Dinghui Yang

A Nearly Analytic Discrete Method for Acoustic and Elastic Wave Equations in Anisotropic Media

We transform the seismic wave equations in 2D inhomogeneous aniso- tropic media into a system of first-order partial differential equations with respect to time t. Based on the transformed equations, a new nearly analytic discrete method (NADM) is developed in this article. Our method enables wave propagation to be simulated in two dimensions through generally anisotropic and heterogeneous mod- els. The space derivatives are calculated by using an interpolation approximation, while the time derivatives are replaced by a truncated Taylor expansion. Our analyses show that the error of the NADM is less than that of the conventional finite-difference method (FDM) and is about 1/60 to 1/100 of that of the FDM. We also demonstrate numerically that the stability of the NADM is higher than that of the FDM. The three- component seismic wave fields in two layered isotropic and transversely isotropic media (TIM) are simulated and compared with the conventional FDM. Again, we show from the three-component seismic wave fields that the NADM has higher ac- curacy, stronger stability, and less numerical dispersion, effectively suppressing the source noises as compared with the FDM.

Optimal Nearly Analytic Discrete Approximation to the Scalar Wave Equation

Recently, we proposed the so-called optimal nearly analytic discrete method (onadm) for computing synthetic seismograms in acoustic and elastic wave problems (Yang et al 2004). In this article, we explore the theoretical properties of the onadm including the stability criteria of the onadm for solving 1D and 2D scalar wave equations, numerical dispersion, theoretical error, and computational efficiency when using the onadm to model the acoustic wave fields. For comparison in the 1D case, we also discuss numerical dispersions and stability criteria of the so- called Lax–Wendroff schemes with accuracy of O (Δ t 4 , Δ x 8 ) and O (Δ t 4 , Δ x 10 ) and the pseudospectral method (psm). We then apply the onadm to the heterogeneous case in synthetic seismograms. Promising numerical results illustrate that the onadm provides a useful tool for large-scale heterogeneous practical problems because it can effectively suppress numerical dispersions caused by discretizing the wave equations when too-coarse grids are used. Numerical modeling also indicates that simultaneously using both the wave displacement and its gradients to approximate the high-order derivatives is important for decreasing the numerical dispersion and source-generated noise caused by the discretization of wave equations because wave- displacement gradients include important seismic information.

An Optimal Nearly Analytic Discrete Method for 2D Acoustic and Elastic Wave Equations

In this article, we present the so-called optimal, nearly analytic, discrete method (ONADM), which is an improved version of the NADM proposed recently (Yang et al. , 2003a). We compare numerically the error of the ONADM with those of the NADM and other finite-difference methods for 1D and 2D cases, and give wavefield modeling in 2D isotropic media. We also discuss the validity of the n -times absorbing boundary condition, when absorbing boundary conditions are incorporated in the ONADM. We show that, compared with the original NADM, the ONADM for the 2D case can significantly reduce storage space and computational cost. The temporal accuracy of the optimal method is also increased from second order in the original NADM to fourth order, and spatial accuracy remains the same as that of the original. Promising numerical results suggest that the ONADM is suitable for large-scale numerical modeling, as it can suppress effectively numerical dispersion caused by discretizing the wave equations when too coarse grids are used. Manuscript received 30 July 2003.

Poroelastic wave equation including the Biot/squirt mechanism and the solid/fluid coupling anisotropy

There is relative motion and inertial coupling between solids and fluids during seismic and acoustic propagation in rocks with fluids. This inertial coupling is anisotropic because of the microvelocity anisotropy of the fluid relative to solid in an anisotropic medium. The Biot mechanism and the squirt-flow mechanism are the two most important mechanisms of solid/fluid interaction in rocks. We extend the Biot/squirt (BISQ) theory to include the solid/fluid coupling anisotropy and develop a general poroelastic wave equation including both mechanisms simultaneously. The new model estimates velocity/frequency dispersion and attenuation of waves propagating in the 2D PTL (periodic thin layers) + EDA (extensive dilatancy anisotropy) medium with fluids. The attenuation and dispersion of the two quasi-P-waves and the quasi-SV-wave, which are related to the solid/fluid coupling density and the permeability tensors, are anisotropic. The anisotropy are simultaneously affected by the anisotropies of the solid skeleton, the permeability, and the solid/fluid coupling effect of the formation. Numerical modeling suggests that variations of attenuation of both the fast quasi-P-wave and the quasi-SV-wave strongly depend on the permeability anisotropy. In the low-frequency range, the maximum attenuation is in the direction of the maximum permeability for the fast quasi-P-wave and in the direction of the minimum permeability for the quasi-SV-wave. The attenuation behaviors of the two waves in the high-frequency range, however, are opposite to those in the low-frequency range. This paper also presents numerically how the attenuation and velocity dispersion of both the fast quasi-P-wave and the quasi-SV-wave are influenced by the anisotropic solid/fluid coupling density. The model results demonstrate when the wave propagation is perpendicular to the direction of maximum solid/fluid coupling density, the wave motion exhibits maximum attenuation (Q −1 ) and maximum velocity dispersion for the fast quasi-P-wave, and minimum attenuation (Q −1 ) and minimum velocity dispersion for the quasi-SV-wave. These phenomena may be applied in extracting anisotropic permeability and further determining the preferential directions of fluid flow in a reservoir containing fluid-filled cracks from attenuation and dispersion data derived from sonic logs and crosswell seismics.

