Seismic Theory 14: Elastic Wave Propagation TheoryThursday a.m., Nov. 6

ST 14.1Modeling 3D acoustic waves in the presence of free surface by the half-spacegeneralized screen propagatorsShengwen Jin* and Ru-Shan Wu; Institute of Tectonics, University of California, SantaCrux, CA 95064, USA

SUMMARY

We present a new 3D modeling method for acous-tic waves in heterogeneous media in the presence offree surface. The method based on the newly devel-oped half-space generalized screen propagators (Wu etal., 1996) is implemented by a dual domain techniquewhich shuttles the calculations between the space andwavenumber domain.The formulations of wide-angleapproximation (pseudo-screen) and small-angle approx-imation (phase-screen) are given in the paper. Numeri-cal examples show that the results from the new methodare in good agreement with those of the finite differencemethod but with much faster computation speed thanthe latter one. The method can also be effectively usedin modeling 3D effects of VSP and cross-hole seismicwavefields.

INTRODUCTION

Several numerical techniques have been developed forsimulating waves propagation in complex heteroge-neous media. The most general of these are the fi-nite difference method, finite element method and thepseudospectral method. These methods are capableof modeling waves in arbitrary heterogeneous media.The major shortcomings of these numerical schemes arethe requirments of huge computation time and memorystorage, especially for 3D problems. The screen propa-gators, such as split-step Fourier method (Stoffa et al,1990), generalized screen propagators (GSP) methodincluding the phase-screen, complex screen and wide-angle screen methods (Wu, 1994; Wu and Xie, 1994;Wu and Huang, 1995) have been proposed recently tocalculate the one-way wave propagation for both acous-tic and elastic cases. But only a few detailed studieshave been made on the implementation of boundarycondition with respect to the one-way wave equations(Charara and Tarantola, 1996; Wu et al, 1996). Toaddress this problem, Wu, Jin and Xie (1996) have re-cently derived a new dual-domain formulation of thehalf-space screen propagators for the 2D elastic SHwave case that properly takes into account the free-surface boundary condition. In this work, we adoptthe same approach and further develop the method tothe 3D acoustic wave case. In the following, we firstsummarize the theory and formulations, then give two

examples to show the validity and application poten-tial of the method. One compares the results obtainedfrom the finite-difference method with those from theimplementation of the half-space GSP method. Theother models the 3D propagating effects of a low veloc-ity reservoir in crosshole and VSP surveys.

THEORY AND METHODS

The acoustic pressure can be described by a scalarwavefield in a heterogeneous propagationmedium of density p(r) and bulk modulus sat-isfying the equation in frequency domain

where is the frequency, = y, is a 3D posi-tion. If we consider the acoustic parameters and thewavefields can be decomposed into

where and are the parameters of the backgroundmedium, and SK are the corresponding perturba-tions, is the primary field and is the scatteredfield. Then can be rewritten as

where kmedium,

isand F(r)

the wavenumber in the backgroundis a perturbation operator,

Note that Eq.(3)is a scalar Helmhotz equation, thescattered field U then can be calculated by

where is the Green’s function in the back-ground medium. Under the screen approximation, thespace model can be sliced into thin slabs perpendicu-lar to the main propagating direction of the waves, and

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3D half-space generalized screen propagators

the total field for each step forward can be calculated asthe sum of primary field propagating freely in space andthe scattered field caused by heterogeneities within thethin-slab. Forward scattering approximation has beenused in deriving the scattered field.

Generalized half-space wide-angle formulation

In the presence of free surface boundary, the pressureshould vanish at surface.In this case, the Green’sfunction in Eq.(6) should be replaced by the half-spaceGreen’s function r). In wavenumber domain, is derived as (Wu et al, 1996)

= Ye ( 7 )

where = = and

is the horizontal propagating wavenumber.By defining forward cosine transform as

(8)

we obtain the dual-domain expression of wide-angle ap-proximation for scattered field at the next screen

with =

Half-space small-angle approximation

Under the small-angle approximation, the above ex-pressions can be simplified. In this case, the screenapproximation can be applied to circumvent the vectoroperation and greatly speed up the computation. Thenwe can simplify Eq.( 10) into the form of

where = is called the slowness perturbationscreen function and is defined as

Summing up the primary and scattered fields and in-voking the Rytov approximation result in the expres-sion of phase-screen propagator

This is a dual domainscreen propagator.

