RBF-FD analysis of 2D time-domain acoustic wave propagation in heterogeneous media
Introduction
Numerical modeling is a widely used approach for computational simulation of geological processes. Numerical approximation of acoustic wave equation in complex velocity media is vital to a wide range of investigations in geophysics seismic modeling, reverse-time migration, seismic inversion, etc. To simulate the acoustic waves in a complex representation of the Earth's subsurface, time-domain wave equation is often solved approximately, using mesh or grids to discretize the domain of interest. Over the years, a wide range of numerical methods have been proposed and applied for acoustic wave simulations in geoscience, including Finite Difference Method (Alford et al., (1974); Kelly et al., (1976); Tarantola (1984); Dablain (1986); Williamson and Pratt (1995); Jo et al., (1996); Carcione et al., (2002); Geiger and Daley (2003); Du and Bancroft (2004); Liu and Sen (2011); Virieux et al., (2012); Wang et al., (2016, 2018, 2019); Cai et al., (2018)), Finite Element Method (Marfurt (1984); Emmerich and Korn (1987); De Basabe and Sen (2007); Ham and Bathe (2012)), Spectral Element Method (Seriani and Priolo (1994); Seriani and Oliveira (2007); Shukla et al., (2019); Malovichko et al., (2018)). Finite difference method (FDM) has been frequently preferred over other methods, due to its excellent compromise between accuracy, stability, and computational efficiency. Nevertheless, FDM has its shortcomings. Given the complexity of the Earth model, it is often desirable to use spatially variable discretization, which could potentially also be adaptive to the velocity variations Jastram and Behle (1992); Hayashi et al., (2001); Kang and Baag (2004); Kristek et al., (2010); Chu and Stoffa (2012). FDM does not offer such flexibility, at least not without special treatment.
However, the Radial Basis Function Generated Finite Differences (RBF-FD) method Fornberg (1988), a generalization of FDM, do not require a predefined grid, and therefore offers great flexibility regarding the geometry and of the domain as well as the distribution of nodes. The conceptual difference between FDM and RBF-FD is in the way the nodes are treated. FDM uses a priori knowledge about the nodes and their connectivity with neighbours, as the nodes are organized in a grid that is known in advance. In RBF-FD no a priori knowledge about the nodal topology is required and the support domains are defined in the solution procedure, but at a larger cost to memory, since generally each node has a different local neighbourhood. A direct consequence of higher flexibility regarding the nodal positioning is that RBF-FD is, in contrast to FDM, able to locally modify node configurations by simply placing more points in areas where needed and removing them from areas that are already overpopulated Slak and Kosec (2019a). The RBF-FD method is a popular variant out of many strong-form local meshless methods. It uses finite difference-like collocation weights on an unstructured set of nodes Tolstykh and Shirobokov (2003). The method has been successfully used in several problems and is still actively researched Fornberg and Flyer (2015); Bayona et al., (2017); Slak and Kosec (2019b); Mishra et al., (2019); Slak and Kosec (2019c).
Previous works for modeling acoustic wave equations using weak-form meshfree methods include Jia et al., (2005); Hahn and Negrut (2009); Zhang et al., (2016) and using strong-form meshfree methods include (Takekawa et al., 2015; Takekawa and Mikada, 2016; Liu et al., 2017; Mishra et al., 2017; Takekawa and Mikada, 2018). The strong-form meshfree investigations, mentioned above, implement meshfree computations only in the space-domain (frequency-domain approximation of the acoustic wave equation). Recently, Li et al., (2017) presented a first investigation of application of a mesh-free FD method, based on least squares optimization, for time-domain simulation of acoustic wave equation. Motivated by the success and robustness of RBF-FD Fornberg and Flyer (2015); Fornberg (1988); Slak and Kosec, 2019b, Slak and Kosec, 2019c, it is intriguing to test them on an extended spectra of problems. In this paper, we present an investigation of RBF-FD method for modeling 2D time-domain acoustic wave propagation in heterogeneous Earth's subsurface. In order to suppress the artificial reflections arising from the truncation of the computational domain while mimicking the infinitely large-domain, we couple absorbing boundary conditions with the RBF-FD formulation.
The rest of the paper is structured as follows. In section 2, we discuss the general RBF-FD formulation for solving PDEs and different aspects of its successful implementation. In section 3, we explain the governing equations of the time-domain acoustic wave propagation and the absorbing boundary conditions. In section 4, a series of numerical tests for modeling the wave propagation in (1) homogeneous (2) layered, and (3) highly-heterogeneous Marmousi velocity model of the subsurface have been performed. Standard FD results are provided in first two cases for a heuristic comparison. All examples were computed using the in-house Slak and Kosec (2019d) library. This is followed by the conclusions and some potential future works.
Section snippets
RBF-FD formulation
RBF-FD, as the name suggests, is a generalization of the Finite Difference Method (FDM). Both methods use computational nodes, or points, at which the solution is approximated. Both are also local, meaning only nodes ‘close’ to the selected node can affect the selected node's next value. This neighbourhood of close nodes is commonly referred to as a stencil or the support domain.
Classical FDM approximates differential operators with a weighted combination of neighbouring nodal values, for
Model of acoustic wave propagation in the earth
The standard 2D constant-density approximation of the time-domain acoustic wave equation is given aswhere u is the pressure amplitude or pressure wavefield and is primary wave (P-wave) velocity, which represents the material properties of the subsurface.
In general, the domain of interest is the entire subsurface of the Earth, which can from local point of view be seen as
However, practical
Uniform velocity field (Homogeneous medium)
We first present a basic example of simulation of wave propagation in a homogeneous medium to verify RBF-FD and FDM implementations. Since RBF-FD can mimic FDM when the same grid layout is used, we can compare the solution obtained with RBF-FD and FDM, to analyse both methods and also compare the effect of ABCs.
We define the problem on a square domain with dimensions . The wave velocity is set to and is kept constant, implying a constant nodal spacing of which gives
Conclusions
We have investigated a local strong-form meshless method RBF-FD for numerical solution of 2D time-domain acoustic wave equation in heterogeneous media. The numerical tests performed here have twofold importance: (a) It is one more-step towards the robustness of the current understanding of the RBF-FD by exploring the acoustic wave propagation problem, and (b) the RBF-FD has the potential of being used in large-scale seismic modeling and inversion applications. Followings are some conclusions we
Computer code availability
The computer codes and instruction to reproduce the results in this paper are freely available at https://gitlab.com/e62Lab/2019_p_wavepropagation_code.
Author's contribution
Gregor Kosec and Jure Slak prepared the approximation engines used for numerical solution of the considered problem Jure M. Berljavac, and Pankaj K Mishra prepared numerical solution procedure and analysed results. All authors equally contributed in preparation of the manuscript.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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