Prepared in cooperation with the Maine Department of Transportation

Peak Flow Regression Equations for Small,

Ungaged Streams in Maine: Comparing

Map-Based to Field-Based Variables

Scientific Investigations Report 2015–5049

U.S. Department of the Interior

U.S. Geological Survey
Cover. Gulliver Brook at Monarda, Maine, July 9, 2012. (Photograph by Terrence Talbot, U.S. Geological Survey)
Peak Flow Regression Equations for Small, 

Ungaged Streams in Maine: Comparing 

Map-Based to Field-Based Variables

By Pamela J. Lombard and Glenn A. Hodgkins

Prepared in cooperation with the Maine Department of Transportation

Scientific Investigations Report 2015–5049

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: 2015

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

Suggested citation:

Lombard, P.J., and Hodgkins, G.A., 2015, Peak flow regression equations for small, ungaged streams in Maine—

Comparing map-based to field-based variables: U.S. Geological Survey Scientific Investigations Report 2015–5049, 

12 p., http://dx.doi.org/10.3133/sir20155049.

ISSN 2328-0328 (online)
iii

Acknowledgments

The authors wish to thank Alexander Mann at the Maine Department of Transportation for his 

work in delineating drainage basins by using orthoimagery.

Thanks are also given to Terrence Talbot at the U.S. Geological Survey for his work collecting 

field data and building HEC-RAS models.

v

Contents

Acknowledgments........................................................................................................................................iii

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

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

Previous Studies ...................................................................................................................................1

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

Data Collection...............................................................................................................................................2

Site Identification and Surveying of Field-Based Variables ..........................................................2

Map-Based Determination of Basin Characteristics .....................................................................2

Peak Flow Hydrology.....................................................................................................................................6

Hydraulic Model Analysis Using a Field-Based Rust Line Variable ......................................................6

Regression Equations....................................................................................................................................7

Ordinary Least-Squares Regression With Bankfull Width as a Field-Based Variable..............7

Generalized Least-Squares Regression Analyses..........................................................................8

Limitations of Regression Equations .................................................................................................8

Evaluation of Regression Equations ..................................................................................................9

Summary........................................................................................................................................................11

References Cited..........................................................................................................................................12

Figures

1. Map showing locations of the study area and 40 USGS streamgaging stations with 

small drainage basins in Maine used in this study .................................................................4

2. Photograph showing example of channel with measurable geomorphic indicators 

of bankfull width at Patte Brook near Bethel, Maine .............................................................5

3. Photograph showing example of a culvert rust line at Clark Brook near Wallagrass 

Plantation, Maine..........................................................................................................................5

4. Graph showing two-dimensional range of explanatory variables for the regression 

equations........................................................................................................................................9

5. Graph showing differences in peak flows with 2-year and 100-year recurrence 

intervals, based on Hodgkins (1999) equations and equations from the current study, 

for drainage areas between 1 and 10 square miles and holding percent basin 

wetlands constant at 5 percent................................................................................................10

6. Graph showing differences in peak flows with 2-year and 100-year recurrence 

intervals, based on Hodgkins (1999) equations and equations from the current study, 

for drainage areas between 1 and 10 square miles and holding percent basin 

wetlands constant at 15 percent..............................................................................................10
vi

Tables

1. Basin characteristics at 40 USGS streamgaging stations on small streams 

in Maine..........................................................................................................................................3

2. Rust-forming flows at seven culverts in Maine.......................................................................7

3. Regression equations and their accuracy for estimating peak streamflows for 

small, ungaged streams in Maine that are rural and unregulated.......................................8

4. Comparison of errors of the 1999 peak-flow equations and errors of the equations 

from the current study when each set of equations is applied to basins with 

drainage areas between 1 and 12 square miles....................................................................11

5. Comparison of errors of the 1999 peak-flow equations and errors of the equations 

from the current study when each set of equations is applied to basins with 

drainage areas between 0.3 and 3.6 square miles................................................................11

Conversion Factors

Inch/Pound to SI

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square foot (ft2

) 929.0 square centimeter (cm2)

square foot (ft2

) 0.09290 square meter (m2)

square inch (in2

) 6.452 square centimeter (cm2)

square mile (mi2

) 2.590 square kilometer (km2)

Flow rate

cubic foot per second (ft3

/s) 0.02832 cubic meter per second (m3/s)
Peak Flow Regression Equations for Small, Ungaged 

Streams in Maine: Comparing Map-Based to Field-Based 

Variables

By Pamela J. Lombard and Glenn A. Hodgkins

Abstract

Regression equations to estimate peak streamflows 

with 1- to 500-year recurrence intervals (annual exceedance 

probabilities from 99 to 0.2 percent, respectively) were 

developed for small, ungaged streams in Maine. Equations 

presented here are the best available equations for estimating 

peak flows at ungaged basins in Maine with drainage areas 

from 0.3 to 12 square miles (mi2). Previously developed 

equations continue to be the best available equations for 

estimating peak flows for basin areas greater than 12 mi2. 

New equations presented here are based on streamflow records 

at 40 U.S. Geological Survey streamgages with a minimum 

of 10 years of recorded peak flows between 1963 and 2012. 

Ordinary least-squares regression techniques were used to 

determine the best explanatory variables for the regression 

equations. Traditional map-based explanatory variables were 

compared to variables requiring field measurements. Two 

field-based variables—culvert rust lines and bankfull channel 

widths—either were not commonly found or did not explain 

enough of the variability in the peak flows to warrant inclusion 

in the equations. The best explanatory variables were drainage 

area and percent basin wetlands; values for these variables 

were determined with a geographic information system. 

