(2005) perform pre-stack inversion by starting with the linearized Aki-Richards equation (Aki and Richards, 2002, Richards and Frasier, 1976) which was re-formulated algebraically by Fatti et al. However, the inverse problem of finding the velocities given the amplitudes is much more difficult than the forward problem, given the nonlinear nature of the Zoeppritz equations. This can also be generalized to multiple layers for both the primaries-only case and also the more complete case in which transmitted, converted and multiple reflections are taken into account (Fuchs and Müller, 1971, Kennett, 1983). Using the Zoeppritz equations, and a knowledge of the angle of incidence of the input P-wave as well as the velocities and densities of the two layers (noting that the other angles can be computed using the angle of incidence and the velocities using Snell’s law), we can easily perform forward modeling at the layer boundary to compute the amplitudes of the reflected and transmitted waves. The computation of the amplitudes of the reflected P and Swaves was first solved by Zoeppritz (1919). Mode conversion of an incident P-wave on the boundary between two elastic layers. (2006) shows the fundamental seismic concept behind pre-stack inversion, in which we show the mode-converted reflected and transmitted waves from an incident seismic P-wave at the boundary between two elastic layers in the subsurface of the earth, with an incident angle equal to θ 1. (2006) in a special edition of the Recorder, which was based on the SEG expanded abstract by Hampson et al. One solution to this problem was discussed by Russell et al. The goal of pre-stack seismic inversion is to obtain reliable estimates of P-wave velocity ( V P), S-wave velocity ( V S), and density ( ρ) from which to predict the fluid and lithology properties of the subsurface of the earth. After a description of the algorithm, we then apply our method to both a synthetic and real data set. Given these three assumptions, we show how a final estimate of P-impedance, S-impedance and density can be found by perturbing an initial low-frequency model. The third is that the background trend can be described by a linear relationship between the logarithm of P-impedance and both S-impedance and density. The second is that PP and PS reflectivity as a function of angle can be given by a set of linearized equations. The first is that the linearized approximation for reflectivity holds.

Our algorithm is based on three assumptions.

We present a new approach to the joint simultaneous inversion of PP and PS angle gathers for the estimation of Pimpedance, S-impedance and density.

Colored inversion hampson russell