Elastic parabolic equation solutions for underwater acoustic
problems using seismic sources
Scott D. Franka)
Department of Mathematics, Marist College, 3399 North Road, Poughkeepsie, New York 12601
Robert I. Odom
Applied Physics Laboratory, University of Washington, 1013 North East 40th Street, Seattle,
Washington 98105
Jon M. Collis
Department of Applied Mathematics and Statistics, Colorado School of Mines, 1500 Illinois Street, Golden,
Colorado 80401
(Received 19 March 2012; revised 12 December 2012; accepted 22 January 2013)
Several problems of current interest involve elastic bottom range-dependent ocean environments
with buried or earthquake-type sources, specifically oceanic T-wave propagation studies and
interface wave related analyses. Additionally, observed deep shadow-zone arrivals are not predicted
by ray theoretic methods, and attempts to model them with fluid-bottom parabolic equation solutions suggest that it may be necessary to account for elastic bottom interactions. In order to study
energy conversion between elastic and acoustic waves, current elastic parabolic equation solutions
must be modified to allow for seismic starting fields for underwater acoustic propagation environments. Two types of elastic self-starter are presented. An explosive-type source is implemented
using a compressional self-starter and the resulting acoustic field is consistent with benchmark solutions. A shear wave self-starter is implemented and shown to generate transmission loss levels consistent with the explosive source. Source fields can be combined to generate starting fields for source
types such as explosions, earthquakes, or pile driving. Examples demonstrate the use of source fields
for shallow sources or deep ocean-bottom earthquake sources, where down slope conversion, a known
T-wave generation mechanism, is modeled. Self-starters are interpreted in the context of the seismic
C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4790355]
moment tensor. V
PACS number(s): 43.30.Ma, 43.30.Dr, 43.20.Gp, 43.30.Zk [MS]
I. INTRODUCTION
A class of problems under investigation involve elastic
bottom range-dependent ocean environments with buried or
earthquake-type sources. Earthquake sources in particular
can generate oceanic T-waves, which occur when elastic
energy is converted to acoustic energy at the ocean-bottom
interface. These waves enter into the sound fixing and ranging (SOFAR) channel via down slope conversion1 or ocean
bottom roughness at the interface.2 Due to propagation in the
SOFAR channel, T-waves from even small sources can be
detected at extremely long ranges and are thus an important
aspect of monitoring for the Comprehensive Nuclear-Test
Ban Treaty (CNTBT),3 in addition to earthquake source
identification,4,5 and tsunamigenesis studies.6 Interface
waves resulting from seismic sources represent an additional
source of acoustic signals in the deep ocean.7,8
Several recent experiments involving deep-water acoustic propagation have recorded so-called “deep shadow-zone”
arrivals which have not been predicted by generic ocean
acoustic propagation models. Deep shadow-zone arrivals are
acoustic signals that have been observed at several experiment sites with hydrophones located well below the
a)
Author to whom correspondence should be addressed. Electronic mail:
scott.frank@marist.edu
1358
J. Acoust. Soc. Am. 133 (3), March 2013
Pages: 1358–1367
ray-theoretic turning point. For example, deep shadow-zone
arrivals have been observed off the coast of California9 during the acoustic thermometry of ocean climate experiment10
in the deep Pacific Ocean11 and in the long-range acoustic
propagation experiment (LOAPEX).12 Ray-based solutions
have been unable to predict these signals10,13 and parabolic
equation solutions have been used as an alternate means of
investigation.12,13 Internal wave scattering from the mixed
layer has been proposed as a mechanism for the penetration
of acoustic signals into the shadow zone14 and fluid bottom
parabolic equation solutions have predicted late acoustic
arrivals on hydrophone arrays near the sound channel.13
However, these studies did not address late arrivals that have
been observed on bottom mounted receivers in the deep
ocean where sediment interaction could play a role.10 After
fluid bottom parabolic equation solutions did not predict late
arrivals on deep water hydrophones and bottom mounted
geophones during the LOAPEX experiment, effects due to
elasticity in the bottom were suggested as a possible generating mechanism for these deep shadow-zone arrivals.12
Current parabolic equation solutions for acoustic propagation in elastic sediments are based on the (ur, w) formulation of
elasticity and are stable for a wide range of parameters.15
Recently, this formulation has been used in rotated variable treatments for range-dependent underwater seismo-acoustic problems.16 A single-scattering approximation in this formulation
0001-4966/2013/133(3)/1358/10/$30.00
C 2013 Acoustical Society of America
V
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has been developed for purely elastic environments17 and
has recently been used to simulate propagation in
complex multilayered range-dependent underwater acoustic
environments, including beach and island topography.18,19
Both the rotated variable and single-scattering advancements
are used in the elastic parabolic equation solutions employed
here.
