A New Zonation Algorithm with Parameter Estimation Using Hydraulic Head and Subsidence Observations.
ABSTRACT Parameter estimation codes such as UCODE_2005 are becoming well-known tools in groundwater modeling investigations. These programs estimate important parameter values such as transmissivity (T) and aquifer storage values (Sa ) from known observations of hydraulic head, flow, or other physical quantities. One drawback inherent in these codes is that the parameter zones must be specified by the user. However, such knowledge is often unknown even if a detailed hydrogeological description is available. To overcome this deficiency, we present a discrete adjoint algorithm for identifying suitable zonations from hydraulic head and subsidence measurements, which are highly sensitive to both elastic (Sske ) and inelastic (Sskv ) skeletal specific storage coefficients. With the advent of interferometric synthetic aperture radar (InSAR), distributed spatial and temporal subsidence measurements can be obtained. A synthetic conceptual model containing seven transmissivity zones, one aquifer storage zone and three interbed zones for elastic and inelastic storage coefficients were developed to simulate drawdown and subsidence in an aquifer interbedded with clay that exhibits delayed drainage. Simulated delayed land subsidence and groundwater head data are assumed to be the observed measurements, to which the discrete adjoint algorithm is called to create approximate spatial zonations of T, Sske , and Sskv . UCODE-2005 is then used to obtain the final optimal parameter values. Calibration results indicate that the estimated zonations calculated from the discrete adjoint algorithm closely approximate the true parameter zonations. This automation algorithm reduces the bias established by the initial distribution of zones and provides a robust parameter zonation distribution.
- SourceAvailable from: Martín Hernández Marin[Show abstract] [Hide abstract]
ABSTRACT: Vertical deformation was measured at 14 benchmarks within the urban area of Jocotepec Mexico using first-order leveling methods and then spatially analyzed in relation to land subsidence and soil discontinuity patterns. The study area is located within the western portion of the Chapala basin, middle-west Mexico. Observations of vertical surficial deformation were made at each benchmark (September and November 2012) relative to a fixed station and compared to an initial survey of each benchmark (April 2012). Results indicate that a maximum subsidence of 7.16 cm over the 8-month measurement interval occurs near downtown coincident with the largest levels of drawdown and translates to a maximum subsidence rate of 0.89 cm/month for the sampling period. Two benchmarks located northwest and southeast of the urban area exhibited uplift of 2.8 and 0.76 cm, respectively, suggesting a complex mechanical response between the sedimentary soil units and the factors causing deformation. A potential spatial relationship exists between subsidence patterns and soil discontinuities. Four separate cones of groundwater depression were observed with two being coincident with subsidence bowls downtown and south of the urban area; however, there is no clear relationship between drawdown and subsidence in the remaining areas. Hydrogeologic reconstructions reveal alternating sequences of alluvial aquifers and highly deformable lacustrine aquitards. An analysis of the soil discontinuities reveals that they are directly aligned with the patterns of vertical deformation.Environmental earth sciences 01/2014; 72(5):1491-1501. · 1.45 Impact Factor
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ABSTRACT: The accurate estimation of aquifer parameters such as transmissivity and specific storage is often an important objective during a ground water modeling investigation or aquifer resource evaluation. Parameter estimation is often accomplished with changes in hydraulic head data as the key and most abundant type of observation. The availability and accessibility of global positioning system and interferometric synthetic aperture radar data in heavily pumped alluvial basins can provide important subsidence observations that can greatly aid parameter estimation. The aim of this investigation is to evaluate the value of spatial and temporal subsidence data for automatically estimating parameters with and without observation error using UCODE-2005 and MODFLOW-2000. A synthetic conceptual model (24 separate cases) containing seven transmissivity zones and three zones each for elastic and inelastic skeletal specific storage was used to simulate subsidence and drawdown in an aquifer with variably thick interbeds with delayed drainage. Five pumping wells of variable rates were used to stress the system for up to 15 years. Calibration results indicate that (1) the inverse of the square of the observation values is a reasonable way to weight the observations, (2) spatially abundant subsidence data typically produce superior parameter estimates under constant pumping even with observation error, (3) only a small number of subsidence observations are required to achieve accurate parameter estimates, and (4) for seasonal pumping, accurate parameter estimates for elastic skeletal specific storage values are largely dependent on the quantity of temporal observational data and less on the quantity of available spatial data.Ground Water 01/2008; 46(4):538-50. · 2.13 Impact Factor
