Tu P13 08 A MATLAB Package for Frequency Domain Modeling of Elastic Waves

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1 Tu P13 8 A MATLAB Package for Frequency Domain Modeling of Elastic Waves E. Jamali Hondori* (Kyoto University), H. Mikada (Kyoto University), T.N. Goto (Kyoto University) & J. Takekawa (Kyoto University) SUMMARY We developed a MATLAB package for finite difference frequency domain modeling of elastic waveforms. By using array-processing abilities of MATLAB, we efficiently computed the large impedance matrix for realistic model sizes. SuiteSparse library was used to calculate LU factors of complex impedance matrix with reasonable computational costs and METIS library was used to benefit from nested dissection ordering scheme to accelerate LU factorization. Once the factors have been calculated, several seismic sources could be modeled by solving for multiple right hand sides. Perfectly Matched Layers (PML) suppressed the reflections from the boundaries of computational domain. Attenuation effects have been considered by including complex valued velocities. Synthetic examples of Marmousi dataset show the performance of the developed package. 75 th EAGE Conference & Exhibition incorporating SPE EUROPEC 13 London, UK, 1-13 June 13

2 Introduction Seismic waveform modeling is a key tool to estimate subsurface characteristics, not only for hydrocarbon explorations but also for proper managements of seismic hazards and civil engineering infrastructures. Modeling in frequency domain found to be effective for its numerous advantages compared to that in time domain. Once triangular factors of impedance matrix have been calculated, multiple sources can be processed with the minimum computational cost. Monochromatic and band limited modeling at desired frequencies are implemented in a straightforward manner and the attenuation behavior of elastic media can directly be dealt with considering complex valued elastic parameters. However, discretizing the computational domain requires more grid points to achieve acceptable accuracy and a program with robust algorithm is needed to minimize the modeling time and cost. Jo et al. (1996) defined an optimal nine point finite difference operator for acoustic waves which was followed by Stekl and Pratt (1998) for elastic media. Shin and Sohn (1998) used 5-point finite difference operators rotated at four different angles to solve acoustic wave equation. Min et al. () defined 5-point weighted averaging operators for elastic waveform modeling. Amini and Javaherian (11) developed a MATLAB package for acoustic waveform modeling based on Jo et al. (1996). We used the method of Min et al. () to develop a MATLAB package for finite difference frequency domain modeling of elastic waveforms. By using array-processing abilities of MATLAB, we efficiently computed the large impedance matrix for realistic model sizes. SuiteSparse library (Davis, 4) was used to calculate LU factors of complex impedance matrix with reasonable computational costs. METIS library (Karypis and Kumar, 1999) was used to benefit from nested dissection ordering scheme to accelerate LU factorization. Once the factors have been calculated, several seismic sources could be modeled by solving for multiple right hand sides. Multicore processors speed up the program in a parallel scheme. In order to truncate the computational area we applied Perfectly Matched Layers (PML) on the boundaries. Attenuation effects have been considered by including complex valued velocities according to Kolsky-Futterman model (Toverud and Ursin, 7). Synthetic examples of Marmousi dataset show the performance of the developed package and suggest that this package can be used for real sized waveform modeling and imaging problems. 5-point finite difference operators One can write D elastic wave equation in frequency domain as bellow u v u v u u fx 1a x x z x z x z u v v v u v fz 1b z x z z x x z where, u, v, ω, ρ, λ, and μ are horizontal and vertical displacements in frequency domain, angular frequency, bulk density, and Lame parameters, respectively. f x and f z are horizontal and vertical external forces. We used 5-point finite difference stencil which was introduced by Min et al. () to discretize the wave equation. Second order horizontal spatial derivatives in the wave equation can be calculated by the stencils shown in Figure 1a. In each row, five grid points can be used to form two centered finite difference operators based on colored circles and unfilled circles. Then each operator is averaged by using appropriate coefficients, c for colored circles and d for unfilled circles. Resulted finite difference operators from each row will then be averaged by the coefficients b 1, b, and b 3. For vertical spatial derivatives the similar procedure can be followed as shown in Figure 1b. Mixed spatial derivatives can be discretized by using the finite difference stencil shown in Figure 1c. In this case two operators can be produced and averaged by coefficients e and f. Figure 1d shows the distribution of weighting coefficients which can be used to average mass acceleration terms, ρω u and ρω v. All the 5 points in the stencil contribute to form the mass acceleration terms. For the sake of space in the 75 th EAGE Conference & Exhibition incorporating SPE EUROPEC 13 London, UK, 1-13 June 13

