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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, B04303, doi:10.1029/2008JB005769, 2009
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Thunder-induced ground motions:
1. Observations
Ting-L. Lin1,2 and Charles A. Langston1
Received 28 April 2008; revised 4 February 2009; accepted 4 March 2009; published 22 April 2009.
[1] Acoustic pressure from thunder and its induced ground motions were investigated
using a small array consisting of five three-component short-period surface seismometers,
a three-component borehole seismometer, and five infrasound microphones. We used the
array to constrain wave parameters of the incident acoustic and seismic waves. The
incident slowness differences between acoustic pressure and ground motions suggest that
ground reverberations were first initiated somewhat away from the array. Using slowness
inferred from ground motions is preferable to obtain the seismic source parameters. We
propose a source equalization procedure for acoustic/seismic deconvolution to generate
the time domain transfer function, a procedure similar to that of obtaining teleseismic
earthquake receiver functions. The time domain transfer function removes the incident
pressure time history from the seismogram. An additional vertical-to-radial ground motion
transfer function was used to identify the Rayleigh wave propagation mode of induced
seismic waves complementing that found using the particle motions and amplitude
variations in the borehole. The initial motions obtained by the time domain transfer
functions suggest a low Poisson’s ratio for the near-surface layer. The acoustic-to-seismic
transfer functions show a consistent reverberation series at frequencies near 5 Hz. This
gives an empirical measure of site resonance that depends on the ratio of the layer velocity
to layer thickness for earthquake P and S waves. The time domain transfer function
approach by transferring a spectral division into the time domain provides an alternative
method for studying acoustic-to-seismic coupling.
Citation: Lin, T.-L., and C. A. Langston (2009), Thunder-induced ground motions: 1. Observations, J. Geophys. Res., 114, B04303,
doi:10.1029/2008JB005769.
1. Introduction
[2] Our working hypothesis is that natural thunder can be
used as a seismic source to study near-surface velocity
structure by its induced ground motions (acoustic-to-seismic
coupling). Previous studies have shown that a variety of
atmospheric disturbances such as sonic booms, meteoroid
falls, thunder, and explosions recorded with a seismograph
can be used to study the propagation of an acoustic wave in
the atmosphere or the coupling of the acoustic wave to the
ground [Espinosa et al., 1968; Cates and Sturtevant, 2002;
Brown et al., 2003; Ishihara et al., 2003; Langston, 2004;
Lin and Langston, 2006]. However, most impulsive atmospheric disturbances such as sonic booms, meteoroid falls,
and explosions are few and far between. In contrast, thunder
created by lightning in thunderstorms is a common natural
meteorological phenomenon that is a common source of
atmospheric shock waves [Lin and Langston, 2007]. For
example, Changnon [1988b] analyzed the thunder occur1
Center for Earthquake Research and Information, University of
Memphis, Memphis, Tennessee, USA.
2
Now at Institute of Seismology, National Chung Cheng University,
Chia-Yi, Taiwan.
Copyright 2009 by the American Geophysical Union.
0148-0227/09/2008JB005769$09.00
rence frequency for the data period of 1948 – 1977 from the
National Climate Data Center and showed that on the
average there are 80 annual thunder events in our study
area near Memphis, Tennessee. A thunder event is defined
as a time interval when thunder is heard and ending 15 min
after thunder is last heard [Changnon, 1988a].
[3] Acoustic-to-seismic wave coupling into the ground
depends on the nature of material properties in the near
surface [Espinosa et al., 1968; Sabatier and Raspet, 1988;
Langston, 2004]. Langston [2004] suggested that acoustically induced ground motion can provide independent
insight into the near-surface site structure and seismic wave
response. Langston [2004] showed that an atmospheric
shock wave from a meteor could excite two kinds of
interactions: ‘‘leaky’’ mode P-SV wave and ‘‘locked’’ mode
Rayleigh wave propagation within the near-surface layer of
Mississippi embayment unconsolidated sediments. Sabatier
and Raspet [1988] recorded acoustically coupled seismic
ringing from blasts and artillery and showed that the seismic
ringing is due to resonance in the near-surface ground layer.
They suggested that the ringing is generated by the trapped
seismic pulse reflecting up and down within the soil layer
because the base layer P wave velocity is larger than the
sound speed in the air. Both the dependence of seismoacoustic coupling on the near-surface velocity structure and
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Figure 1. (a) Schematic diagram of the seismo-acoustic array configuration. Solid circles represent the
microphone and L-28 three-component seismic sensors colocated at the surface. The center square shows
the location of the 8-m borehole and L-28 three-component seismic sensor. Lines show the 8-m porous
garden hoses. (b) The system responses for acoustic pressure and the 18 L-28 seismic sensors. Figures 1a
and 1b are modified from Lin and Langston [2007]. (c) An N-shaped impulsive waveform with a
center frequency at 5 Hz used in Figure 1d. (d) The array response function to Figure 1c. The
frequency-wave number plot is in units of dB for normalized power.
high thunder activity in the study area motivated us to
record thunder-induced ground motions.
