vertices (representing branching) but at most one earlier vertex. For each RF pulse, we find its
nearest earlier neighbor and connect them. Additionally, because conductive channels may be reactivated
by subsequent stepped or dart leaders, RF sources that occur along a preexisting conductive channel are
merged into that channel. In this way, conductive channels may be traced backward from their terminal
vertices to their beginnings.
One consequence of this scheme is that conductive channels may overlap in the early sections of the leader.
This is desirable, because current flow through either will correctly imply thunder production from their
overlapped section. In the case of multiple overlapping channels carrying current, the acoustic source
amplitude of their overlapped sections is the linear sum of the source amplitudes of the individual channels.
This follows from thunder in the near-infrasound and audible bands being produced by resistive heating
during current flow [Few, 1969], which must be conserved throughout the channel structure.
The number of conductive channels to consider can be reduced (Figure 2) by eliminating “dead-end” chan-
nels whose independent segments (i.e., segments that do not overlap with longer channels) are short. These
dead ends are leaders that branched from another channel but failed to propagate far and therefore are
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Journal of Geophysical Research: Atmospheres 10.1002/2014JD021624
unlikely to carry current. Removing dead ends is advantageous for two reasons: it reduces the number of
parameters in the model and reduces the computational expense of modeling thunder signals. This process
may decrease the number of channels from hundreds to less than 10, depending on the independent seg-
ment length threshold used for dead-end identification. The threshold needed to eliminate dead ends but
not the main channels depends on the scale of the strike; for each strike studied here, we picked an opti-
mal threshold that reduced strike complexity while preserving core channels. Additionally, for CG events
the entire set of channels formed by downward propagating leaders before the first return stroke must be
considered because of the possibility of charge deposition along the entire structure during the first return
stroke. Therefore, we include that structure (including dead ends) as a potential acoustic source in addition
to the main channels identified by dead-end elimination.
2.2. Acoustic Forward Modeling
Each conductive channel may be regarded as a string of finely spaced (in this work, every 1 m) acoustic point
sources; this approximation is valid as long as the spacing is short compared to the wavelengths studied
(from about 25 m to 60 m for the 6–12 Hz band studied here). For each point source, we use standard ray
tracing equations [Garces et al., 1998; Anderson, 2013] to calculate travel time and arrival amplitude of signals
from each point source. Time of thunder generation is determined by electromagnetic observations; in this
work, we used electrical interference produced by lightning current flow and recorded by our unshielded
sensor cables, but RF pulse timing could be used as well. Typically, multiple strokes are detected during a
flash. However, the errors associated with propagation through complicated atmospheres make the rela-
tively small intervals between strokes unresolvable. Therefore, only a single source time (the mean time of
all strokes) is used in these calculations. We construct an “arrival function” by superposing impulses whose
timing and amplitude correspond to those of arrivals associated with each point source.
To convert this arrival function into a true pressure signal, we would need to convolve it with a source time
function, which is unconstrained. However, the channel heating and expansion that produces thunder
occurs rapidly, and we ultimately band-pass filter these models and recordings to low frequencies, so the
source time function can be treated as being approximately impulsive. Band-pass filtering also eliminates
the need to consider attenuation, which has little effect on low frequencies at these distances.
Phase coherence between recorded thunder and modeled signals is probably weak for these frequencies
because our assumed source time functions and atmospheres are not exact. Because of this, we calculate
the power envelope of recorded and modeled signals before comparing them; this makes signal comparison
less sensitive to small timing errors in modeled signals. This forward modeling procedure is repeated for
each microphone in the network and each conductive channel, giving us a set of signals each microphone
would record for each conductive channel.
2.3. Inversion for Channel Source Amplitudes
Thunder can be treated as a weighted superposition of different channels’ acoustic signatures. To determine
the weight of each channel, we concatenate signals for different microphones (so that all data to be fit are
contained in a single vector) and perform a nonlinear inversion using the Metropolis-Hastings Markov Chain
Monte Carlo (MCMC) method [Hastings, 1970; Asteretal., 2012] to minimize the misfit between modeled
and observed power envelopes. This method has important advantages over other nonlinear inverse meth-
ods, including its ability to return a posterior distribution of model parameters and its robustness against
returning locally (not globally) optimal models. Using the notation Gfor a matrix whose columns are for-
ward modeled thunder signals, mfor a vector of acoustic source amplitudes, rfor a recorded thunder time
series, and Efor the power envelope function, we invert to find mthat minimizes the normalized data misfit
‖E(Gm)−E(r)‖2
‖E(r)‖2
(1)
which is the ratio between the L2norm of the difference between observed and modeled power envelopes
and the L2norm of the observed power envelope.