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ABSTRACT: Subsidence in Las Vegas Valley has been geodetically monitored since 1935, and several generations of maps have depicted more than 1.5 m of total subsidence. This study presents new geodetic data that reveal insights into the spatial distribution and magnitude of subsidence through the year 2000. In particular, synthetic aperture radar interferometry (InSAR) and global positioning system (GPS) studies demonstrate that subsidence is localized within four bowls, each bounded by Quaternary faults. Conventional level line surveys across the faults further indicate that these spatial patterns have been present since at least 1978, and based on the new geodetic data a revised map showing subsidence between 1963 and 2000 has been developed. A comparison of the location of the subsidence bowls with the distribution of pumping in the valley indicates that subsidence is offset from the principal zones of pumping. Although the reasons for this offset are not well understood, it is likely the result of heavy pumping up-gradient from compressible deposits in the subsidence zones. A compilation of subsidence rates based on conventional, InSAR, and GPS data indicates that rates have significantly declined since 1991 because of an artificial recharge program. The rates in the northwest part of the valley have declined from more than 5–6 cm/year to about 2.5–3 cm/year, a reduction of 50 percent; in the central and southern parts of the valley, rates have declined from about 2.5 cm/year to only a few millimeters per year, a reduction of more than 80 percent.Environmental and Engineering Geoscience 01/2002; 8(3):155-174. · 0.63 Impact Factor
A New Zonation Algorithm with Parameter
Estimation Using Hydraulic Head and Subsidence
by Meijing Zhang1, Thomas J. Burbey2, Vitor Dos Santos Nunes3, and Jeff Borggaard4
Parameter estimation codes such as UCODE_2005 are becoming well-known tools in groundwater modeling investigations.
These programs estimate important parameter values such as transmissivity (T) and aquifer storage values (Sa) from known
observations of hydraulic head, flow, or other physical quantities. One drawback inherent in these codes is that the parameter zones
must be specified by the user. However, such knowledge is often unknown even if a detailed hydrogeological description is available.
To overcome this deficiency, we present a discrete adjoint algorithm for identifying suitable zonations from hydraulic head and
subsidence measurements, which are highly sensitive to both elastic (Sske) and inelastic (Sskv) skeletal specific storage coefficients.
With the advent of interferometric synthetic aperture radar (InSAR), distributed spatial and temporal subsidence measurements
can be obtained. A synthetic conceptual model containing seven transmissivity zones, one aquifer storage zone and three interbed
zones for elastic and inelastic storage coefficients were developed to simulate drawdown and subsidence in an aquifer interbedded
with clay that exhibits delayed drainage. Simulated delayed land subsidence and groundwater head data are assumed to be the
observed measurements, to which the discrete adjoint algorithm is called to create approximate spatial zonations of T, Sske, and Sskv.
UCODE-2005 is then used to obtain the final optimal parameter values. Calibration results indicate that the estimated zonations
calculated from the discrete adjoint algorithm closely approximate the true parameter zonations. This automation algorithm reduces
the bias established by the initial distribution of zones and provides a robust parameter zonation distribution.
et al. 2000; Harbaugh 2005) has become an effective tool
for simulating the long-term response of groundwater
pumping and subsequent land subsidence, and therefore
providing an important management tool for water
purveyors. The application of MODFLOW-2005 with
parameter estimation codes, such as UCODE_2005, is
widely becoming a standard inverse tool in groundwater
model calibration and evaluation to simultaneously
estimate multiple parameter values (Hill 1998; Poeter
et al. 2005). UCODE_2005 compares observations with
1Corresponding author: Department of Geosciences, Virginia
Tech, Blacksburg, VA 24061; +1-540-231-2404; fax: +1-540-
2Department of Geosciences, Virginia Tech, Blacksburg, VA
24061; +1-540-231-6696; email@example.com
3Department of Mathematics, Virginia Tech, Blacksburg, VA
24061; +1-540-231-7667; firstname.lastname@example.org
4Department of Mathematics, Virginia Tech, Blacksburg, VA
24061; +1-540-231-7667; email@example.com
Received January 2013, accepted June 2013.
simulated equivalents to obtain a weighted least squares
objective function; then a nonlinear regression algorithm
is used to minimize the objective function with respect
to the parameter values.
