U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2014–5090

Prepared in cooperation with the Virginia Department of Transportation

Methods and Equations for Estimating Peak Streamflow 

Per Square Mile in Virginia’s Urban Basins 

KENTUCKY

WEST VIRGINIA

TENNESSEE

NORTH CAROLINA

MARYLAND

WASHINGTON, D.C.

Chesapeake Bay

ATLANTIC OCEAN
Cover. Map showing selected urban peak-flow study sites used to develop response surface models and estimating equations. 

Taken from figure 1 (p. 4).
Methods and Equations for Estimating Peak Streamflow 

Per Square Mile in Virginia’s Urban Basins 

By Samuel H. Austin

Prepared in cooperation with the Virginia Department of Transportation

Scientific Investigations Report 2014–5090

U.S. Department of the Interior

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

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2014

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as noted in the text. Permission to reproduce copyrighted items must be secured from the copyright owner.

Suggested citation:

Austin, S.H., 2014, Methods and equations for estimating peak streamflow per square mile in Virginia’s urban basins: 

U.S. Geological Survey Scientific Investigations Report 2014–5090, 25 p., http://dx.doi.org/10.3133/sir20145090.

ISSN 2328-0328 (online)

ISSN 2328-031X (print)

ISBN 978-1-4113-3810-4
iii

Acknowledgments

The author thanks the Virginia Department of Transportation, whose continued collaboration and 

assistance made this report possible. George E. Harlow, Jr. of the U.S. Geological Survey is thanked 

for his advice, review, and editing of this report, and the author appreciates Jennifer Krstolic and 

Brian A. Hasty of the U.S. Geological Survey for developing a map for this report.
iv

Contents

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

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

Purpose and Scope ..............................................................................................................................1

Characteristics of Streamflow in Urban Basins ..............................................................................1

Why Predict Peak Flows in Urban Areas?........................................................................................2

Previous Studies ...................................................................................................................................2

Study Objectives ............................................................................................................................................2

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

Site Selection.........................................................................................................................................3

Data Evaluation .....................................................................................................................................3

Data Analysis.........................................................................................................................................3

Model Development .............................................................................................................................5

Model Verification and Diagnostics ..................................................................................................6

Model Limitations..................................................................................................................................8

Urban Peak Flow Per Square Mile Estimates ..........................................................................................8

Examples of Applying the Equations ........................................................................................................17

Example 1 .............................................................................................................................................17

Example 2 .............................................................................................................................................17

Summary........................................................................................................................................................18

Selected References...................................................................................................................................19

Figures

1. Map showing selected urban peak-flow study sites used to develop 

response surface models and estimating equations..............................................................4

 Graphs showing:

2. Summary statistics and frequency distributions of response surface model 

explanatory variables...................................................................................................................5

3. Examples of regressions prepared during initial tests of bivariate relations 

for 0.995, 0.50, and 0.10 urban peak-flow probability categories, showing 

urban basin peak flow per square mile as explained by Log10-basin 

drainage area ................................................................................................................................6

4. Example leverage plots showing significant explanatory power, insignificant 

explanatory power and, evidence of collinearity....................................................................7

5. Plots for the 0.8 model of urban peak flow per square mile showing actualby-predicted values, drainage area explanatory power, and urban area 

explanatory power........................................................................................................................8

6. How to read three-dimensional response surface model plots ...........................................9

7. Urban peak flow per square mile response surface model for 0.995 annual 

exceedance probability streamflow in Virginia.....................................................................11

8. Urban peak flow per square mile response surface model for 0.99 annual 

exceedance probability streamflow in Virginia.....................................................................11
v

9. Urban peak flow per square mile response surface model for 0.95 annual 

exceedance probability streamflow in Virginia.....................................................................11

10. Urban peak flow per square mile response surface model for 0.9 annual 

exceedance probability streamflow in Virginia.....................................................................11

11. Urban peak flow per square mile response surface model for 0.8 annual 

exceedance probability streamflow in Virginia.....................................................................12

12. Urban peak flow per square mile response surface model for 0.67 annual 

exceedance probability streamflow in Virginia.....................................................................12

13. Urban peak flow per square mile response surface model for 0.5 annual 

exceedance probability streamflow in Virginia.....................................................................12

14. Urban peak flow per square mile response surface model for 0.43 annual 

exceedance probability streamflow in Virginia....................................................................12

15. Urban peak flow per square mile response surface model for 0.2 annual 

exceedance probability streamflow in Virginia....................................................................13

16. Urban peak flow per square mile response surface model for 0.1 annual 

exceedance probability streamflow in Virginia....................................................................13

17. Urban peak flow per square mile response surface model for 0.04 annual 

exceedance probability streamflow in Virginia....................................................................13

18. Urban peak flow per square mile response surface model for 0.02 annual 

exceedance probability streamflow in Virginia....................................................................13

19. Urban peak flow per square mile response surface model for 0.01 annual 

exceedance probability streamflow in Virginia....................................................................14

20. Urban peak flow per square mile response surface model for 0.005 annual 

exceedance probability streamflow in Virginia....................................................................14

21. Urban peak flow per square mile response surface model for 0.002 annual 

exceedance probability streamflow in Virginia....................................................................14

22. Leverage plots indicating statistically significant cross-product effects in 

response surface models for 0.995, 0.50 and 0.10 urban peak flow probability 

categories, and insignificant cross-product effects in response surface 

models for 0.01, 0.005, and 0.002 urban peak-flow probability categories........................15

23. Urban peak flow per square mile actual-by-predicted plot for 0.995 annual 

exceedance probability streamflow in Virginia....................................................................15

24. Urban peak flow per square mile actual-by-predicted plot for 0.99 annual 

exceedance probability streamflow in Virginia....................................................................15

25. Urban peak flow per square mile actual-by-predicted plot for 0.95 annual 

exceedance probability streamflow in Virginia....................................................................15

26. Urban peak flow per square mile actual-by-predicted plot for 0.9 annual 

exceedance probability streamflow in Virginia....................................................................16

27. Urban peak flow per square mile actual-by-predicted plot for 0.8 annual 

exceedance probability streamflow in Virginia....................................................................16

28. Urban peak flow per square mile actual-by-predicted plot for 0.67 annual 

exceedance probability streamflow in Virginia....................................................................16

29. Urban peak flow per square mile actual-by-predicted plot for 0.5 annual 

exceedance probability streamflow in Virginia....................................................................16

30. Urban peak flow per square mile actual-by-predicted plot for 0.43 annual 

exceedance probability streamflow in Virginia....................................................................16

31. Urban peak flow per square mile actual-by-predicted plot for 0.2 annual 

exceedance probability streamflow in Virginia....................................................................16
vi

32. Urban peak flow per square mile actual-by-predicted plot for 0.1 annual 

exceedance probability streamflow in Virginia....................................................................16

33. Urban peak flow per square mile actual-by-predicted plot for 0.04 annual 

exceedance probability streamflow in Virginia....................................................................16

34. Urban peak flow per square mile actual-by-predicted plot for 0.02 annual 

exceedance probability streamflow in Virginia....................................................................16

35. Urban peak flow per square mile actual-by-predicted plot for 0.01 annual 

exceedance probability streamflow in Virginia.....................................................................17

36. Urban peak flow per square mile actual-by-predicted plot for 0.005 annual 

exceedance probability streamflow in Virginia.....................................................................17

37. Urban peak flow per square mile actual-by-predicted plot for 0.002 annual 

exceedance probability streamflow in Virginia.....................................................................17

Tables

1. Drainage basins used to estimate Virginia urban peak-flow discharge 

per square mile ...........................................................................................................................21

2. Parameters for regression equations used for estimating Virginia urban 

peak-flow discharge per square mile .....................................................................................24

3. Regression equation prediction expressions for estimating Virginia urban 

peak-flow discharge per square mile .....................................................................................25

Equations

1. Form of a standard least squares regression equation .........................................................6

2 Form of each response surface model prediction expression .............................................6
vii

Conversion Factors

Inch/Pound to SI

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

acre 0.4047 hectare (ha)

square foot (ft2

) 0.09290 square meter (m2)

square mile (mi2

) 259.0 hectare (ha)

Volume

gallon (gal) 0.003785 cubic meter (m3

) 

cubic foot (ft3

) 0.02832 cubic meter (m3) 

acre-foot (acre-ft) 1,233 cubic meter (m3

)

Flow rate

acre-foot per day (acre-ft/d) 0.01427 cubic meter per second (m3

/s)

foot per second (ft/s) 0.3048 meter per second (m/s)

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)

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. 

Methods and Equations for Estimating Peak Streamflow 

Per Square Mile in Virginia’s Urban Basins

By Samuel H. Austin

Abstract 

Models are presented that describe Virginia urban area 

annual peak streamflow per square mile based on basin 

percent urban area and basin drainage area. Equations are 

provided to estimate Virginia urban peak flow per square mile 

of basin drainage area in each of the following annual exceedance probability categories: 0.995, 0.99, 0.95, 0.9, 0.8, 0.67, 

0.5, 0.43, 0.2, 0.1, 0.04, 0.02, 0.01, 0.005, and 0.002 (recurrence intervals of 1.005, 1.01, 1.05, 1.11, 1.25, 1.49, 2.0, 2.3, 

5, 10, 25, 50, 100, 200, and 500 years, respectively). Equations 

apply to Virginia drainage basins ranging in size from no less 

than 1.2 mi2

 to no more than 2,400 mi2 containing at least 

10 percent urban area, and not more than 96 percent urban 

area. A total of 115 Virginia drainage basins were analyzed. 

Actual-by-predicted plots and leverage plots for response 

variables and explanatory variables in each peak-flow annual 

exceedance probability category indicate robust model fits and 

significant explanatory power. Equations for 8 of 15 urban 

peak-flow response surface models yield R-square values 

greater than 0.8. Relations identified in statistical models, 

describing significant increases in urban peak stream discharges as basin urban area increases, affirm empirical relations reported in past studies of change in stream discharge, 

lag times, and physical streamflow processes, most notably 

those detailed for urban areas in northern Virginia.

Introduction 

Anticipating high-flow conditions in Virginia’s urban 

streams is essential to sound management of water resources 

and associated riparian and watershed ecosystems. Reliable 

predictions of urban peak flows inform planning and design, 

improving management of urban basins. Understanding peak 

flows specific to urban basins and their relation to land-use 

changes over time allows anticipation of shifts in peak-flow 

magnitude and frequency. This knowledge can improve the 

design of hydraulic structures and the management of riparian 

systems and the ecology associated with urban landscapes. 

Urban peak-flow estimates also provide a basis for analysis 

of future streamflow response to changes in ecosystem 

and climate.

The U.S. Geological Survey, in cooperation with the 

Virginia Department of Transportation, developed statistical models and equations to predict flow response in 

Virginia’s urban streams. Basin drainage area and basin 

percent urban area were used to identify and predict peak 

urban streamflow, which aids in the understanding of water 

flows and the environmental health of urban basins and their 

associated ecosystems.

Purpose and Scope

This report presents results of a study conducted to 

(1) develop a set of response surface models that describe 

Virginia urban area peak flow based on changes in basin percent urban area and basin drainage area; (2) develop a series 

of equations that may be used to accurately estimate Virginia 

urban peak streamflow per square mile of basin drainage area 

in the following annual exceedance probability (AEP) categories: 0.995, 0.99, 0.95, 0.9, 0.8, 0.67, 0.5, 0.43, 0.2, 0.1, 0.04, 

0.02, 0.01, 0.005, and 0.002 (recurrence intervals of 1.005, 

1.01, 1.05, 1.11, 1.25, 1.49, 2.0, 2.3, 5, 10, 25, 50, 100, 200, 

and 500 years, respectively), and (3) demonstrate how to use 

these equations to estimate urban peak streamflow per square 

mile of basin drainage area. Measurements of basin percent 

urban area and basin square mile drainage area are used as 

explanatory variables in 15 response surface models. Measurements and models are evaluated and iteratively refined to prepare a set of 15 regression equations. Examples are provided 

describing equation applicability and use.

