Seismo-acoustic propagation near thin and low-shear speed
ocean bottom sediments using a massive elastic interface
Jon M. Collisa)
Department of Applied Mathematics and Statistics, Colorado School of Mines, 1500 Illinois Street, Golden,
Colorado 80401
Adam M. Metzler
Applied Research Laboratory, University of Texas at Austin, Austin, Texas 78713
(Received 27 October 2012; revised 1 October 2013; accepted 28 October 2013)
The seafloor is considered to be a thin surface layer overlying an elastic half space. In addition to
layers of this type being thin, they may also have shear wave speeds that can be small (order
100 m/s). Both the thin and low-shear properties, viewed as small parameters, can cause
mathematical and numerical singularities to arise. Following the derivation presented by Gilbert
[Geophys. J. Int. 133, 230–232 (1998)], the surface layer is approximated as a thick, finite-thickness interface, and modified ocean bottom fluid-solid interface conditions are derived as jump
conditions across the interface. The resultant interface conditions are incorporated into a seismoacoustic parabolic equation solution, and this interface-based solution is benchmarked against
existing solutions and previously derived modified fluid-solid interface jump conditions.
Accuracy quantification is given via dimensionless interface thickness parameters.
C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4829531]
V
PACS number(s): 43.30.Ma, 43.30.Dr, 43.30.Ky [JAT]
I. INTRODUCTION
An area of recent focus has been on acoustic interactions
with the complex ocean-bottom. In many seafloors, the layer
at the ocean bottom consists of partially consolidated sediments and can be complex in that it has geologic properties
approaching those of a fluid.1 In some situations, while the
material may be solid, it is marginally so, and effects due to
elasticity are generally small.2 In other scenarios, the layer
may simply be thin and of harder rock. Further, traditional
elastic media treatments assume discrete transitions from
fluid to sediment layers, where in reality this may not be the
case. The seafloor can be unconsolidated, rough, covered in
calciferous deposits, or at various stages of deposition.
Seafloors of this type are relevant because many common
shallow-water ocean sediments have low-shear speeds and
thin layers.3 As an example, the continental shelf of the
northeastern North American coast is primarily composed of
sandy sediments, which have low shear wave speeds (25 to
200 m/s). The nature of this seafloor layer makes it difficult
to model accurately, as the medium is near to the transition
from an elastic solid to a fluid. Another example, in Bass
Strait on the continental shelf between Australia and
Tasmania, there is a thin 1 m thick layer of capstone rock.4
Obtaining solutions for this particular environment can be
very expensive due to grid size requirements needed to capture effects of the thin layer.
A broad class of range-dependent seismo-acoustic
propagation problems can be solved accurately and efficiently using the parabolic equation method. Recent
a)
Author to whom correspondence should be addressed. Electronic mail:
jcollis@mines.edu
J. Acoust. Soc. Am. 135 (1), January 2014
Pages: 115–123
advances include improved treatments for range-dependent
bathymetry,5 and variable thickness sediment layering.6
Other advances have been to extend fluid-bottom treatments to three-dimensional propagation environments.7
While these solutions are robust and able to handle propagation in many real-world environments, it has been
observed both mathematically and numerically that parabolic equation solutions can be difficult to obtain in the
presence of thin or low-shear speed sediment layers.8 A
simple approach for including a thin surface layer in simulations would be to combine it with adjacent, thicker sediment layers in an aggregate or bulk sense.5,9,10 However,
this approach may neglect important physical properties of
the surface layer. An alternative is to treat the layer, which
is typically thin relative to adjacent layers, as a finite thickness interface, and derive modified fluid-solid interface
conditions across the interface. This approach integrates
material properties and interface thickness into the modified interface conditions. In this paper, modified fluid-solid
massive elastic interface (MEI) conditions are derived for
thin and low-shear speed sediment layers and are incorporated into an elastic parabolic equation solution. Results are
obtained for test cases that investigate the limits of this
approximation and establish regimes of accuracy for parameters such as interface thickness, wave speeds, and frequency. Section II gives motivating examples, and
modified interface conditions are derived for an elastic parabolic equation solution in Sec. III.