Although water levels are the most popular data
type to calibrate a groundwater model, they alone are
usually insufficient to obtain an adequate result (Hill
1998; Hill and Tiedeman 2007; Yan and Burbey 2008).
Land subsidence caused by the compaction of sediments
is a global scale problem (Johnson 1991; Barends et al.
1995; Bell et al. 2002; Galloway et al. 1999). Due to
declining water levels, decreasing pore water pressures
within the aquifer system have led to significant increases
in effective stress, which accounted for large-scale com-
paction of mostly fine-grained sediments (Terzaghi 1925;
Poland and Davis 1969; Poland et al. 1972; Poland 1984;
Helm 1975). Subsidence data, when used in conjunction
with sparse and irregularly distributed drawdown data,
can be used to improve groundwater model calibration
of the hydrologic parameters such as elastic and inelastic
skeletal specific storage, the compaction time constant,
hydraulic diffusivity, and the thickness of the compacting
units (Heywood 1997; Burbey 2001; Hoffmann et al.
2001, 2003a; Pavelko 2004; Pope and Burbey 2004).
Yan and Burbey (2008) found that high spatial and
temporal resolution subsidence observations from InSAR
are extremely beneficial for accurately quantifying both
elastic and inelastic skeletal specific-storage values as
well as hydraulic conductivity values, and the resulting
model calibration results are far more accurate than
using only water-levels as observations, or using just a
few random subsidence observations (such as from GPS
benchmarks). Also, they found that storage estimates
are far more sensitive to the deformation of the aquifer
system than to changes in hydraulic head measurements.
Even with current advances in monitoring technol-
ogy and software for simulating subsidence, drawbacks
exist in using inverse parameter estimation techniques
for model calibration. Stochastic algorithms such as
Simulated Annealing and Markov Chain Monte Carlo
methods are computationally time consuming. The pilot
points method is often used in groundwater-model calibra-
tion (de Marsily et al. 1984; Certes and de Marsily 1991;
LaVenue, M., and de Marsily, G., 2001; Doherty 2003,
2010; Alcolea et al. 2006) and involves the perturbation
of hydraulic properties at a small number of selected “pilot
point” locations in an effort to match observational data.
There are guidelines to estimate the number of points to
use as pilot points based on the spacing between them
and the model grid size. Variograms are used to gener-
ate the spatial distribution of hydraulic properties from
the values at the pilot points. The hydraulic properties are
calculated with Kriging interpolation. One inherent weak-
ness in this method is that the number and location of
pilot points is somewhat subjective. Another commonly
used method is zonation, which is used in inverse mod-
els such as UCODE_2005 (Hill 1998; Poeter et al. 2005).
This method involves dividing the entire study area into a
number of zones and unknown parameters are treated as
uniform. However, the parameter zones must be specified
by the user beforehand. It may lead to a good calibra-
tion of parameters with the user-defined parameter zones.
However, such knowledge about optimal zonal distribu-
tion is often unknown even if a detailed hydrogeological
description of the study area is available. To overcome the
deficiency in parameter zonation definitions, we present
a discrete adjoint parameter estimation (APE) algorithm
for automatically identifying suitable parameter zonations
from hydraulic head and subsidence measurements, which
are highly sensitive to both elastic (Sske) and inelas-
tic (Sskv) skeletal specific storage coefficients. Here we
develop a hypothetical model using MODFLOW-2005 in
which observed measurements of land subsidence (includ-
ing hydrodynamic lag) and hydraulic head data are made
at selected locations and times. Using only these observa-
tions, the distributed parameter identification algorithm is
called to create approximate spatial zonations of T, Sske,
and Sskv. Then the approximation of parameter zonations
is compared with the original (true) zonations assigned
in MODFLOW-2005. Finally, UCODE_2005 is used to
obtain the final optimal parameter values. The new APE
algorithm, when combined with UCODE_2005, provides
a new powerful tool for obtaining optimal zonations.
