Two parabolic equations for propagation in layered
poro-elastic media
Adam M. Metzlera)
Applied Research Laboratories, The University of Texas at Austin, Austin, Texas 78713-8029
William L. Siegmann
Rensselaer Polytechnic Institute, Troy, New York 12180
Michael D. Collins
Naval Research Laboratory, Stennis Space Center, Mississippi 39529
Jon M. Collis
Colorado School of Mines, Golden, Colorado 80401
(Received 10 October 2012; revised 26 March 2013; accepted 9 May 2013)
Parabolic equation methods for fluid and elastic media are extended to layered poro-elastic media,
including some shallow-water sediments. A previous parabolic equation solution for one model
of range-independent poro-elastic media [Collins et al., J. Acoust. Soc. Am. 98, 1645–1656 (1995)]
does not produce accurate solutions for environments with multiple poro-elastic layers. First,
a dependent-variable formulation for parabolic equations used with elastic media is generalized
to layered poro-elastic media. An improvement in accuracy is obtained using a second
dependent-variable formulation that conserves dependent variables across interfaces between
horizontally stratified layers. Furthermore, this formulation expresses conditions at interfaces using
no depth derivatives higher than first order. This feature should aid in treating range dependence
because convenient matching across interfaces is possible with discretized derivatives of first order
C 2013 Acoustical Society of America.
in contrast to second order. V
[http://dx.doi.org/10.1121/1.4807826]
I. INTRODUCTION
The parabolic equation method is accurate and efficient
for ocean acoustic propagation when outgoing energy dominates backscattered.1 In many shallow water regions, sediments closely resemble poro-elastic media2–5 for which a
parabolic equation formulation is necessary for accurate and
efficient computations of the full wave field. In this paper,
two new parabolic equation formulations are derived for
layered poro-elastic media. An earlier formulation6 was
developed and benchmarked for a single poro-elastic layer,
but this approach cannot handle multi-layered environments
because the dependent variables are not continuous across
horizontal interfaces between two poro-elastic layers. The
first new formulation is a natural extension of an elasticmedia dependent variable formulation,7 which has improved
accuracy for layered elastic media.8 While the extended formulation improves the original poro-elastic formulation, it
still contains a dependent variable that is discontinuous
across horizontal interfaces in poro-elastic media. A second
new formulation, extending another elastic variable technique,9 is derived with the explicit goal of keeping all
dependent variables continuous across horizontal interfaces.
This formulation provides accurate and efficient solutions
for layered poro-elastic media. Moreover, the approach
a)
Author to whom correspondence should be addressed. Electronic mail:
ametzler@arlut.utexas.edu
246
J. Acoust. Soc. Am. 134 (1), July 2013
Pages: 246–256
removes one potential difficulty associated with the extension necessary for range dependence.
In this paper, the description of poro-elastic media
follows the standard Biot model,10 which postulates that
sediments are composed of porous solids with pores containing flowing fluid. The model treats the individual behavior
of the solid and fluid components as well as their coupling,
including energy loss from the solid and the fluid viscosity.11,12 Biot-model poro-elastic media support three types of
body waves: Fast and slow compressional and shear,13
although other approaches for poro-elastic media may not
allow for a slow compressional wave.14
Section II presents the equations of motion using the
Biot poro-elastic model and develops the form of parabolic
equations for poro-elastic media. In Sec. III, the parabolic
equation for the first new dependent variable formulation is
derived, and examples are presented for both single and
double-layered poro-elastic environments. Section IV derives
the second new dependent variable formulation, illustrates
the improved accuracy of this formulation for rangeindependent layered environments, and discusses the potential benefits for range-dependent environments. Section V
discusses a summary of findings and conclusions.
II. PARABOLIC EQUATIONS FOR THE BIOT
PORO-ELASTICITY MODEL
A generic parabolic equation formulation is developed
in this section for poro-elastic media consisting of one or
0001-4966/2013/134(1)/246/11/$30.00
C 2013 Acoustical Society of America
V
Author's complimentary copy
PACS number(s): 43.40.Fz, 43.30.Ma [TFD]
more horizontal homogeneous layers. Depth variations in the
poro-elastic parameters (sound speeds, densities, etc.) are
approximated by such homogeneous and stratified layers,
with appropriate conditions at the layer interfaces. Although
heterogeneous poro-elastic media can be handled directly,
the numerical implementation is cumbersome and is not
treated here.
A. Biot equations of motion
The Biot formulation provides the equations of motion
for small displacements in poro-elastic media with the
specific development following Ref. 6. Two-dimensional
Cartesian coordinates are range x and depth z, with two
vectors u ðu; wÞ and uf ðuf ; wf Þ representing the displacements of the porous solid and the fluid. The porosity a
is the volume fraction of the medium consisting of pore
spaces, and qs and qf correspond to the solid and pore-space
fluid densities. The three wave speeds are the fast compressional c1 , slow compressional c2 , and shear c3 . Attenuation
is handled by complex wave speeds cj ¼ cj ð1 þ iebj Þ1 for
j ¼ 1; 2; 3, where bj is the attenuation for each wave type in
decibels per wavelength and ¼ ð40p log10 eÞ1 has units to
make the quantity ebj dimensionless.
