U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2006-5270

1970

1980

1990

2000

Forest

 Agriculture

 Urban

Methods for Adjusting U.S. Geological Survey Rural 

Regression Peak Discharges in an Urban Setting
Cover. Land use change in the Northwest Branch Anacostia River near Colesville, Maryland (USGS streamgage 

01650500). The images show the progression of urban development in this watershed over the period 

1970 to 2000.
Methods for Adjusting U.S. Geological 

Survey Rural Regression Peak Discharges 

in an Urban Setting

By Glenn E. Moglen and Dorianne E. Shivers

Scientific Investigations Report 2006-5270

U.S. Department of the Interior

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

Gale A. Norton, Secretary

U.S. Geological Survey

Mark D. Meyers, Director

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

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Telephone: 1-888-ASK-USGS

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Any use of trade, product, or firm names in this publication is for descriptive purposes only and does not imply 

endorsement by the U.S. Government.

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:

Moglen, G.E. and Shivers, D.E., 2006, Methods for Adjusting U.S. Geological Survey Rural Regression Peak Discharges 

in an Urban Setting: Reston, Virginia, U.S. Geological Survey, Scientific Investigations Report 2006-5270, 55 p.
iii

Contents

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

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

Previous Investigations........................................................................................................................2

Overview of Peak Discharge Adjustment Method...................................................................................4

Assumptions of Peak Discharge Adjustment Method....................................................................4

Enumeration of Steps in Peak Discharge Adjustment Method.....................................................4

Analytical Presentation of Adjustment Method .....................................................................5

Adjustment of the Observed Annual Maximum Time Series for Nonstationary 

Urbanization ....................................................................................................................7

The Adjustment Models..............................................................................................................7

Censoring of Streamages with Non-elevated Flood-Frequency Values ............................9

Example—the Calibrated 5-year Simple Imperviousness Adjustment Model ..................9

Methods.........................................................................................................................................................11

Development of Rural Peak Discharge Estimates ........................................................................11

Delineation of Watershed Boundaries ..................................................................................11

Estimation of Watershed Characteristics .............................................................................11

Development of Flood-frequency Estimates from Gage Record ................................................11

Selection of Gages ...................................................................................................................11

Observed Annual Maximum Time Series...............................................................................12

Use of the PeakFQ Program.....................................................................................................12

Skew ................................................................................................................................12

PeakFQ Run-Time Parameters .......................................................................................12

Identifying the High-Outlier Threshold .........................................................................12

Identifying the Historic Period for the Largest Flood in the Time Series.................13

Example Application of Outlier and Historical Period Identification Procedures...........13

Land-Use Change Time Series .........................................................................................................14

Population Density .............................................................................................................................14

Imperviousness...................................................................................................................................16

Estimating Imperviousness from Census Data ....................................................................16

Implementation of Historical Census Data ..................................................................16

Results ...........................................................................................................................................................18

Comparison to Urban Equations Developed by Sauer and others (1983)..................................32

Application of Calibrated Models to Streamgages with Local Urban Equations..............................34

Interpretation of Results.............................................................................................................................35

Model Strengths .................................................................................................................................35

Model Weaknesses ...........................................................................................................................36

Choosing a Model...............................................................................................................................36

Summary and Conclusions ........................................................................................................................37

Notation.........................................................................................................................................................38

Discharge Adjustment Model Definitions:...............................................................................................39

Acknowledgements.....................................................................................................................................40

References....................................................................................................................................................40
iv

Appendix 1.....................................................................................................................................................44

Appendix 2.....................................................................................................................................................48

Appendix 3.....................................................................................................................................................52

Appendix 4.....................................................................................................................................................53

Appendix 5.....................................................................................................................................................54

Appendix 6.....................................................................................................................................................55

Figures

Figure 1. Locations of urban and rural streamgages used in current (2006) study of rural 

regression peak discharges...................................................................................................3

Figure 2. Flow chart for the process for calibrating adjustment equations....................................6

Figure 3. The relation between observed and scaled imperviousness for the scaled vales of 

I*=12.1 and c4=0.2.....................................................................................................................8

Figure 4. Annual maximum time series for U.S. Geological Survey streamgage 02037800, Falling Creek near Midlothian, Virginia.....................................................................................14

Figure 5. Population density in Montgomery County, Maryland, as a function of census tract 

area from the 2000 census....................................................................................................15

Figure 6. Distribution of population density as a fraction of the total watershed area for U.S. 

Geological Survey streamgage 01645000, Seneca Creek at Dawsonville, Maryland.15

Figure 7. Relations of imperviousness to population density as measured by Stankowski 

(1972) and as measured in this study..................................................................................17

Figure 8. Estimated imperviousness from census tract data for U.S. Geological Survey 

streamgage 01645000, Seneca Creek at Dawsonville, Maryland. .................................17

Figure 9. Application of the (A) simple imperviousness model, (B) imperviousness distribution model, and (C) scaled imperviousness model to the ratio of urban to rural 

discharges as a function of return period and imperviousness at U.S. Geological 

Survey streamgage 01645000, Seneca Creek at Dawsonville, Maryland.....................28

Figure 10. Application of the (A) simple density model, (B) density distribution model, and (C) 

scaled density model to the ratio of urban to rural discharges as a function of return 

period and imperviousness at U.S. Geological Survey streamgage 01645000, Seneca 

Creek at Dawsonville, Maryland..........................................................................................29

Figure 11. Application of the (A) simple imperviousness model, (B) imperviousness distribution model, and (C) scaled imperviousness model to the ratio of urban to rural 

discharges as a function of return period and imperviousness at U.S. Geological 

Survey streamgage 01585200, West Branch Herring Run at Idlewylde, Maryland. ...30

Figure 12. Application of the (A) simple density model, (B) density distribution model, and (C) 

scaled density model to the ratio of urban to rural discharges as a function of return 

period and imperviousness at U.S. Geological Survey streamgage 01585200, West 

Branch Herring Run at Idlewylde, Maryland.....................................................................31
Tables

Table 1. Example of the adjustment of the observed annual maximum time series using 

the 5-year simple imperviousness adjustment model for U.S. Geological Survey 

streamgage 01645000, Seneca Creek at Dawson, Maryland..........................................10

Table 2. Number of gages used in the regression analysis as a function of return period......18

Table 3. Goodness-of-fit characteristics for the models evaluated. ............................................19

Table 4. Calibrated values of the adjustment model coefficients and exponents......................20

Table 5. Rankings of model performance based on the goodness-of-fit values shown in 

table 3.......................................................................................................................................21

Table 6. Summed rankings of model calibration statistics based on goodness-of-fit values 

given in table 3........................................................................................................................21

Table 7. Smoothed values of the adjustment model coefficients and exponents......................23

Table 8. Performance of U.S. Geological Survey urban equations from an earlier study and 

urban equations developed in this report. .........................................................................33

Table 9. Relative Standard error (Se /Sy ) for three imperviousness models developed in this 

report as normalized by the relative standard error determined by Sauer and others 

(1983). .......................................................................................................................................33

Table 10. Mean bias, in cubic feet per second, for new urban equations used on data from 

10 U.S. Geological Survey streamgages for which localized urban equations are 

available...................................................................................................................................34

Table 11. Relative standard error (Se /Sy ) for new urban equations used on data from 10 U.S. 

Geological Survey streamgages for which localized urban equations are available. ...

35

Table 12. Summed and ranked overall performance of new urban equations used on data 

from 10 U.S. Geological Survey streamgages for which localized urban equations 

are available............................................................................................................................35
vi

Conversion Factors

Inch/Pound to SI

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

Area

square mile (mi2) 259.0 hectare (ha)

square mile (mi2) 2.590 square kilometer (km2) 

Flow rate

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

Hydraulic gradient

foot per mile (ft/mi) 0.1894 meter per kilometer (m/km)

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

°F=(1.8×°C)+32

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

°C=(°F-32)/1.8

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 

(NAVD 88).

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).

Altitude, as used in this report, refers to distance above the vertical datum.

The following SI unit of measure is given in this report in reference to impervious mapping 

resolution.

Multiply By To obtain

Length

meter (m) 3.281 foot (ft)
Methods for Adjusting U.S. Geological Survey Rural 

Regression Peak Discharges in an Urban Setting

By Glenn E. Moglen and Dorianne E. Shivers1

1

 University of Maryland, Department of Civil and Environmental Engineering, College Park, MD 20742.

Abstract

A study was conducted of 78 U.S. Geological Survey gaged streams that have been subjected to varying degrees of 

urbanization over the last three decades. Flood-frequency analysis coupled with nonlinear regression techniques were used to 

generate a set of equations for converting peak discharge estimates determined from rural regression equations to a set of peak 

discharge estimates that represent known urbanization. Specifically, urban regression equations for the 2-, 5-, 10-, 25-, 50-, 

100-, and 500-year return periods were calibrated as a function of the corresponding rural peak discharge and the percentage 

of impervious area in a watershed. The results of this study indicate that two sets of equations, one set based on imperviousness and one set based on population density, performed well. Both sets of equations are dependent on rural peak discharges, a 

measure of development (average percentage of imperviousness or average population density), and a measure of homogeneity 

of development within a watershed. Average imperviousness was readily determined by using geographic information system 

methods and commonly available land-cover data. Similarly, average population density was easily determined from census 

data. Thus, a key advantage to the equations developed in this study is that they do not require field measurements of watershed 

characteristics as did the U.S. Geological Survey urban equations developed in an earlier investigation.

During this study, the U.S. Geological Survey PeakFQ program was used as an integral tool in the calibration of all equations. The scarcity of historical land-use data, however, made exclusive use of flow records necessary for the 30-year period 

from 1970 to 2000. Such relatively short-duration streamflow time series required a nonstandard treatment of the historical data 

function of the PeakFQ program in comparison to published guidelines. Thus, the approach used during this investigation does 

not fully comply with the guidelines set forth in U.S. Geological Survey Bulletin 17B, and modifications may be needed before 

it can be applied in practice.

Introduction

The U.S. Geological Survey (USGS) maintains and publishes a set of rural regression equations for use in estimating peak 

discharges of varying return periods for each State in the United States (U.S.; Jennings and others, 1994; Ries and Crouse, 

2002). Sauer and others (1983) developed a method for transforming these rural discharge estimates to estimates for urban 

watersheds based on several watershed characteristics, most notably two different indices of urbanization—the basin development factor (BDF) and the percentage of impervious area. A study was conducted to reassess the method of Sauer and others 

(1983) by taking advantage of 20 additional years of streamflow data, using geographic information system (GIS) data and 

techniques, and considering new approaches in the development of a set of urban regression equations. As in the earlier study, 

the goal of this study was to develop flood-frequency equations for urbanized sites by applying urban adjustments to the T-year 

flood values obtained from the regional equations for rural ungaged sites. The T-year flood is the flood that, on average, is 

equaled or exceeded once in any given number (T) of years.

The USGS typically conducts flood-frequency analyses consistent with Bulletin 17B (Interagency Advisory Committee on 

Water Data, 1982). For the study, the USGS flood-frequency analysis program, PeakFQ, was incorporated into an optimization 

program to calibrate several sets of urban regression equations. Because of limited-duration streamflow time series used in the 

analysis, however, the approach used in this study for the historical period input to the PeakFQ program does not comply with 

Bulletin 17B guidelines.
Previous Investigations

The effects of land-use changes on stream discharge are well documented (for example Carter, 1961; James, 1965; Viessman, 1966; Leopold, 1968; Andersen, 1970). The effects of urbanization, in particular, typically are related to characteristic 

changes in watersheds. Urbanization generally is associated with increased areas of impervious surfaces, such as pavement 

and rooftops. These surfaces usually have drainage features that quickly convey water away from structures or road surfaces 

and into nearby streams. In addition, urbanization generally reduces temporary stormwater storage, such as depression storage 

and tree-leaf interception. These urbanization processes create large volumes of stormwater that generally travel swiftly to the 

surface-drainage network and result in high flood peaks, which are the focus of this study.

Sauer and others (1983) presented a regression-equation approach of using rural flood-frequency estimates, such as those 

resulting from the equations summarized by Jennings and others (1994), and scaling these estimates upwards based on several 

measures of urbanization. Such rural flood-frequency estimates are currently (2006) calculated using the USGS National FloodFrequency Program (version 3; Ries and Crouse, 2002).

Although Sauer and others (1983) calibrated several different sets of adjustment equations, their best model included seven 

parameters as presented in the following form:

, (1)

where UQT is the T-year urban peak discharge, in cubic feet per second;

	 A	 is the drainage area, in square miles;

	 SL is the main channel slope, in foot per mile;

	 RI2 is the 2-hour, 2-year rainfall, in inches;

	 ST is the basin storage in percent;

	 BDF is the basin development factor (an urbanization index that quantifies channel improvements, channel linings, 

storm drains/sewers, and curb-and-gutter streets);

	 IA is the impervious area, in percent; and

	 RQT is the T-year rural discharge as predicted from the appropriate USGS rural-regression equation.

(Note: The difficulty in determining the BDF index based solely on remotely sensed data was a motivating factor for conducting 

this study.)

In some States, the USGS has developed localized urban equations that supersede the use of the national urban equations 

developed by Sauer and others (1983; shaded areas in fig. 1). The localized urban equations vary in sophistication, but all 

generally require some measure of urbanization, such as the percentage of impervious or residential area. The general equations

for the States of New Jersey (Stankowski, 1974), Pennsylvania (Stuckey and Reed, 2000), and Wisconsin (Conger, 1986) are 

presented here to illustrate the available range of localized urban equations.

New Jersey:

, (2)

where QT is the T-year urban peak discharge, in cubic fet per second; 

	 A is the drainage area, in square miles; 

	 SL is the channel slope, in foot per mile; 

	 ST is the basin storage, such as lakes and swamps, in percent; and 

	 IA is the impervious area, in percent, based on population density.

Pennsylvania (Region A):

, (3)

where QT is the T-year urban peak discharge, in cubic feet per second; 

	 A	 is the drainage area, in square miles; 

	 F	 is forest cover, in percent; 

	 U is urban development, in percent; 

	 C	 is the watershed, in percent, underlain by carbonate rock; and 

	 CA is the watershed, in percent, controlled by lakes, swamps, or reservoirs.

 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Figure 1. Locations of urban and rural streamgages used in current (2006) study of rural regression peak discharges. [Shaded states 

have urban prediction regression equations that apply either statewide or in selected areas.]

Wisconsin (Milwaukee County urban areas):

, (4)

where QT is the T-year urban peak discharge, in cubic feet per second; 

	 A is the drainage area, in square miles; and 

	 IA is the impervious area, in percent. 

Gages with only rural indicators

Gages with urban indicators

Rural only

Rural and urban

Regression equations available

Heading 1 
Overview of Peak Discharge Adjustment Method

The central premise of the study is that observed peak discharges at USGS streamgages are a function of watershed characteristics that exist at the time of the observed flood. Specifically, urbanization, measured as impervious area, varies over time 

and needs to be properly accounted for when performing flood-frequency analysis on an observed annual maximum time series.

Assumptions of Peak Discharge Adjustment Method

Assumption 1: The USGS rural regression equations for a State or region represent the best estimate of rural discharges 

for that particular location.

Assumption 2: In a watershed undergoing urbanization, the observed flood peak in any given year (t) depends on the 

amount of impervious area in the watershed. Because urbanization is an ongoing process, it may be necessary to obtain 

annual time series of impervious areas in rapidly changing watersheds.

Assumption 3: At streamgages influenced by urbanization, the observed annual maximum time series can be adjusted 

downward to be consistent with the predicted flood frequency obtained by applying the USGS rural regression equations 

for the gage location.

Assumption 4: The equations used to adjust the observed annual maximum time series can be inverted for use in scaling a USGS rural flood-frequency regression equation upward to account for urbanization effects. (Note: The objective 

function of a regression equation can be minimized only in the form in which it was calibrated, and the inversion process 

may cause a loss in optimality of the objective function.)

Assumption 5: Because calibrated coefficients and exponents vary with each return period, separate optimizations are 

required for each return period considered. This assumption is consistent with the approach in Sauer and others (1983), 

which produced different coefficients and exponents for each return period.

Assumption 6: Stormwater-management methods either are not present or do not have a measurable effect on flooding 

caused by urbanization. Although stormwater-management methods can mitigate increased flooding caused by urbanization, much of the urbanization quantified in this report predates the implementation of stormwater-management methods. 

Therefore, this report does not attempt to quantify any effects from implementing stormwater-management methods.

