The goal of geophysical exploration is to estimate maps of subsurface physical properties from remotely sensed data. The associated inverse problem is highly nonlinear; the objective function is characterized by multiple peaks and troughs. Often gradient descent algorithms are employed for this purpose, which require good starting models. Significant effort is made into building starting models, which are found inadequate in many applications. Global optimization methods such as simulated annealing (SA), genetic algorithms and particle swarm optimization (PSO) have been gaining increasing popularity amongst geophysics community. Recently we applied SA and PSO to full waveform inversion. The SA method was used to generate a starting model with sparse parameterization, which was then used in a gradient based FWI. A new gradient-guided PSO was also applied in FWI with encouraging results.
One other aspect of geophysical inversion is ‘uncertainty quantification’ – often this is achieved by constructing the Hessian at the ‘best-fitting’ model. This essentially assumes that the posterior probability density (PPD) is Gaussian. For a PPD that is multi-modal, MCMC method is the preferred choice but the convergence of such methods is very slow. One other important aspect that is often ignored is that uncertainty is introduced in the model from our choice of gridding or the number of model parameters. To address this, we require that the gridding or the number of model parameters be treated as a variable. Such trans-dimensional inverse problems can be addressed by using the reversible jump MCMC, which is computationally very expensive. We have developed a new scheme in which we combine rjMCMC with the Hamiltonian Monte Carlo (HMC). Our rjHMC is a couple of orders of magnitude faster than the standard rjMCMC.
In my talk I will review global optimization methods and uncertainty quantification in geophysical context and show results from our own work on FWI.
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