Prepared in cooperation with the Montana Department of Transportation, Montana Department 

of Environmental Quality, and Montana Department of Natural Resources and Conservation

Adjusted Peak-Flow Frequency Estimates for Selected 

Streamflow-Gaging Stations in or near Montana Based 

on Data through Water Year 2011

Chapter D of

Montana StreamStats 

Scientific Investigations Report 2015–5019–D

Version 1.1, February 2018

U.S. Department of the Interior

U.S. Geological Survey
Cover photograph: Flooding at the Musselshell River near Martinsdale, Montana streamgage. 

Photograph by U.S. Geological Survey personnel, May 23, 2011.
Adjusted Peak-Flow Frequency Estimates 

for Selected Streamflow-Gaging Stations 

in or near Montana Based on Data through 

Water Year 2011

By Steven K. Sando, Roy Sando, Peter M. McCarthy, and DeAnn M. Dutton

Chapter D of

Montana StreamStats

Prepared in cooperation with the Montana Department of Transportation, 

Montana Department of Environmental Quality, and Montana Department of 

Natural Resources and Conservation

Scientific Investigations Report 2015–5019–D

Version 1.1, February 2018

U.S. Department of the Interior

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

RYAN K. ZINKE, Secretary

U.S. Geological Survey

William H. Werkheiser, Deputy Director

 exercising the authority of the Director

U.S. Geological Survey, Reston, Virginia

First release: April 2016 

Revised: February 2018 (ver 1.1)

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

resources, natural hazards, and the environment—visit https://www.usgs.gov or call 1–888–ASK–USGS.

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

visit https://store.usgs.gov.

Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the 

U.S. Government.

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:

Sando, S.K., Sando, Roy, McCarthy, P.M., and Dutton, D.M., 2018, Adjusted peak-flow frequency estimates for 

selected streamflow-gaging stations in or near Montana based on data through water year 2011 (ver. 1.1, February

2018) : U.S. Geological Survey Scientific Investigations Report 2015–5019–D, 12 p., https://doi.org/10.3133/

sir20155019D.

ISSN 2328-0328 (online)
iii

Contents

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

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

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

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

Methods of Analysis......................................................................................................................................4

Procedures for Weighting with Regional Regression Equations..................................................4

Procedures for Mixed-Station Record Extension ...........................................................................4

Definition of Base Periods..........................................................................................................5

Application of the Mixed-Station Maintenance of Variance Type 1 (MOVE.1) 

Analysis to Synthesize Peak-Flow Data .....................................................................5

Procedures for Frequency Analysis of the Combined Recorded and Synthesized 

Peak-Flow Data ..............................................................................................................6

Adjusted Peak-Flow Frequency Estimates for Selected Streamflow-Gaging Stations in or 

near Montana....................................................................................................................................6

Adjusted Peak-Flow Frequency Estimates Determined by Weighting with Regional 

Regression Equations ............................................................................................................6

Considerations for Interpreting Results for Weighting with Regional Regression 

Equations ........................................................................................................................6

Adjusted Peak-Flow Frequency Estimates Determined by Mixed-Station Record 

Extension...................................................................................................................................6

Considerations for Interpreting Mixed-Station Record Extension Results........................7

Summary .........................................................................................................................................................7

References Cited............................................................................................................................................8

Appendix 1. Information on Selected Streamflow-Gaging Stations, Peak-Flow Frequency 

Adjustment Methods, And Adjusted Peak-Flow Frequency Estimates.................................12

Figures

1. Map showing locations of selected streamflow-gaging stations in or near

Montana .........................................................................................................................................3

Appendix Tables

1–1. Information on selected streamflow-gaging stations in or near Montana for which 

at-site peak-flow frequencies were adjusted........................................................................12

1–2. Peak-flow frequency estimates for selected streamflow-gaging stations in or near 

Montana that were adjusted by weighting with regional regression equations.............12

1–3. Documentation regarding the application of the mixed-station maintenance of 

variance type I (MOVE.1) record extension procedure for selected streamflowgaging stations in or near Montana ........................................................................................12
iv

1–4. Documentation regarding analytical procedures for peak-flow frequency analyses 

on combined recorded and synthesized peak-flow data for selected streamflowgaging stations............................................................................................................................12

1–5. Peak-flow frequency estimates for selected streamflow-gaging stations in or near 

Montana that were adjusted by mixed-station record extension......................................12

Conversion Factors

Inch/Pound to International System of Units

Multiply By To obtain

Area

square mile (mi2

) 259.0 hectare (ha)

square mile (mi2

) 2.590 square kilometer (km2) 

Flow rate

cubic foot per second (ft3

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

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

°F = (1.8 × °C) + 32.

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

°C = (°F – 32) / 1.8.

Datums

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

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

(NAVD 88).

Supplemental Information

Water year is the 12-month period from October 1 through September 30 of the following 

calendar year. The water year is designated by the calendar year in which it ends. For example, 

water year 2010 is the period from October 1, 2009, through September 30, 2010.

Abbreviations

AEP annual exceedance probability

MOVE.1 maintenance of variance type I

RRE regional regression equation

SEP standard error of prediction

USGS U.S. Geological Survey

WIE Weighted Independent Estimates
v

Acknowledgments

The authors would like to recognize the U.S. Geological Survey hydrologic technicians involved 

in the collection of the peak-flow data for their dedicated efforts. The authors also would like to 

recognize the valuable contributions to this report chapter from the insightful technical reviews 

by Dan Driscoll and Skip Vecchia of the U.S. Geological Survey.

Special thanks are given to Mark Goodman and Dave Hedstrom of the Montana Department of 

Transportation and Steve Story of the Montana Department of Natural Resources and Conservation for their support of this study.

Adjusted Peak-Flow Frequency Estimates for Selected 

Streamflow-Gaging Stations in or near Montana Based 

on Data through Water Year 2011

By Steven K. Sando, Roy Sando, Peter M. McCarthy, and DeAnn M. Dutton

Abstract

The climatic conditions of the specific time period during 

which peak-flow data were collected at a given streamflowgaging station (hereinafter referred to as gaging station) can 

substantially affect how well the peak-flow frequency (hereinafter referred to as frequency) results represent long-term 

hydrologic conditions. Differences in the timing of the periods 

of record can result in substantial inconsistencies in frequency 

estimates for hydrologically similar gaging stations. Potential 

for inconsistency increases with decreasing peak-flow record 

length. The representativeness of the frequency estimates for a 

short-term gaging station can be adjusted by various methods 

including weighting the at-site results in association with frequency estimates from regional regression equations (RREs) 

by using the Weighted Independent Estimates (WIE) program. 

