Geoscience Reference
In-Depth Information
Although various analytical solutions exist (e.g central limit theorem and transformation of
variables) for the propagation of uncertainty through a model, they are limited to very simple
problems and will not be detailed here. For problems with nonlinear functions of multiple
variables, such solutions become complicated and are impracticable to use. Therefore, sampling
methods, such as Monte Carlo (MC) and Latin hypercube simulations, or approximation
methods, such as the First-Order Second Moment (FOSM) and Rosenblueth's point
estimation (Rosenblueth, 1975) are preferred for practical applications. The sampling
methods provide the estimation of the probability distribution of an output (propagation
of distribution), while the approximate (or point estimate) methods provide only the
moments of the distribution (propagation of moments). Comparative applications of
different sampling methods and approximate or point estimate methods to
a distributed rainfall-runoff model are reported by Yu et al. (2001). The MC simulation
and FOSM are the broadly used methods for uncertainty assessment (Guinot, 1998)
in a number of fields of engineering, including water resources. The present study
also uses these two methods for the probabilistic assessment of uncertainty.
Subsection 3.1.1 presents the principle of the Monte Carlo simulation and Subsection
3.1.2
outlines the principle of the FOSM method.
3.1.1 Sampling method: Monte Carlo simulation
The term Monte Carlo simulation was originally applied to methods of solving de-
terministic computational problems, for example, solving linear equations, differential equations
and integrals, using statistical techniques. Nowadays, it is a term applied to any random
sampling scheme employed by computers (Levine, 1971). It has also been used extensively and
as a standard tool for probabilistic uncertainty assessment. In MC simulation, a model
is run repeatedly to measure the system response of interest under various uncertain parameter
sets generated from probability distributions of the parameters. The procedure for the
application of MC simulation to assess a model output uncertainty is presented as following. Let
the model under consideration consists of X 1 ,…, X n as input random (uncertain) variables
defined by their probability density functions (PDFs) and Y as output random variable.
1. Derive cumulative distribution functions (CDFs), P X
(x i ), of all random
variables ( i =1,…, n ) from their PDFs.
2. Generate n number of random numbers between 0 and 1. Let the random numbers be
u 1 ,…, u n .
3. Determine a set of values of the random variables ( x 1 ,…, x n ) corresponding to the
random numbers generated in Step 2 from their CDFs derived in Step 1. This is
done by the method called inverse transformation technique or inverse CDF
method . In this method, the CDF of the random variable is equated to the
generated random number, that is,
(3.1)
 
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