Staggered-grid Finite-difference Operators Research Articles

Optimised staggered grid finite difference operators based on the combined window function

ABSTRACT The staggered grid finite difference (SGFD) method has been widely utilised in seismic forward modelling. The accuracy of forward modelling results has a great influence on the results of subsequent seismic imaging and seismic inversion. Numerical dispersion is one of the inherent problems in the SGFD method. To suppress dispersion, we here introduce the truncated window function method, which can get optimised finite difference operators after truncating spatial convolution series of the pseudo-spectral method. The empirical criteria of window functions are the main lobe width that should be as narrow as possible and the side lobe attenuation should be as large as possible. According to these criteria, we propose the combined window function method based on the Gaussian window and Hanning window. The result of the amplitude spectrum response shows that the new combined window has narrower main lobe and greater side lobe attenuation, which results in higher accuracy in numerical simulations with SGFD operators. Numerical tests also illustrate the effectiveness of this new method.

High-order temporal and implicit spatial staggered-grid finite-difference operators for modelling seismic wave propagation

High-order temporal and implicit spatial staggered-grid finite-difference operators for modelling seismic wave propagation

A new staggered grid finite difference scheme optimised in the space domain for the first order acoustic wave equation

Staggered grid finite difference (FD) methods are widely used to synthesise seismograms theoretically, and are also the basis of reverse time migration and full waveform inversion. Grid dispersion is one of the key problems for FD methods. It is desirable to have a FD scheme which can accelerate wave equation simulation while still preserving high accuracy. In this paper, we propose a totally new staggered grid FD scheme which uses different staggered grid FD operators for different first order spatial derivatives in the first order acoustic wave equation. We determine the FD coefficient in the space domain with the least-squares method. The dispersion analysis and numerical simulation demonstrated the effectiveness of the proposed method.

SBP–SAT finite difference discretization of acoustic wave equations on staggered block-wise uniform grids

We consider the numerical simulation of the acoustic wave equations arising from seismic applications, for which staggered grid finite difference methods are popular choices due to their simplicity and efficiency. We relax the uniform grid restriction on finite difference methods and allow the grids to be block-wise uniform with nonconforming interfaces. In doing so, variations in the wave speeds of the subterranean media can be accounted for more efficiently. Staggered grid finite difference operators satisfying the summation-by-parts (SBP) property are devised to approximate the spatial derivatives appearing in the acoustic wave equation. These operators are applied within each block independently. The coupling between blocks is achieved through simultaneous approximation terms (SATs), which impose the interface conditions weakly, i.e., by penalty. Ratio of the grid spacing of neighboring blocks is allowed to be rational number, for which specially designed interpolation formulas are presented. These interpolation formulas constitute key pieces of the simultaneous approximation terms. The overall discretization is shown to be energy-conserving and examined on test cases of both theoretical and practical interests, delivering accurate and stable simulation results.

Optimized staggered-grid finite-difference operators using window functions

The staggered-grid finite-difference (SGFD) method has been widely used in seismic forward modeling. The precision of the forward modeling results directly affects the results of the subsequent seismic inversion and migration. Numerical dispersion is one of the problems in this method. The window function method can reduce dispersion by replacing the finite-difference operators with window operators, obtained by truncating the spatial convolution series of the pseudospectral method. Although the window operators have high precision in the low-wavenumber domain, their precision decreases rapidly in the high-wavenumber domain. We develop a least squares optimization method to enhance the precision of operators obtained by the window function method. We transform the SGFD problem into a least squares problem and find the best solution iteratively. The window operator is chosen as the initial value and the optimized domain is set by the error threshold. The conjugate gradient method is also adopted to increase the stability of the solution. Approximation error analysis and numerical simulation results suggest that the proposed method increases the precision of the window function operators and decreases the numerical dispersion.

