Death valley regional ground-water ﬂow model calibration using …directory.umm.ac.id/Data Elmu/jurnal/A/Advances In Water... · 2010-03-29 · Death valley regional ground-water

Death valley regional ground-water ¯ow modelcalibration using optimal parameter estimationmethods and geoscienti®c information systems

Frank A. D'Agnese*, Claudia C. Faunt, Mary C. Hill & A. Keith Turner

US Geological Survey, Water Resources Division, MS 421, Box 25046, Lakewood, CO, USA

(Received 1 June 1997; revised 8 June 1998)

A regional-scale, steady-state, saturated-zone ground-water ¯ow model wasconstructed to evaluate potential regional ground-water ¯ow in the vicinity ofYucca Mountain, Nevada. The model was limited to three layers in an e�ort toevaluate the characteristics governing large-scale subsurface ¯ow. Geoscienti®cinformation systems (GSIS) were used to characterize the complex surface andsubsurface hydrogeologic conditions of the area, and this characterization wasused to construct likely conceptual models of the ¯ow system. Subsurface prop-erties in this system vary dramatically, producing high contrasts and abruptcontacts. This characteristic, combined with the large scale of the model, makezonation the logical choice for representing the hydraulic-conductivity distribu-tion. Di�erent conceptual models were evaluated using sensitivity analysis andwere tested by using nonlinear regression to determine parameter values that areoptimal, in that they provide the best match between the measured and simulatedheads and ¯ows. The di�erent conceptual models were judged based both on the®t achieved to measured heads and spring ¯ows, and the plausibility of the op-timal parameter values. One of the conceptual models considered appears torepresent the system most realistically. Any apparent model error is probablycaused by the coarse vertical and horizontal discretization. Ó 1999 ElsevierScience Ltd. All rights reserved

1 INTRODUCTION

Yucca Mountain on and adjacent to the Nevada TestSite in southwestern Nevada is being studied as a po-tential site for a high-level nuclear waste geologic re-pository. The United States Geological Survey (USGS),in cooperation with the Department of Energy, is eval-uating the hydrogeologic characteristics of the site aspart of the Yucca Mountain Project (YMP). One of themany USGS studies is the characterization of the re-gional ground-water ¯ow system.

This paper describes the calibration of a three-di-mensional (3D), steady-state, ground-water ¯ow modelof the Death Valley regional ground-water ¯ow system

(DVRFS). The details of this modeling study are de-scribed in US Geological Survey Water Resources In-vestigations Report 96-43006. This paper focuses on thejoint use of state-of-the-art optimal parameter estima-tion and geoscienti®c information systems17 (GSIS) todevelop and test di�erent hydrogeologic conceptualmodels. The goal of the paper is to demonstrate thatthese methods can be used to rigorously constrain modelcalibration and aid in model evaluation.

The study area includes about 100 000 km2 and lieswithin the area bounded by latitude 35° and 38° Northand longitude 115° and 118° West (Fig. 1). The semi-arid to arid region is located within the southern GreatBasin, a subprovince of the Basin and Range Physio-graphic Province. The geologic conditions are typical ofthe Basin and Range province: a variety of intrusive andextrusive igneous, sedimentary, and metamorphic rockshave been subjected to several episodes of compressional

Advances in Water Resources Vol. 22, No. 8, pp 777±790, 1999

Ó 1999 Elsevier Science Ltd

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* Corresponding author. Present address: US GeologicalSurvey, Suite 221, 520 N. Park Ave, Tucson, AZ 85719, USA

777

and extensional deformation throughout geologic time.Land-surface elevations range from 90 m below sea levelto 3 600 m above sea level; thus, the region includes agreat variety of climatic regimes and associated rechargeand discharge conditions.

Previous ground-water modeling e�orts in the regionhave relied on 2D distributed-parameter numericalmodels which have prevented accurate simulation of the3D aspects of the system, including the occurrence ofvertical ¯ow components, large hydraulic gradients, andphysical subbasin boundaries21,5,18,20. In contrast, thedistributed-parameter 3D numerical model used in thiswork allows examination of the internal, spatial, andprocess complexities of the hydrologic system.

The 3D modeling techniques employed herein requirean accurate understanding of the processes a�ectingparameter values and their spatial distribution8. Thesemethods also introduce several concerns resulting from:(1) the large quantity of data required to describe thesystem, (2) the complexity of the spatial and processrelations involved, and (3) the large execution times thatcan make a detailed numerical simulation of the systemunwieldy. These problems are e�ectively managed in thepresent work by the use of integrated GSIS techniquesand a parameter-estimation code.

Regional ground-water ¯ow modeling of this systemwas accomplished in the following ®ve stages:

1. integration of 2D and 3D data sets into a GSIS2. development of a number of digital 3D hydrogeo-

logic conceptual models of the ground-water ¯owsystem

3. numerical simulation of the ground-water ¯ow sys-tem by testing combinations of conceptual models

4. calibration of the ¯ow model using parameter-esti-mation methods and

5. evaluation of the ¯ow model by considering model®t and the optimized parameter values.

