Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue8.1999:

Evaluation of con®dence intervals for a steady-state

leaky aquifer model

S. Christensen

a,*

_{& R.L. Cooley}

b a

Department of Earth Sciences, Aarhus University, Ny Munkegade b. 520, 8000 Aarhus C, Denmark

b

Water Resources Division, U.S. Geological Survey, Box 25046, Mail Stop 413, Denver Federal Center, Denver, Colorado 80225, USA

(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 as-sumption 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, regres-sion.

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 es-timated (calibrated) model parameters and model de-pendent variables can be used to express the degree of uncertainty in these quantities, but calculation of con-®dence intervals is not straightforward because the so-lution 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

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 Cooley23and 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 Brookset al.4presented a

nonstatistical variant.

Ó1999 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0309-1708/99/$ ± see front matter

PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 5 5 - 4

*

Corresponding author.

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

in-tervals (for example, Ref.10). As will be shown,

conclu-sions obtained for the present study will apply to the other types of con®dence intervals.

For brevity we will term con®dence intervals calcu-lated using the nonlinear regression model as nonlinear con®dence intervals. The nonlinear regression model can be linearized using a ®rst order Taylor series ex-pansion. Con®dence intervals calculated using the line-arized model are the standard ones found in most

regression texts22and are termed here `linear con®dence

intervals'.

Synthetic case studies9,10,13,17,23show that: (i)

corre-sponding 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

lin-ear 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 ob-tained from ®eld measurements.

2 BACKGROUND

2.1 Regression model and basis for con®dence intervals

The groundwater ¯ow model and accompanying infor-mation on the groundwater system are stated as a nonlinear regression model of the form

yˆf…b† ‡e …1†

where yˆ[yi] is a vector of n observations of the

groundwater system (in this study hydraulic head and

selected model parameters); bˆ[bj] is a vector ofp

un-known, true, model parameters;f(b)ˆ[fi(b)] is a vector

of n model-computed values corresponding to y; and

eˆ[ei] is a vector ofn true errors. The true errors are

treated as random variables assumed to have zero means and be distributed normally as

eN…0;Wÿ1r2† …2†

where Wˆ[Wij] is an n´_{n known, positive-de®nite}

weight matrix andr2 _{is an unknown scalar.}

MatrixWÿ1_r2 _{is the variance±covariance matrix for}

e. In this study this matrix is assumed to be block

di-agonal with one block for hydraulic heads and the other for prior information on selected parameters.

Diagonal entries of Wÿ1r2 _{are variances and}

o€-diag-onal entries are covariances. Because all head obser-vations 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 parametersg(b). In this study the

functiong(b) can represent hydraulic head or a

param-eter, and thus can be the same as f(b). However, in

generalg(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 simulta-neously, and thus can be de®ned by the probability statement

Prob‰g…bL†6g…b†6g…bU†for all linearizableg…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

in-tervals 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 isa that there

will be any (1ÿa)´_{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

inter-vals. 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

non-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 param-eters, whereas the intrinsic nonlinearity is the compo-nent of the nonlinearity that cannot be removed by any parameter transformation. The sum of the two compo-nents is the total nonlinearity, as measured by eqn (4) below, say, and, through some transformation of pa-rameters, 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 measures20are 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

^

l is the linear model

ap-proximation offl, computed from

f0_l ˆ^f‡X b^ l

ÿ^b …5†

s2 _{is the error variance, de®ned as}

s2_ˆS^_b†

nÿp …6†

andmis the number of parameter setsbl used to

com-pute the measure. In eqn (5) X^ is the n ´_{p sensitivity}

and Na€12give a justi®cation of eqn (4) and recommend

usingmˆ2p pointsbl 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 valuesfl and the linearized valuesfol.

The minimum value ofM^b for any transformation of

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

^

XTWX^_d0_lˆX^T_{W flÿ}^_f8†

Thus, the square of Linssen's measure of intrinsic non-linearity is

Note thatflin 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

classi®es a model as disastrously nonlinear if _M^_b>

1=Fa…p;nÿp† and e€ectively linear if M^b<0:01=

Fa…p;nÿp†, where Fa(p,nÿp) is the upper a point of

theFdistribution withpandn ÿp degrees of freedom.

