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Site-Specific Estimation of Peak-Streamflow Frequency Using Generalized Least-Squares Regression for Natural Basins in Texas

## Water-Resources Investigations Report 99-4172 By William H. Asquith and  Raymond M. Slade, Jr.

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## Contents

- Abstract
- Introduction
- Purpose and ScopeHydrologic RegionalizationBasin CharacteristicsGeneralized Least-Squares and Other Regression Methods in Studies of Peak-Streamflow FrequencySite-Specific Estimation of Peak-Streamflow FrequencySite-Specific ApproachComputer ProgramExampleEvaluation of Site-Specific Regionalization of Peak-Streamflow FrequencySummarySelected ReferencesAppendixTables- Table 1.  Summary of site-specific peak-streamflow frequency estimation for example in text 
- Table 2.  Evaluation of site-specific peak-streamflow frequency options
- Table 3.  Weighted root-mean-square errors (RMSE) for options of site-specific peak-streamflow frequency
- Table 4.  Mean of the five largest station errors (M5LM) for options of site-specific peak-streamflow frequency
- Table 5.  Root-mean-square errors between peak-streamflow frequency analysis using the traditional regional regression approach and selected options of the site-specific peak-streamflow frequency approach
Computer Program- For Windows NT:  txfreq.zip
- For UNIX Systems:  txfreq.tar.gz

U.S. DEPARTMENT OF THE INTERIORBruce Babbitt, Secretary
U.S. GEOLOGICAL SURVEYCharles G. Groat, Director
 
Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government.
 
For additional information write to:

District ChiefU.S. Geological Survey8027 Exchange Dr.Austin, TX 78754.4733Email: mailto:[email protected]

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U.S. Geological SurveyBranch of Information ServicesBox 25286Denver, CO 80225.0286Email: [email protected]
ABSTRACT
The U.S. Geological Survey, in cooperation with the Texas Department of Transportation, has developed a computer program to estimate peak-streamflow frequency for ungaged sites in natural basins in Texas. Peak-streamflow frequency refers to the peak streamflows for recurrence intervals of 2, 5, 10, 25, 50, and 100 years. Peak-streamflow frequency estimates are needed by planners, managers, and design engineers for flood-plain management; for objective assessment of flood risk; for cost-effective design of roads and bridges; and also for the desin of culverts, dams, levees, and other flood-control structures. The program estimates peak-streamflow frequency using a site-specific approach and a multivariate generalized least-squares linear regression. A site-specific approach differs from a traditional regional regression approach by developing unique equations to estimate peak-streamflow frequency specifically for the ungaged site. The stations included in the regression are selected using an informal cluster analysis that compares the basin characteristics of the ungaged site to the basin characteristics of all the stations in the data base. The program provides several choices for selecting the stations. Selecting the stations using cluster analysis ensures that the stations included in the regression will have the most pertinent information about flooding characteristics of the ungaged site and therefore provide the basis for potentially improved peak-streamflow frequency estimation. An evaluation of the site-specific approach in estimating peak-streamflow frequency for gaged sites indicates that the site-specific approach is at least as accurate as a traditional regional regression approach.
INTRODUCTION
Peak-streamflow frequency refers to the peak streamflows for recurrence intervals of 2, 5, 10, 25, 50, and 100 years. Peak-streamflow frequency estimates are needed by planners, managers, and design engineers for flood-plain management; for objective assessment of flood risk; for cost-effective design of roads and bridges; and for the design of culverts, dams, levees, and other flood-control structures. In 1991, the U.S. Geological Survey (USGS), in cooperation with the Texas Department of Transportation, began an investigation of floods in Texas. The primary objective of the investigation was to develop techniques and procedures to estimate peak-streamflow frequency for streams with natural drainage basins in Texas. A natural basin is defined as a basin with less than 10 percent impervious cover, less than 10 percent of its drainage area controlled by reservoirs, and no other human-related factors such as land use changes that would measurably affect peak streamflow. Two techniques that accomplish the primary objective are available: The first technique involves "traditional" regional regression equations for 11 geographically defined regions in Texas (Asquith and Slade, 1997). The second technique, referred to as the site-specific approach, is the subject of this report. 
Purpose and Scope
The purpose of this report is to document a computer program for site-specific peak-streamflow frequency estimation for ungaged sites in natural basins of Texas. The computer program regresses peak-streamflow frequency on basin characteristics of selected stations and produces various diagnostic calculations on the regression. The data base for the computer program, with the exception of climate factors, is documented in Asquith and Slade (1997). The data base contains the peak-streamflow frequency estimated with data through the 1993 USGS water year(1), selected basin characteristics, and ancillary information for each of 559 streamflow-gaging stations in Texas and for each of 105 stations in Arkansas, Louisiana, New Mexico, and Oklahoma. Both the in-state and out-of-state stations met the natural-basin criteria at the time that the data included in Asquith and Slade (1997) were collected. All stations have at least 8 years of annual peak-streamflow data from natural basins. Asquith and Slade (1997, pl. 1) provides a map of Texas showing the locations of the stations. Climate factors involved in development of the computer program were derived from Lichty and Karlinger (1990). 
Hydrologic Regionalization
Hydrologic regionalization is the method by which peak-streamflow frequency information for similar stations is combined to (1) provide more information about the flooding characteristics, which improves the accuracy of peak-streamflow frequency estimates over those derived solely from the data for one station, and (2) provide procedures to estimate peak-streamflow frequency for ungaged stream sites. Traditionally, hydrologic regionalization of peak-streamflow frequency is used to establish regional hydrologic regression equations. The regression establishes the statistical relation between peak-streamflow frequency and selected basin characteristics. In regression, the 2-, 5-, 10-, 25-, 50-, and 100-year peak streamflows are used as dependent variables, and selected basin characteristics are used as independent variables. Logarithmic transformations of the dependent and independent variables commonly are done to increase the linearity between the dependent and independent variables. Numerous references are available describing background, theoretical aspects, application, and problems related to statistical peak-streamflow estimation. The reader is directed to the "Selected References" section. Excellent starting points in the literature are Cunnane (1989) and Stedinger and others (1993, p. 18.33).
Basin Characteristics
Eight independent variables (basin characteristics) were selected for investigation of hydrologic regression for Texas: the 2-year 24-hour precipitation, mean annual precipitation, contributing drainage area, basin shape factor, and stream slope; and the 2-, 25-, and 100-year climate factors (Lichty and Karlinger, 1990). The following basin characteristics are discussed in Asquith and Slade (1997, p. 7): 2-year 24-hour precipitation, mean annual precipitation, contributing drainage area, basin shape factor, and stream slope. The 2-year 24-hour precipitation (Hershfield, 1962) and mean annual precipitation (based on 1951-80 data [Larkin and Bomar, 1983]) were determined for the approximate centroid of each basin and are expressed in inches. The 2-year 24-hour precipitation and other precipitation variables for Texas have been updated (Asquith, 1998a), but the updates were too late to be included in the peak-streamflow frequency study by Asquith and Slade (1997) and in the current report. After extensive testing, the authors decided that the precipitation variables and climate factors would not be implemented in the user interface of the computer program for this application. However, they are retained in program logic, evaluated, and shown in tables 2 and 3 to keep the program adaptable for future applications. Some of the extensive testing is documented in "Evaluation of Site-Specific Regionalization of Peak-Streamflow Frequency" in this report. 