II. THIN AND LOW-SHEAR ELASTIC LAYERS
To understand the effects of small sediment parameters
on field calculations, consider a layered sediment environment that is modeled after a beach at Camp Pendleton, CA,
0001-4966/2014/135(1)/115/9/$30.00
C 2014 Acoustical Society of America
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115
depicted in Fig. 1(a) for a f ¼ 25 Hz sound source at
30 m.5,9 Upslope propagation is considered and water depth
varies, from 113 m at the source to 32 m at 2.5 km in range
near the beach. There is an h ¼ 11 m thick sandy surface
layer, over a 63 m thick sediment layer, over an elastic half
space. Compressional and shear wave speeds, ðcp ; cs Þ, of
the two topmost layers are (1650, 660) m/s and (1705, 684)
m/s. Consider the dimensionless parameters, np ¼ hf =cp
and ns ¼ hf =cs as metrics for layer thickness in terms of the
compressional and shear wavelengths in the surface layer.
For the environment shown in Fig. 1(a), np ¼ 0.17 and
ns ¼ 0.42. Transmission loss contours are shown in Figs. 1
and 2 where properties of the surface layer are varied to
demonstrate numerical instability. Figure 1(a) demonstrates
how current models can accurately and stably treat this
environment. If a more realistic value of the shear wave
speed in the surface solid were used, cs ¼ 25 m/s, numerical
instabilities arise, as shown in Fig. 1(b) where
ðnp ; ns Þ ¼ (0.17, 11.0). Note that increasing frequency,
meaning the dimensionless constants would be larger,
results in even more unstable and divergent solutions. This
FIG. 2. (Color online) Transmission loss contours for the beach environment
at Camp Pendleton, CA. The uppermost sediment layer is split into two
layers that are 3 and 8 m thick. The shear wave speed of the three meter
thick layer is varied: (a) 25 m/s and (b) 225 m/s.
FIG. 1. (Color online) Transmission loss contours for the beach environment
at Camp Pendleton, CA. (a) The original environment, based on geologic
survey, is depicted for a 25 Hz source. The shear wave speed in the uppermost sediment layer is 600 m/s. (b) The shear wave speed in the uppermost
sediment layer is 25 m/s.
116
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
can be understood by considering that wavelengths become
smaller and then the layer becomes less thin.
Suppose that a more refined sediment model were used
and consider that the surface layer is approximated by two
smaller layers, the topmost of which is 3 m thick. The shear
wave speed is taken to be 25 m/s in the 3 m thick layer.
This environment is depicted in Fig. 2(a). Solutions are
almost immediately corrupted, where here ðnp ; ns Þ ¼ (0.11,
3.0). Further decreasing the shear wave speed does not
allow for a solution. Increasing the shear wave speed to
225 m/s, meaning ns ¼ 0.33, results in a solution that
diverges more readily than the lower speed example, as
illustrated in Fig. 2(b). This behavior is surprising as it is
expected that solutions would become stable for greater
shear speeds. In this latter case, either the shear wave speed
is still too low or the layer is too thin. This suggests that
instabilities may arise as a combination of both thin and
low-shear layers. Increasing the shear wave speed to
400 m/s (ns ¼ 0.19) and keeping the same layer thicknesses
produces a stable solution. A final note is that solutions presented in this section required very fine depth grid spacing,
sacrificing efficiency for accuracy, and so are not useful for
long-range propagation scenarios.
J. M. Collis and A. M. Metzler: Massive elastic interface parabolic equation
III. MASSIVE ELASTIC INTERFACE
In this section, modified fluid-solid continuity conditions are derived, following the procedure presented by
Gilbert.8 Consider an idealized stratified ocean environmental model in which the water column is separated from an
semi-infinite elastic half space by a thin, elastic sediment
layer of thickness h (shown in Fig. 3). To quantify what is
meant by a thin layer, it is assumed that h is small compared
to compressional and shear wavelengths in adjacent media.