Cooley and Na€12 added an intermediate class

^

Mb<0:09=Fa…p;nÿp†, for which linear Sche€e-type

intervals for parameters were found to approximate the nonlinear ones fairly well in test cases. Large values of

^

Mb indicate that linear con®dence intervals may not be

accurate. However, small values may not always

indi-cate that linear intervals will be accurate becauseM^b is

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).

As shown in Appendix A,_M^_{/ can be used to correct}

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^/

were the true value and not an estimate2. 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 Zee-land 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

er-ror5 and subsequently by nonlinear regression7 using

MODFLOWP18. Choice of model parameters was

based on a detailed description of the hydrogeology5.

parameters in Ref.7are 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

re-duced 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 geo-logical 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 ground-water 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 vari-able 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.

Ground-water 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

re-gression using a modi®ed code of Cooley and Na€12that

uses a combined modi®ed Gauss±Newton and

quasi-Newton method for the optimization11. The model

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.

Fig. 3.Ground-water model: (a) zonation and head data (sis estimated standard deviation); (b) simulated head and location

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 andY7).

The hydraulic heads in wells were observed with an accuracy of a few centimeters. However, due to pro-cesses 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 drift6and

the measured heads. The estimated variances vary

be-tween 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 Christensen7in the following way.

Local-scale transmissivity was estimated by analyzing short-duration 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-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 re-gression are shown in Fig. 4. The estimated trans-missivities lie within the range of transtrans-missivities 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

esti-mates 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(whereeis the residual}

vector which is an estimate of e), have nearly constant

variance; they appear to be unbiased and normally dis-tributed. 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 variances2_{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.

In case (a) the total nonlinearity,M^bis 0.17, and in case (b)M^b is 0.12. These values lie between 1/F0:05(p,n ÿp)

ˆ0.52 and 0.09/F0:05(p,nÿp)ˆ0.05, which means that

there is signi®cant nonlinearity in both cases. The large values suggest that linear con®dence intervals may not be

accurate. The intrinsic nonlinearity,M^/, is 0.06 in case

(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 Beale2and 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

algo-rithm, 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

ex-treme, 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 proce-dure was followed for the calculation of nonlinear in-tervals for the eleven parameters, as well as for the hydraulic head at 20 grid points homogeneously dis-tributed 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 ofZ1±5converged 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

algo-rithm 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 al-gorithm is a fairly robust and ecient method for

cal-culating 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 con-tour 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 in-terval.

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 magni-tude, 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

corre-sponding linear intervals10,17. Absolute di€erences

be-tween 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 lin-ear and nonlinlin-ear intervals are similar for these

pa-rameters. Intervals for parameters without prior

information are nearly una€ected by adding prior in-formation on other parameters.

Fig. 4 shows that the linear con®dence interval onY3

construct the linear con®dence interval for g(b)

corre-sponds 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^bˆb. Because^bin general will di€er fromb, the

linearized variance actually used will generally not cor-respond 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 (Y7andY8). Because the

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) pa-rameters 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 relation-ship is opposite in the northern part. As argued above,

this indicates that the linearized variance used to com-pute 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

re-duces 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

sig-ni®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

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

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. Calcu-lated con®dence limits did not depend on starting pa-rameter 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 in-tervals.

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 al-ways 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

information. Use of prior information reduced the total nonlinearity of the model only slightly.

The present case study compares linear and corre-sponding nonlinear con®dence intervals and establishes geometric characteristics of nonlinear con®dence inter-vals. 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.

