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Advances in Water Resources Vol. 22, No. 8, pp 807±817, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter

Evaluation of con®dence intervals for a steady-state
leaky aquifer model
S. Christensena,* & R.L. Cooleyb
a
Department of Earth Sciences, Aarhus University, Ny Munkegade b. 520, 8000 Aarhus C, Denmark
Water Resources Division, U.S. Geological Survey, Box 25046, Mail Stop 413, Denver Federal Center, Denver, Colorado 80225, USA

b

(Received 20 May 1997; revised 22 December 1997; accepted 20 November 1998)

The fact that dependent variables of groundwater models are generally nonlinear
functions of model parameters is shown to be a potentially signi®cant factor in
calculating accurate con®dence intervals for both model parameters and functions

of the parameters, such as the values of dependent variables calculated by the
model. The Lagrangian method of Vecchia and Cooley [Vecchia, A.V. & Cooley,
R.L., Water Resources Research, 1987, 23(7), 1237±1250] was used to calculate
nonlinear Sche€e-type con®dence intervals for the parameters and the simulated
heads of a steady-state groundwater ¯ow model covering 450 km2 of a leaky
aquifer. The nonlinear con®dence intervals are compared to corresponding linear
intervals. As suggested by the signi®cant nonlinearity of the regression model,
linear con®dence intervals are often not accurate. The commonly made assumption that widths of linear con®dence intervals always underestimate the
actual (nonlinear) widths was not correct. Results show that nonlinear e€ects can
cause the nonlinear intervals to be asymmetric and either larger or smaller than
the linear approximations. Prior information on transmissivities helps reduce the
size of the con®dence intervals, with the most notable e€ects occurring for the
parameters on which there is prior information and for head values in parameter
zones for which there is prior information on the parameters. Ó 1999 Elsevier
Science Ltd. All rights reserved
Key words: con®dence interval, nonlinearity, groundwater ¯ow, model, regression.

model parameters. Cooley and Vecchia13 derived a
general method for the calculation of simultaneous
con®dence intervals for the output from a groundwater

¯ow model when the statistical distribution of model
parameters is known. Vecchia and Cooley23 and Clarke8
independently derived a similar methodology that can
be used to compute simultaneous con®dence intervals
on both the estimated parameters and the output from a
nonlinear regression model with normally distributed
residuals. Vecchia and Cooley23 showed that the same
algorithm can be used both to estimate the parameters
by nonlinear regression and to calculate con®dence
limits from which the desired simultaneous con®dence
intervals are obtained. Beven and Binley3 presented a
Bayesian method that is similar in intent to the method
of Vecchia and Cooley23, and Brooks et al.4 presented a
nonstatistical variant.

1 INTRODUCTION
Estimates of parameters and dependent variables from a
calibrated groundwater model are generally uncertain
because the data used for calibration are uncertain and
because the model never perfectly represents the system

or exactly ®ts the data. Con®dence intervals on the estimated (calibrated) model parameters and model dependent variables can be used to express the degree of
uncertainty in these quantities, but calculation of con®dence intervals is not straightforward because the solution of the groundwater ¯ow equation for hydraulic
heads, and quantities such as ¯ows that are a function of
hydraulic heads, is generally a nonlinear function of the

*

Corresponding author.
807

808

S. Christensen, R. L. Cooley

Con®dence intervals considered here are Sche€e-type
intervals22, which are simultaneous for all linearizable
functions of model parameters and so are the largest of
all simultaneous intervals for these functions. Individual
con®dence intervals and con®dence intervals that are
simultaneous for ®nite numbers of functions, which are

smaller than Sche€e-type intervals, could also have been
calculated, but published theory (for example, Ref.16) is
not as complete as the theory for the Sche€e-type intervals (for example, Ref.10). As will be shown, conclusions obtained for the present study will apply to the
other types of con®dence intervals.
For brevity we will term con®dence intervals calculated using the nonlinear regression model as nonlinear
con®dence intervals. The nonlinear regression model
can be linearized using a ®rst order Taylor series expansion. Con®dence intervals calculated using the linearized model are the standard ones found in most
regression texts22 and are termed here `linear con®dence
intervals'.
Synthetic case studies9,10,13,17,23 show that: (i) corresponding linear and nonlinear con®dence intervals are
often o€set or shifted relative to one another, and the
nonlinear intervals are generally larger10,17; (ii) the
variability in sizes of nonlinear intervals is generally
larger than the variability in sizes of corresponding linear intervals10,17; (iii) the di€erences between the sizes of
corresponding nonlinear and linear con®dence intervals
increase as the sizes of the intervals increase9; (iv) use of
prior information can have a signi®cant e€ect on the
sizes of con®dence intervals10. In the present study we
compute and analyze both linear and nonlinear con®dence intervals using ®eld data to test the calculation
procedures and the concepts derived from the synthetic

