Ivo Oprsal, Jiri Zahradnik
Robust finite-difference modelling of complex structures
Paper No. 15 in Proc. of Int. Symposium on
High Performance Computing in Seismic Modelling
June 2-4, 1997, Univ. of Zaragoza, Spain
F. J. Seron, and F., J. Sabadell (editors)
Robust nite-dierence modelling of complex
structures1
I. Oprsal, J. Zahradnk
Department of Geophysics, Faculty of Mathematics and Physics, Charles University
V Holesovickach 2, 180 00 Praha, Czech Republic; e-mail: [email protected]
Keywords Finite-dierence method, Irregular grids, Seismic modelling
Abstract The recently developed 2D elastic nite-dierence schemes have successfully passed
through testing in extreme situations. Those involved the non-planar surface topography,
underground cavities, water basins embedded in solid medium, and stochastic material perturbations. The schemes solve elastodynamic equations in displacements. They are based on
geometrically averaged material parameters, allowing correct positions of the interfaces between
grid lines, while (stair-case) free surfaces are treated by an extremely simple vacuum formalism.
Stair-case artifacts are removed by grid renements close to the prominent free-surface features.
This is accomplished by spatially irregular rectangular grids. The grid irregularity has no spurious numerical eects, and yields considerable (30 - 90 %) saving of computer memory and time.
Method Building new computer codes for 3D nite-dierence modelling of seismic waves
requires robust algorithms. A useful laboratory for testing the algorithms is 2D models with
extreme situations. By extreme situations we understand, for example, the following: Irregular
grids with abrupt changes between their ne and coarse parts, models involving fast spatial
variations of the material parameters, models involving non-planar surface topographies and/or
underground cavities, models including solid-liquid contacts, models with stochastic material
perturbations, etc. The objective of this paper is to prove robustness of the recently developed
nite-dierence schemes for the listed situations. Therefore, numerical features are accentuated,
while physical implications are left for another paper.
The nite-dierence schemes used in this paper are PS2 (Zahradnk, 1995; Zahradnk and
Priolo, 1995) and PSi2 (Oprsal, 1996; Oprsal and Zahradnk, submitted). For formulas and
complete bibliography, see the cited papers. The schemes are 2nd-order accurate in homogeneous regions, solving the elastodynamic equations in displacements. They are based on
geometrically averaged material parameters, allowing correct positions of the interfaces passing
between grid lines, while (stair-case) free surfaces are treated by an extremely simple vacuum
formalism. The PS2 and PSi2 are regular- and irregular-grid versions. Advantages of the irregular grid are twofold: Stair-case free-surface artifacts are removed by the grid renements,
and oversampling of the high-velocity regions is avoided. Therefore, the irregular-grid calculations save about 30 - 90 % of computer memory and time as compared to the calculations
of same accuracy on spatially regular grids. The grid irregularity has no spurious numerical
eects, allowing even abrupt spatial changes of the grid step. As compared to most published
nite-dierence schemes, our method has been theoretically validated for consistency with the
Proc. of HIGH PERFORMANCE COMPUTING IN SEISMIC MODELLING, An International Symposium, Zaragoza, Spain, June 2-4, 1997, F. J. Seron, and F., J. Sabadell (editors)
1
Ivo Oprsal, Jiri Zahradnik
Robust finite-difference modelling of complex structures
Paper No. 15 in Proc. of Int. Symposium on
High Performance Computing in Seismic Modelling
June 2-4, 1997, Univ. of Zaragoza, Spain
F. J. Seron, and F., J. Sabadell (editors)
traction continuity; including the vanishing traction at non-planar surfaces, where its 2nd-order
accuracy falls to 1st order.
Ridge topography A ridge model is investigated on two grids: a regular coarse grid, and
an irregular grid rened along the nonplanar surface. The regular coarse grid model (Fig. 1a
left) has space steps x = z = 33:33 m, and time step t = 0:008 s. The irregular grid
model (Fig. 1a right) has x = 6:66 m for all the gridsteps with horizontal coordinate x less
than 1100 m, otherwise x = 33:33 m, and z = 6:66 m for all the gridsteps with vertical
coordinate z less than 50 m under the planar surface, otherwise z = 33:33 m; time step
t = 0:0016 s. The medium parameters are = 2000 m=s, = 1000 m=s, and = 1000 kg=m3
for both models. The surface shape is s(x) = h(1:0 , a)exp(,3a) (Bouchon, 1989); a = (x=l)2,
with h = 375 m, l = 1000 m, where h and l denote the height and the half-width of the
ridge. The equation is applied for x 2 (0; l), while s(x) = 0 for x > l. For x > l the model
continues with at surface up to x = 2400 m, and the maximum depth is of z = 2400m. The
symmetry and antisymmetry in the vertical and horizontal components w and u are applied at
x = 0, respectively. The line sources are realized by a vertical body force whose time history
2
is as follows: f (t) = ,exp(b) fmax [fmax (t , t0) cos c + sin c]; b = , 21 fmax
(t , t0)2,
c = fmax (t , t0), with t0 = 0:136 s, and the maximum frequency is fmax = 22 Hz. Twelve
receivers are located on the free surface with constant x-increments 166:6 m. The comparison
of seismograms for these two models shows (Fig. 1b) that the coarse-grid model is producing
spurious diraction, while the grid renement leads to a signicant improvement. Moreover,
in the irregular grid, the abrupt changes of the gridstep (from 33:33 m to 6:66 m) do not
produce any spurious eects. To verify this, we computed also three other models on regular grids with x = z equal to 10 m; 6:66 m; 3:33 m respectively, and all of them agreed
(within the width of line) with irregular-grid model. The same seismograms were also obtained
when we changed the gridstep not abruptly, but in small steps, e.g. 20% change between the
neighbouring gridlines. For irregular-grid model we needed only 10% of computer time and 5%
of memory compared to the same model computed on the whole-ne grid of x = z = 6:66 m.
