U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2005-5001

A Near-optimum Procedure for Selecting 

Stations in a Streamgaging Network

A Near-optimum Procedure for Selecting 

Stations in a Streamgaging Network

By Kenneth J. Lanfear

U.S. Geological Survey Scientific Investigations Report 2005-5001

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

Gale A. Norton, Secretary

U.S. Geological Survey

Charles G. Groat, Director

U.S. Geological Survey, Reston, Virginia: 2005

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Although this report is in the public domain, permission must be secured from the individual copyright owners to 

reproduce any copyrighted materials contained within this report.

Suggested citation:

Lanfear, K.J., 2005, A Near-optimum Procedure for Selecting Stations in a Streamgaging Network: Reston, Va., U.S. 

Geological Survey, Scientific Investigations Report 2005-5001, 10 p. 
v

Contents

ABSTRACT ................................................................................................ 1

Introduction ................................................................................................ 1

Federal Interests .................................................................................... 1

Satisfying Goals .................................................................................... 2

Finding Basic Feasible Incremental Solutions ................................................... 2

Costs and Benefits ................................................................................. 3

Solution Algorithm ....................................................................................... 3

Alternative solution procedures .................................................................. 3

Initial Conditions .................................................................................... 4

Example Problem .......................................................................................... 4

Data Sets ............................................................................................. 4

Expert Rules for Selecting BFIS .................................................................. 4

Example Costs and Benefits ..................................................................... 5

Determining the Example BFIS .................................................................. 6

Active Stations – Initial Conditions ............................................................... 6

Programming Implementation ..................................................................... 6

Solution for All Goals .............................................................................. 6

Is the Solution Optimal? ........................................................................... 7

Earlier Applications ....................................................................................... 8

Conclusions ................................................................................................ 8

Literature Cited .......................................................................................... 8

Tables

Table 1. Criteria for satisfying each type of Federal goal. .......................................... 5

Table 2. List of relative location costs assigned to solutions. ....................................... 6

Table 3. List of the number of goals of each type and the total number of BFIS’s that 

satisfy each goal type. ........................................................................ 6

Table 4. Percent of each type of Federal goal satified by currently active stations. ............ 6

Table 5. Example of how the algorithm can make a non-optimal choice. ........................ 8

Figures

Figure 1. Graph of the number of goals satisfied as a function of the number of stations 

added. .......................................................................................... 7

Figure 2. Map of stations meeting example Federal goals. Active stations are shown in light 

gray and reactive stations in black. ......................................................... 8
ABSTRACT

Two questions are fundamental to Federal gov- ernment goals for a network of streamgages which 

are operated by the U.S. Geological Survey: (1) 

how well does the present network of streamagag- ing stations meet defined Federal goals and (2) what 

is the optimum set of stations to add or reactivate 

to support remaining goals? The solution involves 

an incremental-stepping procedure that is based 

on Basic Feasible Incremental Solutions (BFIS’s) 

where each BFIS satisfies at least one Federal 

streamgaging goal. A set of minimum Federal goals 

for streamgaging is defined to include water mea- surements for legal compacts and decrees, flooding, 

water budgets, regionalization of streamflow characteristics, and water quality. Fully satisfying all 

these goals by using the assumptions outlined in this 

paper would require adding 887 new streamgaging 

stations to the U.S. Geological Survey network and 

reactivating an additional 857 stations that are currently inactive.

Introduction

Since 1889, the U.S. Geological Survey (USGS) 

has operated a streamgaging network to collect 

information about the Nation’s water resources. 

It is a multipurpose network funded by the USGS 

and many other Federal, State, and local agencies. 

Individual streamgaging stations are supported for 

specific purposes such as water allocation, reservoir 

operations, or regulating permit requirements, but 

the data are used for many other purposes. Thomas 

and Wahl (1993) surveyed cooperators and data 

users to identify uses of data in 9 categories. They 

found that uses of data from a typical streamgaging 

station fall into an average of 2.6 different data-use 

categories. The USGS recently examined the network to see how it is meeting Federal goals (U.S. 