Generalized least-squares regression was used with these two 

variables to determine the equation coefficients and estimates 

of accuracy for the final equations.

Introduction

The Maine Department of Transportation (MaineDOT) 

and other State and town agencies tasked with the construction 

and maintenance of drainage infrastructure in or near rivers 

and streams need methods for estimating streamflows at 

ungaged locations on small streams throughout Maine. 

Equations for estimating peak-flow statistics at streams with 

1-, 2-, 5-, 10-, 25-, 50-, 100-, 200-, and 500-year recurrence 

intervals (equivalent to annual exceedance probabilities 

[AEPs] of 99-, 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent, 

respectively) can be useful for culvert design. A flood with 

a 1-percent AEP has a 1-in-100 chance of being equaled or 

exceeded in any given year and has an average recurrence 

interval of 100 years and is referred to as the 100-year flood. 

Statistical methods currently are available to estimate peak 

flows for ungaged streams at basins in Maine with drainage 

areas from 1 to 1,700 square miles (mi2) (Hodgkins, 1999) 

and monthly mean and median flows at basins with drainage 

areas from 10 to 1,500 mi2 (Dudley, 2004). Methods are not 

available, however, for estimating peak flows for very small 

basins (drainage areas less than 1 mi2) or for estimating mean 

and median monthly flows for small basins (drainage areas 

between 1 and 10 mi2).The exception is for June and August 

median flows in some parts of the State as documented in 

Lombard and others (2003) and Lombard (2004, 2010). 

Although current peak-flow regression equations can be used 

for basins with drainage areas as small as 1 mi2, the resulting 

estimates may not be robust for basins this small because 

the equations were developed using few sites with drainage 

areas less than 3 mi2. In 2011, the U.S. Geological Survey 

began a cooperative investigation with the MaineDOT to test 

regression-equation explanatory variables for estimating peak 

flows on small (less than 12 mi2) basins in Maine.

Previous Studies

In 2007, the accuracy of various methods used for estimating peak flows for small and very small drainage basins 

in Maine was explored (Hodgkins and others, 2007). For very 

small basins, the simple hydrologic models TR-20 and the 

Rational Method were found to have very large errors (root 

mean square errors of –98.0 to 5,010 percent and –69.7 to 

230 percent, respectively). It was not possible to test regression equations for estimating peak flows on very small basins 

due to the lack of very small basins with sufficient data. 

Regression equations were found to be more accurate than 

TR-20, the Rational Method, and the Probabilistic Rational 

Method for estimating peak flows at small basins (Hodgkins 

and others, 2007). 
2 Peak Flow Regression Equations for Small, Ungaged Streams in Maine: Comparing Map-Based to Field-Based Variables

The possibility for testing regression equations for 

computing peak flows at very small, ungaged basins now 

exists because of data collected in the last decade by the 

U.S. Geological Survey (USGS) in cooperation with 

MaineDOT and others. In addition, easy-to-obtain field 

indicators of peak streamflows at culverts, such as culvert rust 

lines and bankfull channel geometry, may allow for improved 

accuracy of regression equations for estimating peak flows 

at small basins in Maine. Many culvert rust lines have been 

observed by MaineDOT that seem to correspond with high 

flows through the culverts. Bankfull channel geometry often 

corresponds to defined high flows (Leopold and Maddock, 

1953; Dunne and Leopold, 1978) and may be a good 

explanatory variable for peak flows.

Purpose and Scope

The purpose of this report is to present the results of 

a study conducted to develop new regression equations for 

estimating peak flows at small (less than 12 mi2), ungaged 

streams in Maine and to determine if field indicators can 

improve the accuracy of these equations. The objectives of 

the study described in this report include (1) computing peak 

flows for selected recurrence intervals at all small, gaged 

basins in Maine, including using peak-flow data that were 

collected during the last 15 years; (2) determining whether two 

field indicators of peak flows—culvert rust lines and bankfull 

channel geometry—are generally available and improve the 

accuracy of peak-flow regression equations compared with 

equations developed solely using basin characteristics such 

as drainage area; and (3) comparing the best new equations 

for small basins with previously published statewide 

peak-flow equations. 

Data Collection

Data collection for this study involved determining which 

small drainage basins in Maine have both streamflow gaging 

data and usable field indicators, surveying and measuring 

channel and culvert geometry at usable sites, and using a 

geographic information system (GIS) to delineate basins and 

calculate basin characteristics for each site. Sufficient field 

data were collected to build hydraulic models for sites with 

established culvert rust lines. The basin characteristics were 

analyzed to aid in the development of regression equations for 

computing flood frequency at ungaged sites.

Site Identification and Surveying of Field-Based 

Variables

Forty USGS streamgaging stations in Maine with 

drainage areas less than 12 mi2 and a minimum of 10 years 

of annual peak-flow record (table 1; fig. 1) were used in the 

study. Note that 10 years of record is generally considered 

the minimum amount of data needed for estimating peak 

flows on small streams. Flood-frequency statistics would be 

expected to be more accurate at stations with longer periods 

of record. Eight of the 40 stations have continuous daily flow 

data, and the remaining 32 stations have annual peak-flow data 

that were measured with crest-stage gages, a cost-effective 

method of measuring peak flows that exceed a set threshold. 

Field reconnaissance at each of the 40 sites determined that 

23 sites had measurable geomorphic indicators of bankfull 

width (fig. 2), and 7 sites had well-defined culvert rust lines 

(fig. 3). The majority of these sites were located near culverts, 

although only seven had well-defined rust lines. 

