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Advances in Water Resources Vol. 22, No. 8, pp 777±790, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter
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 properties 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 distribution. 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 optimal 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
(DVRFS). The details of this modeling study are described in US Geological Survey Water Resources Investigations Report 96-43006. This paper focuses on the
joint use of state-of-the-art optimal parameter estimation and geoscienti®c information 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 semiarid to arid region is located within the southern Great
Basin, a subprovince of the Basin and Range Physiographic 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
1 INTRODUCTION
Yucca Mountain on and adjacent to the Nevada Test
Site in southwestern Nevada is being studied as a potential site for a high-level nuclear waste geologic repository. The United States Geological Survey (USGS),
in cooperation with the Department of Energy, is evaluating 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 regional ground-water ¯ow system.
This paper describes the calibration of a three-dimensional (3D), steady-state, ground-water ¯ow model
of the Death Valley regional ground-water ¯ow system
*
Corresponding author. Present address: US Geological
Survey, Suite 221, 520 N. Park Ave, Tucson, AZ 85719, USA
777
778
F. A. D'Agnese et al.
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 hydrogeologic conceptual models of the ground-water ¯ow
system
3. numerical simulation of the ground-water ¯ow system by testing combinations of conceptual models
4. calibration of the ¯ow model using parameter-estimation 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, surfacewater 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. 1. Map showing location of the Death Valley regional
ground-water ¯ow system, Nevada and California, 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.
Fig. 2. Flow chart showing logical movement of modeling data
through various GSIS packages.
Death valley regional ground-water ¯ow model calibration
This integration allowed the investigators ease of data
manipulation and aided in development of the conceptual and numerical models.
3 DEVELOPMENT OF DIGITAL 3D
HYDROGEOLOGIC CONCEPTUAL MODELS
A digital 3D hydrogeologic conceptual model is a representation 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 hydraulic properties of the materials that control
ground-water ¯ow
2. characterizing the surface and subsurface hydrologic conditions that aect ground-water movement and
3. evaluating various hypotheses about the ¯ow system 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-
779
tion models (DEMs), hydrogeologic maps and sections,
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 specialized 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 properly located in 3D coordinate space to de®ne locations of each hydrogeologic unit in the subsurface
3. Surface and subsurface data were interpolated to
de®ne the top of each hydrogeologic unit, incorporating 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 relationships (Fig. 3).
GSIS procedures were utilized to develop framework
model attributes describing hydraulic properties. For
each hydrogeologic unit, the value of hydraulic conductivity was initially assigned based on log-normal probability 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
Fig. 3. Perspective view of 3D hydrogeologic framework model.
780
F. A. D'Agnese et al.
of magnitude. Within individual hydrogeologic units,
potential values of hydraulic conductivity range over 3±7
orders of magnitude. The large range of hydraulic conductivity within hydrogeologic units indicates substantial, 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 occurring between adjacent hydrogeologic units. Therefore,
this work concentrates on developing and testing conceptual 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 characterization 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 components 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 applied 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 empirical 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, wateruse 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 investigators 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 uncertainty, (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 feasibility of the multiple conceptual models for the ¯ow system by displaying data control on interpreted products.
4 NUMERICAL SIMULATION OF REGIONAL
GROUND-WATER FLOW
Because of the numerous factors controlling groundwater ¯ow in this region, even the relatively coarsegridded DVRFS model is necessarily large and complex.
Calibration of this model by strictly trial-and-error
methods was judged to be both ineective and inecient; 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 parameters 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, denselyspaced 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 conductivity occurring along or within regional features.
The approach to using nonlinear regression methods
presented here is described in more detail by Hill14 who
suggests that the method is best suited to determining
large-scale variations in systems. From a stochastic
viewpoint, the approach can be thought of as identifying
a mean that cannot be assumed to be a constant or a
simple function. Once the large-scale variations are adequately characterized, stochastically-based methods
may be employed to characterize smaller-scale varia-
Death valley regional ground-water ¯ow model calibration
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
T
S b y ÿ y0 W y ÿ y0
1
where,
b is an np ´ 1 vector containing parameter values
np is the number of parameters estimated by regression
y and y0 are n ´ 1 vectors with elements equal to measured and simulated (using b) values, respectively, of
hydraulic heads and spring ¯ows
n is the number of measured or simulated hydraulic
heads and ¯ows
y ÿ y0 is a vector of observed minus simulated values,
which are called residuals
W is an n ´ n weight matrix
W1=2 y ÿ y0 is 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 determined 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 MODFLOWP, 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= n ÿ np1=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 allows these spatially distributed physical quantities to be
represented using zones over which the parameter is
constant, or to be de®ned using more sophisticated interpolation methods. In either case, multipliers or multiplication 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.
