U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2016–5024

Prepared in cooperation with the Alaska Department of Transportation and Public Facilities, 

Alaska Department of Natural Resources, and U.S. Army Corps of Engineers

Estimating Flood Magnitude and Frequency at Gaged and 

Ungaged Sites on Streams in Alaska and Conterminous 

Basins in Canada, Based on Data through Water Year 2012
Cover: Richardson Highway in Keystone Canyon during a flood of the Lowe River in 

October 2006. (Photograph by Mike Isaacs, Alaska Department of Transportation and 

Public Facilities, October 10, 2006.)
Estimating Flood Magnitude and Frequency at 

Gaged and Ungaged Sites on Streams in 

Alaska and Conterminous Basins in Canada, 

Based on Data through Water Year 2012

By Janet H. Curran, Nancy A. Barth, Andrea G. Veilleux, and Robert T. Ourso

Prepared in cooperation with the Alaska Department of Transportation and 

Public Facilities, Alaska Department of Natural Resources, and 

U.S. Army Corps of Engineers

Scientific Investigations Report 2016–5024

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, Director

U.S. Geological Survey, Reston, Virginia: 2016

For more information on the USGS—the Federal source for science about the Earth, its natural and living 

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For an overview of USGS information products, including maps, imagery, and publications, 

visit http://www.usgs.gov/pubprod/.

This report was funded in part with qualified outer continental shelf oil and gas revenues by the Coastal Impact 

Assistance Program, Fish and Wildlife Service, U.S. Department of the Interior. The views and conclusions contained 

in this document should not be interpreted as representing the opinions or policies of the U.S. Department of Interior, 

Fish and Wildlife Service.

Although this information product, for the most part, is in the public domain, it also may contain copyrighted materials 

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

Suggested citation:

Curran, J.H., Barth, N.A., Veilleux, A.G., and Ourso, R.T., 2016, Estimating flood magnitude and frequency at gaged 

and ungaged sites on streams in Alaska and conterminous basins in Canada, based on data through water year 2012: 

U.S. Geological Survey Scientific Investigations Report 2016–5024, 47 p., http://dx.doi.org/10.3133/sir20165024.

ISSN 2328-0328 (online)
iii

Contents

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

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

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

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

Description of Study Area ............................................................................................................................3

Data Compilation............................................................................................................................................7

Peak-Flow Data.....................................................................................................................................7

Trend Analysis ..............................................................................................................................8

Physical and Climatic Basin Characteristics ...................................................................................8

Flood Magnitude and Frequency at Gaged Sites ...................................................................................11

Log-Pearson Type III Frequency Analysis .....................................................................................12

Expected Moments Algorithm .........................................................................................................12

Systematic Record and Nonstandard Censored Flow ........................................................13

Historical Flood Information.....................................................................................................13

Multiple Grubbs-Beck Test for Detecting Multiple Potentially Influential Low Floods ..........15

Statistical Analysis of Regional Skew.............................................................................................16

Estimating Flood Magnitude and Frequency at Ungaged Sites...........................................................18

Elimination of Redundant and Other Non-Eligible Sites from Regression Analysis ................18

Exploratory Regression Analysis .....................................................................................................19

Regionalization of Flood-Frequency Estimates..............................................................................20

Regional Regression Equations........................................................................................................21

Accuracy and Limitations..................................................................................................................23

Variance of Prediction ..............................................................................................................24

Standard Error of Prediction....................................................................................................24

Pseudo Coefficient of Determination .....................................................................................24

Prediction Intervals ...................................................................................................................25

Limitations...................................................................................................................................26

Application of Methods for Estimating Flood Magnitude and Frequency .........................................27

Regression Estimate and Prediction Interval.................................................................................27

Weighted Estimate for a Gaged Site................................................................................................27

Estimate for an Ungaged Site near a Streamgage........................................................................28

Estimate for a Site on the Yukon River ............................................................................................29

Cook Inlet Basin Streamstats ....................................................................................................................30

Summary and Conclusions.........................................................................................................................31

Acknowledgments.......................................................................................................................................32

References Cited..........................................................................................................................................32

Appendix A. Basin Characteristics for Selected Streams in Alaska and Conterminous Basins 

in Canada.........................................................................................................................................37

Appendix B. Regional Skewness Regression Analysis........................................................................39
iv

Figures

1. Maps showing physical features and streamgages used in regional skew and

regression analyses for Alaska and conterminous basins in Canada ................................4

2. Graph showing example of annual peak-flow series containing peaks collected

outside the systematic gaging record, U.S. Geological Survey streamgage

15208000, Tonsina River at Tonsina, Alaska............................................................................14

3. Example of a Log-Pearson Type III flood frequency curve for U.S. Geological

Survey streamgage 15208000, Tonsina River at Tonsina, Alaska, showing the

Expected Moments Algorithm (EMA) with a multiple Grubbs-Beck (MGB) test

representation for two peaks collected outside the systematic gaging record,

water years 1995 and 2006 ........................................................................................................14

4. Log-Pearson Type III fit when multiple Grubbs-Beck (MGB) test found multiple

potentially influential low floods (PILFs) for U.S. Geological Survey streamgage

15476400, Dry Creek near Dot Lake, Alaska ...........................................................................15

5. Map showing regional skew areas for Alaska and conterminous basins in Canada.....16

6. Map showing streamflow analysis regions previously presented for Alaska and

conterminous basins in Canada...............................................................................................20

7. Graph showing relation of discharge to drainage area for selected annual

exceedance probabilities for the Yukon River at and downstream of Carmacks,

Yukon.............................................................................................................................................23

8. Graph showing relation between 1-percent annual exceedance probability

discharges computed for observed streamflow and predicted from regression

equations for streamgages in Alaska and conterminous basins in Canada ....................25

Tables

1. Description of streamgages used in flood frequency analysis and considered for

use in regional skew and regional regression analysis for Alaska and conterminous

basins in Canada...........................................................................................................................7

2. Basin characteristics considered in regional skew analysis and peak flow

regression analysis for Alaska and conterminous basins in Canada..................................9

3. P-percent annual exceedance probabilities and corresponding T-year recurrence

intervals for flood frequency flow estimates .........................................................................11

4. Flood frequency statistics for streamgages in Alaska and conterminous basins in

Canada with at least 10 years of record through water year 2012.....................................12

5. Definition of regional skew areas for Alaska and conterminous basins in Canada,

by hydrologic unit code (HUC)..................................................................................................17

6. Regional skew and summary statistics for regions in Alaska and conterminous

basins in Canada.........................................................................................................................18

7. Regional regression equations for estimating annual exceedance-probability

discharges for unregulated streams in Alaska and conterminous basins in Canada ....22

8. Values used to determine prediction intervals for the regional flood frequency

regression equations for Alaska and conterminous basins in Canada.............................26

9. Variance estimates for station, regression, and weighted estimates of flood

frequency statistics for streamgages in Alaska and conterminous basins in Canada...28
v

Conversion Factors

Inch/Pound to International System of Units

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2

) 259.0 hectare (ha)

square mile (mi2

) 2.590 square kilometer (km2) 

Volume

cubic foot (ft3

) 28.32 cubic decimeter (dm3) 

cubic foot (ft3

) 0.02832 cubic meter (m3) 

Flow rate

cubic foot per second (ft3

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

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

°C=(°F-32)/1.8.

Datums

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).

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

Acronyms

AEP annual exceedance probability

ASC Alaska Science Center

AVP average variance of prediction

EMA expected moments algorithm

GIS geographic information system

GLS generalized least squares

HUC hydrologic unit code

MGB multiple Grubbs-Beck

MSE mean square error

NHD National Hydrography Dataset

NWIS National Water Information System

OLS ordinary least squares

PRISM Parameter-Elevation Regressions on Independent Slopes Model

USGS U.S. Geological Survey

WBD Watershed Boundary Dataset

WLS weighted least squares

WREG weighted-multiple-linear regression
Estimating Flood Magnitude and Frequency at Gaged and 

Ungaged Sites on Streams in Alaska and Conterminous 

Basins in Canada, Based on Data through Water Year 2012

By Janet H. Curran, Nancy A. Barth, Andrea G. Veilleux, and Robert T. Ourso

Abstract

Estimates of the magnitude and frequency of floods are 

needed across Alaska for engineering design of transportation 

and water-conveyance structures, flood-insurance studies, 

flood-plain management, and other water-resource purposes. 

This report updates methods for estimating flood magnitude 

and frequency in Alaska and conterminous basins in Canada. 

Annual peak-flow data through water year 2012 were 

compiled from 387 streamgages on unregulated streams 

with at least 10 years of record. Flood-frequency estimates 

were computed for each streamgage using the Expected 

Moments Algorithm to fit a Pearson Type III distribution to 

the logarithms of annual peak flows. A multiple Grubbs-Beck 

test was used to identify potentially influential low floods 

in the time series of peak flows for censoring in the flood 

frequency analysis.

For two new regional skew areas, flood-frequency 

estimates using station skew were computed for stations with 

at least 25 years of record for use in a Bayesian least-squares 

regression analysis to determine a regional skew value. The 

consideration of basin characteristics as explanatory variables 

for regional skew resulted in improvements in precision 

too small to warrant the additional model complexity, and 

a constant model was adopted. Regional Skew Area 1 in 

eastern-central Alaska had a regional skew of 0.54 and an 

average variance of prediction of 0.45, corresponding to an 

effective record length of 22 years. Regional Skew Area 2, 

encompassing coastal areas bordering the Gulf of Alaska, had 

a regional skew of 0.18 and an average variance of prediction 

of 0.12, corresponding to an effective record length of 

59 years. Station flood-frequency estimates for study sites in 

regional skew areas were then recomputed using a weighted 

skew incorporating the station skew and regional skew. In 

a new regional skew exclusion area outside the regional 

skew areas, the density of long-record streamgages was too 

sparse for regional analysis and station skew was used for all 

estimates. Final station flood frequency estimates for all study 

streamgages are presented for the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, 

and 0.2-percent annual exceedance probabilities.

Regional multiple-regression analysis was used to 

produce equations for estimating flood frequency statistics 

from explanatory basin characteristics. Basin characteristics, 

including physical and climatic variables, were updated 

for all study streamgages using a geographical information 

system and geospatial source data. Screening for similar-sized 

nested basins eliminated hydrologically redundant sites, and 

screening for eligibility for analysis of explanatory variables 

eliminated regulated peaks, outburst peaks, and sites with 

indeterminate basin characteristics. An ordinary least-squares 

regression used flood-frequency statistics and basin 

characteristics for 341 streamgages (284 in Alaska and 57 in 

Canada) to determine the most suitable combination of basin 

characteristics for a flood-frequency regression model and 

to explore regional grouping of streamgages for explaining 

variability in flood-frequency statistics across the study area. 

The most suitable model for explaining flood frequency used 

drainage area and mean annual precipitation as explanatory 

variables for the entire study area as a region. Final regression 

equations for estimating the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 

0.2-percent annual exceedance probability discharge in Alaska 

and conterminous basins in Canada were developed using a 

generalized least-squares regression. The average standard 

error of prediction for the regression equations for the various 

annual exceedance probabilities ranged from 69 to 82 percent, 

and the pseudo-coefficient of determination (pseudo-R2

) 

ranged from 85 to 91 percent.

The regional regression equations from this study were 

incorporated into the U.S. Geological Survey StreamStats 

program for a limited area of the State—the Cook Inlet 

Basin. StreamStats is a national web-based geographic 

information system application that facilitates retrieval of 

streamflow statistics and associated information. StreamStats 

retrieves published data for gaged sites and, for user-selected 

ungaged sites, delineates drainage areas from topographic 

and hydrographic data, computes basin characteristics, 

and computes flood frequency estimates using the regional 

regression equations.
2 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

Introduction

Flooding in Alaska has caused millions of dollars 

of damage to towns and villages and has disrupted major 

transportation links. Additionally, riverine infrastructure 

inadequately designed for flood flows has resulted in impaired 

aquatic biota and aquatic habitat. To minimize the damage 

from floods, protect human health and safety, and conserve 

wildlife habitat, reliable estimates of flood frequency and 

magnitude are essential. Federal, State, regional, and local 

agencies rely on these estimates to effectively plan and 

manage land use and water resources, protect lives and 

property, administer flood insurance programs, and conserve 

habitat. Streamflow statistics compiled from records 

maintained at streamgages provide a basis for the design of 

infrastructure, such as roads and bridges; the management 

of flood risk; the protection of aquatic species; and other 

engineering and environmental analyses. Flood frequency 

statistics can be computed directly for a particular streamgage 

with a suitable length of record. For short-record streamgages 

or ungaged sites, regression equations developed from 

flood frequency statistics and basin characteristics for a 

regional group of streamgages can provide estimates of flood 

frequency statistics.

The U.S. Geological Survey (USGS) has published 

reports that provide methods for estimating streamflow 

statistics at gaged and ungaged sites in States across the 

United States, including Alaska. These studies benefit from 

periodic updates to incorporate new streamflow information, 

which improve how well the equations represent the 

hydrology of the region, and improved measurements of basin 

characteristics or improved computational techniques, which 

can strengthen the statistical quality of the equations. Updates 

are of particular interest for Alaska, where the gaging network 

is spatially sparse, ranking near the bottom of all major 

hydrologic regions of the United States for percentage of area 

gaged except for very large basins. Individual streamgage 

record lengths in Alaska also are relatively short. Alaska 

reference-quality streamgages have a median record length of 

28 years, the lowest among the 50 States (Kiang and others, 

2013). The most recent streamflow statistics reports for Alaska 

consist of a flood frequency update (Curran and others, 2003) 

and a high-flow/low-flow statistics study (Wiley and Curran, 

2003), both based on streamflow data through water year 

1999. This present update of methods for estimating flood 

magnitude and frequency was undertaken to incorporate 

13 additional years of streamflow data, comprehensively 

updated basin characteristics using new digital geospatial 

datasets, and new computational techniques including 

Bayesian weighted least-squares regression for analysis 

of generalized (regional) skew and Expected Moments 

Algorithm (EMA) with a multiple Grubbs-Beck (MGB) test 

for estimation of flood frequency statistics. The update also 

provides a pilot of a StreamStats model for Alaska, facilitating 

the delivery of streamflow statistics for the Cook Inlet Basin.

Purpose and Scope

This report presents results of a study conducted in 

cooperation with the Alaska Department of Transportation and 

Public Facilities, the Alaska Department of Natural Resources, 

and the U.S. Army Corps of Engineers to update methods for 

estimating the magnitude and frequency of floods in Alaska and 

conterminous basins in Canada. The selected areas of Canada 

were included to facilitate estimates in trans-boundary basins 

and improve estimates for near-border basins. 

This report describes methods for determining the 

magnitude of floods that have annual exceedance probabilities 

(AEP) of 50, 20, 10, 4, 2, 1, 0.5, and 0.2 percent, and presents 

discharge estimates for these AEP for 387 streamgages in the 

study area. The report describes methods for revising regional 

skew and presents an updated regional skew for selected areas 

in Alaska and a new regional skew exclusion area for sparsely 

gaged areas. For estimating flood magnitude and frequency at 

ungaged basins, the report presents methods for developing 

regional regression equations using AEP discharge estimates at 

341 streamgages and updated basin characteristics and describes 

the accuracy and limitations of these equations. Example 

applications for estimating flood frequency and magnitude in 

the study area are provided. This report also documents the 

development of a USGS StreamStats Web‐based application 

for obtaining basin characteristics and estimating streamflow 

statistics, including flood magnitude and frequency, for a 

selected area of the State—the Cook Inlet Basin. StreamStats 

is intended to streamline procedures for estimating streamflow 

statistics by providing data and computation tools to minimize 

user error and the need for specialized software for managing 

large datasets. This report supersedes previous reports, most 

recently Curran and others (2003), that describe methods for 

estimating flood frequency and magnitude for Alaska.

Previous Studies

Early analyses of annual peak-flow statistics for Alaska 

include Berwick and others (1964), Childers (1970), Lamke 

(1978), and Parks and Madison (1985), all of which maximized 

use of the State’s small but growing network of streamgages 

by including some or all streamgages with at least 5 years 

of peak‐flow record. Beginning with Childers (1970), all 

studies applied a log-Pearson Type III analysis to annual 

peaks and adopted a multiple-regression technique using basin 

characteristics as independent, or explanatory, variables for 

estimating peak-flow statistics. All studies used a regional 

approach, resulting in a suite of equations for estimating 

streamflow magnitude and frequency in ungaged watersheds in 

each region. Regions varied by study, but commonly identified 

some part of the southern coastal areas of Alaska that border the 

Gulf of Alaska as a distinct hydrologic region. Lamke (1978) 

expressed a distinction between areas with autumn and winter 

rains and areas more commonly with spring and summer floods.
Description of Study Area 3

Subsequent peak-flow studies gradually increased the 

number of years of record required for inclusion in the study. 

Jones and Fahl (1994) included streamgages with at least 

8 years of record instead of the recommended 10 years of 

record in an effort to include small streams, and Curran and 

others (2003) included streamgages with at least 10 years of 

record except for streamgages used in Jones and Fahl (1994) 

for which no additional records were available. The number of 

streamgages included in the regression analysis increased from 

200 in Alaska in the 1985 study (Parks and Madison, 1985) to 

355 in Alaska and conterminous basins in Canada in the 2003 

study (Curran and others, 2003).

Parks and Madison (1985) presented a statewide 

regression equation for estimating flood frequency statistics 

and introduced six hydrologic regions in Alaska, presenting 

reliable regression equations for three of the regions. All 

regression equations contained drainage area and mean 

annual precipitation as explanatory variables. Jones and Fahl 

(1994) used three to six variables in regression equations for 

five regions in Alaska and conterminous basins in Canada, 

always including drainage area, mean annual precipitation, 

and area of lakes and ponds, and selectively including mean 

minimum January temperature, mean basin elevation, or area 

of forests to the equations. Curran and others (2003) updated 

peak-flow statistics with data through water year 1999 and 

slightly modified the region boundaries established by Jones 

and Fahl (1994) by splitting the northern part of the State into 

an interior and a far-north region. The 2003 study presented 

equations using one to four variables in six regions in Alaska 

and conterminous basins in Canada, always including drainage 

area and selectively including area of lakes and ponds, mean 

annual precipitation, area of forest, mean minimum January 

temperature, and elevation. Data included in previous reports 

included maps of climate characteristics that provide the 

required datasets of basin characteristics for regression 

analysis (Lamke, 1978; Jones and Fahl, 1994) and envelope 

curves developed from maximum known floods, most recently 

presented in Jones and Fahl (1994).

Description of Study Area

The area considered for flood frequency analysis in this 

study, referred to in this report as the “study area,” consists 

of the State of Alaska and Canadian basins in Yukon and 

British Columbia that drain to Alaska, an area encompassing 

750,000 mi2 (fig. 1). The study area excludes the Aleutian 

Islands and several other islands off the west coast of Alaska 

that lack sufficient streamgages to characterize streamflow 

for regression analysis. A small part of Alaska, the Cook 

Inlet Basin, which encompasses a land area of 39,000 mi2

in a horseshoe shape around Cook Inlet (fig. 1 inset A), 

was included in the StreamStats application developed for 

this study.

The study area encompasses diverse physical and climatic 

settings. Mountainous areas arc in two major bands across 

the study area and include 23 summits exceeding 4,000 m 

(13,176 ft) in elevation. An extensive high-elevation mountain 

system consisting of the Coast Mountains, the Alaska Range, 

and the Aleutian Range spans the southern part of the study 

area. The Brooks Range, an extension of the Rocky Mountain 

system, extends across the study area near the Arctic Circle. 

The intermontane plateau separating these systems includes 

hills and gentler mountains in Interior Alaska and the 

expansive, low-relief, and lake-rich Yukon-Kuskokwim River 

delta. The cold, dry Arctic coastal plains extend north from the 

Brooks Range to the Arctic Ocean.