Water basin in solid medium The model consists of a rectangular water basin embedded
in a homogeneous elastic solid (Fig. 2a). For this model we used a regular grid with gridstep
z = x = 0:66 m, and the timestep was t = 0:00018 s. The parameters for water were
= 1500 m=s, formally = 0:01 m=s, and = 1000 kg=m3; for the surrounding medium
= 2000 m=s, = 500 m=s, and = 2200 kg=m3 . The sources are realized by a vertical
body force placed 20 m under the free surface. The left side of the model is a vertical plane of
symmetry and antisymmetry in the vertical and horizontal components u and w, respectively.
The time function is a Gabor's wavelet: g(t) = (,exp(,(! (t , ts)= )2)) cos(! (t , ts)+ =2) ;
whose predominant frequency is fp = 75 Hz, and ! = 2fp ; ts = 0:45 =fp ; = 3. The
non-linear amplitude compression (gain control) was applied to resulting synthetics in Fig. 2b,c.
Underground cavity and coal seam The model contains a horizontal coal seam and
cavity (Fig. 3a left). The waveeld is excited by a line array of vertical forces acting along the
free surface, with timefunction of Gabor's wavelet with prevailing frequency 75 Hz. To represent correctly the structural details, we employ an irregular grid, with gridstep x = 0:25 m
Ivo Oprsal, Jiri Zahradnik
Robust finite-difference modelling of complex structures
Paper No. 15 in Proc. of Int. Symposium on
High Performance Computing in Seismic Modelling
June 2-4, 1997, Univ. of Zaragoza, Spain
F. J. Seron, and F., J. Sabadell (editors)
for the whole model, z = 0:25 m in the coal-seam layer and around the cavity (Fig. 3a right),
while elsewhere in large remaining parts of the model there is z = 0:66 m. The seismograms
are displayed in Fig. 3b,c. We also computed synthetics for exactly the same model geometry
and source, while the cavity was lled with water with formal shear wave velocity = 0:01 m=s
(Fig. 3d,e). For both the empty and lled cavity models the 1-D solution (that without the
cavity) was subtracted from the vertical component. Only 37% of computing time and memory
was needed compared to regular-grid models of the same accuracy.
Homogeneous model with perturbed parameters Stochastic perturbations of the ma-
terial parameters bring some sort of scattering eects. The model consists of a homogeneous
halfspace = 2000 m=s, = 500 m=s, and = 2200 kg=m3 excited by a P-wave vertically
incident from below. The 50 receivers are located on the free surface with regular increment of
6 m. The source was realized by Gabor wavelet of predominant frequency 75 Hz. The grid was
regular; space gridsteps z = x = 0:66 m, timestep t = 0:00016 s. We studied a box 200 m
wide and 120 m deep. The planes of symmetry in w and antisymmetry in u were prescribed on
vertical sides of the model. We perturbed separately Lame's parameters ; and density .
The perturbation was a 10% white noise, further subjected to a low-pass ltration in both the
vertical and horizontal directions (models with the non-ltered white noise were unconditionally unstable). An example of the density perturbation along one of the gridlines is shown in
Fig. 4a. The corresponding non-ltered and ltered spatial spectrum of the density is in Fig.
4b (thin and thick lines respectively). Two dierent lters were used (the left and right parts
of Fig. 4). One of them yields unacceptable numerical artifacts after 0:25 s, which is 1 560
time levels. Just a slight change in the perturbation spectrum shape stabilizes the solution
(Fig. 4c, right). This solution is shown only up to 0:3 s. However, a very long run (34 000
time levels, not shown)was equally quite stable. Other shapes of the perturbation spectra were
tested, too. The perturbations with narrow, fast-decaying spectra do not make problems, but
broader spectra may be critical, in particular those with local maximum at the wavenumbers
around 0:5=gridpoint.
Acknowledgement The work has been supported by NATO SfS GR-COAL grant, the
Czech Republic Agency grant No. 205/1743, and the Inco-Copernicus grants ISMOD and
COME.
References
Oprsal, I., Zahradnk, J. (submitted). Elastic nite-dierence scheme on irregular grids.
Geophysics.
Oprsal, I. (1996). Elastic nite-dierence scheme for topography models on irregular grids.
Diploma thesis, Charles University, Prague.
Zahradnk, J. (1995). Simple elastic nite-dierence scheme. Bull. Seism. Soc. Am., 85,
1879{1887.
Zahradnk, J., Priolo, E. (1995). Heterogeneous formulations of elastodynamic equations
and nite-dierence schemes. Geophys. J. Int., 120, 663{676.
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