Enumeration of Steps in Peak Discharge Adjustment Method

The general steps in the adjustment method steps are described here. Because this is a calibration exercise, the process 

outlined here is iterative and is dependent on the minimization of an objective function that quantifies the difference between 

the observed rural flood-frequency values and the flood-frequency results obtained by using the PeakFQ program and adjusted 

annual maximum discharge data for urban streams.

Step 1. Assume (a) a functional form for an adjustment equation and (b) initial trial values for the coefficients and exponents of this adjustment equation. 

Step 2. Adjust the annual maximum time series for the selected gage (i) as a function of the observed annual maximum 

time series and the impervious area time series. This produces an adjusted annual maximum time series.

Step 3. Write the adjusted annual maximum time series in the USGS National Water Data Storage and Retrieval System 

(WATSTORE) format.

Step 4. Use the PeakFQ program to determine the flood-frequency distribution at the selected gage (i) for the adjusted time 

series.

Step 5. Scan the output of the PeakFQ program for the T-year flood and incrementally adjust the objective function by the 

squared difference between the PeakFQ T-year flood and the rural regression value for the same flood-frequency. 

(Note that these differences will be in the logarithmic space to give similar weight to both small and large watersheds.)

Step 6. Modify the trial coefficients and exponents assumed in Step 1.

•

•

•

•

•

•

 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Steps 2 through 6 are repeated for all gages in the database and then performed iteratively until the objective function 

described in Step 5 is minimized. The entire process is conducted separately for the 2-, 5-, 10-, 25-, 50-, 100-, and 500-year 

flood-frequencies. A detailed flow chart of the general process described here is shown in figure 2. (Note: Steps 2-6 are automated by a nonlinear optimization program modified from McCuen (1993). The objective function that is minimized is the sum 

of the squared errors between the PeakFQ output for the T-year flood and the equivalent T-year flood estimated by the rural 

regression equation.)

After an equation is calibrated for each return period, the calibrated urban equation is determined in the following step (7):

Step 7. Substitute the calibrated coefficients from the urban-to-rural adjustment function (f-inverse) into the original ruralto-urban function (f).

From the user’s standpoint, there likely is only an interest in the actual use of the function f from Step 7. The user’s experience will be limited to the following two steps:

Using the National Flood-Frequency (NFF) program (Ries and Crouse, 2002) or direct application of the USGS 

rural regression equation, estimate the rural flood-frequency for the ungaged location; and

Use the rural-to-urban function (f), transform rural T-year floods (from the rural regression equations previously 

determined) into corresponding T-year urban floods.

A critical challenge is the determination of the appropriate annual maximum discharge adjustment equation. The approach 

used during this study was to explore several functional forms and select the form that produced the best goodness-of-fit statistics across all frequencies.

Analytical Presentation of Adjustment Method

Pursuant to Bulletin 17B (Interagency Advisory Committee on Water Data, 1982), FF(x, Qo, HP, SS) is defined as the 

flood-frequency operator. This operator returns the flood-frequency vector, (Qff for the 2-, 5-, 10-, 25-, 50-, 100-, and 500-year 

floods) for a time series of annual maximum floods (x), for an identified high-outlier discharge threshold (Qo), for a historical 

return period associated with the largest flood exceeding Qo (HP), and assuming the station skew option (SS). An individual 

member of the flood-frequency vector, say the T-year flood, is identified as Qff(T) or QT.

The observed annual maximum time series is defined as Qobs(t). The adjusted annual maximum time series is defined as 

Qadj(t). Performing flood-frequency analyses on these two time series produced the flood-frequency vectors, Qff,obs and Qff,adj, 

respectively. Thus, the resulting equations take the following forms:

, (5)

and

, (6)

where HPobs and HPadj are the return periods associated with the largest floods in the respective annual maximum time series 

exceeding Qo. For simplification, the dependency of the flood-frequency operator on a high-outlier threshold, a specified historical period, and a skew option was omitted from the notation. Thus, equations 5 and 6, respectively, reduce to simpler forms: 

, (7)

and

 . (8)

Two other flood-frequency vectors were relevant to this study—Qff,rural and Qff,urban, which are the rural and urban floodfrequency vectors, respectively. Because these vectors are determined from regression equations, they exist only as flood-frequency values and have no time series directly associated with them.

1)

2)

Overview of Peak Discharge Adjustment Method 
Figure 2. Flow chart for the process for calibrating adjustment equations.

Start

Initialize gage information. For each gage, i, enter:

1. Rural flood frequency (from USGS rural equations)

2. Annual maximum time series (from peak flow file)

3. Annual imperviousness series (determined from census data)

Set P=1 (first return period)

[return periods considered: 2, 5, 10, 25, 50 100, 500 years]

Initialize adjustment equation

coefficients and exponents

for return period, P

Set S=0 (sum is 0)

Set i=1 (first gage)

Adjust annual maximum

time series for gage i

Write WATSTORE file

for gage i

Execute PeakFQ

Read PeakFQ output: extract

flood frequency for gage i

For return period, P:

S=S+(Qadj,P – RQP)2

All gages

Calculated?

S minimized?

All return periods,

P, determined?

Increment to next gage:

i=i + 1

(loop over all gages)

Revise adjustment equation

coefficients and exponents

(optimize goodness-of-fit)

Increment to next return period:

P=P + 1

(loop over all return periods)

Record adjustment equation

coefficient and exponent values

and goodness-of-fit statistics

End

Yes

No

No

Yes

No

Yes

   Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Adjustment of the Observed Annual Maximum Time Series for Nonstationary Urbanization

A central premise in the study by Sauer and others (1983) was that the observed flood-frequency record for a selected 

streamgage was stationary; in othet words, the record was unaffected by changes in urbanization over the period of record 

analyzed. To compensate for this premise, Sauer and others (1983) applied the rule that a streamgage record was usable if the 

amount of impervious area in the watershed had not increased more than 50 percent during the period of record.

In the current (2006) study, a different approach was taken. As discussed previously, an annual time series of impervious 

area, IA(t), was developed for each gage studied in order to relate a unique value of impervious area to each observed annual 

maximum flood value.

The observed and adjusted annual maximum time series are related by a set of nonlinear functions, f

T

(.), such that

, (9)

where Qadj,T(t) is the adjusted time series of annual maximum flows that is keyed to the T-year flood. Each of the seven equations for the 2-, 5-, 10-, 25-, 50-, 100-, and 500-year floods (f

2

, f

5

, f

10, f25, f50, f100, and f500, respectively) must be calibrated.

When the flood-frequency operator is applied to each adjusted time series, it produces the rural flood-frequency value for 

the return period, (T):

. (10)

Once all f

T

(.) are determined and assumption 4 was applied, these functions were inverted such that the independent variables 

were the rural flood-frequency values and impervious area, and the dependent variable was the urban flood-frequency value:

, (11)

where (.) is the inverted form of f

T

(.).

The Adjustment Models

Several forms of f

T

(.) were investigated—the null model, the simple imperviousness model, the simple density model, the 

imperviousness distribution model, the density distribution model, the scaled imperviousness model, and the scaled density 

model. These model forms are represented by the following equations (12 - 18, respectively).

Null model:

. (12)

Simple imperviousness model:

. (13)

Simple density model:

. (14)

Imperviousness distribution model:

. (15)

Density distribution model:

. (16)

Overview of Peak Discharge Adjustment Method 
Scaled imperviousness model:

. (17)

Scaled density model:

. (18)

The “distribution” models (eqs. 15 and 16) deserve a brief discussion. The terms ΔIA(t) and ΔPD(t) represent the difference between the 10th and 90th percentiles of the impervious area and population density, respectively. These terms quantify the 

homogeneity or uniformity of development within a watershed. Watersheds with relatively small values of either of these terms 

are considered to be uniform. Consideration of uniformity of development is helpful in predicting flood behavior.

The “scaled” models (eqs. 17 and 18) also deserve a brief explanation. The term “scaled imperviousness” varies nonlinearly from 1 to 100 as IA increases from 0 to 100 percent. For IA = I*, the scaled imperviousness equals 50.5. The scaled 

imperviousness function for typical parameter values is shown in figure 3. Scaled imperviousness is used because it could be 

argued that each incremental increase in imperviousness may not have an equivalent effect on flood-frequency. In figure 3, the 

effect of imperviousness increases most rapidly within the range of 5 to 15 percent imperviousness. This is consistent with studies of the effects of imperviousness on ecological health (for example, Schueler, 1987).

100

90

80

70

60

50

40

30

20

10

0

Scaled Imperviousness, in percent

Observed Imperviousness, in percent

0 5 10 15 20 25 30 35 40

Range of observed imperviousness

Figure 3. The relation between observed and scaled imperviousness for the scaled vales of I*=12.1 and c4=0.2.

 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Censoring of Streamages with Non-elevated Flood-Frequency Values

Using PeakFQ, a preliminary flood-frequency analysis was performed for the observed annual maximum time series at 

each of the 78 streamgages. These analyses were conducted identically to those that are conducted in the optimization model 

with respect to identifying high outliers and determining the length of the historical record (discussed later). The flood-frequency results from these preliminary analyses were compared with the rural regression flood-frequency values described 

later in this report. If, at a specific streamgage, the flood-frequency from the observed annual maximum time series produced 

estimates for the T-year flood such that

(19)

this gage was eliminated from the database for the development of the T-year regression equation. The rationale for such data 

elimination is that if a streamgage does not have an increase in flood-frequency values relative to the rural regression equations 

for peak flows, then it does not make sense to use such a gage to calibrate an urban-scaling factor. The number of gages censored from the analysis will be presented later in the “Results” section.

Example—the Calibrated 5-year Simple Imperviousness Adjustment Model

As an example of the flow-adjustment process, consider the calibrated form for the 5-year, simple imperviousness adjustment model determined in this study:

. (20)

Included in table 1 are the imperviousness time series, IA(t), the observed annual maximum time series, Qobs(t), and the 5-year 

adjusted annual maximum time series, Qadj,5(t) for USGS streamgage 01645000, Seneca Creek at Dawsonville, Maryland. The 

far right column of the table contains the ratios of the adjusted discharges to the observed discharges. Notice that the ratios are 

less than 1 for all by the three largest discharges in 1971, 1972, and 1975, indicating that the adjusted annual maximum time 

series generally is less than the observed annual maximum time series, which would be expected if the adjusted series approximates the comparatively small rural discharges that would have occurred in these years had urbanization not occurred. The 

adjustment ratios also are clearly smaller for smaller discharges and approach (and slightly exceed) 1 as the discharges become 

large. This indicates that relatively small discharges that correspond to more frequent 2- and 5-year flows are subject to adjustment from the effects of imperviousness. Such adjustments are consistent with the expectation that urbanization has the greatest 

effect on flood flows from storms of small magnitude. Further, in general, note that the adjustment ratios decrease with time, 

indicating the need to adjust the more recent discharges that correspond to greater urbanization in the watershed.

The behavior of the adjustment equation can be observed clearly by comparing years 1973 and 1999 (table 1). In 1973, the 

peak discharge was 3,020 ft3/s, and imperviousness was about 5.6 percent. The adjusted discharge for 1973 is 2,397 ft3/s. In 

comparison, the peak discharge in 1999 was 3,060 ft3/s, and imperviousness was 12.8 percent. The adjusted discharge for 1999 

is 2,144 ft3/s. Although the observed discharge in 1999 is only slightly greater (40 ft3/s) than the observed discharge in 1973, 

the adjusted discharge for 1973 is greater by 253 ft3/s than the adjusted discharge for 1999. This clearly indicates the role of the 

annual imperviousness time series in this study. It is worth noting that imperviousness in the watershed between 1970 and 2000 

increased from 4.8 percent to almost 13 percent, which represents more than a doubling of impervious area. For this reason, 

USGS streamgage 01645000 would have been eliminated from the Sauer and others (1983) study because their threshold of a 

50-percent increase in imperviousness.

Finally, it is instructive to look at the 5-year flood-frequency values and compare these to the 5-year rural regression 

equation value of 5,876 ft3/s determined from Dillow (1996). While the observed 5-year peak discharge is 1.27 times the rural 

regression estimate, the adjusted 5-year peak is only 1.05 times this estimate. The similarity of the adjusted 5-year peak to the 

rural regression estimate is indicative of the success of the calibrated adjustment equation in quantifying the effect of imperviousness on the annual maximum discharge. 

Overview of Peak Discharge Adjustment Method 
Table 1. Example of the adjustment of the observed annual maximum time series using the 5-year simple imperviousness adjustment 

model for U.S. Geological Survey streamgage 01645000, Seneca Creek at Dawson, Maryland.

[%, percent; ft3/s, cubic feet per second]

Year Imperviousness

(%)

Observed

Q (ft3/s)

Adjusted

Q (ft3/s)

Qadj,5(t)

/Qobs(t)

1970 4.84 2,200 1,702 0.774

1971 5.10 25,900 28,787 1.111

1972 5.37 26,100 28,831 1.105

1973 5.63 3,020 2,397 0.794

1974 5.90 3,160 2,508 0.794

1975 6.16 16,000 16,094 1.006

1976 6.42 4,900 4,101 0.837

1977 6.69 3,770 3,016 0.800

1978 6.95 7,850 6,969 0.888

1979 7.22 16,000 15,716 0.982

1980 7.48 10,800 9,946 0.921

1981 7.85 1,340 896 0.668

1982 8.23 3,160 2,385 0.755

1983 8.60 3,260 2,455 0.753

1984 8.98 3,010 2,225 0.739

1985 9.35 3,620 2,734 0.755

1986 9.73 1,070 669 0.625

1987 10.10 4,950 3,871 0.782

1988 10.48 7,410 6,120 0.826

1989 10.85 8,250 6,886 0.835

1990 11.23 2,270 1,553 0.684

1991 11.40 5,120 3,947 0.771

1992 11.58 1,750 1,146 0.655

1993 11.76 3,350 2,412 0.720

1994 11.93 9,160 7,651 0.835

1995 12.11 2,080 1,388 0.667

1996 12.28 11,000 9,399 0.854

1997 12.46 3,880 2,829 0.729

1998 12.64 5,280 4,023 0.762

1999 12.81 3,060 2,144 0.701

2000 12.99 1,910 1,244 0.651

Q5 7,435 6,157

10 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Methods

The database used for this investigation contains watershed characteristics for 78 USGS streamgages across the United 

States (Appendix 1). The two most critical elements in this database are the flood-frequency values obtained directly from the 

gage record and the flood-frequency values obtained from the current USGS rural discharge equations that apply at each gage 

site. In this section, the method for determining these two sets of values for the gages used in this investigation is discussed. 

Other ancillary data, such as outlier values, development of impervious-cover time series from census data, and determination of 

watershed characteristics also are discussed.

Development of Rural Peak Discharge Estimates 

The premise of the current study is that the USGS rural flood-frequency regression equations for each State represent the 

best estimate of the nonurbanized flood-frequency at any gage analyzed in the study. Thus, it is necessary to apply these equations to each gage included in the study and use the flood-frequency results in the development of the urban adjustment methods. 

Delineation of Watershed Boundaries 

For each gage used in the study, the digital elevation model (DEM) covering the extent of the gaged watershed was 

obtained from the National Elevation Dataset (NED; U.S. Geological Survey, 2004a). Flow directions were constrained by 

“burning in” stream locations from the 1:100,000-scale National Hydrography Dataset (NHD; U.S. Geological Survey and U.S. 

Environmental Protection Agency, 2004). Standard GIS techniques were used to fill depressions and determine flow directions 

using a D8 algorithm (Jenson and Domingue, 1988). The watershed boundaries were delineated automatically by indicating the 

location of the streamgage to the GIS and initiating the delineation algorithm. An automated delineation is considered acceptable and consistent with USGS standards if the delineated drainage area is within 10 percent of the drainage area reported by the 

USGS (U.S. Geological Survey, 2004b). 

Estimation of Watershed Characteristics 

Using the watershed boundaries generated as described in the previous section other watershed characteristics were determined as needed from additional interpretation of the DEM (for example, calculating the channel slope). If additional landuse and land-cover descriptors were needed, such as percentage of forest cover, these data were determined by using the most 

current land-use and land-cover data available. In most cases this meant using the 1992 National Land Cover Dataset (NLCD; 

Vogelmann and others, 1998a, 1998b; U.S. Geological Survey, 2005), although the 2001 NLCD (Homer and others, 2004; 

Multi-Resolution Land Characterization Consortium, 2005) was available for some locations during this study.