[4] A colocated microphone and seismic sensor array was
built including one borehole seismometer to record incident
thunder acoustic pressure and ground motions during thunderstorms. A total of 19 thunder events with impulsive
waveforms are used here as seismic sources to study the
characteristics of thunder-induced ground motions. A procedure that is similar to producing the teleseismic receiver
function [Langston, 1979] was used to yield a time domain
transfer function that removes the complexity of the incident
acoustic pressure time history from the seismogram through
the deconvolution technique.
2. Configuration of the Seismo-acoustic Array
[5] The seismo-acoustic array was built to determine the
wave slowness and azimuth of the incident atmospheric
acoustic wave and to study the character of impulsive and
continuous thunder signals. The array (Figure 1a) is located
at one of the authors’ rural residence near the small town of
Moscow, Tennessee, at the top of a low flat hill. The array
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Figure 2. Examples of the beamed and stacked (a) acoustic pressure waveforms and (b) vertical ground
velocity seismograms. Thick lines in the pressure and surface velocity waveforms show the beamed and
stacked waveforms. Thin gray lines represent each individual trace.
consists of a five-element infrasound array with colocated
three-component surface seismic sensors buried at a depth
of 0.2 m and an 8 m deep three-component borehole
seismometer located at the center. The aperture of the array
is about 50 m. Each infrasound microphone element
includes a six-port manifold connected to 8-m-long porous
garden hoses for the purposes of wind noise reduction
[Stump, 2000a, 2000b]. Figure 1b shows the frequency
responses for the acoustic pressure and seismic velocity
recording systems of the array. The cutoff frequencies
defined at 3 dB amplitude reduction of the passband are
3 Hz to 28 Hz for acoustic and 4 Hz to 27 Hz for seismic
recording systems. Both microphone and geophone data
streams are sampled at 200 Hz and digitized with a 16-bit
resolution digitizer. The array response function (Figure
1d) to an N-shaped impulsive waveform (Figure 1c) with
a center frequency at 5 Hz shows that the array is uniformly
sampled in space and the main lobe is concentrated and
distinguished from the side lobes. An N-shaped impulsive
waveform was chosen to approximately match recorded
impulsive thunder infrasound pressure waveforms. The
characteristics of the array response function and the study
by Lin and Langston [2007] suggest that the array is capable
of determining incident horizontal slowness and azimuth of
infrasound signals from thunder. Daily thunderstorm events
were recognized by local weather forecast, field observations, and an AM lightning detector [Lin and Langston,
2007] built into the array. Unfortunately, the sensitivity of
the detector made it difficult to infer the times of particular
lightning strikes for the thunder events because so many
events are seen in a storm. Data from the detector was most
useful in scanning daily records to identify the thunderstorm
events from other acoustic sources. We refer to Lin and
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Figure 3. (a) Four examples out of 19 thunder events showing stacked acoustic pressure and ground
motion waveforms. Thick black and thin gray lines show the surface and borehole ground motion
waveforms, respectively. The bracket lines denote examples of reverberations. All waveforms have been
corrected by removing the instrument responses shown in Figure 1c. (b) Acoustic pressure time histories
of the 19 thunder events used in this study.
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Figure 3. (continued)
Langston [2007] for additional information about the configuration of the seismo-acoustic array.
3. Data Processing
[6] The acoustic pressure and seismic velocity data were
first corrected for the system instrument responses and
filtered using a zero phase, four-pole Butterworth bandpass filter with corners at 1 and 50 Hz for acoustic
pressure waveforms and at 1 and 30 Hz for seismic
velocity waveforms. Frequency-wave number beam forming (D. O. ReVelle et al., Discrimination of earthquakes
and mining blasts using infrasound, paper presented at
Table 1. Recording Time, Incident Slowness, and Azimuth
Recording
Pressure
Event
Date
Time (LT)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
11 Mar 2006
3 Apr 2006
8 Apr 2006
20 Apr 2006
21 Apr 2006
21 Apr 2006
21 Apr 2006
21 Apr 2006
21 Apr 2006
26 Apr 2006
4 May 2006
4 May 2006
4 May 2006
17 Jun 2006
18 Jun 2006
18 Jun 2006
18 Jun 2006
21 Aug 2006
29 Aug 2006
0900
0100
0300
1300
0400
0500
0500
0500
0800
0700
0700
0800
0800
1700
0000
2000
2100
0000
0000
Slowness (s/km) (Ground Velocity)
2.78
2.32
2.93
2.78
2.70
2.53
2.95
3.02
3.05
2.48
2.86
2.88
2.86
2.86
2.61
0.74
2.99
1.69
3.08
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
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0.43
0.50
0.33
0.37
0.43
0.51
0.53
0.30
0.46
0.55
0.32
0.20
0.24
0.23
0.47
0.32
0.37
0.71
0.56
(1.93
(2.05
(2.40
(2.38
(2.34
(1.81
(1.86
(2.29
(1.93
(2.31
(2.43
(2.33
(2.51
(2.32
(2.48
(0.75
(2.13
(2.00
(2.11
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
0.51)
0.49)
0.37)
0.43)
0.43)
0.51)
0.53)
0.43)
0.49)
0.46)
0.37)
0.40)
0.40)
0.41)
0.44)
0.23)
0.45)
0.50)
0.50)
Azimuth (deg) (Ground Velocity)
170 ± 15 (161 ± 12)
302 ± 17 (309 ± 14)
130 ± 13 (121 ± 9)
322 ± 12 (326 ± 10)
336 ± 13 (339 ± 9)
131 ± 20 (124 ± 15)
161 ± 22 (150 ± 15)
212 ± 10 (224 ± 11)
223 ± 21 (232 ± 15)
29 ± 18 (22 ± 10)
295 ± 11 (300 ± 9)
305 ± 6 (305 ± 10)
10 ± 6 (15 ± 7)
160 ± 7 (153 ± 10)
30 ± 14 (31 ± 11)
258 ± 34 (285 ± 27)
161 ± 13 (152 ± 11)
276 ± 48 (301 ± 14)
145 ± 19 (142 ± 13)
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Figure 4. Distribution of incident slownesses and azimuths of all 19 thunder events shown by
numbered solid circles in a geographic coordinate system. Six concentric circles show the slowness from
0.5 to 3.0 s/km. Solid and open circles represent the incident slowness and azimuth inferred from acoustic
pressure and vertical ground velocity at the surface, respectively, and the lines connect the same event.
