An investigation of uncertainty in velocity model building problems

Travel time tomography deals with the estimation of subsurface parameters affecting the wave velocity through the medium. It is well-known this is a strongly ill-posed and ill-conditioned inverse problem, which exhibits great structural uncertainties and ambiguities about the true velocity model. Since the estimation of velocities significantly affects depth image analysis and drilling decisions, performing an investigation of model uncertainty becomes paramount in seismic processing workflow. The uncertainty issue is typically treated by relating the algebraic features of the forward operator, which links the acquired measurements to subsurface parameters, with its singular value decomposition. However, when facing geophysical inversions, this strategy becomes infeasible because of the large dimensioning of data. We propose an alternative approach for uncertainty analysis based on Monte Carlo methods. Specifically, we will show how it is possible to evaluate ambiguities related to tomographic inversion by randomly sampling the space of the forward operator. Our method offers a viable alternative solution to singular value decomposition for small-scale problems, at the same time allowing a complete problem characterization for large-scale inversions. In order to show the strength and effectiveness of this proposal, we report a tutorial Velocity Model Building problem, exploring the role of Monte Carlo methods and regularization for uncertainty analysis.