3D acoustic wave modeling with a time-space domain dispersion-relation-based Finite-difference scheme

Transcription

1 P-8 3D acoustic wave odeling with a tie-space doain dispersion-relation-based Finite-difference schee Yang Liu * and rinal K. Sen State Key Laboratory of Petroleu Resource and Prospecting (China University of Petroleu, Beijing), China The Institute for Geophysics, John A. and Katherine G. Jackson School of Geosciences, The University of Texas at Austin, U.S.A. Suary Spatial finite-difference (FD) stencils designed in the space doain are usually eployed in wave equation odeling. In this paper, we adopt a spatial FD stencil which is devised in the tie-space doain based on a dispersion relation to iprove the accuracy of FD odeling for 3D acoustic wave equation. In addition, we eploy a new hybrid absorbing boundary condition (ABC) to eliinate edge reflection due to finite coputational doain. Dispersion analysis and odeling results deonstrate that the tie-space doain dispersion-relation-based spatial FD stencils can indeed iprove the odeling accuracy. The odeling accuracy can be iproved further by using a truncated FD ethod. The new hybrid ABC can effectively absorb boundaries reflections. The odeling schee presented in this paper is ore accurate and efficient than the conventional one and can be used routinely. Introduction The finite-difference (FD) ethod is an iportant ethod for the nuerical solution of partial differential equations and has been widely utilized in seisic odeling and igration (e.g., Kelly et. al, 976; Dablain, 986; Liu and Sen, 9a). A high-order spatial FD is a coon approach to increase odeling accuracy (e.g., Dablain, 986; Fornberg, 987; Etgen and O Brien, 7). A loworder FD algorith uses a shorter operator but needs ore grid points for discretization. A high-order FD algorith uses a longer operator but needs fewer grid points. Since a conventional explicit high-order teporal FD is usually unstable in the wave equation odeling, spatial derivatives are used to replace high-order teporal derivatives (e.g., Dablain, 986) to increase the accuracy of teporal derivatives with additional coputational cost. Generally, ost FD ethods deterine the FD stencils for spatial derivatives only in the space doain. However, the seisic wave propagation calculation is done both in space and tie doains. To iprove the accuracy of conventional FD ethods, a new tie-space doain FD ethod was proposed to derive the spatial FD coefficients in the joint tie-space doain (Finkelstein and Kastner, 7). The key idea of the ethod is that the dispersion relation is copletely satisfied at designated frequencies and thus the spatial FD coefficients are frequency dependent. This ethod was developed further for D, D and 3D acoustic wave odeling using a plane wave theory and the Taylor series expansion (Liu and Sen, 7b). These new spatial FD coefficients, dependent on the Courant nuber and space point nuber, are frequency independent though they lead to a frequency dependent nuerical solution. In this paper, we report on the nuerical results in 3D using FD that replaces the conventional 3D spatial FD coefficients with these new coefficients to iprove the odeling accuracy without increasing the calculation aount. A truncated FD ethod is also used to enhance the odeling accuracy further. A hybrid absorbing boundary condition (ABC) is also developed for 3D acoustic odeling. Conventional finite-difference schee for the 3D acoustic wave equation We start with the 3D acoustic wave equation given by p p p p + + =, () x y z v t * China University of Petroleu, State Key Laboratory of Petroleu Resource and Prospecting, Changping, Beijing, 49, China. Eail: wliuyang@vip.sina.co.

