Prepared in cooperation with the Iowa Department of Transportation and the 

Iowa Highway Research Board (Project TR-519)

Methods for Estimating Annual Exceedance-Probability 

Discharges for Streams in Iowa, Based on Data through 

Water Year 2010

Scientific Investigations Report 2013–5086

U.S. Department of the Interior

U.S. Geological Survey
Cover photograph. View looking southeast at Oakville, Iowa, June 15, 2008; main channel 

of the Iowa River is just out of view in the foreground. Photograph by www.kenpurdy.com.
Methods for Estimating Annual 

Exceedance-Probability Discharges for 

Streams in Iowa, Based on Data through 

Water Year 2010

By David A. Eash, Kimberlee K. Barnes, and Andrea G. Veilleux

Prepared in cooperation with the Iowa Department of Transportation and the 

Iowa Highway Research Board (Project TR-519)

Scientific Investigations Report 2013–5086

U.S. Department of the Interior

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

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2013

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Suggested citation:

Eash, D.A., Barnes, K.K., and Veilleux, A.G., 2013, Methods for estimating annual exceedance-probability 

discharges for streams in Iowa, based on data through water year 2010: U.S. Geological Survey Scientific 

Investigations Report 2013–5086, 63 p. with appendix.
iii

Contents

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

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

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

Description of Study Area ...................................................................................................................2

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

Methods for Dataset Development for Streamgages..............................................................................6

Basin Characteristics...........................................................................................................................7

Geographic Information System Measurements ..................................................................7

Annual Peak-Discharge Data .............................................................................................................8

Trend Analyses.............................................................................................................................9

Annual Exceedance-Probability Analyses .....................................................................................10

Bulletin 17B/GB Analyses.........................................................................................................10

Expected Moments Algorithm (EMA/MGB) Analyses.........................................................11

Multiple Grubbs-Beck (MGB) Test for Detecting Low Outliers..........................................13

Regional Skew Analysis ...........................................................................................................14

Regional Regression Analyses to Estimate Annual Exceedance-Probability Discharges for 

Ungaged Stream Sites...................................................................................................................16

Definition of Flood Regions ...............................................................................................................17

Development of Regional Regression Equations ..........................................................................20

Multiple-Linear Regression......................................................................................................20

Ordinary-Least-Squares Regression.............................................................................20

Generalized-Least-Squares Regression.......................................................................23

Final Regression Equations...............................................................................................................25

Accuracy and Limitations of Regression Equations .....................................................................27

Prediction Intervals ...................................................................................................................27

Application of Regression Equations ..............................................................................................29

Example 1 ....................................................................................................................................29

Example 2 ....................................................................................................................................29

Weighted Method to Estimate Annual Exceedance-Probability Discharges for Streamgages.....30

Example 3 ....................................................................................................................................31

Weighted Methods to Estimate Annual Exceedance-Probability Discharges for Ungaged 

Sites on Gaged Streams................................................................................................................31

Regression-Weighted Estimates for Ungaged Sites on Gaged Streams ..................................31

Example 4 ....................................................................................................................................32

Area-Weighted Estimates for Ungaged Sites on Gaged Streams..............................................32

Example 5 ....................................................................................................................................33

Estimates for Ungaged Sites on Gaged Streams Between Two Streamgages........................33

Weighted Method to Estimate Annual Exceedance-Probability Discharges for Ungaged 

Sites Draining More Than One Flood Region ............................................................................33

Region-of-Influence Method to Estimate Annual Exceedance-Probability Discharges for 

Ungaged Stream Sites...................................................................................................................34

Comparison of Annual Exceedance-Probability Discharges ...............................................................35

Estimates from Annual Exceedance-Probability Analyses .........................................................35
iv

Estimates from Regional Regression Equations............................................................................38

StreamStats ..................................................................................................................................................40

Maximum Floods in Iowa............................................................................................................................40

Summary........................................................................................................................................................42

Acknowledgments.......................................................................................................................................44

References Cited..........................................................................................................................................44

Appendix........................................................................................................................................................51

Regional Skewness Regression.......................................................................................................52

Figures

1. Map showing location of flood regions and streamgages evaluated for use in the 

regional skew analysis and annual exceedance-probability regressions for Iowa .........3

2. Map showing soil regions in Iowa.............................................................................................4

 3. Map showing landform regions in Iowa ...................................................................................5

4. Graph showing annual exceedance-probability curves for Bloody Run Tributary 

near Sherrill, Iowa (05414605), showing the difference between expected moments 

algorithm (EMA/MGB) and Bulletin 17B/GB annual exceedance-probability analyses 

for a crest-stage gage (CSG) with four annual peak discharges below the minimum 

recording threshold....................................................................................................................12

5. Graph showing annual exceedance-probability curves for West Branch Floyd River 

near Struble, Iowa (06600300), showing the effects of including or censoring potentially 

influential low flows identified from the multiple Grubbs-Beck test and of using the 

updated regional-skew-coefficient constant or the superseded regional skew-coefficient value....................................................................................................................................13

6. Map showing location of basin centroids for 240 streamgages used for regional 

skew analysis for Iowa ..............................................................................................................15

7. Graphs showing relation between one-percent annual exceedance-probability discharges and drainage area for A, eight initial; B, three combined; and C, six final 

flood regions defined for study area .......................................................................................19

8. Graphs showing relation between one-percent annual exceedance-probability discharges and drainage area for 127 streamgages in flood region 3 for A, log 10 transformed drainage area and B, power-transformed drainage area......................................22

9. Screen capture of the weighted-multiple-linear regression program (WREG) 

smoothing function for generalized-least-squares (GLS) correlation of the time 

series of annual peak discharges as a function of distance between 176 streamgages in flood region 2 with 30 years of concurrent discharge .........................................23

10. Graphs showing relation between one-percent annual exceedance-probability discharges computed from observed streamflow and those predicted from regression 

equations for flood regions in Iowa.........................................................................................28

11. Graph showing relative percentage change, by drainage area and type of streamgage, between expected moments algorithm (EMA/MGB) and standard Bulletin 

17B/GB analyses computed using the updated regional-skew-coefficient constant 

for one-percent annual exceedance-probability discharges for 283 streamgages 

in Iowa ..........................................................................................................................................37

12. Graph showing relative percentage change, by drainage area and type of streamgage, between expected moments algorithm (EMA/MGB) analyses computed using 

the updated regional-skew-coefficient constant and Bulletin 17B/GB analyses computed using superseded regional skew coefficients for one-percent annual exceedance-probability discharges for 283 streamgages in Iowa.................................................38
v

13. Graph showing relative percentage change by drainage area between one-percent 

annual exceedance-probability discharges computed using regional regression 

equations developed in this study and those developed in the previous study for 

185 streamgages in Iowa...........................................................................................................39

14. Graphs showing relation between maximum flood discharge and drainage area 

for streams in A, flood region 1, B, flood region 2, and C, flood region 3 ..........................41

Tables

1. Description of streamgages located in Iowa and in neighboring States within a 

50-mile buffer of Iowa that were evaluated for use in the regional skew analysis 

and annual exceedance-probability regressions for Iowa ...................................................6

2. Basin characteristics tested for significance in developing regression equations..........7

3. Selected basin-characteristic measurements for streamgages evaluated in study........8

4. Annual exceedance-probability discharges for streamgages evaluated in study 

based on data through the 2010 water year...........................................................................10

 5. Annual exceedance probability and equivalent flood recurrence interval for 

selected probabilities.................................................................................................................10

 6. Annual exceedance-probability-discharge comparison data for 283 streamgages 

in Iowa based on data through the 2010 water year ............................................................12

7. Streamgages removed from regional-regression analyses................................................16

8. Significant explanatory variables and predictive accuracies of preliminary 

statewide regression equations...............................................................................................17

9. Regression equations for estimating annual exceedance-probability discharges 

for unregulated streams in flood region 1 of Iowa................................................................24

10. Regression equations for estimating annual exceedance-probability discharges 

for unregulated streams in flood region 2 of Iowa................................................................24

11. Regression equations for estimating annual exceedance-probability discharges 

for unregulated streams in flood region 3 of Iowa................................................................24

12. Range of basin-characteristic values used to develop annual exceedanceprobability regression equations for unregulated streams in Iowa...................................25

13. Values needed to determine the 90-percent prediction intervals for estimates 

obtained from regional regression equations using covariance matrices in Iowa.........29

14. Variance of prediction values for 394 streamgages included in this study that were 

weighted using expected moments algorithm (EMA/MGB) and regional-regressionequation estimates of annual exceedance-probability discharges...................................30

15. Regional exponents determined from regional regression of log-10 drainage area 

for area-weighting method to estimate annual exceedance-probability discharges 

for ungaged sites on gaged streams.......................................................................................33

16. Significant explanatory variables and predictive accuracies of preliminary regionof-influence equations in Iowa.................................................................................................34

17. Relative percentage change between annual exceedance-probability discharge 

estimates based on data through the 2010 water year for 283 streamgages in Iowa 

using expected moments algorithm (EMA/MGB) and Bulletin 17B/GB analyses 

and using updated and superseded regional skew coefficient values.............................36

18. Relative percentage change from regional-regression-equation estimates 

computed in the previous study to those computed in this study for 185 streamgages in Iowa ..............................................................................................................................39
vi

Conversion Factors 

Inch/Pound to SI

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2

) 2.590 square kilometer (km2)

square mile per mile (mi2

/mi) 1.609 square kilometer per kilometer 

(km2

/km)

Flow rate

cubic foot per second (ft3

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

Hydraulic conductivity

inch per second (in/s) 25,400 micrometers per second (μm/s)

Water year is the 12-month period from October 1 through September 30. The water year 

is designated by the calendar year in which the water year ends and that includes 9 of the 

12 months. Thus, the water year ending September 30, 2010, is called the “2010 water year.”

Abbreviations and Acronyms

ΔA Absolute value of the difference between the drainage areas of a 

streamgage and an ungaged site

Adj-R2 Adjusted coefficient of determination

AEPD Annual exceedance-probability discharge

A(g) Drainage area for a streamgage

A(u) Drainage area for an ungaged site

AVP Average variance of prediction

b Exponent of drainage area from table 15

B17B/GB Bulletin 17B annual exceedance-probability analysis with standard GrubbsBeck low outlier test

BSHAPE Shape factor measure of basin shape

Bulletin 17B/GB Bulletin 17B annual exceedance-probability analysis with standard GrubbsBeck low outlier test

B-GLS Bayesian generalized-least-squares regression

B-WLS Bayesian weighted-least-squares regression

CCM Constant of channel maintenance

CSG Crest-stage gage

DEM Digital elevation model

DESMOIN Percent of basin within Des Moines Lobe landform region

DRNAREA GIS-determined drainage area

EMA/MGB Expected moments algorithm annual exceedance-probability analysis with 

multiple Grubbs-Beck low-outlier test
vii

FEMA Federal Emergency Management Agency

GB Grubbs-Beck low-outlier test

GIS Geographic Information System

GLS Generalized-least-squares regression

HUC Hydrologic unit code

I24H10Y Maximum 24-hour precipitation that happens on average once in 10 years

IDOT Iowa Department of Transportation

KSATSSUR Average saturated hydraulic conductivity of soil

LP3 log-Pearson Type III

MEV Model error variance

MGB Multiple Grubbs-Beck low-outlier test

MSE Mean-square error

MSEG EMA/MGB analysis estimates of the MSE of skew

NHD National hydrography dataset

NRCS Natural Resources Conservation Service 

NWIS National Water Information System

OLS Ordinary-least-squares regression

PILF Potentially influential low flow

PRESS Predicted residual sums of squares

PRISM Parameter-elevation Regressions on Independent Slopes Model

Pseudo-R2 Pseudo coefficient of determination

Q50% Annual exceedance-probability discharge of 50 percent (2-year recurrenceinterval flood discharge)

Q20% Annual exceedance-probability discharge of 20 percent (5-year recurrenceinterval flood discharge)

Q10% Annual exceedance-probability discharge of 10 percent (10-year 

recurrence-interval flood discharge)

Q4% Annual exceedance-probability discharge of 4 percent (25-year recurrenceinterval flood discharge)

Q2% Annual exceedance-probability discharge of 2 percent (50-year recurrenceinterval flood discharge)

Q1% Annual exceedance-probability discharge of 1 percent (100-year 

recurrence-interval flood discharge)

Q0.5% Annual exceedance-probability discharge of 0.5 percent (200-year 

recurrence-interval flood discharge)

Q0.2% Annual exceedance-probability discharge of 0.2 percent (500-year 

recurrence-interval flood discharge)

Qhist Historical flood discharge

QP(g)r RRE estimate of flood discharge for AEPD for a streamgage

QP(g)w WIE estimate of flood discharge for AEPD for a streamgage

QP(u)aw Area-weighted estimate of flood discharge for AEPD for an ungaged site

QP(u)r RRE estimate of flood discharge for AEPD for an ungaged site

QP(u)rw Regression-weighted estimate of flood discharge for AEPD for an ungaged 

site
viii

RMSE Root mean square error, also referred to as SEE

RoI Region of influence

RRE Regional regression equation

SEE Average standard error of estimate, also referred to as RMSE

SEM Standard error of model

SEP Average standard error of prediction

SSURGO NRCS Soil Survey Geographic database

StreamStats USGS Web-based GIS tool (http://water.usgs.gov/osw/streamstats/index.

html)

U Covariance matrix

USACE U.S. Army Corps of Engineers

USDA U.S. Department of Agriculture

USGS U.S. Geological Survey

VIF Variance inflation factor

WBD Watershed boundary dataset

WIE Weighted independent estimates

WREG Weighted-multiple-linear regression program
Methods for Estimating Annual Exceedance-Probability 

Discharges for Streams in Iowa, Based on Data through 

Water Year 2010

By David A. Eash, Kimberlee K. Barnes, and Andrea G. Veilleux

Abstract

A statewide study was performed to develop regional 

regression equations for estimating selected annual exceedance-probability statistics for ungaged stream sites in Iowa. 

The study area comprises streamgages located within Iowa 

and 50 miles beyond the State’s borders. Annual exceedanceprobability estimates were computed for 518 streamgages by 

using the expected moments algorithm to fit a Pearson Type 

III distribution to the logarithms of annual peak discharges for 

each streamgage using annual peak-discharge data through 

2010. The estimation of the selected statistics included a 

Bayesian weighted least-squares/generalized least-squares 

regression analysis to update regional skew coefficients for the 

518 streamgages. Low-outlier and historic information were 

incorporated into the annual exceedance-probability analyses, and a generalized Grubbs-Beck test was used to detect 

multiple potentially influential low flows. Also, geographic 

information system software was used to measure 59 selected 

basin characteristics for each streamgage. 

Regional regression analysis, using generalized leastsquares regression, was used to develop a set of equations 

for each flood region in Iowa for estimating discharges for 

ungaged stream sites with 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 

0.2-percent annual exceedance probabilities, which are equivalent to annual flood-frequency recurrence intervals of 2, 5, 10, 

25, 50, 100, 200, and 500 years, respectively. A total of 394 

streamgages were included in the development of regional 

regression equations for three flood regions (regions 1, 2, and 

3) that were defined for Iowa based on landform regions and 

soil regions. 

Average standard errors of prediction range from 31.8 to 

45.2 percent for flood region 1, 19.4 to 46.8 percent for flood 

region 2, and 26.5 to 43.1 percent for flood region 3. The 

pseudo coefficients of determination for the generalized leastsquares equations range from 90.8 to 96.2 percent for flood 

region 1, 91.5 to 97.9 percent for flood region 2, and 92.4 to 

96.0 percent for flood region 3. The regression equations are 

applicable only to stream sites in Iowa with flows not significantly affected by regulation, diversion, channelization, backwater, or urbanization and with basin characteristics within the 

range of those used to develop the equations. 

These regression equations will be implemented within 

the U.S. Geological Survey StreamStats Web-based geographic information system tool. StreamStats allows users to 

click on any ungaged site on a river and compute estimates 

of the eight selected statistics; in addition, 90-percent prediction intervals and the measured basin characteristics for 

the ungaged sites also are provided by the Web-based tool. 

StreamStats also allows users to click on any streamgage in 

Iowa and estimates computed for these eight selected statistics 

are provided for the streamgage.

Introduction 

Reliable estimates of annual exceedance-probability 

discharges (AEPDs) are essential for the economic planning 

and safe design of bridges, dams, levees, and other structures 

located along rivers and streams, and for the effective management of flood plains. Knowledge of AEPDs allows engineers 

and planners to standardize risk factors. For example, 1- and 

0.2-percent AEPDs are used in the design for, and estimate of, 

scour at bridges (Arneson and others, 2012; Fischer, 1995) and 

to manage development on flood plains through the National 

Flood Insurance Program, administered by the Federal Emergency Management Agency (FEMA; Federal Emergency 

Management Agency, 2002). Methods that are as accurate as 

possible, yet easy to apply, are needed to estimate AEPDs at 

ungaged stream sites in Iowa because long-term annual peakdischarge data are available at few gaged sites.

Streamgages operated by the U.S. Geological Survey 

(USGS) are the primary source of long-term annual peakdischarge data in Iowa. Regression analyses performed on 

AEPDs computed from annual peak-discharge data collected 

at streamgages are used to develop equations to estimate 

AEPDs at ungaged sites. The equations are developed by statistically relating AEPDs to significant basin characteristics for 

selected streamgages. AEPDs computed for streamgages are 

statistics that can change as more annual peak-discharge data 

become available. Statistics become more reliable as longerterm data are collected and used in the computations. 

In response to the need to update and improve the predictive accuracy of estimates of AEPDs for ungaged stream sites 
2 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

in Iowa, the USGS, in cooperation with the Iowa Department 

of Transportation (IDOT) and the Iowa Highway Research 

Board, initiated a statewide study in 2006. This study updates 

AEPD estimation equations for ungaged stream sites in Iowa, 

and AEPDs for streamgages in Iowa, with data collected 

through September 30, 2010. Major accomplishments of the 

study included (1) performing a Bayesian weighted leastsquares/generalized least-squares regression (B-WLS/GLS) 

analysis to update regional skew coefficients for Iowa; 

(2) computing eight selected AEPDs using a new annual 

exceedance-probability analysis method, named expected 

moments algorithm (EMA), at 518 streamgages within Iowa 

and adjacent States with at least 10 years of annual peakdischarge record, based on data through September 30, 2010; 

(3) measuring 59 basin characteristics for each streamgage; 

(4) defining three flood regions for the State and developing 

24 regional regression equations based on basin characteristics 

to estimate the eight selected AEPDs at ungaged stream sites; 

(5) calculating weighted AEPDs at 394 streamgages using 

the weighted independent estimates (WIE) method; and 

(6) calculating AEPD relative percentage change for 

streamgages in Iowa between estimates from different annual 

exceedance-probability analyses based on data through the 

2010 water year (October 1, 2009 through September 30, 

2010) and between regional regression equations developed in 

this study and a previous study (Eash, 2001). A water year is 

the period October 1 through September 30 and is designated 

by the year in which it ends.

Purpose and Scope

Regression equations for estimating AEPDs were developed for use in Iowa and are described in this report. The 

regression equations relate AEPDs to physical and climatic 

characteristics of drainage basins. In addition, the regression 

equations developed from this study also will be included in 

the USGS StreamStats Web-based geographic information system (GIS) tool (http://water.usgs.gov/osw/streamstats/index.

html). StreamStats allows users to obtain selected streamflowstatistic estimates, upstream drainage-basin characteristics, 

and other information for user-selected stream sites. Using a 

GIS-based interactive map of Iowa, the user can “point and 

click” on a stream site and StreamStats will delineate the basin 

boundary upstream from the selected site. The user also can 

“point and click” on USGS streamgages and receive selected 

streamflow statistics and other streamgage information. 

This report presents 24 regional regression equations 

(RREs) that can be used to estimate eight selected annual 

exceedance-probability statistics for ungaged sites on unregulated streams in Iowa. The eight selected annual exceedanceprobability statistics are flood discharges that have probabilities of 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent, which 

are equivalent to annual flood-frequency recurrence intervals 

of 2, 5, 10, 25, 50, 100, 200, and 500 years, respectively; 

hereafter, these statistics are referred to as Q50-percent (%), Q20%, 

Q10%, Q4%, Q2%, Q1%, Q0.5%, and Q0.2%, respectively. This report 

also presents the results of a regional skew analysis performed 

for Iowa to develop updated regional skew coefficients for all 

streamgages in the State. 

The regional regression equations were developed using 

AEPDs computed for streamgages unaffected by regulation, 

diversion, channelization, backwater, or urbanization that 

are located in Iowa and in adjacent States within a 50-mile 

(mi) buffer of Iowa (all gaged drainage basins are within the 

buffer). AEPDs computed for 518 streamgages using the new 

EMA annual exceedance-probability analysis are presented in 

this report. AEPDs for these 518 streamgages were computed 

using annual peak-discharge data collected through September 30, 2010, and were computed using 10 or more years of 

record. The accuracy and limitations of the regression equations and the methodology used to develop the equations are 

described in the report.

Description of Study Area

The study area (fig. 1) includes the entire State of 

Iowa and adjacent areas within a 50-mi buffer of Iowa in 

the neighboring states of Illinois, Minnesota, Missouri, 

Nebraska, South Dakota, and Wisconsin. A map of Iowa soil 

regions created by the National Cooperative Soil Survey 

and the Natural Resources Conservation Service (NRCS) is 

shown in figure 2 (ftp://ftp-fc.sc.egov.usda.gov/IA/technical/

IowaSoilRegionsMap.html). Oschwald and others (1965) 

present a detailed description of soils in Iowa. There are 

10 landform regions in Iowa, each having distinctive topography and geology (fig. 3). A brief description of the landform 

regions in Iowa is presented in Eash and Barnes (2012) and 

a detailed description is presented by Prior (1991). Prior and 

others (2009) describe updates to landform regions in Iowa.

Most precipitation in the study area results from storms 

moving inland primarily from the Gulf of Mexico and secondarily from the Pacific Ocean (Soenksen and Eash, 1991). 

Annual precipitation, which is mostly rain, ranges from 

26 inches (in.) in the extreme northwest to as much as 38 in. 

in the southeast; the statewide average is around 34 in. 

(National Climatic Data Center, 2012). About 75 percent of 

the annual precipitation is received during April through September. Typically, during August through February, streamflow 

in most unregulated streams in the study area is base flow; 

during March through July, streamflow is significantly greater, 

primarily as a result of snowmelt during late February through 

early April and rainfall during May through July. Annual 

maximum streamflows are typically during April through July.

Previous Studies 

This is the seventh in a series of reports that describe 

flood characteristics for Iowa streams. The first report 

(Schwob, 1953) contained information on AEPDs for 

55 continuous-record streamgages in Iowa using annual 
Introduction 3

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1

40 °

SOUTHDAKOTA

NEBRASKA

MISSOURI

WISCONSIN

ILLINOIS

MINNESOTA

0 30 60 MILES

0 30 60 KILOMETERS

Missou ri

River

Mis sissipp i River

Des Moines River

Chariton River

Iowa River

Base from U.S. Geological Survey digital data, 2006, 1:24,000Universal Transverse Mercator projection, Zone 15

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9910 0

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2

Study area boundary Flood-region boundary  and map numberU.S. Geological Survey  streamgage and map  number

EXPLANATION

1

#470

96° 97° 98° 95° 94° 93° 92° 91° 90° 89°

44°43°42°41°

Figure 1. Location of flood regions and streamgages evaluated for use in the regional skew analysis and annual exceedance-probability regressions for Iowa.
4 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

4

6

8

2

1 5

2 1

1 7

1 2

1 1

9

10

2 2

1

1 4

5

2 0

19

3 7

1 6

23

1 3

1 8

0 30 60 MILES

0 30 60 KILOMETERS

96°

43°42°41°

95° 94° 93° 92° 91° 90°

1 Loess over Glacial Till-low rainfall2 Northwest Iowa Loess3 Tazewell Glacial Till4 Loamy Wisconsin Glacial Till5 Clayey Lake Deposits

6 Loamy Glacial Till7 Shallow to Bedrock8 Loess with Bedrock Outcrops9 Shallow Loess over Glacial Till10 Loess Ridges and Sideslopes

11 Loess with Exposures of Glacial Till12 Very Deep Loess13 Missouri River Bluffs14 Missouri River Alluvium15 Loess Ridges/Glacial Till-Southwest  Iowa

16 Loess, Shale, and Glacial Till17 Loess Ridges/Glacial Till-Southeast Iowa18 Eolian Sand 19 Loess-Timbered20 Alluvium

21 Loess Ridges/Glacial Till Sideslopes22 Loess Ridges/Clay Paleosol23 WaterFlood-region boundary

Soil region

EXPLANATION

Missouri River

Mis sissipp i River

Base from U.S. Geological Survey digital data, 2006, 1:24,000Universal Transverse Mercator projection, Zone 15

Figure 2. Soil regions in Iowa.
Introduction 

5

0 30 60 MILES

0 30 60 KILOMETERS

96°

43°42°41°

Base from U.S. Geological Survey digital data, 2006, 1:24,000Universal Transverse Mercator projection, Zone 15

95° 94° 93° 92° 91° 90°

Des Moines LobeEast-Central Iowa Drift PlainIowa-Cedar Lowland

Iowan SurfaceLoess HillsMississippi River Alluvial Plain

EXPLANATION

Missouri River Alluvial PlainNorthwest Iowa PlainsPaleozoic Plateau

Southern Iowa Drift PlainFlood-region Boundary

Landform regions

Missouri

R

iver

Mississipp i River

Figure 3. Landform regions in Iowa.
6 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

peak-discharge data collected through the 1950 water year. 

Schwob (1953) defined eight flood regions for Iowa using 

34 of the 55 streamgages with AEPD information and presented a method for estimating AEPDs for ungaged sites with 

drainage areas greater than about 100 square miles (mi2). 

The method used graphs developed for each region to relate 

drainage area to mean annual floods to estimate a mean annual 

flood-discharge statistic for an ungaged site, and then used 

a graph relating recurrence interval to the ratio of the mean 

annual flood to determine a ratio value for a recurrence interval. AEPDs for as large as the Q2% flood discharge were then 

estimated by multiplying the mean annual flood discharge for 

an ungaged site by the ratio for a recurrence interval. Predictive accuracies were not determined for the AEPD estimates. 

The second report (Schwob, 1966) contained information on AEPDs for 147 continuous-record and crest-stage 

(partial-record) streamgages using annual peak-discharge 

data collected through the 1965 water year. Schwob (1966) 

defined two flood regions for Iowa using 147 streamgages and 

presented a two- or three-variable regression equation for each 

region for estimating a mean annual flood-discharge statistic for ungaged sites with drainage areas greater than 1 mi2

. 

A graph relating recurrence interval to the ratio of the mean 

annual flood was used to determine a ratio value for a recurrence interval. AEPDs for as large as the Q2% flood discharge 

were then estimated by multiplying the mean annual flood 

value for an ungaged site by the ratio for a recurrence interval. 

