87° 86° 85° 84°

43°

42°

INDIANA OHIO

MICHIGAN

Lake

Michig

a

n

U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2010–5020

National Water Availability and Use Pilot Program

Application of AFINCH as a Tool for Evaluating the Effects 

of Streamflow-Gaging-Network Size and Composition on 

the Accuracy and Precision of Streamflow Estimates 

at Ungaged Locations in the Southeast Lake Michigan 

Hydrologic Subregion

Application of AFINCH as a Tool for Evaluating 

the Effects of Streamflow-Gaging-Network 

Size and Composition on the Accuracy and 

Precision of Streamflow Estimates at Ungaged 

Locations in the Southeast Lake Michigan 

Hydrologic Subregion

By G.F. Koltun and David J. Holtschlag

National Water Availability and Use Pilot Program

Scientific Investigations Report 2010–5020

U.S. Department of the Interior

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

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2010

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reproduce any copyrighted materials contained within this report.

Suggested citation:

Koltun, G.F., and Holtschlag, D.J., 2010, Application of AFINCH as a tool for evaluating the effects of 

streamflow-gaging-network size and composition on the accuracy and precision of streamflow estimates at 

ungaged locations in the Southeast Lake Michigan Hydrologic Subregion: U.S. Geological Survey Scientific 

Investigations Report 2010–5020, 14 p.

Cover: Illustration of the network of gages operated between 1971 and 2003 in the Southeast Lake Michigan 

hydrologic subregion.
iii

Contents

Abstract...........................................................................................................................................................1

Introduction.....................................................................................................................................................2

Purpose and Scope ..............................................................................................................................2

Analytical Approach......................................................................................................................................2

Results .............................................................................................................................................................8

Summary and Conclusions.........................................................................................................................12

References Cited..........................................................................................................................................14

Figures

1. Map of the 0405 (Southeast Lake Michigan) hydrologic subregion and 

National Hydrography Dataset Plus (NHDPlus) stream network .........................................3

2. Drawing of hypothetical basin illustrating flow-line types....................................................4

3–4. Diagrams showing:

3. Schematic of steps leading to computation of the pseudo coefficients 

of variation and apparent percent errors ........................................................................6

4. Schematic of “missing gage” analysis steps .................................................................7

5–9. Graphs showing:

5. Average pseudo coefficient of variation ratios for water-year flows 

as a function of censoring level for selected flow lines in the 

0405 hydrologic subregion .................................................................................................9

6. Average pseudo coefficients of variation of monthly flows for 

30-percent censoring level as a function of calendar month for 

selected flow lines in the 0405 hydrologic subregion..................................................10

7. Average apparent percent errors in water-year flows as a 

function of censoring level for selected flow lines in the 

0405 hydrologic subregion ...............................................................................................11

8. Relation between censoring level and probability of estimating 

monthly flow within 10 percent of measured value for the

water year 1999..................................................................................................................12

9. Relation between censoring level and probability of estimating 

monthly flow within 10 percent of measured value for the 

water year 2000..................................................................................................................13

Table

1. Flow-line characteristics.............................................................................................................8
iv

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

kilometer (km) 0.6214 mile (mi)

Area

square mile (mi2) 2.590 square kilometer (km2) 

Flow rate

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as follows: 

°F=(1.8×°C)+32

Conversion Factors 

Abbreviations

Water year is the 12-month period from October 1 through September 30. The water year is 

designated by the calendar year in which it ends and which includes 9 of the 12 months. For 

example, the year ending September 30, 1992 is called the “1992 water year.”

AFINCH Analysis of Flows in Networks of Channels

APE apparent percent error

CDF cumulative distribution function

CVp pseudo coefficient of variation

NHDPlus National Hydrography Dataset Plus

USGS U.S. Geological Survey
Abstract

Bootstrapping techniques employing random subsampling were used with the AFINCH (Analysis of Flows In 

Networks of CHannels) model to gain insights into the effects 

of variation in streamflow-gaging-network size and composition on the accuracy and precision of streamflow estimates 

at ungaged locations in the 0405 (Southeast Lake Michigan) 

hydrologic subregion. AFINCH uses stepwise-regression 

techniques to estimate monthly water yields from catchments 

based on geospatial-climate and land-cover data in combination with available streamflow and water-use data. Calculations are performed on a hydrologic-subregion scale for each 

catchment and stream reach contained in a National Hydrography Dataset Plus (NHDPlus) subregion. Water yields from 

contributing catchments are multiplied by catchment areas and 

resulting flow values are accumulated to compute streamflows 

in stream reaches which are referred to as flow lines. AFINCH 

imposes constraints on water yields to ensure that observed 

streamflows are conserved at gaged locations.

Data from the 0405 hydrologic subregion (referred to as 

Southeast Lake Michigan) were used for the analyses. Daily 

streamflow data were measured in the subregion for 1 or more 

years at a total of 75 streamflow-gaging stations during the 

analysis period which spanned water years 1971–2003. The 

number of streamflow gages in operation each year during 

the analysis period ranged from 42 to 56 and averaged 47. 