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

max-imum and minmax-imum values of g(b) over the standard

likelihood con®dence region forb. 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

regres-sion 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)ˆg(a), whereg(a) is approximately a linear function

ofa. In addition the sum of squared errors functionS(b)

can be written as S(a) using g(a) in place of f(b), and

S(b)ˆS(a) 22 (c.f. pp. 195). Because the transformed

model is nearly linear, the assumption of normally dis-tributed errors (2) leads to

…S…b† ÿS…^b††=p S…^b†=…nÿp† ˆ

…S…/† ÿS…^a††=p

S…^a†=…nÿp† F…p;nÿp†

…A:1†

where /is the true set of transformed parameters

cor-responding to the true setbof original parameters, ^ais

the set of regression estimates of/corresponding to the

set of regression estimates^bofb, andF(p,nÿp) signi®es

the F distribution with p and n ÿp degrees of

free-dom14_{. Based on eqn (A.1), a (1 ÿa)´100% likelihood}

con®dence region forb is given by

S…b† ÿS…^b†6ps2Fa…p;nÿp† ˆda2 …A:2†

whereFa(p,n ÿp) is the uppera point ofF14.

As demonstrated subsequently, a Sche€e-type

con®-dence interval is given by

min

b g…b†; maxb g…b†

…A:3†

subject to the constraint Fig. 8.Reduction of size of nonlinear con®dence interval on

S…b† ÿS…^b†6d_a2 …A:4†

If there are no stationary points of g(b) (where

o_g=o_{bˆ0) within the con®dence region given by}

eqn (A.4), or if any stationary point produces an ext-remum less extreme than one on the boundary of eqn (A.4), then eqn (A.4) can be replaced by the

equality constraint S…b† ÿS…^b† ˆd2

a. In this case the

limits given by eqn (A.3) can be computed by ®nding extreme values of

L…b;k0† ˆg…b† ‡k0hd_a2ÿS…b† ‡S…^b†i …A:5†

with respect tob and Lagrange multiplierk023. An

iter-ative solution for extreme values of eqn (A.5) is given by

Vecchia and Cooley23.

It is straightforward to show, using a proof

analo-gous to that used by Cooley10 for a Bayesian Sche€

e-type interval, that eqn (A.5) gives a Sche€e-type

con®-dence interval for all functionsg(b) that are linearizable

using Taylor series. However, this can also be

demon-strated graphically for pˆ2 by referring to Fig. 9. To

interpret the ®gure, which is a plot in parameter space,

note that g(b)ˆconstant describes a curve. Also note

that g(b1), g(b2), g(bL) and g(bU) are constants. The

con®dence region boundaryS…b† ÿS…^b† ˆd_a2is a closed

surface (here a closed curve) comprising all possible

values forS(b) on the boundary. Now ifb actually lies

outside of the con®dence region forb, such asb1, then

for at least one function of the formg(b)ˆg(b1), such as

the one shown, g(b1) lies outside of its con®dence

in-terval given by (g(bL), g(bU)). Thus, the entire curve

g(b)ˆg(b1) lies outside of the region bounded by

g(b)ˆg(bL) andg(b)ˆg(bU). However, ifbactually lies

within the con®dence region, such as b2, then all

line-arizable functions of the form g(b)ˆg(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

simulta-neously with the same relative frequency, 1ÿa, thatbis

contained within its con®dence region. Although it cannot be graphically portrayed, this argument

gener-alizes for any p. We also argue that, because of the

physical nature of groundwater models, only lineariz-able functions are of interest and thus need be con-sidered.

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 1ÿa. Beale2 _{gave an approximate}

correction to obtain a conservative (1ÿa)´ _100%

con®dence region. With the correction,d_a2 is given by

d_a2ˆ1‡ n

where N/ is a theoretical measure of intrinsic

nonlin-earity obtained by letting setsblbe normally distributed,

and then lettingm®_{1in (9)}2_{. (Actually, Beale}2

made

an approximation that Guttman and Meeter15found to

yield signi®cant error and Linssen20 corrected. The

corrected version is used here.) In this study, we follow

Linssen's20suggestion and replaceN/byM^/as given by

eqn (9). The signi®cance of the intrinsic nonlinearity is

simply measured by the magnitude of the correcteddaas

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€

e-type intervals.