Bankfull width was determined at locations with a clear 

break between the channel and the flood plain, well-defined 

changes in vegetation, and (or) the top of the inside bend of a 

point bar (alluvium that has accumulated on the inside bend of 

a meandering stream). The first two indicators were the most 

common. Five measurements of bankfull width were recorded 

at each of the sites where bankfull width could be adequately 

defined. The five widths were taken at locations along the 

channel that were a minimum distance of 10 culvert diameters 

upstream or downstream from the culvert and a maximum 

distance of 1,000 feet (ft) upstream or downstream from the 

culvert. The width measurements along the channel were made 

at a distance of at least one culvert diameter from each other. 

At the locations where bankfull widths were measured, there 

were no issues with unstable banks, exposed bedrock, nearby 

boulders, artificial encroachments, channel obstructions, 

braided channels, or backwater from downstream constrictions 

(which could include the nearby culvert). Bankfull widths 

were not measured near falls or sharp bends. A median bankfull width was computed from the five measured widths at 

each site; median bankfull widths near the 23 culverts ranged 

from 6.8 to 32.3 ft (table 1).

For sites with well-defined culvert rust lines, culvertgeometry and cross-section data were obtained from field 

surveys conducted by personnel from the USGS New England 

Water Science Center during the summer of 2012. Surveys 

used differential leveling techniques to tie elevations to a local 

datum. Characteristics of each culvert were determined, and 

structural dimensions were measured. 

Map-Based Determination of Basin 

Characteristics

Basin characteristics were determined using a GIS. Basin 

delineation errors can result in large overall errors in basin 

characteristics, especially for sites with small basins. Thus, it 

was especially important to have accurate measurements of 

drainage areas for this study. Several techniques were used to 

obtain drainage areas on the small sites. In 12 Maine basins for 

which light detection and ranging (lidar) data were available 

in 2014, a digital elevation model (DEM) derived from the 

lidar data was used to delineate the basins by using ArcHydro 

within ArcGIS. ArcHydro is a set of data models and tools that 
Data Collection 3

Table 1. Basin characteristics at 40 USGS streamgaging stations on small streams in Maine.

[USGS, U.S. Geological Survey; mi2, square miles; ME, Maine; NA, not available. Crest stage gages measure peak flow only]

USGS 

streamgage 

number

Streamgage name Gage type

Years 

of 

record

Drainage 

area 

(mi2)

Basin 

wetlands 

(percent)

 Average 

basin 

slope 

(percent)

Median 

bankfull 

width 

(feet)

01012895 Clark Brook near Wallagrass Plantation, ME (Eagle Lake) Crest stage 13 1.02 10.6 55.4 NA

01012970 Unnamed Brook near Wallagrass Plantation, ME 

(Soldier Pond)

