In cooperation with the Texas Department of Transportation 

Regression Equations for Estimation of Annual 

Peak-Streamflow Frequency for Undeveloped 

Watersheds in Texas Using an L-moment-Based, 

PRESS-Minimized, Residual-Adjusted Approach 

Scientific Investigations Report 2009–5087
 

(Texas Department of Transportation Research Report 0–5521–1)
 

U.S. Department of the Interior 

U.S. Geological Survey 
 

 

 

 

 

 

 

 

 

 

 

Front cover: “View looking upstream of the Colorado River below the mouth of the Concho River at the 

Miller Ranch, September [18,] 1936. Photograph by Miller at the peak stage. Discharge is 356,000 cubic 

feet per second by slope-area method and is greatest then known by local residents.” Other note “near 

Leaday, Texas.” Undated notes on back of photograph by Harold W. Albert. 

Additional information: The photograph appears to have been taken about 1 mile south of Leaday, Texas, 

looking toward the north. The closest downstream U.S. Geological Survey (USGS) streamflow-gaging 

station is 08136700 Colorado River near Stacy, Texas, which had a peak streamflow on September 18, 

1936, of 356,000 cubic feet per second at a gage height of 64.59 feet, although the period of record begins 

in 1969 to present (2009). A substantial part of the photograph appears to now be inundated by O.H. Ivie 

Reservoir monitored by USGS reservoir station 08136600 O.H. Ivie Reservoir near Voss, Texas. A brief 

history of O.H. Ivie is provided on inside of the back cover. 
— Technical Report Documentation Page — 

1. Report No. 

FHWA/TX–09/0–5521–1 

2. Government Accession No. 3. Recipient’s Catalog No. 

4. Title and Subtitle 5. Report Date 

Regression Equations for Estimation of Annual Peak-Streamflow Frequency 

for Undeveloped Watersheds in Texas Using an L-moment-Based, PRESSMinimized, Residual-Adjusted Approach 

June 2009 

7. Authors(s) 

William H. Asquith and Meghan C. Roussel 

8. Performing Organization Report No. 

USGS SIR 2009–5087 

9. Performing Organization Name and Address 

U.S. Geological Survey 

8027 Exchange Drive 

Austin, Texas 78754 

10. Work Unit No. (TRAIS) 

11. Contract or Grant No. 

Project 0–5521 

12. Sponsoring Agency Name and Address 

Texas Department of Transportation 

Research and Technology Implementation Office 

P.O. Box 5080 

Austin, Texas 78731 

13. Type of Report and Period Covered 

Technical report on research from 2007 

to 2009 

14. Sponsoring Agency Code 

15. Supplementary Notes 

Project conducted in cooperation with the Texas Department of Transportation and the Federal Highway Administration 

16. Abstract 

Annual peak-streamflow frequency estimates are needed for flood-plain management; for objective assessment of flood risk; for 

cost-effective design of dams, levees, and other flood-control structures; and for design of roads, bridges, and culverts. Annual 

peak-streamflow frequency represents the peak streamflow for nine recurrence intervals of 2, 5, 10, 25, 50, 100, 200, 250, and 

500 years. Common methods for estimation of peak-streamflow frequency for ungaged or unmonitored watersheds are regression equations for each recurrence interval developed for one or more regions; such regional equations are the subject of this 

report. The method is based on analysis of annual peak-streamflow data from U.S. Geological Survey streamflow-gaging stations (stations). Beginning in 2007, the U.S. Geological Survey, in cooperation with the Texas Department of Transportation 

and in partnership with Texas Tech University, began a 3-year investigation concerning the development of regional equations to 

estimate annual peak-streamflow frequency for undeveloped watersheds in Texas. The investigation focuses primarily on 638 stations with 8 or more years of data from undeveloped watersheds and other criteria. The general approach is explicitly limited to 

the use of L-moment statistics, which are used in conjunction with a technique of multi-linear regression referred to as PRESS 

minimization. The approach used to develop the regional equations, which was refined during the investigation, is referred to as 

the “L-moment-based, PRESS-minimized, residual-adjusted approach.” For the approach, seven unique distributions are fit to the 

sample L-moments of the data for each of 638 stations and trimmed means of the seven results of the distributions for each recurrence interval are used to define the station-specific, peak-streamflow frequency. As a first iteration of regression, nine weightedleast-squares, PRESS-minimized, multi-linear regression equations are computed using the watershed characteristics of drainage 

area, dimensionless main-channel slope, and mean annual precipitation. The residuals of the nine equations are spatially mapped, 

and residuals for the 10-year recurrence interval are selected for generalization to 1-degree latitude and longitude quadrangles. 

The generalized residual is referred to as the OmegaEM parameter and represents a generalized terrain and climate index that 

expresses peak-streamflow potential not otherwise represented in the three watershed characteristics. The OmegaEM parameter 

was assigned to each station, and using OmegaEM, nine additional regression equations are computed. Because of favorable diagnostics, the OmegaEM equations are expected to be generally reliable estimators of peak-streamflow frequency for undeveloped 

and ungaged stream locations in Texas. The mean residual standard error, adjusted R-squared, and percentage reduction of PRESS 

by use of OmegaEM are 0.30log10, 0.86, and −21 percent, respectively. Inclusion of the OmegaEM parameter provides a substantial reduction in the PRESS statistic of the regression equations and removes considerable spatial dependency in regression 

residuals. Although the OmegaEM parameter requires interpretation on the part of analysts and the potential exists that different 

analysts could estimate different values for a given watershed, the authors suggest that typical uncertainty in the OmegaEM estimate might be about ±0.10log10. Finally, given the two ensembles of equations reported herein and those in previous reports, 

hydrologic design engineers and other analysts have several different methods, which represent different analytical tracks, to 

make comparisons of peak-streamflow frequency estimates for ungaged watersheds in the study area. 

17. Key Words 18. Distribution Statement 

No restrictions 

19. Security Classif. (of report) 

Unclassified 

20. Security Classif. (of this page) 

Unclassified 

21. No. of Pages 

48 

22. Price 

Form DOT F 1700.7 (8-72) Reproduction of complete page authorized 
Blank Page 
Regression Equations for Estimation of Annual 

Peak-Streamflow Frequency for Undeveloped 

Watersheds in Texas Using an L-moment-Based, 

PRESS-Minimized, Residual-Adjusted Approach 

By William H. Asquith and Meghan C. Roussel 

In cooperation with the Texas Department of Transportation 

Scientific Investigations Report 2009–5087

(Texas Department of Transportation Research Report 0–5521–1)

U.S. Department of the Interior 

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

KEN SALAZAR, Secretary 

U.S. Geological Survey 

Suzette M. Kimball, Acting Director 

U.S. Geological Survey, Reston, Virginia: 2009 

For product and ordering information:

World Wide Web: http://www.usgs.gov/pubprod

Telephone: 1-888-ASK-USGS

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endorsement by the U.S. Government.

Although this report is in the public domain, permission must be secured from the individual copyright owners to

reproduce any copyrighted materials contained within this report.

Suggested citation:

Asquith, W.H., and Roussel, M.C., 2009, Regression equations for estimation of annual peak-streamflow frequency 

for undeveloped watersheds in Texas using an L-moment-based, PRESS-minimized, residual-adjusted approach: 

U.S. Geological Survey Scientific Investigations Report 2009–5087, 48 p. [ http://pubs.usgs.gov/sir/2009/5087 ] 

Typeset on June 16, 2009
iii 

Contents 

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

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

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

Identification of Annual Peak-Streamflow Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

An L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

L-moment-Based, Station-Specific Peak-Streamflow Frequency Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Sample L-moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Sampling Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Probability Distributions for Peak-Streamflow Frequency Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Trimmed Mean Estimates of Peak-Streamflow Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Modeling Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Combining Sampling and Modeling Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Weighted-Least-Squares, Multi-Linear Regression Analysis and PRESS Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Residual-Adjusted Regression and the OmegaEM Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Regression Equations in Texas Using an L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach . . . . . . . . . . 11

Regression Equations without OmegaEM Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

OmegaEM Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Regression Equations with OmegaEM Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Example Computations for a Single Station and Comparison to Estimates from Previously Published 

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Comparison of Regional Equations to Those of Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Comparability and Applicability of Peak-Streamflow Estimates from OmegaEM and Previous Equations . . . . 19

Preference for OmegaEM Equations over Previous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Considerations for Application of Regression Equations to Gaged and Ungaged Watersheds . . . . . . . . . . . . . . . . . . . 23

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Appendix 1—Supplemental Information and External Data Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Appendix 2—Results of Weighted-Least-Squares, Multi-Linear Regression Analysis Using PRESS Minimization 

of Peak-Streamflow Frequency to Watershed Characteristics of Drainage Area, Mean Annual 

Precipitation, and Dimensionless Main-Channel Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Appendix 3—Results of Weighted-Least-Squares, Multi-Linear Regression Analysis Using PRESS Minimization 

of Peak-Streamflow Frequency to Watershed Characteristics of Drainage Area, Mean Annual 

Precipitation, Dimensionless Main-Channel Slope, and OmegaEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
iv 

Figures 

1. Graph showing example of peak-streamflow frequency curves with superimposed axis labels and explanation in real and logarithmic space rendered from the R environment (R Development Core Team, 2008)

using functions of Asquith (2008) for seven probability distributions fit to sample L-moments of annual 

peak-streamflow data available in file Appendix1_677annpks.txt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 

2. Map showing hill-shade relief in Texas with superimposed values of OmegaEM parameter that represents a generalized terrain and climate index for regionalization of peak-streamflow frequency . . . . . . . . . . . . . 13 

3. Map showing rivers in Texas with superimposed values of OmegaEM parameter that represents a 

generalized terrain and climate index for regionalization of peak-streamflow frequency . . . . . . . . . . . . . . . . . . . . 14 

4. Map showing level II and III ecoregions in Texas with superimposed values of OmegaEM parameter that

represents a generalized terrain and climate index for regionalization of peak-streamflow frequency . . . . . . . . 15 

5. Graph showing comparison by recurrence interval of peak-streamflow frequency estimates for the 

638 stations using equations listed in table 2 (vertical axis) and using equations originally shown in 

Asquith and Thompson (2005, table 5) (horizontal axis, AT2005) that are based on station-specific, peakstreamflow frequency based from broad interpretation of Interagency Advisory Committee on Water 

Data (1982) (17B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 

6. Graph showing comparison by recurrence interval of peak-streamflow frequency estimates for the 

638 stations using equations listed in table 3 (vertical axis) and using equations originally shown in 

Asquith and Thompson (2005, table 5) (horizontal axis, AT2005) that are based on station-specific, peakstreamflow frequency based from broad interpretation of Interagency Advisory Committee on Water 

Data (1982) (17B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 

Tables 

1. Summary statistics of the watershed characteristics for 638 U.S. Geological Survey streamflow-gaging 

stations used to develop regional equations for estimation of peak-streamflow frequency . . . . . . . . . . . . . . . . . . 4 

2. Summary of weighted-least-squares, PRESS-minimized, regional regression equations using drainage 

area, dimensionless main-channel slope, and mean annual precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 

3. Summary of weighted-least-squares, PRESS-minimized, regional regression equations using drainage 

area, dimensionless main-channel slope, mean annual precipitation, and OmegaEM . . . . . . . . . . . . . . . . . . . . . . . 17 
v 

Conversion Factors and Datum 

Inch/Pound to SI 

Multiply By To obtain 

Length 

inch (in.) 25.4 millimeter (mm) 

mile (mi) 1.609 kilometer (km) 

Slope 

foot per mile (ft/mi) .1894 meter per kilometer (m/km) 

Area 

square mile (mi2) 2.590 square kilometer (km2) 

Flow 

cubic foot per second (ft3/s) .02832 cubic meter per second (m3/s) 

SI to Inch/Pound

Multiply By To obtain 

Length 

meter (m) 3.281 foot (ft) 

Slope 

meter per kilometer (m/km) 5.28 foot per mile (ft/mi) 

Area 

square kilometer (km2) .3861 square mile (mi2) 

Flow 

cubic meter per second (m3/s) 35.31 cubic foot per second (ft3/s) 

Datum 

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83). 
Blank Page 
Regression Equations for Estimation of Annual 

Peak-Streamflow Frequency for Undeveloped Watersheds 

in Texas Using an L-moment-Based, PRESS-Minimized, 

Residual-Adjusted Approach

By William H. Asquith and Meghan C. Roussel 

Abstract 

Annual peak-streamflow frequency estimates are 

needed for flood-plain management; for objective assessment of flood risk; for cost-effective design of dams, 

levees, and other flood-control structures; and for design of 

roads, bridges, and culverts. Annual peak-streamflow frequency represents the peak streamflow for nine recurrence 

intervals of 2, 5, 10, 25, 50, 100, 200, 250, and 500 years. 

Common methods for estimation of peak-streamflow 

frequency for ungaged or unmonitored watersheds are 

regression equations for each recurrence interval developed 

for one or more regions; such regional equations are the 

subject of this report. The method is based on analysis 

of annual peak-streamflow data from U.S. Geological 

Survey streamflow-gaging stations (stations). Beginning 

in 2007, the U.S. Geological Survey, in cooperation 

with the Texas Department of Transportation and in 

partnership with Texas Tech University, began a 3-year 

investigation concerning the development of regional 

equations to estimate annual peak-streamflow frequency 

for undeveloped watersheds in Texas. The investigation 

focuses primarily on 638 stations with 8 or more years 

of data from undeveloped watersheds and other criteria. 

The general approach is explicitly limited to the use 

of L-moment statistics, which are used in conjunction 

with a technique of multi-linear regression referred to 

as PRESS minimization. The approach used to develop 

the regional equations, which was refined during the 

investigation, is referred to as the “L-moment-based, 

PRESS-minimized, residual-adjusted approach.” For the 

approach, seven unique distributions are fit to the sample 

L-moments of the data for each of 638 stations and 

trimmed means of the seven results of the distributions 

for each recurrence interval are used to define the stationspecific, peak-streamflow frequency. As a first iteration of 

regression, nine weighted-least-squares, PRESS-minimized, 

multi-linear regression equations are computed using the 

watershed characteristics of drainage area, dimensionless 

main-channel slope, and mean annual precipitation. The 

residuals of the nine equations are spatially mapped, and 

residuals for the 10-year recurrence interval are selected 

for generalization to 1-degree latitude and longitude 

quadrangles. The generalized residual is referred to as 

the OmegaEM parameter and represents a generalized 

terrain and climate index that expresses peak-streamflow 

potential not otherwise represented in the three watershed 

characteristics. The OmegaEM parameter was assigned 

to each station, and using OmegaEM, nine additional 

regression equations are computed. Because of favorable 

diagnostics, the OmegaEM equations are expected to be 

generally reliable estimators of peak-streamflow frequency 

for undeveloped and ungaged stream locations in Texas. 

The mean residual standard error, adjusted R-squared, and 

percentage reduction of PRESS by use of OmegaEM are 

0.30log10, 0.86, and −21 percent, respectively. Inclusion 

of the OmegaEM parameter provides a substantial reduction in the PRESS statistic of the regression equations and 

removes considerable spatial dependency in regression 

residuals. Although the OmegaEM parameter requires 

interpretation on the part of analysts and the potential 

exists that different analysts could estimate different values 

for a given watershed, the authors suggest that typical 

uncertainty in the OmegaEM estimate might be about 

±0.10log10. Finally, given the two ensembles of equations 

reported herein and those in previous reports, hydrologic 

design engineers and other analysts have several different 

methods, which represent different analytical tracks, to 

make comparisons of peak-streamflow frequency estimates 

for ungaged watersheds in the study area. 
2 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

Introduction 

Annual peak-streamflow frequency estimates are 

needed for flood-plain management; for objective assessment of flood risk; for cost-effective design of dams, levees, 

and other flood-control structures; and for design of roads, 

bridges, and culverts. Annual peak-streamflow frequency 

represents the peak streamflow for nine recurrence intervals 

of 2, 5, 10, 25, 50, 100, 200, 250, and 500 years. Common 

methods for estimation of peak-streamflow frequency for 

ungaged or unmonitored stream watersheds are regression 

equations for each recurrence interval developed for one 

or more regions (not strictly geographic); such regional 

equations are the subject of this report. The method 

is based on analysis of annual peak-streamflow data 

from U.S. Geological Survey (USGS) streamflow-gaging 

stations (stations). 

Beginning in 2007, the USGS, in cooperation with 

the Texas Department of Transportation (Research Project 

0–5521) and in partnership with Texas Tech University, 

began a 3-year investigation concerning the development 

of regional equations to estimate annual peak-streamflow 

frequency for undeveloped watersheds in Texas. The general approach was explicitly limited to use of L-moment 

statistics, which are used in conjunction with a technique 

of multi-linear regression referred to as PRESS minimization (Asquith and Thompson, 2008). The approach used to 

develop the regional equations, which was refined during 

the investigation, is referred to as the “L-moment-based, 

PRESS-minimized, residual-adjusted approach.” 

The study area for this investigation includes Texas and 

selected parts of neighboring states and essentially is the 

same as that considered by Asquith and Slade (1997, 1999). 

These two studies also represent previous research concerning estimation of peak-streamflow frequency for ungaged 

watersheds in Texas. Asquith (2001) provides analysis of 

peak-streamflow frequency in Texas using L-moment statistics and primarily reports the effects of regulation (flood 

control) on those statistics. 

The background and discussion provided by Asquith 

and Thompson (2008), which is based heavily on Asquith 

and Thompson (2005), represents applicable discussion, 

analysis, and ideas that greatly influenced the line of 

research described in this report. To establish context, 

the major new contributions of this report relative to 

Asquith and Thompson (2005, 2008) are the additions of 

L-moment statistics, with subsequent use of numerous 

probability distributions, and the residual adjustment. 

These additions (topics or themes) are described in the “An 

L-moment-Based, PRESS-Minimized, Residual-Adjusted 

Approach” section of this report. 

