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
Prior to testing, design calculations should be conducted by means of synthetic data inversions to evaluate the ability of
the proposed experiment to estimate the parameters of inter- est. Performing synthetic inversions reduces the risk of pro-
ducing an ill-posed inverse problem when analyzing the experimental data. Design calculations provide insight
into the sensitivity of each potential observation with respect to the parameters. Synthetic model calibrations
can be performed to assess whether the inverse problem is well-posed. A number of criteria have been proposed
to
measure the
performance of
an experimental
design.
22,23,44,45
In this section we discuss the design criteria used to evaluate the suitability of multi-step laboratory out-
flow experiments for the determination of unsaturated hydraulic properties. As a general rule, the experiment
should provide sufficiently sensitive data so that the param- eters can be determined with an acceptably low estimation
uncertainty. Furthermore, the inverse problem should be well-posed, leading to a unique and stable solution.
It is obvious that highly sensitive data provide the most valuable information about the model parameters. We intro-
duce a dimensionless sensitivity measure for parameter j that consists of a sum of the absolute values for the sensi-
tivity coefficients ]y
i
]p
j
, scaled by the measurement error j
y
i
and the expected parameter variation j
p
j
: Qj ¼
X
M i ¼
1
] y
i
] p
j
· j
p
j
j
y
i
16 This aggregate measure of sensitivity is similar to the one
proposed by Kool and Parker.
24
However, we use the sum rather than an integral for both space and time, and scale
the sensitivity coefficients by j
p
j
: Note that j
y
i
can be viewed as an actual or required measurement accuracy,
and j
p
j
as the expected or attainable parameter uncertainty. We believe that using the discretized form for Q
j
better reflects the situation where discrete data are available
from individual sensors at different locations, and that cali- bration occurs at selected points in time. In other words, we
use the same information that will be available for estima- tion, which allows us to study how repositioning sampling
points affects the overall sensitivity. The scaling of the sensitivity coefficients is necessary to compare the
434 S. Finsterle, B. Faybishenko
contribution of different data types to the solution of the inverse problem, and to evaluate the sensitivity of
parameters that vary over different scales depending on their measurement units and physical nature.
It is important to realize that a sensitivity analysis does not address the question whether the parameters of interest
can be determined independently and with sufficient accu- racy. High sensitivity is only a necessary, but not sufficient
condition for a well-posed inverse problem for a detailed discussion of this point see Finsterle and Persoff
16
. It is essential to perform synthetic inversions to address issues
such as uniqueness, stability, and estimation uncertainty.
4,24
Here, we focus on the information provided by the covar- iance matrix C
pp
, which contains the variances and correla- tion structure of the estimated parameter set. A measure of
overall estimation uncertainty is given by the trace of the covariance matrix.
Since strong parameter correlations usually lead to an ill- posed inverse problem with large estimation uncertainties,
an experiment should be designed such that the key parameters can be identified as independently as possible.
To analyze the overall parameter correlation, we define j
p p
as the conditional standard deviation of a single parameter, i.e. the uncertainty of a parameter assuming
that all other parameters are fixed, and j
p
as the joint standard deviation, i.e. the square-root of the diagonal
element of C
pp
. The conditional standard deviation of parameter j is the inverse of the jth diagonal element
of the Fisher information matrix F ¼ s
¹ 2
J
T
C
¹ 1
yy
J . We
propose to interpret the ratio k
j
¼ j
p
p j
p j
17 as an aggregate measure of parameter correlation, i.e. of
how independently parameter j can be estimated. A value close to one signifies an independent estimate,
whereas small values indicate a loss of parameter identifl- ability as a result of its correlation to other uncertain
parameters.
As mentioned earlier, inverse modeling provides the opti- mum parameter set for the given conceptual model without
indicating whether the model adequately describes the sali- ent features of the flow system. If competing models have
been developed and matched to the data, a criterion is needed to decide which of the alternatives is preferable. A
number of tests for model discrimination have been described in the literature.
4,39
The simplest test is based on the goodness-of-fit measure given by eqn 15. However,
since the match can always be improved by adding more fitting parameters, the criterion should contain the number
of parameters to guard against overparameterization. We will use the Akaike information criterion AIC for model
discrimination tests.
4
For normally distributed residuals, AIC can be written as
AIC ¼ M ¹ N
s
2
þ log
lC
yy
l þ M·log 2p
þ 2N:
18 A test should be designed so it produces data that allow
discriminating among a set of model alternatives.
3 MATERIALS AND METHODS
Various outflow experiment designs for determining unsatu- rated hydraulic properties have been proposed. Experiments
with flow along the core axis were first described by Richards,
35
Richards and Moore
36
and Gardner.
18
An experiment with radial flow geometry was proposed by
Richards et al.,
38
and was further developed by several investigators.
19,21,37
Gardner
19
developed three methods for the determination of the unsaturated hydraulic conduc-
tivity using radial flow experiments. They include: 1 the constant flux method, in which pressure is recorded in time;
2 the constant pressure method, in which the flow rate is measured; and 3 a method that requires instantaneous
injection or removal of a small quantity of water and mon- itoring the associated pressure change. For these methods,
Gardner derived analytical solutions, assuming a ratio of a sample radius to a cup radius of at least 10. Richards and
Richards
37
and Klute et al.