The time-harmonic equations of motion using the Biot
model are
@rxz @rzz
þ
þ x2 ð1 aÞqs w þ aqf wf ¼ 0;
@x
@z
@r
s g
þ qf x2 uf þ U þ ix U ¼ 0;
@x
a
j
@r
s
g
þ qf x2 wf þ W þ ix W ¼ 0;
@z
a
j
(1)
@u
þ Cf;
@x
@u @w
rxz ¼ l
þ
;
@z @x
(3)
(5)
(6)
(7)
r ¼ CD þ Mf;
(8)
where
J. Acoust. Soc. Am., Vol. 134, No. 1, July 2013
and poro-elastic moduli parameters k, l, C, and M represent
medium properties. Equations (1)–(9b) are Biot’s equations
of motion for a poro-elastic medium. In this paper poroelastic media are specified by a, s, qf , qs , c1 , c2 , c3 , and the
fluid sound speed cw , which under our assumptions are constant. Relationships among these parameters, the four poroelastic moduli, and bulk moduli are found in Appendix A.
Substitution of Eqs. (5)–(8) into Eqs. (1)–(4) and using
U to eliminate uf results in the system
lr2 u þ ðk þ lÞrD þ Crf þ qx2 u þ qf x2 U ¼ 0;
(10)
CrD þ Mrf þ qf x2 u þ qc x2 U ¼ 0;
(11)
where
q ¼ ð1 aÞqs þ aqf ;
qc ¼
(12a)
qf ð1 þ sÞ
g
:
þi
xj
a
(12b)
For a ¼ 0; Eq. (10) simplifies to the elastic equations of
motion.17 Equations (10) and (11) are equivalent to the system described by Eq. (6) in Ref. 18 where H k þ 2l,
e D, and w U.
To be suitable for the parabolic equation method,
Eqs. (10) and (11) must be rewritten in the form
@2
(13)
K 2 þ L q ¼ 0;
@x
(4)
@w
rzz ¼ kD þ 2l
þ Cf;
@z
@u @w
þ
;
D¼ru
@x @z
(9b)
B. Generic parabolic equations
(2)
where U ðU; WÞ ¼ aðuf uÞ is relative displacement of
the porous medium weighted by porosity, and x is circular
frequency. Viscosity g and permeability j correspond to
Darcy’s law for flow through a porous medium,15 and s is an
added mass correction accounting for pore-space geometry.13 Quantities rxx , rxz , and rzz represent stresses, and r
is fluid pressure. Constitutive equations defining the stresses
and fluid pressure are16
rxx ¼ kD þ 2l
@U @W
þ
;
@x
@z
(9a)
where q is a vector of dependent variables and matrices K
and L contain the poro-elastic parameters and depth operators. The dependent variables ðu; w; U; WÞ do not produce a
system of type Eq. (13) because first-order range derivatives
occur, so other choices of variables in q must be chosen. For
such formulations, Eq. (13) is factored to yield
@
@
(14)
þ iðK 1 LÞ1=2
iðK 1 LÞ1=2 q ¼ 0;
@x
@x
where factors in Eq. (14) correspond to outgoing and incoming energy. Assuming outgoing energy dominates, the parabolic equation is
@q
¼ iðK 1 LÞ1=2 q:
@x
(15)
A solution to Eq. (15) can be written as
1=2
qðx þ Dx; zÞ ¼ eik0 DxðIþXÞ qðx; zÞ;
(16)
where X ¼ k02 ðK 1 L k02 IÞ, k0 is a reference wave number,
and I is the identity operator. Equation (16) marches the
solution forward in range by Dx, and the exponential is
Metzler et al.: Poro-elastic parabolic solutions
247
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@rxx @rxz
þ
þ x2 ð1 aÞqs u þ aqf uf ¼ 0;
@x
@z
f¼rU
III. DEPENDENT VARIABLES q^ 5ðux ; w ; fÞT
~ formulation, like its elastic counterpart q
~ E , may
The q
be inaccurate for layered poro-elastic environments, because
both D and f are not continuous across horizontal interfaces
between poro-elastic layers. Guided by success with elastic
media, a poro-elastic dependent-variable formulation corre^ ¼ ðux ; w; fÞT , in which ux is
^ E is derived: q
sponding to q
^ formulation
continuous across these interfaces. The q
provides a significant improvement in accuracy and stability
~ for layered poro-elastic environments.
over q
A. Derivation
Following the derivation for elastic media, the first of
three equations is obtained by substituting the x-component
of Eq. (11) into the x-component of Eq. (10) and taking a
range derivative, yielding
2
3
qf
qf
@ ux
@ w
þ
k
þ
l
C
k þ 2l C
2
qc
@x
qc
@x2 @z
!