Purpose and Scope 

The purpose of this report is to present regression 

equations from the L-moment-based, PRESS-minimized, 

residual-adjusted approach for estimation of annual peakstreamflow frequency for undeveloped watersheds in Texas 

and restricted mainly to ungaged watersheds. The approach, 

which required a complex computational framework, has 

three thematic components: 

• L-moment statistics of the annual peak-streamflow 

data, by station, are used to fit seven probability distributions to the data. From these seven distributions, 

representative values (estimates) of station-specific, 

peak-streamflow frequency were extracted for each of 

the nine recurrence intervals; 

• Weighted-least-squares, multi-linear regression analysis using the station-specific, peak-streamflow frequency and selected watershed characteristics is 

used to develop regression equations. The regression included a technique to minimize the PRediction 

Error Sum of Squares (PRESS) statistic by power 

transformation of drainage area; and 

• An adjustment based on regression residuals is created that represents a generalized correction for 

climate, terrain, and other variables not otherwise 

expressed by the selected watershed characteristics. 

The report is limited to the annual peak-streamflow 

data for 677 and 638 selected stations. Two distinct 

station counts are used in this report as explained in 

the next section. The data for these watersheds are 

provided in the two text files Appendix1_677annpks.txt

and Appendix1_638annpks.txt, which are described in 

appendix 1. The report also is limited to three selected 

watershed characteristics (contributing drainage area, 

dimensionless main-channel slope, and mean annual 

precipitation), which previously have been shown to be 

important predictors of peak-streamflow frequency in 

Texas (Asquith and Slade, 1997; Asquith and Thompson, 

2008) and are identified in the next section. 

Identification of Annual Peak-Streamflow 

Data 

From an initial candidate station count of 1,030 stations that had 1 or more years of annual peak-streamflow 

data in the study area, the number of stations used for this 

report were reduced by review of the data, watershed conditions, and a minimum record-length criterion. 
Introduction 3 

Assessment of undeveloped conditions by peakstreamflow qualification codes in the authoritative USGS 

“peak value” database (U.S. Geological Survey, 2008) for 

the 1,030 stations was made along with historical and 

retrospective analysis concerning development in each 

watershed (such as the years of construction of substantial 

flood-control reservoirs or growth of municipalities). The 

assessment was further augmented by graphical depiction 

and review of a time series for each of the 1,030 stations. 

To complete the assessment, intervals of acceptable record 

from the beginning of record through a station-specific 

terminal year were identified. 

Stations considered for this report include those with 

at least 8 years (through the 2006 water year1) of annual 

peak-streamflow data that are representative of undeveloped streamflow conditions in Texas (acceptable record 

in the previous paragraph). Undeveloped conditions, in 

regard to annual peak streamflow, are watershed conditions 

representative of ideals such as natural, rural, unregulated, 

and unurbanized. Furthermore, surface-water diversions or 

return flows are anticipated to be unsubstantial relative to 

the typical magnitude of individual annual peak-streamflow 

values. 

After the assessment of undeveloped conditions, time 

series, and minimum record-length criterion, 677 stations 

remained. The data for these 677 stations were used 

for a specific component of the analysis described in 

the “Sampling Error” section of this report. The annual 

peak-streamflow data for these stations are available in 

text file Appendix1_677annpks.txt, which is described in 

appendix 1. 

The selected watershed characteristics, which will be 

important in later regression analysis, are derived from 

Asquith and Slade (1997) and augmented with digital processing and verification (David B. Thompson and Lucia S. 

Barbato, Texas Tech University, written commun., 2007). 

The watershed characteristics of contributing drainage area 

(drainage area), dimensionless main-channel slope, and 

mean annual precipitation are used. Assessments of the 

reliability of these watershed characteristics, which were 

obtained in early stages of this investigation, were made. 

The primary watershed characteristics used for this 

investigation are summarized as follows: 

• DRAINAGE AREA is the horizontal projection of 

the area that directs water to the streamflow-gaging 

station and is measured in square miles; 

1A water year is the 12-month period between October 1 and September 30 and 

is designated by the calendar year in which it ends. Thus, the year ending September 

30, 2006, is the “2006 water year.” 

• DIMENSIONLESS MAIN-CHANNEL SLOPE is defined 

as the magnitude of the change in elevation in feet 

between the two end points of the main channel 

divided by the MAIN-CHANNEL LENGTH in feet. 

Main-channel length is defined as the length in 

stream-course miles of the longest defined channel 

from the approximate watershed headwaters to the 

outlet. Although not used directly in the regression 

equations, these lengths are needed for computation 

of dimensionless main-channel slope. Because of 

the wide range in drainage area, various methods for 

length estimation have been used and are represented 

in the database. Asquith and Slade (1997) provide 

a specific definition based on 1:100,000-scale maps. 

For this investigation digital verification, through 

30-meter or 90-meter digital elevation models and 

digital line graphs, was used to define the main channel. Manual techniques were occasionally used as 

station-specific circumstances required; and 

• MEAN ANNUAL PRECIPITATION is the arithmetic 

mean of a suitably long period of time of total annual 

precipitation in inches. The mean annual precipitation was assigned based on the approximate center of 

the watersheds. Asquith and Slade (1997) generally 

provide the values used for this report. The period 

1951–80 was used by Asquith and Slade because 

those authors felt that most of the data (measured in 

then active stations) in the database were within this 

time period. Given the many sources of uncertainty, 

the authors of the current (2009) report consider 

that any general and authoritative source of mean 

annual precipitation for any suitably long period (perhaps 30 years) is sufficient for substitution into the 

regional regression equations reported here. 

Another assessment or restriction for the purpose of the 

regression analysis is that stations with drainage areas less 

than 10,000 square miles were used. This threshold was 

chosen from review of diagnostics of exploratory regression analysis—most watersheds having drainage areas this 

large are currently (modern times) considered regulated— 

and most design needs of the Texas Department of Transportation are for much smaller watersheds in Texas. 

After the assessment of watershed characteristics and 

the less-than-10,000-square-mile criterion, 638 stations 

remained. The watershed characteristics for these stations are available in file Appendix1_638wtrshdchr.txt, 

which is described in appendix 1. The annual peakstreamflow data for these stations are available in file 

Appendix1_638annpks.txt, which also is described in 
4 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

Table 1. Summary statistics of the watershed characteristics for 638 U.S. Geological Survey streamflow-gaging stations used to develop regional equations for estimation of 

peak-streamflow frequency. 

[A, drainage area in square miles; S, dimensionless main-channel 

slope; and P, mean annual precipitation in inches] 

Characteristic 

Minimum 

1st 

quartile Median 3rd quartile 

Maximum 

A 0.100 6.403 97.40 499 9,329 

100 ×S .023 .152 .269 .657 7.03 

P 8 23 31 41 57 

appendix 1. Summary statistics of the selected watershed 

characteristics for the 638 stations are listed in table 1. 

Acknowledgments 

The authors recognize the contributions of many 

Texas Department of Transportation engineers including 

Amy Ronnfeldt, Design Division, 0–5521 Project Director; 

David Zwernemann, Project Advisor; and George Herrmann, San Angelo District, former Project Director and 

Advisor. The authors also acknowledge the contributions 

of our three research supervisors for this project at Texas 

Tech University: Professors David B. Thompson (formerly 

Texas Tech University, 2007), Ken Rainwater (2008), and 

Theodore G. Cleveland (2009). The authors also acknowledge the geographic information system contributions of 

Lucia S. Barbato, Center for Geospatial Technology, Texas 

Tech University. 

An L-moment-Based, PRESSMinimized, Residual-Adjusted 

Approach 

A generalized overview of the L-moment-based, 

PRESS-minimized, residual-adjusted approach (referred to 

as the “Approach”) used for this investigation is described 

in this section. The Approach is complex, highly technical, 

and challenging to succinctly describe. Furthermore, 

the Approach relied on numerous single-purpose, highly 

specialized scripts, computer programs, and integration 

with features of the host-operating system as well as 

considerable judgement and iterative refinement on the 

part of the authors. The computational framework was 

ultimately implemented as a relatively turnkey process 

from start to finish, but nearly 6 months of effort were 

required to build, test, and refine the process to produce 

the equations reported in the “Regression Equations in 

Texas Using an L-moment-Based, PRESS-Minimized, 

Residual-Adjusted Approach” section of this report. 

The generalized overview of the Approach consists 

of three thematic elements. Furthermore, several R code 

listings are presented to demonstrate critical computations. First, the method of computing station-specific, 

peak-streamflow frequency estimates using L-moment 

statistics is described as well as methods of obtaining 

estimates of uncertainty. Second, the use of weighted-leastsquares, multi-linear regression in the context of PRESS 

minimization is described. Third and finally, the residual 

adjustment and subsequent recomputation of the regression 

equations incorporating a special watershed characteristic 

known as the Ω parameter (also the term “OmegaEM”2 f )

is described. 

L-moment-Based, Station-Specific PeakStreamflow Frequency Estimates 

L-moments (Hosking, 1990) summarize samples and 

can be used to fit probability distributions. L-moments are 

based on linear combinations of differences of the expectations of order statistics as opposed to the more familiar 

product moments, which are based on powers (exponents) 

of differences from the mean. This distinction results in 

favorable sampling properties including unbiasness, robustness, and reliable measures of distribution shape. The 

mathematical theory and comparisons between L-moments 

and product moments are available in several sources and 

the references therein (Hosking, 1990; Stedinger and others, 1993; Hosking and Wallis, 1997; Asquith, 2001, 2006;

Asquith and others, 2006). Furthermore, in a peak streamflow application, treatment for low outliers caused by 

typical log10 transformation (Stedinger and others, 1993, 

p. 18.5) of the data prior to statistical processing is not 

needed. Hosking (2006, p. 194) reports that L-moments 

“are now widely used in hydrology to summarize data and 

fit flood frequency [peak-streamflow frequency] distributions” and provides several citations of such practice. 

In brief, L-moments were used for this investigation 

for station-specific computations of peak-streamflow frequency because the method removes or at least relaxes 

the need to consider concepts such as low-outlier thresholds, high-outlier thresholds, historical information, and 

2A textual representation of the mathematical nomenclature also is needed (actually required) by the portable document format (PDF) standard for the headings, 

bookmarks, and other features of the PDF version of this report. 
An L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach 5 

generalized skew as described in other methods for peakstreamflow frequency computation such as those in the 

Interagency Advisory Committee on Water Data (1982) 

guidelines. 

For this report, the algorithms of Asquith (2008) were 

used within the R environment (R Development Core Team, 

2008). These algorithms are based on many years of iterative development from various authentic and authoritative 

sources such as Hosking (1990, 1996) and Hosking and 

Wallis (1997). Specific functions from Asquith (2008) are 

identified in the following sections. The use of function 

names in the description of the L-moment-based portion 

of the Approach permits the exclusion of mathematics and 

succinct demonstration or examples of critical-to-document 

computations. 

Sample L-moments 

The sample L-moments represent specific metrics of 

“distribution geometry” for lack of a better term. The first 

L-moment is the arithmetic mean and the second L-moment 

is analogous, but not numerically equal, to the standard 

deviation. Higher-order L-moments are measures of distribution skewness (symmetry), kurtosis, and other concepts. 

For the Approach, the first five sample L-moments 

were computed for each of the 677 stations using the 

lmoms() function of Asquith (2008). These L-moments, 

respectively, are the mean, L-scale, L-skew, L-kurtosis, and 

Tau5. The coefficient of L-variation or L-CV is defined as 

the ratio of L-scale to the mean. The sample L-moments 

by station were considered results of intermediate computations and are not reported here. 

An example of the computation method of the 

sample L-moments for station 07153500 Dry Cimarron 

River near Guy, New Mex., is shown in the following 

R code listing. This station is the first listed in file 

Appendix1_677annpks.txt and has 33 years of annual 

peak-streamflow data identified for this investigation. This 

station will be used repeatedly in the example computations 

described herein. 

> library(lmomco) # load in the lmoms() function

# annual peaks in cubic feet per second for

# station 07153500

> peaks <- c( 8200, 7120, 1440, 2960, 3000,

980, 2140, 6880, 4350, 3350,

1330, 3950, 2800, 8100, 8500,

5200, 1820, 1610, 1190, 3270,

4920, 475, 3930, 2070, 46100,

22500, 4310, 3660, 987, 962,

2890, 723, 510) # the peaks

> length(peaks) # returns 33 years as the total

# count of annual peak streamflow

[1] 33

> lmr <- lmoms(peaks) # compute sample L-moments

> str(lmr) # show the sample L-moments

List of 6

$ lambdas : num [1:5] 5219 3011 1782 1463 1261

$ ratios : num [1:5] NA 0.577 0.592 0.486 0.419

$ trim : num 0

$ leftrim : NULL

$ rightrim: NULL

$ source : chr "lmoms"

The listing shows that the mean and L-scale values are 

about 5,219 and 3,011 cubic feet per second, respectively. 

The listing also shows that the L-skew, L-kurtosis, and 

Tau5 values (dimensionless) are 0.592, 0.486, and 0.419, 

respectively. (The value 0.577 is L-CV.) These sample 

L-moments also will be used in the “Probability Distributions for Peak-Streamflow Frequency Curves” of this 

report. 

Sampling Error 

For the weighted-least-squares regression analysis, 

weight factors representing sampling error and modeling 

errors as distinct components of uncertainty are used. In 

this section, the concept and method of computation of 

a sampling error of peak streamflow for each recurrence 

interval by station is described. It is favorable to have 

stations with more record (smaller sampling error, more 

certainty) to have more weight in regression analysis. 

For this report, sampling error is the concept that the 

peak-streamflow frequency for a given station is dependent 

on the number of years of available record. Stations having long periods of record (measured in many decades) 

have more defined and presumably more reliable peakstreamflow frequency values than the values for stations 

having short periods of record (measured in less than about 

2 decades). Thus, sample size in years for a station acts to 

progressively decrease sampling error. 

For this report, weighted.mean() values of the first five 

dimensionless sample L-moments of the 677 stations were 

computed by the so-named function in R. The years of 

record for each station in file Appendix1_677annpks.txt

were used as weight factors.3 The term dimensionless in 

this context implies a unit mean and the second L-moment 

equal to the L-CV. The weighted-mean, dimensionless 

L-moments are 1, 0.5055, 0.3939, 0.2496, and 0.1590 for 

the mean, L-scale, L-skew, L-kurtosis, and Tau5, respectively. These L-moments are referred to as the dimensionless regional L-moments, and these weighted-mean values 

provide an approximation of the geometry of the “parent” 

peak-streamflow frequency distribution for the study area. 

3These weights are not to be confused with the weight factors used in the regression analysis reported herein. 
6 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

The dimensionless regional L-moments were used 

to estimate a dimensionless regional Wakeby distribution 

(Landwehr and others, 1979) using the parwak() function of 

Asquith (2008). This fitted Wakeby distribution, because 

of its immense flexibility with five parameters, is assumed 

to provide a reasonable first-order approximation to the 

underlying (dimensionless) structure of peak-streamflow 

frequency for the study area. The use of the dimensionless 

regional Wakeby distribution is described in the remainder 

of this section. 

Using statistical simulation and the genci() function of 

Asquith (2008) for each unique number of years of record 

for the 677 stations and the dimensionless regional Wakeby 

distribution, the L-CV for each of the nine recurrence 

intervals were estimated. An example of the computation 

method for station 07153500, which has 33 years of record, 

and the 2-year recurrence interval is shown in the following 

R code listing: 

> library(lmomco) # load the vec2lmom(),

# parwak(), T2prob(), and genci() functions

# Manually set the dimensionless regional

# L-moments

> L <- vec2lmom(c(1,0.505,0.394,0.250,0.159))

# Fit a dimensionless Wakeby distribution to

# the regional L-moments in variable L

> W <- parwak(L)

# the algebraic form of distribution is shown

# in the text

# Compute nonexceedance probability (0.5)

# for 2-year event

> F <- T2prob(2)

> n <- 33; # sample size, years of record

> nsim <- 10000 # the num. of simulations and

# a large value to gain numerical accuracy

> genci(W, n, F=F, nsim=nsim)

nonexceed_prob lower true upper

lscale lcv

1	 0.500 0.4725784 0.6667878 0.9042445

0.07403137 0.1110269

The regional dimensionless Wakeby distribution fit to 

the L-moments shown in the previous R code listing is 

1.10(1−(1−F)

6.10 Q(F) = −0.0266 + ) 6.10

0.692 (1−(1−F)

−0.206 −	 ), (1) 0.206 

where Q(F) is dimensionless peak streamflow (unit mean) 

for nonexceedance probability F = 1−1/T for recurrence

interval T. 

In brief, the simulation process is as follows: For 

a simulation run of 10,000 iterations, a random sample 

of size n (33 years for the example) for each iteration 

was drawn from the regional dimensionless Wakeby distribution. Subsequently, for each iteration, “new” sample 

L-moments of the random sample were computed and a 

“new” Wakeby distribution fit to the new L-moments. If 

a Wakeby could not be fit, a generalized Pareto distribution was used instead; Hosking and Wallis (1997) provide 

details of the algorithm. The value of the distribution at a 

given T-year recurrence interval (2-year for the example) 

is stored. After the completion of the 10,000 iterations, the 

L-CV of the 10,000 stored estimates of the 2-year event 

was computed. This value represents the model error for a 

sample of size n = 33. 

The previous R code listing shows that the number 

of interest is the value listed under the lcv heading. For 

the example, this value is about 0.111 (the last value in 

the output). Conceptually this value represents the relative (dimensionless) uncertainty in the estimation of the 

2-year peak-streamflow value. Considerable computational 

effort was expended to estimate the L-CV values for each 

of nine selected recurrence intervals and 71 unique sample sizes (number of years of record) for the 677 stations. 

Thus, a total of 6.39 million simulations were conducted 

(9×71×10,000). The computation of modeling error was 

based on statistics of the 677 stations. From this point forward in the discussion of the Approach and results, the 

638 stations become the subject. 

In summary, these 639 L-CV values (9×71) represent 

the relative uncertainty in the peak-streamflow estimate at a 

given recurrence interval and for a given sample size. The 

uncertainty is relative to the “mean” estimate of the peakstreamflow magnitude. Discussion of this mean estimate is 

provided in the next section. 

Probability Distributions for Peak-Streamflow 

Frequency Curves 

For this report, as many as seven probability distributions are fit to the sample L-moments for each station 

and resultant frequency curves were graphically depicted 

and inspected by the authors in real and log10-transformed 

space. 