case studies. Linear and nonlinear con®dence intervals
are compared for two di€erent sets of data used for
calibration: (i) observed heads and (ii) observed heads
and prior information on the parameters that is obtained from ®eld measurements.
2 BACKGROUND
2.1 Regression model and basis for con®dence intervals
The groundwater ¯ow model and accompanying information on the groundwater system are stated as a
nonlinear regression model of the form
y ˆ f…b† ‡ e
…1†
where y ˆ [yi ] is a vector of n observations of the
groundwater system (in this study hydraulic head and
selected model parameters); b ˆ [bj ] is a vector of p unknown, true, model parameters; f(b) ˆ [fi (b)] is a vector
of n model-computed values corresponding to y; and
e ˆ [ei ] is a vector of n true errors. The true errors are
treated as random variables assumed to have zero means
and be distributed normally as

e  N …0; Wÿ1 r2 †
…2†

where W ˆ [Wij ] is an n ´ n known, positive-de®nite
weight matrix and r2 is an unknown scalar.
Matrix Wÿ1 r2 is the variance±covariance matrix for
e. In this study this matrix is assumed to be block diagonal with one block for hydraulic heads and the
other for prior information on selected parameters.
Diagonal entries of Wÿ1 r2 are variances and o€-diagonal entries are covariances. Because all head observations are assumed to be uncorrelated here, all o€diagonal entries are zero in the hydraulic head block.
The prior information on parameters is assumed to be
correlated. Vectors y, f(b), and e are partitioned to
conform with the blocks in W. Weights are initially
estimated, and can be reestimated as the regression
progresses if analysis of residuals indicates that they are
incorrect12. Thus, although the weights are assumed to
be known, it is apparent that they are approximate. To
this extent, the computed con®dence intervals are also
approximate. The unknown scalar r2 is estimated by
the regression.
Sche€e-type con®dence intervals are to be obtained
on some function of parameters g(b). In this study the
function g(b) can represent hydraulic head or a parameter, and thus can be the same as f(b). However, in
general g(b) can also represent other nonlinear or linear

functions of parameters for which a con®dence interval
is desired. As indicated earlier, Sche€e-type intervals are
on all linearizable (see Appendix A) functions simultaneously, and thus can be de®ned by the probability
statement
Prob ‰g…bL † 6 g…b† 6 g…bU † for all linearizable g…b†Š
ˆ1ÿa

…3†

where g(bL ) and g(bU ) are the lower and upper con®dence limits, respectively, and sets bL and bU are in
general di€erent for each function, g(b). These limits are
computed as minimum and maximum values of g(b)
over the parameter con®dence region. Sche€e-type intervals should be contrasted with the more common
individual con®dence intervals that are on a single
function. A good way of visualizing the di€erence is to
note that, from eqn (3), the probability is a that there
will be any (1 ÿ a) ´ 100% Sche€e-type interval that
does not contain the true value, whereas the probability
is a that the individual con®dence interval does not
contain the true value. Because of this di€erence,

Sche€e-type intervals are larger than individual intervals. Calculation of Sche€e-type con®dence intervals is
discussed in Ref.23 and in Appendix A.
2.2 Measures of model nonlinearity
Nonlinearity of the regression model results when the
model function f(b) is a nonlinear function of model
parameters b. There are two components of this non-

Evaluation of con®dence intervals for a steady-state leaky aquifer model
linearity, parameter e€ects nonlinearity and intrinsic
nonlinearity2. Parameter e€ects nonlinearity is the
component of the nonlinearity that can in theory be
removed by a suitable transformation of model parameters, whereas the intrinsic nonlinearity is the component of the nonlinearity that cannot be removed by any
parameter transformation. The sum of the two components is the total nonlinearity, as measured by eqn (4)
below, say, and, through some transformation of parameters, it has a minimum achievable value equal to the
intrinsic nonlinearity. As described later, nonlinearity
a€ects the Sche€e-type intervals.
Measures of nonlinearity have been developed by
Beale2, Linssen20, and Bates and Watts1, among others.
Linssen's measures20 are easily computed and are useful
to gauge the degree of total and intrinsic nonlinearity of

the model. The square of Linssen's measure20 of total
nonlinearity can be stated as12
m ÿ
P