Development of Flood-frequency Estimates from Gage Record

Flood-frequency estimates were developed from the observed annual maximum time series recorded as peak streamflow 

by the USGS (U.S. Geological Survey, 2004b). In this section, the methods will be discussed for selecting the gages used in the 

analysis, obtaining the observed annual maximum time, and using the PeakFQ program (Flynn and others, 2005) to analyze the 

data and produce the flood-frequency values.

Selection of Gages 

The following list of criteria was used for selecting gages for this study:

The gage must have been active for 20 or more years during 1970-2000, the first full period for which census population 

data are available in digital format.

The gage must not be subject to flow regulation.

The gage must be equipped to report measured peak discharges, not just stage measurements.

The current peak-flow regression equations for the gage cannot be dependent on a measure of urbanization for flood 

prediction.

•

•

•

•

Methods 11
The drainage area for the gage, as determined by GIS, must be within plus or minus 10 percent of the reported USGS 

drainage area for the gage, thus providing a high degree of confidence in the application of GIS methods for determining 

the rural discharge estimates.

The gage record must not have a large proportion (approximately 20 percent) of streamflow values identified as outliers 

by the PeakFQ program.

The gage drainage area should contain more than 5 percent impervious area as quantified by the NLCD 2001 (MultiResolution Land Characteristics Consortium, 2005).

Applying these criteria resulted in the final selection of 78 gages in 12 States for the study analyses. The States in which 

the gages are located are California, Florida, Illinois, Maryland, Massachusetts, Michigan, Mississippi, Nevada, New York, 

Rhode Island, Virginia, and West Virginia (fig. 1).

Observed Annual Maximum Time Series

The observed annual maximum time series was obtained from the USGS (U.S. Geological Survey, 2004b). Observed peak 

discharge values for each study gage were obtained for all recorded floods occurring between 1970 and 2000. Data were collected in WATSTORE format, which can be read directly by the PeakFQ program.

Use of the PeakFQ Program

The USGS developed the PeakFQ computer program (Flynn and others, 2005) to conduct flood-frequency analyses based 

on the guidelines in Bulletin 17B (Interagency Advisory Committee on Water Data, 1982). The PeakFQ program uses the sample-moment method to fit the Pearson Type III frequency distribution to the logarithms of annual flood peaks. The skew that is 

used may be (a) user-developed generalized skew for a region, (b) the skew map from Bulletin 17B, (c) computed from the data, 

or (d) weighted between the generalized skew and station skew computed from the data. Adjustments can be made for high and 

low outliers and historic information. Qualification codes can be used to censor data from the analysis.

Skew

The PeakFQ program offers several options for defining a skew value to use in developing a flood-frequency estimate. In 

developing rural flood-frequency estimates for a given State, a regionalized skew typically is developed that is weighted with the 

local skew at the gaging station being analyzed. For urban estimates, this method is problematic because of fewer urban watersheds from which to develop a regionalized value and the varying extent of urbanization. In this study, therefore, each gaging 

station was considered individually and the “station skew” option was used throughout the analyses presented in this report.

PeakFQ Run-Time Parameters 

Because of the relatively short periods of record (31 years or less) for the flood-frequency analysis, there is a greater possibility of time-sampling errors than may be expected when examining longer periods of record. To reduce the potential effects 

of these errors, a systematic method for applying the high-outlier option of the PeakFQ program was applied. To apply this 

option, however, two quantities must be defined—the discharge threshold that defines the minimum high-outlier discharge and 

the return frequency of the largest discharge in the systematic record analyzed. Because of the nature of the model calibration 

process, the procedures outlined here were performed repeatedly (for each iteration, for each gage, for each return period examined) in calibrating various trial-adjustment models. 

Identifying the High-Outlier Threshold 

The high-outlier threshold is the largest discharge that is considered within the normal range of discharge for a particular 

time series being analyzed. Discharges greater than this threshold were not attributed a probability using the Weibull plotting position formula in the analyses. The high-outlier discharge threshold was determined by the following method: given an 

annual maximum time series of n discharges at a particular gage, it was arbitrarily decided that these n values would represent 

the flood-frequency characteristics for that location for a period four times as long. Although this is an arbitrary assumption, it 

adequately produced the desired effect of filtering out large extreme floods that far exceeded the periods of record being analyzed while allowing the consideration of moderate-sized floods. Thus, the approach for identifying outlier discharges for this 

study is not consistent with the approach outlined in Bulletin 17B; however, this outlier-identification approach was a repeatable 

criterion that worked effectively and could be applied uniformly.

•

•

•

12  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
The discharges determined from the rural flood-frequency regression equations were used to determine the high-outlier 

threshold. The flood-frequency distribution was treated as a piecewise log-normal distribution, and the high-outlier threshold 

was determined by linear interpolation within the bounds of these values by applying the following equation:

, (21)

where Tl < 4n ≤ Tu and z(Tx) is the standard normal deviate corresponding to a flood of return period, Tx. Thus, the high-outlier 

threshold (Qo), is determined simply as

. (22)

Identifying the Historic Period for the Largest Flood in the Time Series

In a similar manner, the largest flood in the systematic record was assigned a value for the historic period that it represents 

given the rural regression flood-frequency equation estimates that apply to a specific watershed. The largest flood was compared 

to the rural flood-frequency distribution, and a return period was determined by interpolating from this distribution, thus treating 

the flood-frequency distribution as a piecewise log-normal distribution. If the largest flood in the systematic record exceeded 

the 500-year rural flood, a return period of 500 years was arbitrarily assigned to the flood. The interpolation was performed as 

follows:

, (23)

where Ql < Qo ≤ Qu and z(Qx) is the standard normal deviate corresponding to a flood of magnitude, Qx. With z(Qo) known, the 

cumulative probability (P(Qo)) associated with this value was determined from the standard normal distribution. Finally, the 

return period (T(Qo)) was determined by using

. (24)

Example Application of Outlier and Historical Period Identification Procedures

To illustrate the application of the outlier and historical period identification procedures, the procedures were applied to the 

record of USGS streamgage 02037800, Falling Creek near Midlothian, Virginia. The period of record for this gage is 1951-

2003, with several breaks in the record. Because of limited census data, only the period 1970-2000 was considered. Further, the 

gage was inactive in 1978 and again from 1994 to 2000; therefore, only the records for the periods 1970-1977 and 1979-1993 

were available for this study. The largest flood during these two periods had a magnitude of 5,170 ft3/s and occurred in 1979. 

The next largest flood had a magnitude of 1,400 ft3/s and occurred in 1985 (fig. 4).

The rural peak discharges determined from Bisese (1995) for the Southern Piedmont region produced peak-discharge 

estimates for the streamgage at Falling Creek of 380 ft3/s, 647 ft3/s, 862 ft3/s, 1,217 ft3/s, 1,552 ft3/s, 1,923 ft3/s, and 3,054 ft3/s 

for the 2-, 5-, 10-, 25-, 50-, 100-, and 500-year floods, respectively. To identify the high-outlier threshold (Qm) and determine 

the return period corresponding to the largest flood, the following criteria were applied. The length of the time series used in the 

analysis was n=23 years, representing the 1970-1977 and 1979-1993 periods of record. Using 4n=92 years, the log-normally 

interpolated 92-year flood from equations 21 and 22 is 1,876 ft3/s. No interpolation is necessary to determine the return period 

corresponding to the largest flood for either the observed or adjusted (discussed later) annual maximum time series because the 

observed (5,170 ft3/s) and adjusted (4,220 ft3/s) values exceed the rural 500-year flood of 3,054 ft3/s. A 500-year return period is 

arbitrarily attached to the largest flood during the periods of record analyzed. Thus, the PeakFQ program was executed for both 

the observed and adjusted annual maximum time series using station skew, high-outlier threshold (1,876 ft3/s), and historical 

period (500 years).

Methods 13
Land-Use Change Time Series 

This study draws heavily on the idea of developing a time series of changes in the spatial characteristics of the landscape. 

As a single descriptor of land-use change, imperviousness is the most simple characteristic that can be identified as having a 

strong influence on flood magnitude. Historical records of imperviousness are scarce, however, in both the spatial and temporal 

dimensions. For this reason, historical census-derived population-density estimates were used as a surrogate of imperviousness 

and as a predictor of imperviousness from which to develop the necessary space-time estimates of the changing landscape. 

Population Density 

Census data are readily available in 10-year increments and digital format for 1970 through 2000. Assuming linear changes 

in population density between incremental census data, it is possible to develop space-time estimates of population density for 

any location in the United States at the resolution of a census tract. Census tracts vary considerably in scale depending on population density. For this study, the focus was on urban areas for which the census data tend to delineate small-scale census tracts. 

For example, Montgomery County, Maryland, is a densely populated county directly north of Washington, D.C. In the 2000 

census, this county was composed of 177 census tracts (GeoLytics, 2003), ranging in size from 0.13 mi2 to 70 mi2 and averaging 

2.86 mi2. Population density is not uniformly distributed among these tracts, however, and smaller tracts tend to have greater 

population densities, as shown in figure 5.

Landscape changes, such as increases in imperviousness, introduction of curb and gutter drainage, and channelization can 

vary with population density. Using a single, average population-density value and the landscape changes for which it is a surrogate to quantify human influences within a watershed may be a poor model when a watershed is large or if human influences 

vary considerably. To quantify such variability or nonhomogeneity within a watershed, the distribution of population densities 

was quantified as a fraction of the total watershed area as shown in the example for USGS streamgage 01645000, Seneca Creek 

at Dawsonville, Maryland (fig. 6). The population densities that correspond to the 10th and 90th percentiles of the watershed area 

were tagged, and the difference between these densities was recorded. This quantity (∆PD) was defined as follows: 

. (25)

1960 1970 1980 1990 2000

DATES: 06/01/1951 to 09/18/2003

Streamflow, in cubic feet per second

5000

4000

3000

2000

1000

0

Figure 4. Annual maximum time series for U.S. Geological Survey streamgage 02037800, Falling Creek near Midlothian, Virginia.

14  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Figure 5. Population density in Montgomery County, Maryland, as a function of census tract area from the 2000 census.

12

10

8

6

4

2

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fraction of Area

Population Density, in thousands per square mile

∆PD

Figure 6. Distribution of population density as a fraction of the total watershed area for U.S. Geological Survey streamgage 01645000, 

Seneca Creek at Dawsonville, Maryland. [∆PD is the difference in population density between the 10th and 90th percentiles of the 

distribution of population density in the watershed.]

102

101

100

10-1

10-2

10-1 100 101 102

Population Density, in thousands per square mile

Census Tract Area, in square miles

Methods 15
The more homogeneous the distribution of population within a watershed, the smaller ∆PD will be. ∆PD is a potentially 

useful predictor, as discussed in the earlier section on Adjustment Models. 

Imperviousness

The 2001 NLCD includes imperviousness mapping product at 30-meter resolution. Each pixel reports a value of imperviousness as a percentage of the total area of the pixel. The determination of the percentage of imperviousness across an entire 

watershed is straightforward and requires simply calculating the average imperviousness from the set of pixels within the watershed.

Estimating Imperviousness from Census Data 

A time series of historical land-use (or imperviousness) data covering several decades and uniformly available at the 

national scale does not exist. Because of the need to develop estimates of imperviousness for each year for each watershed 

being studied, it was necessary to develop a procedure that could be uniformly applied to each watershed.

The imperviousness layer from the 2001 NLCD was obtained for the central region of Maryland because it was available 

and because this region includes two highly urbanized areas—Baltimore, Maryland, and Washington, D.C. Similarly, tract data 

from the 2000 census were obtained. For each census tract, the population density (in thousands of people per square mile) 

was determined based on the reported total population and the area of the tract as determined by using GIS. Similarly, the tract 

outlines were used to sample the 2001 NLCD imperviousness layer and to determine the average imperviousness for each tract. 

Several different models were investigated for describing the relation between population density and imperviousness. A simple 

power model was selected to provide the best compromise between statistical power and model rationality. The calibrated equation took the following form: 

, (26)

where IA is imperviousness, in percent; and 

	 PD is population density in thousands of people per square mile. 

This equation was determined for 0.0002 < PD < 176.4 and 0.08 < IA < 96.66. The median PD and IA values were 3.87 

and 21.8, respectively. The regression was performed using data from 998 census tracts. The ratio of the standard error (Se) of 

equation 26 to the standard deviation (Sd) of observed imperviousness was Se/Sd = 0.6163, with an explained variance of 62.1 

percent. 

The regression equation developed by Stankowski (1972) to estimate imperviousness from population density in New Jersey had a different form from that of equation 26:

, (27)

where IA is imperviousness, in percent; and 

	 PD is populatin density, in persons (not thousands of persons) per square mile. 

The results of both equations, however, are quite comparable, as shown in figure 7.

In focusing on predicted imperviousness between 0 and 50 percent, which is the typical range observed in most watersheds, 

the maximum departure between the two methods is about 1.2 percent (fig. 7). The estimated imperviousness obtained by using 

equation 27 is 31.0 percent and 29.8 percent by using equation 26 for a population density of approximately 5,660 persons per 

square mile. While the difference is small, it is even less significant considering that most of the watersheds in this study had 

lower population densities and, thus, smaller estimated imperviousness. The typical departure in estimates of imperviousness 

between the two methods for the watersheds included in this study was less than 1.0 percent.

Implementation of Historical Census Data 

Historical census tract data were obtained for 1970, 1980, 1990, and 2000. Population density and estimated imperviousness were determined, using equation 26, for each census tract in each study watershed for each census period. The watershed 

boundary was then used to select all intersecting census tracts, and an area-weighted average of imperviousness for each census 

period was determined for each watershed. Imperviousness was assumed to vary linearly between each census measure.

16 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
In the event of decreasing population density over time, the estimate of imperviousness naturally would decline (eq. 26). 

Although population could decrease, imperviousness was assumed to be a strictly nondecreasing function with time and, therefore, was assumed to remain at the previous value. Of the 78 watersheds studied, 10 had decreasing population estimates and 

nonvarying estimates of imperviousness during the entire gaged period. 

To illustrate the use of historical census data in estimating imperviousness, census tract data were obtained for the watershed of USGS streamgage 01645000, Seneca Creek at Dawsonville, Maryland (fig. 8). Equation 26 was applied to population 

densities for 1970, 1980, 1990, and 2000. The resulting imperviousness estimations for these years were 4.84 percent, 7.48 

percent, 11.23 percent, and 12.99 percent, respectively.

Figure 7. Relations of imperviousness to population density as measured by Stankowski (1972) and as measured in this study. [Note: 

Stankowski’s (1972) equation was based on density in persons per square mile while this study’s equation was based on density 

in thousands of persons per square mile. For consistency in this figure, Stankowski’s (1972) equation was revised to be based on 

thousands of persons per square mile.]

80

70

60

30

40

50

10

20

0

0 5 10 15 20 25 30

Imperviousness, in percent

Population Density, in thousands per square mile

Stankowski (1972)

This Study

Figure 8. Estimated imperviousness from census tract data for U.S. Geological Survey streamgage 01645000, Seneca Creek at 

Dawsonville, Maryland. [Imperviousness in this figure ranges from 0 to 40 percent. Darker shades correspond to high degrees of 

imperviousness.]

1970 1980 1990 2000

Methods 17
Results

Previously, in the “Methods” section the need to censor gages with non-elevated flood-frequency was discussed. The actual 

numbers of streamgages that were used in the calibration of the adjustment equations as a function of the flood-frequencies 

being considered are given in table 2.

Table 2. Number of gages used in the regression analysis as a function of return period.

Return period 

(years) 2 5 10 25 50 100 500

Number of 

streamgages 

used

52 52 50 48 43 41 28

In total, information was obtained from 61 of the 78 streamgages (identified by italics in appendix 2) for at least one of 

the seven frequencies for which urban regression equations were calibrated in this study. In other words, of the 78 streamgages 

originally selected for use in this study, observed flood-frequency estimates for 17 of the streamgages were no greater than the 

rural regression estimates for any of the seven frequencies examined. The explanation for this is attributed collectively to error 

inherent in the rural regression equations, possible error in the discharge measurement or corresponding rating curve, error in the 

flood-frequency analysis, and other inherent characteristics (such as manmade storage or missing basin attributes) that may have 

served to reduce the flood magnitudes as predicted by the rural regression equations.

The goodness-of-fit characteristics for the model calibrations are presented in table 3, and the calibrated coefficients are 

shown in table 4. Table 3 presents standard error (Se) calculated as

, (28)

where n is the number of observations, 

	 m is the number of coefficients or exponents being calibrated, 

	 y is the observed discharge (from the PeakFQ output), and 

	 ŷ is the predicted output calibrated by the nonlinear regression tool. 