The authors explicitly chose to avoid selection of a single form of a probability distribution to model the stationspecific, peak-streamflow frequency through simultaneous 

use of a substantial number of three-parameter and more 

distributions. Further, by use of more than one probability 

distribution, estimates for modeling error (see next section) 

could be obtained. 

The probability distributions considered and parameter and quantile functions of Asquith (2008) used for this 
An L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach 7 

report are as follows: 

• Generalized extreme value distribution—supported 

by functions pargev() and quagev(); 

• Generalized logistic distribution—supported by functions parglo() and quaglo(); 

• Generalized normal distribution—supported by functions pargno() and quagno(); 

• Generalized Pareto distribution—supported by functions pargpa() and quagpa(); 

• Kappa distribution (if solution available), otherwise the generalized lambda distribution (if solution available)—supported by functions parkap()

and quakap() for kappa and functions pargld() and 

quagld() for generalized lambda; 

• Pearson Type III distribution—supported by functions parpe3() and quape3(); and 

• log-Pearson Type III distribution, which is the same 

as Pearson Type III but the L-moments of the log10 

values of data are used instead to fit the distribution. 

The seven quantile functions for the itemized probability distributions are internally called from the qlmomco()

function, which technically was used in algorithms by the 

authors. 

In summary, for almost all stations, seven distributions 

were fit and seven estimates of peak streamflow for each 

of the nine recurrence intervals were stored for later use. 

An example of the computation method for only the generalized extreme value and generalized logistic distributions 

(two of the seven distributions), which are accessed using 

the qlmomco() function, for station 07153500 is shown in 

the following R code listing: 

> library(lmomco) # load the lmoms(), qlmomco(),

# and lmom2par() functions

# The variable peaks was generated in the section

# ‘‘Sample L-moments’’

> lmr <- lmoms(peaks) # compute sample L-moments

# Compute the 2-year peak streamflow using two

# probability distributions: generalized extreme

# value and generalized logistic distributions

> Q2gev <- qlmomco(0.5, lmom2par(lmr, type="gev"))

> Q2glo <- qlmomco(0.5, lmom2par(lmr, type="glo"))

# Show the estimates in cubic feet per second

> print(Q2gev) # 2-year generalized extreme value

[1] 2724.746

> print(Q2glo) # 2-year generalized logistic

[1] 2754.674

The listing shows that the 2-year peak streamflow from the 

generalized extreme value is about 2,725 cubic feet per second, and the 2-year peak streamflow from the generalized 

logistic distribution is about 2,755 cubic feet per second. 

The difference is a natural and an expected result of the 

form of the fitted distributions. Generally, as recurrence 

interval increases, differences in estimates from different 

distributions tend to increase. These two values will be 

used with five others from five additional distributions in 

the next section. 

For station 07153500, the seven probability distributions are shown in figure 1. Each graph depicts the peakstreamflow frequency curves in real and logarithmic units 

(A and B, respectively). The horizontal axis is rendered in 

standard normal deviates4 and hence is a normal probability 

axis although not labeled in units of probability. Also on the 

figure, the individual plots are rendered with the respective 

station number to ensure consistency with the figure caption. The seven estimates, by distribution, by recurrence 

interval, and by station were considered results of intermediate computations and are not reported here. However, the 

more than 677-page, portable document format (PDF) file 

Appendix1_677freqcurves.pdf described in appendix 1 is 

available. This file provides graphical documentation of the 

computations described in this section and repeats figure 1 

for completeness. 

Trimmed Mean Estimates of PeakStreamflow Frequency 

For each of the nine recurrence intervals, a symmetrically trimmed L-moment (trimmed mean) was computed 

using the TLmoms(QT,trim=1) function of Asquith (2008) 

based on trimmed L-moments (Elamir and Seheult, 2003; 

Hosking, 2007), where QT is a vector that represents as 

many as (generally) seven unique values of peak streamflow for a given recurrence interval. The use of trimming, 

which computationally is more complex than simply dropping the largest and smallest values and computing an arithmetic mean, is anticipated to provide a more robust measure of central location for the formal estimation of stationspecific, peak-streamflow frequency. It is important to note 

that log10 transformation of the peak-streamflow estimates 

was not made prior to computation of the trimmed mean. 

The computation of the trimmed mean is demonstrated 

using results for station 07153500 for the 2-year peak 

streamflow. In the following R code listing, the estimates 

for each of the seven distributions are set into variables and 

4Standard normal deviate—A normally distributed random variable (or nonexceedance probability for the plots) with an expected value of 0 and a variance of 1. 
−3 −2 −1 0 1 2 3

3.0 3.5 4.0 4.5 5.0

U.S. Geological Survey station 07153500

8 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

−3 −2 −1 0 1 2 3

0 50000 100000 150000

U.S. Geological Survey station 07153500

(A) 

STANDARD NORMAL DEVIATES 

PEAK STREAMFLOW, IN CUBIC FEET PER SECOND LOG10 OF PEAK STREAMFLOW, IN CUBIC FEET PER SECOND 

(B) 

STANDARD NORMAL DEVIATES 

EXPLANATION

Pearson Type III distribution

Generalized extreme value distribution (if solution available)

Generalized normal distribution (if solution available)

Kappa distribution (if solution available)

Annual peak-streamflow value

Generalized logistic distribution

log-Pearson Type III distribution

Generalized Pareto distribution

Figure 1. Example of peak-streamflow frequency curves with superimposed axis labels and explanation in real and logarithmic 

space rendered from the R environment (R Development Core Team, 2008) using functions of Asquith (2008) for seven probability 

distributions fit to sample L-moments of annual peak-streamflow data available in file Appendix1_677annpks.txt for station 

07153500 Dry Cimarron River near Guy, New Mexico. 
An L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach 9 

then concatenated into the variable Q2. The arithmetic and 

trimmed means are computed using the TLmoms() function 

of Asquith (2008). The values for gev and glo were computed in the “Probability Distributions for Peak-Streamflow 

Frequency Curves” section of this report. 

> library(lmomco) # load the TLmoms() function

# Seven unique estimates for 2-year recurrence

# interval in cubic feet per second

> gev <- 2725 # from generalized extreme value

> glo <- 2755 # from generalized logistic

> gno <- 2507 # from generalized normal

> gpa <- 2590 # from generalized pareto

> kap <- 3023 # from kappa

> pe3 <- 2037 # from pearson type III

> pe3log <- 2820 # from log-pearson type III

> Q2 <- c(gev,glo,gno,gpa,kap,pe3,pe3log)

# Now compute the usual mean (not used)

> TLmoms(Q2,nmom=1, trim=0)

$lambdas

[1] 2636.714

# Now compute the symmetrically trimmed mean

> TLmoms(Q2,nmom=1, trim=1)

$lambdas

[1] 2683.429

The listing shows that the usual arithmetic mean of the 2­

year peak streamflow for the station is about 2,637 cubic 

feet per second. The trimmed mean is about 2,683 cubic 

feet per second and was used. The authors experimented 

with the arithmetic mean, median, and trimmed mean in the 

iterative development of the Approach; the authors judged 

that the trimmed mean provided an appropriate statistic to 

estimate station-specific, peak-streamflow frequency. 

In summary, the trimmed mean was used to estimate 

the station-specific peak streamflow for each of the nine 

recurrence intervals. The trimmed means are used in 

the regression analysis described in the “Weighted-LeastSquares, Multi-Linear Regression Analysis and PRESS Minimization” section of this report. The trimmed means by 

recurrence interval and by station were considered intermediate computations, but these results are available in 

file Appendix1_677trimmedQTs.txt, which is described in 

appendix 1. 

Modeling Error 

Because seven distributions (models) were fit for each 

station, seven unique estimates of peak streamflow were 

obtained for each recurrence interval. For a given recurrence interval, these estimates have a distribution; the relative variability of this distribution is treated as a measure 

of modeling error. Stations with more variability in peakstreamflow estimates, which is attributable to distribution 

choice, have more uncertainty than those stations with less

variability. Therefore, it is favorable for stations with more 

uncertainty to have less weight in regression analysis. 

The computation of the modeling error is demonstrated 

using results for station 07153500 for the 2-year peak 

streamflow. In the following R code listing, the estimates 

for each of the seven distributions are available in the variable Q2 set in the previous code listing. The L-moments 

(not trimmed) are computed using the lmoms() function of 

Asquith (2008). 

> library(lmomco) # to access lmoms() function

# Compute sample L-moments of seven distributions

# for the 2-year recurrence interval

> lmoms(Q2,nmom=2) # no trimming

$lambdas

[1] 2636.7143 178.5238

$ratios

[1] NA 0.06770692

# Thus, after rounding off the decimal

# Lambda_1 (mean) = 2637; Lambda_2 (L-scale) = 179

The listing shows that the L-scale of the seven fits is about 

179 cubic feet per second. An approximate variance in R

code for the model fit, and representative of model error, is 

thus (179*sqrt(pi))^2 or 100,660. (The sqrt() function is 

the square root.) 

Combining Sampling and Modeling Errors 

Combination of the modeling error with sampling 

error is straightforward. The standard deviation of a peakstreamflow frequency estimate can be estimated by the 

Pythagorean distance of the variances of the sampling and 

modeling errors. 

In the “Sampling Error” section of this report, station 

07153500 has a sampling error measured in L-CV of about 

0.111. If the trimmed mean estimate for the station is 

2,683 cubic feet per second for the 2-year event (see section “Trimmed Mean Estimates of Peak-Streamflow Frequency”), then the sampling error has an approximate variance in R code of (0.111*(2683)*sqrt(pi))^2 or 278,636. 

The previous section shows that the modeling error variance is about 100,660. 

These two variances can be aggregated in R code 

by sqrt(278636 + 100660) for a combined error of about 

616 cubic feet per second. Hence, the standard deviation of 

the 2,683 cubic feet per second estimate of the 2-year peak 

streamflow for station 07153500 is 616 cubic feet per second, which can be expressed as a relative variability as the 

ratio 616/2683 or about 0.230. This relative error, actually 

its inverse, was used as one of the 5,742 weight factors in 
10 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

weighted-least-squares regression analysis. The weight factor count is the product of 638 stations and nine recurrence 

intervals (638 ×9). 

Weighted-Least-Squares, Multi-Linear 

Regression Analysis and PRESS Minimization 

For this report, weighted-least-squares, multi-linear 

regression analysis is used to develop the statistical relation between the station-specific, trimmed-mean, T-year 

peak-streamflow values and the three watershed characteristics (explanatory variables previously described). The 

sampling and modeling errors for each station were combined as described in the “Combining Sampling and Model 

Errors” section of this report. The combined error is used to 

derive weight factors for the weighted-least-squares regression between peak-streamflow frequency and the watershed characteristics of drainage area, dimensionless mainchannel slope, mean annual precipitation, and, when used, 

the fΩ parameter described in the next section. 

PRESS minimization is formally described by Asquith 

and Thompson (2008) and is only summarized in this report. 

The PRESS computation used for this investigation was 

not weighted by the regression weight factors as used by 

Asquith and Thompson (2008). The PRESS statistic generally is regarded as a measure of regression performance 

when the model is used to predict new data (Montgomery 

and others, 2001, p. 153). Prediction of new data is what 

analysts and hydrologic engineers do when they estimate 

peak streamflow from a regression equation. Regression 

equations with small PRESS values are desirable; thus, 

PRESS minimization is an appropriate goal. Helsel and 

Hirsch (2002, p. 247) state that, “Minimizing PRESS means 

that the equation produces the least error when making new 

predictions.” Conceptually, PRESS minimization identifies 

the appropriate transformation of drainage area to “press” 

the bias (residual curvature) out of the equations. 

Other variables were not “pressed” because curvature 

in the residuals is sufficiently removed by use of drainage 

area. Further, considerable computational complexities are 

introduced in investigation of simultaneous multi-transform 

optimization. 

Because the PRESS statistic is an overall measure of 

regression fit (like residual standard error) and is a validation statistic (unlike residual standard error), minimization 

of PRESS is desirable. The most “valid” regression is produced when the PRESS statistic is minimized. The following transformation on drainage area was used: 

A� = Aλ , (2) 

where A� is the transformed value of drainage area for 

the regression, A is drainage area, and λ is a real number. For this report specific values of λ were determined 

by exhaustive search to three significant figures for each 

regression equation. Thus, values for λ can be thought 

of as regression-specific parameters, but are not formally 

counted as such in diagnostic statistics of the regression. 

The λ values computed by the PRESS minimization are 

reported in the “Regression Equations in Texas Using an 

L-moment-Based, PRESS-Minimized, Residual-Adjusted 

Approach” section of this report. 

Residual-Adjusted Regression and the 

OmegaEM Parameter 

The terminal steps towards computation of the final 

(preferred by authors) regression equations reported here 

involved an iterative reprocessing through the development 

of a generalization of regression residuals. The generalized 

regression residual has been previously identified herein as 

the fΩ parameter, and fΩ can be thought of as a special 

watershed characteristic, which is analogous to some other

unknown, but spatially varying variable that expresses peakstreamflow potential in the study area. 

The residual-adjusted, regression process comprises 

the following steps: 

1. For each of the nine recurrence intervals, weightedleast-squares regression analysis using PRESS minimization was made using only the watershed characteristics of drainage area, dimensionless mainchannel slope, and mean annual precipitation for 

the 638 stations. The raw outputs from scripts operating in the R environment for each recurrence 

interval are shown in appendix 2. The information 

contained therein is interpreted and formally presented in the “Regression Equations in Texas Using 

an L-moment-Based, PRESS-Minimized, ResidualAdjusted Approach” section of this report; 

2. The residuals for each of the nine regression equations were computed. Maps depicting the spatial 

variation of the residuals with symbols determined 

by the magnitude and direction of the residual were 

created and are formally presented in appendix 1 

of this report. Spatial dependency is evident for all 
Regression Equations in Texas Using an L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach 11 

recurrence intervals. For each recurrence interval, the 

median residual was computed for each square of 

1-degree quadrangle of latitude and longitude in the 

study area containing at least one station; 

3. The authors, through exploratory analysis and judgement, selected the median values of the residuals to 

pool or combine into 1-degree quadrangles for the 

10-year recurrence interval. The authors, through 

manual smoothing, consultation of various geologic and ecological region maps, interpretation of 

regional topographic maps, and resident familiarity 

with the study area, estimated the magnitude and 

sign of the 10-year recurrence interval residuals for 

each 1-degree quadrangle in the study area. These 

estimates are referred to as the fΩ parameter, are in 

units of log10(streamflow), and are formally in the 

“OmegaEM Parameter” section of this report; 

4. For each station, the fΩ parameter was assigned 

based on the 1-degree quadrangle containing the station. In practice, the fΩ parameter could be assigned 

similarly or by judgement considering the location 

and spatial extent of the watershed in question; 

5. For each of the nine recurrence intervals, weightedleast-squares regression using PRESS minimization is 

made using the watershed characteristics of drainage 

area, dimensionless main-channel slope, mean annual 

precipitation, and fΩ parameter for the 638 stations. 

The raw output from scripts operating in the R environment for each recurrence interval are shown in 

appendix 3. The information contained therein is 

interpreted and formally presented in the “Regression Equations in Texas Using an L-moment-Based, 

PRESS-Minimized, Residual-Adjusted Approach” of 

this report. New power transformations by recurrence 

interval on drainage area are computed by the PRESS 

minimization. Further, for each recurrence interval a 

separate regression coefficient on the fΩ parameter is 

computed; and 

6. As in the second step, the residuals of the regressions 

using the fΩ parameter were computed and mapped. 

Considerable reduction in spatial dependency is evident. The authors interpret that the residuals from the 

regressions using the fΩ parameter are reasonably 

spatially invariant. 

Introduction of a final regression diagnostic is needed. 

Akaike Information Criterion (AIC) statistic is reported here 

and is a measure of information content of a regression 

model. The statistic accounts for a tradeoff between number of parameters and the fit of the model; small values are 

sought. 

Regression Equations in Texas 

Using an L-moment-Based, PRESSMinimized, Residual-Adjusted 

Approach 

The final results of the L-moment-based, PRESSminimized, residual-adjusted approach are described in 

this section. This section is organized as follows. First, 

the regional equations using drainage area, dimensionless 

main-channel slope, and mean annual precipitation are 

formally presented and discussed. Second, maps that 

depict the fΩ parameter are presented and discussed. Third, 

the regional equations using drainage area, dimensionless 

main-channel slope, mean annual precipitation, and the 

fΩ parameter also are formally presented and discussed. 

Fourth, example computations and comparison to estimates 

from the regression equations to those of previous studies is 

provided and interpretations are made. Finally, this section 

ends with example computations and discussion about 

considerations for application of the regional equations to 

gaged and ungaged watersheds. 

Regression Equations without OmegaEM 

Parameter 

Nine weighted-least-squares, PRESS-minimized regression equations were computed using the watershed characteristics of drainage area, dimensionless main-channel 

slope, and mean annual precipitation and each of the 638 

trimmed-mean, peak-streamflow estimates for each of the 

nine recurrence intervals. The multi-linear regression was 

made using the lm() function of R Development Core Team 

(2008). The weights for the regression equations were 

derived from the sampling and modeling errors described 

in previous sections. The PRESS-minimization was made 

using a user-guided search algorithm. Separate PRESSminimizations are made and a unique exponent on drainage 

area is determined for each recurrence interval. 

The final regression equations are listed in table 2 with 

important diagnostic statistics. Additional computational 

details for each equation are listed in appendix 2, in which 

potentially informative statistics that should be archived 
12 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

include summary statistics of the residuals, the p-values 

for the coefficients, maximum leverage statistics, and the 

inverted-covariance matrix. Finally, the residuals for each 

station and for each 1-degree quadrangle by recurrence 

interval are available in file Appendix1_residualmaps.pdf

described in appendix 1. 