The waveforms show the comparison of events of similar azimuth and slowness. The acoustic-to-vertical
ground velocity transfer functions are shown grouped by their similar pressure slownesses and azimuths
determined from the vertical ground motion.
26th Seismic Research Review, Trends in Nuclear Explosion Monitoring, Air Force Research Laboratory, National
Nuclear Security Administration, Orlando, Florida, 21– 23
September 2004) was then applied to both the acoustic and
seismic waveforms to determine the azimuth and horizontal slowness of the incident wavefield and, for comparison,
the time domain grid search methods [Lin and Langston,
2007] was also used in the acoustic waveforms to find the
azimuth and horizontal slowness because of its impulsive
waveforms. The frequency-wave number computation used
to derive the slowness and azimuth is in units of dB for
normalized power. The error estimation in the frequencywave number computation is based on the standard deviation within the range of 1 dB power ratio, which
corresponds to a 0.80 amplitude ratio. The use of the
frequency-wave number method on both acoustic and
seismic records allows us to rule out that seismic signals
were induced by background noise. The consistency of
derived azimuth and slowness between frequency-wave
number and time domain grid search methods suggests
that there is only one high signal-to-noise ratio incident
acoustic pressure wave impinging into the array within the
frequency band range used in the frequency-wave number
computation. After calculating the azimuth and slowness,
the acoustic pressure waveforms and ground velocity
seismograms were beamed and stacked (Figure 2). We
then rotated the E-W and N-S beam-formed seismograms
to the radial and transverse directions according to the
azimuth.
4. Data From the Array
[7] Nineteen thunder events (Figure 3), short-duration
impulsive claps, were chosen to study the seismo-acoustic
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Figure 5. The comparisons between (a) incident azimuth and (b) slowness inferred from acoustic
pressure and vertical ground velocity. The slownesses and azimuths are tabulated in Table 1. The legend
is retained in both plots.
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Figure 6. (a) Peak-to-peak amplitude of incident pressure and scatterplots for peak-to-peak amplitude
of (b) acoustic total pressure against surface ground velocity and (c) surface against borehole ground
velocity. The line fits in the scatterplots were obtained assuming a zero intercept. The spread in the line
fits are represented by the standard deviations. The dashed lines extend the linear regression fits to the
extrapolated values of event 17.
coupling because they have high signal-to-noise ratios and
are isolated in time (at least 5 s) from other thunder events.
Figure 3a shows examples of stacked pressure and ground
motion time histories from four thunder events and
Figure 3b shows stacked pressure waveforms for all 19
thunder events used in this study. Their time of occurrence
and inferred incident pressure slowness and azimuth are
listed in Table 1. Most of the selected events have similar
short-duration N wave waveforms indicating atmospheric
sonic booms [Cook et al., 1972]. Figure 4 shows the
distribution of incident pressure slowness and azimuth for
all 19 events. Most thunder events have slownesses between
2.5 and 3.0 s/km (phase velocity between 333 m/s and
400 m/s) except for events 16 and 18, which have slownesses of 1.69 and 0.74 s/km, respectively, indicating
steeper angles of incidence. However, the incident pressure
slowness and azimuth of event 18, which has the largest
standard deviation values, were computed by four microphone sensors out of five because one microphone sensor
malfunctioned.
[8] Slowness and azimuth were also computed by applying
the frequency-wave number method to the vertical ground
velocity waveforms at the surface (Table 1). Figures 4 and 5
compare the slownesses and azimuths inferred from acoustic
pressure with that from the vertical ground velocity. The
inferred azimuths from the acoustic pressure and vertical
ground velocity are generally consistent. Event 16, with
nearly vertical incidence, and event 18, with only four
available pressure sensors, have the largest variations as
expected (Figure 5). The consistency of azimuth suggests
that the ground motions were induced by the incident thunder
pressure not by background noise (e.g., other close lightning
strikes, directional winds, or tree vibrations).