Standard deviation (Sy) is calculated as

, (29)

where is the mean of the discharges for the return period (T). 

Explained variance (R2) is calculated as

, (30)

where

(31)

in which is the mean of the predicted discharges for the return period (T).

As shown in table 3, the null model performed almost as well as the more sophisticated models, especially in regard to 

the standard error ratio measure. It must be noted, however, that only urban streamgages were used in this study. In particular, 

18  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
the calibrated equations were developed only for urban gages where the observed flood-frequency was associated with greater 

discharges than those predicted by the rural regression equations. Given such a dataset from which to calibrate an equation, the 

null model was certain to perform well simply because the data were already screened for the tendency to produce elevated flood 

magnitudes. The null model could be expected to have a poorer performance if both rural and urban streamgages were in the 

dataset. Further, the major shortcoming of the null model is that it will not predict larger urban discharges in relation to larger 

amounts of urbanization (imperviousness or density). The value of the null model is to provide some information about the 

additional predictive power of the models that include imperviousness in the flow-adjustment scheme.

Provided that a model that accounts quantitatively for urbanization is needed, the question now turns to which model of 

those examined performs “best.” An absolute best model is not indicated in table 3 because only seven models were examined, 

and it is impossible to prove logically that the very best mathematical form was investigated. Nevertheless, the models examined encompass several different approaches to adjusting flows, and their performance probably is not far removed from the 

hypothetical best adjustment model. Further, the relative performance of the seven models varied, depending on the measure 

(explained variance or relative standard error) and the flood-frequency being considered (table 3). The goodness-of-fit values in 

table 3 were ranked within each measure and return period and the best performing model received a ranking of “1” with subsequent rankings increasing as model performance decreased (table 5). The rankings then were summed across both the explanatory measures and flood-frequencies, and the model producing the overall smallest sum was considered to be the best overall 

model among those considered. These summed rankings are presented in table 6. 

This ranking scheme indicates that the best overall performing model was the imperviousness distribution model (table 6). 

This assertion can be confirmed by a closer examination of table 3 in which the imperviousness distribution model produced the 

largest explained variance in three of the seven flood-frequencies examined and was among the top three performing models in 

this measure. With respect to relative standard error, the imperviousness distribution model is the best model in two of the seven 

flood-frequencies and was always among the top three performing models in this measure. In ranking overall performance, 

the imperviousness distribution model performed the best (slightly better than the density distribution model) with regard to 

explained variance and relative standard error (table 5).

In regard to the remaining models, one could ask two questions. First, does population density serve as a stronger predictor than imperviousness? Second, does the “simple” approach, such as simple imperviousness or simple density, outperform the 

“distribution” or “scaled” approach?

With regard to the first question, the tabulated results (table 5) indicate a modest advantage for imperviousness over density as a predictor. The imperviousness distribution model ranked first overall, whereas the density distribution model ranked 

second. The simple imperviousness model ranked fourth overall, whereas the simple density model ranked fifth. The oppoTable 3. Goodness-of-fit characteristics for the models evaluated.

[All statistics presented are for log-transformed discharges: log10(Q). Bold values are best values for the return period.]

Model

Explained variance (R 2) Standard error ratio (Se /Sy )

Return period (years) Return period (years)

2 5 10 25 50 100 500 2 5 10 25 50 100 500

Null 0.821 0.859 0.864 0.902 0.899 0.889 0.814 0.390 0.378 0.341 0.265 0.261 0.277 0.446

Simple 

imperviousness

0.818 0.866 0.879 0.909 0.902 0.884 0.801 0.392 0.389 0.348 0.281 0.277 0.310 0.480

Simple density 0.818 0.866 0.877 0.907 0.897 0.881 0.804 0.397 0.363 0.328 0.257 0.298 0.311 0.478

Imperviousness 

distribution

0.832 0.867 0.874 0.909 0.901 0.897 0.855 0.385 0.374 0.338 0.292 0.268 0.271 0.356

Density 

distribution

0.833 0.870 0.874 0.902 0.893 0.889 0.779 0.383 0.368 0.335 0.278 0.289 0.297 0.526

Scaled 

imperviousness 

0.822 0.873 0.869 0.904 0.900 0.884 0.810 0.409 0.396 0.365 0.288 0.280 0.310 0.528

Scaled density 0.823 0.861 0.873 0.904 0.900 0.887 0.821 0.409 0.414 0.361 0.289 0.287 0.307 0.464

Results 19
Table 4. Calibrated values of the adjustment model coefficients and exponents.

[Variables for all models are defined in equations 12-18.]

Model Variable Return period (years)

2 5 10 25 50 100 500

Null c1 0.275 0.489 0.305 0.330 0.305 0.330 0.330

c2 1.10 1.04 1.10 1.10 1.10 1.10 1.10

Simple 

imperviousness

c1 0.298 0.331 0.33 0.33 0.228 0.225 0.263

c2 1.16 1.15 1.16 1.16 1.15 1.15 1.12

c3 0.177 0.173 0.176 0.176 0.0383 0.0446 0.035

Simple densiy

c1 0.27 0.315 0.314 0.368 0.35 0.35 0.308

c2 1.10 1.10 1.10 1.10 1.10 1.10 1.10

c3 0.05 0.0987 0.0977 0.0642 0.172 0.172 0.0474

Imperviousness 

distribution

c1 0.33 0.42 0.405 0.435 0.363 0.293 0.251

c2 1.10 1.10 1.10 1.10 1.10 1.10 1.10

c3 0.134 0.159 0.135 0.101 0.0819 0.0148 0.0229

c4 0.0687 0.025 0.0166 0.0425 0.0388 0.0666 0.108

Density 

distribution

c1 0.27 0.313 0.312 0.315 0.304 0.345 0.332

c2 1.10 1.10 1.10 1.10 1.10 1.10 1.10

c3 0.171 0.228 0.111 0.155 0.140 0.136 0.198

c4 0.11 0.062 0.0264 0.0739 0.0963 0.126 0.113

Scaled 

imperviousness

c1 0.288 0.315 0.3 0.3 0.3 0.315 0.285

c2 1.15 1.15 1.15 1.15 1.15 1.15 1.15

c3 0.11 0.148 0.0997 0.1 0.1 0.109 0.0898

c4 0.131 0.29 0.2 0.167 0.105 0.149 0.204

I* 13.9 20.1 9 12.5 7.54 10.2 13.1

Scaled

density

c1 0.287 0.3 0.3 0.3 0.3 0.296 0.284

c2 1.15 1.15 1.15 1.15 1.15 1.15 1.15

c3 0.11 0.1 0.125 0.0944 0.108 0.108 0.109

c4 0.271 0.505 0.765 0.213 0.000488 0.00469 0.00907

PD* 1.05 1.2 3.04 0.960 0.284 0.369 0.519

20 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Table 5. Rankings of model performance based on the goodness-of-fit values shown in table 3.

Model

Explained variance (R 2) Standard error ratio (Se /Sy )

Return period (years) Return period (years)

2 5 10 25 50 100 500 2 5 10 25 50 100 500

Null 5 6 7 6 5 3 3 3 4 4 4 1 2 2

Simple 

imperviousness

6 4 1 1 1 5 6 4 5 5 5 3 5 5

Simple density 6 4 2 3 6 7 5 5 1 1 1 7 7 4

Imperviousness 

distribution

2 3 3 1 2 1 1 2 3 3 3 2 1 1

Density 

distribution

1 2 3 6 7 2 7 1 2 2 2 6 3 6

Scaled 

imperviousness 

4 1 6 4 3 5 4 6 6 7 7 4 5 7

Scaled density 3 7 5 4 3 4 2 6 7 6 6 5 4 3

Table 6. Summed rankings of model calibration statistics based on goodness-of-fit values given in table 3.

[Bold values are best values for each column.]

Model Explained variance, 

R2 sum

Standard error ratio, 

Se /Sy sum

Overall sum 

(Overall rank)

Null 35 20 55 (3)

Simple 

imperviousness

24 32 56 (4)

Simple density 33 26 59 (5)

Imperviousness 

distribution

13 15 28 (1)

Density 

distribution

28 22 50 (2)

Scaled 

imperviousness 

27 42 69 (7)

Scaled density 28 37 65 (6)

Results  21
site trend occurred for the scaled models—the imperviousness-based model ranked seventh (last) and the density-based model 

ranked sixth. Thus, the results slightly favor imperviousness over population density as a predictor of flood-frequency magnitude.

With regard to the second question, the “distribution” approach (eqs. 15, 16) ranked first, followed by the “simple” 

approach (eqs. 13, 14); the “scaled” models (eqs. 17, 18) performed the worst. This result is interesting because the simple models require the calibration of three parameters; the distribution models require four parameters, and the scaled models require 

five parameters. Generally, a more complex model would be expected to perform best because it has more flexibility to conform to the observed data. In this case, the moderately complex model performs best. The ∆PD or ∆IA term of the distribution 

model structure apparently is an effective way of quantifying urban heterogeneity within a watershed and, therefore, is useful in 

predicting flood behavior.

The calibration results for all models are presented in table 4. (Appendix 3 contains plots for comparing predicted and 

observed discharges for the calibrated simple imperviousness model, which is fairly representative of other model calibrations 

as well.) While the values shown in table 4 represent optimum calibration results, a brief examination of this table reveals 

somewhat erratic trends in many calibration parameters as return period varies. It would be expected that these parameters 

vary smoothly from one return period to the next. Towards this end, a secondary smoothing was performed for each calibration 

parameter using a linear regression model:

, (32)

where T is the return period and α and β are determined by regressing the calibrated values of cx,T on log10(T).

These smoothed parameter values were used in all subsequent analyses, because it was felt these values were more representative of the true relations between rural and urban flood magnitudes (table 7). Appendix 4 contains the calibrated and 

smoothed coefficients plotted against return periods for each adjustment model.

As stated previously, the functions, f

n

(.) can be rearranged to express the urban peak discharge as a function of the rural 

peak discharge and imperviousness. For simplicity, the equation 11 can be revised as follows:

. (33)

Thus, the null model is rearranged as

, (34)

the simple imperviousness model is rearranged as

, (35)

the simple density model is rearranged as

, (36)

the imperviousness distribution model is rearranged as

, (37)

the density distribution model is rearranged as

22  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Table 7. Smoothed values of the adjustment model coefficients and exponents.

[Variables for all models are defined in equations 12-18.]

Model Variable Return period (years)

2 5 10 25 50 100 500

Null c1 0.289 0.297 0.303 0.311 0.317 0.324 0.338

c2 1.10 1.10 1.10 1.10 1.10 1.10 1.10

Simple 

imperviousness

c1 0.327 0.312 0.301 0.287 0.276 0.264 0.239

c2 1.16 1.16 1.16 1.15 1.15 1.14 1.13

c3 0.200 0.170 0.147 0.117 0.095 0.072 0.019

Simple densiy

c1 0.305 0.312 0.318 0.325 0.330 0.336 0.349

c2 1.10 1.10 1.10 1.10 1.10 1.10 1.10

c3 0.0856 0.0910 0.0950 0.100 0.104 0.108 0.118

Imperviousness 

distribution

c1 0.414 0.393 0.378 0.357 0.341 0.325 0.289

c2 1.10 1.10 1.10 1.10 1.10 1.10 1.10

c3 0.161 0.136 0.118 0.0927 0.0740 0.0552 0.0115

c4 0.0269 0.0361 0.0431 0.0523 0.0592 0.0662 0.0823

Density 

distribution

c1 0.289 0.297 0.304 0.313 0.320 0.326 0.342

c2 1.10 1.10 1.10 1.10 1.10 1.10 1.10

c3 0.166 0.165 0.164 0.163 0.162 0.161 0.159

c4 0.066 0.073 0.079 0.087 0.093 0.098 0.112

Scaled 

imperviousness

c1 0.303 0.302 0.301 0.300 0.300 0.299 0.298

c2 1.15 1.15 1.15 1.15 1.15 1.15 1.15

c3 0.123 0.117 0.113 0.108 0.104 0.100 0.0911

c4 0.189 0.185 0.182 0.178 0.175 0.172 0.165

I* 14.4 13.7 13.1 12.3 11.8 11.2 9.88

Scaled

density

c1 0.298 0.297 0.296 0.295 0.295 0.294 0.292

c2 1.15 1.15 1.15 1.15 1.15 1.15 1.15

c3 0.108 0.108 0.108 0.108 0.108 0.108 0.107

c4 0.512 0.418 0.347 0.253 0.182 0.111 -0.0539

PD* 1.67 1.45 1.28 1.06 0.893 0.725 0.336

, (38)

the scaled imperviousness model is rearranged as

, (39)

and the scaled density model is rearranged as

, (40)

where IA is the impervious area, in percent, for the watershed conditions at which the urban discharge estimate is desired.

Results  23
The coefficients and (or) exponents, cx,T, refer to the coefficients outlined in the models presented in equations 12-18 for the 

T-year prediction equation. Note that equations 12-18 refer to adjustment models that convert urbanized discharges to equivalent rural ones, whereas equations 33-40 convert rural discharges to urban ones.

Based on the smoothed coefficients and (or) exponents given in table 7, seven sets of urban equations are given based on the 

calibrated null, simple imperviousness, simple density, imperviousness distribution, density distribution, scaled imperviousness, 

and scaled density adjustment models. For the null model, the resulting urban equations are as follows:

, (41)

, (42)

, (43)

, (44)

, (45)

, (46)

and

. (47)

The calibrated urban equations for the simple imperviousness model are as follows:

, (48)

, (49)

, (50)

, (51)

, (52)

, (53)

and

. (54)

The calibrated urban equations for the simple density model are as follows:

, (55)

24  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
, (56)

, (57)

, (58)

, (59)

, (60)

and

. (61)

The calibrated urban equations for the imperviousness distribution model are as follows:

, (62)

, (63)

, (64)

, (65)

, (66)

, (67)

and

. (68)

The calibrated urban equations for the density distribution model are as follows:

, (69)

, (70)

, (71)

, (72)

, (73)

Results  25
, (74)

and

. (75)

For the scaled imperviousness model, the calibrated urban equations are as follows:

, (76)

, (77)

, (78)

, (79)

, (80)

, (81)

and

. (82)

Finally, for the scaled density model, the calibrated urban equations are as follows:

, (83)

, (84)

, (85)

, (86)

, (87)

, (88)

and

. (89)

A positive exponent on any predictor variable indicates a positive relation between that predictor and the urban discharge; 

a negative exponent indicates a negative relation. All of the model forms have positive exponents on all predictors except the 

∆IA and ∆PD terms (eqs. 62-75), which have negative exponents. (One other exception, which will be discussed later, occurred 

for c4 in equation 89.) It can be concluded quickly that all exponents are rational because the urban flood magnitude is expected 

26 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
to increase with increasing rural flood magnitude, increasing imperviousness, and increasing population density. Although 

the negative sign on the ∆IA and ∆PD terms is less obvious, it indicates that as the watershed becomes more varied spatially in 

urbanization, the predicted discharge decreases. In essence, the ∆IA and ∆PD terms serve to temper the IA or PD terms, which 

convey only the average imperviousness or population density in a watershed but not variability. Finally, the positive signs on 

the two parameters that control the scaled sigmoid function presented in figure 8 are worth noting.

Trends in the parameter values as return period increases also are important to examine. Generally, as the return period 

increases, one might expect to see diminishing effects of urbanization on flood magnitude. For example, the 2-year flood would 

be expected to have greater dependence on urbanization than the 100-year flood, and trends in the parameter values would be 

expected to support this. The following discussion focuses on the rationality of the trends for each model:

Null model: The evaluation of trends in this model is not applicable.

Simple imperviousness model: Trends in this model are entirely rational. The model has a negative trend in the IA

exponent.

Simple density model: The positive trend in the PD exponent for this model is not rational. This is of concern.

Imperviousness distribution model: Trends in this model are entirely rational. The model has a positive trend in ∆IA

exponent and a negative trend in the IA exponent.

Density distribution model: The trends in this model follow the same pattern as in the imperviousness distribution 

model, although the negative trend in the PD exponent is weak.

Scaled imperviousness: Trends in this model are entirely rational. The model has a negative trend in the scaled IA

exponent. It is not clear whether a positive or negative trend in the two scaled imperviousness terms is expected but both 

terms have negative trends, which is consistent with the trends in the analogous terms in the scaled density model.

Scaled density: This model has a weak positive trend in the scaled PD exponent. This is not rational, but because the 

trend is so weak, the non-rationality similarly is weak. The negative trend in c4 actually leads to a slightly negative value 

for the 500-year flood. The physical interpretation of this result is not clear, and the trends for this model are of concern. 