Six sets (one set for each censoring level), each composed of 

30 random subsets of the 75 streamflow gages, were created 

by censoring (removing) approximately 10, 20, 30, 40, 50, and 

75 percent of the streamflow gages (the actual percentage of 

operating streamflow gages censored for each set varied from 

year to year, and within the year from subset to subset, but 

averaged approximately the indicated percentages). 

Streamflow estimates for six flow lines each were aggregated by censoring level, and results were analyzed to assess 

(a) how the size and composition of the streamflow-gaging 

Application of AFINCH as a Tool for Evaluating the Effects 

of Streamflow-Gaging-Network Size and Composition 

on the Accuracy and Precision of Streamflow Estimates 

at Ungaged Locations in the Southeast Lake Michigan 

Hydrologic Subregion

By G.F. Koltun and David J. Holtschlag

network affected the average apparent errors and variability of 

the estimated flows and (b) whether results for certain months 

were more variable than for others. The six flow lines were 

categorized into one of three types depending upon their network topology and position relative to operating streamflowgaging stations. 

Statistical analysis of the model results indicates that 

(1) less precise (that is, more variable) estimates resulted from 

smaller streamflow-gaging networks as compared to larger 

streamflow-gaging networks, (2) precision of AFINCH flow 

estimates at an ungaged flow line is improved by operation of 

one or more streamflow gages upstream and (or) downstream 

in the enclosing basin, (3) no consistent seasonal trend in estimate variability was evident, and (4) flow lines from ungaged 

basins appeared to exhibit the smallest absolute apparent 

percent errors (APEs) and smallest changes in average APE 

as a function of increasing censoring level. The counterintuitive results described in item (4) above likely reflect both the 

nature of the base-streamflow estimate from which the errors 

were computed and insensitivity in the average model-derived 

estimates to changes in the streamflow-gaging-network size 

and composition. Another analysis demonstrated that errors 

for flow lines in ungaged basins have the potential to be much 

larger than indicated by their APEs if measured relative to 

their true (but unknown) flows.

“Missing gage” analyses, based on examination of 

censoring subset results where the streamflow gage of interest was omitted from the calibration data set, were done to 

better understand the true error characteristics for ungaged 

flow lines as a function of network size. Results examined 

for 2 water years indicated that the probability of computing 

a monthly streamflow estimate within 10 percent of the true 

value with AFINCH decreased from greater than 0.9 at about 

a 10-percent network-censoring level to less than 0.6 as the 

censoring level approached 75 percent. In addition, estimates 

for typically dry months tended to be characterized by larger 

percent errors than typically wetter months.
2 AFINCH as a Tool for Evaluating the Effects of Network Size and Composition on Streamflow Estimates

Introduction

The U.S. Geological Survey (USGS) streamflow-gaging 

network changes over time as gages are added or removed. 

Although there are several streamflow-gaging stations that are 

fully funded by the USGS, the vast majority of streamflow 

gages operated by the USGS are funded, in part or in whole, 

by other Federal, State, local, or tribal agencies. The streamflow-gaging network has continually changed (both in size and 

in composition), over time, to meet the data needs and fiscal 

realities of its supporters owing to the diverse and distributed 

nature of streamflow-gage funding. At times, those changes 

have resulted in appreciable reductions in the numbers of 

streamflow gages operated (Mason and Yorke, 1997).

A computer application called AFINCH (Analysis of 

Flows In Networks of CHannels) (Holtschlag, 2009) was 

developed recently (2009) by the USGS as part of the National 

Water Availability and Use Program—Great Lakes Basin Pilot. 

In short, AFINCH was designed to make use of, and extend, 

the limited available site-specific streamflow and water-use 

information to estimate streamflows and water yields throughout large regions.

AFINCH uses geospatial-climate and land-cover data sets 

along with available streamflow and water-use data to develop 

stepwise-regression models for estimating water yields and 

streamflows on a monthly basis. Calculations are performed 

on a hydrologic-subregion scale and water yields and streamflows are estimated for each catchment and stream reach 

in the hydrologic subregion that is defined in the National 

Hydrography Dataset Plus (NHDPlus) data set (Horizon 

Systems, 2009). Water yields from contributing catchments are 

multiplied by catchment areas, and resulting flow values are 

accumulated to compute streamflows in downstream stream 

reaches, which are referred to as flow lines in the NHDPlus 

data set. AFINCH makes use of the NHDPlus stream topology 

and value-added attributes to adjust and constrain estimated 

water yields and streamflows based on observed streamflows 

and water uses (withdrawals, augmentations, and diversions) 

so that observed streamflows are conserved at gaged locations. 

Although AFINCH was developed primarily as a tool for 

estimating streamflows, it will be used herein to gain insights 

into the effects of variation in streamflow-gaging-network size 

and composition on the accuracy and precision of streamflow 

estimates from AFINCH. 

Purpose and Scope

The purpose of this report is to present the methods and 

results of a study in which the AFINCH model was used with 

bootstrapping techniques employing random subsampling to 

compute selected statistical characteristics of monthly flow 

estimates as a function of streamflow-gaging-network size and 

composition. This was done to gain insights into the effects of 

variation in streamflow-gaging-network size and composition 

on the accuracy and precision of streamflow estimates from 

AFINCH. Analyses were limited to the 0405 hydrologic subregion (fig. 1) and are based on streamflow data collected in the 

subregion for 33 water years ranging from 1971 to 2003. 

