The role of seismic inversion has grown steadily over the last decade, since this mathematical tool is potentially the most complete means of extracting petro-physical information from recorded data. The inverse problem is stated as a non-linear optimal control problem where the objective is to determine the physical parameters that minimize the misfit between the true and the simulated response of the medium. In many cases the parameters of interest are impedance, velocity and density, but also Poisson's ratio, incompressibility, shear modulus, rigidity, porosity or even sand/shale ratios and gas saturation. A theory exists to prescribe how data should be processed to reproduce the model parameters, but, for a number of reasons, it is often difficult to apply. For instance, a rough model can hardly reproduce the complexity of real data, which, in addition, are contaminated by uncertainty on the instrument response. Further, the simulated response is weakly sensitive to perturbations of the medium parameters, while the inverse problem solution shows a strong sensitivity to perturbation of input data (ill-posed problem). Also, since in a realistic experiment the amount of recorded data cannot carry enough information, the solution is seldom unique. In general, the inversion consists of two steps. In the first, the direct problem is solved to obtain the simulated medium response, so that discrepancies between measurements and simulated data can be detected and back propagated to solve the adjoint problem. In the second step, direct and adjoint fields are used to compute a descent direction in the parameter space so as to minimize the data misfit and update the inverse problem solution. The same numerical scheme is used for both direct and adjoint problems. Continuous improvements in computer science have enabled the implementation of increasingly more sophisticated mathematical formulations, closer to the real behaviour of the subsurface, as well as the processing of very large volumes of data.
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