Formulation of the APE Algorithm
The spatial variability of the storage and conductiv-
ity properties for aquifer systems are generally so complex
that the investigator could not possibly identify the param-
eter zones in an adequate or realistic way to describe
the optimal distribution of hydraulic parameters. There-
fore, an algorithm which can automatically determine the
parameter zonations is necessary to produce an accurate
and optimal model conceptualization. Here, we present
a discrete adjoint algorithm for identifying a suitable
zonation scheme from hydraulic head and subsidence
measurements, which are highly sensitive to both elas-
tic and inelastic skeletal specific storage coefficients as
well as transmissivity. The automatically identified param-
eter zonations will then be implemented into the synthetic
model using MODFLOW-2005 and UCODE_2005.
Adjoint methods are widely used in areas such
as optimal control theory, design optimization, and
sensitivity analysis (Duffy 2009). In our study, we
minimize the difference between observed and simulated
groundwater levels and land subsidence. The objective
function can be written as:
J (h,q) = f (h,q) =1
||h(q) − hobs(i)||2
||Sub(q) − Subobs(j)||2+ α||q − q0||2
q = (Sske,Sskv,T)
and represents the parameter vector to be optimized, Sske
is the elastic skeletal specific storage of the interbed,
Sskv is the inelastic skeletal specific storage of the
interbed, T is the transmissivity of the aquifer, h(q)
represents the calculated water level, which is a function
of q, hobs represents the observed water level, Sub
represents the calculated subsidence, which is a function
of q, and Subobsrepresents the observed subsidence, N1is
the number of observed water level, and N1is the number
of observed subsidence. The last term of Equation 1
represents a penalty term, where α is penalty parameter;
q0is the initial guess or mean value of q. Equation 1 is
subject to the groundwater flow equations
where hais simulated hydraulic head of the aquifer, hi
is hydraulic head in the interbed, b is thickness of the
aquifer system, T is transmissivity, Ssis specific storage
2M. Zhang et al. GroundwaterNGWA.org
of the aquifer, Kvis vertical conductivity, Svis storage
coefficient of the interbed, and W is a source term.
After discretization of Equations 3 and 4 we obtain:
G(q) = A(q)hm(q) − hm−1(q) − W = 0 (5)
where G(q) is the governing equation, A(q) is the matrix
of parameter values, time step length, and grid cell length,
m represents the time step.
The adjoint method solves the following equation
Then J?(q) is computed
According to Taylor’s expansion,
∇J (q1) = J
?(q0) + J
When the cost function (Equation 1) reaches a
∇J (q1) = 0 (9)
Then (Equation 9) is substituted into (Equation 8) to
q1= q0− J
where J??can be estimated with the BFGS method (named
after Broyden, Fletcher, Goldfarb, and Shanno). The
BFGS method is a well-known Quasi-Newton algorithm
which is used for solving unconstrained nonlinear opti-
mization problems. From Equation 10, we can calculate
the parameter vector q1from the initial parameter guess
q0. Then, parameter vector q can be updated with the
Newton method described by Equation 11 until the maxi-
mum fractional change of q evaluated after three iterations
is less than 0.01(step 4 of Figure 1).
qk=qk −1−J??(qk −1)−1J?(qk −1).
A procedural outline of the APE algorithm for
calculating the parameter zonations is shown in Figure 1.
In step 6, a “sufficient result” means that the difference
between the simulated and observed water levels and
subsidence is small (a value set by the user).