A staggered-grid convolutional differentiator for elastic wave modelling

The computation of derivatives in governing partial differential equations is one of the most investigated subjects in the numerical simulation of physical wave propagation. An analytical staggered-grid convolutional differentiator (CD) for first-order velocity-stress elastic wave equations is derived in this paper by inverse Fourier transformation of the band-limited spectrum of a first derivative operator. A taper window function is used to truncate the infinite staggered-grid CD stencil. The truncated CD operator is almost as accurate as the analytical solution, and as efficient as the finite-difference (FD) method. The selection of window functions will influence the accuracy of the CD operator in wave simulation. We search for the optimal Gaussian windows for different order CDs by minimizing the spectral error of the derivative and comparing the windows with the normal Hanning window function for tapering the CD operators. It is found that the optimal Gaussian window appears to be similar to the Hanning window function for tapering the same CD operator. We investigate the accuracy of the windowed CD operator and the staggered-grid FD method with different orders. Compared to the conventional staggered-grid FD method, a short staggered-grid CD operator achieves an accuracy equivalent to that of a long FD operator, with lower computational costs. For example, an 8th order staggered-grid CD operator can achieve the same accuracy of a 16th order staggered-grid FD algorithm but with half of the computational resources and time required. Numerical examples from a homogeneous model and a crustal waveguide model are used to illustrate the superiority of the CD operators over the conventional staggered-grid FD operators for the simulation of wave propagations.

Acoustic Reverse-time Migration using Optimal Staggered-grid Finite-difference Operator Based on Least Squares

Reverse-time migration (RTM) directly solves the two-way wave equation for wavefield propagation; therefore, how to solve the wave equation accurately and quickly is very important for RTM. The conventional staggered-grid finite-difference (SFD) operators are usually based on the Taylor-series expansion theory. If they are used to solve wave equation on a larger frequency content, a strong dispersion will occur, which directly affects the seismic image quality. In this paper, we propose an optimal SFD operator based on least squares to solve acoustic wave equation for prestack RTM, and obtain a new antidispersion RTM algorithm that can use short spatial difference operators. The synthetic and real data tests demonstrate that the least squares SFD (LSSFD) operator can mitigate the numerical dispersion, and the acoustic RTM using the LSSFD operator can effectively improve image quality comparing with that using the Taylor-series expansion SFD (TESFD) operator. Moreover, the LSSFD method can adopt a shorter spatial difference operator to reduce the computing cost.

Seismic modeling by optimizing regularized staggered-grid finite-difference operators using a time-space-domain dispersion-relationship-preserving method

The staggered-grid finite-difference (FD) method is widely used in numerical simulation of the wave equation. With stability conditions, grid dispersion often exists because of the discretization of the time and the spatial derivatives in the wave equation. Therefore, suppressing grid dispersion is a key problem for the staggered-grid FD schemes. To reduce the grid dispersion, the traditional method uses high-order staggered-grid schemes in the space domain. However, the wave is propagated in the time and space domain simultaneously. Therefore, some researchers proposed to derive staggered-grid FD schemes based on the time-space domain dispersion relationship. However, such methods were restricted to low frequencies and special angles of propagation. We have developed a regularizing technique to tackle the ill-conditioned property of the symmetric linear system and to stably provide approximate solutions of the FD coefficients for acoustic-wave equations. Dispersion analysis and seismic numerical simulations determined that the proposed method satisfies the dispersion relationship over a much wider range of frequencies and angles of propagation and can ensure FD coefficients being solved via a well-posed linear system and hence improve the forward modeling precision.

Finite-Difference Simulation of Elastic Wave with Separation in Pure P- and S-Modes

Elastic wave equation simulation offers a way to study the wave propagation when creating seismic data. We implement an equivalent dual elastic wave separation equation to simulate the velocity, pressure, divergence, and curl fields in pure P- and S-modes, and apply it in full elastic wave numerical simulation. We give the complete derivations of explicit high-order staggered-grid finite-difference operators, stability condition, dispersion relation, and perfectly matched layer (PML) absorbing boundary condition, and present the resulting discretized formulas for the proposed elastic wave equation. The final numerical results of pure P- and S-modes are completely separated. Storage and computing time requirements are strongly reduced compared to the previous works. Numerical testing is used further to demonstrate the performance of the presented method.