Geological Survey, 1998). The evaluation defined 

key Federal goals for the network and established a 

set of quantitative metrics that measure the extent to 

which those goals are being achieved.

The evaluation technique in this report takes 

earlier analyses by the U.S. Geological Survey 

(1998; 1999) of how well the existing network 

meets goals to the next logical phase of determining 

how to improve the network by proposing an incre- mental-stepping procedure to select streamgaging 

stations that meet additional Federal goals. Key to 

this technique is finding sets of stations or potential 

new station locations that efficiently meet individual goals. For any individual goal – for example, 

knowing the flow of a river at particular location 

– a limited number of practical configurations of 

stations can achieve the desired accuracy. By choosing from among limited sets of stations, rather than 

individual stations, we greatly reduce the number of 

possible solutions and ensure that the procedure will 

reach a near-optimum solution.

Federal Interests

Federal interests in streamgaging, which are 

defined in U.S. Geological Survey (1998; 1999), 

include supporting Federal programs, resolving 

disputes among states, and managing Federal lands. 

These broad interest categories were reduced to spe- cific types of Federal goals with quantitative mea- sures for determining success in meeting the goals. 

The list below includes important goals represen- tative of some major Federal interests. It does not 

represent a full compilation of Federal interests but 

is presented to demonstrate the analysis techniques.

• Compacts and Decrees—Streamflow data 

for interstate and international water transfers 

are required to satisfy legal compacts or court 

decrees. 

• Flooding—The National Weather Service 

(NWS) provides forecasts of flooding condi- tions at 3,116 service locations on riverine 

systems where discharge measurements are 

required. 

• Water Budget—Long-term, nationally consistent data on the inflows and outflows of 

major river basins plays a fundamental role in 

national water policies and planning. A goal, 

therefore, is to be able to provide estimates 

of the water budgets for each of the 329 river 

basins that cover the conterminous United 

States.

• Regionalization of Streamflow Characteristics—Ecoregions are based on perceived patterns of causal and integrative factors, including land use, land-surface form, potential 

natural vegetation, and soils (Omernik, 1987). 

Representative streams within each ecoregion 

need to be monitored to detect changes in 

streamflow that would result from changes 

in climate, changes in land-use practices, 
2 A Near-optimum Procedure for Selecting Stations in a Streamgaging Network

or changes in ground-water withdrawals. 

The term “representative streams” refers to 

streams whose watershed lies entirely within 

the ecoregion, as opposed to streams that simply flow through the ecoregion. 76 ecoregions 

cover the conterminous United States. Intersecting the ecoregions with the polygons of 

329 river basins leads to 802 ecoregion-basin 

combinations. Stations on representative 

streams in each of these areas would provide 

a network for detecting changes in streamflow 

and describing regional streamflow characteristics.

• Quality-Impaired Watersheds—The U.S. 

Environmental Protection Agency provided a 

list (Karen Klima, EPA, written communica- tion, January 26, 1999) of 533 watersheds 

(cataloging units) in which fewer than 50 percent of the assessed rivers meet State or tribal 

water-quality requirements for all designated 

uses.

• USGS Water-Quality Stations—Discharge 

measurements should complement mea- surements of a large suite of water-qual- ity parameters at National Stream Quality 

Accounting Network (NASQAN) stations 

(Ficke and Hawkinson, 1975), which are 

located primarily on major rivers; at Bench- mark stations (Lawrence, 1987), which are 

located primarily on streams unaffected by 

human influences; and at stations that are 

used by the National Water Quality Assess- ment (NAWQA) Program (Hirsch and others, 

1988) for long-term, low-intensity sampling. 

These 3 sources comprise a total of 123 

reaches.

Satisfying Goals

Represent N potential streamgaging stations 

with the N x 1 vector U, where each ui equals 1 

if station i is active and 0 if not active. Using the 

terminology of Yankowitz (1982), Ut is the “control 

set” of the streamgaging network at stage t. Every 

stream reach can have one or more active or inactive 

stations, or is a candidate for a new station. Thus, 

there are at least as many possible station locations 

as there are reaches. 

