Frequency domain modeling requires solving linear systems whose bandwidths (the determinant factor of memory cost) are very sensitive to the order of accuracy of the numerical operators. Consequently, the speed-accuracy tradeoff is probably more pronounced for frequency domain modeling than for time domain modeling. In this work, we propose to use implicit spatial finite difference operators to alleviate this difficulty. Unlike time domain modeling where implicit operators often cause efficiency penalties, frequency domain modeling benefits from implicit operators since it requires solving linear systems no matter whether explicit or implicit operators are employed for spatial derivative discretizations. We show through dispersion analyses and numerical examples that implicit operators improve both the accuracy and the efficiency of seismic wave simulations in the frequency domain.
Seismic wave propagation can be described both in the time domain and in the frequency domain. Frequency domain modeling is a very useful tool for some problems that may be difficult to deal with in the time domain. For example, it is often more convenient to formulate the attenuation effects in the frequency domain than in the time domain. In certain cases, frequency domain simulations have been proved to be more efficient than the computations in the time domain. One of the classic examples is 2D multi-shot modeling. For applications where only a limited number of frequency domain forward solutions are required, the frequency domain method is also a preferred choice over the time domain method (Pratt, 1990). Frequency domain modeling involves solving large scale linear systems. Usually, direct solvers (typically, the LU decomposition method) are preferred, especially for multi-shot simulation problems. Numerical methods that are accurate but do not degrade the sparsity of the impedance matrix after LU decompositions are ideal choices. Traditionally, the compact (optimized) finite difference schemes have been the popular method of choice for frequency domain modeling (Jo et al., 1996; Shin and Sohn, 1998; ? Stekl and Pratt, 1998; Min et al., 2000; Hustedt et al., 2004). Different from the conventional high-order finite difference schemes, compact methods lead to narrow banded matrices but generally have better dispersion properties. In this work, we propose an alternative and better approach to solve this problem: using implicit spatial finite difference operators. For time domain modeling, implicit operators are hard to be made as efficient as explicit operators that have the same order of accuracy because we have to solve linear equations when using implicit operators. For frequency domain model modeling, this situation changes. The final formulation of frequency domain modeling is always a linear system no matter whether we use implicit operators or explicit operators. Most importantly, implicit operators lead to LU decomposed matrices that are close to or smaller than those by explicit operators with similar order of accuracy, as we will demonstrate through dispersion analyses and using numerical examples.
Figure 3 to Figure 8 display the acoustic shot records calculated by six finite difference operators for a homogeneous medium.