The elastic self-starter has been used for purely elastic
range-independent problems20 and for purely elastic rangedependent problems with multiple layers.17 However, solutions have not been implemented for underwater acoustic
propagation scenarios. The capability to model a source in
the sediment is a necessary tool for the use of elastic parabolic equation solutions in the study of elastic wave propagation and acoustic conversion from elastic waves. This will
aid investigations of the previously mentioned T-waves,
interface waves, deep shadow-zone arrivals, or pile driving
sources which involve elastic sediments.21,22 In this paper
we apply previously derived strictly elastic self-starters to
environments that are relevant to studies in underwater
acoustics. Specifically, we demonstrate that these elastic
self-starters can represent either shallow buried sources in
the ocean bottom, or an earthquake source within the Earth’s
crust that transmits energy into the ocean water column. In
addition, elastic parabolic equation solutions exhibit interface wave phenomena and the formation of acoustic signals
via down slope conversion, a key generating mechanism of
T-waves.
In Sec. II the elastic parabolic equation solution is outlined. Section III details the elastic self-starters. Examples of
seismic source parabolic equation solutions that use the
rotated variable treatment and the single-scattering approximation are shown in Sec. IV. In Sec. V the self-starter fields
are related to the seismic moment tensor, which is often used
to describe seismic sources by the geophysical community.
II. PARABOLIC EQUATION METHOD
Assuming a time-harmonic point source, we describe
the parabolic equation solution in an axially symmetric,
two-dimensional coordinate system, where the range r is the
horizontal distance from the source and z is the depth below
the ocean surface. We factor the elliptic Helmholtz equation
into a product of parabolic operators that represent outgoing
and incoming energy. By assuming that backscattered
energy is negligible, only the outgoing factor is retained.
Current versions of the elastic parabolic equation solve for
the horizontal derivative of the horizontal displacement ur
and the vertical displacement w in the frequency domain
using the formulation15
@ ur
@u
1=2 ur
1
¼ iðL MÞ
; ur ¼
;
(1)
w
w
@r
@r
where L and M are matrices containing depth-dependent
operators that incorporate the compressional wave speed cp,
shear wave speed cs, and density q via the Lame parameters
of the elastic medium. Attenuation in the elastic medium
is incorporated using complex wave speeds, where ap
J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013
represents decibels (dB) of loss per unit wavelength of compressional waves and as is dB of loss per unit wavelength of
shear waves. This approach is accurate and stable in part
because interface conditions at the fluid-solid boundary are
explicitly enforced and depth discretization is effectively
handled using Galerkin’s method.15
For numerical implementations, Eq. (1) is written as
@
@r
ur
w
pffiffiffiffiffiffiffiffiffiffiffi ur
¼ ik0 I þ X
;
w
(2)
where X is a matrix of depth operators defined by
X¼
1 1
ðL M k02 Þ:
k02
(3)
I is the identity matrix and k0 is the reference wave number.
Writing the formal solution to this first-order differential
equation gives
pffiffiffiffiffiffiffiffi
ur ik0 Dr I þ X ur ¼
e
;
(4)
w r þ Dr
w r
for range step size Dr. The exponential operator is approximated by a rational-linear Pade series to give the split-step
Pade solution
n
Y
I þ ai;n X ur ur ik0 Dr
¼
e
;
w rþDr
I þ bi;n X w r
i¼1
(5)
where ai,n and bi,n are Pade coefficients calculated by applying accuracy and stability constraints.23 The solution is then
discretized in depth using Galerkin’s method and in range
using a Crank-Nicolson scheme.24 Given the field at range r
the discretized system is represented by a hepta-diagonal
matrix that can be efficiently inverted using an LU factorization to give the field at range r þ Dr. In order to march the
field out in range, what is needed is an initial condition at
range r ¼ 0 that is provided by the parabolic equation selfstarter which will be discussed in the next section.