2
q2f
qf
@ f
@ 2 ux
2
ux ¼ 0:
þl 2 þx q
þ C M
qc
@z
qc
@x2
(17)
Substituting the z-component of Eq. (11) into the z-component
of Eq. (10) produces the second equation,
248
J. Acoust. Soc. Am., Vol. 134, No. 1, July 2013
l
2
qf
qf
@2w
@ux
@ w
þ
k
þ
2l
þ
k
þ
l
C
C
qc
@z
qc
@x2
@z2
!
q2f
qf
@f
2
¼ 0:
(18)
wþ C M
þx q @z
qc
qc
The third equation is derived by subtracting C=ðk þ 2lÞ
times the divergence of Eq. (10) from the divergence of
Eq. (11), giving
2
C2
@ f
C
2
M
q
ux
þ
x
q
f
k þ 2l @x2
k þ 2l
2
C
@w
C2
@ f
q
þ M
þ x2 qf k þ 2l @z2
k þ 2l
@z
C
q f ¼ 0:
þ x2 qc (19)
k þ 2l f
Equations (17)–(19) have the form of Eq. (13) for dependent
^ with matrices K^ and L^ defined in Appendix B.
variables q
The interface conditions in Ref. 6 can be expressed in terms
^ variables through substitution of D. In this variable
of the q
formulation, the conditions contain second-order depth
derivatives of some dependent variables.
B. Examples
In this section, results from three range-independent
environments containing poro-elastic media show why it is
necessary to use a dependent variable formulation that can
handle multiple poro-elastic layers. All environments consist
of a fluid layer containing a point source, overlying one or
two poro-elastic layers with parameter values in Table I.
Environment A is taken from Ref. 6 and demonstrates the
^ (and q
~ ) formulation to handle single poroability of the q
elastic layers. Environment B contains two poro-elastic
^ formulation provides a reasonably accurate
layers, and the q
solution when compared with the benchmark solution. While
^
environment C also contains two poro-elastic layers, the q
formulation does not give an accurate solution when compared to the benchmark. For both environments B and C, the
~ solutions are not accurate. The benchmark solutions in
q
environments B and C are obtained from the wave number
integration code OASES.22
Environment A has a 25 Hz point source at a depth of
25 m within a 200 m thick fluid layer overlying a single
poro-elastic layer. Transmission loss results at 50 m depth
~ and q
^ formulations.
are shown in Fig. 1 for both the q
Convergent results are obtained in this example, and in all to
follow, in the usual manner for parabolic equation calculations by reducing the horizontal and vertical grid sizes until
no differences are visible in transmission loss curves. For
this example, the computational steps in the horizontal and
vertical are Dr ¼ 20 m and Dz ¼ 2 m, and eight terms are
used in the Pade approximation of the exponential in
~
Eq. (16). This example was used in Ref. 6 to show that the q
solution agreed with a benchmark. The two solutions are in
^
excellent agreement, illustrating that results from the q
Metzler et al.: Poro-elastic parabolic solutions
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approximated using a Pade approximation.19 Depth operators in X are discretized using Galerkin’s method, and the
resulting system is solved by Gaussian elimination.
For layered environments, interface conditions are explicitly applied to propagate energy accurately between
layers. At a horizontal fluid/poro-elastic interface, conditions
correspond to continuity of fluid flow, normal stress, and
fluid pressure and vanishing tangential stress. The conditions
at an elastic/poro-elastic interface correspond to continuity
of horizontal and vertical solid displacements and normal
and tangential stresses, and vanishing fluid flow.20 At an
air/poro-elastic interface, fluid pressure, normal stress, and
tangential stress all vanish. At an interface between two
poro-elastic layers, both solid displacements, both stresses,
fluid flow, and fluid pressure are all continuous.
The original poro-elastic parabolic equation6 used
~ ¼ ðD; w; fÞT
a
dependent-variable
formulation
q
which exploits a redundant equation in the system Eqs.
(10) and (11). As a consequence these equations reduce to
a system of three coupled equations having the form of
~ is an extension of the use of
Eq. (13). This formulation q
~ E ¼ ðD; wÞT in the initial development of the elastic paraq
bolic equation.17,21 For some layered elastic environments,
~ E formulation produces inaccurate solutions because the
the q
variable D is not continuous across horizontal interfaces.8
Consequently, an elastic parabolic equation employing the
^ E ¼ ðux ; wÞT , where ux @u=@x, was
dependent variable q
8
developed and shown to be accurate for layered elastic
environments because the variable ux is continuous across
^ E variables to
horizontal interfaces. An extension of the q
poro-elastic media is presented in Sec. III.
TABLE I. Parameters for environments A–C. The parameter qf ¼ 1:0 g/cm3
for all poro-elastic layers.
