U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2013–5230

Version 1.1, August 2014

Prepared in cooperation with the Texas Water Development Board

Trend Analysis and Selected Summary Statistics of Annual 

Mean Streamflow for 38 Selected Long-Term U.S. Geological 

Survey Streamgages in Texas, Water Years 1916–2012

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Trend Analysis and Selected Summary 

Statistics of Annual Mean Streamflow for 

38 Selected Long-Term U.S. Geological 

Survey Streamgages in Texas, Water Years 

1916–2012

By William H. Asquith and Dana L. Barbie

Prepared in cooperation with the Texas Water Development Board

Scientific Investigations Report 2013–5230

Version 1.1, August 2014

U.S. Department of the Interior

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

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 

First release: 2014

Revised: August 2014 (ver. 1.1) 

Suggested citation:

Asquith, W.H., and Barbie, D.L., 2014, Trend analysis and selected summary statistics of annual mean streamflow for 

38 selected long-term U.S. Geological Survey streamgages in Texas, water years 1916–2012 (ver. 1.1, August 2014): 

U.S. Geological Survey Scientific Investigations Report 2013–5230, 16 p., http://dx.doi.org/10.3133/sir20135230. 

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

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U.S. Government.

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

ISSN 2328-031X (print)

ISSN 2328-0328 (online)

ISBN 978-1-4113-3760-2
iii

Contents

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

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

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

Purpose and Scope .............................................................................................................................5

Methods .................................................................................................................................................5

Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow ...............................6

Trend Analysis of Annual Mean Streamflow ...................................................................................7

Selected Summary Statistics of Annual Mean Streamflow .........................................................7

An Example Application of Regional Analysis of Annual Mean Streamflow Frequency .........9

Summary .......................................................................................................................................................14

References ...................................................................................................................................................14

Figures

1. Map showing locations of 38 selected long-term U.S. Geological Survey 

streamgages in Texas providing streamflow data anticipated to represent 

natural and unregulated watershed conditions .....................................................................3

2. Graph showing comparison of periods of operation for 38 selected long-term 

U.S. Geological Survey streamgages in Texas by sequence number listed in 

table 1 and identification of three stations with a statistically significant trend in 

annual mean streamflow by the nonparametric Kendall’s tau statistical test ..................6

3. Graph showing comparison of three annual mean frequency curves in 

probability-density (inset) and quantile-function (main plot) form for U.S. 

Geological Survey streamgage 08123800 Beals Creek near Westbrook, Texas 

(table 1), associated with example application ....................................................................12

Tables

1. Summary of data from 38 selected long-term U.S. Geological Survey 

streamgages in Texas providing streamflow data anticipated to represent 

natural and unregulated watershed conditions .....................................................................4

2. Summary of mean annual streamflow, Kendall’s tau and associated p-value 

(probability value), and declaration of statistical significance with respect to 

trend direction for the selected long-term U.S. Geological Survey streamgages 

in Texas ..........................................................................................................................................8

3. Summary of first six L-moments of base-10 logarithms of offset annual mean 

streamflow and sampling variances computed by bootstrap simulation of 10,000 

replications for the selected long-term U.S. Geological Survey streamgages 

in Texas ........................................................................................................................................10
iv

Conversion Factors

Inch/Pound to SI

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2) 2.590 square kilometer (km2) 

Volume

cubic foot (ft3) 0.02832 cubic meter (m3) 

acre-foot (acre-ft) 1,233 cubic meter (m3)

Flow rate

foot per second (ft/s) 0.3048 meter per second (m/s)

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

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

(NAVD 88).

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

A water year is the 12-month period October 1 through September 30 designated by the 

calendar year in which it ends.
Trend Analysis and Selected Summary Statistics of 

Annual Mean Streamflow for 38 Selected Long-Term 

U.S. Geological Survey Streamgages in Texas, Water 

Years 1916–2012 

By William H. Asquith and Dana L. Barbie

Abstract

In 2013, the U.S. Geological Survey (USGS) operated 

more than 500 continuous streamgages (streamflow-gaging 

stations) in Texas. In cooperation with the Texas Water 

Development Board, the USGS evaluated mean annual 

streamflow data for 38 selected streamgages that were active 

as of water year 2012. The 38 streamgages have annual 

mean streamflow data considered natural and unregulated. 

Collected annual mean streamflow data for a single 

streamgage ranged from 49 to 97 cumulative years. The 

nonparametric Kendall’s tau statistical test was used to detect 

monotonic trends in annual mean streamflow over time. The 

monotonic trend analysis detected 2 statistically significant 

upward trends (0.01 one-tail significance), 1 statistically 

significant downward trend (0.01 one-tail significance level), 

and 35 instances of no statistically significant trend (0.02 

two-tailed significance level). The Theil slope estimate of a 

regression slope of annual mean streamflow with time was 

computed for the three stations where trends in streamflow 

were detected: 2 increasing Theil slopes were measured (+0.40 

and +2.72 cubic feet per second per year, respectively), and 

1 decreasing Theil slope (−0.24 cubic feet per second per year) 

was measured.

Selected summary statistics (L-moments) and estimates 

of respective sampling variances were computed for the 

35 streamgages lacking statistically significant trends. From 

the L-moments and estimated sampling variances, weighted 

means or regional values were computed for each L-moment. 

An example application is included demonstrating how the 

L-moments could be used to evaluate the magnitude and 

frequency of annual mean streamflow. 

Introduction

Streamflow resources are inherently vital to the 

ecological and economic viability of Texas. In 2013, 

the U.S. Geological Survey (USGS) operated more than 

500 continuous streamgages (streamflow-gaging stations) in 

Texas; the data and ancillary information are stored in the 

USGS National Water Information System (NWIS) database 

(U.S. Geological Survey, 2013). Long-term streamflow data, 

which are anticipated to be representative of natural and 

unregulated watershed conditions, are useful for evaluation of 

changes (trends) in streamflow attributable to systematic landuse changes, changes in groundwater gains or losses, climate 

cycles and changes, statistical vagaries, and other influencing 

factors. Long-term data are defined for this investigation as 

having at least 49 years of cumulative record. Magnitude and 

frequency analysis of extreme streamflow conditions and 

events (rare high-, low-, and drought-condition streamflows) is 

improved (uncertainty reduced) with statistics of large sample 

sizes provided by long-term streamflow datasets. The terms 

“natural and unregulated” do not indicate that the annual mean 

streamflow data are not to a minor degree affected by land-use 

changes, municipality development, flood-control regulation, 

or other water resources development infrastructure. Often 

there are diversions upstream for ranch or farm use, and 

municipalities exist within each of the watersheds. 

Additionally, streamgages flagged as “regulated” as of 

2013 in the annual peak streamflow database within NWIS 

(http://nwis.waterdata.usgs.gov/tx/nwis/peak) were not 

included in the analyses. As described by Asquith (2001), two 

commonly used classification schemes consider regulated 

stations or individual peak streamflows as (1) those stations 

where at least 10 percent of the contributing drainage area of 

the basin is affected or controlled by reservoirs, or (2) those 

peak annual streamflows affected to an appreciable degree. 

The later scheme has an aspect of hydrologic judgment. 

USGS hydrologists assign a numeric code of “6” in the annual 

peak streamflow database to a streamgage to designate peak 

streamflows affected by regulation. As a matter of general 

logic and database consistency, once the annual peaks at a 

station are classified with a code “6” that code remains in 

place for all subsequent annual peaks. Although used for this 

investigation in part for streamgage eligibility, it is important 

to note that the numeric code of “6” is primarily associated 

with classification of annual instantaneous peak streamflows in 

contrast to annual mean streamflow.
2 Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow for 38 Selected Long-Term USGS Streamgages

About two-thirds of the total freshwater usage in Texas 

consists of surface water (Kenny and others, 2009, p. 6). 