Evaluation of the APE Algorithm with
The APE algorithm is designed to automatically
identify suitable parameter zonations from hydraulic head
Figure 1. APE algorithm to calculate parameter zonations.
and subsidence measurements. To evaluate the effective-
ness of automated parameter zonation using the APE
algorithm an areal two-dimensional hypothetical model
modified from Yan and Burbey (2008) is developed
using MODFLOW-2005.The model is represented as a
19×29km one-layer confined aquifer, with each cell size
of 1×1km (Figure 2). This is a transient state model
which simulates groundwater flow and land subsidence
for 15years. Each year is divided into two 6-month stress
periods. The simulated aquifer thickness is 200m. A
poorly permeable but highly compressible clay interbed
of variable thickness (from 9 to 130m) is distributed
within the permeable aquifer (Figure 3). The periph-
eral boundaries are set as no-flow conditions. For the
entire region the initial hydraulic head is 800m and
the preconsolidation (previous minimum head value
in the aquifer) head is 795m. Five wells are pumped
at a constant rate in 6-month intervals (6months on
during the summer and 6months off during the winter)
The SUB package (Hoffmann et al. 2003b) is used to
calculate subsidence at each model cell. For the aquifer
that is composed of relatively coarse-grained sand, land
subsidence is simulated to occur instantaneously when
groundwater levels decline. The interbed is assumed to
be areally far more extensive than its thickness, and the
hydraulic conductivity of the interbed is considerably
lower than the aquifer, so the direction of groundwater
flow within the interbed can be treated as vertical.
Groundwater flow from the interbed to the aquifer occurs
when the head in the aquifer declines, with the head
change in the lens lagging that of the aquifer.
NGWA.org M. Zhang et al. Groundwater3
Figure 2. Areal view of the conceptual model showing the
19×29km model grid (1×1km cells), well locations, and
In the context of interbed compaction and land
subsidence, the time delay caused by slow dissipation of
transient overpressures is often given in terms of the time
constant, which is the time during which about 93% of the
ultimate compaction for a given decrease in head occurs
(Riley 1969). The time constant can be expressed as
where b0/2 is one-half the thickness of the interbed, Ssk
is the skeletal specific storage of the interbed, K?vis the
vertical hydraulic conductivity of the interbed. Laboratory
consolidation tests indicate that the compressibility, and
thus the skeletal specific storage, can vary greatly
depending on whether the effective stress exceeds the
previous maximum effective stress, which is termed as
the preconsolidation stress (Johnson et al. 1968; Jorgensen
1980). Inelastic skeletal specific storage Sskvis used when
the water level in the interbed is less than its previous
minimum value, whereas elastic skeletal specific storage
Sske is invoked when the drawdown in the interbed is
higher than the previous minimum values.
The study area is divided into seven transmissivity
zones T1 to T7 (Figure 4). Both the aquifer and interbed
Figure 3. Conceptual model of the aquifer system containing
a variably thick contiguous clay interbed with delayed
are treated as compressible. The storage coefficient of
the aquifer is assumed to be 0.002 for the entire model
region. Three separate zones are used to express the elastic
(Sske1, Sske2, and Sske3) and inelastic (Sskv1, Sskv2, and
Sskv3) skeletal specific storage of the interbed. These
values and the zonation distribution are shown in Figure 5.
The vertical hydraulic conductivity for the interbed was
assumed to be 0.00006m/d.
An initial forward simulation using MODFLOW-
2005 was conducted with known pumping rates and all
true hydraulic property values described above. Hydraulic
heads and subsidence values obtained from this simulation
are treated as the true observation values. We assume
that high temporal and spatial resolution land subsidence
data are available at each grid cell and treat them
as the fictitious subsidence rates available from InSAR
interferograms (Figure 6).
The APE algorithm can now be applied to calculate
the zonations for transmissivity and elastic and inelastic
skeletal specific storage of the interbed. For the very
first call of the APE algorithm, the initial guess for the
parameter P(0)(step 1 of Figure 1) is estimated as follows:
1. Incorporate the observed subsidence data to estimate
an initial elastic and inelastic skeletal specific storage
and set them as Sske(0)and Sskv(0);
4M. Zhang et al. Groundwater NGWA.org
Figure 4. Transmissivity zonations and values for the syn-
2. Develop initial guesses for the transmissivity T(0)using
In Equation 13, we use the observed hydraulic head,
h, only when the pumping rate is zero. The Newton
method converges locally and inheritably so does the
Quasi-Newton method, which is discussed thoroughly
by Ito and Kunisch (2008). Thus, an initial parameter
guess that is close to the local minimum will almost
always guarantee convergence. In this perspective the
choice of the initial guess from Equation 13 is made,
since it takes advantage of the information given by
Equation 3 when there is no pumping. Although this
generally provides a good estimate, it is sensitive to the
complexity of the problem at hand so if the zonation
is highly complex, the likelihood of convergence might
Implementing UCODE_2005 with the APE Algorithm
Once the automatically identified parameter zona-
tions have been estimated using the APE algorithm, we
implement these zones with the initial parameter values
into the synthetic model using MODFLOW-2005 and
UCODE_2005. UCODE_2005 is a nonlinear parameter
estimation program, whichcomparesobservations
Figure 5. Zonations and values for the elastic and inelastic
specific storage parameters.
