Mathematical Reformulation of Highly Nonlinear Large-scale Inverse Problem in Metric Space

Solving a highly nonlinear and often time-dependent large-scale inverse problem is still challenging due to the computing costs, various types and scales of data, nonlinearity between the model and the data. Based on the observation that in most inverse problem of flow the (geological) model is very complex while the response under investigation (the data) is of much lower dimension, we propose to reformulate the inverse problem in metric space. Once we know the distance between any two (geological) model realizations, any such model (whether a structure or property) can be mapped into a metric space, which is non-dimensional but can be represented as its projection to low-dimensional (typically 2D-5D) space through multi-dimensional scaling. Knowing the differences between the responses of the models and the data (= response from the true Earth), the location of this truth can be identified. Therefore, the inverse problem is reformulated by finding the ensemble of models which is mapped into the location of the truth. We propose a methodology to solve this reformulated inverse problem by a series of mathematical techniques. We apply the proposed method to a realistic reservoir history matching problem, which contains various types of constraints and requires large computational costs.