This MCMC implementation consists of 50,000 iterations in which a random model parameter is perturbed
by multiplication by a positive random number drawn from a lognormal distribution centered at 1. Because
negative parameter values indicate unrealistic rarefactions instead of compressions radiating from lightning
channels, we require that all model parameters be nonnegative. This method of parameter perturbation
accomplishes that. In each iteration, misfit between modeled and recorded data is calculated. The proposed
ANDERSON ET AL. ©2014. American Geophysical Union. All Rights Reserved. 13,291
Journal of Geophysical Research: Atmospheres 10.1002/2014JD021624
Figure 3. Map of sensor deployment used in this experiment in central New Mexico, USA. (a) The LMA consisted of a
“regional” component consisting of six antennas spaced around the Magdalena Mountains at distances of more than
10 km and a “local” component including three antennas deployed close together high in the mountains. (b) The acous-
tic network included three broadband microphones deployed in the same area as the local part of the LMA (within the
black square). The MGTM station is used as the origin of the map.
model parameter is always accepted if it decreases the misfit. Additionally, the proposed model parameter
could be accepted if it increases the misfit, with probability of acceptance decreasing with higher differ-
ences in misfit. In practice, our acceptance ratios varied between 0.1 and 0.7. We have no prior information
about the acoustic energy release of the lightning channels and therefore use an uninformative prior model.
After running 50,000 iterations, we consider the first 25,000 to be a “burn-in” period in which the influence
of the initial model has not been completely lost and discard them. Then, to reduce correlation between
successive iterations, only every tenth iteration is sampled. The models in these remaining iterations reflect
the posterior distribution of the model and show the likelihoods of acoustic source amplitude values and
the covariance among different conductive channels.
3. Experiment
3.1. Deployment and Data
We analyze data from a 2009 instrument deployment in the Magdalena Mountains in central New Mexico.
Twelve broadband (<0.1Hz to a Nyquist frequency of 500 Hz) Infra-NMT microphones with flat frequency
responses [Marcillo et al., 2012] were deployed in three arrays (MGTM, MKVH, and MLAN) consisting of four
microphones each (Figure 3). Microphone arrays were in a triangular configuration with three peripheral
and one central microphone connected by cable to a RefTek RT-130 data logger recording 1000 samples
per second at 24 bits. Because of the close spacing of the microphones within the arrays, acoustic data
from the central microphones only are considered here (although, for determining strike timing, electrical
interference recorded on all acoustic channels is considered). Recordings were converted to overpressure
units before analysis. Additionally, we used electromagnetic data from nine LMA sensors, with three form-
ing a local component to the array (within 2 km of the acoustic network center) and another six forming a
regional component (within 30 km of the acoustic network center). The LMA and microphone arrays both
received precise timing information from GPS antennas.
We present results from several flashes during a storm that occurred on 24 July 2009. We examine in particu-
lar detail a CG flash that occurred at 19:42:13 UTC. This strike included 2555 RF sources that were connected
to form 431 conductive channels. The vast majority of these channels are probably inactive dead ends
because of their short independent lengths. In order to simplify the strike and remove these channels from
consideration, all channels whose independent length fell below a threshold of 4500 m were ignored;
ANDERSON ET AL. ©2014. American Geophysical Union. All Rights Reserved. 13,292
Journal of Geophysical Research: Atmospheres 10.1002/2014JD021624
Figure 4. (a) Overlay of modeled and recorded thunder power envelopes for the CG flash at 19:42:13. (b) Map of
inverted channel energy densities. (c) Posterior distributions of channel energy densities of thunder-producing channels.