The analyses and results primarily are intended to illustrate 

the effects of varying streamflow-gaging-network size and composition on the accuracy and precision of AFINCH streamflow 

estimates for stream reaches with varying stream- and gagenetwork topological characteristics. The analyses and results are 

not intended to be exhaustive in scope, nor are they intended 

necessarily to be representative of all flow lines in the 0405 

hydrologic subregion or other hydrologic subregions.

Analytical Approach

The AFINCH model is written in the MATLAB programming language (MathWorks, 2009) and makes use of the 

MATLAB Statistics Toolbox. The AFINCH code was modified for this study to facilitate repetitive unattended runs with a 

predefined set of potential explanatory variables. The potential 

explanatory variables used for all runs were monthly total precipitation (in inches) and monthly mean temperature (in degrees 

Celsius). Precipitation and temperature data were derived from 

geospatial climate data sets prepared by the PRISM Group 

(http://www.prism.oregonstate.edu/). A p‐value criterion of 0.01 

was used for variable selection and retention in the monthly 

stepwise regressions. Data were analyzed for water years 1971–

2003. No water uses were specified for any locations during the 

analysis period owing to a lack of detailed water-use data.

Data from the 0405 (Southeast Lake Michigan) hydrologic 

subregion, which covers an area of approximately 12,800 mi2, 

were used for the analyses, primarily because they already 

had been compiled during the development of the AFINCH 

model. Daily streamflow data were measured in the subregion 

for 1 or more years during the analysis period at a total of 75 

streamflow-gaging stations (hereafter referred to as gages). 

The number of gages in operation each year during the analysis 

period ranged from 42 to 56 and averaged 47. A program was 

developed to create 30 random subsets of the 75 gages, each 

with a specified number of gages removed from the full-station 

complement. Each subset was created starting with the full 

complement of 75 gages; consequently, there is the potential 

(albeit unlikely) of having 2 or more identical subsets in a group 

of 30 subsets. There also is the potential that different subsets 

result in what effectively are identical gage networks for certain 

years. This can occur if omitted gages were not operated in 

the given year (and so the omission does not affect the modelcalibration data set relative to other potential subsets where 

the gage was not omitted). Thirty subsets each, for 6 censoring levels, were created in a bootstrapping fashion (Davidson 

and Hinkley, 1997) by removing approximately 10, 20, 30, 40, 

50, and 75 percent of the gages. The various percent removals 

henceforth will be referred to as censoring levels. For example, 

analyses with approximately 10 percent of the gages removed 

will be referred to as those corresponding to the 10-percent 

censoring level.
Analytical Approach 3

Figure 1. The 0405 (Southeast Lake Michigan) hydrologic subregion and National Hydrography Dataset Plus (NHDPlus) stream network.
4 AFINCH as a Tool for Evaluating the Effects of Network Size and Composition on Streamflow Estimates

All AFINCH processing steps (except plotting of trends 

and duration curves) were run, first using the full complement of gages and then for each censored gage subset. Each 

run resulted in monthly accumulated flow estimates (hereafter referred to as flow estimates) for all of the more than 

113,000 flow lines contained in the NHDPlus data set for the 

0405 hydrologic subregion. These flow lines represent stream 

reaches of varying length ranging from 0.001 to 17.896 km. 

Flow lines are uniquely identified in NHDPlus by a numeric 

code referred to as a COMID. In all, AFINCH was run first 

using all 75 gages and then using 180 subsets (30 subsets each 

for 6 censoring levels) of the 75 gages, resulting in a total of 

5,973 water-year sets of flow estimates for each flow line. 

Estimates were aggregated by censoring level, and results for 

selected flow lines were extracted for statistical analysis.

Three types of flow lines were examined. Figure 2 shows 

a hypothetical basin with 4 gaging stations (identified as A–D) 

and 17 flow lines numbered 1–17 and color coded by type. 

The first type (type 1) of flow line corresponds to one in basins 

where no gages had been operated during the analysis period 

either upstream or downstream from the given flow line. Flow 

lines 2 and 5 in figure 2 are type 1 flow lines. Flow estimates 

for type 1 flow lines are free of gage-based constraints and 

vary solely as a function of explanatory variables in the regression models developed from the calibration-gage network. 

A type 2 flow line corresponds to one in basins where 

one or more gages are operated upstream and (or) downstream from the given flow line. Type 2 flow lines can be 

located upstream from a gage (shown as “Type 2 up” in fig. 2), 

downstream from a gage (shown as “Type 2 down” in fig. 2), 

or between an upstream and downstream gage (shown as 

“Type 2 up/down” in fig. 2). Flow estimates for type 2 flow 

lines are constrained by streamflow data from the closest 

downstream gage (if present) and (or) partially constrained by 

streamflow data from upstream gages (if present). For example, flow lines 13 and 14 are constrained by gage D on flow 

line 12; if the summation of predicted flows from flow lines 

13 and 14 and the flow contributing directly to flow line 12 

were 10-percent larger than the actual flow measured at gage 

D, then the predicted flows for flow lines 13 and 14 and the 

contribution to flow line 12 would be reduced proportionally 

so that the flows sum to the flow measured at the gage. 