Crest stage 13 0.42 1.60 38.3 8.6

010147001 Factory Brook near Madawaska, ME Crest stage 11 5.97 2.01 38.2 25.0

01015700 Houlton Brook near Oxbow, ME Crest stage 11 6.15 20.9 11.3 17.0

01017300 Nichols Brook near Caribou, ME Crest stage 11 3.87 7.41 14.9 NA

01017550 Williams Brook at Phair, ME Continuous 14 3.62 22.2 11.5 NA

01017900 Marley Brook near Ludlow, ME Continuous 18 1.57 0.88 19.0 NA

01021300 Wiggins Brook near West Lubec, ME Crest stage 10 4.98 11.6 19.3 NA

01021470 Libby Brook near Northfield, ME Continuous 13 6.73 13.4 10.2 NA

01021600 Middle River near Machias, ME Crest stage 10 8.05 14.0 17.7 19.2

01021890 Unnamed Brook near Crawford, ME Crest stage 13 0.70 5.57 19.6 NA

01022700 Forbes Pond Brook near Prospect Harbor, ME Crest stage 10 8.53 20.1 19.4 NA

01024200 Garland Brook near Mariaville, ME Continuous 18 9.75 6.68 29.9 32.3

01026800 Frost Pond Brook near Sedgwick, ME Crest stage 10 6.26 17.9 22.1 NA

01030300 Trout Brook near Danforth, ME Crest stage 10 4.29 14. 9 20.7 NA

01030400 Gulliver Brook near Monarda, ME Crest stage 11 11.3 19.9 12.8 28.0

01031470 Brewster Brook near Parkman, ME Crest stage 14 0.65 2.89 23.1 NA

01031600 Morrison Brook near Sebec Corners, ME Continuous 13 4.10 9.62 24.7 22.0

01034900 Coffin Brook near Lee, ME Crest stage 11 2.04 15.3 23.2 13.7

01036380 Unnamed Brook near Bradley, ME Crest stage 14 0.66 10.1 12.3 9.6

01037200 Shaw Brook near Northern ME Junction, ME Crest stage 11 2.93 4.13 14.5 NA

01037430 Goose River at Rockport, ME Crest stage 11 8.24 5.87 43.0 NA

01041900 Mountain Brook near Lake Parlin, ME Crest stage 11 3.99 2.81 55.5 NA

01047860 Unnamed Brook near New Sharon, ME Crest stage 14 0.58 4.64 10.3 11.0

010488401 Unnamed Brook near Newport, ME Crest stage 14 0.42 7.13 7.5 6.8

01049100 Hall Brook at Thorndike, ME Crest stage 11 4.78 3.22 24.2 24.0

01049130 Johnson Brook at South Albion,ME Continuous 11 2.42 15.4 16.5 10.6

01049180 Unnamed Brook near Albion, ME Crest stage 12 0.31 2.25 23.7 9.0

01049300 North Branch Tanning Brook near Manchester, ME Continuous 20 1.14 7.65 28.3 8.8

010496901 Unnamed Brook near Dresden, ME Crest stage 14 0.44 2.25 29.3 12.0

01049700 Gardiner Pond Brook at Dresden Mills, ME Crest stage 10 8.02 20.6 23.4 NA

01050490 Unnamed Brook near Rangeley, ME Crest stage 13 0.44 0.00 53.9 13.0

01050900 Four Ponds Brook near Houghton, ME Crest stage 11 3.23 20.2 36.4 18.3

01054135 Unnamed Brook near Gilead, ME Crest stage 14 0.34 0.02 50.6 7.4

010553001 Bog Brook near Buckfield, ME Crest stage 11 10.5 13.1 29.5 NA

01062700 Patte Brook near Bethel, ME Crest stage 10 3.78 5.68 56.7 17.8

01063310 Stony Brook at East Sebago, ME Continuous 18 1.55 18.5 26.2 8.6

01064200 Mill Brook near Old Orchard Beach, ME Crest stage 10 2.30 5.13 15.1 NA

01066100 Pease Brook near Cornish, ME Crest stage 11 5.01 3.54 49.7 17.3

01069700 Branch Brook near Kennebunk, ME Crest stage 18 9.65 9.15 19.1 19.5

1

These four sites were identified as having high influence in the peak-flow regression analyses (see section “Ordinary Least-Squares Regression With Bankfull 

Width as a Field-Based Variable”).
4 Peak Flow Regression Equations for Small, Ungaged Streams in Maine: Comparing Map-Based to Field-Based Variables

70° 68°

46°

44°

MAINE

NEW HAMPSHIRE

NEW

BRUNSWICK

QUEBEC

ATLANTIC

OCEAN

01014700

01012970

01012895

01017300

01017550

01015700

01017900

01030400

01030300

01041900

01034900

01031600

01031470 01021890

01050490

01048840 01036380 01050900 01021470

01037200

01021300

01021600

01024200

01047860

01049100

01049130 01049180

01022700

01054135 01026800

01049300

01062700

01055300 01037430

01049690

01049700

01063310

01066100

01064200

01069700

EXPLANATION

U.S. Geological Survey streamgaging

 station with number

0 20 40 60 MILES

0 20 40 60 KILOMETERS

Base map from U.S. Geological Survey National Atlas,

June 2002,source scale 1:2,000,000.

Data points from U.S. Geological Survey NWIS database.

Figure 1. Locations of the study area and 40 USGS streamgaging stations with small drainage basins in Maine used in 

this study.
Data Collection 5

Figure 2. Example of channel 

with measurable geomorphic 

indicators of bankfull width at 

Patte Brook near Bethel, Maine.

Figure 3. Example of a culvert rust line at Clark Brook near Wallagrass Plantation, Maine.
6 Peak Flow Regression Equations for Small, Ungaged Streams in Maine: Comparing Map-Based to Field-Based Variables

operates within ArcGIS to support geospatial and temporal 

data analyses. For 16 of the sites for which lidar data were not 

available, a digitized stereoscope method was used with ArcMap to georeference the scanned orthophotos or topographic 

maps on which basin delineations were drawn. Once georeferenced, the delineations were digitized into polygons by tracing 

the drawn delineations on the scanned photographs. Twelve 

sites remained for which there were no available lidar data 

and no available orthophotos suitable for stereoscoping. For 

these sites, GIS with 10 meter (m) DEMs was used to manually delineate the drainage basins. The 28 sites for which lidar 

and (or) ortho stereoscoping methods were used ranged in size 

from 0.3 to 8.5 mi2. The 12 sites for which only 10 m DEMs 

were available ranged in size from 1.5 to 11.3 mi2. 

Because manual delineation using 10 m DEM was available for all sites, the accuracy of the 10 m DEM method could 

be determined as compared to the accuracy of the lidar and 

(or) stereoscoping. Of the 28 sites for which either lidar or 

stereoscoping methods were used with 10 m DEM delineations, 22 had differences between the methods of less than 

10 percent, 5 had differences of 10–35 percent, and 1 had a 

difference greater than 35 percent. Basin characteristics of 

average basin slope and percent basin wetlands were tested 

for use in the peak-flow regression equations. Percent basin 

wetlands was selected for use in the equations based on its 

relation to peak flows demonstrated in statewide peak-flow 

equations in Maine (Hodgkins, 1999). Average basin slope 

was selected as a potential variable based on its importance 

in the Manning’s formula for calculating flows (Maidment, 

1993) and the hypothesis that slope may be a better explanatory variable than basin wetlands for small, steep basins that 

tend to have little wetland area. Both variables were calculated 

for all 40 basins in the study (table 1). To determine average 

basin slope, the slope tool in ArcGIS was used; this tool calculates the maximum rate of change in value from neighboring 

cells for each grid cell. This means that the steepest gradient 

is identified between a cell and its neighbors, and the cell 

is assigned a slope value of the rise over run. Note that this 

method computes slope over the entire basin rather than slope 

of the flowline only. The GIS slope grid used in this study 

was produced from the National Elevation Dataset (NED) 

(Gesch and others, 2002) based on the best available data as of 

March 2014. The slope grid units were feet per feet multiplied 

by 100. The data used for this method were mostly based on 

10 m resolution grids with sections of NED-approved lidar. 

Average basin slopes ranged from 7.5 to 56.7 percent for sites 

in the study (table 1).