2 THREE-DIMENSIONAL DATA INTEGRATION

Extensive integration of regional-scale data was requiredto characterize the hydrologic system, including pointhydraulic-head data and other spatial data such asgeologic maps and sections, vegetation maps, surface-water maps, spring locations, meteorologic data, andremote-sensing imagery. Data were converted into aconsistent digital format using various traditional 2DGIS products10. To integrate these 2D data with 3Dhydrogeologic data, several commercially available andpublic-domain software packages were utilized (Fig. 2).

Fig. 2. Flow chart showing logical movement of modeling datathrough various GSIS packages.

Fig. 1. Map showing location of the Death Valley regionalground-water ¯ow system, Nevada and California, USA.

778 F. A. D'Agnese et al.

This integration allowed the investigators ease of datamanipulation and aided in development of the concep-tual and numerical models.

3 DEVELOPMENT OF DIGITAL 3DHYDROGEOLOGIC CONCEPTUAL MODELS

A digital 3D hydrogeologic conceptual model is a rep-resentation of the physical ground-water ¯ow systemthat is organized in a computerized format. Conceptualmodel development involves:

1. constructing a hydrogeologic framework modelthat describes the geometry, composition, and hy-draulic properties of the materials that controlground-water ¯ow

2. characterizing the surface and subsurface hydro-logic conditions that a�ect ground-water move-ment and

3. evaluating various hypotheses about the ¯ow sys-tem to develop a conceptual model suitable forsimulation.

3.1 Construction of a digital hydrogeologic framework

model

Construction of the hydrogeologic framework modelbegan with the assembly of primary data: digital eleva-

tion models (DEMs), hydrogeologic maps and sections,and lithologic well logs. DEMs and hydrogeologic mapswere manipulated by standard GIS techniques; however,the merging of these four primary-data types to form asingle coherent 3D digital model required more spe-cialized GSIS software products. Construction of a 3Dframework model involved four steps:

1. DEM data were combined with hydrogeologicmaps to provide a set of points representing theoutcropping surfaces of each hydrogeologic unit

2. Hydrogeologic sections and well logs were proper-ly located in 3D coordinate space to de®ne loca-tions of each hydrogeologic unit in the subsurface

3. Surface and subsurface data were interpolated tode®ne the top of each hydrogeologic unit, incorpo-rating the o�sets along major faults and

4. A hydrogeologic framework model was developedby integrating hydrogeologic unit surfaces utilizingappropriate stratigraphic principles to accuratelyrepresent natural stratigraphic and structural rela-tionships (Fig. 3).

GSIS procedures were utilized to develop frameworkmodel attributes describing hydraulic properties. Foreach hydrogeologic unit, the value of hydraulic conduc-tivity was initially assigned based on log-normal proba-bility distributions developed for the Great Basin byBedinger and others2,6. Hydraulic conductivities of rocksoccurring in the Death Valley region vary over 14 orders

Fig. 3. Perspective view of 3D hydrogeologic framework model.

Death valley regional ground-water ¯ow model calibration 779

of magnitude. Within individual hydrogeologic units,potential values of hydraulic conductivity range over 3±7orders of magnitude. The large range of hydraulic con-ductivity within hydrogeologic units indicates substan-tial, and likely important, variability that may a�ectregional ¯ow. Clearly, a regional-scale evaluation of thisvariability should be conducted in terms of the large andabrupt di�erences in average hydraulic conductivity oc-curring between adjacent hydrogeologic units. Therefore,this work concentrates on developing and testing con-ceptual models related to these larger scale variations inaverage hydraulic conductivity; it is intended that theresulting model can form a basis by which variabilitywithin each unit can be evaluated in future studies.

3.2 Characterization of hydrologic conditions

The regional ground-water ¯ow system is a�ected byinteractions among all the natural and anthropogenicmechanisms controlling how water enters, ¯owsthrough, and exits the system. In the DVRFS, quanti-®cation of these system components required charac-terization of ground-water recharge through in®ltrationof precipitation and ground-water discharge throughevapotranspiration (ET), spring ¯ow, and pumpage.

Maps describing the recharge and discharge compo-nents of the ground-water ¯ow system were developedusing remote sensing and GIS techniques7. Multispectralsatellite data were evaluated to produce a vegetationmap. The vegetation map and ancillary data sets werecombined in a GIS to delineate areas of ET, includingwetland, shrubby phreatophyte, and wet playa areas.Estimated water-use rates for these areas were then ap-plied to approximate likely discharge.

Ground-water recharge estimates were developed byincorporating data related to varying soil moistureconditions (including elevation, slope aspect, parentmaterial, and vegetation) into a previously used empir-ical method7. GIS methods were used to combine thesedata to produce a map describing recharge potential ona relative scale. This map of recharge potential was usedto describe ground-water in®ltration as a percentage ofaverage annual precipitation.

Quanti®cation of spring discharge was achieved bydeveloping a point-based GIS map containing springlocation, elevation, and discharge rate. Likewise, water-use records for the region, which are maintained bysurface-water basin and type of water use, were used todevelop a spatially distributed water-extraction mapdescribing long-term average withdrawals.