The third type of flow line (type 3) corresponds to one 

that contains a gage (that is, a gage is located on the flow 

line). Flow lines 6, 7, 10, and 12 in figure 2 are type 3 flow 

lines. Flow estimates for type 3 flow lines are constrained to 

match the flows reported for the gage during periods that the 

gage is operated. If the gage is not operated during a particular 

period, then a type 3 flow line, in effect, will revert to a type 1 

or type 2 flow line, depending upon the presence and relative location of other gages in the basin. For example, if gage 

A (fig. 2) was not operated in a given year, then flow line 6 

would revert to type 1 because there would be no upstream or 

downstream gages with flow data that can be used to constrain 

estimates at the flow line.

D

C

B

A

1

2

3

4

5

6

7 8

9

10

11

12

13

14

15

16

17

Type 1

Type 2 up

Type 2 up/down

Type 2 down

Type 3

Basin boundary

Streamflow gage

Type 3

Basin boundary

Streamflow gage

Figure 2. Hypothetical basin illustrating flow-line types.
Analytical Approach 5

The flow estimates for selected flow lines were analyzed 

to assess (a) how the size (a function of the censoring level) 

and composition of the gage network affected the average 

errors and variability of the estimated monthly flows and 

(b) whether estimates for certain months were more variable 

than for others. 

AFINCH computes estimates of average flow, in cubic 

feet per second, for all flow lines for each month of each 

water year in the specified analysis period. The monthly flow 

estimates subsequently were used to compute water-year 

average flow estimates1, also in cubic feet per second. For a 

given flow line and censoring level, the variance and mean 

of the flow estimates were computed for each month of each 

year in the analysis period and for each water year based on 

AFINCH output from the 30 gage subsets. Two additional statistics, a pseudo coefficient of variation (CVp) and an apparent 

percent error (APE), also were computed from the variances 

and means of the flow estimates. A schematic of the analysis 

steps leading to the computation of these statistics is shown in 

figure 3. 

The pseudo CVp for a given month (or year) was 

calculated as

where 

is the variance of the monthly (or annual) AFINCH 

flow estimates for a given water year determined 

from the 30 gage subsets for a given censoring 

level, and 

is the monthly (or annual) flow estimate for the same 

water year determined from an AFINCH analysis 

using the full-gage complement. 

The APE for a given water year was calculated as

where 

generally is the mean of the annual AFINCH flow 

estimates for the water year based on the 30 gage 

subsets for the given censoring level, and 

is the annual flow estimate for the same water year 

determined from an AFINCH analysis using the 

full-gage complement.

1

 Water-year average flow estimates were computed by determining a 

weighted sum of the monthly flow estimates, where each month’s weight was 

set to the number of days in the month divided by the number of days in the 

year.

This statistic is referred to as the “pseudo” coefficient of variation because, in most cases, the CVps are not computed relative 

to the true monthly (or annual) flows at the flow line, which 

frequently are not known. Instead, the CVps are computed relative to a base “best estimate” of streamflow as determined with 

AFINCH using the full-gage complement for model development. That “best estimate” of streamflow for a given water 

year is certain to be the true streamflow only when a gage is 

operated in the flow line during that year because AFINCH 

constrains the estimate to equal the gaged flow. In other 

cases, the “best estimate” of streamflow will vary in accuracy 

depending upon the quality of the regression model and on 

the numbers and locations of other gages operated in the basin 

whose streamflow data partially can constrain the estimate.

Additional analyses were done by examining AFINCH 

estimates for all flow lines that contained a gage to better 

examine the true error characteristics for ungaged flow lines 

(flow lines corresponding to a stream reaches that do not 

contain a stream gage) as a function of network size. These 

analyses were done by examining only those monthly flow 

estimates for each gaged flow line that resulted from censoring 

subsets for which the given gage was omitted from the network. These analyses will be referred to as the “missing gage” 

analyses because they are based on flow estimates derived 

from network subsets where the gage is missing from the 

calibration network. The difference between the estimated and 

observed monthly streamflows were computed and expressed 

as a percentage of the observed streamflows based on the 

following equation:

where, for a given gage and year, 

is the percent error in monthly streamflow for month 

i (where i=1 to 12), 

is the observed monthly streamflow for month i, and 

is the estimated monthly streamflow for month i

based on a network that excludes the gage in the 

flow line. 

The monthly percent errors for all of the gaged flow lines 

were aggregated by censoring level and were rank ordered; 

then, nonexceedance probabilities were estimated by means 

of the Cunnane (1978) plotting-position formula and used to 

construct empirical cumulative distribution functions (CDFs). 

The CDFs were then used to compute the probabilities that the 

estimated monthly flows were within plus or minus 10 percent 

of the observed flows as a function of censoring level. 

A schematic of the “missing gage” analysis steps is shown 

in figure 4.

PEm Q Q m Q i i m m i i = − 100 * ( ˆ ) /

Qmi

PEmi

Qˆ

mi

(1)

(2)

(3)

CVp  S Qf 100 2 * /

S2

Qf

APE = − 100 * (Q Q s f f ) / Q

Qs

Qf
6 AFINCH as a Tool for Evaluating the Effects of Network Size and Composition on Streamflow Estimates

Figure 3. Schematic of steps leading to computation of the pseudo coefficients 

of variation and apparent percent errors.