Average standard errors of estimate for the two regression 

equations ranged from 30.4 to 37.9 percent. Three basin characteristics were measured manually for each streamgage.

The third report (Lara, 1973) contained information on 

AEPDs for 136 continuous-record and crest-stage streamgages using annual peak-discharge data collected through the 

1972 water year. Lara (1973) defined two flood regions for 

Iowa using the 136 streamgages and presented one- or twovariable regression equations for each region for estimating six 

annual exceedance-probability statistics for as large as the Q1%

flood discharge for ungaged sites with drainage areas greater 

than 2 mi2. Average standard errors of estimate for the regression equations ranged from 26 to 44 percent. Thirteen basin 

characteristics were manually measured for each streamgage.

The fourth report (Lara, 1987) contained information 

on AEPDs for 263 continuous-record and crest-stage streamgages using annual peak-discharge data collected through the 

1984 water year. Lara (1987) defined five flood regions for 

Iowa using 251 of the 263 streamgages and presented onevariable regression equations for each region for estimating six 

annual exceedance-probability statistics for as large as the Q1%

flood discharge for ungaged sites with drainage areas greater 

than 0.04 mi2. Average standard errors of estimate for the 

30 regression equations ranged from 20 to 61 percent. Two 

basin characteristics were measured manually for each 

streamgage.

The fifth report (Eash, 1993) contained information on 

AEPDs for 188 continuous-record and crest-stage streamgages 

using annual peak-discharge data collected through the 1990 

water year. Eash (1993) defined one flood region for Iowa 

using 164 streamgages and presented three-variable, statewide, 

drainage-basin-characteristic regression equations for estimating six annual exceedance-probability statistics for as large as 

the Q1% flood discharge for ungaged sites with drainage areas 

greater than 0.34 mi2. Average standard errors of prediction for 

the drainage-basin regression equations ranged from 38.6 to 

50.2 percent. A GIS program developed by the USGS, named 

Basinsoft (Harvey and Eash, 1996), was used to automate 

measurements of 26 basin characteristics for each streamgage. 

Eash (1993) also defined two flood regions for Iowa using 

157 streamgages and presented two sets of one- or two-variable channel-geometry-characteristic regression equations for 

either region for estimating six annual exceedance-probability 

statistics for as large as the Q1% flood discharge for ungaged 

sites with bankfull widths greater than 9.6 feet (ft). Average 

standard errors of prediction for the channel-geometry regression equations ranged from 30.3 to 70.0 percent. The channelgeometry-characteristic regression equations required the collection of channel-geometry measurements at ungaged sites.

The sixth report (Eash, 2001) contained information on 

AEPDs for 291 continuous-record and crest-stage streamgages 

using annual peak-discharge data collected through the 1997 

water year. Eash (2001) defined three flood regions for Iowa 

using 241 of the 291 streamgages and presented one-, two-, or 

three-variable regression equations for each region for estimating eight annual exceedance-probability statistics as large as 

the Q0.2% flood discharge for ungaged sites with drainage areas 

greater than 1.30 mi2. Average standard errors of prediction for 

the regression equations ranged from 30.8 to 42.7 percent. The 

Basinsoft program was used to measure 38 basin characteristics for each streamgage. A regional skew analysis was performed as part of this study using 239 of the 291 streamgages 

and an updated regional-skew-coefficient constant was used 

for computing annual exceedance-probability analyses for 

streamgages in Iowa.

Methods for Dataset Development for 

Streamgages

Data used in this report were collected for 523 active 

and inactive continuous-record and crest-stage streamgages 

located in Iowa and within a 50-mi buffer of Iowa in the 

neighboring States of Illinois, Minnesota, Missouri, Nebraska, 

South Dakota, and Wisconsin (fig. 1; table 1, link to Excel 

file). Streamgages with at least 10 years of annual peak 

discharges and unaffected by regulation or diversion initially 

were selected for evaluation in the study, which included 

284 streamgages in Iowa and 239 streamgages in neighboring States. Streamgages from neighboring States were used to 

improve the representativeness of annual exceedance-probability statistics and basin characteristics in Iowa border areas 

and to provide better estimates of the error of the regression 

equations for ungaged sites near the State border. Of these 

Table 1. Description of streamgages located in Iowa and in neighboring States within a 50-mile 

buffer of Iowa that were evaluated for use in the regional skew analysis and annual exceedanceprobability regressions for Iowa.
Methods for Dataset Development for Streamgages 7

original 523 streamgages, five of them included for evaluation in the regional skew analysis for Iowa subsequently were 

removed resulting in a total of 518 streamgages evaluated for 

use in annual exceedance-probability regressions for Iowa. See 

the sections Regional Regression Analyses to Estimate Annual 

Exceedance-Probability Discharges for Ungaged Stream Sites 

and Definition of Flood Regions for more information on the 

removal of the five streamgages from the original dataset of 

523 (table 1, link to Excel file).

Basin Characteristics

Physical processes controlling floods vary from one 

region to another region and from one stream site to another 

stream site, but they generally are related to storm events (precipitation intensity) and drainage area. Flood peaks generated 

by snowmelt have a different impetus than those generated by 

rainfall. Peak discharges are a function of many interrelated 

factors that include runoff response to geology, soils, slope, 

and land cover; surface storage such as wetlands, lakes, and 

flood plains; and routing related to drainage density, basin 

shape, channel length, and slope (C.P. Konrad, U.S. Geological Survey, written commun., 2009). Basin characteristics 

investigated in this study as potential explanatory variables 

in the regression analysis were selected on the basis of their 

theoretical relation to peak discharges, results of previous 

studies in similar hydrologic areas, and the ability to quantify 

the basin characteristics using GIS technology and digital 

datasets. The use of GIS enables the automation of the basincharacteristic measurements and solution of the RREs using 

StreamStats.

Using GIS technology, 59 basin characteristics were 

measured for each of the 518 streamgages evaluated for use in 

the development of the regression equations. Table 2 (link to 

Excel file) provides a brief description of each basin characteristic and the data source used to measure the characteristic. 

Basin-characteristic names used in this study were selected to 

maintain consistency with the names of explanatory variables 

in the USGS StreamStats Web-based GIS tool (http://water.

usgs.gov/osw/streamstats/bcdefinitions1.html). 

The basin characteristics can be separated into four 

categories: morphometric (physical or shape) characteristics, 

hydrologic characteristics, pedologic (soils)/geologic/landuse characteristics, or climatic characteristics. Morphometric 

characteristics were measured from one to three data 

sources, which are described in the following section Geographic Information System Measurements. Hydrologic characteristics initially were computed as observed values for 

208 continuous-record streamgages using daily mean discharge data and subsequently were mapped using a kriging 

procedure to compute interpolated values for a low-flow 

estimation study performed for Iowa (Eash and Barnes, 2012). 

A list of the 208 streamgages included in the low-flow study, 

descriptions of the hydrologic-characteristic computations and 

kriging procedure, and isoline maps created from kriged grids 

for three of the five hydrologic characteristics are presented 

in Eash and Barnes (2012). The pedologic/geologic/land-use 

characteristics were computed from the NRCS Soil Survey 

Geographic (SSURGO) Database (Soil Survey Staff, 2012) 

for seven soil characteristics, from the Iowa Geological and 

Water Survey Des Moines Lobe landform region boundary for 

the Des Moines Lobe geologic characteristic (Prior and others, 

2009), and from the Multi-Resolution Land Characteristics 

Consortium 2001 National Land Cover Database (MultiResolution Land Characteristics Consortium, 2012) for the 

land-use characteristic that measured percent area of row crops 

(http://www.mrlc.gov/index.php; Homer and others, 2004). 

The climatic characteristics were computed from Oregon State 

University Parameter-elevation Regressions on Independent 

Slopes Model (PRISM) datasets (Parameter-Elevation Regressions on Independent Slopes Model Climate Group, 2008) and 

from Midwest Climate Center Bulletin 71(Huff and Angel, 

1992). 

Geographic Information System Measurements 

Three primary GIS-data layers were processed to 

produce the Iowa StreamStats data layers. These data layers 

were needed to delineate accurate stream networks and basin 

boundaries, and the layers were used to measure 26 morphometric basin characteristics (table 2, link to Excel file). The 

three primary GIS-data layers include the 1:24,000-scale 

USGS National Hydrography Dataset (NHD) (http://nhd.

usgs.gov/; Simley and Carswell, 2009), the 1:24,000-scale 

U.S. Department of Agriculture (USDA)/NRCS Watershed 

Boundary Dataset (WBD) (http://datagateway.nrcs.usda.

gov/; USGS and NRCS, 2009) using 12-digit hydrologic unit 

codes (HUCs), and the 10-meter (m) (32.81 ft) USGS National 

Elevation Dataset (http://ned.usgs.gov/; Gesch, 2007). 

Several preprocessing steps were needed for each of the 

three data layers to facilitate rapid determination of basin characteristics. Preprocessing of the NHD included removing flowline paths that represent man-made features (a stream network 

that only represents natural streams is needed) and selection of 

the primary flow path in those areas where the NHD indicated 

split flow (such as might happen when flow splits around an 

island in a river or with a braided channel). The NHD and 

WBD had to be verified that the stream from the NHD only 

crossed the watershed boundary (from the WBD) at the outlet 

(unless the watershed is downstream from another watershed, 

in which case the main stem stream will enter the watershed at 

one place); and watershed outlets should align exactly to the 

confluences of the streams. For the national elevation dataset, 

downloaded blocks were mosaicked into one tile, data were 

extracted for a 4-kilometer (km) (2.5 mi) buffer area around 

each 8-digit HUC, and projected from decimal degrees to Universal Transverse Mercator Zone 15. A hydro-corrected digital 

elevation model (DEM) was then developed by filling depressions or sinks, using the basin boundaries from the WBD to 

conserve known drainage divides, and using the streams from 

the NHD to create well-defined flow paths through the elevation data. 

Table 2. Basin characteristics tested for significance in developing 

regression equations.
8 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

ArcHydro Tools, version 1.3, a set of utilities developed 

to operate in the ArcGIS, version 9.3, environment (Environmental Systems Research Institute, 2009) was used to process 

58 HUCs consisting of eight digits to create StreamStats data 

layers for the entire State. To calculate basin characteristics to 

develop the RREs for estimating AEPDs for Iowa, additional 

data layers were generated. These primary base-grid data 

layers include catchments, flow accumulation, flow direction, and an artificial flow-path grid used to delineate drainage basins. These additional layers then were used to create 

layers that control the StreamStats delineation of a watershed, 

subwatersheds, and stream networks within these watersheds, 

including the created layers named AdjointCatchment, Catchment, DrainageLine, DrainagePoint, LongestFlowPathCat, 

and LongestFlowPathAdjCat. Once processing was complete 

for all 58 processing units, a global geodatabase was created 

to direct StreamStats as to how all units relate to each other. 

In addition, the DEM was resampled to 150 m for use in the 

basin-length calculations. All 59 basin characteristics listed in 

table 2 (link to Excel file) were measured using ArcHydro 

Tools or Spatial Analyst tools in ArcGIS, version 9.3 (Environmental Systems Research Institute, 2009).

To measure basin characteristics for streamgages located 

outside of Iowa, similar preprocessing steps were performed 

on GIS data layers for 27 additional 8-digit HUCs located in 

neighboring states. These 27 HUCs are not part of the GIS 

data layers used by StreamStats for Iowa. Because certified 

WBD data were not available at the time in adjacent states, the 

preprocessing of these 27 HUCs did not include the “walling” 

of basin boundaries using WBD, but did include the “burning” 

of streams from the NHD into the national elevation dataset; 

however, a global geodatabase was not created for these 

27 HUCs because none of the streamgages within these HUCs 

accumulated flow from more than one HUC.

GIS measurements of four hydrologic basin characteristics (table 2, link to Excel file) were interpolated by areaweighting values for streamgage watershed boundaries from 

grids that were created using a kriging procedure (Eash and 

Barnes, 2012). GIS measurements of seven soil characteristics (table 2, link to Excel file) were made using a three-step 

process. First, the NRCS Soil Data Viewer tool, built as an 

extension of ArcMap (Environmental Systems Research Institute, 2009), was used to create four 8-digit HUC data layers 

for the soil characteristics. Second, a shapefile was created 

for the hydrosoils data layer (includes the four hydrologic soil 

types A, B, C, and D; (table 2, link to Excel file), and a grid 

was created for each of the SAND, CLAY, and KSATSSUR 

data layers. Third, the ArcMap attribute selection tool was 

used to calculate a percent-area value for each hydrologic 

soil type, and the Spatial Analyst tool was used to calculate 

area-weighted values for SAND, CLAY, and KSATSSUR for 

each streamgage watershed boundary. The geologic characteristic DESMOIN, the land-use characteristic ROWCROP, 

and 19 of the 20 climatic characteristics (table 2, link to Excel 

file) were all measured from grids as area-weighted values 

for each streamgage watershed (PRC5_7 was calculated for 

each watershed as the mean of May, June and July mean 

precipitation).

Table 3 (link to Excel file) lists two drainage area values 

for each streamgage included in the study. Each streamgage 

has a drainage area that is listed in the USGS National Water 

Information System (NWIS) data base (U.S. Geological 

Survey, 2012), which is referred to as the “published” drainage area. Published drainage areas were determined primarily 

from 1:24,000-scale topographic maps by manual planimetering or GIS digitizing methods at the time streamgage 

operation began. Drainage area values listed in table 3 (link 

to Excel file) as “GIS” drainage areas, for the basin characteristic DRNAREA, were measured as part of this study using a 

two-step process within ArcHydro Tools. First, a streamgage 

location was selected using the point generation tool; second, 

one of the watershed delineation tools (such as Batch Watershed Delineation) was used to automatically delineate the 

watershed boundary using hydro-corrected DEM data. The 

watershed delineation process in the second step delineates 

the basin boundary from the DEM data proceeding from 

the streamgage location until an existing basin boundary is 

reached within the WBD data, and then the delineation follows the WBD boundary for the remainder of the watershed 

delineation. For some streamgages with small drainage areas 

that are located completely within a 12-digit HUC, the entire 

watershed delineation was made from the DEM data. 

GIS delineations of watershed boundaries were inspected 

for streamgages with drainage area differences greater than 

5 percent from published values. Basin boundaries of several 

GIS-delineated watersheds were edited where the delineation did not match well with digital raster graphics elevation 

contours. Most edits made only a small difference in the drainage area value for the watershed. If the GIS-delineated basin 

boundary was accurate according to the 8-digit HUC, WBD 

line work, and digital raster graphics contour lines, then the 

GIS delineation was accepted even if it exceeded a 5-percent 

difference from the published drainage area. GIS delineations 

generally are believed to be more accurate than the published 

drainage areas. Most of the GIS watershed delineations are 

using part of the WBD boundaries, which have been certified 

by NRCS, and use of the WBD data accounts for some of the 

differences between GIS and published values of drainage 

areas. GIS measurements of drainage area (DRNAREA) were 

used to develop the regression equations because StreamStats 

will use the same GIS data layers and delineation methods 

for determining watershed boundaries and drainage areas for 

ungaged stream sites. Drainage areas of the 518 streamgages 

ranged from 0.05 to 7,783 mi2 (table 3, link to Excel file).

Annual Peak-Discharge Data

A standard, continuous-record streamgage typically 

records stage (the gage height or water-surface elevation) 

every 15 minutes throughout the course of a water year. A 

crest-stage gage (CSG) is a partial-record streamgage that only 

provides information on the highest stage since the streamgage 

Table 3. Selected basin-characteristic measurements for streamgages 

evaluated in study.
Methods for Dataset Development for Streamgages 9

was last visited. CSGs are the primary source of annual 

peak-discharge data for small drainage basins in Iowa (U.S. 

Geological Survey, Iowa Water Science Center, Flood Information at Selected Bridge and Culvert Sites; http://ia.water.

usgs.gov/projects/ia006.html). Annual peak discharges are 

computed for continuous-record streamgages and CSGs by 

use of a stage-discharge relation (Rantz and others, 1982). 

The stage-discharge relation, or rating, is used to determine 

discharges for all recorded stages at streamgages. The largest 

discharge during a water year is the annual peak discharge, 

and the annual peak-discharge record is the compilation of 

all recorded annual peak discharges. Annual peak-discharge 

records collected through the 2008 water year (through September 30, 2008) initially were retrieved for 330 streamgages 

for use in computing station skew coefficients and the mean 

square error (MSE) of station skews for a regional skew 

analysis performed for Iowa that is described in a following section, Regional Skew Analysis. Annual peak-discharge 

records collected through the 2010 water year (through 

September 30, 2010) subsequently were retrieved for 

518 streamgages for use in computing annual exceedanceprobability analyses that are described in the section Expected 

Moments Algorithm (EMA/MGB) Analysis. All annual 

peak-discharge records analyzed in this study were retrieved 

from the USGS NWIS database (U.S. Geological Survey, 

2012; http://nwis.waterdata.usgs.gov/usa/nwis/peak). Annual 

peak-discharge data were reviewed to eliminate data affected 

by regulations or diversions from biasing the computation of 

AEPDs. Annual peak-discharge records were reviewed by 

using the PFReports computer program described by Ryberg 

(2008).

Trend Analyses

Annual peak-discharge records retrieved for the 518 

streamgages were analyzed for the entire period of record for 

trends using the Kendall’s tau hypothesis test in the PeakFQ 

program (Flynn and others, 2006). Trends in the annual 

peak-discharge data could introduce a bias into the annual 

exceedance-probability analyses because a major assumption of probability analyses is annual peak discharges are 

independent and stationary with time. The Kendall’s tau test 

computes the monotonic relation between peaks (discharge) 

and time (water years) (Helsel and Hirsch, 2002). A p-value 

threshold of 5 percent (α=0.05) was used in this study for the 

Kendall’s tau test, and p-values less than or equal to 5 percent 

were flagged as having statistically significant trends (positive 

or negative). Results of the Kendall’s tau tests (table 1, link to 

Excel file) indicated statistically significant trends for 58 of the 

518 streamgages tested using the entire period of record, for 

which 22 of the trends were negative and 36 of the trends were 

positive. 

Wahl (1998) describes how Kendall’s tau test results 

may be sensitive to multiyear sequences of larger or smaller 

discharges if the sequences happen near the beginning or 

end of the period of record used in the test. Although trend 

results are relatively insensitive to individual outliers, multiyear sequences of extremes near either end of the record can 

have a significant effect on the test results, but may imply 

no systematic change. Annual peak-discharge records for the 

58 streamgages initially indicated to have significant trends 

were retested using the Kendall’s tau test after a few annual 

peak discharges were removed, consecutively, from either the 

beginning or the end of the record, or from the beginning and 

the end of the record. A record-length threshold of 94 percent 

was used for the retesting of the trend analysis. Initially, a 

95-percent threshold was tested on the basis of the assumption that if a significant trend is not identified using 95 percent 

of the record, then there probably is not a trend in the data; 

because of rounding for several streamgages, a 94-percent 

threshold subsequently was selected for the retesting. For 

example, a streamgage with a record length of 50 years could 

have no more than 3 years of record removed for the retest. 

Results of the Kendall’s tau retests indicated statistically 

significant trends for 25 of the 58 streamgages retested using 

94 percent of the entire record length (table 1, link to Excel 

file). A review of the annual peak-discharge records for these 

25 sites indicated that 22 of them have either short records of 

less than 15 years or broken records with sequences of missing 

years in their records because of (1) intermittent gage operation, or (2) annual peak discharges that were censored because 

they were lower than the recordable level of the CSG. Because 

of the short or broken records for these 22 streamgages and 

because the remaining 3 streamgages are isolated and not 

supported by trends at other nearby streamgages, there is 

uncertainty about whether the records of these 25 streamgages 

represent actual trends or are anomalies. Therefore, the 

25 streamgages that recorded significant trends were included 

in the regression analyses and were removed only if they were 

indicated to be a significant outlier in the regression model.

The number of uncensored annual peak-discharge record 

lengths used in the study for the 518 streamgages ranged 

from 9 to 108 years with a mean of 35.4 years and a median 

of 28 years (table 1, link to Excel file). Uncensored annual 

peak discharges include all systematic annual peak discharges 

collected at continuous-record streamgages and include all 

systematic annual peak discharges collected at CSGs that 

were higher than the minimum recordable level of the CSG. 

Annual peak-water levels that did not reach the bottom of the 

CSG were recorded with a “less-than discharge value,” which 

is the minimum recordable discharge threshold of the CSG. 

These “less-than discharge values,” or Qless-than, are censored 

annual peak-discharge data; estimates for these censored 

discharges bounded between 0 and Qless-than can be used in the 

log-Pearson Type III exceedance-probability analysis. For the 

CSG Drainage Ditch 97 tributary near Britt, Iowa (streamgage 

0548065350, map number 270, fig. 1), the one streamgage 

in table 1 (link to Excel file) that lists 9 years of uncensored 

data, the inclusion of seven censored annual peak discharges 

(7 years for which the discharge did not reach the minimum 

recordable discharge) with the nine uncensored annual peak 

discharges (9 years for which the discharge exceeded the 
10 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

minimum recordable discharge) provides a total of 16 years of 

annual peak-discharge record for the annual exceedance-probability analysis. This streamgage was included in this study 

because its annual peak-discharge record included at least 

10 years of record.

Annual Exceedance-Probability Analyses

To estimate AEPDs at continuous-record streamgages 

and CSGs for this study, such as the Q1-percent (%) flood discharge, 

EMA exceedance-probability analyses were performed. EMA 

analyses provide a new alternative method to standard Bulletin 

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

exceedance-probability analyses. To specify the different lowoutlier tests used by either exceedance-probability analysis 

method in this study, hereafter EMA analyses will be referred 

to as EMA/MGB and Bulletin 17B analyses will be referred to 

as Bulletin17B/GB or B17B/GB. MGB refers to the multiple 

Grubbs-Beck test for detecting low outliers and GB refers 

to the standard Grubbs-Beck test for detecting low outliers. 

Because the Hydrologic Frequency Analysis Work Group now 

recommends use of the multiple Grubbs-Beck (MGB) test for 

detecting low outliers for exceedance-probability analyses 

(http://acwi.gov/hydrology/Frequency/minutes/Minutes_

HFAWG_meeting_mar19_2012_040212.pdf) and because the 

EMA analysis method needs a consistent statistical test (MGB) 

to identify potentially influential low flows in an annual peakdischarge series to properly reduce the effect of low outliers 

(N.A. Barth, U.S. Geological Survey, written commun., 2012), 

it is important to specify for this study that all EMA analyses 

included the MGB test and all standard Bulletin 17B analyses 

included the standard Grubbs-Beck test. Additional information on both types of Grubbs-Beck tests are presented in the 

section Multiple Grubbs-Beck (MGB) Test for Detecting Low 

Outliers.

For this report, annual exceedance probabilities were 

estimated for the Q50%, Q20%, Q10%, Q4%, Q2%, Q1%, Q0.5%, and 

Q0.2% flood discharges using EMA/MGB for 518 streamgages 

listed in table 4 (link to Excel file). The annual magnitude and 

probability of flood discharges or AEPDs for a streamgage 

are computed from an exceedance-probability analysis that 

relates observed annual peak discharges to annual exceedance 

probability. Annual exceedance probability is an estimate of 

the likelihood of a flood of a specific magnitude happening in 

any 1 year. 

Annual exceedance probabilities formerly were reported 

as flood recurrence intervals expressed in years. For example, 

a flood magnitude that has a 1-percent chance (annual exceedance probability=0.01) of being exceeded during any particular year is expected to be exceeded on average once during 

any 100-year period (recurrence interval). Percent probability 

is the inverse of the recurrence interval multiplied by 100. 

Because of widespread confusion caused in recent years by 

two or more “100-year floods” happening in a period of much 

less than 100 years, the scientific and engineering community 

has begun expressing the annual likelihood of occurrence of 

flood discharges as a probability. Selected annual exceedance 

probabilities and equivalent flood recurrence intervals are 

listed in table 5. Although the annual probability is an estimate 

of the likelihood of a flood discharge of a specific magnitude 

happening in any 1 year, more than one flood discharge with a 

specific magnitude and annual probability could happen in the 

same year.

Annual exceedance-probability analyses were not 

computed for regulated annual peak-discharge records of 

streamgages located downstream from U.S. Army Corps of 

Engineers (USACE) reservoirs located on the Chariton, Des 

Moines, Iowa, or Missouri Rivers or downstream from locks 

and dams located on the Mississippi River. Table 1 (link to 

Excel file) lists one streamgage located downstream from the 

USACE reservoir on the Iowa River (streamgage 05454500, 

map number 170, fig. 1) and four streamgages located downstream from the USACE reservoir on the Chariton River 

(streamgages 06903900, 06904000, 06904500, and 06905000; 

map numbers 518, 520, 521, and 523, fig. 1, respectively) for 

which annual exceedance-probability analyses were computed 

for unregulated annual peak-discharge records. Information on 

regulated flow-frequency studies for the Iowa and Des Moines 

Rivers are available from U.S. Army Corps of Engineers 

(2009, 2010).

Bulletin 17B/GB Analyses

The Interagency Advisory Committee on Water Data has 

established standard methods for estimating annual exceedance probabilities for streamgages by fitting a log-Pearson 

Type III (LP3) distribution to the logarithms (base 10) of the 

annual peak discharges as described in Bulletin 17B (Interagency Advisory Committee on Water Data,1982). Before 

this study, exceedance-probability analyses for streamgages 

in Iowa were computed using standard Bulletin 17B/GB 

analyses (Flynn and others, 2006). Standard Bulletin 17B/GB 

and EMA/MGB analyses use a LP3 distribution to compute 

AEPDs. Fitting the LP3 distribution requires calculating the 

mean, standard deviation, and skew coefficient of the logarithms of the annual peak-discharge record, which describe 

Table 4. Annual exceedance-probability discharges for streamgages 

evaluated in study based on data through the 2010 water year.

Table 5. Annual exceedance probability and equivalent flood 

recurrence interval for selected probabilities.

Annual exceedance 

probability (percent)

Recurrence interval 

(years)

50 2

20 5

10 10

4 25

2 50

1 100

0.5 200

0.2 500
Methods for Dataset Development for Streamgages 11

the midpoint, slope, and curvature of the annual exceedanceprobability curve, respectively (Gotvald and others, 2012). 

Estimates of AEPDs are computed using these three statistics 

from the LP3 distribution in the following equation:

logQp = X + KpS (1)

where

Qp is the P-percent AEPD, in cubic feet per 

second (ft3/s);

X is the mean of the logarithms of the annual 

peak discharges;

Kp is a factor based on the skew coefficient 

and the given percent annual exceedance 

probability and is obtained from appendix 

3 in Bulletin 17B (Interagency Advisory 

Committee on Water Data,1982); and

S is the standard deviation of the logarithms 

of the annual peak discharges, which is a 

measure of the degree of variation of the 

annual values about the mean value.

