Dispersion analysis of an average-derivative optimal scheme for Laplace-domain scalar wave equation

Geophysics ◽  
2014 ◽  
Vol 79 (2) ◽  
pp. T37-T42 ◽  
Author(s):  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Dispersion Analysis ◽  
Domain Modeling ◽  
Scalar Wave ◽  
Laplace Domain ◽  
Optimal Scheme ◽  
Point Scheme ◽  
Sampling Intervals ◽  
Grid Points ◽  
Average Derivative

Laplace-domain modeling is an important foundation of Laplace-domain full-waveform inversion. However, dispersion analysis for Laplace-domain numerical schemes has not been completely established. This hampers the construction and optimization of Laplace-domain modeling schemes. By defining a pseudowavelength as a scaled skin depth, I establish a method for Laplace-domain numerical dispersion analysis that is parallel to its frequency-domain counterpart. This method is then applied to an average-derivative nine-point scheme for Laplace-domain scalar wave equation. Within the relative error of 1%, the Laplace-domain average-derivative optimal scheme requires four grid points per smallest pseudowavelength, whereas the classic five-point scheme requires 13 grid points per smallest pseudowavelength for general directional sampling intervals. The average-derivative optimal scheme is more accurate than the classic five-point scheme for the same sampling intervals. By using much smaller sampling intervals, the classic five-point scheme can approach the accuracy of the average-derivative optimal scheme, but the corresponding cost is much higher in terms of storage requirement and computational time.

An average-derivative optimal scheme for frequency-domain scalar wave equation

Geophysics ◽  
2012 ◽  
Vol 77 (6) ◽  
pp. T201-T210 ◽  
Author(s):  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Frequency Domain ◽  
Waveform Inversion ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Optimal Scheme ◽  
Phase Velocity Dispersion ◽  
Point Scheme ◽  
Sampling Intervals ◽  
Average Derivative

Forward modeling is an important foundation of full-waveform inversion. The rotated optimal nine-point scheme is an efficient algorithm for frequency-domain 2D scalar wave equation simulation, but this scheme fails when directional sampling intervals are different. To overcome the restriction on directional sampling intervals of the rotated optimal nine-point scheme, I introduce a new finite-difference algorithm. Based on an average-derivative technique, this new algorithm uses a nine-point operator to approximate spatial derivatives and mass acceleration term. The coefficients can be determined by minimizing phase-velocity dispersion errors. The resulting nine-point optimal scheme applies to equal and unequal directional sampling intervals, and can be regarded a generalization of the rotated optimal nine-point scheme. Compared to the classical five-point scheme, the number of grid points per smallest wavelength is reduced from 13 to less than four by this new nine-point optimal scheme for equal and unequal directional sampling intervals. Three numerical examples are presented to demonstrate the theoretical analysis. The average-derivative algorithm is also extended to a 2D viscous scalar wave equation and a 3D scalar wave equation.

An affine generalized optimal scheme with improved free-surface expression using adaptive strategy for frequency-domain elastic wave equation

Geophysics ◽  
2022 ◽  
pp. 1-71
Author(s):  
Shu-Li Dong ◽  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Free Surface ◽  
Elastic Wave ◽  
Rotation Angle ◽  
Surface Expression ◽  
Coordinate Systems ◽  
Elastic Wave Equation ◽  
Point Scheme ◽  
Sampling Intervals ◽  
Grid Points

Effective frequency-domain numerical schemes were central for forward modeling and inversion of the elastic wave equation. The rotated optimal nine-point scheme was a highly used finite-difference numerical scheme. This scheme made a weighted average of the derivative terms of the elastic wave equations in the original and the rotated coordinate systems. In comparison with the classical nine-point scheme, it could simulate S-waves better and had higher accuracy at nearly the same computational cost. Nevertheless, this scheme limited the rotation angle to 45°; thus, the grid sampling intervals in the x- and z-directions needed to be equal. Otherwise, the grid points would not lie on the axes, which dramatically complicates the scheme. Affine coordinate systems did not constrain axes to be perpendicular to each other, providing enhanced flexibility. Based on the affine coordinate transformations, we developed a new affine generalized optimal nine-point scheme. At the free surface, we applied the improved free-surface expression with an adaptive parameter-modified strategy. The new optimal scheme had no restriction that the rotation angle must be 45°. Dispersion analysis found that our scheme could effectively reduce the required number of grid points per shear wavelength for equal and unequal sampling intervals compared to the classical nine-point scheme. Moreover, this reduction improved with the increase of Poisson’s ratio. Three numerical examples demonstrated that our scheme could provide more accurate results than the classical nine-point scheme in terms of the internal and the free-surface grid points.

An average-derivative optimal scheme for modeling of the frequency-domain 3D elastic wave equation

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T209-T234 ◽  
Author(s):  
Jing-Bo Chen ◽  
Jian Cao
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Frequency Domain ◽  
Poisson’S Ratio ◽  
Poisson's Ratio ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Wave Phase Velocity ◽  
Point Scheme ◽  
Grid Points

Because of its high computational cost, we needed to develop an efficient numerical scheme for the frequency-domain 3D elastic wave equation. In addition, the numerical scheme should be applicable to media with a liquid-solid interface. To address these two issues, we have developed a new average-derivative optimal 27-point scheme with arbitrary directional grid intervals and a corresponding numerical dispersion analysis for the frequency-domain 3D elastic wave equation. The novelty of this scheme is that its optimal coefficients depend on the ratio of the directional grid intervals and Poisson’s ratio. In this way, this scheme can be applied to media with a liquid-solid interface and a computational grid with arbitrary directional grid intervals. For media with a variable Poisson’s ratio, we have developed an effective and stable interpolation method for optimization coefficients. Compared with the classic 19-point scheme, this new scheme reduces the required number of grid points per wavelength for equal and unequal directional grid intervals. The reduction of the number of grid points increases as the Poisson’s ratio becomes larger. In particular, the numerical S-wave phase velocity of this new scheme becomes zero, whereas the classic 19-point scheme produces a spurious numerical S-wave phase velocity, as Poisson’s ratio reaches 0.5. We have performed numerical examples to develop the theoretical analysis.

Modeling of frequency-domain elastic-wave equation with an average-derivative optimal method

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. T339-T356 ◽  
Author(s):  
Jing-Bo Chen ◽  
Jian Cao
Keyword(s):  
Wave Equation ◽  
Phase Velocity ◽  
Elastic Wave ◽  
Frequency Domain ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Classical Scheme ◽  
Wave Phase Velocity ◽  
Grid Points ◽  
Average Derivative

Based on an average-derivative method, we developed a new nine-point numerical scheme for a frequency-domain elastic-wave equation. Compared with the classic nine-point scheme, this scheme reduces the required number of grid points per wavelength for equal and unequal directional spacings. The reduction in the number of grid points increases as the Poisson’s ratio becomes larger. In particular, as the Poisson’s ratio reaches 0.5, the numerical S-wave phase velocity of this new scheme becomes zero, whereas the classical scheme produces spurious numerical S-wave phase velocity. Numerical examples demonstrate that this new scheme produces more accurate results than the classical scheme at approximately the same computational cost.

An optimal 125-point scheme for 3D frequency-domain scalar wave equation

2022 ◽  
Author(s):  
Lingyun Yang ◽  
Guochen Wu ◽  
Yunliang Wang ◽  
Qingyang Li ◽  
Hao Zhang
Keyword(s):  
Wave Equation ◽  
Frequency Domain ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Point Scheme
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