Starting with 75 gages, 

randomly censor gages 

at x percent level to 

create subset and 

estimate monthly flows 

for the analysis period 

with AFINCH.

Estimate monthly flows 

for the analysis period 

with AFINCH based 

on the full 75-gage 

complement

Repeat 30 times for each 

censoring level

Extract monthly flow-estimate 

results for selected flow lines

Compute annual flow 

estimates from monthly flow 

estimates for the individual 

flow lines

Compute the pseudo 

coefficient of variation (CVp) 

and apparent percent error 

(APE) of monthly and annual 

flow estimates as a function 

of censoring level for the 

individual flow lines

Compute the mean and 

variance of the monthly and 

annual flow estimates as a 

function of censoring level 

for the individual flow lines
Analytical Approach 7

October

November

December

January

February

March

April

May

June

July

August

September

WATER YEAR 1999

AVERAGE CENSORING LEVEL, IN PERCENT

PROBABILITY THAT ESTIMATE IS WITHIN +/- 10 PERCENT OF MEASURED FLOW 0.4

10 20 30 40 50 60 70 80

0.5

0.6

0.8

0.9

1

0.7

Starting with 75 gages, 

censor gages at x percent 

level to create subset and 

estimate monthly flows for 

the analysis year with 

AFINCH.

Use empirical CDFs to 

estimate probability of 

estimated monthly flows 

being within 10 percent 

of observed as a function 

of censoring level

Repeat 30 times for each 

censoring level

By month, rank order the 

computed PEm values

Compute the percent error 

in the monthly streamflow 

estimates (PEm) relative to 

observed streamflow

Compute Cunnane plotting 

position and generate 

monthly empirical 

cumulative distribution 

functions (CDFs) of percent 

error as a function of 

censoring level

Extract gaged flow line 

results from those analyses 

where the gage was 

omitted from the 

calibration data set

Figure 4. Schematic of “missing gage” analysis steps.
8 AFINCH as a Tool for Evaluating the Effects of Network Size and Composition on Streamflow Estimates

Results

Data from six flow lines initially will be used to discuss 

and illustrate the results. The flow lines were chosen to represent a variety of types (as discussed earlier) and drainage areas 

(table 1). The results are interpreted based on the assumption that the AFINCH model produces the most accurate 

flow estimates when all available gage data are used in the 

model, an assumption that may or may not be true. Based on 

that assumption, (in most cases) variance and error measures 

initially are scaled relative to flow estimates obtained when the 

full complement of gages is used. The flow line for COMID 

9019553 had a gage located on it, and as such was treated as a 

special case, as discussed below.

A pseudo coefficient of variation (CVp) was computed for 

each flow line and censoring level. To facilitate comparisons 

of results between flow lines, each CVp for a given flow line 

was divided by the CVp for the 10-percent censoring level to 

compute a pseudo coefficient of variation ratio (CVp ratio). 

Consequently, the CVp ratios for the various censoring levels 

represent multiples of the CVp for the 10-percent censoring 

level. 

Average CVp ratios for water-year flows (determined by 

summing the annual CVp ratios for each water year and dividing by 33 (the number of water years)) are plotted as a function of censoring level in figure 5. COMID 9019553 received 

special treatment (as mentioned previously), and data for that 

flow line were computed two ways. As discussed earlier, when 

a gage is operated in a reach corresponding to a flow line (as 

was true for COMID 9019553), AFINCH constrains the flow 

estimates to be equal to the flows observed at the gage. Consequently, whenever the gage was included in the network for a 

given run, the estimated flow for the coincident flow line was 

constrained to equal the observed (gaged) flow. Because it is 

statistically likely that the gage will be retained in several, but 

not all, of the subsets at a given censoring level, the result is a 

reduction in the apparent pseudo coefficient of variation relative to a comparable flow line without a gage. To account for 

this, the pseudo coefficients of variation were computed based 

upon results from all 30 subsets for a given censoring level 

(reported as 9019553A) and based upon only those results for 

which the gage data were omitted from the calibration data set 

(reported as 9019553). 

Figure 5 shows that average CVp ratios for water-year 

flows tended to increase as the network size decreased. For 

example, the average CVp ratio corresponding to a 75-percent 

censoring level ranged from about 2 to 5 times the CVp ratio at 

the 10-percent censoring level. This indicates that less precise 

(more variable) estimates resulted from smaller networks as 

compared to larger networks. In fact, for flow lines 9005805, 

9019381, and 9019553, estimate variability was about 5 times 

greater at the 75‐percent censoring level than at the 10-percent 

censoring level. Each of these flow lines effectively represents 

reaches in basins without gages (that is, type 1 flow lines). The 

remaining four flow lines (3468071, 3470628, 9019203, and 

9019553A) also show a tendency toward increasing average 

CVp ratios for annual flows with increasing censoring; however, the CVp ratios tended to be smaller than comparable CVp

ratios for the type 1 flow lines. This phenomenon likely is due 

to variance-stabilizing effects resulting from model constraints 

associated with having one or more gages operated in those 

basins. This result indicates that the precision of AFINCH flow 

estimates at ungaged given flow lines is improved by operation of one or more gages upstream and (or) downstream in 

the enclosing basin.