There are approximately 200 reservoirs with a substantial 

flood-storage capacity of 3,000 acre-feet (acre-ft) or more 

throughout Texas (Dowell, 1964; Asquith, 2001; Texas State 

Historical Association, 2013). Total storage capacity at these 

reservoirs is approximately 25 million acre-ft. There are 

many permitted and nonpermitted diversion- and return-flow 

points throughout watersheds and river basins in Texas (Texas 

Commission on Environmental Quality, 2013), a fact that 

complicates the task of identifying natural and unregulated 

annual peak streamflows unaffected by substantial reservoiror other water-development activities in Texas (Asquith, 

2001). Although hundreds of streamgages were active in Texas 

in 2013 (fig. 1), many were located in watersheds with obvious 

upstream regulation by large-scale flood-control infrastructure, 

large metropolitan areas, or other urbanized land uses.

Streamflow statistics for unregulated streams in Texas are 

useful in resource assessments (such as water-supply studies) 

and to water managers, hydrologists, biologists, and ecologists 

for various types of hydrologic studies. For example, longterm streamflow data that are representative of natural and 

unregulated watershed conditions are useful for evaluation 

of changes (upward or downward trends) in streamflow 

attributable to systematic land-use changes, changes in 

groundwater gains or losses, assessments of climate cycles, 

and investigations of climate-related changes in streamflow. 

Annual mean streamflow data for unregulated watersheds 

can be used to compute mean annual streamflow over long 

periods of time and represent the expected annual volume of 

water measured at a streamgage. Magnitude and frequency 

analyses of extreme streamflow conditions and events 

(rare high-, low-, and drought-condition streamflows) are 

enhanced (uncertainty reduced) with statistics of large sample 

sizes provided by long-term datasets of natural unregulated 

streamflow.

Accordingly, the USGS, in cooperation with the Texas 

Water Development Board, evaluated annual mean streamflow 

data for 38 selected, long-term USGS streamgages that were 

active as of water year 2012 and considered natural and 

unregulated (fig. 1, table 1). Individually, the 38 streamgages 

have at least 49 and as many as 97 years of annual mean 

streamflow data; the average period of record is 68 years. 

The minimum contributing drainage area (CDA) is about 43 

square miles (mi2), the mean CDA is about 788 mi2, and the 

maximum CDA is about 4,190 mi2. Streamgages throughout 

the entire State were considered; the streamgages are located 

throughout the central and eastern parts of Texas as shown in 

figure 1. 

Previous Studies

Retrospective studies of Texas streamflow data 

encompass a considerable body of work from which 

subsequent investigations have been and can be based. 

Statistical and historical summaries of streamflow data in 

Texas within the past 15 years have been published, including 

Lanning-Rush (2000), Asquith (2001), Asquith and others 

(2007a, b), Asquith and Heitmuller (2008), Asquith and 

Roussel (2009), Barbie and Wehmeyer (2012), Barbie and 

others (2012), and Winters (2013). Each of these studies 

considers a broader spectrum of streamflow characteristics 

(statistics) and stations (specific study areas) than does this 

investigation. The previous studies provide considerable 

graphical depictions of streamflow history and augment time 

series graphics and tables that are readily obtained by public 

users of the USGS NWIS interface known as NWISWeb 

(http://waterdata.usgs.gov). 

Lanning-Rush (2000) provided regional regression 

equations to estimate mean annual streamflow (arithmetic 

mean of the annual mean streamflows) as well as mean 

seasonal (spring, summer, fall, winter) streamflow for 

unmonitored stream sites throughout Texas. The regression 

equations by Lanning-Rush (2000) are based on the statistical 

relation between mean annual streamflow (as well as seasonal 

streamflow), contributing drainage area, and mean annual (or 

seasonal) precipitation. Lanning-Rush (2000) evaluated mean 

streamflow values in Texas.

Asquith and Roussel (2009) provided regional regression 

equations for estimation of annual peak streamflow quantiles, 

such as the 0.5 or 0.9 annual nonexceedance probabilities, 

which can be interpreted as cumulative percentiles, of the 

annual peak streamflows (for example, the 0.9 probability 

peak streamflow would be representative of substantial storm 

flow that has on average a 1 in 10 chance of being equaled or 

exceeded in any 1 year). Asquith and Roussel (2009) provided 

a thematic template for regionalization of other streamflow 

characteristics in Texas; as of 2013, a similar study of annual 

mean streamflow quantiles in Texas has not been published. 

Retrospective studies similar to the 2013 investigation 

include Barbie and Wehmeyer (2012), which evaluated trends 

in selected streamflow statistics for 19 long-term streamgages 

assumed to represent outflows from Texas, and Barbie and 

others (2012), which evaluated trends in selected streamflow 

statistics for one west Texas river basin. Asquith (2001) did 

a statewide study of the effects of flood-control regulations 

on statistics of annual peak streamflow for more than 300 

hundred streamgages in Texas. As a part of the Asquith and 

Roussel (2009) study, periods of natural and unregulated 

annual peak streamflow records were identified. Data files 

by station from Asquith and Roussel (2009) were consulted 

as a component of screening of streamgages for inclusion in 

this investigation. 

Winters (2013) provided historical perspective of the 

2011 drought in Texas to the drought of 1951–56 for 19 

selected unregulated streamgages in Texas. Fifteen of those 

19 streamgages are also part of this investigation (table 1). 

Table 1 lists those streamgages in common with Winters 

(2013), streamgages not included in Winters (2013) because 

the streamflow is not generally considered perennial, and those 

not eligible for inclusion in Winters (2013) because of the start 

of data record.
Introduction 3

Figure 1. Locations of 38 selected long-term U.S. Geological Survey streamgages in Texas providing streamflow data anticipated to 

represent natural and unregulated watershed conditions.

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07299670

1

07343000

2

07346070

3

08017200

4

08017300

5

08029500

6

08041500

7

08068500

8

08070000

9

08070500

10

08078000

11

08079600

12

08082700

13

08084800

14

08101000

15

08115000

16

08117500

17

08123800

18

08134000

19

08150000

20

08150800

21

08151500

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08153500

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08164000

24

08167500

25

08171000

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08175000

27 08186000

28

08189500

29

08190000

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08190500

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08192000

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08200000

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08205500

37 08208000

38

Sandies Creek

Neches River

Guadalupe River

Little Cypress Creek

Brazos River

LavacaRiver

East Fork 

San Jacinto

River

Trinity River

Colorado River

Red River

Sulphur

Nueces River Medina River

Rio Grande

Sabine River

River

Concho River

GULF OF MEXICO

Blanco River

Creek

Village Creek

Big Cow

Creek

Big Creek

San Bernard

River

Cowhouse

Millers Creek

Llano River

River Pedernales 

Mission River

Frio River

Dry Frio River

Sabinal

River

Hondo

Creek

Caney Creek

Creek

Spring

South Fork

Sabine River

Cowleech Fork

Sabine River

Groesbeck Creek

California Creek

Beals Creek

Creek Cibolo 

Nueces

North Concho River

San Antonio River

Ata scosa River

Greenville

Quinlan

Alvin

Cleveland

Needville

Splendora

Johnson City

Wimberley

Llano

Quanah

Jefferson

Kountze

Stamford

Junction

Newton

Edna

Mason

Uvalde

Falls City

Westbrook

Cooper

Refugio

Munday

Sabinal Brackettville

Houston

Spring

Justiceburg

Pidcoke

Boling

Carlsbad

Spring Branch

Laguna

Reagan Wells

Tarpley

Derby

Whitsett

Concan

Westhoff

Fort

Worth Dallas

Austin

San

Antonio

08195000

33

08196000

34

08198000

35 Chocolate

Bayou

Beaver Creek

94°

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100°

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28°

West Nueces

River

River

Base modified from U.S. Geological Survey 

1:250,000 scale digital data

Texas Centric Mapping System

Lambert Conformal Conic projection

North American Datum of 1983 

TEXAS

Area enlarged

0 25 50 100 MILES

0 25 50 75 100 KILOMETERS

75

EXPLANATION

U.S. Geological Survey streamgage, station number

 and map identifier with long-term unregulated record—Table 1

U.S. Geological Survey streamgage currently (2013) active in Texas

 and within the inset area that did not meet the long-term unregulated criteria 

Upward trend by the nonparametric Kendall’s tau statistical test—Table 2

Downward trend by the nonparametric Kendall’s tau statistical test—Table 2

08070000

9
Table 1. Summary of data from 38 selected long-term U.S. Geological Survey streamgages in Texas providing streamflow data anticipated to represewatershed conditions. [mi2, square miles; --, not eligible, no record back to 1951–56; °, degrees; ′, minutes; ′′, seconds; SH, State Highway; DMF, Double Mountain Fork] Sequence number Station number Station name Period of analyzed record Latitude Longitude Total drainage area (mi2) Co