2 3D Acoustic odeling with Dispersion-relation-based FD ethod Where, p = p( x, y, z, t) is a scalar wave field, and v is velocity. The following nd-order FD is usually used for the tie derivatives, p δ p = p,, + ( p,, + p,, ) t δt τ, () where, n p, l, j = p( x + h, y + lh, z + jh, t + nτ ), (3) τ is tie step, h is grid size. Generally, the odeling accuracy is iproved by high-order FD on the space derivatives given by p δ p p,, a ( p,, p,, ) x δ x h + +, (4a) p δ p p,, a ( p,, p,, ) y δ y h + +, (4b) p δ p p,, a ( p,, p,, ) z δ z h + +. (4c) A Taylor series expansion is generally used to deterine the FD coefficients (Fornberg, 987; Dablain, 986). The FD coefficients in Equations (4a), (4b) and (4c) are as follows (Liu and Sen, 9a) a a ( ) + n = ( =,,..., ), (5a) n < n, n a =. (5b) It can be proved that when a -order FD space stencil and a nd-order FD tie stencil are used to solve the 3D acoustic wave equation, nuerical odeling has nd-order accuracy. It is noteworthy that increasing ay help reduce the agnitude of the conventional FD error without increasing the accuracy order. Tie-space doain dispersion-relation-based finitedifference schee for the 3D acoustic wave equation For the tie-space doain dispersion-relation-based FD schee, spatial FD coefficients can be derived by using the following steps (Liu and Sen, 9b). First, derive dispersion relation of FD odeling by using plane wave theory and Equations ()-(4). Second, apply the Taylor series expansion for trigonoetric functions in the dispersion relation equation. Then, copare coefficients of power series and obtain the following equations ( cos θ cos φ cos θ sin φ sin θ ) + + a = r j j j j j j j ( j =,,..., ), (6) where, r = vτ / h, θ is the plane wave propagation angle easured fro the horizontal plane perpendicular to z axis, φ is the aziuth of the plane wave. Note that these equations indicate that the coefficients a are a function of θ and φ. We solve Equation (6) to obtain a by using an optial direction with θ = and φ = π /8 (Liu and Sen, 9b), which akes FD odeling attain the highest ()th-order accuracy along 48 directions. Absorbing boundary condition Here, we adopt the new hybrid schee developed by Liu and Sen (9c) to absorb reflections fro the odel boundaries in nuerical solutions of wave equations (WEs). This schee divides the coputational doain into boundary, transition and inner areas. The wavefields within the inner and the boundary areas are coputed by the WE and the one-way wave equation (OWWE) respectively. The wavefields within the transition area are deterined by a weighted cobination of the wavefields coputed by WE and the OWWE to obtain a sooth variation fro the inner area to the boundary via the transition zone. We develop this hybrid ABC for 3D acoustic wave odeling. For 3D odeling, boundaries include six sides, twelve edges and eight corners. Second-order OWWEs are adopted for these sides, and first-order OWWEs for these edges and corners (Clayton and Engquist, 977). Ten grid points are used to define the transition area to absorb boundary reflections. Dispersion analysis The dispersion of FD for 3D odeling is described by the following expression (Liu and Sen, 9b), ( kh θ ) ( ) ( kh θ φ ), + sin cos cos / (7) sin sin / vfd sin sin δ = = r a + kh cos θ sin φ / v rkh

3 3D Acoustic odeling with Dispersion-relation-based FD ethod Where, k is the wavenuber. If δ equals, there is no dispersion. If δ is far fro, a large dispersion will occur. Since kh is equal to π at the Nyquist frequency, when calculating δ, kh only ranges fro to π. Figure illustrates the variation of dispersion paraeter δ ( φ, θ ) with kh along nine directions, which deonstrates that the accuracy of the tie-space ethod is greater than that of the conventional ethod. v FD /v v FD /v kh (a) Conventional ethod kh Exact (, ) (, π/8) (, π/8) (π/8, π/8) (π/8, π/8) (π/8, 3π/8) (π/8, π/8) (π/8, π/8) (π/8, 3π/8) Exact (, ) (, π/8) (, π/8) (π/8, π/8) (π/8, π/8) (π/8, 3π/8) (π/8, π/8) (π/8, π/8) (π/8, 3π/8) (b) Tie-space ethod Figure : Plot of dispersion curves of the conventional and the tie-space FD ethods for 3D acoustic wave equation odeling. =, v = 3/s, τ =.s, h =. Nuerical odeling exaples First, both the conventional and the tie-space FD ethods are used to siulate 3D acoustic wave propagation in a hoogeneous acoustic ediu under the sae discretization. The odel and siulation paraeters are listed in the caption of Figure. Coputed snapshots respectively by the conventional and the tie-space FD ethods are shown in Figures (a) and (b). Coparing these two figures, we can see that the tie-space ethod aintains the wavefor better than the conventional ethod and thus has greater precision. To iprove the odeling accuracy further, we introduce the truncated FD ethod into the tie-space FD ethod. It is known that with the increase of the nuber of grid points involved in FD discretization, the accuracy increases but the coputational cost also increases. Liu and Sen (9d) found that there exist soe very sall coefficients for highorder FD coefficients and with the increase of order the nuber of these sall coefficients increases but their values decrease sharply. They also deonstrated that oitting these sall coefficients can aintain approxiately the sae level of accuracy of FD but reduce coputational cost significantly. Figure (c) shows snapshots by the truncated tie-space ethod, it follows that the dispersion decreases further copared with Figure (b). Next, we adopt the truncated tie-space FD ethod to perfor nuerical odeling. Figure 3 displays the seisogras coputed for a horizontally layered acoustic odel with the Clayton-Engquist ABC (Clayton and Engquist, 977) and the new hybrid ABC. The odel and siulation paraeters are given in the figure caption. The figure suggests that the boundary reflections are still strong copared with the reflections fro the true reflectors for the Clayton-Engquist ABC. The new hybrid ABC achieves nearly perfect absorption. Conclusions We have developed a tie-space doain dispersionrelation-based FD schee cobined with the truncated FD ethod and the hybrid ABC for 3D acoustic wave equation odeling. Dispersion analysis and nuerical odeling results deonstrate that this schee has greater accuracy and can effectively suppress dispersion and boundary reflections. Acknowledgents This research is partially supported by NSFC under contract No and the National 863 Progra of China under contract No. 7AA6Z8. 3