Range-dependent environments can arise in the form of
range-dependent bathymetry or variable sound-speed profiles in the water. Either type of environment has historically
been handled in parabolic equation solutions by splitting
them into a sequence of range-independent regions, known
as a stair-step approximation. Increased accuracy for rangedependent bathymetry is achieved by using the rotated variable solution where the coordinate system is rotated in a
sloping environment so that the primary direction of propagation is aligned with the fluid-elastic interface, becoming
range-independent.16,25 Range dependence in the rotated domain (from the water’s surface and from variable thickness
sediment layering) is handled by stair-stepping.26 Additional
accuracy is obtained by applying a single-scattering correction at each vertical interface in the elastic sediment, which
involves solving a scattering problem and retaining only the
transmitted wave.17,18
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1359
III. ELASTIC SELF-STARTERS
The parabolic equation self-starter efficiently generates
a stable starting field for the parabolic equation solution and
has been derived for fluid,27,28 elastic,20,23 and poro-elastic
media.29 For elastic media, the starting field for a purely
compressional source with frequency f in a cylindrical geometry is15
ur
1=2
¼ qb ðL1 MÞ1=4 exp ir0 ðL1 MÞ1=2
w c
!
kp2 dðz zs Þ þ d00 ðz zs Þ
;
(6)
0
d ðz zs Þ
where kp ¼ x/cp is the compressional wave number, x ¼ 2pf
is the angular frequency, qb is the density in the elastic medium, and cp is the compressional wave speed. Also,
d(z zs) represents the Dirac delta function with d0 (z zs)
and d00 (z zs) its first and second derivatives. This source
field represents an explosive source in an elastic sediment,
with equal force in all directions. This type of source could
be used to represent shallow, buried explosions, such as detonations or nuclear explosions deep in the ocean bottom, and
is relevant to CNTBT monitoring.3
Since earthquakes tend to involve shearing action of
elastic materials, an initial source field that is derived using a
delta function in the vertical direction would be useful for
earthquake localization or tsunamigenesis studies.5,6 The
parabolic equation self-starter for a strictly shearing initial
field is15
ur
1=2
¼ qb ðL1 MÞ1=4 expðir0 ðL1 MÞ1=2 Þ
w s
!
0
d ðz zs Þ
:
(7)
dðz zs Þ
To avoid numerical instabilities associated with the delta
functions, these starting fields are smoothed by a differential
operator before applying Eq. (5).28
To model realistic earthquake or pile-driver type sources, which will generate both compressional and shear
energy,21 it is necessary to add these two source fields
together
!
u
u
ur
¼ a0 r
þ a1 r ; (8)
w
w c
w s
0
where a0 and a1 are weighting factors for the proportion of
compressional and shear energy generated by the source. By
weighting the relative contributions appropriately, a broad
range of complicated seismic sources are represented by
Eq. (8).
IV. BENCHMARK TESTS AND EXAMPLES
Transmission loss is the standard measure of change in
signal strength in underwater acoustics. For typical scenarios, it is defined as the log magnitude of the acoustic pressure
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J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013
at a receiver relative to that at the source, both assumed to be
in the water. Care must be taken when the source or receiver
are in elastic media. For problems featuring a source in the
water column, transmission loss in elastic media is calculated by obtaining the dilatation, which is proportional to
pressure, from the elastic field solution and scaling it relative
to the pressure at the source.19 For scenarios featuring a
source in elastic media, transmission loss in the water is calculated using the dilatation near the source as the reference
for the acoustic pressure:
pw 2pw ;
TL ¼ 20 log10 ¼ 20 log10 p0
qb c2b D0 @w
;
(9)
D0 ¼ u r þ
@z
where p0 is a reference pressure 1 m from the source and pw
is the acoustic pressure at the receiver in the water. The parameters qb and cb represent density and compressional
speed in the elastic layer containing the source. The quantity
qb c2b D0 , is scaled to 1 kg/m3 near the source.
If the source and receiver are both in elastic media,
transmission loss is obtained from the ratio of the pressure at
the receiver and the source, noted by D and D0, respectively.
These are scaled with elastic parameters specific to the layers
they are in, so that transmission loss is
p
q c2 D TL ¼ 20 log10 ¼ 20 log10 b 2b ;
(10)
p0
q0 c0 D0
where q0, c0, qb, and cb represent density and compressional
speed in the source and receiver elastic layers.