Optimally Accurate Nearly Analytic Discrete Scheme for Wave-Field Simulation in 3D Anisotropic Media

We present a new numerical method for elastic wave modeling in 3D isotropic and anisotropic media, which is called the 3D optimal nearly analytic dis- crete method (ONADM) in this article. This work is an extension of the 2D ONADM (Yang et al., 2004) that models acoustic and elastic waves propagating in 2D media. The formulation is derived by using a multivariable truncated Taylor series expansion and high-order interpolation approximations. Our 3D ONADM enables wave propa- gation to be simulated in three dimensions through isotropic and anisotropic models. Promising numerical results show that the error of the ONADM for the 3D case is less than those of the conventional finite-difference (FD) method and the so-called Lax-Wendroff correction (LWC) schemes, measured quantitatively by the root- mean-square deviation from analytical solution. The seismic wave fields in the 3D isotropic and anisotropic media are simulated and compared with those obtained by using the fourth-order LWC method and exact solutions for the acoustic wave case. We show that, compared with the conventional FD method and the LWC schemes, the ONADM for the 3D case can reduce significantly the storage space and compu- tational costs. Numerical experiments illustrate that the ONADM provides a useful tool for the 3D large-scale isotropic and anisotropic problems and it can suppress effectively numerical dispersions caused by discretizing the 3D wave equations when too-coarse grids are used, which is the same as the 2D ONADM. Numerical modeling also implies that simultaneously using both the wave displacement and its gradients to approximate the high-order derivatives is important for both decreasing the nu- merical dispersion caused by the discretization of wave equations and compensating the important wave-field information included in wave-displacement gradients.

n-Times Absorbing Boundary Conditions for Compact Finite-Difference Modeling of Acoustic and Elastic Wave Propagation in the 2D TI Medium

This article presents decoupling n-times absorbing boundary conditions designed to model acoustic and elastic wave propagation in a 2D transversely iso- tropic (TI) medium. More general n-times boundary conditions with absorbing pa- rameters are also obtained by cascading first-order differential operators with param- eters. These boundary conditions are approximated with simple finite-difference schemes for numerical simulations. The numerical results show that the absorbing for the reflection waves strengthens with increasing the absorbing times n and the discretization boundary formulas are stable. Specially, the n-times absorbing bound- ary condition with absorbing parameters is better than that without the absorbing parameters under the case of same absorbing order. Elastic wave fields and three- component synthetic seismograms, generated by using the compact finite-difference and the decoupling n-times absorbing boundary, also illustrate that the n-times ab- sorbing boundary condition can eliminate effectively the spurious numerical reflec- tions in the acoustic and elastic wave modeling for the TI medium case.

An improved nearly analytical discrete method: an efficient tool to simulate the seismic response of 2-D porous structures

The nearly analytic discrete method (NADM) for acoustic and elastic waves in porous elastic media is a perturbation method recently proposed by Yang et al (2006a Commun. Comput. Phys. 1 528–47). This method uses the truncated Taylor series expansion to approximate the time derivatives and the local high-order interpolation to approximate the spatial high-order derivatives by simultaneously using the displacements and its gradients and the velocity. As a result, it can suppress effectively numerical dispersions caused by the discretizing the wave equations when too-coarse grids are used. In this paper, we present an improved nearly-analytic discrete method (INADM) for the porous case. We compare numerically the error of the INADM with those of the original NADM and the so-called Lax–Wendroff correction (LWC) schemes for 1-D and 2-D cases, and give the wave-field modelling in 2-D porous isotropic and anisotropic media. We show that, compared with the original NADM, the INADM for the 2-D case can reduce significantly the storage space and increase time accuracy, while the space accuracy remains the same as that of the original one. Numerical experiments show that the error of the INADM for the porous case is less than those of the NADM and the fourth-order LWC scheme. The three-component seismic wave-fields in the 2-D porous isotropic medium are compared with those obtained by using the NADM, the LWC method, and exact solutions. Several characteristics of waves propagating in porous anisotropic media, computed by the INADM, are also reported in this study. Promising numerical results illustrate that the INADM provides a useful tool for large-scale porous problems and it can effectively suppress numerical dispersions.