Percent basin wetlands was calculated as the percentage 

of wetland cells in each drainage basin by using a grid derived 

from the National Wetland Inventory (NWI) dataset, which 

was produced by the U.S. Fish and Wildlife Service at a scale 

of 1:24,000 (U.S. Fish and Wildlife, 2008). All categories 

of wetlands in the NWI, including ponds and lakes, were 

included in the basin-wetland calculations. Percent basin 

wetlands ranged from 0 to 22.2 percent for sites in this study 

(table 1). 

Peak Flow Hydrology

Statistical methods were used to compute peak flows for 

selected recurrence intervals at streamflow gages for use in 

the regression equations. Eight of the 40 streamflow gages 

used in this study have continuous streamflow records with 11 

to 20 years of data collected between 1963 and 2012, and the 

remaining crest-stage gages (for which only peak flows are 

measured) have from 10 to 18 recorded annual peak streamflows between 1963 and 2012 (U.S. Geological Survey, 2013; 

table 1). Discharges for the 1-, 2-, 5-, 10-, 25-, 50-, 100-, 200-, 

and 500-year floods (equivalent to the 99-, 50-, 20-, 10-, 4-, 2-, 

1-, 0.5-, and 0.2-percent AEPs, respectively) were computed 

at the 40 sites. The at-site estimates for all of the streamgages 

were determined by the standard log-Pearson type III method 

described in Bulletin 17B of the Hydrology Subcommittee 

(Interagency Advisory Committee on Water Data, 1982) and a 

modification of this method called the expected moments algorithm (EMA) (Cohn and others, 1997, 2001; Griffis and others, 

2004) based on 10 to 20 years of peak flows. Flow frequencies 

were run with a generalized skew of 0.029 as developed for 

Maine in Hodgkins (1999). The flood-frequency calculations 

were performed using the U.S. Geological Survey software 

PeakFQ (U.S. Geological Survey, 2014).

Hydraulic Model Analysis Using a 

Field-Based Rust Line Variable

Hydraulic models for the seven culverts with definable 

rust lines were prepared for this study using the stepbackwater model, Hydrologic Engineering Centers River 

Analysis System (HEC-RAS), version 4.1.0 (U.S. Army 

Corps of Engineers, 2010), in order to estimate streamflows 

consistent with measured rust lines. The HEC-RAS analyses 

were done using the steady-state flow computation option. 

Subcritical (tranquil) flow regime was assumed for the 

simulations. Flows were modeled in the channel upstream 

from the culvert, in the culvert, and in the channel downstream 

from the culvert. Starting water-surface elevations were set 

as the normal depth based on downstream slope estimated at 

each of the sites. Hydraulic analyses require the estimation of 

energy losses that result from frictional resistance exerted by 

a channel on streamflow. These energy losses are quantified 

by the Manning’s roughness coefficient (“n” value) and were 

selected on the basis of field observations and high-resolution 

aerial photographs. The final “n” values ranged from 0.04 

to 0.05 for the main channels, 0.07 to 0.08 for the overbank 

areas, and 0.019 to 0.034 within the culverts. Entrance losses 

for the culverts ranged from 0.7 to 0.9, and exit losses were 

typically 1. These are standard values as suggested in the 

HEC-RAS Hydraulic Reference Manual (U.S. Army Corps of 

Engineers, 2010). The HEC-RAS models were run with peak 

flows for selected recurrence intervals. In addition, flows were 

determined that best matched measured rust lines. 
Regression Equations 7

Of the seven culverts with definable rust lines that were 

surveyed and modeled, four were calibrated using both the 

upstream and downstream rust lines. The flows that formed the 

rust lines at these four culverts did not appear to be affected 

by downstream backwater. Flood-frequency analyses and 

hydraulic modeling indicated that the rust-forming flow at 

one of these culverts (station 01012895) corresponds to a flow 

with a recurrence interval slightly greater than 1 year. Rust 

lines at two of the culverts (stations 01021300 and 01034900) 

were likely formed by flows with recurrence intervals 

slightly less than 1 year. The rust line at the remaining culvert 

(station 01037430) was most likely formed by a flow with a 

recurrence interval much less than 1 year (table 2).

The downstream rust lines on the remaining three 

culverts were clearly formed by backwater conditions. The 

rust lines on the downstream ends were higher than would be 

expected due to culvert slope and friction losses and can most 

easily be explained by backwater caused by ice conditions, 

downstream channel constrictions, or temporary obstructions. 

This was especially evident at Hall Brook at Thorndike 

(USGS station number 01049100), where there was a drop at 

the downstream end of the culvert indicating an inlet control, 

but where the rust lines had likely been formed by backwater. 

The four sites with backwater-free rust lines were not a 

sufficient number for testing the elevation of the rust lines or 

the area beneath rust lines as explanatory variables in a peakflow regression equation. Furthermore, the low percentage of 

available culverts with definable rust lines indicates that this 

variable is not useful for computing peak flows at ungaged 

basins in Maine. Culvert rust lines were thus dropped from 

further analyses in this study. 

Regression Equations

Regression equations were created for estimating peak 

flows with 1-, 2-, 5-, 10-, 25-, 50-, 100-, 200-, and 500-year 

recurrence intervals at small (less than 12 mi2) ungaged basins 

in Maine. A limited set of potential explanatory variables was 

tested for use in the regression equations using ordinary leastsquares (OLS) linear regression in the R statistical package 

(R core team, 2014). These variables included drainage area, 

average basin slope, percent basin wetlands developed using 

a GIS, and median bankfull width, which was measured in the 

field. The best explanatory variables for the regression equations were determined by choosing variables that resulted in 

the lowest regression errors. If errors were similar, variables 

that were easier to compute and ones that best explained peakflow variability across the entire range of recurrence intervals 

(from 1 to 500 years) were favored. After choosing explanatory variables with OLS, final equation coefficients were 

computed using generalized least-squares (GLS) regression 

analyses (Stedinger and Tasker, 1985) using the WeightedMultiple-Linear Regression (WREG) program (Eng and others, 2009).

