Elastic parabolic equation solutions for underwater ...inside.mines.edu/~jcollis/JAS001358.pdf ·...

Elastic parabolic equation solutions for underwater acousticproblems using seismic sources

Scott D. Franka)

Department of Mathematics, Marist College, 3399 North Road, Poughkeepsie, New York 12601

Robert I. OdomApplied Physics Laboratory, University of Washington, 1013 North East 40th Street, Seattle,Washington 98105

Jon M. CollisDepartment 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 solu-

tions 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 environ-

ments. 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 solu-

tions. A shear wave self-starter is implemented and shown to generate transmission loss levels con-

sistent 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

moment tensor. VC 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4790355]

PACS number(s): 43.30.Ma, 43.30.Dr, 43.20.Gp, 43.30.Zk [MS] Pages: 1358–1367

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 rang-

ing (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 acous-

tic 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 experi-

ment sites with hydrophones located well below the

ray-theoretic turning point. For example, deep shadow-zone

arrivals have been observed off the coast of California9 dur-

ing 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 generat-

ing mechanism for these deep shadow-zone arrivals.12

Current parabolic equation solutions for acoustic propaga-

tion 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 treat-

ments for range-dependent underwater seismo-acoustic prob-

lems.16 A single-scattering approximation in this formulation

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]

1358 J. Acoust. Soc. Am. 133 (3), March 2013 0001-4966/2013/133(3)/1358/10/$30.00 VC 2013 Acoustical Society of America

Downloaded 11 Mar 2013 to 138.67.22.94. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

mailto:[email protected]

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 range-

dependent problems with multiple layers.17 However, solu-

tions 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 para-

bolic equation solutions in the study of elastic wave propaga-

tion 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 inter-

face 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 out-

lined. Section III details the elastic self-starters. Examples of

seismic source parabolic equation solutions that use the

rotated variable treatment and the single-scattering approxi-

mation 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

@

@r

ur

w

� �¼ iðL�1MÞ1=2 ur

w

� �; ur ¼

@u

@r; (1)

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 Lam�e parameters

of the elastic medium. Attenuation in the elastic medium

is incorporated using complex wave speeds, where ap

represents decibels (dB) of loss per unit wavelength of com-

pressional 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

� �¼ ik0

ffiffiffiffiffiffiffiffiffiffiffiI þ Xp ur

w

� �; (2)

where X is a matrix of depth operators defined by

X ¼ 1

k20

ðL�1M � k20Þ: (3)

I is the identity matrix and k0 is the reference wave number.

Writing the formal solution to this first-order differential

equation gives

ur

w

� ��rþDr

¼ eik0DrffiffiffiffiffiffiffiffiIþXp ur

w

� ��r

; (4)

for range step size Dr. The exponential operator is approxi-

mated by a rational-linear Pad�e series to give the split-step

Pad�e solution

ur

w

� ��rþDr

¼ eik0DrYn

i¼1

I þ ai;nX

I þ bi;nX

ur

w

� ��r

; (5)

where ai,n and bi,n are Pad�e coefficients calculated by apply-

ing 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 rthe discretized system is represented by a hepta-diagonal

matrix that can be efficiently inverted using an LU factoriza-

tion to give the field at range rþDr. In order to march the

field out in range, what is needed is an initial condition at

range r¼ 0 that is provided by the parabolic equation self-

starter which will be discussed in the next section.

Range-dependent environments can arise in the form of

range-dependent bathymetry or variable sound-speed pro-

files 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 range-

dependent bathymetry is achieved by using the rotated vari-

able solution where the coordinate system is rotated in a

sloping environment so that the primary direction of propa-

gation is aligned with the fluid-elastic interface, becoming

range-independent.16,25 Range dependence in the rotated do-

main (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 correc-

tion at each vertical interface in the elastic sediment, which

involves solving a scattering problem and retaining only the

transmitted wave.17,18

J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013 Frank et al.: Parabolic equations with seismic sources 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 geom-

etry is15

ur

w

� �c

¼ q1=2b ðL�1MÞ�1=4

exp�

ir0ðL�1MÞ1=2�

�k2

pdðz� zsÞ þ d00ðz� zsÞ�d

0 ðz� zsÞ

!; (6)

where kp¼x/cp is the compressional wave number, x¼ 2pfis the angular frequency, qb is the density in the elastic me-

dium, 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 det-

onations 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

w

� �s

¼ q1=2b ðL�1MÞ1=4

expðir0ðL�1MÞ1=2Þ

� �d0 ðz� zsÞ

dðz� zsÞ

!: (7)

