A. M. Metzler and J. M. Collis: JASA Express Letters
[http://dx.doi.org/10.1121/1.4794348]
Published Online 12 March 2013
Computationally efficient parabolic equation
solutions to seismo-acoustic problems involving
thin or low-shear elastic layers
Adam M. Metzlera)
Applied Research Laboratories, University of Texas at Austin, Austin, Texas 78713
ametzler@arlut.utexas.edu
Jon M. Collis
Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden,
Colorado 80401
jcollis@mines.edu
Abstract: Shallow-water environments typically include sediments
containing thin or low-shear layers. Numerical treatments of these types
of layers require finer depth grid spacing than is needed elsewhere in the
domain. Thin layers require finer grids to fully sample effects due to
elasticity within the layer. As shear wave speeds approach zero, the governing system becomes singular and fine-grid spacing becomes necessary
to obtain converged solutions. In this paper, a seismo-acoustic parabolic
equation solution is derived utilizing modified difference formulas using
Galerkin’s method to allow for variable-grid spacing in depth.
Propagation results are shown for environments containing thin layers
and low-shear layers.
C 2013 Acoustical Society of America
V
PACS numbers: 43.30.Ma [AL]
Date Received: November 30, 2012
Date Accepted: February 19, 2013
1. Introduction
Parabolic equation solutions are accurate and efficient techniques for long-range ocean
acoustic propagation. Originally these techniques were developed for deep-water environments with weak range dependence,1 but have been extended to include modeling
the ocean sediment as an elastic medium.2 Propagation using the parabolic equation
method is achieved through a marching technique in range and depth-dependent operators which are approximated using a rational linear Pade approximation.3 The depthdependent operators are discretized through the use of Galerkin’s method with a
uniform-depth grid with spacing in a single dimension.2,4 A uniform-depth grid is sufficient for some seismo-acoustic environments; however, shallow-water environments
may contain thin sediment layer or low-shear layers where using uniform grids sacrifices efficiency for accuracy or vice versa.
A thin layer, defined as a sediment layer where the ratio of the layer thickness
to the compressional wavelength is less than one, may require more depth grid points
than is necessary throughout the rest of the domain to sufficiently sample the propagation within the layer. Similarly, low-shear layers, such as mud, clay, sand, and silt,5
contain small shear wavelengths which require a finer-grid sampling to accurately capture effects due to elasticity within the layer. In both cases, failure to finely sample
these regions typically leads to errors spreading throughout the domain yielding inaccurate solutions. Applying the required fine-grid spacing in a uniform grid provides an
accurate solution, but at a significant cost to efficiency, since the majority of the
a)
Author to whom correspondence should be addressed.
EL268 J. Acoust. Soc. Am. 133 (4), April 2013
C 2013 Acoustical Society of America
V
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A. M. Metzler and J. M. Collis: JASA Express Letters
[http://dx.doi.org/10.1121/1.4794348]
Published Online 12 March 2013
computational domain will be oversampled. In this paper, a variable-grid spacing
procedure6–8 applied to the parabolic equation method is presented, allowing for finegrid spacing to be used in the regions where it is necessary and coarser-grid spacing to
be applied elsewhere, resulting in both an accurate and efficient solution.