Environmental Protection Agency Level III Ecoregions 

(Gallant and others, 2010) and climate divisions (Bieniek and 

others, 2012) of Alaska define various regional divisions but 

generally recognize some part of temperate, wet south-east 

Alaska and cold, dry Arctic Alaska, respectively, as markedly 

different from the rest of the State, and generally divide 

the remaining area into interior, western, and south-central 

regions. Mean annual precipitation in the study area ranges 

from more than 300 in. in south-east Alaska to less than 9 in. 

on the Arctic coastal plain (Spatial Climate Analysis Service 

at Oregon State University, 2002; Gibson, 2009a). Average 

annual temperatures generally are correlated with latitude and 

range from 44 °F in southern areas to 10 °F in northern areas 

(Spatial Climate Analysis Service at Oregon State University, 

2002; Gibson, 2009b). As a result of cold winter temperatures, 

winter precipitation can fall as snow onto frozen ground at 

high elevations throughout the study area and as low as sea 

level at many latitudes.

Processes generating peak flows in the study area can 

be inferred from inspection of the dates of occurrence of the 

peak flow and from previous studies by Wiley and Curran 

(2003) and Curran (2012). These processes include snowmelt, 

rainfall, and glacier-related melt. Although considered a single 

population for the purpose of flood frequency analysis, floods 

from these various processes respond to different forcing 

mechanisms. In most study streams, with the exception of 

those in basins bordering the Gulf of Alaska, spring snowmelt 

produces a prominent increase in mean daily discharge (see 

examples in Wiley and Curran, 2003, and Curran, 2012). This 

pulse of increased flow also can produce the annual peak 

flow, particularly in Interior and northern Alaska streams. 

In basins covered by a considerable area of glaciers, mean 

daily streamflow often continues to increase into summer as 

glacier and high-elevation snowmelt and glacier melt augment 

streamflow. Streamflow in glacierized basins can peak from 

these glacier-related melt processes, from spring snowmelt, 

or from rainfall. The glacier-related melt contribution to 

flow is a separate process from glacier outbursts, in which 

water impounded by or stored within the glacier is released 

suddenly. Glacier outbursts, which occur annually in some 

basins, can exceed floods from non-outburst processes. 
4 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

tac16-5245_fig01

#

Mackenzie River

Liard River Tanana River

Pelly River

Colville River

Stewart R vi er

Kuskokw mi River

Koyukuk River

Yukon River

Noatak River

Kobuk River

Peel River

St ki ni e River

Alsek River

N s u gah Rka vi er

Porcupine River

Fort Nelson River

Yukon River

IliamnaLake

Becharof Lake

Teshekpuk Lake

"

"

"

"

7

94 93

92

90

91

89

88

87

77

387

386

385 384

383

382

381

380

379

378

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375

374 373

372 371

369

368 367 366

364

365 363

362

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350 349

348

347 346

345 344

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339

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336 335334

333 332

322 321

317

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305

296

295 291

289

286

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274 273

272 271

270

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268 267

265 263

262 261

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250

248 247

246

245 244

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238 237

236

232 231 230

221

220

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214 210

209

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205

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110

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108

370

361 360

338

320

319

318

316

315 313

306

290

288

287

280

266

264

242

213

211

212

CANADA

Barrow

RUSSIA

YUKON

BRITISH 

COLUMBIA

Aleutian Islands

C o a s t M o u n t a i n s

ALASKA

Carmacks

Juneau

Anchorage

Arctic Circle

Area of Inset B

Area of Inset A

l A a s k a R a n g e

A l e u t i a n R a n g e

B r o o k s

R a n g e

Base from ASTER GDEM version 2 30-meter digital elevation model. Alaska Albers Equal-Area Conic projectionHydrography from North American Atlas-Hydrography, 2006, 1:10,000,000. International boundary from U.S. Geological Survey digital dataArctic Circle from ESRI World GeoReference Lines layer package

130° 120° 110° 140° 150° 160° 170° 180°

65°60°55°

0 100 200 300 400 MILES

0 100 200 300 400 KILOMETERS

EXPLANATION

Streamgage and map identification No.

Used in regional skew and regressionUsed in regional regressionNot used in regional skew or regressionYukon River estimator site

Study area

Flood frequency study areaCook Inlet Basin StreamStats study area

128

130

149

342

Figure 1. Physical features and streamgages used in regional skew and regression analyses for Alaska and conterminous basins in Canada. Inset A, South-central Alaska; 

Inset B, South-east Alaska.
Description of Study Area 5

tac16-5245_fig01 InsetA

Susitna River

Copper River

Kuskokwim River

Iliamna 

Lake

PACIFIC OCEAN

Gulf of Alaska Cook Inlet

Anchorage C o a s t M o u n t a i n s

K e n a i P e n i n s u l a

Kodiak Island

A l a s k a R a n g e

99 98

96

95

330

333 329

328

327

326 325

324323

312 311 310

309 308

307

304 303 302 301

300 299 298

297

294

293

292 218 216 182

181

179 180

207 206

205

200

199

198

197

196

195

194

168 169

167

166

165

164

162 163 161 160 159

158 157 156 155 154

153

152

151 150

149

148

147

146 144

143

140 142

139

138

137

136

135

134 133

132

131

130

129

128

127

126

123

122

145

141

97

94

193

192 191

190

189

188

187

186

185

184 183

178

177

176

175

174

173 172 171 170

125

124

121

120

119 118

117

116

115

114 113

112 111

107

106

105

104

102

101

100

103

Base from ASTER GDEM version 2 30-meter digital elevation model

Alaska Albers Equal-Area Conic projection

Hydrography from North American Atlas-Hydrography, 2006, 1:10,000,000

155° 150° 145°

64°

62°

60°

58°

0 25 50 75 100 MILES

0 25 50 75 100 KILOMETERS

EXPLANATION

Streamgage and map identification No.

Used in regional skew and regression

Used in regional regression

Not used in regional skew or regression

Cook Inlet Basin StreamStats study area

Flood frequency study area

149

130

128

INSET A, South-central Alaska

Figure 1.—Continued
6 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

tac16-5245_fig01 Inset B

Gulf of Alaska

Stikine River 9 8

7

6

5

4

3

2

1

86 85

84

83

82 80 81 79 78

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43

41 42

40

39

38

37

36

33

32 31

30

28

27

26

25

24

23 22

21

20

18

17 16

15

14

13

12

11

10

235

234

233 229

228

227

226

225

224

223

222

35

34

29 19

YUKON

BRITISH 

COLUMBIA

Juneau

CANADA

ALASKA

INSET B, South-east Alaska

Base from ASTER GDEM version 2 30-meter digital elevation model

Alaska Albers Equal-Area Conic projection

Hydrography from North American Atlas-Hydrography, 2006, 1:10,000,000

International boundary from U.S. Geological Survey digital data

0 25 50 75 100 MILES

0 25 50 75 100 KILOMETERS

136° 134° 132° 130° 128°

60°

58°

56°

EXPLANATION

Streamgage and map identification No.

Used in regional skew and regression

Used in regional regression

Not used in regional skew or regression

Flood frequency study area

24

62

59

Figure 1.—Continued
Data Compilation 7

Rainfall-related flooding is most common in autumn in most 

of the study area, or in autumn and winter in basins bordering 

the Gulf of Alaska, where winter precipitation can fall as rain 

instead of snow or as rain on snow. Atmospheric rivers, or 

narrow corridors of moisture in the lower atmosphere (Ralph 

and Dettinger, 2011), often form over the Pacific Ocean and 

flow towards the Pacific coast of North America and can 

bring heavy rainfall that generates notable regional flooding 

in Alaska.

Ice-jam flooding, which occurs when large blocks of 

winter stream ice cover break free during spring melt and 

partially dam streams, can create local high water levels from 

backwater generally not associated with large magnitudes of 

discharge. Although these floods generally do not constitute 

the maximum discharge of the year and are therefore not 

considered in this statistical study, ice-jam flooding is an 

important consideration for public safety. 

Notable non ice-jam floods since 1949, when widespread 

streamgaging began in Alaska, include floods in Interior 

Alaska in 1964, near Fairbanks in 1967 (Childers and others, 

1972), in south‐central Alaska in 1971 (Lamke, 1972) and 

1986 (Lamke and Bigelow, 1988), along the Copper River 

in 1981 (Brabets, 1997), along the middle Koyukuk River in 

1994 (Meyer, 1995), and in south-central Alaska and the Kenai 

Peninsula in 1995 (Stauffer, 2010; National Weather Service 

Alaska-Pacific River Forecast Center, 2015), 2006 (Joling, 

2006; Stauffer, 2010), and 2012 (Homer Tribune, 2012), and 

on the Kenai Peninsula in 2002 (Eash and Rickman, 2004). 

A list of maximum known floods, which can include floods 

other than those considered for flood frequency analysis, for 

selected stations in Alaska and conterminous basins of Canada 

was presented by Jones and Fahl (1994) and was not updated 

for this study.

Data Compilation

Peak-flow records from the USGS and from the Water 

Survey of Canada, the branch of Environment Canada that 

is responsible for streamflow monitoring for Canada, were 

screened for suitability for flood frequency analysis and 

eligibility for regression analysis. Selection considerations for 

flood frequency analysis included record length and the effect 

of any streamflow regulation or diversion, urbanization, or 

natural damming and release of water on peak flow. Peak-flow 

data for the selected sites were then reviewed to assure the 

quality of the records and tested for homogeneity or presence 

of trends over time, which could invalidate the assumptions of 

the analyses. For use in development of regression equations, 

basin characteristics were determined for all streamgages in 

the study.

Peak-Flow Data

Records considered for analysis in this study consisted of 

annual series of peak-flow data at least 10 years in length from 

generally unregulated and non-urbanized USGS streamgages 

in Alaska or Water Survey of Canada streamgages in 

conterminous basins in Canada. The streamgages considered 

included 387 streamgages in Alaska and Canada that were 

selected for analysis (table 1). Alaskan peak-flow data 

were obtained from the USGS National Water Information 

System (NWIS) peak-flow file (U.S. Geological Survey, 

2015) and Canadian peak-flow data were obtained from the 

HYDAT database via the Environment Canada Data Explorer 

(Environment Canada, 2015). Streamgages used in this study 

included continuous-record streamgages that documented 

streamflow on a daily basis and crest-stage partial-record 

streamgages that documented only the annual peak flow. 

No seasonal partial-record streamgages, which operate for 

only part of a year, were used in this study. Regardless of 

the type of streamgage, the annual peak flow consisted of 

the maximum instantaneous discharge for the water year 

(October 1 through September 30) and maximum daily 

discharge for selected streamgages, when the maximum 

instantaneous discharge was not available.

Table 1. Description of streamgages used in flood frequency 

analysis and considered for use in regional skew and regional 

regression analysis for Alaska and conterminous basins in 

Canada.

[Table 1 is a Microsoft© Excel file and can be downloaded at http://dx.doi.

org/10.3133/sir20165024]

All analyses used data compiled by water year. Although 

USGS data are published on a water-year basis, Canadian 

data are published on a calendar-year basis. Conversion of 

Canadian data to a water-year basis occasionally encountered 

two calendar-year peaks in a single water year, requiring 

omission of the smaller peak and an accompanying reduction 

of the available record length. The water years for peaks 

omitted from the analysis during the conversion of Canadian 

data are shown in table 1. For years when maximum 

instantaneous discharge was not available for part of the 

Canadian record but annual maximum daily mean discharge 

was available, an assessment of comparability was made 

using years when these two measures were available. For 

records where maximum daily mean discharge was within 

5–10 percent of the maximum instantaneous peak discharge 

for other years in the record, the annual maximum daily mean 

discharge was used as a surrogate for maximum instantaneous 

peak discharge. This bias toward smaller discharge for 

selected stations is expected to be minor relative to other 

errors in the analysis. The Canadian records as modified for 

this study are available in the NWIS peak-flow file (U.S. 

Geological Survey, 2015).
8 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

The Alaska peak-flow records for streamgages in this 

study were inspected for anomalous values, and qualification 

codes associated with the peaks were reviewed for accuracy. 

Revisions were applied in NWIS prior to a final extraction 

of streamflow data for this study. Peak flows for which a 

discharge qualification code was published in the peak-flow 

file were included or omitted in the flood frequency analysis 

according to default procedures described for USGS flood 

frequency analysis program PeakFQ (Flynn and others, 2006) 

except for selected cases described here. Published USGS 

records containing a large number of instances of use of the 

annual maximum daily mean discharge (peak-flow discharge 

code 1) as a surrogate for the maximum instantaneous peak 

discharge were reviewed to ensure that daily and peak values 

were similar to each other for years when both were available. 

Where the daily and peak values differed by more than 

5–10 percent, the annual maximum daily mean discharges 

were omitted from the record for analysis, as noted in table 1. 

Regulated or diverted sites (peak-flow discharge codes 5 or 6 

in the peak-flow file) were reviewed individually to determine 

the effect of the unknown (code 5) or known (code 6) degree 

of regulation or diversion on peak flows and whether the 

peak‐flow data were suitable for flood frequency analysis. 

Similarly, sites having a peak-flow discharge code 3, when 

the basin was known to be subject to glacier outburst floods, 

were reviewed individually and included in flood frequency 

analysis when appropriate. The water year and condition for 

all instances where a published peak was omitted from flood 

frequency analysis is shown in table 1.

Trend Analysis

Records used in flood frequency analysis are assumed 

to contain a series of independent discharges free of trends 

from changes in basin or climate conditions. Kendall’s tau, a 

statistic in a nonparametric test for whether values increase 

or decrease over time, can test the strength and significance 

of trends in streamflow data (Helsel and Hirsch, 2002). For 

the purposes of determining suitability for flood frequency 

analysis, the Kendall tau test was applied to each record in the 

study, regardless of length and regardless of long gaps in the 

record. Statistically significant trends in peak flow can occur 

for many reasons, but a trend indicating a changing basin 

condition, such as a change in land cover or urbanization, can 

violate the assumptions of flood frequency analysis. When 

the condition creating the trend is understood, it might be 

appropriate to adjust the record for trend and estimate flood 

frequency using the adjusted record. Detection of trends in 

peak flow across a region or the entire State poses a different 

question that would require selection of a suitable subset 

of these records to evaluate and is beyond the scope of 

this report.

Kendall’s tau depends on the rank rather than the 

magnitude of the values, making it effective for identifying 

trends in streamflow where extreme values and skewness 

(skew) are present. The test is conducted by comparing the 

rank of each peak-flow value to the rank of all other values 

in the record. A positive tau value indicates that the number 

of pairs increasing in value exceeds the number of pairs 

decreasing in value, and a negative tau value indicates the 

opposite. Tau ranges from -1 to 1 and approaches 0 when no 

trend is likely to exist. For this study, a trend was considered 

to be significant if the probability (p) value, or probability that 

a true null hypothesis of no trend is erroneously rejected, was 

less than or equal to 0.05.

Statistically significant trends were detected at 43 of the 

387 streamgages in the study (table 1). Of the streamgages 

with significant trends, 22 streamgages showed upward trends, 

and 21 streamgages showed downward trends. No underlying 

cause of any trend was obvious when considering spatial 

distribution, regulation, land-use changes, and urbanization. 

Although a cursory consideration of climate as a variable in 

peak-flow trends suggested no obvious patterns, a thorough 

assessment of any correlation of significant peak-flow trends at 

individual sites to temporal changes in climate was beyond the 

scope of this report. Because adjustments to compensate flood 

frequency estimates for trends or exclude records showing 

trends from analysis might result in a spurious correction 

when the underlying cause is not well understood, all records 

having statistically significant trends were included in the 

analysis without adjustment.

Physical and Climatic Basin Characteristics

Peak flow in unregulated basins generally varies 

as a function of basin size and the physical and climatic 

characteristics of the basin that govern the quantity and rate 

of water delivery to the stream. Basin characteristics that can 

be correlated to peak flow in gaged basins provide convenient 

explanatory variables for estimating peak flows in ungaged 

basins. Basin characteristics tested in the regional skew 

and regional regression analyses as potential explanatory 

variables are listed in table 2. Values for basin characteristics 

for streamgages used in the analyses or presented in Cook 

Inlet Basin StreamStats are shown in appendix A and in a 

geodatabase (.zip file can be downloaded at http://dx.doi.

org/10.3133/sir20165024). The basin characteristics included 

topographic variables; land cover characteristics including 

various vegetation cover categories and glacier and permafrost 

cover; hydrologic characteristics such as coverage by 

lakes; and climate variables summarizing temperature and 

precipitation.

Values for all basin characteristics were updated 

from the most recent geospatial datasets available as of 

2012. For topographic, climatic, and glacier coverage 

variables, suitable datasets were available that spanned 

both the United States and Canadian parts of the study area. 
Data Compilation 

9

Table 2. Basin characteristics considered in regional skew analysis and peak flow regression analysis for Alaska and conterminous basins in Canada.[Data source: ASTER GDEM, Advanced Spaceborne Thermal Emission and Reflection Radiometer Global Digital Elevation Map; NHD, National Hydrography Dataset; NLCD, National Land Cover Dataset; NOAA, National Oceanic and Atmospheric Administration; PRISM, parameter-elevation regressions on independent slopes model; USGS, U.S. Geological Survey; WBD, Watershed Boundary Dataset. Data source citation and link: NALCMS, North American Land Change Monitoring System; NASA, National Aeronautics and Space Adminstration; NHD, National Hydrography Dataset; SCAS/OSU, Spatial Climate Analysis Service, Oregon State University. –, not applicable]

Abbreviation compatible 

with StreamStats version 3 Description Unit Data source Data source citation and link

Geographic 

coverage for 

datasets not 

available for full 

study area

Primary

DRNAREA Area that drains to a point on 

a stream

square miles USGS 1:63,360 topographic maps and 

WBD (Alaska); Natural Resources Canada 1:50,000 topographic maps (Canada)

The National Map (http://nationalmap.

gov) (Alaska); GeoGratis (http://www.nrcan.gc.ca/earth-sciences/geography/topographic-information) (Canada)

–

LAT_CENT Latitude of the basin centroid degrees USGS 1:63,360 topographic maps and 

WBD (Alaska); Natural Resources Canada 1:50,000 topographic maps (Canada)

The National Map (http://nationalmap.

gov) (Alaska); GeoGratis (http://www.nrcan.gc.ca/earth-sciences/geography/topographic-information) (Canada)

–

ELEV Mean basin elevation feet ASTER GDEM Version 2 (30 m) NASA (2011) (http://reverb.echo.nasa.

gov/reverb/)

–

PRECPRIS00 Basin average mean annual 

precipitation for 1971–2000 from the PRISM climate dataset

inches PRISM: 1971–2000 (Alaska), 

1961–1990 (Canada)

Gibson (2009a) (Alaska) (https://irma.

nps.gov/App/portal/Home); SCAS/OSU (2002) (Canada) (http://www.climatesource.com)

–

JANMINTMP Mean minimum January basin 

temperature

degrees 

Fahrenheit

PRISM: 1971–2000 (Alaska), 

1961–1990 (Canada)

Gibson (2009b) (Alaska) (https://irma.

nps.gov/App/portal/Home); SCAS/OSU (2002) (Canada) (http://www.climatesource.com)

–

LAKEAREA Percentage of basin covered by 

lakes and ponds

percent NHD (Alaska) NHD (http://nhd.usgs.gov) Alaska

GLACIER Percentage of basin covered by 

glaciers

percent Randolph Glacier Inventory 

Version 2.0

Arendt and others (2012) (http://www.

glims.org/RGI/index.html)

–

LC01FOREST Percentage of basin covered by 

forest

percent NLCD 2001, classes 41–43 Homer (2007) (http://www.mrlc.gov/

nlcd2001.php)

Alaska

BROADLEAF1 Percentage of basin covered by 

broadleaf vegetation

percent 2005 Land Cover of North America, 

classes 1–6 and 15–20

NALCMS (2013) (http://www.cec.org/

naatlas)

–

PERMAFROST_LOWUP1 Percentage of basin in upland 

and lowland areas covered by permafrost

percent Ferrians (1998), classes Perm 21–25 Ferrians (1998) (http://nsidc.org/data/

ggd320.html)

Alaska
10 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada Abbreviation compatible with StreamStats version 3 Description Unit Data source Data source citation and link Geographic coverage for datasets not available for full study area Alternate LC01WETLND Percentage of basin covered by wetlands percent NLCD 2001, classes 90 and 95 Homer (2007) (http://www.mrlc.gov/ nlcd2001.php) Alaska LC01SNOIC1 Percentage of basin covered by snow and ice percent NLCD 2001, class 12 Homer (2007) (http://www.mrlc.gov/ nlcd2001.php) Alaska I24H2Y 24-hour, 2 year precipitation, or maximum 24-hour precipitation that occurs on average once in 

2 years

inches NOAA atlas Perica and others (2012) (http://hdsc.nws.

noaa.gov/hdsc/pfds/pfds_gis.html)

Alaska

LC01SCRUB1 Percentage of basin covered by 

shrub or scrub

percent NLCD 2001, classes 51 and 52 Homer (2007) (http://www.mrlc.gov/

nlcd2001.php)

Alaska

LC01BARE Percentage of basin covered by 

barren land

percent NLCD 2001, class 31 Homer (2007) (http://www.mrlc.gov/

nlcd2001.php)

Alaska

PERMAFROST_MTN Percentage of basin in mountainous 

areas covered by permafrost

percent Ferrians (1998), classes Perm 11–13 Ferrians (1998) (http://nsidc.org/data/

ggd320.html)

Alaska

NEEDLELEAF1 Percentage of basin covered by 

needleleaf vegetation

percent 2005 Land Cover of North America, 

classes 7–12 and 21–26

NALCMS (2013) (http://www.cec.org/

naatlas)

–

MIXBROADNEEDLE1 Percentage of basin covered by 

mixed broadleaf and needleleaf vegetation

percent 2005 Land Cover of North America, 

classes 13, 14, 27, and 28

NALCMS (2013) (http://www.cec.org/

naatlas)

–

1Informal name; basin characteristic is not available in StreamStats.