Environment
f (Hz)
Fluid layer:
Upper poro-elastic layer:a
Lower poro-elastic layer:
B
C
25
1500
200
–
–
–
–
–
–
–
–
–
–
0.3
0.0
2.0
2400
1000
1200
0.25
1.0
0.5
200
1500
50
0.3
1.25
1.3
1600
900
400
0.2
5.0
0.4
25
0.2
0.5
1.5
2400
900
500
0.3
2.5
0.5
25
1500
100
0.35
1.0
1.25
1700
800
450
0.1
2.5
0.2
25
0.25
0.5
1.75
1900
850
525
0.25
2.5
0.5
Environment A has no upper poro-elastic layer.
formulation are accurate for environments containing a
single poro-elastic layer.
Environment B features a 200 Hz source at 25 m depth
within a 50 m thick fluid layer overlying a 25 m thick poroelastic layer, which overlies a poro-elastic half space.
Transmission loss results at 25 m depth are shown in
Fig. 2(a), comparing the benchmark solution and the result
~ formulation. The q
~ result is sufficiently different
using the q
from the benchmark, both in amplitude and pattern phase,
that it does not provide an appropriate solution. Figure 2(b)
compares transmission loss between the benchmark and the
^ formulation. The q
^ result does very well
solution using the q
in matching the pattern-phase peak and null locations of the
FIG. 1. Transmission loss for environment A at 50 m depth. The dashed
curve represents the solution using the q~ formulation, and the solid curve
represents the solution using the q^ formulation.
J. Acoust. Soc. Am., Vol. 134, No. 1, July 2013
FIG. 2. Transmission loss for environment B at 25 m depth. The dashed
curves represent the benchmark solution. The solid curves represent the
parabolic equation solution using (a) the q~ formulation, and (b) the q^
formulation.
benchmark. While small amplitude discrepancies occur
^ formulation produces a solubetween the two curves, the q
tion in reasonable agreement with the benchmark and is
~ solution.
significantly improved over the q
Environment C contains a 25 Hz source at 75 m depth
within a 100 m thick fluid layer overlying a 25 m thick
poro-elastic layer, which overlies a poro-elastic half space.
Transmission loss results at 75 m depth are depicted in
Fig. 3(a), which compares the benchmark solution to the result
~ solution. For this environment, the q
~ solution is difrom the q
vergent, and no adjustment of computational parameters produces a convergent solution. [In Fig. 3(a), the number of Pade
coefficients is eight, the depth step is 1 m, and the range step
is 10 m.] Transmission loss comparisons between the bench^ solution are shown in Fig. 3(b). This
mark solution and the q
formulation produces a stable solution, but it has significant
differences from the benchmark both in amplitude and pattern
^ formulation does not produce an accuphase. Therefore the q
rate result for this environment, and an improved poro-elastic
dependent variable formulation is needed.
There are at least two possible sources for the differences
shown in the curves of Fig. 3(b). First, the input parameter
structure of the parabolic equation and OASES are different.
The parabolic equation requires the sound speeds and
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249
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a
cw (m/s)
Thickness (m)
a
s
qs (g/cm3 )
c1 (m/s)
c2 (m/s)
c3 (m/s)
b1 (dB/k)
b2 (dB/k)
b3 (dB/k)
Thickness (m)
a
s
qs (g/cm3 )
c1 (m/s)
c2 (m/s)
c3 (m/s)
b1 (dB/k)
b2 (dB/k)
b3 (dB/k)
A
interfaces. The variables are related to those in a poroacoustic environment23 as well as recent work for elastic
media.9 Two new dependent variables C and K are defined
C
rzz
2l @w C
¼Dþ
þ f;
k
k @z k
(20)
K
r
M
¼ D þ f:
C
C
(21)
Because C and K are multiples of normal stress and fluid
pressure, both of which are required to be continuous across
poro-elastic interfaces, they and w are chosen for the de . Consequently, results using
pendent variable formulation q
should be more accurate for layered poro-elastic environq
^ formulation.
ments than the q
A. Derivation
satisfies a system of equations in the
To show that q
form of Eq. (13), a derivation similar to that in Sec. III A is
useful. The first of three equations is obtained by using Eqs.
(20) and (21) in Eq. (10) and subtracting a depth derivative
of the z-component of Eq. (10) from a range derivative of its
x-component,
k
FIG. 3. Transmission loss for environment C at 75 m depth. The dashed
curves represent the benchmark solution. The solid curves represent the parabolic equation solution using (a) the q~ formulation, and (b) the q^
formulation.
H þ 2Ml @ 2 C
H þ Ml @ 3 w
C2 @ 2 K
2l
4l
2
2
H @x2
H
@x
H @x @z
2
Mq Cqf
@ C
C
k 2 þ x2 k
H
@z
!
q2f
H þ Ml
Cl @w
2
þ 2x
þ qf
q
qc
H
H @z
þ2
qf @ 2 K
Cq kqf
K ¼ 0;
C
x2 C
qc @z2
H
(22)
where H ¼ Mk C2 . Substitution of Eqs. (20) and (21) into
the z-component of Eq. (10) produces the second equation
attenuations shown in Table I. In contrast, the benchmark
requires the bulk moduli and environmental parameters such
as viscosity, permeability, density, and grain size. An outline
of the procedure for calculating the physical parameters from
the complex sound speeds is given in Appendix A, and further details are in the appendixes of Ref. 6 and in Ref. 22.