Let the M x 1 vector X represent the “state” of 

M goals. Each xj represents a single goal. Exam- ples of goals are: a NWS Service Location where 

streamflow must be determined; a basin for which 

inflow and outflow must be measured to determine the water budget; a basin within an ecoregion 

where at least one representative sampling station 

is required; or a long-term water-quality sampling 

location which requires estimates of streamflow. Let 

xj equal 1 if the goal is met, 0 if it is not met. A goal 

is either met or unmet; there is no partial satisfaction. Goals can be coincident. For example, a NWS 

Service Location might fall on an interstate crossing. Each, however, would be treated as a separate 

goal.

Define the “dynamics function,” f, which determines if a goal is met, as equation 1:

xj = f

j

(U) . (1)

Each f

j

 is determined by examining the net- work and finding those stations or combinations 

of stations that would satisfy goal j. Assuming 

that every goal has at least one solution, then there 

exists at least one Uj such that f

j

(Uj) equals 1. Now 

impose another condition to eliminate solutions 

that contain redundant stations: discard a Uj if any 

of its stations can be removed without its dynamics 

function becoming zero. A Uj that fulfils these two 

conditions for any goal j is called a Basic Feasible 

Incremental Solution (BFIS). That is, a set of one 

or more stations (new, reactivated, or existing) that 

satisfies a goal, with all stations in the set being necessary to satisfy the goal, is a Basic Feasible Incremental Solution (BFIS) for that goal. A station can 

belong to more than one BFIS.

Finding Basic Feasible Incremental 

Solutions

Streamflow information commonly is estimated 

from nearby stations on the stream network; it is not 

necessary to have a streamgaging station exactly 

where you need to know the streamflow. Stations 

can be located upstream or downstream from the 

point of interest, and, in many cases, discharge 

values from multiple stations can be added or sub- tracted. Streamflow characteristics at ungaged sites 

can be estimated from regression analysis of nearby 

gaged sites (Moss and Karlinger, 1974, Moss and 

others, 1982, Moss and Tasker, 1991).

Determining an appropriate set of existing or 

new stations for estimating streamflow has been the 

subject of considerable research. Benson and Mata- las (1967) showed how basin characteristics could 

be used to estimate streamflow characteristics at 

ungaged sites. Stedenger and Tasker (1985) showed 

how generalized least squares could provide more 

accurate parameter estimates. Sharp (1971), looking 

to optimally place stations to measure the effects of 

known pollutant sources, developed a method for 
SIR 2005-5001 3

selecting sampling locations based on the topology of the river network. Sanders and others (1983) 

applied a similar technique for allocating waterquality sampling stations. Karasev (1968) looked at 

minimum distance criteria.

With multiple objectives, no single method may 

suffice for selecting the BFIS. A goal of estimating 

regional streamflow parameters may require BFIS 

based on regression analysis. Selecting stations for 

a flood-warning goal would have to consider the 

stream network topology and travel times. Moreover, any BFIS does not have to yield the most 

accurate estimate of streamflow; a BFIS need only 

provide sufficient accuracy to meet the purposes 

of the goal. The actual selection method for BFIS 

can be any combination of regression, analysis of 

stream network topology, examination of actual 

field conditions, or expert judgment.

Costs and Benefits

Let C be an N x 1 vector of costs associated 

with each station. Each ci is the cost of activating 

and operating station i. For any Ut,

Costt = C • Ut . (2)

For benefits, let W be an M x 1 vector representing the benefits of each goal. Each wj

 is the 

benefit associated with fulfilling goal j. Then,

Benefitt = W • Xt = W • [f

j

(Ut)] . (3)

Solution Algorithm

A forward-stepping process adds one BFIS at a 

time in an efficient manner to find a set of stations 

that satisfies all goals. The BFIS chosen in each step 

is that which provides the greatest ratio of incre- mental benefits to incremental cost. 

The solution algorithm adds a BFIS in each 

step until there are no more goals to satisfy. Choose 

the optimal BFIS on the basis of its incremental 

cost because some of the stations in the BFIS may 

already have been added through other BFIS, and 

incremental benefits because a BFIS may happen to 

satisfy additional goals through common stations.

