GEOPHYSICS.
VOL.
53,
NO.
5 (MAY
The pseudospectral
in elastic
1988):
P. h2S 637.
19 FIGS.,
I TABLE
method : Accurate
representation
of interfaces
wave calculations
Bengt Fornberg”
smoothly varying coefficients and with interfaces aligned with
the grid, it was found that the required grid spacings in each
space dimension satisfied ratios of 16:4 : 1 for the pseudospectral method, fourth-order finite differences, and secondorder finite differences. These ratios were derived theoretically
from an analysis of dispersion errors in homogeneous media
and were confirmed in more general test calculations. Issues
relevant to practical implementations and not directly addressed in my earlier work include the following:
ABSTRACT
When finite-difference methods are used to solve the
elastic wave equation in a discontinuous medium, the
error has two dominant components. Dispersive errors
lead to artificial wave trains. Errors from interfaces lead
to circular wavefronts emanating from each location
where the interface appears *‘jagged” to the rectangular
grid.
The pseudospectral method can be viewed as the limit
of finite d&lerences with infinite order of accuracy. With
this method, dispersive errors are essentially eliminated.
The mappings introduced in this paper also eliminate
the other dominant error source. Test calculations contirm that these mappings significantly enhance the already highly competitive pseudospectral method with
only a very small additional cost.
Although the mapping method is described here in
connection with the pseudospectral method. it can also
be used with high-order finite-difference approximations.
(i) For the extremely coarse grids which are sufficient
to carry traveling pulse solutions in the pseudospectral
method. some technique must be found to specify interfact locations more accurately than to the nearest grid
point. The roughness in the discretization of a smooth
interface not aligned with the grid acts as a source of
unphysical noise.
(ii) The performance of the pseudospectral method
for surface and interfacial waves must be examined.
(iii) Nonreflecting boundary conditions which are
compatible with the periodicity in space implicit in the
pseudospectral method must be devised.
(iv) time integration techniques must be implemented in a way which minimizes computer time and
storage.
INTRODUCTION
The pseudospectral method was introduced in the early
1970s (Kreiss and Oliger. 1972; Orszag, 1972; Fornberg,
1975). It has since gained wide acceptance in a variety of
areas, including weather forecasting, nonlinear waves, and turbulence modeling The pseudospectral method did not impa~~m
the field of seismic modeling until about 10 years after its
introduction into other areas (Kosloff and Haysal, 1982; Johnson, 19X4), most likely because, initially, neither nonperiodic
domains nor discontinuous interfaces appeared practical with
the method. However. these difficulties are gradually being
overcome.
Fornberg (1987) discussesthe basic features of the pseudospectral method and compares the method to finite-difference
methods for the 2-D elastic wave equation. In the case of
Adequate, but most likely not optimal. solutions to the last
two problems have been presented elsewhere. One way to
address problem (iii) is to extend the computational region
with absorbing boundaries. Cerjan et al. (1985), KoslofS and
Kosioil‘( 1986). and Loewenthal et al. ( 1987) present test results
of such implementations. Regarding problem (iv), note that
straightforward central ditrerencing in time (leap-frog) is only
accurate to second order and requires two time levels of storage. assuming the 2-D equations are formulated as five coupled first-order equations. Higher order accuracies can be
achieved by the “method of lines” approach. Splitting methods (e.g., Bayliss et al., 1986) are limited to second-order accuracy, but can reduce the storage requirement to one time level.
Manuscriptreceivedby the Edltor March 9, 1987;revisedmanuscriptreceivedSeptember29, 1987
*Exxon Researchand EnrmeeringComoanv.AnnandaleNJ OXROI.
(‘ 11)8R
Societyof Explor&n Ge%phys&t~.All rightsreserved.
625
Pseudospectral
Interface
and
Representation
627
Table 1. Number of arithmetic operations for each time step
with the different methods.
J’(<, rl + 1) = Yk 11)+ 1.
Number of arithmetic
operations per time step
(leap frog, N x N grid)
Derivatives xg, yc, x, , and Y,, are calculated analytically from
these splines. The derivatives 5,. E,, q,, and 9, required in
equations (3) are obtained by differentiating x = ~(5, q) and
Without mapping
Y = Y($ q), giving
Second-order
finite differences
36N2
Fourth-order
finite differences
60N2
X<& + X0$ = 0,
Y<L+Yt,%=O,
(5)
Pseudospectral
and
(40 log, N + 16)N*
With mapping
Pseudospectral
(50 log,N + 42)N’
i.e.,
5, = J * Yq
&,= --J*x,
‘1,=
--J*_vs
(6)
and
111,=J*xc,
with J = ,‘’ (‘-“< yq- x1 y,).