Because this procedure involves solving a system of nonlinear equations, numerical inaccuracies may be introduced into
some parameters. It is possible that for some cases, such as
environment A, this source of error is negligible, whereas for
others, it may account for observable differences. Another
more fundamental reason is the variable f, which like D, is
not conserved across interfaces between poro-elastic layers
because it contains the depth derivative of the vertical
component of a displacement. For accurate solutions in
environments containing layered poro-elastic media, it is
hypothesized that a variable formulation is needed with all
dependent variables continuous across horizontal interfaces
between poro-elastic layers.
The third equation is derived by substituting a depth derivative of the z-component of Eq. (11) into a horizontal derivative of its x-component,
5ðC; w ; KÞT
IV. DEPENDENT VARIABLES q
B. Interface grid spacing
A second parabolic equation formulation is developed
with all dependent variables continuous across horizontal
Parabolic equation methods are designed for calculating
propagation in range-dependent environments.1 The most
J. Acoust. Soc. Am., Vol. 134, No. 1, July 2013
C
@2w
H þ Ml @C
H þ 2Ml @ 2 w
l
þ
k
@x2
H ! @z
H
@z2
2
qf
Cl qf @K
þ
¼ 0:
wC
þ x2 q qc
H qc @z
Mqf Cqc
Mqf Cqc @w
@2K
C 2x2 l
þ x2 k
2
H
H
@x
@z
Cqf kqc
@2K
2
K ¼ 0:
þC 2 x C
H
@z
(23)
(24)
Equations (22)–(24) are in the form of Eq. (13) for
with matrices K and L defined in
dependent variables q
Appendix B.
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250
l
widely used method for treating sloping ocean bottoms is the
stair-step approximation for which the environment is
approximated by a series of range-independent segments.24
Specific conditions are applied to march the solution accurately into the next segment, with two common choices
based on energy conservation24–26 and single scattering.27,28
Other approaches for solving sloping bathymetries that avoid
stair steps include coordinate rotation29 and mapping solutions,30 but these will not be considered in this paper. While
single-scattering methods produce accurate solutions for
many elastic propagation problems,31–33 approximate
energy-conservation conditions are more convenient numerically for environments containing several types of media,
including those in this paper. For any procedure using stairsteps, we discuss below a computational complication that
^ variable formulation, because of secondmay arise for the q
order depth derivatives in the horizontal interface condi formulation is that this
tions.9 An added advantage of the q
complication is avoided.
The numerical implementation of horizontal interface
conditions are discretized by
vj þ vjþ1
;
2
(25)
vj þ vjþ1
dv ;
¼
2Dz
dz z¼zi
(26)
vj 2vi þ vjþ1
d2 v ¼
;
2
Dz2
dz z¼zi
(27)
where vk vðzk Þ is a dependent variable, zi is the interface
depth, zj < zi < zjþ1 , and zk kDz with Dz being the depth
grid increment. Equations (25) and (26) require v only at
grid points, unlike Eq. (27) which requires vi to be known. If
zi happens to occur off a grid point, then non-uniform grid
spacing is created. This never occurs for range-independent
environments when stair-steps are used. However, it can
occur for range-dependent environments because vertical
interfaces may lead to non-uniform grid spacing. Suppose
first that Eq. (27) is required. In Fig. 4(a), advancing left to
right across the vertical interface, a depth shift occurs for a
grid point on the horizontal interface. For energy-conserving
corrections, grid points on both sides of the vertical interface
must be associated, and this requirement produces an ambiguity in the circled grid points. In contrast, a variable formulation that requires Eq. (26), as opposed to Eq. (27), does not
produce ambiguity, as shown in Fig. 4(b). That is, with grid
points not required on the interface, the grid remains uniform
throughout the domain, and an energy-conserving correction
can be applied.
formulaTo show that the interface conditions for the q
tion contain no second-order depth derivatives, they are
rewritten in terms of C, w, and K. At a poro-elastic/fluid
interface, for subscripts P and F denoting poro-elastic and
fluid layers,
qf
C @K
k @D
2
1 wþ
¼
;
(28)
x
qc
qc @z P
q @z F
J. Acoust. Soc. Am., Vol. 134, No. 1, July 2013
FIG. 4. Representation of grid spacing for an environment with a rangedependent interface between layers. Solid line represents an interface, asterisks represent discretized grid points. (a) Variable formulations with secondorder depth derivatives in interface conditions require grid points on the
interfaces, producing a locally non-uniform grid and an ambiguity in associating the circled grid points across the interface. (b) Variable formulations
with depth derivatives no higher than first order do not require grid points
on the interface, creating a uniform grid so that circled grid points are easily
associated.
fkCgP ¼ fkDgF ;
(29)
fCKgP ¼ fkDgF ;
(
)
!