Table 1. Flow-line characteristics.

[NHDPlus, National Hydrography Dataset Plus; mi2, square mile]

COMID

of

flow line

Site type Stream name

NHDPlus

contributing

drainage area

(mi2)

Comment

3468071 2 Kalamazoo River 1,533 Several gages operated upstream and downstream.

3470629 2 South Branch Kalamazoo River 86 Several gages operated downstream, none upstream.

9005805 1 Crockery Creek 161 Ungaged.

9019203 2 South Branch Black River 136 One gage operated upstream during entire analysis 

period.

9019381 1 Brandywine Creek 16 Ungaged.

a

9019553 3 South Branch Black River 87 One gage in flow line operated during entire analysis 

period, no other constraining gages upstream or 

downstream.

a Flow line referred to as 9019553A when computations were based on results from all 30 subsets for a given censoring level. Flow line referred to as 

9019553 when computations were based on results for which the gage data were omitted from the calibration data set.
Results 9

0 10 20 30 40 50 60 70 80

CENSORING LEVEL, IN PERCENT

AVERAGE PSEUDO COEFFICIENT OF VARIATION RATIO FOR WATER-YEAR FLOWS

3468071 (type 2)

3470629 (type 2)

9005805 (type 1)

9019203 (type 2)

9019381 (type 1)

9019553 (type 1)

9019553A (type 3)

3

2

1

4

5

6

Figure 5. Average pseudo coefficient of variation ratios for water-year flows as a function of censoring level for 

selected flow lines in the 0405 (Southeast Lake Michigan) hydrologic subregion.

Monthly average CVp values corresponding to a 

30-percent censoring level are plotted in figure 6 for the flow 

lines listed in table 1. Although it is reasonable to expect 

that more variable estimates might be associated with certain 

months or seasons of the year, figure 6 shows no clear-cut 

indication of seasonal trend in variability that is universal 

across all of the flow lines. A 30-percent censoring level was 

chosen for this illustration; however, other censoring levels 

yielded similar results. The three flow lines generally exhibiting the highest monthly average CVp values effectively represent type 1 flow lines. Again, the lower variability associated 

with the remaining flow lines from gaged basins likely is due 

to the variance-stabilizing effects of flow constraints imposed 

by the AFINCH model.

Average APEs for water-year flows (determined by summing the annual APEs for each water year and dividing by 

33 (the number of water years)) are plotted as a function of 

censoring level in figure 7. COMID 9019553 received special 

treatment (as was true with the average CVp ratio analyses), 

and data for that flow line were computed two ways. Average APEs for annual streamflows were computed based 

upon results from all 30 subsets for a given censoring level 

(reported as 9019553A) and based on only those results for 

which the gage data were omitted from the model (reported as 

9019553).
10 AFINCH as a Tool for Evaluating the Effects of Network Size and Composition on Streamflow Estimates

Average APEs for annual flows tended to increase in 

absolute value with increasing censoring level (fig. 7). For 

most of the flow lines listed in table 1, average APEs for 

annual flows tended to become increasingly negative as the 

censoring level increased. The two flow lines (9005805 and 

9019381) that exhibited the smallest absolute average APEs 

and smallest changes in average APEs as a function of increasing censoring level were both in ungaged basins (that is, type 1 

flow lines). This counterintuitive result likely reflects both the 

nature of the base-streamflow estimate from which the APEs 

were computed and an insensitivity in the average estimates to 

changes in the gage-network size and composition (rather than 

the unlikely alternate hypothesis that lower estimate errors can 

be achieved in sparse gage networks by not operating gages 

in a basin). The larger APEs and increases in absolute average APEs with increased censoring observed for flow lines in 

gaged basins likely reflects a bias in the estimate (relative to 

the observed streamflow) resulting from relaxation of streamflow constraints when one or more gages in the basins were 

omitted from the network.

Flow line 9019553, which represents only those results 

where the gage that is located on that flow line (the only gage 

in the basin) was omitted from the analysis, has an average 

APE curve with gentle-slope characteristics similar to those 

for flow lines in ungaged basins (fig. 7); however, the average APE curve lies much farther from the zero error line 

than those of other basins. Unlike the other flow lines in 

figure 7, error results for flow-line 9019553 are reported as a 

3468071 (type 2)

3470629 (type 2)

9005805 (type 1)

9019203 (type 2)

9019381 (type 1)

9019553 (type 1)

9019553A (type 3)

MONTH

AVERAGE PSEUDO COEFFICIENT OF VARIATION OF MONTHLY FLOWS

FOR 30 PERCENT CENSORING LEVEL

Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep

1.0

1.5

2.0

2.5

3.5

3.0

4.0

Figure 6. Average pseudo coefficients of variation of monthly flows for 30-percent censoring level as a function of 

calendar month for selected flow lines in the 0405 (Southeast Lake Michigan) hydrologic subregion.
Results 11

percentage of the gaged flow and therefore represent the true 

average error in the AFINCH estimates (as opposed to other 

error estimates that are dependent upon the assumption that 

the AFINCH flow estimate for the full-gage complement is 

accurate). This result illustrates that the true average percent 

errors for the other flow lines have the potential to be much 

larger than presently indicated by the APEs when computed 

relative to the true, but unknown, flows.

