Metin M. Ozbek

An Application of Dempster-Shafer Theory to Hydraulic Conductivity

While uncertainty is an integral part of the mathematical representation of the environment, behavior forecasting requires the use of mathematical models that require the specification of physically based parameters descriptive of the environment. In subsurface hydrology, for example, the hydraulic conductivity (a measure of soil permeability) must be specified in equations descriptive of groundwater flow. Traditionally probability is used to characterize uncertainty in hydraulic conductivity (K). It seeks to describe uncertainty arising from a lack of knowledge regarding concepts that are inherently crisp and well defined. However, classical probability itself is not applicable to situations where the concepts themselves are vague. In this situation one must consider other avenues for assessing the uncertainty. It is our intention to use a Dempster-Shafer Theory framework to merge probabilistic and fuzzy (subjective) information in an effort to improve our ability to fully define a hydraulic conductivity field. The advantages over using probability theory alone include 1) being able to use all available data to analyze hydraulic conductivity uncertainty (outliers are kept in the analysis) and 2) not having to make assumptions about distribution functions (e.g. typically a lognormal distribution is used to describe hydraulic conductivity of a site). Successful combination of subjective and empirical information will improve our ability to properly describe subsurface heterogeneity and would result in improved models of subsurface environments.

A new perspective on modeling groundwater contaminant-driven health risk with subjective information

Fuzzy rule-based systems provide an efficient environment for the modeling of expert information in the context of risk management for groundwater contamination problems. In general, their use in the form of conditional pieces of knowledge has been either as a tool for synthesizing control laws from data (i.e., conjunction-based models), or in a knowledge representation and reasoning perspective in Artificial Intelligence (i.e., implication-based models), where only the latter may lead to coherence problems (e.g., input data that leads to logical inconsistency when added to the knowledge base). We implement an extension to an implication-based groundwater risk model [7] by incorporating sufficient conditions for a coherent knowledge base. The original model assumes statements of public health professionals for the characterization of the exposed individual and the relation of dose and pattern of exposure to its carcinogenic effects. We demonstrate the utility of the extended model in that it: 1) identifies inconsistent statements and sets coherence in the knowledge base, and 2) minimizes the burden of knowledge elicitation from the experts thereby utilizing existing knowledge in an optimal fashion.

A fuzzy-petri net formalization of expert information for groundwater risk management

The characterization and management of risk in groundwater contamination problems can be improved using subjective expert information. Petri Nets are graphical and mathematical modeling tools for describing information processing systems whereas Fuzzy Set Theory provides a quantitative environment for representing subjective information. The framework used herein is a Fuzzy-Petri Net (FPN), a synergy of Petri Nets and Fuzzy Set Theory. We demonstrate features of the FPN model used for managing a knowledge base available for the solution of a health risk based design problem of benzene contaminated groundwater. Statements of public-health professionals on the toxic behavior of benzene constitute the FPNs knowledge base ( Ozbek and Pinder, 2000 ). First, the net representation of the knowledge base is utilized to construct a subnet which is based on expert information concerning the inquiry, risk-informed concentration constraints for an optimization problem in this case. The resulting subnet is then exploited to characterize the concentration constraints.

Subsurface Characterization Using Geophysical and Geohydrological Data Fusion

We are introducing a novel technology applicable to the robust interpretation of the spatial distribution of hydraulic conductivity in heterogenous formations. The evidence theory approach is based on a combination of directly measured hydrogeological and geophysical data together with expert opinion. The approach first utilizes fuzzy-set based approximate reasoning to quantify subjective expert opinion, especially when data are scarce, to create a fuzzy prior characterization of the hydraulic conductivity field. Subjective information includes, but is not limited to opinions of groundwater professionals regarding the relationship between soil type and grain size or on the grain size to hydraulic conductivity relationship. Utilization of qualitative insights into the effects of geological processes as interpreted by the groundwater professional on hydraulic conductivity properties at a site is also a good example. Secondly, we propose to provide a new framework for integrating available kriged borehole data as well as geophysical information with the prior estimated conductivity field. The resulting evidence theory based approach has its core strengths in 1) enabling the simultaneous use of probabilistic uncertainty and other (non-additive) representations of uncertainty (e.g., fuzzy or possibilistic), 2) integrating expert opinion with objective information, 3) solving the delicate problem of choosing an appropriate prior estimate of hydraulic conductivity for the conditioning process, and 4) enabling a complementary (rather than sequential) use of geophysical data during the characterization process. The framework described herein allows a sensitivity analysis of the resultant characterization with respect to available data which makes it possible to assess the value (in the sense of consistency or conflict) of information. A site application demonstrates the approach.

Fuzzy Inference for Hydraulic Conductivity Estimation

A representation of hydraulic conductivity is most vital to the accurate modeling of groundwater flow. However, in practical applications, hydraulic conductivity measurements are few, while information on soil type and grain size is relatively abundant. The relationship between hydraulic conductivity and the above soil parameters is imprecise and should be modeled as such using fuzzy logic. A fuzzy inference system is proposed whereby grain size distributions and field observations are used to estimate hydraulic conductivity values. To rectify the shortcomings of preferential spatial sampling, spatial fuzzy geologic relationships are defined to infer the location of stratigraphic units from the locations of soil samples. Such relationships, founded upon an expert understanding of hydrogeology, increase the amount of available soil type data, which, in turn, increases the number of hydraulic conductivity estimates. The result is a thoroughly “sampled” domain and consequent hydraulic conductivityy field for input to a numerical simulator.