3.3 Evaluation of 3D hydrogeologic data

Once completed, the 3D data sets describing thehydrogeologic system were integrated and compared todevelop representations of the DVRFS suitable forsimulation. The various con®gurations of the resulting

digital 3D hydrogeologic conceptual model helped in-vestigators during the modeling process to (1) determinethe most feasible interpretation of the system given theavailable data base, (2) determine the location and typeof additional data that will be needed to reduce uncer-tainty, (3) select potential physical boundaries to the¯ow system, and (4) evaluate hypotheses about thehydrogeologic framework.

Conceptual model con®gurations typically included(1) descriptions of the 3D hydrogeologic framework, (2)descriptions of system boundary conditions, (3) estimatesof the likely average values of hydraulic properties of thehydrogeologic units, (4) estimates of ground-watersources and sinks, (5) hypotheses about regional andsubregional ¯ow paths, and (6) a water budget. GSIStechniques also aided modelers in evaluating the feasi-bility of the multiple conceptual models for the ¯ow sys-tem by displaying data control on interpreted products.

4 NUMERICAL SIMULATION OF REGIONAL

GROUND-WATER FLOW

Because of the numerous factors controlling ground-water ¯ow in this region, even the relatively coarse-gridded DVRFS model is necessarily large and complex.Calibration of this model by strictly trial-and-errormethods was judged to be both ine�ective and ine�-cient; therefore, nonlinear-regression methods were usedto estimate parameter values that produce the best ®t tosystem observations. GSIS techniques minimized thee�ort required to develop the required input arrays forthe selected parameter-estimation code-MODFLOWP.

4.1 MODFLOW and optimal parameter estimation

The MODFLOWP computer code is documented inHill12, and uses nonlinear regression to estimate pa-rameters of simulated ground-water ¯ow systems and isbased on the USGS 3D, ®nite-di�erence modular model,MODFLOW11,15. Because the Death Valley regiondominantly contains rocks bearing numerous, densely-spaced fractures, the porous media representation of theMODFLOW code is assumed to reasonably representregional ground-water ¯ow. Where necessary, largefracture zones were represented explicitly to allow forsigni®cant increases or decreases in hydraulic conduc-tivity occurring along or within regional features.

The approach to using nonlinear regression methodspresented here is described in more detail by Hill14 whosuggests that the method is best suited to determininglarge-scale variations in systems. From a stochasticviewpoint, the approach can be thought of as identifyinga mean that cannot be assumed to be a constant or asimple function. Once the large-scale variations are ad-equately characterized, stochastically-based methodsmay be employed to characterize smaller-scale varia-


tions. An e�ort of this kind was beyond the scope of thepresent work.

4.1.1 Nonlinear regression methodsNonlinear regression determines parameter values thatminimize the sum of squared, weighted residuals, S�b�,which is calculated as

S�b� � �yÿ y0�TW�yÿ y0� �1�where,

b is an np ´ 1 vector containing parameter valuesnp is the number of parameters estimated by regres-siony and y0 are n ´ 1 vectors with elements equal to mea-sured and simulated (using b) values, respectively, ofhydraulic heads and spring ¯owsn is the number of measured or simulated hydraulicheads and ¯owsyÿ y0 is a vector of observed minus simulated values,which are called residualsW is an n ´ n weight matrixW1=2�yÿ y0� is a vector of weighted residuals andT superscripted indicates the transpose of the vector.

In this work, the weight matrix is diagonal, with thediagonal entries equal to the inverse of subjectively de-termined estimates of the variances of the observationmeasurement errors. If the variances and the model areaccurate the weighting will result in parameter estimateswith the smallest possible variance1,12,14,19. In MOD-FLOWP, initial parameter values are assigned and thenare changed using a modi®ed Gauss±Newton methodsuch that eqn (1) is minimized. The resulting values arecalled optimal parameter values. A commonly usedstatistic used in this approach that summarizes model ®tis the standard error of the regression, which equals�S�b�=�nÿ np��1=2

.

4.1.2 Parameter de®nitionWith MODFLOWP, parameters may be de®ned torepresent most physical quantities of interest, such ashydraulic conductivity and recharge. MODFLOWP al-lows these spatially distributed physical quantities to berepresented using zones over which the parameter isconstant, or to be de®ned using more sophisticated in-terpolation methods. In either case, multipliers or mul-tiplication arrays can be used to spatially vary the e�ectof the parameter.

4.1.3 Parameter sensitivitiesSensitivities calculated as part of the regression re¯ecthow important each measurement is to the estimation ofeach parameter. Sensitivities can, therefore, be used toevaluate (1) whether the available data are likely to besu�cient to estimate the parameters of interest and (2)what additional parameters probably can be estimated.Sensitivities are calculated by MODFLOWP as oy

0i/obj,

the partial derivative of the ith simulated hydraulic head

or ¯ow, y0i, with respect to the jth estimated parameter,bj, using the accurate sensitivity-equation method12.Because the ground-water ¯ow equations are nonlinearwith respect to many parameters, sensitivities calculatedfor di�erent sets of parameter values will be di�erent.