21
developed analytical solutions for radial flow experiments, taking into account the actual
ratio between the core radius and porous cylinder radius, as well as the membrane impedance. These solutions, which
are analogous to Gardner’s solution for the constant pres- sure method, assumed a linear relationship between the
moisture content and pressure for each pressure step, thus limiting the step size that can be taken without loss of accu-
racy. Doering
7
investigated both a one-step procedure and small increments of pressure changes. Valiantzas
47
pro- posed correction terms to Gardner’s solution for a one-
step experiment for several forms of the diffusivity function. It should be noted that gravity was neglected in all the
proposed analytical solutions.
Elrick and Bowman
13
found that a small amount of air could diffuse through the porous membrane when the pres-
sure exceeds 10 kPa. If the extraction cylinder is water filled, the appearance of air changes the flow rate, and this
air must be removed by means of a flushing procedure, potentially affecting the boundary conditions.
Klute et al.
21
discussed the advantages of using a radial instead of the more common axial flow geometry. They
pointed out that soil shrinkage during drying is signifi- cantly reduced in a design with a central porous cylinder,
thus avoiding loss of contact between the sample and the extracting cylinder. Furthermore, the air trapped in the
porous cylinder can be easily removed with mininimal disturbance of the boundary condition. As a result of the
shorter flow distance, a larger sample volume can be tested in shorter times.
19
Fig. 1 shows a schematic of a flow cell apparatus for radial, single- and multi-step desaturation experiments on
soil samples. Faybishenko
14
and Dzekunov et al.
10
have tested the design and performed a series of radial flow
experiments using the central porous cylinder for both Inverse modeling of multistep outflow experiment
435
injection and extraction of water. A second porous cylinder is used as a monitoring tensiometer. They conducted wet-
ting and drying experiments by applying one-step, multi- step, and continuously changing boundary pressures under
isothermal and nonisothermal conditions using cores of different sizes.
The experimental procedure can be described as follows. Soil cores 22 cm long and 15–18 cm in diameter are cut in
the field using a special cylindrical knife which minimizes the disturbance of the sample.
10,14
The cores are conserved in a solid metal or plastic cylinder, and the annulus is filled
in with a paraffin–tar mixture. In the laboratory, a hole with a diameter of 2.3 cm is drilled co-axially at the center of the
core, in which a ceramic cylinder with an air entry pressure of about 1 bar is inserted. The top and bottom surfaces of the
core are covered with end caps. The central porous cylinder is connected to a burette to measure the cumulative water
discharge. The burette is connected to a vacuum pump. In order to inhibit air accumulation in the cylinder, which
would affect the outflow measurements,
13,21
the opening of the porous cylinder is directed downward, and the inner
void space is kept in an air-filled rather than water-filled state. Such a design allows the extracted water to freely
flow into the measuring burette. A port allows air exchange between the core and the atmosphere.
A tensiometer for water potential measurements is inserted near the outer wall of the flow cell. The radial
flow geometry allows vertical installation of a tensiometer. As confirmed by our numerical simulations, the pressure
measured by the tensiometer is very close to the vertically averaged pressure at any radial distance, i.e. the system
behavior can be accurately described by a one-dimensional, radial model. Also note that the flexibility offered by inverse
modeling does not require that the average pressure be measured by the tensiometer. For example, for soil cores
with higher permeability and weaker capillarity, where gravity effects may become significant, a two-dimensional
r–z representation of the core can be used in the forward and inverse models, correctly representing the conditions
encountered by the vertical tensiometer. The samples inves- tigated were saturated under vacuum in order to minimize
the effect of entrapped air.
50
4 RESULTS AND DISCUSSION 4.1 Evaluation of experimental design
In this section, we discuss numerical simulations of a multi- step, radial outflow experiment and assess the suitability of
the design for the estimation of unsaturated hydraulic prop- erties. We model a synthetic multi-step desaturation experi-
ment where the pressure at the center of a cylindrical core is reduced in discrete steps from ¹2 to ¹10, ¹20, ¹30,
¹
60 and ¹90 kPa after 1, 2, 3, 5, and 10 days, respectively. It is assumed that the cumulative outflow is recorded as a
function of time, and that a tensiometer is installed for capillary pressure measurements. The standard errors of
the outflow and pressure measurements are assumed to be 2 of the measured value.
Design calculations have to rely on preliminary informa- tion about the system. If prior knowledge is poor, the con-
clusions regarding the optimum design have to be assessed for a number of alternative conceptual models, and a robust
design has to be chosen that comprises a wide range of conditions possibly encountered during the experiment. As
an example, we present here only the results obtained with the BCB model, which turned out to be a likely candidate
model for the soils investigated. The performance of the proposed radial flow experiment is analyzed assuming that
three parameters are to be estimated based on capillary pressure and cumulative outflow data. The three parameters
are the logarithm of the absolute permeability, logk
abs
, the pore size distribution index, l, and the logarithm of the air
entry pressure, logp
e
. Fig. 2 shows the simulated system behavior calculated
with the base-case parameter set given in Table 1. The out- flow and capillary pressure curves represent potential data to
be used in an inversion for determining the three parameters of interest. While the pressure prescribed at the central
extraction cylinder is reduced in discrete steps, the capillary pressure at the outer wall of the flow cell decreases rather
smoothly. The cumulative outflow through the extraction cylinder reaches 900 ml at the end of the experiment, drain-
ing the sample to about 50 of its initial water content.
