Using Bayesian statistics to capture the effects of modelling errors in inverse problems

When the parameters of a numerical model are adjusted, so that the predictions of the model match measurements from the real system, we need to take account of two sources of errors. These being measurement errors and modelling errors. Measurement errors are commonly considered, and a number of different approaches are in general usage, the most common being the weighted sum of squares method. In this paper the standard Bayesian equation, used for inverse problems, is reformulated so as to make it more intuitive to use. This allows the inclusion of both a modelling error and correlations between measurements to be carried out easily. The results are tested on a simple one-parameter numerical model and a cross-sectional model of a petroleum reservoir. In the first case the proposed error model appears to work well. In the second case it appears that the objective function is multimodal, leading to multiple acceptable solutions. The results of this paper are important to those whose numerical models are thought to contain significant modelling error. This encompasses many areas of modelling related to earth science and engineering.