Assume that the properties of the thin layer are homogeneous and the dependent variables of the governing elastic
equations of motion, representing physical quantities (such
as displacements), are held to be constant within the layer,
effectively meaning that the layer moves as a unit; i.e., a
very thick or “massive” elastic interface, that can be thought
of as a rigid elastic plate.11 Therefore the elastic equations of
motion are used to derive fluid-solid jump conditions
between the ocean and elastic half space by treating the layer
as an interface. This approximation means there is no propagation within the layer, although there can be interface wave
propagation along the interface at the compressional and
shear wave speeds.8 The assumption of constant dependent
variables within the interface necessarily excludes certain
physical phenomena such as Scholte and Stoneley interface
waves and shear wave resonances.12–14 Moreover, solidsolid interface conditions are not necessary as this type of
interface is no longer present: Modified fluid-solid continuity
conditions will be enforced across the MEI to couple the
ocean layer to the elastic half space.
An axially symmetric cylindrical coordinate geometry is
assumed, the z axis positive downward, with interfaces and
boundaries planar and parallel. The water layer has constant
compressional wave speed cw , density qw , and depth H. A
time-harmonic point source of angular frequency x is
assumed in the water at depth zs , with time dependence
exp(ixt). Elastic media are assumed to be isospeed with
density qb , compressional wave speed cp , and shear wave
speed cs . Note that in this derivation material properties
within the interface are assumed to be constant (i.e., homogeneous medium); however, this is not required of surrounding layers, which allows for sound speed and geoacoustic
parameter profiling.
The elastic equations of motion in the (ur , w) formulation of elasticity are given by12
@ 2 ur @
@ur
þ qx2 ur
l
ðk þ 2lÞ 2 þ
@r
@z
@z
@3w
@l @ 2 w
þ ðk þ lÞ 2 þ
¼0
(1)
@r @z @z @r2
and
@2w @
@w
ðk þ 2lÞ
þ qx2 w
l 2þ
@r
@z
@z
@ur @k
þ ur ¼ 0;
þ ðk þ lÞ
@z
@z
(2)
for u and w, the horizontal and vertical particle displacements in the r and z coordinate directions, k and l, Lame parameters describing the medium, and ur ¼ @u=@r. The Lame
parameters are defined in terms of medium wave speeds and
densities by c2p ¼ ðk þ 2lÞ=q and c2s ¼ l=q. Loss in the
bottom layers is included by using complex wave speeds
Cp ¼ cp =ð1 þ igap Þ and Cs ¼ cs =ð1 þ igas Þ for ap and as the
compressional and shear wave attenuations in decibels per
wavelength, and g ¼ ð40p log10 eÞ1 .15 To convert these
attenuation values to those more typically used by seismologists, dimensionless Q and 1/Q, the reader is referred to Aki
and Richards.16
The displacements must satisfy a pressure-release
boundary condition at z ¼ 0 and continuity conditions at the
ocean bottom interface, z ¼ H, with the elastic half space.
The traction scalars T3 and T1 are the normal and tangential
stresses rzz and rrz and are defined as
T3 ¼ rzz ¼ ðk þ 2lÞ
@w
þ kur
@z
(3)
and
@u @w
þ
:
T1 ¼ rrz ¼ l
@z @r
(4)
At an interface between fluid and elastic media, conditions
that must be satisfied are continuity of vertical particle displacement, normal stress, and tangential stress. To aid in the
derivation, these are expressed as jump conditions, where
the jump in a quantity ½C is defined as the difference in that
quantity between values on either side of the interface,
Ca Cb , where a and b denote fluid and solid media. The
continuity conditions correspond to requiring the quantities
½w, ½T3 , and ½T1 to vanish.
Written out in nonstandard form for use in an elastic
parabolic equation solution,12,17 the fluid-solid continuity
conditions are
FIG. 3. The idealized sediment environment: A fluid layer; over a thin MEI;
over an elastic half space.
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
@
ðka Da Þ þ qa x2 wb ¼ 0;
@z
J. M. Collis and A. M. Metzler: Massive elastic interface parabolic equation
(5)
117
ka Da ¼ kb ður Þb þ ðkb þ 2lb Þ
@wb
;
@z
½T3 ¼ qM x2 hw
(6)
and
and
@
@
@wb
þ qb x2 wb ¼ 0;
ðkb ður Þb Þ þ
ðkb þ 2lb Þ
@z
@z
@z
½T1r ¼ qM x2 hur :
(7)
where the dilatation D in the farfield is D ¼ @u=@r þ @w=@z.