The mean, standard deviation, and skew coefficient can 

be estimated from the available sample data (observed annual 

peak discharges). The station skew coefficient measures the 

asymmetry of the probability distribution of a set of annual 

peak discharges (Gotvald and others, 2009; Feaster and others, 

2009; Weaver and others, 2009). Large positive station skew 

coefficients can result from steep basin and channel slopes, 

low infiltration rates, fast conveyance through systems, and 

(or) one or more extremely large peak discharges (high outliers). Conversely, large negative station skew coefficients can 

result from low mean basin slopes, high infiltration rates, high 

channel losses, a substantial percentage of a basin controlled 

by lakes or swamps, and (or) one or more extremely low peak 

discharges (low outliers) (Ahearn, 2003). The station skew 

coefficient is sensitive to extreme events; therefore, the station 

skew coefficient for short records may not provide an accurate 

estimate of the population or true skew coefficient (Gotvald 

and others, 2009; Feaster and others, 2009; Weaver and others, 2009). Thus, guidelines in Bulletin 17B (Interagency 

Advisory Committee on Water Data, 1982) recommend that 

the skew coefficient calculated from streamgage data (station 

skew) needs to be weighted with a generalized, or regional, 

skew determined from an analysis of selected long-term 

streamgages in the study area (Gotvald and others, 2012). The 

weighted skew is determined by weighting the station skew 

and the regional skew inversely proportional to their respective mean square errors, as shown in the following equation 

(Interagency Advisory Committee on Water Data, 1982):

Gw = , MSER(Gs

) + MSEs(GR) 

MSER + MSEs

(2)

where

Gw is the weighted skew,

Gs is the station skew,

GR is the regional skew, and

MSER and MSEs are the mean square error of the regional and 

station skew, respectively.

An annual peak-discharge record at a streamgage may 

include extremely small or extremely large discharge values 

that are statistically determined to be low or high outliers 

in the record. The peak-discharge record also may include 

information about historic floods that happened outside of 

the period of streamgage operation, or systematic record. 

Historic floods typically are considered to be the largest peak 

discharges during an extended period of time that is longer 

than the systematic record. Bulletin 17B (Interagency Advisory Committee on Water Data, 1982) provides guidelines 

for detecting outliers and interpreting historical floods, and 

provides computational methods for appropriate adjustments 

to the LP3 distribution to account for outliers and historical flood information (Gotvald and others, 2012). Although 

these adjustments generally improve estimates of AEPDs at a 

streamgage, the EMA/MGB method integrates censored discharges (low and high outliers) and historical flood discharges 

more efficiently (Cohn and others, 1997) than the guidelines 

provided in Bulletin 17B. The station estimates of mean, 

standard deviation, and skew for streamgages included in this 

study were computed using EMA/MGB. 

Expected Moments Algorithm (EMA/MGB) 

Analyses

In this study, the EMA/MGB method was used to 

compute LP3 exceedance-probability analyses for all 518 

streamgages evaluated for use in the development of regression equations and for all 330 streamgages evaluated for use 

in the regional skew analysis for Iowa. PeakfqSA versions 

0.960 and 0.974, an EMA/MGB program developed by Cohn 

(2011), were used to compute all EMA/MGB annual exceedance-probability analyses for this study. Identical results were 

obtained using either version of PeakfqSA for the streamgages 

tested. For streamgages that have systematic annual peakdischarge records for complete periods, no low outliers, and 

no historical flood information, the EMA/MGB method provides identical log estimates of the three LP3 statistics (mean, 

standard deviation, and skew coefficient) as the standard LP3 

method described in Bulletin 17B (Gotvald and others, 2012). 

The EMA/MGB method allows for the integration of censored 

and interval peak-discharge data in the analysis. For historical 

periods that extend outside of the period of systematic record 

or between periods of systematic record, censored data may 

be expressed in terms of discharge perception thresholds. 

Thus, there are two types of censored data: (1) annual peak 

discharges collected at CSGs for which the discharge is only 

known to be less than the minimum recordable discharge 

threshold, or (2) in the case of historical periods, annual 

peak discharges that are only known not to have exceeded a 

recorded historical flood discharge. For example, the Thompson River at Davis City (streamgage 06898000, map number 

501, fig. 1) has historic information that indicates a recorded 
12 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

hist, of 30,000 ft3

/s in 1885 was 

the largest during the historical period (1885–1917) before 

streamgage operation began in 1918 and during the historical 

period (1927–40) when streamgage operation was discontinued before reactivation in 1941. Each missing annual peak 

discharge from 1885–1917 and from 1927–40 can be characterized as a censored discharge for which the value is assumed 

not to have exceeded the Qhist; estimates for those censored 

discharges bounded between 0 and Qhist were used in the LP3 

exceedance-probability analysis. As listed in table 1 (link to 

Excel file), the lower bound of the perception threshold 

for the EMA/MGB analysis for this streamgage was set at 

29,500 ft3/s, or just below the historic 1885 peak discharge. 

The basis for the assumption of setting the lower bound of 

the perception threshold just below Qhist for the two historical periods, is that if a flood larger than Qhist had happened 

during the historical period, it would have been documented. 

Streamgages that used lower perception thresholds for missing 

years for historic periods outside of the systematic period of 

streamgage operation, or for missing years during periods of 

streamgage operation, are noted in table 1 (link to Excel file).

The EMA/MGB method also allows use of interval 

discharges to characterize peak discharges that are known to 

be greater or less than some specific value or that can only 

be reliably estimated within a specific range in discharge 

(Gotvald and others, 2012). Interval discharges commonly are 

used by the EMA/MGB method to characterize missing data 

during periods of systematic streamgage operation. For example, for the Bloody Run Tributary near Sherrill, Iowa, CSG 

(streamgage 05414605, map number 85, fig. 1), an exact peak 

discharge was not determined for 4 years (1994, 2001, 2003, 

and 2006) of the 20 years of annual peak-discharge record 

because the water level did not reach the gage base (bottom) 

of the CSG, which is the minimum recording threshold (thus 

producing a censored data record). The missing peaks for 

these 4 years can be characterized as interval discharges with 

a range that is bounded by zero and the discharge associated 

with the elevation of the minimum recording threshold. 

The discharge for the minimum recording threshold was 

45.0 ft3

/s during 1994 and 2001, 33.0 ft3/s during 2003, and 

44.6 ft3/s during 2006. The EMA/MGB analysis allows the use 

of multiple discharge intervals to accommodate the changing 

minimum recording threshold of a CSG. The standard 

Bulletin 17B/GB analysis sets the gage base discharge at the 

largest minimum recording threshold (45.0 ft3

/s) and all 

minimum recording threshold values and observed point discharge values less than 45.0 ft3

/s are truncated in the 

computation of AEPDs. Tables 4 and 6 (links to Excel files) 

lists AEPDs and figure 4 shows annual exceedance-probability 

curves computed for this CSG using EMA/MGB and 

historical flood discharge, Q

Table 6. Annual exceedance-probability-discharge comparison data for 283 

streamgages in Iowa based on data through the 2010 water year.

EXPLANATION

Recorded data

90 95 98 99 20 30 40 50 60 70 80 0.5 1 2 5 10 0.2

Annual exceedance probability, in percent

1,000

1,00

10

1

10,000

Discharge, in cubic feet per second

EMA/MGB analysis using updated regional skew 

 coefficient of -0.400; no low outliers detected 

Bulletin 17B/GB analysis using updated regional 

 skew coefficient of -0.400; no low outliers detected

Figure 4. Annual exceedance-probability curves for Bloody Run Tributary near Sherrill, Iowa (05414605), showing the 

difference between expected moments algorithm (EMA/MGB) and Bulletin 17B/GB annual exceedance-probability 

analyses for a crest-stage gage (CSG) with four annual peak discharges below the minimum recording threshold.
Methods for Dataset Development for Streamgages 13

Bulletin 17B/GB exceedance-probability analyses. CSGs that 

used lower perception thresholds for minimum recordable 

discharges are noted in table 1 (link to Excel file).

Multiple Grubbs-Beck (MGB) Test for Detecting 

Low Outliers

The standard Grubbs-Beck (GB) test is recommended 

in Bulletin 17B (Interagency Advisory Committee on Water 

Data, 1982) for detecting low outliers that depart substantially 

from the trend of the rest of the annual peak-discharge data. As 

with high outliers, the retention, modification, or deletion of 

low outliers can significantly affect the three statistics (mean, 

standard deviation, and skew coefficient) computed from the 

LP3 distribution, which can have a large influence on the 

extreme magnitude flood events that we are most interested in 

estimating. Therefore, identifying the unusually small potentially influential low flows (PILFs) is critical (T.A. Cohn, U.S. 

Geological Survey, written commun., 2012). The standard GB 

test uses the annual peak-discharge data for a streamgage to 

calculate a one-sided, 10-percent significance-level critical 

value based on a log-normal distribution of the data. Annual 

peak discharges identified as low outliers by the standard GB 

test are truncated from the record and a conditional probability 

adjustment is applied in the annual exceedance-probability 

analysis (Interagency Advisory Committee on Water Data, 

1982). Although more than one recorded annual peak discharge may be below the standard GB critical value, typically 

only one nonzero recorded peak discharge is identified from 

the test as being a low outlier (Gotvald and others, 2012). 

EMA/MGB exceedance-probability analyses computed 

for this study used the MGB test for the development of the 

regional skew analysis and the regional regression equations. 

The MGB test also calculates a one-sided, 10-percent significance-level critical value based on a log-normal distribution of the data. The MGB test is performed so that groups 

of ordered data are examined (for example, the six smallest 

values in the record) and excluded from the dataset when the 

critical value is calculated. If the critical value is greater than 

the sixth smallest value in the example, then all six values are 

determined to be low outliers (Gotvald and others, 2012). The 

MGB test can identify low outliers for as much as 50 percent 

of the annual peak-discharge record (T.A. Cohn, U.S. Geological Survey, written commun., 2011). The number of low outliers identified by the MGB test for each of the 518 streamgages 

evaluated for use in the development of regression equations is 

listed in table 1 (link to Excel file). 

Figure 5 shows the resulting annual exceedance-probability curves computed for the West Branch Floyd River near 

Struble (streamgage 06600300, map number 386, fig. 1) using 

Figure 5. Annual exceedance-probability curves for West Branch Floyd River near Struble, Iowa (06600300), showing 

the effects of including or censoring potentially influential low flows identified from the multiple Grubbs-Beck test and 

of using the updated regional-skew-coefficient constant or the superseded regional skew-coefficient value.

90 95 98 99 20 30 40 50 60 70 80 0.5 1 2 5 10 0.2

Annual exceedance probability, in percent

1,000

100

10,000

100,000

Discharge, in cubic feet per second

Recorded data

Low outliers identified using the multiple Grubbs-Beck 

(MGB) test

EMA/MGB censoring low outliers and using updated regional 

skew coefficient of -0.400; 25 low outliers detected

EMA/MGB censoring low outliers and using superseded 

regional skew coefficient of -0.286; 25 low outliers detected

B17B/GB including low outliers and using updated regional 

skew coefficient of -0.400; no low outliers detected

B17B/GB including low outliers and using superseded 

regional skew coefficient of -0.286; no low outliers detected 

EXPLANATION
14 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

EMA/MGB and standard Bulletin 17B/GB analyses. The Bulletin 17B/GB analysis excluded six annual peak discharges 

because they were below the gage base (349 ft3/s). Unfortunately, four of the six peaks are actual observed discharges 

less than 349 ft3/s. The critical low-outlier value calculated 

by the standard GB test was 204.8 ft3/s and no low outliers 

were detected below the critical value. In contrast, the critical 

low-outlier value calculated by the MGB test was 2,390.0 ft3/s; 

25 PILFs (almost one-half of the 54 annual peak discharges) 

were detected below the critical value. As shown in figure 5, 

retention of the PILFs has a large influence on the fitting of the 

right-hand tail of the annual-exceedance-probability distribution. Censoring flows below the low-outlier threshold value 

with the EMA/MGB method leads to a better fit of the upper 

portion of the distribution. For this streamgage, use of EMA/

MGB lowered the Q1% estimate by almost 20 percent and the 

Q0.2% estimate by about 24 percent when compared to the Bulletin 17B/GB estimates (tables 4 and 6, links to Excel files).

Regional Skew Analysis

During the development of flood-frequency techniques 

in the late 1970s and early 1980s, a nationwide generalizedskew study was performed and documented in Bulletin 17B 

(Interagency Advisory Committee on Water Data, 1982). 

Station skew coefficients for long-term streamgages throughout the Nation were computed and used to produce a map of 

isolines of generalized skew; however, the map was prepared 

at a national scale using data and methods that are now more 

than 30 years old. To generate more accurate generalized-skew 

coefficients, three methods are described in Bulletin 17B for 

developing generalized skews using skew coefficients computed from streamgages with long-term annual peak-discharge 

records (25 or more years of record): (1) plot station skew 

coefficients on a map and construct skew isolines, (2) use 

regression techniques to develop a skew-prediction equation 

relating station skew coefficients to some set of basin characteristics, or (3) use the arithmetic mean of station skew coefficients from long-term streamgages in the area (Interagency 

Advisory Committee on Water Data, 1982). 

A generalized-skew coefficient analysis using method 2 

was performed as part of this study to update regional skew 

coefficients for streamgages in the study area. Reis and others 

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

(2008) developed a Bayesian generalized least-squares 

(B-GLS) regression model for determining regional skew. 

Recently, a Bayesian weighted least-squares/generalized leastsquares (B-WLS/B-GLS) methodology that relates observed 

skewness coefficient estimators to basin characteristics in 

conjunction with diagnostic statistics was developed as an 

extension of the B-GLS methodology (Veilleux and others, 

2012; Veilleux, 2011). B-WLS/B-GLS has been shown to be 

effective in two California studies (Parrett and others, 2011; 

Lamontagne and others, 2012). B-WLS/B-GLS uses B-WLS 

to generate stable estimators of model parameters and B-GLS 

to estimate the precision of those B-WLS regression parameters, as well as the precision of the model. The Iowa regional 

skew study described here uses the B-WLS/B-GLS methodology, which is described in detail in the appendix.

Annual peak discharges of basins are cross-correlated 

because a single large storm event can cause the annual peak 

in several basins. One advantage of using a GLS method 

is that it takes this cross-correlation among the basins into 

account. The GLS statistical analysis depends on the estimated 

cross-correlations of the peak discharges at different pairs of 

streamgages. The cross-correlation generally is estimated as a 

function of distances between the centroids of gaged basins. 

Details on the cross-correlation model can be found in the 

appendix of this report.

If watersheds of two streamgages are nested such that one 

is contained within the other, the cross-correlation between 

the concurrent peak discharges would be larger than if the 

basins were not nested. This leads to errors in the estimation of 

cross-correlations for non-nested basins, resulting in incorrect 

model errors. A screening metric was developed to determine 

the redundant streamgage pairs that represent the same watershed. Details on the screening metric used can be found in the 

appendix of this report.

After running the screening metric on the 330 

streamgages initially evaluated for inclusion in the regional 

skew study for Iowa, 55 streamgages subsequently were 

removed from the generalized-skew coefficient regression analysis because of redundancy. An additional two 

streamgages were removed because of backwater effects. 

Also, an additional 33 streamgages with EMA/MGB estimates 

of the MSE of skew (MSEG) greater than 0.4 were removed 

because of the inability of EMA/MGB to correctly estimate 

MSEG when a streamgage has a short record length and a 

large percentage of censored observations. Thus, a total of 

240 streamgages with at least 20 years of recorded, annual 

peak discharges were used for the final B-WLS/B-GLS 

regional skew regression analysis. Figure 6 shows the location 

of the basin centroids for these 240 streamgages. Details on 

the data analysis and streamgage removal can be found in the 

appendix of this report. 

Based on the B-WLS/B-GLS regression analysis, a 

constant generalized-skew value of -0.40 was determined to 

be the best model to predict the generalized skew in the study 

area. More complicated B-WLS/B-GLS models with additional explanatory variables were evaluated, but resulted in 

modest improvements in accuracy. A detailed description of 

the accuracy of these models is available in the appendix. The 

modest improvements in the more complicated models were 

not justified because of the increased complexity associated 

with the additional explanatory variables. The MSE associated with the constant generalized-skew model is 0.16, and 

is equivalent to a record length of 50 years. This is a significant improvement over the skew map MSE value of 0.302 in 

Bulletin 17B, which is equivalent to 17 years of record. The 

generalized-skew value of -0.40 with an associated MSE of 
Methods for Dataset Development for Streamgages 15

Mis sissipp i River

Missou ri

River

SOUTHDAKOTA

NEBRASKA

MISSOURI

WISCONSIN

ILLINOIS

MINNESOTA

Base from U.S. Geological Survey digital data, 2006, 1:24,000Universal Transverse Mercator projection, Zone 15

0 30 60 MILES0 30 60 KILOMETERS

Study area boundary Centroid location of U.S. 

Gelogical Survey streamgage watershed and map number

EXPLANATION

477

96° 97° 98° 95° 94° 93° 92° 91° 90° 89°

44°43°42°41°

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Figure 6. Location of basin centroids for 240 streamgages used for regional skew analysis for Iowa.
16 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

0.16 was used to compute the weighted skew value, shown in 

equation 2, for the 518 streamgages with 10 or more years of 

record. Annual exceedance-probability curves computed for 

streamgage 06600300 (map number 386, fig. 1) using superseded (-0.286) and updated (-0.400) regional skew coefficients 

are shown in figure 5. For this example (fig. 5), use of the 

updated regional-skew-coefficient constant decreased the Q1%

flood discharge by almost 3 percent using EMA/MGB analyses (tables 4 and 6, links to Excel files). Percentage differences 

in estimates of AEPDs vary from site-to-site between annual 

exceedance-probability analyses computed using superseded 

and updated regional skew coefficients, and figure 5 only illustrates the difference for one site. Tables 4 and 6 list AEPDs 

computed for Iowa streamgages using superseded and updated 

regional skew coefficients, and the section Comparison of 

Annual Exceedance-Probability Discharges summarizes relative percentage changes between AEPDs computed for Iowa 

streamgages using superseded and updated regional skew 

coefficients. The appendix of this report provides additional 

details of the generalized-skew coefficient regression analysis 

used for this study.

Regional Regression Analyses 

to Estimate Annual ExceedanceProbability Discharges for Ungaged 

Stream Sites

In a regional regression study, subdividing a large study 

area into subregions that are more homogeneous in terms of 

flood hydrology typically helps to reduce error in the regression equations. Because different flood regions have been 

determined for Iowa in previous studies and different GIS data 

layers, basin-characteristic measurement methods, and annual 

exceedance-probability analyses were used in this study, preliminary statewide regression equations were initially developed for the Q1% flood discharge using 510 streamgages. 

Table 7 lists 13 streamgages that were removed from the 

original 523 streamgages listed in table 1 (link to Excel file)

because of significant trends, urbanization, channelization, 

replacement of a discontinued streamgage with a nearby active 

streamgage, backwater, or questions concerning accuracy or 

determination of some annual peak discharges in the record. 

Table 7 also lists the five streamgages (map numbers 92, 

331, 364, 484, and 496, fig. 1) that initially were included for 

evaluation in the regional skew analysis, but subsequently 

were removed before the regional regression analysis. As 

noted in the previous section, Regional Skew Analysis, redundant streamgages and streamgages with EMA/MGB estimates 

of the MSEG greater than 0.4 were removed from the regional 

Table 7. Streamgages removed from regional-regression analyses.

[no., number; USGS, U.S. Geological Survey. Streamgage locations are shown in figure 1]

Map 

no.

USGS 

streamgage 

number

Streamgage name

Streamgage 

removed prior to 

regional-regression analyses

Reason for removal of streamgage 

from regression analyses

81 05414400 Middle Fork Little Maquoketa River 

near Rickardsville, Iowa

No Significant trend.

92 05417000 Maquoketa River near Manchester, Iowa Yes Used active streamgage 05416900.

118 05422470 Crow Creek at Bettendorf, Iowa No Urbanization.

206 05464130 Fourmile Creek near Lincoln, Iowa No Channelized/significant trend.

207 05464133 Half Mile Creek near Gladbrook, Iowa No Channelized.

208 05464137 Fourmile Creek near Traer, Iowa No Channelized/significant trend.

284 05482800 Happy Run at Churdan, Iowa No Significant trend.

294 05484800 Walnut Creek at Des Moines, Iowa No Urbanization.

331 05501000 North River at Palmyra, Missouri Yes Backwater, some peaks in peak-flow file need to be 

reviewed for accuracy.

364 06482933 Chanarambie Creek near Edgerton, 

Minnesota

Yes Discharge not determined for highest peak stage of 

record.

457 06807470 Indian Creek near Emerson, Iowa No Significant trend.

484 06817500 Nodaway River near Burlington Junction, MissouriYes Some peaks in peak-flow file need to be reviewed 

for accuracy.

496 06897000 East Fork Big Creek near Bethany, MissouriYes Backwater, some peaks in peak-flow file need to be 

reviewed for accuracy.
Regional Regression Analyses to Estimate Annual Exceedance-Probability Discharges for Ungaged Stream Sites 17

skew analysis because their inclusion was believed to significantly affect the results of the analysis (see appendix). For the 

regional regression analysis to estimate annual exceedanceprobability discharges for ungaged sites, streamgages were 

not removed because of redundancy or MSEG greater than 

0.4. Although redundancy could have been addressed in the 

regional regression analysis, the inclusion of potentially redundant sites is not believed to significantly affect the development of regional regression equations for estimating AEPDs 

for ungaged stream sites in Iowa. In this study, the Q1% flood 

discharge was selected for optimizing predictive accuracies for 

the development of regression equations because this AEPD is 

used most often by water managers, engineers, and planners. 

The same explanatory variables determined to be the most 

significant for the development of Q1% flood-discharge regression equations also were used for the other seven exceedance 

probabilities for the development of statewide and regional 

regression equations to minimize the possibility of predictive inconsistencies between estimates of different probabilities. Predictive inconsistencies result when the discharge 

estimate for a larger probability is greater than the discharge 

estimate for a smaller probability; for example, when a Q2% 

flood-discharge estimate is greater than a Q1% flood-discharge 

estimate. Table 8 lists the significant variables identified and 

the predictive accuracies obtained for preliminary statewide 

regression equations initially developed for each of the eight 

annual exceedance probabilities using ordinary least-squares 

(OLS) and subsequently finalized using generalized leastsquares (GLS) multiple-linear regression analyses (see following sections for further discussion of OLS and GLS regression). The preliminary statewide AEPD equations provided 

base-level predictive accuracies that RREs can be compared 

against to evaluate improvement in accuracy. Because RREs 

provided improved accuracies, the statewide equations were 

not developed further and are not listed in this report; they are 

summarized in table 8 to provide a reference indicating the 

improvement obtained through regionalization. 

Definition of Flood Regions

Residual values (differences between AEPDs computed 

from observed annual peak discharges and those predicted 

from the regression equations) from the preliminary statewide 

regression analyses were mapped at streamgage locations 

to identify spatial trends in the predictive accuracy of the 

regression equations. Differences in plotted residual values 

for the streamgages were grouped to define general flood 

regions within the study area. Streamgages were grouped into 

regression subsets on the basis of the flood regions, and OLS 

multiple-linear regression analyses were performed for each 

region. Because of the amount of variability in the residual 

mapping for the Q1% flood discharge, a cluster analysis method 

also was used to help define flood regions. A cluster analysis 

method, called partitioning around medoids using Spotfire 

S+ statistical software (TIBCO Software Inc., 2008), was 

used to help define flood regions in Iowa. Cluster analysis is 

a statistical technique that was used to partition streamgages 

into groups (clusters) with similar streamflow or basin characteristics. The cluster analyses were based on the basin characteristics previously identified as significant variables in the 

preliminary statewide regression equations developed for the 

Q1% flood discharge (table 8). Drainage area was not included 

in the analyses because it is not a unique characteristic for any 

one cluster. The partitioning around medoids method of cluster 

analysis uses medoids instead of centroids to form groups for 

which average dissimilarity of basin-characteristic values in 

each group are minimal (http://www.unesco.org/webworld/

idams/advguide/Chapt7_1_1.htm). Cluster analyses and the 

Table 8. Significant explanatory variables and predictive accuracies of preliminary statewide regression 

equations.

[SEP, average standard error of prediction; Pseudo-R2, pseudo coefficient of determination; %, percent; DRNAREA, geographic information system drainage area; (+), explanatory variable has positive relation to the response variable; DESMOIN, 

percent of basin within Des Moines Lobe landform region; (-), explanatory variable has negative relation to the response variable; CCM, constant of channel maintenance]

Preliminary generalized least-squares regression analyses results

Pseudo-R2 Annual exceed- (percent)

ance probability

Most significant explanatory variables identified 

for the preliminary equation and explanatoryvariable relation signsSEP (percent)

510 streamgages used in regression analyses

Q50% DRNAREA0.011 (+), DESMOIN (-), CCM0.55 (-) 53.9 90.6

Q20% DRNAREA-0.011 (+), DESMOIN (-), CCM0.55 (-) 42.5 92.9

Q10% DRNAREA-0.023 (+), DESMOIN (-), CCM0.55 (-) 41.3 92.7

Q4% DRNAREA-0.035 (+), DESMOIN (-), CCM0.55 (-) 42.0 91.9

Q2% DRNAREA-0.043 (+), DESMOIN (-), CCM0.55 (-) 43.1 91.2

Q1% DRNAREA-0.050 (+), DESMOIN (-), CCM0.55 (-) 45.8 89.9

Q0.5% DRNAREA-0.056 (+), DESMOIN (-), CCM0.55 (-) 47.7 88.8

Q0.2% DRNAREA-0.064 (+), DESMOIN (-), CCM0.55 (-) 51.0 87.0
18 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

residual mapping defined eight initial flood regions for the 

study area.

Streamgages were then grouped into eight regional 

datasets. Three of the regions (Des Moines Lobe, Paleozoic 

Plateau, and Loess Hills) were defined primarily on the basis 

of Iowa’s landform regions (fig. 3) and a fourth region was 

defined on the basis of a soil region, which was used to define 

a hydrologic region in a previous study (region 3 in Eash, 

2001). These four flood regions (three landform regions and 

one soil region) initially were defined with the cluster analysis 

and then subsequently defined as flood regions by the boundaries of their respective landform and soil regions. Analysisof-covariance regression (Helsel and Hirsch, 2002) was used 

to test each flood region for statistically significant differences 

by comparing the intercept for each region’s regression model 

to that for the rest of the study area by assigning a location 

variable for each region. Each location-indicator variable 

was set at one if the streamgage was in a particular region, or 

zero if the streamgage was not in a particular region. A twovariable OLS regression analysis that included drainage area 

and the location-indicator variable was performed statewide 

for each of the eight exceedance probabilities for each of the 

eight flood regions being tested. Statistical significance for 

each region was determined using a 95-percent confidence 

level. Statistical significance for the location-indicator variable indicates a difference in the regression intercept between 

streamgages in that region and streamgages in the rest of the 

study area. All eight flood regions were determined to be 

significantly different from each other, and preliminary RREs 

were developed for the Q1% flood discharge for each of the 

eight regions.