Table 2. Basin characteristics considered in regional skew analysis and peak flow regression analysis for Alaska and conterminous basins in Canada.—Continued
Flood Magnitude and Frequency at Gaged Sites 11

Selected hydrographic and land-cover datasets available for 

Alaska that had no Canadian counterpart having comparable 

resolution or categorical structure are shown in table 2. 

Although variables obtained from these Alaska-only datasets 

could be used for analysis of Alaska regions, alternate 

variables, generally obtained from study-wide databases with 

lower resolution, were required for use in analysis of Canadian 

regions or the full study area.

A few potential basin characteristics were of insufficient 

quality for this study. Minor artifacts in the digital elevation 

model (DEM) used for topographic variables artificially 

inflated values for extreme measures of topography, 

preventing use of maximum elevation or relief. Inconsistent 

or low resolution digital topographic and hydrographic data 

across the study area precluded use of basin characteristics 

such as channel length and basin slope.

A drainage area boundary, or delineation of land area 

draining to a streamgage location, was used to clip digital 

basin characteristics datasets to obtain basin characteristics for 

each streamgage. For the Alaska sites in this study, drainage 

area boundaries previously prepared by the USGS Alaska 

Science Center (ASC) were synchronized with the Watershed 

Boundary Dataset (WBD) for Alaska (available within 

the USGS National Hydrography Dataset [NHD] at http://

nhd.usgs.gov/). Where ASC boundaries could improve the 

WBD, edits were reconciled with the Alaska WBD steward. 

New drainage area boundaries required for this study were 

delineated from the streamgage location to the nearest WBD 

boundary on the basis of digital topographic maps, and then 

followed the WBD for the upstream part of the basin. Final 

drainage area boundaries matched the WBD at the time of 

final analysis; however, the WBD is continually edited and 

future revisions will not be reflected in the study data. For the 

Canadian sites in this study, draft drainage area boundaries 

provided by Environment Canada (Judy Kwan, Meteorological 

Service of Canada, written commun., 2013) or from previous 

USGS studies were reviewed for general agreement with 

digital topographic maps and the WBD (for areas in Alaska). 

Because the Canadian drainage area boundaries were not 

being prepared for publication with this study, edits were made 

only where desired changes appeared to potentially affect 

drainage area by about 5 percent or more.

Drainage area and basin centroid latitude and longitude 

were computed in a Geographic Information System 

(GIS) from the drainage area boundary alone. Drainage 

area computed for this study was used to update the value 

published in NWIS for the streamgage. For all other basin 

characteristics, summary statistics computed in a GIS for the 

respective clipped digital datasets produced the value of the 

basin characteristic.

Flood Magnitude and Frequency at 

Gaged Sites

The frequency analysis of annual peak-flow data 

collected at a streamgage provides an estimate of the flood 

magnitude and frequency for that particular stream site. 

Previously, flood-frequency estimates commonly were 

described as the “T-year” floods based on the recurrence 

interval for the flood statistics (for example, the “100-year 

flood”). The use of the recurrence-interval terminology is 

shifting to annual-exceedance-probability terminology to 

avoid the common misinterpretation of relating recurrence 

interval to a set length of time between floods of a particular 

magnitude (Holmes and Dinicola, 2010). Flood frequency 

estimates relate the probability of a flood of a given magnitude 

being equaled or exceeded in any given year. For example, 

a 1-percent AEP (formerly known as the “100-year” flood) 

corresponds to the flow magnitude that has a 0.01 (1/T-year 

recurrence interval) probability or 1-percent chance of 

being equaled or exceeded in any given year. The P-percent 

AEPs and the corresponding T-year recurrence intervals for 

flood frequency estimates reported in this study are shown 

in table 3.

Flood-frequency estimates in this report were 

computed using the USGS program PeakFQ, version 7.1 

(Veilleux and others, 2014), which performs statistical 

flood-frequency analyses of annual peak flows following 

procedures recommended in Bulletin 17B of the 

Interagency Advisory Committee on Water Data (1982) 

and modified by the Advisory Committee on Water 

Information, Subcommittee on Hydrology, Hydrologic 

Frequency Analysis Work Group (HFAWG) (http://acwi.

gov/hydrology/Frequency/b17_swfaq/EMAFAQ.html). 

Table 3. P-percent annual exceedance probabilities and 

corresponding T-year recurrence intervals for flood frequency 

flow estimates.

P-percent annual

exceedance probability T-year recurrence interval

50 2

20 5

10 10

4 25

2 50

1 100

0.5 200

0.2 500
12 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

The PeakFQ program and documentation are available 

at http://water.usgs.gov/software/PeakFQ/. The resulting 

flood frequency statistics for streamgages in Alaska and 

conterminous basins in Canada with at least 10 years of record 

through water year 2012 are shown in table 4.

Table 4. Flood frequency statistics for streamgages in Alaska 

and conterminous basins in Canada with at least 10 years of 

record through water year 2012.

[Table 4 is a Microsoft© Excel file and can be downloaded at http://dx.doi.

org/10.3133/sir20165024]

Log-Pearson Type III Frequency Analysis 

Flood-frequency estimates for streamgages are computed 

by fitting the base-10 logarithms of the series of annual peak 

flows to a known statistical distribution. The flood magnitude 

and frequency estimates for this study were computed using 

the log-Pearson Type III (LP3) distribution as recommended 

in Bulletin 17B (Interagency Advisory Committee on Water 

Data, 1982). The fitting of this distribution requires calculating 

the three statistics—the mean, standard deviation, and 

skew of the logs of annual peak flows, which describe the 

midpoint, slope, and curvature of the peak-flow frequency 

curve, respectively. The estimates of the P-percent AEP 

flows are computed by inserting the three statistics into the 

following equation:

log Q X KS P P = + (1)

where

QP is the P-percent AEP flow, in cubic feet per 

second;

X is the mean of the logarithms of annual peak 

flow;

is a factor based on the skew coefficient and 

the AEP as obtained from Bulletin 17B 

(Interagency Advisory Committee on 

Water Data, 1982, appendix 3); and

S is the standard deviation of the logarithms of 

annual peak flow.

The skew coefficient is reflected in the curvature of the 

flood-frequency curve. A positively skewed distribution curves 

convexly such that floods are unbounded. A negatively skewed 

distribution has a concave curve such that floods have some 

upper bound. The skew coefficient can be estimated from 

the series of annual peak flows (considered the sample data), 

but tends to be an unreliable estimator of the population for 

streamgages with short periods of record. Guidelines from 

Bulletin 17B of the Interagency Advisory Committee on 

Water Data (1982) recommend weighting the station skew 

with a generalized, or regional, skew to improve the accuracy 

of the station skew estimator. The regional skew is assumed 

to be an unbiased and independent estimate, which improves 

the uncertainty of the skew estimate when weighted with the 

station skew. The station skew coefficients for streamgages 

within new regional skew areas, which are discussed in 

section, “Statistical Analysis of Regional Skew,” were 

weighted with the updated regional skew developed for this 

study when computing the LP3 distribution.

Expected Moments Algorithm 

The 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent 

AEP flows for each streamgage in the study (table 4) were 

computed using the Expected Moments Algorithm (EMA), 

as recommended by HFAWG (http://acwi.gov/hydrology/

Frequency/b17_swfaq/EMAFAQ.html). The EMA method 

applies the LP3 distribution for estimating streamflow 

frequency recommended in Bulletin 17B (Interagency 

Advisory Committee on Water Data, 1982) but more 

efficiently incorporates censored flows and historical flows 

(Cohn and others, 1997, 2001). Censored data includes 

flows that are below or above a monitoring threshold, 

potentially influential low floods (PILFs, formerly referred 

to as low outliers in Bulletin 17B), zero flows, and uncertain 

(non-point discharge) observations; historical data include 

non-exceedance flood information (Gotvald and others, 2012). 

For sites that have no PILFs, no historical information, and 

no other censored information, flood frequency estimates are 

identical to those values obtained when using the conventional 

method of moments described in Bulletin 17B. 

An EMA flood frequency analysis requires two types of 

information to describe the flows for every year in the period 

of record—(1) flow intervals that describe the magnitude of 

the annual peak flow and (2) measurement capabilities of the 

streamgaging techniques, known as perception thresholds. 

Flow intervals are defined by a lower bound, QL, and an upper 

bound, QU. Perception thresholds are defined by the minimum 

flow to the maximum flow that a streamgage could record, 

TL

 to TU, respectively. A recorded point discharge value of 

300 ft3/s from a standard continuous streamgage, for example, 

is described in EMA by a flow interval from 300 ft3/s (QL) to 

300 ft3/s (QU), with perception thresholds set from 0 ft3/s (TL) 

to infinity (TU).

Some streamflow records span a longer historical 

period than just the peaks documented as part of a regular 

streamgaging program, or systematic peaks. Additional 

(historic) peaks and associated historical information can 

be used to extend the period of record beyond the period of 

systematic record or fill in gaps in the period of systematic 

record. Because EMA uses a more general description of flood 

information, all types of flood data can be incorporated in 

the analysis.
Flood Magnitude and Frequency at Gaged Sites 13

Systematic Record and Nonstandard 

Censored Flow

Systematic peaks are those documented routinely, 

regardless of magnitude, as part of a regular streamgaging 

program. Like most USGS science centers, the USGS Alaska 

Science Center (ASC) streamgaging program has focused 

on the routine collection of streamflow data at continuous 

and crest-stage streamgages since it began widespread 

streamgaging in 1949. The resulting systematic records 

include many intermittent and short records because of 

fluctuations in funding, and because of streamgaging strategies 

that balanced funding long-term sites against funding many 

short-term sites in order to bolster Alaska’s relatively sparse 

streamgaging network. The Water Survey of Canada records 

used in this study consisted only of standard systematic 

streamflow data, which could be continuous or intermittent.

Within the USGS systematic streamflow record, data 

can include nonstandard, censored flow data that describes 

a recorded interval. For example, a crest-stage gage (CSG) 

partial-record site records the highest water level above a 

minimum recordable elevation (gage base) since the station 

was last visited. In some cases, an annual peak flow is known 

to not have exceeded the gage base at the CSG. This belowgage-base peak flow is properly described in EMA as a flow 

interval with (QL

) = 0 ft3/s and QU = QGage base, and perception 

thresholds from TQGage base to infinity. In some cases, the opposite

censored flow conditions could arise. For example, it might 

be known that an annual peak flow overtopped a streamgage 

and left no visible high water marks to estimate the magnitude 

of that flow event. The user can set a flow interval in EMA 

from the maximum value recordable at the gage, QGage max, (QL) 

to infinity (QU), respectively, with the perception thresholds 

set from 0 ft3/s to TQGage max. EMA allows the information from 

these nonstandard, censored flow data to be applied in the 

analysis rather than ignored or crudely estimated as would be 

the case for a conventional analysis using Bulletin 17B of the 

Interagency Advisory Committee on Water Data (1982) such 

as Curran and others (2003).

Historical Flood Information

At many locations, flood plain observation over an 

extended period or preserved evidence of floods can be used 

to inform a longer historical period about large floods and 

about peak flows that were not recorded but were known to 

be below some value. The longer historical period consists of 

the envelope of all years for which streamflow information is 

known, including the systematic period of record, the years 

for which peak flows were collected outside the systematic 

record, and non-gaged years for which it was known no peak 

equaled or exceeded some value. Particularly large floods 

can be recorded within the systematic record or, in the case 

of USGS streamflow records, as annual peaks collected 

before, after, or in breaks in the systematic record. Collected 

for some purpose other than documenting the annual peak 

regardless of its magnitude, the non-systematic peaks belong 

to a separate, biased population. Because the occurrence 

of particularly large floods led to the collection of many of 

these non-systematic peaks, they have been conventionally 

referred to as historic peaks in the USGS peak-flow file. The 

term historic can be misleading because it infers that the peak 

must be particularly large if collected outside the systematic 

record; in fact, as found in many Alaska streamgage records, 

peak flow data collected outside the systematic streamgaging 

record can be any magnitude. For example, as part of routine 

collection of streamflow data associated with major regional 

rainstorms, ASC opportunistically recorded flood data at 

inactive (unfunded for regular operation) streamgages as 

requested by a funding agency, particularly if damage to 

infrastructure occurred. The streamgaging resulted in annual 

peaks collected outside the systematic record that may or may 

not have been particularly large, depending on factors such 

as infrastructure damage disproportionate to flood magnitude. 

Because the peak-flow file does not designate the motivation 

for collecting non-systematic peaks, all peaks coded as historic 

in the peak-flow file for this study were first reviewed to 

determine if the peak was large enough to potentially inform 

periods of missing record. Peaks coded as historic but omitted 

from analysis because they could not be established as part 

of the population of non-random, large peaks that could 

inform missing periods of record were maintained as part of 

the peak-flow file to provide important information for other 

studies. For records containing particularly large historic peaks 

or particularly large systematic peaks, historical information 

was then researched to ascertain, where possible, that the 

record contained the full population of peaks above some 

value in the longer historical period, and that non-recorded 

peak flows were known to not have equaled or exceeded 

that value. Historical flood information to support extending 

the historical period included published reports or historical 

accounts, observer interviews, high water marks, damage 

to infrastructure, and other marks or observations from the 

particular site in question. 

Flow data for historic peaks used in the analysis was 

included in EMA as QHistoric (QL

) to QHistoric (QU). Flow data for 

missing record in the longer historical period was represented 

as 0 ft3/s (QL) to QHistoric or QLarge systematic (QU) for records having 

a historic peak or large systematic peak, respectively. The 

perception threshold was set from TQ Historic or TQ Large systematic

(TL), as appropriate, to infinity (TU), for both the water year 

of the historic peak, if present, and the missing period in the 

longer historical period. As an example, figure 2 shows the 

annual peak-flow series for streamgage 15208000 Tonsina 

River at Tonsina, Alaska, which included two peaks collected 

outside the systematic gaged record. The 7,000 ft3/s water year 

1995 peak, which was exceeded during the period of gaged 

record, was deemed too small to inform periods of missing 

record and thus was not part of a population appropriate for 

flood frequency analysis. The 14,000 ft3

/s water year 2006 

peak was determined from highway damage reports and 
14 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

reports from local residents to represent a non-exceedance 

threshold that implied that all floods greater than or equal to 

that magnitude would be known for the historical period. In 

this case, the historical period was extended forward to 2012, 

the latest water year for all data in the study, and perception 

thresholds for water years 1983 to 2012 were set as 14,000 

(TL) to infinity (TU). The corresponding at-site flood frequency 

LP3 curve is shown in figure 3.

For each streamgage in the study, table 1 lists the water 

years for the period of record used, the historic period length 

(number of water years in the period of record used), the 

water years omitted from the frequency analysis, the number 

of peaks in the record (as reported in output from the PeakFQ 

software), the perception thresholds set for every year in the 

period of record (if different from the default 0 ft3/s-to-infinity 

for the systematic gaged record), and interval discharge 

ranges (QINT) for censored flows. The value in the “number 

of peaks in record” column is equal to the number of peaks 

in the NWIS peak-flow file minus the number of peaks listed 

in the table as omitted that fall outside the years listed in 

“period of record used.” The number of peaks that are actually 

used in the analysis equals the value given for “number of 

peaks in record” minus the number of peaks listed in the 

table as omitted that fall within the years listed in “period of 

record used.”

tac16-5245_fig02

15208000 Tonsina River at Tonsina, Alaska

1940 1950 1960 1970 1980 1990 2000 2010 2020

Annual peak flow, in cubic feet per second 

1,000

10,000

100,000

Peak flow collected outside the 

systematic gaged record

Figure 2. Example of annual peak-flow series 

containing peaks collected outside the systematic gaging 

record, U.S. Geological Survey streamgage 15208000, 

Tonsina River at Tonsina, Alaska.

99 98 95 90 80 70 50 30 20 10 5 2 1 0.5 0.2

Peak flow, in cubic feet per second

1,000

10,000

100,000

Annual exceedance probability, in percent

15208000 Tonsina River at Tonsina, Alaska

Fitted EMA-MGB flood-frequency-curve

Confidence limits on EMA-MGB 

flood-frequency curve

Recorded data 

Historic peak

Threshold for historic peak

EXPLANATION

Figure 3. Example of a Log-Pearson Type III flood frequency curve for U.S. Geological Survey streamgage 15208000, 

Tonsina River at Tonsina, Alaska, showing the Expected Moments Algorithm (EMA) with a multiple Grubbs-Beck (MGB) test 

representation for two peaks collected outside the systematic gaging record, water years 1995 and 2006. The water year 1995 

peak was omitted from the analysis on the basis of a lack of at-site information to establish the value of the peak as a nonexceedance threshold. At-site information established the value of the water year 2006 peak as a non-exceedance threshold 

for the missing period of record through the end of the study period.
Flood Magnitude and Frequency at Gaged Sites 15

Multiple Grubbs-Beck Test for Detecting 

Multiple Potentially Influential Low Floods 

To prevent zero-flow and very small annual peaks 

from having undue influence on the fit of the distribution 

for large annual peaks, Bulletin 17B (Interagency Advisory 

Committee on Water Data, 1982) recommends the use of 

the Grubbs‐Beck test (Grubbs and Beck, 1972) to detect low 

outliers in flood series that deviate from the trend of the data. 

