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An analytic technique for stochastic analysis in environmental models
Author(s) -
Tumeo Mark A.,
Orlob Gerald T.
Publication year - 1989
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr025i012p02417
Subject(s) - probability density function , moment (physics) , stochastic process , random variable , monte carlo method , moment generating function , mathematics , stochastic differential equation , probability distribution , mathematical optimization , continuous time stochastic process , statistical physics , gaussian , flexibility (engineering) , fokker–planck equation , computer science , partial differential equation , statistics , mathematical analysis , physics , classical mechanics , quantum mechanics
This paper describes the development and application of a new mathematical technique to include stochasticity in environmental models. The technique, named the probability density function/moment (PDF/M) technique by the authors, is based on a two‐tiered process. First, the basic governing equations are expanded to include stochastic terms. Stochastic terms are separated from nonfluctuating terms, and the resulting set of equations is solved simultaneously. Solutions are used to calculate the moments of the output variables. Second, the moments are used in conjunction with the Fokker‐Planck equation to produce an analytical solution for the probability density functions of the dependent variables. Because the approach produces analytical solutions, it offers greater flexibility than a Monte Carlo approach in treating complex environmental situations. Unlike the stochastic differential equation approach, it is not necessary to assume Gaussian distributions in the solution technique, complex random functions of time and space may be included, and solutions are possible for higher‐dimensioned problems and cases with stochastic variations in stream velocity. The PDF/M technique represents a new and potentially powerful tool for extending the capabilities of computer models in management and decision analysis by providing analytical solutions for the probability density functions and associated moments of important environmental variables.