We first perform a sensitivity analysis to determine the optimal location of the tensiometer. The location within the
core sample yielding the maximum total sensitivity provides the most information about the model parameters to be esti-
mated. Simulations were performed using the parameter set in Table 1, and the sensitivity measure eqn 16 was eval-
uated for different potential locations of the monitoring ten- siometer along the radius of the core. The dependence of
sensitivity on sensor location results from the lack of capil- lary equilibrium during the transient phase of the
Fig. 1.
Schematic of apparatus for radial flow experiment.
436 S. Finsterle, B. Faybishenko
experiment. Fig. 3 shows the sensitivity measure as a func- tion of radial distance of the tensiometer from the core axis.
Recall that a pressure boundary condition is prescribed at the ceramic cylinder in the center of the core. Consequently,
the sensitivity is zero at the extraction cylinder. Data sensi- tivity increases and reaches a maximum if the tensiometer is
placed at a radial distance of about 0.028 m from the core axis, before it starts to decrease towards the wall of the flow
cell. Note, however, that the decrease of sensitivity with radial distance is very minor. Because of the non-linearity
of unsaturated flow problems, the curves shown in Fig. 3 depend on the base-case parameter set, i.e. the point of
maximum sensitivity may shift if soil properties vary. For example, if the permeability of the sample is higher than
expected, the zone of low sensitivity around the core axis is larger. While an optimum design requires the tensiometer to
be located relatively close to the center of the core, a robust design suggests its installation near the outer wall of the
flow cell to avoid the low sensitivity zone in the case of higher perrneability. If permeability happens to be lower
than expected, the sensitivity at the outer boundary decreases slightly, but remains at an acceptable level. Instal-
lation of the tensiometer at the cell wall has the additional advantage of minimizing its impact on the water flowing
towards the center of the core.
As mentioned earlier, strong correlations among the parameters may severely affect the quality of the inverse
modeling results if random or systematic errors are present. Because parameter correlations are not addressed by a stan-
dard sensitivity analysis, design calculations should also include synthetic data inversions to examine the potential
estimation uncertainty and correlation structure. The design should then be modified to minimize the trace of the
resulting covariance matrix or some other uncertainty measures.
44
For problems with few unknown parameters, contouring the objective function eqn 12 is a means to visualize the
well- or ill-posedness of the inverse problem. Points of equal objective function lie on continuous surfaces in the
parameter space. Fig. 4 shows contour plots in the three parameter planes: a logk
abs
¹ logp
e
; b logk
abs
¹l; and c l ¹ logp
e
. The top and middle row of panels show, respectively, the objective function obtained when only flow
rate or only pressure measurements are available. The bottom row results from combining the two types of obser-
vations. The shape, size, orientation, and convexity of the minimum provides information about the uniqueness and
stability of the inversion, and represents the uncertainty and correlation structure of the estimated parameter set.
Furthermore, the presence of local minima can readily be detected. The planes shown in Fig. 4 intersect the global
minimum.
In the absence of measurement errors, the objective func- tion visualized in Fig. 4 is devoid of local minima. The
central panel in Fig. 4 reveals that the joint estimation of logk
abs
and l is likely to be unstable if only pressure measurements were available. The combination of pressure
and flow rate data yields a well-defined global minimum. Note that the orientation of the minima from the flow rate
data tend to be orthogonal to those from the pressure data, i.e. when combined, a well-developed minimum results. In
this example, the topology shown in the bottom row in Fig. 4 is similar to that shown in the middle row, because pressure
Fig. 2.
Simulated cumulative outflow through central extraction cylinder and capillary pressure near wall of flow cell. The total pore volume is 1820 ml.
Table 1. Base case parameter set and expected parameter variation
Parameter Base case
value j
p
logk
abs
m
2
¹ 13.00
1.00 pore size distribution index l
0.25 0.25
logp
e
Pa 3.00
0.50
Inverse modeling of multistep outflow experiment 437
data have a higher relative sensitivity given the assumed measurement accuracy of 2.
Synthetic inversions demonstrate that the global mini- mum is accurately identified by the Levenberg–Marquardt
minimization algorithm. Recall that the value of the objec- tive function is usually evaluated only at a few points in the
parameter space along the path taken by the minimization algorithm. However, information about the structure of the
objective function can be obtained from an analysis of the covariance matrix, which relies on a local examination of
the objective function curvature at the minimum. In this case, the covariance matrix calculated at the minimum indi-
cates that it should be feasible to obtain accurate parameter estimates using the multi-step desaturation experiment.
4.2 Analysis of multi-step experiment