The Grubbs-Beck test provides an objective identification of 

low outliers but does not consider the condition of multiple 

outliers. As described by Cohn and others (2013), the 

multiple Grubbs‐Beck (MGB) test is a generalization of the 

Grubbs-Beck method that allows for a standard procedure 

for identifying multiple PILFs. In flood-frequency analysis, 

PILFs are annual peaks that meet three criteria—(1) their 

magnitude is much smaller than the flood statistic of interest; 

(2) they occur below a statistically significant break in 

the flood-frequency data plot; and (3) they have excessive 

influence on the estimated frequency of large floods (Veilleux 

and others, 2014). PILFs may constitute one-half or more 

of the observations and are believed to arise from physical 

processes that are not relevant to the processes associated with 

large floods. Because the magnitudes of PILFs are expected to 

explain little about the upper right-hand tail of the frequency 

distribution (largest floods), the best estimates for the typical 

suite of AEP flows can be obtained by censoring the PILFs 

(see Subcommittee on Hydrology, Hydrologic Frequency 

Analysis Work Group Frequently Asked Questions at http://

acwi.gov/hydrology/Frequency/B17bFAQ.html#low). When 

an observation is identified as a PILF, all values smaller than 

that flood magnitude also are categorized as PILFs in the 

EMA analysis. Identifying PILFs and recoding them and all 

smaller peaks as censored peaks can greatly improve estimator 

robustness with little or no loss of efficiency. Thus, the use 

of the MGB test can improve the fit of the small annual 

exceedance probabilities (right hand tail of the distribution), 

while minimizing lack-of-fit due to unimportant PILFs in an 

annual peak series (Cohn and others, 2013). An example of a 

flood-frequency curve for a streamgage with multiple PILFs 

is shown in figure 4. Sites for which the default MGB test was 

overridden by a visual inspection, the PILF threshold below 

which all flows are recoded as censored values in the EMA 

analysis, and the corresponding number of PILFs that were 

censored are shown in table 1.

15476400 Dry Creek near Dot Lake, Alaska

Annual exceedance probability, in percent

Peak flow, in cubic feet per second

10

100

1,000

10,000

100,000

Fitted EMA-MGB flood-frequency-curve

Confidence limits on EMA-MGB 

flood-frequency curve

Recorded data

Potentially influential low flood identifed 

by the multiple Grubbs-Beck test

EXPLANATION

99 98 95 90 80 70 50 30 20 10 5 2 1 0.5 0.2

Figure 4. Log-Pearson Type III fit when multiple Grubbs-Beck (MGB) test found multiple potentially influential low floods (PILFs) 

for U.S. Geological Survey streamgage 15476400, Dry Creek near Dot Lake, Alaska.
16 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

Statistical Analysis of Regional Skew

Bulletin 17B (Interagency Advisory Committee on 

Water Data, 1982) guidelines recommend using a weighted 

average of the station skew and a regional skew to improve 

the accuracy of the skew estimator used to estimate the AEP 

discharges. As presented in Bulletin 17B, this is computed 

from the station and regional skew weighted in inverse 

proportion to their mean square errors (MSE):

G

G G

w

G s G r

G G

r s

r s

= ( ) + ( )

+

MSE MSE

MSE MSE (2)

where

 Gw is the weighted skew;

 Gs is the station skew; and 

 Gr is the regional skew; and 

MSEGr

 and MSEGs are the mean square error of the

 regional and station skew, 

 respectively.

Bulletin 17B (Interagency Advisory Committee on 

Water Data, 1982) supplies a national map of regional skew 

but also encourages hydrologists to develop more specific 

local relations. Curran and others (2003) present regional skew 

coefficients for various streamflow analysis regions across 

the study area determined as an average of the station skew 

for long-record streamgages. More recently, Reis and others 

(2005), Gruber and others (2007), and Gruber and Stedinger 

(2008) developed a Bayesian generalized least-squares (GLS) 

regression model for regional skew analyses. The Bayesian 

methodology allows for the computation of a posterior 

distribution of both the regression parameters and the model 

error variance. Due to complications introduced by the use 

of the EMA with MGB censoring of low outliers (Cohn and 

others, 1997) and large cross-correlations between annual peak 

discharges at pairs of streamgages, the Bayesian weighted 

least-squares/Bayesian generalized least-squares (B-WLS/BGLS) regression framework was developed to provide both 

stable and defensible results for regional skew (Veilleux, 2011; 

Lamontange and others, 2012; Veilleux and others, 2012). 

For this study, the B-WLS/B-GLS analysis was applied 

only to areas where gaging density provided an adequate 

dataset of long-record streamgages. The regional skew 

analysis used streamgages with at least 25 years of pseudo 

record length, a measure computed from the record length 

and mean square error of the historical and systematic 

parts of the record. Two areas of Alaska contained a 

density of streamgages meeting these requirements (fig. 5). 

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EXPLANATION

Regional Skew Area 1

Regional Skew Area 2

Regional skew exclusion area

Figure 5. Regional skew areas for Alaska and conterminous basins in Canada.
Flood Magnitude and Frequency at Gaged Sites 17

Regional Skew Area (RSA) 1 in Interior Alaska contains the 

eastern part of Curran and others (2003) Streamflow Analysis 

Regions 4 and 6. The boundary for RSA 1 was delineated 

from the WBD as the hydrologic unit code-8s (HUC-8s) 

containing sites used in regional skew analysis, plus adjacent 

HUC-8s needed to make a continuous area. In several places, 

the HUC-8s were split to omit non-applicable areas. Regional 

Skew Area 2 contains 2003 Streamflow Analysis Regions 1 

and 3 and was modified from Curran and others (2003) to 

match current HUC-8 boundaries. The HUC-8s, and HUC10s where partial HUC-8s were used, contained in each RSA, 

are shown in table 5. The areas outside of the RSAs can be 

considered regional skew exclusion areas, where low gage 

density limited the ability to establish regional patterns and 

only station skew applies to flood frequency analysis.

For a representative assessment of regional skew, sites 

used in the analysis should have reasonably similar basin 

characteristics. For RSA1, sites on the Yukon River, which are 

considerably larger in drainage area than any other sites, were 

excluded. Regional skew should only be applied to weighting 

with station skew for sites physically located within a regional 

skew area and within the range of drainage area for the sites 

that were used for analysis (table 6).

HUC-8 HUC-10s within the selected HUC-8 

that are included in Regional Skew Area

Regional Skew Area 1

19020101 All

19020102 All

19020103 All

19020104 1902010401, 1902010402, 1902010403, 1902010404

19020301 1902030101, 1902030102, 1902030103, 1902030104, 

1902030105, 1902030106, 1902030107, 

1902030108, 1902030109, 1902030110

19020302 All

19020401 All

19020402 All

19020501 All

19020502 All

19020503 All

19020505 All

19080301 All

19080302 All

19080303 All

19080304 All

19080305 All

19080306 All

19080307 All

19080308 All

19080309 All

19080401 All

19080402 All

19080403 All

19080404 All

19090101 All

19090102 All

HUC-8 HUC-10s within the selected HUC-8 

that are included in Regional Skew Area

Regional Skew Area 2

19010102 All

19010103 All

19010104 All

19010105 All

19010106 1901010601, 1901010602, 1901010603

19010107 1901010701

19010204 All

19010206 All

19010207 1901020701, 1901020702, 1901020703

19010208 All

19010209 All

19010210 All

19010211 All

19010212 All

19010301 All

19010302 All

19010303 All

19010304 1901030404

19010402 All

19010404 1901040402, 1901040403, 1901040404, 1901040405

19010405 All

19010406 All

19020104 1902010405, 1902010406, 1902010407, 1902010408, 

1902010409, 1902010410, 1902010411, 

1902010412, 1902010413, 1902010414, 

1902010415, 1902010416, 1902010417, 

19020104218

19020201 All

19020202 All

19020203 All

19020301 1902030111, 1902030112

19020602 1902060208, 1902060209, 1902060210, 1902060211, 

1902060212, 1902060213

19020701 All

19020702 All

19030101 All

19030102 1903010201, 1903010202, 1903010203, 1903010204, 

1903010206

Table 5. Definition of regional skew areas for Alaska and conterminous basins in Canada, by hydrologic unit code (HUC).
18 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

The updated regional skew values and associated 

statistics and performance metrics for RSA1 and RSA2 are 

summarized in table 6. For both RSAs, no basin characteristics 

were found that improved model precision substantially 

and warranted the added model complexity, resulting in the 

adoption of a constant model for regional skew. The average 

variance of prediction (AVP) for a site not in the analysis can 

be substituted for MSEGr in equation 2. The AVP for

RSA1 and RSA2 corresponded to an effective record length 

of 22 and 59 years, respectively. The new regional skews 

and their associated standard errors were used in the flood 

frequency analysis for all sites in the study that were located in 

the new RSAs and met drainage-area requirements. Additional 

details of the methods and results of the regional skew analysis 

are provided in appendix B.

Estimating Flood Magnitude and 

Frequency at Ungaged Sites

A regional regression analysis was used to develop a 

set of equations for estimating the magnitude and frequency 

of floods for ungaged sites in Alaska and conterminous 

basins in Canada. These equations relate the 50-, 20-, 10-, 

4-, 2-, 1-, 0.5-, and 0.2-percent AEP flows computed from 

peak-flow records for streamgages (table 4) to measured 

basin characteristics of the associated drainage basins. After 

elimination of streamgages ineligible for regression because 

of the occurrence of annual glacier outburst floods, the lack 

of a defined basin, or a close similarity to other streamgages 

in the same basin, 341 streamgages (284 in Alaska and 57 in 

Canada) were eligible for regional regression. The regression 

process consisted of exploratory analysis to determine the 

most suitable explanatory variables and their transformations, 

consideration of whether the equations could be improved 

by grouping sites into regions, and the use of more robust 

regression methods to develop the final equations and the 

uncertainty of their estimates.

Elimination of Redundant and Other 

Non-Eligible Sites from Regression Analysis

Regional regression analysis requires that streamflow 

data for all sites be correlated to basin characteristics in 

a comparable manner and that streamflow at a site be 

reasonably independent of streamflow at other sites. Review 

of streamflow records for these criteria resulted in exclusion 

of a number of sites eligible for flood frequency analysis but 

considered ineligible for the purpose of developing predictive 

regression equations from basin characteristics. These include 

sites having non-eligible streamflow populations or basin 

conditions, sites on the Yukon River with very large drainage 

areas, and sites considered hydrologically redundant.

Four streamgages included in this study had streamflow 

records that were suitable for fitting to a frequency distribution 

but included peaks controlled by glacier outbursts (releases of 

water from glacier-controlled storage), when those outbursts 

occur annually. In certain glacierized basins, annual glacier 

outburst peaks are routinely larger in magnitude than annual 

peaks from similarly sized basins not subject to outbursts. 

These large glacier outbursts are considered a separate 

population from snowmelt, rainfall, or glacier-affected melt 

peaks and cannot be used to develop regression equations 

for non-outburst floods. Peak qualification codes in the 

NWIS peak-flow file provided an initial screening tool 

for identifying basins subject to outbursts. Indeterminate 

drainage areas, which result in indeterminate relations to 

basin characteristics, included one site (USGS streamgage 

15485500) where a portion of the stream bypassed the 

streamgage and one discontinued site (USGS streamgage 

15201900) having ambiguous documentation of streamgage 

location. On the Yukon River, drainage areas for sites 

at and downstream of Carmacks, Yukon, are larger than 

any other gaged locations in Alaska. The 10 Yukon River 

streamgages at and downstream of Carmacks (drainage 

areas greater than 31,100 mi2) were excluded from the 

development of regression equations and considered 

separately in section, “Estimate for a Site on the Yukon River.” 

Table 6. Regional skew and summary statistics for regions in Alaska and conterminous basins in Canada.

[Abbreviation: mi2, square mile]

Location

Number of stations 

with at least 

25 non-censored 

peaks

Regional

skew

Average variance 

of prediction 

(for a site not 

in the analysis)

Standard 

error of the 

regional skew

Applicable range 

of drainage area 

(mi2

)

1

Regional Skew Area 1 75 0.54 0.45 0.67 1.2–25,560 1

Regional Skew Area 2 28 0.18 0.12 0.34 1.7–123

Sites outside regional skew areas, and 

sites in regional skew areas but not 

within the applicable range of 

drainage area

No regional skew available. Use station skew.

1

Where basin drainage area is within applicable range of drainage area.
Estimating Flood Magnitude and Frequency at Ungaged Sites 19

Flood frequency estimates for records excluded from 

regression analysis on the basis of non‐eligible streamflow 

populations or basin conditions did not use weighted skew 

and were not weighted with regression estimates. A usage 

designation that distinguishes these station-only sites from 

sites used in regional regressions is included in table 1.

Nested basins having substantially similar size and 

basin characteristics are considered redundant for regression 

analysis. Including redundant basin pairs would unduly 

weight strongly related flows (Gruber and Stedinger, 2008). A 

screening tool developed for identifying potentially redundant 

basins (Veilleux, 2009) uses the drainage area ratio (DAR) 

between the basins and the normalized distance (ND) between 

the basin centroids. For this study, basin pairs with DAR 

less than 5 and ND less than 0.5 were considered redundant 

unless a visual inspection noted that they were not nested 

or had substantially different characteristics likely to affect 

streamflow, such as a large lake in one basin. Within each 

redundant pair, the site with the longest record generally 

was retained for use in regressions. Where record lengths 

were similar, one site was selected randomly for retention. 

For redundant sites in Alaska, station skew was weighted 

with regional skew as appropriate, but the resulting flood 

frequency estimates were not used to develop the regression 

equations. Final flood frequency estimates for redundant sites 

in Alaska were prepared from the station estimate weighted 

with the regression estimate as for non-redundant sites. The 

usage column in table 1 shows the 30 sites included in the 

flood frequency analysis but omitted from development of 

regression equations as redundant. Sites located in Canada 

that were initially considered for use but omitted from the 

study as redundant are not shown in the table; these were the 

streamgages with USGS site identification Nos. 15024098, 

15024300, 15024640, 15024695, 15120730, 15305030, 

15305100, 15305260, 15305400, 15305406, 15305418, 

15305520, 15305582, 15305590, 15305620, 15355000, 

and 15388944.

Exploratory Regression Analysis

Multiple-linear regression analysis is commonly used to 

develop equations that relate two or more physical or climatic 

basin characteristics to a specific streamflow statistic. After 

developing these empirical equations for an area from eligible 

streamgages and a particular basin characteristics dataset, the 

equations can be applied to estimation of flow at ungaged sites 

in that area using basin characteristics for the ungaged site 

obtained from the same or comparable basin characteristics 

dataset. Multiple-linear regression models the relation 

between multiple independent, or explanatory, variables (basin 

characteristics, such as drainage area) and a single dependent 

variable (a streamflow statistic, such as Q1%) by fitting a linear 

equation to the data. 

Ordinary least-squares (OLS) regression is a simple 

form of multiple-linear regression that assumes that the 

peak-flow values at streamgages are independent and 

that each streamflow record has similar variance, which 

is influenced by the length of the record. OLS is a useful 

technique for identifying the most statistically important 

explanatory variables, testing different combinations of 

variables, and determining the general form of the equations. 

R, a programming language and environment for statistical 

computing and graphics, and USGS functions in the R 

package USGSwsStats (renamed smwrStats in 2015) were 

used for OLS regression analysis for this study. 

To achieve a linear relation and improve the spread of 

data between the explanatory variables and the dependent 

variable, many variables associated with streamflow analysis 

require a data transformation. A log transformation commonly 

improves the linearity of most streamflow statistics and many 

basin characteristics, and was applied to all variables in 

this study except mean basin elevation and mean minimum 

January temperature. The logarithm of zero or negative 

values is undefined, but datasets containing these values 

can be first transformed by the addition of a constant, then 

log transformed. For this study, all variables expressed as 

a percentage of basin area were transformed by adding a 

constant 1 percent to all values, and mean minimum January 

temperature was transformed by adding a constant 32 °F to 

all values.

Highly correlated explanatory variables create a 

condition called multicollinearity, which restricts the 

ability of regression analysis to evaluate the importance of 

the respective variables and increases the variance of the 

regression coefficient for the explanatory variables. Obvious 

multicollinearity is generated by redundant metrics for a 

similar basin property, such as the percentage of the basin 

covered by glaciers from the Randolph Glacier Inventory and 

the percentage of the basin covered by snow and ice from the 

National Land Cover Dataset dataset. For this study, redundant 

variables included many land cover variables, metrics for 

amount and intensity of precipitation, and monthly versus 

annual mean values of climate variables. For these cases, 

a preferred dataset was chosen for initial analysis on the 

basis of dataset quality, extent of coverage of the study area, 

and hydrologic judgment of the strength of the explanatory 

nature of the variable (table 2). Selected alternate variables 

to these primary variables (table 2) were tested in multiple 

regression analysis by first removing the primary variable 

from the multiple regression. Correlation coefficients and 

inspection of correlation plots helped eliminate other highly 

correlated explanatory variables, such as mean annual 

precipitation and mean minimum January temperature, or 

either of those variables coupled with latitude of the basin 

centroid, from consideration as potential variables together in 

a single equation.
20 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

Selection of the most suitable explanatory variables for 

all study sites together and for each of six streamflow analysis 

regions designated by Curran and others (2003) and shown in 

figure 6–Regions 1 and 3 together, and Regions 2, 4, 5, 6, and 

7–was based on all-possible-subsets regressions. Explanatory 

variables selected for final models were statistically 

significant, improved the coefficient of determination (R2), 

and minimized the standard error of the estimate. Other 

considerations included whether the sign and magnitude of 

the regression coefficient was hydrologically reasonable, 

the ease of computation of the explanatory variable, and the 

effect of adding the variable on the Mallow’s Cp statistic and 

the variance inflation factor (VIF). For all study sites together 

and for all 2003 regions, logDRNAREA formed the strongest 

explanatory variable. The variable logPRECPRIS00, which 

was statistically significant, considerably improved the fit and 

error of the equation for all study sites together and for some, 

but not all, 2003 regions. For certain AEPs in some regions, 

a third variable, such as ELEV or logGLACIER+1, was 

statistically significant and did not appreciably increase the 

VIF, but also did not considerably improve the fit or error of 

the equation. 

Regionalization of Flood-Frequency Estimates

The streamflow analysis regions from Curran and others 

(2003) (fig. 6) provided an initial suite of regions for testing 

geographic groupings of sites and were supplemented by 

(1) proposed combinations of the 2003 regions and (2) a single 

study-wide region. The 2003 analysis presented seven regions 

for various streamflow analyses, two of which (Regions 1 

and 3) were combined for the purposes of peak-flow analysis. 

The 2003 regions followed general climate patterns and major 

hydrologic basins, including a southern coastal area (Regions 

1 and 3 combined) where precipitation is relatively high and 

temperatures moderate, a band of regions to the north (Region 

4, including south-central Alaska, and Region 2, encompassing 

mostly large streams in Canada that drain to southeast 

Alaska), regions splitting the Yukon River Basin (Region 5 in 

the mostly Canadian upper Yukon River Basin, and Region 

6 in Interior Alaska), and a region spanning the colder and 

drier northern and northwestern part of the State (Region 7). 

Consideration of streamgage density (average streamgages 

per square mile, spatial clustering of streamgages, and total 

streamgages in a region) was a primary factor in seeking 

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7

6

5

2 4

3

1

2003 streamflow analysis regions

EXPLANATION

5

Figure 6. Streamflow analysis regions previously presented for Alaska and conterminous basins 

in Canada (from Curran and others, 2003, fig. 1).
Estimating Flood Magnitude and Frequency at Ungaged Sites 21

alternatives to the 2003 regions. In particular, Regions 2 and 7 

contained only 20 and 29 streamgages eligible for regression 

in this study, respectively, limiting the statistical strength 

of the equations and the representation of large land areas. 

Various combinations of Regions 4, 6, and 7, and of Regions 2 

and 5, were considered, as was a Cook Inlet Basin region to 

match the boundary of the companion StreamStats project. A 

single study-wide region containing all streamgages eligible 

for regression analysis also was proposed to overcome the 

issues of streamgage density in selected areas of the study. 

