Non-Linear Joint AVO Inversion of PP and PS Waves in a VTI Medium

Abstract

This paper aims to estimate elastic parameters such as P-wave and S-wave velocity, Thomsen's anisotropy parameters (d, e), density, and layer thickness of a layered medium using pre-stack seismic data and some prior information. Three different algorithms have been developed in an object oriented environment: Travel time inversion, waveform inversion, and joint traveltime and reflectivity inversion. These algorithms can use compressional or joint compressional and converted shear waves data respectively. We have implemented exact parametric travel time equations in a VTI medium for a stack of either anisotropic or isotropic homogeneous layers. Reflection and transmission coefficients for the whole ray path have been calculated. A convolution model is used to calculate synthetic seismograms. To minimize an objective function a non-linear conjugate gradient (CG) algorithm has been implemented. All of the partial derivatives have been calculated analytically which results in a fast and robust differentiation of traveltimes and reflection and transmission coefficients. A numerical example of a stack of 10 layers has been tested for different elastic parameters and thicknesses. Compressional and shear waves velocities and thicknesses have been varied 25 percent from true values while d and e have been varied 100 percent from their true values. Joint traveltime inversion of P ® P & P ® Sv has successfully estimated the true values of the parameters. Waveform inversion is highly non-linear; hence only anisotropic parameters and density while varying up to 100 percent are estimated. Waveform inversion results in a full estimation of the true values of d and e and a fairly good estimate of density.