Figure 7A shows the relation between 1-percent AEPDs 

and drainage area for the eight initial flood regions defined 

for the study area. Figure 7B shows that the relation for flood 

regions 3A (27 streamgages located in the Paleozoic Plateau 

landform region), 3B (50 streamgages located in the 

Loess Hills landform region and southeastward), and 3C 

[50 streamgages located in the region 3 area from Eash 

(2001)] are similar. A comparison of preliminary OLS regression analyses for the Q1% flood discharge between a combined 

flood region 3 with 127 streamgages and three separate flood 

regions (3A, 3B, and 3C) indicated little difference in predictive accuracies regarding average standard errors of estimate 

or adjusted coefficients of determination; therefore, these 

three initial regions were combined into a single flood region. 

Because the analysis-of-covariance tests compared the regression intercept for each of these three regions individually to 

the regression intercept for the rest of the study area, each of 

these three regions were determined individually to be statistically significant compared to the rest of the study area. 

Figure 7C shows the relation between 1-percent AEPDs and 

drainage area for the six final flood regions defined for the 

study area. Analysis-of-covariance regression tests determined 

that all six final flood regions were significantly different from 

each other and figure 7C shows that the relation for these six 

final regions is unique.

The goal of the regionalization analyses was to define the 

best overall regions for the Q1% flood discharge and to have 

an adequate number of streamgages (preferably, at least 30) in 

each regional dataset for the regression analyses. Streamgages 

flagged as outliers (high leverage or high influence points) in 

the GLS regression analyses, using a weighted-multiple-linear 

regression program (WREG) (Eng and others, 2009), were 

reviewed for inaccurate data and for possible effects that may 

bias the development of regression equations. In addition to 

the five streamgages previously removed from the original 

dataset of 523 streamgages (table 1, link to Excel file), 

another eight streamgages that were flagged as outliers in 

the GLS regression analyses also were removed from the 

518 streamgages included in the regression datasets (tables 3

and 4, links to Excel files) because of significant trends, urbanization, or channelization (table 7). All other streamgages that 

were flagged as outliers were kept in the regression datasets 

because there was no justification for removing them. Thus, 

a total of 510 streamgages were considered to have unaltered 

annual peak discharges for the regional regression analyses. 

All 510 of these streamgages have at least 10 years of record 

that can be used for the development of the eight selected 

annual exceedance-probability equations.

Three flood regions (regions 1, 2, and 3, fig. 1) were 

defined for Iowa after testing a number of different regional 

regression-model combinations to define the initial eight, 

and then the final six, flood regions for the study area 

(fig. 7). Because three of the final six flood regions defined 

for the study area are completely outside of Iowa, regression 

equations were not developed further for flood regions 4, 5, 

and 6. Flood regions 4–6 are not shown in figures 1–3, but 

table 1, (link to Excel file) lists streamgages located in flood 

regions 4–6 and figure 1 shows the location of all streamgages 

listed in table 1 (link to Excel file). Region 4 includes 

57 streamgages located in Illinois, Wisconsin, and southeastern Minnesota; region 5 includes 16 streamgages located 

in South Dakota and southwestern Minnesota; and region 6 

includes 43 streamgages located in Nebraska (tables 1, 3, and 

4). Although predictive accuracies were optimized by subdividing the 510 streamgages in the study area into the final six 

flood regions that are more homogeneous in terms of flood 

hydrology, it is important to note that flood-region boundaries 

are not actually distinct lines, but the boundaries are transition zones where the hydrologic characteristics of one region 

transition to the hydrologic characteristics of another region. 

Figure 1 shows the three flood regions defined for Iowa for 

the development of final RREs. Flood-region boundaries 

were defined along 12-digit WBD HUC boundaries to avoid 

drawing a flood-region boundary through a HUC polygon. 

For a 12-digit HUC that overlies a landform region boundary, 

the flood-region boundary was drawn to include the landform 

region that comprises the most of the 12-digit HUC area. 

Flood region 1 is defined primarily by the Des Moines 

Lobe landform region and contains approximately 23 percent 

of the total land area of the State (figs. 1–3). Flood region 1 

generally comprises low-relief terrain, accentuated by natural 
Regional Regression Analyses to Estimate Annual Exceedance-Probability Discharges for Ungaged Stream Sites 19

100

1,000

10,000

100,000

100

1,000

10,000

100,000

One-percent annual exceedance-probability discharge, in cubic feet per second

Drainage area, in square miles

0.01 0.1 1 10 100 1,000 10,000

100

1,000

10,000

100,000

A

B

C

EXPLANATION

Region 1 (91 sites)

Region 2 (176 sites)

Region 3A (27 sites)

Region 3B (50 sites)

Region 3C (50 sites)

Region 4 (57 sites)

Region 5 (16 sites)

Region 6 (43 sites)

EXPLANATION

Region 1 (91 sites)

Region 2 (176 sites)

Region 3 (127 sites)

Region 4 (57 sites)

Region 5 (16 sites)

Region 6 (43 sites)

EXPLANATION

Region 3A (27 sites)

Region 3B (50 sites)

Region 3C (50 sites)

Figure 7. Relation between one-percent annual exceedance-probability discharges and drainage area for A, eight 

initial; B, three combined; and C, six final flood regions defined for study area.
20 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

lakes, potholes, and marshes, where surface-water drainage 

typically is poorly defined and sluggish. For a given size of 

drainage area, estimates of AEPDs from region 1 generally are 

lower than those from flood regions 2 and 3 (fig. 7C). 

Flood region 2 is the largest flood region in the State 

and contains approximately 45 percent of the total land area 

(figs. 1–3). Flood region 2 partially is defined by the Des 

Moines Lobe and Paleozoic Plateau landform-region boundaries, although flood region 2 only includes small areas of 

either landform region. Region 2 includes most of the Iowan 

Surface, the Northwest Iowa Plains, and the East-Central Iowa 

Drift Plain landform regions; the entire Iowa-Cedar Lowland 

landform region; and nearly one-half of the Southern Iowa 

Drift Plain landform region. Region 2 was defined as the 

remaining area following the definition of flood regions 1 and 

3, and it generally comprises gently rolling topography and 

well-established drainage systems. For a given size of drainage area, estimates of AEPDs from region 2 are generally in 

between those from flood regions 1 and 3 (fig. 7C). 

Flood region 3 is a combined regression-model region 

(fig. 7B) and contains approximately 32 percent of the total 

land area of the State (figs. 1–3). Flood region 3 includes two 

non-contiguous areas of the State, one that includes most 

of the Paleozoic Plateau landform region and the other that 

includes most of the Loess Hills landform region and more 

than one-half of the Southern Iowa Drift Plain landform 

region. As noted earlier in this section, flood region 3 was 

formed by combining three initial regression-model flood 

regions: (1) the Paleozoic Plateau landform region, (2) the 

Loess Hills landform region and areas of deep loess southeast of the Loess Hills, and (3) hydrologic region 3 in Eash 

(2001). Region 3 generally comprises steeply to gently rolling 

topography and well-established drainage in the area of the 

Southern Iowa Drift Plain landform region; a bedrock-dominated, erosional topography of integrated drainage networks 

with steep gradients, and deeply entrenched valleys in the area 

of the Paleozoic Plateau landform region; and topography 

that is sharp-featured, with a dense drainage network forming 

tight hollows, narrow ravines, and steep gullies in the area of 

the Loess Hills landform region; and an area of low soilpermeability rates in the southeast area of the State (fig. 2, soil 

regions 21 and 22; Eash, 2001). For a given size of drainage 

area, estimates of AEPDs from region 3 generally are greater 

than those from flood regions 1 and 2 (fig. 7C). 

Development of Regional Regression Equations

GLS multiple-linear regression analyses, weighted on 

the basis of streamgage record length and the variance and 

cross-correlation of the annual peak discharges, were used to 

develop equations to estimate AEPDs for ungaged stream sites 

in Iowa. 

Multiple-Linear Regression

Multiple-linear-regression analysis is the most common method used to develop equations for the estimation of 

streamflow statistics at ungaged sites. Multiple-linear regression models the relation between two or more basin characteristics (called explanatory or independent variables) and a 

streamflow statistic (called a response or dependent variable) 

by fitting a linear equation to the data. Every value of each 

basin characteristic is associated with the value of the streamflow statistic. Upon the development of regression equations, 

measurements of the basin characteristics at ungaged stream 

locations can be used to estimate the streamflow statistic. 

The general form of equations developed from multiplelinear-regression analysis is shown in the following equation:

Yi = b0 + b1X1 + b2X2 +....+ bnXn + ei (3)

where

Yi is the response variable (estimate of the 

streamflow statistic computed from 

observed streamflow) for site i, 

X1 to Xn are the n explanatory variables (basin 

characteristics) for site i,

b0 to bn are the n+1 regression model coefficients, and

ei is the residual error (difference between 

the observed and predicted values of the 

response variable) for site i.

Assumptions for the use of regression analyses are 

(1) the model adequately describes the linear relation between 

the response and explanatory variables, (2) the mean of ei

is zero, (3) the variance of ei is constant and independent of 

the values of Xn, (4) the values of ei are distributed normally, 

and (5) the values of ei are independent of each other (Iman 

and Conover, 1983). Because streamflow data are naturally 

correlated spatially and temporally, the last assumption is not 

completely satisfied with the use of OLS. As a result, GLS 

regression was used to develop the final equations for estimating AEPDs for Iowa. A general overview of the OLS and 

GLS multiple-linear regression techniques used to develop the 

initial and final equations is presented in the following two 

sections.

Ordinary-Least-Squares Regression

OLS regression analyses were used to develop initial 

multiple-linear regression equations, or models, for estimating statewide and regional annual Q1% flood discharges for 

Iowa. Final equations were developed using GLS regression 

procedures. OLS regression analyses were used to identify the 

best combinations of basin characteristics to use as explanatory variables in the development of regression models and to 

define the flood regions. 

Logarithmic transformations (base 10) were performed 

for all response variables and for selected explanatory variables used in the OLS and GLS regression analyses. Data 

transformations, other than logarithmic transformations, also 
Regional Regression Analyses to Estimate Annual Exceedance-Probability Discharges for Ungaged Stream Sites 21

were used for selected explanatory variables to obtain a more 

constant variance of the residuals about the regression line and 

to linearize the relation between the response variable and the 

explanatory variables. The response variable is assumed to be 

a linear function of one or more explanatory variables. The 

form of a base-10 logarithmic transformation is shown in the 

following equation:

logYi = b0 + b1logX1 + b2logX2 +...+ bnlogXn + ei (4)

When equation 4 is retransformed back to its original 

units, it is algebraically equivalent to the following equation:

 (5)

Several basin characteristics were deleted from the 

original regression dataset of 59 basin characteristics because 

of multicollinearity. Multicollinearity is the condition wherein 

at least one explanatory variable is related closely to (that is, 

not independent of) one or more other explanatory variables. 

Regression models that include variables with multicollinearity may be unreliable because coefficients in the models may 

be unstable. Correlation coefficients greater than 0.5, or less 

than -0.5, and plots of the data were used as guides in identifying variables with multicollinearity. The hydrologic validity 

of variables with multicollinearity in the context of annual 

peak discharge was the principal criterion used in determining 

which basin characteristics were deleted from the dataset.

OLS regression analyses were performed using Spotfire 

S+ statistical software (TIBCO Software Inc., 2008). Initial 

selections of significant explanatory variables for the OLS 

regression models were performed using the Efroymson 

stepwise-selection method (Efroymson, 1960). The Efroymson method is an automatic procedure for regression model 

selection when there are a large number of potential explanatory variables. The procedure is similar to forward selection, 

which tests basin characteristics one by one and identifies 

those that are statistically significant, except as each new basin 

characteristic is identified as being significant, partial correlations are checked to see if any previously identified variables 

can be deleted (Ahearn, 2010). When basin characteristics 

were determined to be highly correlated to each other, only 

one basin characteristic at a time was tested in the Efroymson 

selection process.

The Efroymson analyses produced a subset of potential 

significant basin characteristics for the Q1% flood discharge 

for each flood region. Each subset of basin characteristics was 

then iteratively tested using standard OLS regression analyses 

to identify several sets of the best equations (regression models) that contained no more than three significant explanatory 

variables (basin characteristics). A limit of three explanatory 

variables per equation was used to minimize overfitting of the 

regression models. Results of the OLS models were evaluated to determine their adequacy, including graphical relations 

and residual plots, variance inflation factor (VIF), Cook’s D 

statistic (Cook, 1977; Helsel and Hirsch, 2002), high-leverage 

Yi = 10b0 X1

b1 X2b2 ... Xnbn10ei

points, the average standard error of estimate (SEE), and the 

adjusted coefficient of determination (adj-R2) (Helsel and 

Hirsch, 2002). The selection of explanatory variables, and the 

signs and magnitudes of their respective regression coefficients, were each evaluated to ensure hydrologic validity in the 

context of AEPDs. This criterion takes precedence for all other 

criteria. All explanatory variables selected by OLS regression 

in this study were statistically significant at the 95-percent 

confidence level. Explanatory variables were selected to 

minimize SEE and to maximize the adj-R2. SEE is a measure 

of the fit of the observed data to the regression model (difference between the value of the observed AEPD and the value of 

the predicted AEPD) and of the error inherent in the regression 

model; SEE also is referred to as the root mean square error 

(RMSE). Adj-R2 is a measure of the proportion of the variation in the response variable that is explained by the explanatory variables and adjusted for the number of streamgages 

and explanatory variables used in the analysis. Correlation 

between explanatory variables and VIF (Marquardt, 1970; 

Helsel and Hirsch, 2002) was used to assess multicollinearity in the regression models. Multicollinearity problems were 

identified with a regression-diagnostics tool implemented in 

the USGS library version 4.0 (Lorenz and others, 2011) for 

Spotfire S+ statistical software (TIBCO Software Inc., 2008) 

by checking each explanatory variable for VIF greater than 

two. 

A logarithmic graphical relation between observed Q1%

flood discharges and drainage area fitted with a one-variable 

OLS regression equation indicates a discernible curvilinear 

relation for flood region 3 and a poor fit of the data with 

the regression line for drainage areas less than about 1 mi2 

(fig. 8A). Traditionally, log10-transformations have been used 

for all explanatory variables (basin characteristics) for the 

development of annual exceedance-probability regression 

equations for which the equations are of the form shown by 

equation 5 (Eash, 2001); however, for some regional datasets, 

exclusive use of log10-transformations does not fully linearize 

the relations between the variables. As a result, some systematic bias remains because of a discernible curvilinear relation 

between AEPDs and drainage area in log10-space (fig. 8A). A 

procedure developed for Texas (Asquith and Thompson, 2008) 

was followed in this study to determine whether a log10-transformation or a power transformation of drainage area provided 

the best fit of the data for each of the three flood regions 

defined for Iowa. The Texas procedure is a statistical framework based on the minimization of the predicted residual sums 

of squares (PRESS) statistic through a test of power transformations of drainage area. The PRESS statistic is regarded as a 

measure of regression performance when the model is used to 

predict new data (Montgomery and others, 2001). The PRESS 

statistic is a validation-type statistic (Helsel and Hirsch, 2002) 

and optimal power transformations were selected on the basis 

of minimizing the PRESS statistic. Minimization of PRESS 

is appropriate for annual exceedance-probability analysis 

because the regression equations are used in hydrologic engineering practice to predict new data (Asquith and Thompson, 
22 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

Drainage area, in square miles

100

1,000

10,000

100,000

Drainage area-0.097, in square miles

0.4

0.04 0.1 1 10 100 1,000 4,000

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

100

1,000

10,000

100,000

A

B

One-percent annual exceedance-probability discharge, in cubic feet per second

Figure 8. Relation between one-percent annual exceedance-probability discharges and drainage area for 

127 streamgages in flood region 3 for A, log 10 transformed drainage area and B, power-transformed drainage 

area.
Regional Regression Analyses to Estimate Annual Exceedance-Probability Discharges for Ungaged Stream Sites 23

2008). Compared to equations using log10-transformations of 

drainage area, lower values for PRESS statistics and average 

standard error of predictions (SEPs) were obtained from equations using power transformations of drainage area derived 

from the PRESS minimization procedure for two of the three 

flood regions tested. Use of a power transformation of drainage area, instead of a log10-transformation of drainage area, 

improved the fit of the regression model for flood regions 2 

and 3, and removed the discernible curvilinear relation for 

flood region 3 (fig. 8B). The PRESS-minimization procedure 

computed a different optimal power transformation of drainage area for each of the eight annual exceedance probabilities 

for flood regions 2 and 3. Because there was no difference 

between the PRESS statistic and SEP values for either the 

log10 or power transformation of drainage area for the Q50% 

flood-discharge equation for flood region 3, the log10-transformation was selected for the final regression model.

Generalized-Least-Squares Regression

GLS multiple-linear regression was used to develop 

24 regression equations for estimating AEPDs for the three 

flood regions defined for Iowa. GLS regression analyses were 

performed using the WREG program (Eng and others, 2009). 

GLS regression, as described by Stedinger and Tasker (1985), 

Tasker and Stedinger (1989), and Griffis and Stedinger (2007), 

is a method that weights streamgages in the regression according to differences in streamflow reliability (record lengths) and 

variability (record variance), and according to spatial cross 

correlations of concurrent streamflow among streamgages. 

Compared to OLS regression, GLS regression provides 

improved estimates of streamflow statistics and improved 

estimates of the predictive accuracy of the regression equations (Stedinger and Tasker, 1985). The correlation smoothing 

function used by the WREG program to compute a weighting 

matrix for the 176 streamgages included in the development 

of the GLS regression equation for estimating AEPDs for 

flood region 2 with 30 years of concurrent flow is shown 

in figure 9. The smoothing function relates the correlation 

between annual peak-discharge time series at two streamgages 

to the geographic distance between the streamgages for every 

paired combination of the 176 streamgages with 30 years of 

concurrent flow. Strong evidence of cross correlation is shown 

in figure 9 because of the abundance of paired points for 

30 years of concurrent flow that form the tail extending 

towards the right side of the graph. Final GLS regression 

models were selected primarily on the basis of minimizing 

values of the standard error of model (SEM) and the SEP, and 

maximizing values of the pseudo coefficient of determination 

(pseudo-R2

).

Figure 9. Screen capture of the weighted-multiple-linear regression program (WREG) smoothing function for generalized-leastsquares (GLS) correlation of the time series of annual peak discharges as a function of distance between 176 streamgages in flood 

region 2 with 30 years of concurrent discharge.
24 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

Table 9. Regression equations for estimating annual exceedance-probability discharges for unregulated 

streams in flood region 1 of Iowa.

[SEP, average standard error of prediction; Pseudo-R2, pseudo coefficient of determination; SEM, average standard error of model; 

AVP, average variance of prediction; %, percent; DRNAREA, geographic information system drainage area; I24H10Y, maximum 

24-hour precipitation that happens on average once in 10 years; CCM, constant of channel maintenance]

Annual exceedance-probability equation SEP 

(percent)

Pseudo-R2

(percent)

SEM 

(percent)

AVP

 (log ft3

/s)2

91 streamgages used to develop equations

Q50%=DRNAREA0.67510(-0.355+0.601 x I24H10Y-0.595 x CCM0.55) 41.6 94.8 39.4 0.030

Q20%=DRNAREA0.62610(0.094+0.594 x I24H10Y-0.609 x CCM0.55) 32.6 96.2 30.4 0.019

Q10%=DRNAREA0.60410(0.340+0.588 x I24H10Y-0.636 x CCM0.55) 31.8 96.2 29.4 0.018

Q4%=DRNAREA0.58510(0.604+0.579 x I24H10Y-0.679 x CCM0.55) 33.2 95.6 30.4 0.020

Q2%=DRNAREA0.57410(0.769+0.573 x I24H10Y-0.712 x CCM0.55) 35.6 94.7 32.6 0.023

Q1%=DRNAREA0.56610(0.917+0.567 x I24H10Y-0.742 x CCM0.55) 38.0 93.8 34.7 0.025

Q0.5%=DRNAREA0.55910(1.06+0.560 x I24H10Y-0.771 x CCM0.55) 41.0 92.6 37.5 0.029

Q0.2%=DRNAREA0.55310(1.22+0.550 x I24H10Y-0.808 x CCM0.55) 45.2 90.8 41.5 0.035

Table 10. Regression equations for estimating annual exceedance-probability discharges for unregulated 

streams in flood region 2 of Iowa.

[SEP, average standard error of prediction; Pseudo-R2, pseudo coefficient of determination; SEM, average standard error of model; 

AVP, average variance of prediction; %, percent; DRNAREA, geographic information system drainage area; DESMOIN, percent of 

area within Des Moines Lobe landform region; BSHAPE, measure of basin shape]

Annual exceedance-probability equation SEP

 (percent)

Pseudo-R2

(percent)

SEM 

(percent)

AVP 

(log ft3

/s)2

176 streamgages used to develop equations

Q50%=10(-49.0+51.2 x DRNAREA0.005) x (DESMOIN+1)-0.069 46.8 91.5 45.7 0.037

Q20%=10(30.4-27.8 x DRNAREA-0.009 -0.001 x DESMOIN-0.016 x BSHAPE) 25.7 96.7 24.4 0.012

Q10%=10(17.4-14.6 x DRNAREA-0.017 -0.002 x DESMOIN -0.019 x BSHAPE) 20.8 97.7 19.3 0.008

Q4%=10(13.7-10.7 x DRNAREA-0.023 -0.002 x DESMOIN -0.022 x BSHAPE) 19.4 97.9 17.6 0.007

Q2%=10(12.2-9.10 x DRNAREA-0.027 -0.002 x DESMOIN -0.024 x BSHAPE) 20.4 97.6 18.5 0.008

Q1%=10(11.1-7.92 x DRNAREA-0.031 -0.002 x DESMOIN -0.025 x BSHAPE) 22.3 96.9 20.3 0.009

Q0.5%=10(10.5-7.20 x DRNAREA-0.034 -0.002 x DESMOIN -0.026 x BSHAPE) 24.9 96.0 22.9 0.011

Q0.2%=10(9.95-6.60 x DRNAREA-0.037 -0.002 x DESMOIN -0.028 x BSHAPE) 28.2 94.7 26.1 0.014

Table 11. Regression equations for estimating annual exceedance-probability discharges for unregulated 

streams in flood region 3 of Iowa.

[SEP, average standard error of prediction; Pseudo-R2, pseudo coefficient of determination; SEM, average standard error of model; 

AVP, average variance of prediction; %, percent; DRNAREA, geographic information system drainage area; KSATSSUR, average 

saturated hydraulic conductivity of soil; BSHAPE, measure of basin shape]

Annual exceedance-probability equation SEP 

(percent)

Pseudo-R2

(percent)

SEM 

(percent)

AVP 

(log ft3

/s)2

127 streamgages used to develop equations

Q50%=DRNAREA0.600 x 10(2.43-0.009 x KSATSSUR-0.017 x BSHAPE) 43.1 92.4 41.6 0.032

Q20%=10(10.6-7.78 x DRNAREA-0.035-0.009 x KSATSSUR-0.024 x BSHAPE) 30.4 95.3 29.0 0.017

Q10%=10(8.14-5.17 x DRNAREA-0.054-0.008 x KSATSSUR-0.028 x BSHAPE) 27.0 96.0 25.6 0.013

Q4%=10(7.07-3.92 x DRNAREA-0.073-0.008 x KSATSSUR-0.031 x BSHAPE) 26.5 95.9 24.8 0.013

Q2%=10(6.65-3.39 x DRNAREA-0.086-0.009 x KSATSSUR-0.033 x BSHAPE) 27.8 95.3 26.0 0.014

Q1%=10(6.41-3.06 x DRNAREA-0.097-0.009 x KSATSSUR-0.035 x BSHAPE) 29.1 94.7 27.2 0.015

Q0.5%=10(6.23-2.80 x DRNAREA-0.108-0.009 x KSATSSUR-0.037 x BSHAPE) 30.5 94.1 28.4 0.017

Q0.2%=10(6.10-2.59 x DRNAREA-0.119-0.009 x KSATSSUR-0.039 x BSHAPE) 33.7 92.6 31.5 0.020
Regional Regression Analyses to Estimate Annual Exceedance-Probability Discharges for Ungaged Stream Sites 25

Final Regression Equations

Final regression equations developed for the three flood 

regions defined for Iowa are listed in tables 9–11, along 

with the number of streamgages included in each regression 

analysis and several performance metrics. A total of 

394 streamgages were included in the development of the final 

regression equations for flood regions 1–3. StreamStats variable names are used for the explanatory variables in the final 

regression equations (tables 9–11); definitions of the variables 

and the units of measure are listed in tables 2 and 3. Six basin 

characteristics are used as explanatory variables in the final 

regression equations (table 12); these include three morphometric characteristics [DRNAREA, constant of channel maintenance (CCM), and a measure of basin shape (BSHAPE)], 

two geologic/pedologic characteristics [percent of basin within 

the Des Moines Lobe landform region (DESMOIN) and average saturated hydraulic conductivity of soil (KSATSSUR)], 

and one climatic characteristic [maximum 24-hour precipitation that happens on average once in 10 years (I24H10Y)] 

(table 2, link to Excel file). GIS software is required to measure the basin characteristics included as explanatory variables 

in the final regression equations. All explanatory variables 

included in the final regression equations were statistically 

significant at the 95-percent confidence level and were not 

correlated with other explanatory variables used in the same 

equation. The performance metrics in tables 9–11 indicate the 

predictive accuracy of the final regression equations. 

Data transformations were tested for each explanatory 

variable included in the final regression equations. As previously noted, logarithmic and power transformations were 

tested and used for DRNAREA. Power transformations for 

DRNAREA for flood regions 2 and 3 are negative for all 

annual exceedance probabilities, with the exception of Q50% 

(tables 10 and 11). Logarithmic and nonlogarithmic data 

transformations were tested for each of the other explanatory 

variables in the regression equations (tables 9–11) to linearize 

the relation between the response variable and the explanatory 

variables. For flood region 1, an exponent of 0.55 for CCM 

was determined to provide the best data transformation. A 

range of exponent values were tested for the transformation of 

CCM and the relation between log Q1% and CCM was plotted 

for each exponent value tested. The data transformation that 

visually provided the most linear relation for Q1% and Q10% 

was then modeled with OLS and GLS regression to verify 

the selected data transformation also was providing the best 

predictive accuracy.

For the 24 GLS regression equations developed for 

estimating AEPDs for flood regions 1–3, an SEP (in percent), 

a pseudo-R2 (in percent), an SEM (in percent), and an average 

variance of prediction [AVP, in (log ft3/s)2 units] are reported 

in tables 9–11. SEP represents the sum of the model error 

and the sampling error. SEP is the square root of the GLS 

Table 12. Range of basin-characteristic values used to develop annual exceedance-probability regression equations for 

unregulated streams in Iowa.

