Modeling Uncertainty of Complex Earth Systems in Metric Space

Modeling the subsurface of the Earth has many characteristic challenges. Earth models reflect the complexity of the Earth subsurface and contain many complex elements of modeling, such as the subsurface structures, the geological processes of growth and/or deposition, and the placement, movement, or injection/extraction of fluid and gaseous phases contained in rocks or soils. Moreover, due to the limited information provided by measurement data, whether from boreholes or geophysics, and the requirement to make interpretations at each stage of the modeling effort, uncertainty is inherent to any modeling effort. As a result, many alternative (input) models need to be built to reflect the ensemble of sources of uncertainty. On the other hand, the (engineering) purpose (in terms of target response) of these models is often very clear, simple, and straightforward: do we clean up or not, do we drill, where do we drill, what are oil and gas reserves, how far have contaminants traveled, etc. The observation that models are complex but their purpose is simple suggests that input model complexity and dimensionality can be dramatically reduced, not by itself, but by means of the purpose or target response. Reducing dimension by only considering the variability between all possible models may be an impossible task, since the intrinsic variation between all input models is far too complex to be reduced to a few dimensions by simple statistical techniques such as principal component analysis (PCA). In this chapter, we will define a distance between two models created with different (and possibly randomized) input parameters. This distance can be tailored to the application or target output response at hand, but should be chosen such that it correlates with the difference in target response between any two models. A distance defines then a metric space with a broad gamma of theory. Starting from this point of view, we redefine many of the current Cartesian-based Earth modeling problems and methodologies, such as inverse modeling, stochastic simulation and estimation, model selection and screening, model updating, and response uncertainty evaluation in metric space. We demonstrate how such a redefinition greatly simplifies as well as increases effectiveness and efficiency of any modeling effort, particularly those that require addressing the matter of model and response uncertainty.