Approximate-vs. full-Hessian in FWI: 1D analytical and numerical experiments

Raul Cova and Kris Innanen


Feasibility of using Full Waveform Inversion (FWI) to build velocity models has been increasing as the computational power and more comprehensive forward modelling approaches have arose. Traditionally, the objective function used during the FWI process has been minimized by the means of steepest-descent methods by conditioning the gradient function. Quasi-and full-Newton methods use the Hessian, or an approximate to it, as a gradient conditioning. For a 1D scalar medium, we derived an analytical expression for the approximate-Hessian suggesting that it brings the model update to within a first order approximation of the exact reflection coefficient for a single interface. In its functional form the Hessian is represented by delta functions into the integration for computing the model update. Compared to the approximate-Hessian, we found that the full-Hessian provides additional scaling information at the depth of the interface, improving the accuracy of the inversion. These ideas were also tested using a numerical example displaying how both Hessians move very fast toward the actual velocity model. It is also shown that the full-Hessian leads to a very accurate inversion in the presence of large velocity contrasts superior to the approximate-Hessian. Hence, the full-Hessian may achieves a faster convergence and accurate inversion while providing amplitude information. For large velocity contrasts, or in a 2D case, where strong AVO effects may be present, the application of the full-Newton FWI might be a good candidate.

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