Abstract
Seismic imaging is a major challenge in geophysics with broad applications. It involves solving wave propagation equations with absorbing boundary conditions (ABC) multiple times. This drives the need for accurate and efficient numerical methods. This study examines a collection of exponential integration methods, known for their good numerical properties on wave representation, to investigate their efficacy in solving the wave equation with ABC. The purpose of this research is to assess the performance of these methods. We compare a recently proposed Exponential Integration based on Faber polynomials with well-established Krylov exponential methods alongside a high-order Runge-Kutta scheme and low-order classical methods. Through our analysis, we found that the exponential integrator based on the Krylov subspace exhibits the best convergence results among the high-order methods. We also discovered that high-order methods can achieve computational efficiency similar to low-order methods while allowing for considerably larger time steps. Most importantly, the possibility of undertaking large time steps could be used for important memory savings in full waveform inversion imaging problems.
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Acknowledgements
This research was carried out in association with the ongoing R&D project registered as ANP20714-2 STMI - Software Technologies for Modelling and Inversion, with applications in seismic imaging (USP/Shell Brasil/ANP). It was funded in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, and in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) - Brasil. Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) grant 2021/06176-0 is also acknowledged. It has also partially received funding from the Federal Ministry of Education and Research and the European High-Performance Computing Joint Undertaking (JU) under grant agreement No 955701, Time-X. The JU receives support from the European Union’s Horizon 2020 research and innovation programme and Belgium, France, Germany, and Switzerland.
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A Appendix
A Appendix
1.1 A.1 Approximations at the free-surface
We present the finite difference approximations of 8th order for the required derivatives of the functions at the points near the free surface. To simplify the notation, we define \(u_i=u(x,-i\varDelta x)\), and \(w_i=w_y\left( x,-(i+\frac{1}{2})\varDelta x\right) \). Since we are considering a uniform grid, we have that \(\varDelta y=\varDelta x\), and so, only \(\varDelta x\) will be used.
1.2 A.2 Homogeneous medium
This section complements the results in Section 4. First, we show the convergence, dispersion, and dissipation errors associated with the eighth-order spatial discretization scheme using \(\varDelta x=10m\) (Fig. 10). Additionally, we present how varying the peak frequencies as \(f_M={10,\;15,\;20,\;25}\), impact the maximum allowable time-step \(\varDelta t_{\text {max}}\) and the number of matrix-vector operations (MVOs) for different schemes and approximation degrees.
1.2.1 A.2.1 Dispersion results
From Fig. 11, we perceive that the general behavior is maintained independent of the peak frequencies. With the difference that when the peak frequency increases, the results for the Krylov method are more oscillatory, and the high-degree approximations using Faber polynomials suffer from more round-off errors.
In Fig. 12, we still observe that the Leap-frog algorithm requires the least amount of MVOs. The FA and HORK methods share a similar number of computations independent of the peak frequency.
1.2.2 A.2.2 Dissipation results
A similar trend of Fig. 11 is observed in Fig. 13, as with the dispersion error. The Krylov method still has the worst performance for the different peak frequencies. However, it is noteworthy that the RK9-7 method (red triangle) displays an even better performance concerning the dissipation error.
Regarding computational efficiency in the analysis of the dispersion error, the RK9-7 scheme still maintains an efficient computational performance. The FA and HORK exhibit similar behavior among the high-order methods, with a decline in efficiency for high-order Faber polynomials as the peak frequency increases. Nonetheless, the Krylov method exhibits the best performance in general, but with a very marked oscillatory behavior (Fig. 14).
1.3 A.3 Convergence and computational efficiency
In this section, we complement the results of the numerical experiments of Section 5. First, we show the error graphics using the minimum time-step of \(\varDelta t=\frac{\varDelta x}{8c_{\text {max}}}\), where \(c_\text {max}\) is the medium maximum velocity. These graphs account for all the methods discussed in Section 3 and several approximation degrees for the high-order schemes. Following that, we present the graphics of the estimation of \(\varDelta t_\text {max}\), the computational efficiency, and the memory utilization.
Based on Fig. 15, we observe an approximation error in all the numerical examples that do not decrease with the order of the method or with the selected method. This error is independent of the time integration strategy and is produced by the spatial discretization operator. While the dependence of the spatial error on the numerical experiment is weak, it is important to estimate it accurately for a reliable computation of \(\varDelta _\text {max}\), as quantified in Table 2.
Table 2 contains two key columns of information. The first column, labeled “Spatial error”, represents the error stemming from the spatial discretization. Meanwhile, the second column, labeled “Error tolerance”, accounts for the error tolerance of 150% of the spatial error we defined for the numerical experiment.
For the minimum error using the seismogram data, we have the respective error graphics and tolerance for each numerical test (Fig. 16). Additionally, Table 3 summarizes the error values.
1.3.1 A.3.1 Computational efficiency and memory consumption
Figure 17 displays each time the integrator’s computational cost and memory utilization for the numerical tests Corner Model, Santos Basin, and SEG/EAGE. Although there are some variations between the experiments, the general behavior remains consistent. High-order methods require significantly less memory; in some cases, they are competitive with low-order methods, such as the Leap-Frog scheme.
The relationship between the number of MVOs and the quantity of stored solution vectors concerning the polynomial degree is illustrated for the Corner Model (first line), Santos Basin (second line), and SEG/EAGE (third line) numerical tests. As the number of stages increases, there is a stabilization in the number of computations, and memory usage decreases.
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Ravelo, F.V., Schreiber, M. & Peixoto, P.S. High-order exponential integration for seismic wave modeling. Comput Geosci (2024). https://doi.org/10.1007/s10596-024-10319-5
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DOI: https://doi.org/10.1007/s10596-024-10319-5