Since dilatation is related to the compressional potential
/ as D ¼ r2 /, it is analogous to calculate transmission loss
which is related to
of the elastic rotation in the xz plane, x,
¼ uz wx ¼ r2 w.23,30
elastic shear potential w as 2x
Transmission loss is calculated as
x
TLs ¼ 20 log10 ;
0
x
(11)
0 is the reference rotation 1 m from the source.
where x
We now consider some benchmark examples to demonstrate the accuracy of the self-starters in Eqs. (6) and (7).
The wave number integration model OASES is known to
give accurate results for range-independent seismo-acoustic
problems, including those with explosive sources and can be
used for a benchmark test of the compressional self-starter.31
Wave number integration solutions directly evaluate integral
transforms of the wave equation using numerical quadrature,
and are primarily applicable in range-independent environments.32 Figure 1 compares transmission loss curves from
parabolic equation solutions using the compressional selfstarter (solid curve) against those from OASES (dashed
curve) for a range-independent environment featuring a
500 m water column with c ¼ 1500 m/s and a receiver in the
water. The elastic bottom is a halfspace with cp ¼ 1700 m/s,
cs ¼ 800 m/s, and qb ¼ 2.0 g/cm3. Compressional and shear
wave attenuations are given by ap ¼ 0.1 dB/k and
Frank et al.: Parabolic equations with seismic sources
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FIG. 1. (Color online) Transmission
loss curves from an elastic parabolic
equation solution (solid curve) using
a compressional self-starter and
OASES (dashed curve) with an explosive source normalized to unit
pressure at 1 m distance. Source
location was 100 m below the fluidelastic interface. There is excellent
agreement between the two solutions
over 5 km range for complicated
modal patterns for a broad frequency
range. Curves are shown for a receiver depth of 200 m for (a) 15 Hz
and (b) 50 Hz sources, as well as
near the interface at depth 499 m for
(c) 15 Hz and (d) 50 Hz sources.
as ¼ 0.2 dB/k. The source is located in the elastic layer
100 m below the fluid-elastic interface. Figures 1(a) and 1(b)
show comparisons at 15 and 50 Hz, for a receiver near the
middle of the water column at 200 m depth. Figures 1(c) and
1(d) show comparisons for a receiver in the water, very close
to the water-sediment interface at 499 m depth. There is
excellent agreement between the two solutions in all cases,
especially in the far field. The small variations at short
ranges are likely due to evanescent modes, which are included
in the parabolic equation self-starter.27 In addition, these comparisons demonstrate excellent agreement at different frequencies. These comparisons involve a frequency-dependent shift
to account for the different source normalizations between the
elastic parabolic equation self-starter and OASES.33 At this
time a comparative solution does not exist for benchmarking
the shear self-starter.
Example A demonstrates how solutions generated using
the shear self-starter tend to those generated by the compressional self-starter (which has been benchmarked above) in a
smooth manner as the source approaches the interface. In
this example, we demonstrate that a Scholte interface wave
is excited by both types of seismic sources. The environment
is range-independent and consists of a 2 km water column
with soundspeed of 1500 m/s, overlying an elastic half space
cs ¼ 1700 m/s,
ap ¼ 0.05 dB/k,
with
cp ¼ 2400 m/s,
as ¼ 0.1 dB/k, and q ¼ 2.7 g/cm3. Figure 2(a) shows the compressional wave transmission loss field for a 15 Hz compressional source (such as that resulting from an explosion)
located 10 m below the fluid-solid interface. Elastic energy
J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013
is converted to acoustic at ranges close to the source. The
acoustic energy in the water column that appears to have
nearly horizontal direction after about 6 km is reflected off
the surface, while nearly horizontal acoustic energy past
8 km range and below about 1500 m in depth includes both
reflected acoustic energy and energy transmitted from the
sediment.
