Death valley regional ground-water ¯ow model
calibration using optimal parameter estimation
methods 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 was constructed to evaluate potential regional ground-water ¯ow in the vicinity of Yucca Mountain, Nevada. The model was limited to three layers in an eort to evaluate the characteristics governing large-scale subsurface ¯ow. Geoscienti®c information systems (GSIS) were used to characterize the complex surface and subsurface hydrogeologic conditions of the area, and this characterization was used to construct likely conceptual models of the ¯ow system. Subsurface prop-erties in this system vary dramatically, producing high contrasts and abrupt contacts. This characteristic, combined with the large scale of the model, make zonation the logical choice for representing the hydraulic-conductivity distribu-tion. Dierent conceptual models were evaluated using sensitivity analysis and were tested by using nonlinear regression to determine parameter values that are optimal, in that they provide the best match between the measured and simulated heads and ¯ows. The dierent 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 to represent the system most realistically. Any apparent model error is probably caused by the coarse vertical and horizontal discretization. Ó 1999 Elsevier Science Ltd. All rights reserved
1 INTRODUCTION
Yucca Mountain on and adjacent to the Nevada Test Site 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 as part of the Yucca Mountain Project (YMP). One of the many 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 model of 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 the joint use of state-of-the-art optimal parameter estima-tion and geoscienti®c informaestima-tion systems17 (GSIS) to develop and test dierent hydrogeologic conceptual models. The goal of the paper is to demonstrate that these methods can be used to rigorously constrain model calibration and aid in model evaluation.
The study area includes about 100 000 km2 and lies within the area bounded by latitude 35°and 38°North
and longitude 115° and 118° West (Fig. 1). The
semi-arid to semi-arid region is located within the southern Great Basin, a subprovince of the Basin and Range Physio-graphic Province. The geologic conditions are typical of the Basin and Range province: a variety of intrusive and extrusive igneous, sedimentary, and metamorphic rocks have been subjected to several episodes of compressional 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 3 - 0
*Corresponding author. Present address: US Geological
Survey, Suite 221, 520 N. Park Ave, Tucson, AZ 85719, USA
and extensional deformation throughout geologic time. Land-surface elevations range from 90 m below sea level to 3 600 m above sea level; thus, the region includes a great variety of climatic regimes and associated recharge and discharge conditions.
Previous ground-water modeling eorts in the region have relied on 2D distributed-parameter numerical models which have prevented accurate simulation of the 3D aspects of the system, including the occurrence of vertical ¯ow components, large hydraulic gradients, and physical subbasin boundaries21,5,18,20. In contrast, the distributed-parameter 3D numerical model used in this work allows examination of the internal, spatial, and process complexities of the hydrologic system.
The 3D modeling techniques employed herein require an accurate understanding of the processes aecting parameter values and their spatial distribution8. These methods also introduce several concerns resulting from: (1) the large quantity of data required to describe the system, (2) the complexity of the spatial and process relations involved, and (3) the large execution times that can make a detailed numerical simulation of the system unwieldy. These problems are eectively managed in the present work by the use of integrated GSIS techniques and a parameter-estimation code.
Regional ground-water ¯ow modeling of this system was accomplished in the following ®ve stages:
1. integration of 2D and 3D data sets into a GSIS 2. development of a number of digital 3D
hydrogeo-logic conceptual models of the ground-water ¯ow system
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 required to characterize the hydrologic system, including point hydraulic-head data and other spatial data such as geologic maps and sections, vegetation maps, surface-water maps, spring locations, meteorologic data, and remote-sensing imagery. Data were converted into a consistent digital format using various traditional 2D GIS products10. To integrate these 2D data with 3D hydrogeologic data, several commercially available and public-domain software packages were utilized (Fig. 2).
Fig. 2.Flow chart showing logical movement of modeling data through various GSIS packages.
This integration allowed the investigators ease of data manipulation and aided in development of the concep-tual and numerical models.