[9] Interestingly, the inferred slownesses measured using
acoustic pressure are consistently larger (slower horizontal
phase velocity) than that from vertical ground velocity
except for events 16 and 18 (Figure 5). Event 16, with
nearly vertical incidence, is the only event with identical
slowness computed by acoustic pressure and vertical ground
velocity. The differences between acoustic pressure and
vertical ground velocity slownesses imply that reverberations on seismic sensor, that follow the direct initial motion
on acoustic sensor, were not initiated at the array but
somewhat away from the array (within hundreds of meters)
except for event 16.
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Figure 7. Normalized amplitude spectra of acoustic pressure and induced vertical ground velocity.
Thick and thin lines in the velocity spectra represent the surface and borehole observations, respectively.
[10 ] The peak-to-peak amplitudes of total pressure
recorded by the microphones have average amplitudes of
0.17 Pa excluding event 17, which has the largest peak-topeak amplitude of 0.73 Pa (Figure 6a). These peak-to-peak
pressure levels are consistent with observations made by
Bhartendu [1971] and Bohannon et al. [1977] in previous
studies of thunder. The amplitude spectra of the pressure
data (Figure 7) have maxima at a frequency below 10 Hz,
which is similar to the spectral observations on thunder
made by Bhartendu [1964], Balachandran [1979], and
Holmes et al. [1971].
[11] The induced ground velocity waveforms for the
horizontally propagating acoustic waves show consistent
clear reverberations at a frequency between 4 and 7 Hz
(Figures 3 and 7). In contrast, ground velocities for event 16
were induced by an incident acoustic wave with a nearvertical incidence angle. These waveforms show a shortduration impulsive waveform without much reverberation
(Figure 3). The amplitude spectra of event 16 are broader than
the others because of its impulsive waveform (Figure 7).
[12] Using amplitude scatterplots, we first investigated
peak-to-peak amplitude correlations between pressure and
ground motions, and second the correlation between surface
and borehole ground motions (Figures 6b and 6c). The
linear regression fits in the scatterplots were extrapolated to
the values of event 17 since event 17 data were not used.
The scatterplots of acoustic pressure versus surface ground
velocity show that an acoustic pressure of 0.1 Pa can give
rise to about 1.2 to 1.3 mm/s surface ground velocity while
for event 17 (the largest peak-to-peak acoustic pressure), the
corresponding induced surface ground velocity is about 2.9
to 3.5 mm/s. However, because of the lack of high-pressure
thunder events (>0.4 Pa), this single high-amplitude-in-
duced ground motion probably is not sufficient to be
explained by nonlinear interaction of the acoustic wave at
the ground surface [Lu, 2005]. Lu [2005] proposed that
acoustic propagation in a soil is a hysteretic process, which
might be a major source of the acoustic nonlinearity of soils.
The amplitude spectrum of acoustic pressure for event 17
does not show apparent greater high-frequency content than
the other events (Figure 7). Acoustically induced ground
motions propagating from the surface downward into the
borehole show a clear linear relation and extrapolated values
fit event 17 well. Overall ground velocity amplitudes of
radial and vertical components at the surface are similar.
However, for event 16 with nearly vertical incidence its
radial velocity is significantly smaller than the vertical
component at the surface (Figure 3a). The transverse ground
velocity at the surface has much smaller amplitude than the
radial component in all events except for event 16 and is
consistent with radial polarization of the acoustical pressure
wave (Figure 3a).
[13] The 8 m depth of the borehole is relatively small
compared with the 80 m wavelength of a 5 Hz seismic
wave. However, the observations (Figures 3a and 6c) show
that the peak-to-peak velocity amplitude for radial and
vertical motions are significantly different over this vertical
distance. Figure 6c shows that radial peak-to-peak amplitudes at the borehole decrease to about 64% of the surface
value while the vertical motions are amplified by about
20%. Most particle motion plots of the reverberations in the
radial-vertical plane at the surface indicate clear retrograde
elliptical motion [Lin and Langston, 2007]. The major axis
of the ellipse lengthens in the vertical plane for the borehole
observations because of the larger amplitude decrease of the
radial component. Ground velocity waveforms of the bore-
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hole instrument generally retain the frequency content
(Figure 7) and waveshape (Figure 3a) from the surface
observations in the frequency range below about 20 Hz [Lin
and Langston, 2007].
5. Acoustic Source Equalization
[14] First, we calculated the spectral ratio between incident acoustic pressure and ground velocity at the surface,
which is analogous to the acoustic-to-seismic transfer function of Bass et al. [1980], Sabatier et al. [1986a, 1986b],
and Sabatier and Raspet [1988]. The spectral division
serves to remove the incident pressure source time function.