Again, the trends in both terms in the scaled density model are negative, consistent with the analogous terms in the scaled 

imperviousness model.

The results of the rationality analysis are summarized as follows:

All models had rational signs in the urbanization (imperviousness (IA) or population density (PD)) exponents.

All imperviousness models (simple, distribution, and scaled) had rational trends in the exponents.

Two population density models (simple and scaled) had one or more non-rational trends in the exponents. The results 

for these two models should be used with caution, if at all.

Each of the above equations (eqs. 41-89) was divided by RQT to produce the ratio of urban to rural discharge (UQ/RQ). 

Assuming a vector of rural flood-frequency values (Qff,rural), the UQ/RQ was determined as a function of varying imperviousness. 

To visualize results of this approach, each model was applied to values in two watersheds, one fairly large and one fairly small. 

The fairly large watershed (102 mi2) was represented by USGS streamgage 01645000, Seneca Creek at Dawsonville, Maryland; 

the fairly small watershed (2.2 mi2) was represented by USGS streamgage 01585200, West Branch Herring Run at Idlewylde, 

Maryland. The UQ/RQ ratios are shown in figures 9-12.

The three imperviousness models that were investigated have several common characteristics (figs. 9, 11). First, all models 

indicate a general decline in UQ/RQ, approaching 1, as the return period increases. Because the effects of urbanization are most 

profound for small magnitude and high-frequency flooding events, the greatest adjustment ratios are required for the small return 

periods. Also, all models had a greater UQ/RQ value as imperviousness increased. This was not surprising because, all things 

being equal, higher values of imperviousness would be expected to lead to greater urban discharges. In most cases, the null 

model results were roughly centered within the spread of imperviousness values and were considered representative of average 

imperviousness across the datasets evaluated (figs. 9, 11).

The three calibrated population density models were applied to discharge data from the two selected gaging stations to 

obtain the UQ/RQ values across all return periods (figs. 10, 12). The irrationality in the trend in the population density exponent 

in the simple density model is shown in figure 10A by the increasing spread of the constant population density traces in relation 

to return period increases. The tight clustering of the constant population density traces in figure 10C indicates that the calibrated parameters for the scaled population density term do not lead to greatly varying estimates of scaled population density. 

This is not irrational, but it diminishes the credibility of the scaled population density model.

•

•

•

•

•

•

•

1.

2.

3.

Results  27
2 5 10 25 50 100 500

0.5

1

1.5

2

2.5

A

2 5 10 25 50 100 500

0.5

1

1.5

2

2.5

Ratio of urban to rural discharge

B

2 5 10 25 50 100 500

0.5

1

1.5

2

2.5

Return Period, in years

C

Figure 9. Application of the (A) simple imperviousness model, (B) imperviousness distribution model, and (C) scaled imperviousness 

model to the ratio of urban to rural discharges as a function of return period and imperviousness at U.S. Geological Survey streamgage 

01645000, Seneca Creek at Dawsonville, Maryland. [The heavy dashed line shows the ratio for the “null” model. The light lines 

indicate imperviousness increasing in 2-percent increments starting from 0 percent for the lowest trace.]

28  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
2 5 10 25 50 100 500

0.5

1

1.5

2

2.5

3 A

2 5 10 25 50 100 500

0.5

1

1.5

2

2.5

3

Ratio of urban to rural discharge

B

2 5 10 25 50 100 500

1

1.5

2

2.5

3

Return Period, in years

C

Figure 10. Application of the (A) simple density model, (B) density distribution model, and (C) scaled density model to the ratio of 

urban to rural discharges as a function of return period and imperviousness at U.S. Geological Survey streamgage 01645000, Seneca 

Creek at Dawsonville, Maryland. [The heavy dashed line shows the ratio for the “null” model. The light lines indicate population 

density increasing in increments of 1,000 persons per square mile starting from 0 persons per square mile for the lowest trace.]

Results  29
2 5 10 25 50 100 500

0.5

1

1.5

2

2.5

A

2 5 10 25 50 100 500

0.5

1

1.5

2

2.5

Ratio of urban to rural discharge

B

2 5 10 25 50 100 500

0.5

1

1.5

2

2.5

Return Period, in years

C

Figure 11. Application of the (A) simple imperviousness model, (B) imperviousness distribution model, and (C) scaled imperviousness 

model to the ratio of urban to rural discharges as a function of return period and imperviousness at U.S. Geological Survey streamgage 

01585200, West Branch Herring Run at Idlewylde, Maryland. [The heavy dashed line shows the ratio for the “null” model. The light 

lines indicate imperviousness increasing in 2-percent increments starting from 0 percent for the lowest trace.]

30 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
2 5 10 25 50 100 500

0.5

1

1.5

2

2.5

3 A

2 5 10 25 50 100 500

0.5

1

1.5

2

2.5

3

Ratio of urban to rural discharge

B

2 5 10 25 50 100 500

1

1.5

2

2.5

3

Return Period, in years

C

Figure 12. Application of the (A) simple density model, (B) density distribution model, and (C) scaled density model to the ratio of 

urban to rural discharges as a function of return period and imperviousness at U.S. Geological Survey streamgage 01585200, West 

Branch Herring Run at Idlewylde, Maryland. [The heavy dashed line shows the ratio for the “null” model. The light lines indicate 

population density increasing in increments of 1,000 persons per square mile starting from 0 persons per square mile for the lowest 

trace.]

Results  31
By comparing the UQ/RQ ratios for imperviousness (figs. 9, 11) and population density (figs. 10, 12) in the two watersheds, the effect of watershed scale is apparent between the two datasets. Herring Run is roughly two orders of magnitude 

smaller than Seneca Creek, which results in a greater range in UQ/RQ ratios for this smaller watershed. In comparing figures 

9A and 11A for the 2-year return period, a UQ/RQ range is observed of 0.8 to 1.6 for Seneca Creek and approximately 1.2 to 

2.2 for Herring Run, respectively. The greater range in UQ/RQ ratios for the smaller watershed is because the exponent on RQ, 

which is positive but less than 1, has the effect of the UQ/RQ ratio varying more widely for small discharges associated with 

small watersheds.

Comparison to Urban Equations Developed by Sauer and others (1983)

Application of the urban equations in Sauer and others (1983) to the database assembled for this current study was not possible because all of the equations developed in the earlier work included many watershed characteristics that were not gathered 

for the current study, most notably the basin development factor (BDF). It is possible, however, to return to the dataset from 

Sauer and others (1983) and apply three of the equations generated in the current study.

The three equations represent the null model, the simple imperviousness model, and the scaled imperviousness model. The 

adjustment models that included population density and (or) imperviousness distribution as predictors could not be compared 

because these data were not measured or presented in the dataset generated by Sauer and others (1983). Another model was 

added, however; the “Sauer (null)” model is the analog of the null model presented in the current study. The Sauer (null) model 

was calibrated as a simple power model of the rural discharge,

, (90)

using a logarithmic transformation of the rural and urban discharges and calibrating a linear model through least-squares regression. The purpose of the Sauer (null) model was to provide some perspective on the merits of the seven-parameter Sauer and 

others (1983) model relative to the simplest possible model.

The mean bias and relative standard error were examined for the seven-parameter model (Sauer and others, 1983), the 

Sauer (null) model, and the three models developed in the current study as applied to the seven return periods (table 8). For the 

purposes of this report, mean bias was calculated as the average difference between predicted discharges (from the equations 

developed in this report) and observed urban flood-frequency (reported by Sauer and others, 1983).

In general, the equations from Sauer and others (1983) performed best in predicting urban discharges across the 203 

streamgages in the study, especially in terms of standard error (table 8). This would be expected because Sauer and others 

(1983) used these gages to calibrate their equations. Although the three new models calibrated in the current study did not make 

use of the database from Sauer and others (1983), the prediction ability of these new models was respectable as evidenced by 

similar, albeit slightly poorer, bias and standard error values (table 8). This is somewhat surprising in that this comparison is not 

exactly fair given the possibility for systematic differences in the way imperviousness was measured in the two studies and the 

fact that this is the calibration set for the Sauer and others (1983) equations. Further, the Sauer (null) model shows comparable 

relative standard error performance for the 25- through 500-year return periods. The complexity of the seven-parameter model 

from Sauer and others (1983) seems to yield meaningfully better predictions relative to the other models only for the 2-, 5-, and 

10-year return periods. Finally, the performance of the new models developed in this study is even more impressive when considering the use of only two predictors (rural discharge and imperviousness) in these models.

To further quantify the performance of the three models developed in this (current) study, the relative standard errors of 

each of the new models (null, simple, and scaled) were normalized by the relative standard error from the seven-parameter Sauer 

and others (1983) model for the same return period. Each of the three new models performed similarly across all return periods 

(table 9). On average, the standard error of the three new models was about 6 to 9 percent greater than the equivalent standard 

error from Sauer and others (1983). Standard error is not uniform across return periods, however, but tends to decrease as the 

return period increases. 

This finding mirrors the performance of the Sauer (null) model, which produced poorer relative standard errors for the 2-, 

5-, and 10-year return periods and then was essentially equivalent in relative standard error to the seven-parameter model (Sauer 

and others, 1983) for all larger flood events. 

32  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Table 8. Performance of U.S. Geological Survey urban equations from an earlier study and urban equations developed in this report.

[Bold values indicate overall best model performance. Italic values indicate best model performance developed in this report. ft3/s, cubic feet per second]

Mean bias (ft3/s) Relative standard error: Se

 /Sy

Return 

period 

(years)

Sauer 

(null) Sauer* Null Simple Scaled Sauer (null) Sauer* Null Simple Scaled

2 -127 -111 -33 -111 32 0.462 0.381 0.486 0.461 0.461

5 -171 -178 -80 -214 6 0.463 0.425 0.466 0.477 0.467

10 -235 -141 -174 -285 -65 0.474 0.449 0.474 0.496 0.481

25 -357 -289 -388 -455 -241 0.493 0.490 0.496 0.523 0.507

50 -537 -281 -553 -644 -377 0.520 0.518 0.521 0.550 0.534

100 -675 -300 -867 -734 -658 0.532 0.533 0.540 0.573 0.555

500 -1,200 -1,150 -1,748 -1,106 -1,458 0.600 0.600 0.600 0.586 0.612

*

Sauer and others, 1983.

Table 9. Relative Standard error (Se /Sy ) for three imperviousness models developed in this report as normalized by the relative 

standard error determined by Sauer and others (1983).

Return 

period 

(years)

Null Simple Scaled Mean

2 1.244 1.218 1.205 1.223

5 1.094 1.146 1.106 1.115

10 1.047 1.120 1.078 1.082

25 1.004 1.073 1.040 1.039

50 1.002 1.052 1.034 1.029

100 1.016 1.048 1.048 1.037

500 1.000 0.976 1.020 0.999

Mean 1.058 1.090 1.076 1.075

Results  33
Application of Calibrated Models to Streamgages with Local Urban Equations

A total of 10 USGS streamgages were identified in New Jersey (Stankowski, 1974), Pennsylvania (Stuckey and Reed, 

2000), and Wisconsin (Walker and Krug, 2003) that were subject to substantial urbanization. A set of USGS peak flow regression equations that include an imperviousness or urbanization descriptor is available for each of these States. For each of the 10 

urban streamgages, watershed characteristics were determined and regression equations were applied twice—first, to obtain the 

best estimate of the current urban flow, and second, to set the urban descriptor (either imperviousness or urban area) at zero to 

develop an equivalent rural flow. The watershed characteristics for the 10 streamgages are provided in appendix 5, and the rural 

and urban flood-frequency results are provided in appendix 6. 

The following example is given for USGS streamgage 01393450, Elizabeth at Ursino Lake at Elizabeth, New Jersey. The 

2-year USGS regression equation for New Jersey is as follows:

, (91)

where the constants of 1 are added by default on the chance that either of these quantities would be zero for a particular watershed being studied. For USGS streamgage 01393450, the drainage area is 17.6 mi2

, the channel slope is 22.4 feet per mile, the 

storage is 1.6 percent, and the imperviousness in 2000 was 41.9 percent. The result of applying equation 91 to these values is 

an urban 2-year discharge of 1,071 ft3/s. If imperviousness is set at zero (actually 1, as applied in the National Flood-Frequency 

program), the equivalent rural discharge is 550 ft3/s. 

To test the quality of the urban equation models calibrated in this study, the urban equations were applied to the equivalent 

rural discharge, and the result was compared with the urban value of 1,071 ft3/s. Continuing this example, the simple imperviousness model was applied as follows:

. (92)

The result of 1,127 ft3/s was compared with the estimated 1,071 ft3/s from equation 91. Similar comparisons were made for the 

other nine streamgages, and the mean bias results are presented in table 10 and the relative standard error results are presented in 

table 11.

The performance of the models was ranked in the same way as the model calibrations. The sums of ranks for both the 

mean bias and relative standard error for the performance of the models for the 10 streamgages with local urban equations are 

given in table 12. Here, mean bias is defined as the average difference between the predicted discharges determined by applying 

the equations developed in this study and the discharges determined by applying the appropriate (published) State urban regression equations. The strongest performing models were models based on population density; however, all models outperform the 

null model, which indicates the genuine predictive capabilities that these models offer beyond a simplistic scaling up of a rural 

discharge.

Table 10. Mean bias, in cubic feet per second, for new urban equations used on data from 10 U.S. Geological Survey streamgages for 

which localized urban equations are available.

[Bold values indicate overall best model performance.]

Return period 

(years) Null

Simple Distribution Scaled

Imperviousness Density Imperviousness Density Imperviousness Density

2 -481 -428 -391 -390 -331 -403 -359

5 -587 -431 -417 -506 -392 -491 -383

10 -592 -503 -345 -541 -380 -473 -305

25 -557 -418 -191 -575 -349 -404 -135

50 -495 -288 -10 -584 -305 -317 64

100 -385 -77 248 -569 -231 -180 341

500 62 803 1165 -406 55 336 -1,708

34  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Table 11. Relative standard error (Se /Sy ) for new urban equations used on data from 10 U.S. Geological Survey streamgages for which 

localized urban equations are available.

[Bold values indicate overall best model performance.]

Return period 

(years) Null

Simple Distribution Scaled

Imperviousness Density Imperviousness Density Imperviousness Density

2 0.486 0.454 0.409 0.417 0.378 0.500 0.428

5 0.361 0.284 0.252 0.311 0.249 0.376 0.280

10 0.298 0.262 0.178 0.271 0.200 0.311 0.202

25 0.234 0.192 0.139 0.238 0.166 0.243 0.138

50 0.201 0.160 0.174 0.221 0.162 0.209 0.148

100 0.177 0.156 0.236 0.202 0.162 0.184 0.196

500 0.218 0.307 0.439 0.184 0.209 0.222 0.361

Table 12. Summed and ranked overall performance of new urban equations used on data from 10 U.S. Geological Survey streamgages 

for which localized urban equations are available.

[Smaller sums and lower ranks indicate best performance.]

Performance measure Null

Simple Distribution Scaled

Imperviousness Density Imperviousness Density Imperviousness Density

Sum of mean bias ranks 41 29 22 40 17 28 19

Sum of relative standard 

error ranks

34 25 25 33 14 42 23

Overall Sum 75 54 47 73 31 70 42

Overall Rank 7 4 3 6 1 5 2

Interpretation of Results

The equations developed in this study appear to be comparable in predictive capability to those developed by Sauer and 

others (1983). Three models developed during this current study—the null, simple imperviousness, and scaled imperviousness models—were compared with the best model from the Sauer and others (1983) study. The simple and scaled imperviousness models performed, on average, within about 6 to 9 percent of the results obtained by Sauer and others (1983). The model 

performance for the more frequent return periods, however, was not as strong. The relative standard error was about 20 to 24 

percent greater for the 2-year return period, 10 to 15 percent greater for the 5-year return period, and 5 to 12 percent greater for 

the 10-year return period that the results obtained by Sauer and others (1983). These results are considered comparable because 

the statistics from Sauer and others (1983) were based on performance using the same dataset from which the equations were 

calibrated, and the new models were calibrated using a completely different dataset.

Model Strengths 

The models developed in this study have two key strengths

Simplicity: The equations require three predictors or less. The rural discharge is a predictor in all models. Imperviousness or population density is used as a predictor in all but the null model. The quantities ∆IA and ∆PD are used as third 

predictors in the distribution models to quantify the heterogeneity of urbanization within a watershed. Although ∆IA and 

∆PD qualify as third predictors, they are based directly on the same quantities as either imperviousness or population 

•

Interpretation of Results  35
density. A user of any of these models needs access only to the rural discharges and a representation of either imperviousness or population density.

