An explicit method to calculate implicit spatial finite differences

Abstract

The finite-difference (FD) solution of the second-order acoustic wave equation is a fundamental algorithm in seismic exploration for seismic forward modeling, imaging, and inversion. Unlike the standard explicit FD (EFD) methods that usually suffer from the so-called “saturation effect,” the implicit FD methods can obtain much higher accuracy with relatively short operator length. Unfortunately, these implicit methods are not widely used because band matrices need to be solved implicitly, which is not suitable for most high-performance computer architectures. We have introduced an explicit method to overcome this limitation by applying explicit causal and anticausal integrations. We can prove that the explicit solution is equivalent to the traditional implicit lower-upper-decomposition method in analytical and numerical ways. In addition, we also compare the accuracy of the new methods with the traditional EFD methods up to 32nd order, and numerical results indicate that the new method is more accurate. In terms of the computational cost, the newly proposed method is standard eighth-order EFD plus two causal and anticausal integrations, which can be applied recursively, and no extra memory is needed. In summary, compared with the standard EFD methods, the new method has a spectral-like accuracy; compared with the traditional lower-upper-decomposition implicit methods, the new method is explicit. It is more suitable for high-performance computing without losing any accuracy.