0
Sensitivities are calculated by MODFLOWP as oyi /obj ,
the partial derivative of the ith simulated hydraulic head
781
or ¯ow, y0i , with respect to the jth 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 information that the measurements contain toward the
estimation of that parameter. Composite scaled sensitivity for parameter j, cj , is calculated as
("
#, )0:5
X
2 2
0
cj
wi oyi =obj bj
2
n
i1;n
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
smaller cj values may be more dicult or impossible to
estimate. For some parameters, the available measurements may not provide enough information for estimation, 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. Composite 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 unestimated parameter can subsequently be estimated.
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, linear 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 conductivity) 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 measurements, it can be concluded that the available measurements 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 parameters are favored over complex models that are
equally valid in all other ways.
4.2 Model development and calibration
Prior to numerical simulation, the 3D hydrogeologic
data sets, which were discretized at various grid-cell
resolutions ranging from 100 m to 500 m, were resampled into a uniform 1500-m grid for input to
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F. A. D'Agnese et al.
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®cation 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 involved do not signi®cantly impact regional ¯ow estimates. 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 utilizes 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 reformatted 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) hydraulicconductivity 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. Subsequently, composite scaled sensitivities were used to determine 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 recon®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, extinction depth, and maximum ET rate15. Each of these
values was speci®ed as a 2D array generated from GISbased 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 percentage (Fig. 7). A parameter de®ned for each zone
represents the percentage of average annual precipitation that in®ltrates. A multiplication array is used to
represent the more predictable variation of average annual precipitation.
Fig. 4. Location of the model grid.
4.2.4 Conceptual model evolution
A number of conceptual models were evaluated using
the regression methods in MODFLOWP. Because of
the simpli®ed nature of the initial simulations, succeeding conceptual models often involved adding
complexities to the ¯ow model. For the DVRFS model,
Death valley regional ground-water ¯ow model calibration
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.
783
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F. A. D'Agnese et al.
Fig. 7. Recharge zone array for the DVRFS model.
changes to the conceptual models involved modi®cations 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 hydrogeologic 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 system). 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-produced distribution tended to ``smooth out'' some important 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 conductivity 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 trending 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 hydraulic 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 advantage of calibrating the DVRFS model using nonlinear regression is the existence of a substantial
methodology for model evaluation that facilitates a
better understanding of model strengths and weaknesses. 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.
Death valley regional ground-water ¯ow model calibration
5.1 Evaluation of hydraulic heads and spring ¯ows
The observations used by the regression initially included 512 measured hydraulic heads and 63 measured
spring ¯ows. During calibration, 12 of the hydraulichead observations were removed from the data set because 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 locally-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 measured 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 precise the measurement was thought to be. This statistic
was used to calculate the weights of eqn (1). More precise 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 surveyed for location and elevation. Less precise measurements typically included water-level measurements made
with unspeci®ed methods from wells located on smallerscale topographic maps. Weighting of the hydraulichead and ¯ow observations was initially assigned by
calculating the needed estimates of variances from assumed head standard deviations of 10 m and assumed
¯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, depending somewhat on data quality, model accuracy.
Consideration of unweighted residuals is intuitively appealing because the values have the dimensions of the
observations, and indicate, for example, that a hydraulic
785
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 precision, or noise, of the data. They are less intuitively appealing 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 gradients are within 60 percent of the gradients 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 simulated spring ¯ows is 51 700 m3 /day; the sum of estimated
regional spring ¯ows is 125 400 m3 /day. The total simulated ¯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.
When weighted as described above, the sum of
squared, weighted residuals objective function was
11 050, with heads contributing 9 500 and ¯ows contributing 1 650. Considering the number of heads and
¯ows involved, these numbers indicate that, on average, the head and ¯ow weighted residuals are similar in
magnitude, as needed for a valid regression. The
standard error of regression equals 4.5, which indicates
that overall model ®t is consistent with hydraulic head
standard deviations that are 4.5 times the standard
deviations used to calculate the weights of eqn (1).
