hydromechanical processes. It is obvious that the values for S
lr
do not have to coincide in the two different curves. It is important to realize that the parameters estimated by
inverse modeling are not intrinsic properties of the soil, but are model parameters strictly related to the specific formu-
lation as outlined in this section. Transferring parameters from one model to another may lead to conceptual and
thus numerical prediction inconsistencies. The same prob- lem arises when using ‘directly’ measured parameters
values in an analytical or numerical prediction model. On the other hand, the parameters estimated by inverse model-
ing can be considered optimal for the specific forward model used to simulate the experiment.
2.2 Formulation of the inverse problem
Solving the inverse problem is usually defined as the esti- mation of parameters by calibrating a model against the
observed data. In the broader sense of model identification, however, inverse modeling also requires identifying the
most suitable conceptual model, which includes the func- tional form of the characteristic curves. In this section we
focus on the parameter estimation procedure.
We follow the standard procedure and minimize some measure of the differences between the observed and pre-
dicted system responses, which are assembled in the resi- dual vector r with elements
r
i
¼ y
p i
¹ y
i
p 11
Here, y
p i
is an observation at a given point in space and time, and y
i
is the corresponding model prediction, which
depends on the vector p of the unknown model parameters. In inverse modeling, the distribution of the final residuals is
expected to be consistent with the distribution of the measurement errors, provided that the true system response
is correctly identified by the model. If the error structure is assumed to be Gaussian, the objective function to be mini-
mized can be infened from maximum-likelihood considera- tions to be the sum of the squared residuals weighted by the
inverse of the covariance matrix C
yy
:
20
Z
p ¼
r
T
C
¹ 1
yy
r
12 An iterative procedure is needed to minimize eqn 12. The
Levenberg–Marquardt modification of the Gauss–Newton algorithm
1
was found to be suitable for our purposes. Under the assumption of normality and linearity, a
detailed error analysis of the final residuals and the esti- mated parameters can be conducted.
17
For example, the covar- iance matrix of the estimated parameter set is given by:
C
pp
¼ s
2
J
T
C
¹ 1
yy
J
¹ 1
13
where J is the Jacobian matrix at the solution. Its elements are the sensitivity coefficients of the calculated system
response with respect to the parameters:
J
ij
¼ ¹ ]
r
i
] p
j
¼ ]
y
i
] p
j
14 In eqn 13, s
2
is the estimated error variance, a goodness- of-fit measure given by
s
2
¼
r
T
C
¹ 1
yy
r
M ¹ N 15
where M is the number of observations, and N is the num- ber of parameters. The inverse modeling formulation out-
lined above is implemented in a computer program named Itough2.
15
The selection of the most appropriate model given a set of candidate models will be discussed
in Section 2.3.
2.3 Design criteria