d

1 07299670 Groesbeck Creek at SH 6 near Quanah, Tex. 1963–2012 34° 21′ 16′′ 99° 44′ 24′′ 303.0 2 07343000 North Sulphur River near Cooper, Tex. 1950–2012 33° 28′ 29′′ 95° 35′ 15′′ 276.0 3 07346070 Little Cypress Creek near Jefferson, Tex. 1947–2012 32° 42′ 46′′ 94° 20′ 45′′ 675.0 4 08017200 Cowleech Fork Sabine River at Greenville, Tex. 1960–2012 33° 07′ 58′′ 96° 04′ 36′′ 81.0 5 08017300 South Fork Sabine River near Quinlan, Tex. 1960–2012 32° 53′ 52′′ 96° 15′ 11′′ 78.7 6 08029500 Big Cow Creek near Newton, Tex. 1953–2012 30° 49′ 08′′ 93° 47′ 08′′ 128.0 7 08041500 Village Creek near Kountze, Tex. 1940–2012 30° 23′ 52′′ 94° 15′ 48′′ 860.0 8 08068500 Spring Creek near Spring, Tex. 1940–2012 30° 06′ 37′′ 95° 26′ 10′′ 409.0 9 08070000 East Fork San Jacinto River near Cleveland, Tex. 1940–2012 30° 20′ 11′′ 95° 06′ 14′′ 325.0 

10 08070500 Caney Creek near Splendora, Tex. 1945–2012 30° 15′ 34′′ 95° 18′ 08′′ 105.0 

11 08078000 Chocolate Bayou near Alvin, Tex. 1960–2012 29° 22′ 09′′ 95° 19′ 14′′ 87.7 

12 08079600 DMF Brazos River at Justiceburg, Tex. 1963–2012 33° 02′ 18′′ 101° 11′ 50′′ 1,466.0 13 08082700 Millers Creek near Munday, Tex. 1964–2012 33° 19′ 45′′ 99° 27′ 53′′ 104.0 14 08084800 California Creek near Stamford, Tex. 1963–2012 32° 55′ 51′′ 99° 38′ 32′′ 478.0 15 08101000 Cowhouse Creek at Pidcoke, Tex. 1951–2012 31° 17′ 05′′ 97° 53′ 05′′ 455.0 16 08115000 Big Creek near Needville, Tex. 1948–49, 1953–2012 29° 28′ 35′′ 95° 48′ 45′′ 42.8 17 08117500 San Bernard River near Boling, Tex. 1955–96, 1998–2012 29° 18′ 48′′ 95° 53′ 37′′ 727.0 18 08123800 Beals Creek near Westbrook, Tex. 1959–2012 32° 11′ 57′′ 101° 00′ 49′′ 9,802.0 19 08134000 North Concho River near Carlsbad, Tex. 1925–2012 31° 35′ 33′′ 100° 38′ 12′′ 1,266.0 20 08150000 Llano River near Junction, Tex. 1916–92, 1998–2012 30° 30′ 15′′ 99° 44′ 03′′ 1,849.0 21 08150800 Beaver Creek near Mason, Tex. 1964–2012 30° 38′ 36′′ 99° 05′ 44′′ 215.0 22 08151500 Llano River at Llano, Tex. 1940–2012 30° 45′ 04′′ 98° 40′ 10′′ 4,197.0 23 08153500 Pedernales River near Johnson City, Tex. 1940–2012 30° 17′ 30′′ 98° 23′ 57′′ 901.0 24 08164000 Lavaca River near Edna, Tex. 1939–2012 28° 57′ 35′′ 96° 41′ 10′′ 817.0 25 08167500 Guadalupe River near Spring Branch, Tex. 1923–2012 29° 51′ 37′′ 98° 23′ 00′′ 1,315.0 26 08171000 Blanco River at Wimberley, Tex. 1925, 1929–2012 29° 59′ 39′′ 98° 05′ 19′′ 355.0 27 08175000 Sandies Creek near Westhoff, Tex. 1931–34, 1960–2012 29° 12′ 54′′ 97° 26′ 57′′ 549.0 28 08186000 Cibolo Creek near Falls City, Tex. 1931–90, 1992–2012 29° 00′ 50′′ 97° 55′ 48′′ 827.0 29 08189500 Mission River at Refugio, Tex. 1940–2012 28° 17′ 30′′ 97° 16′ 44′′ 690.0 30 08190000 Nueces River at Laguna, Tex. 1924–2012 29° 25′ 42′′ 99° 59′ 49′′ 737.0 31 08190500 West Nueces River near Brackettville, Tex. 1940–1950, 1957–2012 29° 28′ 52′′ 100° 14′ 21′′ 694.0 32 08192000 Nueces River below Uvalde, Tex. 1940–2012 29° 07′ 25′′ 99° 53′ 40′′ 1,861.0 33 08195000 Frio River at Concan, Tex. 1925–29, 1931–2012 29° 29′ 18′′ 99° 42′ 16′′ 389.0 34 08196000 Dry Frio River near Reagan Wells, Tex. 1953–2012 29° 30′ 16′′ 99° 46′ 52′′ 126.0 35 08198000 Sabinal River near Sabinal, Tex. 1943–2012 29° 29′ 27′′ 99° 29′ 33′′ 206.0 36 08200000 Hondo Creek near Tarpley, Tex. 1953–2012 29° 34′ 12′′ 99° 14′ 52′′ 95.6 37 08205500 Frio River near Derby, Tex. 1916–2012 28° 44′ 11′′ 99° 08′ 40′′ 3,429.0 38 08208000 Atascosa River at Whitsett, Tex. 1933–2006, 2008–12 28° 37′ 19′′ 98° 16′ 52′′ 1,171.0 
Introduction 5

Purpose and Scope

The purpose of this report is (1) to describe the results 

of statistical tests for monotonic trends in annual mean 

streamflow statistics from natural and unregulated watersheds 

in Texas that were computed by using the nonparametric 

Kendall’s tau statistical test to evaluate for independence 

with time or quantify whether statistically significant 

monotonic changes in a given streamflow statistic have 

occurred over time, and (2) to provide select summary 

statistics (L-moments) of the annual mean streamflow data 

that provide specific details of the “geometry” and “shape” 

of probability distributions of annual mean streamflow 

for streamgages where statistically significant trends in 

streamflow are not detected. Trend analyses and summary 

statistics for the 38 streamgages are provided. This report also 

provides an example application demonstrating how selected 

summary statistics (L-moments) of annual mean streamflow 

can be used to enhance the regional understanding of the 

magnitude and frequency of annual mean streamflow “events” 

(quantiles) through the use of a dimensionless regional 

frequency curve.

Methods

The underlying data for this investigation were daily 

mean streamflow for each station (365 or 366 values per 

year). USGS NWIS databases provided time series of annual 

mean streamflow for each of the streamgages. The dataset 

was restricted to the selected 38 long-term streamgages 

that each had a minimum of 49 years of cumulative record 

and were considered to be representative of natural and 

unregulated watershed conditions because they were coded 

in the NWIS database as recording peak streamflows that are 

not substantially affected (less than 10 percent of the CDA 

is controlled by dams and others structures that regulate 

streamflow). The streamgages were selectively screened 

for inclusion in this investigation from the overall USGS 

streamgage network in Texas based on information or 

foreknowledge of Texas hydrology obtained from prior USGS 

studies (Asquith and Slade, 1997; Lanning-Rush, 2000; Slade 

and others, 2001). The data for this investigation encompass 

the period of available record of annual mean streamflow. 

Many of the 38 streamgages included in this study have gaps 

in their records as listed in table 1. The gaps are illustrated 

in figure 2 where the sequence numbers listed in table 1 

provide streamgage identification. Gaps in record can result 

from disruptions in the cooperative agreements between the 

USGS and the Federal, State, and local cooperators that share 

responsibility for operation of the streamgage. Major roadway 

construction also can disrupt continuous operation because 

streamgages often are located on bridges and along roads and 

highways.

