Wave-Equation-Based Amplitude Vs Offset (WEB-AVO ) inversion solves the full elastic wave equation both for properties and for the total wavefield. It is a non-linear inversion technique that accounts for multiple scattering and mode conversions inside the target interval. When prior geological information interpreted from well logs is incorporated, stochastic inversion can be performed by honouring Bayes' theorem for probability density functions. The posterior function is proportional to the product of the likelihood function and the prior probability density function.

The prior probability function is built from well logs and is a complex mixture of Gaussians that account for thicknesses, property values and their corresponding standard deviations.

The likelihood function is built from the maximum likelihood estimator, the result of the deterministic inversion, and from the Hessian derived from the inversion kernel, scaled by the variance of the noise in the data.

The present work proposes that the best estimate of the noise in the data can be extracted from the residual of the seismic-to-well match.
The inaccuracy of the method can be quantified by taking the second derivative of the posterior function at the Maximum a Posteriori estimate. The present work also proposes that an additional source of inaccuracy is the intrinsic uncertainty, or non-uniqueness, of the method. It can be estimated with the help of random starting models on a perfect data set (synthetic data).

The stochastic WEB-AVO inversion is a natural extension of the already existing deterministic WEB-AVO inversion workflow. The inversion result is constrained by the prior to honour the true geology observed in the wells.