Ordinary Least-Squares Regression With 

Bankfull Width as a Field-Based Variable

Only 23 of the 40 sites had usable measurements of 

bankfull width; thus, initial OLS regression analyses were 

done using this subset of 23 sites with combinations of 

drainage area, percent basin wetlands, average basin slope, 

and bankfull width as explanatory variables and flows with 

1- to 500-year recurrence intervals as response variables. 

Regression diagnostics during exploratory analyses included 

residual plots that checked for linearity, homoscedasicity 

(constant variance), normality, and the presence of outliers. 

Residual plots indicated whether explanatory variables showed 

bias over their ranges and whether variables needed to be 

transformed to ensure a linear relation between explanatory 

and response variables, a requirement for satisfactory 

regressions. A base-10 logarithmic transformation of the 

drainage-area explanatory variable and all peak-flow response 

variables was tested and adopted because this transformation 

Table 2. Rust-forming flows at seven culverts in Maine.

[U.S. Geological Survey; ME, Maine; ft3/s, cubic feet per second]

USGS 

streamgage 

number

Streamgage name

Flow with a 1-year 

recurrence interval, 

(ft3/s)

Rust-forming flow, 

(ft3/s)

01012895 Clark Brook near Wallagrass Plantation, ME (Eagle Lake) 10.6 15.0

01021300 Wiggins Brook near West Lubec, ME 88.5 67.0

01021600 Middle River near Machias, ME 100 Rust line influenced by backwater

01024200 Garland Brook near Mariaville, ME 119 Rust line influenced by backwater

01034900 Coffin Brook near Lee, ME 21.8 17.0

01037430 Goose River at Rockport, ME 135 28.0

01049100 Hall Brook at Thorndike, ME 37.1 Rust line influenced by backwater
8 Peak Flow Regression Equations for Small, Ungaged Streams in Maine: Comparing Map-Based to Field-Based Variables

improved the fit of the regression equations. Multicollinearity 

among the explanatory variables (variables that are highly 

correlated) was measured using the variance inflation factor, 

and no problems with multicollinearity were noted between 

the final variables. 

Influence of individual stations on the regression 

equations was measured using the Cook’s D statistic. High 

influence means that a site is both an outlier and has high 

leverage. Although the Cook’s D statistic indicated that four 

sites exerted high influence on the final regression equations 

(table 1), these sites were included in the final analyses 

because there were no obvious problems with the data that 

would warrant their removal. Alternatively, if these four sites 

had a common basin characteristic, all sites with that basin 

characteristic could be removed and the equations would not 

apply to any sites with this same basin characteristic—or 

the common basin characteristic could indicate the need 

for additional explanatory variables. No common basin 

characteristic was found for these four sites. 

The 23-site model with drainage area and percent basin 

wetlands explained more variability in the peak flows than 

did drainage area and average basin slope, drainage area by 

itself, or bankfull width by itself as based on the R2 values 

of the models. The model with drainage area and percent 

basin wetlands explained 75 to 84 percent of the variability 

for peak flows with 1- to 500-year recurrence intervals which 

was slightly better than the model that included drainage area 

combined with bankfull width that explained 73 to 83 percent 

of the variability for these peak flows. Since bankfull width is 

more time consuming to calculate (involving a field visit) and 

is not available for all sites, bankfull width was removed from 

further analyses. 

Ordinary least-squares regressions were computed again 

using the remaining variables for all 40 sites. Once again, 

the model with drainage area and percent basin wetlands 

explained the most variability; explaining 75 to 84 percent of 

the variability in peak flows with 1- to 500-year recurrence 

intervals. Drainage area by itself explained almost as much 

variability as drainage area combined with percent basin wetlands for the 1-year recurrence interval but much less of the 

variability for the 500-year recurrence interval, and thus basin 

wetlands was retained in all of the models for consistency. The 

model with slope and drainage area explained 68 to 79 percent 

of the variability in peak flows with 1- to 500-year recurrence 

intervals and thus did not perform as well as the model with 

wetlands and drainage area. 

Generalized Least-Squares Regression 

Analyses

Final equation coefficients and accuracy were computed 

with GLS regression analyses using the WREG program (Eng 

and others, 2009) and are presented in table 3. The equations 

are of the form

Q1...500 = bAa10-w(W)

where Q1 through Q500 are discharges with 1- to 500-year 

recurrence intervals, A is drainage area, W is basin wetlands, 

and the coefficients b, a, and w vary with recurrence interval.

Stedinger and Tasker (1985) found that GLS regression 

equations are more accurate and provide better estimates of 

uncertainty than OLS regression equations when streamflow 

records are of different and widely varying lengths and when 

concurrent peak flows are correlated. Concurrent peak flows 

across Maine can be correlated because rainstorms can affect 

large areas and snowmelt-related runoff tends to be regionally 

coherent. Less weight is given to GLS regression techniques 

used at stations where the streamflow record is more highly 

correlated with the concurrent record of other stations. 