Characteristics of Streamflow in Urban Basins

Streams in urban areas serve important functions. They 

are sources of water withdrawals for industrial, agricultural, 

and domestic uses, and they are essential to the maintenance 

and functioning of the physical and ecological processes 

associated with the surrounding urban landscape and within 

the larger basin. Streamflow through urban streams maintains 

channel morphology, channel functions, and movement of natural sediments. This transport of rainfall and runoff also helps 

to prevent flooding. These same flows provide habitat and 

food sources for aquatic organisms, as well as transportation 
2 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

corridors and water sources essential to aquatic organisms and 

terrestrial plants and animals.

The effects of urbanization upon stream systems are 

significant. In their book “Water in Environmental Planning,” 

Dunne and Leopold (1978) describe the changes to stream 

systems that occur as basin urban area increases. Changes in 

runoff processes result from covering parts of the basin with 

impervious roofs, sidewalks, roadways, and parking lots. The 

infiltration capacity of these areas drops to zero, and the volume and rate of overland flow is increased. Because urban gutters, drains, and storm sewers are laid out in a manner meant 

to convey runoff rapidly to stream channels, and natural channels are often straightened, deepened, or lined with concrete 

to make them hydraulically smoother, channel efficiencies are 

increased and flood waters are transmitted downstream more 

quickly and with less storage in the channel than would be 

available in a natural stream system. Stormwaters accumulate 

downstream more quickly than in natural stream systems, 

producing higher flood peaks. Increased water velocities in 

stream channels decrease the lag between rainfall and runoff. 

Decreased lag times ensure that peak-flow rates must increase, 

even if total water volumes remain constant. In fact, total 

water volumes actually increase because net infiltration of precipitation is reduced as a consequence of impervious surfaces. 

This increase in total water volume drives flood peaks even 

higher than they would be if water volumes remained constant.

The increased peak flows in urbanizing basins lead to significant consequences that must be identified and anticipated 

by infrastructure planners, designers, and managers. Overbank 

flooding, as peak flows exceed the capacity of natural channels, may cause damage to structures or landscapes, or may 

disrupt traffic. Areas near streams may suffer erosion and 

reduced property values. Culverts and bridges may be overtaxed and may eventually fail if not designed with sufficient 

capacity to handle larger water flows. Because water conveyed 

during increased peak flows is not available to infiltrate and 

recharge groundwater supplies, low flows during dry months 

are often reduced, resulting in degraded water quality. Less 

water is available for withdrawal and use, and the efficacy of 

stream processes, such as transfer of sediments, is limited.

Why Predict Peak Flows in Urban Areas?

Predictions of urban streamflows allow us to better 

protect against floods, anticipate water availability during critical periods, and facilitate improved planning and management 

of water resources. The benefits of accurate flow predictions 

include (1) extended lead-times for designing and building 

urban infrastructure and flow control structures, (2) improved 

knowledge and prediction of high flows during storms, 

(3) anticipation of future high flows as functions of change in 

urban land area, regional and local precipitation, water withdrawals, land-use management, and climate.

Previous Studies

Previous studies have investigated the effects of urban 

development on streamflows in Virginia. Anderson (1968 and 

1970) prepared empirical relations projecting the effects of 

urban development on floods in northern Virginia. Dunne 

and Leopold (1978) described generalized runoff processes 

in urban areas and conditions associated with urban runoff. 

Focazio and others (1993) described the effects of wetlands on 

water quality downstream from an urbanized area in Virginia. 

Leopold (1968) summarized the general effects of urbanization on hydrologic factors. Rice (2000) identified the effects of 

urban sprawl on the water resources of northern Virginia. With 

the exception of Dunne and Leopold’s generalized description 

of urban runoff conditions, each of these studies focused on 

aspects of urban streamflow at relatively local scales for areas 

composed of several basins or several counties. 

Study Objectives

This study estimates peak streamflows per square mile 

associated with urban basins in Virginia. Urban response 

surface models and equations are developed from analysis of 

Virginia peak-flows and basin data, specifically basin percent 

urban area and basin drainage area, published by Austin and 

others (2011b). A set of response surface models that describe 

Virginia urban area peak streamflow per square mile of basin 

drainage area is presented along with a set of equations that 

may be used to accurately estimate Virginia urban peak 

streamflow per square mile of basin drainage area. Examples 

are provided demonstrating how to use the equations, with the 

intent of facilitating awareness, planning, and management of 

streams within Virginia’s urbanizing basins.

Analytical Approach

Response surface models and regression equations were 

developed for each peak-flow annual exceedance probability 

category to estimate urban peak streamflow per square mile 

from percentage of basin urban area and Log10-transformed 

basin drainage area. Response surface models fit continuous 

parameters using the principle of least squares to describe 

a response (y) as a function of several explanatory (x) variables. A response surface was created illustrating the threedimensional nature of relations among the variables. Percent 

urban area and square mile drainage area were compiled 

for stream-gaging stations associated with published peakflow characteristics (Austin and others, 2011b; Hayes and 

Wiegand 2006; Homer and others, 2004). These urban-area 

and drainage-area data were evaluated. Provisional response 

surface models were prepared, evaluated, and modified. Final 

response surface models were developed using regression 

analysis. Final regression equations were prepared for use in 
Analytical Approach 3

estimating urban peak streamflow per square mile of drainage 

area. These response surface models and regression equations 

facilitate estimating peak flows in urban basins, simulating 

peak-flow effects of increasing or decreasing basin urban area, 

and simulating peak-flow response to rates of change in basin 

land use.

Site Selection

Study sites were identified and selected by using peakflow estimates and basin characteristics published in U.S. Geological Survey Scientific Investigations Report 2011–5144, 

“Peak-Flow Characteristics of Virginia Streams” (Austin and 

others 2011b). Tables 5 and 6 of “Peak-Flow Characteristics 

of Virginia Streams” list Virginia basin percent urban area, 

defined as the fraction of basin land area classified as urban 

area, and basin drainage area defined as the published values 

of surface drainage area for each basin in the study. Basin 

percent urban area values were prepared from land cover 

classification data (Homer and others, 2004). Land cover data 

from four land cover classes were combined to determine 

percent urban area. The four land cover classes are open 

space, low-intensity development, medium-intensity development, and high-intensity development, and range from areas 

where impervious surfaces account for less than 20 percent of 

total land cover to areas where impervious surfaces account 

for 80 to nearly 100 percent of total land cover (Homer and 

others, 2004). A ratio of land area occupied by the combined 

four urban land cover classes to the total basin drainage area 

was calculated to yield basin percent urban area. Basin drainage area values previously published by Hayes and Wiegand 

(2006) were used in the analysis. These values are also available as basin characteristics listed in table 6 of Austin and 

others (2011b). Krstolic (2006) provides the digital boundaries 

associated with these basins.

Site selection began with more than 3,500 sites in 

Virginia for which records of water discharge measurements 

have been collected. Of these, 3,226 may be identified as sites 

with urban land cover associated with at least 1 percent of the 

upstream basin, and 2,073 may be identified as sites for which 

urban land cover is associated with at least 5 percent of the 

upstream basin. There are 1,068 sites that may be identified 

as having urban land cover associated with at least 10 percent 

of the upstream basin. Exploratory analyses undertaken using 

the three datasets associated with 1 percent, 5 percent, and 

10 percent basin urban land cover identified meaningful relations between urban peak flow per square mile and basin percent urban area when at least 10 percent of basin drainage area 

is occupied by urban land. The dataset containing 1,068 sites 

for which urban land cover is associated with at least 10 percent of each upstream basin was thus selected for additional 

analysis. Interestingly, the 10 percent urban land cover minimum threshold of meaningful influence on urban peak flows, 

though identified independently, coincides with urban land 

cover minimum thresholds of meaningful influence at or near 

10 percent as reported in other studies of urban peak flows 

(Booth and Jackson, 1997; Klein, 1979; Landers and others, 

2007; Mallin and others, 2000). Of the 1,068 sites for which 

urban land cover is associated with at least 10 percent of 

each upstream basin, 115 sites were identified as also having 

records of continuous stream-gaging measurements that were 

extensively evaluated and used to compute peak-flow statistics 

from which peak-flow characteristics were determined and 

published in Austin and others (2011b). These 115 sites were 

selected for use in this study.

Selection of these 115 sites constrained urban response 

surface models to basin urban area explanatory variable input 

values between 10 percent and 96 percent. The 115 selected 

study sites, for which urban land cover is associated with at 

least 10 percent of the upstream basin area, are listed in table 1

and shown in fgure 1.

Data Evaluation

The selected urban peak-flow study sites are located 

throughout Virginia. Basin drainage area associated with 

the 115 study sites ranges from 0.07 square mile (mi2

) to 

2,404 mi2

. The mean basin drainage area is 143.6 mi2, and the 

median basin drainage area is 8.26 mi2

. Basin drainage area 

data were Log10-transformed to facilitate use as an explanatory variable as suggested by Tukey’s bulging rule (Mosteller 

and Tukey, 1977). The Log10-transformed distribution of basin 

drainage area values for the selected study sites is normally 

distributed, with a minimum value of –1.16 and a maximum 

value of 3.38. The mean Log10-transformed basin drainage 

area is 1.08, and the median Log10-transformed basin drainage area is 0.92. Basin percent urban area associated with 

the 115 study sites ranges from 10 percent to 96 percent. 

The mean basin percent urban area is 39.01 percent, and the 

median basin percent urban area is 25.75 percent (fig. 2).

Data Analysis

Data analysis began with thorough, iterative evaluation and screening of relations among response variables and 

explanatory variables. First, percent urban area, basin drainage 

area, and mathematical transformations of each were screened 

for applicability and explanatory power. Then, bivariate relations were explored. Single explanatory variables were tested 

to discover their potential efficacy in determining a response. 

Bivariate relations were tested in each of 15 response categories: urban basin peak flow per square mile for annual 

exceedance probabilities of 0.995, 0.99, 0.95, 0.9, 0.8, 0.67, 

0.5, 0.43, 0.2, 0.1, 0.04, 0.02, 0.01, 0.005, and 0.002 (recurrence intervals of 1.005, 1.01, 1.05, 1.11, 1.25, 1.49, 2.0, 2.3, 

5, 10, 25, 50, 100, 200, and 500 years, respectively). From 

these initial iterative screenings and tests of bivariate relations, 

it soon became apparent that (1) basin percent urban area and 

(2) Log10-basin drainage area are each potentially useful predictors of urban basin peak flow per square mile.
4 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

EXPLANATION

Land Use Type

Water

Salt marsh

Rooftop

Residential/industrial

Forest

Pavement

Barren

Grassland

Forest harvest

Crop

84° 83° 82° 81° 80° 79° 78° 77° 76° 75°

40°

39°

38°

37°

36°

KENTUCKY

WEST VIRGINIA

TENNESSEE

NORTH CAROLINA

MARYLAND

WASHINGTON, D.C.

Chesapeake Bay

ATLANTIC OCEAN

Site type

Urban (percent)

10–30

31–50

51–75

76–96

Drainage area (mi2

)

0–75

76–250

251–750

751–1,500

1,501–2,500

0 25 50 Miles

0 40 80 Kilometers

Base from U.S. Geological Survey Digital Line Graph, 1:2,000,000 (1987)

Projection: Albers Equal Area, NAD 83,

Land cover: Fry and others, 2011,

Completion of the 2006 National Land Cover Database for the Conterminous United States:

Photogrammetric Engineering and Remote Sensing, v. 70, no. 9, p. 858–864.

Drainage area: Hayes, D.C. and Wiegand, U. 2006, 

Drainage Areas of Selected Streams of Virginia: U.S. Geological Survey 

Open-File Report 2006-1308, p. 51.

Figure 1. 115 selected urban peak-flow study sites used to develop response surface models and estimating 

equations. For each site, urban land cover is associated with at least 10-percent of the up-stream basin. Each 

dot represents one study site. Dot color indicates the percentage of urban area in the basin. Dot size indicates 

relative basin square mile drainage area.