2. Elastic parabolic equation
Assuming a time-harmonic source, the governing equations for a two-dimensional elastic medium, where x and z are range and depth coordinates in an axially symmetric cylindrical coordinate system, are9
@2u @
@u
@ 2 w @l @w
l
þ qx2 u þ ðk þ lÞ
þ
¼ 0;
(1)
ðk þ 2lÞ 2 þ
@x
@z
@z
@x@z @z @x
@2w @
@w
@2u
@k @u
ðk þ 2lÞ
þ qx2 w þ ðk þ lÞ
þ
¼ 0;
(2)
l 2þ
@x
@z
@z
@x@z @z @x
for horizontal and vertical displacements u and w, density q, angular frequency x, and
Lame parameters k and l where k ¼ qðCp2 2Cs2 Þ and l ¼ qCs2 . Attenuation is
included through complex compressional and shear sound speeds Cp and Cs where
Cj ¼ cj =ð1 þ igbj Þ for j ¼ p; s, with compressional and shear attenuations bp and bs and
g ¼ ð40p log eÞ1 . Differentiating Eq. (1) with respect to x yields the system of equations
@
(3)
L 2 þ M q ¼ 0;
@x
for
"
L¼
k þ 2l
#
@ @l
þ
@z @z ;
0
ðk þ lÞ
l
2
3
@ @
2
l þ qx
0
6
7
M ¼ 4 @z @z
5;
@ @k @
@
2
ðk þ 2lÞ þ qx
ðk þ lÞ þ
@z @z @z
@z
ux
:
q¼
w
(4)
Equation (3) can be factored into a product of incoming and outgoing wave operators, and
by assuming that outgoing energy dominates, the parabolic approximation is obtained:
pffiffiffiffiffiffiffiffiffiffiffiffiffi
@q
¼ ik0 I þ X q;
@x
(5)
where X ¼ k02 ðL1 M k02 I Þ contains depth operators. A solution to Eq. (5) that
marches the field in range is constructed by using a Pade approximation3
qðx þ DxÞ ¼ eik0 Dx
n
Y
I þ aj;n X
qðxÞ;
I þ bj;n X
j¼1
(6)
where Dx is the range step and aj;n and bj;n are Pade coefficients determined by stability and accuracy constraints on the solution and its derivative.
J. Acoust. Soc. Am. 133 (4), April 2013
A. M. Metzler and J. M. Collis: Variable-grid parabolic equation EL269
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A. M. Metzler and J. M. Collis: JASA Express Letters
[http://dx.doi.org/10.1121/1.4794348]
Published Online 12 March 2013
The operator matrix X is discretized using Galerkin’s method which involves
representing the depth-dependent terms of Eqs. (1) and (2) via the equation2
ð
wi Q½h; / dz
ð
;
(7)
Q ½h; /jz¼zi ¼
wi dz
where / is a dependent variable, h is a physical parameter (such as density), Q½h; / is
an operator on h and /, and wi is a “hat” basis function defined by
(
jz zi j
wi ðzÞ ¼ 1 Dz ; jz zi j < Dz
(8)
0;
jz zi j Dz;
where Dz ¼ zi zi1 is a uniform spacing between depth grid points. Variables h and
/ in Eq. (7) are discretized by
X
X
hðzÞ ’
hi wi ðzÞ; /ðzÞ ’
/i wi ðzÞ;
(9)
i
i
where hi hðzi Þ and /i /ðzi Þ. When fully discretized according to Eq. (7), X is represented using a heptadiagonal matrix which is inverted efficiently using an LUdecomposition to solve for qðx þ DxÞ in Eq. (6), the solution at the next range point.
To propagate energy through layers of differing media, continuity conditions
must be satisfied at interfaces between media. Continuity of vertical displacement and
normal stress and vanishing tangential stress are conditions that are required to be met
across a horizontal interface between a fluid and elastic layer. These conditions can be
expressed as
1 @
2 ðkDÞ
¼ fwgE ;
(10)
qx @z
F
@w
fkDgF ¼ kux þ ðk þ 2lÞ
;
(11)
@z E
@
@
@w
ðkux Þ þ
ðk þ 2lÞ
þ qx2 w ;
(12)
0¼
@z
@z
@z
E
for the fluid dilatation D, and where subscripts F and E denote fluid and elastic
components.
3. Depth-dependent variable-grid spacing
To obtain an accurate solution using a uniform grid for environments containing either
thin layers or low-shear layers, a finely sampled grid must be used throughout the computational domain, resulting in a lack of efficiency. A mechanism to address this loss
of computational efficiency is to allow for variable-depth grid spacing in parabolic
equation solutions. In regions where a greater number of grid points are required, such
as in thin layers and in low-shear media, fine-grid spacing is used, and a coarser grid
can be employed where the fine grid is unnecessary.
