A highly efficient implicit finite difference scheme for acoustic wave propagation

Highlights

• Highly efficient 2D/3D implicit finite difference scheme of 10th order is presented.

• The modified implicit scheme becomes almost twice efficient.

• A comparison of explicit and implicit FD operator and resulting schemes is provided.

• Application on a realistic model has been demonstrated.

Abstract

The accuracy of a numerical derivative has a significant effect on any numerical simulation. Long stencils can provide high accuracy as well as reduce the numerical anisotropy error. However, such a long stencil demands extensive computational resources and with their growing size, such derivatives may become physically non-realistic since contributions from very far offset whereas the derivative is local in nature. Further, the application of such long stencils at boundary points may introduce errors. In this paper, we present a very efficient, accurate and compact size numerical scheme for acoustic wave propagation using implicit finite difference operator, which utilizes a lesser number of points to estimate derivatives in comparison to the conventional central difference derivative operator. The implicit derivative operator, despite its several advantages, is generally avoided due to its high computational cost. Therefore in this paper, we discuss a method which can dramatically reduce the computational cost of this scheme to almost half. This strategy is useful particularly for 2D and 3D case. Spectral characteristics of the derivative operator and the numerical scheme are compared with several other central difference schemes. We have also demonstrated an application of this scheme for seismic wave propagation in 2D and 3D acoustic media.

Introduction

Numerical seismic simulations have much importance in seismic workflows, and among its numerous applications, it includes algorithms, such as reverse time migration and full waveform inversion. It is always desired to have an accurate, stable and computationally efficient numerical scheme which can be used to solve governing (and constitutive) relations. This scheme involves both time and space derivatives, thus its accuracy chiefly depends on following three factors: (1) the step size chosen for space and time discretization (i.e. Δx,Δt), (2) number of neighborhood points used for estimating derivative, and (3) methods used for estimating spatial derivatives and to carry out time integration. Each factor affects the simulation cost in terms of memory, number of operations and computational time. For example, when a smaller size is employed for space and time discretization, the results would be more accurate but at the expense of increased computational cost due to the increased number of nodes. Increasing the derivative stencil size/length much can violate the local nature as it have a contribution from the nodes at far offset, which may make this derivative estimations non-local in nature. Also after a certain number of nodes, further increment in derivative stencil length will increase only formal accuracy (defined in terms of truncation error) whereas the spectral accuracy gets saturated (Kosloff et al., 2010). Thus a derivative stencil longer than saturation length, would only increase the computational cost but accuracy. Considering the above factors, it is desirable to carry out a numerical simulation using a scheme that can produce highly accurate numerical derivatives using only a small number of nodes for a given grid spacing without adding a significant computational cost.

As mentioned before, the accuracy of a numerical scheme is directly linked with the accuracy of the numerical derivative operator. Several types of numerical derivative operators, which have been utilized for wave propagation can be found in the literature. For example, the classical central derivative (CD) operator, which is based upon Taylor series expansion, can be used for simulating the seismic wavefield on the collocated grid (Alterman and Karal, 1968; Kelly et al., 1976, etc.). The CD operator over the staggered grid provides decoupling of even-odd modes as well as higher formal accuracy than conventional operators (Harlow and Welch, 1965). It has also been widely used in the geophysics community (Madariaga, 1976; Virieux, 1986; Levander, 1988, etc.). These derivative operators can be made to yield spectrally more accurate derivative estimates with proper choice of coefficients. These coefficients can be determined by matching the spectral characteristics of the numerical derivative to the exact in the wavenumber domain (Holberg, 1987; Tam and Webb, 1993). Following this approach, many researchers have also proposed N-points Laplacian stencils (Jo et al., 1996; Shin and Sohn, 1998) as well as their optimized versions (Gosselin-Cliche and Giroux, 2014; Liu et al., 2015; Wang et al., 2016; Fan et al., 2017) for acoustic wave propagation in earth. With the increasing accuracy of FD operator, the error in phase velocity due to numerical anisotropy also reduces. Some more reduction (2–3%) in this error can be achieved with the help of the rotated staggered-grid scheme (Saenger and Bohlen, 2004; Virieux et al., 2011), however, it comes at the cost of the extra cross derivative term (Štekl and Pratt, 1998).

Another type of operator, known as implicit FD operator, is based on Padé schemes (Collatz, 1960). This operator utilizes the value of the function as well as the value of the derivative, which are provided at the given point with some of its neighborhood points for estimating derivative of a function at a given node. An improved version of the implicit FD operator was presented by Lele (1992) which provides better resolution at higher wavenumbers. This scheme was further optimized by (Kim and Lee, 1996) to achieve better performance at higher wavenumbers. Despite its advantage of high accuracy, it also shows increased computational cost due to its implicit nature which requires solving a tridiagonal or pentadiagonal matrix. This computational overburden can be reduced by prefactorization of the banded matrix (Ashcroft and Zhang, 2003; Zhou and Zhang, 2011), so that, the matrix can be solved using the simpler (L + U) operation.

In this paper, we used a sixth order accurate, tridiagonal, implicit type spatial derivative operator for both, analysis and simulations in 2D & 3D domain. To reduce its computational cost, we have presented a method based upon LDM^T decomposition followed by a pre-decomposition technique. This scheme is very much efficient in reducing the computational cost for 2D and 3D domains. Further, we have compared this scheme with various other schemes for 2D and 3D models and discussed (i) the accuracy of derivative operators, (ii) the numerical anisotropy error, and (iii) the associated computational cost. Finally, we have demonstrated the application of implicit finite difference scheme to simulate the seismic wave propagation in 2D and 3D acoustic media to verify its feasibility.