Figure 1. Selected urban peak-flow study sites used to develop response surface models and estimating 

equations. For each of the 115 sites, urban land cover is associated with at least 10 percent of the upstream 

basin. Each dot represents one study site. Dot color indicates the percentage of urban area in the basin. Dot size 

indicates relative basin drainage area, in square miles.
Analytical Approach 5

(a) Basin percent urban area

Quantiles

100.0% maximum 95.96

99.5% 95.96

97.5% 91.98

90.0% 78.11

75.0% quartile 64.51

50.0% median 25.75

25.0% quartile 14.42

10.0% 11.12

2.5% 10.14

0.5% 10.01

0% minimum 10.01

Summary Statistics

Mean 39.01

Std Dev 27.62

Std Err Mean 2.58

Upper 95% Mean 44.11

Lower 95% Mean 33.90

N 115

(b) Log10 -basin drainage area (mi2) 

Quantiles

100.0% maximum 3.38

99.5% 3.38

97.5% 3.22

90.0% 2.63

75.0% quartile 1.75

50.0% median 0.92

25.0% quartile 0.36

10.0% –0.17

2.5% –0.62

0.5% –1.16

0.0% minimum –1.16

Summary Statistics

Mean 1.08

Std Dev 1.01

Std Err Mean 0.09

Upper 95% Mean 1.26

Lower 95% Mean 0.89

N 115

(c) Basin drainage area (mi2)

Quantiles

100.0% maximum 2,404

99.5% 2,404

97.5% 1,649

90.0% 431.7

75.0% quartile 55.81

50.0% median 8.26

25.0% quartile 2.28

10.0% 0.68

2.5% 0.25

0.5% 0.07

0% minimum 0.07

Summary Statistics

Mean 143.6

Std Dev 388.6

Std Err Mean 36.24

Upper 95% Mean 215.4

Lower 95% Mean 71.81

N 115

Figure 2. Summary statistics and frequency distributions of response surface model explanatory variables: Basin percent urban 

area (a), and Log10 Basin drainage area (b). Summary statistics and frequency distribution for Non-Log10-transformed basin 

drainage area (c) are shown for comparison.

Figure 2. Summary statistics and frequency distributions of response surface model explanatory variables: (a) basin percent 

urban area, and (b) Log10-basin drainage area. Summary statistics and frequency distribution for (c) non-Log10-transformed basin 

drainage area are shown for comparison.

Model Development

Bivariate relations using linear regression, as shown 

in fgure 3, were used to test potential explanatory variables 

including basin percent urban area, Log10-basin drainage 

area, and basin area encompassing discrete ranges of percent 

impervious surface in the following categories: 0 percent 

impervious surface, 1 to 5 percent impervious surface, 6 to 

10 percent impervious surface, 11 to 25 percent impervious 

surface, 26 to 50 percent impervious surface, 51 to 75 percent 

impervious surface, and 76 to 100 percent impervious surface. 

These tests clearly identified (1) overall basin percent urban 

area and (2) Log10-basin drainage area as the most informative 

and potentially useful predictors of urban basin peak flow per 

square mile over the range of peak-flow response categories. 

These relations were further investigated. Preliminary models 

were developed by using standard least squares regression to 

predict urban basin peak flow per square mile in each response 

category by a combination of two explanatory variables, 

basin percent urban area and Log10-basin drainage area. These 

regressions combining two explanatory variables for each 

response curve were used to develop response surface models. 

Response surface models fit continuous parameters using the 

principle of least squares to describe a response (y), as a function of several explanatory (x) variables. A response surface is 

created to illustrate and document the three-dimensional nature 

of relations between a single response variable—basin peak 

flow per square mile—and two explanatory variables—basin 

percent urban area and Log10-basin drainage area. Response 

surface models follow the form of standard least squares 

regression (eq. 1). Cross-product effects in response surface 

models yield final equation prediction expressions (eq. 2). 

Each datapoint used in the fitting of a response surface model 

is an observation. The 115 study sites suggest the potential 

availability of 115 observations for each response surface 

model. There are instances when a meaningful response (y) 

value for a single study site is not determined for 1 of the 

15 annual exceedance probability response categories used 

in response surface models. This is due to lack of sufficient 

data to describe a particular probability response for the single 

study site. In these instances, a response surface model is 

developed without the observation—probability response (y) 

value and associated explanatory (x) values—particular to the 

single study site. Parameters for each regression equation are 
6 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

Q 1 Q 2 Q 10

10 2 10 2 10 2

Figure 3. Examples of regressions prepared during initial tests of bivariate relations for 0.995, 0.50 and 0.10 urban peak 

flow probability categories (recurrence intervals of 1-year, 2-years, and 10-years, respectively), showing urban basin 

peak flow per square mile as explained by Log10-basin drainage area. Red lines and shading describe all basins, green 

lines and shading describe urban basins, blue lines and shading describe rural basins.

Log10 drainage area (mi2) Log10 drainage area (mi2) Log10 drainage area (mi2)

Log10 (Peak Q per mi2 (0.995))

Log10 (Peak Q per mi2 (0.50))

Log10 (Peak Q per mi2 (0.10))

Figure 3. Examples of regressions prepared during initial tests of bivariate relations for 0.995, 0.50, and 0.10 urban peak-flow 

probability categories [recurrence intervals of 1 year (Q1), 2 years (Q2), and 10 years (Q10), respectively], showing urban basin 

peak flow per square mile as explained by Log10-basin drainage area. Red lines and shading describe all basins, green lines 

and shading describe urban basins, blue lines and shading describe rural basins (Q denotes discharge).

listed in table 2. Regression equation prediction expressions 

for each response surface model, estimating Virginia urban 

peak flow per square mile, are listed in table 3 along with the 

number of observations used in each response surface model.

(1)

Equation 1: Form of a standard least squares regression equation where y is the response variable, x1

 is the first 

explanatory variable; x2 is the second explanatory variable; β0, 

β1, and β2 are parameters for the intercept, the first explanatory 

variable, and the second explanatory variable, respectively; 

and ε is an independent and normally distributed error term.

y = β0 + β1x1 + β2x2 + ε

Log10 ( y) = β0 + (x1 – β1) × ((Log10 (x2) – β2) × β3) + 

β4 x1 + β5 x2 + ε 

(2)

Equation 2: Form of each response surface model prediction expression where Log10 ( )y is the response variable, x1

is the first explanatory variable, x2 is the second explanatory 

variable, β0 is an intercept parameter, β1, β2, and β3, are parameters for the cross-product terms, β4

 is a parameter multiplied 

by the first explanatory variable, β5 is a parameter multiplied 

by the second explanatory variable, and ε is an independent 

and normally distributed error term.

Response surface models illustrate the nature of relations 

between three model parameters by using a two-dimensional 

kite-shaped surface that may be oriented, curved, and skewed 

in three dimensions to describe interaction between three variables: one (dependent) response variable and two (independent) explanatory variables. These interactions are illustrated 

in relief and can be rendered, animated, and explored as virtual 

three-dimensional objects using computer software.

Model Verification and Diagnostics

Evaluation of candidate models, using R-square, root 

mean square error (RMSE), and Akaike’s Information Criterion adjusted for small samples (AICc), provided insights into 

the most appropriate candidate model choices for each regression equation describing discharge per square mile for each 

urban annual exceedance probability category. The R-square 

is a measure of the proportion of the total variance explained 

by the model. The R-square value indicated the proportion 

of variability in each relation that was accounted for by the 

statistical model—in this case a multiple regression model. 

R-square values approaching 1 suggested increased explanatory power. The RMSE is an estimate of the error variance. 

The RMSE value indicated the differences between values 

predicted by each model and the values actually observed or 

measured. The RMSE helped to aggregate individual residual 

differences into a single measure of predictive power. The 

AICc measures the goodness of fit of an estimated statistical 

model. The AICc evaluates the model fit relative to the number 

of model covariates. The AICc value indicated the goodness of 

fit of each regression model, relative to other models, providing a tool for model selection. AICc values are grounded 

in the concept of entropy, offering a relative measure of the 

information lost when a given model is used. AICc describes 

the difference between model bias and model variance, or 

the tradeoff between model precision and model complexity. 

Models may be ranked by their AICc values. Lower AICc values were considered better than higher values (SAS Institute 

Inc., 2012).

Leverage plots (Sall, 1990), also known as “partialregression residual leverage plots” (Belsley and others, 1980), 

or “added variable plots” (Cook and Weisberg, 1982), were 

used to further investigate and screen candidate regression 

models. Leverage plots allow one to view the significance 

of each model parameter in the candidate regression model 

within the context of the larger model. The effect of a model 

parameter upon the larger model is displayed on the leverage 

plot by comparing the sum of squared residuals of the whole 
Analytical Approach 7

model to the sum of squared residuals of the model with the 

effect of interest removed. Residual errors that are smaller 

when the effect of a particular parameter is included in the 

model help confirm that the effect contributes significantly 

to the model fit. The leverage plot graphically displays this 

potential significance for each candidate explanatory variable, 

showing for each point in the model what the residual would 

be with and without the effect in the model. In the leverage 

plot, the distance from a point to the line of fit describes the 

actual residual. The distance from a point to the horizontal 

line of the mean describes the potential residual error if the 

effect were not part of the model. In other words, the mean 

line of the leverage plot represents the model if the value of 

the parameter effect were constrained to zero. The idea is 

to determine if the line of fit on the leverage plot for each 

explanatory variable carries each point on the plot significantly better than does the horizontal line of the mean shown 

on the plot. Leverage plots were used to identify (1) regression 

variables with significant explanatory power, (2) regression 

variables with borderline or insignificant explanatory power, 

and (3) regression variables exhibiting significant collinearity 

within the model. A leverage plot indicates that a regression 

variable has significant explanatory power if the slope of the 

line of fit is statistically distinct from the slope of the mean 

line, as shown in fgure 4A. A leverage plot indicates that a 

regression variable has borderline or insignificant explanatory 

power if the slope of the line of fit is not statistically distinct 

from the slope of the mean line, as shown in fgure 4B. A 

leverage plot indicates that a regression variable is exhibiting 

significant collinearity within the model if plotted points collapse toward a centroid, and the slope of the line of fit is not 

statistically distinct from the slope of the mean line, as shown 

in fgure 4C.

Analyses of leverage plots for each potential explanatory 

variable in a candidate regression model afforded the opportunity to see the characteristics of each candidate variable, 

the influential strength of the variable, and its consequences 

within the larger model. The picture provided by each leverage 

plot enabled a thorough, visual vetting of individual candidate 

model variables and a measure of confidence in variable selection that could not be obtained from non-graphical indices 

of collinearity and variable inflation, preparing a means to 

construct robust regressions. A leverage plot was constructed 

and evaluated for each model parameter in each candidate 

regression model. Models containing variables with significant 

explanatory power and limited collinearity were preferred as 

final candidates.

Selection of (1) Log10-basin drainage area, and (2) 

basin percent urban area as explanatory variables yielded 

a robust regression in each model. Eight of the 15 urban 

peak-flow response surface models describing annual 

exceedance probability discharge per square mile in basins 

containing urban land, as a function of basin drainage area 

and basin percent urban area, yielded R-square values greater 

than 0.8. Actual-by-predicted plots were produced for model 

response variables, and leverage plots were produced for 

model explanatory variables in each peak-flow response 

category (fig. 5). Review of actual-by-predicted plots showed 

significant explanatory power (α = 0.05) and robust, outlierresistant model fits for each of the 15 response surface models. 

