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Keywords: expansion coefficient

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Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, October 11–16, 2020

Paper Number: SEG-2020-3407568

... suppress numerical dispersion. Presentation Date: Tuesday, October 13, 2020 Session Start Time: 9:20 AM Presentation Time: 11:25 AM Location: Poster Station 12 Presentation Type: Poster machine learning fd scheme convergence speed artificial intelligence

**expansion****coefficient**numerical...
Abstract

In this paper, we derive an improved QPSO algorithm and develop a new FD scheme based on this improved QPSO algorithm. The improved QPSO algorithm has obvious advantages of convergence speed, which can be converged in the 200th generation. Under the same condition, the convergence speed of QPSO algorithm is much lower than that of improved QPSO algorithm. Numerical dispersion analysis shows that, the optimized FD scheme based on the improved QPSO algorithm has a larger spectral coverage, and the accuracy error is controlled within a valid range, which means the improved QPSO algorithm has better ability to search for accurate global solutions. Finally, numerical modelling of elastic wave equations is performed using the optimized FD scheme based on the improved QPSO algorithm. Numerical modelling results indicate that the optimized FD scheme based on the improved QPSO algorithm can effectively suppress numerical dispersion. Presentation Date: Tuesday, October 13, 2020 Session Start Time: 9:20 AM Presentation Time: 11:25 AM Location: Poster Station 12 Presentation Type: Poster

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2012 SEG Annual Meeting, November 4–9, 2012

Paper Number: SEG-2012-0811

... schlumberger-doll research efficient estimation upstream oil & gas formula artificial intelligence estimation quadrature rule main menu polynomial chaos

**expansion****coefficient**polynomial chaos coefficient integration sparse polynomial chaos proxy quadrature point simulation quadrature...
Abstract

Summary We investigate the use of sparse grid methods in computing polynomial chaos (PC) proxies for forward stochastic problems associated with numerically-expensive simulators. These are problems where some input parameters are random with known distributions, and stochastic properties of the simulator output are desired. The bottleneck for PC proxy construction is the estimation of the coefficients, which typically require computationally intensive forward simulations and multi-dimensional integration. To minimize the number of simulations, we compare two methods for computing polynomial coefficients using sparse quadrature integration: generalized Fejér quadrature (FQ), and sparse reduced quadrature (RQ). We compare the efficiency (as determined by the number of quadrature points needed to accurately estimate coefficients) of these methods for a 5-dimensional stochastic electromagnetic problem. Paradoxically, we find that for general weight functions, sparse FQ requires very high degree exactness to accurately estimate proxy coefficients, which makes this scheme very inefficient. In contrast, RQ requires the minimum number of quadrature points for a pre-defined polynomial exactness. By using the sparse reduced quadrature approach, PC can apply to problems with arbitrary input PDFs and high-dimensional spaces. The trade-off is that sparse FQ has nested abscissae allowing for adaptive refinement of integration degree, while RQ does not.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010

Paper Number: SEG-2010-3288

... implementation with implicit finite difference. The

**expansion****coefficients**are obtained through a nonlinear optimization of the dispersion error cost function. The strong-amplitude evanescent energy is attenuated with a frequency-dependent phase shift at each depth step. We test and verify our method...
Abstract

Summary This paper presents an implicit finite difference algorithm for 3D TTI media. We first give the detailed derivation of 3-D TTI dispersion relation equation. We approximate the 3-D TTI dispersion equation with four-direction splitting partial fractional expansion to enable the cascading implementation with implicit finite difference. The expansion coefficients are obtained through a nonlinear optimization of the dispersion error cost function. The strong-amplitude evanescent energy is attenuated with a frequency-dependent phase shift at each depth step. We test and verify our method and implementation with few synthetic examples. Introduction Anisotropy exists in many physical rocks such as shales due to the intrinsic anisotropic properties of rock mineral components and fine-layered sandstones with preferable principal directions. Anisotropy has been observed in quite different geological regions from North Sea to the Gulf of Mexico, from onshore survey (foothills) to offshore survey. It is important to incorporate anisotropy in seismic depth imaging algorithms to get high quality structurally-correct images. In present seismic exploration practice, anisotropy is either described by a VTI model or by a TTI model. TTI model provides better description to real life anisotropy and TTI image techniques have been increasingly demanded. Due to the amazing advances both in seismic imaging algorithms and high performance computing techniques, TTI anisotropic imaging has been made possible and been widely used in field data processing in industries in the last few years. As imaging in isotropic media, there are three technical classes for TTI imaging: 1. Kirchhoff-type imaging based on the high frequency asymptotic integral solution of wave equation with ray tracing (Jiao et al, 2005; Zhu et al, 2006); 2. Downward extrapolation imaging based on one-way wave equation (Shan, 2005, 2007; Bale et al, 2007; Valenciano et al., 2009a, 2009b); and 3. Reverse time migration (RTM) by directly solving two-way wave equation (Du et al, 2007; Zhang et al, 2008; Fletcher et al, 2008, 2009). On one hand, in many cases, wave equation migration is superior to Kirchhoff migration because it can naturally handle multi-pathing issues for complex velocity models and provides better images than Kirchhoff migration. On the other hand, despite the successes of TTI reverse time migration (RTM), one-way migrations remain important imaging tools in industries: One-way propagator is much faster than RTM,. it is useful in iterative velocity updating (Shen et al, 2005, Fei et al, 2007) and wide bandwidth data imaging where finer grid space must be used and higher frequency has to be migrated. One-way TTI migration is complicated to solve because it involves more model parameters (four in 2-D and five in 3- D) than isotropic migration (only one parameter – velocity) and TTI dispersion relation equation is in implicit form (difficult to be analytically solved, especially for 3-D). Algorithms applied in isotropic migration have been extended to TTI model. Shan and Biondi (2005) developed a 3-D TTI wave field extrapolation method using an implicit isotropic operator with an explicit anisotropic correction. Bale et al (2007) adapted isotropic PSPI and Fourier split-step operators for TTI media.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2009 SEG Annual Meeting, October 25–30, 2009

