MODELING OF WATER QUALITY - Water Quality Engineering in Natural Systems - page 388

classified into the following three categories: direct mea-
surement, direct fitting with model expression, and indi-
rect fitting with whole model. The approach to be taken
depends on the scale at which the model is to be applied,
and the availability of measured data for the region
studied. These approaches to estimating the model
parameters are described in more detail below.

Calibration involves varying parameter values within
reasonable ranges until differences between observa-
tions and predictions are minimized. For each variable
of interest, the performance of a model for various
parameter sets is typically evaluated using error statis-
tics that measure the discrepancy between the observed
data, y j , and model output, y j . Commonly used error
statistics are the average relative error, coefficient of
determination, and the model efficiency; these error sta-
tistics are defined later in this chapter. The error statis-
tic, or combination of error statistics, used to measure
model performance is commonly called the objective
function . Some calibration objectives involve maximiz-
ing some error statistics, such as the Nash-Sutcliffe coef-
ficient, while simultaneously minimizing other statistics,
such as bias (e.g., Tolson and Shoemaker, 2007). Calibra-
tion does not necessarily yield a unique set of parame-
ters, and, as a result, best professional judgement is
often a crucial element in determining what is accept-
able and what is not. The phenomenon that equally
good model simulations might be obtained with differ-
ent combinations of parameters or that different models
can produce equally good results is called equifinality, ,
and this phenomenon is commonly encountered in
hydrologic modeling (Beven, 1993, 2006; Seibert, 2001).
The calibration period should reflect system stresses
that are normally encountered in the system, otherwise,
the calibrated parameters will likely depend on the par-
ticular stresses encountered during calibration (e.g.,
Bakker et al., 2008). In hydrologic models, calibration
periods of at least 5 years have been shown to capture
most of the temporal variability (Merz et al., 2009).
In contrast to the above-described approach of iden-
tifying an optimal parameter set based on defined objec-
tive functions, some calibration procedures treat model
parameters as being inherently uncertain and character-
ized by uniform probability distributions between an
upper and lower bound. For example, a calibration pro-
cedure used by Abbaspour et al. (2007) starts by assum-
ing a large parameter uncertainty so that the observed
data initially falls within the 95% prediction uncertainty
(95PPU) calculated at the 2.5 and 97.5% levels of the
cumulative distribution of output variables obtained by
Latin hypercube sampling and Monte Carlo analysis.
This model uncertainty is decreased in steps by reducing
parameter uncertainty in steps until two rules are satis-
fied: (1) the 95PPU brackets “most of the observations”
and (2) the average distance between the upper (at
97.5% level) and the lower (at 2.5% level) parts of the
95PPU is small.
Calibration utilizes a data set over a range of condi-
tions that ultimately establishes the site-specific limits
of the model. Strictly speaking, this restricts the use of
the model to area(s) where calibration occurred and

Direct Measurement. Model parameters are obtained
by independent measurements, without having to
run the model. For example, saturated hydraulic
conductivity, soil water retention functions, bulk
density, soil organic carbon content, soil texture,
and cation exchange capacity can all be measured
using core samples of soil at the site being modeled.
In field-scale models, in situ field scale measure-
ments give a greater level of accuracy in model
performance than laboratory core sample mea-
surements of soil hydraulic properties (Hills et al.,
1988; Wierenga et al., 1991).
Direct Fitting. Parameters can be estimated by direct
fitting of the data from the study site to a process
equation used in the model. This approach has the
advantage that only the process equation corre-
sponding to a particular conceptual model is used
to find the best fitting parameters, not the whole
model.
Indirect Fitting. In this approach, the modeler adjusts
a few relevant parameters until model predictions
agree well with the measured results being used in
the calibration. A disadvantage of indirect fitting
is that there is no guarantee that the adjustment
of particular process parameters results in true
values for those parameters. The resulting param-
eter estimates are simply those that optimize the
match between the predictions and the measure-
ments. A second disadvantage of indirect fitting is
that there are multiple input parameters that are
uncertain or cross-correlated, and the modeler
often does not know which parameter to adjust to
improve the fit between model predictions and
experimental measurements. The sensitivity analy-
sis will assist the modeler in deciding which model
input parameters have the greatest influence on
model predictions, and which parameters can be
omitted from the calibration process. The param-
eters with the greatest influence on model predic-
tions are those that produce the greatest percent
change in model predictions for a given level of
change in the input parameter. These parameters
are the ones that need to be specified with the
greatest accuracy, because a small error in specify-
ing the value of these input parameters will result
in relatively large errors in model predictions.