Using finite-difference approximations of moderate orders
of accuracy, say up to four or six, time stepping with equations (3) would be significantly more costly than with equation
(i), becausetwice as many terms are present. However, for the
pseudospectral method, the cost is dominated by the calculation of the spatial derivatives required in each time step. The
number of different derivatives has only increased from 8 to
10.
Table 1 compares operation counts for the different methods employed (for one leap-frog time step). The methods were
implemented as described in Fornberg (1987). In vectorized
32-bit precision on a two-pipe CDC Cyber 205, typical computational rates were between 200 and 300 Mflop (out of a
theoretical machine maximum of 400 Mflop) for the finitedifference methods and just below 200 Mflop for the basic
pseudospectral method. Although the mapping increases the
scalar operation count for the pseudospectral method by
about 25 percent, the increase in time on a CDC Cyber 205 is
much less. In the fast Fourier transform package, the lengths
of vectors running across the simultaneous cases rather than
the number of vector operations are increased.
In all test cases below, the initial condition, i.e., the incidence wave, is a plane, horizontal P-wavefront moving
down in the 4’ direction [given by o = h/3”’ = 3”“f=
1/[(40~ - 30)’ + I]‘, IA= y = 0; Figure I]. For calculations in
the mapped 5, IJ domain, cubic splines were used initially for
transforming the initial condition and finally for returning the
results for graphical display.
DESCRIPTION OF TEST CASES
reduced by a factor of l/4. The variable S at the final time
same in both cases, should look similar to what is shown in
Figures 3a and 3b. The reflected P wave is followed by a
weaker, reflected S wave and by some still weaker other features. The transmitted wave is much stronger than these reflected ones. In order to obtain better graphical resolution of
the reflected parts only, Figures 4a and 4b show only the
upper three-quarters of each solution from Figures 3a and 3b,
-with tie contrast renormalized. Figures Sa and 5b show the
approximations to Figures 4a and 4b obtained by applying
the different numerical schemesmentioned above with various
grid sizes.
For second-order and Fourth-order finite-difference methods
with fairly coarse grids, dispersive errors dominate errors
caused by the interface discretization. When the grid density
in each spatial direction is doubled, dispersive errors decrease
by factors of 4 and !h, respective!y, whereas itice
discretl7ation errors in all cases decrease only by a factor of 2.
Although smaller at first, interface discretiration errors ulti-
Yt
.8
-
-65
+
Test case 1
0
Figure 2 shows the geometry of the medium, the initial
conditions, and the 5, q grid in test case lb. Test case la is
identical, apart from the amplitude of the syncline which is
1
Amplitude
FIG. I. Structure of a P-wavefront used as the initial condition.
The level of resolution on a 32 x 32 grid is indicated.
626
Fornberg
mately dominate. Their structure is most clear in the results
with the pseudospectral method without mappings, since the
dispersive errors in this case are absent. In test case la, the
interface on the square x. y grid appears to be piecewise
straight with just two jumps up to, and including, the
128 x 128 grid density. In each constant section, an accurate
reflection-transmission occurs. However, each jump between
sections becomes a source for a disturbance which radiates in
all directions. Each disturbance contains both P-wave and
S-wave components as can be seen in the double wavefronts
with the same centers. As x. 4’ grids get still finer or the interface gets less straight as in test case la on grids above
128 x 128 or test case lb above 32 x 32, more interface jumps
occur and their ring patterns overlap. The errors now resemble background noise. Again, disturbances of this type are
absent when the mapped pseudospectral method is used.
For strongly curved interfaces. the mappings required to
straighten the interfaces might become severely distorted or
impractical altogether. The following test cases serve to illus-
Y
1
trate to what degree the mapping method will remain advantageous under such conditions.
Test case 2
In this test case. the plane P-wavefront hits a curved interface which also has several fault-like translations. The curvilinear grid I in Figure 6a follows most of the interfaces. Results in Figure 7, the upper three-quarters of the computational domain, again show a significant improvement over
the pseudospectral method without mappings and over finitedifli-rence methods as well.