The composite scaled sensitivity (cj) is a statisticwhich summarizes all the sensitivities for one parameter,and, therefore, indicates the cumulative amount of in-formation that the measurements contain toward theestimation of that parameter. Composite scaled sensi-tivity for parameter j, cj, is calculated as

cj �Xi�1;n

wi�oy0i=obj�2b2j

" #( ,n

)0:5

�2�

and is dimensionless. Parameters with large cj valuesrelative to those for other parameters are likely to beeasily estimated by the regression; parameters withsmaller cj values may be more di�cult or impossible toestimate. For some parameters, the available measure-ments may not provide enough information for esti-mation, and the parameter value will need to be set bythe modeler or more measurements will need to beadded to the regression. Parameters with values set bythe modeler are called unestimated parameters. Com-posite scaled sensitivities can be calculated at any stageof model calibration. The values calculated for di�erentsets of parameter values will be di�erent, but are rarelydi�erent enough to indicate that a previously unesti-mated parameter can subsequently be estimated.

Sensitivities calculated for the optimal parametervalues are used in this work to calculate con®denceintervals on the estimated parameter values using linear(®rst-order) theory. Because linear theory is used, lin-ear con®dence intervals are only approximations formodels which are nonlinear; however, they can still beused to identify potentially unneeded parameters. If,for example, a model input (such as hydraulic con-ductivity) is speci®ed using four parameters, but theregression yields parameter estimates that are withineach others' con®dence intervals, it is likely that fewerparameters are adequate. If the regression using fewerparameters yields a similar model ®t to the measure-ments, it can be concluded that the available mea-surements are insu�cient to distinguish between themodel with the four parameters and the model withfewer parameters. This approach applies the principleof parsimony, in which simpler models with fewer pa-rameters are favored over complex models that areequally valid in all other ways.

4.2 Model development and calibration

Prior to numerical simulation, the 3D hydrogeologicdata sets, which were discretized at various grid-cellresolutions ranging from 100 m to 500 m, were resam-pled into a uniform 1500-m grid for input to


MODFLOWP. This process inevitably resulted in thefurther simpli®cation of the ¯ow-system conceptualmodel.

4.2.1 Model grid and boundary conditionsThe DVRFS model is oriented north-south and isareally discretized into 163 rows and 153 columns(Fig. 4). The model is vertically discretized into threelayers of constant thickness that represent a simpli®ca-tion of the material properties described in the 3Dhydrogeologic framework model at 0±500 m, 500±1 250m, and 1 250±2 750 m below an interpreted water tabledeveloped from hydraulic-head data. Thus, the layersare not ¯at. The three model layers were intended torepresent the local, subregional and regional ¯ow pathsrespectively. The three layers are considered to be theminimum number required to reasonably representthree-dimensional ¯ow for this system. The resultingsimulation is likely to su�er from some inaccuracies inareas of signi®cant vertical ¯ow. This limitation wasconsidered acceptable at this stage of the investigation,in which large-scale features were being characterized.

The lower boundary of the ¯ow system (2 750 mbelow the water table) is assumed to be the depth whereground-water ¯ow is dominantly horizontal and moves

with such small velocities that the volumes of water in-volved do not signi®cantly impact regional ¯ow esti-mates. Lateral boundaries are dominantly no-¯ow withconstant-head boundaries speci®ed for regions whereinterbasinal ¯ux is believed to occur in the north andnortheastern parts of the model boundary or whereperennial lakes occur in Death Valley (Fig. 5).

4.2.2 Flow parameter discretizationThe cellular data structure of the 3D hydrogeologicframework model allows it to be easily recon®gured foruse in MODFLOWP. The GSIS used in this study uti-lizes a resampling function that produces ``slices'' fromthe 3D framework model. In the case of the DVRFSmodel, these ``slices'' represent the material propertiesfor each numerical model layer. These slices were re-formatted into three 2D GIS maps. To start modelcalibration with a simple system representing only thedominant subsurface characteristics, these maps initiallywere simpli®ed to four zones representing high (K1),moderate (K2), low (K3), and very low (K4) hydraulic-conductivity values. The resulting initial zones were notcontiguous; each zone included cells distributed throughthe model (Fig. 6). Using such a small number of zonesat the beginning of the calibration allowed for a clearevaluation of gross features of the subsurface. Subse-quently, composite scaled sensitivities were used to de-termine whether zones could be subdivided to produceadditional parameters that could be estimated with theavailable data.

4.2.3 Spatially-distributed source/sink parametersThe GIS-based in®ltration and ET maps also were re-con®gured into arrays for use in MODFLOWP. In thecase of ET, a series of three maps were used to de®neinputs. In MODFLOWP, ET is expressed in terms of alinear function based on land-surface elevation, extinc-tion depth, and maximum ET rate15. Each of thesevalues was speci®ed as a 2D array generated from GIS-based data sets and resampled to a 1 500 m grid.

Ground-water recharge is likewise speci®ed using twogrid-based GIS maps. To de®ne ground-water recharge,the recharge percentage map was reclassi®ed into asmany as four zones representing high (RCH3), moderate(RCH2), low (RCH1), and zero (RCH0) recharge per-centage (Fig. 7). A parameter de®ned for each zonerepresents the percentage of average annual precipita-tion that in®ltrates. A multiplication array is used torepresent the more predictable variation of average an-nual precipitation.

4.2.4 Conceptual model evolutionA number of conceptual models were evaluated usingthe regression methods in MODFLOWP. Because ofthe simpli®ed nature of the initial simulations, suc-ceeding conceptual models often involved addingcomplexities to the ¯ow model. For the DVRFS model,Fig. 4. Location of the model grid.