Figure 6. Cyclical pumping and resulting simulated land
with simulated equivalents to obtain a weighted least
squares objective function. Then it employs a modified
Gauss-Newton method to iteratively solve a general
nonlinear regression problem (Hill 1998; Poeter et al.
2005). UCODE_2005 can be used to analyze sensitivity,
and calculate confidence and prediction intervals. The
weighted least-squares objective function M(q) is defined
NGWA.orgM. Zhang et al. Groundwater5
Figure 7. Estimated specific storage zonations using the APE algorithm (A) after 1st iteration, (B) after 2nd iteration, (C)
after 3rd iteration compared with the (D) true specific storage zonations calculated by MODFLOW-2005.
as follows (from Hill 1998):
where q is a vector containing values of each of the
parameters being estimated, and in this case
q = (Sske,Sskv,T)
where ND is the number of observations, ωiis the weight
for the ith observation, yi is the ith observation being
matched by the regression, and yi
value which corresponds to the ith observation.
UCODE_2005 yields a new set of estimated param-
eter values based on the zonations from the APE
algorithm. The estimated parameter values calculated
from UCODE_2005 are then returned to the APE algo-
rithm as the initial guess for the parameter P(0)(step
1 of Figure 1). This iterative procedure between the
APE and UCODE_2005 is continued until the simu-
lated heads and subsidence values accurately approach the
true heads and subsidence values. Generally, less than 10
iterations are required to achieve convergence.
?(q) is the simulated
Results and Discussion
The APE algorithm (Figure 1) was applied using a
portion of the water-level observations and subsidence
data produced by the synthetic model. No known
information about the distribution of known parameter
values (divided into specific zones where each zone
represents a constant parameter value) was provided to
the APE algorithm.
Initial estimates of interbed elastic and inelastic skele-
tal specific storage zones obtained from the APE algorithm
are provided as a starting point in the iteration sequence.
After estimating the distribution of Sskeand Sskvaquifer
hydraulic transmissivity (T) zones are estimated. Then
UCODE_2005 is used to obtain a new set of estimated
parameter values based on the zonations and initial values
from the APE algorithm. Then the new set of estimated
parameter values calculated from UCODE_2005 are
returned to the APE algorithm to recalculate Sskeand Sskv
and then to obtain new estimates for T. The parameter-
estimation iterations stop if the maximum fractional
change in the sum-of-squared weighted residuals over
three parameter-estimation iterations is less than 0.01.
Generally, conversion will occur after about six
iterations between the APE algorithm and UCODE_2005.
The estimated Sske and Sskv zonations using the APE
algorithm along with the true zonations that were
developed from the synthetic model using MODFLOW-
2005 are shown in Figure 7. Similarly, the estimated
aquifer transmissivity zonations using the APE algorithm,
along with the true zonations that were developed from the
synthetic model using MODFLOW-2005, are shown in
Figure 8. The number of zones after each iteration is listed
in Table 1. These results show that after several iterations
between the APE algorithm and UCODE_2005 the
distributed parameter identification algorithm appears to
accurately match the true spatial distributions of the zones.
Figure 9 shows the estimated parameter errors after
each iteration where100% means the simulated value
differs from the true value by a factor of 2 and a value of
one means the result comes from the 1st iteration. It can
be seen that calibrated transmissivity and inelastic skeletal
specific storage have lower errors than the calibrated elas-
tic skeletal specific storage. Composite scaled sensitivities
(CSS) are used to measure the overall sensitivity of the
observations to the parameters. It can also been seen from
Figure 10 that transmissivity and inelastic skeletal specific
storage have higher CSS than the elastic skeletal spe-
cific storage. This means that it is easier to attain accurate
transmissivity and inelastic skeletal specific storage values
and it is more elusive to obtain elastic skeletal specific val-
ues. This is because the relatively small amount of elastic
subsidence is masked by the delayed drainage of the
interbed and by the relatively large inelastic subsidence.