Also, for gaging stations that cannot be adjusted by using the 

WIE program because of regulation or drainage areas too 

large for application of RREs, frequency estimates might be 

improved by using record extension procedures, including 

a mixed-station analysis using the maintenance of variance 

type I (MOVE.1) procedure. The U.S. Geological Survey, in 

cooperation with the Montana Department of Transportation 

and the Montana Department of Natural Resources and Conservation, completed a study to provide adjusted frequency 

estimates for selected gaging stations through water year 2011.

The purpose of Chapter D of this Scientific Investigations Report is to present adjusted frequency estimates for 

504 selected streamflow-gaging stations in or near Montana 

based on data through water year 2011. Estimates of peak-flow 

magnitudes for the 66.7-, 50-, 42.9-, 20-, 10-, 4-, 2-, 1-, 0.5-, 

and 0.2-percent annual exceedance probabilities are reported. 

These annual exceedance probabilities correspond to the 1.5-, 

2-, 2.33-, 5-, 10-, 25-, 50-, 100-, 200-, and 500-year recurrence 

intervals, respectively. 

The at-site frequency estimates were adjusted by weighting with frequency estimates from RREs using the WIE 

program for 438 selected gaging stations in Montana. These 

438 selected gaging stations (1) had periods of record less 

than or equal to 40 years, (2) represented unregulated or minor 

regulation conditions, and (3) had drainage areas less than 

about 2,750 square miles. 

The weighted-average frequency estimates obtained by 

weighting with RREs generally are considered to provide 

improved frequency estimates. In some cases, there are substantial differences among the at-site frequency estimates, the 

regression-equation frequency estimates, and the weightedaverage frequency estimates. In these cases, thoughtful 

consideration should be applied when selecting the appropriate 

frequency estimate. Some factors that might be considered 

when selecting the appropriate frequency estimate include 

(1) whether the specific gaging station has peak-flow characteristics that distinguish it from most other gaging stations 

used in developing the RREs for the hydrologic region; and 

(2) the length of the peak-flow record and the general climatic 

characteristics during the period when the peak-flow data were 

collected. For critical structure-design applications, a conservative approach would be to select the higher of the at-site frequency estimate and the weighted-average frequency estimate.

The mixed-station MOVE.1 procedure generally was 

applied in cases where three or more gaging stations were 

located on the same large river and some of the gaging stations 

could not be adjusted using the weighted-average method 

because of regulation or drainage areas too large for application of RREs. The mixed-station MOVE.1 procedure was 

applied to 66 selected gaging stations on 19 large rivers.

The general approach for using mixed-station record 

extension procedures to adjust at-site frequencies involved 

(1) determining appropriate base periods for the gaging stations on the large rivers, (2) synthesizing peak-flow data for 

the gaging stations with incomplete peak-flow records during 

the base periods by using the mixed-station MOVE.1 procedure, and (3) conducting frequency analysis on the combined 

recorded and synthesized peak-flow data for each gaging 

station. Frequency estimates for the combined recorded and 

synthesized datasets for 66 gaging stations with incomplete 

peak-flow records during the base periods are presented. The 

uncertainties in the mixed-station record extension results are 

difficult to directly quantify; thus, it is important to understand the intended use of the estimated frequencies based on 
2 Adjusted Peak-Flow Frequency Estimates for Selected Streamflow-Gaging Stations, Montana

analysis of the combined recorded and synthesized datasets. 

The estimated frequencies are considered general estimates of 

frequency relations among gaging stations on the same stream 

channel that might be expected if the gaging stations had 

been gaged during the same long-term base period. However, 

because the mixed-station record extension procedures involve 

secondary statistical analysis with accompanying errors, the 

uncertainty of the frequency estimates is larger than would be 

obtained by collecting systematic records for the same number 

of years in the base period. 

Introduction

Reliable information on peak-flow characteristics at 

specific sites is essential for many water-resources applications including effective planning and management of water 

resources and flood plains, protection of lives and property in 

flood-prone areas, determination of actuarial flood-insurance 

rates, and design of highway infrastructure. Peak-flow data are 

readily available at sites that are monitored by U.S. Geological Survey (USGS) streamflow-gaging stations (hereinafter 

referred to as gaging stations) and can be downloaded through 

the USGS National Water Information System (U.S. Geological Survey, 2014). The gaged streamflow data can be statistically analyzed to estimate peak-flow frequencies (that is, peakflow magnitudes for the 66.7-, 50-, 42.9-, 20-, 10-, 4-, 2-, 1-, 

0.5-, and 0.2-percent annual exceedance probabilities [AEPs]). 

These AEPs correspond to the 1.5-, 2-, 2.33-, 5-, 10-, 25-, 50-, 

100-, 200-, and 500-year recurrence intervals, respectively. 

Sando, McCarthy, and Dutton (2016) reported at-site peakflow frequency estimates (hereinafter usually referred to as 

frequency estimates, or simply as frequencies) for 725 gaging 

stations in Montana based on data through water year 2011; 

a water year is the 12-month period from October 1 through 

September 30 and is designated by the year in which it ends. 

The USGS, in cooperation with the Montana Department 

of Transportation and the Montana Department of Natural 

Resources and Conservation, completed a study to provide 

adjusted frequency estimates for selected gaging stations 

through water year 2011. Adjustments to at-site frequency 

estimates can be appropriate for several reasons. The climatic 

conditions of the specific time period during which the data 

were collected at a given gaging station can substantially 

affect how well the frequency estimates represent long-term 

hydrologic conditions. Differences in the timing of the periods 

of record can result in substantial inconsistencies in frequency 

estimates for hydrologically similar gaging stations. Potential 

for inconsistency increases with decreasing peak-flow record 

length. The representativeness of the frequency estimates 

for a short-term gaging station can be adjusted by various 

methods including weighting the at-site frequency estimates in 

association with frequency estimates from regional regression 

equations (RREs) by using the Weighted Independent Estimates (WIE) program as described by Cohn and others (2012). 

