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This chapter addresses a range of topics that hold considerable promise for future developments. We start by considering nested-inversions that allow definition of heterogeneity across a wide range of length scales from local through regional to global. This is followed by discussion of adaptive numerical gridding, exploitation of data redundancy, the development of efficient random sampling methods for inversion, and the use of Hamiltonian Monte-Carlo techniques for efficient searching of high-dimensional spaces.
We here establish basic inversion framework in a Bayesian context, with introduction of measures of data fit and model suitability. We introduce Bayes’ theorem and identify the conditional probability with posterior probability distribution for model parameters through a composite misfit combining the match between observations and simulations and assumptions about the nature of acceptable models. We discuss Monte Carlo techniques and the assessment of model resolution, leading into the formulation of the non-linear inversion process in terms of optimisation of a measure of misfit.
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