nlei Chu

3D Elastic Wave Modeling Using Modified High-Order Time Stepping Schemes with Improved Stability Conditions

Summary We present two Lax-Wendroff type high-order time stepping schemes and apply them to solving the 3D elastic wave equation. The proposed schemes have the same format as the Taylor series expansion based schemes, only with modified temporal extrapolation coefficients. We demonstrate by both theoretical analysis and numerical examples that the modified schemes significantly improve the stability conditions.

Acoustic Anisotropic Wave Modeling Using Normalized Pseudo-Laplacian

With a simple modification to the pseudo-Laplacian operator, we derive the normalized pseudo-Laplacian operator which can be used to extend the pseudo-analytical method to solve a broader range of problems than pseudo-Laplacian. As a demonstration example, we apply the normalized pseudoLaplacian operator to solving acoustic anisotropic wave equations in this work. Like pseudo-Laplacian, normalized pseudoLaplacian effectively compensates for the second-order time stepping errors which enables the pseudo-analytical method to produce more accurate results than the pseudospectral method.

Frequency domain modeling using implicit spatial finite difference operators

SUMMARY 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.

An Accurate And Stable Wave Equation For Pure Acoustic TTI Modeling

One fundamental shortcoming of the conventional pseudoacoustic approximation is that it only prevents shear wave propagation along the symmetry axis of anisotropy and not in other directions. This problem leads to the presence of unwanted shear waves in P-wave simulation results and brings artifacts into P-wave RTM images. More significantly, the pseudo-acoustic wave equations become unstable for anisotropy parameters e < δ and for heterogeneous models with highly varying dip and azimuth angles in tilted transversely isotropic (TTI) media. Pure acoustic anisotropic wave equations completely de-couple the P-wave response from the elastic wavefield and naturally solve all the above-mentioned problems. In this work, we propose a new pure acoustic TTI wave equation and compare it with the conventional coupled pseudo-acoustic wave equation. We derive finite difference solutions to this new equation and use numerical examples to demonstrate that it produces highly accurate P-wave results, very close to results produced by coupled pseudo-acoustic wave equations, but completely free from shear wave artifacts.

Compensating for time stepping errors locally in the pseudo‐analytical method using normalized pseudo‐Laplacian

The pseudo-analytical method relies on pseudo-Laplacians to compensate for time stepping errors caused by the secondorder time stepping scheme. Pseudo-Laplacian slowly varies with the compensation velocity which makes it well suited for models with mild velocity variations. For models with high velocity variations, the pseudo-analytical method becomes difficult because high compensation velocities cause over-compensations to wavefields in low velocity areas which can bring significant artifacts into the simulation results. To tackle this problem, I propose to use spatially varying normalized pseudo-Laplacians, which are determined by actual velocity variations in space, to locally compensate for time stepping errors. This new implementation of the pseudoanalytical method involves two steps. The first step applies local compensations using adaptive normalized pseudoLaplacians, computed either in wavenumber domain or in space domain. The second step carries out the second-order time marching computations, which can be realized by any numerical schemes and not limited to the wavenumber domain method. I use numerical experiments to demonstrate that the proposed method can produce highly accurate results with relaxed stability conditions compared to the conventional pseudospectral method.

High-Order Rotated Staggered Finite Difference Modeling of 3D Elastic Wave Propagation In General Anisotropic Media

Summary We analyze the dispersion properties and stability conditions of the high-order convolutional finite difference operators and compare them with the conventional finite difference schemes. We observe that the convolutional finite difference method has better dispersion properties and becomes more efficient than the conventional finite difference method with the increasing order of accuracy. This makes the high-order convolutional operator a good choice for anisotropic elastic wave simulations on rotated staggered grids since its enhanced dispersion properties can help to suppress the numerical dispersion error that is inherent in the rotated staggered grid structure and its efficiency can help us tackle 3D problems cost-effectively.

Derivation And Numerical Analysis of Implicit Time Stepping Schemes

We demonstrate that implicit time integration methods for second-order-in-time wave equations can be derived from rational expansions to the cosine function of pseudo-differential operators. Using the scalar wave equation as an example, we give complete stability condition and grid dispersion analysis results for general implicit time integration methods. Furthermore, we propose an optimization method to develop unconditionally stable implicit time stepping schemes.

A pseudospectral‐finite difference hybrid approach for large‐scale seismic modeling and RTM on parallel computers

We present numerical comparisons between the finite difference method and the Fourier pseudospectral method on the 3D SEG/EAGE salt model to investigate the potentials of the two methods for large scale seismic modeling and reverse-time migration problems on distributed-memory parallel systems. We compare the accuracy of the seismic modeling results by the two methods. We discuss the parallel implementations and present the runtime and speedup rate data on a Linux cluster. Based on these comparison results, we propose a pseudospectral-finite difference hybrid method for largescale seismic modeling and RTM problems on massively distributed-memory parallel systems.

Parallel Seismic Modeling Using the Pseudospectral Method on Marmousi2

We present parallel pseudospectral seismic modeling results for the elastic, variable-density acoustic and constant-density acoustic wave equations on the Marmousi2 model. We focus on the parallel implementation on distributed-memory systems to explore the potentials of the pseudospectral method for large scale seismic simulations. The numerical results demonstrate superior accuracy of the pseudospectral method and good scalability on a Linux cluster, which suggest the promising extensions of the current work to solving 3D seismic modeling and imaging problems on supercomputers.