Ease of Use/GIS friendly: These models are applied easily by using available data from remote locations. The water, 

environmental, and ecological communities increasingly recognize the value of imperviousness as a strong predictor of 

human effects; simultaneously, the availability and quality of GIS coverages of imperviousness are increasing (Homer 

and others, 2004). Similarly, GIS coverages of census data are available commonly and render the determination of 

population density and ∆PD to an easily automated task. In contrast, the equations from Sauer and others (1983) require 

the determination of seven predictors, of which at least one (BDF) requires a physical reconnaissance of the watershed 

for accurate determination.

Model Weaknesses 

The principal weakness of the models developed in this study is the relatively mild dependency of flood magnitude 

on imperviousness or population density. This mild dependency is underscored by the good performance of the calibrated 

null model and on the non-smoothness of the calibrated adjustment model parameters shown in table 6. If this method were 

expanded to more gages or repeated with longer time series as additional data become available, it may be the case that the functional dependency on urbanization measures would become more clear and the need for calibrated model parameter smoothing 

(shown in table 7) would diminish.

Another weakness of the models, which is apparent from the comparison to data from Sauer and others (1983; see table 8) 

and to the locally available urban equations (table 10), is a clearly negative bias in the errors of all models for the small return 

periods. A shift to a positive bias occurred for the largest return periods (table 10). The origin of this bias is not known with 

certainty, although it may be a result of the model calibration method which depends on a logarithmic transformation of the 

observed rural discharges. Because calibration takes place in the logarithmic space, positive and negative errors measured in the 

arithmetic space are unequal, and a negative bias can result.

Choosing a Model

Ultimately, the questions one faces are these: “Which is the best model form to use?” and “Are there models that are 

clearly superior or inferior to the others?”

Population density approach: Among the models based on population density, the density distribution model (eqs. 

69-75) was the best performer during calibration and testing with the local urban equations. Further, the other two sets of 

density-based equations had irrational exponent values or trends, which cast doubt on their application.

Imperviousness approach: The choice of the best imperviousness model is less certain. The imperviousness distribution model (eqs. 62-68) performed the best overall during model calibration, but the performance of this model in 

testing the local urban equations was not as strong as for the simple imperviousness model (eqs. 48-54). This is perhaps 

attributable to the small sample size (10 streamgages) used for testing. Because of the clearly strong performance during 

model calibration and consistency with the population density approach, the imperviousness distribution models appear 

to be good choices if imperviousness data are used.

•

•

•

36 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Summary and Conclusions 

In this study, a new approach was developed, using data from 78 urbanized streamgages across the United States, for 

calibrating a set of USGS urban regression equations. The rural flood-frequency was determined for each streamgage by using 

the best available GIS data. The annual maximum time series for the 31-year period, 1970–2000, formed the observed peak 

discharge portion of the dataset. GIS techniques were used to develop an annual time series of imperviousness over this period 

using census-derived population density as a surrogate and predictor of imperviousness. The PeakFQ program1 was embedded 

in a modified nonlinear optimization program, and several adjustment equations were investigated for calibrating adjustment 

equations that could scale-back the observed annual maximum time series to an equivalent rural annual maximum time series 

closely representative of the rural regression equation flood frequency. Seven adjustment models of varying degrees of complexity were calibrated. These models then were compared with earlier USGS urban equations and with urban flood-frequency 

equations for individual States.

The results indicate that the urban equations developed in this study are comparable in performance to the earlier USGS 

urban equations but are simpler in that they are dependent only on rural discharge and measures of either imperviousness or 

population density. Further, the new urban equations can be applied readily by using only remotely sensed data rather than 

requiring data that can only be measured during site visits to the watershed of interest.

The highest performing models emerging from this study are the imperviousness distribution models (eqs. 62-68) and the 

population density distribution models (eqs. 69-75). These models depend on three predictors each—rural discharge, imperviousness or population density, and delta imperviousness or delta population density. The imperviousness or population density 

predictor serves to scale up the rural discharge, and the delta imperviousness or delta population density predictor scales down 

the discharge. This delta predictor quantifies the homogeneity of the development in a watershed and is calculated readily by 

GIS techniques using the same spatial data used in determining the imperviousness or population density predictor. 

Several directions are worth pursuing in future efforts to develop urban equations.

Expand the model calibration database: The data collection for 78 gages used in this study was labor intensive but 

crucial to the development of nationally applicable regression equations. In contrast, the database for the Sauer and 

others (1983) urban equation study was slightly in excess of 200 streamgages. Although expanding the current database may be impractical (or impossible), more streamgages could be identified that have been active in recent (1970 to 

present) times in States for which no local urban equations have been developed. Alternatively, this dataset likely will 

improve with time as periods of record for urbanization data increase.

Expand the test application database: Data for only 10 streamgages were used to compare locally available urban 

equations to predictions from the urban equations developed in this study. The results of this application are, therefore, 

only as sound as the representativeness of these 10 streamgages and the respective local urban equations. Local urban 

equations have been developed for approximately 13 States. Using data from streamgages in all of these States may lead 

to more definitive or possibly different results from those documented here.

Investigate additional trial adjustment models: As previously stated, it is a logical impossibility to consider all possible models for relating urban floods to rural floods. In this study, a creative attempt was made in this regard. Other 

models, however, could be investigated and may prove more effective.

Recalibrate model in compliance with Bulletin 17B: It is advisable that additional investigative work, using the 

embedded PeakFQ-optimization approach developed in this study, be done in compliance with the guidelines set forth in 

Bulletin 17B (Interagency Advisory Committee on Water Data, 1982).

Investigate different optimization and calibration algorithms: The tendency for calibrated models (developed in this 

study and in the Sauer and others (1983) urban equation study) to have a negative bias for small return periods and a 

neutral or positive bias for large return periods indicates a weakness in the current log-transformation method. Using a 

different model calibration approach that avoids this log-transformation method may produce unbiased calibrations and 

ultimately smaller standard errors.

•

•

•

•

•

1

 In this study, a nonstandard treatment of the historical period was used in the PeakFQ program, and this approach does not fully comply with the procedures 

set forth in Bulletin 17B, as perviously noted. Therefore, modifications may be needed before this approach can be applied in practice.

Summary and Conclusions   37
Notation

A drainage area of watershed (in square miles)

BDF basin development factor – a number between 0 and 12 (used in the Sauer and others (1983) urban equations) 

that represents the degree of channel improvements, channel linings, storm drains or storm sewers, and curb 

and gutter streets within the upper, middle, and lower third of the watershed

C percentage of watershed underlain by carbonate rock (in Pennsylvania Region A flood regression equations (in 

percent))

CA percentage of the basin controlled by lakes, swamps, or reservoirs (in Pennsylvania Region A flood regression 

equations (in percent))

cx,T calibrated coefficients or exponents corresponding to parameter x and the T-year flood in one of the adjustment 

equations

F forest cover (in percent)

FF(.) flood-frequency operator in Bulletin 17B—produces a vector of flood-frequency discharge estimates (in units of 

cubic feet per second)

f

T

(.) a discharge adjustment model for return period T

(.) a discharge adjustment model for return period T

HP historical period used as input in Bulletin 17B flood-frequency analysis (in years)

IA impervious area within the watershed (in percent)

IAx impervious area corresponding to the xth percentile of the distribution of imperviousness within the watershed 

(in percent)

IA(t) impervious area within the watershed in year t (in percent)

∆IA difference between the 10th and 90th percentile imperviousness values (in percent)

∆IA(t) difference between the 10th and 90th percentile imperviousness values in year t (in percent)

threshold impervious area for return period T in equation 39 (in percent)

m the number of coefficients and (or) exponents being calibrated in a discharge adjustment model

n number of years in a gage record (in years)

n number of observations (gages) used in the calibration of a discharge adjustment model

P(Qx) cumulative probability associated with having an annual maximum flood of magnitude Qx

PD population density (thousands of people per square mile)

PDx population density corresponding to the xth percentile of the distribution of population density within the 

watershed (thousands of people per square mile)

PD(t) population density in year t (thousands of people per square mile)

threshold population density for return period T in equation 40 (thousands of people per square mile)

∆PD difference between the 10th and 90th percentile population density values (thousands of people per square mile)

∆PD(t) difference between the 10th and 90th percentile population density values in year t (thousands of people per 

square mile)

PT cumulative probability associated with the T-year annual maximum flood

Qadj(t) the generic adjusted annual maximum flood in year t (in cubic feet per second)

Qadj,T(t) the T-year adjusted annual maximum flood in year t (in cubic feet per second)

Qff(T) the T-year flood from a flood-frequency vector (in cubic feet per second)

Qff,rural the rural flood-frequency vector (2-, 5-, 10-, 25-, 50-, 100-, and 500-year floods) (in cubic feet per second)

Qff,urban the urban flood-frequency vector (2-, 5-, 10-, 25-, 50-, 100-, and 500-year floods) (in cubic feet per second)

Qff, x the x (rural or urban) flood-frequency vector (2-, 5-, 10-, 25-, 50-, 100-, and 500-year floods) (in cubic feet per 

second)

Qff(T) the T-year flood discharge from flood-frequency analysis (in cubic foot per second)

QT the T-year flood discharge (in cubic foot per second)

Qobs(t) the observed annual maximum flood in year t (in cubic foot per second)

Ql lower bound flood-frequency discharge used in piecewise log-normal interpolation of high-outlier discharge 

threshold (in cubic foot per second)

Qmax the largest annual maximum flood in the systematic record of a streamgage (in cubic feet per second)

Qo in flood-frequency analysis, the high-outlier threshold flood discharge (in cubic feet per second)

Qu upper-bound flood-frequency discharge used in piecewise log-normal interpolation of high-outlier discharge 

threshold (in cubic feet per second)

R2 explained variance of a calibrated regression equation

38  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
RI2 the 2-year, 2-hour rainfall depth (used in the Sauer and others (1983) urban equations in inches)

RQT rural discharge for the T-year return period (in cubic feet per second)

Sd standard deviation of imperviousness estimates from a regression equation (in percent)

Sy standard deviation of discharge estimates from a regression equation (in cubic feet per second)

Se standard error of discharge (or imperviousness) estimates from a regression equation (in cubic feet per second or 

in percent)

Se/Sd relative standard error of imperviousness estimated from population density

SL the main channel slope measured between the points that are 10 percent and 85 percent of the main channel 

length upstream from the study site (in feet per mile)

SS station skew option—an input to a Bulletin 17B flood-frequency analysis

ST basin storage—the percentage of the drainage basin occupied by lakes, reservoirs, swamps, and wetlands (in 

percent)

T return period of flood in question (in years)

Tl lower bound return period used in piecewise log-normal interpolation of high-outlier discharge threshold (in 

years)

Tu upper bound return period used in piecewise log-normal interpolation of high-outlier discharge threshold (in 

years)

T(Qx) the return period or frequency of a flood of magnitude Qx (in years)

U urban development (in Pennsylvania Region A flood regression equations (in percent))

UQT urban discharge for the T-year return period (in cubic feet per second)

UQ/RQ ratio of urban peak discharge to rural peak discharge (A value greater than 1 indicates urban amplification of 

flood peak)

x the time series of annual maximum floods reported at a streamgage (reported in cubic feet per second)

y a single, rural flood-frequency value from a statewide rural regression equation calculated for an arbitrary 

streamgage and return period (in cubic feet per second)

ŷ a single, predicted value from an adjusted annual maximum time series calculated for an arbitrary streamgage 

(in cubic feet per second)

the mean of the set of rural flood-frequency values from statewide rural regression equations calculated across 

all such values for a single return period (in cubic feet per second)

the mean of the set of predicted flood-frequency values from adjusted annual maximum time series calculated 

across all such values for a single return period (in cubic feet per second)

z(Tx) the standard normal deviate corresponding to a flood of return period Tx

z(Qx) the standard normal deviate corresponding to a flood of magnitude Qx

Discharge Adjustment Model Definitions:

Null: Urban discharges are a power function of rural discharges. The null model has no dependency on imperviousness, population density, or any other watershed characteristic. Because the model does not depend on any watershed characteristics, it is 

of little practical value by itself but is useful in comparison to other models being calibrated.

Simple Imperviousness: Urban discharges are a power function of rural discharges and percent imperviousness in a watershed.

Simple Density: Urban discharges are a power function of rural discharges and population density in a watershed.

Imperviousness Distribution: Urban discharges are a power function of rural discharges, percentage of imperviousness, and 

the difference in percentage of imperviousness between the 10th and 90th percentiles of the watershed area. This last quantity 

captures the homogeneity of the distribution of imperviousness within a watershed. A watershed with a large difference in percentages of imperviousness between these two percentiles has a wide range of urbanization within its domain. A watershed with 

a small difference is fairly uniform in degree of urbanization.

Density Distribution: Urban discharges are a power function of rural discharges, population density, and the difference in 

population density between the 10th and 90th percentiles of a watershed area. This last quantity captures the homogeneity of the 

distribution of population density within a watershed. A watershed with a large difference in population density between these 

Discharge Adjustment Model Definitions:  39
two percentiles has a wide range of urbanization within its domain. A watershed with a small difference is fairly uniform in 

degree of urbanization.

Scaled Imperviousness: Urban discharges are a power function of rural discharges and scaled imperviousness. This model 

assumes a nonlinear effect of imperviousness such that the marginal effect of increasing imperviousness is greatest close to the 

value .

Scaled Density: Urban discharges are a power function of rural discharges and scaled population density. This model assumes 

a nonlinear effect of population density such that the marginal effect of increasing population density is greatest close to the 

value .

Sauer (null): Analogous to the “null” model described above, urban discharges are a power function of rural discharges for the 

dataset assembled by Sauer and others (1983).

Acknowledgements

The senior author conducted much of the research presented in this report during 2003-04 while on sabbatical at the 

USGS in Reston, Virginia. The author thanks the USGS for financial support during this sabbatical. The investigative study 

summarized in this report benefited from many helpful discussions with current and former USGS personnel, including Steve 

Blanchard, Timothy A. Cohn, Kate Flynn, John R. Gray, William H. Kirby, Harry F. Lins, and Will Thomas. We are especially 

thankful for the insightful technical reviews provided by Kenneth Eng and William H. Kirby, the thorough editorial reivews 

performed by Becky Deckard and Sandy Cooper, and the careful documentation preparation of Annette Ledford.

References

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40 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
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References  41
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42  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
References  43
Appendix 1

Appendix 1: Watershed characteristics of streamgages with only rural indicators. 