Let {B} be the set of all BFIS for all goals. The 

problem is to select a subset of {B} that optimally 

meets all goals. Starting with no selections, we will 

do this by adding one Bt at each stage, t, such that 

each Bt fulfils at least one new goal. In calculating 

benefits, we must account for all additional goals 

that Bt fulfils and assign the costs of any stations 

needed to complete Bt that are not already in the 

solution. The incremental costs and benefits of each 

stage are computed as follows.

For any Bt added, 

Ut = Ut-1 ∪ Bt , (4)

∆Costt-(t-1) = C • (Ut – Ut-1) , (5)

and

∆Benefitt-(t-1) = W • (Xt – Xt-1) . (6)

In moving from stage t-1 to t, the gain, 

expressed as a benefit-cost ratio will be

Gt-(t-1) = ∆Benefitt-(t-1) / ∆Costt-(t-1) . (7)

The objective in each step is to select a Bk to 

maximize the benefit-cost ratio of that step,

k | maxGt-(t-1) = W • ([f

j

(Ut-1 ∪ Bk)] – Xt-1) / 

C • ((Ut-1 ∪ Bk) – Ut-1) . (8)

This is where the BFIS play an important role. 

Since, by definition, each BFIS satisfies at least one 

goal, we are guaranteed that for some number of 

stages T where T ≤ M,

maxGt-(t-1) > 0 for t ≤ T (9)

and

maxGt-(t-1) = 0 for t > T . (10)

This ensures that the algorithm will proceed to a 

solution that satisfies all goals.

Alternative solution procedures

A simpler procedure for selecting an appro- priate solution might be to add one station per 

step, rather than one BFIS. Maddock (1974) used 

a technique of removing stations one-by-one in 

looking at how to reduce the number of stations in 

a network. Tasker (1986) used a backward-stepping technique to select stations that minimized the 

average sampling mean-square error of a regional 

regression equation. The problem with adding one 

station per step is that, at some step, all remaining 

goals might require two or more stations each. 
4 A Near-optimum Procedure for Selecting Stations in a Streamgaging Network

This will set all incremental benefit-cost ratios 

maxGt-(t-1) equal to zero before all the goals are 

satisfied, which will cause the algorithm to stall 

before it achieves a solution. Tasker (1986) did not 

face this problem because he considered a single 

goal involving regional regression. A way around 

this local minimum is an n-step algorithm – that is, 

trying all combinations of N stations. This method, 

however, becomes computationally infeasible with a 

large number of stations. Selecting from among the 

BFIS, as proposed here, effectively implements an 

n-step technique with a variable n being the number 

of stations in each BFIS.

Another variant would be to allow partial goal 

solutions. That is, allow f to take on any real num- ber 0 ≤ f ≤ 1. While this eliminates the problem of 

stalling at local minima, it introduces the possibility 

of selecting stations that ultimately don’t contribute 

to satisfying a goal. A solution that is not a union of 

BFIS’s can not be optimal since it will include sta- tions that contribute nothing to the solution.

The solution can not be solved by linear pro- gramming because the BFIS can share stations 

and, thus, are not independent of each other. Fiering (1965) used nonlinear integer programming to 

decide which sites in a network should be gaged 

further. His approach bears many similarities to the 

one proposed here, but it is based on individual stations, not sets of stations.

Burn and Goulter (1991) and Yang and Burn 

(1994) used clustering techniques to identify groups 

of similar stations, then selected a single station 

from each group. Their method is applicable to 

regionalization, but does not address the problem of 

multiple goals.

Initial Conditions

One set of initial conditions is to accept the 

currently active stations as a given. Goals satisfied 

by the set of active stations are eliminated from the 

solution before beginning the iteration procedure. 

Depending on how successful the active stations are 

in meeting the goals, this approach can substantially 

reduce the computation time for selecting BFIS to 

satisfy the remaining goals. Of course, the selected 

set of stations will be optimized only for the added 

goals, not the entire set of goals. 

Example Problem

Data Sets

An analysis of this scope would not be possible 

without a substantial infrastructure of geospatial 

data sets:

• Traces of 60,000 stream segments that constitute 1 million kilometers of rivers. Attributes 

of the reaches include information on how the 

reaches connect to each other, so that the network can be traced upstream or downstream.