T ÿ


f l ÿ f 0l W f l ÿ f 0l
^ b ˆ ps2 lˆ1
M

 2
T 
m 
P
0
0
^
^
fl ÿ f W fl ÿ f

…4†

lˆ1

where f l ˆ f…bl †; ^f ˆ f…^
b†; f 0l is the linear model approximation of f l , computed from


^ bl ÿ ^
b
…5†
f 0l ˆ ^f ‡ X
s2 is the error variance, de®ned as
s2 ˆ S…^b†
nÿp

…6†

and m is the number of parameter sets bl used to com^ is the n ´ p sensitivity

pute the measure. In eqn (5) X
matrix given by


^ ˆ @f i
…7†
X
@bj bˆ^b
and in eqn (6) S…^b† is the sum of squared errors objective
T
b††. Cooley
b†† W…y ÿ f…^
function for ^b, given by …y ÿ f…^
12
and Na€ give a justi®cation of eqn (4) and recommend
using m ˆ 2p points bl at the maximum and minimum
linear Sche€e-type con®dence limits for the parameters.
From eqn (4) it can be seen that Linssen's measure is a
scaled length of the discrepancy between the correct
model values fl and the linearized values fol .
^ b for any transformation of
The minimum value of M
parameters is the intrinsic nonlinearity and is computed
using sets of transformed parameters d0l for which the
numerator of eqn (4) is a minimum. Linssen20 shows
that each set can be found by solving the set of linear
equations


^ T W f l ÿ ^f
^ 0 ˆX
^ T WXd
X
…8†
l
Thus, the square of Linssen's measure of intrinsic nonlinearity is

m 
P

T



W f l ÿ f 0l
^ / ˆ ps2 lˆ1
M
2
T 
m 
P
f 0l ÿ ^f W f 0l ÿ ^f
f l ÿ f 0l

809

…9†

lˆ1

where
^ 0
f 0l ˆ ^f ‡ Xd
l

…10†

Note that fl in eqn (9) is computed using parameter sets
bl .
Degrees of total nonlinearity have been classi®ed by
Beale2 and Cooley and Na€12. For example, Beale2
^b >
classi®es a model as disastrously nonlinear if M
^
1=Fa …p; n ÿ p† and e€ectively linear if M b < 0:01=
Fa …p; n ÿ p†, where Fa (p,n ÿ p) is the upper a point of
the F distribution with p and n ÿ p degrees of freedom.
Cooley and Na€12 added an intermediate class
^ b < 0:09=Fa …p; n ÿ p†, for which linear Sche€e-type
M
intervals for parameters were found to approximate the
nonlinear ones fairly well in test cases. Large values of
^ b indicate that linear con®dence intervals may not be
M
accurate. However, small values may not always indi^ b is
cate that linear intervals will be accurate because M
an average value that may not measure the relevant
degree of nonlinearity as it a€ects a particular interval,
and because of the possible in¯uence of nonlinearity of
the function g(b).
^ / can be used to correct
As shown in Appendix A, M
a Sche€e-type interval for intrinsic nonlinearity. The
resulting interval is theoretically conservative to the
^/
degree of approximation used in its derivation, if M
2
were the true value and not an estimate . In practice, the
correction should be regarded as approximate and the
resulting intervals as probably conservative.

3 FIELD CASE
The test case is a steady-state groundwater ¯ow model
of a 450 km2 leaky aquifer in Quaternary deposits of
glacial till and ¯uvioglacial sand and gravel. The aquifer
is located in the western part of the Danish island Zeeland within the catchment of the stream Tude aa
(Fig. 1). The aquifer system is outlined in Fig. 2. The
aquifer is overlain by 10±80 m of till. In high-lying areas,
the aquifer is recharged by downward leakage from a
phreatic aquifer in the more permeable upper zone of
the till, whereas in low-lying stream valleys and areas
near the coast, the aquifer is discharged by upward
leakage. More information about the hydrogeology is
given by Christensen5.
The model was originally calibrated by trial and error5 and subsequently by nonlinear regression7 using
MODFLOWP18. Choice of model parameters was
based on a detailed description of the hydrogeology5.
However, the estimates and linear individual con®dence
intervals of several of the 19 groundwater-model

810

S. Christensen, R. L. Cooley

Fig. 3. Ground-water model: (a) zonation and head data (s is
estimated standard deviation); (b) simulated head and location
of head con®dence intervals.