3 DEVELOPMENT OF DIGITAL 3D
HYDROGEOLOGIC CONCEPTUAL MODELS
A digital 3D hydrogeologic conceptual model is a rep-resentation of the physical ground-water ¯ow system that is organized in a computerized format. Conceptual model development involves:
1. constructing a hydrogeologic framework model that describes the geometry, composition, and hy-draulic properties of the materials that control ground-water ¯ow
2. characterizing the surface and subsurface hydro-logic conditions that aect ground-water move-ment and
3. evaluating various hypotheses about the ¯ow sys-tem to develop a conceptual model suitable for simulation.
3.1 Construction of a digital hydrogeologic framework model
Construction of the hydrogeologic framework model began with the assembly of primary data: digital
eleva-tion models (DEMs), hydrogeologic maps and seceleva-tions, and lithologic well logs. DEMs and hydrogeologic maps were manipulated by standard GIS techniques; however, the merging of these four primary-data types to form a single coherent 3D digital model required more spe-cialized GSIS software products. Construction of a 3D framework model involved four steps:
1. DEM data were combined with hydrogeologic maps to provide a set of points representing the outcropping 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 to de®ne the top of each hydrogeologic unit, incorpo-rating the osets along major faults and
4. A hydrogeologic framework model was developed by integrating hydrogeologic unit surfaces utilizing appropriate stratigraphic principles to accurately represent natural stratigraphic and structural rela-tionships (Fig. 3).
GSIS procedures were utilized to develop framework model attributes describing hydraulic properties. For each hydrogeologic unit, the value of hydraulic conduc-tivity was initially assigned based on log-normal proba-bility distributions developed for the Great Basin by Bedinger and others2,6. Hydraulic conductivities of rocks occurring in the Death Valley region vary over 14 orders
of magnitude. Within individual hydrogeologic units, potential values of hydraulic conductivity range over 3±7 orders of magnitude. The large range of hydraulic con-ductivity within hydrogeologic units indicates substan-tial, and likely important, variability that may aect regional ¯ow. Clearly, a regional-scale evaluation of this variability should be conducted in terms of the large and abrupt dierences 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 in average hydraulic conductivity; it is intended that the resulting model can form a basis by which variability within each unit can be evaluated in future studies.
3.2 Characterization of hydrologic conditions
The regional ground-water ¯ow system is aected by interactions among all the natural and anthropogenic mechanisms controlling how water enters, ¯ows through, and exits the system. In the DVRFS, quanti-®cation of these system components required charac-terization of ground-water recharge through in®ltration of precipitation and ground-water discharge through evapotranspiration (ET), spring ¯ow, and pumpage.
Maps describing the recharge and discharge compo-nents of the ground-water ¯ow system were developed using remote sensing and GIS techniques7. Multispectral satellite data were evaluated to produce a vegetation map. The vegetation map and ancillary data sets were combined in a GIS to delineate areas of ET, including wetland, 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 by incorporating data related to varying soil moisture conditions (including elevation, slope aspect, parent material, and vegetation) into a previously used empir-ical method7. GIS methods were used to combine these data to produce a map describing recharge potential on a relative scale. This map of recharge potential was used to describe ground-water in®ltration as a percentage of average annual precipitation.
Quanti®cation of spring discharge was achieved by developing a point-based GIS map containing spring location, elevation, and discharge rate. Likewise, water-use records for the region, which are maintained by surface-water basin and type of water use, were used to develop a spatially distributed water-extraction map describing long-term average withdrawals.
3.3 Evaluation of 3D hydrogeologic data
Once completed, the 3D data sets describing the hydrogeologic system were integrated and compared to develop representations of the DVRFS suitable for simulation. The various con®gurations of the resulting
digital 3D hydrogeologic conceptual model helped in-vestigators during the modeling process to (1) determine the most feasible interpretation of the system given the available data base, (2) determine the location and type of additional data that will be needed to reduce uncer-tainty, (3) select potential physical boundaries to the ¯ow system, and (4) evaluate hypotheses about the hydrogeologic framework.