3 abstract, we refer the reader to Min et al. () for the detailed finite difference operators and weighting coefficients. (a) (b) (c) (d) Figure 1: 5-point finite difference stencils for spatial derivatives and mass acceleration terms. For an n n finite difference grid, with n equal to number of grid points, equation (1) can be rewritten in matrix from as Su f where, S is the complex impedance matrix with size of n n, u and f are displacement and source function frequency domain representations, respectively, both of them a column vector of length n. In order to solve the system of equations one may factorize impedance matrix S and then use forward/backward substitution to achieve horizontal and vertical displacements in u. Since the impedance matrix has a band structure and very sparse pattern using efficient ordering schemes to reduce fill in during factorization is necessary. In our developed MATLAB code we used METIS library (Karypis and Kumar, 1999) together with SuiteSparse library for sparse LU factorization. METIS uses a multilevel nested dissection algorithm to calculate a fill reducing ordering. This brings a superior performance to the program. SuitSparse which is an open source library available for public use includes several factorization and solution modules, such as UMFPACK, SparseQR, and CHOLMOD, for sparse matrixes and linear system of equations. We used UMFPACK and SparseQR modules in our modeling code for problems with different sizes. Perfectly matched layers boundary conditions Reflections from truncated boundaries appear in the solutions of the wave equation which must be suppressed adequately. Perfectly matched layers (PML), which was first developed for the electromagnetic waves (Berenger, 1994), found to be the most effective method to eliminate boundary reflections. To implement PML in the elastic wave modeling one needs to transform the wave equation to the complex coordinate system by defining complex stretching functions (Drossaert and Giannopoulos, 7) x z x 1, z 1 3 i i where the attenuation function σ represents the damping behavior of the PML zone, and then calculate partial derivatives for the elastic wave equation in the PML region as bellow 1 1, 4 x x x z z z where x and z are horizontal and vertical complex coordinate variables, respectively. Attenuation of seismic waves In order to consider attenuation effects for the seismic wave one can use complex valued velocities which are directly implemented in frequency domain. We used the method of Kolsky-Futterman 75 th EAGE Conference & Exhibition incorporating SPE EUROPEC 13 London, UK, 1-13 June 13

4 (Toverud and Ursin, 7) which is the most popular model for attenuation of the seismic waves. The complex velocity could be calculated as bellow r sgn( ) ln i 5 c( ) c cq cq where, c, ω, ω r, and Q are the velocity, angular frequency, reference frequency, and quality factor, respectively. Note that in equation (5) complex velocity depends on the frequency and this implies that the dispersive behavior of the wave which is required for retaining the causality is preserved. Numerical results We have tested our developed package with various simple and complex geological models. Here, results of Marmousi (Martin et al., ) example are shown to represent the efficiency and accuracy of the MATLAB code. P and S wave velocity models are shown in Figure, we cropped the original model to focus on the more complex area in the center of the geological model. The final model is 66 m long and 3 m deep. Since the Marmousi includes very low S wave velocities with Poisson s ratio near.5 we have modified the S wave velocities such that the Poisson s ratio ranges between. and.3, so the grid intervals of 1 m let the maximum frequency to be Hz while avoiding grid dispersion. We used an explosive source on the right side of the model at depth of 5 m with receivers located on the surface with receiver intervals of m. The source signature is a Ricker wavelet with dominant frequency of 8 Hz. Figure 3 shows the vertical and horizontal components of the displacement recorded at the surface. As obvious in horizontal displacement component (Figure 3), strong Rayleigh waves appear in the seismograms and travel near surface with low velocity. Snapshots of wave propagation indicate that different events including surface waves, diffractions, and multiples are recorded by the receivers, note the high amplitude low velocity surface wave (Figure 4). P wave Velocity S wave Velocity 4 6 Figure : Marmousi P wave and S wave velocity models (cropped for our modeling). Vertical Displacement Horizontal Displacement Time (ms) Time (ms) Figure 3: Vertical and horizontal displacements recorded at the surface Conclusions Using finite difference frequency domain modeling we could develop an efficient MATLAB package which can handle real size complex geological models. Array processing ability of MATLAB provides a tool for calculating large complex impedance matrix. SuiteSparse and METIS libraries speed up the calculation by sparse LU factorization and fill reducing ordering. Perfectly matched 75 th EAGE Conference & Exhibition incorporating SPE EUROPEC 13 London, UK, 1-13 June 13

5 layers could suppress artificial boundary reflections and attenuation of the seismic wave has been considered by complex valued velocities. Based on the results of Marmousi data and several other models which already tested the program, the developed MATLAB package can be used for fast and accurate elastic waveform modelling. t = 75. ms 4 6 x t = 35. ms 4 6 Figure 4: snapshots of elastic waves propagating in Marmousi model. Surface waves with high amplitude and low velocity travel beneath surface. References Amini, N. and Javaherian, A., 11, A MATLAB-based frequency-domain finite difference package for solving D visco-acoustic wave equation. Waves in Random and Complex Media, 1: 1, Berenger, J. P., 1994, A perfectly matched layer for absorption of electromagnetic waves. Journal of Computational Physics, 114, 185. T.A. Davis, Algorithm 83: UMFPACK, an unsymmetric-pattern multifrontal method, 4, ACM Transactions on Mathematical Software, 3, Library available at: Drossaert, F. H. and Giannopoulos, A., 7, A non-split complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves. Geophysics, 7,, T9 T17. Jo, C. H., Shin, C. and Suh, J. H., 1996, An optimal 9-point finite difference frequency space -D scalar wave extrapolator. Geophysics, 61, G. Karypis and V. Kumar,1999, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM Journal of Scientific Computing,, Min, D., Shin, C., Kwon, B. and Chung, S.,, Improved frequency-domain elastic wave modeling using weighted-averaging difference operators. Geophysics, 65, 3, Martin, G. S., K. J Marfurt, and S. Larsen,, Marmousi-: an updated model for the investigation of AVO in structurally complex areas, 7 nd Annual International Meeting, SEG, Expanded Abstract, Shin, C. and Sohn, H., 1998, A frequency-space -D scalar wave extrapolator using extended 5-point finite-difference operators. Geophysics, 63, Stekl, I. and Pratt, R. G., 1998, Accurate viscoelastic modeling by frequency-domain finite differences using rotated operators. Geophysics, 63, T. Toverud and B. Ursin, 7, Comparison of seismic attenuation models using zero-offset vertical seismic profiling (VSP) data, Geophysics 7, F17 F5. x th EAGE Conference & Exhibition incorporating SPE EUROPEC 13 London, UK, 1-13 June 13

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