• Locations of 17,823 active and historical 

USGS streamgaging stations within the conterminous United States. About 6,600 stations 

were active in 1996; each station is related to 

a stream reach.

• Locations of 3,116 National Weather Service 

locations where flood forecasts are provided 

for riverine systems. Each location is related 

to a stream reach, and about two-thirds are 

related to a specific USGS streamgaging sta- tion.

• Boundaries of 329 river basins (accounting 

units) and 2,079 watersheds (cataloging units) 

in the conterminous United States.

• Boundaries of 76 ecoregions (Omernik, 1987) 

in the conterminous United States.

Expert Rules for Selecting BFIS

Many hydrologic judgments can be expressed 

as rules that apply to a network of streams as represented by a GIS. Rules for finding sets of BFIS can 

address: 

• Percent of the basin that must be gaged. That 

is, what percent of the target basin must be 

gaged by stations upstream, or what percent 

additional area may be added by using stations downstream;

• How far away in stream distance a station 

may be located upstream or downstream;

• The number of stations and the sizes of their 

drainage areas;

• Whether stations upstream or downstream of 

reservoirs may be used;

• Whether stations on large mainstem rivers 
SIR 2005-5001 5

may be included; or

• Whether requirements such as satisfying legal 

compacts require an exact match to an existing USGS streamgaging station.

The rules applied for selecting stations that 

satisfy the Federal goals are summarized in table 1. 

A team of hydrologists experienced in streamgaging 

determined each rule. The rules were incorporated 

into computer code that operated on the data sets to 

find solutions. Using a geographic information system to view the results, the team modified the rules 

in several rounds of adjustments until they produced 

logical, practical alternatives. That is, for any goal, 

the rules would produce one or more alternatives 

for streamgaging station locations that would be 

comparable to the choices made by an experienced 

hydrologist. In effect, the rules are an expert sys- tem for choosing streamgaging solutions. Although 

these rules are generally applicable for stations in 

the conterminous United States, no final decision 

about a particular streamgaging situation should be 

made without examining local site conditions.

Example Costs and Benefits

Costs fall into two classes: (1) station costs or 

the relative costs of installing and operating a particular type of streamgaging station, and (2) “location” costs that actually are a penalty assigned to a 

set of stations that satisfies a goal in a less desirable 

manner. Estimating streamflow from an upstream 

station, for example, might be less desirable than 

having a station exactly on the reach where a 

streamflow estimates is required, and this could be 

expressed as a cost penalty.

A detailed solution would require assigning an 

accurate cost to each station. To illustrate the solution technique, an arbitrary relative cost of 10 was 

assigned to active stations, and 15 to inactive stations that could be reactivated. New stations were 

assigned a cost of 20. The important point is not 

absolute costs, but relative costs: with these relative 

values, using an active station is preferable to reac- tivating a station, but reactivating a single station is 

better than using two active stations. Location costs 

were arbitrarily assigned a value much lower than 

Table 1. Criteria for satisfying each type of Federal goal.

Goal Principle Criteria1

Compacts and Decrees Each compact or decree is associated with a specific USGS station.

Current NWS FloodForecast SitesMust include 90 to 110 percent of the service location’s drainage area and be within 20 

km, measured along the streams, of the service location. A solution may have no more 

than 2 stations, and each must have a drainage area at least 20 percent the size of the 

service location.

Accounting-Unit 

Water Budgets

Must include 75-125 percent of the accounting unit drainage, with no more than 25 per- cent of the drainage outside the accounting unit. Large mainstream rivers flowing through 

the basin are not included in the totals. Where possible, use only reaches with existing 

(active or inactive) streamgaging stations, but accept new stations if necessary. If possible, 

the number of stations in a solution should be limited to 3. However, if no solution is 

found with 3 or fewer stations, then as many as 4 stations may be accepted.

Regionalization One station for each intersection of ecoregions with accounting units. Each station must 

have a drainage area of less than 100 mi2, or 500 mi2 if it is a Hydroclimatic Data Network (HCDN) station (Slack and Landwehr, 1992), and the drainage must be entirely 

within the ecoregion-accounting unit intersection.