Actually, the elastic skeletal specific storage has a high
dimesionless scaled sensitivity (DSS) to land subsidence,
but has low DSS to drawdown. Both inelastic skeletal
6 M. Zhang et al. GroundwaterNGWA.org
Figure 8. Estimated transmissivity zonations using the APE algorithm (A) after 1st iteration, (B) after 2nd iteration, (C) after
3rd iteration compared to the (D) true transmissivity zonations calculated by MODFLOW-2005.
Number of Calculated Zones After Each Iteration
IterationTransmissivity (T) Specific Storage (SS)
specific storage and transmissivity have high DSS to both
land subsidence and drawdown. Thus subsidence data are
highly sensitive to elastic (Sske), inelastic (Sskv) skeletal
specific storage coefficients and transimissivity (T), which
indicates that high spatial and temporal resolution InSAR
data are required to accurately calibrate parameter values.
The size of the parameter zones also influences the
calibrated result. For example, the 1st iteration zone T55
covers only one grid cell and it leads to the largest
calibrated parameter error (88%) among all the trans-
missivity zones (Figure 9, a1). Also for the 1st iter-
ation zone Ssk12 covers only five grid cells leading
to the largest calibrated parameter error (106%) among
all the specific storage zones (Figure 9, b1). The require-
ment of further delineating small zones could easily be
the result of high spatial variability of the parameters
that cannot be simulated with a single value, but should
be simulated with a finer representation of the spatial
variability of hydraulic properties. Hence, it is important
for the APE algorithm to divide the zonation boundaries
into new zones after each iteration (step 3 of Figure 1).
On the other hand the requirement of further delineat-
ing smaller zones may also be an indication of combing
zones with similar magnitude after each iteration (step 5 of
Figure 1). This modification to the APE algorithm is still
under investigation. Nonetheless, the APE algorithm com-
bined with UCODE_2005 is able to provide reasonable
and stable results. If some parameters have CSS that are
less than about 0.01 times the largest CSS, it is likely
that the regression will not converge (Anderman et al.
1996; Hill 1998). In this model all the optimal parame-
ters have CSS that are larger than 0.022 times the largest
CSS (Figure 10), indicating that the parameters will likely
be accurately estimated. Parameter correlation coefficients
can be used to indicate whether the estimated parameter
values are likely to be unique. Absolute values of parame-
ter correlation coefficients close to 1 indicate a high degree
of correlation. Thus, changing the parameter values in a
linearly coordinated manner will result in the same value
of the objective function. In this model, most of the param-
eter correlation coefficients are on the order of 10−2-10−4
and the largest value is 0.58, suggesting that uniqueness
was not a problem.
The final simulated drawdown and subsidence dis-
tributions and the true hydraulic heads and subsidence
distributions are shown in Figures 11 and 12. Nash-
Sutcliffe efficiency (NSE) is chosen here to measure the
overall fit of the hydrographs (Nash and Sutcliffe 1970).
NSE is computed as
NSE = 1 −
where Yiobsis the ith observation value, Yiobsis the ith
simulated value, Ymeanis the mean of observed data, and n
is the total number of observations. NSE ranges between
−∞ and 1. Generally values between 0.0 and 1.0 are
acceptable, with NSE=1.0 being the optimal value. In
our case NSE is 0.9997 for drawdown and 0.9813 for
subsidence, which indicates that the simulated drawdown
values more closely reflect the observed values than does
the simulated subsidence distribution. One reason for this
is that the elastic skeletal specific storage values, which
control land subsidence, are less accurately estimated than
other parameters. Another reason is the delayed land
subsidence mechanism makes computation quite complex,
so that a small error in the estimation of the parameters
will lead to large differences in calculated land subsidence.