4 3D Acoustic odeling with Dispersion-relation-based FD ethod (a) Conventional ethod (a) Snapshots at t=5s by the Clayton-Engquist ABC (left) and (b) Tie-space ethod (b) Snapshots at t=7s by the Clayton-Engquist ABC (left) and (c) Snapshots at t=s by the Clayton-Engquist ABC (left) and 7 3 Trace no Tie (s) (c) Truncated tie-space ethod Figure : Snapshots coputed by finite-difference odeling for a 3D hoogeneous acoustic odel respectively with conventional, tie-space and truncated tie-space FD ethods. Each figure includes panels with snapshots at 5s and 5s fro left to right. The odel velocity is 3/s, grid size is, grid diensions are, grid point coordinates range fro (,,) to (99,99,99), tie step is s, =. The sae discretization is involved in all the three ethods, but their spatial FD coefficients are different. A source pulse of 5Hz sine function with one period length is located at the center of the odel. The free surface condition and ABC are not included here. Top, front and right surfaces of snapshots are recorded at z=, y=5 and x=99 respectively (d) Seisogras with the Clayton-Engquist ABC 4

5 3D Acoustic odeling with Dispersion-relation-based FD ethod Tie (s) Trace no (e) Seisogras with the hybrid ABC Figure 3: Snapshots and seisogras coputed by truncated tiespace FD odeling for 3D horizontally layered acoustic odel respectively with the Clayton-Engquist ABC and the hybrid ABC. Each figure of seisogras includes 3 receiver lines. The odel has 6 layers, whose velocities are velocity are 5/s, 3/s, 35/s, 3/s, 36/s and 4/s fro shallow to deep, 5 interfaces depth are 3, 5, 8, and 4. Grid size is, grid diensions are, grid coordinates range fro (,,) to (99,99,99), tie step is s. =. A source pulse of Hz sine function with one period length is located (99, 99, ). Receivers are located on the surface, and 3 receiver lines shown in this figure are located at y=99, 49 and 99 fro left to right. Trace no. of seisogra for each receiver line ranges fro to. Traces whose no. varies fro to 9 are shown since grids of width are used for the hybrid ABC. The free surface condition is included here. Top, front and right surfaces of snapshots are recorded at z=, y= and x=99 respectively. References Clayton, R. W., and B. Engquist, 977, absorbing boundary conditions for acoustic and elastic wave equations: Bulletin of the Seisological Society of Aerica, 6, Dablain,. A., 986, the application of high-order differencing to the scalar wave equation: Geophysics, 5, Etgen, J. T., and. J. O Brien, 7, Coputational ethods for large-scale 3D acoustic finite-difference odeling: A tutorial: Geophysics, 7, S3 S3. Fornberg B., 987, the pseudospectral ethod - coparisons with finite differences for the elastic wave equation: Geophysics, 5, Finkelstein B., and R. Kastner, 7, Finite difference tie doain dispersion reduction schees: Journal of Coputational Physics,, Kelly, K. R., R. Ward, W. S. Treitel, and R.. Alford, 976, Synthetic seisogras: A finite-difference approach: Geophysics, 4, 7. Liu Y., and. K. Sen, 9a, A practical iplicit finitedifference ethod: exaples fro seisic odeling: Journal of Geophysics and Engineering, 6, Liu Y., and. K. Sen, 9b, A new tie-space doain high-order finite-difference ethod for the acoustic wave equation, Journal of Coputational Physics, doi:.6/j.jcp Liu Y., and. K. Sen, 9c, A hybrid schee for absorbing edge reflections in nuerical odeling of wave propagation: Geophysics, in revision. Liu Y., and. K. Sen, 9d, Nuerical odeling of wave equation by a truncated high-order finite-difference ethod: Earthquake Science, ():