Chen, Hanming (China University of Petroleum, Beijing) | Zhou, Hui (China University of Petroleum, Beijing) | Zhang, Qingchen (China University of Petroleum, Beijing) | Zhang, Qi (China University of Petroleum, Beijing)
Two staggered-grid finite-difference (SGFD) methods with fourth- and sixth-order accuracy in time have been developed recently based on two new SGFD stencils. The SGFD coefficients are determined by Taylor-series expansion (TE), which is accurate only nearby zero wavenumber. We adopt the new two SGFD stencils and optimize the SGFD coefficients by minimizing the errors between the wavenumber responses of the SGFD operators and the first-order k (wavenumber)-space operator in a least-squares (LS) sense. We solve the LS problems by performing weighted pseudo-inverse of nonsquare matrices to obtain the SGFD coefficients, and to yield two LS based SGFD methods. Dispersion analysis and numerical examples demonstrate that our LS based SGFD methods can preserve the original fourth- and sixth-order temporal accuracy and achieve higher spatial accuracy than the existing TE based time-space domain SGFD methods.
The staggered-grid finite-difference (SGFD) (Virieux, 1984) method has been widely used in seismic wave propagation modeling. Most of the SGFD applications adopt the traditional (2M, 2) scheme, which uses 2M-order Taylorseries expansion (TE) based FD operator to discretize spatial derivatives, and 2nd-order TE based FD operator to discretize temporal derivative. Although high-order spatial accuracy can be achieved by using a long stencil length, the temporal accuracy is only second-order. When a coarse time step is used, the traditional scheme suffers from obvious temporal dispersion during long time wave propagation.
Recently, Tan and Huang (2014a) propose two new SGFD methods with fourth-order and sixth-order accuracy in time respectively by incorporating a few of off-axial grid points into the standard SGFD stencil. The two methods are denoted as (2M, 4) and (2M, 6). The FD coefficients are determined in the time-space domain using TE approach. Althouth high-order temporal accuracy has been achieved, the TE based (2M, 4) and (2M, 6) methods still suffer from obvious spatial disperion when a large grid size or a short stencil length is adopted. Tan and Huang (2014b) continue to improve the spatial accuracy by using a nonlinear optimization to seek the optimal FD coefficients. However, the optimization requires repeated iterations, and the procedure may be time-consuming.
In this paper, the Lattice Spring Model (LSM) is adopted in forward modeling of elastic waves propagation in solid medium by combination with the Verlet Algorithm. Different from the traditional methods, such as Finite Difference Method (FDM), Finite Element Method (FEM) etc., LSM is a new method which is not based on the wave equations, but on the microcosmic mechanism that causes wave propagation. Firstly, the origin and history of LSM is introduced. Secondly, the theoretical framework of LSM is elaborated and a stability condition for the evolution of this system is deduced. Then, some numerical results of LSM are demonstrated and they are compared with the wave fields obtained by FDM. Finally, a brief conclusion is drawn based on the previous discussions.
First devised by Grest and Webman in 1984, Lattice Spring Model (LSM) is a collection of linear springs connected at nodes distributing on a cubic lattice used for describing solid medium (Grest and Webman, 1984; Hassold and Srolovitz, 1989). In order to model materials of different Poisson’s ratios, angular springs are added to the original linear spring system (Wang, 1989). Ladd and Kinney (1997) developed this model by taking the idea of elastic element to improve its calculation precision. Such a simple model is sufficient to simulate heterogeneous elastic medium, and its application can be seen in modeling deformation and failure (Ladd and Kinney, 1997; Buxton et al., 2001; Zhao et al., 2011).
As is known to all, extensive research has been performed to solve the dynamic problems involving waves, and FDM is the most frequently used numerical method, which solves the wave equation by finite difference approximation of its partial derivative (Toomey and Bean, 2000). Yim and Sohn (2000) adopted a model similar to LSM for visualization of ultrasonic waves, but the evolution of wave fields are calculated by FDM. Pazdniakou and Adler (2012) made a further introduction of LSM and laid the foundation for its potential application in wave propagation in porous media in the low frequency band. Xia et al. (2014) modeled P waves from low frequencies (seismic frequency) to high frequencies (sonic log frequency) by importing a stability conditional for LSM dynamics.
Zhang, Qingchen (China University of Petroleum) | Zhou, Hui (China University of Petroleum) | Wang, Jie (SINOPEC Geophysical Research Institute) | Zuo, Anxin (China University of Petroleum) | Xia, Muming (China University of Petroleum)
Due to the gradient calculation requiring cross-correlation of the forward wavefields and back-propagated residual wavefields at each time step, the great storage amount becomes an obstacle of practical application of full-waveform inversion, especially in three-dimensional elastic case in time domain. In this paper we extend the efficient boundary storage to the time domain three-dimensional elastic full-waveform inversion on multi-GPU. Based on the efficient boundary storage strategy, the storage amount can be reduced dramatically. As a result, we can save the partial forward wavefields directly on the GPU memory and reconstruct the full forward wavefields synchronized with back-propagated residual wavefields along the reverse time direction. This strategy avoids frequent CPU-to-GPU or GPU-to-CPU memory copy (extremely time-consuming) at the cost of the recomputation (little time-consuming) of the forward wavefields. Our forward simulation tests show that the GPU’s supercomputing effect can be fully exploited with this strategy. In addition, we perform a three-parameter simultaneous inversion of P-, S-wave velocities and density. The favorable inversion results verify that our algorithm is feasible and efficient.