Channel 4, which is the direct channel to ground, produces the most thunder. Channels 2 and 5, which connect
channel 4 to higher parts of the strike, produce little thunder.
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Journal of Geophysical Research: Atmospheres 10.1002/2014JD021624
Figure 5. The evolution of the CG flash at 19:42:13 with time. Color of plotted points corresponds to VHF pulse
times. Black vertical lines indicate times of major low-frequency radio pulses from return strokes; gray lines indicate
smaller radio pulses. Most of the upper channels formed after the return strokes, making them unlikely to produce
significant thunder.
this reduced the number of potential current-carrying channels to seven. These seven include one that
extends all the way to ground, one that extends downward but fails to reach ground, two that propagate
from above to meet the top of the ground strike, and three that extend horizontally away from the ground
strike at high elevation (Figure 2).
Low-frequency radio interference from five significant current pulses (as well as many more smaller cur-
rent pulses) was detected during this flash (Figure 5). However, the time intervals separating these pulses
(0.05–0.2 s) are short compared to the duration of thunder signals and to the likely timing errors associated
with propagation through complex atmospheres. Therefore, we find the average time of all strokes and use
it as the sole source time in these calculations.
For each flash, we performed a grid search over many windless atmospheres with constant vertical sound
speed gradients to find an optimal fit. Sound speed gradient varied from −0.006 to −0.0036 s−1, and
ground-level (3000 m above sea level) sound speed varied from 336 to 354 m s−1. These correspond to
temperature gradients of roughly 6–10◦Ckm
−1and ground-level temperatures of roughly 9–41◦C. Each
atmosphere was tested using the inversion method described in the previous section. After identifying the
atmosphere with the best fit, we performed a more detailed inversion on that atmosphere using 200,000
iterations instead of 50,000 and a burn-in period of 100,000 instead of 25,000. This was done in order to
better characterize the posterior distributions of the source models.
Additionally, we tested five frequency bands for the flash that occurred at 19:42:13. The low corner of these
bands ranged from 3 to 12 Hz, and the high corner was set to twice the low corner. We found that using the
band from 6 to 12 Hz resulted in the lowest misfit for this flash. Additionally, synthetic data presented in the
next section show that this band is the least susceptible to wind-related errors. Therefore, we picked this
frequency band for analyzing other flashes.
We test many atmospheres to find an optimal fit between modeled and observed thunder, and it is worth
noting that the atmosphere with the best fit is not necessarily the most correct atmosphere but rather the
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Figure 6. (a) Overlay of modeled and recorded thunder power envelopes for the IC flash at 19:06:36. (b) Map of inverted
channel energy densities for the flash. (c) Posterior distributions of channel energy densities of thunder-producing
channels. Channels 1 and 2 are somewhat correlated; this is a potential source of ambiguity in the results.
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Figure 7. (a) Overlay of modeled and recorded thunder power envelopes for the 19:14:41 CG flash. (b) Map of inverted
channel energy densities. The ground strike has the highest energy density; however, thunder is also detected from
two upper channels. (c) Posterior distributions of channel energy densities. Energy densities of two upper channels are
correlated, making distinguishing thunder from them difficult.
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Figure 8. (a) Overlay of modeled and recorded thunder power envelopes for the 19:32:51 IC flash. (b) Map of inverted
channel energy densities. Most energy is released by a high-elevation channel, but small amounts are also produced by
other channels. (c) Posterior distributions of channel energy densities of thunder-producing channels.
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Figure 9. (a) Overlay of modeled and recorded thunder power envelopes for the 19:50:08 IC flash. (b) Map of inverted
channel energy densities. (c) Posterior distribution of channel energy density of the thunder-producing channel (only
one channel was found to produce any thunder in this flash).
very simplified atmosphere whose acoustic propagation effects best matched those of the very compli-
cated true atmosphere. We describe best fit atmospheres of each flash in Table 1 but are reluctant to infer
properties of the true atmosphere from them.
3.2. Results
We show results for six flashes (Figures 4–10). For each flash, the fit of modeled and recorded power
envelopes is shown, along with the maximum-likelihood source amplitudes of the conductive channels
constituting the flash and their posterior distributions.
