(untitled) (Communications in Computational Physics, 2020)
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.