Fig. 5. Model boundary conditions: constant heads, springs, wells.

Fig. 6. Example of hydraulic conductivity zone array generated for layer one of the DVRFS model.


changes to the conceptual models involved modi®ca-tions to (1) the location and type of lateral ¯ow systemboundary conditions, (2) the de®nition of the extent ofareas of recharge, and (3) the con®guration of hydro-geologic framework features. The types of ¯ow systemboundaries were adjusted in the north and northeastparts of the model area, where some boundaries wereconverted from constant head (simulating ¯ux into themodel area) to no-¯ow (simulating a closed ¯ow sys-tem). The con®guration of recharge areas was changedfrom a ®xed, single percentage of precipitation to acombination of 4 zones with varying percentages. Mostof the conceptual model variations involved changes tothe hydrogeologic framework because the GSIS-pro-duced distribution tended to ``smooth out'' some im-portant hydrogeologic features. For example, areas ofvery low hydraulic conductivity were delineated intonew distinct zones including (1) NW±SE trending faultzones, (2) clastic shales, (3) metamorphosed quartzites,and (4) isolated terrains of shallow Precambrian schistsand gneisses. The location, extent and hydraulic con-ductivity of these zones were critical to accuratelysimulate existing large hydraulic gradients. Areas ofhigh hydraulic conductivity also were delineated as newdistinct zones. These typically included NE±SW trend-ing zones of highly fractured and faulted terrains.Many of these zones control dominant regional ¯owpaths and large-volume ¯ows to spring discharge areas.

All changes to the hydrogeologic framework weresupported by hydrogeologic information existing in thedatabase; no changes were made simply to produce abetter model ®t.

For each conceptual model MODFLOWP was usedto adjust parameter values to obtain a best ®t to hy-draulic head and spring ¯ow observations. Afterward,model ®t and estimated parameters were evaluated tohelp determine model validity. Modi®cations then weremade to the existing conceptual model, observation datasets, or weighting, always maintaining consistency withthe 3D hydrogeologic data base.

5 MODEL EVALUATION

After calibration, the DVRFS model was evaluated toassess the likely accuracy of simulated results. An ad-vantage of calibrating the DVRFS model using nonlin-ear regression is the existence of a substantialmethodology for model evaluation that facilitates abetter understanding of model strengths and weakness-es. A protocol exists to evaluate the likely accuracy ofsimulated results, associated con®dence intervals, andother measures of parameter and prediction uncertainty.Such information was not available for previous modelsof the Death Valley region, which were calibratedwithout nonlinear regression.

Fig. 7. Recharge zone array for the DVRFS model.


5.1 Evaluation of hydraulic heads and spring ¯ows

The observations used by the regression initially in-cluded 512 measured hydraulic heads and 63 measuredspring ¯ows. During calibration, 12 of the hydraulic-head observations were removed from the data set be-cause of recording errors found in the database, thus500 head observations were used in the ®nal regression.Five of these remaining hydraulic-head observationsare questionable because they appear to represent lo-cally-perched water levels rather than regional waterlevels.

The spring ¯ows were represented as head-dependentboundaries connected to either the top or bottom modellayers, depending on whether the temperature andchemistry of the spring indicated a shallow or deepsource. Accounting for the initial 63 individual mea-sured spring ¯ows during calibration proved to be dif-®cult; therefore they were combined into 16 groupsbased on proximity and likely depth from which thesprings originate. The combined ¯ows are consistentwith the simulated model scale.

For the regression, each of the observed head andspring-¯ow values were assigned an estimated standarddeviation or coe�cient of variation based on how pre-cise the measurement was thought to be. This statisticwas used to calculate the weights of eqn (1). More pre-cise measurements were assigned a greater weight(smaller standard deviation or coe�cient of variation;less precise measurements were assigned a lesser weight(greater standard deviation or coe�cient of variation).More precise measurements typically included accuratewater-level measurements at wells that had been sur-veyed for location and elevation. Less precise measure-ments typically included water-level measurements madewith unspeci®ed methods from wells located on smaller-scale topographic maps. Weighting of the hydraulic-head and ¯ow observations was initially assigned bycalculating the needed estimates of variances from as-sumed head standard deviations of 10 m and assumed¯ow coe�cients of variation of 10%. The ®nal weightswere variable for each type of data: standard deviationsfor hydraulic heads included values of 10, 30, 100, or 250m, with all but 24 standard deviations being 10 m, andthe 100 and 250 m values used only in vicinity of largehydraulic gradients. Coe�cients of variation for ¯owsincluded values of 5%, 10%, 33%, and 100%, with thelarger values being applied to small springs of uncertainrelation to the regional ¯ow system.

5.1.1 Evaluation of model ®tUnweighted and weighted residuals (de®ned aftereqn (1)) are important indicators of model ®t and, de-pending somewhat on data quality, model accuracy.Consideration of unweighted residuals is intuitively ap-pealing because the values have the dimensions of theobservations, and indicate, for example, that a hydraulic

head is matched to within 10 m. Unweighted residualscan, however, be misleading because observations aremeasured with di�erent precision.