Assuming an incident acoustic pressure plane wave at the
surface, we can write the induced velocity ground motion
and total pressure field at the surface as
RðtÞ ¼ SðtÞ*ER ðtÞ
PðtÞ ¼ SðtÞ*EP ðtÞ
EP ðtÞ ffi KdðtÞ
ð2Þ
where K is a constant containing the reflection coefficient
and d (t) is a Dirac delta function. Since the Fourier
transform of the Delta function is unity, the spectrum of
EP(t) is a white spectrum. The spectral ratio between ground
velocity and acoustic total pressure at the surface then can
be expressed as
RðwÞ
¼ CER ðwÞ
PðwÞ
ð3Þ
where C is a constant. In order to stabilize the spectral ratio
by avoiding small or zero values of the denominator, a water
level criterion [Clayton and Wiggins, 1976] was used to
replace small spectrum values in the denominator with a
fraction of the maximum value of the denominator.
[15] By taking the inverse Fourier transform of the
complex spectral division between the ground motion and
acoustic pressure, the resulting waveform can be conceptually considered as an empirical Green’s function or the
impulse response of the ground to incident acoustic pressure
as suggested by equation (3). Before taking the inverse
Fourier transform, the spectral quotient was multiplied by
the Fourier transform of the time derivative of a Gaussian
pulse to serve as a band-pass filter. The Gaussian time
derivative filter is given by
GðwÞ ¼ ðiwÞe
w2
=4a2
where a is the parameter determining the frequency content
of the Gaussian time derivative filter and will consequently
affect the absolute amplitude (in time domain) of the inverse
Fourier transform of the filtered spectral quotient.
[16] The choice of a Gaussian time derivative rather than
a Gaussian pulse usually employed in teleseismic receiver
function studies [e.g., Langston, 1979] is due to its N wavelike waveform and that its resulting spectrum is close to
many acoustic pressure spectra.
[17] It is useful to examine the effect of background noise
on the deconvolution. An inspection of equation (1) shows
that spectral division between ground velocity and acoustic
total pressure can be expressed as
RðwÞ SðwÞER ðwÞ þ nR ðwÞ
¼
¼
PðwÞ SðwÞEP ðwÞ þ nP ðwÞ
ð1Þ
where R(t) and P(t) are ground velocity and total pressure,
respectively, which have been corrected for the instrument
responses, S(t) is the incident pressure source time function,
and ER (t) and EP (t) are ground velocity and total pressure
impulse responses from an incident pressure wave,
respectively. The reflected pressure field is nearly identical
to the incident pressure field owing to the very large
acoustic impedance at the surface at seismic wave
frequencies [Langston, 2004]. Thus, EP (t) can be
approximately written as
ð4Þ
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nR ðwÞ
SðwÞ
nP ðwÞ
EP ðwÞ þ
SðwÞ
ER ðwÞ þ
ð5Þ
where nR and nP represent background noises for ground
motion and total pressure, respectively. A filter is applied to
stabilize the spectral division such that equation (5)
becomes
RðwÞ
FðwÞ ¼ PðwÞ
nR ðwÞ
SðwÞ
nP ðwÞ
1
EP ðwÞ þ
SðwÞ FðwÞ
ER ðwÞ þ
ð6Þ
where F(w) is the Fourier transform of the filter. The second
term in the denominator on the right-hand side can be
further expressed as
nP ðwÞ 1
1
¼
SðwÞ
SðwÞ FðwÞ
FðwÞ
nP ðwÞ
ð7Þ
where S(w)/nP(w) is signal-to-noise ratio of incident
pressure. Equation (7) implies that a higher signal-to-noise
ratio for incident pressure after filtering will improve the
spectral division. Figure 8 shows that using a Gaussian time
derivative filter rather than a Gaussian pulse filter is more
appropriate for examining the main part of the signal band.
[18] Taking the spectral ratio between the radial and
vertical components gives
RðwÞ jRðwÞjeifR
¼
V ðwÞ jV ðwÞjeifV
ð8Þ
where f is the phase spectrum. Radial and vertical motions
have similar amplitude spectra that differ, roughly, by a
scaling factor as seen in both recorded and synthetic data.
Therefore, the spectral ratio can be approximated by
RðwÞ
eifR
k if
e V
V ðwÞ
ð9Þ
where k is a scaling constant.
[19] The particle motion plots of reverberations indicate
clear retrograde elliptical motion. Consequently, the propa-
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Figure 8. Signal-to-noise ratios of two incident pressures compared with the spectra of Gaussian time
derivative and Gaussian pulse filters. The frequency sampling used in generating the S/N ratio is at 0.8 Hz.
gation mode appears to be an air-coupled Rayleigh wave
excitation. We can apply the Rayleigh wave phase characteristic that says the Rayleigh wave radial and vertical
motions have a 90° phase shift in the frequency domain
[Boore and Toksöz, 1969]. This gives
p
RðwÞ
eifR
p
p
¼ K iðf pÞ ¼ Kei2 ¼ K cos þ i sin
¼ Ki
R
2
V ðwÞ
2
2
e
ð10Þ
where K can be interpreted as an ellipticity constant, the
ratio of the two semiaxes of the elliptic trajectory of particle
motion. A circular particle motion plot has an ellipticity of
one and an ellipticity of zero corresponds to linear
polarization.
