Providing for uncertainty in non-linear inversions of geophysical data

A method of obtaining a model of a sub-surface region, the model comprising a three-dimensional matrix of sub-surface material property values, and comprising:

• obtaining an upgoing energy wave data set drec associated with a set of known acquisition parameters including a source location and a plurality of spaced apart recording locations within, at or above the surface, the upgoing energy waves depending on downgoing energy waves and the sub-surface material properties;
• providing a first approximation of said model;
• performing simulations using said first approximation of said model to generate (i) a reference upgoing energy wave data set d0, where the acquisition parameters used in the simulation are said set of known acquisition parameters, and (ii) a plurality of perturbed upgoing energy wave data sets di, where the acquisition parameters are obtained by randomly perturbing said set of known acquisition parameters;
• forming a matrix D, the columns of which are the differences di−d0 between 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;
• selecting the most important vectors from said set of vectors to obtain a subset of vectors that describe said matrix D to an approximation;
• and
• performing an iterative procedure 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 drec and 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 drec.