Reducing error accumulation of optimized finite-difference scheme using the minimum norm

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. T275-T291
Author(s):  
Zhongzheng Miao ◽  
Jinhai Zhang
Keyword(s):  
Finite Difference ◽  
Objective Function ◽  
Constant Coefficient ◽  
Total Error ◽  
Error Tolerance ◽  
Seismic Exploration ◽  
Numerical Dispersion ◽  
Effective Bandwidth ◽  
Coefficient Optimization ◽  
Error Accumulation

The finite-difference (FD) scheme is popular in the field of seismic exploration for numerical simulation of wave propagation; however, its accuracy and computational efficiency are restricted by the numerical dispersion caused by numerical discretization of spatial partial derivatives using coarse grids. The constant-coefficient optimization method is used widely for suppressing the numerical dispersion by tuning the FD weights. Although gaining a wider effective bandwidth under a given error tolerance, this method undoubtedly encounters larger errors at low wavenumbers and accumulates significant errors. We have developed an approach to reduce the error accumulation. First, we construct an objective function based on the [Formula: see text] norm, which can constrain the total error better than the [Formula: see text] and [Formula: see text] norms. Second, we translated our objective function into a constrained [Formula: see text]-norm minimization model, which can be solved by the alternating direction method of multipliers. Finally, we perform theoretical analyses and numerical experiments to illustrate the accuracy improvement. The proposed method is shown to be superior to the existing constant-coefficient optimization methods at the low-wavenumber region; thus, we can obtain higher accuracy with less error accumulation, particularly at longer simulation times. The widely used objective functions, defined by the [Formula: see text] and [Formula: see text] norms, could handle a relatively wider range of accurate wavenumbers, compared with our objective function defined by the [Formula: see text] norm, but their actual errors would be much larger than the given error tolerance at some azimuths rather than axis directions (e.g., about twice at 45°), which greatly degrade the overall numerical accuracy. In contrast, our scheme can obtain a relatively even 2D error distribution at various azimuths, with an apparently smaller error. The peak error of the proposed method is only 40%–65% that of the [Formula: see text] norm under the same error tolerance, or only 60%–80% that of the [Formula: see text] norm under the same effective bandwidth.

Acoustic VTI Modeling Using an Optimal Time-Space Domain Finite-Difference Scheme

2016 ◽  
Vol 24 (04) ◽  
pp. 1650016 ◽  
Author(s):  
Hongyong Yan ◽  
Lei Yang ◽  
Xiang-Yang Li ◽  
Hong Liu
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Taylor Series Expansion ◽  
Transversely Isotropic ◽  
High Accuracy ◽  
Seismic Exploration ◽  
Numerical Dispersion ◽  
Optimal Time ◽  
Space Domain ◽  
Time Space

Finite-difference (FD) schemes have been used widely for solving wave equations in seismic exploration. However, the conventional FD schemes hardly guarantee high accuracy at both small and large wavenumbers. In this paper, we propose an optimal time-space domain FD scheme for acoustic vertical transversely isotropic (VTI) wave modeling. The optimal FD coefficients for the second-order spatial derivatives are derived by approaching the time-space domain dispersion relation of acoustic VTI wave equations with the combination of the Taylor-series expansion and the sampling interpolation. We perform numerical dispersion analyses and acoustic VTI modeling using the optimal time-space domain FD scheme. The numerical dispersion analyses show that the optimal FD scheme has smaller dispersion than the conventional FD scheme at large wavenumbers, and also preserves high accuracy at small wavenumbers. The acoustic VTI modeling examples also demonstrate that the optimal time-space domain FD scheme has greater accuracy compared with the conventional time-space domain FD scheme for the same modeling parameters.

scholarly journals Low-Frequency Expansion Approach for Seismic Data Based on Compressed Sensing in Low SNR

2021 ◽  
Vol 11 (11) ◽  
pp. 5028
Author(s):  
Miaomiao Sun ◽  
Zhenchun Li ◽  
Yanli Liu ◽  
Jiao Wang ◽  
Yufei Su
Keyword(s):  
Reflection Coefficient ◽  
Compressed Sensing ◽  
Objective Function ◽  
Low Frequency ◽  
Original Data ◽  
Seismic Exploration ◽  
High Signal ◽  
Inversion Process ◽  
Frequency Information ◽  
Low Snr