Several general observations about the equations listed 

in table 2 can be made. First, there are 634 degrees of freedom in each equation—this is a substantially large number 

relative to many other hydrologic equations such as those 

by Asquith and Slade (1997), Asquith (2001), or Asquith 

and Roussel (2007). The mean of the nine residual standard errors is 0.34log10. The residual standard errors are 

generally larger than about 1/3log10, which by the authors’ 

experience is difficult to eclipse in many hydrostatological 

models. The adjusted R-squared values are substantially 

large, which indicate that a reasonably large amount of 

collective variation in peak-streamflow frequency for the 

638 stations is explained by the equations. The exponent 

on drainage area increases with increasing recurrence interval, which suggests that increasing curvature in the relation 

between T-year peak streamflow and drainage area exists 

(Asquith and Thompson, 2005, 2008). 

OmegaEM Parameter 

The fΩ parameter represents a generalized terrain 

and climate index that expresses relative differences in 

peak-streamflow potential across the study area. The 

fΩ parameter is interpreted as an expression of peakstreamflow potential not represented in the watershed 

characteristics of drainage area, dimensionless mainchannel slope, and mean annual precipitation. Maps 

depicting, by 1-degree quadrangle and plotted at the quadrangle center, the fΩ parameter superimposed on maps 

of hill-shade relief, Texas rivers (U.S. Geological Survey, 

2003), and ecoregions (Commission for Environmental 

Cooperation, 1997), respectively, are shown in figures 2–4. 

The fΩ parameter was derived from interpretive analysis of the spatial distributions of residuals from the first 

regression analysis (table 2). The spatial interpretation of 

the residuals, in particular, was focused on those for the 

10-year recurrence interval. The 10-year recurrence interval was chosen because either the PRESS statistic is at a 

minimum or the residual standard error is at a minimum 

(see table 2). This is not a new finding because the authors 

observe similar patterns in other studies (Asquith and Slade, 

1997). 

The authors interpret the minimum (PRESS = 297) at 

the 10-year recurrence interval as representing maximum 

predictability of peak-streamflow potential in relation to the 

three watershed characteristics. In particular, for recurrence 

intervals greater than about 10 years the authors envision 

that such peak streamflows are produced by storms of sufficient size (depth, volume, duration) that the watershed 

characteristics used in this study (dominated by drainage 

area) explain a maximum fraction of the variability of peak 

streamflow. 

As recurrence interval increases beyond 10 years, the 

three watershed characteristics still play a primary role 

in production of peak streamflow; however the increasing magnitude of the PRESS statistics (decreasing adjusted 

R-squared, increasing residual standard error) is explained 

by increasing error in the peak-streamflow frequency estimates because quantile estimations are being made in 

increasingly more distal, right-hand regions of the probability distributions. In other words, regional equations for 

the 10-year recurrence interval represent a situation of some 

sort of maximum of information content. 

The median residuals by 1-degree quadrangle are 

shown in file Appendix1_residualmaps.pdf described in 

appendix 1. These medians subsequently were spatially 

interpreted and further generalized through “moving” 

weighted-mean values—that is, overlapping averaging—of 

two or more neighboring 1-degree quadrangles. When 

weighted means were computed, the number of stations for 

each quadrangle were used as weights. These averaged or 

generalized residuals are plotted in figures 2–4 and labeled 

as the fΩ parameter. The {30–31°, 104–105°} quadrangle 

shows two fΩ values. Because of the mountainous 

region in the approximately northern one-half of the quad, 

separate values were selected as better representation of 

the parameter than a single value. 

The authors explicitly acknowledge that the development of the fΩ parameter represents an ad hoc process and 

that alternative and potentially more optimal means of residual generalization could exist. As shown in the next section, 

the use of the fΩ parameter as another variable in regional 

regression development provides a marked reduction in 

the PRESS statistic with associated increases (decreases) in 

adjusted R-squared and residual standard error. 

Regression Equations with OmegaEM 

Parameter 

Nine weighted-least-squares, PRESS-minimized regression equations were computed using the watershed char­
Regression Equations in Texas Using an L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach 13

107° 106° 105° 104° 103° 102° 101° 100° 99° 98° 97° 96° 95° 94° 93°

36°

35°

34°

33°

32°

31°

30°

29°

28°

27°

26°

−0.232

−0.232 −0.232 −0.232

−0.232 −0.232 −0.070

−0.384 0.169 0.004 −0.413 −0.273 0.197

− 0.286 0.386 0.351 −0.398 0.398 −0.167 −0.134 0.192 0.147 0.140 0.330

− 0.157 0.175 0.211 0.503

−0.102

−0.503

−0.102 − −0.309 0.071 0.253 0.112 0.125 0.013 0.345

− 0.025 0.025 −0.001 −0.200 0.146 − −0.102 0.102 − −0.159 −0.127 0.120

−0.106 0.161 0.258 0.071

− −0.153 −0.104 −0.076 −0.212 0.242 − −0.294 −0.015 0.230

− −0.058 0.175 0.121 0.166

− −0.091 −0.103 −0.083 −0.103 0.144 −0.152 0.152 0.003

− 0.015 0.061 0.236 0.005

− −0.141 −0.157 −0.063 −0.138 0.207 0.042 −0.037 0.138 0.123 −0.027 0.058 0.123

− −0.083 −0.060 −0.006 −0.077 0.089 0.018

−0.127 −0.125 −0.159 −0.054 −0.138 −0.042

0

0

125 250 MILES

200 400 KILOMETERS

Base from Texas Natural Resources Information System digital data

Scale 1:7,920,000

Albers equal-area projection, datum NAD 83

Standard parallels 27°30' and 35°00', latitude of origin 31°00', central meridian -100°00'

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

Figure 2. Hill-shade relief in Texas with superimposed values of OmegaEM parameter that represents a generalized terrain 

and climate index for regionalization of peak-streamflow frequency. 
14 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 107° 106° 105° 104° 103° 102° 101° 100° 99° 98° 97° 96° 95° 94° 93°

36°

35°

34°

33°

32°

31°

30°

29°

28°

27°

26°

−0.232

−0.232 −0.232 −0.232

−0.232 −0.232 −0.070

−0.384 0.169 0.004 −0.413 −0.273 0.197

− 0.286 0.386 0.351 −0.398 0.398 −0.167 −0.134 0.192 0.147 0.140 0.330

− 0.157 0.175 0.211 0.503

−0.102

−0.503

−0.102 − −0.309 0.071 0.253 0.112 0.125 0.013 0.345

− 0.025 0.025 −0.001 −0.200 0.146 − −0.102 0.102 − −0.159 −0.127 0.120

−0.106 0.161 0.258 0.071

− −0.153 −0.104 −0.076 −0.212 0.242 − −0.294 −0.015 0.230

− −0.058 0.175 0.121 0.166

− −0.091 −0.103 −0.083 −0.103 0.144 −0.152 0.152 0.003

− 0.015 0.061 0.236 0.005

− −0.141 −0.157 −0.063 −0.138 0.207 0.042 −0.037 0.138 0.123 −0.027 0.058 0.123

− −0.083 −0.060 −0.006 −0.077 0.089 0.018

−0.127 −0.125 −0.159 −0.054 −0.138 −0.042

0

0

125 250 MILES

200 400 KILOMETERS

Base from Texas Natural Resources Information System digital data

Rivers from U.S. Geological Survey, 2003

Scale 1:7,920,000

Albers equal-area projection, datum NAD 83

Standard parallels 27°30' and 35°00', latitude of origin 31°00', central meridian -100°00'

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

Figure 3. Rivers in Texas with superimposed values of OmegaEM parameter that represents a generalized terrain and climate 

index for regionalization of peak-streamflow frequency. 
Regression Equations in Texas Using an L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach 15

9.4.1

9.4.3

9.4.7

9.4.6

10.2.4

9.5.1

9.6.1

9.4.6

9.4.7

8.3.8

13.1.1

8.3.7

9.4.1

9.4.5

9.4.3

9.4.2

107° 106° 105° 104° 103° 102° 101° 100° 99° 98° 97° 96° 95° 94° 93°

36°

35°

34°

33°

32°

31°

30°

29°

28°

27°

26°

EXPLANATION

Level II Ecoregion

Southeastern USA Plains

South-Central Semi-Arid Prairies

Texas-Louisiana Coastal Plain

Tamaulipas-Texas Semi-Arid Plain

Sonoran and Mohave Deserts

Upper Gila Mountains

Level III Ecoregion Boundaries

Level III Ecoregions

8.3.7 South Central Plains

8.3.8 East Central Texas Plains

9.4.1 High Plains

9.4.2 Central Great Plains

9.4.3 Southwestern Tablelands

9.4.5 Cross Timbers

9.4.6 Edwards Plateau

9.4.7 Texas Blackland Prairies

9.5.1 Western Gulf Coastal Plain

9.6.1 Southern Texas Plains

10.2.4 Chihuahuan Desert

13.1.1 Arizona/New Mexico Mountains −0.232

−0.232 −0.232 −0.232

−0.232 −0.232 −0.070

−0.384 0.169 0.004 −0.413 −0.273 0.197

− 0.286 0.386 0.351 −0.398 0.398 −0.167 −0.134 0.192 0.147 0.140 0.330

− 0.157 0.175 0.211 0.503

−0.102

−0.503

−0.102 − −0.309 0.071 0.253 0.112 0.125 0.013 0.345

− 0.025 0.025 −0.001 −0.200 0.146 − −0.102 0.102 − −0.159 −0.127 0.120

−0.106 0.161 0.258 0.071

− −0.153 −0.104 −0.076 −0.212 0.242 − −0.294 −0.015 0.230

− −0.058 0.175 0.121 0.166

− −0.091 −0.103 −0.083 −0.103 0.144 −0.152 0.152 0.003

− 0.015 0.061 0.236 0.005

− −0.141 −0.157 −0.063 −0.138 0.207 0.042 −0.037 0.138 0.123 −0.027 0.058 0.123

− −0.083 −0.060 −0.006 −0.077 0.089 0.018

−0.127 −0.125 −0.159 −0.054 −0.138 −0.042

0

0

125 250 MILES

200 400 KILOMETERS

Base from Texas Natural Resources Information System digital data

Ecoregions from Commission for Environmental Cooperation, 1997

Scale 1:7,920,000

Albers equal-area projection, datum NAD 83

Standard parallels 27°30' and 35°00', latitude of origin 31°00', central meridian -100°00'

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

Figure 4. Level II and III ecoregions in Texas with superimposed values of OmegaEM parameter that represents a generalized 

terrain and climate index for regionalization of peak-streamflow frequency. 
16 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

Table 2. Summary of weighted-least-squares, PRESS-minimized, regional regression equations using drainage area, dimensionless main-channel slope, and mean annual precipitation. 

[RSE, residual standard error in log10-units of cubic feet per second; Adj., adjusted; AIC, Akaike Information Criterion; PRESS, PRediction Error Sum of Squares; QT , peak streamflow for T-year recurrence interval in cubic feet per second; P, mean annual precipitation in 

inches; S, dimensionless main-channel slope; and A, drainage area in square miles] 

Regression equation RSE Adj. R-squared AIC statistic PRESS statistic 

Q2 = P1.562 S0.385 ×10[38.64−37.94A−0.0080 ] 0.32 0.81 396 77.4 

Q5 = P1.491 S0.500 ×10[16.35−15.04A−0.0228 ] .30 .84 314 65.2 

Q10 = P1.377 S0.530 ×10[13.67−11.99A−0.0299 ] .30 .85 297 63.7 ← minimum 

Q25 = P1.316 S0.574 ×10[12.00−9.992 A−0.0380 ] .31 .84 348 67.1 

Q50 = P1.285 S0.608 ×10[11.40−9.195 A−0.0429 ] .33 .83 414 73.5 

Q100 = P1.257 S0.640 ×10[11.07−8.673 A−0.0470 ] .35 .82 493 82.8 

Q200 = P1.221 S0.665 ×10[10.86−8.282 A−0.0506 ] .37 .80 586 95.3 

Q250 = P1.209 S0.676 ×10[10.83−8.181 A−0.0517 ] .38 .79 617 100 

Q500 = P1.181 S0.705 ×10[10.67−7.846 A−0.0554 ] .41 .77 713 117 

acteristics of drainage area, dimensionless main-channel 

slope, mean annual precipitation, and fΩ (figs. 2–4) as well 

as each of the 638 trimmed-mean, peak-streamflow estimates for each of the nine recurrence intervals. The weights 

for the regression equations were the same as those used for 

the regression equations listed in table 2. Separate PRESSminimizations were made for each recurrence interval and 

a unique exponent on drainage area was determined. These 

exponents differ from those listed in table 2. 

The final regression equations are listed in table 3 with 

important diagnostic statistics. Additional computational 

details for each equation are listed in appendix 3, in which 

potentially informative statistics that should be archived 

include the summary statistics of the residuals, p-values for 

the coefficients, maximum leverage statistics, and invertedcovariance matrix. The residuals for each station and for 

each 1-degree quadrangle for each recurrence interval are 

available in file Appendix1_residualmaps.pdf. 

Several general observations about the equations listed 

in table 2 can be made and are similar to those made 

for the equations listed in table 3. The mean of the nine 

residual standard errors is 0.30log10. The mean adjusted 

R-squared is about 0.86, and the mean percentage reduction in PRESS is about −21 percent. The residual standard 

errors are arguably smaller than 1/3log10, which was a professional goal of the authors for the Approach. The adjusted 

R-squared values are substantially large, which indicate a 

reasonably large amount of variation in peak-streamflow 

frequency for the 638 stations is explained. The exponent 

on drainage area increases with increasing recurrence interval, which suggests that increasing curvature in the relation 

between T-year peak streamflow and drainage area exists 

(Asquith and Thompson, 2005, 2008). 

Of the two suites of regional equations shown in this 

report, the authors prefer the use of those based on fΩ . Further, by noting the spatial dependence of residuals for the 

equations in table 2 (equations not based on fΩ ) and, by 

generality, such dependence for the equations in Asquith 

and Thompson (2005, 2008) must exist, the authors conclude that equations in table 3 are preferred over those in the 

two reports (Asquith and Thompson) because the equations 

in table 3 generally lack substantial spatial dependency in 

their residuals. 

Example Computations for a Single Station 

and Comparison to Estimates from Previously Published Equations 

This section provides example computations that 

demonstrate how the regional regression equations 

(tables 2 and 3) could be used in practice, and a comparison to estimates from previously published equations. 

The focus of the computations is on the 100-year peak 

streamflow, but application to other recurrence intervals 

is straightforward. For the computations, the watershed 

represented by station 08190000 Nueces River at Laguna, 

Tex., was arbitrarily chosen. 

The suggested equation for estimation of the 100-year 

peak streamflow (see table 3) has the form 

Ω +a+bAλ ] Q100 = Pc Sd ×10[e� , (3) 
Regression Equations in Texas Using an L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach 17 

Table 3. Summary of weighted-least-squares, PRESS-minimized, regional regression equations using drainage area, dimensionless main-channel slope, mean annual precipitation, and OmegaEM. 

[RSE, residual standard error in log10-units of cubic feet per second; Adj., adjusted; AIC, Akaike Information Criterion; PRESS, PRediction Error Sum of Squares; Percent change, percent change from PRESS in table 2 to PRESS listed to the left; QT , peak streamflow for 

T-year recurrence interval in cubic feet per second; P, mean annual precipitation in inches; S, dimensionless main-channel slope; �Ω ,

OmegaEM parameter in figures 2–4; and A, drainage area in square miles] 

Regression equation RSE Adj. R-squared AIC statistic PRESS statistic Percent change 

Ω +50.98−50.30A−0.0058 

= ] P1.398 S0.270 ×10 Q [0.776� 2 0.29 0.84 273 64.6 −16.5 

Ω +16.62−15.32A−0.0215 

= ] P1.308 S0.372 ×10 Q [0.885� 5 .26 .88 122 49.1 −24.7 

Ω +13.62−11.97A−0.0289 ] Q10 = P1.203 S0.403 ×10[0.918� .25 .89 86.5 46.6 −26.8 

Ω +11.79−9.819A−0.0374 ] Q25 = P1.140 S0.446 ×10[0.945� .26 .89 140 49.5 −26.2 

Ω +11.17−8.997A−0.0424 ] Q50 = P1.105 S0.476 ×10[0.961� .28 .87 220 55.6 −24.4 

Ω +10.82−8.448A−0.0467 

= ] P1.071 S0.507 ×10 Q [0.969� 100 .30 .86 320 64.8 −21.7 

Ω +10.61−8.058A−0.0504 

= ] P1.034 S0.531 ×10 Q [0.975� 200 .33 .84 436 77.2 −19.0 

Ω +10.56−7.943A−0.0516 

= ] P1.021 S0.541 ×10 Q [0.977� 250 .34 .83 474 81.9 −18.1 

Ω +10.40−7.605A−0.0554 

= ] P0.988 S0.569 ×10 Q [0.976� 500 .37 .81 591 98.7 −15.6 

where a, b, c, d, and e are regression coefficients (listed 

in this sentence in the order as produced by the R software) specific for the 100-year recurrence interval, λ is 

a power determined by iterative PRESS-minimization for 

the 100-year recurrence interval, P is mean annual precipitation in inches, S is dimensionless main-channel slope, 

A is drainage area in square miles, and the fΩ parameter is 

a generalized terrain and climate index (figs. 2–4), which 

was derived from the residuals of the 10-year equation in 

table 2. 

For the example computations, the 100-year regression 

equation (table 3) is 

= P1.071 S0.50710[0.969� Q100 

Ω +10.82−8.448A−0.0467]

, (4) 

for which the residual standard error is 0.30log10 and 

adjusted R-squared is 0.86. 

Station 08190000 has a latitude and longitude of 

29°25'42" and 99°59'49", respectively, with the following 

estimates from Asquith and Slade (1997, table 1) of 

the three watershed characteristics: P = 24.5 inches, 

S = 0.00326 (or about 17.21 feet per mile), A = 737 square 

miles. The value of fΩ = 0.33 is derived from figures 2–4 

given the latitude and longitude of the station. 

Solving the equation for Q100 produces an estimate of 

Ω about 145,000 cubic feet per second (Q�

100). Using the Q100 

equation without the fΩ parameter in table 2, the Q100 is 

about 73,400 cubic feet per second (QAR 

100). For reference, 

the station-specific, trimmed-mean, Q100 streamflow for 

this station is about 302,000 cubic feet per second (QLM 

100). 