Using Eqs. (1) and (2), the jump conditions can be expressed
in the form given in Eqs. (5)–(7). To enforce interface conditions in terms of the dependent variable formulation ður ; wÞ,
a nonstandard form of the tangential stress condition is used,
@ðT1 Þb
@T1
@ur @ 2 w
¼ lb
þ 2
¼ 0:
(8)
¼
@r
@r
@z
@r b
Equation (8) is the same nonstandard interface condition
that has been enforced in past works, allowing for the condition to be enforced and expressed solely in terms of vertical derivatives.12,17 Implementations which enforce this
interface condition have been benchmarked against laboratory experimental data and shown to be highly accurate.18,19 To simplify notation, define T1r ¼ @T1 =@r.
Assuming homogeneous media, Eq. (2) is written as
@T3
@ur @w
¼ qx2 w l
þ 2 ;
(9)
@z
@z @r
Where previously these jumps were held to be zero there is
now a correction term associated with the thick interface.
The modified interface conditions written out are
ka Da ¼ kb ður Þb þ ðkb þ 2lb Þ
@
@
@wb
ðkb ður Þb Þ þ
ðkb þ 2lb Þ
@z
@z
@z
þ qb x2 wb ¼ qM x2 hur :
(10)
To understand the singularities that arise when the shear
modulus l is small, Eqs. (3), (8), (9), and (10) are written as
a linear system of the form
0
1
0
1
ur
ur
C
B
C
@ B
B w C ¼ AB w C;
(11)
@ T3 A
@z @ T3 A
T1r
T1r
where
@2
2
@r
0
k
k þ 2l
@
l
@z
qx2 ðk þ 2lÞ
0
qx2 l
@2
@r 2
(13)
The condition on the vertical displacement remains
unchanged. Note that this result does not include information
such as the wave speeds in the interface, only the density.
To derive improved MEI conditions that incorporate
geophysical parameters of the interface, an r-dependent form
of the solution is assumed in the farfield, kr 1. This
assumption is made only in deriving modified interface conditions and not on the general solution to the system.
Suppose that r-dependence is in the form of a plane wave,
ur ðr; zÞ ¼ eikr u^r ðzÞ;
@T1r
@3w
@ 2 ur
¼ qx2 ur k 2 ðk þ 2lÞ 2 :
@z
@r
@r @z
B
B
B
B
B
A¼B
B
B
B
B
@
(12)
and
and similarly, Eq. (1) is given by
0
@wb
qM x2 hw
@z
k
@2
@r 2
@3
@r 2 @z
1
1
0
lC
C
C
1
0C
k þ 2l C
C:
C
0
0C
C
C
A
0
0
or, alternatively that the r-dependence is a cylindrical wave
(a Hankel function) in asymptotic form in the farfield
eikr
H01 ðkrÞ ’ pffiffiffiffiffi ;
kr
with
!
@ 2 ur
ik2
3k2
1
2
’ H0 k þ
u^r ðzÞ:
@r 2
ðkrÞ1=2 4ðkrÞ3=2
(14)
To leading order these assumptions (plane or cylindrical
waves) give the same result after applying the farfield
assumption. Integrating through the layer gives
2
Þhw
½T3 ¼ ðqM x3 þ lM kM
and
2
½T1r ¼ ðqM x2 ðkM þ 2lM ÞkM
Þhur ;
with MEI interface conditions
As l ! 0 the term 1=l ! 1 suggests a singularity. This
term comes from the continuity of tangential stress condition, which only applies to elastic media.
To derive modified fluid-solid MEI conditions, ur and w
are held constant and Eqs. (9) and (10) are integrated
through the interface with respect to z, yielding
118
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
@wb
@z
2
þ ðqM x2 þ lM kM
Þhw
ka Da ¼ kb ður Þb þ ðkb þ 2lb Þ
(15)
and
J. M. Collis and A. M. Metzler: Massive elastic interface parabolic equation
TABLE I. Elastic properties of the layers in examples A through D, where Cp and Cs are compressional and shear wave speeds, is density, and ap and as are
compressional and shear attenuations. The layers are numbered from the seafloor.
