Estimates of Median Flows for Streams on the Kansas Surface Water Register

The Kansas State Legislature, by enacting Kansas Statute KSA 82a-2001 et. seq., mandated the criteria for determining which Kansas stream segments would be subject to classification by the State. One criterion for the selection as a classified stream segment is based on the statistic of median flow being equal to or greater than 1 cubic foot per second. As specified by KSA 82a-2001 et. seq., median flows were determined from U.S. Geological Survey streamflow-gaging-station data by using the most-recent 10-years of gaged data (KSA) for each streamflow-gaging station. Median flows also were determined by using gaged data from the entire period of record (all-available hydrology, AAH).

Least-squares multiple regression techniques were used, along with Tobit analyses, to develop equations for estimating median flows for uncontrolled stream segments. The drainage area of the uncontrolled gaging stations used in the regression analyses ranged from 2.06 to 12,004 square miles. A logarithmic transformation of the data was needed to develop the best linear relation for computing median flows. In the regression analyses, the significant climatic and basin characteristics, in order of importance, were drainage area, mean annual precipitation, mean basin permeability, and mean basin slope. Tobit analyses of KSA data yielded a root mean square error of 0.285 logarithmic units, and the best equations using Tobit analyses of AAH data had a root mean square error of 0.247 logarithmic units.

These equations and an interpolation procedure were used to compute median flows for the uncontrolled stream segments on the Kansas Surface Water Register. Measured median flows from gaging stations were incorporated into the regression-estimated median flows along the stream segments where available. The segments that were uncontrolled were interpolated using gaged data weighted according to the drainage area and the bias between the regression-estimated and gaged flow information. On controlled reaches of Kansas streams, the median flow information was interpolated between gaging stations using only gaged data weighted by drainage area.

Of the 2,232 total stream segments on the Kansas Surface Water Register, 30 percent of the segments had an estimated median streamflow of less than 1 cubic foot per second when the KSA analysis was used. When the AAH analysis was used, 40 percent of the segments had an estimated median streamflow of less than 1 cubic foot per second.

The purpose of this report is to document the methods and results of a study designed to estimate the median flow (50-percent flow duration) for the downstream end of each stream segment listed on the Kansas Surface Water Register. Median flow for each stream segment was determined from gaged-location streamflow records or was estimated from statewide regression models. This report documents development of regression models to estimate median flow from climatic and basin characteristics. The report describes application of the drainage-area ratio method and the regression models to estimate the median flows for Kansas Surface Water Register stream segments, the interpolation of estimates for ungaged segments, and the Internet dissemination of results and a geographic-information-system (GIS) database.

Two different statistical analysis were performed on uncontrolled flows measured at 149 gaging stations. According to language in KSA 82a-2001 et. seq., only the most-recent 10 years of streamflow data for each gaging station were to be used for statistical analysis. This analysis was termed the KSA analysis. The entire period of record also was used for analysis of median flows, and this analysis was termed the all-available hydrology (AAH) analysis.

The information contained in this report can be used by State agencies and others to help in the effective management of Kansas surface-water resources. Optimal reservoir operations, legally distributed in-stream withdrawals, and water-quality concerns are issues directly linked to median streamflows. The methods described herein can be applied nationwide using USGS streamflow data that are available throughout the United States.

Previous low-flow and flow-duration studies for Kansas include an investigation by Furness (1959) who developed a method for estimating flow-duration curves for ungaged sites that was based on regionalized flow-duration data from 122 continuous-record, streamflow-gaging stations with drainage areas of between 100 and 3,000 mi² for the period 1921-56. Maps were developed showing a variety of statewide low and mean streamflow maps. Furness (1959) also noted that the low-flow parts of the flow-duration curves could be verified or improved by relating base-flow measurements at the ungaged site to base-flow measurements at a nearby, index streamflow-gaging station.

Jordan (1983) updated the maps developed by Furness by including additional streamflow-gaging stations and data for the period 1957-76. Jordan's study included a map that depicted the areas of Kansas where the median streamflow for a 500-mi² basin was greater than 0.1 ft³/s.