Low-frequency information can reflect the basic trend of a formation, enhance the accuracy of velocity analysis and improve the imaging accuracy of deep structures in seismic exploration. However, the low-frequency information obtained by the conventional seismic acquisition method is seriously polluted by noise, which will be further lost in processing. Compressed sensing (CS) theory is used to exploit the sparsity of the reflection coefficient in the frequency domain to expand the low-frequency components reasonably, thus improving the data quality. However, the conventional CS method is greatly affected by noise, and the effective expansion of low-frequency information can only be realized in the case of a high signal-to-noise ratio (SNR). In this paper, well information is introduced into the objective function to constrain the inversion process of the estimated reflection coefficient, and then, the low-frequency component of the original data is expanded by extracting the low-frequency information of the reflection coefficient. It has been proved by model tests and actual data processing results that the objective function of estimating the reflection coefficient constrained by well logging data based on CS theory can improve the anti-noise interference ability of the inversion process and expand the low-frequency information well in the case of a low SNR.

Controlling Numerical Dispersion by Variably Timed Flux Updating in Two Dimensions

1982 ◽  
Vol 22 (03) ◽  
pp. 409-419 ◽  
Author(s):  
R.G. Larson
Keyword(s):  
Finite Difference ◽  
Pressure Gradient ◽  
Single Point ◽  
Companion Paper ◽  
Two Dimensions ◽  
Independent Component ◽  
Numerical Dispersion ◽  
Coarse Mesh ◽  
Fixed Concentration ◽  
Order Of Magnitude

Abstract The variably-timed flux updating (VTU) finite difference technique is extended to two dimensions. VTU simulations of miscible floods on a repeated five-spot pattern are compared with exact solutions and with solutions obtained by front tracking. It is found that for neutral and favorable mobility ratios. VTU gives accurate results even on a coarse mesh and reduces numerical dispersion by a factor of 10 or more over the level generated by conventional single-point (SP) upstream weighting. For highly unfavorable mobility ratios, VTU reduces numerical dispersion. but on a coarse mesh the simulation is nevertheless inaccurate because of the inherent inadequacy of the finite-difference estimation of the flow field. Introduction A companion paper (see Pages 399-408) introduced the one-dimensional version of VTU for controlling numerical dispersion in finite-difference simulation of displacements in porous media. For linear and nonlinear, one- and two-independent-component problems, VTU resulted in more than an order-of-magnitude reduction in numerical dispersion over conventional explicit. SP upstream-weighted simulations with the same number of gridblocks. In this paper, the technique is extended to two dimensional (2D) problems, which require solution of a set of coupled partial differential equations that express conservation of material components-i.e., (1) and (2) Fi, the fractional flux of component i, is a function of the set of s - 1 independent-component fractional concentrations {Ci}, which prevail at the given position and time., the dispersion flux, is given by an expression that is linear in the specie concentration gradients. The velocity, is proportional to the pressure gradient,. (3) where lambda, in general, can be a function of composition and of the magnitude of the pressure gradient. The premises on which Eqs. 1 through 3 rest are stated in the companion paper. VTU in Two Dimensions The basic idea of variably-timed flux updating is to use finite-difference discretization of time and space, but to update the flux of a component not every timestep, but with a frequency determined by the corresponding concentration velocity -i.e., the velocity of propagation of fixed concentration of that component. The concentration velocity is a function of time and position. In the formulation described here, the convected flux is upstream-weighted, and all variables except pressure are evaluated explicitly. As described in the companion paper (SPE 8027), the crux of the method is the estimation of the number of timesteps required for a fixed concentration to traverse from an inflow to an outflow face of a gridblock. This task is simpler in one dimension, where there is only one inflow and one outflow face per gridblock, than it is in two dimensions, where each gridblock has in general multiple inflow and outflow faces. SPEJ P. 409^

An explicit method to calculate implicit spatial finite differences

Geophysics ◽  
2021 ◽  
pp. 1-71
Author(s):  
Hongwei Liu ◽  
Yi Luo
Keyword(s):  
Finite Difference ◽  
High Performance ◽  
Computational Cost ◽  
New Method ◽  
Seismic Exploration ◽  
Explicit Method ◽  
Lu Decomposition ◽  
Implicit Methods ◽  
Band Matrices ◽  
Explicit Finite Difference

The finite-difference solution of the second-order acoustic wave equation is a fundamental algorithm in seismic exploration for seismic forward modeling, imaging, and inversion. Unlike the standard explicit finite difference (EFD) methods that usually suffer from the so-called "saturation effect", the implicit FD methods can obtain much higher accuracy with relatively short operator length. Unfortunately, these implicit methods are not widely used because band matrices need to be solved implicitly, which is not suitable for most high-performance computer architectures. We introduce an explicit method to overcome this limitation by applying explicit causal and anti-causal integrations. We can prove that the explicit solution is equivalent to the traditional implicit LU decomposition method in analytical and numerical ways. In addition, we also compare the accuracy of the new methods with the traditional EFD methods up to 32nd order, and numerical results indicate that the new method is more accurate. In terms of the computational cost, the newly proposed method is standard 8th order EFD plus two causal and anti-causal integrations, which can be applied recursively, and no extra memory is needed. In summary, compared to the standard EFD methods, the new method has a spectral-like accuracy; compared to the traditional LU-decomposition implicit methods, the new method is explicit. It is more suitable for high-performance computing without losing any accuracy.