Thus, eective model ®t for most wells is 45 m, which
is consistent with the size of the unweighted hydraulic
head residuals described above. For spring ¯ow, overall model ®t is consistent with 4.6 times the coecients
of variation used to calculate the weights. Thus, eective model ®t for most spring ¯ows is 46%, which,
again, is consistent with the ®t to spring ¯ows described above.
786
F. A. D'Agnese et al.
Fig. 8. Weighted residuals against weighted simulated values.
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 than ÿ14.1. 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 residuals, 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, negative weighted residuals for spring ¯ows indicate that
the observed ¯ows are larger in magnitude than the
simulated ¯ows. These residuals indicate that the regional 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 nonlinearity
The normality of the weighted residuals and model linearity 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 eectively 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 parameter values are presented.
The normal probability graph of the weighted residuals 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 regression-derived correlations, but some of the curving related to extreme values might be explained. This analysis
indicates that weighted residuals are not normally distributed.
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 eectively 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
Fig. 9. Normal probability plot of weighted residuals.
787
Death valley regional ground-water ¯ow model calibration
Fig. 10. Normal probability plot of correlated normal random
deviates.
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 indicators of simulation uncertainty, even in the presence of
some model nonlinearity. The nonnormal weighted residuals 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 indicators of the uncertainty in estimated parameter values.
5.2 Evaluation of estimated parameter values
The set of parameters estimated by regression in the
DVRFS model includes all of the most important system 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 estimated 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 information about the parameters being estimated, it is reasonable 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 information16.
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 interval on the parameter estimate also excludes reasonable 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 standard 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 characterization 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
Log-transformed
for regression
Estimated value
Coecient of
variation a
95% Linear con®dence
upper/lower limits on
the estimate b
Expected upper/lower
range of reasonable
values
K1 (m/d)
K2 (m/d)
Yes
Yes
0.275
0.443 ´ 10ÿ1
0.149
0.113
100.0; 0.1
0.1; 0.0004
K3 (m/d)
Yes
0.562 ´ 10ÿ2
0.181
K4 (m/d)
Yes
0.856 ´ 10ÿ4
0.263
K5 (m/d)
K9 (m/d)
ANIV3
RCH2 (percent)
RCH3 (percent)
Yes
Yes
Yes
No
No
21.2
0.159
164
3.02
22.7
0.499
0.479
0.518
0.107
0.0518
0.369; 0.205
0.554 ´ 10ÿ1 ;
0.354 ´ 10ÿ1
0.801 ´ 10ÿ2 ;
0.394 ´ 10ÿ2
0.146 ´ 10ÿ3 ;
0.501 ´ 10ÿ4
0.50 ´ 102 ; 0.889 ´ 101
0.367 ´ 100 ; 0.686 ´ 10ÿ1
399; 67.2
3.66; 2.37
25.0; 20.3
Parameter label
(units)
a
0.02; 0.0001
1 ´ 10ÿ4 ; 2 ´ 10ÿ7
100.0; 8.0
1.0; 0.01
1000.0; 1.0
8.0; 1.0
30.0; 15.0
For parameters that were log-transformed for regression, these are calculated as sB /B, where B is the untransformed estimated
value, sB 2 exp(s2ln B + 2(ln B))(exp(s2ln 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.
788
F. A. D'Agnese et al.
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 parameter value is within its expected range, though most
of the hydraulic conductivity estimates tend to be in the
upper end of the reasonable range.
Fig. 11. Estimated hydraulic conductivity parameters, their
95% linear con®dence intervals, and the ranges of reasonable
values.
No prior information was included in the sum-ofsquares 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 information has the advantage of allowing a more direct
test of the model using the observation data (the hydraulic heads and ¯ows). In this approach, the available information on reasonable parameter values is
used to evaluate the estimated parameter values. For
the DVRFS model, this evaluation revealed no indication 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 produced 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 high cj values often
were included in the estimation process, while parameters 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 correlation coecients9,14 with another parameter of higher
sensitivity. In these cases, the parameter of lower sensitivity 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 of cj
change somewhat as the parameter values change. As a
result, the evaluation of cj values was repeated frequently. Composite scaled sensitivity values for estimated 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 important parameters. Exceptions occur for parameters
that were correlated with parameters with larger
Death valley regional ground-water ¯ow model calibration
Fig. 12. Composite scaled sensitivities for estimated parameters in the ®nal calibrated model.