As a rule of thumb, only the strongest of the fairly horizontal (or vertical) interfaces should be considered for rectification. To illustrate this, Figure 6b gives a second mapping for
the same problem in which all interfaces are rectified. As the
bottom row in Figure 7 shows, errors have actually increased
becausegrid 2 is considerably more nonuniform than grid 1.
1
condition:
Initial
p--wave
Straight
1
front
1
t
Medium
interface
0
-c
FIG. 2. Initial condition and medium parameters for test case 1b displayed on the mapped 32 x 32 grid
0.26
0.P6
“1
0
(4
x
1
0
_
(a)
1.
I
d*
@)
1.
x
x
(b)
Frti. 3. Accurate results for the solution in test cases la and lb,
respectively.
FIG. 4. Upper three-quarters of each diagram in Figure 3 with
the contrast, proportional to the amplitude. scaled up to display more structures in the reflected part of the solutions.
Pseudospectral
32132
64*64
interface Representation
128,128
629
256*256
512*512
2nd
order
FD
4th
order
FD
X,Y - coordinates
PS
(without mapping)
3,~ - coordinates
PS
(with mapping)
32’32
64*64
126928
256*256
512’512
2nd
order
FD
4th
order
FD
PS
(without mapping)
--PS
f,g - coordinates
(with mapping)
FIG 5. (a) Results of test case la. Variable .f displayed. (Axes as in Figure 4.) (b) Results of test case 1b. Variable f
displayed.(Axes as in Figure 4.)
Fornberg
Y
--7
1
_
4
_ Initial
condition:
Straight
P wavefront
4
c
Medium
interface
c
0
-r
r
d
1
x
(4
Y
--I1
- -
Straight
4
condition:
P wavefront
4
C
0
- Initial
Medium
interface
I
0
FIG. 6. Initial conditions and medium parameters in test case 2. (a) and (b) show two different mappings, denoted grid 1
and grid 2, respectively. for the same problem.
Pseudospectral
32*32
64*64
Interface
631
Representation
256*256
126*126
512*512
2nd
order
FD
4th
order
FD
KY - coordinates
PS
(withoutmapping)
---
--------___
--
5,~ - coordinates
PS
(with mapping)
GRID 1
FS
GRID 2
FIG. 7. Results of test case 2. Variable
L’displayed.
(Axes as in Figure 4.)
_
_
Initial
condition:
Straight
P wavefront
1
c
Medium
interfaces
c
FIG. 8. Initial
condition
and medium
parameters
for test case 3 displayed
on top of the mapped
32 x 32 grid.
632
Fornberg
Test case 3
The geometry in this case (shown in Figure 8) includes
pinchouts and a boundary which cannot conveniently be
aligned with a grid. To illustrate the influence of this trapped
iL = 0.6 region in the resulting wave pictures, Figures 9a and
Yb compare the results in this case with a case when this
region has been removed, i.e., h was changed from 0.6 to 0.8
within it. Fourth-order finite differences on the 512 x 512 grid
0.
@I
1.
x
0.
0)
1.
x
FIG. 9. Solution (13component) in test case 3 (a) and the solution when the trapped region is eliminated, i.e., h = 0.6 replaced by h = 0.8 (b).
32’32
64*64
were used for Figures 9a and 9b. The differences between
Figures 9a and 9b are the key features to look for in assessing
the results in Figure IO. Clearly, the mapping of the main
interfaces has sufficiently defined the overall picture so as to
make more obvious the influence of the trapped region.
CONCLUSIONS
Numerical errors from using finite dilTerenceson the elastic
wave equation with discontinuous media have been found to
have two dominant components. Dispersive errors lead to
wave trains following traveling pulses; errors from interfaces
lead to circular wavefronts emanating from each interface irregularity sensed on a rectangular grid.
Dispersive errors are essentially absent with the pseudospectral method. The mappings introduced here also eliminate
the second source of error at only a small additional cost. In
the earlier study (Fornberg, 1987),it was noted that the resolution of the pseudospec:ral method on an .N x N grid typically
cyuals that of a fourth-order method on a 4N x 4N grid and
a second-order method on a 16N x 16N grid, if curved interfacts are not present. The present mapping method significantly lessensthat limitation.
128*128
256*256
512912
2nd
order
FD
4th
order
FD
KY - coordinates
PS
_-PS
(without mapping)
-----------__
5, p - reef&a tes(with mapping)
FIG. LO.Results of test case 3. Variable r displayed. (Axes as in Figure 9.)