Also, for gaging stations that cannot be adjusted by using the 

WIE program because of regulation or drainage areas too 

large for application of RREs, frequency estimates might be 

improved by using record extension procedures, including a 

mixed-station analysis using the maintenance of variance type 

I (MOVE.1) procedure (Alley and Burns, 1983). Application 

of the mixed-station MOVE.1 procedure to peak-flow records 

for South Dakota is described by Sando and others (2008). 

Purpose and Scope

The study described in Chapter D of this Scientific Investigations Report is part of a larger study to develop a StreamStats application for Montana, compute streamflow characteristics at streamflow-gaging stations, and develop regional 

regression equations to estimate streamflow characteristics at 

ungaged sites (as described fully in Chapters A through G of 

this Scientific Investigations Report). The purpose of Chapter D is to present adjusted frequency estimates for selected 

streamflow-gaging stations (fig. 1, table 1–1 in appendix 1 

at the back of this report chapter (available at https://doi.

org/10.3133/sir20155019D); map numbers assigned according 

to McCarthy and others [2016]) in or near Montana based on 

data through water year 2011. Adjustments are presented for 

504 of the 725 gaging stations for which frequency estimates 

were reported by Sando, McCarthy, and Dutton (2016). 

Adjusted estimates of peak-flow magnitudes for the 66.7-, 

50-, 42.9-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEPs are 

reported. These AEPs correspond to the 1.5-, 2-, 2.33-, 5-, 

10-, 25-, 50-, 100-, 200-, and 500-year recurrence intervals, 

respectively. 

The at-site frequency estimates (reported by Sando, 

McCarthy, and Dutton [2016]) were adjusted by weighting 

with frequency estimates from RREs (reported by Sando, 

Roy, and others [2016]) for 438 of the selected gaging stations 

(fig. 1, tables 1–1 and 1–2 in appendix 1 at the back of this 

report chapter). For 66 of the selected gaging stations 

(fig. 1, appendix tables 1–1 and 1–3 in appendix 1 at the back 

of this report chapter, available at https://doi.org/10.3133/

sir20155019D) that are located on 19 large rivers with multiple gaging stations, the at-site frequency estimates were 

adjusted by using mixed-station record extension procedures. 

The RRE weighting and mixed-station record extension 

procedures are documented in this report chapter. Unadjusted 

at-site frequency estimates for 27 long-term streamflowgaging stations (fig. 1, tables 1–1 and 1–3) that are located on 

19 large rivers also are included to assist in assessing adjusted 

frequency estimates for other gaging stations located on the 

large rivers. 
Introduction 3

Cr

Fork

Two

Redwater

River

South

Fork

River

Yaak

River

Kootenai

River

Clark

Fork

Flathead

 Lake

North

Fork

Middle

Cut Bank Creek

Medicine River Marias

River

Birch

Milk

River

Lodge Creek

Battle

Creek

Frenchman River

Milk

River

Poplar

River

Missouri River

Medicine

 Lake

River

Fort Peck

Reservoir

Big

Dry Creek

Boxelder Creek

River

Judith

Missouri

River

Sun River

Smith

River

River

Teton

River Swan

Flathead

River

Bitterroot

River

Blackfoot

River

Creek

Rock

Clark

Fork

Musselshell

River

Yellowstone

Powder

River

River

Tongue

Bighorn

River

Bighorn

 Lake

Little

Bighorn

River

Yellowstone

Boulder

River

Clarks

Fork

Gallatin

River

Madison

River

Ruby

River

River

Jefferson

R

Beaverhead

Big

Hole

River

Yellowstone

 Lake

Dearborn River

 Lake Koocanusa

Creek

Armells

St Mary 

River

Flathead R.

Milk River

Tenmile Cr.

Stillwater

River

Glendive

Sidney

Havre Shelby

Great

 Falls

Lewistown

Miles City

Billings

Butte

Dillon

Malta

CANADA

UNITED STATES

Wyoming

DIVIDE

CONTINENTAL

DIVIDE

Glasgow

Troy

Libby

Hamilton

Ashland

Broadus

Hardin

Livingston

Big Timber

Mosby

Jordan

Pryor

Harlowton

116o

115o

113

49o

48o

47o

46o

45o

114 o o

Kalispell

Helena

Missoula

CONTINENTAL

Bozeman

112o 111o 110o 109o 108o 107o 106o 105o

Idaho

SOUTH DAKOTA

NORTH DAKOTA

ALBERTA

BRITISH

COLUMBIA

MONTANA

0 10 20 30 40 50 MILES

0 10 20 30 40 50 KILOMETERS

EXPLANATION

75

1

28

Base modified from U.S. Geological Survey state base map, 1968

North American Datum of 1983 (NAD 83)

Streamflow-gaging station (number corresponds 

 to map number in table 1–1)

 

Long-term streamflow-gaging station for which 

peak-flow frequencies were not adjusted. 