Review of 45 explanatory variable leverage plots, 3 associated with each of the 15 response surface models, showed 

significant explanatory power (α = 0.05) in 39 instances with 

no evidence of significant collinearity. The urban peak-flow 

response surface model describing 0.995 annual exceedance 

probability (1-year recurrence interval) discharge per square 

mile yielded an R-square value of 0.84. The urban peak-flow 

response surface model describing 0.99 annual exceedance 

probability (1.01-year recurrence interval) discharge per 

square mile yielded an R-square value of 0.84. The urban 

peak-flow response surface model describing 0.95 annual 

exceedance probability (1.05-year recurrence interval) 

discharge per square mile yielded an R-square value of 

0.84. The urban peak-flow response surface model describing 0.9 annual exceedance probability (1.11-year recurrence 

interval) discharge per square mile yielded an R-square value 

Figure 4. Example leverage plots showing (A) significant explanatory power, (B) insignificant explanatory power and, 

(C) evidence of collinearity. The solid red lines represent the significance of each explanatory model parameter, shown 

on the x-axis, within the context of the larger model. The dashed red lines represent the confidence interval of the 

significance of each explanatory model parameter. The dashed blue lines represent the horizontal line of the mean, 

describing the larger model if the effect of each explanatory model parameter were constrained to zero. The idea 

behind each leverage plot is to determine if the line of fit for each potential explanatory model variable carries each 

point on the plot significantly better than does the horizontal line of the mean shown on the plot.

0.5

1.0

1.5

2.0

2.5

3.0

Response variable:

Leverage Residuals

Response variable:

Leverage Residuals

Response variable:

Leverage Residuals

1.3 1.4 1.5 1.6 1.7 1.8

Explanatory variable leverage, P<0.0001

0

0.5

1.0

1.5

2.0

2.5

3.0

1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45

Explanatory variable leverage, P=0.2322

0

5,000

10,000

15,000

20,000

0 500 1,000 1,500 2,000 2,500 3,000 3,500

Explanatory variable leverage, P=0.2066

A B C

Figure 4. Example leverage plots showing (A) significant explanatory power, (B) insignificant explanatory power and, (C) evidence 

of collinearity. The solid red lines represent the significance of each explanatory model parameter, shown on the x-axis, within 

the context of the larger model. The dashed red lines represent the confidence interval of the significance of each explanatory

model parameter. The dashed blue lines represent the horizontal line of the mean, describing the larger model if the effect of each 

explanatory model parameter were constrained to zero. The idea behind each leverage plot is to determine if the line of fit for each 

potential explanatory model variable carries each point on the plot significantly better than does the horizontal line of the mean 

shown on the plot.
8 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

A B C

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0.6 1 1.2 1.6 2 2.2 2.6

Log10(Peak Q per mile2 (0.80))

Predicted P<0.0001 RSq=0.83

RMSE=0.2252

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

–0.4 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

Log10 Drainage area

(mi2

) Leverage, P<0.0001

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

Log10(Peak Q per mi2

(0.8) Leverage Residuals

10 20 30 40 50 60 70 80 90

Urban (%) Leverage, P=0.0168

Figure 5. Plots for the 0.8 (1.25-year recurrence interval) model of urban peak flow per square mile showing (A) actual 

by predicted values, (B) drainage area explanatory power, and (C) urban area explanatory power. Log10(Peak Q per mi2 (0.8)) Actual Log10(Peak Q per mi2 (0.8)) Leverage Residuals

Figure 5. Plots for the 0.8 (1.25-year recurrence interval) model of urban peak flow per square mile showing (A) actual-by-predicted 

values, (B) drainage area explanatory power, and (C) urban area explanatory power (Q denotes discharge).

of 0.83. The urban peak-flow response surface model describing 0.8 annual exceedance probability (1.25-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.83. The urban peak-flow response surface model describing 0.67 annual exceedance probability (1.5-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.82. The urban peak-flow response surface model describing 0.5 annual exceedance probability (2-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.81. The urban peak-flow response surface model describing 0.43 annual exceedance probability (2.33-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.81. The urban peak-flow response surface model describing 0.2 annual exceedance probability (5-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.76. The urban peak-flow response surface model describing 0.1 annual exceedance probability (10-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.74. The urban peak-flow response surface model describing 0.04 annual exceedance probability (25-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.69. The urban peak-flow response surface model describing 0.02 annual exceedance probability (50-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.65. The urban peak-flow response surface model describing 0.01 annual exceedance probability (100-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.57. The urban peak-flow response surface model describing 0.005 annual exceedance probability (200-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.55. The urban peak-flow response surface model describing 0.002 annual exceedance probability (500-year recurrence 

interval) discharge per square mile yielded an R-square value 

of 0.44. Urban peak-flow response surface regression equations are found in table 3.

Model Limitations

These models provide meaningful estimates when 

applied to the area for which they are developed, using values 

of explanatory variables within the range of data values used 

for model development (figs. 1, 2). They are not meant for use 

with the high or low endpoints of each explanatory variable 

range, or values outside of these limits. Application of urban 

peak-flow response surface models is limited to all drainage 

basins contained within the boundaries of the Commonwealth 

of Virginia that range in size from no less than 1.2 mi2 to no 

more than 2,400 mi2 and contain no less than 10 percent and 

no more than 96 percent urban area.

Urban Peak Flow Per Square Mile 

Estimates 

Urban peak-flow response surface models provide a 

three-dimensional representation of change in peak flow per 

square mile, as a function of basin drainage area and basin 

percent urban land area. In each figure (figs. 6–21), a kiteshaped model response surface is shown in green. Black 

circles plot the data used to fit each model. In each figure, the 

relation among the three variables—two explanatory variables, Log10-basin drainage area (in square miles) and basin 

urban area (in percent), and one response variable, Log10-peak 

flow per square mile—may be traced along the green kiteshaped surface. Each response surface changes in a non-linear 

fashion relative to changes in basin drainage area and basin 

urban area.

The best available estimates of basin square mile drainage area are recommended for Log10-transformation and use 

as the first of two explanatory variables. Accurate basin square 

mile drainage area values are important because estimates 

of urban area annual peak flow per square mile may not be 

true if inaccurate explanatory values are used. One source of 

these estimates is U.S. Geological Survey Open-File Report 
Urban Peak Flow Per Square Mile Estimates 9

2006–1308, “Drainage areas of selected streams in Virginia” 

(Hayes and Wiegand, 2006). Hayes and Wiegand provide a 

list of drainage area values for selected Virginia basins. These 

same basin drainage area values are also available as basin 

characteristics listed in table 6 of Austin and others (2011b). 

Krstolic (2006) provides the digital boundaries associated with 

these basins.

Urban land cover data developed by Homer and others 

(Homer and others, 2004) are recommended as source data 

for calculating basin percent urban area, the second of two 

explanatory variables. This is because accurate urban land 

cover values are required to ensure true estimates of urban 

area annual peak flow per square mile. These urban land cover 

data are labeled as NLCD2001 Land Cover (Version 2.0) and 

are available from the National Land Cover Database hosted 

by the Multi-Resolution Land Characteristics Consortium Web 

site (http://www.mrlc.gov/nlcd2001.php, accessed September 20,

2013). Land cover data from four combined land cover classes 

within this database were summarized to determine percent 

urban area. These four land cover classes are: (1) Class 21: 

developed, open space; (2) Class 22: developed, low intensity; 

(3) Class 23: developed, medium intensity; and (4) Class 24: 

developed, high intensity. A ratio of land area occupied by 

the combined four urban land cover classes to the total basin 

drainage area was calculated to yield basin percent urban area 

(Austin and others 2011b). Because models were developed 

using the basin urban area percentages prepared from these 

data, it is recommended that any new basin urban area percentages be developed using the most up-to-date iteration of this 

dataset. Doing so will help ensure accurate estimates of urban 

area annual peak flow per square mile. At the time this report 

was written (2013), the most up-to-date dataset was available 

online from the Multi-Resolution Land Characteristics Consortium Web site (http://www.mrlc.gov/nlcd06_data.php). An 

alternative to preparing new basin urban area percentages is to 

use the values previously reported by the U.S. Geological Survey and used to develop the models and equations listed in this 

report. These previously reported basin urban area percentages 

may be found in table 6 of Austin and others (2011b).

Figure 6 illustrates how to read each three-dimensional 

response surface model plot. Enter the plot along the top, 

(or bottom), axis of each of the two explanatory variables, 

Log10-basin drainage area and basin urban area. Find the 

explanatory variable values of interest on each explanatory 

!

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 6. How to read three-dimensional response surface model plots: 

Find the explanatory variable values of interest on each explanatory 

variable axis, then trace across the top grid (or bottom grid) to locate the 

point at which these values converge. From the point of convergence 

project a line downward, (or upward), to an intersection point on the green 

response surface. From the intersection point read across to intersect the 

grid associated with the axis of the response variable, Log10-peak flow per 

square mile. Project a line from this point to the response variable axis, and 

read the result. This pathway is illustrated in red.

Figure 6. How to read three-dimensional response surface model plots: 

Find the explanatory variable values of interest on each explanatory 

variable axis, then trace across the top grid (or bottom grid) to locate the 

point at which these values converge. From the point of convergence, 

project a line downward (or upward) to an intersection point on the green 

response surface. From the intersection point, read across to intersect 

the grid associated with the axis of the response variable, Log10-peak 

flow per square mile. Project a line from this point to the response 

variable axis, and read the result. This pathway is illustrated in red 

(Q denotes discharge).
10 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

variable axis, then trace across the top grid (or bottom grid) 

to locate the point at which these values converge. From the 

point of convergence, project a line downward (or upward) 

to an intersection point on the green response surface. From 

the intersection point, read across to intersect the grid associated with the axis of the response variable, Log10-peak flow 

per square mile. Project a line from this point to the response 

variable axis, and read the result. The response surfaces for 

each of 15 models describing urban discharge per square mile 

annual exceedance probabilities are illustrated in figures 7–21. 

The equations in table 3 may be used to determine urban 

discharge per square mile annual exceedance probabilities. 

Example calculations are provided in the report section titled 

Examples of Applying the Equations.

Response surface models for each urban basin peakflow probability category indicate strong correlation between 

(1) basin peak flow per square mile and basin percent urban 

area, and (2) basin peak flow per square mile and Log10-basin 

drainage area. The shapes of response surface models also 

suggest that interaction between basin percent urban area 

and Log10-basin drainage area may influence basin peak flow 

per square mile. The curvilinear shape of each kite in each 

response surface model plot suggests a synergistic relation 

associated with the interaction of basin percent urban area 

and Log10-basin drainage area. This relation yields non-linear 

increases in urban peak flows per square mile with increasing 

basin percent urban area and decreasing Log10-basin drainage 

area. This relation may reflect the reinforcing consequences of 

structural changes in basin surface and sub-surface flow paths 

combining the increasing density of impervious surfaces, subsurface piping, storm drains, and sewer installations, and the 

decreasing precipitation infiltration rates associated with urban 

development. Reduced infiltration rates, accelerated rates of 

water flow across impervious surfaces, and rapid concentration of water flow through pipes, storm drains, and sewers 

combine to increase net water volume and shorten water travel 

distances and travel times, increasing water flow rates.

These effects of urban change on surface-water volumes, water travel times, and water discharge rates have 

also been identified and characterized in previous studies, 

most notably in Virginia by Anderson (1968, 1970). In his 

1970 report, Anderson illustrates the dramatic increase in 

urban peak flows and equally dramatic decrease in lag-times 

associated with urban hydrographs, as urban basins transition from natural channels with natural basin surfaces, to 

completely sewered basins having natural basin surfaces, to, 

finally, completely sewered basins having highly impervious 

basin surfaces (Anderson, 1970, page C4, fig. 2). While the 

empirical equations describing the process-based interactions 

of precipitation, impervious surface, channel straightening, 

and piping of water flows as described by Anderson may not 

be directly comparable to the statistical relations identified 

in response surface models, both methods of analysis identify significant increases in urban peak stream discharges as 

basin urban area grows. The statistical relations identified in 

response surface models support the empirically determined 

characterizations of change in physical processes reported 

by Anderson (figs. 7–21). Leverage plots of response surface 

model crossed-product effects that multiply basin percent 

urban area by Log10-basin drainage area affirm these reinforcing responses, showing a statistically significant cross-product 

interaction between basin percent urban area and Log10-basin 

drainage area in response surface models for 11 of the 15 peak 

flow per square mile probability categories, and no evidence of 

explanatory variable cross-product collinearity in the 15 peakflow probability response surface models (fig. 22). Actual-bypredicted plots, shown in figures 23–37, illustrate the strength 

of predicted outcomes for each response surface model.

