7.1 Development and Application of the SRSM

One of the main contributions of this research is the development of computationally efficient methods for uncertainty propagation, specifically, the development of the Stochastic Response Surface Method (SRSM) (Chapter 3). The SRSM approximates the model inputs and the outputs through a series of ``well behaved'' standard random variables; the series expansions of the outputs contain unknown coefficients which are calculated by a method that uses the results of a limited number of model simulations. The SRSM is based on the Stochastic Finite Element Method [83] and the Deterministic Equivalent Modeling Method [198].

The SRSM has been implemented as a modular and readily portable stand-alone tool that can be used by researchers as a black-box tool, without requiring the mathematical details of the SRSM. In fact, the SRSM has been implemented as a web-based tool, that can be accessed through a web browser, as shown towards the end of Chapter 3. This implementation includes modules for (a) the transformation of non-standard distributions into functions of the srvs, (b) the construction of polynomial chaos approximations for the model outputs, (c) the identification of sample points for model runs, (d) the estimation of the unknown coefficients in the approximations, and (e) the subsequent calculation of the statistical properties of the model outputs. Further, mathematical formulae have been developed and presented for the extension of this method for dealing with arbitrary empirical distributions (with correlated and uncorrelated inputs), and for the estimation of correlations between two model outputs, and between a model input and a model output.

The development and the application of the SRSM meets the need for fast and accurate estimates of the uncertainties associated with the model outputs, and also the need for identification of inputs that contribute to the output uncertainties the most. The main impact of this research is the facilitation of fast uncertainty analyses of complex, computationally intensive models, and the estimation of the relative contribution of each model input to the uncertainties in the outputs.

The SRSM has been evaluated for case studies involving the following models:

(a)

a biological model (a Physiologically Based PharmacoKinetic (PBPK) model, i.e., a lumped system described by a set of stiff ODEs) describing the uptake and metabolism of perchloroethylene in humans,

(b)

a two-dimensional atmospheric photochemical plume model that calculates pollutant concentrations downwind of point sources, the Reactive Plume Model (RPM),

(c)

a three-dimensional urban/regional scale air quality model, the Urban Airshed Model (UAM), version IV, and

(d)

a one-dimensional ground water model with discontinuous probability density functions, the EPA's Composite Model for leachate Migration and Transformation Products (EPACMTP).

In the first two cases, the performance of the SRSM was significantly superior to both the Monte Carlo and the Latin Hypercube Sampling methods - the SRSM application required up to an order of magnitude fewer simulations for estimating the uncertainties in the model outputs. In the third case study, involving the Urban Airshed Model, it was impractical to perform hundreds of Monte Carlo simulations, whereas, the SRSM required just twenty (20) simulations to estimate the output uncertainties: this demonstrates the potential of the SRSM to address problems that are beyond the scope of conventional methods (in terms of computational and time limitations). In the last case study involving the ground water model, EPACMTP, the SRSM, as before, required significantly fewer model runs compared to the Monte Carlo method.