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 sour-

ces, which will generate both compressional and shear

energy,21 it is necessary to add these two source fields

together

ur

w

!0

¼ a0ur

w

� �c

þ a1ur

w

� �s

;

(8)

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 scenar-

ios, it is defined as the log magnitude of the acoustic pressure

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 calcu-

lated 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 cal-

culated using the dilatation near the source as the reference

for the acoustic pressure:

TL ¼ �20 log10

�� pw

p0

�� ¼ �20 log10

�� 2pw

qbc2bD0

��;D0 ¼ ur þ

@w

@z; (9)

where p0 is a reference pressure 1 m from the source and pw

is the acoustic pressure at the receiver in the water. The pa-

rameters qb and cb represent density and compressional

speed in the elastic layer containing the source. The quantity

qbc2bD0, 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

TL ¼�20 log10

p

p0

�� ¼ �20 log10

qbc2bD

q0c20D0

��; (10)

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

of the elastic rotation in the xz plane, �x, which is related to

elastic shear potential w as 2�x ¼ uz � wx ¼ r2w.23,30

Transmission loss is calculated as

TLs ¼�20 log10

�x�x0

��; (11)

where �x0 is the reference rotation 1 m from the source.

We now consider some benchmark examples to demon-

strate 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 environ-

ments.32 Figure 1 compares transmission loss curves from

parabolic equation solutions using the compressional self-

starter (solid curve) against those from OASES (dashed

curve) for a range-independent environment featuring a

500 m water column with c¼ 1500 m/s and a receiver in the

water. The elastic bottom is a halfspace with cp¼ 1700 m/s,

cs¼ 800 m/s, and qb¼ 2.0 g/cm3. Compressional and shear

wave attenuations are given by ap¼ 0.1 dB/k and

1360 J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013 Frank et al.: Parabolic equations with seismic sources


as¼ 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 com-

parisons demonstrate excellent agreement at different frequen-

cies. 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 compres-

sional 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

with cp¼ 2400 m/s, cs¼ 1700 m/s, ap¼ 0.05 dB/k,as¼ 0.1 dB/k, and q¼ 2.7 g/cm3. Figure 2(a) shows the com-

pressional wave transmission loss field for a 15 Hz compres-

sional source (such as that resulting from an explosion)

located 10 m below the fluid-solid interface. Elastic energy

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 transmis-

sion 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

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 ex-

plosive source normalized to unit

pressure at 1 m distance. Source

location was 100 m below the fluid-

elastic 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 re-

ceiver 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.



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 ener-

getic 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 transmis-

sion loss values of the surface reflections and the nearly hori-

zontal rays past 6 km. A strong Scholte interface wave is

also present in this example. These results, in the water col-

umn and at the interface, suggest the validity of the shear

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 trans-

mission loss field is shown in Fig. 2(d). Despite some varia-

tions 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 propa-

gating waves that result from interaction of acoustic waves

with the fluid-solid interface.

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 result-

ing from a shear self-starter. (d) Shear field for same source as (c).



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. Transmis-

sion loss curves are shown in Fig. 3 for receiver depth

zr¼ 1000 m and source depths (a) zs¼ 2050 m, (b)

zs¼ 2025 m, and (c) zs¼ 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

zr¼ 1950 m and (d) zs¼ 2050 m, (e) zs¼ 2025 m, and (f)

zs¼ 2010 m. In Fig. 3(d) differences on the order of 5 dB

appear between the curves, in particular after about 7 km.

