On the inverse problem of reflection Green’s function seismograms over stratified acoustic media: Estimation of background models

Abstract

A theory is presented for estimating the background velocity and density of an acoustic stratified medium by iterative least-squares waveform inversion in the frequency-horizontal slowness domain of low-frequency precritical reflection incidence seismograms of time length [Formula: see text]. The initial model is constant. The prerequisites for the method are that the reflection seismograms should be Green’s function seismograms and that the fundamental frequency component [Formula: see text] is present. Then, the gradients of the objective function provide the low-wavenumber trend of the medium. A matrix formulation for the model update is expressed mathematically by the classic seismogram residual, Jacobian, gradient, and Hessian in the Levenberg-Marquardt approximation. The first iteration, which is equal to a constant-parameter migration inversion (CPMI), is thoroughly analyzed, and expressions for band-limited gradients and block Hessians are found. For primary precritical reflection incidence seismograms of infinite bandwidth, it is shown theoretically that the partial gradients in the CPMI model become a reflection strength-weighted sum of shifted discrete sign functions, typical of step or staircase functions, which provide interface locations in Born depth and amplitudes that can be mapped to velocity and density information. For frequency-band-limited primary reflection seismograms, the partial gradients become a reflection strength-weighted sum of wavenumber-band-limited discrete sign functions. When the fundamental frequency component in the seismograms is present, the band-limited discrete sign functions are oscillatory but keep the information of the step function characteristic of the partial gradient. When the fundamental frequency component in the seismograms is absent, the band-limited discrete sign functions keep information of where the steps are located but lose the information of the amplitudes of the steps. The Hessian elements are nonstandard with the Hessian modeled over a broader frequency range than the frequencies of the observed low-frequency seismogram to avoid it becoming close to singular. The main mathematical findings are illustrated by a simple model and seismograms, for which the background models are found after two iterations. For the sake of completeness, the background models are classically used as initial models in a Levenberg-Marquardt least-squares inversion scheme to estimate the layer velocities and densities from broadband seismograms.