Weighted residuals demonstrate model ®t relative towhat is expected in the calibration based on the preci-sion, or noise, of the data. They are less intuitively ap-pealing because they are dimensionless quantities thatequal the number of standard deviations or coe�cientsof variation needed to equal the unweighted residual.Maps of both weighted and unweighted residuals wereconstructed and analyzed for the DVRFS model6.

Unweighted hydraulic-head residuals tended to belarger in areas with steep hydraulic gradients than inareas with ¯at gradients, so these two types of areas arediscussed separately. In areas of relatively ¯at hydraulicgradients, the largest unweighted residuals have absolutevalues less than 75 m and are commonly less than 50 m.In areas of large hydraulic gradients, the di�erencesbetween simulated and observed heads are sometimeslarger (as large as 150 m); however, all simulated gra-dients are within 60 percent of the gradients evidentfrom the data. The match is good considering the 2 000m di�erence in hydraulic head across the model domainand the size of the grid relative to the width of some ofthe steep hydraulic gradients.

Matching spring ¯ows was di�cult but extremelyimportant to model calibration. The sum of all simu-lated spring ¯ows is 51 700 m3/day; the sum of estimatedregional spring ¯ows is 125 400 m3/day. The total sim-ulated ¯ow through the system is 405 000 m3/day, sothat the di�erence is 13%. The di�erence between thetotal estimated and simulated spring ¯ows is large;however, from the perspective of a regional model beingable to simulate a process with many local e�ects likespring ¯ow, the match is considered to be quite good. Inaddition, the ET in many of these areas is larger thanexpected; therefore, the total ¯ux from both ET andspring ¯ow is matched more closely.

When weighted as described above, the sum ofsquared, weighted residuals objective function was11 050, with heads contributing 9 500 and ¯ows con-tributing 1 650. Considering the number of heads and¯ows involved, these numbers indicate that, on aver-age, the head and ¯ow weighted residuals are similar inmagnitude, as needed for a valid regression. Thestandard error of regression equals 4.5, which indicatesthat overall model ®t is consistent with hydraulic headstandard deviations that are 4.5 times the standarddeviations used to calculate the weights of eqn (1).Thus, e�ective model ®t for most wells is 45 m, whichis consistent with the size of the unweighted hydraulichead residuals described above. For spring ¯ow, over-all model ®t is consistent with 4.6 times the coe�cientsof variation used to calculate the weights. Thus, e�ec-tive model ®t for most spring ¯ows is 46%, which,again, is consistent with the ®t to spring ¯ows de-scribed above.


5.1.2 Distribution weighted residuals relative to weightedsimulated valuesTo evaluate model results for systematic model error orerrors in assumptions concerning observations andweights, weighted residuals are plotted against weightedsimulated values4,9. Ideally, weighted residuals varyrandomly about zero regardless of the simulated value.

Fig. 8 shows that most of the weighted residuals forhydraulic heads in the DVRFS model vary randomlyabout a value of zero, but there are some large positivevalues. Positive residuals indicate that the simulatedhead is lower than the observed head. Nine values aregreater than +14.1, which is three times the regressionstandard error of 4.6; no values are less than ÿ14.1. Fornormally distributed values only 3 in 1000, on average,would be so di�erent from the expected value. Thus, thisdistribution is distinctly biased by the large positivevalues. Evaluation of these residuals indicates that manyof the measurements occur where perched conditions aresuspected, so that the bias may result from misclassi®-cation of the data instead of model error.

The weighted residuals for spring ¯ows shown inFig. 8 are mostly negative. For two of the weighted re-siduals, the values are less than ÿ14.1, which is morethan three regression standard errors from the expectedmean of 0.0. Because of the sign convention used, neg-ative weighted residuals for spring ¯ows indicate thatthe observed ¯ows are larger in magnitude than thesimulated ¯ows. These residuals indicate that the re-gional model probably is not representing some theprocesses related to spring ¯ows correctly. Whether ornot this is an important model error probably needs tobe judged in the context of the total ¯ux at the dischargeareas, which includes ET. As mentioned above, the total¯uxes match more closely, especially at the large volumesprings.

5.1.3 Normality of weighted residuals and model nonlin-earityThe normality of the weighted residuals and model lin-earity are important to the use of measures of parameter

and prediction uncertainty, such as linear con®denceintervals. Speci®cally, the weighted residuals need to benormally distributed and the model needs to be e�ec-tively linear for the calculated linear con®dence intervalson estimated parameters and predicted heads and ¯owsto accurately represent simulation uncertainty3,13,19. Inthis report, only con®dence intervals on estimated pa-rameter values are presented.

The normal probability graph of the weighted resid-uals of the ®nal model is shown in Fig. 9. The pointswould be expected to fall along a straight line if theweighted residuals were both independent and normallydistributed. Clearly, the points do not fall along astraight line. One possibility is that the residuals arenormally distributed, but they are correlated instead ofbeing independent. Correlations are derived from the®tting of the regression.