Fourth‐order finite‐difference P-SV seismograms

Geophysics ◽  
1988 ◽  
Vol 53 (11) ◽  
pp. 1425-1436 ◽  
Author(s):  
Alan R. Levander
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Fourth Order ◽  
Equations Of Motion ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Order System ◽  
Surface Boundary ◽  
Lamb’S Problem

I describe the properties of a fourth‐order accurate space, second‐order accurate time, two‐dimensional P-SV finite‐difference scheme based on the Madariaga‐Virieux staggered‐grid formulation. The numerical scheme is developed from the first‐order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga‐Virieux staggered‐grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic‐elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free‐surface or within a layer and to satisfy free‐surface boundary conditions. Benchmark comparisons of finite‐difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite‐difference and reflectivity solutions for elastic‐elastic and acoustic‐elastic layered models.

Acoustic and elastic modeling by optimal time-space-domain staggered-grid finite-difference schemes

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. T17-T40 ◽  
Author(s):  
Zhiming Ren ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Stability Condition ◽  
Computational Cost ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Equation Modeling ◽  
Optimal Time ◽  
Finite Difference Schemes ◽  
Space Domain ◽  
Time Space

Staggered-grid finite-difference (SFD) methods are widely used in modeling seismic-wave propagation, and the coefficients of finite-difference (FD) operators can be estimated by minimizing dispersion errors using Taylor-series expansion (TE) or optimization. We developed novel optimal time-space-domain SFD schemes for acoustic- and elastic-wave-equation modeling. In our schemes, a fourth-order multiextreme value objective function with respect to FD coefficients was involved. To yield the globally optimal solution with low computational cost, we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least-squares (LS) to derive the optimal SFD coefficients by minimizing the relative error of time-space-domain dispersion relations over a given frequency range. To ensure the robustness of our schemes, a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold. Moreover, the hybrid absorbing boundary condition was applied to remove artificial boundary reflections. We compared our optimal SFD with the conventional, TE-based time-space-domain, and LS-based SFD schemes. Dispersion analysis and numerical simulation results suggested that the new SFD schemes had a smaller numerical dispersion than the other three schemes when the same operator lengths were adopted. In addition, our LS-based time-space-domain SFD can obtain the same modeling accuracy with shorter spatial operator lengths. We also derived the stability condition of our schemes. The experiment results revealed that our new LS-based SFD schemes needed a slightly stricter stability condition.

A simple and exact acoustic wavefield modeling code for data processing, imaging, and interferometry applications

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. F17-F27 ◽  
Author(s):  
Erica Galetti ◽  
David Halliday ◽  
Andrew Curtis
Keyword(s):  
Finite Difference ◽  
Synthetic Data ◽  
Acoustic Modeling ◽  
Numerical Dispersion ◽  
Seismic Interferometry ◽  
Numerical Errors ◽  
Matlab Code ◽  
Modeling Code ◽  
Seismic Images ◽  
Methods Numerical

Improvements in industrial seismic, seismological, acoustic, or interferometric theory and applications often result in quite subtle changes in sound quality, seismic images, or information which are nevertheless crucial for improved interpretation or experience. When evaluating new theories and algorithms using synthetic data, an important aspect of related research is therefore that numerical errors due to wavefield modeling are reduced to a minimum. We present a new MATLAB code based on the Foldy method that models theoretically exact, direct, and scattered parts of a wavefield. Its main advantage lies in the fact that while all multiple scattering interactions are taken into account, unlike finite-difference or finite-element methods, numerical dispersion errors are avoided. The method is therefore ideal for testing new theory in industrial seismics, seismology, acoustics, and in wavefield interferometry in particular because the latter is particularly sensitive to the dynamics of scattering interactions. We present the theory behind the Foldy acoustic modeling method and provide examples of its implementation. We also benchmark the code against a good finite-difference code. Because our Foldy code was written and optimized to test new theory in seismic interferometry, examples of its application to seismic interferometry are also presented, showing its validity and importance when exact modeling results are needed.

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