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 estimated parameter values include the aspects of the system 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-estimation techniques utilized in this study materially
assisted in modeling the complex DVRFS. Three-dimensional hydrogeologic framework modeling combined 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 conceptual 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
789
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 constrains the alternative conceptual models; (2) testing
possible conceptual models by determining the parameter 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 techniques and optimal parameter estimation by nonlinear
regression was essential to achieving these objectives.
REFERENCES
1. Bard, Y., Nonlinear parameter estimation. Academic Press,
New York, 1974.
2. Bedinger, M.S., Sargent, K.A., Langer, W.H., Studies of
geology and hydrology in the Basin and Range Province,
Southwestern United States, for isolation of high-level
radioactive waste ± characterization of the Death Valley
region, Nevada and California. US Geological Survey
Professional Paper 1370-F, 1989.
3. Christensen, S. and Cooley, R.L., Simultaneous con®dence
intervals for a steady-state leaky aquifer groundwater ¯ow
model. In: K. Kovar and Paul van der Heidje, (Eds.),
Calibration and Reliability in Groundwater Modeling (Proceedings of the ModelCARE '96 Conference held at
Golden Colorado, September 1996). IAHS Publ. no. 237
(1996) 561±569.
4. Cooley, R.L. and Na, R.L., Regression modeling of
ground-water ¯ow. US Geological Survey Techniques of
Water-Resources Investigations, Book 3, Chapter B4,
1990.
5. Czarnecki, J.B. and Waddell, R.K., Finite-element simulation of ground-water ¯ow in the vicinity of Yucca
Mountain, Nevada-California. US Geological Survey,
Water Resources Investigations Report, 1984, 84-4349.
6. D'Agnese, F.A., Faunt, C.C., Turner, A.K., Hill, M.C.,
Hydrogeologic evaluation and numerical simulation of the
Death Valley regional ground-water ¯ow system, Nevada
and California. US Geological Survey Water Resources
Investigations Report 96±4300, 1997.
7. D'Agnese, F.A., Faunt, C.C., Turner, A.K., Using remote
sensing and GIS techniques to estimate discharge and
recharge ¯uxes for the Death valley regional ground-water
¯ow system, Nevada and California, USA. In: K. Kovar
and H.P. Nachtnebel (Eds.) HydroGIS '96: Application of
Geographic Information Systems in Hydrology and Water
Resources Management, (Proceedings of the Vienna Conference, April 1996). IAHS Publ. no. 235 (1996) 503±511.
8. Domenico, P.A., Concepts and models in groundwater
hydrology. McGraw-Hill, New York, 1972.
790
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9. Draper, N. and Smith, H., Applied regression analysis, 2nd
edition. Wiley, New York, 1981.
10. Faunt, C.C., D'Agnese, F.A. and Turner, A.K.,
Characterizing the three-dimensional hydrogeologic
framework model for the Death Valley region, southern Nevada and California, USA. In: K. Kovar and
H.P. Nachtnebel (Eds.), HydroGIS '93: Application
Advances in Water Resources Vol. 22, No. 8, pp 777±790, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter
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 properties 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 distribution. 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 optimal 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
(DVRFS). The details of this modeling study are described in US Geological Survey Water Resources Investigations Report 96-43006. This paper focuses on the
joint use of state-of-the-art optimal parameter estimation and geoscienti®c information 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 semiarid to arid region is located within the southern Great
Basin, a subprovince of the Basin and Range Physiographic 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
1 INTRODUCTION
Yucca Mountain on and adjacent to the Nevada Test
Site in southwestern Nevada is being studied as a potential site for a high-level nuclear waste geologic repository. The United States Geological Survey (USGS),
in cooperation with the Department of Energy, is evaluating 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 regional ground-water ¯ow system.
