Data Series 248

In cooperation with the Texas Commission on Environmental Quality

Summary of Annual Mean, Maximum, Minimum, and L-scale Statistics of Daily Mean Streamflow for 712 U.S. Geological Survey Streamflow-Gaging Stations in Texas Through 2003

By William H. Asquith, Joseph Vrabel, and Meghan C. Roussel

Abstract

Analysts and managers of surface-water resources might have interest in selected statistics of daily mean streamflow for U.S. Geological Survey (USGS) streamflow-gaging stations in Texas. The selected statistics are the annual mean, maximum, minimum, and L-scale of daily mean streamflow. Annual L-scale of streamflow is a robust measure of the variability of the daily mean streamflow for a given year. The USGS, in cooperation with the Texas Commission on Environmental Quality, initiated in 2006 a data and reporting process to generate annual statistics for 712 USGS streamflow-gaging stations in Texas. A graphical depiction of the history of the annual statistics for most active and inactive, continuous-record gaging stations in Texas provides valuable information by conveying the historical perspective of streamflow for the watershed. Each figure consists of four time-series plots of the annual statistics of daily mean streamflow for each streamflow-gaging station. Each of the four plots is augmented with horizontal lines that depict the mean and median annual values of the corresponding statistic for the period of record. Monotonic trends for each of the four annual statistics also are identified using Kendall’s τ. The history of one or more streamflow-gaging stations could be used in a watershed, river basin, or other regional context by analysts and managers of surface-water resources to guide scientific, regulatory, or other inquiries of streamflow conditions in Texas.

Introduction

Analysts and managers of surface-water resources might have interest in selected statistics of daily mean streamflow for U.S. Geological Survey (USGS) streamflow-gaging stations in Texas. To facilitate information transfer, this report provides graphical and statistical summaries for most of the active and inactive, continuous-record streamflow-gaging stations in Texas. The statistics selected for this report are the annual mean, maximum, minimum, and L-scale. These are collectively referred to as the "annual statistics." Annual L-scale of streamflow is a robust measure of the variability of the daily mean streamflow for a given year.

A graphical depiction of the annual statistics for streamflow-gaging stations in Texas provides valuable information by conveying the historical characteristics of streamflow for the watershed, in particular, and Texas, in general. Therefore, in 2006 the USGS, in cooperation with the Texas Commission on Environmental Quality (TCEQ), initiated a data and reporting process to generate station-specific histories of the annual statistics of daily mean streamflow in Texas. This report includes graphical depictions of the annual statistics for 712 USGS streamflow-gaging stations in Texas with at least 1 year of record through water year 2004 (fig. 1).

Streamflow-gaging stations that monitor spring flow or stage (water level) only were not used. Further, partial-record streamflow-gaging stations were not used because the full range of streamflow at each station was needed. Partial-record streamflow-gaging stations are sites where discrete measurements of streamflow are obtained over a period of time without continuous data being recorded. These stations intrinsically do not monitor the entire range of streamflow. The 712 stations considered in this report were analyzed in Asquith and others (2006) for an evaluation of the drainage-area ratio method in Texas. A listing of the station numbers, station names, and ancillary information is available in Asquith and others (2006, table 1). A companion report (Asquith and others, 2007) considers percentages of zero daily mean streamflow for the same stations and streamflow data.

The data for the 712 stations were obtained from the USGS National Water Information System (U.S. Geological Survey, 2005). The stations have at least 1 year of daily mean streamflow record through the 2004 water year. A water year is the 12-month period between October 1 and September 30. The water year is designated by the calendar year in which it ends. Thus, the year ending September 30, 2004, is called the "2004 water year." The data were trimmed to the last complete calendar year (2003). The earliest year of streamflow record is 1898. For the analysis reported here the calendar year context is used for statistical computations. Therefore, the last date of daily mean streamflow is December 31, 2003. The total number of daily values processed is 7,748,449.