Interestingly, in response surface models that characterize 

the four highest-flow annual exceedance probability categories 

0.02, 0.01, 0.005, and 0.002 (recurrence intervals of 50, 100, 

200, and 500 years, respectively), absence of statistically significant cross-product interactions is also consistent with the 

nature of urban basin streamflow. During these highest flows 

basin soils are often saturated and stream channels are often 

overwhelmed by water. In these instances most of an urban 

basin drainage system may be inundated. Any interaction 

effects between basin percent urban area and basin square mile 

drainage area may be less meaningful as predictors of urban 

area peak-flow response during these events.

Response surface models correctly signal this shift. For 

the four highest-flow annual exceedance probability categories 

0.02, 0.01, 0.005, and 0.002 (recurrence intervals of 50, 100, 

200, and 500 years, respectively) response surface models 

show that basin percent urban area and basin square mile 

drainage area are statistically significant predictors, exhibiting 

significant explanatory power (α = 0.05) in each of these four 

response categories. Cross-product interactions between basin 

percent urban area and basin square mile drainage area, however, demonstrate insignificant explanatory power (α = 0.05) 

in each of these four highest-flow response categories. Visual 

cues to this shift are evident when comparing the forms of the 

green kite-shaped surfaces in each response surface model 

plot. Response surface model plots for annual exceedance 

probabilities 0.02, 0.01, 0.005, and 0.002 (figs. 18–21) are 

more planer in shape (less curved) than plots for the remaining 

annual exceedance probability categories (figs. 7–17), signaling cross-product interactions with less explanatory power 

than those shown in the other plots.
Urban Peak Flow Per Square Mile Estimates 11

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 7. Urban peak flow per square mile response surface 

model for 0.995 annual exceedance probability (1-year 

recurrence interval) streamflow.

Figure 7. Urban peak flow (Q) per square mile response 

surface model for 0.995 annual exceedance probability 

(1-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 8. Urban peak flow per square mile response surface 

model for 0.99 annual exceedance probability (1.01-year 

recurrence interval) streamflow.

Figure 8. Urban peak flow (Q) per square mile response 

surface model for 0.99 annual exceedance probability 

(1.01-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 9. Urban peak flow per square mile response surface 

model for 0.95 annual exceedance probability (1.05-year 

recurrence interval) streamflow.

Figure 9. Urban peak flow (Q) per square mile response 

surface model for 0.95 annual exceedance probability 

(1.05-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 10. Urban peak flow per square mile response 

surface model for 0.9 annual exceedance probability 

(1.11-year recurrence interval) streamflow.

Figure 10. Urban peak flow (Q) per square mile response 

surface model for 0.9 annual exceedance probability 

(1.11-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).
12 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 11. Urban peak flow per square mile response 

surface model for 0.8 annual exceedance probability 

(1.25-year recurrence interval) streamflow.

Figure 11. Urban peak flow (Q) per square mile response 

surface model for 0.8 annual exceedance probability 

(1.25-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 12. Urban peak flow per square mile response 

surface model for 0.67 annual exceedance probability 

(1.5-year recurrence interval) streamflow.

Figure 12. Urban peak flow (Q) per square mile response 

surface model for 0.67 annual exceedance probability 

(1.5-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 13. Urban peak flow per square mile response 

surface model for 0.5 annual exceedance probability 

(2-year recurrence interval) streamflow.

Figure 13. Urban peak flow (Q) per square mile response 

surface model for 0.5 annual exceedance probability 

(2-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 14. Urban peak flow per square mile response 

surface model for 0.43 annual exceedance probability 

(2.33-year recurrence interval) streamflow.

Figure 14. Urban peak flow (Q) per square mile response 

surface model for 0.43 annual exceedance probability 

(2.33-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).
Urban Peak Flow Per Square Mile Estimates 13

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 15. Urban peak flow per square mile response 

surface model for 0.2 annual exceedance probability 

(5-year recurrence interval) streamflow.

Figure 15. Urban peak flow (Q) per square mile response 

surface model for 0.2 annual exceedance probability 

(5-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 16. Urban peak flow per square mile response 

surface model for 0.1 annual exceedance probability 

(10-year recurrence interval) streamflow.

Figure 16. Urban peak flow (Q) per square mile response 

surface model for 0.1 annual exceedance probability 

(10-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 17. Urban peak flow per square mile response 

surface model for 0.04 annual exceedance probability 

(25-year recurrence interval) streamflow.

Figure 17. Urban peak flow (Q) per square mile response 

surface model for 0.04 annual exceedance probability 

(25-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 18. Urban peak flow per square mile response 

surface model for 0.02 annual exceedance probability 

(50-year recurrence interval) streamflow.

Figure 18. Urban peak flow (Q) per square mile response 

surface model for 0.02 annual exceedance probability 

(50-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).
14 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 19. Urban peak flow per square mile response 

surface model for 0.01 annual exceedance probability 

(100-year recurrence interval) streamflow.

Figure 19. Urban peak flow (Q) per square mile response 

surface model for 0.01 annual exceedance probability 

(100-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 20. Urban peak flow per square mile response 

surface model for 0.005 annual exceedance probability 

(200-year recurrence interval) streamflow.

Figure 20. Urban peak flow (Q) per square mile response 

surface model for 0.005 annual exceedance probability 

(200-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).

Log10 drainage area (mi2

)

Urban (%)

Log10 (Peak Q per mi2

)

Log10 (Peak Q per mi2

)

Log10 drainage area (mi2

)

Urban (%)

Figure 21. Urban peak flow per square mile response 

surface model for 0.002 annual exceedance probability 

(500-year recurrence interval) streamflow.

Figure 21. Urban peak flow (Q) per square mile response 

surface model for 0.002 annual exceedance probability 

(500-year recurrence interval) streamflow in Virginia 

(Q denotes discharge).
Urban Peak Flow Per Square Mile Estimates 15

A B C

D E F

0.2

0.6

1.0

1.4

1.8

2.2

2.6

Log10(Peak Q per mi2

(0.995)) Leverage Residuals

1.00 1.10 1.20 1.30 1.40 1.50 1.60

Urban (%)*log10 Drainage

area (mi2

) Leverage, P<0.0001

0.8

1.2

1.6

2.0

2.4

2.8

3.2

Log10(Peak Q per mi2

(0.5)) Leverage Residuals

1.65 1.75 1.85 1.95 2.05 2.15 2.25

Urban (%)*log10 Drainage

area (mi2

) Leverage, P<0.0001

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Log10(Peak Q per mi2

(0.1)) Leverage Residuals

2.10 2.20 2.30 2.40 2.50

Urban (%)*log10 Drainage

area (mi2

) Leverage, P=0.003

1.5

2.0

2.5

3.0

3.5

4.0

Log10(Peak Q per mi2

(0.01)) Leverage Residuals

2.50 2.55 2.60 2.65 2.70 2.75 2.80

Urban (%)*log10 Drainage

area (mi2

) Leverage, P=0.1597

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Log10(Peak Q per mi2

(0.005)) Leverage Residuals

2.65 2.70 2.75 2.80 2.85 2.90

Urban (%)*log10 Drainage

area (mi2

) Leverage, P=0.3327

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Log10(Peak Q per mi2

(0.002)) Leverage Residuals

2.93 2.94 2.95 2.96 2.97 2.98

Urban (%)*log10 Drainage

area (mi2

) Leverage, P=0.8798

Figure 22. Example leverage plots indicating statistically significant cross-product effects in response surface models 

for (a) 0.995, (b) 0.50 and (c) 0.10 urban peak flow probability categories (recurrence intervals of 1-year, 2-years, and 

10-years respectively), and insignificant cross-product effects in response surface models for (d) 0.01, (e) 0.005, and 

(f) 0.002 urban peak flow probability categories (recurrence intervals of 100-years, 200-years, and 500-years respectively). Each leverage line in plots (a), (b), and (c), is statistically distinct from the blue-dashed horizontal line of the 

mean. Each leverage line in plots (d), (e), and (f), is not statistically distinct from the blue-dashed horizontal line of 

the mean. Dispersion of points along each leverage line signals no evidence of collinearity among cross-product 

explanatory variables.

Figure 22. Leverage plots indicating statistically significant cross-product effects in response surface models for (A) 0.995, (B) 

0.50 and (C) 0.10 urban peak flow probability categories (recurrence intervals of 1 year, 2 years, and 10 years, respectively), and 

insignificant cross-product effects in response surface models for (D) 0.01, (E) 0.005, and (F) 0.002 urban peak-flow probability 

categories (recurrence intervals of 100 years, 200 years, and 500 years, respectively). Each leverage line in plots A, B, and C is 

statistically distinct from the blue-dashed horizontal line of the mean. Each leverage line in plots D, E, and F is not statistically 

distinct from the blue-dashed horizontal line of the mean. Dispersion of points along each leverage line signals no evidence of

collinearity among cross-product explanatory variables (Q denotes discharge).

0

0.5

1

1.5

2

2.5

3

Log10(Peak Q per mi2

(0.995)) Actual

0 0.5 1 1.5 2 2.5 3

Log10(Peak Q per mi2 (0.995))

Predicted P<0.0001 RSq=0.84

RMSE=0.2641

Figure 23. Urban peak flow per square mile 

actual-by-predicted plot for 0.995 annual 

exceedance probability (1-year recurrence 

interval) streamflow.

Figure 23. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.995 

annual exceedance probability (1-year 

recurrence interval) streamflow in Virginia 

(Q denotes discharge).

0.5

1

1.5

2

2.5

3

Log10(Peak Q per mi2

(0.99)) Actual

0.5 1 1.5 2 2.5 3

Log10(Peak Q per mi2 (0.99))

Predicted P<0.0001 RSq=0.84

RMSE=0.2561

Figure 24. Urban peak flow per square mile 

actual-by-predicted plot for 0.99 annual 

exceedance probability (1.01-year recurrence 

interval) streamflow.

Figure 24. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.99 

annual exceedance probability (1.01-year 

recurrence interval) streamflow in Virginia 

(Q denotes discharge).

0.5

1

1.5

2

2.5

3

Log10(Peak Q per mi2

(0.95)) Actual

0.5 1 1.5 2 2.5 3

Log10(Peak Q per mi2 (0.95))

Predicted P<0.0001 RSq=0.84

RMSE=0.2343

Figure 25. Urban peak flow per square mile 

actual-by-predicted plot for 0.95 annual 

exceedance probability (1.05-year recurrence 

interval) streamflow.

Figure 25. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.95 

annual exceedance probability (1.05-year 

recurrence interval) streamflow in Virginia 

(Q denotes discharge).
16 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

0.5

1

1.5

2

2.5

3

Log10(Peak Q per mi2

(0.9)) Actual

0.5 1 1.5 2 2.5 3

Log10(Peak Q per mi2 (0.9))

Predicted P<0.0001 RSq=0.83

RMSE=0.2311

Figure 26. Urban peak flow per square mile 

actual-by-predicted plot for 0.9 annual 

exceedance probability (1.11-year recurrence 

interval) streamflow.

Figure 26. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.9 annual 

exceedance probability (1.11-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

0.5

1

1.5

2

2.5

3

Log10(Peak Q per mi2

(0.8)) Actual

0.5 1 1.5 2 2.5 3

Log10(Peak Q per mi2 (0.8))

Predicted P<0.0001 RSq=0.83

RMSE=0.2252

Figure 27. Urban peak flow per square mile 

actual-by-predicted plot for 0.8 annual 

exceedance probability (1.25-year recurrence 

interval) streamflow.

Figure 27. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.8 annual 

exceedance probability (1.25-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

0.5

1

1.5

2

2.5

3

Log10(Peak Q per mi2

(0.67)) Actual

0.5 1 1.5 2 2.5 3

Log10(Peak Q per mi2 (0.67))

Predicted P<0.0001 RSq=0.82

RMSE=0.2255

Figure 28. Urban peak flow per square mile 

actual-by-predicted plot for 0.67 annual 

exceedance probability (1.5-year recurrence 

interval) streamflow.