Using the Asquith and Thompson (2005, table 5; 2008, 

table 7) equation for Q100, which is 

= P0.9883 S0.6295 ×107.307−6.714A−0.0601

Q100 , 

and solved for the watershed characteristics 

= 24.50.9883 ×17.210.6295 Q100 (5) 

×107.307−6.714×737−0.0601 

, 

or about 139,000 cubic feet per second (QAT 

100). 

Asquith and Slade (1997, table 1) report for station 

08190000 that the Q100 streamflow was then (circa 1997) 

estimated as 336,000 cubic feet per second (Q17B 

100 ). This 

station resides in region 5 of Asquith and Slade (1997) and 

the regional equation provided by those authors (Asquith 

and Slade, 1997, table 2) is Q100 = 9180 A0.594 H−0.420 , 

which when solved for the watershed characteristics5 is 

Q100 = 9180 ×7370.594 ×5.8−0.420 or about 223,000 cubic 

feet per second (QAS 

100). 

A discussion of the example computations for the arbitrarily selected watershed follows; unfortunately, the scope 

of the investigation is far too large for an effective discussion to be provided for other watersheds. However, the 

discussion is intended to help guide analysts in making 

comparisons between various methods of peak-streamflow 

frequency estimation. 

5The basin-shape factor is H = 5.80 as reported by Asquith and Slade (1997, 

table 1). 
18 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

The example computations can be summarized according to two different analytical tracks. Each track is summarized in separate itemized lists: 

ANALYTICAL TRACK 1—For the results using stationspecific, peak-streamflow frequency based on broad interpretation of Interagency Advisory Committee on Water 

Data (1982) made by Asquith and Slade (1997), the three 

Q100 estimates are: 

• The station-specific Q17B 

100 estimate from Asquith and 

Slade (1997, table 1) using data through about 1994 

for the station is about 336,000 cubic feet per second; 

• The regional-equation QAS 

100 estimate from Asquith 

and Slade (1997, table 2) is about 223,000 cubic feet 

per second; and 

• The regional-equation QAT 

100 estimate from alternative interpretation of the regional regression method 

based on PRESS minimization, drainage area, dimensionless main-channel slope, and mean annual precipitation by Asquith and Thompson (2005, 2008) is 

about 139,000 cubic feet per second. 

ANALYTICAL TRACK 2—For the results using 

station-specific, peak-streamflow frequency based on the 

L-moment approach described in this report, the Q100 

estimates are: 

• The station-specific QLM 

100 estimate from the current 

investigation by the trimmed mean of the seven distributions using data through about 2006 for the station 

is about 302,000 cubic feet per second; 

• The regional-equation QAR 

100 estimate (table 2) from 

the current investigation using the equation lacking 

the fΩ parameter is about 73,400 cubic feet per second; and 

Ω • The regional-equation Q�

100 estimate (table 3) from 

the current investigation using the equation with the 

fΩ parameter is about 145,000 cubic feet per second. 

The computations of the first analytical track can be 

compared or interpreted in several ways. First, QAS 

100 and 

QAT are each smaller than the station-specific estimate 

of Q17B Hence, the watershed represented by station 100 . 

08190000 has larger peak-streamflow frequency relative to 

many other stations in either region 5 (QAS 

100) or statewide 

(QAT 

100). The QAS The QAS 

100 estimate is larger than QAT 

100. 100 

is closer to the Q17B than QAT 

100 is. These differences can 100 

be attributed in part to the many fewer degrees of freedom 

for the regressions used in region 5 of Asquith and Slade 

(1997) than in Asquith and Thompson (2008). 

The computations of the second analytical track also 

can be interpreted in several ways. First, QAR Ω

100 and Q�100 are 

each less than QLM 

100. Hence, again, it can be concluded that 

the watershed represented by station 08190000 has greater 

peak-streamflow frequency relative to many other stations. 

The differences between the estimates appear substantial 

but can be generally said to be within about 0.40log10. 

The conclusion that the station has greater peakstreamflow frequency than estimated by the regional 

models considered can be stated in another way—the 

peak-streamflow data for this station appear to represent 

comparatively greater peak values for the watershed characteristics relative to similarly characterized watersheds 

in the study area. In fact, many of the watersheds in the 

general region of Texas containing the station are well 

known to produce some of the largest peak-streamflow 

values for their respective drainage areas in Texas as well 

as the nation (Burnett, 2008). The fΩ maps in figures 2–4 

clearly support this conclusion by the positive values for 

the fΩ parameter. 

Continuing, the values for Q17B and QLM 

100 are within 100 

0.05log10 of each other—some validation of the basic computational steps used in this report thus is provided. This is 

an important observation, which is generalized in the next 

section. 

Comparison of Regional Equations to Those 

of Previous Studies 

A comparison of the regional equations reported in 

tables 2 and 3 to those of previous studies is made in this 

section. Two substantially important questions must be 

asked to further frame the comparisons and are answered 

in succession in the following two sections: 

1. How are the station-specific, peak-streamflow estimates based on the trimmed mean of the seven distributions fit to the sample L-moments comparable 

and applicable to more established estimates based 

on broad interpretation of Interagency Advisory Committee on Water Data (1982) and documented by 

Asquith and Slade (1997)? 

2. Should the fΩ equations in table 3 (authors prefer 

those in table 3 over those lacking fΩ in table 2) 

be preferred over equations in Asquith and Slade 

(1997)? Asquith and Thompson (2005, 2008) did 

100 
Regression Equations in Texas Using an L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach 19 

not directly ask this question of their equations in 

comparison to those of Asquith and Slade (1997), but 

only identify the Asquith and Thompson equations as 

“alternative.” 

Comparability and Applicability of PeakStreamflow Estimates from OmegaEM and 

Previous Equations 

To facilitate the comparisons, the watershed characteristics of the 638 stations will be used to make computations. 

To explore the answer to the first question, the relations 

between estimates of peak streamflow for the 2-, 5-, 10-, 

25-, 50-, and 100-year recurrence intervals using the equations in table 2 and the equations in Asquith and Thompson 

(2005, table 5) are shown in figure 5. The vertical axis in 

the plots represents the regional model-derived features of 

the Approach including L-moments, weights based on sampling and modeling errors, and PRESS minimization. The 

horizontal axis in the plots represents a similarly parameterized regional model by Asquith and Thompson (2005) 

using equivalent years of record as weights (Asquith and 

Slade, 1997) based on broad interpretation of the Interagency Advisory Committee on Water Data (1982) guidelines. An equal value line is superimposed on the plots. 

By inspection of figure 5, the estimates are close to 

parallel—that is, show remarkable similarity—to the equal 

value line and generally show increasing divergence (offset 

to the right) as recurrence interval increases. 

An encompassing conclusion can be drawn, but 

unfortunately it is dependent on technical familiarity with 

peak-streamflow frequency analysis. The conclusion is 

that the use of L-moments, multiple distributions, and 

trimmed mean produces a regional model (table 2) of 

peak-streamflow frequency that is congruent6 with the 

regional model (Asquith and Thompson, 2005, 2008) having a foundation on broad interpretation of the Interagency

Advisory Committee on Water Data (1982) guidelines. 

Several natural extensions to this conclusion can be drawn 

and address long-held research questions of the senior 

author (Asquith) that date back to the senior author’s lead 

in application of the Interagency Advisory Committee on 

Water Data (1982) guidelines for the Asquith and Slade 

(1997) report. These extensions are: 

1. LOW-OUTLIER THRESHOLDS—The authors of 

Asquith and Slade (1997) had a first priority to use 

6Congruent—in agreement or harmony. 

a custom equation7 to estimate low-outlier thresholds, which have the effect of adjusting the fit of the 

log-Pearson Type III distribution. For many fits of the 

distribution, the authors then abandoned their own 

equation and selected low-outlier thresholds in an ad 

hoc fashion to improve the visual fit of the distribution. The use of low-outlier thresholds to make the 

procedures of Interagency Advisory Committee on 

Water Data (1982) work for Texas data was judged 

critical for Asquith and Slade (1997) to produce reliable estimates. 

The effect of the low-outlier threshold often is to 

twist or rotate the fitted distribution to acquire generally reduced values of peak streamflow in the right 

tail of the distribution. The senior author (Asquith) 

and associates have long held concerns about the 

arbitrary and capricious application of a low-outlier 

threshold. Yet, Texas peak-streamflow analysis generally requires low-outlier treatment when computations occur within the Interagency Advisory Committee on Water Data (1982) framework. The use of 

the L-moments, which avoid log10 transformation of 

the data and the resultant magnification of low outliers on peak-streamflow frequency relations can be 

thought of as a technique to avoid low-outlier treatment altogether; 

2.HIGH OUTLIERS AND HISTORICAL RECORD—The 

Interagency Advisory Committee on Water Data 

(1982) guidelines provide considerable emphasis on 

refinement of statistical computations using historical 

information. For example, a historical peak streamflow is a peak that has a historically documented 

nonexceedance interval greater than the period of 

record represented by the station. Such historical 

information can be difficult to interpret but can also 

enhance peak-streamflow estimation at particular stations. For the use of the L-moments as implemented 

in the Approach, such adjustment was not performed. 

The effect of historical adjustment, when available, 

is to dilate or stretch the far right-hand side (large 

recurrence interval) of the peak-streamflow relation 

and thus generally reduce values of peak streamflow. 

A conclusion, based on the fact that the two regional 

models in figure 5 are so similar, is that historical 

peak-streamflow information as represented for the 

study area through Asquith and Slade (1997) and 

7The equation is log10(L) = 1.09μ − 0.584σ + 0.14γ − 0.799, where L is the 

low-outlier threshold in cubic feet per second; and μ, σ, and γ are the respective 

product moment mean, standard deviation, and skew of the annual peak streamflow 

for a given station in log10(cubic feet per second) with exception of γ (dimensionless). 
20 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

20

50

100

200

500

1,000

2,000

5,000

10,000

20,000

50,000

100,000

200,000

300,000

PEAK STREAMFLOW FROM L-MOMENT-BASEDQT , PRESS-MINIMIZED EQUATIONS, IN CUBIC FEET PER SECOND

20

50

100

200

500

1,000

2,000

5,000

10,000

20,000

50,000

100,000

200,000

300,000

PEAK STREAMFLOW FROM 17B-BASED-QT , PRESS-MINIMIZED EQUATIONS BY AT2005, IN CUBIC FEET PER SECOND

20 100 1,000 10,000 100,000

20

50

100

200

500

1,000

2,000

5,000

10,000

20,000

50,000

100,000

200,000

300,000

20 100 1,000 10,000 100,000

2-year recurrence interval 5-year recurrence interval

10-year recurrence interval 25-year recurrence interval

50-year recurrence interval 100-year recurrence interval

EQUAL VALUE LINEEQUAL VALUE LINE

EQUAL VALUE LINEEQUAL VALUE LINE

EQUAL VALUE LINEEQUAL VALUE LINE

Figure 5. Comparison by recurrence interval of peak-streamflow frequency estimates for the 638 stations using equations 

listed in table 2 (vertical axis) and using equations originally shown in Asquith and Thompson (2005, table 5) (horizontal axis, 

AT2005) that are based on station-specific, peak-streamflow frequency based from broad interpretation of Interagency Advisory 

Committee on Water Data (1982) (17B). 
Regression Equations in Texas Using an L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach 21 

references therein provides little overall information 

to the regional model of peak-streamflow frequency 

for the study area. The authors consider this conclusion to be consistent with the opinion of Hosking and 

Wallis (1997, p. 161), who, although in the context of 

a structurally different regional model (index-flood 

procedure), are “skeptical about the practical utility 

of historical information;” and 

3. GENERALIZED SKEW—For the use of the 

L-moments, the skewness (and kurtosis) of the 

distribution were estimated solely on the data on 

a per station basis. There was no treatment for a 

spatially varying parameter (generalized skew) 

supposedly representing distribution symmetry (Judd 

and others, 1996). The authors interpret, should 

generalized skew exist, that the influence of this 

skew as well as other characteristics influencing 

peak-streamflow potential of the watershed have been 

lumped into the fΩ parameter. 

On further interpretation of figure 5, it is seen that 

the divergence from the equal value line is offset to the 

right as opposed to the left. This suggests that the Asquith 

and Thompson (2005) equations (those based on Asquith 

and Slade [1997] results) produce larger estimates of peak 

streamflow, up to about 0.10log10, more than those derived 

from table 2. Thus, the equations in table 2 produce 

slightly smaller estimates. The authors ask further questions: Are the Asquith and Slade (1997) estimates of 

station-specific, peak-streamflow frequency then too large? 

Are the L-moment estimates too low? Are the differences related to sample size? The apparently larger values occur even with the considerable treatment of low outliers and high outliers in the Asquith and Slade (1997) 

analysis. However, the differences could be thought of 

as slight. Specifically, what differences in station-specific, 

peak-streamflow frequency can be attributed to the addition of more record between about 1994 and 2006 for some 

stations? The two suites of regional equations represented 

in figure 5 differ slightly by degrees of freedom—that is, 

stations included in the regression analysis—does this contribute to the divergence (or accidental convergence) in the 

figure? The answers to these questions are not known and 

exploration is beyond the scope of this report. 

Preference for OmegaEM Equations over 

Previous Equations 

The discussion of the first question provides an authoritative account that the trimmed mean L-moment-based estimates of station-specific, peak-streamflow frequency are 

reliable. 

The regional regression equations listed in this report 

that are preferred by the authors are the fΩ equations listed 

in table 3 and thus use of the fΩ parameter is suggested. 

These equations should be preferred over the equations 

listed in table 2 as justified in part by the large percentage 

reduction in the PRESS statistics. 

To have some parallelism with the discussion in the 

previous section, a comparison of the 638 estimates from 

the fΩ equations to those in Asquith and Thompson (2005, 

table 5) is made in figure 6. The data points scatter around 

the equal value line. In part, this shows that the inclusion 

of the fΩ parameter does not have a deleterious8 effect on 

the regression analysis. The scatter does not represent and 

should not be interpreted as a lack-of-fit, but rather demonstrates that the fΩ parameter has the effect of pulling the 

data points in the vertical direction according to the sign of 

the parameter. 

Perhaps the most important question, which will 

undoubtedly be asked of the authors and many of the 

acknowledged colleagues, deserves repeating: Should the 

fΩ	 equations in table 3 be preferred over equations in 

Asquith and Slade (1997)? 

The authors suggest that the “OmegaEM equations” 

listed in table 3 should be preferred over the equations and 

procedures of Asquith and Slade (1997) for the following 

reasons and supporting ancillary discussion: 

1. The fΩ equations have solutions for 200-, 250-, and 

500-year recurrence intervals, which are not available

in Asquith and Slade (1997); 

2. The fΩ equations have nearly 18 times more degrees 

of freedom than those in Asquith and Slade (1997). 

The equations would be expected to have broader 

applicability across a wider range of watersheds; 

3. There are six fΩ equations compared to 96 in 

Asquith and Slade (1997)—the three greater than 

100-year recurrence intervals in table 3 cannot be 

included in this particular comparision; 

4. The use of L-moment statistics for the fΩ equations 

has removed: 

(a)the largely ad hoc process of low-outlier adjustment to station-specific, peak-streamflow frequency estimates; 

8Deleterious—causing harm or damage. 
22 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

20

50

100

200

500

1,000

2,000

5,000

10,000

20,000

50,000

100,000

200,000

300,000

PEAK STREAMFLOW FROM L-MOMENT-BASEDQT , PRESS-MINIMIZED,

—Ω -CORRECTED EQUATIONS, IN CUBIC FEET PER SECOND

20

50

100

200

500

1,000

2,000

5,000

10,000

20,000

50,000

100,000

200,000

300,000

PEAK STREAMFLOW FROM 17B-BASED-QT , PRESS-MINIMIZED EQUATIONS BY AT2005, IN CUBIC FEET PER SECOND

20 100 1,000 10,000 100,000

20

50

100

200

500

1,000

2,000

5,000

10,000

20,000

50,000

100,000

200,000

300,000

20 100 1,000 10,000 100,000

2-year recurrence interval 5-year recurrence interval

10-year recurrence interval 25-year recurrence interval

50-year recurrence interval 100-year recurrence interval

EQUAL VALUE LINEEQUAL VALUE LINE

EQUAL VALUE LINEEQUAL VALUE LINE

EQUAL VALUE LINEEQUAL VALUE LINE

Figure 6. Comparison by recurrence interval of peak-streamflow frequency estimates for the 638 stations using equations 

listed in table 3 (vertical axis) and using equations originally shown in Asquith and Thompson (2005, table 5) (horizontal axis, 

AT2005) that are based on station-specific, peak-streamflow frequency based from broad interpretation of Interagency Advisory 

Committee on Water Data (1982) (17B). 
Regression Equations in Texas Using an L-moment-Based, PRESS-Minimized, Residual-Adjusted Approach 23 

(b) the need to use the sometimes difficult-tointerpret or inconsistent station-to-station historical record; and 

(c)the concept of generalized skew, at least as 

implemented in the Approach. 

5. The fΩ equations generally have similar residual standard errors and adjusted R-squared values as those in 

Asquith and Slade (1997), but more importantly, the 

fΩ equations have minimal PRESS statistics; 

6. The minimization of PRESS for the fΩ equations 

should be expected to have enhanced extrapolation 

characteristics to small watersheds (Asquith and 

Thompson, 2005, 2008); 

7. The maps in figures 2–4 of the fΩ parameter supplant 

the need for separate and sometimes arbitrary regions 

of Asquith and Slade (1997). Although it certainly 

can be argued that the 1-degree quadrangles of the 

fΩ parameter have themselves become “microregions of arbitrary size and extent,” the authors fully 

acknowledge that complications and ambiguities to 

the selection of the fΩ will arise. The authors conclude that the smoother regional structure of the map 

relative to the regions of Asquith and Slade (1997) 

elucidates9 previously suspected, but inadequately 

documented, relative peak-streamflow potential in 

Texas. All other variables being equal (drainage area, 

dimensionless main-channel slope, mean annual precipitation), peak-streamflow potential is relatively 

less in far east Texas and far west Texas, greater 

in a swath through much of the central part of the 

state (the region demarked by positive fΩ values), 

and highest along the Balcones escarpment in southcentral Texas. O’Connor and Costa (2003, p. 9) identify this region (Balcones escarpment) of the nation 

as having “concentrations of large floods.” 

