Nonlinear Fréchet derivative and its De Wolf approximation

Wu, Ru‐Shan; Zheng, Yingcai

doi:10.1190/segam2012-1468.1

Cited by 3 publications

(2 citation statements)

References 28 publications

(24 reference statements)

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“…This leads to the transmission case in which the incident and scattered slowness vectors are parallel. This corresponds to the forward scattering of the De Wolf approximation (De Wolf ; Wu and Zheng ). We shall also consider a perturbation only smoothed in the horizontal direction that leads to the reflection case over a flat horizontal reflector where the incident and scattered slowness vectors are related through Snell's law.…”

Section: Introductionmentioning

confidence: 99%

Analysis of different parameterisations of waveform inversion of compressional body waves in an elastic transverse isotropic Earth with a vertical axis of symmetry

Plessix

2016

Geophysical Prospecting

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In a multi‐parameter waveform inversion, the choice of the parameterisation influences the results and their interpretations because leakages and the tradeoff between parameters can cause artefacts. We review the parameterisation selection when the inversion focuses on the recovery of the intermediate‐to‐long wavenumbers of the compressional velocities from the compressional body (P) waves. Assuming a transverse isotropic medium with a vertical axis of symmetry and weak anisotropy, analytical formulas for the radiation patterns are developed to quantify the tradeoff between the shear velocity and the anisotropic parameters and the effects of setting to zero the shear velocity in the acoustic approach. Because, in an anisotropic medium, the radiation patterns depend on the angle of the incident wave with respect to the vertical axis, two particular patterns are discussed: a transmission pattern when the ingoing and outgoing slowness vectors are parallel and a reflection pattern when the ingoing and outgoing slowness vectors satisfy Snell's law. When the inversion aims at recovering the long‐to‐intermediate wavenumbers of the compressional velocities from the P‐waves, we propose to base the parameterisation choice on the transmission patterns. Since the P‐wave events in surface seismic data do not constrain the background (smooth) vertical velocity due to the velocity/depth ambiguity, the preferred parameterisation contains a parameter that has a transmission pattern concentrated along the vertical axis. This parameter can be fixed during the inversion which reduces the size of the model space. The review of several parameterisations shows that the vertical velocity, the Thomsen parameter δ, or the Thomsen parameter ε have a transmission pattern along the vertical axis depending on the parameterisation choice. The review of the reflection patterns of those selected parameterisations should be done in the elastic context. Indeed, when reflection data are also inverted, there are potential leakages of the shear parameter at intermediate angles when we carry out acoustic inversion.

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Section: Introductionmentioning

confidence: 99%

Analysis of different parameterisations of waveform inversion of compressional body waves in an elastic transverse isotropic Earth with a vertical axis of symmetry

Plessix

2016

Geophysical Prospecting

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“…Zheng (2012, 2014) introduced the higher order Fréchet derivatives and the theory of nonlinear partial derivative (NLPD) operator for the acoustic wave equation. Our previous work (Wu and Zheng 2012, Wu et al 2013 have reported the renormalization procedure using De Wolf series and its approximation to improve the convergence of forward scattering series. In this paper we will report the progress in removing the divergence of inverse Born T-series by renormalization procedure and the derivation of the inverse thin-slab propagator (ITSP).…”

Section: Introductionmentioning

confidence: 99%

Renormalized nonlinear sensitivity kernel and inverse thin-slab propagator in T -matrix formalism for wave-equation tomography

Wang

2015

Inverse Problems

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We derived the renormalized nonlinear sensitivity operator and the related inverse thin-slab propagator (ITSP) for nonlinear tomographic waveform inversion based on the theory of nonlinear partial derivative operator and its De Wolf approximation. The inverse propagator is based on a renormalization procedure to the forward and inverse transition matrix scattering series. The ITSP eliminates the divergence of the inverse Born series for strong perturbations by stepwise partial summation (renormalization). Numerical tests showed that the inverse Born T-series starts to diverge at moderate perturbation (20% for the given model of Gaussian ball with a radius of 5 wavelength), while the ITSP has no divergence problem for any strong perturbations (up to 100% perturbation for test model). In addition, the ITSP is a non-iterative, marching algorithm with only one sweep, and therefore very efficient in comparison with the iterative inversion based on the inverse-Born scattering series. This convergence and efficiency improvement has potential applications to the iterative procedure of waveform inversion.

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Nonlinear sensitivity operator and inverse thin-slab propagator for tomographic waveform inversion

Wu¹,

Hu²,

Wang³

2014

SEG Technical Program Expanded Abstracts 2014

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We derived the nonlinear sensitivity operator and the related inverse thin-slab propagator (ITSP) for nonlinear tomographic waveform inversion based on the theory of nonlinear partial derivative operator. The inverse propagator is based on a renormalization procedure to the forward and inverse T-matrix series. The inverse thin-slab propagator solves the divergence problem of the inverse series for strong perturbations by stepwise partial summation (renormalization). Numerical tests showed that the inverse Born T-series starts to diverge at 20% perturbation (for the given model), while the inverse thinslab propagator has no convergence problem for up to 50% perturbation. This convergence improvement has potential applications to the iterative procedure of waveform inversion.

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Nonlinear Fréchet derivative and its De Wolf approximation

Cited by 3 publications

References 28 publications

Analysis of different parameterisations of waveform inversion of compressional body waves in an elastic transverse isotropic Earth with a vertical axis of symmetry

Analysis of different parameterisations of waveform inversion of compressional body waves in an elastic transverse isotropic Earth with a vertical axis of symmetry

Renormalized nonlinear sensitivity kernel and inverse thin-slab propagator in T -matrix formalism for wave-equation tomography

Nonlinear sensitivity operator and inverse thin-slab propagator for tomographic waveform inversion

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