The source of correlation can be investigated usingthe graphical procedures described by Cooley and Na�4.Normally distributed random numbers generated to beconsistent with the regression derived correlations arecalled correlated normal random deviates, and areshown in Fig. 10. These plots show that most of thecurvilinearity in Fig. 9 cannot be attributed to regres-sion-derived correlations, but some of the curving re-lated to extreme values might be explained. This analysisindicates that weighted residuals are not normally dis-tributed.

Model linearity was tested using a statistic referred toas the modi®ed Beale's measure4, which is calculatedusing the computer program BEALEP13. The modi®edBeale's measure calculated for the DVRFS model equals0.42, which is between the critical values of 0.05 and 0.5.If Beale's measure is less than 0.05 the model is e�ec-tively linear. If Beale's measure is greater than 0.5 themodel is highly nonlinear. Thus, the ®nal model is closeto being highly nonlinear.

The lack of normality of the weighted residuals andthe moderately high degree of nonlinearity of the

Fig. 9. Normal probability plot of weighted residuals.

Fig. 8. Weighted residuals against weighted simulated values.


DVRFS model indicate that linear con®dence intervalsare likely to be inaccurate. It can be concluded fromprevious work by Christensen and Cooley3, that linearcon®dence intervals often can be used as rough indica-tors of simulation uncertainty, even in the presence ofsome model nonlinearity. The nonnormal weighted re-siduals indicate a greater degree of potential error in thelinear con®dence intervals. Despite this problem, linearcon®dence intervals are used in this work as rough in-dicators of the uncertainty in estimated parameter val-ues.

5.2 Evaluation of estimated parameter values

The set of parameters estimated by regression in theDVRFS model includes all of the most important sys-tem characteristics, as indicated by evaluating compositescaled sensitivities. This analysis helps to ensure that themeasures of prediction uncertainty calculated using the

model will re¯ect most of the uncertainty in the system,because all measures of prediction uncertainty presentlyavailable mostly propagate the uncertainty of the esti-mated parameter values. Uncertainty in other aspects ofthe model are not propagated into the uncertaintymeasures as thoroughly.

If a model represents a physical system adequately,and the observations used in the regression (heads and¯ows for the DVRFS model) provide substantial infor-mation about the parameters being estimated, it is rea-sonable to think that the parameter values that producethe best match between the measured and simulatedheads and ¯ows would be realistic values. Thus, modelerror would be indicated by unreasonable estimates ofparameters for which the data provide substantial in-formation16.

A measure of the amount of information provided bythe observations for any parameter is the compositescaled sensitivity discussed earlier and the linear con®-dence interval on the parameter. Generally, a parameterwith a large composite scaled sensitivity will have asmall con®dence interval relative to a parameter with asmaller composite scaled sensitivity. If an estimatedparameter value is unreasonable and the data provideenough information that the linear 95% con®dence in-terval on the parameter estimate also excludes reason-able parameter values, the problem is less likely to belack of data or insensitivity, and more likely to be modelerror or misinterpreted hydraulic-conductivity data.

Table 1 shows the estimated parameter values for theDVRFS model, their coe�cients of variation (the stan-dard deviation of the estimate divided by the estimatedvalue), 95% linear con®dence intervals, and the range ofvalues thought to be reasonable based on informationgathered as part of the regional hydrogeologic charac-terization but not used in the regression. The hydraulic-

Table 1. Estimated values, coe�cients of variation, and the 95% linear con®dence intervals for the parameters of the ®nal calibratedmodel, and the range of reasonable values, with the range of reasonable values

Parameter label(units)

Log-transformedfor regression

Estimated value Coe�cient ofvariation a

95% Linear con®denceupper/lower limits on

the estimate b

Expected upper/lowerrange of reasonable

values

K1 (m/d) Yes 0.275 0.149 0.369; 0.205 100.0; 0.1K2 (m/d) Yes 0.443 ´ 10ÿ1 0.113 0.554 ´ 10ÿ1;

0.354 ´ 10ÿ10.1; 0.0004

K3 (m/d) Yes 0.562 ´ 10ÿ2 0.181 0.801 ´ 10ÿ2;0.394 ´ 10ÿ2

0.02; 0.0001

K4 (m/d) Yes 0.856 ´ 10ÿ4 0.263 0.146 ´ 10ÿ3;0.501 ´ 10ÿ4

1 ´ 10ÿ4; 2 ´ 10ÿ7

K5 (m/d) Yes 21.2 0.499 0.50 ´ 102; 0.889 ´ 101 100.0; 8.0K9 (m/d) Yes 0.159 0.479 0.367 ´ 100; 0.686 ´ 10ÿ1 1.0; 0.01ANIV3 Yes 164 0.518 399; 67.2 1000.0; 1.0RCH2 (percent) No 3.02 0.107 3.66; 2.37 8.0; 1.0RCH3 (percent) No 22.7 0.0518 25.0; 20.3 30.0; 15.0

a For parameters that were log-transformed for regression, these are calculated as sB/B, where B is the untransformed estimatedvalue, sB

2� exp(s2ln B + 2(ln B))(exp(s2

ln B) ÿ 1.), and s2ln B is the variance of the log-transformed value estimated by regression.

b The con®dence intervals are not symmetric about the estimated value for parameters that were log-transformed for the regression.

