Srikanta Mishra

Model averaging techniques for quantifying conceptual model uncertainty.

In recent years a growing understanding has emerged regarding the need to expand the modeling paradigm to include conceptual model uncertainty for groundwater models. Conceptual model uncertainty is typically addressed by formulating alternative model conceptualizations and assessing their relative likelihoods using statistical model averaging approaches. Several model averaging techniques and likelihood measures have been proposed in the recent literature for this purpose with two broad categories--Monte Carlo-based techniques such as Generalized Likelihood Uncertainty Estimation or GLUE (Beven and Binley 1992) and criterion-based techniques that use metrics such as the Bayesian and Kashyap Information Criteria (e.g., the Maximum Likelihood Bayesian Model Averaging or MLBMA approach proposed by Neuman 2003) and Akaike Information Criterion-based model averaging (AICMA) (Poeter and Anderson 2005). These different techniques can often lead to significantly different relative model weights and ranks because of differences in the underlying statistical assumptions about the nature of model uncertainty. This paper provides a comparative assessment of the four model averaging techniques (GLUE, MLBMA with KIC, MLBMA with BIC, and AIC-based model averaging) mentioned above for the purpose of quantifying the impacts of model uncertainty on groundwater model predictions. Pros and cons of each model averaging technique are examined from a practitioners perspective using two groundwater modeling case studies. Recommendations are provided regarding the use of these techniques in groundwater modeling practice.

Global sensitivity analysis techniques for probabilistic ground water modeling.

Global sensitivity analysis techniques are better suited for analyzing input-output relationships over the full range of parameter variations and model outcomes, as opposed to local sensitivity analysis carried out around a reference point. This article describes three such techniques: (1) stepwise rank regression analysis for building input-output models to identify key contributors to output variance, (2) mutual information (entropy) analysis for determining the strength of nonmonotonic patterns of input-output association, and (3) classification tree analysis for determining what variables or combinations are responsible for driving model output into extreme categories. These techniques are best applied in conjunction with Monte Carlo simulation-based probabilistic analyses. Two examples are presented to demonstrate the applicability of these methods. The usefulness of global sensitivity techniques is examined vis-a-vis local sensitivity analysis methods, and recommendations are provided for their applications in ground water modeling practice.

An Evaluation of Sharp Interface Models for CO2 -Brine Displacement in Aquifers.

Understanding multiphase transport within saline aquifers is necessary for safe and efficient CO2 sequestration. To that end, numerous full-physics codes exist for rigorously modeling multiphase flow within porous and permeable rock formations. High-fidelity simulation with such codes is data- and computation-intensive, and may not be suitable for screening-level calculations. Alternatively, under conditions of vertical equilibrium, a class of sharp-interface models result in simplified relationships that can be solved with limited computing resources and geologic/fluidic data. In this study, the sharp-interface model of Nordbotten and Celia (2006a,2006b) is evaluated against results from a commercial full-physics simulator for a semi-confined system with vertical permeability heterogeneity. In general, significant differences were observed between the simulator and the sharp-interface model results. A variety of adjustments were made to the sharp-interface model including modifications to the fluid saturation and effective viscosity in the two-phase region behind the CO2 -brine interface. These adaptations significantly improved the predictive ability of the sharp interface model while maintaining overall tractability.

Chapter 9 – Concluding Remarks

This chapter provides a recapitulation of the main topics of the book, discusses the style adopted along with the intended use of the book and the contents of an online resource. This is followed by some key takeaways regarding variable and model selection, limitations of data-driven modeling and extrapolating from the past, and fitting versus over fitting. The chapter ends with some thoughts on the road ahead.

Chapter 6 – Uncertainty Quantification

The focus of this chapter is uncertainty quantification, which involves translating the uncertainty in the inputs of a model into the corresponding uncertainty in model outputs. To this end, we present a systematic approach regarding how to characterize the uncertainties, propagate them through the system model of interest into uncertainties in model predictions, and analyze the relative importance of various sources of uncertainty.

Chapter 1 – Basic Concepts

Statistics is the science of acquiring and utilizing data. It provides us with the tools for data collection, summarization, and interpretation, with the goal of identifying the underlying structure, trends, and relationships inherent in the data. This is how we convert data into information.

Regression Modeling and Analysis

Regression modeling is one of the most widely used tools for exploring and exploiting the relationship between dependent (response) and independent (predictor) variables. When the relationship can be expressed using linear equations (i.e., straight lines and their generalizations in multiple dimensions), it is called a linear regression. In the petroleum geosciences, a very broad class of problems can be addressed using linear regression and its variations. Such variations often involve simple transformation of the response and/or predictor variables (e.g., logarithmic) to linearize their relationship. With little loss of generality, regression modeling concepts for continuous data can also be applied to categorical data such as geologic facies. Furthermore, using arbitrary smooth functions ( scatterplot smoothers ) for data transformation, we can extend the regression modeling to identify inherent nonlinear relationship between response and predictor variables. The generalized linear models and alternating conditional expectation are examples of such generalization using data transformations.

Chapter 7 – Experimental Design and Response Surface Analysis

Numerical models are widely used in engineering and scientific studies. Making a number of simulation runs at various input configurations is what we call a computer experiment. The design problem is the choice of inputs for efficient analysis of data. Experimental design is an intelligent way to pick the choice of input combinations for minimizing the number of computer model runs for the purpose of data analysis, inversion problems, and input uncertainty assessment. One way to carry the above tasks on experimental design results is to build a response surface. A response surface is an empirical fit of computed responses as a function of input parameters.

Chapter 5 – Multivariate Data Analysis

In the previous chapter, we introduced multivariate regression techniques involving two or more variables. Before embarking on an analysis involving large number of variables, we might want to first examine if there are any underlying data structure or patterns that we can exploit to improve and sometimes simplify the analysis. A common approach will be to graphically visualize the data cloud that is limited to three variables. Often, a fourth dimension can be added by varying the type and size of symbols, but that is our limit for graphic visualization. For high-dimensional datasets, an alternative approach is to reduce the dimensionality of the data with minimum loss of important attributes, for example, data variance. Multivariate data analysis techniques allow us to accomplish these goals. Essentially, we define a smaller number of linear combination of the original data, called principal components that allow for data visualization and pattern recognition in a reduced dimensional space. The pattern recognition or classification techniques can be either “supervised” or “unsupervised.” In the unsupervised classification techniques, commonly known as cluster analysis, we partition the data into relatively “homogeneous” entities based on the characteristics of the data, without resorting to prior information. In the supervised pattern-recognition method, also known as discriminant analysis, we assign group membership to a given dataset based on a prior classification. Multivariate data analysis by itself is a vast topic, and several excellent references are available on this topic. In this chapter, we limit our discussion to three important elements of multivariate data analysis, namely, principal component analysis, cluster analysis, and discriminant analysis, in the context of data partitioning and pattern recognition for multiple regression. After introducing the concepts using a simple example, we discuss in detail the application of these techniques to the Salt Creek field data introduced in the previous chapter.

Chapter 2 – Exploratory Data Analysis

Exploratory data analysis, which is concerned with summarizing and visualizing data as a starting point for more detailed analyses, is the subject of this chapter. We restrict ourselves to numerical data (as opposed to text or images) and note that: (a) data can be univariate or multivariate, data can be categorical or numerical, (c) random variables can have more than one value, and (d) distributions capture the values taken by variables, and the frequency with each specific value occurs.