There were other qualified long-term streamgages that 

were not included in this report because substantial cross 

correlation in the streamflow measured at these streamgages 

with the streamflow measured at other streamgages on 

the same stream is either likely or has been previously 

documented (Slade and others, 2001). For example, if two 

long-term streamgages reside on the same stream course and 

are comparatively close to each other (distance not defined but 

rather based on ad hoc considerations including the size of the 

watershed upstream from the gage), then their records lack 

statistical independence. 

The nonparametric Kendall’s tau statistical test was 

used to detect monotonic trends in annual mean streamflow 

over time (Hollander and Wolfe, 1973; Helsel and Hirsch, 

2002). Specifically, the test uses the relation between time 

and ranked streamflow, rather than streamflow magnitude, for 

computations; thus, the trend analyses in this report (unless 

otherwise noted) refer to the direction but not the magnitude 

of streamflow change. A statistically significant positive 

Kendall’s tau value indicates an upward streamflow trend; 

conversely, a statistically significant negative tau indicates a 

downward streamflow trend. The probability value (p-value) 

of Kendall’s tau is a measure of the statistical significance of 

the trend assuming a null hypothesis that no trend is present. 

For this investigation, a p-value less than 0.02 indicated the 

presence of a statistically significant trend (alpha = 0.02); 

conversely, a p-value greater than 0.02 indicated the absence 

of a statistically significant trend. If an inquiry into the actual 

direction of the trend was needed, then the p-values required 

a division by 2 because the Kendall’s tau statistical test is 

inherently a two-tailed test.

To compute the Kendall’s tau statistics and p-values, 

the R environment for statistical computing (R Development 

Core Team, 2012) was used. Specifically, the R function 

cor.test(..., method=”kendall”), and its associated 

features were used. The documentation within the R 

environment provides further information. Generally, if 

trends are apparent, they might be caused by factors such 

as (1) landscape and land-use modifications and changes, 

(2) changes in groundwater gains or losses, (3) surfacewater use and return flows from facilities such as municipal 

wastewater treatment plants, and (4) potential trends in 

precipitation patterns and other general climatic shifts, cycles, 

and changes.

When monotonic trends in annual mean streamflow were 

detected, the nonparametric Theil slope of the streamflow 

was used to quantify change in streamflow in cubic feet per 

second per year. The Theil slope is a nonparametric estimate 

of a regression slope of annual mean streamflow with time. 

The Theil slope is defined as the median of all unique pairwise 

slopes (there are n • (n − 1)/2 unique slopes, with n being 

the record length) from the available record for an individual 

streamgage (Hollander and Wolfe, 1973, p. 205–206).

L-moments (Hosking, 1990) are especially suitable 

for estimation of the parameters of probability distributions 

(Hosking and Wallis, 1997; Asquith, 2011). A fitted 

probability distribution facilitates the extrapolation to 
6 Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow for 38 Selected Long-Term USGS Streamgages

extreme and otherwise rare low- or high-flow conditions 

or events outside or beyond the available length of record 

(Hosking and Wallis, 1997). L-moments have been 

successfully used for regional assessment of the magnitude 

and frequency of various hydrometeorological phenomena in 

Texas (Asquith, 2001; Asquith and Roussel, 2004; Asquith and 

others, 2006; Asquith and Roussel, 2009). A general overview 

of regionalization in hydrology is provided by Stedinger and 

others (1993). Selected summary statistics of annual mean 

streamflow (L-moments) for the 35 streamgages for which 

no trends in streamflow were detected. The sample variances 

for the L-moments of each streamgage also were estimated 

by bootstrap resampling (Verzani, 2005; Rizzo, 2008; 

Ugarte and others, 2008). These sampling variances were 

subsequently used to compute inverse-variance, weighted 

mean values of each of the L-moments. The weighted mean 

values of the L-moments represent “regional expectations” for 

the study area.

Trend Analysis and Selected Summary 

Statistics of Annual Mean Streamflow

This section (1) summarizes the trend analysis of annual 

mean streamflow supported by Kendall’s tau, (2) reports 

a nonparametric Theil slope of the streamflow change per 

year for streamgages having statistically significant trends, 

and (3) presents selected summary statistics of annual mean 

streamflow for the 35 streamgages for which Kendall’s tau 

did not detect a statistically significant trend in streamflow for 

the period of record. Because published graphical and tabular 

Figure 2. Comparison of periods of operation for 38 selected long-term U.S. Geological Survey (USGS) streamgages in Texas by 

sequence number listed in table 1 and identification of three stations with a statistically significant trend in annual mean streamflow by 

the nonparametric Kendall’s tau statistical test.

1915 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2015

Water year

Sequence number listed in table 1

1

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

USGS station 07299670 Groesbeck Creek at State Highway 6 

near Quanah, Tex.

USGS station 08134000 North Concho River near 

Carlsbad, Tex.

USGS station 08068500 Spring 

Creek near Spring, Tex.

Blue and red lines and named streamgage 

indicate that a statistically significant 

trend (upward or downward) at the 0.02 

significance level, and gaps in horizontal 

lines indicate gaps in streamgage 

operation.
Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow 7

depictions of streamflow time series information in Texas are 

readily available (Asquith and others, 2007a, b; Asquith and 

Heitmuller, 2008; Winters 2013), and because time series 

graphics and tables can now be produced with ease by using 

the NWISWeb interface, graphical depictions of streamflow 

time series are not presented in this report.

Trend Analysis of Annual Mean Streamflow

A summary of the Kendall’s tau analysis for each of 

the 38 streamgages is listed in table 2. The results of the 

analysis show that the majority of the streamgages do not have 

statistically significant trends with respect to annual mean 

streamflow. In other words, the general annual volumetric 

production of surface water from the corresponding 

watersheds appears to have not appreciably changed during 

the respective periods of record for the individual streamgages. 

Such an observation does not mean that one or more parts 

of the hydrologic spectrum, such as flood peaks or annual 

minimum streamflows, are themselves “trendless.”

Three stations have statistically significant trends. The 

monotonic trend analysis detected 2 statistically significant 

upward trends (0.01 one-tail significance), 1 statistically 

significant downward trend (0.01 one-tail significance level), 

and 35 instances of no statistically significant trend (0.02 

two-tailed significance level). Figure 2 identifies the three 

streamgages that had statistically significant trends in annual 

mean streamflow. These results were not surprising given that 

the streamgages were screened to include only those where the 

streamflow was thought to be natural and unregulated for the 

entire period of record at each station. Possible effects from 

potential long-term changes in groundwater levels along the 

stream channel or any changes in precipitation amounts over 

time were not considered. 

Theil slopes for the three streamgages for which 

statistically significant trends in streamflow were identified 

are as follows: station 07299670 has an estimated increase of 

+0.40 cubic feet per second (ft3/s) per year, station 08068500 

has an estimated increase of +2.72 ft3/s per year, and station 

08134000 has an estimated decrease of −0.24 ft3/s per year. 

The Theil slope values are listed in table 2.

Although not statistically significant, the results in table 2 

show that a positive Kendall’s tau value was determined for 

25 of the 35 streamgages for which no trend was detected. 

The binomial test (Hollander and Wolfe, 1973, p. 15–22) 

was used to assess the likelihood of such a high number of 

positive values for Kendall’s tau and to assess if there were 

an equal likelihood of numerically negative or positive 

values in the absence of any streamflow trends. If 25 were 

considered the “number of successes” from a sample size of 

35 with an anticipated probability of 0.5, the binomial test, 

which was performed by using the binom.test() function 

with the respective arguments from the R environment (25, 

35, p=0.5), yields a p-value of 0.02. This also is a small 

p-value and not related to the alpha value of 0.02 used to 

assign statistical significance for the Kendall’s tau tests. Thus, 

it may be concluded that there is a statistically significant 

likelihood that in aggregate, annual mean streamflow has 

increased slightly to an undocumented degree for 25 of the 

35 streamgages although these 25 streamgages individually do 

not have statistically significant trends. Furthermore, analysis 

of long-term statewide precipitation totals could be made to 

bolster (or counter) the finding of a statistically significant 

likelihood that in aggregate, the annual mean streamflow has 

increased slightly for 25 of the 35 streamgages for which 

individual trends were not detected. 