Pseudospectral
Interface
REFERENCES
Aki, K., and Richards. P. G., 1980, Quantitative seismology. Theory
and methods, I : W. H. Freeman and Co.
Bayliss, A., Jordan, K. E., Le Mesurier, B. J., and Turkel, E., 1986, A
fourth order accurate finite difference scheme for the computation
ofelastic waves: Bull., Seis. Sot. Am., 76, 1115~11?2.
Cerjan. C., KosloH: D.. Koslof, R.. and Reshef. M., 1985, A nonrefleeting boundary condition for discrete acoustic and elastic wave
equations: Geophysics, 50, 705~708.
Fornberg, B., 1975, On a Fourier method for the integration of hyperbolic equations: Sot. Industr. Appl. Math., J. Numer. Anal., 12,
509-528.
1987. The pseudospectral method: Comparisons with finite
APPENDIX
Representation
din‘ercnceq for the elaslic wave equation: Geophysics, 52, 483-501.
Johnson, 0. G.. 1984. Three-dimensional wave equation computations on vector computers: Proc., Inst. Electr. Electron. Eng., 72,
W95.
Koslotl’, D., and Baysal. E., 1982, Forward modeling by a Fourier
method: Geophysics, 47, 1402-1412.
KoslofX R.. and Kosloff, D., 1986, Absorbing boundaries for wave
propagation problrms: _I.Camp. Phys.. 63, 363-376.
Kreiss. H.-O., and Oliger, J., 1972, Comparison of accurate methods
for the integration of hyperbolic equations: Tellus, 24, 199-215.
Loewenthal, D.. Stoffa. P. L.. and Faria, E. L., 1987, Suppressing the
unwanted reflections of the full wave equation: Geophysics, 52,
1007~1012.
Orazag, S. A., 1972, Comparison of pseudospectral and spectral approximatlon: Stud. Appl. Math., 51,253%259.
A
TEST OF REFLECTION-TRANSMISSION
In Fornberg
solutions
(1987), I demonstrated
were obtained
transmissions. Although
more
important
than
that qualitatively
in cases involving
amplitude
amplitude
information
versus oll‘set studies, full-waveform
and
is essential in amplitude
model-based
Figures A-la
(e.g., most structural
I
l-r
and A-2a schematically
P-WAVE
show the structure of
lr
\ = k\ = 1.
TOP MEDIUM
P-WAVE
BOTTOMMEDIUM
)
q
p=
.5
L
‘
1
0
INITIAL
x
STATE
END STATE
(b)
6-d
FIG. A-l.
Structure of the medium and of the initial and end states in test case A-l
BOTTOMMEDIUM
i=p:
.5
1
0
INITIAL
(a)
STATE
inversion, and
other applications.
clarity and timing of signals are often
their
COEFFICIENTS
modeling),
clean
reflections
633
x
0
1
END STATE
@I
FIG. A-2. Structure of the medium and of the initial and end states in test case A-2.
x
Fornberg
634
ORDER
2ND
FD
128*128
64*64
4-i-H
ORDER
FD
_.
64*64
32*32
PSEUDOSPECTRAL
FIG. A-3. Results of test case A-l.
Pseudospectral
@l.
2ND
ORDER
ORDER
636
Representation
128’128
FD
61.
4TH
Interface
64*64
128*128
FD
64’64
PSEUDOSPECTRAL
32*32
FIG. A-4. Results of test cast A-2.
636
Fornberg
the medium and the initial conditions: P-wavefronts striking
the interfaces at 0’ and 45 incidence, respectively. Figures
A-lb and A-2b show the end states. Exact results for the
directions and amplitudes of the resulting waves can be calculated (e.g., Aki and Richards. 1980, chapter 5). In the first case,
0 angle of incidence, the P component of the incoming wave
has amplitude 1. The same component !in the two outgoing
waves, in order of increasing r. have amplitudes I. 1716 and
0.1716, respectively. In the second case. the .rl component of
the incoming wave has amplitude 1. For the outgoing waves,
the corresponding amplitudes are 0.6371. 0.2780, -0.1652,
and 0.0803, respectively.