Conceptual model con®gurations typically included (1) descriptions of the 3D hydrogeologic framework, (2) descriptions of system boundary conditions, (3) estimates of the likely average values of hydraulic properties of the hydrogeologic units, (4) estimates of ground-water sources and sinks, (5) hypotheses about regional and subregional ¯ow paths, and (6) a water budget. GSIS techniques 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-error methods was judged to be both ineective and ine-cient; therefore, nonlinear-regression methods were used to estimate parameter values that produce the best ®t to system observations. GSIS techniques minimized the eort required to develop the required input arrays for the selected parameter-estimation code-MODFLOWP.
4.1 MODFLOW and optimal parameter estimation
The MODFLOWP computer code is documented in Hill12, and uses nonlinear regression to estimate pa-rameters of simulated ground-water ¯ow systems and is based on the USGS 3D, ®nite-dierence modular model, MODFLOW11,15. Because the Death Valley region dominantly contains rocks bearing numerous, densely-spaced fractures, the porous media representation of the MODFLOW code is assumed to reasonably represent regional ground-water ¯ow. Where necessary, large fracture zones were represented explicitly to allow for signi®cant increases or decreases in hydraulic conduc-tivity occurring along or within regional features.
varia-tions. An eort of this kind was beyond the scope of the present work.
4.1.1 Nonlinear regression methods
Nonlinear regression determines parameter values that minimize the sum of squared, weighted residuals,S b, which is calculated as
S b yÿy0T
W yÿy0 1
where,
bis annp´1 vector containing parameter values npis the number of parameters estimated by regres-sion
yandy0aren´1 vectors with elements equal to
mea-sured and simulated (usingb) values, respectively, of hydraulic heads and spring ¯ows
nis the number of measured or simulated hydraulic heads and ¯ows
yÿy0is a vector of observed minus simulated values,
which are called residuals Wis ann´ nweight matrix
W1=2 yÿy0is a vector of weighted residuals and
T superscripted indicates the transpose of the vector. In this work, the weight matrix is diagonal, with the diagonal entries equal to the inverse of subjectively de-termined estimates of the variances of the observation measurement errors. If the variances and the model are accurate the weighting will result in parameter estimates with the smallest possible variance1,12,14,19. In MOD-FLOWP, initial parameter values are assigned and then are changed using a modi®ed Gauss±Newton method such that eqn (1) is minimized. The resulting values are called optimal parameter values. A commonly used statistic used in this approach that summarizes model ®t is the standard error of the regression, which equals S b= nÿnp1=2.
4.1.2 Parameter de®nition
With MODFLOWP, parameters may be de®ned to represent most physical quantities of interest, such as hydraulic conductivity and recharge. MODFLOWP al-lows these spatially distributed physical quantities to be represented using zones over which the parameter is constant, 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 eect of the parameter.
4.1.3 Parameter sensitivities
Sensitivities calculated as part of the regression re¯ect how important each measurement is to the estimation of each parameter. Sensitivities can, therefore, be used to evaluate (1) whether the available data are likely to be sucient to estimate the parameters of interest and (2) what additional parameters probably can be estimated. Sensitivities are calculated by MODFLOWP asoy0i/obj,
the partial derivative of theith simulated hydraulic head
or ¯ow, y0
i, with respect to thejth estimated parameter,
bj, using the accurate sensitivity-equation method12. Because the ground-water ¯ow equations are nonlinear with respect to many parameters, sensitivities calculated for dierent sets of parameter values will be dierent.
The composite scaled sensitivity (cj) is a statistic
which summarizes all the sensitivities for one parameter, and, therefore, indicates the cumulative amount of in-formation that the measurements contain toward the estimation of that parameter. Composite scaled sensi-tivity for parameterj,cj, is calculated as
cj
and is dimensionless. Parameters with large cj values
relative to those for other parameters are likely to be easily estimated by the regression; parameters with smallercjvalues may be more dicult or impossible to
estimate. For some parameters, the available measure-ments may not provide enough information for esti-mation, and the parameter value will need to be set by the modeler or more measurements will need to be added to the regression. Parameters with values set by the modeler are called unestimated parameters. Com-posite scaled sensitivities can be calculated at any stage of model calibration. The values calculated for dierent sets of parameter values will be dierent, but are rarely dierent enough to indicate that a previously unesti-mated parameter can subsequently be estiunesti-mated.