Selected Summary Statistics of Annual Mean 

Streamflow

This section describes selected summary statistics 

(L-moments) and sampling variances from replacementbootstrap simulation for the 35 streamgages that individually 

lack statistically significant trends. These statistics of annual 

mean streamflow for each streamgage represent a collection 

of different numerical values that together describe the 

location, scale, and various metrics of the shape of the 

observed data. 

Logarithmic transformations are commonly used in 

statistical analyses of streamflow data (Stedinger and others, 

1993). For example, logarithmic transformations of strictly 

positive hydrologic data are done to avoid conditional 

probability adjustment for the zero values; values equal 

to zero must be offset to avoid using a logarithm of zero. 

A mathematical benefit of using logarithmic transformation 

is that probability distributions with infinite lower and upper 

limits become applicable. Logarithmic transformations of 

annual mean streamflows at each of the 35 unregulated 

streamgages were done to compute summary statistics 

that have utility in other applications. An arbitrary value 

of 10 ft3/s was added to the streamflows for each of the 35 

streamgages lacking statistically significant trends prior to 

logarithmic transformation to accommodate mean annual 

streamflows equal to zero (no flow). These data are referred 

to as the “offset annual mean streamflow.” The offsetting 

along the real-number line permits direct use of logarithmic 

transformations without the added complexity of conditional 

probability adjustment for zero values in magnitude and 

frequency analyses (Hosking and Wallis, 1997, p. 151–152; 

Asquith, 2011, p. 283–285).
8 Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow for 38 Selected Long-Term USGS Streamgages

Table 2. Summary of mean annual streamflow, Kendall’s tau and associated p-value (probability value), and declaration of statistical 

significance with respect to trend direction for the selected long-term U.S. Geological Survey streamgages in Texas.

[ft3/s, cubic feet per second; p-value, probability value; --, not applicable because trend detected; (number), indicates the Theil slope in cubic feet per second per 

year]

Sequence

number

Station 

number

Mean annual 

streamflow

(ft3/s)

Station 

record length

(years)

Kendall’s 

tau

p-value of 

Kendall’s tau

Statistically 

significant at 0.02 

significance level

1 07299670 -- 50 0.340 <0.001 yes (+0.40)

2 07343000 260.5 63 0.100 0.245 no

3 07346070 529.8 66 −0.004 0.965 no

4 08017200 65.5 53 −0.019 0.842 no

5 08017300 78.9 53 0.044 0.645 no

6 08029500 130.3 60 0.184 0.038 no

7 08041500 877.5 73 0.065 0.418 no

8 08068500 -- 73 0.203 0.011 yes (+2.72)

9 08070000 235.5 73 0.040 0.614 no

10 08070500 81.3 68 0.113 0.172 no

11 08078000 115.2 53 −0.032 0.736 no

12 08079600 29.3 50 0.015 0.880 no

13 08082700 7.3 49 −0.012 0.911 no

14 08084800 34.8 50 −0.012 0.900 no

15 08101000 98.2 62 0.021 0.813 no

16 08115000 35.9 62 0.094 0.280 no

17 08117500 528.1 57 0.065 0.474 no

18 08123800 23.2 54 −0.191 0.042 no

19 08134000 -- 88 −0.412 <0.001 yes (−0.24)

20 08150000 194.8 92 0.034 0.632 no

21 08150800 19.1 49 −0.043 0.674 no

22 08151500 371.5 73 0.010 0.901 no

23 08153500 198.7 73 0.065 0.418 no

24 08164000 375.1 74 0.124 0.117 no

25 08167500 369.5 90 0.147 0.040 no

26 08171000 141.4 85 0.138 0.061 no

27 08175000 130.7 57 −0.016 0.863 no

28 08186000 141.9 81 0.079 0.296 no

29 08189500 126.0 73 0.115 0.150 no

30 08190000 160.9 89 0.077 0.288 no

31 08190500 33.1 67 0.036 0.665 no

32 08192000 137.2 73 0.124 0.121 no

33 08195000 122.9 87 0.127 0.082 no

34 08196000 34.8 60 −0.008 0.924 no

35 08198000 63.5 70 0.149 0.069 no

36 08200000 43.1 60 0.032 0.716 no

37 08205500 145.3 97 0.049 0.481 no

38 08208000 119.1 79 −0.141 0.066 no
Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow 9

L-moments are a type of statistic used to quantify 

the location, scale, and shape of probability distributions 

(Hosking, 1990; Hosking and Wallis, 1997; Asquith, 2011). 

The first six sample L-moments of the base-10 logarithms of 

the offset annual mean streamflow were computed by using 

the lmoms(..., nmom=6) function in the lmomco package for 

R by Asquith (2013). To clarify, the nmom=6 in the argument to 

the lmoms() function is a command to R to compute the first 

six L-moments. The sample L-moments are listed in table 3. 

The L-moments are not listed for the three streamgages for 

which statistically significant trends were detected because 

of a concern that the year-over-year distribution of the data 

shows a monotonic trend. The columns titled “L-scale” and 

“L-CV” can each be computed from the other by the equation 

L-CV = L-scale / mean, where L-scale is analogous (but not 

numerically equivalent) to standard deviation, and L-CV is the 

coefficient of L-variation (table 3). Viglione (2010, p. 2231) 

provides the following explanation of the L-CV coefficient 

and its utility in hydrologic studies: 

* * * the sample coefficient of variation (CV), that 

is, the ratio of standard deviation to the mean of a 

series of data, exhibits substantial bias and variance 

when samples are small or belong to highly skewed 

populations (Vogel and Fennessey, 1993). This is the 

problem that is normally encountered in hydrology 

when dealing with floods or extreme rainfall 

events. The coefficient of L-variation (L-CV) is 

another—more efficient in many cases—measure 

of data dispersion introduced by Hosking (1990). It 

has hence replaced the conventional CV in various 

applications of statistical hydrology.

The sampling variances of each corresponding L-moment 

were used to compute regional or study-area values for the 

L-moments through weighted mean computation. Wang 

and Hutson (2013) describe an exact analytical bootstrap 

method for computation of sampling variance of L-moments. 

Unfortunately, the available years of record for each of the 

35 streamgages is so large as to produce severe numerical 

problems in matrices in their method. In order to compute 

sampling variances for each of the sample L-moments for 

each streamgage, replacement bootstrap simulation was 

done by using the sample(..., replace=TRUE) function 

in R, which shows that replacement sampling was used. For 

estimation of the sampling variance, 10,000 replications for 

the corresponding sample size for each streamgage were 

used. An overview of bootstrap simulation is found in Rizzo 

(2008, p. 183–190), Ugarte and others (2008), and Verzani 

(2005). The documentation within the R environment provides 

further information. For each bootstrap sampling, the first six 

L-moments were retained, and subsequently the variances 

of the 10,000 replicates of each of the six L-moments were 

computed. These sampling variances of the L-moments are 

listed in table 3.

The following values represent the weighted means 

of the L-moments (the regional L-moments) found by 

using the inverse of their respective variances as weight 

factors: mean = 1.9679, L-scale = 0.1706, L-CV = 0.0857, 

L-skew = −0.0200, L-kurtosis = 0.0759, Tau5 = −0.0030, 

and Tau6 = 0.0264. In turn, these regional L-moments can 

provide a basis for parameter estimation of a probability 

distribution to model the magnitude and frequency of annual 

mean streamflow as described by the example in the section 

that follows.

An Example Application of Regional Analysis of 

Annual Mean Streamflow Frequency

An example application of the L-moments listed in 

table 3 is provided in this section. In this example, the 

regional L-moments of annual mean streamflow in Texas 

are used to quantify the scale and shape of a dimensionless 

regional probability distribution. The adjective “regional” 

indicates that the weighted mean L-moments for the region 

loosely defined by the spatial scale that is congruent with 

the locations of the 35 streamgages and the respective 

watersheds for each streamgage. The structure of the 

probability distribution (Hosking and Wallis, 1997) is useful 

in regional magnitude and frequency analyses (Asquith and 

Roussel, 2009). The 4-parameter asymmetric exponential 

power (AEP) distribution was chosen from more than 

20 possible distribution candidates (Asquith, 2011, 2013). 