A High‐Order Stereo‐Modeling Method for Solving Wave Equations

Abstract In this paper, we propose a high‐order stereo‐modeling method (STEM) to approximate the high‐order spatial derivatives included in the wave equations using simultaneously wave‐field displacements and their gradients and propose a two‐step method for time marching, which is called the two‐step STEM in brief. The two‐step STEM has a higher‐order accuracy in space than conventional finite‐difference (FD) methods when the same number of spatial grid points in a wavelength is used. For example, the stereo‐modeling method that uses five points in one spatial direction can achieve an eighth‐order accuracy in space, whereas other FD methods such as conventional FD methods, Lax–Wendroff correction (LWC) methods, and other methods only have a fourth‐order accuracy. Theoretical properties of the two‐step STEM including stability and errors are analyzed for 1D and 2D cases. The numerical dispersion relationship provided by the two‐step STEM for 1D and 2D cases are also investigated in this study. Meanwhile, we present numerical results computed by the two‐step STEM and compare with the eighth‐order LWC method, the eighth‐order staggered‐grid FD method, and the fourth‐order staggered‐grid method. Numerical results show that the two‐step STEM can effectively suppress numerical dispersion caused by discretizing the wave equations when coarse spatial grids are used or models have strong velocity contrasts between adjacent layers. In contrast to other high‐order FD methods such as the eighth‐order LWC, the eighth‐order staggered‐grid FD, and the fourth‐order staggered‐grid method, the new method has substantially less computational time and requires less memory because large spatial and time increments can be used. Thus, the two‐step STEM can be potentially used to solve large‐scale wave‐propagation problems and seismic tomography.

An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations

Abstract A new explicit differentiator series method based on the implicit Runge–Kutta method, called the IRK-DSM in brief, is developed for solving wave equations. To develop the new algorithm, we first transform the wave equation, usually described as a partial differential equation (PDE), into a system of first-order ordinary differential equations (ODEs) with respect to time t . Then we use a truncated differentiator series method of the implicit Runge–Kutta method to solve the semidiscrete ordinary differential equations, while the high-order spatial derivatives included in the ODEs are approximated by the local interpolation method. We analyze the theoretical properties of the IRK-DSM, including the stability criteria for solving the 1D and 2D acoustic-wave equations, numerical dispersion, discretizing error, and computational efficiency when using the IRK-DSM to model acoustic-wave fields. For comparison, we also present the stability criteria and numerical dispersion of the so-called Lax–Wendroff correction (LWC) methods with the fourth-order and eighth-order accuracies for the 1D case. Promising numerical results show that the IRK-DSM provides a useful tool for large-scale practical problems because it can effectively suppress numerical dispersions and source-noises caused by discretizing the acoustic- and elastic-wave equations when too-coarse grids are used or the models have a large velocity contrast between adjacent layers. Theoretical analysis and numerical modeling also demonstrate that the IRK-DSM, through combining both the implicit Runge–Kutta scheme with good stability condition and the approximate differentiator series method, is a robust wave-field modeling method.

Effects of the Biot and the squirt-flow coupling interaction on anisotropic elastic waves

Considering the velocity anisotropy of the solid/fluid relative motion and employment of the BISQ theory[1] based on the one-dimensional porous isotropic case, we establish a two-phase anisotropic elastic wave equation to simultaneously include the Biot and the squirt mechanisms in terms of both the basic principles of the fluid’s mass conservation and the elastic-wave dynamical equations in the two-phase anisotropic rock. Numerical results, while the Biot-flow and the squirt-flow effects are simultaneously considered in the transversely isotropic (TI) poroelastic medium, show that the attenuation of the quasi P-wave and the quasi SV-wave strongly depend on the permeability anisotropy, and the attenuation behavior at low and high frequencies is contrary. Meanwhile, the attenuation and dispersion of the quasi P-wave are also affected seriously by the anisotropic solid/fluid coupling additional density.