The posterior distributions of the model parameters (diagonal subplots of Figures 4c, 6c, 7c, 8c, 9c, and 10c)
are obtained from the detailed inversion; these show the likelihood distributions of the acoustic source
amplitudes of different conductive channels and the covariance among them. A narrow distribution means
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Figure 10. (a) Overlay of modeled and recorded thunder power envelopes for the 19:54:19 IC flash. (b) Map of inverted
channel energy densities. (c) Posterior distributions of channel energy densities of thunder-producing channels. An
upper channel 3 produces most thunder; contributions from the lower channel 2 are smaller and poorly resolved.
that the source amplitude has been resolved with high precision; a wide distribution means that the source
amplitude could not be resolved very precisely. The off-diagonal subplots in those figures show relation-
ships between inverted source amplitudes of different conductive channels; when a strong correlation
exists, the source amplitudes of the two channels can trade off in a way that is difficult for the inversion
to resolve.
3.3. Flash at 19:42:13
Detected thunder in this flash came from three mainly vertical channels extending toward ground (Figure 4).
One of these is the original channel that extends from the initial breakdown to the ground; this produced
the most thunder. Additionally, two subsequent channels formed from later breakdowns and connected to
the top of the main ground strike; these produced much less thunder.
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Interestingly, no thunder was detected from the long upper channels that propagated mainly horizontally.
Considering the time of channel formation with respect to current pulse times (detected by low-frequency
RF interference) can explain this finding (Figure 5). Glitches caused by RF interference synchronous with
dart leaders are seen throughout the flash. More than half of these occur before the upper channels begin
to form, and nearly all occur before the upper channels are fully formed. This indicates that these chan-
nels might not be major thunder sources because during most of the thunder-producing return strokes,
the channels did not yet exist. On the other hand, the three channels that were found to produce thunder
formed early in the strike.
3.4. Flash at 19:06:36
This IC flash includes two downward propagating channels, one channel that extends horizontally away
from the initial breakdown and one that extends upward (Figure 6). All of these except one of the lower
channels were found to produce thunder, and the midlevel, horizontal channel was the most energetic.
Recovered energy densities of the lower and upper channels have a moderate negative correlation of −0.3,
meaning that fit to recorded thunder may be roughly preserved by increasing energy density of one and
decreasing it for the other. As a result, it is difficult to determine exactly how acoustic energy is partitioned
between those two channels.
3.5. Flash at 19:14:41
This CG flash consists of a channel going to ground, along with a few horizontally propagating upper chan-
nels (Figure 7). Thunder from this flash is fit relatively well by models (normalized RMS misfit of 0.73). The
ground strike is the most energetic channel, but two upper channels also produce substantial thunder. How-
ever, the recovered energy densities of the two upper channels are correlated (r=−0.39), meaning that
energy density can be allocated to either and have a relatively small effect on model misfit.
3.6. Flash at 19:32:51
This IC flash consists of one main low-level channel, two upper level channels, and a single connection
between the layers (Figure 8). One of the upper channels is calculated to produce the most thunder. A sec-
ond upper channel and the low channel also produce measurable thunder. Fit of models to observations is
relatively poor (RMS misfit of 0.9), meaning that much of the thunder cannot be explained by this method.
3.7. Flash at 19:50:08
This IC flash includes two high-elevation channels and two lower channels (Figure 9). Of these, only one
channel (an upper channel) is found to produce any thunder. However, the misfit is high (0.936), meaning
that some thunder-producing channels are not being identified because of unmodeled effects on waves
they produce.
3.8. Flash at 19:54:19
This IC flash also consists of two upper channels and two lower channels (Figure 10). Thunder is detected
prominently from one of the upper channels and ambiguously from a lower channel. The fit between
modeled and recorded thunder is moderate (0.806).
4. Sensitivity of Method to Sources of Error
Monte Carlo simulations were performed in order to assess the importance of potential sources of error. The
main sources of error are location uncertainty of RF pulses located by the LMA (addressed in section 4.1) and
the unknown structure of the atmosphere (section 4.2). In each simulation, synthetic thunder was modeled
for a realistic lightning strike (using the geometry of the 19:42:13 flash) and atmosphere. We then attempted
to invert this signal for channel amplitude after adjusting the strike or atmosphere to reflect the source
of error. We tested 12 frequency bands and source-of-error intensities, performed 20 trials for each com-
bination, and calculated expected values for normalized data misfit (equation (1)) and normalized source
model error
‖mTrue −mInv ‖2
N⋅‖mTrue ‖2
,(2)
where mTrue is the true source model, mInv is an inverted source model, and Nis the number of
model parameters.
