U.S. Department of the Interior

U.S. Geological Survey

Fact Sheet 2011–3014

May 2011

National Water Availability and Use Pilot Program

Using Models for the Optimization of Hydrologic Monitoring

Introduction

Hydrologists are often asked what kind of monitoring 

network can most effectively support science-based waterresources management decisions. Currently (2011), hydrologic 

monitoring locations often are selected by addressing observation gaps in the existing network or non-science issues such 

as site access. A model might then be calibrated to available 

data and applied to a prediction of interest (regardless of how 

well-suited that model is for the prediction). However, modeling 

tools are available that can inform which locations and types of 

data provide the most “bang for the buck” for a specified prediction. Put another way, the hydrologist can determine which 

observation data most reduce the model uncertainty around a 

specified prediction. 

An advantage of such an approach is the maximization of 

limited monitoring resources because it focuses on the difference in prediction uncertainty with or without additional collection of field data. Data worth can be calculated either through 

the addition of new data or subtraction of existing information 

by reducing monitoring efforts (Beven, 1993). The latter generally is not widely requested as there is explicit recognition that 

the worth calculated is fundamentally dependent on the prediction specified. If a water manager needs a new prediction, the 

benefits of reducing the scope of a monitoring effort, based 

on an old prediction, may be erased by the loss of information 

important for the new prediction. 

This fact sheet focuses on the worth or value of new data 

collection by quantifying the reduction in prediction uncertainty 

achieved be adding a monitoring observation. This calculation of worth can be performed for multiple potential locations 

(and types) of observations, which then can be ranked for their 

effectiveness for reducing uncertainty around the specified 

prediction. This is implemented using a Bayesian approach with 

the PREDUNC utility in the parameter estimation software suite 

PEST (Doherty, 2010). 

The techniques briefly described earlier are described in 

detail in a U.S. Geological Survey Scientific Investigations 

Report available on the Internet (Fienen and others, 2010; 

http://pubs.usgs.gov/sir/2010/5159/). This fact sheet presents a 

synopsis of the techniques as applied to a synthetic model based 

on a model constructed using properties from the Lake Michigan Basin (Hoard, 2010).

A Highly Parameterized Approach to Achieving 

Model Parsimony 

An important aspect of designing a model is deciding 

the structure and number of parameters used to represent a 

natural world of unknowable complexity (Hunt and others, 

2007; Hunt and Welter, 2010). The level of parameterization 

can have important ramifications for the ability of the model 

to take full advantage of the information encapsulated in the 

field data. Perhaps less intuitively, parameterization also can 

affect the ability of the model to assess effects on predictions 

which, in turn, affects the ability of the model to determine 

the worth of future data collection (Fienen and others, 2010). 

A highly parameterized model does not necessarily mean that 

the model will be highly heterogeneous (or have unrealistic 

distributions of properties or suffer from instability and nonuniqueness; e.g., Fienen and others, 2009). This is because 

mathematical methods (regularized inversion) are available to 

constrain insensitive and correlated parameters with subjective information—a modeler’s soft-knowledge—and provide 

the ability to solve the problem with a reduced set of linear 

combinations of model inputs unconditionally informed by 

the observations (Hunt and others, 2007; Doherty and Hunt, 

2010a). The mathematical tools collapse the highly parameterized problem to something “as simple as possible,” and 

high numbers of parameters help ensure that the twin requirement of parsimony—“but not simpler”—also is met (Doherty 

and Hunt, 2010b). 

Using a Model to Optimize Hydrologic Monitoring

To demonstrate the utility of the methods described herein, 

a synthetic model, based on a local inset model constructed 

using properties from the Lake Michigan Basin (Hoard, 2010), 

is used to determine the optimal future data-collection locations 

needed to optimize network design. The specified prediction of 

interest is the amount of groundwater-level change in an area of 

interest; for example, an area with an endangered plant species. The ultimate goal of the monitoring is to assess the effect 

of a new high-capacity pumping well (500 gallons per minute 

(gal/min)) in this area, which also is near a headwater stream.