Two studies by Studley (2000, 2001) evaluated the application of the Furness method to ungaged stream sites in Kansas using nearby streamflow-gaging stations as index stations. The results of these two recent studies indicated that the Furness method continues to be a useful tool for estimating flow-duration curves for ungaged sites and that the method could be used for sites with drainage areas less than 100 mi².

Many studies have been conducted to evaluate low flow from regression equations that relate low flow to basin characteristics. In a recent USGS study (Ries and Friesz, 2000), basin characteristics were determined from digital map data, and flow statistics were computed for individual stream segments using GIS techniques. Ries and Friesz (2000) used the drainage-area ratio method to compute streamflow characteristics for stream segments that had between 0.5 and 1.5 times the drainage area of streamflow-gaging stations on the same stream. Many States have used regression analysis to regionalize low-flow frequency statistics including New Hampshire, Rhode Island, and Vermont (Johnson, 1970); Pennsylvania and New York (Ku and others, 1975); Maine (Parker, 1977); Massachusetts (Male and Ogawa, 1982; Vogel and Kroll, 1990; Risley, 1994; Ries and Friesz, 2000); Montana (Parrett and Hull, 1985); Indiana (Arihood and Glatfelter, 1991); and central New England (Wandle and Randall, 1994).

FACTORS AFFECTING STREAMFLOW

Physiographically, Kansas is located almost entirely within the Interior Plains as described by Schoewe (1949). A description of the hydrologic characteristics of the physiographic provinces within the Interior Plains is beyond the scope of this report, but the fact that there are significant variations denotes the complex nature of and difficulty in attempting to define flow characteristics across Kansas.

The climate of Kansas is affected by the movement various air masses of tropical and continental origin over the open, inland plains, and seasonal precipitation extremes are common. About 70 percent of the mean annual precipitation falls from April through September. Precipitation during early spring and late fall occurs in association with frontal air masses that produce low-intensity rainfall of regional coverage. During the summer months, the weather is dominated by warm, moist air from the Gulf of Mexico or by hot, dry air from the Southwest. Summer precipitation generally occurs as high-intensity thunderstorms.

Climatic and basin characteristics were used in the analyses of median flows at gaged and ungaged sites on controlled and uncontrolled streams. For this study, ARC/INFO GIS software was used to estimate climatic and basin characteristics. Many spatial data sets were available for this task, including: (1) 30-year (1961-90) mean annual precipitation data (Daly and others, 1997), (2) 30-m gridded elevation data (U.S. Geological Survey, 1998) for determining drainage area, mean basin slope, and mean basin elevation, and (3) STATSGO soil-permeability data (U.S. Department of Agriculture, 1994).

The USGS has established standard methods for estimating flow duration (Searcy, 1959) for streamflow-gaging stations. The computer software programs IOWDM, ANNIE, and SWSTAT were used to format input data, manage and display data, and complete the flow-duration statistical analyses (Lumb and others, 1990; Flynn and others, 1995). These programs are available on the World Wide Web at http://water.usgs.gov/software/surface_water.html

Daily mean flows for all complete water years of record were used to determine flow-duration statistics for continuous-record, streamflow-gaging stations. The water year begins on October 1 and ends on September 30 of the following year. Daily mean flows for USGS streamflow-gaging stations in Kansas are available on the World Wide Web at http://waterdata.usgs.gov/ks/nwis/

A flow-duration curve is a graphical representation of the percentage of time streamflows for a given time step (usually daily) are equaled or exceeded over a specified period (usually the complete period of record) at a stream site. Flow-duration curves usually are constructed by first ranking all of the daily mean discharges for the period of record at a gaging station from largest to smallest, next computing the probability for each value of being equaled or exceeded, then plotting the discharges against their associated exceedance probabilities (Loaiciga, 1989, p. 82). The daily mean discharges are not fit to an assumed distribution. Flow-duration analysis can be done by use of the USGS software described previously or by use of commercially available statistical software.