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

zs¼ 2025 m. However, for zs¼ 2010 m the compressional

source is more efficiently converting energy into the inter-

face 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 col-

umn is a 300 m thick sediment layer with cp¼ 1650 m/s,

cs¼ 700 m/s, ap¼ 0.05 dB/k, as¼ 0.1 dB/k, and q¼ 2.1 g/cm3.

Beneath the sediment layer is an elastic half space with

cp¼ 2400 m/s, cs¼ 1700 m/s, ap¼ 0.05 dB/k, as¼ 0.1 dB/k,and q¼ 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 para-

bolic 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 trans-

mission of elastic wave energy in the sediment into the

water. These waves propagate as T-waves and can be seen to

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 zr¼ 1000 m and (a) zs¼ 2050 m, (b) zs¼ 2025 m, (c) zs¼ 2010 m. Bottom panels show

zr¼ 1950 and (d) zs¼ 2050 m, (e) zs¼ 2025 m, (f) zs¼ 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.



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) associ-

ated with acoustic reflections in the water column in Fig.

4(a). There is also shear wave ducting in the sediment layer

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

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) Com-

pressional field from a deep shear source. Note reduced transmission loss at the sediment interface near 1 km that corresponds to a compressional wave intro-

duced 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).



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 envi-

ronment 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 c¼ 1550 m/s at the ocean surface

and c¼ 1480 m/s at 500 m depth is used. The sediment

halfspace has cp¼ 3400 m/s, cs¼ 1700 m/s, ap¼ 0.1 dB/k,as¼ 0.2 dB/k, and q¼ 1.8 g/cm3. A 25 Hz source is located

at 1500 m depth and is given by Eq. (8) with equal weight-

ings. 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

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 THESELF-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 elas-

tic medium to obtain medium displacements at a point dis-

tant 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 analy-

sis 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 magni-

tudes of nine possible force couples, Mij,

M ¼Mxx Mxy Mxz

Myx Myy Myz

Mzx Mzy Mzz

0@

1A; (12)

where each couple represents two equal magnitude forces

pointing in opposite directions along coordinate direction iand separated by a small distance in coordinate j, in a Carte-

sian 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 repre-

sented by the self-starter. An explosion in an elastic medium

is represented in seismology with three equal magnitude or-

thogonal 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 posi-

tive 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 Mc¼M0I3, 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 gen-

erates a dipole field in the vertical coordinate which is

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 com-

pressional 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 pro-

file is evident as range increases.



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 z¼ zs with slip in the xdirection as36

fxðx; y; zÞ / dðxÞdðyÞd0 ðz� zsÞ;fyðx; y; zÞ / 0;

fzðx; y; zÞ / d0 ðxÞdðyÞdðz� zsÞ ; (13)

where fx, fy, and fz represent forces in the subscripted direc-

tions. Allowing the range coordinate r in cylindrical coordi-

nates 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 two-

dimensional 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 de-

rivative 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 do-

main that slips in the r direction. The moment tensor repre-

sentation for this type of source is found in texts to be35

Ms ¼ M0

0 0 1

0 0 0

1 0 0

0@

1A: (14)

The field associated with this type of double-couple seismic

source is composed of dipole fields in both the r and zdirections.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 direc-

tion and the other in the z direction, but the lobes corre-

sponding 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 convert-

ing 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 range-

dependent underwater acoustic environments. The acoustic

field in the water column generated by a compressional elas-

tic self-starter has been benchmarked with wave number

integration solutions at several frequencies. Examples dem-

onstrate that elastic parabolic equation solutions produce

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 self-

starter. 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 result-

ing from pile driving activities.21,22

These new source capabilities improve parabolic equa-

tion 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 nec-

essary 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 ana-

lyze 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 sim-

ulations in deep-water environments at higher acoustic fre-

quencies 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 self-

starters 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 Met-

zler (University of Texas at Austin Applied Research Labora-

tory) for his expertise regarding OASES.

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