Wave‐equation phase inversion in the frequency domain

1991 ◽  
Author(s):  
Gerard T. Schuster
Keyword(s):  
Wave Equation ◽  
Frequency Domain ◽  
Phase Inversion

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 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.

Stability results for an elastic–viscoelastic wave equation with localized Kelvin–Voigt damping and with an internal or boundary time delay

2020 ◽  
pp. 1-57
Author(s):  
Mouhammad Ghader ◽  
Rayan Nasser ◽  
Ali Wehbe
Keyword(s):  
Wave Equation ◽  
Time Delay ◽  
Frequency Domain ◽  
Viscoelastic Damping ◽  
One Dimensional ◽  
Viscoelastic Wave ◽  
Smooth Coefficients ◽  
The Stability ◽  
Stability Results ◽  
Dimensional Wave

We investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin–Voigt type and with boundary or localized internal delay feedback. The main novelty in this paper is that the Kelvin–Voigt and the delay damping are both localized via non smooth coefficients. Under sufficient assumptions, in the case that the Kelvin–Voigt damping is localized faraway from the tip and the wave is subjected to a boundary delay feedback, we prove that the energy of the system decays polynomially of type t − 4 . However, an exponential decay of the energy of the system is established provided that the Kelvin–Voigt damping is localized near a part of the boundary and a time delay damping acts on the second boundary. While, when the Kelvin–Voigt and the internal delay damping are both localized via non smooth coefficients near the boundary, under sufficient assumptions, using frequency domain arguments combined with piecewise multiplier techniques, we prove that the energy of the system decays polynomially of type t − 4 . Otherwise, if the above assumptions are not true, we establish instability results.

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.

scholarly journals Traveltime calculations from frequency‐domain downward‐continuation algorithms

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1380-1388 ◽  
Author(s):  
Changsoo Shin ◽  
Seungwon Ko ◽  
Wonsik Kim ◽  
Dong‐Joo Min ◽  
Dongwoo Yang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Frequency Domain ◽  
Downward Continuation ◽  
Calculation Algorithm ◽  
Near Surface ◽  
Absolute Value ◽  
Head Waves ◽  
First Arrival ◽  
The One ◽  
Derivatives Of

We present a new, fast 3D traveltime calculation algorithm that employs existing frequency‐domain wave‐equation downward‐continuation software. By modifying such software to solve for a few complex (rather than real) frequencies, we are able to calculate not only the first arrival and the approximately most energetic traveltimes at each depth point but also their corresponding amplitudes. We compute traveltimes by either taking the logarithm of displacements obtained by the one‐way wave equation at a frequency or calculating derivatives of displacements numerically. Amplitudes are estimated from absolute value of the displacement at a frequency. By using the one‐way downgoing wave equation, we also circumvent generating traveltimes corresponding to near‐surface upcoming head waves not often needed in migration. We compare the traveltimes computed by our algorithm with those obtained by picking the most energetic arrivals from finite‐difference solutions of the one‐way wave equation, and show that our traveltime calculation method yields traveltimes comparable to solutions of the one‐way wave equation. We illustrate the accuracy of our traveltime algorithm by migrating the 2D IFP Marmousi and the 3D SEG/EAGE salt models.

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