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Figure 11. Contours showing the effect of LMA RF pulse location noise on inversions of synthetic data. Monte Carlo
simulations revealed effects on (a) normalized misfit between prenoise and postnoise synthetic thunder and (b) errors
between true and inverted amplitude (expressed as RMS residual between true and inverted amplitudes). Normalized
thunder misfits increase with both the RMS location error and the corner frequency. However, errors in inverted channel
amplitudes depend more strongly on RMS location error than on corner frequency. The true noise level for this LMA
configuration is shown by the dashed horizontal line.
4.1. Sensitivity to RF Pulse Location Noise
RF pulse locations provided by the LMA inevitably contain some noise. In this configuration, horizontal and
vertical location errors have standard deviations of 10 m and 30 m, respectively [Thomas et al., 2004]. These
errors affect travel times of modeled acoustic waves and alter thunder signals accordingly. Here we quantify
the effect of RF pulse location noise on thunder signal recovery.
Twenty independent iterations were run for each frequency band and location noise level. In each iteration,
thunder was forward modeled for the original RF pulse locations using the main downward reaching chan-
nel as the sole source. Then, RF pulses were offset by random errors distributed according to the standard
deviations being tested. Finally, the modeled thunder and offset RF pulse locations were inverted using the
frequency band being tested. We tested 12 location standard deviations and 12 frequency bands. Horizontal
standard deviations ranged from 1.5 m to 100 m, with vertical standard deviation set to 3 times the horizon-
tal standard deviation in each case. Central frequencies of the frequency bands ranged from 3.5 Hz to 24 Hz;
in each case, the band’s high corner was twice the low corner.
Figure 11 shows the dependence of misfit on frequency band and location standard deviation. The normal-
ized misfit (equation (1)) was calculated for each simulation, and mean normalized misfits were calculated
for each frequency standard deviation pair.
Additionally, we examine the effect of RF pulse location noise on accuracy of inverted channel amplitudes.
Normalized error (equation (2)) between true and inverted channel amplitudes is strongly dependent on
RF pulse location noise, and, to a lesser extent, corner frequency. These errors are generally between 0.1
and 0.2 for near-infrasound corner frequencies and realistic RF pulse location noise values. Notably, a high
normalized misfit can correspond to a fairly low-error result; for example, in some cases, the normalized
misfit was around 0.7 while the normalized error was around 0.1.
4.2. Sensitivity to Atmospheric Simplification
Thunderstorm atmospheric structure is complex, dynamic, and typically unknown. It is obviously impossible
to test every possible atmospheric structure; with computation time being the main constraint, it is practical
to only test windless atmospheres with constant sound speed gradients limited to the range in which thun-
derstorms typically develop. As a result, the difference between the tested atmosphere and true atmosphere
is an important source of error.
We use a similar procedure to the previous test. The wind structure of the atmosphere from which thunder
is generated is allowed to vary (with 12 different windiness levels), with a realistic and fixed sound speed
ANDERSON ET AL. ©2014. American Geophysical Union. All Rights Reserved. 13,301
Journal of Geophysical Research: Atmospheres 10.1002/2014JD021624
Figure 12. Effect of atmospheric wind intensity on (a) normalized misfit between true synthetic and inverted synthetic
thunder and (b) accuracy of inverted channel amplitudes. Thunder misfit increases with corner frequency and maximum
surface wind. However, true channel amplitudes can still be recovered reasonably accurately.
structure (surface sound speed of 343 m s−1and vertical sound speed gradient of −0.005 s−1). The windiest
scenario tested has a surface wind of 11 m s−1and a shear of 0.0275 s−1. Winds during thunderstorms are
highly variable, and we did not measure wind speeds or shear during the storm, but we consider this range
to be illustrative of typical wind-related errors. For each windiness level, 12 frequency bands were tested
with 20 iterations each. In each simulation, surface wind in a random direction was set to a random speed,
with random wind shears chosen from an interval that depended on the windiness level being tested.
