Estimation of seismic parameters from multifold reflection seismic data by generalized linear inversion of Zoeppritz equations

An inversion method is developed to estimate the P- and S-wave velocities and density ratio of two elastic, isotropic, and homogeneous media separated by a plane, horizontal boundary from P-wave reflection amplitude-versus-offset (AVO) data recorded at the surface. The method has for its basis the inversion of the plane wave Zoeppritz. equations by generalized linear inversion (GLI) and bootstrapping. The Zoeppritz equations are converted into the time-offset domain by using Snell’s law, common mid-point (CMP) geometry, and two-way travel (twt) time. The equations in the time-offset domain have five independent variables that enable estimation of P- and S-wave velocities and density ratio for the upper and lower layers. The linearity and uniqueness of the inversion are investigated by residual function maps (RFMs). The RFMs show closed elliptical contours around the true values of the seismic parameter pairs except in the case of S-wave velocity pair for which the open contours imply a linear correlation. However, the RFMs of S-wave velocities with the other model parameters show well defined minima, indicating the uniqueness of the inverse problem in the absence of noise. The estimation of seismic parameters is constrained by physical considerations and the results are enhanced statistically by bootstrapping to obtain the most likely solutions, i.e., the mode values of the distribution functions of solutions, and the confidence limits of the most likely solutions.

The inversion method is tested using model AVO data with and without random noise. The tests show that the model parameters are exactly recovered when offset-to-depth (O/D) ratio 1s about 2 or larger, depending on the contrast among the seismic parameters of the media. The results for small O/D ratios (< 1) diverge from the true values, especially for S-wave velocities, and indicate the importance of the O/D ratio in the AVO data inversion. The parameters are not recovered correctly in the case of noisy model AVO data because of the degrading effect of noise in the inversion. However, the model parameters fall into the confidence limits of the estimated parameters when tight constraints are imposed on the solutions, and the signal to noise (S/N) ratio is high. The inversion method is sensitive to auxiliary parameters such as the root-mean-square (rms) velocity and zero-offset twt time which are used in the adjustments of observed or calculated reflection amplitudes to compensate for the effects of wave propagation. Because the plane wave Zoeppnitz equations define the variation in reflection amplitude with offset for a single boundary, the method is limited to isolated reflections in the CMP gathers.

The AVO inversion is applied to field data from the Atlantic Coastal Plain in South Carolina to show the feasibility of the method. The first example is from Charleston, S.C. where the estimated seismic parameters from adjacent CMP gathers are in close agreement demonstrating the stability of the AVO inversion. The second example is a data set that crosses the border fault of the Dunbarton Triassic basin, S.C. For this data set common offset stacked CMP gathers are used to increase the S/N ratio and minimize the surface coupling effects. The inversion results show that the seismic parameters are greater north of the border fault indicating crystalline basement while smaller parameters to the south represent the Triassic basin. P-wave velocities estimated for the crystalline basement (6.4 km/s) and the Triassic basin (4.8 km/s) are in good agreement with the computed refraction velocities and support the interpreted location of the Dunbarton Triassic border fault.