The Role of “Potential Observations”

Evaluation of the worth of additional data does not 

require that we actually know the observed values at the proposed monitoring locations. Rather, it requires only that we 

know the proposed observations’ sensitivity to parameter perturbation in the model. Such information is easily calculated 

using a parameter-estimation approach and is encapsulated 

in the “Jacobian matrix” (a matrix that reports the response 

of each observation to a perturbation of each parameter). 

Moreover, the Jacobian matrix can be calculated at any stage 

of the calibration process; thus, this approach is applicable to 

precalibration or other early stages of an investigation. 
The original local model inset was constructed to evaluate 

the effects of groundwater pumping on groundwater /surfacewater interaction (Hoard, 2010) and is described briefly herein. 

The model uses a telescopic mesh-refinement approach where a 

local model was constructed (cell size = 21.8 meters (m)) within 

an intermediate model (cell size = 152.4 m), which was, in turn, 

inset within a regional model (cell size ranging from 1,524 m 

to more than 21,000 m) of the Lake Michigan Basin (Feinstein 

and others, 2010). The objective of the local-scale modeling 

involved the water-balance question of effects of pumping on 

the flow to a headwater stream. Figure 1 shows the features 

and locations of the regional, intermediate, and local models. 

The model contains six layers; the two shallowest layers are of 

principal interest in this investigation. Recharge and fixed-head 

lateral boundaries, simulated with the RCH and BAS packages within MODFLOW-2005, Version 1.6 (Harbaugh, 2005), 

combine with surface-water features modeled through the 

streamflow routing (SFR) package to represent water inflows 

and outflows. Further details about the model features and 

implementation are discussed in Hoard (2010).

To demonstrate the effect of parameterization on dataworth analyses, three parameterization resolutions are considered (fig. 2)1

: (1) a hydraulic conductivity (K) layer-multiplier 

(“1 Zone”) approach in which a single multiplier is applied 

to all horizontal (Kh) and vertical (Kv) hydraulic-conductivity 

values in each of the 6 layers resulting in a 12-parameter model; 

(2) a 25-zone version of the model (“25 Zones”) in which 25 Kh

and 25 Kv zones are specified in each of the 6 layers resulting 

in a 300-parameter model; and (3) a pilot-point (“400 Pilot 

Points”) approach in which a 20 by 20 grid of pilot points 

(Doherty, 2003) represents both Kh and Kv in each of the 6 

layers resulting in a 4,800-parameter model. In the “400 Pilot 

Points” approach, the estimated parameter values are interpolated to the model grid in areas between the pilot points using 

Kriging. 

It is important to note that only the parameter flexibility 

(specified for the data-worth analysis) was being varied in the 

three cases described above; the actual hydraulic conductivity 

values input into the model were exactly the same in all three 

cases and were equal to those inherited from the calibrated 

regional model. 

1

 The terminology used here to describe the three parameterizations is modified for clarity from the more extensive analysis of Fienen and others (2010).

Wisconsin

Michigan

Illinois

Michigan

Indiana Ohio

Lake Michigan

Lake Superior

Lake 

Erie

Lake

 St. Clair

90°0' 87°0' 84°0'

48°0'

45°0'

42°0'

EXPLANATION

Stream network

0 5 10 2.5 Kilometers

0 2.5 5 10 Miles

0 100 200 50 Miles

0 100 20050 Kilometers

Lake Michigan Basin

Lake Huron

Local model extent

Intermediate model extent

Regional model extent

Figure 1.  The location and features of the regional, intermediate, and local models (modified from Hoard, 2010).
25 Zones 400 Pilot Points