2
q
q
@C
@K
f
f
k
þ x2 q w C
¼ 0;
qc
qc @z
@z
(30)
(31)
P
where D with subscript F is dilatation in the fluid.
Conditions at a poro-elastic/solid interface, for subscript E
denoting an elastic layer, are
(32)
fwgP ¼ fwgE ;
Mk
H þ 2Ml @w C2
k þ 2l @w
C
K ¼ C
;
H
H
@z H
k @z E
P
(33)
fkCgP ¼ fkCgE ;
(
)
!
q2f
qf @K
@C
2
þx q
w C
k
qc
qc @z
@z
P
@C
þ qx2 w ;
¼ k
@z
E
@K
q f x2 w þ C
¼ 0;
@z P
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(34)
(35)
(36)
251
Author's complimentary copy
vjz¼zi ¼
where C with subscript E has the same form as Eq. (20) with
C ¼ 0. Conditions at an air/poro-elastic interface, or an
interface between two poro-elastic layers, can be extracted
from the poro-elastic terms in the preceding expressions.
C. Examples
TABLE II. Parameters for environments D and Ei . The parameter
qf ¼ 1:0 g/cm3 for all poro-elastic layers.
Environment
f (Hz)
Fluid layer:
Upper poro-elastic layer:
Lower layer:b
cw (m/s)
Thickness (m)
a
s
qs (g/cm3 )
c1 (m/s)
c2 (m/s)
c3 (m/s)
b1 (dB/k)
b2 (dB/k)
b3 (dB/k)
Thickness (m)
a
s
qs (g/cm3 )
c1 (m/s)
c2 (m/s)
c3 (m/s)
b1 (dB/k)
b2 (dB/k)
b3 (dB/k)
D
Ei
25
1500
100
0.2
0.0
1.3
1700
1250
800
0.1
1.0
0.2
20
0.2
0.0
1.7
2040
1250
950
0.3
5.0
0.5
1000
1500
10
ai a
0.75
2.65
1698.1
1023.3
118.9
0.76
10.18
1.46
1
–
–
3.0
2400
–
1200
0.3
–
0.5
a
Varies for each i; a1 ¼ 0:4, a2 ¼ 0:2, a3 ¼ 0:02.
b
Lower layer is poro-elastic for environment D and elastic for environment E.
252
J. Acoust. Soc. Am., Vol. 134, No. 1, July 2013
FIG. 5. Transmission loss for environment B at 25 m depth. The dashed
curve represents the benchmark solution, and the solid curve represents the
solution using the q formulation.
conversions noted at the end of Sec. III B. Next the result
formulation.
from environment C is obtained using the q
Transmission loss at 75 m depth is shown in Fig. 6 for the
solution, and the two curves
benchmark solution and the q
are in excellent agreement over the 5 km propagation range.
The amplitude and pattern phase differences depicted in
formulation.
Fig. 3(b) have been corrected using the q
Environment D has a 25 Hz source at 98 m depth within
a 100 m thick fluid layer overlying a 20 m thick poro-elastic
layer, which overlies a poro-elastic half space. The proximity of the source to the fluid/poro-elastic interface produces
an interface wave, which is analogous to a Scholte wave that
occurs at a fluid/elastic interface. The interface wave is seen
in the transmission loss contours of Fig. 7(a) propagating
both above and below the upper interface. The interface
wave structure is very similar to that for a Scholte wave near
a fluid/elastic interface. The corresponding horizontal wave
number spectrum at receiver depth 98 m is shown in
Fig. 7(b). The largest peak on the left corresponds to the fast
compressional sound speed in the upper poro-elastic layer,
FIG. 6. Transmission loss for environment C at 75 m depth. The dashed
curve represents the benchmark solution, and the solid curve represents the
solution using the q formulation.
Metzler et al.: Poro-elastic parabolic solutions
Author's complimentary copy
In this section, the accuracy of the parabolic equation
formulation is described for environments conusing the q
taining multiple poro-elastic layers. For single poro-elastic
formulayered environments, such as environment A, the q
lation produces results of the type in Fig. 1. For multilayered poro-elastic environments, such as environments B
are significantly improved when
and C, solutions using q
^ formulation, and the results
compared to those from the q
agree extremely well with benchmarks. Two additional
layered poro-elastic environments are also examined with
parameter values given in Table II. Environment D shows
interface wave propagation, an important feature of
seismo-acoustic wave propagation. The final example
examines several related environments that are typical in
shallow water regions and that contain fluid, elastic, and
poro-elastic layers. The influence of varying porosity is
explored.
formulation.