Evaluation of the 2003 regions, various combinations 

of the 2003 regions, and the study-wide region consisted of 

inspection of performance metrics for the best all-possiblesubsets OLS regressions for each region followed by an 

analysis of covariance. The Q1% and Q50% were chosen as 

representative flow statistics for assessing potential regions. 

Results from a single-variable model using logDRNAREA 

or a two-variable model using logDRNAREA and 

logPRECPRIS00, where both were statistically significant, 

showed that grouping sites by regions or combinations of 

regions produced a range of fit and associated error that 

straddled the fit and error produced by grouping all study 

sites together. Performance metrics were within generally 

acceptable limits for all regions and combinations tested.

Analysis of covariance (Helsel and Hirsch, 2002) using 

an indicator variable can be applied to test the statistical 

significance of the difference between any two regions. 

The indicator variable is set to 1 for all streamgages in a 

particular region and 0 for all other streamgages in the other 

region or regions being compared, and is then included in the 

regression as an explanatory variable. Statistical significance 

for the indicator variable indicates a difference in the 

regression intercept between streamgages in that region and 

the other streamgages in the test. An OLS regression was 

conducted using logDRNAREA and the indicator variable, 

and logDRNAREA, logPRECPRIS00, and the variable, 

for the Q1% flow for the 2003 regions and various alternate 

combinations of the 2003 regions. Although this analysis 

of covariance confirmed the statistical significance of the 

pairing of Regions 1 and 3, for example, the independence of 

many other regions varied depending on how regions were 

combined, suggesting ambiguity in the delineation of the 2003 

region boundaries. The 2003 regions and various combinations 

of the 2003 regions, although not invalidated as suitable 

streamflow analysis regions, were not strongly supported 

by the analysis of variance. The comparable performance 

metrics of the group of all study sites together relative to the 

performance metrics of various combinations of the 2003 

regions, coupled with ambiguous results of the analysis of 

covariance between regions, led to the selection of the entire 

study area except sites on the Yukon River at and downstream 

from Carmacks, Yukon as the single region for analysis for 

this study.

Use of a single study-wide region equivalent to the 

study area increased the statistical strength of the regression 

equations as compared to use of the 2003 regions in part by 

substantially increasing the number of sites used to develop 

the equation. Replacing multiple sets of equations in regions 

based largely on climate with a statistically stronger, single 

set of equations based in part on a climate variable provided 

comparable flow estimates having a more direct relation to 

basin characteristics and greater simplicity in application. The 

fit of a single set of equations for the study area suggests that 

this relation of streamflow statistics to basin characteristics 

is applicable over a wide range of basins, reducing, although 

not eliminating, the concern about the applicability of the 

regression equations over the large areas of the study area 

where no streamgages are present.

Regional Regression Equations

Streamflow data are naturally correlated spatially and 

temporally, making the assumptions of OLS regression 

incompletely satisfied. A more sophisticated technique, 

generalized least-squares (GLS) analysis, improves the 

equations by accounting for time-sampling error, which is a 

function of record length, and cross-correlation of annual peak 

flows between streamgages (Stedinger and Tasker, 1985). If 

two streamgages are in close proximity and flooding is caused 

by regional rainstorms or other basin climate conditions, the 

annual series of peak flow will be largely similar (correlated) 

at both streamgages and cannot be considered independent 

information for the purposes of the regression. GLS assigns 

different weights to each observation on the basis of its 

contribution to total variance. The USGS weighted-multiplelinear-regression (WREG) computer program, version 1.05 

(Eng and others, 2009) incorporates GLS techniques and 

metrics developed by Stedinger and Tasker (1985), Tasker and 

Stedinger (1989), Martins and Stedinger (2002), Griffis and 

Stedinger (2007, 2009), and was used for GLS analysis of the 

final equations for this study.

Following the OLS regression analysis to develop 

a preferred multiple-linear regression model for relating 

each flood frequency statistic to basin characteristics and 

regionalization, which established the entire study area as 

the preferred regional structure, a GLS regression analysis 

was used to produce final equations. The resulting equations 

were evaluated using performance metrics and diagnostic 

tools in the WREG program. The significance of regression 

coefficients for each basin characteristic was checked to 

confirm the results of the OLS regression. Diagnostic plots of 

all residuals (difference of the predicted and observed values 

for a streamgage) against the predicted flows and against 

each explanatory variable in the equation were examined to 

ensure points were generally randomly distributed around 

zero, the assumed condition for linear regression (Helsel and 

Hirsch, 2002). 
22 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

WREG calculates leverage and influence statistics 

for the GLS analysis (Eng and others, 2009), which serve 

as a regression diagnostic for the effects of an individual 

streamgage on the regional regression models. Leverage is 

a measure of how much the values of explanatory variables 

at a streamgage vary from the values of those variables at all 

other streamgages. Because unusual values for a streamgage, 

that is, values with high leverage, might or might not have a 

significant impact on the regression equation, the influence 

metric is computed to indicate how strongly the values for a 

streamgage influenced the estimated regression parameters. 

A streamgage can have high leverage, indicating that its 

independent variables are substantially different from those 

at all other streamgages, but not have a large influence on the 

fitted regression relation. Conversely, a streamgage with high 

influence might not have high leverage. Streamgages that had 

leverage or influence metrics that exceeded the thresholds 

calculated by WREG, especially those that had both high 

leverage and high influence, were evaluated for potential 

erroneous data reporting or conditions that would make the 

streamgage ineligible for regression. If no such errors existed, 

high leverage or influence metrics alone were insufficient 

justification for removing the streamgage from the regression 

analysis, and these streamgages were kept in the model.

The final regional regression equations for the 50-, 20-, 

10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEP flows for a single 

region encompassing Alaska basins, except the Aleutian 

Islands and other islands off the western coast of Alaska 

(fig. 1) and sites on the Yukon River with very large basins, 

together with those major basins in Canada that drain to 

Alaska are shown in table 7. The combination of drainage 

area and mean annual precipitation as explanatory variables 

produced the best fit and lowest error while minimizing 

the disadvantages of adding variables to the equation. The 

streamgage-specific values of drainage area and mean annual 

precipitation for the 341 streamgages used in the analysis are 

given in table 1, and ranges of these values are presented with 

the regional regression equations in table 7.

Sites along the Yukon River downstream of USGS 

streamgage 15305350 in Carmacks, Yukon have drainage 

areas that exceed that of all other rivers in the study. These 

large basins (drainage areas greater than 31,100 mi2

) might 

not respond to basin characteristics as do other, smaller basins 

and were not included in the regression analysis. Regression 

analysis would not be well-suited for this small collection 

of streamgages along a single river. Instead, a method for 

simple drainage-area-based interpolation of flow statistics 

between long-term sites is presented. The graphical relation 

of peak flow to drainage area for Yukon River sites with at 

least 20 years of record is shown in figure 7 and may be used 

to estimate flood frequency statistics for sites along the Yukon 

River in Alaska and Canada with very large basins. USGS 

streamgage 15305700 had about one-half the record length 

of nearby USGS streamgage 15356000 and was removed to 

improve the continuity of the curve. The estimate of flood 

frequency flows may be obtained mathematically by a linear 

interpolation between the logarithms of the flood frequency 

flows in table 4 for the nearest designated long-term upstream 

and downstream Yukon River estimator streamgages shown in 

figure 7 using the logarithms of the drainage areas.

Table 7. Regional regression equations for estimating annual exceedance-probability discharges for 

unregulated streams in Alaska and conterminous basins in Canada.

[Regional regression equation: DRNAREA, drainage area, in square miles; PRECPRIS00, basin average mean annual 

precipitation, in inches, for 1971 to 2000 from the PRISM climate dataset. AVP: Average variance of prediction. SEP: Average 

standard error of prediction. 2 Rpseudo: pseudo coefficient of determination]

Percent annual 

exceedance 

probability

Regional regression equation for estimating

annual exceedance probability discharge,

 in cubic feet per second1,2

AVP

(log units)

SEP 

(percent)

Rpseudo

2

(percent)

50 0.944 (DRNAREA)0.836(PRECPRIS00)1.023 0.077 70.8 91.1

20 2.47 (DRNAREA)0.795(PRECPRIS00)0.916 0.074 69.1 90.6

10 4.01 (DRNAREA)0.775(PRECPRIS00)0.865 0.074 69.2 90.0

4 6.53 (DRNAREA)0.755(PRECPRIS00)0.816 0.077 71.2 89.0

2 8.79 (DRNAREA)0.743(PRECPRIS00)0.787 0.080 72.8 88.2

1 11.4 (DRNAREA)0.732 (PRECPRIS00)0.764 0.083 74.6 87.4

0.5 14.3 (DRNAREA)0.723(PRECPRIS00)0.744 0.089 77.4 86.3

0.2 18.7 (DRNAREA)0.712(PRECPRIS00)0.721 0.097 81.9 84.7

1

Equations are valid for DRNAREA between 0.4 and 1,000 mi2 with PRECPRIS00 between 8 and 280 in. and for DRNAREA 

greater than 1,000 and less than 31,100 mi2 with PRECPRIS00 between 10 and 111 in.

2

Equations are not suitable for use in the Aleutian Islands and other islands outside the study area.
Estimating Flood Magnitude and Frequency at Ungaged Sites 23

Figure 7. Relation of discharge to drainage area for selected annual exceedance probabilities for the Yukon 

River at and downstream of Carmacks, Yukon.

Accuracy and Limitations

The regression equations presented are empirical models 

that relate AEP flood flows to a particular dataset of physical 

and climatic basin attributes. These statistical relations must 

be interpreted and applied within the limits of the data and 

with the understanding that the results are best-fit estimates 

with an associated variance. As with all models, the regression 

equations have an associated measure of quality indicating 

tac16-5245_fig07

15564800

15565447

15453500

15356000

15305450

15305350

EXPLANATION

Drainage area, in square miles

10,000 100,000 1,000,000

Discharge, in cubic feet per second

100,000

1,000,000

10,000,000

50

20

10

4

2

1

0.5

0.2

Annual exceedance probability, in percent

USGS streamgage No.

how well the predicted values represent the true values, and 

a reported uncertainty. For an individual regression equation 

estimate for a site, the variance of prediction, the standard 

error of prediction, and prediction intervals describe the 

accuracy and uncertainty of the estimate. The regression 

equations can be evaluated by the average variance of 

prediction and average standard error of prediction for 

the sites used to develop the equations, and by the pseudo 

coefficient of determination.
24 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

Variance of Prediction

For a site i, the variance of prediction, Vreg,i, can be 

thought of as a measure of the uncertainty of the regression 

model predictions. The Vreg,i is the sum of the model error 

variance and the sampling error variance and can be computed 

as:

Vreg i, , = + σ σ δ η i

2 2 , (3)

where

σδ

2

is the model error variance; and

ση,i

2

is the sampling mean square error for site i.

The variance of prediction is computed in WREG for 

each site used in the regression from the regression 

covariance matrix and the site-specific basin characteristics. 

For a regression equation, assuming that the explanatory 

variables for the streamgages used to develop the equation are 

representative of all sites in the region, the average accuracy 

of prediction for a regression equation can be determined 

by computing the average variance of prediction, AVP, for n

number of streamgages:

AVP = + 



 





=

σ σ δ η ∑ 2

1

1 2

n i

n

,i . (4)

Standard Error of Prediction

The standard error of prediction, SEP, is an alternate way 

to express the accuracy of the regression equations. The SEP 

is simply the square root of the Vreg, transformed from log units 

to percent. For a regression equation, the average standard 

error of prediction, SEPavg, as a percentage of the respective 

AEP flow, can be computed from the AVP in log units using 

the following transformation:

SEPavg

AVP = − e





 



 ( ) 100 1 10 2 1 2

ln

/

(5)

About two-thirds of the estimates obtained from a 

regression equation for ungaged sites will have errors less than 

the SEPavg for the equation (Helsel and Hirsch, 2002).

Pseudo Coefficient of Determination

A measure of the percentage of the variability in 

the dependent variable (AEP flow) explained by the 

explanatory variables in OLS regressions is the coefficient of 

determination, R2. For GLS regression, a more appropriate 

performance metric is the pseudo coefficient of determination, 

or Rpseudo

2 (Griffis and Stedinger, 2007). The Rpseudo

2 is 

a measure of the variability in the dependent variable 

explained by the regression after removing the effect of the 

time sampling error and is computed as:

R

k

pseudo

2

2

2 1

0 = − ( )

( )

σ

σ

δ

δ

(6)

where

σδ

2 ( ) k is the model error variance from a GLS 

regression with k explanatory variables; 

and

σδ

2 ( ) 0 is the model error variance from a GLS 

regression with no explanatory variables.

The average variance of prediction, average standard 

error of prediction, and Rpseudo

2 for the final regression 

equations are given in table 7. The average standard error of 

prediction ranged from 69 to 82 percent for equations for the 

various AEPs, and the Rpseudo

2 ranged from 85 to 91 percent. 

These study performance metrics indicate that the flood 

frequency regression equations explained a fair amount of 

the variation in the AEP flows. In similar studies in other 

States, regression equations for many regions had lower 

errors and higher Rpseudo

2 . However, other States have noted 

that considerably greater errors and lower Rpseudo

2 can exist in 

areas of extreme flow variability (Gotvald and others, 2012; 

Paretti and others, 2014). The relation between observed 

and predicted flood discharges for the 1-percent AEP is 

shown as an example in figure 8. A slight bias appears in the 

distribution of points at the upper end of the plot, where the 

regression equation tends to under-predict the 1-percent AEP 

for streamflows on the order of 100,000 ft3

/s or more. The 

uncertainty of the regression estimates can be seen graphically 

as a greater scatter of plotted observed to predicted points 

along the 1:1 line.
Estimating Flood Magnitude and Frequency at Ungaged Sites 25

tac16-5245_fig08

Predicted 1-percent annual exceedance probability flow, in cubic feet per second

10 100 1,000 10,000 100,000 1,000,000

Observed 1-percent annual exceedance probability flow, 

in cubic feet per second 

10

100

1,000

10,000

100,000

1,000,000

Figure 8. Relation between 1-percent annual exceedance probability discharges computed 

for observed streamflow and predicted from regression equations for streamgages in Alaska 

and conterminous basins in Canada.

Prediction Intervals

For a particular P-percent AEP flow at a streamgage, a 

site-specific uncertainty can be expressed as the confidence 

interval of a prediction, or prediction interval. The prediction 

interval is the range in values of an estimated dependent 

variable over which the true value of the dependent variable 

occurs with some stated probability. For example, for a 

90-percent prediction interval for an estimated flow value, 

the probability that the true flow value lies within that 

interval is 90 percent. Tasker and Driver (1988) determined 

that a 100 (1-α) prediction interval for a streamflow statistic 

estimated at an ungaged site from a regression equation can be 

computed as follows:

Q

C

< < Q QC (7)

where

Q is the P-percent AEP flow computed from the 

regional regression equation; and

C is computed as:

C t n p i = − 

  10 ( / α 2, )SEP , (8)

where

t

(α/2,n–p) is the critical value from the Student’s 

t-distribution at a particular alpha-level 

and degrees of freedom (n-p) where n

is the number of streamgages included 

in the regression analysis and p is the 

number of parameters in the equation, 

including the intercept coefficient, 

and is equal to 1.65 for an α of 0.10, 

corresponding to a 90-percent prediction 

interval, and degrees of freedom 

computed from 341 streamgages and 3 

regression parameters; and

SEPi is the standard error of prediction for site i

and is computed as:

 SEPi i

T

i

T = + ( ) 



 



 − −

σδ

2 1 1 0 5

X X Λ X X

.

 (9)

where

σδ

2 is the model error variance (computed 

using WREG and presented in table 8);

Xi is a row vector of the explanatory variables 

for site i, augmented by 1 as the first 

element;

X X TΛ− −

( ) 1 1

is the covariance matrix for the regression 

coefficients (computed using WREG 

and presented in table 8); and

Xi

T is the transpose of Xi

 (Ludwig and Tasker, 

1993).
26 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

An example computation is provided in section, 

“Application of Methods for Estimating Flood Magnitude 

and Frequency.” A variety of tools are available online and 

within common spreadsheet software for computation of 

matrix algebra.

Limitations

The regression equations are valid for use in unregulated 

basins in Alaska and conterminous basins in Canada except 

for the Aleutian Islands and other islands off the west coast 

of Alaska (fig. 1) and for locations along the Yukon River at 

and downstream of Carmacks, Yukon (Map ID 242, fig. 1). 

No estimates are available for the Aleutian Islands, where 

insufficient streamgages are available for analysis. Separate 

methods are available for estimating flood frequency statistics 

Table 8. Values used to determine prediction intervals for the regional flood frequency 

regression equations for Alaska and conterminous basins in Canada.

[σδ

2, the regression model error variance; X X T Λ− −

( ) 1 1

, the covariance matrix; Intercept, y-axis

intercept of regression equation; DRNAREA, drainage area, in square miles; PRECPRIS00, basin 

average mean annual precipitation, in inches, for 1971 to 2000 from the PRISM climate dataset]

P-percent annual 

exceedance 

probability

σδ

2 ( ) 1 T 1 − − X X Λ

Intercept DRNAREA PRECPRIS00

50 0.076 Intercept 7.26E-3 -7.56E-4 -3.42E-3

DRNAREA -7.56E-4 2.22E-4 2.12E-4

PRECPRIS00 -3.42E-3 2.12E-4 1.86E-3

20 0.073 Intercept 7.42E-3 -7.70E-4 -3.47E-3

DRNAREA -7.70E-4 2.20E-4 2.18E-4

PRECPRIS00 -3.47E-3 2.18E-4 1.88E-3

10 0.073 Intercept 7.82E-3 -8.09E-4 -3.64E-3

DRNAREA -8.09E-4 2.27E-4 2.30E-4

PRECPRIS00 -3.64E-3 2.30E-4 1.96E-3

4 0.076 Intercept 8.64E-3 -8.91E-4 -4.01E-3

DRNAREA -8.91E-4 2.45E-4 2.55E-4

PRECPRIS00 -4.01E-3 2.55E-4 2.15E-3

2 0.079 Intercept 9.29E-3 -9.56E-4 -4.30E-3

DRNAREA -9.56E-4 2.60E-4 2.74E-4

PRECPRIS00 -4.30E-3 2.74E-4 2.30E-3

1 0.082 Intercept 9.95E-3 -1.02E-3 -4.60E-3

DRNAREA -1.02E-3 2.76E-4 2.94E-4

PRECPRIS00 -4.60E-3 2.94E-4 2.46E-3

0.5 0.087 Intercept 1.08E-2 -1.11E-3 -4.98E-3

DRNAREA -1.11E-3 2.97E-4 3.19E-4

PRECPRIS00 -4.98E-3 3.19E-4 2.66E-3

0.2 0.095 Intercept 1.20E-2 -1.23E-3 -5.54E-3

DRNAREA -1.23E-3 3.29E-4 3.55E-4

PRECPRIS00 -5.54E-3 3.55E-4 2.96E-3

for sites on the Yukon River with particularly large drainage 

areas, considered to be those at and downstream of Carmacks, 

Yukon, where drainage area exceeds that of any other river in 

the study. Users should be cautioned that regression estimates 

are not exact and should be accompanied with estimates 

of uncertainty.