implementation of thephase

NUMERICAL SIMULATIONS

In this section, we show two examples of our half-spaceGSP method applied to the numerical simulations ofVSP and crosshole surveys. To verify the theory andmethods of screen propagators, first we present a 2Dlayered model(see Fig.1) to compare the synthetic seis-mograms from the half-space GSP method with theresults from the fourth-order FD algorithm. In themodel, a pressure source was located beneath the freesurface. We take the background parameters of density2.8g/cm3 and velocity 3500m/s as in the first layer.Figure 2 and 3 illustrate the vertical section received inthe well and the horizontal section recorded at surface,respectively. In the figures, thick solid line indicatesthe results from GSP, and the thin line from the FDmethod. Excellent agreements can be seen clearly intiming, phase and amplitude for the direct arrivals, up-going waves reflected from interface, downgoing wavesreflected from free surface, multiples between free sur-face and interface, and transmitted waves. In addition,the computation by our method is about 2 orders ofmagnitude faster than that of the FD method.Due to a variety of constraints, 2D models are oftenused as approximations for structures which are knownto be 3D. This can be a problem because it is difficultto fully understand the 3D effects, especially in VSPand crosshole seismic surveys where the target of inter-est is a finite object with geometric features. Here wepresent an example of idealized 3D portion of the earth(Mufti,1995), which consists of a reservoir formationwith low velocity 3000m/s and density 2.5g/cm3 indi-cated by dotted line situated in a background mediumwith higher velocity 4500m/s and density 3.3g/cm3(seeFig.4). Figure 5 shows the 3-D numerical results in theform of the shot record at the receiving holes. The ear-liest events received in the central hole-2 represent theenergy which bypasses the reservoir and finds its way tothe receiving hole as the result of diffraction. From thetime section received in hole-3 the 3D effects are shownclearly, where the earliest events are the direct arrivalsin the background medium, followed by reflected wavesby the reservoir walls. In Figure 6, we can see the pat-terns of the wavefields on the receiver plane for a giventime slice at 340ms, which indicates the spatial distri-butions of energy as a result of diffractions, reflections,

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3D half-space generalized screen propagators

refractions and interferences. Note that the spatial pat-tern is strongly influenced by the shape of the reservoir.In Figure 6(b), the strong reflections from free surfacecan been seen.

CONCLUSIONS

We have presented a new numerical technique in which3D acoustic waves in heterogeneous media are modeledwith free surface boundary conditions. The method isdeveloped based on the half-space generalized screenpropagators and is implemented in dual domain. Byusing the forward marching algorithm, the screen prop-agator needs only to store 2D data arrays for 3D prob-lems. Therefore, this approach is attractive becauseit requires significantly less memory with computationspeed about two orders of magnitude faster than thefinite difference method. Numerical examples showedgood agreements in timing, phase and amplitude withthe results from the fourth-order finite-difference algo-rithm. It demonstrates that the GSP method can ef-fectively model 3D effects in heterogeneous media forVSP and crosshole seismic surveys.

ACKNOWLEDGEMENTS

We appreciate the helps from Dr. Xiao-Bi Xie and LingZhang. This work is supported by the ACT1 project ofUCSC granted from the United States Department ofEnergy administered by the Los Alamos National Labo-ratory. The facility support from the W.M. Keck Foun-dation is also acknowledged. Contribution number 324of the Institute of Tectonics, University of California atSanta Cruz.

References

Charara, M., and Tarantola, A., 1996, Boundary conditionsand the source term for one-way acoustic depth extrapo-lation, Geophysics, 61, 244-249.

Mufti, I.R., 1995, Pitfalls in crosshole seismic interpretationas a result of 3-D effects, Geophysics, 60, 821-833

Stoffa, P.L., Fokkema, J.T., de Luna Freire, R.M., andKessinger, W.P., 1990, Split-step Fourier migration, Geo-physics, 55, 410-421.

Wu, R.S., 1994. Wide-angle elastic wave one-way propaga-tion in heterogeneous media and an elastic wave complex-screen method, J. Geophys. Res., 99, 751-766.

Wu, R.S.,and Huang, L.J.,1995, Reflected wave modeling inheterogeneous acoustic media using the de Wolf approx-imation: in S.Hassanzadeh, Mathematical Methods inGeophysical Imaging III, Proc. SPIE 2571, 176-186.

Wu, R.S., and Xie, X.B., 1994, Multi-screen backpropa-gator for fast 3D elastic prestack migration, in S. Has-sanzadeh, Editor, Mathematical Methods in GeophysicalImaging II, Proc. SPIE 2301, 181-193..

Wu, R.S., Jin S.,and Xie,X.B.,1996, Synthetic seismogramsin heterogeneous crust al waveguides using screen propa-gators, Proc. 18th Seismic Research Symposium,291-300.

Figure 1: The schematic illustration for a 2-D layeredmodel

Figure 2: Comparison of synthetic seismograms alonga vertical profile by the half-space GSP method (thickline) and the fourth-order finite-difference method (thinline). The dotted line is the travel time calculated bvray theory.

Figure 3: Comparison of synthetic seismograms re-ceived at surface.

1 8 3 2

3D half-space generalized screen propagators

Figure 4: An idealized 3D crosshole model which consists of a low-velocity and low-density reservoir. S and Rindicate the source in hole 1 and the receivers in hole 2 respectively. S is located at (0,640,640).

Figure 5: Cross-hole shot records at the receiver hole plane.

Figure 6: Time slices of the 3-D crosshole simulation time (T=340ms) at the receiving plane.

which show the spatial distributions of the energy for a given

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