The AEP distribution and its L-moments are thoroughly 

described by Asquith (2014). The mathematical definitions 

for the AEP distribution are complex and are not listed in 

this report because they are not directly germane to the 

example application. The AEP distribution best reproduces the 

dimensionless regional L-moments with the minor change 

of setting the mean to unity and setting the regional L-scale 

to equal the L-CV, which removes dimension from the 

distribution (Hosking and Wallis, 1997). The dimensionless 

regional L-moments, which are used to fit the AEP, are 

mean = 1, L-scale = 0.0857, L-skew = −0.0200, and 

L-kurtosis = 0.0759. 

The parameters for the dimensionless regional AEP 

distribution defined per nomenclature of Asquith (2014) 

are xi = 1.028 (location parameter), alpha = 0.245 (scale 

parameter), kappa = 1.121 (shape parameter), and h = 3.064 

(shape parameter). These parameters can be written in short 

hand as AEP(1.028, 0.245, 1.121, 3.064) or alternatively as 

AEP(Θreg) where Θ is a vector of the four parameters. This 

alpha parameter nomenclature is the same as the nomenclature 

of Asquith (2014) and elsewhere in L-moment literature for 

scale parameters (Hosking and Wallis, 1997) but is not the 

same as the statistical significance level defined in the context 

of Kendall’s tau.
10 Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow for 38 Selected Long-Term USGS Streamgages

Table 3. Summary of first six L-moments of base-10 logarithms of offset annual mean streamflow and sampling variances computed by 

bootstrap simulation of 10,000 replications for the selected long-term U.S. Geological Survey streamgages in Texas.

[ft3/s, cubic feet per second; log, base-10 logarithm; “10”, 10 cubic feet per second added; L-CV, coefficient of L-variation (L-scale/mean); L-skew, L-moment 

measurement of skew; L-kurtosis, L-moment measurement of kurtosis; Tau5, unnamed L-moment; Tau6, unnamed L-moment; --, not applicable because trend 

detected]

Sequence

number

Station

number

L-moments (Hosking, 1990) of base-10 logarithms of annual mean streamflow plus “10” cubic feet per second

Mean log10

(ft3/s)

L-scale log10

(ft3/s) L-CV L-skew L-kurtosis Tau5 Tau6

1 07299670 -- -- -- -- -- -- --

2 07343000 2.3671 0.1446 0.0611 −0.1454 0.1130 −0.0418 0.0154

3 07346070 2.6462 0.1690 0.0639 −0.1749 0.1025 −0.0335 0.0313

4 08017200 1.8163 0.1398 0.0769 −0.1386 0.1498 −0.0310 0.0276

5 08017300 1.8907 0.1358 0.0718 −0.1429 0.1444 −0.0713 0.0488

6 08029500 2.1062 0.1119 0.0531 −0.0643 0.0536 −0.0345 0.0704

7 08041500 2.8710 0.1565 0.0545 −0.1024 0.1013 −0.0508 0.0524

8 08068500 -- -- -- -- -- -- --

9 08070000 2.2813 0.1898 0.0832 −0.1166 0.0754 −0.0160 0.0263

10 08070500 1.8954 0.1415 0.0747 −0.0442 0.0674 −0.0062 −0.0232

11 08078000 2.0445 0.1261 0.0617 −0.0707 0.1415 −0.0091 0.0768

12 08079600 1.5276 0.1403 0.0918 0.0043 0.0700 0.0033 0.0368

13 08082700 1.1804 0.1038 0.0880 0.3865 0.1496 0.0659 0.0354

14 08084800 1.5509 0.1693 0.1091 0.0726 0.0644 −0.0271 0.0036

15 08101000 1.8438 0.2408 0.1306 0.0120 0.0798 −0.0024 0.0074

16 08115000 1.6087 0.1278 0.0794 −0.0280 0.0567 0.0031 0.0286

17 08117500 2.6346 0.1788 0.0679 −0.1486 0.0967 −0.0330 0.0683

18 08123800 1.4528 0.1368 0.0941 0.0769 0.0917 0.0180 0.0326

19 08134000 -- -- -- -- -- -- --

20 08150000 2.2223 0.1602 0.0721 0.0104 0.1223 −0.0050 0.0104

21 08150800 1.3960 0.1294 0.0927 0.1909 0.1109 0.0314 −0.0027

22 08151500 2.4846 0.1705 0.0686 −0.0412 0.1462 −0.0136 0.0311

23 08153500 2.1612 0.2268 0.1049 −0.0824 0.0924 0.0249 0.0221

24 08164000 2.4218 0.2353 0.0972 −0.0974 0.0895 −0.0292 0.0423

25 08167500 2.4246 0.2196 0.0906 −0.0517 0.0829 0.0272 0.0375

26 08171000 2.0403 0.2126 0.1042 −0.0683 0.0401 0.0476 0.0266

27 08175000 1.9628 0.2455 0.1251 −0.0088 0.0054 −0.0448 0.0246

28 08186000 2.0409 0.2044 0.1002 0.0328 0.0440 −0.0224 0.0111

29 08189500 1.9499 0.2530 0.1297 −0.0960 0.0222 0.0001 0.0102

30 08190000 2.1414 0.1644 0.0768 0.0026 0.0701 −0.0175 0.0583

31 08190500 1.4167 0.2314 0.1634 0.2303 −0.0244 0.0090 0.0317

32 08192000 1.9386 0.2758 0.1423 −0.0176 0.0068 −0.0028 0.0357

33 08195000 2.0204 0.1758 0.0870 −0.0286 0.0962 0.0092 0.0494

34 08196000 1.5754 0.1520 0.0965 −0.0184 0.0331 0.0371 0.0407

35 08198000 1.7184 0.2122 0.1235 −0.0119 0.0757 0.0185 0.0209

36 08200000 1.5783 0.2087 0.1322 0.0289 0.0373 0.0541 0.0213

37 08205500 1.9362 0.2691 0.1390 0.0359 0.1058 0.0215 0.0127

38 08208000 1.9368 0.2245 0.1159 0.0482 0.1019 −0.0520 −0.0080
Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow 11

Table 3. Summary of first six L-moments of base-10 logarithms of offset annual mean streamflow and sampling variances computed by 

bootstrap simulation of 10,000 replications for the selected long-term U.S. Geological Survey streamgages in Texas.—Continued

[ft3/s, cubic feet per second; log, base-10 logarithm; “10”, 10 cubic feet per second added; L-CV, coefficient of L-variation (L-scale/mean); L-skew, L-moment 

measurement of skew; L-kurtosis, L-moment measurement of kurtosis; Tau5, unnamed L-moment; Tau6, unnamed L-moment; --, not applicable because trend 

detected]

