Towards a Theoretical Background for Strong-Scattering Inversion–Direct Envelope Inversion and Gel’fand-Levitan-Marchenko Theory  

作　　者：Ru-Shan Wu 

机构地区：[1]Modeling and Imaging Laboratory,Earth and Planetary Sciences,University of California,Santa Cruz,95060,CA,USA

出　　处：《Communications in Computational Physics》2020年第6期41-73,共33页计算物理通讯（英文）

摘　　要：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 Schr̈odinger inverse scattering(inverse problem for Schr̈odinger equation)by GLM(Gel’fand-Levitan-Marchenko)the-ory and the direct envelope inversion(DEI)using reflection data.The difference between wave equation and Schr̈odinger 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 Schr̈odinger 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 multi-scale direct envelope inversion(MS-DEI)and its connection to the inverse Schr̈odinger scattering.It is conceivable that the combination of strong-scattering inversion(inverse Schr̈odinger 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.

关 键 词：Strong-scattering strong nonlinear inversion GLM theory envelope inversion 

分 类 号：O17[理学—数学]