[GIS, geographic information system; DRNAREA, GIS drainage area; I24H10Y, maximum 24-hour precipitation that happens on average once in 10 years; 

CCM, constant of channel maintenance; mi2, square mile; mi, mile; DESMOIN, percent of area within Des Moines Lobe landform region; BSHAPE, measure 

of basin shape, dimensionless; KSATSSUR, average saturated hydraulic conductivity of soil; μm/s, micrometer per second; NA, not applicable, basin characteristic not used to develop regional regression equations]

GIS drainage area, 

DRNAREA (mi2

) I24H10Y (inches) CCM (mi2

/mi) DESMOIN

(percent) BSHAPE KSATSSUR (μm/s)

Region 1

Minimum 0.06 3.580 0.110 NA NA NA

Maximum 5,463.88 4.500 2.220 NA NA NA

Mean 353.74 4.175 0.906 NA NA NA

Median 40.72 4.115 0.875 NA NA NA

Number of sites 91 91 91 NA NA NA

Region 2

Minimum 0.08 NA NA 0 0.806 NA

Maximum 7,782.62 NA NA 100.000 13.941 NA

Mean 467.84 NA NA 8.175 3.826 NA

Median 45.55 NA NA 0 3.303 NA

Number of sites 176 NA NA 176 176 NA

Region 3

Minimum 0.05 NA NA NA 0.339 1.883

Maximum 2,809.05 NA NA NA 13.523 33.572

Mean 323.29 NA NA NA 4.514 7.867

Median 52.26 NA NA NA 4.019 6.153

Number of sites 127 NA NA NA 127 127
26 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

AVP (Tasker and Stedinger, 1989; Eng and others, 2009). 

The pseudo-R2 is a measure of the percentage of the variation 

explained by the basin characteristics (explanatory variables) 

included in the model. The pseudo-R2 value is calculated on 

the basis of the degrees of freedom in the regression. Griffis 

and Stedinger (2007) describe how the pseudo-R2 is more 

appropriate than the traditional R2 or adjusted R2 in measuring 

the true variation explained by the explanatory variables in 

the GLS model. SEM measures the error of the model itself 

and does not include sampling error. SEM is the square root of 

the GLS model error variance (MEV) (Tasker and Stedinger, 

1989). AVP is a measure of the average accuracy of prediction 

for all sites used in the development of the regression model 

and assumes that the explanatory variables for the streamgages 

included in the regression analysis are representative of all 

streamgages in the region (Verdi and Dixon, 2011). 

The performance metrics pseudo-R2 and SEM indicate 

how well the equations perform on the streamgages used in 

the regression analyses. SEP and AVP are measures of the 

accuracy that GLS regression models can predict AEPDs at 

ungaged sites. The same explanatory variables were used to 

develop all eight annual exceedance-probability equations 

for each region to minimize the possibility of predictive 

inconsistencies between estimates of different probabilities, 

so that estimates will increase with decreasing probabilities. 

For example, maintaining the same regression-model form 

(same explanatory variables) helps to maximize the chances 

an estimate for the Q1% flood discharge is greater than an 

estimate for the Q2% flood discharge, an estimate for the Q2%

flood discharge is greater than an estimate for the Q4% flood 

discharge, and so forth for all eight exceedance probabilities. 

The one exception to maintaining the same regression-model 

form for all eight annual exceedance probabilities is the Q50% 

regression equation for flood region 2. A two-variable regression model was developed instead of a three-variable model 

for this annual exceedance probability because the basin characteristic BSHAPE was not a statistically significant explanatory variable at the 95-percent confidence level and the three 

explanatory variables DRNAREA, DESMOIN, and BSHAPE 

provided the best GLS regression model for the seven other 

annual exceedance probabilities. Flood-discharge estimates 

computed for streamgages using the two-variable Q50% and

three-variable Q20% regression equations for flood region 2 

listed in table 10 indicated that Q20% estimates should exceed 

Q50% estimates.

Output from the WREG program (Eng and others, 2009) 

for GLS regression models identifies streamgages that are 

possible outliers in the dataset as plotted points or tabulated 

values that exceed a leverage threshold value or an influence 

threshold value. Leverage points (Eng and others, 2009) are 

outliers that may unduly influence the estimation of regression 

constants and are a measure of how large or small explanatory 

variables (basin characteristics) are for a specific streamgage 

as compared to the centroid of values of the same explanatory 

variables at all other streamgages. Influence points are measured by Cook’s D statistic (Cook, 1977; Helsel and Hirsch, 

2002), and these are outliers that have unduly influenced the 

estimation of regression constants. As previously noted in an 

earlier section Definition of Flood Regions, eight streamgages 

identified by the WREG program as outliers were removed 

from the regression datasets because of significant trends, 

urbanization, or channelization (table 7). 

DRNAREA, drainage area, is the most significant 

explanatory variable for each set of RREs and is related 

positively to AEPDs; ungaged basins with larger drainage 

areas will produce greater estimates of AEPDs and ungaged 

basins with smaller drainage areas will produce lower estimates of AEPDs. The second RRE variable for flood region 

1, I24H10Y, also is related positively to AEPDs. I24H10Y 

values for flood region 1 decrease in a northerly direction and 

basins within the southern part of the Des Moines Lobe will 

produce greater estimates of AEPDs than basins within the 

northern part of the Des Moines Lobe given the same values 

for DRNAREA and CCM. The third RRE variable for flood 

region 1, CCM, is related negatively to AEPDs. CCM is a 

measure of drainage density calculated as a ratio of drainage area divided by the total length of all mapped streams 

(1:24,000-scale) in the basin. Because most of flood region 

1 is within the Des Moines Lobe landform region, an area 

where surface-water drainage typically is defined poorly, 

watersheds with fewer miles of stream length per square mile 

of drainage area will result in larger values for CCM and 

lower estimates of AEPDs. The second RRE variable for flood 

region 2, DESMOIN, is related negatively to AEPDs. Basins 

with areas within the Des Moines Lobe landform region will 

produce lower estimates of AEPDs than basins outside the Des 

Moines Lobe or than basins with a smaller percentage of area 

within the Des Moines Lobe (fig. 3). The third RRE variable 

for flood regions 2 and 3, BSHAPE, also is related negatively 

to AEPDs. BSHAPE is calculated as a ratio of basin length 

squared divided by drainage area. The more elongated the 

shape of a basin, the larger the value for BSHAPE and the longer the lag-time for tributary flows to peak at the basin outlet, 

which will produce lower estimates of AEPDs; conversely, 

the more rounded the shape of the basin, the smaller the value 

for BSHAPE and the shorter the lag-time for tributary flows 

to peak at the basin outlet, which will produce greater estimates of AEPDs. The second RRE variable for flood region 3, 

KSATSSUR, is related negatively to AEPDs. KSATSSUR, or 

saturated hydraulic conductivity, is a quantitative measure of 

a saturated soil’s ability to transmit water when subjected to a 

hydraulic gradient, which can be thought of as the ease with 

which pores of a saturated soil permit water movement (http://

soils.usda.gov/technical/technotes/note6.html). Basins with 

smaller average values of KSATSSUR have lower saturated 

hydraulic-conductivity and infiltration rates resulting in more 

surface-water runoff and larger estimates of AEPDs; conversely, basins with larger average values of KSATSSUR have 

higher saturated hydraulic-conductivity and infiltration rates 

leading to less surface-water runoff and lower estimates of 

AEPDs.
Regional Regression Analyses to Estimate Annual Exceedance-Probability Discharges for Ungaged Stream Sites 27

Accuracy and Limitations of Regression 

Equations

The RREs developed in this study only apply to stream 

sites in Iowa where peak discharges are not affected significantly by regulation, diversion, channelization, backwater, or 

urbanization. The applicability and accuracy of the regional 

equations depend on if the basin characteristics measured 

for an ungaged stream site are within the range of the characteristic values used to develop the regression equations. 

The acceptable range of basin-characteristic values used to 

develop each RRE (tables 9–11) are tabulated as minimum and 

maximum values in table 12. The applicability of the regional 

equations is unknown when any characteristic value measured 

for an ungaged site is outside the acceptable range. In addition, 

basin-characteristic measurements at ungaged sites should 

be computed using the same GIS datasets and measurement 

methods used in this study; the USGS StreamStats Web-based 

GIS tool includes the same GIS data layers and measurement 

methods as used to develop the regression equations in this 

study. 

The AEPD regression equations presented in this report 

should be used with caution for ungaged stream sites with 

basin-characteristic values approaching the minimum or 

maximum limits (table 12) because the predictive errors of 

the equations increase with distance from the mean or median 

values of the explanatory variables and inconsistencies in the 

estimates may result. For different annual exceedance probabilities, the AEPD estimate for a larger probability may be 

greater than the AEPD estimate for a smaller probability; for 

example, a Q2% flood discharge estimate may be greater than 

a Q1% flood discharge estimate. Although no inconsistencies 

in RRE estimates resulted for any of the eight AEPDs for the 

394 streamgages listed in table 4 (link to Excel file), there 

is the possibility that inconsistencies in RRE estimates may 

result for ungaged sites. If inconsistencies in RRE estimates 

are obtained for an ungaged stream site, a comparison of 

all AEPDs for the site and a check of streamgage data or 

other published data may help to determine which AEPD is 

inconsistent. 

In general, predictive accuracies tend to be the best for 

flood region 2, second best for flood region 3, and poorest 

for flood region 1. For the AEPD equations, SEP for flood 

region 1 ranges from 31.8 to 45.2 percent (table 9), SEP for 

flood region 2 ranges from 19.4 to 46.8 percent (table 10), 

and SEP for flood region 3 ranges from 26.5 to 43.1 percent 

(table 11). The percentage of variation in the response variables explained by the explanatory variables (pseudo-R2

) for 

the AEPD equations developed for flood region 1 ranges from 

90.8 to 96.2 percent, for flood region 2 ranges from 91.5 to 

97.9 percent, and for flood region 3 ranges from 92.4 to 

96.0 percent. Of the eight AEPD equations developed for 

each region, the Q10%, Q4%, and Q2% flood-discharge regression 

equations generally have the best predictive accuracy and the 

Q50% and Q0.2% flood-discharge equations generally have the 

poorest accuracy. The natural variability of streamflow may be 

an important factor associated with the predictive accuracy of 

AEPD equations. Estimation of AEPDs that have greater variability will have poorer predictive accuracies than estimation 

of AEPDs with less variability. 

The regression equations also should be used with 

caution for streams within the Mississippi River and Missouri River Alluvial Plains landform regions (fig. 3) because 

streamgage data representing these landform regions were 

not included in the development of the regression equations. 

Although two streamgages are located within the Missouri 

River Alluvial Plain landform region, West Fork Ditch at 

Hornick, Iowa (streamgage 06602020, map number 394, 

fig. 1), and Little Sioux River near Kennebec, Iowa 

(streamgage 06606700, map number 407, fig. 1), neither streamgage is considered representative of this landform region because nearly the entire watershed for each 

streamgage is located outside of the Missouri River Alluvial 

Plain landform region. 

Because the precision of response- and explanatory-variable data used to develop the equations was often limited to 

three significant figures, AEPDs estimated from the regression 

equations also should be limited to three significant figures. 

Figure 10 shows the relation between observed and 

predicted flood discharges for Q1% for each of the three flood 

regions. The uncertainty of regression estimates can be seen 

graphically as a greater scatter of observed in relation to predicted points along the 1:1 line. 

Prediction Intervals

Although regression equations presented in tables 9–11 

can be used to estimate AEPDs, the true values of the AEPDs 

are unknown. A measure of the uncertainty associated with 

the regression estimate of an AEPD is the prediction interval. The interval is the estimated value plus or minus a given 

margin of error. A prediction interval is the probability that 

the actual value of the estimated AEPD will be within this 

margin of error (Helsel and Hirsch, 2002). The prediction 

interval determines the range of discharge values predicted for 

selected statistics given a confidence level and the SEP. For a 

90-percent prediction interval, the true AEPD has a 90-percent 

probability of being within the margin of error. The USGS 

StreamStats Web-based GIS tool (http://water.usgs.gov/osw/

streamstats/index.html) uses the 90-percent prediction interval 

estimates as part of the computation of AEPD estimates for 

ungaged stream sites. The following equation, modified from 

Tasker and Driver (1988), can be used to compute the 90-percent prediction interval for the true value of an AEPD for an 

ungaged site:

Q < Q < QT T (6)

where

Q is the AEPD predicted for the ungaged site 

from the regression equation, and the 
28 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

100

1,000

10,000

100,000

100

1,000

10,000

100,000

Observed one-percent annual exceedance-probability discharge, in cubic feet per second

100 1,000 10,000 100,000

100

1,000

10,000

100,000

Region 1

Region 3

Region 2

1:1 line

1:1 line

1:1 line

Map no. 204 omitted from plot

Observed = 101,000 cubic feet per second

Predicted = 80,400 cubic feet per second

Predicted one-percent annual exceedance-probability discharge, in cubic feet per second

Figure 10. Relation between one-percent annual exceedance-probability discharges computed from observed 

streamflow and those predicted from regression equations for flood regions in Iowa.
Regional Regression Analyses to Estimate Annual Exceedance-Probability Discharges for Ungaged Stream Sites 29

following equation can be used to compute 

T:

T = 10[t(α ⁄ 2,n–p)Si

] (7)

where

t

(α/2,n-p)	 is the critical value, t, from the student’s 

t-distribution at alpha level α [α=0.10 for 

90-percent prediction intervals, critical 

values may be obtained in many statistics 

textbooks, Iman and Conover (1983), or 

from the World Wide Web];

n-p is the degrees of freedom with n streamgages 

included in the regression analysis and p

parameters in the equation (the number of 

explanatory variables plus one); and

Si is the standard error of prediction for site i, 

and the following equation can be used to 

compute Si:

Si = [MEV + XiUXi']

0.5 (8)

where 

MEV is the model error variance from GLS 

regression equations developed in this 

study;

Xi is the row vector for the streamgage i, 

starting with the number 1, followed 

by the logarithmic values of the basin 

characteristics used in the regression;

U is the covariance matrix for the regression 

coefficients; and

Xi' is the matrix algebra transpose of Xi (Ludwig 

and Tasker, 1993; Ries and Friesz, 2000).

Similar to the SEP, Si represents the sum of the model error 

and the sampling error for a single site i. The XiUXi' term in 

equation 8 also is referred to as the sampling error variance. 

The values of t

(α/2,n-p) and U needed to determine prediction 

intervals for estimates obtained by the regression equations in 

tables 9–11 are presented in table 13 (link to Excel file).

Application of Regression Equations

Methods for applying the RREs listed in tables 9–11 are 

described in the following examples:

Example 1

This example is a calculation of the Q0.2% flood discharge 

(500-year recurrence-interval flood discharge) for a stream 

site in flood region 1. Figure 1 shows the location of the 

streamgage 05482500 North Raccoon River near Jefferson, 

Iowa, as map number 283. This watershed is located entirely 

within flood region 1. Using the USGS StreamStats Webbased GIS tool, DRNAREA (drainage area) is measured as 

1,609.35 mi2, I24H10Y (maximum 24-hour precipitation 

that happens on average once in 10 years) is measured as 

4.321 inches, and CCM (constant of channel maintenance) 

is measured as 1.067 square miles per mile (mi2/mi) (table 3, 

link to Excel file). Because all three basin-characteristic values 

are within the range of values listed in table 12, the GLS 

regression equation is applicable for estimating the Q0.2% flood 

discharge. The GLS regression equation for estimating the 

Q0.2% flood discharge from table 9 is

Q0.2%=DRNAREA0.55310(1.22+0.550 x I24H10Y-0.808 x CCM0.55)

Q0.2%=1,609.350.55310(1.22+0.550 x 4.321-0.808 x 1.0670.55)

Q0.2%=59.33 10(1.22+2.377-0.8373)

Q0.2%= 34,100 ft3/s

To calculate a 90-percent prediction interval for this Q0.2% 

flood-discharge estimate using equation 6, the Xi vector is

Xi = {1, log10 (1,609.35), 4.321, 1.0670.55},

the MEV from table 13 (link to Excel file) is 0.029998, 

and the following table lists the covariance matrix (U):

Using matrix algebra, the product of XiUXi' is determined 

in two steps: (1) by multiplying Xi' (the transpose of Xi) by the 

covariance matrix, U, to obtain UXi' ; and (2) by multiplying 

UXi' by Xi. In this example, the value of XiUXi' is 0.00327303.

The standard error of prediction for this site as computed 

from equation 8 is

Si = [0. 029998 + 0.00327303]0.5 = 0.182403, 

and T from equation 7 is

T= 10(1.6626)(0.182403) = 2.0103,

where the critical value (t

(α/2,n-p)

) from the student’s 

t-distribution for the 90-percent prediction interval is 1.6626 

(table 13, link to Excel file).

The 90-percent prediction interval is estimated from 

equation 6 as

34,100/2.0103<Q0.2%<(34,100)(2.0103), or

17,000 < Q0.2% < 68,600 ft3/s. 

Example 2

This example is a calculation of the Q1% flood discharge 

(100-year recurrence-interval flood discharge) for a stream 

site in flood region 2. Figure 1 shows the location of the 

streamgage 05455500 English River at Kalona, Iowa, as 

map number 181. This watershed is located entirely within 

flood region 2. Using the USGS StreamStats Web-based GIS 

tool, DRNAREA (drainage area) is measured as 574.10 mi2

, 

DESMOIN (percent area of basin within the Des Moines Lobe 

Table 13. Values needed to determine the 90-percent 

prediction intervals for estimates obtained from regional 

regression equations using covariance matrices in Iowa.

Intercept DRNAREA I24H10Y CCM

Intercept 0.579267620 -0.000733524 -0.141069630 0.014150952

DRNAREA -0.000733524 0.000744762 0.000042403 -0.001270299

I24H10Y -0.141069630 0.000042403 0.035474003 -0.007395614

CCM 0.014150952 -0.001270299 -0.007395614 0.020494642
30 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

landform region) is measured as 0.000 percent, and BSHAPE 

(measure of basin shape) is measured as 6.155 (table 3, link 

to Excel file). Because all three basin-characteristic values 

are within the range of values listed in table 12, the GLS 

regression equation is applicable for estimating the Q1% flood 

discharge. The GLS regression equation for estimating the Q1%

flood discharge from table 10 is

Q1% = 10(11.1-7.92 x DRNAREA-0.031 -0.002 x DESMOIN -0.025 x BSHAPE)

Q1% = 10(11.1-7.92 x 574.10-0.031 -0.002 x 0.000 -0.025 x 6.155)

Q1% = 10(11.1-6.504 -0.000 -0.1539)

Q1% = 27,700 ft3/s

To calculate a 90-percent prediction interval for this Q1% 

flood-discharge estimate using equation 6, the Xi vector is

Xi = {1, 574.10-0.031, 0.000, 6.155},the MEV from table 13

(link to Excel file) is 0.007617, and the following table lists 

the covariance matrix (U):

Using matrix algebra, the product of XiUXi' is determined 

in two steps: (1) by multiplying Xi' (the transpose of Xi) by the 

covariance matrix, U, to obtain UXi'; and (2) by multiplying 

UXi' by Xi. In this example, the value of XiUXi' is 0.000902081.

The standard error of prediction for this site as computed 

from equation 8 is

Si = [0. 007617 + 0.000902081]0.5 = 0.0922989, and T 

from equation 7 is

T= 10(1.6538)(0.0922989) = 1.4212,

where the critical value (t

(α/2,n-p)) from the student’s 

t-distribution for the 90-percent prediction interval is 1.6538 

(table 13, link to Excel file).

The 90-percent prediction interval is estimated from 

equation 6 as

27,700/1.4212 < Q1% < (27,700) (1.4212), or

19,500 < Q1% < 39,400 ft3/s. 

Weighted Method to Estimate Annual 

Exceedance-Probability Discharges 

for Streamgages

The Interagency Advisory Committee on Water Data 

(1982) recommends that improved estimates of AEPDs at 

streamgages can be obtained by weighting the annual exceedance-probability LP3 estimate (EMA/MGB) with the RRE 

estimate using the variance of prediction for each of these two 

estimates. The variance of prediction can be thought of as a 

measure of the uncertainty in either the EMA/MGB estimate 

or the RRE estimate. If the two estimates are assumed to be 

independent and are weighted inversely proportional to their 

associated variances, the variance of the weighted estimate 

will be less than the variance of either of the independent 

estimates. Optimal weighted estimates of AEPDs were computed for this study using the Weighted Independent Estimates 

(WIE) computer program available at http://water.usgs.gov/

usgs/osw/swstats/freq.html. Information on this computer 

program is presented by Cohn and others (2012).

The variance of prediction corresponding to the EMA/

MGB estimate from the LP3 analysis is computed using the 

asymptotic formula given in Cohn and others (2001) with 

the addition of the MSE of the generalized skew (Griffis 

and others, 2004). This variance varies as a function of the 

length of record, the fitted LP3 distribution parameters (mean, 

standard deviation, and weighted skew), and the accuracy of 

the procedure used to determine the regional skew component 

of the weighted skew (Verdi and Dixon, 2011). The variance of prediction for the EMA/MGB LP3 estimate generally 

decreases with increasing record length and the quality of the 

LP3 distribution fit. The variance of prediction values for the 

EMA/MGB LP3 estimates for 394 streamgages included in 

this study are listed in table 14 (link to Excel file). The variance of prediction from the RREs is a function of the regression equations and the values of the explanatory variables 

(basin characteristics) used to compute the AEPDs from the 

regression equations. This variance generally increases as 

the values of the explanatory variables move further from the 

mean or median values of the explanatory variables. The variance of prediction values for the RREs used in this study also 

are listed in table 14 (link to Excel file).

Once the variances have been computed, the two independent discharge estimates can be weighted using the following equation (Verdi and Dixon, 2011; Cohn and others, 2012; 

Gotvald and others, 2012):

logQP(g)w = , VPP(g)r

logQP(g)s + VPP(g)slogQP(g)r

VPP(g)s +VPP(g)r (9)

where

QP(g)w	 is the WIE estimate of flood discharge for 

the selected P-percent annual exceedance 

probability for a streamgage, g, in cubic 

feet per second;

VPP(g)r	 is the variance of prediction at the streamgage 

derived from the applicable RREs for the 

selected P-percent annual exceedance 

probability (from table 14, link to Excel 

file), in log units;

QP(g)s	 is the estimate of flood discharge at the 

streamgage from the EMA/MGB LP3 

analysis for the selected P-percent annual 

exceedance probability (from table 4, link 

to Excel file), in cubic feet per second;

VPP(g)s	 is the variance of prediction at the streamgage 

Table 14. Variance of prediction values for 394 streamgages included in this study that were weighted 

using expected moments algorithm (EMA/MGB) and regional-regression-equation estimates of annual 

exceedance-probability discharges.

Intercept DRNAREA DESMOIN BSHAPE

Intercept 0.041672405 -0.045784820 -0.000039051 -0.000518851

DRNAREA -0.045784820 0.052558399 0.000038686 0.000436101

DESMOIN -0.000039051 0.000038686 0.000000287 0.000000161

BSHAPE -0.000518851 0.000436101 0.000000161 0.000025378
Weighted Methods to Estimate Annual Exceedance-Probability Discharges for Ungaged Sites on Gaged Streams 31

from the EMA/MGB LP3 analysis for 

the selected P-percent annual exceedance 

probability (from table 14, link to Excel 

file), in log units; and

QP(g)r	 is the flood-discharge estimate for the selected 

P-percent annual exceedance probability at 

the streamgage derived from the applicable 

RREs (from table 4, link to Excel file), in 

cubic feet per second.

When the variance of prediction corresponding to one of 

the estimates is high, the uncertainty also is high, for which 

the weight of the estimate is relatively small. Conversely, 

when the variance of the prediction is low, the uncertainty 

also is low, for which the weight is correspondingly large. The 

variance of prediction associated with the weighted estimate, 

VPP(g)w, is computed using the following equation (Verdi and 

Dixon, 2011; Gotvald and others, 2012):

VPP(g)w = , VPP(g)s

VPP(g)r

VPP(g)s+ VPP(g)r (10)

Table 4 (link to Excel file) lists the improved AEPDs 

that were weighted using equation 9 along with the variance 

of prediction values from table 14 (link to Excel file) for 

394 streamgages included in this study.

Example 3

This example is a calculation of a weighted estimate of 

the Q1% flood discharge (100-year recurrence-interval flood 

discharge) for a discontinued streamgage in flood region 3 

with only 15 years of annual peak-discharge record available 

for computing AEPDs using an EMA/MGB LP3 analysis. 

Figure 1 shows the location of the streamgage 06610520 

Mosquito Creek near Earling, Iowa, as map number 425. 

This watershed is located entirely within flood region 3. The 

estimate for the Q1% flood discharge from the EMA/MGB 

LP3 analysis is 14,400 ft3/s and from the RRE is 10,400 ft3/s 

(table 4, link to Excel file). The variance of prediction from 

the EMA/MGB LP3 analysis is 0.0160 and from the RRE is 

0.0146 (table 14, link to Ecxel file). A WIE estimate is calculated for this streamgage using equation 9 as

logQP(g)w = , VPP(g)r

logQP(g)s + VPP(g)slogQP(g)r

VPP(g)s +VPP(g)r 

logQP(g)w = , 0.0146 log14,400 + 0.0160 log10,400

0.0160 + 0.0146 

log QP(g)w = 4.084 or QP(g)w = 12,100 ft3/s.

The weighted variance is calculated for this streamgage 

using equation 10 as

VPP(g)w = , VPP(g)s

VPP(g)r

VPP(g)s+ VPP(g)r 

VPP(g)w = , 0.0160 x 0.0146

0.0160 + 0.0146 

VPP(g)w = 0.0076.

Because of the short record for the streamgage in this 

example, the variance of the RRE estimate is slightly lower 

than the variance of the EMA/MGB LP3 estimate and slightly 

more weight is given to the RRE estimate in the WIE calculation in equation 9 than to the EMA/MGB LP3 estimate. The 

variance of prediction is lower for the WIE estimate than for 

either the EMA/MGB LP3 or RRE estimates indicating that 

the uncertainty in the estimate of the Q1% flood discharge is 

reduced.

Weighted Methods to Estimate Annual 

Exceedance-Probability Discharges for 

Ungaged Sites on Gaged Streams

AEPDs at ungaged sites located on gaged streams can 

be improved by weighting AEPDs from a nearby streamgage. 

Two methods for weighting AEPDs from a nearby streamgage are applicable. Both methods require the measurement 

of drainage area (DRNAREA) for the ungaged site and 

a weighted AEPD QP(g)w from a nearby streamgage (see 

equation 9 in previous section). The first weighting method, 

presented in the following section Regression-Weighted 

Estimates for Ungaged Sites on Gaged Streams, requires a 

regional regression estimate for the ungaged site. The second 

weighting method, presented in the following section AreaWeighted Estimates for Ungaged Sites on Gaged Streams,

does not require a regional regression estimate for the 

ungaged site. AEPDs calculated from the regression-weighted 

method are considered to provide better predictive accuracies 

for ungaged sites than estimates calculated from the areaweighted method.