Sensitivities calculated for the optimal parameter values are used in this work to calculate con®dence intervals on the estimated parameter values using linear (®rst-order) theory. Because linear theory is used, lin-ear con®dence intervals are only approximations for models which are nonlinear; however, they can still be used to identify potentially unneeded parameters. If, for example, a model input (such as hydraulic con-ductivity) is speci®ed using four parameters, but the regression yields parameter estimates that are within each others' con®dence intervals, it is likely that fewer parameters are adequate. If the regression using fewer parameters yields a similar model ®t to the measure-ments, it can be concluded that the available mea-surements are insucient to distinguish between the model with the four parameters and the model with fewer parameters. This approach applies the principle of parsimony, in which simpler models with fewer pa-rameters are favored over complex models that are equally valid in all other ways.
4.2 Model development and calibration
MODFLOWP. This process inevitably resulted in the further simpli®cation of the ¯ow-system conceptual model.
4.2.1 Model grid and boundary conditions
The DVRFS model is oriented north-south and is areally discretized into 163 rows and 153 columns (Fig. 4). The model is vertically discretized into three layers of constant thickness that represent a simpli®ca-tion of the material properties described in the 3D hydrogeologic framework model at 0±500 m, 500±1 250 m, and 1 250±2 750 m below an interpreted water table developed from hydraulic-head data. Thus, the layers are not ¯at. The three model layers were intended to represent the local, subregional and regional ¯ow paths respectively. The three layers are considered to be the minimum number required to reasonably represent three-dimensional ¯ow for this system. The resulting simulation is likely to suer from some inaccuracies in areas of signi®cant vertical ¯ow. This limitation was considered acceptable at this stage of the investigation, in which large-scale features were being characterized.
The lower boundary of the ¯ow system (2 750 m below the water table) is assumed to be the depth where ground-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 with constant-head boundaries speci®ed for regions where interbasinal ¯ux is believed to occur in the north and northeastern parts of the model boundary or where perennial lakes occur in Death Valley (Fig. 5).
4.2.2 Flow parameter discretization
The cellular data structure of the 3D hydrogeologic framework model allows it to be easily recon®gured for use in MODFLOWP. The GSIS used in this study uti-lizes a resampling function that produces ``slices'' from the 3D framework model. In the case of the DVRFS model, these ``slices'' represent the material properties for each numerical model layer. These slices were re-formatted into three 2D GIS maps. To start model calibration with a simple system representing only the dominant subsurface characteristics, these maps initially were simpli®ed to four zones representing high (K1), moderate (K2), low (K3), and very low (K4) hydraulic-conductivity values. The resulting initial zones were not contiguous; each zone included cells distributed through the model (Fig. 6). Using such a small number of zones at the beginning of the calibration allowed for a clear evaluation of gross features of the subsurface. Subse-quently, composite scaled sensitivities were used to de-termine whether zones could be subdivided to produce additional parameters that could be estimated with the available data.
4.2.3 Spatially-distributed source/sink parameters
The GIS-based in®ltration and ET maps also were re-con®gured into arrays for use in MODFLOWP. In the case of ET, a series of three maps were used to de®ne inputs. In MODFLOWP, ET is expressed in terms of a linear function based on land-surface elevation, extinc-tion depth, and maximum ET rate15. Each of these values 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 two grid-based GIS maps. To de®ne ground-water recharge, the recharge percentage map was reclassi®ed into as many as four zones representing high (RCH3), moderate (RCH2), low (RCH1), and zero (RCH0) recharge per-centage (Fig. 7). A parameter de®ned for each zone represents the percentage of average annual precipita-tion that in®ltrates. A multiplicaprecipita-tion array is used to represent the more predictable variation of average an-nual precipitation.