Figure 28. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.67 annual 

exceedance probability (1.5-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

1

1.5

2

2.5

3

Log10(Peak Q per mi2

(0.5)) Actual

1 1.5 2 2.5 3 3.5

Log10(Peak Q per mi2 (0.5))

Predicted P<0.0001 RSq=0.81

RMSE=0.2266

Figure 29. Urban peak flow per square mile 

actual-by-predicted plot for 0.5 annual 

exceedance probability (2-year recurrence 

interval) streamflow.

Figure 29. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.5 annual 

exceedance probability (2-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

1

1.5

2

2.5

3

Log10(Peak Q per mi2

(0.43)) Actual

1 1.5 2 2.5 3 3.5

Log10(Peak Q per mi2 (0.43))

Predicted P<0.0001 RSq=0.81

RMSE=0.226

Figure 30. Urban peak flow per square mile 

actual-by-predicted plot for 0.43 annual 

exceedance probability (2.33-year recurrence 

interval) streamflow.

Figure 30. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.43 annual 

exceedance probability (2.33-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

1

1.5

2

2.5

3

3.5

Log10(Peak Q per mi2

(0.2)) Actual

1 1.5 2 2.5 3 3.5

Log10(Peak Q per mi2 (0.2))

Predicted P<0.0001 RSq=0.77

RMSE=0.2385

Figure 31. Urban peak flow per square mile 

actual-by-predicted plot for 0.2 annual 

exceedance probability (5-year recurrence 

interval) streamflow.

Figure 31. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.2 annual 

exceedance probability (5-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

1

1.5

2

2.5

3

3.5

Log10(Peak Q per mi2

(0.1)) Actual

1 1.5 2 2.5 3 3.5

Log10(Peak Q per mi2 (0.1))

Predicted P<0.0001 RSq=0.74

RMSE=0.2503

Figure 32. Urban peak flow per square mile 

actual-by-predicted plot for 0.1 annual 

exceedance probability (10-year recurrence 

interval) streamflow.

Figure 32. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.1 annual 

exceedance probability (10-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

1.5

2

2.5

3

3.5

Log10(Peak Q per mi2

(0.04)) Actual

1.5 2 2.5 3 3.5

Log10(Peak Q per mi2 (0.04))

Predicted P<0.0001 RSq=0.68

RMSE=0.2832

Figure 33. Urban peak flow per square mile 

actual-by-predicted plot for 0.04 annual 

exceedance probability (25-year recurrence 

interval) streamflow.

Figure 33. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.04 annual 

exceedance probability (25-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

1.5

2

2.5

3

3.5

Log10(Peak Q per mi2

(0.02)) Actual

1.5 2 2.5 3 3.5

Log10(Peak Q per mi2 (0.02))

Predicted P<0.0001 RSq=0.63

RMSE=0.314

Figure 34. Urban peak flow per square mile 

actual-by-predicted plot for 0.02 annual 

exceedance probability (50-year recurrence 

interval) streamflow.

Figure 34. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.02 annual 

exceedance probability (50-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).
Examples of Applying the Equations 17

1.5

2

2.5

3

3.5

4

Log10(Peak Q per mi2

(0.01)) Actual

1.5 2 2.5 3 3.5 4

Log10(Peak Q per mi2 (0.01))

Predicted P<0.0001 RSq=0.58

RMSE=0.3493

Figure 35. Urban peak flow per square mile 

actual-by-predicted plot for 0.01 annual 

exceedance probability (100-year recurrence 

interval) streamflow.

Figure 35. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.01 annual 

exceedance probability (100-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

1.5

2

2.5

3

3.5

4

4.5

Log10(Peak Q per mi2

(0.005)) Actual

1.5 2 2.5 3 3.5 4 4.5

Log10(Peak Q per mi2 (0.005))

Predicted P<0.0001 RSq=0.55

RMSE=0.3601

Figure 36. Urban peak flow per square mile 

actual-by-predicted plot for 0.005 annual 

exceedance probability (200-year recurrence 

interval) streamflow.

Figure 36. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.005 annual 

exceedance probability (200-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

1.5

2

2.5

3

3.5

4

4.5

5

Log10(Peak Q per mi2

(0.002)) Actual

1.5 2 2.5 3 3.5 4 4.5 5

Log10(Peak Q per mi2 (0.002))

Predicted P<0.0001 RSq=0.48

RMSE=0.4262

Figure 37. Urban peak flow per square mile 

actual-by-predicted plot for 0.002 annual 

exceedance probability (500-year recurrence 

interval) streamflow.

Figure 37. Urban peak flow (Q) per square 

mile actual-by-predicted plot for 0.002 annual 

exceedance probability (500-year recurrence 

interval) streamflow in Virginia (Q denotes 

discharge).

Examples of Applying the Equations

Two step-by-step examples are presented that demonstrate how to perform calculations using the equations in 

this report to estimate peak flow per square mile in urban 

areas. The first example demonstrates how to estimate the 

1.11-year (0.9 annual exceedance probability) urban peak 

flow per square mile for a drainage basin with an upstream 

area of 10 mi2

, 60 percent of which is classified as urban 

area. The second example demonstrates how to estimate the 

10-year (0.10 annual exceedance probability) urban peak flow 

per square mile for a drainage basin with an upstream area 

of 95 mi2, 15 percent of which is classified as urban area.

Example 1

Estimate the 1.11-year (0.9 annual exceedance probability) urban peak flow per square mile for a drainage basin with 

an upstream area of 10 mi2, 60 percent of which is classified as 

urban area.

Given: 1. Drainage area (DA) of 10.0 mi2

 for the 

upstream area of this basin, measured from a 1:24,000-scale 

topographic map. 2. Urban area (URBAN) upstream of the 

measurement point is estimated as 60 percent of the upstream 

basin drainage area, from data developed by Homer and others 

(2004) and available from the Multi-Resolution Land Characteristics Consortium Web site (http://www.mrlc.gov).

Solution: The estimated urban peak flow per square mile 

for the site is computed by using the appropriate regression 

equation prediction expression in table 3. For 1.11 years (0.9 

annual exceedance probability), equation 4 in table 3 is used to 

compute urban peak flow per square mile as follows:

Log10(0.9 AEP peak per square mile) = 1.869 + 

(URBAN – 42.339) × ((Log10(DA) – 1.340) × –0.00523) + 

0.00323 × URBAN + –0.446 × Log10(DA) 

(equation 4 from table 3).

The 10-mi2

 drainage area (DA) and the 60 percent urban 

area (URBAN) are entered into the equation as follows:

Log10(0.9 AEP peak per square mile) = 1.869 + 

(60 – 42.339) × ((Log10(10) – 1.340) ×

–0.00523) + 0.00323 × 60 + –0.446 × Log10(10).

The equation then simplifies to

Log10(0.9 AEP peak per square mile) = 1.648.

verted to a non-Log10 value as an exponent of 10, as follows:

The 1.11-year (0.9 annual exceedance probability) 

urban peak flow per square mile for a drainage basin with an 

upstream area of 10 mi2, 60 percent of which is classified as 

urban area, is estimated as 44.46 ft3/s.

Example 2

Estimate the 10-year (0.1 annual exceedance probability) 

urban peak flow per square mile for a drainage basin with an 

upstream area of 95 mi2, 15 percent of which is classified as 

urban area.

Given: 1. Drainage area (DA) of 95.0 mi2 for the 

upstream area of this basin, measured from a 1:24,000-scale 

topographic map. 2. Urban area (URBAN) upstream of the 

measurement point is estimated as 15 percent of the upstream 

basin drainage area, from data developed by Homer and others 

(2004) and available from the Multi-Resolution Land Characteristics Consortium Web site (http://www.mrlc.gov). 

This Log10(0.9 AEP peak per square mile) value is con-

= 101.648 = 44.46 ft3

/s.
18 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

Solution: The estimated urban peak flow per square mile 

for the site is computed by using the appropriate equation 

in table 3. For 10 years (0.1 annual exceedance probability), 

equation 10 in table 3 is used to compute urban peak flow per 

square mile as follows:

Log10(0.1 AEP peak per square mile) = 2.373 +

(URBAN – 38.706) × ((Log10(DA) – 1.103) ×

–0.00313) + 0.00470 × URBAN + –0.334 × Log10(DA) 

(equation 10 from table 3).

The 95-mi2 drainage area (DA) and the 15 percent urban 

area (URBAN) are entered into the equation as follows:

Log10(0.1 AEP peak per square mile) = 2.373 + 

(15 – 38.706) × ((Log10(95) – 1.103) ×

–0.00313) + 0.00470 × 15 + –0.334 × Log10(95).

 The equation then simplifies to

Log10(0.1 AEP peak per square mile) = 1.848.

This Log10(0.1 AEP peak per square mile) value is converted to a non-Log10 value as an exponent of 10, as follows:

The 10-year (0.1 annual exceedance probability) urban 

peak flow per square mile for a drainage basin with an 

upstream area of 95 mi2, 15 percent of which is classified as 

urban area, is estimated as 70.47 ft3/s.

= 101.848 = 70.47 ft3

/s.

Summary

Anticipating high-flow conditions in Virginia’s urban 

streams is essential to sound management of water resources 

and associated riparian and watershed ecosystems. Reliable 

predictions of urban peak flows inform planning and design, 

improving management of Virginia’s urban basins. The U.S. 

Geological Survey, in cooperation with the Virginia Department of Transportation, has developed a set of 15 response 

surface models that describe Virginia urban area peak streamflow per square mile based on changes in basin percent urban 

area and basin drainage area. A total of 115 Virginia drainage 

basins were selected for analysis, each containing at least 

10 percent urban area, and not more than 96 percent urban 

area, and ranging in size from 0.07 to 2,404 square miles (mi2).

Equations for each response surface model may be used 

to estimate Virginia urban peak streamflow per square mile 

of basin drainage area for annual exceedance probabilities 

of 0.995, 0.99, 0.95, 0.9, 0.8, 0.67, 0.5, 0.43, 0.2, 0.1, 0.04, 

0.02, 0.01, 0.005, and 0.002 (recurrence intervals of 1.005, 

1.01, 1.05, 1.11, 1.25, 1.49, 2.0, 2.3, 5, 10, 25, 50, 100, 200, 

and 500 years, respectively). Equations provide meaningful 

estimates when applied using values of explanatory variables 

inside the range of data values used for model development. 

They are not meant for use outside of these limits. Application 

of urban peak-flow response surface models and equations 

is limited to drainage basins contained within the boundaries 

of the Commonwealth of Virginia that range in size from no 

less than 1.2 mi2

 to no more than 2,400 mi2 and contain no 

less than 10 percent and no more than 96 percent urban area. 

Examples are provided demonstrating use of the equations.

Streams in urban areas serve important functions, and the 

effects of urbanization upon stream systems are significant. 

Changes in runoff processes result from covering parts of a 

basin with impervious roofs, sidewalks, roadways, and parking lots. Stormwaters accumulate downstream more quickly 

than in natural stream systems, producing higher flood peaks. 

Increased water velocities in stream channels decrease the lag 

between rainfall and runoff. Decreased lag times ensure that 

peak flow rates must increase, even if total water volumes 

remain constant, but total water volumes actually increase 

since net infiltration of precipitation is reduced as a consequence of impervious surfaces. This increase in total water 

volume drives flood peaks even higher than they would be if 

water volumes remained constant.

Increased peak flows in urbanizing basins lead to significant consequences that must be identified and anticipated 

by urban planners, designers, and managers. Predictions of 

urban streamflows provide knowledge for better protection 

from floods, anticipation of water availability during critical 

periods, and improved planning and management of water 

resources.