The regression equations are intended for use with 

basin characteristics obtained using the methods and datasets 

described in this report. Substituting basin characteristics 

obtained through alternate methods or datasets might have 

unpredictable results. Although the drainage area might vary 

only slightly depending on the topographic data available for a 

basin, the mean annual precipitation is likely to be sensitive to 

the method used to obtain it. In particular, users are cautioned 

to consider potential future changes in the processes that 

generate streamflow when exploring application of estimates 
Application of Methods for Estimating Flood Magnitude and Frequency 27

of future precipitation from climate models in these regression 

equations. Similarly, although site-specific data collection 

can improve the estimate of mean annual precipitation for 

a particular basin, the equations were developed from a 

regional-scale dataset that did not include such values. For 

example, regional precipitation datasets such as the PRISM 

model data used in this study have a known scarcity of data 

from high elevations, where few weather stations exist. Using 

a more accurate basin-specific mean annual precipitation 

developed from site-specific data in the regression will 

generate an estimate that has unknown error because the 

PRISM model did not consider such data.

The regression equations are intended for use for basins 

having values for drainage area and precipitation within the 

ranges shown in table 7. Applying the equations to sites in 

basins having values of explanatory variables outside the 

ranges of those used to develop the equations could result in 

prediction errors considerably greater than those indicated in 

table 7. The regression equations are not applicable for sites 

where peak-flow magnitudes are affected substantially by flow 

regulation, urbanization (as determined by impervious area), 

or glacier outbursts.

Application of Methods for Estimating 

Flood Magnitude and Frequency 

Within the limitations previously described, the equations 

and streamflow statistics in this report can be used to estimate 

the 50-, 20, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEP flows 

(see table 3 for corresponding recurrence intervals) for gaged 

and ungaged streams in the study area. The best estimates of 

flood frequency statistics for a site typically are obtained by 

properly weighting independent estimates produced by more 

than one method (Interagency Advisory Committee on Water 

Data, 1982). Estimates from an EMA with MGB analysis for a 

gaged site can be weighted with estimates from the regression 

equations in this report in inverse proportion to their variance 

for an improved, weighted estimate (Cohn and others, 2012). 

Estimates at ungaged sites near a gaged site on the same 

stream can be improved by weighting with estimates for the 

nearby gaged site.

Regression Estimate and Prediction Interval

The regression estimate of flood frequency statistics, 

QP%(reg), for a gaged or ungaged site can be computed from 

the equations in table 7 by inserting the values of the 

basin characteristics for the site. The following example 

demonstrates the use of the regression equations to obtain an 

estimate of the Q1%(reg)

 and the use of error equations to obtain 

the corresponding 90-percent prediction interval and standard 

error of prediction for USGS streamgage 15129500 Situk 

River near Yakutat, Alaska. 

1. Using GIS tools, obtain drainage area for the basin and 

determine mean annual basin precipitation from the 

Gibson (2009a) PRISM dataset (DRNAREA=35.5 mi2, 

and PRECPRIS00=170 in.);

2. If the computed basin characteristics are within the range 

of values listed in table 7, apply the regression equation 

in table 7 for estimating the flood frequency statistic 

(Q1%(reg)=11.4×35.50.732×1700.764 = 7,870 ft3/s);

3. Compute the standard error of prediction using matrix 

algebra to solve equation 9 where the model error σδ

2

is retrieved from table 8, the Xi vector is in the format 

Xi ={1, log10DRNAREA, log10PRECPRIS00}and the

 covariance matrix X X TΛ− −

( ) 1 1

 is retrieved from table 8,

 (σδ

2=0.082, X15129500 = {1, log10(35.5), log10(170)},

 SEP15129500 =(0.082 + 0.001202)0.5 =0.2884);

4. Compute C from equation 8 (C = 101.65×0.2884=2.992);

5. Compute the 90-percent prediction interval from 

equation 7 (Q/C)<Q1%(reg)<QC, or 2,630 ft3/s < Q1%(reg)

< 23,500 ft3/s, meaning that one can be 90 percent 

confident that the true value of the estimate for site 

15129500 lies between 2,630 and 23,500 ft3/s.

The regression estimate, prediction interval, and standard 

error of prediction also may be computed using the application 

tool provided as a Microsoft© Excel file at http://dx.doi.

org/10.3133/sir20165024. For an ungaged site, the regression 

estimate as described above will likely be the final estimate 

unless it is near a long-term gaged site (see section, “Estimate 

for an Ungaged Site near a Streamgage” for this procedure) 

and should be reported as an estimate with the associated 

prediction interval and standard error of prediction for a 

particular AEP. Estimates using the regression equations in 

table 7 may vary slightly from the estimates shown in table 4, 

which were computed using WREG, because of rounding 

differences in equation parameters. For a gaged site, the 

regression estimate as described above is weighted with a 

station estimate from the station annual peak data as described 

in the following section.

Weighted Estimate for a Gaged Site

Flood frequency estimates at streamgages can be 

improved by computing a weighted average of the streamgage 

estimate obtained by log-Person Type III analysis of peak 

flows, here referred to as the station estimate, and the 

estimate from the regression equation (Interagency Advisory 

Committee on Water Data, 1982). Optimal weighted flow 

estimates can be obtained if the variance for each of the two 

estimates is known or can be estimated accurately. If the two 

flow estimates can be assumed to be independent and are 

weighted in inverse proportion to their associated variances, 

the variance of the weighted estimate will be less than the 

variance of either of the independent estimates. 
28 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

For streamgages meeting eligibility requirements 

addressed in section, “Limitations,” the station and regression 

estimates can be computed using the methods described in 

this report, and then weighted using their respective variances 

using the following equation for a particular P-percent AEP:

log

log log Q V Q V Q

V V wtd

reg sta sta reg

sta reg

= +

+

(10)

where

Qwtd is the weighted estimate of the peak flow for a 

streamgage, in cubic feet per second;

Vreg is the variance of prediction for the regression 

estimate of peak flow for the streamgage, 

in log units (from table 9 for sites in the 

study or from the average variance of 

prediction for sites not in this study);

Qsta is the station estimate of peak flow for the 

streamgage, in cubic feet per second (from 

table 4 for sites in this study or from 

software such as PeakFQ for a site not in 

this study);

Vsta is the variance of the station estimate of peak 

flow, in log units (from table 9 for sites in 

the study or from software such as PeakFQ 

for a site not in this study); and

Qreg is the regression estimate of peak flow for the 

streamgage, in cubic feet per second (from 

the equations in table 7).

Weighted flow estimates computed for streamgages in the 

study are presented in table 4. The variance associated with 

the weighted estimate, Vwtd, is computed as:

V

V V

V V wtd

sta reg

sta reg

= + . (11)

The variance of the station, regression, and weighted estimates 

of the P-percent AEP flows are presented in table 9.

Table 9. Variance estimates for station, regression, and 

weighted estimates of flood frequency statistics for streamgages 

in Alaska and conterminous basins in Canada.

[Table 9 is a Microsoft© Excel file and can be downloaded at http://dx.doi.

org/10.3133/sir20165024]

An example of the application of the procedure for 

obtaining a weighted estimate of a streamflow statistic for 

a gaged site is the following computation of the weighted 

50-percent AEP flow for USGS streamgage 15297475, Red 

Cloud River tributary near Kodiak, Alaska. Slight differences 

between values listed below and in the respective tables are 

the result of rounding.

1. Obtain the station estimate of the 50-percent AEP flow 

at the streamgage using log-Pearson type III techniques 

and weighting the station skew with the appropriate 

regional skew if the site is located in a regional skew 

area (table 5, fig. 5). For streamgages in the study, the 

station estimate was computed in PeakFQ and is listed in 

table 4 (Q50%(sta)=386 ft3/s);

2. Obtain explanatory variables drainage area and basin 

average mean annual precipitation for 1971–2000 from 

the PRISM climate dataset. Table 1 contains values for 

these basin characteristics for sites in this study and 

table 2 provides links to data sources. (DRNAREA = 

1.7 mi2, PRECPRIS00 = 85 in.);

3. Compute the regression estimate by inserting the 

explanatory variables in the equation for the 50-percent 

AEP from table 7 (Q50%(reg) = 0.944 (1.70.836 × 851.023) = 

138 ft3/s);

4. Obtain the variance for the station estimate from output 

from software such as PeakFQ, or from table 9 for 

streamgages in this study (V50%(sta) = 0.001);

5. Obtain the variance of prediction for the regression 

estimate from table 9 for gages used to develop the 

regression equation or from the AVP for other gages 

(V50%(reg) = 0.077);

6. Compute the weighted 50-percent AEP flow for 

the streamgage using equation 10 (logQ50%(wtd) = 

(0.077(log(386)) + 0.001(log(138)))/(0.001+0.077) = 

2.5786 and Q50%(wtd) = 381 ft3/s);

7. Compute the weighted variance of prediction for 

the weighted flow estimate for the streamgage using 

equation 11 (V50%(wtd) = (0.001 × 0.077)/(0.001+0.077) = 

0.001).

Estimate for an Ungaged Site near a Streamgage

For ungaged sites near a gaged site on the same stream, 

an improved estimate can be obtained from the regression 

estimate for the ungaged site weighted with an estimate based 

on the weighted estimate for the gaged site and a drainage-area 

based multiplier (Sauer, 1974; Ries, 2007). The sites are 

considered near if the drainage area of the ungaged site is 

within 50—150 percent of the drainage area of a gaged site. 

To obtain a weighted flood frequency estimate for the ungaged 

site, Q(u)wtd, the weighted flow estimate for a particular AEP 

for an upstream or downstream gage, Q(g)wtd , must first be 

computed from the equation given in the previous section for 

gaged sites. The estimate for the gaged site is then scaled by 
Application of Methods for Estimating Flood Magnitude and Frequency 29

a drainage-area-ratio based multiplier to obtain a gage-based 

estimate for the ungaged site, Q(u)g, from the equation:

Q

A

A u g Q u

g

b

( ) g wtd

( )

( ) ( ) =

















, (12)

where

A(u) is the drainage area of the ungaged site;

A(g) is the drainage area of the gaged site; and

b is the exponent of the drainage area variable 

in the regional regression equation 

(DRNAREA, table 7).

The regression estimate for the ungaged site, Q(u)reg, is 

computed from the regional regression equations as described 

in a previous section and then weighted with the gage-based 

estimate for the ungaged site in the following equation:

Q A

A Q AA u wtd Q

g

u reg

g

( ) ( ) ( ) u g = ∆

+ −

 ∆















2 1 2 , (13)

where

ΔA is the absolute value of the difference between 

the drainage areas of the gaged and 

ungaged site, |Ag-Au|.

The weighting procedure gives full weight to the 

regression estimates when the drainage area for the gaging 

station is less than 0.5 or greater than 1.5 times the drainage 

area for the ungaged site and increasing weight to the 

gage-based estimates as the drainage area ratio approaches 1. 

This procedure for weighting a regression-based and 

gage-based estimate does not account for the length of the 

streamgage record. For gaged sites with short records having 

a large difference in drainage area from the ungaged site, the 

regional regression equations may produce a better estimate.

An example of the application of the procedure for 

obtaining an estimate of an AEP discharge for an ungaged 

site near a gaged site on the same stream is the following 

computation of the weighted 10-percent AEP flow for USGS 

streamgage 15291700 Susitna River above Tsusena Creek 

near Chulitna, Alaska, which had only 3 years of record at the 

time of this report and will be treated here as an ungaged site. 

USGS streamgage 15292000 Susitna River at Gold Creek, 

Alaska is a nearby streamgage that is included in this study 

and is used in this example to provide a weighted estimate 

between the gaged and ungaged estimates for the ungaged site.

1. Obtain the value of Q(g)wtd for the desired AEP (from 

table 4 for the 10-percent AEP, Q(g)wtd = 66,900 ft3/s);

2. Obtain the drainage areas and mean annual precipitation 

for the gaged and ungaged sites (from table 1, 

DRNAREAg = 6,130 mi2 and PRECPRIS00g = 25 in.; 

from GIS computations using a drainage area boundary 

and the mean annual precipitation dataset (Gibson, 

2009a), DRNAREAu = 5,160 mi2, PRECPRIS00u = 

24 in.);

3. Determine the ratio of the ungaged site drainage area 

to the gaged site drainage area and confirm that it lies 

between 0.5 and 1.5 (Au/Ag = 5,160/6,130 = 0.842, 

indicating the gaged site is near the ungaged site);

4. Compute Q(u)g using the results of steps 1 and 3 

and table 7 in equation 12 (for the 10-percent AEP, 

b = 0.775; Q(u)g = (0.842)0.775 × 66,900 = 58,552 ft3/s);

5. Compute Q(u)reg from the drainage area and mean 

annual precipitation for the ungaged site and the 

regression equation in table 7 for the desired AEP (for 

the 10-percent AEP, Q(u)reg = 4.01× DRNAREAu

0.775 × 

PRECPRIS00u

0.865 = 4.01 × 5,1600.775 × 240.865 = 

47,242 ft3/s);

6. Compute the absolute value of the difference in drainage 

areas (ΔA = 6,130 - 5,160 = 970 mi2); and

7. Compute the weighted estimate for the desired AEP for 

the ungaged site, Q(u)wtd, using equation 13, 

Q( ) u wtd = × + −  ×



 





=

2 970

6130

47 242 1 2 970

6130

58 552

0 316 47 24

, ,

. , 2 0 684 58 552

55 000

( ) + ( )

=

. ,

, . ft3/s

In this case, the estimate for the ungaged site obtained 

from the data for gaged site was weighted more heavily than 

the estimate obtained from the regression equation.

Estimate for a Site on the Yukon River

Gaged or ungaged sites on the Yukon River upstream 

of Carmacks, Yukon that meet the eligibility requirements 

addressed in section, “Limitations” are eligible for 

computation of the regression estimate of flood frequency 

statistics as in section, “Regression Estimate and Prediction 

Interval.” Gaged sites on the Yukon River upstream of 

Carmacks, Yukon also are eligible for computation of a 

weighted station and regression estimate. Downstream of 

Carmacks, Yukon, however, drainage areas for sites on 

the Yukon River exceed 31,100 mi2

, the maximum size 

eligible for application of the regression equations. For 

gaged sites exceeding this drainage area limit, station 

estimates can be computed using the methods described 

in this report but may not be weighted with regression 

estimates. For ungaged sites with drainage areas greater 

than 31,100 mi2, flood frequency statistics may be estimated 

by assuming a linear relation between the nearest upstream 
30 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

and downstream streamgages. The estimate can be obtained 

graphically from figure 7 or more accurately by linear 

interpolation from the logarithms of the drainage areas and flood 

frequency flows for the sites shown in figure 7. The estimate for 

an ungaged large Yukon River site, QP%lgYukon can be expressed in 

terms of the drainage area and QP% for the nearest upstream (up) 

and downstream (down) streamgages

log log

log log

lo

Q Q

DRNAREA DRNAREA

P lgYukon P up

site up

% % = ( )

+

( ) − ( )

g log

log log .

DRNAREA DRNAREA

Q Q

down up

P down P up

( ) − ( )

×( ) ( ) % % − ( ) (14)

An example of a linear interpolation to estimate the 

2-percent AEP flow for a hypothetical ungaged site near 

Fort Yukon, Alaska, with a drainage area of 177,000 mi2 is 

shown below.

1. Determine the nearest eligible upstream and downstream 

long-term USGS streamgages from figure 7 or from 

table 1 (15356000 is the nearest upstream gage, and 

15453500 is the nearest downstream gage);

2. Obtain the streamgage drainage areas from table 1

(DRNAREAup=111,600 mi2, DRNAREAdown = 

194,000 mi2) and the streamgage Q2% from table 4 (Q2%up= 

501,000 ft3/s, Q2%down= 760,000 ft3/s);

3. Estimate the Q2% by linear interpolation using equation 14 

(logQ2%=log(501,000)+ ((log(177,000) – log(111,600)) × 

(log(760,000)-log(501,000))/(log(194,000)-log(111,600)) 

= 5.8508 and Q2%=709,000 ft3/s).

Cook Inlet Basin Streamstats

StreamStats (http://water.usgs.gov/osw/streamstats/index.

html) is a web-based GIS application developed by the USGS 

and ESRI, Inc. that facilitates retrieval of streamflow statistics 

and associated information. StreamStats allows users to 

easily obtain selected streamflow-statistic estimates, upstream 

drainage-basin characteristics, and other information for a 

user-selected point on a stream. Using a GIS-based interactive 

map, the user can select a point on a stream and StreamStats 

will delineate the drainage area boundary upstream of the 

selected point. StreamStats will then use the boundary to obtain 

basin characteristics and solve streamflow‐statistics estimation 

equations. The user also can select USGS streamgages to obtain 

selected streamflow statistics. Ries and others (2008) provide 

a detailed description of the StreamStats application. Although 

designed to eventually be a national application, StreamStats is 

being implemented on a state-by-state basis, typically through 

cooperative funding agreements between the USGS and local 

partners. A StreamStats application for the Cook Inlet Basin 

of Alaska was implemented as the initial application for 

Alaska to test the functionality and utility of the relatively low 

resolution geospatial data available for Alaska at the time of 

this report for use in StreamStats.

StreamStats delineates drainage areas for user-selected 

sites from topographic and hydrographic data within limits 

enforced by pre-defined major watershed boundaries. Three 

GIS-data layers were processed to produce the Cook Inlet 

Basin StreamStats data layers: The 1:63,360-scale USGS 

National Hydrography Dataset (NHD) (http://nhd.usgs.gov/; 

Simley and Carswell, 2009) for Alaska, the 1:63,360-scale 

Watershed Boundary Dataset (WBD) (stored within the 

NHD geodatabase), and a digital elevation model (DEM) 

modified from the 60-m (197-ft) USGS National Elevation 

Dataset (NED) (http://ned.usgs.gov/; Gesch and others, 

2009). The NHD is a digital vector dataset representing the 

locations of streams, lakes, and other surface-water features 

organized into a streamflow network. The WBD is a digital 

nested hierarchy of hydrologic units defining the areal extent 

of surface water drainage to a point. Hydrologic units in the 

WBD are delineated such that a 2-digit hydrologic code, or 

HUC-2, represents a major geographic region in the United 

States, and is then subsequently broken down into smaller 

units (HUC-4s, 8s, 10s, and 12s), each defining a smaller area 

(U.S. Geological Survey and U.S. Department of Agriculture, 

Natural Resources Conservation Service, 2009). The Cook 

Inlet Basin StreamStats incorporated HUC-12s as interior 

divisions within the eleven HUC-8s in the Basin. StreamStats 

for Cook Inlet Basin used a DEM built from the 60-m NED, 

and then resampled to 30 m (98 ft), as part of the National 

Water Quality Assessment Program for the Cook Inlet Basin 

(Brabets and others, 1999).

Several preprocessing steps were needed for each of 

the three data layers to facilitate rapid determination of the 

drainage area to a selected point and subsequent computation 

of basin characteristics. Preprocessing of the NHD included 

removing flowline paths disconnected from the stream 

network and selection of the primary flow path in those areas 

where the NHD indicated split flow (such as might happen 

when flow diverges around an island in a river or with a 

braided channel). Preprocessing also verified that any stream 

from the NHD only crossed the HUC boundary at the inlet 

and outlet of the HUC and that HUC outlets aligned exactly 

to the confluences of the streams. Streamgage locations were 

adjusted, or snapped, from their published coordinates to 

a new location on the corresponding NHD flowline for the 

purposes of network functionality. Snapped locations were 

reviewed to ensure their placement on the correct stream 

and edited manually as required. A hydro-corrected DEM 

was developed by filling depressions or sinks using the basin 

boundaries from the WBD to conserve known drainage 

divides and using the streams from the preprocessed NHD to 

create well-defined flow paths through the elevation data.
Summary and Conclusions 31

ArcHydro Tools, version 1.3, a set of utilities 

developed to operate in the ArcGIS v. 9.3.1 environment 

(Environmental Systems Research Institute, 2015) was 

used to process StreamStats data layers for Cook Inlet. As 

the project progressed, the migration to ArcGIS version 

10.2.2 and ArcHydro Tools version 2.0 was adopted as a 

programmatically sound transition that would not adversely 

affect work already accomplished. 