Computational results are displayed in Figures A-3aPA-3f
and A-4aaA-4f. In the later case, the calculation was performed in a periodic square [0, 21 x [0, 21 with only the
[O. I J x [O. I] section displayed. The solid dots along the
wavefront curves in each of Figures A-& A-3f and A-4amAm4f
mark where the peak of each outgoing wave should be
APPENDlX
SMOOTHING
OF INTERI;ACES
TO SIMIJLATE
The basis for the present paper is that the pseudospectral
method has proven exceptionally effective in the case of
straight interfaces located halfway between adjacent grid lines,
One might think that other interface positions between (or at]
grid points could be simulated by replacing the discontinuous
medium with a smoothly varying medium. In that case, even
small shifts in the interface position would be felt by the numerical method, giving a smooth transition between otherwise
discrete interface positions. To illustrate an intrinsic problem
with this approach. it suffices to consider a I-D model problem. With no .Ydependence, the system (I) simplifies to (setting
P=
11
(B-l)
FIG. B- 1. I-D tests or rellcction-transmission. test case B-1.
located. Some observations includee:
(i) Based on these tests and on more tests performed
but not reproduced here, it would appear that a lack of
accuracy d reflection-transmission coefficients is not a
difficulty with any of the methods studied. The amplitudes are already quite accurate when the resolution has
been increased soffictentiy to distinguish ihc waves from
the background noise.
(ii) In the finite-difference cases, but not with the
pseudospectral method. some very high-frequency noise
from the reelection-transmission processshows up under
continued grid refinement. Although the frequency is so
high that it probably can be filtered out selectively, the
presence of this noise on finite-difference plots indicates
that the pseudospectral method possessessome not understood but particularly advantageous properties relative to sharp interfaces.
B
“IN-BETWEEN
GRID
POINT”
LOCATIONS
Consider now a medium with h = 1 above a sharp interface
and i, = 0.25 below it. From above, a P-wave is sent down to
the interface. A part of the wave is transmitted and the rest is
reflected. Every one of the vertical traces in Figures B-l to B-6
represents a snapshot at the final time (same in all cases) of
such a I-D experiment. The 64 I-D cases in each strip correspend to the interface being shirtcd down one-sixteenth of the
distance between adjacent grid points (total vertical resolution
was 32 points). In each of the figures, three such strips of
traces arc given. They difl‘er in that the initial pulse has been
made increasingly sharper. The graph to the left in each figure
shows the prccisc intctface representation employed.
In case B-l (Figure B-l), no attempt was made to smooth
the interhtace(see the solid curve in the diagram to the left). As
this interface is translated down, no change occurs at any grid
point (and hence in the results) until the interface actually
passesa grid point.
Figure B-2 shows the simplest case of a smoothed (piecewise
linear) interface. As this medium (solid line to the left in the
C‘L H-2. 1-II tests of reflection-transmission, test ease B 2.
Pseudospeciral
Interface
Representation
FIG. B-3. I-D tests of reflection-transmission, test case B-3.
FK;. B-5. 1-D tests of renection-transmission, test case B-5.
FIG. B-4. 1-D tests of refiection-transmission, test case B-4.
TIC;.B-6. I-D tests of reflection-transmission, test case B-6.
Figure B-2) is gradually shifted down, the value of 1, is shifted
one grid point at a time from the value at the bottom
(h = 0.25) to the value at the top (i, = I). For a wide incident
pulse. the only one the grid could handle if finite differences
rather than the pseudospectral method were used. the top
strip shows a fairly acceptable result. However, the bottom
two cases illustrate the fundamental problem with this whole
approach. The reflected signal takes on an entirely wrong
structure. Ascending fringes appear, instead of a descending
straight line.
Cases B-3 and B-3 show that the problem noted in case B-2
is general. In case B-3, the smoothing is still linear but is
extended over three grid spaces instead of one. In case B-4, the
interface is represented by the interpolating trigonometric
polynomjs)~to !he two discrete constant s:atc3. This suggests
that using Fourier space to represent the medium in a manner
similar to this space being used for the solution in the pseudospectral or spectral methods is unlikely to be of any help in
locating interPaces.
As a curiosity, cases B-5 and B-6 show that good timing
and crisp shape for both transmitted and reflected signals can
be achieved by some direct alterations at the interface. Although this approach might possibly find some application in
1-D situations, it has unacceptable shortcomings in the present application:
(i) The artificial peaks at the interface must be made
larger if the interface gets weaker;
(ii) serious errors occur in reflection-transmission
COefficient5 (for sharp pulses); and
(iii) in 2-LY situations, osciilations tendsto occur near
interfaces treated in this ad hoc way.