However, the streamflow-gaging station was 

included in this report chapter to assist in 

assessing adjusted peak-flow frequency 

estimates for other streamflow-gaging 

stations located on the same large river 

Streamflow-gaging station for which peak-flow 

frequencies were adjusted by weighting with 

regression equations

Streamflow-gaging station for which peak-flow 

frequencies were adjusted by mixed-station 

record extension procedure

SASKATCHEWAN

4

3

1

99 98

94

92

91 89

86

85

84

83

82

78

77

76

75

74

73

72 71

70

69

66

65 64

63

59

58

55 54

52

50

48

47 46

45

43 44

42

41

40 39

38

37

36

35

34 33

32

31

30

29

28

27

26 25

23 24

20 19 18 16 15

10

689

688

687

686

683 684

682 681 680

679

678 677

676

675 674

673

672

671

669

664 662 661

660

659 658

657 656

652

649

647

646

645

643

639

637

636

634 635

633

632

629

627

625 623

622 620

619

618

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616

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614

612 613

611

610

609

605

603 602

600

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590

589

588 587 586

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583

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578 577

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571

570

569

568

566

565

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561

560

559

558

557

556

555

554

553 552

551

549

548

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546

545

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543

542

541

539

538 537

536

535

534

533

532

531

530

529

527

526

525

524

523 522

521

520

519

518 517

516

515 514

513

512

511

509

508

507

504

503

502

501

500

499 497

495

494

493

492

491

490

489

487 484

483

482

480

479

478 476

475

473 472

471

469

466 465

464

462

755

753

752

751

750

747

746 745

744 743

742 741

740

738

737

735

734

733

732

731

730

729

727

726

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724

719

718

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715

714 713

712

711

708

707

706

705

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697

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693 692

691

690

459

458

457

451 452

447

446

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436 437

435

434

431

430

428 429

426

425

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417

415

414

413

411

410

408

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401

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397 398

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394

390

389

385 383 384

382

379

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367

365

364

363

362

360

358

357

354

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351

346

345

344 343 342

341

331

328

326

324 323

320

319

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312

294

293

292

290 287

285

283 284

282

281

280

276 269

267

266

265

264

262

261

260

259

258

257 256

254

253 252

247 248 251

246

245

244

243

238

236

235

234 233 232

231

230 228

227 226

225

223

222

221

219

218 217

216

214 215

213

211

208 206

205

203

200 202

199

198

197

196

195 193 194

191

190

189188

187 186

185 184

182

181

180

178

177

176

175 174

172

171

170

169 167 168

166 165

163

159

158

157

156

152 148

147

146

145

144 143

142 141

138

137

134 133

130

129

128 127

126 125

124

123 122

121

120

119 118

117

116

112 114

111 110

109

107

106

105

104

102

100

81

685

667

665

655

654

650

638

628

624

608

604

567

510

754

710

709

375

212

108

Figure 1. Locations of selected streamflow-gaging stations in or near Montana.
4 Adjusted Peak-Flow Frequency Estimates for Selected Streamflow-Gaging Stations, Montana

Methods of Analysis

At-site frequency estimates for 504 selected gaging stations (fig. 1, table 1–1) were adjusted by RRE weighting and 

mixed-station record extension procedures. Details on the 

procedures are presented herein.

Procedures for Weighting with Regional 

Regression Equations

Regional regression equations typically are used to 

estimate frequencies for ungaged sites; however, RREs also 

can be used to adjust at-site frequency estimates for gaging 

stations, which can reduce uncertainty of frequency estimates 

(Cohn and others, 2012). For gaging stations with characteristics that allow estimation of frequencies using RREs, the 

at-site frequencies can be adjusted by weighted averaging with 

results from RREs by using the WIE program (Cohn and 

others, 2012). For 438 selected gaging stations in Montana 

(fig. 1, tables 1–1 and 1–2), the at-site frequency estimates 

were adjusted by weighting with results from RREs (reported 

by Sando, Roy, and others [2016]). The 438 selected gaging 

stations (1) had periods of record less than or equal to 

40 years, (2) represented unregulated or minor regulation 

conditions (based on criteria for determination of regulation 

status described by McCarthy and others [2016]), and (3) had 

drainage areas less than about 2,750 square miles (mi2). 

The WIE program weights the at-site frequency estimates and the RRE frequency estimates based on the meansquare errors of the two estimates. The variance of the at-site 

frequency estimate used in the weighting procedure was 

determined using the expected moments algorithm (Cohn and 

others, 2001). In determining the variance of the RRE frequency estimate for a given station, the at-site values of basin 

characteristics (explanatory variables) used in the RRE were 

analyzed in relation to the matrix form of the applicable generalized least squares regression equation (Tasker and Stedinger, 

1989; Sando, Roy, and others, 2016) or weighted least-squares 

regression equation (Tasker, 1980; Sando, Roy, and others, 

2016). Thus, for each gaging station, the distance of the at-site 

basin characteristic values from the center of the joint distribution of all values used in developing the regression equation 

was incorporated into the weighting process.

Procedures for Mixed-Station Record Extension

Mixed-station record extension generally was applied in 

cases where three or more gaging stations were located on the 

same large river and some of the gaging stations could not 

be adjusted using the weighted-average method because of 

regulation or drainage areas too large (greater than about 

2,750 mi2) for application of RREs. Mixed-station record 

extension was applied to 66 selected gaging stations 

(table 1–3) on the following 19 large rivers: 

1. the Beaverhead River (3 gaging stations); 

2. the Ruby River (4 gaging stations); 

3. the Big Hole River (4 gaging stations); 

4. the Jefferson River (2 gaging stations); 

5. the Madison River (5 gaging stations); 

6. the Missouri River (3 gaging stations); 

7. the Marias River (1 gaging station); 

8. the Musselshell River (7 gaging stations); 

9. the Yellowstone River (6 gaging stations); 

10. the Little Bighorn River (3 gaging stations); 

11. the Tongue River (4 gaging stations); 

12. the Powder River (3 gaging stations); 

13. the Kootenai River (2 gaging stations); 

14. the Clark Fork (7 gaging stations); 

15. the Bitterroot River (3 gaging stations); 

16. the North Fork Flathead River (2 gaging stations); 

17. the Middle Fork Flathead River (4 gaging stations); 

18. the South Fork Flathead River (2 gaging stations); and 

19. the Flathead River (1 gaging station).

For most of the large rivers, in addition to the gaging stations with incomplete peak-flow records during the base periods (described in the section “Definition of Base Periods”), 

one or more long-term gaging stations with complete datasets 

during the base periods were available. For all of the 19 large 

rivers combined, a total of 27 long-term gaging stations with 

complete datasets during the base periods were available. 

These 27 long-term streamflow-gaging stations were included 

in this report chapter to assist in assessing adjusted frequency 

estimates for other gaging stations located on the large rivers.

The general approach for using mixed-station record 

extension procedures to adjust at-site frequencies involved 

(1) determining appropriate base periods for the gaging stations on the large rivers, (2) synthesizing peak-flow data for 

the gaging stations with incomplete peak-flow records during 

the base periods by using the mixed-station MOVE.1 procedure, and (3) conducting frequency analysis on the combined 

recorded and synthesized peak-flow data for each gaging 

station. Details on the procedures are presented in the following sections: “Definition of Base Periods,” “Application of the 

Mixed-Station Maintenance of Variance Type 1 (MOVE.1) 

Analysis to Synthesize Peak-Flow Data,” and “Procedures for 

Frequency Analysis of the Combined Recorded and Synthesized Peak-Flow Data.”
Methods of Analysis 5

Definition of Base Periods

For each large river (or subreach for several of the rivers), 

the base period typically started with the water year of earliest 

systematic peak-flow data collection for that river or subreach 

and extended to 2011. For some of the large rivers, some 

gaging stations had sporadic peak-flow data collected before 

the start of the defined base period. In those cases, peak-flow 

data could not be synthesized for all of the gaging stations on 

a given large river; thus, the base period did not include the 

early years of sporadic data collection. 