Basin characteristics for the purpose of developing 

flood frequency regression equations were obtained external 

to StreamStats. Mean annual precipitation from the PRISM 

precipitation dataset, developed by the PRISM Climate Group 

and published for Alaska by Gibson (2009a), was selected as a 

variable in flood frequency regression equations in this study 

and was built into StreamStats as the basis for obtaining basin 

characteristics. The regression equations for estimating flood 

frequency statistics published in the report will be available 

online in Cook Inlet Basin StreamStats immediately following 

publication of this report. Estimates of flood frequency 

statistics obtained using StreamStats to determine basin 

characteristics should be very close to estimates obtained by 

GIS methods external to StreamStats.

The NHD, WBD, and NED are national datasets that 

are subject to change as improvements are made. Edits 

to the NHD or WBD required for this study were derived 

from interpretations of various sources such as topographic 

maps, aerial photography, and satellite imagery. Edits to the 

WBD were reconciled with the Alaska WBD steward prior 

to implementation of StreamStats. Edits to the NHD were 

tracked for submission to the Alaska NHD steward. Any edits 

to the NHD or WBD subsequent to July 2011 or December 

2012, respectively, will not be reflected in StreamStats for the 

Cook Inlet Basin.

Summary and Conclusions

This report, prepared by the U.S. Geological Survey in 

cooperation with the Alaska Department of Transportation and 

Public Facilities, the Alaska Department of Natural Resources, 

and the U.S. Army Corps of Engineers, updates methods for 

determining flood magnitude and frequency at streamgages 

and ungaged sites in Alaska and conterminous basins in 

Canada. An annual series of peak-flow data through water 

year 2012 was compiled for each of 387 streamgages that 

were not substantially affected by regulation or urbanization 

and that had at least 10 years of record. Flood-frequency 

estimates were computed for each streamgage using the 

Expected Moments Algorithm with a multiple Grubbs-Beck 

test for detecting multiple potentially influential low floods 

to fit a log-Pearson type III distribution to the logarithms 

of the annual peak flows. Evaluation of historical and 

censored flood information facilitated incorporation of 

these types of non‐standard streamgage data into the flood 

frequency analysis where appropriate. A Bayesian leastsquares regression regional skew analysis using station skew 

coefficients for streamgages having at least 25 years of record 

produced a new constant model for regional skew in two new 

regional skew areas covering parts of Alaska. For streamgages 

in the regional skew areas, the station skew coefficients were 

weighted with the new regional skew for computation of final 

station estimates of the magnitude of the 50-, 20-, 10-, 4-, 

2-, 1-, 0.5-, and 0.2-percent annual exceedance probability 

flows. For streamgages in locations outside the regional 

skew areas, considered regional skew exclusion areas, station 

skew coefficients were used for computation of final station 

estimates of these annual exceedance probability flows.

Basin characteristics consisting of physical and 

hydrologic properties of the streamgage basins were obtained 

using GIS methods and used as explanatory variables in an 

exploratory all-subsets ordinary least-squares regression. 

Regression analysis used 341 streamgages that did not 

have an ineligible relation to basin characteristics, such as 

glacier outbursts or indeterminate basin, were not considered 

redundant with a nested basin, and were not a site on the 

Yukon River with a drainage area greater than 31,100 mi2

. 

A separate method for streamgages on the Yukon River with 

very large drainage areas was defined. Regression analysis for 

the eligible streamgages included testing various combinations 

of streamflow analysis regions and basin characteristics to 

determine the simplest model of statistically significant and 

hydrologically reasonable variables that best explained the 

variation in the flood frequency flows. Drainage area and 

mean annual precipitation together in a single study-wide set 

of equations performed as well as the best sets of variables 

within previously published streamflow analysis regions, 

collectively, and eliminated concerns associated with the 

small number of streamgages within a region. Final regional 

regression equations were developed using generalized leastsquares techniques to account for cross correlation between 

streamgage locations and concurrent records. Average standard 

errors of prediction for the regression equations for the various 

annual exceedance probabilities ranged from 69 to 82 percent, 

and the pseudo coefficient of determination ranged from 85 to 

91 percent. 

Techniques for applying the regional regression equations 

for gaged and ungaged sites and for weighting the station 

and regression estimates for gaged sites are presented along 

with final estimates for streamgages used in the study. 

Simultaneously with this study, a StreamStats pilot application 

for the state of Alaska was developed for the Cook Inlet Basin 

that incorporates the new regional regression equations and 

facilitates access to published flood frequency and basin 

characteristic statistics and estimates for ungaged sites. 

StreamStats is a web-based application that provides published 

data for streamgages and delineates basins, computes basin 

characteristics, and obtains flood frequency estimates for 

ungaged sites.
32 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

Acknowledgments

Data and assistance provided by Environment Canada, 

particularly Lynne Campo (Water Survey of Canada) and 

Judy Kwan (Meteorological Service of Canada); by the 

National Weather Service Alaska-Pacific River Forecast 

Center; and by streamside residents and other river observers 

is gratefully acknowledged. This study benefitted from 

discussions of concepts and methods of flood frequency 

analysis with Timothy Cohn and Julie Kiang of the USGS 

Office of Surface Water and discussions of preliminary results 

and interpretations with many Alaska hydrologists. Sheryl 

Boyack of the USGS coordinated the WBD edits necessary for 

preparation of drainage area boundaries, and Donna Knifong 

of the USGS provided a computational tool for obtaining basin 

characteristics. The authors also thank Katharine Kolb, Alan 

Rea, Peter Steeves, and Lovina Turney of the USGS for their 

assistance with StreamStats.

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36 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

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Appendix A 37

Appendix A. Basin Characteristics for Selected Streams in Alaska and

Conterminous Basins in Canada

Appendix A is a Microsoft© Excel file and can be downloaded at http://dx.doi.org/10.3133/sir20165024.
38 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

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Appendix B 39

Appendix B. Regional Skewness Regression Analysis

By Andrea G. Veilleux

Introduction to Statistical Analysis of 

Regional Skew

For the log-transformation of annual peak discharges, 

Bulletin 17B (Interagency Advisory Committee on Water 

Data, 1982) recommends using a weighted average of the 

station skew coefficient and a regional skew coefficient to help 

improve estimates of annual exceedance probability (AEP) 

discharges (eq. 2). Bulletin 17B supplies a national map but 

also encourages hydrologists to develop more specific local 

relations. Since the first map was published in 1976, some 

40 years of additional information has accumulated and better 

spatial estimation procedures have been developed (Stedinger 

and Griffis, 2008). 

Tasker and Stedinger (1986) developed a weighted 

least-squares (WLS) procedure for estimating regional 

skew coefficients based on sample skew coefficients for the 

logarithms of annual peak-discharge data. Their method 

of regional analysis of skew estimators accounts for the 

precision of the skew-coefficient estimate for each streamgage 

or station, which depends on the length of record for each 

streamgage and the accuracy of an ordinary least-squares 

(OLS) regional mean skew. More recently, Reis and others 

(2005), Gruber and others (2007), and Gruber and Stedinger 

(2008) developed a Bayesian generalized least-squares (GLS) 

regression model for regional skew analyses. The Bayesian 

methodology allows for the computation of a posterior 

distribution of both the regression parameters and the model 

error variance. As shown in Reis and others (2005), for cases 

in which the model error variance is small compared to the 

sampling error of the station estimates, the Bayesian posterior 

distribution provides a more reasonable description of the 

model error variance than both the GLS method-of-moments 

and maximum likelihood point estimates (Veilleux, 2011). 

Although WLS regression accounts for the precision of the 

regional model and the effect of the record length on the 

variance of skew-coefficient estimators, GLS regression also 

considers the cross-correlations among the skew-coefficient 

estimators. In some studies, the cross-correlations have had a 

large impact on the precision attributed to different parameter 

estimates (Feaster and others, 2009; Gotvald and others, 

2009;Weaver and others, 2009; Parrett and others, 2011).

Due to complications introduced by the use of 

the expected moments algorithm (EMA) with multiple 

Grubbs‐Beck censoring of low outliers (Cohn and others, 

1997) and large cross-correlations between annual peak 

discharges at pairs of streamgages, an alternate regression 

procedure was developed to provide both stable and defensible 

results for regional skew (Veilleux, 2011; Lamontange and 

others, 2012; Veilleux and others, 2012). This alternate 

procedure is referred to as the Bayesian WLS/Bayesian GLS 

(B-WLS/B-GLS) regression framework (Veilleux, 2011; 

Veilleux and others, 2011, 2012). It uses an OLS analysis to fit 

an initial regional skew model; that OLS model is then used to 

generate a stable regional skew-coefficient estimate for each 

site. That stable regional estimate is the basis for computing 

the variance of each station skew-coefficient estimator used 

in the WLS analysis. Then, B-WLS is used to generate 

estimators of the regional skew-coefficient model parameters. 

Finally, B-GLS is used to estimate the precision of those WLS 

parameter estimators, to estimate the model error variance 

and the precision of that variance estimator, and to compute 

various diagnostic statistics.

The Alaska regional skew study described here used the 

Expected Moments Algorithm to estimate the station skew and 

its mean square error. Because EMA allows for the censoring 

of potentially influential low floods (PILFs), as well as the 

use of estimated interval discharges for missing, censored, 

and historic data, it complicates the calculations of effective 

record length (and effective concurrent record length) used to 

describe the precision of sample estimators because the peak 

discharges are no longer solely represented by single values. 

To properly account for these complications, the new B-WLS/

B-GLS procedure was used. 
40 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

Methodology for Regional Skew Model

This section provides a brief description of the B-WLS/

B-GLS methodology (as it appears in Veilleux and others, 

2012). Veilleux (2011) and Veilleux and others (2011) provide 

a more detailed description.

Ordinary Least-Squares Analysis

The first step in the B-WLS/B-GLS regional skew 

analysis is the estimation of a regional skew model using OLS. 

The OLS regional regression yields parameters β

OLS and a 

model that can be used to generate unbiased and relatively 

stable regional estimates of the skew for all streamgages:

  y X OLS OLS = β (B1)

where 

X is an (n × k) matrix of basin characteristics;

y OLS are the estimated regional skew values;

n is the number of streamgages; and

k is the number of basin parameters including a 

column of ones to estimate the constant. 

These estimated regional skew values y OLS are then used 

to calculate unbiased station-regional skew variances using 

the equations reported in Griffis and Stedinger (2009). These 

station-regional skew variances are based on the regional OLS 

estimator of the skew coefficient instead of the station skew 

estimator, thus making the weights in the subsequent steps 

relatively independent of the station skew estimates.

Weighted Least-Squares Analysis

A B-WLS analysis is used to develop estimators of the 

regression coefficients for each regional skew model (Veilleux, 

2011; Veilleux and others, 2011). The WLS analysis explicitly 

reflects variations in record length, but intentionally neglects 

cross correlations, thereby avoiding the problems experienced 

with GLS parameter estimators (Veilleux, 2011; Veilleux and 

others, 2011). 

Generalized Least-Squares Analysis

After the regression model coefficients,β

WLS , are 

determined with a WLS analysis, the precision of the fitted 

model and the precision of the regression coefficients are 

estimated using a B-GLS analysis (Veilleux, 2011; Veilleux 

and others, 2011). Precision metrics include the standard 

error of the regression parameters, SE WLS β

( ), the model 

error variance, σδ,B G− LS

2 , pseudo coefficient of determination, 

pseudo-Rδ

2

, and the average variance of prediction at a new 

streamgage that is not used in the regional model, AVPnew. 

Data Analysis

This regional skew study is based on annual peakdischarge data from 103 streamgages in Alaska (table 1). 

Streamgages included in the skew study were screened 

as described in the main section of the report to omit 

streamgages on the same river close enough to be considered 

hydrologically redundant. The annual peak-discharge data 

were downloaded from the USGS National Water Information 

System (NWIS) database. In addition to the peak-discharge 

data, 11 basin characteristics for each of the 103 sites were 

available as explanatory variables in the regional study. The 

basin characteristics available included hydrologic regions 

(Curran and others, 2003), as well as the more standard 

morphometric parameters (basin centroid latitude and 

longitude, drainage area, and mean basin elevation), climate 

variables (mean annual precipitation and mean minimum 

January temperature), and land cover variables (percentage 

of basin forested, percentage of basin covered by lakes and 

ponds, percentage of basin covered by glaciers, and percentage 

of basin covered by permafrost).

Due to the sparsity of streamgages in parts of Alaska and 

conterminous basins in Canada, a regional skew analysis could 

not be performed for the entire study area. Instead, two regions 

of Alaska were identified which each contained a sufficient 

density of gages from which to build regional skew models. 

These regions are shown in figure 5 of the report. Regional 

Skew Area 1 (RSA1) (75 streamgages) covers 108,000 mi2

and encompasses a swath along the road corridors near the 

more populated areas of the interior part of the State. Regional 

Skew Area 2 (RSA 2) (28 streamgages) covers 72,800 mi2

and encompasses the coastal areas of the State bordering the 

Gulf of Alaska. The boundaries of these regions are generally 

defined by HUC 8 borders, split where necessary to omit nonapplicable areas. 

RSA1 is situated in the Interior of Alaska, which 

experiences a continental climate having warmer summers, 

colder winters, and less precipitation than RSA2. The 

median value for mean annual precipitation for the study 

basins in RSA1 is 25 in., and winter precipitation generally 

falls as snow. Mean basin elevations generally are higher 

than elevations in coastal basins. Annual peak flow can be 

generated by snowmelt or rainfall, or by glacier-related melt in 

glacierized basins.

The temperate, moist climate of RSA2 reflects the 

maritime influence of the Gulf of Alaska. Mean annual 

precipitation is much higher than for RSA1; the median value 

of the mean annual precipitation for study basins is 145 in. 

for RSA2. Flooding is more commonly generated by rainfall, 

particularly in autumn and winter. Most of the gaged basins in 

RSA2 are small relative to the median study basin size, which 

in part mirrors the physiography of short drainages extending 

from the mountain ranges surrounding the Gulf of Alaska to 

the coast. 
Appendix B 41

The statistical analysis of the data requires several steps. 

This section describes the calculations for pseudo record 

length for each site given the number of censored observations 

and concurrent record lengths, as well as the development 

of the model of cross-correlations of concurrent annual peak 

discharges.

Station Skew Estimators

To estimate the station logarithm base10 (log) skew 

coefficient, G, and its mean square error, MSEG, the skew 

study used the results of the EMA analysis described in 

section, “Flood Magnitude and Frequency at Gaged Sites” 

(Cohn and others, 1997; Griffis and others, 2004). EMA 

provides a straightforward and efficient method for the 

incorporation of historical information and censored data, 

such as those from a CSG, contained in the record of annual 

peak discharges for a streamgage. For this analysis, PeakFQ 

(Veilleux and others, 2014), which combines EMA with the 

multiple Grubbs-Beck test for PILFs, was used. Use of the 

most current PeakFQ version available at the time resulted in 

a small subset of study sites being affected by computational 

discrepancies in version 7.0 before version 7.1, which 

corrected processing of selected conditions, became available. 

Documentation for PeakFQ is available at http://water.usgs.

gov/software/PeakFQ/. PeakFQ was used to generate the 

station log estimates of G and the corresponding MSEG, 

assuming a log-Pearson Type III distribution and generally 

using a multiple Grubbs-Beck test for PILF screening. EMA 

estimates, based on annual peak-discharge data through 

September 30, 2012, of G and MSEG are listed in table 4 of 

the report for the 103 streamgages evaluated for the Alaska 

regional skew study.

Pseudo Record Length

Because the dataset includes censored data and historical 

information, the effective record length used to compute 

the precision of the skew estimators is no longer simply the 

number of annual peak discharges at a streamgage. Instead, a 

more complex calculation was used to take into account the 

availability of historical information and censored values. 

Although historical information and censored peaks provide 

valuable information, they often provide less information 

than an equal number of years with systematically recorded 

peaks (Stedinger and Cohn, 1986). The following calculations 

provide a pseudo record length, PRL, associated with skew, 

which appropriately accounts for all peak-discharge data types 

available for a site. 

The PRL is defined in terms of the number of years of 

systematic record that would be required to yield the same 

mean square error of the skew (MSE G ( ))as the combination 

of historical and systematic record actually available at a 

streamgage. Thus, the PRL of the skew is a ratio of the MSE 

of the at-site skew when only the systematic record is analyzed 

(MSE(GS )  ) versus the MSE of the at-site skew when all data, 

including historical and censored data, are analyzed 

(MSE(GC )  ). 

P

P G

G

RL

s S

C

= ( )

( )

*MSE

MSE



 (B2)

where 

PRL is the pseudo record length for the entire 

record at the streamgage; 

Ps is the number of systematic peaks in the 

record;

MSE GS 

( ) is the estimated MSE of the skew when only 

the systematic record is analyzed; and

MSE GC 

( ) is the estimated MSE of the skew when all 

data, including historical and censored 

data, is analyzed. 

As the PRL is an estimate, the following conditions also 

must be met to ensure a valid approximation. PRL must be 

non-negative. If PRL is greater than PH (length of the historical 

period), then PRL should be set to equal PH. Also, if PRL is less 

than PS, then PRL is set to PS. This ensures that the pseudo 

record length will not be larger than the complete historical 

period or less than the number of systematic peaks.

For the Alaska skew, only sites with PRL greater than or 

equal to 25 years of record were used in the analysis.

Unbiasing the Station Estimators

The station skew estimates were unbiased by using the 

correction factor developed by Tasker and Stedinger (1986) 

and employed in Reis and others (2005). The unbiased station 

skew estimator using the pseudo record length is

γ

i

RL i

i P

= + G

















1 6

,

(B3)

where

γ

i is the unbiased station sample skew estimate 

for site i,

PRL,i is the pseudo record length for site i as 

calculated in equation B2; and 

Gi is the traditional biased station skew estimator 

for site i from EMA.
42 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

The variance of the unbiased station skew includes the 

correction factor developed by Tasker and Stedinger (1986):

Var P i Var G

RL i

i γ

  = +

















1 [ ] 6

2

,

(B4)

where 

Var[Gi] is calculated using (Griffis and Stedinger, 

2009).

Var G

P

a P

b P G c P

RL

RL

RL RL





( ) = + ( ) 



 





+ + ( ) 



 



 + + ( )

6

1 9

6

15

48

2

* 



 



 



 



G  4

 (B5)

where

a P

P P RL

RL RL

( ) = − +

17 75 50 06

2 3

. .

;

b P

P P P RL

RL RL RL

( ) = − +

3 92 31 10 34 86

03 06 09

. . .

. . .

; and

c P

P P P RL

RL RL RL

( ) = − + −

7 31 45 90 86 50

0 59 1 18 1 77

. . . . . . .

Estimating the Mean Square Error of the 

Skew Estimator

There are several possible ways to estimate MSEG. 

The approach used by EMA (Cohn and others, 2001, eq.55) 

generates a first-order estimate of the MSEG, which should 

perform well when interval data are present. Another option is 

to use the Griffis and Stedinger (2009) formula in equation B5 

(variance is equated to the MSE), using either the systematic 

record length or the length of the whole historical period. 

However, this method does not account for censored data, and 

thus can lead to inaccurate and underestimated MSEG. This 

issue has been addressed by using the pseudo record length 

instead of the length of the historical period; the pseudo record 

length reflects the impact of the censored data and the number 

of recorded systematic peaks. Thus, the unbiased Griffis 

and Stedinger (2009) MSEG was used in the regional skew 

model because it is more stable and relatively independent 

of the station skew estimator. This methodology was used 

in previous regional skew studies (Eash and others, 2013; 

Southard and Veilleux, 2014).

