Methods and guidelines for effective model calibration

Linearized true nonlinear model. A linearized true nonlinear model is defined here for the 
purposes of this discussion. The model is linearized using a Taylor series expansion about the true 
parameter values and has the form of C1, within a constant additive vector, but the 
X
j
are deriva-
tives of the nonlinear model with respect to the parameters, evaluated at the true parameter values. 
Linearized models are further discussed below.
Approximate linear model. The approximate model is the model being developed to rep-

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resent the system of concern, and is the model to be calibrated. A linear approximate model can be 
represented as:
y = b
0
+ b
1
X
1
+ b
2
X
2
+ . . .
β
j
X
j
. . . 
β
n
X
n
+ e = y
’
+ e
(C2)
where 
y is a measurement of the dependent variable (here, hydraulic heads, flows, and so on), as above;
b
j
are the estimated parameter values;
X
j
are the independent variables (here, location, depth, time, etc.);
n
is the number of terms in the approximate model;
e is the true error; and
y
’
is the simulated equivalent of the measured dependent variable.
Approximate nonlinear model. As for the true model, the approximate nonlinear model 
can not be represented as in C1, and requires the more general form presented after equation 1 -- 
that is, using vector notation, y= f(b,
ξ
) + e, where f repesents the form of the unknown nonlinear 
function, 
ξ 
represents the independent variables, and the other symbols are as defined for equation 
C2.
Linearized approximate nonlinear model. The linearized approximate nonlinear model is 
produced using a Taylor series expansion about a defined set of parameter values, b*. Within an 
additive vector that is constant for any b* (this vector is needed to derive the interative equation 
4a, but is not important to the present discussion), the linearized approximate nonlinear model can 
be expressed in the form of equation C2. In this situation however, the X
j
are no longer simply in-
dependent variables, but equal the derivatives of the approximate linear model with respect to the 
parameter values, evaluated at b*. These derivatives were defined for equation 8 and have the fol-
lowing characteristics:
1. Like the X
j 
for linear problems, the derivatives include the independent variables; but they also 
include the effects of other aspects of the nonlinear model.
2. Because of model nonlinearity, the values of the derivatives depend on the parameter values in 
b*. 
3. The derivatives generally are called sensitivities because the represent the senstivity of the sim-
ulated value to a change in the parameter value.
Linearized models reproduce the same simulated value at b* as the nonlinear model, by 
definition, and often closely mimick the nonlinear model for values of b near b*. As the linearized 
model is evaluated for values further from b*, simulated values will vary from those of the approx-
imate nonlinear model depending on its degress of nonlinearity. This deviation is apparent in the 
sum-of-squared residuals surfaces of figure 2, which shows an objective-function surface calculat-
ed using the Theis equation as the approximate nonlinear model, and two objective-function sur-

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faces calculated using a linearized approximate model. The linearized surfaces closely mimic the 
nonlinear surface near the b* values, marked by an x, and mimick it less well, and even poorly, for 
increasingly different sets of parameter values.