For some of the large rivers, all of the gaging stations are 

affected by the same major dam or canal regulation structure 

(as described by McCarthy and others [2016] and Sando, 

McCarthy, and Dutton [2016]). In such cases, the base period 

was restricted to the period after the start of the regulation. 

For some of the large rivers, some reaches are unregulated 

and some reaches are regulated. In such cases, different base 

periods for different reaches were defined to accommodate the 

variability in unregulated and regulated conditions.

Application of the Mixed-Station Maintenance of 

Variance Type 1 (MOVE.1) Analysis to Synthesize 

Peak-Flow Data

The mixed-station MOVE.1 procedure (Alley and Burns, 

1983) is based on correlation of concurrent peak-flow records 

for a target gaging station (a gaging station with incomplete 

peak-flow records during the base period) and one or more 

index stations (gaging stations that have peak-flow records 

for one or more of the missing years of the target station). 

The MOVE.1 fitting procedure, though analogous to ordinary 

least squares regression, results in an extended peak-flow 

record with a variance comparable to that of the unextended 

peak-flow record of the target station (Cary and Parrett, 1996). 

The use of multiple potential index stations allows additional 

important peak-flow events or periods of record to be available for record extension that could not be achieved by using a 

single index station. Also, the mixed-station MOVE.1 procedure can be applied to cases for which an appropriate index 

station has a shorter record length than the target station. The 

mixed-station MOVE.1 procedure synthesizes peak-flow data 

for individual years of missing record for the target station. 

Errors in estimated peak flows for individual years might be 

relatively large with use of the mixed-station procedure. 

Results from the mixed-station MOVE.1 procedure were 

reviewed for consistency among gaging stations on the same 

large river with emphasis placed on specific key years with 

unusually high or low peak-flow conditions. In some cases, 

the initial mixed-station MOVE.1 results indicated inconsistencies among some gaging stations for some of the key 

years. In these cases, the initial mixed-station MOVE.1 results 

for some of the key years were adjusted using drainage-area 

ratio methods described by Sando, Roy, and others (2016), as 

documented in table 1–3. For a given gaging station, use of 

the drainage-area ratio methods accounted for a small amount 

of the combined recorded and synthesized data (generally less 

than 3 percent) and probably had only minimal effect on the 

appropriate maintenance of variance within the dataset. Also, 

use of the drainage-area ratio methods probably had a minimal 

effect on the reported standard error of prediction (SEP) of the 

method. However, because the drainage-area ratio methods 

were used to synthesize some peak flows, the calculation of 

the SEP for the mixed-station MOVE.1 results was not always 

precise and is reported as “estimated” in table 1–3.

Documentation regarding the application of the mixedstation MOVE.1 procedure is presented in table 1–3. For a 

given target station, table 1–3 includes the index stations, the 

weighted-average Pearson correlation coefficient, and the 

SEP for the mixed-station MOVE.1 procedure. The weightedaverage Pearson correlation coefficient was determined by 

multiplying the number of peak flows synthesized based on an 

index gaging station times the Pearson correlation coefficient 

for the index gaging station for each index gaging station. The 

resultant products then were summed and divided by the total 

number of synthesized peak flows.

In general, confidence in the mixed-station MOVE.1 

results increases with increasing weighted-average Pearson 

correlation coefficient, and confidence also increases with 

decreasing SEP. In some cases, however, there are inconsistencies in relations between the weighted-average Pearson 

correlation coefficients and the SEPs. For some target stations, 

the weighted-average Pearson correlation coefficient is somewhat low (less than about 0.75), but the SEP also is somewhat 

low (less than about 25 percent). The low weighted-average 

Pearson correlation coefficient indicates that the index station 

did not strongly explain the variability of the target station; 

however, the low SEP indicates that variability in the targetstation recorded peak flows was small and also that variability 

in the synthesized peak flows using the index station(s) was 

small. In other cases, for some target stations the weightedaverage Pearson correlation coefficient is high (greater than 

about 0.85), but the SEP also is somewhat high (greater than 

about 40 percent). The high weighted-average Pearson correlation coefficient indicates that the index station explained most 

of the variability of the target station; however, the high SEP 

indicates that variability in the target-station recorded peak 

flows was large and also that variability in the synthesized 

peak flows using the index station(s) was large. For the cases 

with inconsistencies in relations between the weighted-average 

Pearson correlation coefficients and the SEPs, the mixedstation MOVE.1 results were considered to provide reasonable 

representation of annual peak flows. 

The uncertainties in the mixed-station MOVE.1 results 

are difficult to directly quantify; thus, it is important to understand the intended use of the estimated frequencies based on 

analysis of the combined recorded and synthesized datasets. 

The estimated frequencies are considered general estimates of 

frequency relations among gaging stations on the same stream 

channel that might be expected if the gaging stations had 

been gaged during the same long-term base period. However, 
6 Adjusted Peak-Flow Frequency Estimates for Selected Streamflow-Gaging Stations, Montana

because the mixed-station MOVE.1 procedure involves 

secondary statistical analysis with accompanying errors, the 

uncertainty of the frequency estimates is larger than would be 

obtained by collecting systematic records for the same number 

of years represented by the base period.

Procedures for Frequency Analysis of the 

Combined Recorded and Synthesized Peak-Flow 

Data

Frequency analyses of the combined recorded and synthesized peak-flow data for the selected gaging stations 

(table 1–3) were conducted using procedures described in 

detail by Sando, McCarthy, and Dutton (2016), with documentation provided in table 1–4 in appendix 1 at the back 

of this report chapter (available at https://doi.org/10.3133/

sir20155019D). A brief overview of the procedures is presented herein. 