Actual-by-predicted plots for model response variables 

and leverage plots for model explanatory variables showed 

significant explanatory power (α = 0.05) and robust model 

fits for 15 response surface models and significant explanatory power (α = 0.05) for explanatory variables in 39 of 45 

instances, with no evidence of explanatory variable collinearity. Equations for 8 of 15 urban peak-flow response surface 

models yield R-square values greater than 0.8. Urban peak 

flow response surface models describing annual exceedance 

probabilities of 0.995, 0.99, 0.95, 0.9, 0.8, 0.67, 0.5, 0.43, 0.2, 

0.1, 0.04, 0.02, 0.01, 0.005, and 0.002 (recurrence intervals of 

1.005, 1.01, 1.05, 1.11, 1.25, 1.49, 2.0, 2.3, 5, 10, 25, 50, 100, 

200, and 500 years, respectively), yield R-square values of 

0.84, 0.84, 0.84, 0.83, 0.83, 0.82, 0.81, 0.81, 0.76, 0.74, 0.69, 

0.65, 0.57, 0.55, and 0.44, respectively.
Selected References 19

Selected References

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176 p. 

Anderson, D.G., 1968, Effects of urban development on floods 

in northern Virginia: U.S. Geological Survey Open-File 

Report, 26 p.

Anderson, D.G., 1970, Effects of urban development on floods 

in northern Virginia: U.S. Geological Survey Water-Supply 

Paper 2001-C, 27 p.

Austin, S.H., Krstolic, J.L., and Wiegand, Ute, 2011a, Lowflow characteristics of Virginia streams: U.S. Geological 

Survey Scientific Investigations Report 2011–5143, 122 p. 

+ 9 tables on CD. (Also available online at http://pubs.usgs.

gov/sir/2011/5143/.)

Austin, S.H., Krstolic, J.L., and Wiegand, Ute, 2011b, Peakflow characteristics of Virginia streams: U.S. Geological 

Survey Scientific Investigations Report 2011–5144, 106 p. 

+ 3 tables and 2 appendixes on CD. (Also available online 

at http://pubs.usgs.gov/sir/2011/5144/.)

Austin, S.H., and Wiegand, U., 2009, Annual maximum stages 

and discharges of selected streams in Virginia through 2007: 

U.S. Geological Survey Open-File Report 2009–1007, 733 p.,

accessed September 20, 2013, at http://pubs.water.usgs.gov/

ofr2009–1007. 

Belsley, D.A., Kuh, E., and Welsch, R.E., 1980, Regression 

diagnostics: New York, John Wiley and Sons, 292 p. 

Benson, M.A., 1962, Factors influencing the occurrence of 

floods in a humid region of diverse terrain: U.S. Geological 

Survey Water-Supply Paper 1580-B, 64 p.

Benson, M.A., 1964, Factors affecting the occurrence of floods 

in the southwest: U.S. Geological Survey Water-Supply 

Paper 1580-D, 72 p.

Berenbrock, C., 2002, Estimating the magnitude of peak flows 

at selected recurrence intervals for streams in Idaho: U.S. 

Geological Survey Water-Resources Investigations Report 

02–4170, 59 p.

Booth, D.B., and Jackson, R.C., 1997, Urbanization of aquatic 

systems—Degradation thresholds, stormwater detection, 

and the limits of mitigation: American Water Resources 

Association, Water Resources Bulletin, v. 33, no. 5, 

p. 1077–1090.

Cook, R.D., and Weisberg, S., 1982, Residuals and influence 

in regression: N.Y., Chapman and Hall, 229 p. 

Cushing, E.M., Kantrowitz, I.H., and Taylor, K.R., 1973, 

Water resources of the Delmarva Peninsula: U.S. Geological Survey Professional Paper 822, 58 p.

Dillow, J.A., 1996, Technique for estimating magnitude and 

frequency of peak flows in Maryland: U.S. Geological Survey Water-Resources Investigations Report 95–4154, 55 p.

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and frequency of floods in Indiana: U.S. Geological Survey 

Water-Resources Investigations Report 84–4134, 110 p.

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and frequency of floods in Kentucky: U.S. Geological Survey Water-Resources Investigations Report 76–62, 70 p.

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selected streams in Virginia: U.S. Geological Survey OpenFile Report 2006–1308, 51 p.

Hayes, D.C., and Young, R.L., 2005, Comparison of peak discharge and runoff characteristic estimates from the rational 

method to field observations for small basins in Central 

Virginia: U.S. Geological Survey Scientific Investigations 

Report 2005–5254, 44 p.

Helsel, D.R., and Hirsch, R.M., 2002, Statistical methods in 

water resources: U.S. Geological Survey Techniques of 

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510 p.

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Homer and others, 2004, Development of a 2001 National 

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20 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

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Guidelines for determining flood flow frequency: Bulletin 

17B of the Hydrology Subcommittee, Office of Water Data 

Coordination: U.S. Geological Survey, Reston, Va., Gilroy, 

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Tables 21

Table 1. Drainage basins used to estimate Virginia urban peak-flow discharge per square mile.—Continued

[mi2, square miles; %, percent]

Station number Station name Basin drainage 

area (mi2

)

Basin urban 

area (%)

01613900 Hogue Creek at Route 614 near Hayfield, Va. 15.95 10.59 

01615000 Opequon Creek above Route 7 near Berryville, Va. 58.16 18.78 

01616000 Abrams Creek above Route 659 near Winchester, Va. 17.00 55.88 

01621450 Blacks Run tributary 1 at Route 11 near Harrisonburg, Va. 0.67 15.41 

01623000 Bell Creek at St. Pauls Chapel, near Staunton, Va. 0.67 14.05 

01623500 Bell Creek along Route 612 near Staunton, Va. 3.85 18.99 

01624000 Bell Creek at Route 732 at Franks Mill near Staunton, Va. 9.81 16.94 

01624800 Christians Creek at Route 794 near Fishersville, Va. 73.03 15.15 

01625000 Middle River at Route 769 near Grottoes, Va. 372.73 13.51 

01626000 South River at Route 664 near Waynesboro, Va. 126.92 11.22 

01626500 South River (at Basic City) at Lyndhurst Road at Waynesboro, Va. 133.49 13.66 

01626850 South River at Hopeman Parkway near Dooms, Va. 147.77 15.60 

01627500 South River at Route 778 at Harriston, Va. 211.65 14.84 

01628500 South Fork Shenandoah River below Lower Lewis Run near Lynnwood, Va. 1,079.32 11.65 

01629500 South Fork Shenandoah River at Route 211 near Luray, Va. 1,372.27 10.97 

01631000 South Fork Shenandoah River below Route 619 at Front Royal, Va. 1,634.30 10.55 

01636210 Happy Creek at Kerfoot Avenue at Front Royal, Va. 14.01 10.71 

0163626650 Manassas Run at Route 645 near Front Royal, Va. 11.22 15.52 

01638350 South Fork Catoctin Creek at Route 698 near Waterford, Va. 31.62 10.48 

01643805 North Fork Goose Creek at Route 729 near Lincoln, Va. 38.12 11.39 

01644280 Broad Run at Route 7 near Leesburg, Va. 76.13 38.70 

01644290 Stave Run at Parking Lot Road at Reston, Va. 0.07 91.40 

01644291 Stave Run below Parking Lot Road near Reston, Va. 0.08 91.43 

01644295 Smilax Branch at Dulles Airport Road at Reston, Va. 0.33 75.03 

01644300 Sugarland Run at Route 606 at Herndon, Va. 3.39 81.35 

01645700 Difficult Run at Route 665 near Fairfax, Va. 4.36 47.91 

01645750 South Fork Little Difficult Run at Fox Mill Road near Fairfax, Va. 1.58 24.68 

01645784 Snakeden Branch at Soapstone Drive at Reston, Va. 0.79 62.38 

01645800 Piney Branch above railroad at Vienna, Va. 0.27 77.69 

01645900 Colvin Run at Hunter Mill Road at Reston (near Herndon), Va. 5.04 50.81 

01645975 Rocky Run at Towlston Road near Great Falls, Va. 3.19 29.52 

01646000 Difficult Run below Route 193 near Great Falls, Va. 57.83 35.77 

01646200 Scott Run at Old Georgetown Pike near McLean, Va. 4.67 61.73 

01646600 Pimmit Run at Great Falls Road near Falls Church, Va. 2.88 74.53 

01646700 Pimmit Run below Kirby Road at Arlington, Va. 8.26 67.85 

01646750 Little Pimmit Run tributary above Little Falls Road at Arlington, Va. 0.43 90.74 

01646800 Little Pimmit Run at Route 689 at Arlington, Va. 2.31 64.33 

01652400 Long Branch at South Carlyn Springs Road at Arlington, Va. 0.94 91.00 

01652430 Doctors Run at 8th Street at Arlington, Va. 0.93 91.80 

01652470 Lucky Run at Dinwiddie Street at Arlington, Va. 1.30 93.64 

01652500 Fourmile Run at Shirlington Road at Alexandria, Va. 12.59 86.50 

01652600 Holmes Run at Route 50 at Merrifield, Va. 2.73 71.83 
22 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

Table 1. Drainage basins used to estimate Virginia urban peak-flow discharge per square mile.—Continued

[mi2, square miles; %, percent]

Station number Station name Basin drainage 

area (mi2

)

Basin urban 

area (%)

01652610

01652620

01652645

01652650

01652710

01652810

01652910

01653000

01653210

01653447

01653900

01653950

01654000

01654500

01655000

01655310

01655350

01655370

01655380

01655390

01656800

01656960

01657000

01657415

01657500

01657655

01657800

01667600

01673500

01673550

02019400

02027700

02030800

02031000

02032700

02033500

02034000

02034050

02037800

02038000

02038500

02042000

02042287

 Holmes Run at Sleepy Hollow Road near Annandale, Va. 

 Tripps Run above South Washington Street at Falls Church, Va. 

 Tripps Run tributary at Holmes Run Road near Falls Church, Va. 

 Tripps Run at Sleepy Hollow Road near Falls Church, Va. 

 Backlick Run at Leesville Boulevard at Springfield, Va. 

 Turkeycock Run at Shirley Route at Alexandria, Va. 

 Backlick Run below South Van Dorn Street at Alexandria, Va. 

 Cameron Run below railroad bridge at Alexandria, Va. 

 Pike Branch at Route 611 at Alexandria, Va. 

 Penn Daw outfall at Huntington Avenue at Alexandria, Va. 

 Accotink Creek at Pickett Street at Fairfax, Va. 

 Long Branch at Lee Route at Vienna, Va. 

 Accotink Creek above Route 620 near Annandale, Va. 

 Long Branch (Little Run) at Route 620 near Annandale, Va. 

 Accotink Creek at Army Engineer Road near Accotink Station, Va. 

 Rabbit Branch at Guinea Road near Burke, Va. 

 Pohick Creek at Hooes Road near Springfield, Va. 

 Middle Run at Route 640 near Lorton, Va. 

 South Run at Hooes Road near Lorton, Va. 

 Pohick Creek at Lorton Road at Lorton, Va. 

 Cub Run at Route 50 near Chantilly, Va. 

 Cub Run at Route 658 near Bull Run, Va. 

 Bull Run at Route 616 near Manassas, Va. 

 Bull Run above Yates Ford near Clifton, Va. 

 Occoquan River (Creek) near Occoquan (site now in Occoquan Reservoir), Va

 Hooes Run above Route 641 near Occoquan, Va. 

 Giles Run at Route 1 near Woodbridge, Va. 

 Cedar Run tributary at Route 522 near Culpeper, Va. 

 Totopotomoy Creek at Route 301 near Atlee, Va. 

 Totopotomoy Creek at Route 606 near Studley, Va. 

 Mill (Looney Mill) Creek at Route 636 near Buchanan, Va. 

 Buffalo River tributary at Route 60 near Amherst, Va. 

 Stockton Creek at Route 6 near Afton, Va. 

 Mechums River at Route 614 near White Hall (Ivy), Va. 

 Schenks Branch above Route 250 at Charlottesville, Va. 

 Rivanna River below Moores Creek near Charlottesville, Va. 

 Rivanna River at Route 15 at Palmyra, Va. 

 Hunters Branch at Route 15 near Palmyra, Va. 

 Falling Creek at Route 653 near Midlothian, Va. 

 Falling Creek at Route 651 near Chesterfield, Va. 

