# 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.

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- Martin Hernandez-Marin, Jesus Pacheco-Martinez, Alejandro Ramirez-Cortes, Thomas J. Burbey, Jose A. Ortiz-Lozano, Mario E. Zermeño-de-Leon, Jacobo Guinzberg-Velmont, German Pinto-Aceves[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 - SourceAvailable from: Thomas J. Burbey[Show abstract] [Hide abstract]

**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 - SourceAvailable from: nbmg.unr.edu[Show abstract] [Hide abstract]

**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

Page 1

A New Zonation Algorithm with Parameter

Estimation Using Hydraulic Head and Subsidence

Observations

by Meijing Zhang1, Thomas J. Burbey2, Vitor Dos Santos Nunes3, and Jeff Borggaard4

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.

Introduction

Groundwater

MODFLOW-2000,

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

modeling

MODFLOW-2005

softwaresuch

(Harbaugh

as

1Corresponding author: Department of Geosciences, Virginia

Tech, Blacksburg, VA 24061; +1-540-231-2404; fax: +1-540-

231-3386; zhangmj@vt.edu

2Department of Geosciences, Virginia Tech, Blacksburg, VA

24061; +1-540-231-6696; tjburbey@vt.edu

3Department of Mathematics, Virginia Tech, Blacksburg, VA

24061; +1-540-231-7667; vitor@vt.edu

4Department of Mathematics, Virginia Tech, Blacksburg, VA

24061; +1-540-231-7667; jborggaard@vt.edu

Received January 2013, accepted June 2013.

©2013,NationalGroundWaterAssociation.

doi: 10.1111/gwat.12102

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).

NGWA.orgGroundwater1

Page 2

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:

?N1

i=1

N2

?

J (h,q) = f (h,q) =1

2

?

||h(q) − hobs(i)||2

+

j=1

||Sub(q) − Subobs(j)||2+ α||q − q0||2

⎞

⎠

(1)

where,

q = (Sske,Sskv,T)

(2)

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

Ssb∂ha

∂t

= ∇·?T∇ha?+

Sv∂hi

∂t

?

j

Kvj

dhi

dz

j

− W

(3)

=

∂

∂z

?

Kvj

∂

∂zhi

j

?

(4)

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

Page 3

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

for λ.

∂f

∂h+ λ∂G

∂h= 0(6)

Then J?(q) is computed

J

?(q) =∂f

∂q+ λT∂G

∂q.

(7)

According to Taylor’s expansion,

∇J (q1) = J

?(q0) + J

??(q0)(q1− q0)

(8)

When the cost function (Equation 1) reaches a

minimum,

∇J (q1) = 0 (9)

Then (Equation 9) is substituted into (Equation 8) to

yield:

q1= q0− J

??(q0)−1J

?(q0)

(10)

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).

(11)

Evaluation of the APE Algorithm with

MODFLOW-2005

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)

(Figure 2).

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

Page 4

Figure 2. Areal view of the conceptual model showing the

19×29km model grid (1×1km cells), well locations, and

pumping rate.

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

τ0=

?b0

2

?2Ssk

K?

v

(12)

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

drainage.

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

Page 5

Figure 4. Transmissivity zonations and values for the syn-

thetic model.

2. Develop initial guesses for the transmissivity T(0)using

Equation 13.

T(0)(x) =

Ss∂h

∇·∇h(t,x)

∂t(t,x)

(13)

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

be lower.

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

subsidence pattern.

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

Page 6

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):

M

?

q

?

=

ND

?

i=1

ωi

?

yi− y

?

i

?

q

??2

(14)

where q is a vector containing values of each of the

parameters being estimated, and in this case

q = (Sske,Sskv,T)

(15)

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

Page 7

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.

Table 1

Number of Calculated Zones After Each Iteration

IterationTransmissivity (T) Specific Storage (SS)

1st Iteration

2nd Iteration

3rd Iteration

True zones

69

38

31

27

19

7

73

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 −

⎡

⎢

⎢

⎣

⎢

⎢

n

?

n

?

i=1

?Yobs

?Yobs

i

− Ysim

i

?2

i=1

i

− Ymean?2

⎤

⎥

⎥

⎦

⎥

⎥

(16)

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

Page 8

Figure 9. (A) Calibrated transmissivity errors after each iteration and (B) calibrated specific storage errors after each

iteration.

8M. Zhang et al. GroundwaterNGWA.org

Page 9

Figure 10. Composite scaled sensitivity of transmissivity and

specific storage for the last iteration.

Conclusions

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-

2005.

NGWA.orgM. Zhang et al. Groundwater9

Page 10

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.

The finalsimulated

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

subsidence errors.

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.

hydraulicdrawdown and

Acknowledgments

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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