Fig. 1. Location of aquifer.

Fig. 2. Schematic representation of the aquifer system along
the pro®le line in Fig. 1. The arrows give the ¯ow directions in
the aquifer and in the overlying till.

parameters in Ref.7 are nearly the same, which indicates
redundancy in the parameterization.
Deletion of redundant parameters improves the
conditioning for the calculation of nonlinear con®dence
intervals, so the number of model parameters was reduced to a total of 11: the log10 -transmissivity of the
nine zones shown in Fig. 3 …Y1


Y9 †, and the log10 vertical hydraulic conductivity of the semi-con®ning till
layer within zones 1±5 (Z1±5 ) and zones 6±9 (Z6±9 ).
Each aquifer zone is characterized by speci®c geological conditions. In zone 1 the aquifer is thin, and well
logs show that it is absent in some places. The expected
e€ective transmissivity of this zone is therefore small,
which is also indicated by the large head gradients in the
northern part of zone 1 (Fig. 3(b)). In the southern part
of the zone, the head gradients are smaller because the
groundwater ¯uxes are reduced by scattered groundwater withdrawals. The thickness and the transmissivity
of the aquifer varies signi®cantly within zones 2, 6 and 7.
The aquifer is rather thick and continuous within zones
3±5, with observed transmissivities of the order of

10ÿ3 ±10ÿ2 m2 /s. Zones 8 and 9 represent an extremely
heterogeneous area having local sand layers (zone 9)
surrounded by till (zone 8). Fig. 3(b) shows that the
head gradients are very large in this area. (Note that the
observed head ®eld is not shown, but it compares well
with the simulated head ®eld in Fig. 3(b).)
The simulated leakage is a function of the vertical
hydraulic conductivity, the thickness of the semi-con®ning till layer, and of the head di€erence between the
water table in the upper part of the till and the head in
the aquifer. The hydraulic conductivity was estimated by
calibration as mentioned, whereas the thickness of the
semi-con®ning layer was interpolated from well logs and
thus was not a model parameter. The water table in the
till was ®xed at an elevation 3 m below the ground level,
whereas the head in the aquifer was a dependent variable computed by the model.
A no-¯ux condition was used along the entire model
boundary. In some areas, the no-¯ux condition is due to
geologic boundaries, whereas groundwater divides de®ne the boundary condition in other areas5. Groundwater is withdrawn from several major and minor well
®elds throughout the model area. The total withdrawal
is about 8 million m3 per year, and around 40% of the
total withdrawal is withdrawn from the four major well
®elds shown in Fig. 1. The withdrawal corresponds to
approximately 45% of the ground water leaking to the
aquifer, whereas 50% is leaking upward through the till
in the low stream valleys, and 5% is leaking to the sea5.

4 PARAMETER ESTIMATION AND MODEL
NONLINEARITY
The model parameters were estimated by nonlinear regression using a modi®ed code of Cooley and Na€12 that
uses a combined modi®ed Gauss±Newton and quasiNewton method for the optimization11. The model

Evaluation of con®dence intervals for a steady-state leaky aquifer model
parameters were estimated from two di€erent sets of
data: (a) synchronously measured head in 100 wells
(Fig. 3); and (b) these same heads supplemented with
prior information on four zonal log10 -transmissivities
(Y4 , Y5 , Y6 and Y7 ).
The hydraulic heads in wells were observed with an
accuracy of a few centimeters. However, due to processes such as small-scale variability in hydrogeology,
which are unknown and/or are not represented in the
model, the groundwater model can only simulate
the large scale variations (here termed ``the drift'') of the
true head ®eld7. The weights of the head observations
were therefore estimated as the inverses of the variances
between a polynomial approximation of the drift6 and
the measured heads. The estimated variances vary between 1 and 4.4 m2 (Fig. 3(a)), whereas the covariances
are assumed to be zero.
Prior information on zonal log-transmissivities was
obtained by Christensen7 in the following way. Localscale transmissivity was estimated by analyzing shortduration pumping tests in 90 wells. These results were
used to estimate the transmissivity semi-variogram, and
to estimate by kriging the local-scale transmissivity at
regularly spaced grid points within each of the four
zones. The zonal estimates and the corresponding co-