Figure 2(b) shows the shear transmission loss for this
environment and source configuration. Different transmission loss scales are used for the compressional and shear
fields to emphasize aspects of the respective solutions. Were
they plotted over the same dynamic range, say that of the
compressional field, then the shear plots would be visually
empty. This suggests the compressional field has a greater
contribution to the total field, as illustrated by Collins,23
although we do not investigate this here. The two lobes that
occur near the source at approximately 90 dB are due to the
direct shear wave arrival from the self-starter in Eq. (7) and
shear wave energy resulting from an incident compressional
wave at the interface, as reflection of a compressional wave
at an interface results in both shear and compressional
waves.30 After approximately 5 km, there is a thin region of
reduced transmission loss along the interface. This energy
corresponds to the Scholte interface wave and would not be
present if the bottom were treated as a fluid. The presence of
this energy in conjunction with the compressional wave
energy observed at the interface in Fig. 2(a) is required by
interface wave physics34 and is what restricts these types of
waves to environments involving elastic media. Transmitted
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1361
FIG. 2. (Color online) Transmission loss fields for Example A, consisting of 15 Hz seismic sources in an elastic halfspace beneath a 2000 m water column. In
each panel the source is located 10 m below the fluid-elastic interface. (a) Compressional field resulting from a compressional self-starter showing an interface
wave. Acoustic waves radiate horizontally from the interface in the far field. (b) Shear field for the same source showing an interface wave that decays away
from the interface and shear waves that result from transmission of downward propagating acoustic waves across the interface. (c) Compressional field resulting from a shear self-starter. (d) Shear field for same source as (c).
shear wave energy from downward propagating acoustics
interacting with the fluid-elastic interface is evident deeper
in the sediment.
Figure 2(c) shows the compressional wave transmission
loss field for a 15 Hz shear source located 10 m below the
fluid-solid interface. There is conversion to water-column
acoustic energy at the interface, though much less at angles
close to normal incidence on the interface. The most energetic parts of the acoustic field are comparable to the field
resulting from the compressional source in Fig. 2(a), and
notably, there is similarity between the angle and transmission loss values of the surface reflections and the nearly horizontal rays past 6 km. A strong Scholte interface wave is
also present in this example. These results, in the water column and at the interface, suggest the validity of the shear
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J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013
self-starter for underwater acoustic applications. The most
notable difference between Figs. 2(a) and 2(c) can be seen in
the elastic bottom. As expected, the shear source does not
cause substantial compressional wave energy to propagate
and the field attenuates over a shorter range compared to the
compressional source. The interface wave and acoustic field
converted from elastic waves are evident and form a Lloyd
mirror pattern, in particular after 3 km. The shear wave transmission loss field is shown in Fig. 2(d). Despite some variations in the near field, as range increases the field looks very
similar to that from the compressional source in Fig. 2(b),
including energy contributions from the interface wave,
reflected energy in the water-column, and downward propagating waves that result from interaction of acoustic waves
with the fluid-solid interface.
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FIG. 3. (Color online) Compressional field transmission loss curves resulting from a compressional self-starter (solid curves) and a shear self-starter (dashed
curves) for two receiver depths in Example B. Top panels show zr ¼ 1000 m and (a) zs ¼ 2050 m, (b) zs ¼ 2025 m, (c) zs ¼ 2010 m. Bottom panels show
zr ¼ 1950 and (d) zs ¼ 2050 m, (e) zs ¼ 2025 m, (f) zs ¼ 2010 m. Differences between the curves are shaded to emphasize the limiting behavior of the acoustic
field as the different source types approach the water-bottom interface.
Example B demonstrates that elastic parabolic equation
solutions resulting from the shear self-starter are consistent
with those from the compressional self-starter as the source
gets close to the water-sediment interface. Transmission loss
curves for the compressional wave field are shown in Fig. 3
for the compressional source (solid curve) and shear source
(dashed curve) with a shallow source at three different
depths in the same environment as in Example A. Transmission loss curves are shown in Fig. 3 for receiver depth
zr ¼ 1000 m and source depths (a) zs ¼ 2050 m, (b)
zs ¼ 2025 m, and (c) zs ¼ 2010 m. Differences between the
two curves are shaded to emphasize the increasingly similar
acoustic field. Figure 3(a) shows that the shear wave source
has less loss than the compressional wave source, which is
consistent with the finding that strike-slip earthquake sources
with vertically polarized shear components couple efficiently
to acoustic waves in the water column.2,5 The similarity
between the curves in Fig. 3(c) is expected from examination
of Figs. 2(a) and 2(c). A perfect match is not expected due to
the different starting field values in the ur and w variables
from the different sources, but this consistent behavior from
sources near the interface is notable. To examine possible
effects of the interface waves in the water column, the
remaining panels of Fig. 3 show transmission loss curves for
zr ¼ 1950 m and (d) zs ¼ 2050 m, (e) zs ¼ 2025 m, and (f)
zs ¼ 2010 m. In Fig. 3(d) differences on the order of 5 dB
appear between the curves, in particular after about 7 km.
