Finite‐difference modeling with adaptive variable length spatial operators

2010 ◽  
Author(s):  
Yang Liu ◽  
Mrinal K. Sen
Keyword(s):  
Finite Difference ◽  
Variable Length ◽  
Spatial Operators

Finite-difference modeling with adaptive variable-length spatial operators

Geophysics ◽  
2011 ◽  
Vol 76 (4) ◽  
pp. T79-T89 ◽  
Author(s):  
Yang Liu ◽  
Mrinal K. Sen
Keyword(s):  
Finite Difference ◽  
Dispersion Analysis ◽  
Variable Length ◽  
Finite Difference Schemes ◽  
Low Velocity ◽  
Absorbing Boundary ◽  
Local Grid ◽  
Spatial Derivatives ◽  
Spatial Operators ◽  
And Inversion

Most finite-difference simulation algorithms use fixed-length spatial operators to compute spatial derivatives. The choice of length is dictated by computing cost, stability, and dispersion criteria that are satisfied globally. We propose finite-difference schemes with adaptive variable-length spatial operators to decrease computing costs significantly without reducing accuracy. These schemes adopt long operators in regions of low velocity and short operators in regions of high velocity. Two methods automatically determine variable operator lengths. Dispersion analysis, along with 1D and 2D modeling, demonstrates the validity and efficiency of our schemes. In addition, a hybrid absorbing boundary condition helps reduce unwanted reflections from model boundaries. Our scheme is more efficient than those based on variable-grid methods for modeling, migration, and inversion of models with complex velocity structures because the latter require local grid refinement, which usually increases memory requirements and computing costs.

Acoustic and elastic finite-difference modeling by optimal variable-length spatial operators

Geophysics ◽  
2020 ◽  
Vol 85 (2) ◽  
pp. T57-T70 ◽  
Author(s):  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Finite Differences ◽  
Spatial Dispersion ◽  
Finite Difference Methods ◽  
High Order ◽  
Variable Length ◽  
Time Step ◽  
Heterogeneous Model ◽  
Fixed Length ◽  
Temporal Dispersion

Time-space domain finite-difference modeling has always had the problem of spatial and temporal dispersion. High-order finite-difference methods are commonly used to suppress spatial dispersion. Recently developed time-dispersion transforms can effectively eliminate temporal dispersion from seismograms produced by the conventional modeling of high-order spatial and second-order temporal finite differences. To improve the efficiency of the conventional modeling, I have developed optimal variable-length spatial finite differences to efficiently compute spatial derivatives involved in acoustic and elastic wave equations. First, considering that temporal dispersion can be removed, I prove that minimizing the relative error of the phase velocity can be approximately implemented by minimizing that of the spatial dispersion. Considering that the latter minimization depends on the wavelength that is dependent on the velocity, in this sense, this minimization is indirectly related to the velocity, and thus leads to variation of the spatial finite-difference operator with velocity for a heterogeneous model. Second, I use the Remez exchange algorithm to obtain finite-difference coefficients with the lowest spatial dispersion error over the largest possible wavenumber range. Then, dispersion analysis indicates the validity of the approximation and the algorithm. Finally, I use modeling examples to determine that the optimal variable-length spatial finite difference can greatly increase the modeling efficiency, compared to the conventional fixed-length one. Stability analysis and modeling experiments also indicate that the variable-length finite difference can adopt a larger time step to perform stable modeling than the fixed-length one for inhomogeneous models.

Forward modeling with adaptive variable-length spatial operators in pseudodepth domain

2014 ◽  
Author(s):  
Li Qingyang* ◽  
Huang Jianping ◽  
Li Zhenchun ◽  
Zhang Lin ◽  
Li Na
Keyword(s):  
Forward Modeling ◽  
Variable Length ◽  
Spatial Operators

scholarly journals Exact extrapolation and immersive modelling with finite-difference injection

2020 ◽  
Vol 223 (1) ◽  
pp. 584-598
Author(s):  
Dirk-Jan van Manen ◽  
Xun Li ◽  
Marlies Vasmel ◽  
Filippo Broggini ◽  
Johan Robertsson
Keyword(s):  
Finite Difference ◽  
Arbitrary Order ◽  
Extrapolation Methods ◽  
Machine Precision ◽  
Injection Method ◽  
Efficient Calculation ◽  
Long Range Interactions ◽  
Equivalent Sources ◽  
Spatial Operators ◽  
Wavefield Extrapolation

SUMMARY In numerical modelling of wave propagation, the finite-difference (FD) injection method enables the re-introduction of simulated wavefields in model subdomains with machine precision, enabling the efficient calculation of waveforms after localized model alterations. By rewriting the FD-injection method in terms of sets of equivalent sources, we show how the same principles can be applied to achieve on-the-fly wavefield extrapolation using Kirchhoff–Helmholtz (KH)-like integrals. The resulting extrapolation methods are numerically exact when used in conjunction with FD-computed Green’s functions. Since FD injection only relies on the linearity of the wave equation and compactness of FD stencils in space, the methods can be applied to both staggered and non-staggered discretizations with arbitrary-order spatial operators. Examples for both types of discretizations show how these extrapolators can be used to truncate models with exact absorbing or immersive boundary conditions. Such immersive modelling involves the evaluation of KH-type extrapolation and representation integrals in the same simulation, which include the long-range interactions missing from conventional FD injection.

Determination of finite difference coefficients for the acoustic wave equation using regularized least-squares inversion

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Yanfei Wang ◽  
Wenquan Liang ◽  
Zuhair Nashed ◽  
Changchun Yang
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Least Squares ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Spatial Domain ◽  
Variable Length ◽  
Staggered Grid ◽  
Regular Grid ◽  
Low Velocity

AbstractFinite difference (FD) solutions of wave equations have been proven useful in exploration seismology. To yield reliable and interpretable results, the numerically induced error should be minimized over a range of frequencies and angles of propagation. Grid dispersion is one of the key numerical problems and there exist some methods to solve this problem in the literature. Traditionally, the spatial FD operator coefficients are only determined in the spatial domain; however, the wave equation is solved in the temporal and spatial domain simultaneously. Recently, some methods based on the joint temporal-spatial domains have been proposed to address this problem. Variable length coefficients methods are proposed in the literature to improve efficiency while preserving accuracy by using longer operators in the low velocity regions and shorter operators in the high velocity regions. To cope with the ill-conditioning of the linear system induced by long stencil FD operators, we study in this paper a regularizing simplified least-squares model to minimize the phase velocity error in the joint temporal-spatial domain with a variable length of coefficients. Different from our previous study, we determine FD coefficients on the regular grid instead of on the staggered grid. Though the regular grid FD methods are less precise, however, with a little increase of the operator length, the precision can be improved. Stability of the numerical solutions is enhanced by the regularization. Numerical simulations made on one-dimensional to three-dimensional examples show that our scheme needs shorter operators and preserves accuracy compared with the previous methods.

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