MODELING OF WATER QUALITY - Water Quality Engineering in Natural Systems - page 402

set of parameters θ , and g ( θ | Y n , X n ) is the posterior
density of parameters given the historical observations,
also called the likelihood function. Equation (11.51) is
commonly approximated by the gLUE methodology,
even though the gLUE approximation is not entirely
consistent with the fundamentals of Bayesian probabil-
ity theory (Beven et al., 2007, 2008; Mantovan and
Todini, 2006; Mantovan et al., 2007).
The gLUE formulation, which has been described in
detail earlier, mainly differs from the classical Bayesian
approach in the choice of the likelihood function and
that it approximates uncertainty by

11.6.2 Monte Carlo Analysis

Monte Carlo simulation involves assuming a probability
distribution for each uncertain parameter in the model,
repeatedly running the model with the value of each
uncertain parameter being randomly selected from the
predetermined probability distribution, and construct-
ing the probability distributions of model outputs. In
general, there is no serious issue about its application to
uncertainty analysis; however, the application of the
method is affected by: (1) the appropriateness of the
chosen probability distributions for the parameters, and
(2) the number of simulations used in the analysis.
Several investigators in hydrological modeling have
argued that good estimates of the mean and variance of
the model parameters are of greater importance than
the actual form of its probability distribution (Haan et
al., 1998; Lei and Schilling, 1994). Selection of an appro-
priate probability distribution for each uncertain param-
eter is often a tradeoff between theoretical justification
and empirical evidence, whereas the number of simula-
tions required to achieve convergence is often deter-
mined by balancing desired accuracy and affordable
computational burden. The constraint that usually limits
this Monte Carlo approach is the computer time
required to complete enough simulations such that the
output percentile values of interest are not sensitive to
number of simulations. A variety of hybrid Monte Carlo
approaches, such as the dual Monte Carlo method (Wei
et al., 2008), have been proposed to alleviate some of
these limitations.
Monte Carlo simulation is viewed as the most robust
method for propagating uncertainty through either
simple or complex models, and has become the pre-
ferred method for addressing uncertainty in complex
hydrological and water-quality modeling.

= ∫

f y x Y X

( ˆ | , , ) f y x ( ˆ | , θ ) ( g Y X d θ | , ) θ

(11.52)

p p n n p p n n

Θ

where y p is the model output. As a consequence of
using Equation (11.52), uncertainty estimates derived
from using the gLUE approach are properly character-
ized as model output uncertainty due to parameter
uncertainty, rather than prediction uncertainty as given
by Equation (11.51). These two kinds of uncertainties
are often confused and care should be taken in inter-
preting the results from gLUE applications. In most
surface-water hydrology models, prediction uncertainty
is much greater than model output uncertainty (e.g.,
Vrugt et al., 2003), and so the gLUE approach might
not be particularly useful for estimating prediction
uncertainty. However, in groundwater applications,
model output uncertainty due to parameter uncertainty
has been found to be comparable with the prediction
uncertainty (Hassan et al., 2008).
The main steps in the gLUE approach to uncertainty
analysis are: (1) choose parameters from a (subjective)
prior probability distribution; (2) use a subjective likeli-
hood function; (3) select a criteria for behavioral
(i.e., acceptable) parameters; (4) derive the posterior
probability distribution of the parameters via Monte
Carlo sampling; and (5) derive the predictive probabil-
ity distribution. The gLUE approach takes into account
(either explicitly or implicitly) several sources of uncer-
tainty; however, care must be taken to ensure that the
uncertainty in gLUE output is insensitive to the likeli-
hood measure, number of simulations, and definition
of behavioral parameters used in the gLUE approach
(Zheng and Keller, 2007a,b). Some investigations
have shown that the number of observations falling
within predicted gLUE confidence intervals is some-
times much less than predicted, and the number of
simulations required to identify behavioral parameter
sets can be excessive. Modifications to the gLUE meth-
odology to improve these shortcomings have been pro-
posed (e.g., Tolson and Shoemaker, 2008; Xiong and
O'Connor, 2008) but are yet to find widespread use.

11.6.3 Analytical Probability Models

Analytical probability models assume that uncertainty
in model output is fundamentally a result of uncertainty
in the model input, and uncertainty in model output can
theoretically be determined from an analysis of the
propagation of the input pdfs. Models that track the
propagation of pdfs using deterministic functional rela-
tionships are called analytical probability models .
The mathematical background for analytical proba-
bility models can be traced back to the transformation
of variables (e.g., Hahn and Shapiro, 1994). The math-
ematical problem of propagation of pdfs can be illus-
trated as follows: if two random variables x and y are
related such that the function y = g ( x ) relating the two
variables is always increasing or decreasing, and there
is only one value of y for each value of x , then it can be