In conclusion, the fΩ equations are expected to be reliable estimators of peak-streamflow frequency for undeveloped and ungaged stream locations in Texas with watershed 

characteristics within the distribution of those characteristics used to develop the equations. Although the fΩ parameter requires interpretation on the part of analysts and the 

potential exists that different analysts could estimate different values for a given watershed, the authors suggest 

that typical uncertainty in the fΩ estimate might be about 

±0.10log10, which is small relative to other uncertainties, such as those measured by residual standard error, in 

9Elucidates—makes (something) clear; explains. 

hydrostatological models. Finally, given the two ensembles 

of equations reported herein and those in previous reports, 

hydrologic design engineers and other analysts in Texas 

now have several different methods, which were derived 

from differing analytical tracks, to make comparisons of 

peak-streamflow frequency estimates for ungaged stream 

locations in Texas. 

Considerations for Application of Regression Equations to Gaged and Ungaged 

Watersheds 

There are three remaining aspects of regional equation 

use in practice that require discussion: (1) application to 

“gaged” watersheds, (2) practice of watershed subdivision, 

and (3) estimation of prediction limits. 

The first aspect is that the authors observe that the 

equations reported herein will occasionally be applied at 

stream locations coincident with an operational streamflowgaging station or locations with historical annual peakstreamflow data. Further these locations could also have 

peak-streamflow frequency estimates from combinations of 

synthetic or calibrated rainfall and runoff models or other 

hydrologic methods. Procedures or suggestions for combining the various estimates of peak-streamflow frequency 

from two or more methods explicitly are not provided in 

this report. 

The second aspect has not been previously discussed in 

either Asquith and Slade (1997), Asquith (2001), Asquith 

and Thompson (2005), or Asquith and Thompson (2008); 

although the authors have had considerable communication with practitioners on the following topic. Occasionally, 

some users, who often are concerned with equation applicability for “unusual” watersheds, (1) subdivide watersheds 

into one or more subbasins of more “typical” characteristics relative to those used to develop the equations, (2) compute peak-streamflow frequency from regional regression 

equations for each ungaged subbasin, and (3) then combine the estimated values by summation. The authors suggest that users are seeking to enhance prediction accuracy 

by reducing extrapolation to watershed characteristics not 

well represented in the equations. (This is a particular issue 

with the limited degree-of-freedom equations in Asquith 

and Slade[1997].) Such a practice is not optimal. 

The authors observe that annual peak-streamflow 

data from stations represent aggregate or integrated peakstreamflow potential (information) of whole watersheds. 

The regression process seeks to explain the variability of 

peak-streamflow frequency for whole watersheds and not 
24 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

the distributed processes within subbasins. The authors 

suggest that the summation explicitly assumes that the 

joint probability of contemporaneous peak-streamflow 

occurrence within each subbasin is 100 percent. Logical 

extension suggests that the practice of subdivision could 

yield peak-streamflow frequency curves that are generally 

too large and hence poorly reflect the risk level as specified 

by the T-year recurrence interval. 

The third aspect is that the prediction limits can be 

useful for expressing uncertainty when a equation is used. 

A 100 ×(1−α) prediction interval for QT is given by the 

probability 

Pr{↓

QT ≤ QT ≤ ↑QT } ≥ 1−α, (6) 

where the down (up) arrow signifies the lower (upper) prediction limit for QT at the α significance level. The lower 

and upper limits are computed by 

 [QT ]

= 10log10(QT )−t[α/2,df] σ[QT ] ↓Q 1+ho T , and (7) 

 [QT ]

= 10log10(QT )+t[α/2,df] σ[QT ] ↑Q 1+ho T , (8) 

where QT is the prediction from a T-year equation, t[α/2,df]

is the upper tail of the t-distribution for df[QT ] degrees of 

freedom at the α significance level, σ[QT ] is the residual 

standard error of the T-year equation, and h

[

o

QT ]

is the leverage of the prediction for the watershed. The h

[

o

QT ]

requires 

matrix multiplication—the requisite matrixes are listed 

in appendixes 2 and 3 under the INVERTED-COVARIANCE

MATRIX headings. 

The authors observe that a rapid method for estimation of prediction limits can be conceived because of the 

large degrees of freedom for the regressions. The rapid 

method is more straightforward than the method involving 

matrix computation described in Asquith and Slade (1997) 

and mathematically outlined in Asquith and Roussel (2007). 

Other observations follow. First, the number of stations in 

the equations is n = 638 and the number of parameters (variables plus 1 for the regression intercept) are either p = 4 

(table 2) or p = 5 (table 3). Second, the average leverage for 

a regression is about p/n so for this investigation, this ratio 

is less than 0.008. The authors suggest that h

[QT ]

≈ 0.008. √ o

Thus, the term 1+0.008 ≈ 1. Third, again because the 

degrees of freedom (df = n− p) is large, t[α/2,df] can be well 

approximated by the quantile function10 of the standard nor10The quantile function of the standard normal distribution for nonexceedance 

probability F could be approximated for the purposes of this report by Q(F) =

5.063[F0.135 −(1−F)0.135]. 

mal distribution N(F) for nonexceedance probability F. 

= 10log10(QT )−N(1−α/2) σ[QT ] ↓

QT , and (9) 

= 10log10(QT )+N(1−α/2) σ[QT ] ↑

QT . (10) 

The rapid method is demonstrated in the following 

example. Suppose that the 90th-percentile prediction limits need rapid estimation for the 100-year peak streamflow 

estimate by the respective equation in table 3. The estimate 

is QT = 9,000 cubic feet per second and σ[Q100] = 0.30. 

The N(0.95) = 1.6 where 0.95 = (1 − α/2) = (1 − (1 −

0.90)/2) for the 90th-percentile (two-tailed) limits. (In R

code, the quantity 1.6 is qnorm(0.95).) As a result, the 

90th-percentile prediction limits are estimated as 

= 10log10 ↓ (9,000)−1.6×0.30 QT = 2,980, and (11) 

= 10log10 ↑ (9,000)+1.6×0.30 QT = 27,200. (12) 

Thus, the prediction limits for the 100-year peak streamflow 

estimate of Q100 = 9,000 cubic feet per second can be written as 2,980 ≤ Q(90th) ≤ 27,200 cubic feet per second. The 100 

authors, however, suggest the more compact notation of 

Q(90th)

100 = 9,000 ±0.48log10 cubic feet per second be used 

instead where the quantity 0.48 is 1.6×0.30. Finally, the 

authors observe that prediction limits constructed as shown 

will be generally too small (narrow) because of inherent 

uncertainties of the fΩ parameter, which are difficult to 

propagate through the preceding computations. 

Summary 

Annual peak-streamflow frequency estimates are 

needed for flood-plain management; for objective assessment of flood risk; for cost-effective design of dams, levees, 

and other flood-control structures; and for design of roads, 

bridges, and culverts. Annual peak-streamflow frequency 

represents the peak streamflow for nine recurrence intervals 

of 2, 5, 10, 25, 50, 100, 200, 250, and 500 years. Common 

methods for estimation of peak-streamflow frequency for 

ungaged or unmonitored stream watersheds are regression 

equations for each recurrence interval developed for one 

or more regions (not strictly geographic); such regional 

equations are the subject of this report. The method is 

based on statistical analysis of annual peak-streamflow data 

from U.S. Geological Survey (USGS) streamflow-gaging 

stations (stations). 

Beginning in 2007, the USGS, in cooperation with 

the Texas Department of Transportation and in partnership 

with Texas Tech University, began a 3-year investigation 
concerning the development of regional equations to estimate annual peak-streamflow frequency for undeveloped 

watersheds in Texas. The general approach was explicitly 

limited to the use of L-moment statistics, which are used 

in conjunction with a technique of multi-linear regression 

referred to as PRESS minimization. The approach used to 

develop the regional equations, which was refined during 

the investigation, is referred to as the “L-moment-based, 

PRESS-minimized, residual-adjusted approach.” The study 

area for this investigation includes Texas and selected parts 

of neighboring states and essentially is the same as that 

considered in previous studies. 

The primary purpose of this report is to present 

regression equations from the L-moment-based, PRESSminimized, residual-adjusted approach for estimation 

of annual peak-streamflow frequency for undeveloped 

watersheds in Texas and primarily restricted to ungaged 

watersheds. 

The primary scope of the report is limited to the 

annual peak-streamflow data for 677 and 638 selected 

stations. The scope of the report also is limited to three 

selected watershed characteristics, which have previously 

been shown as important predictors of peak-streamflow 

frequency in Texas. The watershed characteristics are 

drainage area, dimensionless main-channel slope, and 

mean annual precipitation. 

From an initial candidate station count of 1,030 stations that had 1 or more years of annual peak-streamflow 

data in the study area, the number of stations used for 

this report were reduced by review of the data, watershed 

conditions, reliability of watershed characteristics, and a 

minimum record-length criteria (8 years), leaving 677 stations. The data for these 677 stations are used for a specific 

component of the analysis. Another assessment or restriction for the purpose of the regression analysis is that stations with drainage areas less than 10,000 square miles are

used. After further assessment of watershed characteristics 

and the less-than-10,000-square-mile criterion, 638 stations 

remained. 

The L-moment-based, PRESS-minimized, residualadjusted approach (Approach) used for this investigation is 

complex and highly technical. Furthermore, the Approach 

relied on numerous single-purpose, highly specialized 

scripts, computer programs, and integration with features 

of the host-operating system as well as considerable 

judgement and iterative refinement on the part of the 

authors. 

L-moments are used for station-specific computations 

of peak-streamflow frequency with no consideration for 

concepts such as low-outliers, high-outliers, historical inforSummary 25 

mation, and generalized skew as described in other methods for peak-streamflow frequency computation. For the 

Approach, the first five sample L-moments were computed 

for each of the 677 stations. These L-moments, respectively, are the mean, L-scale, L-skew, L-kurtosis, and Tau5. 

For the weighted-least-squares regression analysis, 

weight factors representing sampling error and modeling 

errors as distinct components of uncertainty are used. It 

is favorable to have stations with more record (smaller 

sampling error, more certainty) to have more weight in 

regression analysis. 

Sampling error estimates were based on dimensionless 

regional L-moments, which are weighted-mean values for 

the study area, and these weighted-mean values provide 

an approximation of the geometry of the “parent” peakstreamflow frequency distribution for the study area. The 

dimensionless regional L-moments were used to estimate 

a dimensionless regional Wakeby distribution. This fitted 

Wakeby distribution is assumed to provide a reasonable 

first-order approximation to the underlying (dimensionless) 

structure of peak-streamflow frequency for the study area. 

For this report, as many as seven probability distributions are fit to the sample L-moments for each station. The 

authors explicitly chose to avoid selection of a single form 

of a probability distribution to model the station-specific, 

peak-streamflow frequency through use of a substantial 

number of three-parameter and more distributions. For 

almost all stations, seven distributions were fit and seven 

estimates of peak streamflow for each of the nine recurrence 

intervals were computed. 

Because seven distributions were fit for each station, 

seven unique values of peak streamflow were obtained for 

each recurrence interval. At a given recurrence interval, 

these values have a distribution; the relative variability of 

this distribution is treated as a measure of modeling error. 

It is favorable for stations with more variability in peakstreamflow frequency estimates by distribution to have less 

weight in regression analysis because more uncertainty 

exists at such stations. 

For each of the nine recurrence intervals, a symmetrically trimmed L-moment (trimmed mean) was computed 

from each of seven unique values of peak streamflow for 

a given recurrence interval. The use of trimming, which 

computationally is more complex than simply dropping the 

largest and smallest values and computing an arithmetic 

mean, is anticipated to provide a more robust measure of 

central location for the formal estimation of station-specific, 

peak-streamflow frequency. The trimmed mean was used 

to estimate the station-specific, peak streamflow for each of 

the nine recurrence intervals. 
26 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

For this report, weighted-least-squares, multi-linear 

regression analysis is used to develop the statistical relation between the station-specific, trimmed-mean, T-year 

peak-streamflow values and the three watershed characteristics (explanatory variables). The sampling and modeling errors are combined and are used to derive weight 

factors for the weighted-least-squares regression between 

peak-streamflow frequency and the watershed characteristics of drainage area, dimensionless main-channel slope, 

mean annual precipitation, and, when used, the fΩ parameter. 

Nine weighted-least-squares, PRESS-minimized regression equations are computed using the watershed characteristics of drainage area, dimensionless main-channel 

slope, and mean annual precipitation and each of the 638 

trimmed-mean, peak-streamflow estimates for each of the 

nine recurrence intervals. The PRESS-minimization was 

made using a user-guided search algorithm. Separate 

PRESS-minimizations were made and a unique exponent 

on drainage area is determined for each recurrence interval. 

The residuals for each of the nine regression equations 

were computed. Maps depicting the spatial distribution of 

the residuals with symbols determined by the magnitude 

and direction of the residual are shown. Spatial dependency 

is evident for all recurrence intervals. For each recurrence 

interval, the median residual for each square of 1-degree 

quadrangle of latitude and longitude in the study area was 

computed. 

The median values of the residuals were computed for 

1-degree quadrangles for the 10-year recurrence interval 

for generalization. The authors, through manual smoothing, consultation of various geologic and ecological region 

maps, interpretation of regional topographic maps, and resident familiarity with the study area, estimated the magnitude and sign of the 10-year recurrence interval residuals for each 1-degree quadrangle in the study area. These 

estimates, referred to as the fΩ parameter, are in units of 

log10(streamflow); maps of the study area depicting, by 

1-degree quadrangle, the fΩ parameter superimposed on

hill-shade relief, Texas rivers, and ecoregions are provided. 

For each station, the fΩ parameter was assigned based 

on the 1-degree quadrangles containing the station. The 

fΩ parameter represents a generalized terrain and climate 

index that expresses relative differences in peak-streamflow 

potential across the study area. The fΩ parameter is interpreted as an expression of peak-streamflow potential not 

represented in the watershed characteristics of drainage 

area, main-channel slope, and mean annual precipitation. 

Nine weighted-least-squares, PRESS-minimized regression equations were computed using the watershed characteristics of drainage area, dimensionless main-channel 

slope, mean annual precipitation, and fΩ as well as each 

of the 638 trimmed-mean, peak-streamflow estimates for 

each of the nine recurrence intervals. The weights for 

the regression equations were the same as those used for 

the nine regression equations not including fΩ . Separate 

PRESS-minimizations were made for each recurrence interval and a unique exponent on drainage area was determined. 

The mean residual, standard error, adjusted R-squared, and 

mean percentage reduction in PRESS by use of fΩ are 

0.30log10, 0.86, and about −21 percent, respectively. 

Example computations are provided to demonstrate 

how the regional regression equations could be used in practice. A comparison of the regression equations to results 

from a previous study is made, and the results are discussed. 

In brief, a similar regional model or structure of the relation 

between peak-streamflow frequency and the three watershed characteristics in Texas results from the Approach. 

The fΩ equations are expected to be reliable estimators of peak-streamflow frequency for undeveloped and 

ungaged stream locations in Texas. Although the fΩ parameter requires interpretation on the part of analysts and the 

potential exists that different analysts could estimate different values for a given watershed, the authors suggest 

that typical uncertainty in the fΩ estimate might be about 

±0.10log10, which is small relative to other uncertainties, such as those measured by residual standard error, in 

hydrostatological models. Finally, given the two ensembles of equations reported herein and those in previous 

reports, hydrologic design engineers and other analysts in 

Texas now have several different methods, which represent 

different analytical tracks, to make comparisons of peakstreamflow frequency estimates for ungaged stream locations in the study area. 

References 

Asquith, W.H., 2001, Effects of regulation on L-moments 

of annual peak streamflow in Texas: U.S. Geological 

Survey Water-Resources Investigations Report 01–4243, 

66 p. [ http://pubs.usgs.gov/wri/wri014243/ ] 

Asquith, W.H., 2006, L-moments and TL-moments of the 

generalized lambda distribution: Computational Statistics and Data Analysis, v. 51, p. 4,484–4,496. 

Asquith, W.H., 2008, lmomco—L-moments, Trimmed 

L-moments, L-comoments, and many distributions: R package version 0.93.3, February 2, 2008. 

[ http://www.cran.r-project.org/package=lmomco ] 
Asquith, W.H., and Roussel, M.C., 2007, An 

initial-abstraction, constant-loss model for unit 

hydrograph modeling for applicable watersheds in Texas: U.S. Geological Survey Scientific Investigations Report 2007–5243, 82 p. 

[ http://pubs.usgs.gov/sir/2007/5243 ] 

Asquith, W.H., Roussel, M.C., Cleveland, T.G., 

Fang, Xing, and Thompson, D.B., 2006, Statistical characteristics of storm interevent time, 

depth, and duration for eastern New Mexico, Oklahoma, and Texas: U.S. Geological Survey Professional Paper 1725, ISBN 1–411–31041–1, 299 p. 

[ http://pubs.usgs.gov/pp/pp1725/ ] 

Asquith, W.H., and Slade, R.M., 1997, Regional equations for estimation of peak-streamflow frequency 

for natural basins in Texas: U.S. Geological Survey 

Water-Resources Investigations Report 96–4307, 68 p. 

[ http://pubs.usgs.gov/wri/wri964307/ ] 

Asquith, W.H., and Slade, R.M., 1999, Site-specific 

estimation of peak-streamflow frequency using 

generalized least-squares regression for natural 

basins in Texas: U.S. Geological Survey WaterResources Investigations Report 99–4172, 19 p. 

[ http://pubs.usgs.gov/wri/wri994172/ ] 

Asquith, W.H., and Thompson, D.B., 2005, Alternative 

regression equations for estimation of annual peakstreamflow frequency for undeveloped watersheds 

in Texas using PRESS minimization: Lubbock, Texas 

Tech Center for Multidisciplinary Research in Transportation, Texas Tech University, Texas Department 

of Transportation Research Report 0–4405–2, 27 p. 

[ http://library.ctr.utexas.edu/pdf/4405-2.pdf ] 

Asquith, W.H., and Thompson, D.B., 2008, Alternative 

regression equations for estimation of annual peakstreamflow frequency for undeveloped watersheds in 

Texas using PRESS minimization: U.S. Geological Survey Scientific Investigations Report 2008–5084, 40 p. 