4.2.4 Conceptual model evolution
Fig. 5.Model boundary conditions: constant heads, springs, wells.
changes to the conceptual models involved modi®ca-tions to (1) the location and type of lateral ¯ow system boundary conditions, (2) the de®nition of the extent of areas of recharge, and (3) the con®guration of hydro-geologic framework features. The types of ¯ow system boundaries were adjusted in the north and northeast parts of the model area, where some boundaries were converted from constant head (simulating ¯ux into the model area) to no-¯ow (simulating a closed ¯ow sys-tem). The con®guration of recharge areas was changed from a ®xed, single percentage of precipitation to a combination of 4 zones with varying percentages. Most of the conceptual model variations involved changes to the hydrogeologic framework because the GSIS-pro-duced distribution tended to ``smooth out'' some im-portant hydrogeologic features. For example, areas of very low hydraulic conductivity were delineated into new distinct zones including (1) NW±SE trending fault zones, (2) clastic shales, (3) metamorphosed quartzites, and (4) isolated terrains of shallow Precambrian schists and gneisses. The location, extent and hydraulic con-ductivity of these zones were critical to accurately simulate existing large hydraulic gradients. Areas of high hydraulic conductivity also were delineated as new distinct zones. These typically included NE±SW trend-ing zones of highly fractured and faulted terrains. Many of these zones control dominant regional ¯ow paths and large-volume ¯ows to spring discharge areas.
All changes to the hydrogeologic framework were supported by hydrogeologic information existing in the database; no changes were made simply to produce a better model ®t.
For each conceptual model MODFLOWP was used to adjust parameter values to obtain a best ®t to hy-draulic head and spring ¯ow observations. Afterward, model ®t and estimated parameters were evaluated to help determine model validity. Modi®cations then were made to the existing conceptual model, observation data sets, or weighting, always maintaining consistency with the 3D hydrogeologic data base.
5 MODEL EVALUATION
After calibration, the DVRFS model was evaluated to assess the likely accuracy of simulated results. An ad-vantage of calibrating the DVRFS model using nonlin-ear regression is the existence of a substantial methodology for model evaluation that facilitates a better understanding of model strengths and weakness-es. A protocol exists to evaluate the likely accuracy of simulated results, associated con®dence intervals, and other measures of parameter and prediction uncertainty. Such information was not available for previous models of the Death Valley region, which were calibrated without nonlinear regression.
5.1 Evaluation of hydraulic heads and spring ¯ows
The observations used by the regression initially in-cluded 512 measured hydraulic heads and 63 measured spring ¯ows. During calibration, 12 of the hydraulic-head observations were removed from the data set be-cause of recording errors found in the database, thus 500 head observations were used in the ®nal regression. Five of these remaining hydraulic-head observations are questionable because they appear to represent lo-cally-perched water levels rather than regional water levels.
The spring ¯ows were represented as head-dependent boundaries connected to either the top or bottom model layers, depending on whether the temperature and chemistry of the spring indicated a shallow or deep source. Accounting for the initial 63 individual mea-sured spring ¯ows during calibration proved to be dif-®cult; therefore they were combined into 16 groups based on proximity and likely depth from which the springs originate. The combined ¯ows are consistent with the simulated model scale.
For the regression, each of the observed head and spring-¯ow values were assigned an estimated standard deviation or coecient of variation based on how pre-cise the measurement was thought to be. This statistic was used to calculate the weights of eqn (1). More pre-cise measurements were assigned a greater weight (smaller standard deviation or coecient of variation; less precise measurements were assigned a lesser weight (greater standard deviation or coecient of variation). More precise measurements typically included accurate water-level measurements at wells that had been sur-veyed for location and elevation. Less precise measure-ments typically included water-level measuremeasure-ments made with unspeci®ed methods from wells located on smaller-scale topographic maps. Weighting of the hydraulic-head and ¯ow observations was initially assigned by calculating the needed estimates of variances from as-sumed head standard deviations of 10 m and asas-sumed ¯ow coecients of variation of 10%. The ®nal weights were variable for each type of data: standard deviations for hydraulic heads included values of 10, 30, 100, or 250 m, with all but 24 standard deviations being 10 m, and the 100 and 250 m values used only in vicinity of large hydraulic gradients. Coecients of variation for ¯ows included values of 5%, 10%, 33%, and 100%, with the larger values being applied to small springs of uncertain relation to the regional ¯ow system.