Cross-Correlation Models

A critical step for a GLS analysis is estimation of the 

cross-correlation of the skew coefficient estimators. Martins 

and Stedinger (2002) used Monte Carlo experiments to derive 

a relation between the cross-correlation of the skew estimators 

at two stations i and j as a function of the cross-correlation of 

concurrent annual maximum flows, ρij: 

ρ γ , γ ρ ρ   i j ij ij ij

k

( ) = Sign( ) cf (B6)

where

ρ

ij is the cross-correlation of concurrent annual 

peak discharge for two streamgages;

k is a constant between 2.8 and 3.3; and 

cfij is a factor that accounts for the sample size 

difference between stations and their 

concurrent record length, is defined as 

follows:

cfij ij = CY ( ) P P RL i R ( ) L j / , , (B7)

where 

CYij is the pseudo record length of the period of 

concurrent record; and,

PRL,i and PRL,j are the pseudo record length corresponding to 

sites i and j, respectively (see equation B2). 

Pseudo Concurrent Record Length

After calculating the PRL for each streamgage in the study, 

the pseudo concurrent record length between pairs of sites 

can be calculated. Due to the use of censored data and historic 

data, the effective concurrent record length calculation is more 

complex than determining in which years the two streamgages 

both have recorded systematic peaks. 

The years of historical record in common between the 

two streamgages is first determined. For the years in common, 

with beginning year YBij and ending year YEij, the following 

equation is used to calculate the concurrent years of record 

between site i and site j.

CY YE YB

P

P

P

P ij ij ij

RL i

H i

RL j

H j

= − ( ) +































 1 ,

,

,

,

. (B8)

The computed pseudo concurrent record length depends 

on the years of historical record in common between the two 

streamgages, as well as the ratios of the pseudo record length 

to the historical record length for each of the two streamgages.

Alaska Study Area Cross-Correlation Models of 

Concurrent Annual Peak Discharge

Cross-correlation models for the log annual peak 

discharges were developed for both RSA1 and RSA2 in 

Alaska. The cross-correlation model for RSA1 was developed 

using 27 sites with at least 35 years of concurrent systematic 

peaks (zero flows not included). Similarly, the crosscorrelation model for RSA2 was developed using 28 sites with 

at least 20 years of concurrent systematic peaks (zero flows 
Appendix B 43

not included). Various models relating the cross-correlation 

of the concurrent annual peak discharge at two sites, ρij, 

to various basin characteristics were considered. A logit 

model, termed the Fisher Z Transformation (Z = log[(1+r)/

(1–r)] ), provided a convenient transformation of the sample 

correlations rij from the (–1, +1) range to the (-∞ +∞) range. 

The adopted models for estimating the cross-correlations 

of concurrent annual peak discharge at two stations, which 

used the distance between basin centroids, Dij, as the only 

explanatory variable, are

ρij

ij

ij

Z

Z = ( ) −

( ) +

exp

exp

2 1

2 1

(B9)

where

Z(RSA1)ij = exp (0.48 – 0.0082 * Dij), and 

Z(RSA2)ij = exp (0.39 – 0.0078 * Dij). 

For RSA1, an OLS regression analysis based on 

252 station-pairs indicated that this model is as accurate 

as having 67 years of concurrent annual peaks from which 

to calculate cross-correlation. For RSA2, an OLS regression 

analysis based on 157 station-pairs indicated that this model 

is as accurate as having 900 years of concurrent annual peaks 

from which to calculate cross-correlation. The fitted relation 

between Z and distance between basin centroids together 

with the plotted sample data for RSA1 and RSA2 is shown in 

figure B1. The functional relation between the untransformed 

cross correlation and distance between basin centroids 

together with the plotted sample data for RSA1 and RSA2 is 

shown in figure B2. The cross correlation models were used 

to estimate site-to-site cross correlations for concurrent annual 

peak discharges at all pairs of sites in RSA1 and RSA2.

0 100 200 300 400 500 600

−0.5

−0.25

0

0.25

0.5

0.75

1

1.25

1.5

Fisher Z cross−correlation of concurrent 

annual−maximum flows with greater than or 

equal to 35 years of concurrent records

Z = exp(−0.48 − 0.0082*D)

Site pairs

0 100 200 300 400 500 600 700 800 900

−0.5

−0.25

0

0.25

0.5

0.75

1

1.25

1.5

Distance between streamgage centroids, in miles

Fisher Z cross−correlation of concurrent 

annual−maximum flows with greater than or 

equal to 20 years of concurrent records

A. Regional Skew Area 1

B. Regional Skew Area 2

EXPLANATION

Z = exp(−0.39 − 0.0078*D)

Site pairs

EXPLANATION

Figure B1. Relation between Fisher Z transformed cross-correlation of logs of annual peak 

discharge and distance between basin centroids for (A) Regional Skew Area 1 and (B) Regional Skew 

Area 2, Alaska and conterminous basins in Canada.
44 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

Alaska Regional Skew Study Results

The results of the Alaska regional skew study using the 

B-WLS/B-GLS regression methodology are provided in the 

following sections for RSA1 and RSA2. 

Regional Skew Area 1

All available basin characteristics were initially 

considered as explanatory variables in the regression analysis 

for regional skew in region RSA1. The basin characteristics 

that were statistically significant in explaining the streamgageto-streamgage variability in skew were the mean annual 

precipitation (PRECPRIS00), drainage area (DRNAREA), 

percent glacier, and mean minimum January temperature.

The best regional skew model is classified as having the 

smallest model error variance, σδ

2

, and largest pseudo Rδ

2

 while 

minimizing the number of variables, or model complexity. The 

results for the two regional skew models that best met these 

criteria—the constant skew model denoted “CONSTANT” 

and the mean annual precipitation and drainage area model 

“PRECPRIS00+DRNAREA” are shown in table B1.

The pseudo Rδ

2

 describes the estimated fraction 

of the variability in the true skew from streamgage-tostreamgage explained by each model (Gruber and others, 

2007; Parrett and others, 2011). A constant model does 

not explain any variability, so the pseudo Rδ

2

 equals 0 for 

that model. The addition of basin characteristics in the 

PRECPRIS00+DRNAREA model produced a pseudo Rδ

2

equal to 23 percent. This indicates that the inclusion of 

the PRECPRIS00 and DRNAREA basin characteristics as 

explanatory variables in the regression only help to explain 

23 percent of the streamgage-to-streamgage variability in the 

true skew. The addition of the PRECPRIS00 and DRNAREA 

basin characteristics to the RSA1 regional skew model was 

not warranted as it resulted in a small improvement in model 

precision while increasing model complexity. 

tac16-5245_figA2

EXPLANATION

EXPLANATION

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cross-correlation of concurrent annual-maximum 

flows with greater than or equal to 

35 years of concurrent records

0 100 200 300 400 500 600 700 800 900

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance between streamgage centroids, in miles

Cross-correlation of concurrent annual-maximum 

flows with greater than or equal to 

20 years of concurrent records

0 100 200 300 400 500 600

Site pairs

r=(exp(2Z)−1)/(exp(2Z)+1)

Site pairs

r=(exp(2Z)−1)/(exp(2Z)+1)

B. Regional Skew Area 2

A. Regional Skew Area 

Figure B2. Relation between untransformed cross-correlation of logs of annual peak discharge and 

distance between basin centroids for (A) Regional Skew Area 1 and (B) Regional Skew Area 2, Alaska 

and conterminous basins in Canada.
Appendix B 45

Table B1. Regional skewness models for Alaska Regional Skew Area 1 study area.

[Standard deviations are in parentheses. Bayesian plausibility (in percent) are in square brackets. σδ

2, model error variance; ASEV, average sampling error

variance; AVPnew, average variance of prediction for a new site; pseudo R- δ

2, fraction of the variability in the true skews explained by each model (Gruber and 

others, 2007)]

Model Regression parameters

σδ

2 ASEV AVPnew

Pseudo-Rδ

2

β (percent) 1 β2 β3

CONSTANT:

1 γ = β

0.54 0.44 0.013 0.45 0

(0.11) (0.11)

PRECPRIS00+DRNAREA:

γ = β +β 1 2 10 log (PRECPRIS00 log )  + β3 10 ( ) DRNAREA      

-1.8 1.4 0.22 0.33 0.031 0.36 23

(0.68) (0.49) (0.09) (0.09)

[0] [2]

Table B2. Regional skewness models for Alaska Regional Skew Area 2 study area.

[Standard deviations are in parentheses. σδ

2

, model error variance; ASEV, average sampling error variance; AVPnew,

average variance of prediction for a new site; pseudo R- δ

2 , fraction of the variability in the true skews explained by 

each model (Gruber and others, 2007)]

Model

Regression 

parameter σδ

2 ASEV AVPnew

Pseudo-Rδ

2

(percent) β1

CONSTANT:

1 γ = β

0.18 0.10 0.014 0.12 0

(0.12) (0.07)

Thus, the CONSTANT model is chosen as the best 

regional skew model for the Alaska RSA1 study area. The 

posterior mean of the model error variance, σδ

2

, for the 

CONSTANT model is σδ

2

 = 0.44. The average sampling 

error variance (ASEV) in table B1 is the average error 

in the regional skew estimator at the sites in the dataset. 

The average variance of prediction at a new site (AVPnew) 

corresponds to the mean square error (MSE) used in 

Bulletin 17B (Interagency Advisory Committee on Water 

Data, 1982) to describe the precision of the generalized skew. 

The CONSTANT model has an AVPnew, equal to 0.45, which 

corresponds to an effective record length of 22 years.

Drainage areas for the 75 long-term streamgages that 

were used to develop the CONSTANT regional skew model 

for RSA1 ranged from 1.2 to 25,560 mi2. The CONSTANT 

regional skew model is only applicable if the basin drainage 

area in RSA1 is greater than or equal to 1.2 mi2 and less than 

or equal to 25,560 mi2.

Regional Skew Area 2

All available basin characteristics were initially considered 

as explanatory variables in the regression analysis for regional 

skew in region RSA2. None of the basin characteristics 

were statistically significant in explaining the streamgageto-streamgage variability in skew. Thus, the best model, as 

classified by having the smallest model error variance, σδ

2 , and 

largest pseudo Rδ

2

, is the constant model. The final results for 

the constant skew model denoted “CONSTANT” are shown in 

table B2. 

The CONSTANT model is chosen as the best regional 

skew model for the Alaska RSA2 study area as none of the 

available basin characteristics were statistically significant 

in a model. The posterior mean of the model error variance, 

σδ

2

, for the CONSTANT model is σδ

2

= 0.10. The CONSTANT 

model has an AVPnew, equal to 0.12, which corresponds to an 

effective record length of 59 years.

As the 28 long-term streamgages used to develop the 

CONSTANT regional skew model for RSA2 had drainage areas 

ranging from 1.7 to 123 mi2

, the CONSTANT regional skew 

model is only applicable if the basin drainage area in RSA2 is 

greater than or equal to 1.7 mi2 and less than or equal to 123 mi2.
46 Estimating Flood Magnitude and Frequency on Streams in Alaska and Conterminous Basins in Canada

Table B3. Pseudo ANOVA table for the Alaska Regional Skew Area 1 CONSTANT model regional skew and the Alaska Regional Skew 

Area 2 CONSTANT model regional skew.

[EVR, error variance ratio; MBV*, misrepresentation of the beta variance, pseudo R- δ

2 , fraction of variability in the true skews explained by the model]

Source

Degrees of freedom Sum of squares

Equations RSA1 CONSTANT 

model

RSA2 CONSTANT 

model Equations RSA1 CONSTANT model

RSA2 CONSTANT 

model

Model k 0 0 n k σ σ δ δ

2 2 ( ) 0 − ( ) 

 

0 0

Model error n k − −1 74 27 n k σδ

2 ( ) 

 

33 3

Sampling error n 75 28 Var i

n γ

( ) ∑ =1

22 6

Total 2 1 n − 149 55 n k Var i

n σδ

2

1 ( ) 

  ( ) ∑ = γ 55 9

Model diagnostics

 EVR 0.7 2.3

 MBV* 1.3 1.2

Pseudo-Rδ

2

 (percent)

0 0

Bayesian Weighted Least-Squares/Bayesian 

Generalized Least-Squares Regression 

Diagnostics

To determine if a model is a good representation of 

the data and which regression parameters, if any, should be 

included in a regression model, diagnostic statistics have 

been developed to evaluate how well a model fits a regional 

hydrologic data set (Griffis, 2006; Gruber and others, 2008). 

In this study, the goal was to determine the set of possible 

explanatory variables that best fit annual peak discharges for 

the Alaska skew study areas RSA1 and RSA2 affording the 

most accurate skew predictions while also keeping the model 

as simple as possible. This section presents the diagnostic 

statistics for a B-WLS/B-GLS analysis, and discusses the 

specific values obtained for the Alaska RSA1 and RSA2 

regional skew studies. 

A Pseudo Analysis of Variance (Pseudo ANOVA) table 

for the Alaska RSA1 and RSA 2 regional skew analysis is 

shown in table B3, respectively. The table contains regression 

diagnostics/goodness of fit statistics. In particular, the table 

describes how much of the variation in the observations can be 

attributed to the regional model, and how much of the residual 

variation can be attributed to model error and sampling error, 

respectively. Difficulties arise in determining these quantities. 

The model errors cannot be resolved because the values of the 

sampling errors ƞi for each site i, are not known. However, the 

total sampling error sum of squares can be described by

its mean value, Var

i

n

i

=

∑ 

  1

γ . Because there are n equations,

the total variation due to the model error δ for a model with k

parameters has a mean equal to n k σδ

2 ( ). Thus, the residual

variation attributed to the sampling error is Var

i

n

i

=

∑ 

  1

γ , and the

residual variation attributed to the model error is n k σδ

2 ( ).

For a model with no parameters other than the mean 

(that is, the constant skew model), the estimated model error 

variance σδ

2 ( ) 0 describes all anticipated variation in γi

 = μ + 

δ, where μ is the mean of the estimated station sample skews. 

Thus, the total expected sum of squares variation due to model 

error δi and due to sampling error ηi = i − i γ γ  in expectation

should equal n Var i

i

n

σδ

2

1

( ) 0 + ( ) =

∑ γ . Therefore, the expected

sum of squares attributed to a regional skew model with k

parameters equals n k [ ( σ σ ) ( )] δ δ

2 2 0 − , because the sum of the

model error variance n k σδ

2 ( ) and the variance explained by the

model must sum to nσδ

2 ( ) 0 . Table B3 considers a model with 

k = 0 (a constant model).
Appendix B 47

This division of the variation in the observations is 

referred to as a Pseudo ANOVA because the contributions of 

the three sources of error are estimated or constructed, rather 

than being determined from the computed residual errors and 

the observed model predictions, while also ignoring the impact 

of correlation among the sampling errors. 

Table B3 contains the Pseudo ANOVA results for the 

RSA 1 CONSTANT model and the RSA2 CONSTANT 

model. The CONSTANT models do not have any explanatory 

variables, thus the variation attributed to the models is 0. 

The Error Variance Ratio (EVR) is a modeling diagnostic 

used to evaluate if a simple OLS regression is sufficient, or a 

more sophisticated WLS or GLS analysis is appropriate. EVR 

is the ratio of the average sampling error variance to the model 

error variance. Generally, an EVR greater than 0.20 indicates 

that the sampling variance is not negligible when compared 

to the model error variance, suggesting the need for a WLS or 

GLS regression analysis. The EVR is calculated as 

EVR

SS sampling error

SS model error = ( )

( ) = ( )

( ) ∑i=

n Var i

n k

1

2

γ

σδ

(B10)

For the Alaska regional skew study areas, EVR had a 

value of 0.7 for the RSA1 CONSTANT model and 2.3 for 

the RSA2 CONSTANT model. The sampling variability in 

the sample skew estimators was larger than the error in the 

regional model. Thus an OLS model that neglects sampling 

error in the station skew estimators may not provide a 

statistically reliable analysis of the data. Given the variation of 

record lengths from site-to-site, it is important to use a WLS 

or GLS analysis to evaluate the final precision of the model, 

rather than a simpler OLS analysis. 

The Misrepresentation of the Beta Variance (MBV*) 

statistic is used to determine whether a WLS regression is 

sufficient, or if a GLS regression is appropriate to determine 

the precision of the estimated regression parameters (Griffis, 

2006; Veilleux, 2011). The MBV* describes the error 

produced by a WLS regression analysis in its evaluation of 

the precision of b0

WLS

, which is the estimator of the constant 

β0

WLS

, because the covariance among the estimated station 

skews γ

i

 generally has its greatest impact on the precision of 

the constant term (Stedinger and Tasker, 1985). If the MBV* 

is substantially greater than 1, then a GLS error analysis 

should be used. The MBV* is calculated as,

MBV

GLS 

WLS 

*

WLS

WLS =



 



 

= Var b analysis

Var b analysis

w

T 0

0

|

|

Λ

Λ

w

w

w

i

n

i

i

ii

∑ =

=

1

1 where 

 (B11)

For the Alaska regional skew study areas, MBV* had a 

value of 1.3 for the RSA1 CONSTANT model and 1.2 for the 

RSA2 CONSTANT model. For both RSA1 and RSA2, the 

MBV* were larger than 1, indicating that the cross‐correlation 

among the skew estimators had an impact on the precision 

with which the regional average skew coefficient can be 

estimated; if a WLS precision analysis were used for the 

estimated constant parameter in the CONSTANT model, the 

variance would be underestimated by a factor of 0.3 in the 

case of RSA1 and 0.2 in the case of RSA2. Thus, a WLS 

analysis would misrepresent the variance of the constant in 

the CONSTANT model in both RSA1 and RSA2. Moreover, 

a WLS model would have resulted in underestimation of the 

variance of prediction, given that the sampling error in the 

constant term in both models was sufficiently large enough 

to make an appreciable contribution to the average variance 

of prediction.

Leverage and Influence

Leverage and influence diagnostics statistics can be used 

to identify rogue observations and to effectively address lackof-fit when estimating skew coefficients. Leverage identifies 

those streamgages in the analysis where the observed values 

have a large impact on the fitted (or predicted) values (Hoaglin 

and Welsch, 1978). Generally, leverage considers whether 

an observation, or explanatory variable, is unusual, and 

thus likely to have a large effect on the estimated regression 

coefficients and predictions. Unlike leverage, which highlights 

points which have the ability or potential to affect the fit of 

the regression, influence attempts to describe those points 

which do have an unusual impact on the regression analysis 

(Belsley and others, 1980; Cook and Weisberg, 1982; Tasker 

and Stedinger, 1989). An influential observation is one with 

an unusually large residual that has a disproportionate effect 

on the fitted regression relations. Influential observations often 

have high leverage. For a detailed description of the equations 

used to determine leverage and influence for a B-WLS/B-GLS 

analysis see Veilleux (2011) and Veilleux and others (2011).

For the B-WLS/B-GLS CONSTANT regional skew 

models for Alaska RSA1 and RSA2, no sites had high 

leverage. The differences in leverage values for the constant 

models reflect the variation in record lengths among sites. 

In RSA1, six sites in the study (streamgages 15281000, 

15212000, 15546200, 15291100, 15476400, and 15564887) 

have high influence, and thus have an unusual impact on the 

fitted regression relation. These six sites also have the six 

largest magnitude residuals in the study.

In RSA2, only one site in the study (streamgage 

15098000) has high influence, and thus has an unusual impact 

on the fitted regression relation. Streamgage 15098000 has 

the highest influence value due to its large residual, the largest 

magnitude residual in the study; its unbiased station skew of 

1.8 is the largest in the study.

Publishing support provided by the U.S. Geological Survey

Science Publishing Network, Tacoma Publishing Service Center

For more information concerning the research in this report, contact the

 Director, Alaska Science Center

U.S. Geological Survey

4210 University Drive

Anchorage, Alaska 99508-4560

http://alaska.usgs.gov
Curran and others—Estimating Flood Magnitude and Frequency at Sites on Streams in Alaska and Conterminous Basins in Canada—Scientific Investigations Report 2016–5024

ISSN 2328-0328 (online)

http://dx.doi.org/10.3133/sir20165024