Sequence

number

Station

number

Sampling variances of the L-moments shown to left multiplied by 1,000

Mean L-scale L-CV L-skew L-kurtosis Tau5 Tau6

1 07299670 -- -- -- -- -- -- --

2 07343000 1.0268 0.1744 0.0365 2.1661 1.6174 0.9672 0.6098

3 07346070 1.3654 0.2262 0.0391 2.3555 1.3247 0.7549 0.6181

4 08017200 1.1728 0.2143 0.0782 2.8796 2.0802 1.3552 0.9856

5 08017300 1.1371 0.2140 0.0719 3.4830 2.0398 1.6238 1.4941

6 08029500 0.6316 0.0762 0.0189 3.8768 1.5516 1.0166 0.6671

7 08041500 1.0071 0.1705 0.0234 2.8065 1.4442 0.8783 0.7308

8 08068500 -- -- -- -- -- -- --

9 08070000 1.4934 0.1998 0.0466 2.2141 1.1906 0.5595 0.3963

10 08070500 0.8874 0.0991 0.0303 1.6456 1.4558 0.8203 0.3316

11 08078000 0.9281 0.1615 0.0425 4.0349 1.5851 1.1388 0.9835

12 08079600 1.1544 0.1376 0.0637 3.6910 2.0659 0.9786 0.7691

13 08082700 0.8264 0.2289 0.1219 3.6585 3.3076 2.7468 2.1709

14 08084800 1.7267 0.1965 0.0745 3.2731 2.3122 1.2978 1.4558

15 08101000 2.7620 0.3206 0.1068 2.1669 1.3869 0.6844 0.5098

16 08115000 0.7691 0.0835 0.0362 2.7020 1.1507 0.7239 0.6441

17 08117500 1.7246 0.3042 0.0534 3.9754 2.1005 1.3182 0.9087

18 08123800 1.0524 0.1411 0.0612 3.0657 1.6648 0.9377 0.7602

19 08134000 -- -- -- -- -- -- --

20 08150000 0.8529 0.1245 0.0264 1.4044 1.0512 0.5321 0.2925

21 08150800 1.0801 0.1841 0.0732 2.7521 2.6873 1.8102 0.9666

22 08151500 1.2067 0.2080 0.0358 2.0595 1.1255 0.6208 0.6459

23 08153500 2.1687 0.2844 0.0779 2.1140 1.3178 0.7534 0.6204

24 08164000 2.2616 0.3319 0.0707 2.3964 1.2519 0.6907 0.4730

25 08167500 1.5916 0.2124 0.0426 2.0788 1.2042 0.6125 0.3990

26 08171000 1.5763 0.1581 0.0498 2.1868 1.1581 0.5061 0.3648

27 08175000 3.0348 0.2415 0.0779 3.6633 1.3895 0.7950 0.7083

28 08186000 1.5259 0.1565 0.0369 2.0421 1.2120 0.4957 0.4288

29 08189500 2.6382 0.2358 0.0953 2.4360 1.0608 0.5439 0.3429

30 08190000 0.9267 0.1106 0.0249 2.1805 0.7747 0.4459 0.3389

31 08190500 2.5717 0.2882 0.0669 4.4999 1.3656 0.8441 0.4138

32 08192000 2.9885 0.2324 0.0819 2.7588 0.7720 0.5253 0.3627

33 08195000 1.0649 0.1533 0.0404 2.0701 0.8579 0.4403 0.3310

34 08196000 1.1160 0.1035 0.0476 3.2074 1.1673 0.6816 0.4747

35 08198000 1.8678 0.2225 0.0863 2.2118 1.1697 0.5825 0.4580

36 08200000 2.1613 0.2098 0.0895 3.0601 1.3689 0.5828 0.4964

37 08205500 2.2598 0.3152 0.0847 1.3356 0.8439 0.4872 0.2727

38 08208000 1.9158 0.2591 0.0692 1.5705 1.5771 0.6733 0.4149
12 Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow for 38 Selected Long-Term USGS Streamgages

The theoretical L-moments for AEP(Θreg) fit to the 

regional L-moments (Asquith, 2014) are mean = 1, L-scale 

= 0.0857, L-CV = 0.0857, L-skew = −0.0200, L-kurtosis = 

0.0759, Tau5 = −0.00480, and Tau6 = 0.0297. A comparison of 

these theoretical L-moments to the regional L-moments used 

to fit the AEP indicates that the AEP is a suitable distribution 

because of the “closeness” and “correct sign” of Tau5 (the 

5th L-moment) (−0.0030 compared to −0.00480) as well as 

Tau6 (0.0264 compared to 0.0297). The fifth and sixth AEP 

L-moments are almost identical to the regional L-moments 

even though the AEP has just four parameters and directly fits 

the first four L-moments but also well approximates the fifth 

and sixth L-moments, which justifies the choice of the AEP 

over alternative 4-parameter distributions such as the kappa 

distribution (Hosking and Wallis, 1997; Asquith, 2011; Karian 

and Dudewicz, 2010) and the generalized lambda distributions 

(Asquith, 2011, 2014; Karian and Dudewicz, 2010). In fact, 

although the regional AEP in this instance comprises just four 

parameters, it achieves a remarkable degree of fit through the 

sixth regional L-moment.

By using the lmomco software package described by 

Asquith (2013), an AEP(1.028, 0.245, 1.121, 3.064) of the 

regional distribution can be solved for a given nonexceedance 

probability (0.75 or the 75th percentile) with the following 

R code sequence:

library(lmomco)

AEP <- c(1.028, 0.245, 1.121, 3.064)

Q75 <- quaaep4(0.75, vec2par(AEP, type=”aep4”))

The 75th percentile (Q75) is about 1.11 (dimensionless).

The probability density function (PDF) of the regional 

AEP(Θreg) distribution is shown in the inset in figure 3 

as the dashed red line. Inspection of the PDF in the figure 

inset shows that the regional AEP distribution has slight 

asymmetry of about 1 (unity) and a flatter central part than 

that for the site-specific PDF. If the original data were lognormally distributed, then their logarithms would be normally 

distributed and thus would plot as a nearly straight line on a 

logarithmic (vertical axis) and probability (horizontal axis) 

plot. The frequency curves in figure 3 indicate that the data for 

the long-record streamgages have inherent deviance from the 

log-normal distribution.

The average number of years of record (about 69 years) 

for the 35 streamgages lacking trends in streamflow in Texas 

can be used as a weight factor in combining the regional 

Figure 3. Comparison of three annual mean frequency curves in probability-density (inset) and quantile-function (main plot) form for 

U.S. Geological Survey streamgage 08123800 Beals Creek near Westbrook, Texas (table 1), associated with example application.

1.7

Logarithms of annual mean streamflow

0.5 1 1.5

Annual nonexceedance probability

0.2 0.5 1 2 5 10 15 20 30 40 50 60 70 80 85 90 95 98 99 99.5 99.8

Annual mean streamflow, in cubic feet per second

0.2

0.3

0.4

0.5

0.6

0.8

1

2

3

4

5

6

8

10

20

30

40

50

60

80

100

200

Probability density

0

1

2

2.5

INSET Dimensionless

distributions

Unity

Possible probability range of underprediction, 

but increasing sample size in time might 

readily produce a different result.{

The term regional is to encompass a conceptual spatial 

scale that is congruent with the distribution of the 35 

streamgages and their respective watersheds.

AEP() is the asymmetric exponential power distribution 

for the four parameters listed within the parentheses.

EXPLANATION

Weighted curve

Site-specific AEP (1.315, 0.343, 0.688, 2.500) curve

Regional AEP (1.028, 0.245, 1.121, 3.064) curve (dimensionless)

 multiplied by the site-specific (logarithmic) mean

Annual mean streamflow (54 years of record)
Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow 13

AEP distribution with a site-specific distribution from 

the observed annual mean streamflow data for a given 

streamgage. A weighted-frequency curve could enhance the 

reliability of regional annual mean streamflow frequency 

analysis for other streamgages in Texas as well as those in this 

investigation.

For the simplified analysis for this example, the regional 

AEP distribution is assumed applicable throughout the study 

area. A more refined analysis following Hosking and Wallis 

(1997) guidelines might consider the potential for smaller 

regions of similar rainfall, topography, geology, and other 

potential factors; or the analysis might provide for systematic 

explanation of regional (that is, spatial) variation in the 

frequency curve attributable to potential explanatory variables 

as listed.

