High accuracy wave simulation – Revised derivation, numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation

Abstract

a b s t r a c tThe nearly analytic integration discrete (NAID) method for solving the two-dimensional acoustic wave equation has been fully mathematically revised, analyzed and tested. The NAID method is an alternative numerical modeling method for generating synthetic seismograms. The acoustic wave equation is first transformed into a system of first-order ordinary differential equations (ODEs) with respect to time variable t, and then directly integrated at a small time interval of [t n , t n+1 ] to obtain semi-discrete ordinary differential equations. The integral kernel is expanded into a truncated Taylor series, to which the integration operator is explicitly applied. The high-order temporal derivatives involved in the integral kernel are replaced by high-order spatial derivatives, which then are approximately calculated as a weighted linear combination of the displacement, the particle-velocity, and their spatial gradients. In this article, we investigate the theoretical properties of the revised NAID method, including the discrete error and the stability criteria. Numerical results for constant and layered velocity models show that, comparing to the Lax-Wendroff correction (LWC) scheme and the staggered-grid finite difference method, the NAID method can effectively suppress the numerical dispersion and source-noises caused by the discretization of the acoustic wave equation with too-coarse spatial grids or when models have strong velocity contrasts between adjacent layers. The proposed NAID method has been applied in computing the acoustic wavefields for two heterogeneous models -the corner edge model and the Marmousi model. Promising numerical results illustrate that the NAID method provides an encouraging tool for large-scale and complex wave simulation and inversion problems based on the acoustic equation.

Three-dimensional frequency-domain full waveform inversion based on the nearly-analytic discrete method

“…Then, according to Yang et al, (2003, 2006), Tong (2011, 2013) and Lang and Yang, (2017), the discretization of each partial derivative using the NAD method is obtained as:…”

“…The nearly analytic discrete (NAD) method is a finite difference method that uses displacement and its gradient to approximate the high‐order partial derivatives in the wave equation (Yang DH et al, 2003, 2004, 2006; Tong P et al, 2011, 2013). As this method captures more detailed wave equation information, dispersion analysis and numerical experiments have shown that it can solve the wave equation more accurately and efficiently than traditional finite difference methods (Yang DH et al, 2010, 2012).…”

A nearly analytic exponential time difference method for solving 2D seismic wave equations

“…To compute the high-order spatial derivatives in the wave equations (12) and (21), we follow Tong et al (2011) and Yang et al (2006Yang et al ( , 2007 and obtain the approximation formulae. For convenience, we list the formulae used in the NETD method as follows:…”

“…These NAD operators use only three grid points in a spatial direction to achieve the fourthorder spatial accuracy, and operators can efficiently suppress the numerical dispersion. Based on the idea, a range of different effective numerical algorithms have been proposed by Chen et al (2010), Tong et al (2011), and Ma et al (2011).…”

Modelling the wave phenomena in acoustic and elastic media with sharp variations of physical properties using the grid‐characteristic method

“…The three most common types of modelling methods in use in geophysics are direct methods, integral‐equation methods, and asymptotic methods (Carcione, Herman and ten Kroode ). Over the last decade significant progress has been made in all these methods (Etgen and O'Brien ; Wang and Liu ; Hestholm ; Tong, Yang and Hua ; Di Bartolo, Dors and Mansur ; Hobro, Chapman and Robertsson ; Sanyi et al . ).…”