[ http://pubs.usgs.gov/sir/2008/5084 ] 

Burnett, Jonathan, 2008, Flash floods in Texas: College 

Station, Texas A&M Univesity Press, 330 p. 

Commission for Environmental Cooperation, 1997, Ecological regions of North America—Toward a common 

perspective: Montreal, Quebec, Canada, 71 p. 

Elamir, E.A., and Seheult, A.H., 2003, Trimmed 

L-moments: Computational Statistics and Data Analysis, v. 43, p. 299–314. 

References 27 

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

in water resources: U.S. Geological Survey Techniques 

of Water-Resources Investigations, book 4, chap. A3, 

510 p. [ http://pubs.usgs.gov/twri/twri4a3/ ] 

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order 

statistics: Journal Royal Statistical Society B, v. 52, 

no. 1, p. 105–124. 

Hosking, J.R.M., 1996, FORTRAN routines for use 

with the method of L-moments, version 3: Yorktown 

Heights, N.Y., IBM Research Division, T.J. Watson 

Research Center, Research Report RC20525. 

Hosking, J.R.M., 2006, On the characterization of distributions by their L-moments: Journal of Statistical 

Planning and Inference, v. 136, p. 193–198. 

Hosking, J.R.M., 2007, Some theory and practical uses 

of trimmed L-moments: Journal of Statistical Planning 

and Inference, v. 137, p. 3,024–3,039. 

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: 

Cambridge, Cambridge University Press, ISBN 0–521– 

43045–3, 224 p. 

Interagency Advisory Committee on Water Data, 1982, 

Guidelines for determining flood flow frequency: 

Reston, Va., U.S. Geological Survey, Office of Water 

Data Coordination, Hydrology Subcommittee Bulletin 

17B [variously paged]. 

Judd, L.J., Asquith, W.H., and Slade, R.M., 1996, Techniques to estimate generalized skew coefficients of 

annual peak streamflow for natural basins in Texas: 

U.S. Geological Survey Water-Resources Investigations 

Report 96–4117, 28 p. 

Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979, 

Estimation of parameters and quantiles of Wakeby 

distributions: Water Resources Research, v. 15, no. 5, 

p. 1,362–1,379. 

Montgomery, D.C., Peck, E.A., and Vining, G.G., 2001, 

Introduction to linear regression analysis (3rd ed.): New 

York, Wiley, 641 p. 

O’Connor, J.E., and Costa, J.E., 2003, Large floods in the 

United States—Where they happen and why: U.S. Geological Survey Circular 1245, ISBN 0–607–89380–X, 

13 p. 

R Development Core Team, 2008, R—A language and 

environment for statistical computing: Vienna, Aus­
28 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

tria, R Foundation for Statistical Computing, ISBN 3– 

900051–07–0, version 2.7.0, http://www.R-project.org

Stedinger, J.R., Vogel, R.M., and Foufoula-Georgiou, E., 

1993, Frequency analysis of extreme events, chapter 18,

in Maidment, D.A., ed., Handbook of hydrology: New

York, McGraw-Hill, p. 18.1–18.66. 

U.S. Geological Survey, 2003, The national atlas of the 

United States of America—Streams and waterbodies 

of the United States: Available for download at http:

//nationalatlas.gov/atlasftp.html?openChapters=

chpwater#chpwater

U.S. Geological Survey, 2008, National Water Information System: accessed on various dates in 2007 at 

http://nwis.waterdata.usgs.gov/usa/nwis/peak
Appendix 1—Supplemental Information and 

External Data Files 
Blank Page 
Appendix 1 31 

The purpose of this appendix is to provide a central reference to several text or portable document format (PDF) files that 

accompany this report and are referenced. Each of these are listed below and available from the report Web site: 

1. Appendix1_638annpks.txt—This text file is a comma-delimited file, which lists by station, the annual peakstreamflow values used for statistical computation as described in the text. The number of stations is 638 and the 

criteria leading to this count are described in the report in the “Identification of Annual Peak Streamflow Data” section of this report; 

2. File Appendix1_638wtrshdchr.txt—This text file is a comma-delimited file, which lists by station, the watershed 

characteristics of drainage area, dimensionless main-channel slope, and mean annual precipitation that are used for 

regression analyses as described in the text. The number of stations is 638 and the criteria leading to this count are 

described in the report in the “Identification of Annual Peak-Streamflow Data” section of this report. A statistical 

summary of the selected watershed characteristics in this file is provided in table 1 on page 4; 

3. File Appendix1_677annpks.txt—This text file is a comma-delimited file, which lists by station, the annual peakstreamflow values used for statistical computation as described in the text. The number of stations is 677 and the 

criteria leading to this count are described in the report in the “Identification of Annual Peak-Streamflow Data” section of this report; 

4. File Appendix1_677trimmedQTs.txt—This text file is a comma-delimited file, which lists by station, the annual 

peak-streamflow values used for statistical computation as described in the text. The number of stations is 677 and 

the criteria leading to this count are described in the report in the “Trimmed Mean Estimates of Peak-Streamflow 

Frequency” section of this report; 

5. File Appendix1_677freqcurves.pdf—This PDF file provides a graphical archive of the L-moments fits of as many 

as seven probability distributions to the annual peak-streamflow data listed in file Appendix1_677annpks.txt. The 

results reported in this file were used to compute the trimmed mean estimates of station-specific, peak-streamflow 

frequency reported in file Appendix1_677trimmedQTs.txt; and 

6. File Appendix1_residualmaps.pdf—This PDF file provides a graphical archive of the spatial distribution of the 

residuals from the regression equations reported in tables 2 and 3. 

Each file also contains either a heading (text files) or, for the PDF files, a file-specific “Introduction” and “References” 

section. These headings or sections provide further documentation of the respective contents of each file. The file 

Appendix1_677freqcurves.pdf is listed before the other PDF file; therefore, the figures within that file are numbered 

with a “1A” prepended to the sequential figure number. Similarly, figures in file Appendix1_residualmaps.pdf are numbered with a “1B” prepended to the sequential figure number. 
Blank Page 
Appendix 2—Results of Weighted-Least-Squares, 

Multi-Linear Regression Analysis 

Using PRESS Minimization of 

Peak-Streamflow Frequency to 

Watershed Characteristics of 

Drainage Area, Mean Annual 

Precipitation, and Dimensionless 

Main-Channel Slope 
Blank Page 
---

---

Appendix 2 35 

# START NEW EQUATION

[1] "Doing 2 year"

The power is -0.008

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S), weights = mywgts)
 

Residuals:
 

Min 1Q Median 3Q Max

-1.230070 -0.198423 -0.000294 0.202798 1.059189

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 38.63528 1.01911 37.911 < 2e-16 ***

CDA1 -37.94274 1.03466 -36.672 < 2e-16 ***

log10(P) 1.56157 0.10683 14.617 < 2e-16 ***

log10(S) 0.38458 0.05262 7.308 8.17e-13 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.3191 on 634 degrees of freedom

Multiple R-Squared: 0.8099, Adjusted R-squared: 0.809

F-statistic: 900.6 on 3 and 634 DF, p-value: < 2.2e-16

PRESS= 77.3954345522906

[1] "LATEX:$Q_{2} = P^{1.562},S^{0.385}\times 10^{[38.635-37.943,A^{-0.008]}},$ & 0.32 & 0.81 & 77.4 & 395.8\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4]

[1,] 10.2015098 -10.2857298 0.55103002 0.42527734

[2,] -10.2857298 10.5152921 -0.64343158 -0.42570485

[3,] 0.5510300 -0.6434316 0.11210399 0.03740539

[4,] 0.4252773 -0.4257049 0.03740539 0.02719925

[1] "MAXIMUM LEVERAGE"

[1] 0.05328113

# START NEW EQUATION

[1] "Doing 5 year"

The power is -0.0228

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S), weights = mywgts)
 

Residuals:
 

Min 1Q Median 3Q Max

-1.04803 -0.18715 0.02627 0.19603 0.96727

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 16.34951 0.35560 45.98 <2e-16 ***

CDA1 -15.03710 0.35712 -42.11 <2e-16 ***

log10(P) 1.49059 0.09973 14.95 <2e-16 ***

log10(S) 0.49965 0.04934 10.13 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.3029 on 634 degrees of freedom

Multiple R-Squared: 0.8373, Adjusted R-squared: 0.8365

F-statistic: 1087 on 3 and 634 DF, p-value: < 2.2e-16

PRESS= 65.1550224898523

[1] "LATEX:$Q_{5} = P^{1.491},S^{0.5}\times 10^{[16.35-15.037,A^{-0.0228]}},$ & 0.3 & 0.84 & 65.2 & 313.5\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4]

[1,] 1.3781673 -1.3125511 0.13748791 0.15309923
---

---

36 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

[2,] -1.3125511 1.3899001 -0.22737095 -0.15251351

[3,] 0.1374879 -0.2273710 0.10839049 0.03579911

[4,] 0.1530992 -0.1525135 0.03579911 0.02653628

[1] "MAXIMUM LEVERAGE"

[1] 0.05091717

# START NEW EQUATION

[1] "Doing 10 year"

The power is -0.0299

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S), weights = mywgts)
 

Residuals:
 

Min 1Q Median 3Q Max

-1.05995 -0.17900 0.04071 0.19795 0.95176

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 13.67349 0.27797 49.19 <2e-16 ***

CDA1 -11.98655 0.27300 -43.91 <2e-16 ***

log10(P) 1.37713 0.09827 14.01 <2e-16 ***

log10(S) 0.52954 0.04838 10.95 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.2996 on 634 degrees of freedom

Multiple R-Squared: 0.8465, Adjusted R-squared: 0.8458

F-statistic: 1165 on 3 and 634 DF, p-value: < 2.2e-16

PRESS= 63.6909387784546

[1] "LATEX:$Q_{10} = P^{1.377},S^{0.53}\times 10^{[13.673-11.987,A^{-0.0299]}},$ & 0.3 & 0.85 & 63.7 & 297.4\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4]

[1,] 0.86102385 -0.7734697 0.08289624 0.11796275

[2,] -0.77346969 0.8305043 -0.17342680 -0.11704920

[3,] 0.08289624 -0.1734268 0.10760071 0.03496131

[4,] 0.11796275 -0.1170492 0.03496131 0.02607787

[1] "MAXIMUM LEVERAGE"

[1] 0.04964972

# START NEW EQUATION

[1] "Doing 25 year"

The power is -0.038

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S), weights = mywgts)
 

Residuals:
 

Min 1Q Median 3Q Max

-1.10198 -0.18766 0.03567 0.19749 0.92946

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 11.9959 0.2389 50.22 <2e-16 ***

CDA1 -9.9918 0.2286 -43.72 <2e-16 ***

log10(P) 1.3156 0.1021 12.89 <2e-16 ***

log10(S) 0.5740 0.0500 11.48 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.3107 on 634 degrees of freedom

Multiple R-Squared: 0.8415, Adjusted R-squared: 0.8408
---

Appendix 2 37 

F-statistic: 1122 on 3 and 634 DF, p-value: < 2.2e-16

PRESS= 67.1310969481496

[1] "LATEX:$Q_{25} = P^{1.316},S^{0.574}\times 10^{[11.996-9.992,A^{-0.038]}},$ & 0.31 & 0.84 & 67.1 & 347.8\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4]

[1,] 0.59101965 -0.49267483 0.04866039 0.09480674

[2,] -0.49267483 0.54106276 -0.14043067 -0.09393455

[3,] 0.04866039 -0.14043067 0.10786137 0.03469492

[4,] 0.09480674 -0.09393455 0.03469492 0.02589693

[1] "MAXIMUM LEVERAGE"

[1] 0.04922661

# START NEW EQUATION

[1] "Doing 50 year"

The power is -0.0429

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S), weights = mywgts)
 

Residuals:
 

Min 1Q Median 3Q Max

-1.13394 -0.20740 0.02724 0.20993 0.92946

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 11.40360 0.23024 49.53 <2e-16 ***

CDA1 -9.19510 0.21648 -42.48 <2e-16 ***

log10(P) 1.28501 0.10730 11.98 <2e-16 ***

log10(S) 0.60823 0.05255 11.57 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.3265 on 634 degrees of freedom

Multiple R-Squared: 0.8309, Adjusted R-squared: 0.8301

F-statistic: 1039 on 3 and 634 DF, p-value: < 2.2e-16

PRESS= 73.514894167704

[1] "LATEX:$Q_{50} = P^{1.285},S^{0.608}\times 10^{[11.404-9.195,A^{-0.0429]}},$ & 0.33 & 0.83 & 73.5 & 414.2\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4]

[1,] 0.49734992 -0.39486813 0.03482157 0.08539406

[2,] -0.39486813 0.43964842 -0.12689784 -0.08455224

[3,] 0.03482157 -0.12689784 0.10801117 0.03469890

[4,] 0.08539406 -0.08455224 0.03469890 0.02590499

[1] "MAXIMUM LEVERAGE"

[1] 0.04914469

# START NEW EQUATION

[1] "Doing 100 year"

The power is -0.047

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S), weights = mywgts)
 

Residuals:
 

Min 1Q Median 3Q Max

-1.18887 -0.21895 0.02182 0.23338 0.95608

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 11.07336 0.23019 48.10 <2e-16 ***

CDA1 -8.67252 0.21349 -40.62 <2e-16 ***
---

---

38 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

log10(P) 1.25687 0.11400 11.03 <2e-16 ***

log10(S) 0.64039 0.05591 11.45 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.3462 on 634 degrees of freedom

Multiple R-Squared: 0.8159, Adjusted R-squared: 0.815

F-statistic: 936.7 on 3 and 634 DF, p-value: < 2.2e-16

PRESS= 82.8434210751132

[1] "LATEX:$Q_{100} = P^{1.257},S^{0.64}\times 10^{[11.073-8.673,A^{-0.047]}},$ & 0.35 & 0.82 & 82.8 & 493.4\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4]

[1,] 0.44212745 -0.33758804 0.02658168 0.07958441

[2,] -0.33758804 0.38029216 -0.11876646 -0.07889804

[3,] 0.02658168 -0.11876646 0.10844215 0.03500918

[4,] 0.07958441 -0.07889804 0.03500918 0.02608425

[1] "MAXIMUM LEVERAGE"

[1] 0.04942995

# START NEW EQUATION

[1] "Doing 200 year"

The power is -0.0506

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S), weights = mywgts)
 

Residuals:
 

Min 1Q Median 3Q Max

-1.25427 -0.23860 0.01366 0.24947 1.01623

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 10.8633 0.2359 46.059 <2e-16 ***

CDA1 -8.2818 0.2154 -38.444 <2e-16 ***

log10(P) 1.2209 0.1221 9.996 <2e-16 ***

log10(S) 0.6647 0.0600 11.078 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.3712 on 634 degrees of freedom

Multiple R-Squared: 0.7975, Adjusted R-squared: 0.7965

F-statistic: 832.3 on 3 and 634 DF, p-value: < 2.2e-16

PRESS= 95.3359086701712

[1] "LATEX:$Q_{200} = P^{1.221},S^{0.665}\times 10^{[10.863-8.282,A^{-0.0506]}},$ & 0.37 & 0.8 & 95.3 & 586.4\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4]

[1,] 0.40368172 -0.29668113 0.01974134 0.07508040

[2,] -0.29668113 0.33676662 -0.11169444 -0.07431564

[3,] 0.01974134 -0.11169444 0.10824824 0.03499739

[4,] 0.07508040 -0.07431564 0.03499739 0.02612667

[1] "MAXIMUM LEVERAGE"

[1] 0.04934428

# START NEW EQUATION

[1] "Doing 250 year"

The power is -0.0517

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S), weights = mywgts)
 

Residuals:
 
---

---

Appendix 2 39 

Min 1Q Median 3Q Max

-1.25699 -0.23952 0.01588 0.25038 1.05275

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 10.82773 0.23832 45.433 <2e-16 ***

CDA1 -8.18108 0.21707 -37.689 <2e-16 ***

log10(P) 1.20879 0.12510 9.662 <2e-16 ***

log10(S) 0.67555 0.06145 10.993 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.3797 on 634 degrees of freedom

Multiple R-Squared: 0.7903, Adjusted R-squared: 0.7893

F-statistic: 796.5 on 3 and 634 DF, p-value: < 2.2e-16

PRESS= 100.050979279010

[1] "LATEX:$Q_{250}	 = P^{1.209},S^{0.676}\times 10^{[10.828-8.181,A^{-0.0517]}},$ & 0.38 & 0.79 & 100.1 &

617.2\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4]

[1,] 0.39397766 -0.28676632 0.01822596 0.07395883

[2,] -0.28676632 0.32683298 -0.11041327 -0.07332429

[3,] 0.01822596 -0.11041327 0.10856462 0.03515585

[4,] 0.07395883 -0.07332429 0.03515585 0.02619725

[1] "MAXIMUM LEVERAGE"

[1] 0.04953543

# START NEW EQUATION

[1] "Doing 500 year"

The power is -0.0554

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S), weights = mywgts)
 

Residuals:
 

Min 1Q Median 3Q Max

-1.29275 -0.26326 0.01617 0.26951 1.11027

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 10.67132 0.24615 43.35 <2e-16 ***

CDA1 -7.84563 0.22150 -35.42 <2e-16 ***

log10(P) 1.18100 0.13451 8.78 <2e-16 ***

log10(S) 0.70541 0.06626 10.65 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.4073 on 634 degrees of freedom

Multiple R-Squared: 0.7669, Adjusted R-squared: 0.7658

F-statistic: 695.2 on 3 and 634 DF, p-value: < 2.2e-16

PRESS= 116.921491737753

[1] "LATEX:$Q_{500}	 = P^{1.181},S^{0.705}\times 10^{[10.671-7.846,A^{-0.0554]}},$ & 0.41 & 0.77 & 116.9 &

712.7\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4]

[1,] 0.36529409 -0.25706556 0.01343282 0.07057228

[2,] -0.25706556 0.29580421 -0.10558462 -0.07012783

[3,] 0.01343282 -0.10558462 0.10909026 0.03560090

[4,] 0.07057228 -0.07012783 0.03560090 0.02646788

[1] "MAXIMUM LEVERAGE"