First environment B is revisited using the q
Figure 5 compares transmission loss from the benchmark
solution at 25 m depth, and the
solution and from the q
agreement is very good. While there remains slight amplitude differences between the curves, this formulation is a
considerable improvement over the result in Fig. 2(b). It is
likely that the differences are associated with the parameter
FIG. 7. (Color online) Results for environment D. (a) Transmission loss
contours near the fluid and upper poro-elastic layer interface display a propagating Scholte-like interface wave. (b) A horizontal wave number spectrum
at 98 m depth shows a peak on the right corresponding to the Scholte-like
interface wave.
J. Acoust. Soc. Am., Vol. 134, No. 1, July 2013
FIG. 8. Transmission loss for environments Ei at 8 m depth. Dashed curves
represent the solution for environment E0 where the upper-sediment layer is
elastic. Solid curves are solutions for (a) environment E1 with a ¼ 0:4, (b)
Environment E2 with a ¼ 0:2, and (c) environment E3 with a ¼ 0:02.
disparities between the two solutions indicate that the porous
structure of the relatively thin 1 m layer significantly influences the propagation patterns within the fluid. Figures 8(b)
and 8(c) compare transmission loss at 8 m depth for environments E2 and E0 and for environments E3 and E0 . These figures are meant to illustrate effects of varying porosity. As
porosity in the poro-elastic layer decreases, the solutions exhibit properties closer to the E0 case. In fact propagation
Metzler et al.: Poro-elastic parabolic solutions
253
Author's complimentary copy
while the peak on the right corresponds to the interface
wave. The wave number peak of the interface wave yields a
speed of cI 695 m/s. In comparison, Eq. (13) in Ref. 34
gives the speed of a Scholte wave for a corresponding elastic
layer as cSC 654 m/s, which is generally consistent with
the interface wave speed for the poro-elastic environment.
The final example examines environments Ei , each of
which has a 1 kHz source at 5 m depth within a 10 m thick
fluid layer overlying a 1 m thick upper-sediment layer, which
overlies an elastic half space. Environments Ei , for
i ¼ 1; 2; 3, treat the upper-sediment layer as poro-elastic
with varying porosities: a1 ¼ 0:4, a2 ¼ 0:2, and a3 ¼ 0:02.
Environment E0 treats this layer as purely elastic with
parameter values corresponding to analogous ones of the
poro-elastic cases: cp ¼ c1 , cs ¼ c3 , bp ¼ b1 , bs ¼ b3 , and
q ¼ qs . The parameter values for the poro-elastic layer in
environment E1 are taken from the Biot parameters of the
Applied Research Laboratories, The University of Texas at
Austin (ARL:UT) sand tank.35 Figure 8(a) compares transmission loss at 8 m depth for environments E1 and E0 . The
results for environments with poro-elastic layers having
decreasing porosities approach smoothly to the result from
environment E0 . This situation contrasts sharply with the
transition of elastic layers to a fluid by decreasing shear
speeds, which leads to numerical stability issues because of
singularities.
Kr
;
Kb
Kr
1
a 1
Kr
Kf
Kb
C¼M 1
;
Kr
V. CONCLUSION
k¼
ACKNOWLEDGMENTS
Work supported by the Office of Naval Research, and
for the first author by an Ocean Acoustics Graduate
Traineeship Award and through the Internal Research and
Development program of the Applied Research Laboratories
at The University of Texas at Austin.
APPENDIX A: PORO-ELASTIC PARAMETER
CONVERSION
In this appendix, relations are presented among the complex wave speeds, the poro-elastic moduli, and the physical
bulk parameters. The complex wave speeds are used in the
input structure for the parabolic equation, the poro-elastic
moduli are used in the equations of motion, and the bulk parameters are used in the input structure of the benchmark
code OASES. The six relevant equations are6
c23 ¼
254
lqc
;
E
J. Acoust. Soc. Am., Vol. 134, No. 1, July 2013
(A1)
(A2)
(A3)
C2
2
þ Kb l;
M
3
(A4)
F
;
E
(A5)
c21 þ c22 ¼
c21 c22 ¼
G
;
E
(A6)
where
E ¼ qqc q2f ;
(A7)
F ¼ qM þ qc ðk þ 2lÞ 2qf C;
(A8)
G ¼ ðk þ 2lÞ M C2 :
(A9)
The complex quantity Kb is the bulk modulus of the porous
frame formed by the sediment grains. The real quantities Kr
and Kf are the bulk moduli of the sediment grains and the
fluid. It is assumed that the pore spaces are filled with fluid
and that qf and Kf are known.
Equations (A1)–(A6) represent six complex nonlinear
equations that are equivalent to 12 real nonlinear equations.