811

variance matrix were obtained by combining the kriged
estimates as described by Journel and Huijbregts19 (c.f.
pp. 320±324). The inverse covariance matrix was used as
the weight matrix of the prior information.
The estimated parameters from the nonlinear regression are shown in Fig. 4. The estimated transmissivities lie within the range of transmissivities
determined by analysis of numerous pumping tests
within the various aquifer zones5,7, and the estimated
hydraulic conductivities of the con®ning layer are within
the range of values (10ÿ9 ±10ÿ8 m/s) found by analyzing a
smaller number of pumping tests5. The parameter estimates also compare well to those of previous model
studies in the area5,7.
Residual analyses12 show that, in both cases (a) and
(b), the weighted residuals, W1=2 e (where e is the residual
vector which is an estimate of e), have nearly constant
variance; they appear to be unbiased and normally distributed. Thus, there is no reason to suspect that the
weight matrix used is not a good estimate of the true
covariance matrix for the calibration data (head and
prior information). The error variance s2 increases from
1.06 in case (a) to 1.12 in case (b). This may indicate a
weak incompatibility between the ¯ow model and the
prior information.

Fig. 4. Parameter estimates and con®dence intervals.

812

S. Christensen, R. L. Cooley

^ b is 0.17, and in case
In case (a) the total nonlinearity, M
^
(b) M b is 0.12. These values lie between 1/F0:05 (p,n ÿ p)
ˆ 0.52 and 0.09/F0:05 (p,n ÿ p) ˆ 0.05, which means that
there is signi®cant nonlinearity in both cases. The large
values suggest that linear con®dence intervals may not be
^ / , is 0.06 in case
accurate. The intrinsic nonlinearity, M
(a) and 0.05 in case (b). Both values are probably not
signi®cant, but the con®dence region was corrected to
improve the approximation of the con®dence intervals.
We used the correction method of Beale2 and Linssen20,
which is summarized in Appendix A. Finally note that
the prior information slightly reduces the total as well as
the intrinsic nonlinearity of the regression model.

5 CALCULATION OF CONFIDENCE INTERVALS
In this study, linear and nonlinear 95% Sche€e-type
con®dence intervals were calculated on the estimated
parameters and hydraulic head. The linear intervals
were calculated by the standard method22, and the
nonlinear intervals were calculated by the likelihood
method described in Appendix A. The latter intervals
were calculated using a modi®ed Gauss±Newton algorithm, as suggested by Vecchia and Cooley23.
The calculation of each con®dence limit (the upper or
lower bound of the con®dence interval) is a separate
optimization problem in which the bound is obtained as
an extreme value on the edge of the parameter con®dence region23. To guard against obtaining a local extreme, the calculation of each limit was carried out with
nine di€erent sets of starting values for parameters. The
starting values were obtained by adding or subtracting
up to two linearized standard deviations to or from the
values estimated by nonlinear regression. This procedure was followed for the calculation of nonlinear intervals for the eleven parameters, as well as for the
hydraulic head at 20 grid points homogeneously distributed over the model area (Fig. 3).
The variation of parameter starting-values did not
produce signi®cant di€erences in the con®dence limits
for the parameters. Also, for all limits but one, the
calculations converged in 5±14 iterations. In the case
where no prior information was used, the calculation of
the lower bound value of Z1±5 converged more slowly, in
17±25 iterations. The convergence criterion was 0.001
for the relative parameter change from one iteration to
the next.
The variation of parameter starting values also did
not produce signi®cant di€erences in the con®dence
limits for the head at the grid points. For all limits but
one, convergence was achieved in 5±19 iterations. For
the upper limit at point 13, however, convergence was
not achieved for any of the starting parameters. In this
case convergence was achieved by manually adjusting
the value of one parameter (Y1 ) and letting the algorithm estimate the other 10 parameter values.

For con®dence intervals on selected parameters and
heads, we not only searched for extreme values (i.e.
con®dence limits) on the edge of the parameter con®dence region, but also searched the inside of the region.
In all of these cases, parameter sets giving the con®dence
limits were found on the edge of the region. The results
thus strongly indicate that the con®dence intervals are
calculated correctly.
Based on the above results, we conclude that the algorithm is a fairly robust and ecient method for calculating nonlinear con®dence intervals on both
parameters and heads for the present ®eld case. We
therefore calculated the head con®dence interval at 485
grid points evenly spaced throughout the model area. At
400 of these points, the calculations converged in less
than 25 iterations. These 400 points were used to contour characteristics of the con®dence intervals. The
calculation of nonlinear intervals was made on a 50
MHz 80486 PC. The CPU time used was about 5 min
per parameter interval, and about 6 min per head interval.