5.1.1 Evaluation of model ®t
Unweighted and weighted residuals (de®ned after eqn (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 the observations, and indicate, for example, that a hydraulic
head is matched to within 10 m. Unweighted residuals can, however, be misleading because observations are measured with dierent precision.
Weighted residuals demonstrate model ®t relative to what 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 that equal the number of standard deviations or coecients of variation needed to equal the unweighted residual. Maps of both weighted and unweighted residuals were constructed and analyzed for the DVRFS model6.
Unweighted hydraulic-head residuals tended to be larger in areas with steep hydraulic gradients than in areas with ¯at gradients, so these two types of areas are discussed separately. In areas of relatively ¯at hydraulic gradients, the largest unweighted residuals have absolute values less than 75 m and are commonly less than 50 m. In areas of large hydraulic gradients, the dierences between simulated and observed heads are sometimes larger (as large as 150 m); however, all simulated gra-dients are within 60 percent of the gragra-dients evident from the data. The match is good considering the 2 000 m dierence in hydraulic head across the model domain and the size of the grid relative to the width of some of the steep hydraulic gradients.
Matching spring ¯ows was dicult but extremely important to model calibration. The sum of all simu-lated spring ¯ows is 51 700 m3/day; the sum of estimated regional spring ¯ows is 125 400 m3/day. The total sim-ulated ¯ow through the system is 405 000 m3/day, so that the dierence is 13%. The dierence between the total estimated and simulated spring ¯ows is large; however, from the perspective of a regional model being able to simulate a process with many local eects like spring ¯ow, the match is considered to be quite good. In addition, the ET in many of these areas is larger than expected; therefore, the total ¯ux from both ET and spring ¯ow is matched more closely.
5.1.2 Distribution weighted residuals relative to weighted simulated values
To evaluate model results for systematic model error or errors in assumptions concerning observations and weights, weighted residuals are plotted against weighted simulated values4,9. Ideally, weighted residuals vary randomly about zero regardless of the simulated value. Fig. 8 shows that most of the weighted residuals for hydraulic heads in the DVRFS model vary randomly about a value of zero, but there are some large positive values. Positive residuals indicate that the simulated head is lower than the observed head. Nine values are greater than +14.1, which is three times the regression standard error of 4.6; no values are less thanÿ14.1. For normally distributed values only 3 in 1000, on average, would be so dierent from the expected value. Thus, this distribution is distinctly biased by the large positive values. Evaluation of these residuals indicates that many of the measurements occur where perched conditions are suspected, so that the bias may result from misclassi®-cation of the data instead of model error.
The weighted residuals for spring ¯ows shown in Fig. 8 are mostly negative. For two of the weighted re-siduals, the values are less than ÿ14.1, which is more than three regression standard errors from the expected mean of 0.0. Because of the sign convention used, neg-ative weighted residuals for spring ¯ows indicate that the observed ¯ows are larger in magnitude than the simulated ¯ows. These residuals indicate that the re-gional model probably is not representing some the processes related to spring ¯ows correctly. Whether or not this is an important model error probably needs to be judged in the context of the total ¯ux at the discharge areas, which includes ET. As mentioned above, the total ¯uxes match more closely, especially at the large volume springs.
5.1.3 Normality of weighted residuals and model nonlin-earity
The 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®dence intervals. Speci®cally, the weighted residuals need to be normally distributed and the model needs to be eec-tively linear for the calculated linear con®dence intervals on estimated parameters and predicted heads and ¯ows to accurately represent simulation uncertainty3,13,19. In this 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 points would be expected to fall along a straight line if the weighted residuals were both independent and normally distributed. Clearly, the points do not fall along a straight line. One possibility is that the residuals are normally distributed, but they are correlated instead of being independent. Correlations are derived from the ®tting of the regression.
The source of correlation can be investigated using the graphical procedures described by Cooley and Na4. Normally distributed random numbers generated to be consistent with the regression derived correlations are called correlated normal random deviates, and are shown in Fig. 10. These plots show that most of the curvilinearity 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 analysis indicates that weighted residuals are not normally dis-tributed.