[20] Recall that we applied a Gaussian time derivative
filter to stabilize the spectral division such that the spectral
ratio becomes
RðwÞ
p
p
FðwÞ ¼ KijFðwÞjeifF ¼ KjFðwÞjeifF ei2 ¼ KjFðwÞjeiðfF þ2Þ
V ðwÞ
ð11Þ
where F(w) is the Fourier transform of Gaussian time
derivative filter. Transforming frequency to time domain
using the inverse Fourier transform gives
F 1
RðwÞ
FðwÞ ¼ KðH½f ðtÞ
Þ
V ðwÞ
ð12Þ
where H[ ] is the Hilbert transform [Bracewell, 1978] and
f(t) is the Gaussian time derivative. The spectral ratios of
radial and vertical components were also filtered by a
Gaussian for the sole purpose of validating equation (12).
Because the Hilbert transform is a linear operator, the
Gaussian filtered transfer function can be simply computed
from the Gaussian time derivative filtered transfer function
by time integration.
[21] We used the computation algorithm and an Earth
model developed by Langston [2004] to first generate
synthetic seismograms and then produce the corresponding
time domain transfer functions (Figure 9). A propagator
matrix technique [Haskell, 1962] was used to investigate the
response of a plane-layered elastic half-space bounded by
an upper fluid half-space representing the atmosphere to an
incident acoustic wave. A Gaussian time derivative is
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Figure 9. (a) Comparisons between synthetic seismograms and corresponding velocity/pressure time
domain transfer functions. (b) Particle motion plots of the synthetic seismograms and time domain
transfer functions showing consistent retrograde propagation mode. The Earth model and incident
slowness (2.2 s/km) used to generate the synthetic seismograms is the same as Figure 16c of Langston
[2004].
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Figure 10. Velocity/pressure and radial/vertical velocity time domain transfer functions for all 19 events
with the Gaussian time derivative filter. The numbers shown at the end of each trace are event numbers.
The dashed lines indicate the time shift, which is at 0.5 s.
chosen as the source time function in generating synthetic
seismograms (Figure 9) due to its N wave-like waveform
(Figure 8). P and P-to-S wave reverberations are created
because of the upper low-velocity layer and high contrast
interface at depth. All P and P-to-S wave reverberations are
trapped in the upper low-velocity layer since the lower halfspace P and S wave velocities are greater than the horizontal
phase velocity of the acoustic wave. The resulting velocity/
pressure transfer functions have clear reverberation series as
in the seismograms. The initial motions interpreted from the
seismograms and transfer functions are consistent in both
radial and vertical components. The particle motion plots in
Figure 9b both display retrograde elliptical particle motions.
The synthetic transfer functions between the radial and
vertical components show that the waveform shapes are
close to the scaled Hilbert transform of the Gaussian time
derivative (Figures 9a) confirming retrograde elliptical particle motion.
6. Data Analysis
[22] We used the source equalization procedure to analyze velocity/pressure and radial/vertical time series for the
19 thunder events (Figures 4 and 10). Considering the
variations and complexity of the acoustic and seismic
waveforms among the events, the resulting time domain
transfer functions are striking simple and similar. The
transfer functions for event 16 (that has the lowest incident
slowness at 0.74 s/km) show far different waveshape than
the other transfer functions. This difference suggests that the
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Figure 11. Normalized amplitude spectra of the time domain transfer functions given in Figure 10.
impulse response of the ground to acoustic pressure is
affected by the incident slowness of the acoustic wave
[Lin and Langston, 2007] and implies that even with exactly
the same incident acoustic pressure waveshape the induced
ground response can be totally different owing to different
incident slowness.
[23] Figure 11 shows that amplitude spectra of the transfer functions have peak frequencies between 3.5 and 7 Hz
for all events except for event 16. Spectra of the transfer
functions for event 16 are broader and consistently peak at
frequencies near 10 Hz. Overall peak-to-peak amplitudes of
the time domain transfer function (Figure 12) for most
events are generally comparable with each other, which is
expected because of linearity shown in Figure 6b. However,
as before, event 17 having the largest incident pressure, the
radial and vertical transfer functions show the highest
amplitudes, but not the vertical-to-radial transfer function
(Figure 10). Since the event 16 acoustic wave has nearly
vertical incidence, its radial peak-to-peak amplitude is much
smaller than the others and is affected by a relative lower
signal-to-noise ratio compared to the vertical component
(Figure 3a). Peak-to-peak amplitudes of the radial transfer
function are generally the same as the vertical transfer
function; however, the radial peak-to-peak transfer function
amplitudes of event 16 are significantly smaller than the
vertical component (Figure 12). The time domain transfer
functions between the radial and vertical components for all
events (except for event16) show that the waveform shape is
close to the scaled Hilbert transform of the Gaussian time
derivative (Figures 9 and 10) implying retrograde elliptical
particle motion.