Thunder misfit increases rapidly with windiness and, to a lesser extent, corner frequency (Figure 12). How-
ever, these high misfits do not imply high normalized errors; over the range of atmospheres tested, expected
error barely exceeds 0.25.
5. Discussion
Misfit between modeled and observed thunder varied among flashes (Table 1). We calculated normal-
ized RMS misfits as low as 0.723 and as high as 0.936. So for some flashes, thunder could be reproduced
fairly well, while for other flashes, little of the thunder could be modeled. Notably, CG flash thunder was
reproduced much more reliably than IC thunder: misfits for both CG flashes were lower than misfits of any
IC flashes. This could be related to the acoustic sources in IC flashes being higher (and therefore, more
susceptible to atmospheric propagation effects) than those in CG flashes.
Tab le 1. Summary of Lightning Flashes Studieda
Thunder-Producing Normalized Energy Density Total Basal Temperature Temperature Gradient
Time (UTC) Type Channels Misfit (Jm−1) Energy (J) (◦C) (◦Cm
−1)
19:06:36 IC 3 0.841 2.22 ×10−2223 37.0 −9.1×10−3
19:14:41 CG 3 0.73 2.89 ×10−2550 20.1 −1.0×10−2
19:32:51 IC 3 0.902 1.60 12595 31.3 −9.1×10−3
19:42:13 CG 3 0.723 5.12 ×10−13379 22.2 −1.0×10−2
19:50:08 IC 1 0.936 2.94 ×10−13133 26.1 −9.0×10−3
19:54:19 IC 2 0.806 6.35 ×10−21761 30.9 −1.0×10−2
aData are from a 24 July 2009 thunderstorm. RMS misfit, energy density, total energy, best fit ground-level temperature, and best fit temperature gradient
refer to values obtained by analyzing the 6–12 Hz frequency band. Basal temperature and temperature gradient correspond to the simplified windless test
atmosphere for which the best model data fit was found; these values should not be interpreted as properties of the true atmosphere at the time of the flash.
ANDERSON ET AL. ©2014. American Geophysical Union. All Rights Reserved. 13,302
Journal of Geophysical Research: Atmospheres 10.1002/2014JD021624
The geometry of inverted acoustic sources in CG flashes is reasonable. In each CG flash studied, the chan-
nel from the initial breakdown toward ground was the most energetic acoustic source, while some higher
channels produced thunder as well. While upper channels do not necessarily carry current, the channel to
ground necessarily carries current in at least one return stroke in every strike, so this finding is unsurpris-
ing. In the flash at 19:42:13, the ground channel’s energy density is much greater than the sum of the upper
channels’ energy densities. This indicates that most current flowed through the ground channel and that
the return strokes carried by the upper channels were much less energetic. The CG flash at 19:14:41, on the
other hand, has upper channels whose energy density sum is only slightly less than the energy density of
the ground channel. This indicates that the upper channels were active during the most energetic return
strokes, and most current flowed through both an upper and lower channel.
Because IC flashes do not involve a large conductive charge reservoir, their current pulses are less pre-
dictable. Consequently, it is difficult to assess whether a thunder source geometry for an IC event is
reasonable. In the flash at 19:50:08, only a single channel was found to produce any thunder at all. However,
in the 19:06:36 flash, three of the four channels produced substantial thunder.
Normalized misfits between recorded and modeled thunder are somewhat high in these flashes: no flash
has a normalized misfit less than 0.73, and no IC flash has a normalized misfit less than 0.8. However, in the
previous section, we determined that sources of error like VHF pulse location uncertainty and atmospheric
winds could cause substantial thunder misfit while still allowing reasonably accurate estimates of source
energy density. For example, assuming that wind is the main source of error, a ground-level wind of 3 m
s−1and a shear of 7.5×10−3s−1could cause a normalized thunder misfit of 0.7 but a normalized source
amplitude error of only 0.1 (Figure 12). High thunder misfits do not necessarily indicate similarly inaccurate
source energy estimates.