Example
Elastic layer
Cp (m/s)
Cs (m/s)
qb (g/cm3)
ap (dB/wavelength)
as (dB/wavelength)
MEI
half space
MEI
1
half space
MEI
half space
MEI
half space
1650
1800
1650
1800
2400
1700
2400
1650
1800
100
800
400
800
1200
25
1200
600
720
1.5
2.0
1.5
2.0
3.2
1.2
1.5
1.58
2.25
0.1
0.1
0.1
0.1
0.1
0.5
0.5
0.0
0.2
0.05
0.05
0.05
0.05
0.05
0.5
0.5
0.00
0.1
A
B
C
D
@
@
@wb
þ qb x2 wb
ðkb ður Þb Þ þ
ðkb þ 2lb Þ
@z
@z
@z
2
¼ ðqM x2 þ ðkM þ 2lM ÞkM
Þhur :
(16)
These conditions include the geoacoustic parameters and
thickness of the interface. Again the condition on the vertical
displacement remains unchanged. A modified condition on
ur could be derived here; however, as no continuity condition is enforced on that variable, it is not needed for fluidsolid interface conditions, although it would be applied at
solid-solid interfaces. For calculations, since ur and w are
held constant within the interface their values are taken to be
the average of those on either side of the interface. A final
note is that this derivation does not require a half space
below the MEI: A layer of isotropic elastic media would
have yielded the same result.
IV. EXAMPLES
In this section the capabilities of an MEI-incorporated
parabolic equation solution are benchmarked. The MEI
conditions have been incorporated into a (ur , w) elastic parabolic equation solution,12 where existing fluid-solid conditions have been augmented with derived MEI conditions,
Eqs. (15) and (16). Solutions are compared against an elastic
parabolic equation solution RAMS,15 and the wavenumber
integration code OASES.20 The program RAMS, which does
not apply the MEI conditions, can be unstable for problems
with thin layers or low-shear sediments and therefore
requires very fine gridding to capture effects of these types
of layers. Note that OASES provides stable solutions in the
presence of low-shear and thin layers, in range-independent
environments, because it utilizes the direct global matrix
approach, which is unconditionally stable.20,21 Comparisons
are also made against alternate modified fluid-elastic interface conditions, those of Rokhlin and Wang (RW),22
described in the Appendix. Unless otherwise specified, the
basic environment under consideration is a rangeindependent oceanic waveguide with a thin elastic interface
between the water and an elastic half space, as depicted in
Fig. 3. Geoacoustic parameters for each example are given
in Table I. As is standard practice when computing parabolic
FIG. 4. Transmission loss line plots for
example A at a receiver depth of 130 m
for varying MEI thicknesses. The solid
curves represent the MEI solution and
the dashed curves represent the benchmark solution produced by OASES. The
compressional and shear wave-lengths
in the MEI are 82.5 and 5 m, corresponding to dimensionless interface
thicknesses of (a) ðnp ; ns Þ ¼ (6 103,
0.1), (b) ðnp ; ns Þ ¼ (3.6 102, 0.6),
(c) ðnp ; ns Þ ¼ (6.1 101, 1.0), and (d)
ðnp ; ns Þ ¼ (1.21 101, 2.0).
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
J. M. Collis and A. M. Metzler: Massive elastic interface parabolic equation
119
equation solutions, the computational domain is truncated by
placing a highly attenuating absorbing layer at the bottom of
the domain to prevent artificial bottom reflections, thus representing a half space.
To benchmark the MEI solution, it is compared against
OASES where interface thickness is varied. Example A considers a 20 Hz point source at a depth of 75 m located in a
150 m deep fluid layer. Transmission loss curves at a receiver depth of 130 m are shown in Fig. 4 for four thicknesses of the MEI: h ¼ 0.5, 3.0, 5.0, and 10.0 m. At this
frequency, compressional and shear wavelengths are 82.5 m
and 5 m, corresponding to (np , ns ) ¼ (6 103, 0.1),
(3.6 102, 0.6), (6.1 102, 1.0), and (1.2 101, 2.0). In
these plots, the solid curves represent the parabolic equation
solution with MEI conditions and the dashed curve represents the reference solution obtained from OASES. For each
of the cases the MEI-based solution maintains the accuracy
of the solution.