6 RESULTS
6.1 Con®dence intervals on parameters
Fig. 4 shows the calculated con®dence intervals for the
parameters. For the log10 -vertical conductivities, Z, the
sizes of the intervals are about half an order of magnitude, and the linear and nonlinear intervals are quite
similar. For the log10 -transmissivities, Y, the sizes of the
intervals vary from one to more than three orders of
magnitude. The widest intervals are for: (i) the zones
with a small Y-estimate and/or small head gradients
throughout the zone (Y1 , Y2 , Y4 ), or (ii) a zone without
head observations (Y8 ). A comparison of linear and
nonlinear con®dence intervals supports conclusions
from previous synthetic case studies. Nonlinear intervals
are generally larger than the linear intervals, except for
Y3 , and the two types of intervals are often o€set relative
to one another10,17. The di€erences among nonlinear
intervals are greater than the di€erences among corresponding linear intervals10,17. Absolute di€erences between the sizes of nonlinear and linear intervals tend to
increase as the sizes of intervals increase9, but di€erences
relative to the sizes of the corresponding linear intervals
are more nearly constant. Addition of prior information
signi®cantly reduces the sizes of the con®dence intervals
for the parameters with prior information, and the linear and nonlinear intervals are similar for these parameters. Intervals for parameters without prior
information are nearly una€ected by adding prior information on other parameters.
Fig. 4 shows that the linear con®dence interval on Y3
is larger than the corresponding nonlinear interval. This
contradicts the idea that the linearized variance used to

Evaluation of con®dence intervals for a steady-state leaky aquifer model
construct the linear con®dence interval for g(b) corresponds to the Cramer±Rao bound21, and therefore,
should be the lower bound. However, the linearized
variance only corresponds to the Cramer±Rao bound if
it is calculated at the unknown true set of parameters;
i.e., for ^b ˆ b. Because ^
b in general will di€er from b, the
linearized variance actually used will generally not correspond to the Cramer±Rao bound, and linear con®dence intervals may therefore be larger than their
nonlinear counterparts.
6.2 Con®dence intervals on hydraulic head
Fig. 5 shows the calculated con®dence intervals for head
at the 20 grid points, and Fig. 6 shows the contoured
size of the nonlinear con®dence intervals throughout the
model area. The size of the nonlinear intervals generally
ranges from 2±10 m within the largest part of the area.
The size, however, is as large as 40 m in an area within
and east of zones 8 and 9, where head data are lacking.
Head gradients in this area are signi®cant (toward two
well ®elds in zone 9), which results in a large sensitivity
of head to the local parameters (Y7 and Y8 ). Because the

813

estimates of these parameters are uncertain due to the
mentioned lack of data, large con®dence intervals result.
Some reduction in uncertainty of parameter estimates is
expected if head data are collected immediately south
and east of zone 8. Similar conditions explain the large
con®dence intervals around a minor well ®eld in zone 1.
These results show that hydraulic head can vary over a
large range when parameter values vary over their
con®dence region. Thus, when Sche€e-type intervals are
appropriate, as when gauging the uncertainty of a
number of di€erent dependent variables and (or) parameters simultaneously, large uncertainties can result.
Comparison of the linear and nonlinear intervals in
Fig. 5 shows that the two types of intervals are often
o€set from one another. Many of the nonlinear intervals
are skewed, and the skewness tends to increase with the
size of the interval.
Fig. 7 shows the di€erences between the sizes of
nonlinear and linear con®dence intervals. Note that the
sizes of the nonlinear con®dence intervals in general are
larger than the sizes of the linear intervals in the
southern part of the model area, whereas the relationship is opposite in the northern part. As argued above,

Fig. 5. Con®dence intervals for the hydraulic head at grid points in Fig. 3(b).

814

S. Christensen, R. L. Cooley

Fig. 6. Size of nonlinear con®dence intervals on head [m].

Fig. 7. Size of nonlinear minus size of linear con®dence interval on head [m].

this indicates that the linearized variance used to compute the linear con®dence interval does not in general
correspond to the Cramer±Rao bound.
Figs. 5 and 6 show that prior information on Y reduces the width and skewness of the con®dence intervals
somewhat in the zones with the prior information, and
corresponding linear and nonlinear intervals are more
alike in these zones. In contrast to this, the con®dence
intervals are nearly una€ected in zones without prior
information. Fig. 8 shows the reduction in size of the
con®dence intervals due to the use of prior information.
It can be seen that the prior information reduces the
sizes of the intervals by less than 0.5 m in most of the
area, but in minor areas the reduction is of the order of 1
m. Within and east of zones 8 and 9, the reduction is,
however, up to 16 m. The head gradient in this area is

large, and it is sensitive to Y7 . Prior information signi®cantly reduces the uncertainty of this parameter and
thus also reduces the head con®dence interval.