NGWA.org M. Zhang et al. Groundwater7
Figure 9. (A) Calibrated transmissivity errors after each iteration and (B) calibrated specific storage errors after each
8M. Zhang et al. GroundwaterNGWA.org
Figure 10. Composite scaled sensitivity of transmissivity and
specific storage for the last iteration.
Our goal in this investigation involves applying
a fully distributed parameter identification algorithm
to a hypothetical model to produce results that show
that this automation process can remove user bias and
provide a more accurate and robust parameter zonation
distribution. We have outlined an automated parameter
estimation process that can greatly aid the calibration of
groundwater flow models. After analyzing and comparing
the results of the newly developed APE model, we make
the following important conclusions.
With the advent of InSAR, basin-wide coverage
of spatial and temporal subsidence and rebound mea-
surements, which occur in response to cyclical aquifer
pumping, can be obtained where surface deformations can
be expected to occur. Subsidence data are highly sensitive
to both elastic (Sske) and inelastic (Sskv) skeletal specific
storage. High spatial and temporal resolution InSAR data
can help reveal the heterogeneity properties of the aquifer
system in ways that hydraulic head data alone cannot.
Figure 12. Observed vs. simulated (A) final drawdown, and
(B) final subsidence.
The distributed parameter identification algorithm
we applied is verified to be effective. It can be seen
that the estimated zones approach the spatial distribution
of the true parameter zones that are developed from
MODFLOW-2005. This automation process removes
user bias and provides an accurate robust parameter
zonation distribution. The effectiveness of the final
zonation is influenced by the initial calculated zonation
(step 1 of Figure 1). Once an initial estimation of the
parameters is made using UCODE_2005, the specific
storage and transmissivity zonations become simplier to
solve for with the APE algorithm. Thus, the algorithm
presented here for the identification of appropriate zones
Figure 11. (A) Estimated drawdown using the estimated parameter values, (B) the true drawdown developed by MODFLOW-
2005, (C) estimated subsidence using the estimated parameter values, and (D) the true subsidence developed by MODFLOW-
NGWA.orgM. Zhang et al. Groundwater9
establishes the link between improvements on zonation
distribution and the limit where every point in the grid
is a zone. Equation 13 represents the link between these
two. The choice of the initial guess from Equation 13
takes advantage of the information given by Equation 3
when there is no pumping. Although this generally is a
good estimate and typically guarantees convergence, it
is sensitive to the complexity of the problem at hand,
so if the zonation is highly complex the likelihood of
convergence might be lower.
The size of the zone also influences the calibrated
result. Small zones are likely to lead to large calibrated
parameter errors. It is therefore important for the APE
algorithm to divide these small zones into smaller zones
or combine these small zones with larger similar zones
after each iteration. The requirement of further delineating
small zones could easily be the result of high spatial
variability of the parameters that cannot be simulated with
a single value but rather there exists a continuum of values
(as represented by a particular variogram), which points to
one of the weaknesses of zonation method presented here.
Analysis of CSS and parameter correlation coef-
ficients shows that the APE algorithm combined with
UCODE_2005 is able to provide reasonable, unique and
stable results for the model used in this study.
subsidence distribution matches the true observation
distributions quite well. The simulated drawdown val-
ues more closely reflect observed values than do the
simulated subsidence values. The more poorly estimated
elastic skeletal specific storage values coupled with the
mechanisms responsible for complex delayed drainage
are the two main factors leading to larger calculated land
The distributed parameter identification algorithm
developed herein should be useful for the calibration of
all groundwater models using multiple types of obser-
vations. However, there are some limitations that were
identified from this study. One limitation is that this one-
layer model oversimplifies the real-world system. More
challenges will be encountered with complex, multilay-
ered systems. Another limitation is that we use the true
land subsidence and hydraulic drawdown data developed
from MODFLOW-2005 as the observations with no errors
in the observed data; however, errors are impossible to
avoid in real field data collection and processing (partic-
ularly with InSAR), which makes parameter estimation
more difficult. Despite these limitations, this study shows
that the algorithm and iterative process developed in this
study can be an effective method for model calibration.
The authors would like to thank Frank W. Schwartz
and three anonymous reviewers for their insightful
comments, which greatly contributed to the improvement
of this manuscript.
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