Quality-Impaired 

Watersheds

Must include at least 20 percent of the cataloging unit drainage, with no more than 

20 percent of the drainage outside the cataloging unit. Large mainstem rivers flowing 

through the watershed are not included in the totals. Where possible, use only reaches 

with existing (active or inactive) streamgaging stations, but accept new stations if necessary. If possible, only 1 station should comprise a solution. However, if no solution is 

found with 1 station, then 2 stations may be accepted.

Water Quality Stations NASQAN (Ficke and Hawkinson, 1975) and Benchmark stations (Lawrence, 1987) are 

matched to a specific USGS station. NAWQA stations (Hirsch and others, 1988) must 

have a streamgaging station on the same reach.

1 These criteria are not necessarily the same as described in USGS, 1998. Some criteria have been added and others modified 

to reflect better insights or advances in modeling.
6 A Near-optimum Procedure for Selecting Stations in a Streamgaging Network

that of station costs, so that the location cost broke 

ties among solutions having equal station costs. 

The location cost depended upon the placement of 

stations relative to the goal that needs to be satisfied 

(table 2). 

Table 2. List of relative location costs assigned to 

solutions.

Location of solution Location 

cost

Exact match to USGS station required by 

the goal

0.0

On the same reach as the goal requires 0.1

One or more stations upstream of the reach 

the goal requires

0.2

One or more stations downstream of the 

reach the goal requires

0.3

For a regionalization goal, the station is not 

an HCDN station (Slack and Landwehr, 

1992)

0.1

Because the intent of this example run was 

to satisfy all goals, the goals were assigned equal 

benefits by making W the unit vector. This way, the 

algorithm is free to pick the BFIS in any way that 

minimizes the cost of the stations required. 

The analysis tool was used to examine how 

the set of USGS streamgaging stations that were 

active in 1996 met each of the Federal goals. Then, 

starting over with no stations selected, BFIS’s were 

chosen to fully meet all the goals. Remember that 

this analysis uses only streamgaging stations operated by the USGS. There are hundreds of additional 

streamgaging stations operated by local, State, and 

other Federal agencies. Although these non-USGS 

streamgaging stations are not included in this study, 

the USGS plans to incorporate their contribution to 

the defined Federal streamgaging goals when data 

on their locations are compiled.

Determining the Example BFIS

Examining each of the goals yielded a total of 

20,865 BFIS’s (table 3). The number of BFIS per 

goal varies widely and depends on how many ways 

a goal can be satisfied. Only a single specified station can satisfy some goals, such as Compacts and 

Decrees. Many different combinations can satisfy 

others.

Table 3. List of the number of goals of each type 

and the total number of BFIS’s that satisfy each goal 

type.

Goal type No. of 

goals

No. of 

BFIS

Compacts and Decrees 120 120

Flooding 3,116 10,469

Water Budget 329 3,983

Regionalization 802 4,231

Quality-Impaired Watersheds 533 1,902

Water Quality Stations 123 160

TOTAL 5,023 20,865

Active Stations – Initial Conditions

Currently active stations achieved the levels of 

goal satisfaction shown in table 4.

Table 4. Percent of each type of Federal goal satis- fied by currently active stations.

Goal type Percent Satisfied

Compacts and Decrees 100%

Flooding 66%

Water Budget 57%

Regionalization 58%

Quality-Impaired Watersheds 71%

Water Quality Stations 81%

Programming Implementation

Determining the BFIS and solving the optimiza- tion problem were implemented in software using 

the Perl programming language (Wall and others, 

1996). Perl was chosen because its complex data 

structures could easily represent stream network 

topology and the relationship of the BFIS to the 

individual stations. It would be possible to accomplish the same thing with many other languages. 

The programs were used to solve a very specific 

problem and were not written in a generalized form 

for widespread distribution.