Flow-duration statistics are points along a flow-duration curve. For example, the 99-percent duration streamflow is equaled or exceeded 99 percent of the time, whereas the 50-percent duration streamflow is equaled or exceeded 50 percent of the time. Strictly interpreted, flow-duration statistics reflect only the period for which they are calculated; however, when the period of record used to compute the statistics is sufficiently long, the statistics often are used as an indicator of probable future conditions (Searcy, 1959). Median-flow statistics in this report were determined using the Loaiciga (1989) approach.

The drainage-area ratio method assumes that the streamflow at an ungaged site for the same stream is the same per unit area or at least responds in the same fashion as that at a nearby, hydrologically similar streamflow-gaging station used as an index. Drainage areas for the ungaged site and the index station are determined from topographic maps, digital elevation maps (DEMs), or by other GIS methods. Streamflow statistics are computed for the index station, then the statistics are divided by the drainage area to determine streamflows per unit area at the index station. These values are multiplied by the drainage area at the ungaged site to obtain estimated statistics for the site. This method is most commonly applied when the index gaging station is on the same stream as the ungaged site because the accuracy of the method depends on the proximity of the two sites and on similarities in drainage area and on other climatic and basin characteristics of the respective drainage basins.

Several researchers have provided guidelines as to how large the difference in drainage areas can be before use of multiple linear-regression analysis is preferred over use of the drainage-area ratio method. Guidelines have been provided for estimating peak-flow statistics, and usually the recommendation has been that the drainage area for the ungaged site should be within 0.5 and 1.5 times the drainage area of the index station (Choquette, 1988, p. 41; Koltun and Roberts, 1990, p. 6; Lumia, 1991, p. 34; Bisese, 1995, p. 13). One report (Koltun and Schwartz, 1986, p. 32) selected a range of 0.85 to 1.15 times the drainage area of the index station for estimating low flows at ungaged sites in Ohio. None of these researchers provided any scientific basis for use of these guidelines (R.E. Thompson, Jr., U.S. Geological Survey, written commun., 1999). In this report, the median flows at uncontrolled, ungaged locations are estimated by interpolation procedures using weighted drainage-area ratio from gaged sites and Tobit regression estimates at ungaged sites. No limit was placed on the ratios between the drainage area of the index station and the drainage area of the ungaged stream segment.

Multiple linear-regression analysis (regression analysis) has been used by the USGS and other researchers throughout the United States and elsewhere to develop equations for estimating streamflow statistics at ungaged sites. In regression analysis, a streamflow statistic (the dependent variable) for a group of gaging stations is related statistically to the climatic or basin characteristics of the drainage basins for the stations (the independent variables). This results in an equation that can be used to estimate the statistic for sites where no streamflow data are available.

Equations can be developed by use of several different regression analysis algorithms. The various algorithms use different methods to minimize the differences between the values of the dependent variable for the stations used in the analysis (the observed values) and the corresponding values provided by the resulting regression equation (the estimated or fitted values). Choice of one algorithm over another depends on the characteristics of the data used in the analysis and on the underlying assumptions for use of the algorithm. The multiple linear-regression equation takes the general form:

where Y_i is the value of the dependent variable for site i, X₁ to X_n are the n independent variables, b₀ to b_n are the n + 1 regression-model coefficients, and e_i is the error (difference between the observed and estimated values of the dependent variable) for site i. Assumptions for use of regression analysis are (1) equation 1 adequately describes the relation between the dependent and the independent variables, (2) the variance of the e_i is constant and independent of the values of X_n, (3) the e_i are normally distributed for a Tobit regression, and (4) the e_i are independent of each other (Inman and Conover, 1983, p. 367). Tobit regression is discussed in the following paragraph. Regression analysis results must be evaluated to assure that these assumptions are met. Streamflow and basin characteristics used in hydrologic regression usually are log normally distributed; therefore, transformation of the variables to logarithms is usually necessary to satisfy regression assumption 3. Transformation results in a model of the form:

The algebraically equivalent form when logarithms-base 10 (log₁₀) are used in the transformations, and the equation retransformed to original units is:

