Providing for uncertainty in non-linear inversions of geophysical data

Abstract

A method of obtaining a model of a sub-surface region, the model comprising a three dimensional matrix of sub-surface material property values. The method comprises providing a first approximation of said model, and performing simulations using the model to generate (i) a reference upgoing energy wave data set d, where the acquisition parameters used in the simulation are a set of known acquisition parameters, and (ii) a plurality of perturbed upgoing energy wave data sets dwhere the acquisition parameters are obtained by randomly perturbing said set of known acquisition parameters. The method further comprises forming a matrix D, the columns of which are the differences d−dbetween the ith perturbed data set and the reference data set, applying an analysis to the matrix D to describe the matrix using a set of vectors, and selecting the most important vectors from said set of vectors to obtain a subset of vectors that describe said matrix D to an approximation. An iterative procedure is performed to improve said first approximation of said model, or a further approximation model, wherein, at each step of the procedure, a model update Δm is determined, the model update being that model update that minimises a measure of the data misfit between said upgoing energy wave data set dand a data set simulated using the updated model, said measure using said subset of vectors to account for uncertainty in the upgoing energy wave data set d.