Information Theory for Correlation Analysis and Estimation of Uncertainty Reduction in Maps and Models

Abstract

Abstract:The quantification and analysis of uncertainties is important in all cases where maps and models of uncertain properties are the basis for further decisions. Once these uncertainties are identified, the logical next step is to determine how they can be reduced. Information theory provides a framework for the analysis of spatial uncertainties when different subregions are considered as random variables. In the work presented here, joint entropy, conditional entropy, and mutual information are applied for a detailed analysis of spatial uncertainty correlations. The aim is to determine (i) which areas in a spatial analysis share information, and (ii) where, and by how much, additional information would reduce uncertainties. As an illustration, a typical geological example is evaluated: the case of a subsurface layer with uncertain depth, shape and thickness. Mutual information and multivariate conditional entropies are determined based on multiple simulated model realisations. Even for this simple case, the measures not only provide a clear picture of uncertainties and their correlations but also give detailed insights into the potential reduction of uncertainties at each position, given additional information at a different location. The methods are directly applicable to other types of spatial uncertainty evaluations, especially where multiple realisations of a model simulation are analysed. In summary, the application of information theoretic measures opens up the path to a better understanding of spatial uncertainties, and their relationship to information and prior knowledge, for cases where uncertain property distributions are spatially analysed and visualised in maps and models.

Monte Carlo simulation for uncertainty estimation on structural data in implicit 3-D geological modeling, a guide for disturbance distribution selection and parameterization

“…However, it is possible to post-process the results of an MCUE simulation to compare them to other forms of prior knowledge and update accordingly . The MCUE approach is usually applied to geometric modeling engines (Wellmann and Regenauer-Lieb, 2012;Lindsay et al, 2013;Jessell et al, 2010Jessell et al, , 2014a, although it can be applied to dynamic or kinematic modeling engines (Wang et al, 2016;Wellmann et al, 2016). This choice is motivated by critical differences between the three approaches, at both the conceptual and practical level (Aug, 2004).…”

“…Often, the disturbance distribution used to estimate input uncertainty is the same (same type and same parameterization) for all observations of the same nature (Wellmann et al, 2010;Wellmann and Regenauer-Lieb, 2012;Lindsay et al, 2012Lindsay et al, , 2013. Disturbance distribution parameters are defined arbitrarily (Lindsay et al, 2012;Wellmann and RegenauerLieb, 2012) in most cases.…”

“…Monte Carlo simulation for uncertainty propagation (MCUE) has been a widely used uncertainty propagation method in implicit 3-D geological modeling during the last decade (Wellmann and Regenauer-Lieb, 2012;Lindsay et al, 2012;Jessell et al, 2014a;de la Varga and Wellmann, 2016). A similar approach was introduced to geoscience with the generalized likelihood uncertainty estimation (GLUE; Beven and Binley, 1992), which is a non-predictive (Camacho et al, 2015) implementation of Bayesian Monte Carlo (BMC).…”

“…www.solid-earth.net/9/385/2018/ E. Pakyuz-Charrier et al: Monte Carlo Simulation for Uncertainty Estimation 387 been used to express the model uncertainty in MCUE, including information entropy (Shannon, 1948;Wellmann, 2013;Wellmann and Regenauer-Lieb, 2012), stratigraphic variability (Lindsay et al, 2012) and kriging error. The case for reliable uncertainty estimation in 3-D geological modeling has been made repeatedly and this paper aims to further improve several points of MCUE methods at the preprocessing steps (Fig.…”

Integrating deterministic lithostratigraphic models in stochastic realizations of subsurface heterogeneity. Impact on predictions of lithology, hydraulic heads and groundwater fluxes

“…Mays et al (2002) used a metric similar to the one defined in Equation (9) to evaluate the complexity of numerical simulations of infiltration through unsaturated heterogeneous soils. Information entropy has also been applied to quantify uncertainty in the context of structural geological models (Wellmann and Regenauer-Lieb, 2012) and geological maps (Wellmann, 2013;Stafleu et al, 2014), as well as to estimate spatial disorder in synthetic aquifers (Scheibe, 1993;Scheibe and Murray, 1998) and in three-dimensional realizations of distributions of sand and clay (Huang et al, 2012).…”