The contributing drainage area is expressed in square miles. The basin shape factor is the ratio of the square of the stream length to the contributing drainage area, which mathematically represents the ratio of the longest stream length to the mean width of the basin. The stream length for the stream-slope calculation is defined from the longest mapped channel from the station to the headwaters, on the basis of USGS 1:100,000 quadrangle maps. The average stream slope, expressed in feet per mile, is the ratio of (1) the change in elevation of the longest mapped channel from the station to the headwaters to (2) the length of the longest mapped channel.

Maps of 2-, 25-, and 100-year climate factors (Lichty and Karlinger, 1990) were developed for the United States east of the Rocky Mountains and south of approximately latitude 45� N using data from 71 long-term rainfall stations and regression analysis that related rainfall-runoff estimates of peak-streamflow frequency from 50 model calibrations to basin characteristics. The 2-, 25-, and 100-year climate factors were estimated for each of the 664 stations by interpolation of a gridded data base derived from climate-factor contour maps. The interpolation was done using Fortran algorithms provided by Gary D. Tasker (U.S. Geological Survey, written commun., 1996). These algorithms are incorporated in the computer program presented in this report. Only 2-, 25-, and 100-year climate factors are documented by Lichty and Karlinger (1990), but six recurrence intervals are discussed in this report. For program development, the 2-year climate factor was used for hydrologic regression of the 2- and 5-year peak streamflows. The 25-year climate factor was used for the 10- and 25-year peak streamflows, and the 100-year climate factor was used for the 50- and 100-year peak streamflows. However, for the final version of the program, the authors decided not to implement the climate factors in the user interface of the computer program for the application presented here. 
Generalized Least-Squares and Other Regression Methods in Studies of Peak-Streamflow Frequency
Generalized least-squares (GLS) regression (Stedinger and Tasker, 1985; Tasker and Stedinger, 1989) is the basis of the hydrologic regression in this report. GLS has advantages over the more widely known regression methods--ordinary least-squares (OLS) and weighted least-squares (WLS). One advantage of GLS regression over OLS is that stations with more data, hence more reliable peak-streamflow frequency estimates, are given more weight in the regression procedures. Another advantage of GLS regression over OLS and WLS is that further adjustments in weight are done on the basis of cross correlations in the data--thus, the undesirable effects of stations with related data (cross-correlated data) are reduced. Neither OLS or WLS regression considers the effects of cross-correlated data. 