Model linearity was tested using a statistic referred to as the modi®ed Beale's measure4, which is calculated using the computer program BEALEP13. The modi®ed Beale's measure calculated for the DVRFS model equals 0.42, which is between the critical values of 0.05 and 0.5. If Beale's measure is less than 0.05 the model is eec-tively linear. If Beale's measure is greater than 0.5 the model is highly nonlinear. Thus, the ®nal model is close to being highly nonlinear.
The lack of normality of the weighted residuals and the moderately high degree of nonlinearity of the
DVRFS model indicate that linear con®dence intervals are likely to be inaccurate. It can be concluded from previous work by Christensen and Cooley3, that linear con®dence intervals often can be used as rough indica-tors of simulation uncertainty, even in the presence of some model nonlinearity. The nonnormal weighted re-siduals indicate a greater degree of potential error in the linear con®dence intervals. Despite this problem, linear con®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 the DVRFS model includes all of the most important sys-tem characteristics, as indicated by evaluating composite scaled sensitivities. This analysis helps to ensure that the measures of prediction uncertainty calculated using the
model will re¯ect most of the uncertainty in the system, because all measures of prediction uncertainty presently available mostly propagate the uncertainty of the esti-mated parameter values. Uncertainty in other aspects of the model are not propagated into the uncertainty measures 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 produce the best match between the measured and simulated heads and ¯ows would be realistic values. Thus, model error would be indicated by unreasonable estimates of parameters for which the data provide substantial in-formation16.
A measure of the amount of information provided by the observations for any parameter is the composite scaled sensitivity discussed earlier and the linear con®-dence interval on the parameter. Generally, a parameter with a large composite scaled sensitivity will have a small con®dence interval relative to a parameter with a smaller composite scaled sensitivity. If an estimated parameter value is unreasonable and the data provide enough 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 be lack of data or insensitivity, and more likely to be model error or misinterpreted hydraulic-conductivity data.
Table 1 shows the estimated parameter values for the DVRFS model, their coecients of variation (the stan-dard deviation of the estimate divided by the estimated value), 95% linear con®dence intervals, and the range of values thought to be reasonable based on information gathered as part of the regional hydrogeologic charac-terization but not used in the regression. The
hydraulic-Table 1. Estimated values, coecients of variation, and the 95% linear con®dence intervals for the parameters of the ®nal calibrated model, and the range of reasonable values, with the range of reasonable values
Parameter label (units)
Log-transformed for regression
Estimated value Coecient of
variationa 95% Linear con®denceupper/lower limits on
the estimateb
Expected upper/lower range of reasonable
values
K1 (m/d) Yes 0.275 0.149 0.369; 0.205 100.0; 0.1
K2 (m/d) Yes 0.443´10ÿ1 0.113 0.554´10ÿ1;
ANIV3 Yes 164 0.518 399; 67.2 1000.0; 1.0
RCH2 (percent) No 3.02 0.107 3.66; 2.37 8.0; 1.0
RCH3 (percent) No 22.7 0.0518 25.0; 20.3 30.0; 15.0
aFor parameters that were log-transformed for regression, these are calculated ass
B/B, whereBis the untransformed estimated value,sB2exp(s2lnB+ 2(lnB))(exp(s
2
lnB)ÿ1.), ands
2
lnBis the variance of the log-transformed value estimated by regression.
bThe con®dence intervals are not symmetric about the estimated value for parameters that were log-transformed for the regression.
conductivity parameter values, together with their con-®dence intervals and reasonable ranges of values, are also shown in Fig. 11. In all cases, the optimized pa-rameter value is within its expected range, though most of the hydraulic conductivity estimates tend to be in the upper end of the reasonable range.
No prior information was included in the sum-of-squares objective function to restrict the estimation process; only the model design and the observation data in¯uenced parameter estimation. Estimation of the most important parameters without prior infor-mation has the advantage of allowing a more direct test of the model using the observation data (the hy-draulic heads and ¯ows). In this approach, the avail-able information on reasonavail-able parameter values is used to evaluate the estimated parameter values. For the DVRFS model, this evaluation revealed no indi-cation of model error.