Regression-Weighted Estimates for Ungaged 

Sites on Gaged Streams

 Sauer (1974) presented the following regressionweighted method to improve AEPDs for an ungaged site near 

a streamgage, on the same stream, with 10 or more years of 

annual peak-discharge record. To obtain a regression-weighted 

AEPD (QP(u)rw) for P-percent annual exceedance probability at 

the ungaged site, the WIE estimate for an upstream or downstream streamgage (QP(g)w) must first be determined using 

equation 9 presented in the previous section. The regressionweighted AEPD for the ungaged site (QP(u)w) is then computed 

using the following equation (Verdi and Dixon, 2011):

Q , P(u)rw = QP(u)r

QP(g)w

QP(g)r

1 [ ] ( ( ) ) ) ( + –

2ΔA

A(g)

2ΔA

A(g)

 (11)

where

QP(u)rw is the regression-weighted estimate of flood 

discharge for the selected P-percent annual 
32 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

exceedance probability for the ungaged 

site, u, in cubic feet per second;

ΔA is the absolute value of the difference between 

the drainage areas of the streamgage and 

the ungaged site, in square miles;

QP(g)w is described for equation 9 above;

A(g) is the drainage area for the streamgage, in 

square miles; 

QP(g)r is described for equation 9 above; and

QP(u)r is the flood-discharge estimate derived from 

the applicable RREs for the selected 

P-percent annual exceedance probability at 

the ungaged site, in cubic feet per second.

Use of equation 11 gives full weight to the regression equation 

estimates when the drainage area for the ungaged site is equal 

to 0.5 or 1.5 times the drainage area for the streamgage, and 

increasing weight to the streamgage estimates as the drainage 

area ratio approaches 1. The regression-weighted method is 

not applicable when the drainage area ratio for the ungaged 

site and streamgage is less than 0.5 or greater than 1.5 (Verdi 

and Dixon, 2011; Gotvald and others, 2012). The regressionweighted method is not implemented in StreamStats (Ries and 

others, 2008), but this method can easily be calculated outside 

of StreamStats after using StreamStats to compute QP(u)r . The 

StreamStats computation of QP(u)r provides the drainage area 

for the ungaged site from which ΔA can be calculated for use 

in equation 11.

Example 4

This example is a calculation of a regression-weighted 

estimate for the Q2% flood discharge (50-year recurrenceinterval flood discharge) statistic for a stream site in flood 

region 1. Streamgage 06605600 Little Sioux River at Gillett Grove is shown on figure 1 as map number 402; this 

streamgage will be assumed to be an ungaged site for this 

example. Another streamgage 06605850 Little Sioux River 

at Linn Grove, also is shown on figure 1 as map number 404, 

which is located downstream on the same stream; this site will 

be used as the streamgage in this example. This watershed is 

located entirely within flood region 1. Seven steps are required 

to calculate a regression-weighted estimate using equation 11:

Q2%(u)rw = Q2%(u)r

Q2%(g)w

Q2%(g)r

1 [ ] ( ( ) ) ) ( + –

2ΔA

A(g)

2ΔA

A(g)

The first step is to use the USGS StreamStats Web-based 

GIS tool to measure basin characteristics for the ungaged site 

(map number 402, fig. 1). DRNAREA (drainage area) is measured as 1,352.59 mi2

, I24H10Y (maximum 24-hour precipitation that happens on average once in 10 years) is measured as 

4.088 inches, and CCM (constant of channel maintenance) is 

measured as 0.840 mi2/mi (table 3, link to Excel file). Because 

all three basin characteristic values are within the range of 

values listed in table 12, the flood region 1 regression equation 

is applicable for estimating the Q2%(u)r flood discharge for the 

ungaged site. The second step is to calculate the drainage area 

ratio between the ungaged site and the streamgage (map number 404, fig. 1) to determine whether the regression-weighted 

method is applicable for the ungaged site. The drainage area 

(DRNAREA) of the streamgage is listed as 1,567.26 mi2 in 

table 3 (link to Excel file); therefore, the drainage area ratio 

is 0.863 (1,352.59 mi2/1,567.26 mi2), and the regressionweighted method is applicable for the ungaged site because 

the drainage area ratio is not less than 0.5 or greater than 1.5. 

The third step is to calculate the Q2%(u)r flood discharge for the 

ungaged site (map number 402, fig. 1) using the GLS regression equation from table 9:

Q2%(u)r=DRNAREA0.57410(0.769+0.573 x I24H10Y-0.712 x CCM0.55)

Q2%(u)r=1,352.590.57410(0.769+0.573 x 4.088-0.712 x 0.8400.55)

Q2%(u)r=62.70 10(0.769+2.342-0.6469)

Q2%(u)r=18,300 ft3/s

The fourth step is to obtain the value of Q2%(g)w for the 

streamgage (map number 404, fig. 1), which is the WIE estimate listed in table 4 (link to Excel file) as 19,700 ft3

/s. The 

fifth step is to obtain the value of Q2%(g)r for the streamgage, 

which is the RRE estimate listed in table 4 (link to Excel 

file) as 20,500 ft3/s. The sixth step is to calculate ΔA, where 

ΔA=|1,352.59 mi2-1,567.26 mi2|=214.67 mi2. The seventh 

step is to calculate the regression-weighted estimate for the 

ungaged site, Q2%(u)rw, using equation 11:

Q2%(u)rw = 18,300 19,700

20,500 1 [ ] ( ( ) ) ) ( + –

2x214.67

1,567.26

2x214.67

1,567.26

Q2%(u)rw = 17,800 ft3/s.

Area-Weighted Estimates for Ungaged Sites on 

Gaged Streams

A similar, but simpler, calculation is used in StreamStats 

(Ries and others, 2008; http://streamstats.usgs.gov/ungaged2.

html) to area-weight AEPDs on the basis of the drainage area 

ratio between a streamgage and an ungaged site on the same 

stream. The weighting procedure is not applicable when the 

drainage area ratio is less than 0.5 or greater than 1.5, or when 

the flood characteristics significantly change between sites. 

The area-weighting method was presented in Elements of 

Applied Hydrology (Johnstone and Cross, 1949; Zarriello and 

others, 2012) and the original equation from this publication is 

listed on page 11 in Federal Emergency Management Agency 

(2009). To obtain an area-weighted AEPD QP(u)aw for P-percent 

annual exceedance probability at the ungaged site, the WIE 

estimate for an upstream or downstream streamgage QP(g)w

must first be determined using equation 9. The area-weighted 

AEPD for the ungaged site QP(u)aw is then computed using the 

following equation:
Weighted Method to Estimate Annual Exceedance-Probability Discharges for Ungaged Sites Draining More Than One Flood Region 33

( ) QP(u)aw = QP(g)w , A(u)

b

A(g)

(12)

where

QP(u)aw is the area-weighted estimate of flood 

discharge for the selected P-percent annual 

exceedance probability for the ungaged 

site, u, in cubic feet per second;

A(u) is the drainage area of the ungaged site, in 

square miles;

A(g) is described for equation 11 above;

QP(g)w is described for equation 9 above; and

b is the exponent of drainage area from the 

appropriate regional exponent in table 15.

Regional exponents derived from WREG using a GLS 

analysis of log-10 drainage area (DRNAREA) range from 

0.506 to 0.641 for flood region 1, from 0.436 to 0.579 for 

flood region 2, and from 0.438 to 0.579 for flood region 3 

(table 15). The exponent for a selected P-percent annual 

exceedance probability (table 15) is recommended for use for 

exponent b for this study to obtain an area-weighted estimate 

of an AEPD at an ungaged site on a gaged stream, although an 

average exponent b for the range of exceedance probabilities 

(0.554 for flood region 1, 0.484 for flood region 2, and 

0.487 for flood region 3 in this study) is used by some in 

equation 12. 

Example 5

This example is a calculation of an area-weighted estimate for the Q2% flood discharge (50-year recurrence-interval 

flood discharge) for the same ungaged stream site (map 

number 402, fig. 1) and streamgage (map number 404, fig. 1) 

as illustrated in example 4. Values for A(u), A(g), and QP(g)w from 

example 4 are used to solve equation 12:

( ) QP(u)aw = 19,700 1,352.594 0.535

1,567.265

QP(u)aw = 18,200 ft3

/s.

Estimates for Ungaged Sites on Gaged Streams 

Between Two Streamgages

For an ungaged site that is located between two 

streamgages on the same stream, two AEPDs can be estimated 

for the ungaged site using equations 11 or 12 by substituting either streamgage into the equation. StreamStats uses the 

area-weighted estimates (eq. 12) from both streamgages to 

then weight the individual estimates based on the proximity of 

the streamgages to the ungaged site to obtain final weighted 

estimates for the ungaged site (Ries and others, 2008; http://

streamstats.usgs.gov/ungaged2.html). Additional hydrologic 

judgment may be necessary to determine which of the two 

estimates (or an average or some interpolation thereof) is most 

appropriate, including an evaluation of possible differences in 

flood-region characteristics of the two streamgages in comparison to the ungaged site. Other factors that might be considered when evaluating the two estimates include differences in 

the length or quality of annual peak-discharge records between 

the two streamgages, and the hydrologic conditions that 

occurred during the data-collection period for each streamgage 

(for example, were the annual peak discharges collected during a predominately wet or dry period?) (Verdi and Dixon, 

2011; Gotvald and others, 2012).

Weighted Method to Estimate Annual 

Exceedance-Probability Discharges 

for Ungaged Sites Draining More Than 

One Flood Region

For an ungaged stream site with a watershed that drains 

more than one flood region, the RREs can be applied separately for each flood region using basin characteristics for 

the entire watershed upstream from the ungaged site. AEPDs 

computed for each flood region can then be weighted by the 

Table 15. Regional exponents determined from regional regression of log-10 drainage area for area-weighting method to 

estimate annual exceedance-probability discharges for ungaged sites on gaged streams.

[Note, the constant is not used in the area-weighting method (eq. 11), but could be used to estimate annual exceedance-probability discharges at 

ungaged sites from drainage area only]

Annual exceedance probability 

(percent)

Flood region 1 Flood region 2 Flood region 3

Exponent b Constant Exponent b Constant Exponent b Constant

50 0.641 1.66 0.579 2.15 0.579 2.31

20 0.592 2.06 0.525 2.57 0.528 2.69

10 0.569 2.26 0.499 2.78 0.503 2.87

4 0.548 2.45 0.476 2.97 0.479 3.05

2 0.535 2.57 0.463 3.08 0.466 3.16

1 0.524 2.67 0.453 3.18 0.455 3.25

0.5 0.516 2.75 0.445 3.26 0.447 3.33

0.2 0.506 2.85 0.436 3.35 0.438 3.41
34 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

proportion of drainage area within each flood region to calculate final AEPDs for the ungaged site. For example, if 30 percent of the drainage area for an ungaged site is in the upstream 

region and 70 percent is in the downstream region, the AEPD 

computed from an equation for the upstream region is multiplied by 0.30 and is added to 0.70 times the AEPD computed 

from an equation for the downstream region. The variance of 

prediction for this weighted method also can be approximated 

by using the same weighting method based on proportional 

drainage areas (Gotvald and others, 2012). StreamStats uses 

this weighted method to estimate AEPDs and prediction errors 

for ungaged sites draining more than one flood region (Ries 

and others, 2008; http://streamstats.usgs.gov/ungaged2.html).

Region-of-Influence Method to 

Estimate Annual ExceedanceProbability Discharges for Ungaged 

Stream Sites

The region-of-influence (RoI) method has been used to 

estimate AEPDs at ungaged sites by relating basin characteristics to AEPDs for a unique subset of streamgages (Burn, 

1990; Eng and others, 2005, 2007). The RoI method was 

tested as part of this study using WREG (Eng and others, 

2009) to determine if predictive accuracies for AEPDs may 

be improved using RoI compared to the traditional regionalregression method. The RoI method defines a unique subset, 

or region of influence, for each ungaged site determined by 

selecting streamgages with basin characteristics similar to 

those measured for the ungaged site. The RoI is defined as 

the number of streamgages (N) “nearest” to the ungaged 

site, where “nearest” is measured by similarity of basin 

characteristics in Euclidean space. An advantage of this 

method is extrapolation errors tend to be small because the 

predictions naturally result near the center of the space of the 

basin characteristics.

To investigate the RoI method for this study, basin 

characteristics identified as the most significant in the statewide OLS regression analyses were selected and compiled 

into an RoI dataset that included the same 510 streamgages as 

used for the development of statewide regression equations 

(table 8). The RoI method in WREG allows three approaches 

for defining hydrologic similarity among streamgage basins: 

independent or predictor-variable space RoI, geographic space 

RoI, and a combination of predictor-variable and geographic 

spaces called hybrid RoI. Preliminary RoI analyses were 

performed to determine the best combination of three input 

parameters required by the RoI program in WREG: (1) the 

best set of basin characteristics must be selected for use as 

explanatory variables, (2) the number of streamgages (N) must 

be selected to compose the specific region of influence for 

the statewide study area, and (3) the predictor-variable space 

RoI, geographic space RoI, or hybrid RoI approach must be 

selected.

RMSEs were evaluated for the preliminary RoI analyses 

to determine the best combination of the three required input 

parameters for WREG. Table 16 lists the best combinations 

of explanatory variables with the lowest RMSEs that were 

identified statewide, and by flood regions, for each of the eight 

selected annual exceedance probabilities through iterative 

RoI analyses using WREG. Although statewide and regional 

RMSE (RoI method) and SEP (RRE method) performance 

metrics are not directly comparable, overall, RMSEs for RoI 

were poorer than SEPs for statewide GLS regression equations (table 8) or SEPs for RREs (tables 9–11). RMSE (SEE) 

is not appropriate for evaluating GLS regressions because 

of the unequal weighting given to the streamgages in GLS 

regression (Risley and others, 2008). The resulting unequally 

Table 16. Significant explanatory variables and predictive accuracies of preliminary region-of-influence equations in Iowa.

[RoI, region of influence; RMSE, root mean-square error; %, percent; DRNAREA, geographic information system drainage area; (+), explantory variable has 

a positive relation with the response variable; DESMOIN, percent of basin within Des Moines Lobe landform region; (-), explantory variable has a negative 

relation with the response variable; CCM, constant of channel maintenance; GRoI, geographic space RoI]

Statistic

Most significant explanatory variables 

identified for the preliminary equation and 

explanatory-variable relation signs

N, number of 

streamgages 

used to form 

RoI

RoI 

method

Statewide 

RMSE 

(percent)

Region 1 

RMSE 

(percent)

Region 2 

RMSE 

(percent)

Region 3 

RMSE 

(percent)

Preliminary RoI analysis results

Q50% DRNAREA0.011 (+), DESMOIN (-), CCM0.55 (-) 55 GRoI 50.3 46.7 46.9 48.9

Q20% DRNAREA-0.011 (+), DESMOIN (-), CCM0.55 (-) 55 GRoI 44.0 39.7 34.2 42.8

Q10% DRNAREA-0.023 (+), DESMOIN (-), CCM0.55 (-) 55 GRoI 45.3 40.4 33.8 43.7

Q4% DRNAREA-0.035 (+), DESMOIN (-), CCM0.55 (-) 55 GRoI 49.1 43.8 36.8 46.9

Q2% DRNAREA-0.043 (+), DESMOIN (-), CCM0.55 (-) 55 GRoI 52.6 47.0 40.0 49.7

Q1% DRNAREA-0.050 (+), DESMOIN (-), CCM0.55 (-) 55 GRoI 56.3 50.5 43.5 52.7

Q0.5% DRNAREA-0.056 (+), DESMOIN (-), CCM0.55 (-) 55 GRoI 60.0 54.1 47.1 55.8

Q0.2% DRNAREA-0.064 (+), DESMOIN (-), CCM0.55 (-) 55 GRoI 64.9 59.1 51.7 59.9

 Number of streamgages included in RoI analysis 510 91 176 127
Comparison of Annual Exceedance-Probability Discharges 35

weighted GLS residuals produce inflated RMSE values that 

are not comparable to RMSE from the RoI regression analyses. Because RREs provided improved predictive accuracies, 

the RoI method was not developed further and RoI equations 

are not listed in this report but are summarized in table 16 to 

provide a reference for indicating the improvement obtained 

using RREs. 

Comparison of Annual ExceedanceProbability Discharges

To better understand how AEPDs computed using the 

new EMA/MGB analyses used in this study compare to those 

computed using the standard Bulletin 17B/GB analyses, how 

AEPDs computed using the updated regional-skew-coefficient 

constant compared to those computed using the superseded 

regional skew coefficients, and how AEPDs computed using 

the updated RREs developed in this study compare to those 

computed in the previous study (Eash, 2001), relative percentage changes were calculated for these different AEPD 

estimates. Relative percentage change is calculated using the 

following equation: 

( ) RPchange = 100 Qnew – Qold

Qold

(13)

where

RPchange is the relative percentage change, which 

represents the relative change between 

the old AEPD estimate value and the new 

AEPD estimate value;

Qnew is the new AEPD estimate value, in cubic feet 

per second; and

Qold is the old AEPD estimate value, in cubic feet 

per second.

RPchange values that are positive imply that Qnew is greater than 

Qold, and RPchange values that are negative imply that Qold is 

greater than Qnew.

Estimates from Annual Exceedance-Probability 

Analyses

Two new elements were included in the computation 

of AEPDs for this study. First, as described in the section 

Regional Skew Analysis, a regional skew study was performed 

for Iowa from which an updated regional-skew-coefficient 

constant was developed for the study area and was used in 

annual exceedance-probability analyses for all streamgages 

included in this study. Second, as described in the section 

Expected Moments Algorithm (EMA/MGB) Analyses, a new 

annual exceedance-probability analysis method was used to 

compute AEPDs for all streamgages included in this study. 

To better understand the effects of these two new elements in 

the computation of AEPDs for this study, three variations of 

annual exceedance-probability analyses were also computed 

for 283 streamgages (table 6, link to Excel file). These 

283 streamgages are a subset of the 523 streamgages listed in 

table 1, (link to Excel file). All 283 streamgages are located 

in Iowa and were used to develop regional regression equations for this study, as noted in table 4 (link to Excel file) with 

an asterisk next to the acronym RRE. Table 17 lists relative 

percentage change between AEPDs based on data through 

the 2010 water year for four variations of annual exceedanceprobability analyses: (1) EMA/MGB analyses computed 

using the updated regional-skew-coefficient constant (table 4, 

link to Excel file), (2) EMA/MGB analyses computed using 

superseded regional skew coefficients (table 6, link to Excel 

file), (3) standard Bulletin 17B/GB analyses computed using 

the updated regional-skew-coefficient constant (table 6, link 

to Excel file), and (4) Bulletin 17B/GB analyses computed 

using superseded regional skew coefficients (table 6, link to 

Excel file). Table 17 lists relative percentage change calculated 

for five sets of comparisons between different combinations 

of these four sets of AEPD estimates for 283 streamgages in 

Iowa.

The first and second sets of comparisons listed in 

table 17 summarize relative percentage changes from the use 

of superseded regional skew coefficients to the use of the 

updated regional-skew-coefficient constant for Bulletin 17B/

GB and EMA/MGB analyses. Mean and median relative 

percentage changes indicate that AEPDs generally decreased 

statewide with the use of the updated regional-skew-coefficient constant for probabilities of 10 percent and lower (for 

10-year recurrence-interval floods and greater) for Bulletin 

17B/GB and EMA/MGB analyses, and that AEPDs generally 

increased statewide for probabilities greater than 10 percent 

(2- and 5-year recurrence-interval floods). For example, the 

first set of comparisons indicate that mean and median 

differences for Q1% flood discharge estimates are 6.0 and 

5.1 percent lower, respectively, for Bulletin 17B/GB analyses 

using the updated-regional-skew coefficient constant, and the 

second set of comparisons indicate similar results with mean 

and median differences that are 7.1 and 5.7 percent lower, 

respectively, for EMA/MGB analyses using the updated 

regional-skew-coefficient constant. These relative percentage 

changes appear to be reasonable considering that the updated 

statewide regional-skew-coefficient constant of -0.400 is 

lower than superseded regional skew coefficients for most of 

the State (fig. 2 and table 2 (link to Excel file); Eash, 2001). 

Also, mean and median relative percentage changes indicate 

that AEPDs decrease more for the smaller exceedance probabilities using the updated regional-skew-coefficient constant, 

compared to using the superseded regional skew coefficients, 

and the decrease is slightly greater using EMA/MGB analyses compared to using Bulletin 17B/GB analyses. As the first 

and second sets of comparisons listed in table 17 indicate for 

the 283 streamgages in Iowa that were compared, relative 

percentage change from AEPD estimates computed using 

superseded regional skew coefficients to those computed using 
36 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

the updated regional-skew-coefficient constant have decreased 

on average about 4 percent for the eight annual exceedance 

probabilities.

The third and fourth sets of comparisons listed in table 17 

summarize relative percentage changes from the use of Bulletin 17B/GB analyses to the use of EMA/MGB analyses for 

the use of the updated regional-skew-coefficient constant and 

of superseded regional skew coefficients. Mean and median 

relative percentage changes indicate that AEPDs generally 

increased statewide with the use of EMA/MGB analyses for 

probabilities of 20 percent and lower (for 5-year recurrenceinterval floods and larger) for the use of the updated regionalskew-coefficient constant and the use of superseded regional 

skew coefficients, and that AEPDs generally decreased statewide for the 50-percent probability (2-year recurrence-interval 

flood). For example, the third set of comparisons indicate that 

mean and median differences for Q1% flood discharge estimates 

are 13.5 and 3.1 percent greater, respectively, for EMA/MGB 

analyses using the superseded regional skew coefficients, 

and the fourth set of comparisons indicate similar results 

with mean and median differences that are 11.8 and 

2.0 percent greater, respectively, for EMA/MGB analyses 

using the updated regional-skew-coefficient constant.

Figure 11 is a plot of the Q1% flood-discharge data 

used in the fourth set of comparisons. This plot shows 

that relative percentage changes from Bulletin 17B/GB to 

Table 17. Relative percentage change between annual exceedance-probability discharge estimates based on data through the 

2010 water year for 283 streamgages in Iowa using expected moments algorithm (EMA/MGB) and Bulletin 17B/GB analyses and using 

updated and superseded regional skew coefficient values.

Annual exceedance probability*

50-percent 20-percent 10-percent 4-percent 2-percent 1-percent 0.5-percent 0.2-percent

Relative percentage change between Bulletin 17B/GB estimates computed using the updated regional skew coefficient value and 

Bulletin 17B/GB estimates computed using superseded regional skew coefficient values

Maximum 10.9 20.3 22.3 22.7 21.7 19.4 17.6 19.7

Minimum -10.1 -9.9 -14.3 -19.0 -23.6 -28.9 -33.7 -41.5

Mean 1.5 0.4 -1.0 -3.0 -4.5 -6.0 -7.4 -9.3

Median 1.2 0.2 -0.8 -2.5 -3.8 -5.1 -6.5 -8.2

Relative percentage change between EMA/MGB estimates computed using the updated regional skew coefficient value and 

EMA/MGB estimates computed using superseded regional skew coefficient values

Maximum 10.5 8.1 6.1 4.5 8.3 12.1 16.2 21.6

Minimum -4.4 -2.4 -4.4 -13.1 -22.4 -30.8 -38.9 -48.6

Mean 1.9 1.4 -0.3 -2.9 -5.0 -7.1 -9.0 -11.6

Median 1.3 0.8 0.0 -2.1 -4.1 -5.7 -7.4 -9.9

Relative percentage change between EMA/MGB and Bulletin 17B/GB estimates computed using superseded regional 

skew coefficient values

Maximum 43.3 33.1 76.6 141.7 186.9 225.2 259.3 295.4

Minimum -71.7 -42.8 -23.2 -33.9 -39.4 -43.8 -47.0 -50.5

Mean -5.2 2.4 6.2 9.8 11.8 13.5 14.8 16.1

Median -1.3 1.1 2.8 3.1 3.5 3.1 2.8 2.4

Relative percentage change between EMA/MGB and Bulletin 17B/GB estimates computed using the updated regional 

skew coefficient value

Maximum 42.1 33.1 76.0 141.1 186.4 225.7 261.1 298.7

Minimum -73.5 -40.0 -20.7 -29.7 -34.1 -37.4 -39.9 -42.4

Mean -4.8 3.4 6.9 9.8 11.1 11.8 12.3 12.3

Median -0.7 1.8 2.9 2.7 2.7 2.0 1.6 1.0

Relative percentage change between EMA/MGB estimates computed using the updated regional skew coefficient value and 

Bulletin 17B/GB estimates computed using superseded regional skew coefficient values

Maximum 42.8 33.1 76.4 143.2 190.5 232.0 269.9 311.8

Minimum -72.9 -39.8 -23.2 -34.8 -41.1 -45.8 -49.8 -54.1

Mean -3.4 3.8 5.8 6.5 6.0 5.0 3.8 1.8

Median 0.2 2.4 2.2 0.7 -0.9 -2.1 -3.3 -5.6

*

EMA/MGB estimates using the updated regional skew coefficient value developed in this study are listed in table 4 (link to Excel file) and all other 

exceedance-probability estimates are listed in table 6 (link to Excel file).
Comparison of Annual Exceedance-Probability Discharges 37

EMA/MGB analyses for the larger drainage area, continuousrecord streamgages generally are small values, and that differences for a few of these streamgages have negative values 

greater than 20 percent. Figure 11 also shows that relative percentage changes for the smaller drainage-area CSGs generally 

are positive values greater than 20 percent and that differences 

for several of these CSGs are positive values greater than 

50 percent. This plot indicates that the type of streamgage, 

or the type of streamgage record, results in the greatest 

differences between EMA/MGB and Bulletin 17B/GB 

estimates of the Q1% flood discharge for this comparison of 

283 streamgages. Of these 283 streamgages, 147 of them are 

continuous-record streamgages, or streamgages that have 

been operated as a continuous-record streamgage and as a 

CSG, and 136 of them are CSGs. For the fourth set of comparisons, mean and median relative percentage changes for 

Q1% flood discharge estimates for the 147 streamgages that 

have been operated as a continuous-record streamgage, or as 

a continuous-record and CSG streamgage, are -1.1 and 0 percent, respectively, and mean and median relative percentage 

changes for the 136 CSGs are 25.8 and 17.9 percent, respectively. Because most of the CSGs included in this comparison 

have censored data records, the larger AEPDs computed for 

CSGs using EMA/MGB analyses, compared to Bulletin 17B/

GB analyses, are believed to result mainly from the ability of 

the EMA/MGB analysis to use a specific discharge interval 

for data that is censored by the standard Bulletin 17B/GB 

analysis when it is below the largest minimum recordable 

threshold discharge. As the third and fourth sets of comparisons listed in table 17 indicate for the 283 streamgages in Iowa 

that were compared, relative percentage change from AEPD 

estimates computed using standard Bulletin 17B/GB analyses 

to those computed using EMA/MGB analyses have increased 

on average about 8 percent for the eight annual exceedance 

probabilities.