Results of the “missing gage” analyses are shown in 

figures 8 and 9 for water years 1999 and 2000, respectively. 

The nominal censoring levels shown in these figures differ somewhat from each other and from those reported for 

the flow-line analyses because they were computed based 

only on data for the indicated water year. It is evident from 

figures 8 and 9 that the likelihood of AFINCH computing a 

monthly streamflow estimate within 10 percent of the measured value decreased appreciably as the network size was 

reduced. For example, in both 1999 and 2000, the probability 

of computing a monthly streamflow estimate within 10 percent 

of the true (observed) value was greater than 0.9 at about a 

10-percent network-censoring level; however, that probability 

dropped to less than 0.6 as the censoring level approached 

75 percent. The data shown in figures 8 and 9 also indicate 

that estimates for typically dry months (October and November) tended to be characterized by larger percent errors (as evidenced by the fact that those months exhibited smaller probabilities that the estimate was within plus or minus 10 percent 

of the observed value) than typically wetter months. 

Figure 7. Average apparent percent errors in water-year flows as a function of censoring level for selected flow 

lines in the 0405 (Southeast Lake Michigan) hydrologic subregion.

3468071 (type 2)

3470629 (type 2)

9005805 (type 1)

9019203 (type 2)

9019381 (type 1)

9019553 (type 1)

9019553A (type 3)

0 10 20 30 40 50 60 70 80

CENSORING LEVEL, IN PERCENT

AVERAGE APPARENT ERROR FOR WATER-YEAR FLOWS, IN PERCENT -20

-15

-10

-5

0

5
12 AFINCH as a Tool for Evaluating the Effects of Network Size and Composition on Streamflow Estimates

Summary and Conclusions

A computer application called AFINCH (Analysis of 

Flows In Networks of CHannels) was developed by the 

U.S. Geological Survey (USGS) to facilitate the estimation of 

monthly water yields and streamflows at multiple locations 

throughout large regions. AFINCH uses stepwise-regression 

techniques to estimate monthly water yields from catchments 

based on geospatial-climate and land-cover data in combination with available streamflow and water-use data. Water 

yields from contributing catchments are multiplied by their 

contributing areas to compute streamflows, which are accumulated to compute streamflows in stream reaches referred to as 

flow lines. Water yields and flow estimates are constrained by 

AFINCH to ensure that observed streamflows are conserved at 

gaged locations.

October

November

December

January

February

March

April

May

June

July

August

September

WATER YEAR 1999

AVERAGE CENSORING LEVEL, IN PERCENT

PROBABILITY THAT ESTIMATE IS WITHIN +/- 10 PERCENT OF MEASURED FLOW

0.4

10 20 30 40 50 60 70 80

0.5

0.6

0.8

0.9

1

0.7

Bootstrapping techniques employing random subsampling were used with AFINCH to facilitate computation of 

selected statistical characteristics of monthly flow estimates 

at flow lines within the 0405 hydrologic subregion. This was 

done to gain insights into the effects of variation in streamflow-gaging-network size and composition on the accuracy 

and precision of streamflow estimates. Analyses were based 

on data for streamflow gages in the subregion that were operated in 1 or more water years from 1971 to 2003. Although 

AFINCH is capable of accounting for water uses, none were 

specified or considered in the analyses.

A program was developed and used to create 

30 streamflow-gaging-network subsets for each of several censoring levels. Thirty subsets each were created 

by randomly censoring (that is, removing) streamflow 

gages corresponding to removals of approximately 

Figure 8. Relation between censoring level and probability of estimating monthly flow within 10 percent of measured 

value for the water year 1999.
Summary and Conclusions 13

10, 20, 30, 40, 50, and 75 percent of the 75 streamflow gages 

that were operated in 1 or more years during the analysis 

period. AFINCH was then used to compute flow estimates 

based on each subset streamflow-gaging-network and a preselected set of potential explanatory variables (monthly total 

precipitation and monthly mean temperature). Estimates were 

aggregated by censoring level and results for six flow lines 

were analyzed to assess (a) how the size and composition of 

the streamflow-gaging network affected the average apparent 

errors and variability of the estimated flows and (b) whether 

results for certain months were more variable than for others.

Average pseudo coefficient of variation (CVp) ratios 

for water-year flows tended to increase as the network 

size decreased. For example, CVp ratios corresponding to a 

75-percent censoring level ranged from about 2 to 5 times the 

CVp ratio at the 10-percent censoring level. Results such as 

October

November

December

January

February

March

April

May

June

July

August

September

AVERAGE CENSORING LEVEL, IN PERCENT

WATER YEAR 2000

PROBABILITY THAT ESTIMATE IS WITHIN +/- 10 PERCENT OF MEASURED FLOW

0.4

10 20 30 40 50 60 70 80

0.5

0.6

0.8

0.9

1

0.7

Figure 9. Relation between censoring level and probability of estimating monthly flow within 10 percent of measured 

value for the water year 2000.

these indicate that less precise (that is, more variable) estimates resulted from smaller networks as compared to larger 

networks. When streamflow gages are present in the basin, 

CVp ratios tended to be smaller than comparable CVp ratios 

for flow lines in ungaged basins, probably due to variancestabilizing effects resulting from flow constraints imposed 

by AFINCH. Not surprisingly, this result indicates that the 

precision of AFINCH flow estimates at a given flow line will 

be improved by operation of one or more streamflow gages 

upstream and (or) downstream in the enclosing basin.