 Falling Creek at Hopkins Road near Drewrys Bluff, Va. 

 Swift Creek at Bradley Bridge Road near Chester, Va. 

 Chickahominy River at Route 301 near Atlee, Va. 

 7.11 

 1.91 

 0.51 

 4.63 

 2.08 

 2.28 

 13.48 

 33.86 

 2.59 

 0.79 

 6.86 

 1.21 

 23.87 

 3.72 

 37.07 

 3.82 

 15.17 

 3.64 

 6.30 

 31.26 

 7.75 

 49.74 

 145.98 

 184.70 

 570.14 

 3.92 

 4.52 

 0.67 

 5.77 

 25.53 

 29.30 

 0.45 

 2.70 

 95.27 

 1.39 

 503.28 

 662.77 

 1.59 

 18.16 

 33.10 

 53.35 

 140.15 

 62.15 

 63.05 

 67.64 

 70.99 

 71.48 

 63.68 

 64.51 

 73.91 

 70.21 

 78.74 

 81.53 

 73.88 

 71.38 

 62.22 

 59.32 

 61.81 

 64.03 

 64.87 

 68.32 

 30.47 

 54.24 

 51.67 

 52.27 

 29.10 

 29.55 

 15.85 

 45.08 

 41.39 

 12.35 

 31.50 

 14.95 

 13.09 

 13.56 

 17.31 

 11.54 

 95.96 

 12.41 

 11.83 

 13.74 

 66.61 

 58.19 

 59.25 

 16.99 

 23.70 
Tables 23

Table 1. Drainage basins used to estimate Virginia urban peak-flow discharge per square mile.—Continued

[mi2, square miles; %, percent]

Station number Station name Basin drainage 

area (mi2

)

Basin urban 

area (%)

02042426

02042500

02042780

02044400

02055000

02055100

02056000

02056650

02057500

02057700

02059000

02059450

02060500

02061150

02062500

02076400

02076500

03165200

03167300

03167700

03177700

03177710

03474700

03474800

03475600

03475700

03478400

03524500

03525800

03530000

 Upham Brook at Wilkinson Road near Richmond, Va. 

 Chickahominy River at Route 618 near Providence Forge, Va. 

 West Branch Long Hill Swamp at Route 612 near Lightfoot, Va. 

 Hurricane Branch at Route 40 at Blackstone, Va. 

 Roanoke River at Walnut Street at Roanoke, Va. 

 Tinker Creek below railroad bridge near Daleville, Va. 

 Roanoke River below powerplant at Niagara, Va. 

 Back Creek at Route 660 near Dundee, Va. 

 Roanoke River near Toshes, Va. 

 Powder Mill Creek at Route 220 at Rocky Mount, Va. 

 Roanoke River at Route 1109 near Gretna, Va. 

 South Fork Goose Creek at Route 607 at Montvale, Va. 

 Roanoke (Staunton) River at Route 29 (Alternate) at Altavista, Va. 

 Chestnut Branch near Forest at Goode, Va. 

 Roanoke (Staunton) River above Route 501 at Brookneal, Va. 

 Whitethorn Creek tributary at Route 29 at Gretna, Va. 

 Georges Creek at Route 40 near Gretna, Va. 

 Mill Creek tributary at Route 58 at Galax, Va. 

 Mira Fork tributary at Route 221 near Dugspur, Va. 

 Beaverdam Creek at Route 1009 at Hillsville, Va. 

 Bluestone River above wastewater outfall at Bluefield, Va. 

 Bluestone River at Route 717 at Falls Mills, Va. 

 Hutton Creek at Route 11 near Chilhowie, Va. 

 Hall Creek at Route 11 near Glade Spring, Va. 

 Cedar Creek at Route 11 near Meadowview, Va. 

 Spring Creek at Route 11 near Abingdon, Va. 

 Beaver Creek at Route 1405 at (near) Bristol, Va. 

 Guest River at Route 72 at Coeburn, Va. 

 Copper Creek tributary 1 at Route 58 (Alternate) near Dickensonville, Va. 

 South Fork Powell River at Route 23 at Big Stone Gap, Va. 

 37.45 

 251.47 

 2.47 

 1.61 

 383.92 

 11.69 

 509.45 

 55.81 

 1,017.01 

 0.78 

 1,418.33 

 10.97 

 1,781.97 

 1.54 

 2,403.72 

 2.02 

 9.18 

 1.07 

 0.66 

 4.47 

 39.75 

 44.27 

 8.20 

 7.83 

 3.42 

 2.78 

 26.94 

 87.16 

 0.68 

 39.75 

 66.86 

 24.49 

 14.22 

 22.16 

 18.65 

 14.73 

 23.56 

 18.67 

 15.21 

 71.86 

 12.18 

 10.81 

 10.91 

 14.37 

 10.22 

 19.11 

 15.84 

 33.69 

 10.01 

 20.10 

 26.83 

 25.75 

 16.00 

 10.15 

 20.51 

 10.06 

 14.42 

 15.03 

 16.53 

 12.06 
24 Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins

Table 2. Parameters for regression equations used for estimating Virginia urban peak-flow discharge per square mile.

[DA, basin drainage area in square miles; URBAN, basin urban area in percent]

Equation 

number Log10(y) x1 x2 β0 β1 β2 β3 β4 β5

1 Log10(0.995 peak per 

square mile)

URBAN Log10(DA) 1.673 43.179 1.412 –0.006 0.004 –0.512

2 Log10(0.99 peak per 

square mile) 

URBAN Log10(DA) 1.713 43.538 1.400 –0.006 0.004 –0.505

3 Log10(0.95 peak per 

square mile) 

URBAN Log10(DA) 1.764 42.993 1.350 –0.005 0.004 –0.448

4 Log10(0.9 peak per 

square mile) 

URBAN Log10(DA) 1.869 42.339 1.340 –0.005 0.003 –0.446

5 Log10(0.8 peak per 

square mile) 

URBAN Log10(DA) 1.952 42.563 1.325 –0.005 0.003 –0.424

6 Log10(0.67 peak per 

square mile) 

URBAN Log10(DA) 1.984 40.843 1.272 –0.005 0.004 –0.399

7 Log10(0.5 peak per 

square mile) 

URBAN Log10(DA) 2.027 40.290 1.216 –0.004 0.005 –0.366

8 Log10(0.43 peak per 

square mile) 

URBAN Log10(DA) 2.072 40.227 1.205 –0.004 0.005 –0.362

9 Log10(0.2 peak per 

square mile) 

URBAN Log10(DA) 2.229 39.370 1.139 –0.003 0.005 –0.338

10 Log10(0.1 peak per 

square mile) 

URBAN Log10(DA) 2.373 38.706 1.103 –0.003 0.005 –0.334

11 Log10(0.04 peak per 

square mile) 

URBAN Log10(DA) 2.557 39.168 1.083 –0.002 0.004 –0.332

12 Log10(0.02 peak per 

square mile) 

URBAN Log10(DA) 2.697 39.168 1.083 –0.002 0.004 –0.343

13 Log10(0.01 peak per 

square mile) 

URBAN Log10(DA) 2.776 38.765 1.070 –0.002 0.004 –0.342

14 Log10(0.005 peak per 

square mile) 

URBAN Log10(DA) 2.863 39.063 1.057 –0.002 0.005 –0.329

15 Log10(0.002 peak per 

square mile) 

URBAN Log10(DA) 2.961 39.287 0.904 –0.000 0.006 –0.317
Tables 25

Table 3. Regression equation prediction expressions for estimating Virginia urban peak-flow discharge per square mile.

[DA, basin drainage area in square miles; AEP, annual exceedance probability; URBAN, basin urban area in percent]

Equation 

number Regression equation prediction expression R-square Root mean square error

Observations

1 Log10(0.995 AEP peak per square mile) = 1.673 + (URBAN – 43.179) × 

((Log10(DA) – 1.412) × –0.00637) + 0.00372 × URBAN + –0.512 × 

Log10(DA)

0.84 0.26 69

2 Log10(0.99 AEP peak per square mile) = 1.713 + (URBAN – 43.538) ×

((Log10(DA) – 1.400) × –0.00626) + 0.00359 × URBAN + –0.505 × 

Log10(DA)

0.84 0.26 70

3 Log10(0.95 AEP peak per square mile) = 1.764 + (URBAN – 42.993) × 

((Log10(DA) – 1.350) × –0.00518) + 0.00391 × URBAN + –0.448 × 

Log10(DA)

0.84 0.23 80

4 Log10(0.9 AEP peak per square mile) = 1.869 + (URBAN – 42.339) × 

((Log10(DA) – 1.340) × –0.00523) + 0.00323 × URBAN + –0.446 × 

Log10(DA)

0.83 0.23 85

5 Log10(0.8 AEP peak per square mile) = 1.952 + (URBAN – 42.563) × 

((Log10(DA) – 1.325) × –0.00459) + 0.00336 × URBAN + –0.424 × 

Log10(DA)

0.83 0.23 89

6 Log10(0.67 AEP peak per square mile) = 1.984 + (URBAN – 40.843) × 

((Log10(DA) – 1.272) × –0.00479) + 0.00402 × URBAN + –0.399 × 

Log10(DA)

0.82 0.23 97

7 Log10(0.5 AEP peak per square mile) = 2.027 + (URBAN – 40.290) × 

((Log10(DA) – 1.216) × –0.00414) + 0.00468 × URBAN + –0.366 × 

Log10(DA)

0.81 0.23 103

8 Log10(0.43 AEP peak per square mile) = 2.072 + (URBAN – 40.227) × 

((Log10(DA) – 1.205) × –0.00403) + 0.00465 × URBAN + –0.362 × 

Log10(DA)

0.81 0.23 104

9 Log10(0.2 AEP peak per square mile) = 2.229 + (URBAN – 39.370) × 

((Log10(DA) – 1.139) × –0.00346) + 0.00487 × URBAN + –0.338 × 

Log10(DA)

0.76 0.24 110

10 Log10(0.1 AEP peak per square mile) = 2.373 + (URBAN – 38.706) × 

((Log10(DA) – 1.103) × –0.00313) + 0.00470 × URBAN + –0.334 × 

Log10(DA)

0.74 0.25 113

11 Log10(0.04 AEP peak per square mile) = 2.557 + (URBAN – 39.168) × 

((Log10(DA) – 1.083) × –0.00224) + 0.00434 × URBAN + –0.332 × 

Log10(DA)

0.69 0.28 114

12 Log10(0.02 AEP peak per square mile) = 2.697 + (URBAN – 39.168) × 

((Log10(DA) – 1.083) × –0.00219) + 0.00390 × URBAN + –0.343 × 

Log10(DA)

0.65 0.31 114

13 Log10(0.01 AEP peak per square mile) = 2.776 + (URBAN – 38.765) × 

((Log10(DA) – 1.070) × –0.00242) + 0.00434 × URBAN + –0.342 × 

Log10(DA)

0.57 0.37 110

14 Log10(0.005 AEP peak per square mile) = 2.863 + (URBAN – 39.063) × 

((Log10(DA) – 1.057) × –0.00223) + 0.00465 × URBAN + –0.329 × 

Log10(DA)

0.55 0.38 105

15 Log10(0.002 AEP peak per square mile) = 2.961 + (URBAN – 39.287) × 

((Log10(DA) – 0.904) × –0.00049) + 0.00636 × URBAN + –0.317 × 

Log10(DA)

0.44 0.46 65
For additional information regarding this 

publication, contact:

Director

USGS Virginia Water Science Center

1730 East Parham Road 

Richmond, Virginia 23228

email: dc_va@usgs.gov

Or visit the USGS Virginia Water Science 

Center Web site at:

 http://va.water.usgs.gov

Prepared by:

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Austin—Methods and Equations for Estimating Peak Streamflow Per Square Mile in Virginia’s Urban Basins—Scientific Investigations Report 2014–5090

Printed on recycled paper

ISBN 978-1-4113-3810-4

9 7 81411 338104

ISSN 2328-0328 (online)

ISSN 2328-031X (print)

http://dx.doi.org/10.3133/sir20145090