Environmental Engineering Reference
In-Depth Information
conductivity and recharge will be highly cor-
related. In this case, the inverse model will
actually be determining the ratio of recharge
to hydraulic conductivity that provides the
best model fit; any simulation that uses that
same ratio will produce an identical model fit.
Including base-flow or other flux measurements
along with water levels in the calibration process
alleviates this nonuniqueness problem (Sanford,
2002
). Instability refers to the situation where
slight changes in parameter values or observed
data cause large changes in simulation results.
PEST (Doherty,
2005
) and UCODE_2005
(Poeter
et al
.,
2005
) are two widely used universal
model-calibration programs. These programs can
be applied with virtually any hydrologic model;
both of these programs use nonlinear regression
techniques to minimize the objective function.
Alternative inverse-modeling approaches are
available. Adjoint-state methods (Carrera and
Neuman,
1986
; Xiang
et al
.,
1993
; Tarantola,
2005
)
rely on the derivative of the objective function
with respect to parameter value. Global optimi-
zation methods, such as simulated annealing,
genetic algorithms (Wagner,
1995
), and tabu
search (Zheng and Wang,
1996
), are useful for
problems with highly irregular objective func-
tion surfaces. These problems may not be ame-
nable to solution by nonlinear regression.
The ability of inverse models to quantify
uncertainty in estimated recharge rates is a
benefit not available from other recharge-
estimation techniques. The uncertainties are
valid only if the model accurately represents
the actual hydrologic system. Two methods
for quantifying parameter uncertainty are
inferential statistics and Monte Carlo methods
(Hill and Tiedeman,
2007
). Inferential statistics
can be used to calculate linear and nonlinear
confidence intervals for parameter values. An
individual linear confidence interval for a par-
ameter estimate, such as recharge, has a given
probability of containing the true value of that
parameter regardless of whether confidence
intervals on other model parameters include
their true values. Linear confidence intervals
of parameter estimates can be calculated and
printed by PEST and UCODE_2005; the accur-
acy of any of these calculations depends on
the validity of the assumptions invoked for
N
∑
obs
Sb
( )
=
ω
[
y
−
y
'( )]
b
2
(3.1)
i
i
i
i
=
1
where
b
is a vector of the parameter values that
are being estimated,
N
obs
is the number of meas-
ured heads and fluxes, ω
i
is the weight for the
i
ith observation,
y
i
is the observed head or flux,
and
y
i
' is the simulated head or flux. Inverse
models automatically adjust parameter values
so as to minimize the objective function. The
weights, ω
i
, serve a dual purpose. They convert
all observations to a uniform set of units, and
they can be used to more heavily weight the
observations that are the most accurate or the
most important (e.g. a measured groundwater
level is essentially a point estimate in space
and time; base flow, on the other hand, reflects
a response that is integrated over some larger
space and time scales, so in a groundwater-flow
model, measurements of base flow may warrant
larger weights than measurements of ground-
water levels). Use of nonuniform weights in
inverse modeling adds a degree of subjectivity
to the calibration process, so this needs to be
done with careful consideration.
Inverse modeling is a powerful tool for any
hydrologic modeling. Automatic calibration
can save substantial amounts of time relative to
manual model calibration, especially for large,
complex simulations. In addition to determin-
ing parameter values that produce the best
model fit, inverse modeling programs can gen-
erate diagnostic statistics that quantify the
quality of the calibration, point out data short-
comings and needs, determine the sensitivity
of model results to individual parameters, and
quantify reliability of parameter estimates and
predictions (Hill,
1998
).
Potential problems with using inverse mod-
eling include insensitivity, nonuniqueness, and
instability. Insensitivity occurs when the obser-
vations do not contain enough information to
support estimation of all of the parameters.
Nonuniqueness implies that different combin-
ations of parameter values produce equally good
matches to observed data. This occurs when
one model parameter is highly correlated with
another; a steady-state groundwater-flow model
that is calibrated only with water level data will
have a nonunique best fit because hydraulic