To determine an upper bound on acceptable interface
thickness, the MEI solution is compared against RAMS for
example B. In this environment a 5 Hz point source is
located at 150 m in a 300 m deep water column. The fluid
layer overlies the MEI, of variable thickness, which overlies
a sediment layer and a half space. The lower interface
between the sediment layer and half space occurs at a depth
of 100 m below the MEI. Three cases are considered for
which the MEI dimensionless layer thicknesses are (np , ns )1
¼ (103, 1.25 105), (np , ns )2 ¼ (102, 1.25 104), and
(np , ns )3 ¼ (101, 1.25 103). Transmission loss curves for
the three related environments at a receiver depth of 150 m
are given in Fig. 5. Solid curves represent the solution using
the MEI treatment, and dashed curves represent the solution
using RAMS. An extremely small vertical step size is used
in RAMS to ensure numerical convergence. The MEI
FIG. 5. Transmission loss line plots for example B at a receiver depth of
150 m for varying MEI thicknesses. The solid curves represent the MEIincorporated solution and the dashed curves represent the solution using
RAMS. Interface thicknesses are (a) ðnp ; ns Þ1 ¼ (103, 1.25 105), (b)
ðnp ; ns Þ2 ¼ (102, 1.25 104), and (c) ðnp ; ns Þ3 ¼ (101, 1.25 103).
FIG. 6. Transmission loss line plots for example B at a receiver depth of
150 m. The solid curves represent the MEI solution and the dashed
curves represent the RW jump conditions. Corresponding dimensionless
interface thicknesses are (a) ðnp ; ns Þ1 ¼ (103, 1.25 105) and (b)
ðnp ; ns Þ2 ¼ (102, 1.25 104).
120
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
J. M. Collis and A. M. Metzler: Massive elastic interface parabolic equation
solution provides a good approximation for the first two
cases as shown in Figs. 5(a) and 5(b), where the ratios
(np , ns ) are no greater than (102, 1.25 104). For the third
case, shown in Fig. 5(c), the MEI exceeds an acceptable
thickness that can be treated using a MEI, and there is significant disagreement between the two solutions.
Further investigation for example B is done by considering the alternate approximate jump conditions of RW.
Figure 6 gives transmission loss curves for a receiver at
150 m using the MEI conditions described in Sec. III (solid)
against MEI interface conditions from RW (dashed), for
(np , ns )1 and (np , ns )2 specified in the previous paragraph.
Figure 6(a) indicates nearly identical results between the two
sets of interface conditions, but when the MEI thickness is
increased, as in Fig. 6(b), the RW solution does not match
the result already established using the MEI conditions.
The RW conditions are only a good approximation when
the ratios (np , ns ) are approximately (np , ns )1 ¼ (103,
1.25 105) or less, an order of magnitude more restrictive
than the MEI conditions derived in Eqs. (15) and (16).
For example C, consider a range-independent environment consisting of 300 m of water, a 50 m thick interface,
and an elastic half space. Three frequencies are considered:
5 Hz [Figs. 7(a) and 7(d)]; 25 Hz [Figs. 7(b) and 7(e)]; and
100 Hz [Figs. 7(c) and 7(f)]. The MEI has a very low shear
wave speed of 25 m/s. The MEI solution (solid curve) is
compared against RAMS [Figs. 7(a)–7(c)] and OASES
[Figs. 7(d) and 7(e)] at receiver depth zr ¼ 60 m. The solution
FIG. 7. Transmission loss line plots for example C, where the MEI solution (solid curves) is compared against RAMS [(a), (b), and (c)] and OASES [(d) and
(e)] (dashed curves), at receiver depth 60 m. Results are shown at frequencies: 5 Hz [(a) and (d)]; 25 Hz [(b) and (e)]; and 100 Hz [(c) and (f)].
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
J. M. Collis and A. M. Metzler: Massive elastic interface parabolic equation
121
compares well against OASES at 5 Hz and 25 Hz, however
OASES was unable to produce a solution at 100 Hz. The
RAMS solution diverges in all cases and becomes closer to
the MEI solution at 100 Hz, where the interface becomes
larger with regard to wavelength, although the solution is not
converged. The conclusion here is that the MEI-incorporated
solution was able to produce a solution for this challenging
environment.
As a final example D, the utility of the MEI solution is
demonstrated for a range-dependent environment. The environment presented in Sec. II for the Camp Pendleton beach
is revisited, here with MEI conditions used to account for
the surface layer. This example illustrates that the MEI solution can be applied to environments with variably sloping
bottoms, an important capability of range-dependent elastic
parabolic equation solutions. A compressional field transmission loss plot is given in Fig. 8 for a 25 Hz source corresponding to the case attempted in Fig. 1(b). The MEIincorporated parabolic equation produces a solution, where
previously one could not be obtained for the low-shear speed
(25 m/s) case.