Numerous studies of peak-streamflow frequency have been done for various regions of Texas. These studies, although differing in scope, all have used linear regression. The peak-streamflow frequency study for Texas by Schroeder and Massey (1977) used OLS regression. Peak-streamflow frequency studies using WLS have been done for several localities. Such studies include regional equations for the Highland Lakes area (Asquith and others, 1996), the entire State of Texas (Asquith and Slade, 1997), the tributaries of the lower Colorado River (Asquith, 1998b), and the Brazos River Basin (Raines, 1998). A peak-streamflow frequency study using GLS was done for Hays County (Slade and others, 1995). A more basic-research-oriented study of peak-streamflow frequency in Texas by Tasker and Slade (1994) used a GLS regression and focused on the performance of a site-specific regression analysis. Tasker and Slade (1994) demonstrated that the site-specific approach (referred to as "interactive" by Tasker and Slade [1994] and as "region-of-influence" by Hodge and Tasker [1995]) coupled with GLS regression has smaller root-mean-square errors than the traditional geographic regional approach. 
SITE-SPECIFIC ESTIMATION OF PEAK-STREAMFLOW FREQUENCY
Comprehensive discussion of the various approaches and regionalization techniques of peak-streamflow frequency is beyond the scope of this report. The authors assume that the reader has some knowledge and experience with the various aspects of peak-streamflow frequency regionalization. The reader is directed to "Selected References" for further details.
Site-Specific Approach
The site-specific approach for peak-streamflow frequency regionalization, first proposed for Texas by Slade and Smith (1993), differs from the standard or traditional regional regression approach in that equations are developed to estimate peak-streamflow frequency specifically for the ungaged site. The equations are unique to the ungaged site and therefore have limited applicability or transferability to other ungaged sites. In the traditional approach, equations are carefully and purposefully constructed to be applicable for a broad range of sites within a common geographic region. The emphasis in the site-specific approach is on the peak-streamflow prediction and associated errors of prediction rather than the equation that produces them. 

A major element of any hydrologic regionalization technique is how stations to be pooled together, or regionalized, are chosen. In the traditional regional approach, the stations that are pooled together are chosen on the basis of a commonality of geographic region. The region encompassing the stations is delineated on the basis of factors such as channel types, climatic characteristics, elevation, geology, physiography, proximity to regional water bodies, and relative topography. In an effort to develop equations that are applicable to a broad range of ungaged sites within a region, it is common to seek and include stations that provide a large range of basin characteristics. Accordingly, some of the stations can have considerably different basin characteristics. For example, in region 3 of Asquith and Slade (1997, p. 62), drainage areas of the included stations range from 11.8 to 14,635 square miles (mi2). Logic dictates that the annual peak-streamflow generation processes for the station with the 14,635-mi2 basin have little relevance to the annual peak-streamflow generation processes for the station with the 11.8-mi2 basin, even though the 11.8-mi2 station is in the larger basin and is proximate to the 14,635 mi2 station.
The regionalization in the site-specific approach is based on a fundamentally different line of reasoning. The regionalization is based on an informal cluster analysis that compares the basin characteristics of the ungaged site to the basin characteristics of all the stations in the data base. In the site-specific approach, the peak-streamflow frequency and basin characteristics used for the estimation of peak frequency at an ungaged site include only the data from stations whose basin and climatic characteristics are similar to those for the ungaged site.

Because of the restriction that the basin characteristics be similar, the site-specific approach has several advantages. One advantage of the approach is that the stations included in the regression will be hydrologically similar to the ungaged site. The basin characteristics of the ungaged site should lie within the multidimensional explanatory-variable cloud defined by the basin characteristics of the selected stations. Asquith and Slade (1997, p. 16-26) provide numerous two-dimensional representations of the multidimensional cloud as shaded areas encompassing clustered points on graphs of the relations between basin characteristics.

Several important consequences result when the ungaged site basin characteristics lie within and near the middle of the explanatory variable cloud. Because estimations of peak streamflows from the regression are made near the center of the space of the explanatory variables, extrapolation errors, or more specifically extrapolation biases, in the relation between peak-streamflow frequency and basin characteristics are reduced. Additionally, violations of the linearity assumption for the regression are less likely to cause problems for the linear regression (Tasker and Slade, 1994). The issue of linearity between peak-streamflow frequency and basin characteristics for Texas is discussed in Asquith and Slade (1997, p. 8-10). 

Regions are not defined by fixed geographic boundaries. Therefore, the potential for substantially different estimates for similarly characterized watersheds on either side of such a boundary is greatly reduced. Although it is considered an advantage not to have fixed regional geographic boundaries, visualization of the region surrounding a particular ungaged site is sometimes difficult. 

The site-specific approach can be easily updated in the future as more data are collected; and as new statistical tools, such as L-moments (Hosking and Wallis, 1997), improve peak-streamflow frequency estimation and more detailed basin-characteristic assessment becomes possible through sophisticated computer analysis (Randy L. Ulery, U.S. Geological Survey, oral commun., 1998). 

The most important aspect of a site-specific approach is the selection of stations with characteristics similar to those of the ungaged site. Two techniques are identified for selection of suitable stations: (1) a proximity search with a drainage area range restriction and (2) a similarity search. Discussion of the application of these two techniques for Texas is presented later.

The first technique available for selecting suitable stations is a geographic proximity search in which the selected "nearest" stations are within a specified range in drainage area. The geographic distance *Dij (or proximity) of the ungaged site *i* to station *j* is calculated by the following: (1)  ,*