The annual L-scale statistic of daily mean streamflow was computed for each year of record. L-scale is the second L-moment and represents the intra-variation in streamflow. L-scale is analogous, but not equal to, standard deviation; both are similarly interpreted by analysts. L-scale has several well-documented sampling properties such as unbiasedness and efficiency in small samples that make this statistic well suited for streamflow datasets (Hosking, 1990). Although not an exact equality when computed from samples, the theoretical relation between the standard deviation (σ) and L-scale (λ₂) is σ = √πλ₂. Recently, Asquith (2006) summarized the mathematics and theory of L-moments.

Finally, an analysis of monotonic temporal trends in each of the four statistics was done using Kendall’s τ. Kendall’s τ measures the strength of the monotonic relation between time and the annual streamflow statistic. Kendall’s τ is nonparametric, meaning that the statistic is based on the ranks of the data and not the actual data values. Positive τ values indicate that the annual statistic increases with time for the period of record, and negative τ values indicate that the annual statistic decreases with time for the period of record. Perfect monotonically decreasing relations result in τ values of exactly -1; conversely, perfect monotonically increasing relations result in τ values of exactly 1. The p-value is a measure of the strength or statistical significance of the relation; small p-values (p-value ≤ .02 in this report) indicate a strong relation.

Kendall’s τ is visually augmented by a Theil line when a strong relation is indicated. The Theil line, as discussed by Hollander and Wolfe (1973, p. 205) and Helsel and Hirsch (1992, p. 266), is a robust estimator of the slope of a linear relation between time and the streamflow statistic. The slope is estimated as the median of all unique (n x (n - 1)/2) slopes between individual data points. The purpose of the Theil line is to provide a visual cue that a statistically significant trend in the annual statistic was detected.

Summary of Annual Mean, Maximum, Minimum, and L-scale Statistics of Daily Mean Streamflow in Texas

The graphical depiction of the history of daily mean streamflow statistics for each of the 712 stations is provided in figures 2--713 (at end of report). Each figure consists of four time-series plots of annual mean, maximum, minimum, and L-scale values of daily mean streamflow. Above the top graph is the text annotation "U.S. Geological Survey streamflow-gaging station #," where # is the eight-digit station identification number. This annotation is included to confirm that the pairing of the graphs and the figure caption are correct. Some stations have substantial gaps (measured in years) such as those shown in figure 588 on page 592. No special treatment of the streamflow data was applied for stations with gaps in the period of record.

Each statistic type consists of four components. First, the annual statistics (mean, maximum, minimum, and L-scale) are plotted as n open circles (n represents the number of data points) in figures 2--713. Each data point was computed using the number of days of observed record for the calendar year. For a full year of record, the number of days for the year was 365 (or 366 for leap years). Incomplete years are plotted for the figures as well with the requisite change in sample size. For the second and third components, the mean and median of the n data points were computed and are illustrated as solid and dashed horizontal lines, respectively. The numeric values of mean and median annual statistics of streamflow are shown in the explanation of each graph. The fourth component is the assessment of a monotonic trend in the data.

For the fourth component, Kendall’s τ was computed using an integrated statistical computing environment (The MathWorks, 2006) for the n annual statistics. The p-value for τ also is shown. Hollander and Wolfe (1973, p. 185--199) and Helsel and Hirsch (1992, p. 212 and 216) provide the background and details of computation. Both τ and the p-value are shown in figures 2--713 to three significant figures, except when there is a perfect decreasing or increasing monotonic relation, in which case the exact τ value of -1 or 1, respectively, is given. P-values less than .001 are expressed as <.001.

For stations for which Kendall’s τ has a p-value less than or equal to .02, a Theil trend line is superimposed on a graph as a solid grey line. Thus, the Theil line is not drawn on all graphs of the figures. Also, for some graphs, such as the annual minimum streamflow graph for station 07311783 (fig. 46 on page 50), the Theil line is drawn, but not visible--it is masked by the mean annual minimum streamflow line (solid horizontal line). The masking occurs because the median pairwise slope is precisely zero and is most likely to occur with the annual minimum streamflow because of repeated values of zero streamflow.