Paper Number: SEG-2009-2687

... one-way propagator and evaluate two implementations of the amplitude correction. Numerical examples show that the lateral velocity variation related correction term can play a significant role in the asymptotic true-amplitude one-way propagator. Optimization of the

**expansion****coefficients**...
Abstract

Summary Much work has been done on the vertical velocity variation related amplitude correction term in the asymptotic true-amplitude one-way wave equation, but the lateral velocity variation related correction has not received much attention, even being excluded in some asymptotic true-amplitude one-way propagator formulations. Here we investigate the effects of different amplitude correction terms in the asymptotic true-amplitude one-way wave equation, especially the effect related to the lateral velocity variation. We derive a dual-domain wide-angle screen type asymptotic true-amplitude one-way propagator and evaluate two implementations of the amplitude correction. Numerical examples show that the lateral velocity variation related correction term can play a significant role in the asymptotic true-amplitude one-way propagator. Optimization of the expansion coefficients in the asymptotic true-amplitude one-way propagator can improve both the amplitude and phase accuracy for wide-angle waves. Introduction Conventional one-way wave equations do not pay much attention to the correctness of amplitude information and cannot provide accurate amplitudes even at the level of leading order asymptotic WKBJ or ray theory (Zhang et al., 2003). WKBJ amplitudes have long been introduced into the conventional one-way wave propagator (e.g. Clayton and Stolt, 1981). Traditionally the WKBJ solution was derived by asymptotic approximation in smoothly varying v ( z ) media (e.g. Morse and Feshbach, 1953; Aki and Richards, 1980). It has also been obtained by approximately factorizing the full-wave operator into one-way wave operators in heterogeneous media (Zhang, 1993). It was theoretically proved that the obtained one-way wave equation could provide the same amplitude as the full wave equation in heterogeneous media in the sense of high-frequency asymptotics (e.g., Zhang, 1993). In this sense, they are called the “true-amplitude” one-way wave equation (Zhang et al., 2003). To precisely represent the physics, we refer to it as the asymptotic true-amplitude one-way wave equation. The WKBJ solution was also derived from the conservation of energy flux in smoothly varying v ( z ) media and extended to general media using local wavenumber/angle domain propagators by introducing the concepts of a “transparent boundary condition” and a “transparent propagator” (Wu and Cao, 2005; Cao and Wu, 2005, 2006, 2008; Luo et al., 2005). With the asymptotic true-amplitude one-way propagator, improved image amplitude is obtained from single-shot migration in some smoothly varying models (Zhang et al., 2003; Zhang et al., 2005). As for the wavefield amplitude, much attention is paid to the vertical velocity variation related amplitude correction term in the literature, while the lateral velocity variation related correction term has rarely been studied, even having been excluded in some asymptotic true-amplitude one-way propagator formulations. In this paper, we analyze the effects of amplitude correction terms in the asymptotic true-amplitude one-way propagator, especially the effect related to the lateral velocity variation, by comparing the wavefield amplitude from the one-way propagator with that from full-wave modeling. We illustrate these with scalar wave propagation. A finite-difference scheme is used to solve the full wave equation. The implementation of the asymptotic true-amplitude one-way propagator in Zhang et al. (2005) uses a space-domain finite-difference scheme.