PARAMETERIZATION

1 Zone

LAYER 1 

HYDRAULIC CONDUCTIVITY

LAYER 2 

HYDRAULIC CONDUCTIVITY

EXPLANATION

Hydraulic conductivity,

in feet per day

1.1

1.2 - 1.7

1.8 - 2.0

2.1 - 2.3

2.4 - 2.6

2.7 - 3.5

3.6 - 4.3

4.4 - 7.3

7.4 - 11.2

11.3 - 25.9

26.0 - 50.0

50.1 - 70.0

Stream

Pilot-point 

location

Zone

boundary 

N

0 5 10 Kilometers

0 5 10 Miles 2.5

2.5

Figure 2.  Local model domain showing the parameterization and observation network. The left panel is 1 Zone, the middle panel is

25 Zones, and the right panel is the 400 Pilot Points parameterization. The grid in the 25 Zones panel outlines hydraulic conductivity zone

boundaries; the ‘x’ marks on the 400 Pilot Points panel show the pilot-point locations.

model of Hoard (2010) allowed the surface-water features to be 

more refined, but the discretization of aquifer properties is not. 

The 25 Zone case can be considered a more typical case, where 

the parameter flexibility appropriate for the regional model is 

assumed to be appropriate for the data-worth calculation, when 

used in conjunction with the additional surface-water feature 

refinement of the higher grid resolution of the inset model. 

This also can be thought of as the number of zones that might 

be used in a traditional calibration approach. The 400 Pilot 

Point case represents a highly parameterized approach typical 

of regularized inversion methods that aim to interject sufficient 

parameter flexibility such that confounding artifacts associated 

with model simplification of a complex world are minimized. In 

this way, the three parameterization approaches can be considered a continuum of degrees of model-parameter flexibility.

The degree of parameterization is of interest because it 

enters the problem in two ways. First, the more zones used to 

represent the model domain, the smaller the size of each individual zone. Therefore, when each zone is varied independently 

in the sensitivity analysis, the size of the spatial area of the 

model domain perturbed is different, which results in different 

sensitivity results reported in the Jacobian matrix. A second, less 

obvious, effect is that the resolution of the modeler’s expected 

parameter uncertainty will differ with different spatial parameterization. That is, if a piecewise-constant zone represents the 

average value of all properties in that zone, a larger zone may 

have a larger standard deviation around that mean than a smaller 

zone. Therefore, the 1 Zone case represents the end member 

of oversimplification as might happen when the model used 

for assessing data worth is only minimally changed from the 

regional model conceptualization. That is, in this case, the inset 
The prediction includes a new stress added to the refined 

inset model—the addition of one new pumping well, extracting at 500 gal/min from layer 2 at the location indicated in 

figure 3. The prediction, related to the new stress, is the head 

in a location between the pumping well and stream (fig. 3, 

cell H115_259 in layer 1) that might represent an area containing groundwater-dependent endangered species. Using PEST, 

potential head observations were added throughout the model to 

determine the importance of each possible location for reducing 

uncertainty associated with a head prediction at cell H115_259, 

owing to the new stress. 

The results of data-worth calculations are contoured and 

displayed on a map in figure 4. Increasing data worth is indicated by warmer colors, where the values indicate the reduction 

in prediction uncertainty if the proposed additional data were 

used in recalibrating the model. These dimensionless values are 

interpreted relative to one another rather than on an independent 

scale. The extent of the map is the same as the model domain, 

and the panels in figure 4 depict results for the first two layers 

for all three parameterization strategies.

In figure 4, the differences in the displayed values from 

the left panel to the right reflect the progressively more flexible 

parameterization of hydraulic conductivity, from a single-layer 

multiplier on the left (the simplest, 1 Zone scenario), followed 

by a 5 by 5 grid of homogeneous zones (25 Zones), to a 20 

by 20 grid of pilot points (400 Pilot Points) on the right. Two 

features are evident when comparing the parameterization 

scenarios: first, counter-intuitive artifacts are encountered in the 

low (1 Zone) and intermediate (20 Zones) cases of parameter 

flexibility. The areas reported as most important for reducing 

the prediction uncertainty of groundwater levels between the 

well and the headwater stream are counter-intuitive because 

they are distant from both the stress and the related prediction. 