Given the input requirements for the parabolic equation (the
parameters described in Tables I and II), the known quantic j , and
ties in Eqs. (A1)–(A6) are Kf , qf , q, a, <½qc , <½
=½
c j , for j ¼ 1; 2; 3. The unknowns in the system of 12
equations are the real and imaginary parts of M, C, k, l, and
Kb , the real quantity Kr , and the imaginary part of qc . A
method for solving a system of nonlinear equations, such as
trust-region dogleg (as used by the fsolve routine in MATLAB),
can be used to obtain the unknown poro-elastic moduli and
bulk parameters. For an initialization of the method, the lossless case in Eqs. (A1)–(A6) leads to a single quadratic equation for M. Once <½M is known, the remaining real parts
can be obtained from Eqs. (A1)–(A6), providing an initialization with zero imaginary values.
Conversion from the input parameters for OASES to the
input parameters for the parabolic equation is straightforward. The former requires 13 input parameters: qf , Kf , g, qs ,
Kr , a, j, grain size a, <½l, <½Kb , sediment frame shear
attenuation as , sediment bulk attenuation ac , and virtual
mass parameter cm ¼ 1 þ s.22 The poro-elastic moduli and
qc are obtained from Eqs. (A2)–(A4) and (12b). Then the
complex wave speeds are calculated from the square root of
Eq. (A1) and
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F þ F2 4EG
;
(A10)
c1 ¼
2E
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F F2 4EG
c2 ¼
:
(A11)
2E
Metzler et al.: Poro-elastic parabolic solutions
Author's complimentary copy
Two new variable formulations are presented for propagation calculations with the parabolic equation method
through poro-elastic media with homogeneous layers. The
^ ¼ ðux ; w; fÞT , is a natural dependent variable
first, with q
selection based on results for elastic-media parabolic equations. For a single poro-elastic layer, this method produces
transmission loss solutions with benchmark accuracy. For
environments with multiple poro-elastic layers, the results
are mixed. There are significant improvements over the
~ ¼ ðD; w; fÞT formulation, but accuracy is not of benchq
mark quality. It is believed that the primary cause is the
dependent variable f being discontinuous across interfaces
between two poro-elastic layers.
¼ ðC; w; KÞT , is develThe second formulation, with q
oped with the goal of using dependent variables that are all
continuous across poro-elastic interfaces. Examples show
that this leads to excellent accuracy for transmission loss calculations. Furthermore, this formulation contains depth
derivatives not higher than first order in conditions at horizontal interfaces between the same or different types of
media. This permits a uniform numerical grid throughout
range and depth, which is expected to improve accuracy
using energy-conserving techniques for problems in rangedependent environments. Moreover, this formulation is able
to treat shallow water environments that typically contain
interface waves and combinations of fluid, poro-elastic, and
elastic layers. Using the variable formulations in this paper,
the parabolic equation method is accurate and efficient for
solving layered poro-elastic propagation problems.
M¼
2
Cq kqc
‘ 33 ¼ C @ x2 C f
:
2
@z
H
APPENDIX B: OPERATORS OF THE PORO-ELASTIC
PARABOLIC EQUATIONS
0
v
B
K^ ¼ B
@0
0
ðv lÞ
@
@z
l
0
qf 1
M
qc C
C;
A
0
M Cn
C
(B1)
0
1
@2
2
l
þ
x
u
0
0
C
B @z2
B
C
B
2
qf
@
@
@C
L^ ¼ B
C;
v 2 þ x2 u
C M
B ðv lÞ
C
@z
qc
@z
@z A
@
^‘ 32
^‘ 33
^‘ 31
(B2)
where v ¼ k þ 2l ðqf =qc ÞC, n ¼ C=ðk þ 2lÞ, u ¼ q
ðq2f =qc Þ, and
^‘ 31 ¼ x2 ðqf qnÞ;
(B3)
^‘ 32 ¼ x2 ðqf qnÞ @ ;
@z
(B4)
2
^‘ 33 ¼ ðM CnÞ @ þ x2 ðqc qf nÞ:
@z2
(B5)
formulation, K and L derived from Eqs. (22)–(24)
For the q
are
0
1
Ml @
C2
2l
kw
4l
w
B
HC
H @z
B
C
(B6)
K ¼ B
C;
l
0
@ 0
A
0
0
C
0
1
‘ 11
‘ 12
‘ 13
B C
Ml @
@2
B
‘ 23 C
lw 2 þ x2 u
Bk w C
L¼B
C;
@z
H @z
B
C
@
A
Mqf Cqc
Mqf Cqc @
2
2
2x l
xk
‘ 33
H
H
@z
(B7)
where w ¼ ðH þ 2MlÞ=H and
2
Mq Cqf
‘ 11 ¼ k @ þ x2 k
;
@z2
H
"
#
q2f
Ml
Cl
@
2
‘ 12 ¼ 2x
þ qf
;
q w
qc
H
H @z
2
‘ 13 ¼ 2 qf C @ x2 C Cq kqf ;
qc @z2
H
q
‘ 23 ¼ C Cl þ f @ ;
H qc @z
J. Acoust. Soc. Am., Vol. 134, No. 1, July 2013
(B8)
(B9)
(B10)
(B11)
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(17)–(19) are
(B12)
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28