Weighted-frequency analysis recognizes that the 

available record for most streamgages other than those for 

this investigation is comparatively short for reliable distaltail (extreme, and therefore rare, low- and high-streamflow) 

estimation. Weighted-frequency analysis incorporates the 

blending of a regional frequency curve with a site-specific 

curve that is based on the available data at a particular 

streamgage. A general algorithm for weighted-frequency 

analysis based on the AEP(Θreg) can be derived that 

hydrologists and engineers might find useful for investigating 

the annual mean streamflow in Texas for unregulated 

watersheds. The steps for using the algorithm with the data 

from a given streamgage in an unregulated watershed are as 

follows:

1. Acquire the annual mean streamflow data (X) in cubic 

feet per second for the selected streamgage and record the 

length of record (m) for a later step;

2. Add 10 ft3/s (an arbitrary value) to the annual mean 

streamflow data as protection against the potential for 

zero-flow annual values in step 1: X+10;

3. Compute the base-10 logarithmic transformation of 

the offset annual mean streamflow data in step 2: 

X = log10(X+10);

4. Compute the first six L-moments (mean through Tau6, 

[lmoms(X, nmom=6)], Asquith [2013]) of the data in step 

3 and particularly record the “site-specific mean” (mu) for 

a later step;

5. (Optionally perform a subtask of candidate distribution 

evaluation and use the chosen distribution). Fit the AEP or 

other distribution to the L-moments (this is known as the 

method of L-moments [Hosking, 1990; Asquith, 2011]) 

computed in step 4 to create a site-specific frequency 

curve and let this distribution be known as S;

6. Let R be the regional annual mean frequency curve 

computed from AEP(Θreg) multiplied by the site-specific 

mean mu of step 4;

7. Compute a weighted combination Q of the dimensionless 

distributions for nonexceedance probabilities or quantiles 

of interest better stated as the straightforward proration 

Q = (mS+69R) / (m+69), and note that Q remains in 

logarithmic space;

8. Detransform the frequency curve in step 7: Q = 10Q;

9. Subtract 10 ft3/s: Q = Q − 10; and lastly

10. Set all negative values to zero, which is an ad hoc 

truncation to physical constraints, by using a vector 

operation that all Q[Q < 0] = 0.

Quantiles of interest in step 7 might be, for example, the 

0.02 annual nonexceedance probability that represents a year 

of drought or a 0.98 annual nonexceedance probability that 

represents streamflow during a water year having abundant 

rainfall relative to long-term climatic expectations. For 

graphical purposes, a sequence of quantiles for nonexceedance 

probabilities between say 0.01 and 0.99 in increments of 0.01 

would be useful to draw smoothly varying frequency curves.

An example application of the algorithm used with USGS 

streamgage 08123800 Beals Creek near Westbrook, Tex., 

is shown in figure 3, in which the 54 years of annual mean 

streamflow data for this streamgage are depicted.

The site-specific AEP or S (step 5 of the algorithm) 

fit by L-moments of the offset base-10 logarithms is 

AEP(1.315, 0.343, 0.688, 2.500). The site-specific L-moments 

are mean = 1.4528, L-scale = 0.1368, L-CV = 0.0941, 

L-skew = 0.0769, L-kurtosis = 0.0917, Tau5 = 0.0180, and 

Tau6 = 0.0326 (table 3). The site-specific frequency curve 

S is shown as the dashed black line in figure 3 as well as its 

dimensionless PDF (mean = 1, L-scale = 0.0941) in the inset. 

As anticipated, S closely approximates the actual distribution 

of the 54 observed values.

The offset base-10 logarithmic mean of the streamflow 

data for USGS streamgage 08123800 Beals Creek near 

Westbrook, Tex., is 1.452, so R is 1.452 multiplied by 

AEP(Θreg) (table 3). The resulting regional frequency curve 

or R (step 6 of the algorithm) is shown as the dashed red line 

in figure 3 as well as its dimensionless PDF (dimensionless 

meaning that the mean = 1, L-scale = 0.0857) in the inset.

Comparison of the PDFs in the inset in figure 3 shows 

that the two PDFs are reasonably similar to each other. The 

dashed black line results from 54 years of data, whereas the 

dashed red line results from a total of 2,375 total years of data 

(albeit not all of the individual years are truly independent of 

the others).

Regional information about the geometry and shape of 

annual mean streamflow from other streamgages in Texas can 

be incorporated into the algorithm. The weighted-frequency 

curve Q (step 10 of the algorithm) is shown as the black line 

in figure 3 and clearly resides between the two dashed curves 

as expected. The weighed frequency curve might provide 

preferable quantile estimation for years with extremely rare 

annual mean streamflow because of the inclusion or pooling 

of information from the 35 long-term streamgages that lack 

statistically significant trends in annual mean streamflow.
14 Trend Analysis and Selected Summary Statistics of Annual Mean Streamflow for 38 Selected Long-Term USGS Streamgages

The enumerated steps related to the example application 

just described are an overview of a relatively simplified 

method for estimating mean streamflow frequency. The result 

is an outline of a technique that hydrologists and engineers 

might find useful for evaluating annual mean streamflow 

in Texas for unregulated watersheds; for example, the 

dimensionless regional frequency curve shown in figure 3 

(when multiplied by the estimated mean annual streamflow 

from the equations by Lanning-Rush [2000]) could be used 

to evaluate the magnitude and frequency of annual mean 

streamflow. The example application uses a single regional 

frequency curve for illustration purposes. By incorporating 

additional variables to account for spatial variation in 

rainfall (variation in mean annual precipitation for instance), 

topography, geology, and other factors that influence the 

magnitude and frequency of annual mean streamflow, the 

reliability of the method described in this report could be 

enhanced (Hosking and Wallis, 1997; Nuñez and others, 

2011).

Summary

In 2013, the U.S. Geological Survey (USGS) operated 

more than 500 continuous streamgages (streamflow-gaging 

stations) in Texas. In cooperation with the Texas Water 

Development Board, the USGS evaluated mean annual 

streamflow data for 38 selected streamgages that were active 

as of water year 2012. The 38 streamgages have annual mean 

streamflow data anticipated to be representative of watersheds 

through 2013 that can be considered natural and unregulated. 

Long-term streamflow data representative of natural and 

unregulated watershed conditions are useful for evaluation of 

changes (trends) in streamflow attributable to systematic landuse changes, changes in groundwater gains or losses, climate 

cycles and changes, statistical vagaries, and other influencing 

factors. Annual mean streamflow data are especially diagnostic 

because these statistics can be used to compute mean annual 

streamflow over long periods of time and represent the 

expectation of the annual volume of water measured at a 

streamgage. The 38 streamgages have at least 49 and as 

many as 97 cumulative years of annual mean streamflow 

data; the average period of record is 68 years. The minimum 

contributing drainage area (CDA) is about 43 square miles 

(mi2), the mean CDA is about 788 mi2, and the maximum CDA 

is about 4,190 mi2. The streamgages are scattered around the 

central and eastern parts of Texas.

The Kendall’s tau statistical test was used to detect 

monotonic trends in annual mean streamflow over time. Three 

stations have statistically significant trends. The monotonic 

trend analysis detected 2 statistically significant upward 

trends (0.01 one-tail significance), 1 statistically significant 

downward trend (0.01 one-tail significance level), and 

35 instances of no statistically significant trend (0.02 twotailed significance level). These results are not surprising 

given that the streamgages were screened to only include 

those where the streamflow was thought to be natural and 

unregulated for the entire period of record at each station. 

For the three stations where trends in streamflow were 

detected, the Theil slope estimate of a regression slope of 

annual mean streamflow with time was computed in units 

of cubic feet per second (ft3/s). Two increasing Theil slopes 

were measured (+0.40 and +2.72 ft3/s per year, respectively), 

and one decreasing Theil slope (−0.24 ft3/s per year) was 

measured.

Selected summary statistics (L-moments) and sampling 

variances for the 35 streamgages lacking statistically 

significant trends were computed. These statistics of annual 

mean streamflow for each streamgage represent a collection 

of different numerical values that together describe the 

location, scale, and various metrics of shape of the observed 

data. An example application of selected summary statistics 

(regional L-moments) to estimate the magnitude and 

frequency of annual mean streamflow is provided. In this 

example, the regional L-moments of annual mean streamflow 

in Texas are used to quantify the scale and shape of a 

dimensionless regional probability distribution. The term 

“regional” is intended to encompass a conceptual spatial scale 

that is congruent with the distribution of the 35 streamgages 

with natural and unregulated streamflow and lacking 

statistically significant trends. The 4-parameter asymmetric 

exponential power (AEP) distribution was chosen from more 

than 20 possible distribution forms. The example application 

uses a single regional frequency curve for illustration 

purposes. A technique was derived that hydrologists and 

engineers might find useful for evaluating annual mean 

streamflow in Texas for unregulated watersheds, and the steps 

for using the algorithm with the data from a given streamgage 

are described. More refined dimensionless regional frequency 

curves that account for spatial variation in rainfall (variation in 

mean annual precipitation for instance), topography, geology, 

and other potential factors could be developed by following 

the process outlined in the example application. 

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Publishing support provided by

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http://tx.usgs.gov/
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