REFERENCES

1. Bates, D.M. & Watts, D.G., Relative curvature measures of nonlinearity. Journal of the Royal Statistical Society, Series B, 1980,42(1), 1±25,.

2. Beale, E.M.L., Con®dence regions in non-linear estima-tion.Journal of the Royal Statistical Society, series B, 1960,

22(1), 41±76.

3. Beven, K. & Binley, A., The future of distributed models: Model calibration and uncertainty prediction. Hydrolog-ical Processes, 1992,6,279±298.

4. Brooks, R. J., Lerner, D. N. & Tobias, A. M., Determin-ing the range of predictions of a groundwater model which arises from alternative calibrations. Water Resources Research, 1994,30(11), 2993±3000.

5. Christensen, S., Hydrological model for the Tude aa catchment.Nordic Hydrology, 1994,25(3), 145±166. 6. Christensen, S., Calibration data accuracy in a

regional-scale groundwater model.Internal research report, Dept. of Earth Sciences, Aarhus University, Aarhus, 1995, 29 pp. 7. Christensen, S., On the strategy of estimating

regional-scale transmissivity ®elds.Ground Water, 1997,35(1), 131± 139.

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† ÿS…^b† ˆd2

8. Clarke, G.P.Y., Approximate con®dence limits for a parameter function in nonlinear regression.Journal of the American Statistical Association, 1987,82(397), 221±230. 9. Cooley, R.L., Exact Sche€e-type con®dence intervals for

output from groundwater ¯ow models, 1, Use of hydro-geologic information. Water Resources Research, 1993,

29(1), 17±33.

10. Cooley, R.L., Exact Sche€e-type con®dence intervals for output from groundwater ¯ow models, 2, Combined use of hydrogeologic information and calibration data. Water Resources Research, 1993,29(1), 35±50.

11. Cooley, R.L. & Hill, M.C., A comparison of three Newton-like nonlinear least-squares methods for estimat-ing parameters of ground-water ¯ow models. Computa-tional Methods in Water Resources IX, Vol.1,(ed. by T.F. Russell, R.E. Ewing, C.A. Brebbia, W.G. Gray & G.F. Pinder), 1992, 379±386.

12. Cooley, R.L. & Na€, R.L., Regression modelling of ground-water ¯ow.U.S. Geological Survey Techniques of Water-Resources Investigations, 1990, book 3, ch.B4,232 pp.

13. Cooley, R.L. & Vecchia, A.V., Calculation of nonlinear con®dence and prediction intervals for ground-water ¯ow models.Water Resources Bulletin, 1987,23(4), 581±599. 14. Draper, N. & Smith, H.,Applied regression analysis. 2nd

ed., John Wiley, New York, 1981, 709 pp.

15. Guttman, I. & Meeter, D.A., On Beale's measures of non-linearity.Technometrics, 1965,7(4), 623±637.

16. Hamilton, D., & Wiens, D., Correction factors for F ratios in nonlinear regression. Biometrica, 1987, 74(2), 423±425.

17. Hill, M.C., Analysis of accuracy of approximate, simulta-neous, nonlinear con®dence intervals on hydraulic heads in analytical and numerical test cases. Water Resources Research, 1989,25(21), 177±190.

18. Hill, M.C., A computer program (MODFLOWP) for estimating parameters of a transient, three-dimensional ground-water ¯ow model using nonlinear regression.U.S. Geological Survey, Open-File Report 91-484, 1992, 358 pp. 19. Journel, A.G. & Huijbregts, C., Mining geostatistics.

Academic Press, New York, 1978, 600 pp.

20. Linssen, H.N., Nonlinearity measures: A case study.

Statistica Neerlandica, 1975,29,93±99.

21. Mood, A.M., Graybill, F.A. & Boes, D.C.,Introduction to the theory of statistics. 3rd ed., McGraw-Hill, New York, 1974, 564 pp.

22. Seber, G.A.F. & Wild, C.J., Nonlinear regression. John Wiley, New York, 1989, 768 pp.