Inspection of the locations of the most important data worth 

indicates that these areas are associated with zone boundaries 

and intersections—a factor introduced solely by the modeler 

when the parameter flexibility was specified. When the same 

data-worth analysis is performed using the highly parameterized 

(400 Pilot Points) case, locations of higher values of data worth 

are located where intuition indicates—the area near both the 

stress and the prediction. 

Figure 3.  Local model

domain and the locations

of the pumping well and

the head prediction cell

(H115_259).

0 5 10 Kilometers N

0 5 10 Miles 2.5

2.5

Pumping well

H115_259

EXPLANATION

Head-prediction location (H115_259)

Pumping well

MODFLOW stream cells

Figure 4.  Observation data worth

for head-prediction scenario

evaluated for layer 1 (upper 3

panels) and layer 2 (lower 3 panels).

In each row of panels, three

parameterizations are shown: 1

Zone (left panel), 25 Zones (middle

panel), and 400 Pilot Points (right

panel). Note that the scales vary

in each panel to show relative

patterns. Details of the units and

calculations are in Fienen and

others (2010).

0.00

0.20

0.40

0.60

0.80

0.00

0.005

0.009

0.014

0.019

0.00

0.05

0.10

0.15

0.20

0.000

0.004

0.008

0.012

0.016

0.00

0.20

0.40

0.60

0.80

0.00

0.02

0.04

0.06

0.08

1 ZONE 25 ZONES 400 PILOT POINTS

LAYER 1 LAYER 2
Assessing the Importance of Model Objective for 

Quantifying Prediction Uncertainty 

Some objectives of a modeling study do not require a 

great aquifer or system to provide a representative prediction; 

therefore, this complexity can be excluded from the model. 

For example, if the objective of a study is one of water balance, such as defining a well’s contributing recharge area, 

this integrates relatively large areas of an aquifer, and the 

real-world, small-scale distribution of the hydraulic conductivity within that area is less important than the bulk properties. If the modeling objectives are to examine groundwater 

travel time and location in the groundwater system, these 

are directly dependent on spatial heterogeneity of hydraulic properties, such as hydraulic conductivity and effective 

porosity and preferential flow paths. To the extent that oversimplification of the model misrepresents or omits salient 

details of the system simulated, the predicted location and 

time of travel will be in error (Moore and Doherty, 2005). 

Less obvious, perhaps, is the case of a model being parameterized appropriately for one modeling objective, but then 

used without consideration for a different objective. Such 

is the concern of using models constructed for an original 

purpose, then being used in subsequent data-worth analysis. 

This concern arises because simplifications appropriate for 

one objective can confound the insights that might otherwise 

be gained through data-worth analysis. The concern centers 

on the possibility that outcomes of data-worth analysis in 

such oversimplified models are more reflective of parametersimplification devices than of the true information content of 

the hypothetical data collected. 

layer 1, the location of a southwest-northeast trending stream 

near the stress and prediction can be seen as reducing data 

worth for potential head observations, as would be expected 

given the ability of the stream to constrain the sensitivity of 

nearby heads (Hunt, 2002). There is asymmetry in data worth 

about the stream, with potential observations east of the river 

having greater data worth. This is noteworthy because the 

prediction (and pumping) is on the western side of the stream; 

however, the greater importance of the eastern part of the 

domain reflects the ability of this area to constrain the distribution of groundwater flow captured by the stream and that which 

underflows the stream and is captured at the well. The distribution of underflow and stream capture is more easily seen when 

not near the pumping stress. The small data worth for potential 

observation wells located very near the stream itself is a reflection of the dampening effect the stream has on the sensitivity of 

adjacent potential observation heads. In this case, potential head 

observations near groundwater divides have more relative worth 

than potential observations near surface-water features. This is 

consistent with the results of Hunt (2002) and Sheets and others (2005) and can be considered a general result that likely is 

transferable to many monitoring-network design problems. 

Parameterization Implications from MonitoringNetwork Analysis

Parameterization flexibility must be commensurate with 

the model objective (Hunt and Zheng, 1999; Hunt and others, 

2007). Coarse parameterization flexibility appropriate for a 

regional or large-scale water-balance objective that integrates 

large areas of the model domain likely is inappropriate for 

quantifying the worth of potential observation-well locations. 