[dd, decimal degrees; mi2, square miles; ft/mi, feet per mile; ft, feet; thousand ft, thousands of feet; %, percent; mi2/mi2, square miles per square mile]

Streamgage 

number Streamgage name State Hydrologic region

2-year, 

24-hour 

rainfall 

(inch)

Mean 

annual 

precipitation 

(inch)

Latitude 

(dd)

Area 

(mi2)

Contributing area 

(mi2)

Slope 

(ft/mi)

11023330 LOS PENASQUITOS C BL POWAY C NR POWAY CA South Coast 14 31.2767 

11023340 LOS PENASQUITOS C NR POWAY CA South Coast 13.5 42.3671 

11075800 SANTIAGO C AT MODJESKA CA South Coast 23 13.0287 

11162720 COLMA C AT S SAN FRANCISCO CA Central Coast 22.5 10.8595 

11162800 REDWOOD C AT REDWOOD CITY CA Central Coast 25 1.68449 

11166000 MATADERO C AT PALO ALTO CA Central Coast 20 7.05069 

11183600 WALNUT C AT CONCORD CA Central Coast 23 84.1331 

11336580 MORRISON C NR SACRAMENTO CA Central Coast 18.5 46.8109 

02231280 THOMAS C NR CRAWFORD FL Region A 34.9028 15.7775 

02246300 ORTEGA R AT JACKSONVILLE FL Region A 28.1 12.9732 

02246828 PABLO C AT JACKSONVILLE FL Region A 22.6575 15.0657 

03337000 BONEYARD C AT URBANA IL Region 1 3 4.20883 37.931 

05528500 BUFFALO C NR WHEELING IL Region 1 & 2 2.7 23.8388 19.8662 

05529500 MC DONALD C NR MT PROSPECT IL Region 1 2.7 8.47548 21.7277 

05530000 WELLER C AT DES PLAINES IL Region 1 2.7 12.8585 19.2446 

05531500 SALT C AT WESTERN SPRINGS IL Region 1 2.7 114.996 6.41933 

05532000 ADDISON C AT BELLWOOD IL Region 1 2.7 16.3686 17.4477 

05533000 FLAG C NR WILLOW SPRINGS IL Region 1 2.7 16.8802 19.2316 

05533400 SAWMILL C NR LEMONT IL Region 1 2.75 12.1458 27.5769 

05534500 NB CHICAGO R AT DEERFIELD IL Region 1 2.7 19.6466 10.7933 

05535000 SKOKIE R AT LAKE FOREST IL Region 1 2.7 12.7473 14.6833 

05535500 WF OF NB CHICAGO R AT NORTHBROOK IL Region 1 2.7 11.6297 17.5063 

05536000 NB CHICAGO R AT NILES IL Region 1 2.7 99.2577 5.79078 

05536215 THORN C AT GLENWOOD IL Region 1 2.8 18.2409 18.583 

05536255 BUTTERFIELD C AT FLOSSMOOR IL Region 1 2.8 23.3021 12.023 

05536275 THORN C AT THORNTON IL Region 1 2.8 102.905 10.0069 

05536340 MIDLOTHIAN C AT OAK FOREST IL Region 1 2.8 12.8987 18.1147 

05536500 TINLEY C NR PALOS PARK IL Region 1 2.75 11.2544 21.7328 

05539900 WB DU PAGE R NR W CHICAGO IL Region 2 2.75 29.629 12.3176 

05540060 KRESS C AT W CHICAGO IL Region 2 2.8 19.9741 17.5559 

05540160 E BR DU PAGE R NR DOWNERS GROVE IL Region 1 2.75 25.3031 16.0058 

05540500 DU PAGE R AT SHOREWOOD IL Region 2 2.8 328.971 4.45675 

05550500 POPLAR C AT ELGIN IL Region 2 2.75 34.625 13.7089 

01097300 NASHOBA BROOK NR ACTON MA Eastern 11.3877 32.5419 

01105600 OLD SWAMP R NR S WEYMOUTH MA Eastern 4.83577 34.5835 

01581700 WINTERS RUN NR BENSON MD Piedmont 34.7 

01583500 WESTERN RUN AT WESTERN RUN MD Piedmont 60.3 

01584050 LONG GREEN C AT GLEN ARM MD Piedmont 9.3 

01584500 LITTLE GUNPOWDER FALLS AT LAUREL 

BROOK

MD Piedmont 36.1 

44  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Main 

channel 

length 

(mi)

Main 

channel 

elevation (ft)

Main 

channel 

elevation 

(thousand 

ft)

Basin 

elevation (ft)

Basin 

elevation 

(thousand ft)

Basin 

relief 

(ft)

Storage 

(%)

Lake 

area 

(%)

Forest 

cover 

(%)

Limestone 

(%)

Shape 

index 

(mi2/mi2)

Start 

imperviousness 

(%)

End 

imperviousness 

(%)

Period of 

record (years)

0.51706 6.29 11.98 1970-1993 (23)

0.493116 5.74 15.34 1970-2000 (31)

1.08444 1.06 2.17 1970-2000 (31)

0.239556 23.95 30.62 1970-1996 (26)

0.152659 10.76 10.76 1970-1997 (28)

0.195519 10.73 10.73 1970-2000 (30)

0.357578 10.58 14.29 1970-1997 (24)

0.103499 11.91 12.93 1970-2000 (21)

4.79563 2.59 2.59 1970-2000 (28)

1.7356 4.67 7.54 1970-2000 (31)

24.2448 13.17 14.84 1974-2000 (24)

28.68 28.94 1970-2000 (31)

7.06 27.29 1970-2000 (31)

19.78 24.97 1970-2000 (31)

28.31 28.36 1970-2000 (31)

16.49 21.01 1970-2000 (31)

23.44 24.36 1970-2000 (31)

18.91 21.81 1970-2000 (31)

10.42 18.97 1970-2000 (25)

11.3 12.81 1970-2000 (31)

14.43 17.39 1970-2000 (31)

13.29 15.53 1970-2000 (31)

17.37 18.04 1970-2000 (31)

18.16 18.91 1970-2000 (31)

36.75 36.75 1970-2000 (31)

12.67 14.81 1970-2000 (31)

12.51 22.08 1970-2000 (31)

9.85 20.55 1970-2000 (31)

9.95 22.11 1970-2000 (31)

4.52 9.57 1970-2000 (26)

18.32 23.49 1970-2000 (18)

9.57 9.57 1970-2000 (31)

11.64 16.76 1970-2000 (31)

136.09 6.93514 5.06 6.56 1970-2000 (31)

88.4992 6.17764 15.07 15.97 1970-2000 (31)

31.9 4.79 8.64 1970-2000 (31)

33.6 3.25 4.55 1970-2000 (31)

21.2 5.39 5.8 1970-2000 (25)

31.2 4.13 5.86 1970-2000 (17)

Appendix 1  45
Streamgage 

number Streamgage name State Hydrologic region

2-year, 

24-hour 

rainfall 

(inch)

Mean 

annual 

precipitation 

(inch)

Latitude 

(dd)

Area 

(mi2)

Contributing area 

(mi2)

Slope 

(ft/mi)

01585100 WHITEMARSH RUN AT WHITE MARSH MD Western Coastal Plain 7.6 

01585200 W B HERRING RUN AT IDLEWYLDE MD Piedmont 2.2 

01585500 CRANBERRY B NR WESTMINSTER MD Piedmont 3.4 

01586000 N B PATAPSCO R AT CEDARHURST MD Piedmont 56.3 

01589100 E B HERBERT RUN AT ARBUTUS MD Western Coastal Plain 2.4 

01589300 GWYNNS FALLS AT VILLA NOVA MD Piedmont 32.6 

01589330 DEAD RUN AT FRANKLINTOWN MD Piedmont 5.6 

01589440 JONES FALLS AT SORRENTO MD Piedmont 25.2 

01590500 BACON RIDGE B AT CHESTERFIELD MD Western Coastal Plain 6.9 

01591000 PATUXENT R NR UNITY MD Piedmont 34.8 

01591400 CATTAIL C NR GLENWOOD MD Piedmont 22.9 

01591700 HAWLINGS R NR SANDY SPRING MD Piedmont 26.2 

01639500 BIG PIPE C AT BRUCEVILLE MD Blue Ridge and Great 

Valley

102.8 

01645000 SENECA C AT DAWSONVILLE MD Blue Ridge and Great 

Valley

102 

01649500 NE B ANACOSTIA R AT RIVERDALE MD Western Coastal Plain 73.1 

01651000 NW B ANACOSTIA R NEAR HYATTSVILLE MD Western Coastal Plain 49.3 

01653600 PISCATAWAY C AT PISCATAWAY MD Western Coastal Plain 38.6 

02485650 PURPLE C AT JACKSON MS East 6.11212 27.858 

02485700 HANGING MOSS C NR JACKSON MS East 17.1624 22.3212 

02485950 TOWN C AT JACKSON MS East 9.49435 26.0154 

02486100 LYNCH C AT JACKSON MS East 11.9325 28.9894 

09419663 LAS VEGAS WASH TR S OF NELLIS AIR FORCE 

BASE

NV Region 10 36.19444 1.04392 1.04392 

09419670 RED ROCK WASH NR BLUE DIAMOND NV Region 10 36.15833 8.04635 8.04635 

10311200 ASH CANYON C NR CARSON CITY NV Region 5 39.17639 5.38204 5.38204 

01376500 SAW MILL R AT YONKERS NY Region 3 48 25.2228 15.8832 

04214500 BUFFALO C AT GARDENVILLE NY Region 6 38 142.638 10.6801 

04215500 CAZENOVIA C AT EBENEZER NY Region 6 39 135.705 16.0498 

04218518 ELLICOTT C BELOW WILLIAMSVILLE NY Region 6 35 80.5685 11.4892 

04240010 ONONDAGA C AT SPENCER ST, SYRACUSE NY Region 7 37 127.722 27.3073 

04240100 HARBOR BROOK AT SYRACUSE NY Region 7 37 10.0487 71.1392 

01114000 MOSHASSUCK R AT PROVIDENCE RI - 15.5125 

01117000 HUNT R NR E GREENWICH RI - 23.0741 

01652500 FOURMILE RUN AT ALEXANDRIA VA Northern Piedmont 14.0295 14.0295 37.1694 

01653000 CAMERON RUN AT ALEXANDRIA VA Northern Piedmont 33.9277 18.7489 25.8906 

01654000 ACCOTINK C NR ANNANDALE VA Northern Piedmont 23.8425 0.781281 22.5089 

02037800 FALLING C NR MIDLOTHIAN VA Southern Piedmont 18.7191 13.768 22.3087 

02038000 FALLING C NR CHESTERFIELD VA Southern Piedmont 33.8736 6.35192 15.65 

03052500 SAND RUN NR BUCKHANNON WV North Region 14.3806 

03062400 COBUN CREEK AT MORGANTOWN WV North Region 10.9832 

Appendix 1: Watershed characteristics of streamgages with only rural indicators—continued. 

[dd, decimal degrees; mi2, square miles; ft/mi, feet per mile; ft, feet; thousand ft, thousands of feet; %, percent; mi2/mi2, square miles per square mile]

46 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Main 

channel 

length 

(mi)

Main 

channel 

elevation (ft)

Main 

channel 

elevation 

(thousand 

ft)

Basin 

elevation (ft)

Basin 

elevation 

(thousand ft)

Basin 

relief 

(ft)

Storage 

(%)

Lake 

area 

(%)

Forest 

cover 

(%)

Limestone 

(%)

Shape 

index 

(mi2/mi2)

Start 

imperviousness 

(%)

End 

imperviousness 

(%)

Period of 

record (years)

12.7 13.58 23.82 1970-2000 (29)

15.5 30.06 30.06 1970-2000 (20)

23 4.1 7.09 1970-2000 (31)

25.9 4.78 6.98 1970-2000 (31)

8.3 26.52 26.79 1970-2000 (22)

24.3 15.29 20.51 1970-2000 (23)

5 16.88 21.57 1970-2000 (20)

46.5 7.64 11.09 1970-2000 (23)

52.8 7.93 7.93 1970-1990 (21)

38.2 3.52 5.38 1970-2000 (31)

17.9 3.65 4.99 1979-2000 (22)

26.7 6.22 8.37 1979-2000 (22)

292.6 0 3.54 5.06 1970-2000 (31)

253.8 0 4.8 12.89 1970-2000 (31)

28.4 15.84 18.48 1970-2000 (31)

20.8 22.2 24.96 1970-2000 (31)

49.9 7.15 10.23 1970-2000 (31)

7.32016 10.71 12.13 1970-2000 (28)

8.14158 9.79 10.75 1970-2000 (31)

6.89985 20.63 20.91 1970-2000 (27)

6.70905 16.39 17.54 1970-2000 (29)

883.491 0.46 5.53 1970-1998 (28)

1936.4 0.37 0.64 1970-1999 (30)

2312.27 1.74 2.89 1977-2000 (23)

162.236 0.711219 35.7406 21.1296 19.01 19.89 1970-1999 (24)

288.828 0.879581 42.0923 13.356 3.99 5.37 1970-2000 (31)

424.913 0.507045 56.7619 11.9188 5.03 5.2 1970-2000 (31)

356.182 1.21751 33.944 20.8887 8.5 8.5 1973-2000 (28)

393.303 0.762828 51.5949 5.96086 8.22 8.22 1971-2000 (30)

310.797 0.296921 35.3556 5.02514 11.9 12.24 1970-2000 (31)

0.152314 29.8198 22.88 24.31 1970-2000 (31)

0.154381 62.5405 8.07 8.88 1970-2000 (31)

8.22381 122.926 20.4007 33.81 37.05 1970-2000 (30)

12.1626 129.507 32.3356 19.85 19.85 1970-2000 (31)

11.1018 153.17 38.1093 20.53 24.51 1970-2000 (30)

9.67723 133.053 61.0473 3.7 13.32 1970-1993 (23)

14.5847 128.416 61.2425 4.64 13.87 1970-2000 (29)

0 2.07 1970-2000 (31)

0 5.95 1970-2000 (31)

Appendix 1  47
Appendix 2

Appendix 2: Actual and estimated rural discharges for streamgages with only rural indicators.

[ft3/s, cubic feet per second; italics, numbers in italics satisfy equation 19 and indicate their use in the regression equation analyses in this study]

Streamgage 

number Streamgage name

2-year 

actual 

(ft3/s)

2-year 

rural 

(ft3/s)

5-year 

actual 

(ft3/s)

5-year 

rural 

(ft3/s)

11023330 LOS PENASQUITOS C BL POWAY C NR POWAY 775 120 1757 490 

11023340 LOS PENASQUITOS C NR POWAY 1027 141 2352 582 

11075800 SANTIAGO C AT MODJESKA 338 143 1188 578 

11162720 COLMA C AT S SAN FRANCISCO 1775 717 2524 1385 

11162800 REDWOOD C AT REDWOOD CITY 218 277 348 445 

11166000 MATADERO C AT PALO ALTO 400 447 746 872 

11183600 WALNUT C AT CONCORD 4005 3209 7874 6789 

11336580 MORRISON C NR SACRAMENTO 1115 4170 1809 6871 

02231280 THOMAS C NR CRAWFORD 594 552 1095 1061 

02246300 ORTEGA R AT JACKSONVILLE 802 663 1421 1260 

02246828 PABLO C AT JACKSONVILLE 418 144 647 297 

03337000 BONEYARD C AT URBANA 644 451 814 805 

05528500 BUFFALO C NEAR WHEELING 447 540 602 904 

05529500 MC DONALD C NEAR MOUNT PROSPECT 237 322 380 543 

05530000 WELLER C AT DES PLAINES 892 422 1105 708 

05531500 SALT C AT WESTERN SPRINGS 1425 1407 1873 2255 

05532000 ADDISON C AT BELLWOOD 505 488 666 814 

05533000 FLAG C NR WILLOW SPRINGS 779 524 1085 876 

05533400 SAWMILL C NR LEMONT 707 558 1075 956 

05534500 NB CHICAGO R AT DEERFIELD 417 447 583 734 

05535000 SKOKIE RIVER AT LAKE FOREST 277 368 377 612 

05535500 WF OF NB CHICAGO R AT NORTHBROOK 542 373 712 623 

05536000 NB CHICAGO R AT NILES 1270 1192 1632 1905 

05536215 THORN C AT GLENWOOD 1275 721 1752 1225 

05536255 BUTTERFIELD C AT FLOSSMOOR 801 709 1254 1188 

05536275 THORN C AT THORNTON 2024 2099 2884 3473 

05536340 MIDLOTHIAN C AT OAK FOREST 232 541 323 921 

05536500 TINLEY C NR PALOS PARK 661 469 969 796 

05539900 WB DU PAGE R NR WEST CHICAGO 559 419 721 697 

05540060 KRESS CREEK AT WEST CHICAGO 289 412 398 699 

05540160 E BR DU PAGE R NR DOWNERS GROVE 612 768 863 1287 

05540500 DU PAGE R AT SHOREWOOD 3697 1948 4965 3128 

05550500 POPLAR C AT ELGIN 539 499 735 832 

01097300 NASHOBA BROOK NR ACTON 202 191 329 283 

01105600 OLD SWAMP R NR S WEYMOUTH 161 106 249 159 

01581700 WINTERS RUN NR BENSON 2610 1590 4792 2850 

01583500 WESTERN RUN AT WESTERN RUN 2332 2230 4260 3940 

01584050 LONG GREEN C AT GLEN ARM 695 743 1475 1380 

48  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
10-year 

actual 

(ft3/s)

10-year 

rural 

(ft3/s)

25-year 

actual 

(ft3/s)

25-year 

rural 

(ft3/s)

50-year 

actual 

(ft3/s)

50-year 

rural 

(ft3/s)

100-year 

actual 

(ft3/s)

100-year 

rural 

(ft3/s)

500-year 

actual 

(ft3/s)

500-year 

rural 

(ft3/s)