Equation accuracy is quantified by the average standard error of prediction (ASEP), which is a measure of how 

well the regression equations will estimate the flow statistic 

of interest when applied to an ungaged, unregulated, rural 

basin in Maine. The probability that the true value of the flow 

statistic at a site is between the negative- and positive-percent 

ASEP is approximately 68 percent. For example, there is a 

68-percent probability that the true peak-flow value with a 

100-year recurrence interval is between –33.3 and 50.0 percent 

of the 100-year peak flow value computed with the regression 

equation (table 3). 

Limitations of Regression Equations

When estimating peak flows for ungaged basins in Maine 

on the basis of the regression equations described in this 

Table 3. Regression equations and their accuracy for 

estimating peak streamflows for small, ungaged streams in 

Maine that are rural and unregulated.

[ASEP, average standard error of prediction; Q1 through Q500, peak flows 

with 1- to 500-year recurrence intervals in cubic feet per second; A, drainage 

area in square miles; W, basin wetlands in percent]

Regression equation ASEP 

(in percent)

Q1 = 18.836(A).77310-.013(W) -31.0 to 45.0

Q2 = 64.121(A).80310-.015(W) -28.4 to 39.6

Q5 = 102.565(A).80910-.016(W) -28.9 to 40.7

Q10 = 131.522(A).81110-.017(W) -30.0 to 42.9

Q25 = 171.791(A).81410-.017(W) -31.5 to 46.0

Q50 = 204.174(A).81610-.018(W) -32.4 to 48.0

Q100 = 238.781(A).81710-.018(W) -33.3 to 50.0

Q250 = 275.423(A).81810-.019(W) -34.6 to 52.9

Q500 = 327.341(A).82010-.019(W) -35.8 to 55.7
Regression Equations 9

report, it is important that the explanatory variables of drainage area and percent basin wetlands be derived using the same 

or comparable methods as those documented in this report. 

If basin characteristics are derived using different methods, 

estimates of flow computed from the equations will be of 

unknown accuracy. If the values of drainage area and percent 

basin wetlands at a site are outside the two-dimensional range 

of those used to develop these equations (fig. 4, outside shaded 

area), estimates will also be of unknown accuracy. The equations in this report were developed using rural basins in Maine 

and thus should be applied to rural basins in Maine. Hodgkins 

(1999) contains guidance on estimating peak flows for urbanized basins. There were no peak-flow data available for small 

basins in the northwestern part of Maine (fig. 1); therefore, the 

equations may not apply to that area of the State. 

Basins included in this study had no substantial peakflow reservoir regulation; therefore, estimates of peak flows 

at small, ungaged basins, using the equations in this report, 

are unlikely to be accurate for basins with substantial peakflow regulation. Benson (1962) defines substantial reservoir 

regulation as more than 4.5 million cubic feet per square mile 

(or approximately 100 acre-feet per square mile) of usable 

reservoir storage. Usable storage is the volume of water available for release from a reservoir, between the minimum and 

maximum controllable elevations.

Evaluation of Regression Equations

A previous peak-flow study for Maine developed 

regression equations to estimate peak flows on basins with 

drainage areas of 1 to 1,700 mi2 (Hodgkins, 1999). The final 

explanatory variables in Hodgkins (1999) were drainage 

area and percent basin wetlands, the same as in the current 

report; however, many more potential explanatory variables 

were tested in the 1999 report. Basin slope was tested in 

both equations, but wetlands combined with drainage area 

was found to be a better predictor of peak flows than slope 

combined with drainage area in both reports. 

The area of overlap between the 1999 equations and the 

current equations includes basins with drainage areas between 

1 and 12 mi2; both sets of equations were designed to be used 

within this range. The equations can result in very different estimates of peak flows in this range due to the different 

sets of stations from which they were developed. Differences 

between estimates from the two sets of equations are most pronounced for estimates of high recurrence-interval peak flows 

for basins with a low percentage of wetlands (figs. 5 and 6). 

Thirty sites have drainage areas between 1 and 12 mi2

and sufficient streamflow data for testing the accuracy of the 

equations. For each of the 30 basins, differences between the 

predicted peaks from the two sets of regression equations 

were compared to observed peaks. Overall errors between 

predicted and observed peaks were computed with the root 

mean square error (RMSE). Analyses of RMSEs indicate that 

errors are less for the equations in the current study as opposed 

to equivalent equations in the 1999 study. For example, 

RMSEs for peak flows with a 10-year recurrence interval are 

–30.9 to 44.8 percent for the 1999 equations, but are –27.4 to 

37.7 percent for the current equations (table 4). Thus, the 

equations in the current study should be used for ungaged 

basins in Maine with drainage areas from 1 to 12 mi2

 because 

these equations are expected to provide estimates that are 

closer to the true values for these small basins. 

0

5

10

15

20

25

0.1 1 10 100

Basin wetlands, in percentage

Drainage area, in square miles

Figure 4. Two-dimensional range of explanatory variables for the regression equations.
10 Peak Flow Regression Equations for Small, Ungaged Streams in Maine: Comparing Map-Based to Field-Based Variables

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

0 2 4 6 8 10

Discharge, in cubic feet per second

Drainage area, in square miles

5 percent basin wetlands

2-year recurrence interval

2015

2015

1999

1999

100-year recurrence interval

Figure 5. Differences in peak flows with 2-year and 100-year recurrence intervals, based on 

Hodgkins (1999) equations and equations from the current study, for drainage areas between 1 and 

10 square miles and holding percent basin wetlands constant at 5 percent.