V. DISCUSSION
Modified fluid-solid, MEI conditions have been derived
for the seafloor boundary condition to account for a complex
thin layer between the ocean and sub-bottom layering, by
treating it as a finite-thickness interface. By assuming the
r-dependent form of the incident waveform, continuity conditions accounting for the elasticity of the interface have
been derived to give modified fluid-solid interface jump
conditions. The resultant approximate fluid-solid interface
conditions were then incorporated into an elastic parabolic
equation solution. The approximation was benchmarked
against OASES for accuracy. The MEI solution has been
shown to accurately handle environments with thin layers
and layers whose rigidity is small, which were previously
difficult to treat accurately and efficiently. The ratios of the
interface’s thickness to the compressional and shear
wavelengths were considered as dimensionless measures on
interface thickness and these values would need to be less
than O(101) and O(103), respectively, to maintain accuracy. Further, because the MEI conditions have been incorporated into a modern elastic parabolic equation solution, it
can be applied in range-dependent environments.
The MEI-incorporated solution is not applicable for every
ocean bottom study, particularly those that involve interface
waves or shear wave resonances, as propagation is not permitted through the interface and these phenomena are necessarily
excluded. Within the parameter regimes established in this
work, the MEI solution is accurate for low-shear speed problems, and more efficient than existing elastic parabolic equation solutions involving thin layers. Future work will consider
a slightly more complicated sediment model in which there is
a water layer, a surface layer, an intermediate elastic layer
above the so-called R-reflector, and then an elastic half space.
This type of sediment model was presented in the Camp
Pendleton examples and is a more appropriate generic sediment model. Other work would be to incorporate the modified
MEI conditions into normal mode solutions, or an Arctic environmental representation, and also to better account for
neglected physics within the interface.
ACKNOWLEDGMENTS
This work was partially supported through a grant to
A.M.M. by the Internal Research and Development program
of the Applied Research Laboratories at The University of
Texas at Austin. Special thanks to Bob Odom, Applied
Physics Laboratory, University of Washington, for an introduction to the MEI.
APPENDIX: ALTERNATE JUMP INTERFACE
CONDITIONS
Rokhlin and Wang22 derive interface conditions
between two solids where a thin elastic interface with thickness h exists between the solids. The particle displacements
and stresses across the interface are related by a transfer matrix B such that
0
1
0 0 1
u
u
B w C
B w0 C
B
C
B
C
(A1)
@ rzz A ¼ B@ r0 A;
zz
r0xz
rxz
where primed quantities lie in the lower solid and unprimed
quantities lie in the upper solid. The transfer matrix B contains properties of the interface which will contain parameters subscripted with 0. Following the assumptions in Ref.
18, the system of equations reduces to
Kn
;
r0zz
Kt
½u ¼ 0 ;
rxz
½w ¼
FIG. 8. (Color online) Transmission loss color contours for a 25 Hz source,
produced by MEI solution for the range-dependent environment at Camp
Pendleton.
122
J. Acoust. Soc. Am., Vol. 135, No. 1, January 2014
½rzz ¼ x2 mn w0 ;
½rxz ¼ x2 mp u0 ;
(A2)
J. M. Collis and A. M. Metzler: Massive elastic interface parabolic equation
for
l0
;
h
k0 þ 2l0
Kn ;
h
mn q0 h;
"
#
2
k
2
mp mn 1 4
ð1 n0 Þ ;
k0s
Kt (A3)
and n0 ¼ c0s =c0p where c0p and c0s are the compressional
and shear wave speeds within the interface and k0s is the
shear wavenumber. Rewriting these conditions in terms of
the traction scalars T3 and T1r and assuming that the upper
layer is a fluid results in the conditions:
½T3 ¼ qb x2 wh;
½T1r ¼ qb x2 þ 4kM l 1 (A4)
l
kþ2l
ur h;
(A5)
where the subscript M denotes a physical value in the MEI
and kM is taken to be the compressional wavenumber in the
interface.
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