High‐frequency asymptotic solution of the wave equation in an inhomogeneous medium

1991 ◽  
Vol 32 (3) ◽  
pp. 651-655 ◽  
Author(s):  
Sam L. Robinson
Keyword(s):  
Wave Equation ◽  
Asymptotic Solution ◽  
Inhomogeneous Medium ◽  
High Frequency

Parabolic Representations of Scattering Green’s Functions

1999 ◽  
Author(s):  
Paul E. Barbone
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Green’S Function ◽  
Inhomogeneous Medium ◽  
Green's Function ◽  
Free Space ◽  
Green's Functions ◽  
Green’S Functions

Abstract We derive a one-way wave equation representation of the “free space” Green’s function for an inhomogeneous medium. Our representation results from an asymptotic expansion in inverse powers of the wavenumber. Our representation takes account of losses due to scattering in all directions, even though only one-way operators are used.

Ray equations in retarded Snell midpoint coordinates

Geophysics ◽  
1984 ◽  
Vol 49 (12) ◽  
pp. 2100-2108
Author(s):  
Alfonso González‐Serrano ◽  
Mathew J. Yedlin
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Group Velocity ◽  
High Frequency ◽  
Frame Of Reference ◽  
Velocity Analysis ◽  
Acoustic Wave Equation ◽  
Arbitrary Angle ◽  
Frequency Limit ◽  
Velocity Function

Group velocity (ray) equations describe the dynamic behavior of wave‐equation extrapolators in the high‐frequency limit. They are found in general from the dispersion relation of an arbitrary acoustic wave equation. Wave‐equation operators require a background extrapolation velocity. As an application of the group velocity equations, a sensitivity analysis to the background‐operator velocity illustrates the trade‐off between uncertainty in velocity and precision in imaging. Exact wave extrapolators are most useful when the exact velocity function is known. Wave‐equation imaging for velocity analysis in Snell midpoint coordinates requires velocity‐insensitive extrapolation operators. In this frame of reference, approximations of the exact acoustic wave equation are referenced to an arbitrary angle of propagation. Group velocity equations show that in Snell midpoint coordinates, using wide‐reference propagation angles, the fifteen‐degree wave equation gives satisfactory velocity‐independent images. The forty‐five degree wave equation does not appreciably improve the image.

Zeroth‐order asymptotics: Waveform inversion of the lowest degree

Geophysics ◽  
2003 ◽  
Vol 68 (2) ◽  
pp. 614-628 ◽  
Author(s):  
D. W. Vasco ◽  
Henk Keers ◽  
John E. Peterson ◽  
Ernest Majer
Keyword(s):  
Asymptotic Solution ◽  
High Frequency ◽  
Perturbation Technique ◽  
Waveform Inversion ◽  
Time Derivative ◽  
Zero Order ◽  
Bacterial Transport ◽  
Forward Simulation ◽  
Inversion Algorithm ◽  
Transport Site

Sensitivity computation is an integral part of many waveform inversion algorithms. An accurate and efficient technique for sensitivity computation follows from the zero‐order asymptotic solution to the elastodynamic equation of motion. Given the particular form of the asymptotic solution, we show that perturbations in high‐frequency waveforms are primarily sensitive to perturbations in phase. The resulting expression for waveform sensitivity is the time derivative of the synthetic seismogram multiplied by the phase sensitivity. All of the necessary elements for a step in the waveform inversion algorithm result from a single forward simulation. A comparison with sensitivities calculated using a purely numerical perturbation technique demonstrates that zero‐order sensitivities are accurate. Based upon the methodology, we match 330 waveforms from a crosswell experiment at a bacterial transport site near Oyster, Virginia. Each iteration of the waveform inversion takes approximately 18 minutes of CPU time on a workstation, illustrating the efficiency of the approach.

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