@Article{CiCP-28-1,
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 = {<p style="text-align: justify;">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&#39;s local optimization method which is based on weak-nonlinear assumption.
Here I try to illustrate the connection between the Schr<span style="text-align: justify;">ö</span>dinger inverse scattering (inverse problem for Schrödinger equation) by GLM (Gel&#39;fand-Levitan-Marchenko) theory and the direct envelope inversion (DEI) using reflection data. The difference between wave equation and Schr<span style="text-align: justify;">ö</span>dinger 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 Schr<span style="text-align: justify;">ö</span>dinger 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 Schr<span style="text-align: justify;">ö</span>dinger
scattering. It is conceivable that the combination of strong-scattering inversion (inverse Schr<span style="text-align: justify;">ö</span>dinger 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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2018-0108},
url = {https://global-sci.com/article/79671/towards-a-theoretical-background-for-strong-scattering-inversion-direct-envelope-inversion-and-gelfand-levitan-marchenko-theory}
}