Probabilistic Uncertainty of Parameters and Conceptual Models in Geophysical Inversion

Stochastic uncertainty in parameters estimated from geophysical observations has a long history. In the situation where the data model relationship is linear or may be linearized, and data noise can be characterized, then in principle the uncertainty can be estimated in a straightforward manner. In the optimistic case where data noise can be assumed to follow Gaussian errors with known variances and co-variances then much favoured matrix expressions are available that quantify stochastic model uncertainty for linear problems. As the number of data or unknowns increase, nonlinearity and/or non-uniqueness can become severe, or knowledge of data errors itself becomes uncertain, then there are significant practical challenges in the computation and interpretation of uncertainty. These challenges are well known and much effort has recently been devoted to finding efficient ways to quantify uncertainty for such cases. A major aspect of uncertainty that is often acknowledged but seldom addressed is conceptual uncertainty in the inversion process itself. By this we mean assumptions about the physics, chemistry or geology captured in the forward problem, assumptions about the level or type of data noise, and assumptions about the appropriate complexity and form of the model parameterization. Conceptual assumptions are made in building the inference framework in the first place and conceptual uncertainty can have a significant influence on and feedback with uncertainty quantification. This area is receiving increasing attention in the geosciences utilizing techniques from the field of computational Bayesian statistics, where they are referred to as model selection. This presentation will summarize recent, and not so recent, developments in this field, and point to some promising directions.