[24] Another interesting observation of the time domain
transfer function is the polarity of the initial motion. To
ensure that initial motions were not numerically generated
by the application of water level and filtering techniques,
various water level and filter parameters were tried in
addition to reversing the incident pressure or ground motion
polarities. For the acoustic-to-vertical ground motion transfer function, the initial motions consistently show downward motions in both recorded and synthetic data when the
Gaussian time derivative (source signal) is positive, meaning positive incident pressure. The resulting initial motion in
the vertical component is obtained from the fluid/solid
interface boundary conditions requiring continuity of the
vertical motion. For the acoustic-to-radial ground motion
transfer function, the positive incident pressure from the
Gaussian time derivative corresponds to negative radial
motion that is directed back toward the acoustic source.
Recall that there is no continuity of horizontal motion on the
fluid/solid interface boundary. This backward radial initial
motion was also observed by Langston [2004] and his
model suggested that it is theoretically possible for a solid
with a Poisson’s ratio less than about 0.25 to have the
backward radial initial motion.
7. Discussion
[ 25 ] The peak-to-peak amplitudes of total pressure
recorded by the microphones (Figure 6a) have an average
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Figure 12. Peak-to-peak amplitude ratios between acoustic-to-seismic time domain transfer functions
given in Figure 10.
amplitude of 0.17 Pa while event 17 has the largest
amplitude of 0.73 Pa. Generally, the thunder pressure
amplitude of any particular thunder event will be related
to the intensity of the lightning strike, giving rise to the Nwave, the orientation and irregularity of the lightning
source, and, most importantly, the distance to the source.
Wind and temperature might introduce propagation effect
on amplitude as well. However, we do not have control on
lightning distance, geometry, nor wind and temperature data
owing to instrument limitations [Lin and Langston, 2007].
The reconstruction of the lightning geometry and location is
beyond the scope of the present study.
[26] The incident acoustic waveform is basically removed
from the time domain transfer function by means of
deconvolution. As an ideal result, the time domain transfer
functions from several events should be similar if they have
the same incident slowness and if we assume the site
response is independent of azimuth. For instance, we paired
the events 2 and 18, events 4 and 5, and events 11 and 12,
according to their similar incident vertical ground motion
Figure 13. Relation between the dominant frequencies observed in the acoustic-to-vertical ground
motion transfer functions (Figure 10) and the wave parameters inferred from (a) acoustic pressure and
(b) vertical ground velocity. Note the systematic behavior of peak frequency with vertical motion slowness.
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Figure 14. Spectral division of (top) a long-duration
incident acoustic pressure signal and (bottom) its induced
vertical ground motion. A five-point moving average was
used to smooth the spectral ratio and is plotted as the dashed
line. The corresponding spectral ratio of event 15, which has
similar incident slowness and azimuth inferred from vertical
ground velocity, is compared.
slownesses and azimuths (Table 1). Although each has a
very different shape for the incident acoustic pressure, they
show remarkable similarity in their resulting time domain
transfer functions in each pair (Figure 4).
[27] Figure 13a and 13b plot the relation between the
dominant frequencies observed in the acoustic-to-vertical
ground motion transfer functions and the source parameters
inferred from acoustic pressure and vertical ground velocity,
respectively, for all events except for event 16. The dominant frequencies and the incident slownesses inferred from
vertical ground velocity have a systematic relation indicating that larger incident slowness has a higher dominant
frequency in the site response. In contrast, the incident
slownesses inferred from acoustic pressure do not have a
clear relation to the dominant frequencies. Note that in
Figure 13a the dominant frequencies of the acoustic-tovertical ground motion transfer functions are used but not
the dominant frequency in the acoustic pressure. Incident
azimuths inferred from the acoustic pressure or vertical
ground velocity have no notable pattern or relation to the
dominant frequencies implying that the site response is
independent of azimuth at the array site. Results from
Figures 4 and 13 suggest that using incident slowness
inferred from vertical ground velocity is more appropriate
than from acoustic pressure to study the time domain
transfer function or site response. This result suggests that
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the reverberations couple into the ground away from the
array. This observation implies that we might not need an
acoustic array other than a single instrument but instead use
the seismic array to obtain incident wave slowness and
azimuth for this type of experiment scale. However, this
suggestion does not argue that from coupled ground velocity is more robust than those from acoustic pressure in
estimating incident wave slowness and azimuth. The systematic relation between the ground motion slowness and
the dominant frequency of the acoustic-to-seismic transfer
function might serve as part of the medium’s phase velocity
dispersion relation. Such acoustic-induced seismic dispersion relation might be feasible for finding the medium’s
seismic properties.
[28] The spectral division (equation (3)) should be unaffected by the incident pressure source time function. This
was tested using a long-duration thunder event with a
significantly different time history. The resulting spectral
division of the long-duration signal (Figure 14) has a peak
frequency of about 5 Hz and is comparable to that from
horizontally propagating impulsive, short-duration events,
which have peak frequencies between 3.5 and 7 Hz.
Specifically, event 15, with similar incident slowness and
azimuth has comparable spectral division to that of the longduration signal (Figure 14).
[29] At the borehole, ground velocity amplitude in the
radial component decays rapidly with depth but vertical
component amplitudes are unaffected (Figure 6c) consistent
with a Rayleigh wave excitation [Dobrin et al., 1951].
Similarity and consistency of the time domain transfer
functions between radial and vertical motions (Figure 10)
shows the scaled Hilbert transform of the Gaussian time
derivative provides additional supportive evidence for aircoupled Rayleigh wave excitation.