J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013
The field generated by the shear source once again exhibits
less loss than that from the compressional source. Figure
3(e) shows an almost perfect match in the far field for
zs ¼ 2025 m. However, for zs ¼ 2010 m the compressional
source is more efficiently converting energy into the interface wave, since more loss occurs for the shear source. The
observation that the transmission loss curves are different
near the interface but similar near the middle of the water
column suggests that there is conversion of energy from the
Scholte wave into acoustic energy.
Example C demonstrates deep compressional and shear
seismic sources in a range-dependent environment. The
sound speed in the water is 1500 m/s. Below the water column is a 300 m thick sediment layer with cp ¼ 1650 m/s,
cs ¼ 700 m/s, ap ¼ 0.05 dB/k, as ¼ 0.1 dB/k, and q ¼ 2.1 g/cm3.
Beneath the sediment layer is an elastic half space with
cp ¼ 2400 m/s, cs ¼ 1700 m/s, ap ¼ 0.05 dB/k, as ¼ 0.1 dB/k,
and q ¼ 2.7 g/cm3. Range-dependent bathymetry is present in
the form of a seamount. The sediment layer thicknesses are
constant throughout the domain.
Figure 4 shows the (a) compressional and (b) shear field
solutions for this environment obtained by the elastic parabolic equation for a 10 Hz compressional source located
5 km below the fluid-sediment interface. There is a clear
acoustic wave in the water column that results from transmission of elastic wave energy in the sediment into the
water. These waves propagate as T-waves and can be seen to
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1363
FIG. 4. (Color online) (a) Compressional field for Example C resulting from a deep compressional seismic source in a range-dependent environment with an
intermediate seamount. (b) Shear field for the same environment. Initial shear energy at the interface occurs due to reflection of an elastic wave arrival directly
from the source. Shear wave ducting appears in the sediment layer and contributes to the acoustic field. A high number of interactions with the seabed due to
high-order modes excited by the range dependence is evident from the downward propagating energy in the elastic layer at ranges greater than 28 km. (c) Compressional field from a deep shear source. Note reduced transmission loss at the sediment interface near 1 km that corresponds to a compressional wave introduced by the interaction of a shear wave incident upon the interface. The low transmission loss region in the water starting at about 5 km range is likely due to
constructive interference of acoustic waves in the water and those introduced to the water by the interaction of elastic waves with the interface. (d) Shear wave
field for Example C. Note dipole fields at source depth in panels (b) and (c).
be reflecting off the surface and the bottom with increasing
range. A well-defined Lloyd mirror pattern appears in the
elastic basement, resulting from reflections of seismic waves
back into the elastic layer.
Conversion of the acoustic compressional energy into
shear upon interaction with the sediment interface is
observed at several points past 28 km where there are spikes
of downward propagating shear energy in Fig. 4(b) associated with acoustic reflections in the water column in Fig.
4(a). There is also shear wave ducting in the sediment layer
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J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013
that results in the continuous conversion of shear energy into
acoustic energy in the water column.
Figure 4(c) shows the compressional field and Fig. 4(d)
shows the shear field for this environment with a 10 Hz shear
source, also 5 km below the interface. Conversion of elastic
waves into acoustic energy in the water column can be
observed near 5 km where a region of reduced transmission
loss appears in the water column. This is likely the result of
constructive interference between acoustic energy that is
transmitted into the water column by elastic energy
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interacting with the interface at about 5 km and acoustic
energy that was introduced to the water column at shorter
ranges. A contribution from possible shear wave ducting in
the sediment layer also appears to add to the acoustic field.
There is apparent downslope conversion from the face of the
seamount opposite the source. Another interesting feature in
Fig. 4(c) is the small patch of low transmission loss at 1 km
range at the sediment-halfspace interface. This low
transmission loss spike occurs when shear waves generated
at the source are converted into compressional waves upon
reflection from the interface. Figure 4(d) shows strong shear
wave propagation in the sediment layer and the conversion
of acoustic energy into downward propagating shear energy,
visible as the diagonal spikes in the elastic layer at longer
ranges. These spikes are not evident at shorter ranges since
they are at lower intensity levels than the shear field near the
source.