Monthly average CVp values were examined to assess 

whether streamflow estimates at flow lines typically were 

more variable in particular months or seasons of the year. 

No clear-cut indication of a seasonal trend in variability was 

evident. 
14 AFINCH as a Tool for Evaluating the Effects of Network Size and Composition on Streamflow Estimates

Average apparent percent errors (APEs) for annual flows 

tended to increase in absolute value with increasing censoring 

level. The flow lines that exhibited the smallest absolute average APEs and smallest changes in average APE as a function 

of increasing censoring level were both in ungaged basins. 

These counterintuitive results likely reflect both the basestreamflow estimate from which the APEs were determined 

and insensitivity in the average estimates to changes in the 

streamflow-gaging-network size and composition. The larger 

average APEs and increases in average APE with increased 

censoring observed for flow lines in gaged basins likely 

reflects a bias in the estimate (relative to the measured streamflow) resulting from relaxation of streamflow constraints when 

one or more streamflow gages in the basins were omitted from 

the network. 

Data for a flow line (9019553) that contained a streamflow gage were analyzed in a special fashion to assess the 

accuracy and precision of estimates produced only when the 

streamflow gage on the flow line was omitted from the calibration network. Precision characteristics of water-year flow 

estimates associated with the flow line when the streamflow 

gage was omitted from the calibration network were similar 

to those of flow lines in ungaged basins in that the overall 

slope of the relation between censoring level and average 

APE for water-year flows was relatively gentle. However, 

the magnitudes of the errors, in this case measured relative to 

the observed flows, was much larger than for other flow lines 

examined where the errors were measured relative to estimates 

produced using the full-gage complement. This last result 

indicates that the average percent errors for the other flow 

lines have the potential to be much larger than indicated by 

the average APEs were they reported relative to their true (but 

unknown) flows.

“Missing gage” analyses, based on examination of 

censored-subset results where the streamflow gage of interest 

was omitted from the calibration data set, were performed for 

all flow lines that contained a streamflow gage. Those analyses 

were done by examining only those monthly flow estimates 

for each gaged flow line that resulted from censoring subsets 

for which the given streamflow gage was omitted from the 

network. The percentage differences between the estimated 

and observed monthly streamflows (relative to the observed 

streamflows) for all of the flow lines gaged in a given water 

year were aggregated by censoring level and used to compute 

probabilities that the estimated monthly flows were within plus 

or minus 10 percent of the observed flows. Results examined 

for water years 1999 and 2000 indicated that the probability 

of AFINCH computing monthly streamflow estimates within 

10 percent of the true (measured) value decreased from greater 

than 0.9 at about a 10-percent network-censoring level to less 

than 0.6 as the censoring level approached 75 percent. In addition, estimates for typically dry months tended to be characterized by larger percent errors than typically wetter months. 

It is cost prohibitive to operate streamflow gages at all 

locations where data are desired; consequently, models like 

AFINCH are useful for extending the limited site-specific 

streamflow data that are available. Given that models like 

AFINCH will be used, it is important to understand the implications of streamflow-gaging-network size and composition 

on the accuracy and precision of flow estimates. The results 

described in this report are specific to both the AFINCH model 

and the hydrologic subregion analyzed; however, similar patterns in the accuracy and precision of flow estimates might be 

expected from any regression-based model that likewise constrains flow estimates on the basis of streamflow-gage data. In 

basins that contain one or more streamflow gages, the flow and 

water-use constraint features of AFINCH should frequently 

result in flow estimates that are more accurate than would be 

expected from a more traditional regression-based model without flow constraints. Although not tested, AFINCH’s ability to 

account for water uses may facilitate more accurate estimates 

and (or) the use of more streamflow-gage data for model calibration than more traditional regression-based models without 

water-use constraints. 

References Cited

Cunnane, C., 1978, Unbiased plotting-positions—A review: 

Journal of Hydrology, v. 37, p. 205–222.

Davidson, A.C., and Hinkley, D.V., 1997, Bootstrap Methods 

and their Applications: Cambridge University Press, 582 p.

Holtschlag, D.J., 2009, Application guide for AFINCH 

(Analysis of flows in networks of channels) described by 

NHDPlus: U.S. Geological Survey Scientific Investigations 

Report 2009–5188, 107 p., accessed October 14, 2009, at 

http://pubs.usgs.gov/sir/2009/5188/.

Horizon Systems, 2009, National Hydrography Dataset Plus—

NHDPlus home, accessed February 6, 2009, at http://www.

horizon-systems.com/nhdplus/.

Mason, R.R., and Yorke, T.H., 1997, Streamflow information 

for the Nation: U.S. Geological Survey Fact Sheet FS–006–

97, 4 p., accessed October 14, 2009, at http://pubs.usgs.gov/

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mathworks.com/products/matlab/.

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Koltun and Holtschlag—AFINCH as a Tool for Evaluating the Effects of Network Size and Composition on Streamflow Estimates—SIR 2010–5020