Oversimplification by analytical solutions or overly strict 

homogeneous, isotropic assumptions can result in poor representations of the response of natural systems (Hunt and Welter, 

2010). The role of parameterization in uncertainty analysis, and 

the significant cost that can accompany simplifying the natural 

world into models, also is consistent with the findings of Moore 

and Doherty (2006). 

The prediction of interest ultimately defines the data-worth 

analysis objective, and thus the optimal network design. Different predictions may require different levels of parameterization flexibility; however, starting with a high level of flexibility 

helps ensure that a model is suitable for a range of objectives, 

for data-worth analyses of a high-resolution question also will 

be appropriate for data-worth analyses of a low-resolution 

question. Moreover, the converse is not true. Thus, true to the 

highly parameterized approach of Doherty and Hunt (2010a), 

“if in doubt, include it.” That is, the computational savings 

gained during calculating the sensitivity from an oversimplified 

model is more than offset by the often unquantified confounding 

artifacts introduced by oversimplification. As shown here, these 

artifacts can completely obfuscate any insight gained from a 

data-worth analysis. 

Application for Monitoring-Network Design

For the highly parameterized 400 Pilot Points case in 

These confounding artifacts of oversimplification are 

incurred by the sharp (and subjective) regional parameter-zone 

boundaries in the hydraulic conductivity field specified by the 

modeler. Oversimplification adversely affects the identification 

of the optimal potential observation-well location because it 

can eclipse the ability of the method to discern subtle information within the coarser zone, such as one head location versus 

an adjacent head location within the same zone. The parameter 

flexibility afforded the 400 Pilot Point case overcomes the 

confounding effects of oversimplification caused by the hardwired imposition of hydraulic conductivity piecewise-constant 

change that occurs at the zonal boundaries. Thus, the resolution 

of the parameter flexibility required for the model is a direct 

result of the resolution of the question being asked of the model. 

In the case of the original pumping-stream depletion waterbalance objective of Hoard (2010), the regional model level of 

parameterization was appropriate. However, when the model 

objective changed and became smaller scale (ranking the data 

worth of one observation well location over a nearby location), 

a parameter-flexibility level was needed that was commensurate with the spacing of the proposed observation wells, not the 

regional model calibration targets. Note that it is the parameterization flexibility that is required, not different parameter values

specified in the model input (as the actual parameter values 

were identical in all three cases). 
In summary, broad, piecewise-constant zones may represent prior knowledge about the hydrogeologic conceptualization 

of a model and may be appropriate for large-scale model predictions (see Haitjema, 1995, p. 272, 274, and 279). A data-worth 

objective such as comparing the importance of one potential 

observation-well location to another nearby potential location 

requires a parameterization scheme that is commensurate for the 

observation network being tested. Therefore, although existing 

off-the-shelf models provide a useful quantitative framework to 

assess data worth of potential data collection, it is unlikely they 

can be used directly without some additional parameter flexibility, even if the prediction itself can be appropriately simulated 

using coarse parameter representations. That is, the parameterization should reflect the representative scale of the range of 

predictions the model is to evaluate, not necessarily the scale of 

the original prediction of interest.

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on “Two statistics for evaluating parameter identifiability 

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Written by Michael N. Fienen, Randall J. Hunt, John E. Doherty, 

and Howard W. Reeves

For more information, contact: 

Michael N. Fienen, U.S. Geological Survey, 

(608) 821-3894, mnfienen@usgs.gov

Randall J. Hunt, U.S. Geological Survey, (608) 821-3847, 

rjhunt@usgs.gov

John E. Doherty, Watermark Numerical Computing and 

Australian National Centre for Groundwater Research and 

Training, +61 7 3379 1664, johndoherty@ozemail.com.au

Howard W. Reeves, U.S. Geological Survey, 

(517) 887-8914, hwreeves@usgs.gov