Frequency analyses generally were conducted using 

procedures described in Bulletin 17B “Guidelines for Determining Flood Flow Frequency” (U.S. Interagency Advisory 

Council on Water Data, 1982; hereinafter referred to as 

Bulletin 17B). Bulletin 17B uses the log-Pearson III probability distribution, which is fit by using the mean, standard 

deviation, and skew of the logs of the combined recorded and 

synthesized peak flows for a given gaging station. Procedures 

described in Bulletin 17B were used as primary guidelines for 

developing the frequency estimates presented in this report 

chapter. The computer program PEAKFQ, which was developed by the U.S. Geological Survey (Flynn and others, 2006), 

was used to run the frequency analyses.

Frequencies for the combined recorded and synthesized 

datasets initially were analyzed using standard Bulletin 17B 

procedures for fitting the log-Pearson III distribution. In this 

report chapter, standard Bulletin 17B procedures are considered to include the use of weighted skew coefficients, the 

use of the Grubbs-Beck outlier test (Grubbs and Beck, 1972) 

for identifying low outliers, and, where applicable, the use 

of historical adjustment procedures. Fits of the preliminary 

frequency curves with the probability plots of the peak flows 

were then evaluated. In most cases, fits of the standard Bulletin 17B analyses were determined to be satisfactory. 

In other cases, however, the frequency estimates could 

be improved by using alternative procedures for handling 

specific characteristics of the peak-flow records for some gaging stations. The specific characteristics of peak-flow records 

addressed by alternative procedures include (1) regulated 

peak-flow records, (2) mixed-population peak-flow records, 

and (3) atypical low-end peak-flow records. Stations for which 

alternative procedures were applied for any of these reasons 

are noted in table 1–4. The alternative procedures for handling 

the specific characteristics are described in Sando, McCarthy, 

and Dutton (2016).

Adjusted Peak-Flow Frequency 

Estimates for Selected StreamflowGaging Stations in or near Montana

At-site frequency estimates for 504 selected gaging stations (fig. 1, table 1–1) were adjusted by RRE weighting and 

mixed-station record-extension procedures. The results of the 

analyses are presented herein.

Adjusted Peak-Flow Frequency Estimates 

Determined by Weighting with Regional 

Regression Equations

Results of weighting with RREs for the 438 selected gaging stations in or near Montana are presented in table 1–2. For 

each gaging station, the at-site frequency estimates, RRE frequency estimates, and weighted-average frequency estimates 

are included in appendix table 1–2.

Considerations for Interpreting Results for 

Weighting with Regional Regression Equations 

The weighted-average frequencies obtained by weighting with RREs generally are considered to provide improved 

frequency estimates compared to using only at-site frequencies 

(Cohn and others, 2012). In some cases, differences among the 

at-site frequency estimates, the regression-equation frequency 

estimates, and the weighted-average frequency estimates are 

substantial. In these cases, thoughtful consideration should be 

applied when selecting the appropriate frequency estimate. 

Some factors that might be considered when selecting the 

appropriate frequency estimate include (1) whether the specific gaging station has peak-flow characteristics that distinguish it from most other stations used in developing the RREs 

for the hydrologic region; and (2) the length of the peak-flow 

record and the general climatic characteristics during the 

period when the peak-flow data were collected. Information 

on peak-flow temporal trends at long-term gaging stations in 

or near Montana (Sando, McCarthy, and others, 2016) might 

aid in assessing the general climatic characteristics during the 

period when the peak-flow data were collected. For critical 

structure-design and flood-plain mapping applications, a conservative approach would be to select the higher of the at-site 

frequency estimate (reported by Sando, McCarthy, and Dutton 

[2016]) and the weighted-average frequency estimate.

Adjusted Peak-Flow Frequency Estimates 

Determined by Mixed-Station Record Extension

Frequency estimates for the combined recorded and 

synthesized datasets for the 66 selected gaging stations 

(table 1–3) with incomplete peak-flow records during the base 
Summary 7

periods are presented in table 1–5 in appendix 1 at the back 

of this report chapter (available at https://doi.org/10.3133/

sir20155019D). Table 1–5 also includes 27 long-term gaging 

stations with complete datasets during the base periods that 

were included to assist in assessing adjusted frequency estimates for other gaging stations located on the 19 large rivers. 

Thus, frequency estimates are reported for a total of 93 gaging 

stations in table 1–5. For some of the 27 long-term gaging stations with complete datasets during the base periods, the entire 

gaged period of record is longer than the base period. The base 

period frequency estimates are presented for comparison with 

other stations on the same large river; however, for these longterm gaging stations, more reliable frequency estimates (based 

on the entire period of record) are reported in Sando, McCarthy, and Dutton (2016).

Considerations for Interpreting Mixed-Station 

Record Extension Results

The uncertainties for the mixed-station record extension results are difficult to directly quantify because the 

mixed-station record extension procedures involve secondary 

statistical analysis with accompanying errors. The uncertainties for the resultant frequency estimates are larger than would 

be obtained by collecting systematic records for the same 

number of years represented by the base period. However, the 

mixed-station record extension frequency estimates incorporate data from nearby gaging stations (generally on the same 

river) and are considered to be more representative of actual 

streamflow conditions during the base periods than frequency 

estimates derived from the shorter-term, sometimes sporadic, 

gaged records. It is important to understand the intended use 

of the frequency estimates based on analysis of the combined 

recorded and synthesized datasets. The frequency estimates 

are considered general estimates of frequency relations among 

gaging stations on the same stream channel that might be 

expected if the gaging stations had been gaged during the 

same long-term base period. Caution should be used when 

using the frequency estimates for important applications, such 

as structure design. For critical structure-design applications 

based on a given gaging station, a conservative approach 

would be to select the higher of the at-site frequency estimate 

reported by Sando, McCarthy, and Dutton (2016) and mixedstation record extension frequency estimate.

For some gaging stations on some of the large rivers, the 

frequency estimates for some AEPs do not always increase in 

the downstream direction, as might typically be the case for 

many unregulated natural streams. While decreases in peak 

flows in a downstream direction are unusual for unregulated 

rivers, decreases in a downstream direction can be fairly 

common for regulated rivers and can result from effects of 

reservoir operations and irrigation diversions. For cases where 

frequency estimates decreased between an upstream gaging 

station and the next downstream gaging station, concurrent 

periods of record for the gaging stations were investigated 

to verify that the relations between the frequency estimates 

generally were similar to the relations between the recorded 

concurrent peak flows.