This paper describes the calibration of a three-dimensional (3D), steady-state, ground-water ¯ow model
of the Death Valley regional ground-water ¯ow system
*
Corresponding author. Present address: US Geological
Survey, Suite 221, 520 N. Park Ave, Tucson, AZ 85719, USA
777
778
F. A. D'Agnese et al.
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 hydrogeologic conceptual models of the ground-water ¯ow
system
3. numerical simulation of the ground-water ¯ow system by testing combinations of conceptual models
4. calibration of the ¯ow model using parameter-estimation 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, surfacewater 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. 1. Map showing location of the Death Valley regional
ground-water ¯ow system, Nevada and California, 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.
Fig. 2. Flow chart showing logical movement of modeling data
through various GSIS packages.
Death valley regional ground-water ¯ow model calibration
This integration allowed the investigators ease of data
manipulation and aided in development of the conceptual and numerical models.
3 DEVELOPMENT OF DIGITAL 3D
HYDROGEOLOGIC CONCEPTUAL MODELS
A digital 3D hydrogeologic conceptual model is a representation 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 hydraulic properties of the materials that control
ground-water ¯ow
2. characterizing the surface and subsurface hydrologic conditions that aect ground-water movement and
3. evaluating various hypotheses about the ¯ow system 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-
779
tion models (DEMs), hydrogeologic maps and sections,
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 specialized 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 properly located in 3D coordinate space to de®ne locations of each hydrogeologic unit in the subsurface
3. Surface and subsurface data were interpolated to
de®ne the top of each hydrogeologic unit, incorporating 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 relationships (Fig. 3).
GSIS procedures were utilized to develop framework
model attributes describing hydraulic properties. For
each hydrogeologic unit, the value of hydraulic conductivity was initially assigned based on log-normal probability 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
Fig. 3. Perspective view of 3D hydrogeologic framework model.
780
F. A. D'Agnese et al.
of magnitude. Within individual hydrogeologic units,
potential values of hydraulic conductivity range over 3±7
orders of magnitude. The large range of hydraulic conductivity within hydrogeologic units indicates substantial, 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 occurring between adjacent hydrogeologic units. Therefore,
this work concentrates on developing and testing conceptual 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 characterization 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 components 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 applied 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 empirical 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, wateruse 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 investigators 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 uncertainty, (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 feasibility of the multiple conceptual models for the ¯ow system by displaying data control on interpreted products.
4 NUMERICAL SIMULATION OF REGIONAL
GROUND-WATER FLOW
Because of the numerous factors controlling groundwater ¯ow in this region, even the relatively coarsegridded DVRFS model is necessarily large and complex.
Calibration of this model by strictly trial-and-error
methods was judged to be both ineective and inecient; 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 parameters 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, denselyspaced 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 conductivity occurring along or within regional features.
The approach to using nonlinear regression methods
presented here is described in more detail by Hill14 who
suggests that the method is best suited to determining
large-scale variations in systems. From a stochastic
viewpoint, the approach can be thought of as identifying
a mean that cannot be assumed to be a constant or a
simple function. Once the large-scale variations are adequately characterized, stochastically-based methods
may be employed to characterize smaller-scale varia-
Death valley regional ground-water ¯ow model calibration
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
T
S b y ÿ y0 W y ÿ y0
1
where,
b is an np ´ 1 vector containing parameter values
np is the number of parameters estimated by regression
y and y0 are n ´ 1 vectors with elements equal to measured and simulated (using b) values, respectively, of
hydraulic heads and spring ¯ows
n is the number of measured or simulated hydraulic
heads and ¯ows
y ÿ y0 is a vector of observed minus simulated values,
which are called residuals
W is an n ´ n weight matrix
W1=2 y ÿ y0 is 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 determined 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 MODFLOWP, 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= n ÿ np1=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 allows these spatially distributed physical quantities to be
represented using zones over which the parameter is
constant, or to be de®ned using more sophisticated interpolation methods. In either case, multipliers or multiplication 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.
0
Sensitivities are calculated by MODFLOWP as oyi /obj ,
the partial derivative of the ith simulated hydraulic head
781
or ¯ow, y0i , with respect to the jth 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 information that the measurements contain toward the
estimation of that parameter. Composite scaled sensitivity for parameter j, cj , is calculated as
("
#, )0:5
X
2 2
0
cj
wi oyi =obj bj
2
n
i1;n
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
smaller cj values may be more dicult or impossible to
estimate. For some parameters, the available measurements may not provide enough information for estimation, 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. Composite 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 unestimated parameter can subsequently be estimated.