Fig. 10. Normal probability plot of correlated normal randomdeviates.


conductivity parameter values, together with their con-®dence intervals and reasonable ranges of values, arealso shown in Fig. 11. In all cases, the optimized pa-rameter value is within its expected range, though mostof the hydraulic conductivity estimates tend to be in theupper end of the reasonable range.

No prior information was included in the sum-of-squares objective function to restrict the estimationprocess; only the model design and the observationdata in¯uenced parameter estimation. Estimation ofthe most important parameters without prior infor-mation has the advantage of allowing a more directtest of the model using the observation data (the hy-draulic heads and ¯ows). In this approach, the avail-able information on reasonable parameter values isused to evaluate the estimated parameter values. Forthe DVRFS model, this evaluation revealed no indi-cation of model error.

As shown in Table 1 and Fig. 11, the reasonableranges on some of the parameters, and especially thehydraulic-conductivity parameters, are wide, which maysuggest that this evaluation is not very powerful. Duringcalibration, however, many conceptual models pro-duced parameter estimates that violated this seeminglyeasy test, and the evaluation was found to be very useful.

The con®dence intervals on the parameter estimatesshown in Fig. 11 may seem unrealistically small, but thisis largely because they represent the con®dence intervalfor the mean hydraulic-conductivity value. As pointedout by Hill14, con®dence intervals on mean values arerapidly reduced from the entire range of the populationas data is applied to the estimation of the mean. Thevalidity of the idea that the hydrogeologic units haveuniform `mean' or `e�ective' values is, of course, a basichypothesis of the modeling approach used in this work.The ability of a model, developed using this approach,to reproduce the measured hydraulic heads and ¯ows, aswell as is done by the DVRFS model, suggests that theapproach is likely to be valid for this system.

Composite scaled sensitivities (cj for parameter j,eqn (2)) were used during calibration to decide whatparameters to include and exclude from the estimationprocess. Parameters with relatively high cj values oftenwere included in the estimation process, while parame-ters with relatively low cj values were not included. Insome cases, a parameter may have had a high enoughsensitivity to be easily estimated by the regression, butwas correlated (as determined from parameter correla-tion coe�cients9,14 with another parameter of highersensitivity. In these cases, the parameter of lower sen-sitivity typically was left unestimated. At times, thenumber of parameters that were estimated was limitedby the execution time of the computer used.

Partly because of model nonlinearity, the values of cj

change somewhat as the parameter values change. As aresult, the evaluation of cj values was repeated fre-quently. Composite scaled sensitivity values for esti-mated parameters of the ®nal model are shown inFig. 12. The ®nal values changed somewhat, but werestill quite similar to initial values, and generally indicatethat the parameters being estimated were the most im-portant parameters. Exceptions occur for parametersthat were correlated with parameters with larger

Fig. 11. Estimated hydraulic conductivity parameters, their95% linear con®dence intervals, and the ranges of reasonable

values.


composite scaled sensitivities, and for parameters thatmostly in¯uence model ®t to a single observation.

5.3 Signi®cance of model evaluation

The model evaluation results presented suggest that theDVRFS model reproduces the measured hydraulicheads reasonably accurately and the measured spring¯ows with somewhat more error. In addition, the esti-mated parameter values include the aspects of the sys-tem that are most important for steady-state simulationof the observed quantities. Also, the ®nal estimatedparameter values are all within reasonable ranges.

The model used at this stage of model calibration wasable to reproduce major characteristics of the systemquite well considering its simplicity. The simplicity wasbelieved to be crucial to the analysis because it allowed amore thorough analysis of the large-scale aspects of thesystem which would not have been possible with a moredetailed model and accompanying longer executiontimes. Knowledge of the system available from thismodel forms an excellent foundation for more detailedmodel development.

6 CONCLUSIONS

The available state-of-the-art GSIS and parameter-esti-mation techniques utilized in this study materiallyassisted in modeling the complex DVRFS. Three-di-mensional hydrogeologic framework modeling com-bined with geologically realistic interpretation allowedcharacterization of the ``data-sparse'' subsurface, whileintegrated image processing and hydrologic processmodeling using traditional GIS techniques supportsurface-based characterization e�orts. The di�erentcon®gurations of the digital 3D hydrogeologic concep-tual model allows rapid evaluation of various likelyrepresentations of the ¯ow system.

While ground-water inverse problems are generallyplagued by problems of nonuniqueness, this work clearly

demonstrates that, even for a complex ground-watersystem, substantial constraints can be developed fromground-water model calibration. The constraints used inthis work include (1) a geologic framework, which con-strains the alternative conceptual models; (2) testingpossible conceptual models by determining the parame-ter values needed to produce a best ®t to the hydrologicdata (heads and spring ¯ows in this work) using inversemodeling; and (3) testing the validity of the model byconsidering the ®t between the data and the associatedsimulated values, comparing simulated global budgetterms to values estimated from ®eld data, and by testingthe plausibility of optimized parameter values. Becausethis is a complex system, the problem of nonuniquenessis never completely eliminated. By e�ectively satisfyingmore constraints, however, the probability is increasedthat the resulting model more accurately represents thephysical system. The key is development and use of theproper 2D and 3D data sets. Joint use of GSIS tech-niques and optimal parameter estimation by nonlinearregression was essential to achieving these objectives.

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