2639 969 4008 2123 5206 3331 6549 4724 10260 9586

3587 1156 5582 2542 7393 3994 9493 5678 15610 11575

2116 1156 3700 2566 5151 4069 6802 5779 11310 11754

2980 1871 3511 2511 3877 3040 4219 3521 4945 4738

423 553 504 684 555 776 598 866 678 1080

1018 1195 1401 1631 1711 1993 2040 2332 2878 3205

10560 9470 13830 13068 16120 16320 18250 19040 22600 26013

2249 8621 2766 10857 3119 12750 3446 14123 4120 17373

1553 1476 2309 2082 3022 2580 3884 3116 6625 4428

1882 1742 2505 2442 2992 3016 3493 3633 4713 5192

807 427 1018 622 1180 784 1345 961 1744 1392

906 1062 1005 1395 1069 1652 1126 1903 1238 2502

700 1150 820 1456 908 1678 994 1888 1190 2367

494 693 660 880 800 1017 956 1146 1386 1443

1221 901 1346 1142 1427 1318 1500 1483 1645 1863

2202 2806 2655 3486 3021 3961 3413 4410 4442 5426

780 1034 931 1308 1049 1507 1172 1695 1481 2125

1306 1115 1605 1414 1843 1631 2094 1837 2738 2309

1329 1231 1658 1581 1907 1840 2160 2089 2764 2666

692 924 826 1158 924 1325 1020 1482 1242 1839

437 775 504 977 550 1123 593 1260 681 1573

830 792 987 1002 1110 1154 1237 1298 1557 1626

1869 2367 2168 2934 2391 3328 2615 3701 3146 4541

2072 1573 2481 2013 2790 2338 3101 2651 3848 3377

1627 1512 2194 1919 2693 2216 3263 2499 4927 3154

3500 4400 4333 5573 4992 6418 5684 7234 7449 9117

389 1182 481 1512 555 1755 635 1989 843 2530

1179 1022 1450 1305 1655 1515 1863 1714 2360 2176

817 880 926 1108 1001 1269 1072 1421 1221 1765

474 893 574 1136 652 1313 732 1480 932 1864

1062 1640 1354 2084 1603 2408 1882 2718 2669 3431

5649 3884 6371 4821 6823 5466 7215 6077 7954 7457

853 1053 989 1328 1082 1524 1169 1709 1354 2128

429 359 576 471 701 567 838 676 1217 963

308 204 382 270 436 326 490 391 614 564

6478 3970 8828 5690 10710 7210 12690 8970 17680 14200

5852 5440 8227 7740 10260 9760 12530 12100 18810 19000

2207 1960 3416 2870 4550 3680 5903 4630 10080 7490

Appendix 2  49
Streamgage 

number Streamgage name

2-year 

actual 

(ft3/s)

2-year 

rural 

(ft3/s)

5-year 

actual 

(ft3/s)

5-year 

rural 

(ft3/s)

01584500 LITTLE GUNPOWDER FALLS AT LAUREL BROOK 3856 1630 6022 2930 

01585100 WHITEMARSH RUN AT WHITE MARSH 1559 625 2248 1116 

01585200 WB HERRING RUN AT IDLEWYLDE 718 314 1141 606 

01585500 CRANBERRY B NR WESTMINSTER 380 386 845 740 

01586000 NB PATAPSCO R AT CEDARHURST 2685 2250 4232 3960 

01589100 EB HERBERT RUN AT ARBUTUS 663 329 932 602 

01589300 GWYNNS FALLS AT VILLA NOVA 1940 1610 3303 2880 

01589330 DEAD RUN AT FRANKLINTOWN 1587 654 2336 1220 

01589440 JONES FALLS AT SORRENTO 1111 1190 2006 2180 

01590500 BACON RIDGE B AT CHESTERFIELD 196 240 311 457 

01591000 PATUXENT R NR UNITY 1491 1530 2896 2750 

01591400 CATTAIL C NR GLENWOOD 2169 1360 3413 2450 

01591700 HAWLINGS R NR SANDY SPRING 1329 1370 2625 2480 

01639500 BIG PIPE C AT BRUCEVILLE 3318 3864 5544 6874 

01645000 SENECA C AT DAWSONVILLE 4031 3366 7435 5876 

01649500 NE B ANACOSTIA R AT RIVERDALE 4953 2204 7121 4039 

01651000 NW B ANACOSTIA R NR HYATTSVILLE 3645 2063 5594 3670 

01653600 PISCATAWAY C AT PISCATAWAY 975 922 1852 1770 

02485650 PURPLE C AT JACKSON 1091 692 1416 1168 

02485700 HANGING MOSS C NR JACKSON 2771 1527 3693 2638 

02485950 TOWN C AT JACKSON 2725 1008 3288 1718 

02486100 LYNCH C AT JACKSON 3316 1229 3883 2118 

09419663 LAS VEGAS WASH TR S OF NELLIS AIR FORCE BASE 1.40 12.0 24.3 87.0 

09419670 RED ROCK WASH NR BLUE DIAMOND 5.40 40.0 112 291 

10311200 ASH CANYON C NR CARSON CITY 14.3 2.10 38.3 36.0 

01376500 SAW MILL R AT YONKERS 687 852 965 1371 

04214500 BUFFALO C AT GARDENVILLE 6251 4005 7938 5747 

04215500 CAZENOVIA C AT EBENEZER 7115 4541 9333 6674 

04218518 ELLICOTT C BELOW WILLIAMSVILLE 1787 2190 2347 3204 

04240010 ONONDAGA C AT SPENCER ST, SYRACUSE 1907 6132 2643 9086 

04240100 HARBOR BROOK AT SYRACUSE 227 950 342 1453 

01114000 MOSHASSUCK R AT PROVIDENCE 913 369 1116 632 

01117000 HUNT R NR E GREENWICH 434 334 629 564 

01652500 FOURMILE RUN AT ALEXANDRIA 3048 1010 3918 1737 

01653000 CAMERON RUN AT ALEXANDRIA 3908 1221 5077 2093 

01654000 ACCOTINK C NR ANNANDALE 2651 1429 3961 2444 

02037800 FALLING C NR MIDLOTHIAN 534 380 852 647 

02038000 FALLING C NR CHESTERFIELD 830 477 1374 803 

03052500 SAND RUN NEAR BUCKHANNON 858 951 1472 1518 

03062400 COBUN CREEK AT MORGANTOWN 470 782 855 1264 

Appendix 2: Actual and estimated rural discharges for streamgages with only rural indicators—continued.

[ft3/s, cubic feet per second; italics, numbers in italics satisfy equation 19 and indicate their use in the regression equation analyses in this study]

50 Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
10-year 

actual 

(ft3/s)

10-year 

rural 

(ft3/s)

25-year 

actual 

(ft3/s)

25-year 

rural 

(ft3/s)

50-year 

actual 

(ft3/s)

50-year 

rural 

(ft3/s)

100-year 

actual 

(ft3/s)

100-year 

rural 

(ft3/s)

500-year 

actual 

(ft3/s)

500-year 

rural 

(ft3/s)

7367 4080 8930 5850 9994 7410 10970 9210 12990 14600

2736 1545 3385 2220 3892 2836 4419 3560 5738 5816

1458 881 1899 1320 2256 1710 2636 2180 3622 3600

1250 1070 1862 1590 2385 2060 2959 2610 4493 4280

5494 5480 7386 7800 9033 9840 10900 12200 16260 19200

1128 847 1395 1239 1609 1594 1836 2006 2423 3291

4502 4020 6420 5760 8187 7310 10280 9090 16770 14500

2882 1740 3628 2560 4224 3300 4854 4160 6474 6800

2811 3060 4119 4410 5338 5610 6797 6980 11360 11100

388 668 484 1040 553 1410 621 1880 774 3510

4225 3840 6473 5510 8643 6990 11310 8690 20030 13800

4183 3430 5074 4940 5677 6290 6230 7850 7357 12600

3719 3470 5364 5000 6775 6350 8342 7920 12640 12600

7839 9593 12080 13947 16550 17846 22520 22395 45190 36241

10440 8092 15210 11526 19550 14499 24640 18017 40030 28327

8661 5701 10720 8387 12340 10951 14020 13983 18270 23874

7167 5089 9510 7291 11540 9277 13820 11645 20350 18755

2783 2590 4550 4020 6459 5420 9050 7200 19140 13300

1622 1540 1873 1983 2056 2348 2235 2647 2645 3476

4192 3501 4721 4508 5053 5333 5343 6018 5894 7902

3590 2270 3913 2918 4121 3452 4306 3889 4674 5097

4195 2811 4539 3615 4767 4279 4975 4819 5403 6320

89.8 205 318 411 673 607 1262 876 3987 1837

485 729 2106 1551 5179 2386 11260 3583 49570 8164

69.9 181 142 1072 232 3026 372 7003 1043 38288

1167 1795 1441 2433 1660 2959 1891 3531 2488 5142

8897 6940 9968 8480 10680 9696 11340 10974 12680 13982

10680 8154 12280 10065 13390 11583 14460 13181 16800 17015

2738 3898 3255 4786 3658 5478 4076 6194 5125 7844

3153 11177 3823 13775 4341 15704 4874 17595 6195 22190

420 1801 519 2218 594 2507 668 2771 845 3402

1248 694 1416 1015 1541 1385 1667 1830 1966 3219

779 628 993 919 1171 1253 1367 1656 1900 2912

4432 2381 5024 3394 5431 4299 5813 5349 6630 8455

5819 2867 6728 4086 7387 5176 8035 6441 9522 10197

4786 3345 5770 4765 6459 6036 7112 7514 8518 11909

1068 862 1341 1217 1542 1552 1740 1923 2193 3054

1735 1061 2177 1479 2491 1868 2792 2293 3443 3576

2011 1944 2869 2536 3657 3012 4588 3518 7455 4795

1238 1631 1919 2143 2613 2556 3510 2997 6712 4120

Appendix 2  51
Appendix 3

Appendix 3. Plots of predicted and observed flood discharges. Predicted discharges are from calibrated simple imperviousness 

adjustment model. Observed are from appropriate U.S. Geological Survey rural regression equations. 10 0 10 2 10 4 10 0 10 1 10 2 10 3 10 4 2−year 10 1 10 2 10 3 10 4 10 1 10 2 10 3 10 4

5−year

10 2 10 3 10 4 10 5

10 210 310 410 5

10 −year

10 2 10 3 10 4 10 5

10 210 310 410 5

25 −year

Predicted Discharge, in cubic feet per second 10 2 10 3 10 4 10 5 10 2 10 3 10 4 10 5 50 −year 10 2 10 3 10 4 10 5 10 2 10 3 10 4 10 5 100 −year

10 2 10 3 10 4 10 5

10 210 310 410 5

500 −year

Observed Discharge, in cubic feet per second

52  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Appendix 4

Appendix 4. Calibrated (x) and smoothed (o) coefficient and exponents for the discharge adjustment models and return period. 100 101 102 103 0.25 0.3 0.35

c

2 = 1.1

0.40.30.2

1.3

1.21.1 0

0.40.2

0.40.30.2

c = 1.1

2

0

0.20.1

0

0.5

1

c

0

0.20.1

0

0.20.1

0.25

0.3

0.35

c = 1.1

2

0

0.40.2

0

0.20.1

0.28

0.3

0.32

c

2 = 1.15

0

0.20.1

0

0.40.2

0

4020

0.28

0.3

0.32

c

2 = 1.15

0

0.20.1

−1

10

420

100 101 102 103

100 101 102 103

100 101 102 103

100 101 102 103100 101 102 103

100 101 102 103

100 101 102 103

c = 1.1

2

Null

Simple

imperviousness

Simple density

Imperviousness

distribution

Density

distribution

Scaled

imperviousness

Scaled

density

100 101 102 103

100 101 102 103

100 101 102 103

100 101 102 103

100 101 102 103100 101 102 103 100 101 102 103

100 101 102 103

100 101 102 103

100 101 102 103

100 101 102 103100 101 102 103

c

Return period (years)

c

c

c I* or PD* 1 2

3

4

Appendix 4  53
Appendix 5

Appendix 5. Watershed characteristics for 10 U.S. Geological Survay streamgages with localized urban equations. 

[mi2, square miles; ft/mi, feet per mile; %, percent]

Streamgage 

number Streamgage name State Area (mi2)

Slope 

(ft/mi)

Storage 

(%)

Impervious 

area 

(%)

Forest 

cover 

(%)

Urban 

area 

(%)

Carbonate 

area 

(%)

Controlled 

area 

(%)

01393450 ELIZABETH R AT URSINO LAKE 

AT ELIZABETH NJ

NJ 17.6 22.4 1.6 41.9

01394500 RAHWAY R NR SPRINGFIELD NJ NJ 25.3 30.6 1.3 22.2

01407705 SHARK R NR NEPTUNE CITY NJ NJ 9.8 29.4 17.6 9.0

01407760 JUMPING BROOK NR NEPTUNE 

CITY NJ

NJ 6.3 35.2 8.4 10.1

01465798 POQUESSING C AT GRANT AVE 

AT PHILADELPHIA PA

PA 21.1 16.6 66.5 1.1 14.9

01467048 PENNYPACK C AT LOWER 

RHAWN ST BDG, PHILA PA

PA 50.0 29.2 59.4 1.5 6.1

01467087 FRANKFORD C AT CASTOR AVE, 

PHILADELPHIA PA

PA 29.8 17.4 77.2 0.0 0.0

03049800 LITTLE PINE C NR ETNA PA PA 5.7 79.8 8.2 0.0 0.0

04087204 OAK C AT S MILWAUKEE WI WI 25.4 16.8

04087220 ROOT R NR FRANKLIN WI WI 49.8 19.6

54  Methods for Adjusting USGS Rural Regression Peak Discharges in an Urban Setting
Appendix 6

Appendix 6. Rural and urban discharges, in cubic feet per second for 10 U.S. Geological Survey streamgages with localized urban 

equations.

[Entries in italics were not available directly from the regression equations and were determined from log-normal extrapolation.]

Streamgage 

number Streamgage name State Return Period (years)

2 5 10 25 50 100 500

Urban Discharges

01393450 ELIZABETH R AT URSINO LAKE AT 

ELIZABETH NJ

NJ 1,071 1,516 1,992 2,552 2,993 3,528 4,922

01394500 RAHWAY R NR SPRINGFIELD NJ NJ 1,452 2,088 2,784 3,580 4,219 5,016 7,120

01407705 SHARK R NR NEPTUNE CITY NJ NJ 157 244 338 460 568 694 1,039

01407760 JUMPING BROOK NR NEPTUNE CITY NJ NJ 166 254 350 476 584 715 1,075

01465798 POQUESSING C AT GRANT AVE AT 

PHILADELPHIA PA

PA 2,739 3,984 4,845 5,970 7,009 8,140 11,180

01467048 PENNYPACK C AT LOWER RHAWN ST 

BDG, PHILADELPHIA PA

PA 5,019 7,095 8,503 10,312 11,971 13,742 18,381

01467087 FRANKFORD C AT CASTOR AVE, PHILADELPHIA PA

PA 4,520 6,213 7,337 8,761 10,060 11,433 14,968

03049800 LITTLE PINE C NR ETNA PA PA 347 611 821 1,124 1,415 1,752 2,742

04087204 OAK C AT S MILWAUKEE WI WI 941 1,301 1,585 1,912 2,207 2,440 2,991

04087220 ROOT R NR FRANKLIN WI WI 1,847 2,518 3,062 3,691 4,266 4,716 5,777

Rural Discharges

01393450 ELIZABETH R AT URSINO LAKE AT 

ELIZABETH NJ

NJ 550 863 1,217 1,672 2,104 2,674 4,343

01394500 RAHWAY R NR SPRINGFIELD NJ NJ 903 1,411 1,993 2,713 3,386 4,287 6,912

01407705 SHARK R NR NEPTUNE CITY NJ NJ 91 152 220 313 405 517 850

01407760 JUMPING BROOK NR NEPTUNE CITY NJ NJ 97 159 229 327 421 540 896

01465798 POQUESSING C AT GRANT AVE AT 

PHILADELPHIA PA

PA 1,132 2,089 2,879 4,051 5,220 6,626 11,062

01467048 PENNYPACK C AT LOWER RHAWN ST 

BDG, PHILADELPHIA PA

PA 2,237 3,932 5,281 7,233 9,142 11,383 18,204

01467087 FRANKFORD C AT CASTOR AVE, PHILADELPHIA PA

PA 1,676 3,012 4,090 5,670 7,227 9,075 14,791

03049800 LITTLE PINE C NR ETNA, PA PA 303 553 757 1,059 1,415 1,752 2,737

04087204 OAK C AT S MILWAUKEE, WI WI 454 651 792 966 1,095 1,219 1,514

04087220 ROOT R NR FRANKLIN, WI WI 796 1,127 1,360 1,645 1,859 2,062 2,544

Appendix 6  55

Moglen, Glenn E. and Shivers, Dorianne E.—Methods for Adjustuing USGS Rural Regression Peak Discharges—Scientific Investigations Report 2006-5270

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