Volume 28, Issue 1
Towards a Theoretical Background for Strong-Scattering Inversion – Direct Envelope Inversion and Gel'fand-Levitan-Marchenko Theory

Ru-Shan Wu

Commun. Comput. Phys., 28 (2020), pp. 41-73.

Published online: 2020-05

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  • Abstract

Strong-scattering inversion or the inverse problem for strong scattering has different physical-mathematical foundations from the weak-scattering case. Seismic inversion based on wave equation for strong scattering cannot be directly solved by Newton's local optimization method which is based on weak-nonlinear assumption. Here I try to illustrate the connection between the Schrödinger inverse scattering (inverse problem for Schrödinger equation) by GLM (Gel'fand-Levitan-Marchenko) theory and the direct envelope inversion (DEI) using reflection data. The difference between wave equation and Schrödinger equation is that the latter has a potential independent of frequency while the former has a frequency-square dependency in the potential. I also point out that the traditional GLM equation for potential inversion can only recover the high-wavenumber components of impedance profile. I propose to use the Schrödinger impedance equation for direct impedance inversion and introduce a singular impedance function which also corresponds to a singular potential for the reconstruction of impedance profile, including discontinuities and long-wavelength velocity structure. I will review the GLM theory and its application to impedance inversion including some numerical examples. Then I analyze the recently developed multiscale direct envelope inversion (MS-DEI) and its connection to the inverse Schrödinger scattering. It is conceivable that the combination of strong-scattering inversion (inverse Schrödinger scattering) and weak-scattering inversion (local optimization based inversion) may create some inversion methods working for a whole range of inversion problems in geophysical exploration.

  • Keywords

Strong-scattering, strong nonlinear inversion, GLM theory, envelope inversion.

  • AMS Subject Headings

74J20, 86A15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

rwu@ucsc.edu (Ru-Shan Wu)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-28-41, author = {Wu , Ru-Shan }, title = {Towards a Theoretical Background for Strong-Scattering Inversion – Direct Envelope Inversion and Gel'fand-Levitan-Marchenko Theory}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {1}, pages = {41--73}, abstract = {

Strong-scattering inversion or the inverse problem for strong scattering has different physical-mathematical foundations from the weak-scattering case. Seismic inversion based on wave equation for strong scattering cannot be directly solved by Newton's local optimization method which is based on weak-nonlinear assumption. Here I try to illustrate the connection between the Schrödinger inverse scattering (inverse problem for Schrödinger equation) by GLM (Gel'fand-Levitan-Marchenko) theory and the direct envelope inversion (DEI) using reflection data. The difference between wave equation and Schrödinger equation is that the latter has a potential independent of frequency while the former has a frequency-square dependency in the potential. I also point out that the traditional GLM equation for potential inversion can only recover the high-wavenumber components of impedance profile. I propose to use the Schrödinger impedance equation for direct impedance inversion and introduce a singular impedance function which also corresponds to a singular potential for the reconstruction of impedance profile, including discontinuities and long-wavelength velocity structure. I will review the GLM theory and its application to impedance inversion including some numerical examples. Then I analyze the recently developed multiscale direct envelope inversion (MS-DEI) and its connection to the inverse Schrödinger scattering. It is conceivable that the combination of strong-scattering inversion (inverse Schrödinger scattering) and weak-scattering inversion (local optimization based inversion) may create some inversion methods working for a whole range of inversion problems in geophysical exploration.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0108}, url = {http://global-sci.org/intro/article_detail/cicp/16826.html} }
TY - JOUR T1 - Towards a Theoretical Background for Strong-Scattering Inversion – Direct Envelope Inversion and Gel'fand-Levitan-Marchenko Theory AU - Wu , Ru-Shan JO - Communications in Computational Physics VL - 1 SP - 41 EP - 73 PY - 2020 DA - 2020/05 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2018-0108 UR - https://global-sci.org/intro/article_detail/cicp/16826.html KW - Strong-scattering, strong nonlinear inversion, GLM theory, envelope inversion. AB -

Strong-scattering inversion or the inverse problem for strong scattering has different physical-mathematical foundations from the weak-scattering case. Seismic inversion based on wave equation for strong scattering cannot be directly solved by Newton's local optimization method which is based on weak-nonlinear assumption. Here I try to illustrate the connection between the Schrödinger inverse scattering (inverse problem for Schrödinger equation) by GLM (Gel'fand-Levitan-Marchenko) theory and the direct envelope inversion (DEI) using reflection data. The difference between wave equation and Schrödinger equation is that the latter has a potential independent of frequency while the former has a frequency-square dependency in the potential. I also point out that the traditional GLM equation for potential inversion can only recover the high-wavenumber components of impedance profile. I propose to use the Schrödinger impedance equation for direct impedance inversion and introduce a singular impedance function which also corresponds to a singular potential for the reconstruction of impedance profile, including discontinuities and long-wavelength velocity structure. I will review the GLM theory and its application to impedance inversion including some numerical examples. Then I analyze the recently developed multiscale direct envelope inversion (MS-DEI) and its connection to the inverse Schrödinger scattering. It is conceivable that the combination of strong-scattering inversion (inverse Schrödinger scattering) and weak-scattering inversion (local optimization based inversion) may create some inversion methods working for a whole range of inversion problems in geophysical exploration.

Ru-Shan Wu. (2020). Towards a Theoretical Background for Strong-Scattering Inversion – Direct Envelope Inversion and Gel'fand-Levitan-Marchenko Theory. Communications in Computational Physics. 28 (1). 41-73. doi:10.4208/cicp.OA-2018-0108
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