Example D features a seamount in a shallow-water environment with a downward refracting profile, as well as a
seismic source that illustrates the combination of the two
self-starters using Eq. (8) with equal weighting. This source
is representative of a naturally occurring earthquake-type
source which will in general excite both wave types. A linear
sound-speed profile with c ¼ 1550 m/s at the ocean surface
and c ¼ 1480 m/s at 500 m depth is used. The sediment
halfspace has cp ¼ 3400 m/s, cs ¼ 1700 m/s, ap ¼ 0.1 dB/k,
as ¼ 0.2 dB/k, and q ¼ 1.8 g/cm3. A 25 Hz source is located
at 1500 m depth and is given by Eq. (8) with equal weightings. Compressional transmission loss results are shown in
Fig. 5. Note the seamount has depth 0 at 15 km range, and so
no acoustic energy in the water is being transmitted from the
near side of the seamount to the far side. The acoustic waves
in the water column past 30 km in range are generated by
seismic energy from the seismic source interacting with the
FIG. 5. (Color online) Example D showing a shallow-water environment
with a seamount that intersects the water’s surface. The 25 Hz source is
located at 1500 m depth and is a combination of equal weights of the compressional and shear wave self-starters. There are clear acoustic waves in the
water on the far side of the seamount, which are generated by the interaction
of elastic waves with the slope. The effect of the downward refracting profile is evident as range increases.
J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013
far slope, suggesting the downslope conversion mechanism
is being properly simulated. The effects of the downward
refracting profile can be clearly seen in the water past the
seamount, where the bulk of the propagating energy is near
the seafloor.
V. GEOPHYSICAL INTERPRETATION OF THE
SELF-STARTERS
The ability to generate parabolic equation solutions for
seismic sources extends the usefulness of elastic parabolic
equation solutions to geophysical applications. The seismic
moment tensor is often used to represent an elastic source
since it is flexible enough to accommodate a wide range of
source types and parameters, including explosive and
earthquake-type sources. The moment tensor can be used in
conjunction with Green’s tensor for the response of an elastic medium to obtain medium displacements at a point distant from the source.35 Indeed, expressions for each
component of the displacement vector can be obtained from
calculus operations that consist of the product of the moment
tensor and the gradient of the Green’s tensor.34,35 While full
details regarding the seismic moment tensor and its use can
be found in standard texts on seismology,34,35 a short analysis can establish a relationship with the elastic self-starters
presented in Eqs. (6) and (7). This relationship would allow
results from elastic parabolic equation solutions to be used
for analysis of T-wave arrivals in earthquake localization
studies or CNTBT monitoring, for example.
The seismic moment tensor is composed of the magnitudes of nine possible force couples, Mij,
0
1
Mxx Mxy Mxz
(12)
M ¼ @ Myx Myy Myz A;
Mzx Mzy Mzz
where each couple represents two equal magnitude forces
pointing in opposite directions along coordinate direction i
and separated by a small distance in coordinate j, in a Cartesian system. To preserve angular momentum, force couples
always appear in symmetric pairs or double-couples.35 This
means that non-zero off-diagonal elements of Eq. (12) will
appear in such a way that the tensor is symmetric.
The compressional self-starter in Eq. (6) represents an
explosive source, and is benchmarked against OASES in
Fig. 1, which confirms this type of source is properly represented by the self-starter. An explosion in an elastic medium
is represented in seismology with three equal magnitude orthogonal force couples. Specifically, one force couple points
in the positive and negative x directions, one points in the
positive and negative z directions, and one points in the positive and negative y directions (represented perpendicular to
the page for our two-dimensional considerations). The
moment tensor for this arrangement, and thus corresponding
to Eq. (6), is Mc ¼ M0I3, where M0 represents the explosion
magnitude and I3 represents the 3 3 identity matrix. The
physics of the relationship between Mc and Eq. (6) can be
further corroborated by noting that this moment tensor generates a dipole field in the vertical coordinate which is
Frank et al.: Parabolic equations with seismic sources
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1365
clearly evident near the source depth in the shear wave field
shown in Fig. 4(b).35
Such a direct relationship is not as evident for the shear
wave self-starter in Eq. (7). However, a comparison can be
drawn to equivalent body forces that are derived for the point
source equivalent of a fault along z ¼ zs with slip in the x
direction as36
0
fx ðx; y; zÞ / dðxÞdðyÞd ðz zs Þ;
fy ðx; y; zÞ / 0;
0
fz ðx; y; zÞ / d ðxÞdðyÞdðz zs Þ ;
(13)
where fx, fy, and fz represent forces in the subscripted directions. Allowing the range coordinate r in cylindrical coordinates to correspond to the x coordinate very close to the
source, there is a clear correspondence between the above
and Eq. (7) in the r component. Note that both expressions
include the derivative of the delta function in the z direction.