To include zero values in a logarithmic transformation regression analysis, the Tobit regression was used. Tobit regression is a widely accepted method for estimating a regression-like model when there are adjusted data (Tobin, 1958; Judge and others, 1985). Adjusted data are data that are either censored or have had a discrete value delta (δ) added to them. Censored data are values below a threshold and are raised to the censoring value (for example, all values below 0.7 are raised to 0.7). Discrete values of delta (δ) are added to all data before transformation and then subtracted from the final regression model value. By applying these techniques, zero values of data can be transformed logarithmically. The Tobit procedure uses a maximum likelihood estimator. The Survival Regression Procedure in the S-Plus 2000 software package (MathSoft, 1999) was used in this study to fit the Tobit model.

Drainage basins for each stream segment on the Kansas Surface Water Register were determined in the GIS by converting the vector stream-segment coverage into a raster-grid network with a raster size of 492 by 492 ft (150 by 150 m). Euclidean allocation was performed on the rasterized stream network to calculate for each cell the identity of the closest source or stream cell using the Euclidean distance. Euclidean distance is defined as the shortest length between two points in two-dimensional space.

Mean values for climatic and basin characteristics were calculated for stream-segment drainage basins using zonal statistics on basin-characteristic grids with Euclidean allocation zones. Zonal statistics were recorded in an attribute table and included the area and mean of the values of all cells in the basin-characteristic grids that belong to the same Euclidean zone. The climatic and basin characteristics computed included drainage area, mean annual precipitation, mean basin elevation, mean basin permeability, mean basin slope, a Base Flow Index (BFI), and the Gebert and PRISM flow model values. Output zonal statistics tables were relationally joined back to the original vector streams coverage so that each reach had an estimated value for each climatic and basin characteristic.

The Kansas State Legislature, by enacting Kansas Statute 82a-2001 et. seq. (KSA), has mandated the selection of Kansas streams for water-quality classification by the State. One criterion for selecting stream segments for classification is whether stream segments listed on the Kansas Surface Water Register have a median flow equal to or greater than 1 ft³/s. Therefore, information on median flow characteristics is needed for streams in Kansas. Daily streamflow information available for 214 gaging stations within Kansas and in adjacent States were used by the USGS in cooperation with KDHE to compute these statistics at gaged sites and to estimate these statistics at ungaged sites.

Least-squares multiple-regression techniques, along with Tobit analyses, were used to develop equations for estimating median flow (dependent variable) for ungaged, uncontrolled stream segments. Median flows were determined from streamflow-gaging station data using the most-recent 10 years of gaged data as defined by KSA analysis, and from the entire period of record, which is defined in this report as the all-available hydrology (AAH) analysis. Independent variables in the regression equations were the climatic and basin characteristics for streams flowing through Kansas. In the development of the regression equations, the significant climatic and basin characteristics, in order of importance, were drainage area, mean annual precipitation, mean basin permeability, and mean basin slope. Only the 149 gages on uncontrolled streams with at least 10 years of streamflow record were used in the regression analyses. The drainage area of these gages ranged from 2.06 to 12,004 mi².

A logarithmic transformation of the basin characteristics was needed to develop a linear relation for computing median flows. Because there were numerous zero values for median gaging-station flows, the Tobit regression was used to include those zero values in the regression. The resulting regression equations and an interpolation procedure were used to estimate median flows for the uncontrolled stream segments on the Kansas Surface Water Register.

Streamflow-gaging-station data were used to improve the quality of the estimates along the streams that had gages. Median flows for the segments that were uncontrolled were interpolated using gaged data weighted according to the drainage area and the bias between the regression estimate and gaged flow information. On controlled reaches of Kansas streams, the median flow information was interpolated between gaging stations by using only gaged data weighted by drainage area.

Of the 2,232 stream segments on the Kansas Surface Water Register, 30 percent of the segments had an estimated median flow of less than 1 ft³/s when the most-recent 10 years of data (KSA analysis) were used. When all-available data (AAH analysis) were used, which resulted in a regression equation with a lower level of uncertainty when compared to KSA analysis, 40 percent of the stream segments had an estimated median flow of less than 1 ft³/s.

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