The mixed-station record extension frequency estimates 

for Bitterroot River at Bell Crossing near Victor, Montana 

(gaging station 12350250; map number 680) should be 

used with caution. The MOVE.1 analysis for this station 

has a low weighted-average Pearson correlation coefficient 

(0.56; table 1–3), a high SEP (88.8 percent; table 1–3), 

and synthesized peak flows account for a large proportion 

(65 percent; table 1–5) of the combined recorded and synthesized peak-flow data. The hydrology of the Bitterroot River 

near gaging station 12350250 is especially complex and 

affected by canal diversions and return flows.

Summary

The climatic conditions of the specific time period during 

which peak-flow data were collected at a given streamflowgaging station (hereinafter referred to as gaging station) can 

substantially affect how well the peak-flow frequency (hereinafter referred to as frequency) estimates represent long-term 

hydrologic conditions. Differences in the timing of the periods 

of record can result in substantial inconsistencies in frequency 

estimates for hydrologically similar gaging stations. Potential 

for inconsistency increases with decreasing peak-flow record 

length. The representativeness of the frequency estimates for a 

short-term gaging station can be adjusted by various methods 

including weighting the at-site results in association with frequency estimates from regional regression equations (RREs) 

by using the Weighted Independent Estimates (WIE) program. 

Also, for gaging stations that cannot be adjusted by using the 

WIE program because of regulation or drainage areas too 

large for application of RREs, frequency estimates might be 

improved by using record extension procedures, including a 

mixed-station analysis using the maintenance of variance type 

I (MOVE.1) procedure. The U.S. Geological Survey, in cooperation with the Montana Department of Transportation and 

the Montana Department of Natural Resources and Conservation, completed a study to provide adjusted frequency estimates for selected gaging stations through water year 2011.

The purpose of Chapter D of this Scientific Investigations Report is to present adjusted frequency estimates for 

504 selected gaging stations in or near Montana based on data 

through water year 2011. Estimates of peak-flow magnitudes 

for the 66.7-, 50-, 42.9-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent annual exceedance probabilities are reported. These 

annual exceedance probabilities correspond to the 1.5-, 2-, 

2.33-, 5-, 10-, 25-, 50-, 100-, 200-, and 500-year recurrence 

intervals, respectively. 

For gaging stations with characteristics that allow estimation of frequencies using RREs, the at-site frequencies can 

be adjusted by weighted averaging with the results from the 

RREs. The at-site frequency estimates for 438 selected gaging 
8 Adjusted Peak-Flow Frequency Estimates for Selected Streamflow-Gaging Stations, Montana

stations in Montana were adjusted by weighting with results 

from RREs using the WIE program. The 438 selected gaging 

stations (1) had periods of record less than or equal to 

40 years, (2) represented unregulated or minor regulation 

conditions, and (3) had drainage areas less than about 

2,750 square miles. 

The weighted-average frequencies obtained by weighting with RREs generally are considered to provide improved 

frequency estimates. In some cases, differences among the 

at-site frequency estimates, the regression-equation frequency 

estimates, and the weighted-average frequency estimates are 

substantial. In these cases, thoughtful consideration should be 

applied when selecting the appropriate frequency estimate. 

Some factors that might be considered when selecting the 

appropriate frequency estimate include (1) whether the specific gaging station has peak-flow characteristics that distinguish it from most other stations used in developing the RREs 

for the hydrologic region; and (2) the length of the peak-flow 

record and the general climatic characteristics during the 

period when the peak-flow data were collected. For critical 

structure-design applications, a conservative approach would 

be to select the higher of the at-site frequency estimate and the 

weighted-average frequency estimate.

The mixed-station MOVE.1 procedure generally was 

applied in cases where three or more gaging stations were 

located on the same large river, and some of the gaging stations could not be adjusted using the weighted-average method 

because of regulation or drainage areas too large for application of RREs. The mixed-station MOVE.1 procedure was 

applied to 66 selected gaging stations on 19 large rivers. 

The general approach for using mixed-station record 

extension to adjust at-site frequencies involved (1) determining appropriate base periods for the gaging stations on the 

large rivers, (2) synthesizing peak-flow data for the gaging 

stations with incomplete peak-flow records during the base 

periods by using the mixed-station MOVE.1 procedure, and 

(3) conducting frequency analysis on the combined recorded 

and synthesized peak-flow data for each gaging station. Frequency estimates for the combined recorded and synthesized 

datasets for 66 gaging stations with incomplete peak-flow 

records during the base periods are presented. The uncertainties in the mixed-station record extension results are difficult 

to quantify; thus, it is important to understand the intended 

use of the estimated frequencies based on analysis of the 

combined recorded and synthesized datasets. The estimated 

frequencies are considered general estimates of frequency 

relations among gaging stations on the same stream channel 

that might be expected if the gaging stations had been gaged 

during the same long-term base period. However, because the 

mixed-station record extension procedures involve secondary 

statistical analysis with accompanying errors, the uncertainty 

of the frequency estimates is larger than would be obtained 

by collecting systematic records for the same number of years 

represented by the base period. Caution should be used when 

using the frequency estimates for important applications, such 

as structure design. For critical structure-design applications 

based on a given gaging station, a conservative approach 

would be to select the higher of the at-site frequency estimate 

and the mixed-station record extension frequency estimate.

References Cited

Alley, W.M., and Burns, A.W., 1983, Mixed station extension of monthly streamflow records: Journal of Hydraulic 

Engineering, American Society of Civil Engineers, v. 109, 

no. 10, p. 1272–1284. [Also available at http://dx.doi.

org/10.1061/(ASCE)0733-9429(1983)109:10(1272).]

Cary, L.E., and Parrett, Charles, 1996, Synthesis of natural 

flows at selected sites in the upper Missouri River Basin, 

Montana, 1928–89: U.S. Geological Survey WaterResources Investigations Report 95–4261, 109 p. [Also 

available at http://pubs.er.usgs.gov/publication/wri954261.]

