Joint inversion of AVA data for elastic parameters by bootstrapping

Abstract

A joint inversion method is developed to estimate the elastic constants of two elastic, homogeneous, isotropic media separated by a flat horizontal boundary. The method jointly uses P and S-converted wave reflection amplitude-versus-angle (AVA) data and seeks the Poisson's ratios of each layer, ratios of the densities and bulk modulus of the layers. The generalized linear inversion (GLI) method is used as a mathematical tool and the Zoeppritz equations defining the seismic energy partitioning at a boundary are used as the physical model. The P and S-converted wave velocity terms in the Zoeppritz equations were replaced by the bulk modulus ( k 1 , k 2 ) , Poisson's ratios ( σ 1 , σ 2 ) , and densities ( ρ 1 , ρ 2 ) of each layer. After expressing the equations in these six elastic constants, reflection coefficients of P and S-converted waves ( R pp , R ps ) are obtained as functions of ratios of bulk modulus and densities of the lower layer to those of the upper layer ( k 2 / k 1 and ρ 2 / ρ 1 ) and Poisson's ratios of the upper and lower layers ( σ 1 and σ 2 ) . Using the ratios of bulk modulus and densities, the number of unknown parameters is reduced from 6 to 4 and this improves the success of inversion. The other contribution is that the calculation of R pp and R ps and their derivatives with respect to elastic constants and their ratios in the inversion are calculated analytically and coded in the Fortran programming language. In this way, the approach has an important advantage among the other AVA inversion methods, which are mostly based on numerical solutions or approximations to the Zoeppritz equations. A bootstrapping method of statistical analysis is combined with the GLI method to find the most likely elastic parameters and their confidence limits for repeated inversions for a large number of times by rearranging the noise distribution of the AVA data.