To introduce a nonuniform-grid spacing, the basis functions wi in Eq. (7) are
redefined. For nonuniform-grid spacing, Dz is cast as two terms
di ¼ zi zi1 ;
diþ1 ¼ ziþ1 zi ;
EL270 J. Acoust. Soc. Am. 133 (4), April 2013
(13)
A. M. Metzler and J. M. Collis: Variable-grid parabolic equation
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A. M. Metzler and J. M. Collis: JASA Express Letters
[http://dx.doi.org/10.1121/1.4794348]
Published Online 12 March 2013
where di and diþ1 differ, and the basis functions wi are redefined as
8 z zi1
>
; zi1 < z < zi
>
< di
wi ðzÞ ¼ ziþ1 z ; z < z < z
i
iþ1
>
> diþ1
:
0;
otherwise:
(14)
The depth-dependent operators in Eqs. (1) and (2) are discretized using variable-grid
spacing according to Eqs. (7), (9), and (14) as follows:
di ðhi1 þ hi Þ
di hi1 þ 3chi þ diþ1 hiþ1
diþ1 ðhi þ hiþ1 Þ
/i þ
/i1 þ
/iþ1 ;
6c
6c
6c
@/ hi1 2hi
hi1 hiþ1
2hi þ hiþ1
/i1 þ
/i þ
/iþ1 ;
h
¼
3c
3c
3c
@z z¼zi
h/jz¼zi ¼
(15)
(16)
@h @/ hi1 hi
hi1 þ hi hi hiþ1
hi þ hiþ1
/i1 þ
þ
/iþ1 ;
¼
/i þ
di c
di c
diþ1 c
diþ1 c
@z @z z¼zi
(17)
@
@/ hi1 þ hi
hi1 þ hi hi þ hiþ1
hi þ hiþ1
/i1 þ
/i þ
/iþ1 ;
h
¼
di c
di c
diþ1 c
diþ1 c
@z
@z z¼zi
(18)
where c ¼ di þ diþ1 . Note that setting di ¼ diþ1 in Eqs. (15)–(18) results in the difference formulas for the uniform grid found in Ref. 2.
4. Examples
In this section examples involving two environments containing thin or low-shear
speed layers are presented, comparing accuracy and efficiency between parabolic equation solution implementations with traditional, uniform-depth grid spacing and
variable-depth grid spacing. Both examples are short-range, shallow-water environments introduced in Sec. V of Ref. 4.
Example A illustrates an environment containing a thin-layer sediment. A
50 Hz point source is located at a depth of 98 m in a 100 m water column with constant sound speed and density of cw ¼ 1500 m/s and qw ¼ 1:0 g/cm3 . A 2.5 m thick
sediment layer overlies a sediment half space. It is noted that this environment slightly
differs from Ref. 4, which used a 10 m thick thin layer. The thin-layer parameters are:
cp ¼ 1700 m/s, cs ¼ 800 m/s, q ¼ 1:3 g/cm3 , bp ¼ 0:1 dB/k, and bs ¼ 0:2 dB/k, and the
half space parameters are: cp ¼ 2400 m/s, cs ¼ 1200 m/s, q ¼ 1:7 g/cm3 , bp ¼ 0:1 dB/k,
and bs ¼ 0:2 dB/k. Transmission loss curves at a receiver depth of 60 m are shown in
Fig. 1(a) for two uniform-grid spacings: Dz ¼ 0:0625 m (dashed) and Dz ¼ 1:0 m
(solid). There are significant differences between the two curves, especially for ranges
greater than 1.5 km, suggesting that a choice of Dz ¼ 1:0 m is insufficient for accurate
propagation results. In Fig. 1(b) a solution using variable-grid spacing is shown (solid)
which uses a grid spacing of Dz ¼ 0:0625 m inside the thin layer and Dz ¼ 1:0 m everywhere else in the domain. This solution is in excellent agreement with the fine uniformgrid spacing solution which has been verified to be converged. Furthermore, computational efficiency is greatly improved using the variable-grid solution when compared to
the fine uniform-grid solution. In both cases, a maximum depth was set at 2 km (with
an appropriate absorbing layer to simulate a half space) resulting in a total of 32 000
depth-grid points computed per range step for the fine uniform-grid case and 2038
depth-grid points computed per range step for the variable-grid case, a 93% reduction
in the number of grid points calculated.
J. Acoust. Soc. Am. 133 (4), April 2013
A. M. Metzler and J. M. Collis: Variable-grid parabolic equation EL271
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A. M. Metzler and J. M. Collis: JASA Express Letters
[http://dx.doi.org/10.1121/1.4794348]
Published Online 12 March 2013
Fig. 1. Transmission loss for Example A at receiver depth of 60 m. Dashed curves represent the solution with a
uniform-grid spacing of Dz ¼ 0:0625 m. Solid curves represent the solution using (a) a uniform-grid spacing of
Dz ¼ 1:0 m and (b) a variable-grid spacing of Dz ¼ 1:0 m outside the thin layer and Dz ¼ 0:0625 m inside the
thin layer.