The fifth set of comparisons listed in table 17 summarizes the relative percentage changes from Bulletin 17B/GB 

analyses computed using superseded regional skew coefficients to EMA/MGB analyses computed using the updated 

regional-skew-coefficient constant. This comparison includes 

both of the new elements that were applied in the computation 

of AEPDs for this study. Mean relative percentage changes 

indicate that AEPDs generally increased statewide with the 

use of EMA/MGB analyses using the updated regional-skewcoefficient constant for probabilities of 20 percent and lower 

(for 5-year recurrence-interval floods and larger), and that 

estimates generally decreased statewide for the 50-percent 

probability (2-year recurrence-interval flood). Median relative 

percentage changes indicate that AEPDs generally decreased 

statewide with the use of EMA/MGB analyses using the 

updated regional-skew-coefficient constant for the 2-percent 

probability and lower (50-year recurrence-interval floods and 

larger), and that estimates generally increased statewide for the 

4-percent probability and higher (25-year recurrence-interval 

floods and smaller). For example, the fifth set of comparisons 

indicate that the mean difference for Q1% flood discharge 

0.01 0.1 1 10 100 1,000 10,000

Drainage area, in square miles

-50

0

50

100

150

200

250

Relative percentage change

Continuous-record streamgage or both

Crest-stage gage (CSG) 

EXPLANATION

Figure 11. Relative percentage change, by drainage area and type of streamgage, between expected moments algorithm 

(EMA/MGB) and standard Bulletin 17B/GB analyses computed using the updated regional-skew-coefficient constant for onepercent annual exceedance-probability discharges for 283 streamgages in Iowa.
38 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

estimates is 5.0 percent greater, and that the median difference is 2.1 percent lower for EMA/MGB analyses using the 

updated regional-skew-coefficient constant. Figure 12 is a plot 

of the Q1% flood-discharge data used in fifth set of comparisons. This plot shows similar results to figure 11, except there 

are six fewer streamgages with relative percentage changes 

greater than 50 percent in figure 12. The difference between 

figures 11 and 12 is that relative percentage changes from 

superseded to updated regional skew coefficients also are 

shown in figure 12. Similar to figure 11, figure 12 indicates 

that regardless of the regional skew coefficients used in the 

analyses, the greatest differences between EMA/MGB and 

Bulletin 17B/GB estimates of the Q1% flood discharge are 

because of the type of streamgage, or streamgage record, 

for this comparison of 283 streamgages in Iowa. As the 

fifth set of comparisons listed in table 17 indicates for the

283 streamgages in Iowa that were compared, relative percentage change from AEPD estimates computed using standard 

Bulletin 17B/GB analyses and superseded regional skew 

coefficients to those computed using EMA/MGB analyses and 

the updated regional-skew-coefficient constant have increased 

on average about 4 percent for the eight annual exceedance 

probabilities. A comparison of EMA/MGB and standard Bulletin 17B/GB analysis estimates for the Q1% annual exceedance probability for 283 streamgages in Iowa indicated a 

median flood-discharge relative percentage change of zero 

percent for 147 continuous-record streamgages and a median 

flood-discharge relative percentage change increase of 18 percent for 136 crest-stage gages. Because most of the crest-stage 

gages included in this comparison have censored data records, 

the larger annual exceedance-probability discharges computed 

for crest-stage gages using EMA/MGB analyses, compared 

to Bulletin 17B/GB analyses, are believed to result mainly 

from the ability of the EMA/MGB analyses to use a specific 

discharge interval for data that is censored by standard Bulletin 17B/GB analyses when it is below the largest minimumrecording-threshold discharge. 

Estimates from Regional Regression Equations

Table 18 lists relative percentage change from RRE 

estimates computed in a previous study (Eash, 2001) based on 

data through the 1997 water year to RRE estimates computed 

in this study based on data through the 2010 water year for 

185 streamgages in Iowa. Mean and median relative percentage changes indicate that RRE estimates generally increased 

statewide ranging from 4.4 to 13.3 percent and from 3.0 

to 14.1 percent, respectively, with the development of the 

RREs for this study. For example, table 18 shows that mean 

and median relative percentage changes for the Q1% flood 

discharge estimate are 7.9 and 6.4 percent greater using the 

updated RREs developed in this study. In part, the larger RRE 

estimates computed for this study, compared to the previous 

study (Eash, 2001), can be attributed to the additional annual 

0

Continuous-record streamgage or both

Crest-stage gage (CSG) 

EXPLANATION

0.01 0.1 1 10 100 1,000 10,000

Drainage area, in square miles

-50

50

100

150

200

250

Relative percentage change

Figure 12. Relative percentage change, by drainage area and type of streamgage, between expected moments algorithm 

(EMA/MGB) analyses computed using the updated regional-skew-coefficient constant and Bulletin 17B/GB analyses 

computed using superseded regional skew coefficients for one-percent annual exceedance-probability discharges for 

283 streamgages in Iowa.
Comparison of Annual Exceedance-Probability Discharges 39

peak-discharge record collected from 1998 to 2010 that was 

included in annual exceedance-probability analyses computed 

in this study. Large flood events in Iowa during 1998 (Fischer, 

1999), 1999 (Ballew and Fischer, 2000; Ballew and Eash, 

2001), 2002 (Eash, 2004), during 2004 (Eash, 2006), 2008 

(Fischer and Eash, 2010; Linhart and Eash, 2010; Buchmiller 

and Eash, 2010; Holmes and others, 2010), and during 2010 

(Eash, 2012; Barnes and Eash, 2012) have contributed to 

increased AEPDs and RRE estimates computed in this study. 

Relative percentage changes shown in table 18 highlight the 

need to periodically update AEPDs for streamgages and RREs 

for Iowa to obtain reliable estimates of AEPDs for ungaged 

stream sites.

Figure 13 shows relative percentage change by drainage 

area for Q1% flood-discharge estimates computed using RREs 

from a previous study (Eash, 2001) to those computed using 

RREs developed in this study for the same 185 streamgages in 

Iowa as summarized in table 18. Figure 13 shows a fairly uniform relative percentage change by drainage area between the 

two sets of RRE estimates, with a few sites indicating positive 

and negative relative percentage changes greater than 40 percent for drainage areas generally less than about 300 mi2

.This 

plot indicates that there does not appear to be a bias because of 

drainage area between AEPD estimates computed from RREs 

developed in this study compared to AEPD estimates computed from RREs developed in Eash (2001). As table 18 and 

figure 13 indicate for the 185 streamgages in Iowa that were 

compared, relative percentage change from AEPD estimates 

computed using RREs developed in the previous study (Eash, 

2001) to those computed using RREs developed in this study 

have increased on average about 8 percent for the eight annual 

exceedance probabilities. As indicated by the fifth set of 

comparisons in table 17, about 4 percent of this increase likely 

Table 18. Relative percentage change from regional-regression-equation estimates computed in the previous study to those 

computed in this study for 185 streamgages in Iowa.

Relative percentage change for annual exceedance probability1

50-percent 20-percent 10-percent 4-percent 2-percent 1-percent 0.5-percent 0.2-percent

Maximum 68.2 70.0 74.7 83.6 87.7 98.3 109.3 115.4

Minimum -62.7 -63.4 -62.8 -61.7 -61.3 -60.8 -60.1 -60.8

Mean 8.9 6.0 4.4 8.5 8.2 7.9 13.3 6.8

Median 10.6 4.4 3.0 8.1 7.6 6.4 14.1 6.2

1

Regional-regression-equation estimates are listed in table 4 (link to Excel file) for this study and are listed in Eash (2001) for the previous study.

Figure 13. Relative percentage change by drainage area between one-percent annual exceedance-probability discharges 

computed using regional regression equations developed in this study and those developed in the previous study for 

185 streamgages in Iowa.

1 10 100 1,000 10,000

Drainage area, in square miles

-80

-60

-40

-20

0

20

40

60

80

100

Relative percentage change
40 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

is because of the use of EMA/MGB analyses with the updated 

regional-skew-coefficient constant.

During the early 1990’s, 35 new CSGs with drainage 

areas of 11 mi2 or less began operation statewide (May and 

others, 1996) replacing older CSGs with either adequate, or 

inadequate, annual peak-discharge records for the computation 

of AEPDs. These new CSGs were not included in the development of RREs for a previous study (Eash, 2001) because they 

did not have 10 years of annual peak-discharge record through 

the 1997 water year, whereas most of them were included in 

the development of RREs in this study. Thus, although these 

newer CSGs are included in figures 11 and 12 and in table 17, 

they are not included in figure 13 or in table 18. Therefore, 

because figures 11 and 12 indicate that a few of these newer 

CSGs with drainage areas of 11 mi2 or less have large 

positive relative percentage changes, if these newer CSGs 

were included in figure 13 there could be a few more data 

points plotting as positive relative percentage changes above 

40 percent.

StreamStats

StreamStats is a USGS Web-based GIS tool (http://water.

usgs.gov/osw/streamstats/index.html) that allows users to 

obtain streamflow statistics, drainage-basin characteristics, and 

other information for user-selected sites on streams. Users can 

select stream site locations of interest from an interactive map 

and can obtain information for these locations. If a user selects 

the location of a USGS streamgage, the user will receive 

previously published information for the streamgage from a 

database. If a stream site location is selected where no data are 

available (an ungaged site), a GIS program will estimate information for the site. The GIS program determines the boundary 

of the drainage basin upstream from the stream site, measures 

the basin characteristics of the drainage basin, and solves 

the appropriate regression equations to estimate streamflow 

statistics for that site. The results are presented in a table and 

a map showing the basin-boundary outline. The estimates are 

applicable for stream sites not significantly affected by regulation, diversions, channelization, backwater, or urbanization. In 

the past, it could take an experienced person more than a day 

to estimate this information at an ungaged site. StreamStats 

reduces the effort to only a few minutes. 

StreamStats makes the process of computing streamflow statistics for ungaged sites much faster, more accurate, 

and more consistent than previously used manual methods. 

It also makes streamflow statistics for streamgages available 

without the need to locate, obtain, and read the publications in 

which they were originally provided. Examples of streamflow 

statistics that can be provided by StreamStats include the Q1%

flood discharge, the median annual flow, and the mean 7-day, 

10-year low flow. Examples of basin characteristics include 

the drainage area, basin shape, mean annual precipitation, percent of area underlain by hydrologic soil types, and so forth. 

Basin characteristics provided by StreamStats are the physical, 

geologic, and climatic properties that have been statistically 

related to movement of water through a drainage basin to a 

stream site.

Streamflow statistics can be needed at any location along 

a stream and can assist with water-resources planning, management, and permitting; design and permitting of facilities 

such as wastewater-treatment plants and water-supply reservoirs; and design of structures such as roads, bridges, culverts, 

dams, locks, and levees. In addition, planners, regulators, 

engineers, and hydrologists often need to know the physical 

and climatic characteristics (basin characteristics) of the drainage basins upstream from locations of interest to help them 

understand the processes that control water availability and 

water quality at these locations. StreamStats will be a valuable 

tool to assist with these needs.

The regression equations presented in this report will be 

incorporated in the USGS StreamStats Web-based GIS tool 

(http://water.usgs.gov/osw/streamstats/index.html). StreamStats will provide users the ability to estimate eight AEPDs, 

and 90-percent prediction intervals for ungaged stream sites in 

Iowa. 

Maximum Floods in Iowa

For certain high-risk flood-plain developments or for 

evaluation of the reasonableness of unusually large flooddischarge estimates, data on maximum known floods may be 

considered in addition to AEPDs. Maximum floods in Iowa 

and their estimated annual exceedance-probability ranges are 

listed in table 1, (link to Excel file) for streamgages included 

in this study. Figure 14 shows the relation between maximum 

flood discharge and drainage area for each of the three flood 

regions for 516 streamgages in Iowa. A total of 360 of these 

sites are active or discontinued, unregulated streamgages 

with annual peak-discharge records and 156 of these sites are 

ungaged sites. Flood-peak discharges were determined at the 

ungaged sites using indirect measurement methods (Benson 

and Dalrymple, 1967). Regression lines for the Q0.2% flood 

discharge (one-variable equations from table 15) and enveloping curves for the maximum known floods are shown for each 

flood region in figure 14. The enveloping curves indicate maximum flood-discharge potential for a range of drainage areas 

for each flood region. Figure 14 shows that about 115 of the 

516, or about 22 percent, of the data points for streamgages 

and ungaged sites are present between the enveloping curves 

and the regional regression lines for the Q0.2% flood discharge. 

Most of these maximum floods happened as the result of rare 

storm phenomena. Maximum differences between the regression lines and the enveloping curves occur in the drainage 

area range from approximately 5 to 50 mi2 for flood region 1, 

from approximately 20 to 500 mi2 for flood region 2, and from 

approximately 5 to 100 mi2 for flood region 3. These maximum differences may indicate that maximum flood-discharge 

potential, as unit runoff per square mile [(ft3/s)/mi2], may be 

greatest within these drainage area ranges for watersheds in 

these three flood regions.
Maximum Floods in Iowa 41

0.01 0.1 1 10 100 1,000 10,000

Drainage area, in square miles

10 

100

1,000

10,000

100,000

1,000,000

10

100

1,000

10,000

100,000

Discharge, in cubic feet per second

A

B

Flood region 1 regression line: 0.2-percent annual exceedance-probability 

discharge = 102.85 DRNAREA0.506

Enveloping curve for flood region 1

Flood region 1 streamgage

Flood region 1 ungaged site 

EXPLANATION

Flood region 2 regression line: 0.2-percent annual exceedance-probability 

discharge = 103.35 DRNAREA0.436

Enveloping curve for flood region 2

Flood region 2 streamgage

Flood region 2 ungaged site 

EXPLANATION

Figure 14. Relation between maximum flood discharge and drainage area for streams in A, flood region 1, B, flood 

region 2, and C, flood region 3.
42 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

Summary

Reliable estimates of annual exceedance-probability 

discharges (AEPDs) are essential for the economic planning 

and safe design of bridges, dams, levees, and other structures 

located along rivers and streams, and for the effective management of flood plains. In response to the need to update 

and improve the predictive accuracy of estimates of AEPDs 

for ungaged stream sites in Iowa, the U.S. Geological Survey 

(USGS), in cooperation with the Iowa Department of Transportation and the Iowa Highway Research Board, initiated a 

statewide study in 2006. 

Methods described in this report for estimating selected 

AEPDs are applicable to streams in Iowa that are not significantly affected by regulation, diversion, channelization, backwater, or urbanization. Estimation equations were developed 

for eight selected annual exceedance-probability statistics 

for flood discharges with 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 

0.2-percent annual exceedance probabilities, which are equivalent to annual flood-frequency recurrence intervals of 2, 5, 

10, 25, 50, 100, 200, and 500 years, respectively. Major 

accomplishments of the study included (1) performing 

a Bayesian weighted least-squares/generalized least-squares 

regression analysis to update regional skew coefficients 

for Iowa; (2) computing eight selected AEPDs at 

518 streamgages within Iowa and adjacent States with at 

least 10 years of annual peak-discharge record, based on data 

through September 30, 2010; (3) measuring 59 basin characteristics for each streamgage; (4) defining three flood regions 

for the State and developing 24 regional regression equations 

(RREs) to estimate eight selected AEPDs at ungaged stream 

sites based on basin characteristics; (5) calculating weighted 

AEPDs at 394 streamgages using the weighted independent 

estimates method; and (6) calculating AEPD relative percentage change for streamgages in Iowa between estimates from 

different annual exceedance-probability analyses based on 

data through the 2010 water year and between RREs developed in this study and a previous study (Eash, 2001).

Kendall’s tau tests were performed for 518 streamgages 

included in the regression study because trends in annual 

peak-discharge data could introduce a bias into the annual 

exceedance-probability analyses. Results of the Kendall’s 

tau tests indicated statistically significant trends for 

25 streamgages when tested using 94 percent of the entire 

record length. The entire record length was not used because 

of the sensitivity of the Kendall’s tau tests to multiyear 

sequences of larger or smaller discharges if the sequences happen near the beginning or end of the period of record. Twentytwo of these streamgages were included in the regional regression analyses because of uncertainty in the trends because 

of short or broken records. The remaining three streamgages 

represent actual trends or anomalies and were omitted from 

the regression analyses. The number of uncensored annual 

peak-discharge record lengths used in the study for the 

518 streamgages ranged from 9 to 108 years with a mean 

Figure 14. Relation between maximum flood discharge and drainage area for streams in A, flood region 1, B, flood 

region 2, and C, flood region 3.—Continued

Flood region 3 regression line: 0.2-percent annual exceedance-probability 

discharge = 103.41 DRNAREA0.438

Enveloping curve for flood region 3

Flood region 3 streamgage

Flood region 3 ungaged site 

EXPLANATION

0.01 0.1 1 10 100 1,000 10,000

Drainage area, in square miles

10 

100

1,000

10,000

100,000

1,000,000

Discharge, in cubic feet per second

C
Summary 43

of 35.4 years and a median of 28 years. Drainage areas of the 

518 streamgages ranged from 0.05 to 7,783 square miles.

Three regionalization approaches were investigated in 

this study for estimating AEPDs at ungaged sites in Iowa: 

statewide, regional, and region-of-influence regression. 

Regression analyses were used to relate physical and climatic 

characteristics of drainage basins to AEPDs. Data collected 

for 518 streamgages were compiled into statewide, regional, 

and region-of-influence datasets for regression analyses. Root 

mean square errors and average standard errors of prediction 

(SEP) calculated for each equation for each AEPD were compared for each regression to evaluate the predictive accuracy. 

Because regional regression provided the best predictive 

accuracy of the three approaches investigated, preliminary 

equations developed for the statewide and region-of-influence 

methods are not listed in this report. Regional regression analyses included 510 streamgages after 8 additional streamgages 

were removed from the regression dataset. These additional 

eight streamgages were flagged as outliers in the regression 

analyses and were removed because of significant trends, 

urbanization, or channelization.

The study area, which included all of Iowa and adjacent 

areas (within 50 miles of the State border) of neighboring 

States, was divided into three flood regions on the basis of 

landform regions and soil regions. The three flood regions 

(regions 1, 2, and 3) were defined for Iowa after testing a 

number of different regional combinations to define six final 

flood regions for the study area. Because three of the six final 

flood regions defined for the study area are completely outside 

of Iowa, regression equations were not developed further for 

flood regions 4, 5, and 6. Generalized least-squares (GLS) 

multiple-linear regression analyses, weighted on the basis of 

streamgage record length and the variance and cross-correlation of the annual peak discharges, were performed to develop 

the final equations that included 394 streamgages for the three 

flood regions. Preliminary multiple-linear-regression analyses, 

using ordinary-least-squares regression, were performed to 

test for significant differences among the flood regions and to 

identify the most significant basin characteristics for inclusion 

in the GLS regressions. 

Fifty-nine basin characteristics measured for each 

streamgage were determined from digital databases using 

geographic information system (GIS) software. Six basin 

characteristics are used as explanatory variables in the final 

regression equations; these include three morphometric 

characteristics: drainage area (DRNAREA), constant of channel maintenance (CCM), and basin shape (BSHAPE); two 

geologic/pedologic characteristics: percent area within the 

Des Moines Lobe landform region (DESMOIN) and average saturated hydraulic conductivity of soil (KSATSSUR); 

and one climatic characteristic: maximum 24-hour precipitation that happens on average once in 10 years (I24H10Y). 

Predictive accuracies for the annual exceedance-probability 

equations developed for each region are indicated by several 

performance metrics. SEPs range from 31.8 to 45.2 percent for 

flood region 1, from 19.4 to 46.8 percent for flood region 2, 

and from 26.5 to 43.1 percent for flood region 3. The pseudo 

coefficients of determination (pseudo-R2

) for the GLS equations range from 90.8 to 96.2 percent for flood region 1, range 

from 91.5 to 97.9 percent for flood region 2, and range from 

92.4 to 96.0 percent for flood region 3. In general, predictive 

accuracies tend to be the best for flood region 2, second best 

for flood region 3, and poorest for flood region 1. Of the eight 

annual exceedance-probability equations developed for each 

region, the Q10-percent (%), Q4%, and Q2% flood-discharge regression 

equations generally have the best predictive accuracy and the 

Q50% and Q0.2% flood-discharge equations generally have the 

poorest accuracy. 

The regional-regression equations developed in this study 

are not intended for use at ungaged stream sites in which the 

basin characteristics are outside the range of those used to 

develop the equations. Inconsistencies in estimates may result 

for the annual exceedance-probability equations if basin-characteristic values approach the minimum or maximum limits 

of the range. GIS software is required to measure the basin 

characteristics included as explanatory variables in the regression equations. 

To better understand the effects of the new expected 

moments algorithm (EMA), with the multiple Grubbs-Beck 

(MGB) test for detecting low outliers, method of annual 

exceedance-probability analysis and the new updated regionalskew-coefficient constant used in this study have on the 

estimation of AEPDs for Iowa, AEPDs computed using these 

two new elements were compared to AEPDs computed using 

the standard Bulletin 17B/GB annual exceedance-probability 

analysis and superseded regional skew coefficient. Results of 

this comparison for 283 streamgages in Iowa indicate that on 

average for the eight annual exceedance probabilities, 

AEPDs are lower by about 4 percent using the updated 

regional-skew-coefficient constant and are greater by about 

8 percent using the EMA/MGB analysis method, and overall, AEPDs are about 4 percent greater using the EMA/MGB 

analysis method and the updated regional-skew-coefficient 

constant. The larger estimates computed on average for these 

283 streamgages by the EMA/MGB analysis, compared to the 

Bulletin 17B/GB analysis, primarily are because of the type 

of streamgage. Comparison results for the Q1% flood discharge 

indicate on average that AEPDs are about 1 percent lower for 

147 continuous-record streamgages and are about 26 percent 

greater for 136 crest-stage gages (CSGs). Because most of the 

CSGs included in this comparison have censored data records, 

the larger AEPDs computed for CSGs are believed to result 

mainly from the ability of the EMA/MGB analysis to use a 

specific discharge interval for data that is censored by the 

Bulletin 17B/Grubbs-Beck analysis when it is below the largest minimum recordable threshold discharge. A comparison 

between regional-regression-equation estimates computed in 

this study and those computed in a previous 2001 study also is 

presented. Results of this comparison for 185 streamgages in 

Iowa indicate that on average AEPDs are greater by about 

8 percent for the eight annual exceedance probabilities using 

the regional-regression equations developed in this study. 
44 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

About 4 percent of this increase likely is because of the use of 

EMA/MGB analyses with the updated regional-skew-coefficient constant.

All 24 regression equations developed for this study 

are to be included in the USGS StreamStats Web-based GIS 

tool. StreamStats will provide users with a set of AEPDs for 

ungaged stream sites within Iowa in addition to the basin 

characteristics for the sites. Ninety-percent prediction intervals also are automatically calculated. A 90-percent prediction interval denotes there is 90-percent certainty that the true 

value of an AEPD at an ungaged stream site will be within a 

plus or minus interval around the predicted AEPD.

Acknowledgments

The authors gratefully acknowledge the following U.S. 

Geological Survey (USGS) personnel: William Asquith, for 

his assistance with information on how to apply a program to 

optimize power transformations for drainage area on the basis 

of minimization of the predicted residuals sum of squares statistic; Kris Lund, for her work to prepare geographic information system (GIS) base data layers required for StreamStats; 

Rochelle Galer, for her work to measure basin characteristics 

for streamgages, to prepare base GIS data layers required 

for StreamStats and for areas in neighboring States, and to 

prepare Natural Resources Conservation Service Soil Survey 

Geographic database data layers required for use with the Soil 

Data Viewer; Gabe Ritter, for his work to prepare GIS base 

data layers required for StreamStats, and to create programs 

for calculating the streamflow-variability index basin characteristic and for improving the efficiency of calculating the 

base-flow index basin characteristic; Ken Eng, for his assistance with the measurement of the annual base-flow-recession 

time constant basin characteristic; and Ed Fischer, for his work 

to create scripts for automating the computations from the 

weighted independent estimates and hydrograph separation 

and analysis programs, for plotting eight of the figures in this 

report, and for his work to develop statewide region of influence regression equations. 

The authors also would like to express their appreciation to the many other USGS personnel who assisted with 

collection and analysis of flood data used in this study. The 

flood data often were collected during adverse conditions, and 

the efforts of these individuals made this study possible. The 

information contained herein also is based on data collected by 

the U.S. Army Corps of Engineers, the National Weather Service, and several State and local agencies who were involved 

with collection of flood data. 

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Appendix
52 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

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 

discharges (AEPDs) (eq. 2 in report). Bulletin 17B supplies a 

national map, but also encourages the hydrologist to develop 

more specific local relations. Since the first map was published 

in 1976, some 35 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 leastsquares (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 skewness 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 skewness. 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 skewness analyses. The Bayesian methodology allows for the computation of a posterior distribution of 

the regression parameters and the model error variance. As 

documented 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 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 skewcoefficient estimators, GLS regression also considers the 

cross-correlations among the skewn-coefficient estimators. In 

some studies, the cross-correlations have had a large effect on 

the precision attributed to different parameter estimates (Parrett and others, 2011; Feaster and others, 2009; Gotvald and 

others, 2009; Weaver and others, 2009).

Because of complications introduced by the use of the 

expected moments algorithm (EMA/MGB) 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 stable and defensible 

results for regional skewness (Veilleux and others, 2012; 

Veilleux, 2011; Lamontagne and others, 2012). This alternate 

procedure is referred to as the Bayesian WLS/Bayesian GLS 

(B-WLS/B-GLS) regression framework (Veilleux and others, 2012; Veilleux, 2011; Veilleux and others, 2011). It uses 

an OLS analysis to fit an initial regional skewness 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 employed 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.

To provide cost effective peak-discharge data for smaller 

drainage basins in the study area, the U.S. Geological Survey 

(USGS) operates a large network of crest-stage gages (CSGs) 

that only measure discharges above a minimum recording 

threshold (thus producing a censored data record). CSGs are 

different from continuous-record streamgages, which measure 

almost all discharges and have been used in previous B-GLS 

and B-WLS/B-GLS regional skew studies. Thus, although 

the Iowa regional skew study described here did not exhibit 

large cross-correlations between annual peak discharges, it 

did make extensive use of EMA/MGB to estimate the station 

skew and its mean square error. Because EMA/MGB allows 

for the censoring of low outliers, 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 employed. The steps of this alternative procedure 

are described in the following sections.

Methodology for Regional Skewness 

Model

This section provides a brief description of the B-WLS/BGLS methodology (as it appears in Veilleux and others, 2012). 

Veilleux and others (2011) and Veilleux (2011) provide a more 

detailed description.