In regards to the Theil line, the line could represent an over simplification of the relation between time and the corresponding annual statistic for individual watersheds. The actual temporal changes of the annual statistic for each watershed potentially could indicate curvilinear or even cyclical variations caused by natural or anthropogenic sources.

General Discussion

The number of stations summarized in this report is too large for effective discussion and commentary of station-specific results. Conceptually, numerous attributes or factors of a watershed influence the annual statistics of daily mean streamflow. Some of these factors could affect one annual statistic and not another. The factors could include climatic setting, land-use changes, and upstream water use or regulations. As a result, specific discussion of the daily mean streamflow history for a given station is beyond the scope of this report. However, some general comments for each of the four annual statistics can be made to guide analysts.

The trend evaluations on the 712 stations can be summarized as follows: The number of stations with a statistically significant positive trend (increase) in the annual mean streamflow is 69 (about 10 percent). The number of stations with a statistically significant negative trend (decrease) in the annual mean streamflow is 34 (about 5 percent). No significant trends were detected for 609 stations (about 85 percent).

The number of stations with a statistically significant positive trend in the annual maximum streamflow is 21 (about 3 percent). The number of stations with a statistically significant negative trend in the annual maximum streamflow is 52 (about 7 percent). No significant trends were detected for 639 stations (about 90 percent).

The number of stations with a statistically significant positive trend in the annual minimum streamflow is 127 (about 18 percent). The number of stations with a statistically significant negative trend in the annual minimum streamflow is 20 (about 3 percent). No significant trends were detected for 565 stations (about 79 percent).

The number of stations with a statistically significant positive trend in the annual L-scale of streamflow is 47 (about 7 percent). The number of stations with a statistically significant negative trend in the annual L-scale of streamflow is 41 (about 6 percent). No significant trends were detected for 624 stations (about 87 percent).

Finally, each summary provides a compact visual description of the history of selected statistics of daily mean streamflow for the watershed monitored by the USGS streamflow-gaging station. Station-specific interpretations of climatic, hydrologic, anthropogenic, and other processes potentially influencing the magnitude and temporal variations can be made. One or more summaries could be used in a watershed, river basin, or other regional context by analysts and managers of surface-water resources to guide scientific, regulatory, or other inquiries of daily streamflow conditions in Texas.

References

Asquith, W.H., 2006, L- and TL-moments of the generalized lambda distribution: Computational Statistics and Data Analysis, 13 p., corrected proof available online since August 1, 2006, doi: 10.1016/j.csda.2006.07.016.

Asquith, W.H., Roussel, M.C., and Vrabel, Joseph, 2006, Statewide analysis of the drainage-area ratio method for 34 streamflow percentile ranges in Texas: U.S. Geological Survey Scientific Investigations Report 2006--5286, 34 p., 1 appendix. Available Online

Asquith, W.H., Vrabel, Joseph, and Roussel, M.C., 2007, Summary of percentages of zero daily mean streamflow for 712 U.S. Geological Survey streamflow-gaging stations in Texas through 2003: U.S. Geological Survey Data Series 247, 721 p. Available Online

Helsel, D.R., and Hirsch, R.M., 1992, Statistical methods in water resources--- Studies in environmental science 49: Amsterdam, Elsevier, 529 p.

Hollander, Myles, and Wolfe, D.A., 1973, Nonparametric statistical methods: New York, John Wiley, 503 p.

Hosking, J.R.M., 1990, L-moments---Analysis and estimation of distributions using linear combinations of order statistics: Journal Royal Statistical Society B, v. 52, no. 1, p. 105--124.

The MathWorks, 2006, MATLAB version 7.2.0.232 (R2006a): Natick, Mass.

U.S. Geological Survey, 2005, National Water Information System: accessed in July 2005 at http://nwis.waterdata.usgs.gov/nwis/discharge