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, linear 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 conductivity) 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 measurements, it can be concluded that the available measurements 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 parameters are favored over complex models that are
equally valid in all other ways.
4.2 Model development and calibration
Prior to numerical simulation, the 3D hydrogeologic
data sets, which were discretized at various grid-cell
resolutions ranging from 100 m to 500 m, were resampled into a uniform 1500-m grid for input to
782
F. A. D'Agnese et al.
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®cation 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 involved do not signi®cantly impact regional ¯ow estimates. 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 utilizes 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 reformatted 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) hydraulicconductivity 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. Subsequently, composite scaled sensitivities were used to determine 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 recon®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, extinction depth, and maximum ET rate15. Each of these
values was speci®ed as a 2D array generated from GISbased 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 percentage (Fig. 7). A parameter de®ned for each zone
represents the percentage of average annual precipitation that in®ltrates. A multiplication array is used to
represent the more predictable variation of average annual precipitation.
Fig. 4. Location of the model grid.
4.2.4 Conceptual model evolution
A number of conceptual models were evaluated using
the regression methods in MODFLOWP. Because of
the simpli®ed nature of the initial simulations, succeeding conceptual models often involved adding
complexities to the ¯ow model. For the DVRFS model,
Death valley regional ground-water ¯ow model calibration
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.
783
784
F. A. D'Agnese et al.
Fig. 7. Recharge zone array for the DVRFS model.
changes to the conceptual models involved modi®cations 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 hydrogeologic 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 system). 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-produced distribution tended to ``smooth out'' some important 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 conductivity 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 trending 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 hydraulic 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 advantage of calibrating the DVRFS model using nonlinear regression is the existence of a substantial
methodology for model evaluation that facilitates a
better understanding of model strengths and weaknesses. 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.
Death valley regional ground-water ¯ow model calibration
5.1 Evaluation of hydraulic heads and spring ¯ows
The observations used by the regression initially included 512 measured hydraulic heads and 63 measured
spring ¯ows. During calibration, 12 of the hydraulichead observations were removed from the data set because 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 locally-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 measured 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 precise the measurement was thought to be. This statistic
was used to calculate the weights of eqn (1). More precise 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 surveyed for location and elevation. Less precise measurements typically included water-level measurements made
with unspeci®ed methods from wells located on smallerscale topographic maps. Weighting of the hydraulichead and ¯ow observations was initially assigned by
calculating the needed estimates of variances from assumed head standard deviations of 10 m and assumed
¯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, depending somewhat on data quality, model accuracy.
Consideration of unweighted residuals is intuitively appealing because the values have the dimensions of the
observations, and indicate, for example, that a hydraulic
785
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 precision, or noise, of the data. They are less intuitively appealing 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 gradients are within 60 percent of the gradients 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 simulated spring ¯ows is 51 700 m3 /day; the sum of estimated
regional spring ¯ows is 125 400 m3 /day. The total simulated ¯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.
When weighted as described above, the sum of
squared, weighted residuals objective function was
11 050, with heads contributing 9 500 and ¯ows contributing 1 650. Considering the number of heads and
¯ows involved, these numbers indicate that, on average, the head and ¯ow weighted residuals are similar in
magnitude, as needed for a valid regression. The
standard error of regression equals 4.5, which indicates
that overall model ®t is consistent with hydraulic head
standard deviations that are 4.5 times the standard
deviations used to calculate the weights of eqn (1).
Thus, eective model ®t for most wells is 45 m, which
is consistent with the size of the unweighted hydraulic
head residuals described above. For spring ¯ow, overall model ®t is consistent with 4.6 times the coecients
of variation used to calculate the weights. Thus, eective model ®t for most spring ¯ows is 46%, which,
again, is consistent with the ®t to spring ¯ows described above.
786
F. A. D'Agnese et al.
Fig. 8. Weighted residuals against weighted simulated values.
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 than ÿ14.1. 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 residuals, 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, negative weighted residuals for spring ¯ows indicate that
the observed ¯ows are larger in magnitude than the
simulated ¯ows. These residuals indicate that the regional 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 nonlinearity
The normality of the weighted residuals and model linearity 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 eectively 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 parameter values are presented.