[1] 0.05004109
Blank Page 
Appendix 3—Results of Weighted-Least-Squares, 

Multi-Linear Regression Analysis 

Using PRESS Minimization of 

Peak-Streamflow Frequency to 

Watershed Characteristics of 

Drainage Area, Mean Annual 

Precipitation, Dimensionless 

Main-Channel Slope, and OmegaEM 
Blank Page 
---

---

Appendix 3 43 

# START NEW EQUATION

[1] "Doing 2 year"

The power is -0.0058

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S) + OMEGAEM,
 

weights = mywgts)

Residuals:

Min 1Q Median 3Q Max

-1.117311 -0.164565 -0.003824 0.156962 1.112398

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 50.97508 1.27640 39.936 < 2e-16 ***

CDA1 -50.29910 1.29205 -38.930 < 2e-16 ***

log10(P) 1.39817 0.09800 14.266 < 2e-16 ***

log10(S) 0.27018 0.04874 5.543 4.36e-08 ***

OMEGAEM 0.77587 0.06629 11.705 < 2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.2896 on 633 degrees of freedom

Multiple R-Squared: 0.8437, Adjusted R-squared: 0.8427

F-statistic: 854.1 on 4 and 633 DF, p-value: < 2.2e-16

PRESS= 64.601770644145

[1] "LATEX:$Q_{2} = P^{1.398},S^{0.27}\times 10^{[50.975-50.299,A^{-0.0058} + 0.776OmegaEMs] }$ & 0.29 & 0.84 &

64.6 & 273.2\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4] [,5]

[1,] 19.4253954 -19.5922208 0.81164439 0.597916940 -0.109309193

[2,] -19.5922208 19.9044211 -0.90384486 -0.598196650 0.108244879

[3,] 0.8116444 -0.9038449 0.11451891 0.039052288 -0.011122022

[4,] 0.5979169 -0.5981967 0.03905229 0.028323546 -0.007694694

[5,] -0.1093092 0.1082449 -0.01112202 -0.007694694 0.052392588

[1] "MAXIMUM LEVERAGE"

[1] 0.05358737

# START NEW EQUATION

[1] "Doing 5 year"

The power is -0.0215

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S) + OMEGAEM,
 

weights = mywgts)

Residuals:

Min 1Q Median 3Q Max

-0.99501 -0.13423 0.01847 0.14796 0.83026

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 16.61519 0.32417 51.255 <2e-16 ***

CDA1 -15.31993 0.32617 -46.968 <2e-16 ***

log10(P) 1.30764 0.08662 15.096 <2e-16 ***

log10(S) 0.37159 0.04327 8.588 <2e-16 ***

OMEGAEM 0.88461 0.05897 15.000 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.2604 on 633 degrees of freedom

Multiple R-Squared: 0.8799, Adjusted R-squared: 0.8791

F-statistic: 1159 on 4 and 633 DF, p-value: < 2.2e-16
---

44 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

PRESS= 49.1241230911915

[1] "LATEX:$Q_{5} = P^{1.308},S^{0.372}\times 10^{[16.615-15.32,A^{-0.0215} + 0.885OmegaEMs] }$ & 0.26 & 0.88 &

49.1 & 121.6\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4] [,5]

[1,] 1.54962628 -1.48774667 0.15709427 0.16635099 -0.03302225

[2,] -1.48774667 1.56885660 -0.24679239 -0.16563232 0.03206169

[3,] 0.15709427 -0.24679239 0.11064184 0.03735295 -0.01066574

[4,] 0.16635099 -0.16563232 0.03735295 0.02760925 -0.00742009

[5,] -0.03302225 0.03206169 -0.01066574 -0.00742009 0.05128716

[1] "MAXIMUM LEVERAGE"

[1] 0.05118202

# START NEW EQUATION

[1] "Doing 10 year"

The power is -0.0289

Call:

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S) + OMEGAEM,

weights = mywgts)

Residuals:

Min 1Q Median 3Q Max

-1.05107 -0.12910 0.01404 0.13987 0.81665

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 13.62460 0.24348 55.957 <2e-16 ***

CDA1 -11.96593 0.23971 -49.917 <2e-16 ***

log10(P) 1.20257 0.08398 14.320 <2e-16 ***

log10(S) 0.40314 0.04175 9.657 <2e-16 ***

OMEGAEM 0.91773 0.05794 15.838 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.2537 on 633 degrees of freedom

Multiple R-Squared: 0.89, Adjusted R-squared: 0.8893

F-statistic: 1281 on 4 and 633 DF, p-value: < 2.2e-16

PRESS= 46.6043351340521

[1] "LATEX:$Q_{10} = P^{1.203},S^{0.403}\times 10^{[13.625-11.966,A^{-0.0289} + 0.918OmegaEMs] }$ & 0.25 & 0.89

& 46.6 & 86.5\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4] [,5]

[1,] 0.92075433 -0.83429385 0.092911995 0.124886828 -0.023884308

[2,] -0.83429385 0.89247573 -0.183156553 -0.123763614 0.022313589

[3,] 0.09291200 -0.18315655 0.109537106 0.036345323 -0.009971505

[4,] 0.12488683 -0.12376361 0.036345323 0.027067908 -0.007181418

[5,] -0.02388431 0.02231359 -0.009971505 -0.007181418 0.052145254

[1] "MAXIMUM LEVERAGE"

[1] 0.04984506

# START NEW EQUATION

[1] "Doing 25 year"

The power is -0.0374

Call:

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S) + OMEGAEM,

weights = mywgts)

Residuals:

Min 1Q Median 3Q Max
---

---

Appendix 3 45 

-1.09229 -0.14103 0.02173 0.15698 0.78448

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 11.79063 0.20629 57.16 <2e-16 ***

CDA1 -9.81896 0.19772 -49.66 <2e-16 ***

log10(P) 1.13976 0.08735 13.05 <2e-16 ***

log10(S) 0.44556 0.04322 10.31 <2e-16 ***

OMEGAEM 0.94541 0.06013 15.72 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.2637 on 633 degrees of freedom

Multiple R-Squared: 0.886, Adjusted R-squared: 0.8853

F-statistic: 1230 on 4 and 633 DF, p-value: < 2.2e-16

PRESS= 49.4591620341938

[1] "LATEX:$Q_{25} = P^{1.14},S^{0.446}\times 10^{[11.791-9.819,A^{-0.0374} + 0.945OmegaEMs] }$ & 0.26 & 0.89 &

49.5 & 139.5\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4] [,5]

[1,] 0.61180545 -0.51351782 0.054197182 0.098699053 -0.018955130

[2,] -0.51351782 0.56202692 -0.145648642 -0.097592316 0.017195210

[3,] 0.05419718 -0.14564864 0.109693565 0.036021717 -0.009706565

[4,] 0.09869905 -0.09759232 0.036021717 0.026858442 -0.007064779

[5,] -0.01895513 0.01719521 -0.009706565 -0.007064779 0.051983466

[1] "MAXIMUM LEVERAGE"

[1] 0.04938781

# START NEW EQUATION

[1] "Doing 50 year"

The power is -0.0424

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S) + OMEGAEM,
 

weights = mywgts)

Residuals:

Min 1Q Median 3Q Max

-1.11420 -0.16605 0.02835 0.17509 0.87985

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 11.17250 0.20031 55.78 <2e-16 ***

CDA1 -8.99737 0.18862 -47.70 <2e-16 ***

log10(P) 1.10458 0.09286 11.90 <2e-16 ***

log10(S) 0.47647 0.04593 10.37 <2e-16 ***

OMEGAEM 0.96111 0.06364 15.10 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.2801 on 633 degrees of freedom

Multiple R-Squared: 0.8757, Adjusted R-squared: 0.8749

F-statistic: 1115 on 4 and 633 DF, p-value: < 2.2e-16

PRESS= 55.6054009326601

[1] "LATEX:$Q_{50} = P^{1.105},S^{0.476}\times 10^{[11.172-8.997,A^{-0.0424} + 0.961OmegaEMs] }$ & 0.28 & 0.87 &

55.6 & 219.9\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4] [,5]

[1,] 0.51122238 -0.40861290 0.039403326 0.08863528 -0.017386707

[2,] -0.40861290 0.45333014 -0.131154457 -0.08755453 0.015615535

[3,] 0.03940333 -0.13115446 0.109858491 0.03603858 -0.009718802

[4,] 0.08863528 -0.08755453 0.036038576 0.02687713 -0.007077780
---

---

46 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

[5,] -0.01738671 0.01561553 -0.009718802 -0.00707778 0.051603573

[1] "MAXIMUM LEVERAGE"

[1] 0.04930603

# START NEW EQUATION

[1] "Doing 100 year"

The power is -0.0467

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S) + OMEGAEM,
 

weights = mywgts)

Residuals:

Min 1Q Median 3Q Max

-1.15429 -0.19251 0.03230 0.18716 0.98946

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 10.81851 0.20276 53.36 <2e-16 ***

CDA1 -8.44803 0.18823 -44.88 <2e-16 ***

log10(P) 1.07107 0.10032 10.68 <2e-16 ***

log10(S) 0.50665 0.04968 10.20 <2e-16 ***

OMEGAEM 0.96895 0.06848 14.15 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.302 on 633 degrees of freedom

Multiple R-Squared: 0.8601, Adjusted R-squared: 0.8593

F-statistic: 973.2 on 4 and 633 DF, p-value: < 2.2e-16

PRESS= 64.7959688613551

[1] "LATEX:$Q_{100} = P^{1.071},S^{0.507}\times 10^{[10.819-8.448,A^{-0.0467} + 0.969OmegaEMs] }$ & 0.3 & 0.86 &

64.8 & 320\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4] [,5]

[1,] 0.45078440 -0.34598773 0.030317559 0.082225298 -0.016017361

[2,] -0.34598773 0.38848667 -0.122200320 -0.081320632 0.014418496

[3,] 0.03031756 -0.12220032 0.110351187 0.036377618 -0.009880846

[4,] 0.08222530 -0.08132063 0.036377618 0.027065505 -0.007100056

[5,] -0.01601736 0.01441850 -0.009880846 -0.007100056 0.051424464

[1] "MAXIMUM LEVERAGE"

[1] 0.04958827

# START NEW EQUATION

[1] "Doing 200 year"

The power is -0.0504

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S) + OMEGAEM,
 

weights = mywgts)

Residuals:

Min 1Q Median 3Q Max

-1.20847 -0.21382 0.02703 0.20050 1.08574

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 10.60993 0.21114 50.251 <2e-16 ***

CDA1 -8.05788 0.19298 -41.756 <2e-16 ***

log10(P) 1.03433 0.10945 9.451 <2e-16 ***

log10(S) 0.53118 0.05428 9.786 <2e-16 ***

OMEGAEM 0.97538 0.07474 13.051 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1
---

Appendix 3 47 

Residual standard error: 0.3298 on 633 degrees of freedom

Multiple R-Squared: 0.8404, Adjusted R-squared: 0.8394

F-statistic: 833.5 on 4 and 633 DF, p-value: < 2.2e-16

PRESS= 77.1727531680508

[1] "LATEX:$Q_{200} = P^{1.034},S^{0.531}\times 10^{[10.61-8.058,A^{-0.0504} + 0.975OmegaEMs] }$ & 0.33 & 0.84 &

77.2 & 436.3\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4] [,5]

[1,] 0.40989959 -0.30259286 0.022951245 0.077351356 -0.014780835

[2,] -0.30259286 0.34242026 -0.114612001 -0.076376797 0.013236336

[3,] 0.02295124 -0.11461200 0.110139892 0.036347426 -0.009837713

[4,] 0.07735136 -0.07637680 0.036347426 0.027090365 -0.007032369

[5,] -0.01478084 0.01323634 -0.009837713 -0.007032369 0.051356645

[1] "MAXIMUM LEVERAGE"

[1] 0.04948925

# START NEW EQUATION

[1] "Doing 250 year"

The power is -0.0516

Call:

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S) + OMEGAEM,

weights = mywgts)

Residuals:

Min 1Q Median 3Q Max

-1.22113 -0.22338 0.02842 0.20574 1.13024

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 10.55893 0.21427 49.279 <2e-16 ***

CDA1 -7.94317 0.19520 -40.693 <2e-16 ***

log10(P) 1.02134 0.11271 9.061 <2e-16 ***

log10(S) 0.54069 0.05591 9.671 <2e-16 ***

OMEGAEM 0.97717 0.07685 12.716 <2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.3391 on 633 degrees of freedom

Multiple R-Squared: 0.833, Adjusted R-squared: 0.8319

F-statistic: 789.2 on 4 and 633 DF, p-value: < 2.2e-16

PRESS= 81.8535513717814

[1] "LATEX:$Q_{250} = P^{1.021},S^{0.541}\times 10^{[10.559-7.943,A^{-0.0516} + 0.977OmegaEMs] }$ & 0.34 & 0.83

& 81.9 & 474.1\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4] [,5]

[1,] 0.39917437 -0.29156312 0.02125209 0.076122309 -0.014815036

[2,] -0.29156312 0.33128176 -0.11313034 -0.075265114 0.013195683

[3,] 0.02125209 -0.11313034 0.11046049 0.036517920 -0.009856940

[4,] 0.07612231 -0.07526511 0.03651792 0.027175928 -0.007087477

[5,] -0.01481504 0.01319568 -0.00985694 -0.007087477 0.051347383

[1] "MAXIMUM LEVERAGE"

[1] 0.04968222

# START NEW EQUATION

[1] "Doing 500 year"

The power is -0.0554

Call:
 

lm(formula = log10(Q) ~ CDA1 + log10(P) + log10(S) + OMEGAEM,
 
---

 

 

48 Regression Equations for Estimation of Annual Peak-Streamflow Frequency in Texas 

weights = mywgts)

Residuals:

Min 1Q Median 3Q Max

-1.27882 -0.23558 0.02474 0.22946 1.23151

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 10.40251 0.22475 46.284 < 2e-16 ***

CDA1 -7.60456 0.20224 -37.601 < 2e-16 ***

log10(P) 0.98824 0.12329 8.016 5.3e-15 ***

log10(S) 0.56851 0.06132 9.272 < 2e-16 ***

OMEGAEM 0.97583 0.08382 11.643 < 2e-16 ***

Signif. codes: 0:"***", 0.001:"**", 0.01:"*", 0.05:".", 0.1:" ", 1

Residual standard error: 0.3699 on 633 degrees of freedom

Multiple R-Squared: 0.808, Adjusted R-squared: 0.8068

F-statistic: 666 on 4 and 633 DF, p-value: < 2.2e-16

PRESS= 98.6828845882508

[1] "LATEX:$Q_{500} = P^{0.988},S^{0.569}\times 10^{[10.403-7.605,A^{-0.0554} + 0.976OmegaEMs] }$ & 0.37 & 0.81

& 98.7 & 590.9\\"

[1] "INVERTED-COVARIANCE MATRIX"

[,1] [,2] [,3] [,4] [,5]

[1,] 0.36919005 -0.26055941 0.01622655 0.072556520 -0.014143283

[2,] -0.26055941 0.29893745 -0.10809000 -0.071907268 0.012683503

[3,] 0.01622655 -0.10809000 0.11109360 0.037023770 -0.010141912

[4,] 0.07255652 -0.07190727 0.03702377 0.027478471 -0.007203266

[5,] -0.01414328 0.01268350 -0.01014191 -0.007203266 0.051343459

[1] "MAXIMUM LEVERAGE"

[1] 0.05019887

Prepared by the USGS Lafayette Publishing Service Center. 

Information regarding water resources in Texas is available at 

http: //tx.usgs.gov/ 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Back cover: O.H. IVIE RESERVOIR: The O.H. Ivie Reservoir, once called Stacy Reservoir, is impounded 

by the S.W. Freese Dam at the Concho-Coleman county line. It is located in Concho, Coleman, and 

Runnels Counties. In 1938 the U.S. Army Corps of Engineers expressed a desire for a reservoir site near 

the confluence of the Concho and Colorado Rivers. An agreement was finally reached in 1985, when the 

Texas Water Commission granted permission to impound 554,000 acre-feet of water on the Colorado 

River at Stacy, 16 miles below the confluence. The project was delayed by negotiations to preserve the 

endangered Concho water snakes, and to relocate several local family cemeteries. The reservoir was to 

be named for the Stacy settlement, but it was later decided instead to honor the water district’s general 

manager, O.H. Ivie, and to name the dam for Simon W. Freese, a Fort Worth engineer whose firm had 

worked on major reservoir projects since 1949. The lake waters are used for domestic and municipal 

water supply for a number of West Texas cities and towns. The conservation surface area of the lake is 

20,000 surface acres. The reservoir and its 2-mile rolled earthfill dam, constructed by Brown and Root 

USA, were dedicated in 1990 and are owned and operated by the Colorado River Municipal Water 

District. The reservoir is surrounded by a recreation area. 

Bibliography: Dallas Morning News, April 10, 1989. John Peterson, “Trouble Rising behind Stacy Dam,” 

Texas Observer, December 10, 1982. Robert Thomas, “Stacy Dam Gets Approval,” Ranch Magazine, July 

1985. Ed Todd, Cultural Resource Inventory and Assessment of the Proposed Stacy Reservoir, Concho, 

Coleman, and Runnels Counties, Texas (3 volumes, Austin, 1980). 

Handbook of Texas Online, http://www.tshaonline.org/handbook/online/articles/OO/roogh.html (accessed 

April 15, 2009). 

The Handbook of Texas Online is a project of the Texas State Historical Association, 

http://www.tshaonline.org
 

Asquith and Roussel—Regression Equations for Estimation of Annual Peak-Streamflow Frequency for Undeveloped Watersheds in Texas—SIR 2009–5087 

Printed on recycled paper 

Structures believed to be visible 

in historical photograph on the 

cover and inundated by O.H. Ivie 

Reservoir as shown in bottom 

photograph. 

Part of U.S. Geological Survey 1:24,000 

Quadrangle, “Leaday, Tex.,” dated 1967, 

about 1 mile south of Leaday. 

O.H. Ivie Reservoir