Solution for All Goals

Starting the analysis with no stations selected 

and continuing until all Federal goals listed herein 

are satisfied, 2,313 active stations, 857 reactivated 

stations and 887 new stations would be required, for 
SIR 2005-5001 7

a total network of 4,057 stations. (There are about 

4,300 additional active stations in the network that 

address other goals, but they are not required to 

meet the Federal goals considered in this analysis.) 

A graph of the incremental process to select stations 

(figure 1) shows two inflection points. For the first 

1,000 stations, the incremental procedure adds primarily stations that satisfy more than one goal each. 

The next 2,700 stations added satisfy mostly one 

goal each (slope is 1:1). Finally, after adding about 

3,700 stations, the procedure adds stations that deal 

with more complex goals that require more than one 

station per goal.

The relative smoothness of the goal curve 

shown in figure 1 results from all goals having equal 

benefits. Had benefits been differentially assigned, 

more inflection points would have resulted as goals 

were satisfied generally by order of priority. A map 

of the selected stations (figure 2) shows their geo- graphic distribution.

Is the Solution Optimal?

The algorithm exhibits a preference for select- ing stations that solve multiple goals, as indicated 

by the slope greater than 1 for the first 1000 sta- tions in figure 1. This is what is expected in an 

optimal or nearly optimal solution. The solution, 

though very good, is not, however, guaranteed to 

be optimal. Situations can be constructed where the 

algorithm will choose a non-optimal path. Table 

5 shows a situation where cost differences caused 

the algorithm to make the non-optimal choice of 

A and C. The higher cost of choice B caused it to 

be rejected in the first step, even though, looking 

further ahead, it is intuitively the best choice for satisfying all 3 goals. A technique that looked ahead 2 

steps would have made the optimal choice. Perhaps 

an n-step technique, where n is the maximum number of goals met by any BFIS, would be optimal.

Table 5. Example of how the algorithm can make a 

non-optimal choice.

BFIS Meets 

goals

Cost Benefit/Cost 

(Step 1)

Benefit/Cost 

(Step 2)

A 1,2 1 2 (select) n/a

B 1,2,3 1.6 1.875 0.625

C 2,3 1 2 1 (select)

Whether the non-optimal situation described in 

Table 5 actually occurs is unknown. One possibility suggested by reviewers was to select smaller 

parts of the data set and compare the solution to one 

determined from complete enumeration. Performing this test, however, would require a substantial 

amount of additional computer programming and 

could be the subject of another paper.

Figure 1. Graph of the number of goals satisfied as a function of the number 

of stations added.

0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Stations Added

Goals Met
8 A Near-optimum Procedure for Selecting Stations in a Streamgaging Network

Earlier Applications

The example shown in this paper should only 

be regarded as a starting point for discussions, as 

other reasonable assumptions about goals and costs 

could be made. In 1998, the U.S. Geological Survey 

(1998), for example, employed an earlier version 

of this technique to evaluate how well the existing 

streamgaging network met selected Federal goals. 

However, the study was based upon preliminary 

data, a few different goals, and slightly different 

rules for satisfying goals and determining solutions. In 1999, the U.S. Geological Survey (1999) 

employed this technique as a starting point, but 

made simplifying assumptions about goals for water 

quality and long-term change detection.

Conclusions

This paper describes and demonstrates an 

incremental technique for selecting a nearly opti- mal set of streamgaging stations to meet any given 

set of goals for streamflow information, such as 

flood protection, water allocations, water quality, 

and long-term changes. Data sets are available that 

have adequate resolution to apply this technique 

on a scale suitable for the conterminous United 

States for Federal goals concerning: compacts and 

decrees; flooding; water budgets; regionalization 

of streamflow characteristics; quality-impaired 

watersheds; and USGS water-quality stations. The 

technique is sufficiently scalable to deal successfully with the entire network of USGS streamgaging stations.

A preliminary model run indicates that adding 887 new streamgaging stations and reactivating 

857 others could meet some of the most important 

Federal goals for streamgaging in the conterminous United States. This should only be regarded 

as a starting point for discussions, as other reasonable assumptions about goals and costs could be 

made. The analysis technique provides a tool for 

quantitatively evaluating the number and location 

of streamgaging stations for any list of Federal 

streamgaging goals, now, or in a future time with 

different priorities.

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