(Buxton 1989), but not for vulnerability assessments. Methods for evaluating uncertainty associated with data errors can be grouped into the following five categories (Brandstetter and Buxton 1989):

  1. Classical statistical variance component analysis, which can be used to partition the total observed variance in the output to contributing factors.

  2. First-order uncertainty analysis (FOUA) based on Taylor series expansion of the function (model) to evaluate variance of the output as a function of the variance in input parameters.

  3. Statistical sampling methods that utilize a range of likely values for input parameters to assess the probable range of output parameters. Examples are Monte Carlo simulation, Latin hypercube sampling, discrete-event simulation, and boot-strapping methods.

  4. Stochastic modeling approaches that directly incorporate the parameter or process uncertainties into the model itself and provide direct uncertainty estimates of model outputs.

  5. Bayesian methods when uncertainties in input parameters can be specified by either expert judgment, or estimated from existing databases from which input parameter values have been determined.

Of these techniques, only FOUAs statistical sampling methods, and stochastic modeling techniques have been applied to vulnerability assessments. A description of the FOUA technique is presented in Box 3.1 and an example of the use of Monte Carlo methods in Box 3.2. Recent examples of first-order uncertainty analysis applied to a process-based index of vulnerability are reviewed below. Small and Mular (1987) and Jury and Gruber (1989) present examples of the applications of stochastic modeling approaches to evaluate uncertainty associated with climatic and soil variability in assessments of ground water vulnerability.

Applications of FOUA to Vulnerability Assessment

The earliest attempt to utilize spatial modeling techniques for regional-scale assessment of pesticide leaching potential was reported by Khan et al. (1986) and Khan and Liang (1989). They used two simple indices—the Retardation Factor (RF) and the Attenuation Factor (AF) developed by Rao et al. (1985)—as measures of leaching potential, and a GIS database—the Hawaii Natural Resource Information System (HNRIS) developed by Liang and Khan (1986)—in conducting an assessment for the Hawaiian island of Oahu. An example of the vulnerability maps generated by Khan and Liang (1989) is shown in Figure 3.3. The RF index is a measure of the relative time needed for a pesticide pulse to leach past some specified depth when compared to a nonsorbed tracer, whereas the AF index is the fraction of the

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