As shown in Table 1 and Fig. 11, the reasonable ranges on some of the parameters, and especially the hydraulic-conductivity parameters, are wide, which may suggest that this evaluation is not very powerful. During calibration, however, many conceptual models pro-duced parameter estimates that violated this seemingly easy test, and the evaluation was found to be very useful. The con®dence intervals on the parameter estimates shown in Fig. 11 may seem unrealistically small, but this is largely because they represent the con®dence interval for the mean hydraulic-conductivity value. As pointed out by Hill14, con®dence intervals on mean values are rapidly reduced from the entire range of the population as data is applied to the estimation of the mean. The validity of the idea that the hydrogeologic units have uniform `mean' or `eective' values is, of course, a basic hypothesis 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, as well as is done by the DVRFS model, suggests that the approach is likely to be valid for this system.
Composite scaled sensitivities (cj for parameter j,
eqn (2)) were used during calibration to decide what parameters to include and exclude from the estimation process. Parameters with relatively highcj values often
were included in the estimation process, while parame-ters with relatively low cj values were not included. In
some cases, a parameter may have had a high enough sensitivity to be easily estimated by the regression, but was correlated (as determined from parameter correla-tion coecients9,14 with another parameter of higher sensitivity. In these cases, the parameter of lower sen-sitivity typically was left unestimated. At times, the number of parameters that were estimated was limited by the execution time of the computer used.
Partly because of model nonlinearity, the values ofcj
change somewhat as the parameter values change. As a result, the evaluation of cj values was repeated
fre-quently. Composite scaled sensitivity values for esti-mated parameters of the ®nal model are shown in Fig. 12. The ®nal values changed somewhat, but were still quite similar to initial values, and generally indicate that the parameters being estimated were the most im-portant parameters. Exceptions occur for parameters that were correlated with parameters with larger Fig. 11. Estimated hydraulic conductivity parameters, their
composite scaled sensitivities, and for parameters that mostly in¯uence model ®t to a single observation.
5.3 Signi®cance of model evaluation
The model evaluation results presented suggest that the DVRFS model reproduces the measured hydraulic heads 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 simulation of the observed quantities. Also, the ®nal estimated parameter values are all within reasonable ranges.
The model used at this stage of model calibration was able to reproduce major characteristics of the system quite well considering its simplicity. The simplicity was believed to be crucial to the analysis because it allowed a more thorough analysis of the large-scale aspects of the system which would not have been possible with a more detailed model and accompanying longer execution times. Knowledge of the system available from this model forms an excellent foundation for more detailed model development.
6 CONCLUSIONS
The available state-of-the-art GSIS and parameter-esti-mation techniques utilized in this study materially assisted in modeling the complex DVRFS. Three-di-mensional hydrogeologic framework modeling com-bined with geologically realistic interpretation allowed characterization of the ``data-sparse'' subsurface, while integrated image processing and hydrologic process modeling using traditional GIS techniques support surface-based characterization eorts. The dierent con®gurations of the digital 3D hydrogeologic concep-tual model allows rapid evaluation of various likely representations of the ¯ow system.
While ground-water inverse problems are generally plagued by problems of nonuniqueness, this work clearly
demonstrates that, even for a complex ground-water system, substantial constraints can be developed from ground-water model calibration. The constraints used in this work include (1) a geologic framework, which con-strains the alternative conceptual models; (2) testing possible conceptual models by determining the parame-ter values needed to produce a best ®t to the hydrologic data (heads and spring ¯ows in this work) using inverse modeling; and (3) testing the validity of the model by considering the ®t between the data and the associated simulated values, comparing simulated global budget terms to values estimated from ®eld data, and by testing the plausibility of optimized parameter values. Because this is a complex system, the problem of nonuniqueness is never completely eliminated. By eectively satisfying more constraints, however, the probability is increased that the resulting model more accurately represents the physical system. The key is development and use of the proper 2D and 3D data sets. Joint use of GSIS tech-niques and optimal parameter estimation by nonlinear regression was essential to achieving these objectives.
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