Fuzzy-probabilistic calculations of water-balance uncertainty

Abstract

Hydrogeological systems are often characterized by imprecise, vague, inconsistent, incomplete, or subjective information, which may limit the application of conventional stochastic methods in predicting hydrogeologic conditions and associated uncertainty. Instead, predictions and uncertainty analysis can be made using uncertain input parameters expressed as probability boxes, intervals, and fuzzy numbers. The objective of this paper is to present the theory for, and a case study as an application of, the fuzzyprobabilistic approach, combining probability and possibility theory for simulating soil water balance and assessing associated uncertainty in the components of a simple waterbalance equation. The application of this approach is demonstrated using calculations with the RAMAS Risk Calc code, to assess the propagation of uncertainty in calculating potential evapotranspiration, actual evapotranspiration, and infiltration-in a case study at the Hanford site, Washington, USA. Propagation of uncertainty into the results of water-balance calculations was evaluated by changing the types of models of uncertainty incorporated into various input parameters. The results of these fuzzy-probabilistic calculations are compared to the conventional Monte Carlo simulation approach and estimates from field observations at the Hanford site.

Fuzzy-Probabilistic Calculations of Evapotranspiration

“…For example, the application of fuzzy methods is not optimal (i.e., it overestimates uncertainty) when sufficient data are available to construct reliable PDFs needed to perform a Monte Carlo analysis. In a recent paper (Faybishenko 2010), this author demonstrated the application of the fuzzyprobabilistic method using a hybrid approach, with direct calculations, when some quantities can be represented by fuzzy numbers and other quantities by probability distributions and interval numbers (Kaufmann and Gupta 1985;Ferson 2002;Guyonnet et al 2003;Cooper et al 2006). In this paper, the author combines (aggregates) the results of Monte Carlo calculations with multiple E o models by means of fuzzy numbers and p-boxes, using the RAMAS Risk Calc software (Ferson 2002).…”

“…If sufficient information is available, probability density functions (PDFs) of input parameters can be used for stochastic simulations to assess aleatory evapotranspiration uncertainty. In the event of a lack of reliable experimental data, fuzzy numbers can be used for fuzzy or fuzzy-probabilistic calculations of the aleatory evapotranspiration uncertainty (Faybishenko 2010). Epistemic uncertainty arises because of a lack of knowledge or poor understanding, ambiguous, conflicting, or insufficient experimental data needed to characterize coupledphysics phenomena and processes, as well as to select or derive appropriate conceptualmathematical models and their parameters.…”

“…For the sake of simulation simplicity, the input parameters are assumed to be independent variables. Scenarios 0 to 8, described in detail in Faybishenko (2010), are based on the application of a single Penman model for E o calculations, with annual average values of input parameters. Scenario 0 was modeled using input PDFs by means of Monte Carlo simulations, using RiskAMP Monte Carlo Add-In Library version 2.10 for Excel.…”

“…Table 2. Scenarios of input and output parameters used for water-balance calculations (Scenarios 0, and 1-8 are from Faybishenko, 2010). Figure 4a shows cumulative distributions of E o from different models, along with an aggregated p-box, and Figure 4b shows the corresponding FMFs (calculated as normalized PDFs) of E o from different models, along with an aggregated trapezoidal fuzzy E o .…”

“…Note that this work is a further extension of this author's recently published work (Faybishenko, 2007(Faybishenko, , 2010. The structure of this chapter is as follows: Section 2 includes a review of semi-empirical equations describing potential evapotranspiration, and a modified Budyko's model for evaluating evapotranspiration.…”