0

100

200

300

400

500

600

700

800

900

0 2 4 6 8 10

Discharge, in cubic feet per second

Drainage area, in square miles

15 percent basin wetlands

2-year recurrence interval

100-year recurrence interval

2015

2015

1999

1999

Figure 6. Differences in peak flows with 2-year and 100-year recurrence intervals, based on 

Hodgkins (1999) equations and equations from the current study, for drainage areas between 1 and 

10 square miles and holding percent basin wetlands constant at 15 percent.
Summary 11

Table 4. Comparison of errors of the 1999 peak-flow equations 

and errors of the equations from the current study when each set 

of equations is applied to basins with drainage areas between 1 

and 12 square miles.

[RMSE, root mean square error]

Recurrence 

interval

1999 equations RMSE 

(in percent)

Current study equations RMSE 

(in percent)

2 years -27.5 to 37.9 -23.1 to 30.1

5 years -29.3 to 41.4 -25.3 to 33.8

10 years -30.9 to 44.8 -27.4 to 37.7

25 years -33.0 to 49.2 -29.9 to 42.6

50 years -34.5 to 52.7 -31.9 to 46.8

100 years -35.9 to 56.1 -33.6 to 50.7

500 years -39.1 to 64.1 -37.6 to 60.2

Comparisons of RMSE errors between the two sets of 

equations were also examined for the 20 stations with the 

smallest drainage areas in Maine that have sufficient data. 

These stations have drainage areas between 0.3 and 3.6 mi2. 

Although the 1999 equations were not designed to be used 

on basins less than 1 mi2 and because data were not available 

in this range at the time they were developed, the equations 

can now be tested in this range with the newer data. For these 

20 small basins, the current equations perform better than do 

the 1999 equations. For example, RMSEs for peak flows with 

a 500-year recurrence interval are –50.3 to 101.1 percent for 

the 1999 equations, but are –37.6 to 60.3 percent for the current equations (table 5). The equations developed by Hodgkins 

(1999) should continue to be used for all basin areas greater 

than 12 mi2. 

Table 5. Comparison of errors of the 1999 peak-flow equations 

and errors of the equations from the current study when each set 

of equations is applied to basins with drainage areas between 0.3 

and 3.6 square miles.

[RMSE, root mean square error]

Recurrence 

interval

1999 equations RMSE 

(in percent)

Current study equations RMSE 

(in percent)

2 years -33.7 to 50.9 -30.7 to 44.2

5 years -37.4 to 59.7 -31.1 to 45.2

10 years -40.0 to 66.6 -32.1 to 47.2

25 years -43.1 to 75.6 -33.2 to 49.7

50 years -45.0 to 81.9 -34.4 to 52.3

100 years -46.8 to 88.1 -35.3 to 54.5

500 years -50.3 to 101.1 -37.6 to 60.3

Summary

In 2011, the U.S. Geological Survey began a cooperative 

investigation with the Maine Department of Transportation 

(MaineDOT) to test regression-equation explanatory variables 

for estimating peak flows on small (less than 12 square miles

[mi2]) basins in Maine that have 1- to 500-year recurrence 

intervals. These recurrence intervals are equivalent to floods 

that have a 99- to 0.2-percent chance of being exceeded each 

year. Peak-flow data from 40 USGS streamgages that had 

small drainage basins and a minimum of 10 years of recorded 

peak flows were used in this study. A major objective of the 

study was to determine if field-based characteristics, such as 

culvert rust lines or bankfull channel widths, could improve 

peak-flow regression equations. Seven of the 40 sites had 

well-defined rust lines in a culvert near a gage and three of the 

seven had rust lines that were likely created by downstream 

backwater; thus, the rust line variable was determined not to 

be useful for estimating peak flows in Maine. Twenty-three 

of the 40 gages had definable bankfull widths near gages. 

Bankfull width was tested as an explanatory variable in the 

regression equations, but it did not improve the equations over 

easier-to-compute variables such as percent basin wetlands in 

the basin and, thus, was not selected as a variable in the final 

peak-flow equations. 

Ordinary least-squares regression analyses were used to 

determine the variables that best explained peak-flow statistics 

and to examine regression equation diagnostics to ensure that 

regression assumptions were being met. The best variables 

were basin area and percent basin wetlands. Generalized 

least-squares regression analyses using 40 gages were used to 

derive the final regression coefficients for drainage area and 

percent basin wetlands and the measures of uncertainty for the 

regression equations. 

Regression equations presented here are the best 

available equations for estimating peak flows on ungaged 

rural, unregulated basins in Maine with drainage areas of 

0.3 to 12 mi2 and percent basin wetlands of 0 to 22.2 percent 

of the drainage area. Equations developed by Hodgkins (1999) 

continue to be the best available equations for estimating 

peak flows for ungaged basins with areas greater than 12 mi2. 

Hodgkins (1999) also contains guidance on estimating peak 

flows for urbanized basins. Drainage areas and percent basin 

wetland basin characteristics should be derived using the 

same or comparable methods as those documented in this 

report. Using basin characteristics outside the range of those 

used to derive the equations, or using different methods of 

computing the basin characteristics, will lead to estimates of 

unknown accuracy. 
12 Peak Flow Regression Equations for Small, Ungaged Streams in Maine: Comparing Map-Based to Field-Based Variables

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For more information concerning this report, contact:

Director, New England Water Science Center

U.S. Geological Survey 

196 Whitten Road 

Augusta, ME 04330 

dc_nweng@usgs.gov

or visit our Web site at:

http://me.water.usgs.gov

Publishing support by: 

The Pembroke and Raleigh Publishing Service Centers.
Lombard and Hodgkins—Peak Flow Regression Equations for Small, Ungaged Streams in Maine: Comparing Map-Based to Field-Based Variables—SIR 2015–5049

ISSN 2328-0328 (online)

http://dx.doi.org/10.3133/sir20155049