A highly efficient implicit finite difference scheme for acoustic wave propagation
Introduction
Numerical seismic simulations have much importance in seismic workflows, and among its numerous applications, it includes algorithms, such as reverse time migration and full waveform inversion. It is always desired to have an accurate, stable and computationally efficient numerical scheme which can be used to solve governing (and constitutive) relations. This scheme involves both time and space derivatives, thus its accuracy chiefly depends on following three factors: (1) the step size chosen for space and time discretization (i.e. Δx,Δt), (2) number of neighborhood points used for estimating derivative, and (3) methods used for estimating spatial derivatives and to carry out time integration. Each factor affects the simulation cost in terms of memory, number of operations and computational time. For example, when a smaller size is employed for space and time discretization, the results would be more accurate but at the expense of increased computational cost due to the increased number of nodes. Increasing the derivative stencil size/length much can violate the local nature as it have a contribution from the nodes at far offset, which may make this derivative estimations non-local in nature. Also after a certain number of nodes, further increment in derivative stencil length will increase only formal accuracy (defined in terms of truncation error) whereas the spectral accuracy gets saturated (Kosloff et al., 2010). Thus a derivative stencil longer than saturation length, would only increase the computational cost but accuracy. Considering the above factors, it is desirable to carry out a numerical simulation using a scheme that can produce highly accurate numerical derivatives using only a small number of nodes for a given grid spacing without adding a significant computational cost.
As mentioned before, the accuracy of a numerical scheme is directly linked with the accuracy of the numerical derivative operator. Several types of numerical derivative operators, which have been utilized for wave propagation can be found in the literature. For example, the classical central derivative (CD) operator, which is based upon Taylor series expansion, can be used for simulating the seismic wavefield on the collocated grid (Alterman and Karal, 1968; Kelly et al., 1976, etc.). The CD operator over the staggered grid provides decoupling of even-odd modes as well as higher formal accuracy than conventional operators (Harlow and Welch, 1965). It has also been widely used in the geophysics community (Madariaga, 1976; Virieux, 1986; Levander, 1988, etc.). These derivative operators can be made to yield spectrally more accurate derivative estimates with proper choice of coefficients. These coefficients can be determined by matching the spectral characteristics of the numerical derivative to the exact in the wavenumber domain (Holberg, 1987; Tam and Webb, 1993). Following this approach, many researchers have also proposed N-points Laplacian stencils (Jo et al., 1996; Shin and Sohn, 1998) as well as their optimized versions (Gosselin-Cliche and Giroux, 2014; Liu et al., 2015; Wang et al., 2016; Fan et al., 2017) for acoustic wave propagation in earth. With the increasing accuracy of FD operator, the error in phase velocity due to numerical anisotropy also reduces. Some more reduction (2–3%) in this error can be achieved with the help of the rotated staggered-grid scheme (Saenger and Bohlen, 2004; Virieux et al., 2011), however, it comes at the cost of the extra cross derivative term (Štekl and Pratt, 1998).
Another type of operator, known as implicit FD operator, is based on Padé schemes (Collatz, 1960). This operator utilizes the value of the function as well as the value of the derivative, which are provided at the given point with some of its neighborhood points for estimating derivative of a function at a given node. An improved version of the implicit FD operator was presented by Lele (1992) which provides better resolution at higher wavenumbers. This scheme was further optimized by (Kim and Lee, 1996) to achieve better performance at higher wavenumbers. Despite its advantage of high accuracy, it also shows increased computational cost due to its implicit nature which requires solving a tridiagonal or pentadiagonal matrix. This computational overburden can be reduced by prefactorization of the banded matrix (Ashcroft and Zhang, 2003; Zhou and Zhang, 2011), so that, the matrix can be solved using the simpler (L + U) operation.