7 SUMMARY AND CONCLUSIONS
Nonlinear regression was successfully used to estimate
the parameters of a steady-state groundwater ¯ow
model of a leaky aquifer. The residuals are normally
distributed and nonlinear con®dence intervals can
therefore be calculated by the method of Vecchia and
Cooley23. Linssen's measure20 shows that total nonlinearity of the regression model is signi®cant, even when
prior information on some transmissivities is added to
the head data. To a lesser extent, this is also true for the

Evaluation of con®dence intervals for a steady-state leaky aquifer model

815

information. Use of prior information reduced the total
nonlinearity of the model only slightly.
The present case study compares linear and corresponding nonlinear con®dence intervals and establishes
geometric characteristics of nonlinear con®dence intervals. Thus the general conclusions should be valid not
only for Sche€e-type intervals, but also for individual
and other types of con®dence intervals as well.
ACKNOWLEDGEMENTS
This work was partly ®nanced by the Danish Natural
Science Research Council via a research fellowship for
Steen Christensen.

Fig. 8. Reduction of size of nonlinear con®dence interval on
head [m] due to the use of prior information on transmissivity.

intrinsic nonlinearity. We therefore corrected the size of
the parameter con®dence region used to calculate the
nonlinear Sche€e-type intervals as suggested by Beale2
and Linssen20.
The method of Vecchia and Cooley23 was, for the
present case, a fairly robust and ecient algorithm for
the calculation of nonlinear con®dence intervals on both
the hydraulic head and the model parameters. Calculated con®dence limits did not depend on starting parameter values, which indicates that all con®dence limits
are probably unique. One of the main assumptions of
Vecchia and Cooley's23 Lagrangian method, that the
limits of the con®dence interval can be calculated from
parameter sets on the edge of the parameter con®dence
region, is ful®lled for all the calculated con®dence intervals.
As suggested by the signi®cant model nonlinearity,
linear con®dence intervals are often not accurate.
Nonlinear e€ects can cause the nonlinear intervals to be
o€set from, and either larger or smaller than, the linear
approximations. The nonlinear head intervals are largest
in areas of the model with large head gradients, whereas
linear intervals are largest in areas with small gradients.
Thus, widths of linear con®dence intervals do not always underestimate the actual (nonlinear) widths so that
the linearized variance used to compute a linear con®dence interval does not in general correspond to the
Cramer±Rao bound. There is only correspondence if the
linearized variance is calculated at the unknown true set
of parameters.
The general agreement of regression estimates of
parameters with and without prior information suggests
that the parameter and head data are compatible. The
prior information helps reduce and stabilize the con®dence intervals, although the most notable e€ects occur
for the parameters on which there is prior information
and for head values in zones for parameters having prior

APPENDIX A
It is shown here that if the intrinsic nonlinearity is
small, then nearly exact Sche€e-type con®dence intervals
for the function g(b) can be computed by ®nding maximum and minimum values of g(b) over the standard
likelihood con®dence region for b. It is also shown that,
if the intrinsic nonlinearity is signi®cant, then the size of
the con®dence region can be corrected, and the resulting
con®dence intervals are conservative.
Assume ®rst that intrinsic nonlinearity is negligible.
Then there exists some transformation, a, of original
model parameters, b, that nearly linearizes the regression model, as measured by eqn (4) 22 (c.f. pp. 133±136).
In this case, model function f(b) can be written as
f(b) ˆ g(a), where g(a) is approximately a linear function
of a. In addition the sum of squared errors function S(b)
can be written as S(a) using g(a) in place of f(b), and
S(b) ˆ S(a) 22 (c.f. pp. 195). Because the transformed
model is nearly linear, the assumption of normally distributed errors (2) leads to
…S…b† ÿ S…^b††=p …S…/† ÿ S…^a††=p
ˆ
 F …p; n ÿ p†
S…^a†=…n ÿ p†
S…^b†=…n ÿ p†
…A:1†
where / is the true set of transformed parameters corresponding to the true set b of original parameters, ^a is
the set of regression estimates of / corresponding to the
set of regression estimates ^b of b, and F(p,n ÿ p) signi®es
the F distribution with p and n ÿ p degrees of freedom14 . Based on eqn (A.1), a (1 ÿ a) ´ 100% likelihood
con®dence region for b is given by
…A:2†
S…b† ÿ S…^b† 6 ps2 Fa …p; n ÿ p† ˆ da2
where Fa (p,n ÿ p) is the upper a point of F14.
As demonstrated subsequently, a Sche€e-type con®dence interval is given by