Regional Skewness Regression

By Andrea G. Veilleux, U.S. Geological Survey and Jery R. Stedinger, Cornell University, Ithaca, New York
Data Analysis 53

OLS Analysis

The first step in the B-WLS/B-GLS regional skewness analysis is the estimation of a regional skewness model using OLS. 

The OLS regional regression yields parameters ßOLS and a 

model that can be used to generate unbiased and relatively 

stable regional estimates of the skewness for all streamgages:

y

~

OLS =XßOLS (A1)

where

y

~

OLS are the estimated regional skewness values,

X is an (n x k) matrix of basin characteristics,

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 skewness values y

~

OLS are then 

used to calculate unbiased station-regional skewness variances 

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

These station-regional skewness variances are based on the 

regional OLS estimator of the skewness coefficient instead of 

the station skewness estimator, thus making the weights in the 

subsequent steps relatively independent of the station skewness estimates.

WLS Analysis

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

regression coefficients for each regional skewness 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). 

GLS Analysis

After the regression model coefficients, ßWLS, 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( ßWLS), and the model error variance, 

σ2

δ,B-GLS, pseudo R2δ

 as well as the average variance of prediction 

at a streamgage not used in the regional model, AVPnew. 

Data Analysis

The statistical analysis of the data requires several steps. 

This section describes a redundant site analysis, 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.

Data for Iowa Regional Skew Study

This study is based on annual peak-discharge data from 

330 streamgages in Iowa and the surrounding states. The 

annual peak-discharge data were downloaded from the USGS 

National Water Information System (NWIS) database (U.S. 

Geological Survey, 2012). In addition to the peak-discharge 

data, over 65 basin characteristics for each of the 330 sites 

were available as explanatory variables in the regional study. 

The basin characteristics available include percent of basin 

contained within different hydrologic regions, as well as the 

more standard morphometric parameters such as location of 

the basin centroid, drainage area, main channel slope, and 

basin shape among others.

Redundant Sites

Redundancy results when the drainage basins of two 

streamgages are nested, meaning that one is contained inside 

the other and the two basins are of similar size. Then, instead 

of providing two independent spatial observations, depicting how drainage-basin characteristics are related to skew 

(or AEPDs), these two basins will have the same hydrologic 

response to a given storm, and thus represent only one spatial 

observation. When sites are redundant, a statistical analysis 

using both streamgages incorrectly represents the information in the regional dataset (Gruber and Stedinger, 2008). To 

determine if two sites are redundant and thus represent the 

same hydrologic experience, two pieces of information are 

considered: (1) if their watersheds are nested, and (2) the ratio 

of the basin drainage areas.

The standardized distance (SD) is used to determine the 

likelihood the basins are nested. The standardized distance 

between two basin centroids, SD, is defined as 

Dij

0.5(DAi + DAj) SDij = , (A2)

where 

Dij is the distance between centroids of basin i

and basin j, and 

DAi and DAj are the drainage areas at sites i and j.

The drainage area ratio, DAR, is used to determine if 

two nested basins are sufficiently similar in size to conclude 

that they are essentially, or are at least in large part, the same 

watershed for the purposes of developing a regional hydrologic model. The drainage area ratio of two basins, DAR, is 

defined as (Veilleux, 2009)

DAi

DAj

DAR = Max

DAj

DAi

, , (A3) 

where 

DAi and DAj have already been defined in equation A2. 

Two basins might be expected to have possible redundancy if the basin sizes are similar and the basins are nested. 

Previous studies suggest that site-pairs having SD less than or 

equal to 0.50 and DAR less than or equal to 5 likely were to 
54 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

have possible redundancy problems for purposes of determining regional skew. If DAR is large enough, even if the 

sites are nested, they will reflect different hydrologic experiences because they respond to storms of different sizes and 

durations. 

Table A1 (link to Excel file) shows the results of the 

redundant site screening on the Iowa regional skew data. All 

possible combinations of site-pairs from the 330 streamgages 

were considered in the redundancy analysis (including all 

types of streamgages: continuous-record streamgages, CSGs, 

and both continuous-record streamgages/CSGs). To be conservative, all site-pairs with SD less than 0.75 and DAR less than 

8 were identified as possible redundant site-pairs. All sites 

identified as redundant were then investigated to determine 

if in fact one site of the pair was nested inside the other. For 

site-pairs that were nested, one site from the pair was removed 

from the regional skew analysis. Sites removed from the Iowa 

regional skew study because of redundancy are identified in 

table A1 (link to Excel file) as “no - R.”

From the 95 identified possible redundant site-pairs, 87 

were found to be redundant and 55 sites were removed. Two 

sites also were removed because of backwater effects and they 

are identified in table A1 (link to Excel file) as “no - B.” Thus, 

of the 330 sites, 57 have been removed because of redundancy 

and backwater, leaving 273 sites for the Iowa regional skew 

study.

Station Skewness Estimators

The EMA/MGB analysis method was used to estimate 

the station log10 skew coefficient, G, and its mean square 

error, MSEG (Cohn and others, 1997; Griffis and others, 2004). 

EMA/MGB 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. PeakfqSA, an EMA/

MGB software program developed by Cohn (2011), is used 

to generate the station log10 estimates of G and its MSEG, 

assuming a log-Pearson Type III distribution and employing 

a multiple Grubbs-Beck test for low-outlier screening. EMA/

MGB estimates, based on annual peak-discharge data through 

September 30, 2008, of G and its MSEG are listed in table A1

(link to Excel file) for the 330 streamgages evaluated for the 

Iowa regional skew study. [See sections Expected Moments 

Algorithm (EMA/MGB) Analyses and Multiple Grubbs-Beck 

Test for Detecting Low Outliers for more detail regarding 

EMA/MGB.]

Pseudo Record Length

Because the dataset includes censored data and historic 

information, the effective record length used to compute the 

precision of the skewness estimators is no longer simply the 

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

more complex calculation should be used to take into account 

the availability of historic information and censored values. 

Although historic 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, which appropriately accounts 

for all peak-discharge data types available for a site. PRL equals 

the systematic record length if such a complete record is all 

that is available for a site.

The first step is to run EMA/MGB with all available 

information, including historic information and censored 

peaks (denoted EMA/MGBc, for EMA/MGB complete). From 

the EMA/MGB run, the station skewness without regional 

information ĜC and the MSE of that skewness estimator 

MSE(ĜC) are extracted, as well as the year the historical 

period begins, YBC, the year the historical period ends, YEC, 

and the length of the historical period HC. (YBC, YEC, and HC are 

used in equation A12.) 

The second step is to run EMA/MGB with only the 

systematic peaks (denoted EMA/MGBS, for EMA/MGB 

systematic). From the EMA/MGBS analysis, the station skewness without regional information Ĝs

 and the MSE of that 

skewness estimator, MSE(Ĝs), are extracted, as well as the 

number of peaks PS. (PS is used in equation A6.)

The third step is to represent, from EMA/MGBC and 

EMA/MGBS, the precision of the skewness estimators as two 

record lengths, RLC and RLS, based on the estimated skew and 

MSE. The corresponding record lengths are calculated using 

equation A4 below from Griffis and others (2004) and Griffis 

and Stedinger (2009): 

6

RL MSE(Ĝ) = 1 + ( ( 96

15

48 + a(RL) + b(RL)) Ĝ + c(RL)) 2 + Ĝ4 *

(A4)

17.75

RL a 2 (RL) = – 50.06RL3 +

3.93

RL b 0.3 (RL) = 30.97RL0.6

37.1

RL − + 0.9

6.16

RL c 0.56 (RL) = – 36.83RL1.12

66.9

RL1.68 −+

where 

RLC uses ĜC and MSE(ĜC), and

RLS uses ĜS and MSE(ĜS).

Next, the difference between RLC and RLS is employed as 

a measure of the extra information provided by the historic or 

censored information, or both, that was included in the EMA/

MGBc analysis, but not in the EMA/MGBs analysis.

RLdiff = RLC– RLS (A5)

The pseudo record length for the entire record at the 

streamgage, PRL, is calculated using RLdiff from equation A5 

and the number of systematic peaks PS:

PRL = RLdiff + PS (A6)
Data Analysis 55

PRL must be non-negative. If PRL is greater than HC, then 

PRL should be set to equal HC. 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.

Unbiasing the Station Estimators

The station skewness estimates are unbiased by using the 

correction factor developed by Tasker and Stedinger (1986) 

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

skewness estimator using the pseudo record length is

Ŷi = Gi

6

PRL,i

1 + (A7)

where

	 Ŷi is the unbiased station sample skewness 

estimate for site i,

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

calculated in equation A6, and 

Gi is the traditional biased station skewness 

estimator for site i from EMA/MGB.

The variance of the unbiased station skewness includes 

the correction factor developed by Tasker and Stedinger 

(1986):

Var[Ŷi] = Var[Gi] 6

PRL,i

2

1 + (A8)

where

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

2009).

Var(Ĝ) = Ĝ2 + Ĝ4 * ( ( 6

PRL

9

6 + a(PRL) + b(PRL)) 1548 + c(PRL 1+ ))

(A9)

Estimating the Mean Square Error of the 

Skewness Estimator

There are several possible ways to estimate MSEG. The 

approach used by EMA/MGB [taken from eq. 55 in Cohn and 

others (2001)] 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 8 (the variance is equated to the MSE), 

employing 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 effect of the 

censored data and the number of recorded systematic peaks. 

Figure A1 compares the unbiased MSEG estimates from the 

Griffis and Stedinger (2009) approach based on pseudo record 

lengths and regional skewness estimates, and the unbiased 

EMA/MGBC MSEG estimates based on the estimated station 

skewness. 

As shown in figure A1, for those streamgages with MSEG

less than about 0.4, the two methods generate similar MSEG; 

however, for 33 streamgages, EMA/MGB generates unreasonably large MSEG with values greater than about 0.4. For 

these sites, the Griffis and Stedinger (2009) formula does not 

generate a MSEG greater than 0.5. For these 33 streamgages 

EMA/MGB is having trouble estimating the MSEG due at 

least in part to the number of censored observations. Of these 

33 sites with EMA/MGB unbiased MSEG greater than 0.4, 

45-percent of the sites had 50 percent or more of their record 

comprised of censored observations, whereas 81 percent of 

the sites had 20 percent or more of their record comprised of 

censored observations. Also, the average PRL for all 273 sites 

in the Iowa study is 49 years; however, the longest record of 

the 33 sites with EMA/MGB unbiased MSEG greater than 0.4 

is 43 years, with 85 percent of the 33 sites having PRL less than 

or equal to 35 years and 42 percent of the 33 sites have PRL

less than or equal to 25 years. Thus, it appears that for those 

sites with shorter record lengths and a large percentage of 

their record comprised of censored observations, EMA/MGB 

has trouble estimating the MSEG. For this reason, these 33 

sites with EMA/MGB unbiased MSEG greater than 0.4 were 

removed from the analysis. Thus, there are 240 streamgages 

remaining from which to build a regional skewness model for 

the Iowa study area. Figure 6 shows the location of the basin 

centroids for these 240 streamgages. The unbiased Griffis 

and Stedinger (2009) MSEG is used in the regional skewness 

model because it is more stable and relatively independent of 

the station skewness estimator. The 33 sites removed from the 

Iowa regional skew study because of a MSEG greater than 0.4 

as estimated by EMA/MGB are identified in table A1 (link to 

Excel file) as “no – E.”

Cross-Correlation Models

A critical step for a GLS analysis is estimation of the 

cross-correlation of the skewness coefficient estimators. Martins and Stedinger (2002) used Monte Carlo experiments to 

derive a relation between the cross-correlation of the skewness 

estimators at two stations i and j as a function of the crosscorrelation of concurrent annual maximum flows, ρij:

ρˆ(ŷi

 , ŷj) = Sign(ρ

^

ij

)csij|ρ

^

ij

|

k (A10)

where

ρ

^

ij is the cross-correlation of concurrent annual 

17.75

PRL

2

50.06

PRL

a 3 (PRL) = – +

3.92

PRL

0.3

31.10

PRL

0.6

34.86

PRL

b 0.9 (PRL) = – +

7.31

PRL

0.59

45.90

PRL

1.18

86.50

PRL

1.77 c(PRL) = – – +
56 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

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 in the 

following equation:

(PRL,i)(PRL,j cƒ ) ij =CYij / (A11)

where

CYij is the pseudo record length of the period of 

concurrent record, and

PRL,i, PRL,j is the pseudo record length corresponding to 

sites i and j, respectively (see equation A6) 

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. Because of 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 

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.

CYij = (YEij – YBij + 1) ( ) PRL, i

HC, i ( ) PRL, jHC, j

 (A12)

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.

Iowa Study Area Cross-Correlation Model of 

Concurrent Annual Peak Discharge

A cross-correlation model for the log annual peak discharges in the Iowa study area were developed using 53 sites 

with at least 65 years of concurrent systematic peaks (zero 

flows not included). Various models relating the cross-correlation of the concurrent annual peak discharge at two sites, 

pij, 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 model 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, is

ρij = ) exp(2Zij) – 1

exp(2Zij ( )+1 (A13)

where

Zij = exp (0.42 – 0.076 ) Dij

0.46–1 ( 0.46 ) (A14) 

An OLS regression analysis based on 1,164 station-pairs 

indicated that this model is as accurate as having 152 years 

of concurrent annual peaks from which to calculate crosscorrelation. Figure A2 shows the fitted relation between Z and 

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Expected moments algorithm unbiased mean-square error of skew

Griffis and Stedinger unbiased mean-square 

error of skew

Figure A1. Comparison of EMA/MGB and Griffis and Stedinger (2009) MSEG estimates of station skewness 

estimators for each of the 273 streamgages in the Iowa regional skewness study.
Data Analysis 57

Figure A2. Relation between Fisher Z transformed cross-correlation of logs of annual peak discharge and 

distance between basin centroids for 1,164 station-pairs with concurrent record lengths greater than or equal 

to 65 years from 53 streamgages in Iowa and neighboring states.

Figure A3. Relation between un-transformed cross-correlation of logs of annual peak discharge and distance 

between basin centroids based for 1,164 station-pairs with concurrent record lengths greater than or equal to 

65 years from 53 streamgages in Iowa and neighboring states. 

-0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 50 100 150 200 250 300 350 400

Fisher Z cross-correlation of concurrent annual-peak discharges

Distance between streamgage centroids, in miles

Relation between cross correlation and distance 

(greater than or equal to 65 years concurrent records)

Site-pairs

Z = exp(0.42-0.076 * ((D0.46-1)/0.46))

EXPLANATION

-0.2

0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250 300 350 400

Cross-correlation of concurrent annual-peak discharges

Distance between streamgage centroids, in miles

Relation between cross correlation and 

distance (greater than or equal to 65 years 

concurrent records)

Site-pairs

r = (exp(2Z)-1) / (exp(2Z)+1)

EXPLANATION
58 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

distance between basin centroids together with the plotted 

sample data from the 1,164 station pairs of data. Figure A3 

shows the functional relation between the untransformed cross 

correlation and distance between basin centroids together with 

the plotted sample data from the 1,164 station pairs of data. 

The cross correlation model was used to estimate site-to-site 

cross correlations for concurrent annual peak discharges at all 

pairs of sites in the regional skew study.

Iowa Regional Skew Study Results

The results of the Iowa regional skew study using the 

B-WLS/B-GLS regression methodology are provided below. 

All of the available basin characteristics initially were considered as explanatory variables in the regression analysis for 

regional skew. Available basin characteristics include precipitation (mean annual, mean monthly, maximum 24 hours for 

a number of years), soil (hydrologic soil types, soil type, soil 

permeability), stream characteristics (stream density, ruggedness, number of first order streams, total stream length), basin 

measures (drainage area, slope, relief, length, perimeter, shape 

factor), hydrologic parameters (base flow index, base flow 

recession), and hydrologic regions. A few basin characteristics statistically were significant in explaining the site-to-site 

variability in skewness, including slope, drainage area, basin 

length, and the total length of mapped streams in the basin. 

The best model, as classified by having the smallest model 

error variance, σδ

2

, and largest pseudo Rδ

2

, which included a 

constant and a parameter (or combination of parameters), was 

the model that included drainage area. Table A2 provides the 

final results for the constant skewness model denoted “Constant,” and the model that uses a linear relation between skewness and log10 drainage area.

Table A2 includes the pseudo Rδ

2

 value for both models; 

pseudo Rδ

2

 describes the estimated fraction of the variability 

in the true skewness from site-to-site 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. The “DA” model has a pseudo 2 R of 19 percent. 

The posterior mean of the model error variance, σδ

2

, for the 

DA model is 0.12, which is smaller than that for the constant 

model for which σδ

2

= 0.15. This indicates that the inclusion 

of drainage area as an explanatory variable in the regression 

helps explain some of the variability in the true skewness; 

however, this small gain in precision does not warrant the 

increased model complexity. Thus, the constant model is 

selected as the best regional model for Iowa study area skewness. The average sampling error variance (ASEV) in table 

A2 is the average error in the regional skewness 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 to describe the precision of the 

generalized skewness. The constant model has an AVPnew, 

equal to 0.16, which corresponds to an effective record length 

of 50 years. An AVPnew of 0.16 is a marked improvement over 

the Bulletin 17B national skew map, whose reported MSE is 

0.302 (Interagency Committee on Water Data, 1982) for a corresponding effective record length of only 17 years. Thus, the 

new regional model has three times the information content 

(as measured by effective record length) of that calculated for 

the Bulletin 17B map. 

 Figure A4 shows the relation between the unbiased station skewness and drainage area; the marker selected for each 

streamgage represents the station pseudo record length. The 

sites with the largest drainage area generally have the longest 

pseudo record lengths. It is not apparent from the data that the 

upward trend, suggested by the DA model, occurs between 

the unbiased station skewness and drainage area. Thus, for 

this study, the simpler model is selected, in other words the 

constant model. 

B-WLS/B-GLS 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 dataset (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 Iowa 

study area affording the most accurate skew prediction while 

Table A2. Regional skewness models for Iowa study area. 

[σδ

2

 is the model error variance; ASEV, is the average sampling error variance; AVPnew, is the average variance of prediction for a new site; 

Pseudo Rδ

2

 (%), describes the fraction of the variability in the true skews explained by each model (Gruber and others, 2007); %, percent; 

ŷ, unbiased regional skewness estimate; DA, drainage area. Standard deviations are in parentheses]

Regression parameters

Model b1 b2 σδ

2 ASEV AVPnew Pseudo Rδ

2

Constant: ŷ = b1 -0.40 - 0.15 0.01 0.16 0%

(0.09) (0.03)

0%

DA: ŷ = b1

 + b2[log10(DA)] -0.78 0.20 0.12 0.01 0.13 19%

(0.16) (0.05) (0.02)

0% 0%
Iowa Regional Skew Study Results 59

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 Iowa 

regional skew study. 

Table A3 presents a Pseudo Analysis of Variance (Pseudo 

ANOVA) table for the Iowa regional skew analysis containing 

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 ni

 for each site i, are not known; however, the total 

sampling error sum of squares can be described by its mean 

value, Var(ŷi ∑ )

n

i =1

 Because there are n equations, the total variation 

because of the model error δ for a model with k parameters has 

a mean equal to nσδ

2

(k). Thus, the residual variation attributed 

to the sampling error is Var(ŷi ∑ )

n

i =1

, and the residual variation 

attributed to the model error is nσδ

2

(k).

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 of the anticipated variation in yi = μ + 

δi

, where μ is the mean of the estimated station sample skews. 

Thus, the total expected sum of squares variation because 

of model error δi

 and because of sampling error ni = ŷ – yi

in expectation should equal ∑Var(yöi nσ ) δ

2

(0) + n

i=1

. Therefore, the 

expected sum of squares attributed to a regional skew model 

with k parameters equals n[σδ

2

(0) – σδ

2

(k)], because the sum 

of the model error variance nσδ

2

(k) and the variance explained 

by the model must sum to nσδ

2

(0). Table A3 considers models 

with k = 0 and 1.

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 effect 

of correlation among the sampling errors. 

Table A3 compares the Pseudo ANOVA results for 

the constant model and log10(DA) model described in the 

report text. The log10(DA) model contains a constant and 

one explanatory variable, a linear function of drainage area. 

As described previously, the first step of the B-WLS/BGLS regression procedure is to perform an OLS analysis to 

generate smoothed estimates of the mean square error of the 

station skew. Thus, the constant model used a constant OLS 

regression to generate the MSEG, while the log10(DA) model 

contains a constant and log10(DA) as an explanatory variable, 

which indicates the estimates of the MSEG vary between the 

two models.

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Unbiased at-site skew

log10 (drainage area, in square miles)

Greater than 100

80−99

60−79

40−59

25−39

EXPLANATION

Pseudo record length (PRL)

Constant model

DA model

Figure A4. Relations between the unbiased station skew and drainage area for the 240 sites in the Iowa 

regional skew study. Each of the diamonds represents one of the 240 sites with each of the five different 

colors of the diamonds representing five different groupings of station pseudo record length (PRL); 

purple/violet = record length greater than 100 years, pink = record length between 80–99 years, green = record 

length between 60–79 years, brown = record length between 40–59 years, gray = record length between 

25–39 years. The solid black line represents the constant model from table A2, while the dashed black line 

represents the DA model from table A2.
60 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

 The constant model and the log10(DA) model have sampling error variances larger than their model error variances; 

however, it is important to note that the model error attributed 

to the log10(DA) model σδ

2

(1) is about one-sixth of the model 

error variance for the constant model σδ

2

(0). This difference in 

model error is accounted for by the variation in the sample that 

the log10(DA) model appears to explain. Because the constant 

model does not have any explanatory variables, the variation attributed to that model is zero. On the other hand, the 

log10(DA) model has one explanatory variable. The analysis 

attributes a variance of six to the log10(DA) model. This 

accounts for the reduction of the model error variance from the 

constant model to the log10(DA) model; however, the addition 

of the drainage area explanatory variable in the log10(DA) 

model does not significantly improve the ability of the model 

to describe the variation in observed skew coefficients. This 

is reflected in the pseudo Rδ

2

, which in this case has a value 

of only 19 percent because the log10(DA) model explains 

only 19 percent of the estimated variation σδ

2

(0) in the true 

skew from site-to-site. Thus, the constant model is the model 

selected for the Iowa regional skew model.

The Error Variance Ratio (EVR) is a modeling diagnostic 

used to evaluate if a simple OLS regression is sufficient, or if 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) nσδ

2

(k)

∑i=1Var(ŷi

) n

 (A15)

For the Iowa study-area data, EVR had a value of 1.3 

for the constant model and 1.6 for the log10(DA) model. The 

sampling variability in the sample skewness estimators was 

larger than the error in the regional model. Thus, an OLS 

model that neglects sampling error in the station skewness 

estimators may not provide a statistically reliable analysis of 

the data. Given the variation of record lengths from site-tosite, 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 if a WLS regression is sufficient, 

or if a GLS regression is appropriate to determine the precision of the estimated regression parameters (Veilleux, 2011; 

Griffis, 2006). 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 ß0WLS, because the 

Table A3. Pseudo ANOVA table for the Iowa regional skew study for the constant model and 

the log10(DA) model.

[Pseudo ANOVA, pseudo analysis of variance; DA, drainage area; EVR, error variance ratio; MBV*, misrepresentation of the beta variance; Pseudo Rδ

2

, fraction of variability in the true skews explained by the model

Source

Degrees-of-freedom

Equations

Sum of squares

A B A: 

Constant 

B:

DA

Model k 0 1

n[σδ

2

(0) – σδ

2

(k)]

0 6

Model error n-k-1 272 271

n[σδ

2

(0)]

37 30

Sampling error n 273 273

Var(ŷi ∑ )

n

i =1

46 47

Total 2n-1 545 545

Var(ŷi n[σ ) δ

2

(0)] +∑

n

i =1

83 83

EVR Var(ŷi

) EVR = n[σδ

2

(k)]

∑ n

i =1

1.3 1.6

MBV*

wi

MBV* = where wi = wT

 Λw

Λii

1

∑ n

i =1

6.5 6.9

Rδ

2

Rδ

2

 = 1 – σδ

2

(k)

σδ

2

(0)

0% 19%
Iowa Regional Skew Study Results 61

covariance among the estimated station skews ŷi generally has 

its greatest effect on the precision of the constant term (Stedinger and Tasker, 1985). If the MBV* substantially is greater 

than 1, then a GLS error analysis should be employed. The 

MBV* is calculated as

 

MBV* = where wi = = Var[bo

WLS|GLS analysis]

Var[bo

WLS|WLS analysis] ∑i=1

wT Λw 1

√Λii wi

n

(A16)

For the Iowa regional skew study, the MBV* is equal to 

6.5 for the constant model and 6.9 for the log10(DA) model. 

This is a large value indicating the cross-correlation among the 

skewness estimators has had an effect 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 6.5. Thus, a WLS analysis would 

seriously misrepresent the variance of the constant in the 

constant model and in the log10(DA) model of regional skew. 

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 sufficiently was large 

enough to make an appreciable contribution to the average 

variance of prediction. 

Leverage and Influence

Leverage and influence diagnostic 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 effect on the fitted (or predicted) values (Hoaglin 

and Welsch, 1978). Generally, leverage considers if 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 with the 

ability or potential to affect the fit of the regression, influence attempts to describe those points that do have an unusual 

effect 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 

and others (2011) and Veilleux (2011).

Figure A5 displays the leverage and influence values 

for the B-WLS/B-GLS constant regional skew model for the 

Iowa study area. The 15 sites included in the figure have high 

influence, and thus have an unusual effect on the fitted regression relation. The sites are ordered, starting from the left, 

by decreasing influence, as it identifies those sites that had a 

large effect on the analysis. No sites in the regression had high 

0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

43 95 190 187 73 143 250 149 154 206 321 176 100 256 151

Influence/leverage 

Regional skew index number

Influence

Threshold for high influence

Leverage

Threshold for high leverage

Figure A5. Regression Diagnostics: Leverage and Influence for the Iowa study area B-WLS/B-GLS constant 

model. The solid line represents the threshold for high leverage, while the dotted line represents the threshold 

for high influence.
62 Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa

leverage, and the differences in leverage values for the constant model reflect the variation in record lengths among sites. 

Streamgage 05410490 (regional skew index number 43, map 

number 65, fig. 1) has the highest influence value because of 

its large residual, the third largest positive residual in the study 

(in other words the largest positive unbiased station skew = 

1.04), and its large drainage area (700 mi2), which is larger 

than all of the other sites with large influences.

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Publishing support provided by:

Rolla Publishing Service Center

For more information concerning this publication, contact:

Director, USGS Iowa Water Science Center

P.O. Box 1230

Iowa City, IA 52244

(319) 337–4191

Or visit the Iowa Water Science Center Web site at:

http://ia.water.usgs.gov

Eash and others—Methods for Estimating Annual Exceedance-Probability Discharges for Streams in Iowa, Based on Data through Water Year 2010—Scientific Investigations Report 2013–5086