Cohn, T.A., Berenbrock, Charles, Kiang, J.E., and Mason, 

R.R., Jr., 2012, Calculating weighted estimates of peak 

streamflow statistics: U.S. Geological Survey Fact Sheet 

2012–3038, 4 p. [Also available at http://pubs.usgs.gov/

fs/2012/3038/.]

Cohn, T.A., Lane, W.L., and Stedinger, J.R., 2001, Confidence intervals for expected moments algorithm flood 

quantile estimates: Water Resources Research, v. 37, 

no. 6, p. 1695–1706. [Also available at http://dx.doi.

org/10.1029/2001WR900016.]

Flynn, K.M., Kirby, W.H., and Hummel, P.R., 2006, User’s 

manual for PeakFQ, annual flood frequency analysis using 

Bulletin 17B guidelines: U.S. Geological Survey Techniques and Methods, book 4, chap. B4, 42 p. [Also available 

at http://pubs.usgs.gov/tm/2006/tm4b4/.]

Grubbs, F.E., and Beck, Glenn, 1972, Extension of sample 

sizes and percentage points for significance tests of outlying 

observations: Technometrics, v. 14, no. 4, p. 847–854. [Also 

available at http://dx.doi.org/10.2307/1267134.]

McCarthy, P.M., Dutton, D.M., Sando, S.K., and Sando, Roy, 

2016, Montana StreamStats—A method for retrieving basin 

and streamflow characteristics in Montana: U.S. Geological 

Survey Scientific Investigations Report 2015–5019–A, 16 p. 

[Also available at http://dx.doi.org/10.3133/sir20155019A.]

Sando, Roy, Sando, S.K., McCarthy, P.M., and Dutton, D.M., 

2016, Methods for estimating peak-flow frequencies at 

ungaged sites in Montana based on data through water year 

2011: U.S. Geological Survey Scientific Investigations 

Report 2015–5019–F, 30 p. [Also available at http://dx.doi.

org/10.3133/sir20155019F.]
References Cited 9

Sando, S.K., Driscoll, D.G., and Parrett, Charles, 2008, 

Peak-flow frequency estimates based on data through water 

year 2001 for selected streamflow-gaging stations in South 

Dakota: U.S. Geological Survey Scientific Investigations 

Report 2008–5104, 367 p. [Also available at http://pubs.

usgs.gov/sir/2008/5104/.]

Sando, S.K., McCarthy, P.M., and Dutton, D.M., 2016, Peakflow frequency analyses and results based on data through 

water year 2011 for selected streamflow-gaging stations in 

or near Montana: U.S. Geological Survey Scientific Investigations Report 2015–5019–C, 27 p. [Also available at http://

dx.doi.org/10.3133/sir20155019C.]

Sando, S.K., McCarthy, P.M., Sando, Roy, and Dutton, D.M., 

2016, Temporal trends and stationarity in annual peak flow 

and peak-flow timing for selected long-term streamflowgaging stations in or near Montana through water year 

2011: U.S. Geological Survey Scientific Investigations 

Report 2015–5019–B, 48 p. [Also available at http://dx.doi.

org/10.3133/sir20155019B.]

Tasker, G.D., 1980, Hydrologic regression with weighted least 

squares: Water Resources Research, v. 16, no. 6, p. 1107–

1113. 

Tasker, G.D., and Stedinger, J.R., 1989, An operational 

GLS model for hydrologic regression: Journal of Hydrology, v. 111, p. 361–375. [Also available at http://dx.doi.

org/10.1016/0022-1694(89)90268-0.]

U.S. Geological Survey, 2014, National Water Information 

System (NWISWeb): U.S. Geological Survey database, 

accessed November 27, 2014, at http://waterdata.usgs.gov/

nwis/.

U.S. Interagency Advisory Council on Water Data, 1982, 

Guidelines for determining flood flow frequency: Hydrology Subcommittee, Bulletin 17B, appendixes 1–14, 28 p.

Appendix 1
12 Adjusted Peak-Flow Frequency Estimates for Selected Streamflow-Gaging Stations, Montana

Appendix 1. Information on Selected Streamflow-Gaging Stations, Peak-Flow 

Frequency Adjustment Methods, And Adjusted Peak-Flow Frequency Estimates

This appendix presents information on selected streamflow-gaging stations for which at-site peak flow frequencies presented in Sando, McCarthy, and Dutton (2016) were adjusted by various methods (table 1–1). Documentation regarding adjustment methods is presented in table 1–3 for the mixed-station maintenance of variance type I (MOVE.1) record extension procedure and in table 1–4 for the combined recorded and synthesized peak-flow data. Peak-flow frequency estimates for selected 

streamflow-gaging stations are presented in table 1–2 for those adjusted by weighting with regional regression equations and in 

table 1–5 for those that were adjusted by mixed-station record extension procedures. 

An Excel file containing the tables is available at https://doi.org/10.3133/sir20155019D.

Table 1–1. Information on selected streamflow-gaging stations in or near Montana for which at-site peak-flow frequencies were 

adjusted.

Table 1–2. Peak-flow frequency estimates for selected streamflow-gaging stations in or near Montana that were adjusted by 

weighting with regional regression equations.

Table 1–3. Documentation regarding the application of the mixed-station maintenance of variance type I (MOVE.1) record extension 

procedure for selected streamflow-gaging stations in or near Montana.

Table 1–4. Documentation regarding analytical procedures for peak-flow frequency analyses on combined recorded and synthesized 

peak-flow data for selected streamflow-gaging stations. 

Table 1–5. Peak-flow frequency estimates for selected streamflow-gaging stations in or near Montana that were adjusted by mixedstation record extension. 

Publishing support provided by:

Rolla Publishing Service Center

For more information concerning this publication, contact: 

Director, Wyoming-Montana Water Science Center 

U.S. Geological Survey 

3162 Bozeman Ave 

Helena, MT 59601 

(406) 457–5900

Or visit the Wyoming-Montana Water Science Center Web site at: 

https://wy-mt.water.usgs.gov/

Sando and others—Adjusted Peak-Flow Frequency Estimates for Selected Streamflow-Gaging Stations, Montana—Scientific Investigations Report 2015–5019–D, ver. 1.1

ISSN 2328-0328 (online)

https://doi.org/10.3133/sir20155019D