Other sources of error, such as echoes from topography and atmospheric heterogeneities more compli-
cated than one-dimensional linear variations with elevation could potentially reduce the accuracy of this
method. For example, a ground-level inversion could preferentially focus waves from low-elevation sources
to receivers. Wave propagation models would not predict this and would therefore overestimate energy
density of low channels while underestimating high channels. Such structures are common near downdrafts
in storms, so this may be a common source of error for this technique. Similarly, a local vertical wind or tur-
bulent region might distort thunder signals from nearby channels, causing those channels’ source energies
to be underestimated by the inversion. Topographic echoes might have the opposite effect: the coincident
arrival of direct waves from one channel and a topographic echo from a different channel could not be pre-
dicted without a more complex propagation model; lacking one, all arriving energy would be attributed to
the direct waves, and the energy of the channel producing them would be overestimated.
The source energy densities found in this work (between 2×10−2Jm−1and 1.6 Jm−1) may seem low for
lightning strikes carrying current on the order of 3×104A (Table 1). However, several factors must be
considered, first being that the channels are very long (thousands of meters) and energy is radiated along
their entire length, so the total energy released is, in these units, 3–4 orders of magnitude higher than the
energy density. Further, we are looking only at a narrow frequency band (6–12 Hz) that carries only a small
fraction of the total thunder energy. Finally, thunder generation is highly inefficient: most input energy is
dissipated in shock wave decay close to the channel. So these seemingly low thunder energy densities do
not conflict with the enormous amount of energy involved in lightning.
6. Conclusion
We have introduced a new method for locating thunder sources and current flow within a lightning flash
by joint inversion of synchronous thunder recordings and RF pulse catalogs from the LMA. This method
involves connecting RF pulses to reconstruct conductive channels created by leaders, modeling acoustic
signals produced by each conductive channel, and inverting using a Monte Carlo Markov Chain to deter-
mine the source energy density of each conductive channel in order to minimize misfit between modeled
and recorded thunder power envelopes. The returned posterior distribution of channel energy density can
be used to distinguish thunder-producing channels from silent channels.
Sources of error in this method include LMA RF pulse location noise and complications in atmospheres. Typi-
cal levels of LMA RF pulse location noise have little effect on accuracy of recovered channel energy densities.
ANDERSON ET AL. ©2014. American Geophysical Union. All Rights Reserved. 13,303
Journal of Geophysical Research: Atmospheres 10.1002/2014JD021624
The effect of high winds is more significant. Additionally, effects of turbulence and topography could cause
errors that are difficult to quantify, while intrinsic attenuation could potentially cause errors for frequencies
higher than those studied here.
We applied this method to lightning flashes that occurred on 24 July 2009 in the Magdalena Mountains,
New Mexico, USA. Thunder from CG flashes was reproduced more reliably than thunder from IC flashes. The
misfit observed in some flashes is similar to expected values for moderately windy atmospheres, for which
source energy estimate errors are expected to be close to 25%. In most strikes, multiple channels produced
thunder, and in all strikes, at least one channel produced no thunder.
Current flow in lightning is a powerful and often destructive process, and this project enables its study by
combining RF and acoustic observations. The method’s sensitivity to atmospheric structure means that
accuracy could be improved by incorporating meteorological data. Radiosondes, meteorological mod-
els, or other constraints on atmospheric structure near flashes will improve the accuracy of this method.
Additionally, knowing the true atmospheric structure would eliminate the need to test many atmospheres;
however, as the true atmosphere in and around the storm will contain three-dimensional structure, the
need to use a more complex and computationally expensive three-dimensional propagation model would
increase runtimes and memory needs of this method along with its accuracy. Future work might include
current measurements and shock wave modeling to study energy conversion along conductive channels or
determining characteristics of channels that tend to carry current.
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Acknowled gments
E. Badillo, R. Johnson, J. Michnovicz,
and M. Murray participated in field-
work and H. Edens provided RF pulse
locations from the LMA. This work was
funded by NSF grant AGS-0934472.
Thunder and LMA data are available
upon request to the authors.
ANDERSON ET AL. ©2014. American Geophysical Union. All Rights Reserved. 13,304