In this paper, we used a sixth order accurate, tridiagonal, implicit type spatial derivative operator for both, analysis and simulations in 2D & 3D domain. To reduce its computational cost, we have presented a method based upon LDMT decomposition followed by a pre-decomposition technique. This scheme is very much efficient in reducing the computational cost for 2D and 3D domains. Further, we have compared this scheme with various other schemes for 2D and 3D models and discussed (i) the accuracy of derivative operators, (ii) the numerical anisotropy error, and (iii) the associated computational cost. Finally, we have demonstrated the application of implicit finite difference scheme to simulate the seismic wave propagation in 2D and 3D acoustic media to verify its feasibility.
Section snippets
The wave equation
The governing equation for the propagation of pressure field in an acoustic media with a given velocity structure is given bywhere, ∇2 = ∂x2 + ∂y2 + ∂z2 is the Laplacian operator; ψ(x, t) describes the pressure field at a given time instance, t and location, x = (x, y, z) in the given medium; c(x) is the velocity of the medium within the given computational domain. The medium is evenly discretized in all, x, y, and z direction. The subscripts indices i, j and k, denote node in x, y
Theoretical cost (FLOPS)
The computational cost of a numerical operator/scheme can be estimated by computing the total floating point arithmetic operations (FLOPs) which includes basic operations viz. addition, subtraction, multiplication or division of two floating point numbers.
First, we obtain the computational cost in FLOPS for 1D CD operator to estimate h2∂x2f(xi) using the Eq. (A.1). It requires a stencil of length 2L + 1 to achieve an accuracy of order 2L. For this, CPU has to perform 2L additions and 2L + 1
Numerical test
To demonstrate the application of implicit finite difference scheme, we carried out the numerical simulation on two types of models, namely, (1) homogeneous and (2) basalt. Both models are also extended to 3D and used for simulation.
- 1.
Homogeneous model:
The model has a uniform medium velocity of 2000ms−1. The model is discretized with the same grid spacing of dh = 4m along all direction. The length of each side of the model is 1600m and thus have 401 nodes in each horizontal (x or y) and vertical (
Discussion
As we know that, the high accuracy and reduced numerical anisotropy can be achieved at the expense of computational resources (in terms of memory utilized and the number of CPU cycles). In explicit schemes, it happens due to the higher number of nodes calculation, whereas, implicit scheme requires node summation and a matrix inversion. A theoretical FLOPS calculation shows that both the schemes are equivalent but the simulation time shows otherwise. The compactness of the stencil in the
Conclusions
In this paper, we have presented an efficient method for acoustic wave simulation using implicit FD operator. We have compared various aspects of the implicit scheme with various CD schemes of different orders, viz. accuracy of the numerical derivative operator, error in phase velocity due to discretization, and computational cost. The spectral characteristics of 6th order implicit scheme are equivalent to 10th order conventional CD schemes. The error characteristics (i.e., minima and maxima
Data and resources
All data used in this paper came from published sources listed in the references.
Acknowledgment
AM is grateful to CSIR-UGC for awarding him the fellowship to carry out his Ph.D. work. RKT is grateful to Department of Atomic Energy (DAE) for awarding Raja Ramanna Fellowship. Authors are also thankful to AcSIR-NGRI Coordinator & Director NGRI for support and permission to publish the results. This work was carried out under the project MLP-6402-28.
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2022, Journal of Applied GeophysicsCitation Excerpt :Numerical modeling of the seismic wave equation is an effective way to understand the propagation principle of the seismic wavefields in different media, also plays a significant role in the oil and gas exploration. Among all numerical simulation methods for the seismic wave equation, the finite-difference (FD) method has outstanding advantages, such as economical for computational cost and memory storage, as well as easy implementation (Virieux, 1986; Kindelan et al., 1990; Moczo et al., 2007, 2014; Du et al., 2009; Fang et al., 2014; Etemadsaeed et al., 2016; Malkoti et al., 2019). The basic principle of these methods for numerically solving the seismic wave equations is to replace these corresponding temporal and spatial derivatives with discrete difference operators.