1.5 Outline of the Thesis

Chapter 1 has presented a brief introduction to the concepts of uncertainty, and the importance of uncertainty analysis in the context of transport-transformation models. It has also discussed the steps involved in a systematic uncertainty analysis, and the associated limitations. Additionally, Chapter 1 summarizes the objectives and presents a brief overview of this thesis (Figure 1.1).

Chapter 2 presents the relevant background information in the following order:

• classification of uncertainty into natural, model, and parametric/data uncertainties, and into reducible and irreducible uncertainties,
• approaches for representing parametric/data uncertainty, such as, Interval Mathematics, Fuzzy Theory, and Probabilistic Analysis (discussed further in Appendix A), and
• conventional methods for sensitivity and sensitivity/uncertainty analysis, classified here as sensitivity testing methods, analytical methods, sampling based methods, and computer algebra based methods.

Additionally, Chapter 2 presents several applications of the conventional methods to the uncertainty analysis of transport-transformation models. The merits and limitations of each method are discussed, and an argument is made for alternative methods that can be used for combined sensitivity and uncertainty analysis. This argument is made both from the perspective of the computational cost reduction, and from the perspective of obtaining sensitivity information.

Chapter 3 presents the Stochastic Response Surface Method (SRSM), which is developed here as a computationally efficient alternative method for uncertainty analysis of numerical models. The formulation, implementation, and application of the SRSM are discussed in detail. Individual sections elaborate on the steps involved in the application of the SRSM: (a) representation of random inputs in a ``standardized manner'', (b) expression of model outputs in a parametric form (e.g., series expansions), (c) estimation of the parameters of output approximation (e.g., coefficients in a series expansion), (d) calculation of the statistical properties of outputs, and (e) evaluation of the approximation. Chapter 3 also presents an example problem that illustrates the application of the SRSM, explaining all the steps in detail; case studies for the evaluation of the SRSM are presented in Chapter 5. Additionally, Chapter 3 presents a web-interface for the application of the SRSM through a browser, without requiring additional effort in studying the mathematical details of the method.

Chapter 4 presents the coupling of the SRSM with the Automated Differentiation method, ADIFOR (Automatic DIfferentiation of FORtran). ADIFOR is a computer algebra based method that reads the Fortran source code of a model and constructs the code that calculates first order partial derivatives of model outputs with respect to model inputs or parameters. In Chapter 4, first the rationale for coupling the SRSM with ADIFOR is presented, and is followed by a description of ADIFOR. Then, the coupling of the SRSM with ADIFOR is presented, and the application of the coupled method, SRSM-ADIFOR, is illustrated with an example problem.

Chapter 5 presents a comprehensive evaluation of both the SRSM and the SRSM-ADIFOR methods. This evaluation consists of four case studies, covering a wide range of transport-transformation models, as follows:

I:

A zero-dimensional physiologically-based pharmacokinetic model.

II:

A two-dimensional reactive atmospheric plume model.

III:

A three-dimensional urban/regional scale photochemical air quality model.

IV:

A one-dimensional ground water model with discontinuous probability density functions for inputs and parameters.

The SRSM is evaluated for all the four models, whereas the SRSM-ADIFOR is evaluated for the first two models. Each case study description contains the information about the model and its application in exposure/risk assessment. Then, the uncertainties associated with the model are described, and the application of different uncertainty propagation methods is presented. The estimates from the uncertainty propagation methods are then presented graphically, in the form of probability density functions (pdfs), along with a brief discussion at the end.

Chapter 6 addresses the issues involved with model uncertainty, which arises due to uncertainties in model formulation and model structure. This chapter describes how such uncertainty can be addressed in the framework of photochemical air quality modeling. A study is presented involving a regulatory atmospheric photochemical plume model, the Reactive Plume Model (RPM). The development and implementation of the RPM-3D, a three-dimensional version of the RPM, is presented in this Chapter. Subsequently, case studies are presented to compare the estimates of RPM-3D and RPM, in order to characterize uncertainties associated with model resolution.

Chapter 7 presents the conclusions of this thesis, and recommendations for future work. This is followed by bibliography.

Appendix A presents background information on probabilistic analysis that is useful in understanding the SRSM and the SRSM-ADIFOR. Appendix B, Appendix C, and Appendix D present information on the models used in the case studies in Chapter 5. Appendix E lists the programs that are included with the accompanying CDROM.

  

Figure 1.1: Schematic depiction of the outline of this thesis. The shaded areas indicate the contributions of this work