Geoscience Reference
In-Depth Information
4.3 Uncertainty Analysis
Taking into consideration that the dead zone model is deterministic and that
equivalence of many parameter sets occurs, the uncertainty associated with the
model predictions was estimated. Main sources of uncertainty are measure-
ment errors, errors associated with model structure and numerical solution
(Aronica et al.
1998
; Ratto et al.
2001
;Beven
2008
;Blasoneetal.
2008
;
Smith et al.
2008
).
It is often the case that decomposition of a model error is difficult, especially
when it comes to nonlinear hydrological models, and thus it is not possible to
specify the statistical error model (Beven
2006
; Smith et al.
2008
). Because of the
lack of knowledge about error model, the Bayesian Theorem does not apply and
“pseudo-Bayesian” approach (Mantovan and Todini
2006
) is often used (Beven
2006
; Smith et al.
2008
), which is called the Generalized Likelihood Uncertainty
Estimation (GLUE). Comparing to Bayesian approach, in the GLUE method a
performance criterion (informal likelihood function), which does not include an
error model, is used instead of formal likelihood function derived from a stochastic
error model (Smith et al.
2008
). An informal likelihood measure evaluates accept-
ability of the parameter set according to the degree of simulated values fit to
observed data and does not include statistical error model (Smith et al.
2008
). It
is required that a likelihood measure equals zero for unrealistic models and
increases monotonically with the model fit to the data (Romanowicz and Beven
2006
; Beven
2008
).
Some scientists argue that GLUE methodology is not formally Bayesian and
thus not statistically correct (Montanari
2005
; Moradkhani et al.
2005
; Mantovan
and Todini
2006
). However, Stedinger et al. (
2008
) showed that the GLUE meth-
odology gives results consistent with “widely accepted and statistically valid
analyses” when a statistically valid likelihood function is used.
According to the GLUE concept, it is possible to estimate a posterior distribution
of parameters given a prior parameters distribution based on the knowledge of a
modeled system and an informal likelihood measure (Beven
2008
).
In the study, prior distributions were set as follows: for parameters
K
x
and
A
,
normal distributions with a mean equal to the optimal parameter value derived by DE
method and a standard derivation equal to 20% and 10% of the mean, respectively,
for
K
x
and
A
; for the other two parameters, uniform distributions over the following
intervals:
A
s
[0.01; 0.5],
[0.03; 3]. Parameters were sampled from the described
distributions by MC method to create 100,000 parameter sets. Next, for each param-
eter set simulation was computed and the informal likelihood measure, which is
proportional to the Gaussian distribution function, was applied (Romanowicz
and Beven
2006
). According to Stedinger et al. (
2008
), the performance measure
implemented in this study is statistically valid, in contrast to many other widely used
performance measures. Next, 95% confidence intervals were evaluated, which are
illustrated in Fig.
5
.
a