…A:3†
min g…b†; max g…b†
b

b

subject to the constraint

816

S. Christensen, R. L. Cooley

S…b† ÿ S…^b† 6 da2
…A:4†
If there are no stationary points of g(b) (where
og=ob ˆ 0) within the con®dence region given by
eqn (A.4), or if any stationary point produces an extremum less extreme than one on the boundary of
eqn (A.4), then eqn (A.4) can be replaced by the
equality constraint S…b† ÿ S…^
b† ˆ da2 . In this case the
limits given by eqn (A.3) can be computed by ®nding
extreme values of
h
i
b†
…A:5†
L…b; k0 † ˆ g…b† ‡ k0 da2 ÿ S…b† ‡ S…^

with respect to b and Lagrange multiplier k0 23. An iterative solution for extreme values of eqn (A.5) is given by
Vecchia and Cooley23.
It is straightforward to show, using a proof analogous to that used by Cooley10 for a Bayesian Sche€etype interval, that eqn (A.5) gives a Sche€e-type con®dence interval for all functions g(b) that are linearizable
using Taylor series. However, this can also be demonstrated graphically for p ˆ 2 by referring to Fig. 9. To
interpret the ®gure, which is a plot in parameter space,
note that g(b) ˆ constant describes a curve. Also note
that g(b1 ), g(b2 ), g(bL ) and g(bU ) are constants. The
con®dence region boundary S…b† ÿ S…^
b† ˆ da2 is a closed
surface (here a closed curve) comprising all possible
values for S(b) on the boundary. Now if b actually lies
outside of the con®dence region for b, such as b1 , then
for at least one function of the form g(b) ˆ g(b1 ), such as
the one shown, g(b1 ) lies outside of its con®dence interval given by (g(bL ), g(bU )). Thus, the entire curve
g(b) ˆ g(b1 ) lies outside of the region bounded by
g(b) ˆ g(bL ) and g(b) ˆ g(bU ). However, if b actually lies
within the con®dence region, such as b2 , then all linearizable functions of the form g(b) ˆ g(b2 ) must lie
within their con®dence intervals because they must pass
through b2 . Therefore, all linearizable functions are

contained within their con®dence intervals simultaneously with the same relative frequency, 1 ÿ a, that b is
contained within its con®dence region. Although it
cannot be graphically portrayed, this argument generalizes for any p. We also argue that, because of the
physical nature of groundwater models, only linearizable functions are of interest and thus need be considered.
If the intrinsic nonlinearity is signi®cant (which can
be measured by eqn (9)), then f(b) cannot be nearly
linearized and eqn (A.1) may not be nearly correct. As a
result, con®dence region eqn (A.2) may not contain b
with probability 1 ÿ a. Beale2 gave an approximate
correction to obtain a conservative (1 ÿ a) ´ 100%
con®dence region. With the correction, da2 is given by


n
N/ ps2 Fa …p; n ÿ p†; p ˆ 1
da2 ˆ 1 ‡
nÿ1


n…p ‡ 2†
2
N/ ps2 Fa …p; n ÿ p†; p P 2
da ˆ 1 ‡
…n ÿ p†p
…A:6†
where N/ is a theoretical measure of intrinsic nonlinearity obtained by letting sets bl be normally distributed,
and then letting m ® 1 in (9)2 . (Actually, Beale2 made
an approximation that Guttman and Meeter15 found to
yield signi®cant error and Linssen20 corrected. The
corrected version is used here.) In this study, we follow
^ / as given by
Linssen's20 suggestion and replace N/ by M
eqn (9). The signi®cance of the intrinsic nonlinearity is
simply measured by the magnitude of the corrected da as
compared to the uncorrected value.
The same arguments as originally used to justify the
use of eqn (A.5) to compute Sche€e-type intervals can
be used to incorporate the correction eqn (A.6). Because
the con®dence region is conservative, so are the Sche€etype intervals.
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Fig. 9. Relations between a con®dence interval (g(bL ),
g(bU )) and possible true values g(b1 ) and g(b2 ) for g(b).
S…b† ÿ S…^
b† ˆ da2 is the con®dence region boundary.

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