The normal probability graph of the weighted residuals 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 regression-derived correlations, but some of the curving related to extreme values might be explained. This analysis
indicates that weighted residuals are not normally distributed.
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 eectively 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
Fig. 9. Normal probability plot of weighted residuals.
787
Death valley regional ground-water ¯ow model calibration
Fig. 10. Normal probability plot of correlated normal random
deviates.
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 indicators of simulation uncertainty, even in the presence of
some model nonlinearity. The nonnormal weighted residuals 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 indicators of the uncertainty in estimated parameter values.
5.2 Evaluation of estimated parameter values
The set of parameters estimated by regression in the
DVRFS model includes all of the most important system 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 estimated 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 information about the parameters being estimated, it is reasonable 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 information16.
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 interval on the parameter estimate also excludes reasonable 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 standard 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 characterization 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
Log-transformed
for regression
Estimated value
Coecient of
variation a
95% Linear con®dence
upper/lower limits on
the estimate b
Expected upper/lower
range of reasonable
values
K1 (m/d)
K2 (m/d)
Yes
Yes
0.275
0.443 ´ 10ÿ1
0.149
0.113
100.0; 0.1
0.1; 0.0004
K3 (m/d)
Yes
0.562 ´ 10ÿ2
0.181
K4 (m/d)
Yes
0.856 ´ 10ÿ4
0.263
K5 (m/d)
K9 (m/d)
ANIV3
RCH2 (percent)
RCH3 (percent)
Yes
Yes
Yes
No
No
21.2
0.159
164
3.02
22.7
0.499
0.479
0.518
0.107
0.0518
0.369; 0.205
0.554 ´ 10ÿ1 ;
0.354 ´ 10ÿ1
0.801 ´ 10ÿ2 ;
0.394 ´ 10ÿ2
0.146 ´ 10ÿ3 ;
0.501 ´ 10ÿ4
0.50 ´ 102 ; 0.889 ´ 101
0.367 ´ 100 ; 0.686 ´ 10ÿ1
399; 67.2
3.66; 2.37
25.0; 20.3
Parameter label
(units)
a
0.02; 0.0001
1 ´ 10ÿ4 ; 2 ´ 10ÿ7
100.0; 8.0
1.0; 0.01
1000.0; 1.0
8.0; 1.0
30.0; 15.0
For parameters that were log-transformed for regression, these are calculated as sB /B, where B is the untransformed estimated
value, sB 2 exp(s2ln B + 2(ln B))(exp(s2ln 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.
788
F. A. D'Agnese et al.
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 parameter value is within its expected range, though most
of the hydraulic conductivity estimates tend to be in the
upper end of the reasonable range.
Fig. 11. Estimated hydraulic conductivity parameters, their
95% linear con®dence intervals, and the ranges of reasonable
values.
No prior information was included in the sum-ofsquares 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 information has the advantage of allowing a more direct
test of the model using the observation data (the hydraulic heads and ¯ows). In this approach, the available information on reasonable parameter values is
used to evaluate the estimated parameter values. For
the DVRFS model, this evaluation revealed no indication 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 produced 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 high cj values often
were included in the estimation process, while parameters 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 correlation coecients9,14 with another parameter of higher
sensitivity. In these cases, the parameter of lower sensitivity 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 of cj
change somewhat as the parameter values change. As a
result, the evaluation of cj values was repeated frequently. Composite scaled sensitivity values for estimated 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 important parameters. Exceptions occur for parameters
that were correlated with parameters with larger
Death valley regional ground-water ¯ow model calibration
Fig. 12. Composite scaled sensitivities for estimated parameters in the ®nal calibrated model.
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 estimated parameter values include the aspects of the system 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-estimation techniques utilized in this study materially
assisted in modeling the complex DVRFS. Three-dimensional hydrogeologic framework modeling combined 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 conceptual 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
789
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 constrains the alternative conceptual models; (2) testing
possible conceptual models by determining the parameter 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 techniques and optimal parameter estimation by nonlinear
regression was essential to achieving these objectives.
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Southwestern United States, for isolation of high-level
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Professional Paper 1370-F, 1989.
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