Example B considers an environment containing a low-shear speed sediment
layer. Example B has the same acoustical properties as Example A with the following
differences: the low-shear layer is 20 m thick with cs ¼ 100 m/s, q ¼ 1:2 g/cm3 , bp
¼ bs ¼ 0:5 dB/k, and half space parameters q ¼ 1:5 g/cm3 and bp ¼ bs ¼ 0:5 dB/k. The
shear wavelength in the low-shear layer is ks ¼ 2 m, suggesting that the grid spacing in
this layer must be fine. Solutions using a uniform grid with Dz > 0:1 m results in a
numerically unstable solution, requiring at least 20 grid points per shear wavelength in
the thin layer. Transmission loss curves at a receiver depth of 60 m are shown in Fig. 2
for a uniform-grid spacing of Dz ¼ 0:0625 m (dashed) and a variable-grid spacing
(solid) which uses a grid spacing of Dz ¼ 0:0625 m in the low-shear layer and Dz ¼ 1:0
m everywhere else in the domain. The two solutions are in excellent agreement. Again,
efficiency is greatly improved using the variable-grid solution, which requires only 7%
of the depth grid points per range step than the uniform-grid solution.
EL272 J. Acoust. Soc. Am. 133 (4), April 2013
A. M. Metzler and J. M. Collis: Variable-grid parabolic equation
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A. M. Metzler and J. M. Collis: JASA Express Letters
[http://dx.doi.org/10.1121/1.4794348]
Published Online 12 March 2013
Fig. 2. Transmission loss for Example B at receiver depth of 60 m. The dashed curve represents the reference solution from Ref. 4 with a uniform-grid spacing of Dz ¼ 0:0625 m. The solid curve represent the solution using a
variable-grid spacing of Dz ¼ 1:0 m outside the low-shear layer and Dz ¼ 0:0625 m inside the low-shear layer.
5. Conclusions
A two-dimensional elastic parabolic equation solution implemented with a variablegrid Galerkin discretization is presented that produces efficient solutions for environments containing either thin or low-shear speed sediment layers. These sediment layers
require a fine-depth grid to accurately represent propagation within these layers.
Failure to adequately sample these regions leads to inaccurate or divergent solutions.
Variable-grid spacing is implemented into modified Galerkin-difference formulas which
allow each layer to be appropriately sampled to produce an efficient solution. An
extension of this method would be to employ range-dependent variable-grid spacing so
that layers which vary with bathymetry can be efficiently sampled throughout rangedependent environments.
Acknowledgment
Work of the first author supported by the Internal Research and Development program of
the Applied Research Laboratories at The University of Texas at Austin.
References and links
1
F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics (AIP,
New York, 1994), pp. 343–412.
2
M. D. Collins, “A higher-order parabolic equation for wave propagation in an ocean overlying an elastic
bottom,” J. Acoust. Soc. Am. 86, 1459–1464 (1989).
3
M. D. Collins, “A split-step Pade solution for the parabolic equation method,” J. Acoust. Soc. Am. 93,
1736–1742 (1993).
4
W. Jerzak, W. L. Siegmann, and M. D. Collins, “Modeling Rayleigh and Stonely waves and other
interface and boundary effects with the parabolic equation,” J. Acoust. Soc. Am. 117, 3497–3503 (2005).
5
Reference 1, p. 41.
6
A. P. Rosenberg, “A new rough surface parabolic equation program for computing low-frequency acoustic forward scattering from the ocean surface,” J. Acoust. Soc. Am. 105, 144–153 (1999).
7
J. M. Collis, “Low-frequency seismo-acoustic propagation near thin and low shear speed ocean sediment
layers,” Proc. Meet. Acoust. 8, 070001 (2009).
8
W. Sanders, “Use of Glaerkin’s method using variable depth grids in the parabolic equation model,”
J. Acoust. Soc. Am. 132, 1972 (2012).
9
H. Kolsky, Stress Waves in Solids (Dover, New York, 1963), pp. 1–13.
J. Acoust. Soc. Am. 133 (4), April 2013
A. M. Metzler and J. M. Collis: Variable-grid parabolic equation EL273
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