[30] The acoustic-to-seismic transfer functions for ground
motions induced by the slowly propagating pressure wave
show consistent retrograde reverberation series at frequencies mostly around 5 Hz. Two previous studies using the
Center for Earthquake Research and Information (CERI)
Cooperative Seismic Network to record the atmosphere
disturbances [Langston, 2004; Lin and Langston, 2007]
show that one station HBAR of the CERI seismic network
displays consistent retrograde particle motions in both
studies and stations LPAR, LVAR, and TWAR have prograde particle motions as well. These propagation mode
consistencies suggest that our seismo-acoustic array site
structure is similar to station HBAR site structure situated in
the Mississippi embayment.
[31] The consistent measured slowness differences inferred between acoustic pressure and ground motions suggest that the ground reverberations were induced away from
the array. As shown in the observed seismograms (Figures 2
and 3), the major arrival is the ground reverberations and
not the direct air wave. Therefore, the frequency-wave
number analysis of the ground motions (Table 1) was
dominated entirely by the ground reverberations. These
ground reverberations appear to be air-coupled Rayleigh
waves composed of interfering trapped P and SV waves
[Langston, 2004]. Using the near-surface velocity structure
of the array site (one layer over a half-space velocity model)
obtained by Lin and Langston [2009], several travel time-
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Figure 15. Travel time- and ray parameter-distance plots assuming different heights of a point source in
the constant acoustic velocity atmosphere (330 m/s) to a ground surface receiver. The curves for the direct
air wave and reflected P, S, and PS arrivals for one, three, and five multiple reflections are shown. For
each reflected phase the thinnest and thickest lines represent one and five multiple reflections,
respectively. The legend is retained in all plots.
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and ray parameter-distance curves assuming different
heights of a point source in the constant acoustic velocity
atmosphere (330 m/s) were calculated (Figure 15). The
surface layer Vp, Vs, and thickness are 430 m/s, 300 m/s,
and 11 m, respectively. The lower half-space Vp and Vs are
960 and 660 m/s, respectively. Generally speaking, the
reflected seismic phases (PP and PS) will arrive earlier than
a direct air wave from an atmospheric point source not
higher than about 500 m to a ground receiver within a
horizontal distance of a few hundred meters horizontal.
Also, the acoustic ray parameters (horizontal slowness)
are appreciably larger than the seismic ray parameters (P
and PS reflections) at the same horizontal propagating
distance. As the height of an atmospheric point source
increases, the ray parameter deviation between acoustic
and seismic rays at the same horizontal distance decreases.
The average measured acoustic and seismic ray parameters
(discarding events 16 and 18) are at about 2.80 and 2.21 s/
km, respectively (Table 1).
8. Conclusions
[32] The time domain transfer function based on the
deconvolution technique is a useful tool for analyzing the
ground response induced by incident acoustic pressure since
the incident pressure source time function can be removed
from the seismogram. Another advantage of the time
domain transfer function is that the phase information can
be interpreted more easily than from the spectral ratios. The
phase information can be used to define the initial motion of
the induced ground motion, identify the ground motion
propagation mode, and study the details of the reverberation
waveform to quantify a site model.
[33] Incident slowness differences between acoustic pressure and vertical ground velocity suggest that reverberations
are first initiated somewhat away from the array. The
ground motions induced by a horizontally propagating
acoustic wave show consistent reverberations at a frequency
between 4 and 7 Hz. For a nearly vertical incident acoustic
wave, induced ground motions are short in duration and are
impulsive. Ground motion frequency content for surface
and borehole observations are generally similar. However,
the radial peak-to-peak amplitudes in the borehole drop to
about 64% of the surface value, while the vertical motions
are amplified by about 20%.
[34] The time domain transfer functions have consistent
initial motions, comparable overall amplitudes, and a welldefined propagation mode. The initial motions are downward and consistent with a positive incident overpressure.
For acoustic-to-radial ground motion transfer functions, the
initial motions are in the reverse direction of the wave
propagation direction due to the low Poisson’s ratio. Particle
motion plots in the radial-vertical plane, radial and vertical
ground velocity amplitude variations at the borehole, and
radial-to-vertical ground motion time domain transfer functions suggest the observation of an air-coupled Rayleigh
wave where reverberations of P waves and P-to-S conversions are trapped in the near-surface layer. Thunder-induced
air-coupled Rayleigh wave dispersion relation was observed
using the dominant frequencies of the acoustic-to-seismic
ground motion transfer function and incident slownesses
inferred from the seismic ground velocity.
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[35] Acknowledgments. We would like to thank Greg Steiner, Steve
Brewer, and Chris McGoldrick at CERI; Chris Hayward at Southern
Methodist University; and Carrick Talmadge and Shantharam Dravida at
NCPA, University of Mississippi. We want to express our sincere appreciation to them in constructing the array. Frequency-wave number analysis
was computed with the MatSeis software package [Harris and Young,
1997]. We thank our two anonymous reviewers and Brian Stump, the
Associate Editor, for their valuable and constructive comments.
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