The zero in the y direction of the above is also consistent
with Eq. (7) since that equation is derived in the absence of
motion in the y direction, which is outside of the twodimensional grid the elastic parabolic equation solution is
derived in. Finally, the z component above includes the delta
function in the z direction, as does Eq. (7), and also, the derivative in the r direction is accounted for in the derivation.15
This derivative is also expressed in the exponent of 1/4, as
opposed to 1/4 seen in Eq. (6), due to the derivative
expression in Eq. (1). Thus the shear self-starter corresponds
to a fault plane perpendicular to the (r, z) computational domain that slips in the r direction. The moment tensor representation for this type of source is found in texts to be35
0
1
0 0 1
(14)
Ms ¼ M0 @ 0 0 0 A :
1 0 0
The field associated with this type of double-couple seismic
source is composed of dipole fields in both the r and z
directions.34 Figure 4(c) shows a set of dipole lobes in the
compressional field, confirming the physical accuracy of the
self-starter in Eq. (7) and credibility of this analysis. It is
possible there would be a corresponding dipole in the shear
field of Fig. 4(d) oriented with one set of lobes in the r direction and the other in the z direction, but the lobes corresponding to the z direction are likely smoothed during the
initiation of the self-starter. Shear energy does appear to be
immediately interacting with the fluid interface and converting to acoustic energy in the water column, suggesting the
upward oriented lobe of the shear source is accounted for.
VI. CONCLUSION
Two types of elastic parabolic equation self-starters
have been implemented for seismic sources in rangedependent underwater acoustic environments. The acoustic
field in the water column generated by a compressional elastic self-starter has been benchmarked with wave number
integration solutions at several frequencies. Examples demonstrate that elastic parabolic equation solutions produce
1366
J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013
expected transmission and reflection properties of elastic
waves incident on a fluid-solid interface for both explosive
sources using the compressional self-starter and faulting
source types that are better modeled with the shear selfstarter. Combining these two self-starters allows generation
of starting fields for a broad range of seismic sources such as
complicated strike-slip earthquakes or seismic fields resulting from pile driving activities.21,22
These new source capabilities improve parabolic equation solutions as tools to be used for the study of elastic
propagation mechanisms in seismo-acoustic environments.
The ability to isolate the source in an elastic medium is necessary to properly study elastic propagation effects in an
underwater acoustic environment. The proper transmission
of T-waves into the SOFAR channel and the generation of
interface waves by earthquakes or shallow buried sources,
such as ordinance, are two mechanisms that deserve such
investigation. These seismic sources can also be used to analyze effects of environmental parameters on the existence
and amplitudes of potential deep-shadow zone arrivals. A
full study of possible contributions of elastic bottom effects
to explain these types of arrivals will require broadband simulations in deep-water environments at higher acoustic frequencies than those presented here.
The interpretation of the compressional and shear wave
parabolic equation self-starters in the context of the seismic
moment tensor suggests the potential for the self-starter to
be useful in geophysical applications that study earthquake
source localization and characterization, as well as potential
uses for CNTBT monitoring. If parabolic equation selfstarters are implemented in three-dimensional settings,37,38 a
relationship between those fields and the seismic moment
tensor would allow parabolic equation solutions to simulate
results from any type of seismic source.
ACKNOWLEDGMENTS
Work supported by Office of Naval Research (ONR)
grants to Marist College and the Applied Physics Laboratory
of the University of Washington. Some computations were
performed on Intel Bladeservers provided to Marist College
by grants from the National Science Foundation (NSF). The
authors would like to thank William L. Siegmann (Rensselaer
Polytechnic Institute) and Paul A. Martin (Colorado School of
Mines) for interesting and useful discussions, and Adam Metzler (University of Texas at Austin Applied Research Laboratory) for his expertise regarding OASES.
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