Methods for Estimating Flood Frequency in Montana Based on Data through Water Year 1998

By Charles Parrett and D.R. Johnson

U.S. Geological Survey
Water-Resources Investigations Report 03-4308

In cooperation with 

Bureau of Indian Affairs, Bureau of Land Management,
Confederated Salish and Kootenai Tribes,
Montana Department of Natural Resources and Conservation,
Montana Department of Transportation, and the
U.S. Department of Agriculture Forest Service 

Abstract

Annual peak discharges having recurrence intervals of 2, 5, 10, 25, 50, 100, 200, and 500 years (T-year floods) were determined for 660 gaged sites in Montana and in adjacent areas of Idaho, Wyoming, and Canada, based on data through water year 1998. The updated flood-frequency information was subsequently used in regression analyses, either ordinary or generalized least squares, to develop equations relating T-year floods to various basin and climatic characteristics, equations relating T-year floods to active-channel width, and equations relating T-year floods to bankfull width. The equations can be used to estimate flood frequency at ungaged sites. Montana was divided into eight regions, within which flood characteristics were considered to be reasonably homogeneous, and the three sets of regression equations were developed for each region.

A measure of the overall reliability of the regression equations is the average standard error of prediction. The average standard errors of prediction for the equations based on basin and climatic characteristics ranged from 37.4 percent to 134.1 percent. Average standard errors of prediction for the equations based on active-channel width ranged from 57.2 percent to 141.3 percent. Average standard errors of prediction for the equations based on bankfull width ranged from 63.1 percent to 155.5 percent. In most regions, the equations based on basin and climatic characteristics generally had smaller average standard errors of prediction than equations based on active-channel or bankfull width. An exception was the Southeast Plains Region, where all equations based on active-channel width had smaller average standard errors of prediction than equations based on basin and climatic characteristics or bankfull width.

Methods for weighting estimates derived from the basin- and climatic-characteristic equations and the channel-width equations also were developed. The weights were based on the cross correlation of residuals from the different methods and the average standard errors of prediction. When all three methods were combined, the average standard errors of prediction ranged from 37.4 percent to 120.2 percent. Weighting of estimates reduced the standard errors of prediction for all T-year flood estimates in four regions, reduced the standard errors of prediction for some T-year flood estimates in two regions, and provided no reduction in average standard error of prediction in two regions. A computer program for solving the regression equations, weighting estimates, and determining reliability of individual estimates was developed and placed on the USGS Montana District World Wide Web page. A new regression method, termed Region of Influence regression, also was tested. Test results indicated that the Region of Influence method was not as reliable as the regional equations based on generalized least squares regression.

Two additional methods for estimating flood frequency at ungaged sites located on the same streams as gaged sites also are described. The first method, based on a drainage-area-ratio adjustment, is intended for use on streams where the ungaged site of interest is located near a gaged site. The second method, based on interpolation between gaged sites, is intended for use on streams that have two or more streamflow-gaging stations.

Contents

Abstract
Introduction
    Purpose and scope
    Acknowledgements
    General flood conditions
Methods for estimating flood frequency
    Flood-frequency analysis
        Flood-frequency data
        Regional skew map
        Mixed-population flood-frequency analysis
    Regional flood-frequency relations
        Basin-characteristics regression analysis
        Channel-width regression analysis
        Limitations of regression equations
        Comparison of regional relations with those from previous study
        Maximum recorded floods and envelope lines
        Weighted regression estimates
        Region of Influence regression
    Estimating flood frequency on gaged streams
        Ungaged sites on gaged streams
        Weighted estimates at gaged sites
Example applications (examples 1, 2, and 3)
Summary
References cited
Appendix
Data

Figures

1.  Map showing hydrologic regions and streamflow-gaging stations having compiled flood-frequency characteristics, Montana and adjacent areas in Idaho, Wyoming, and Canada (PLATE 1).
2.    Box plots showing variability in monthly occurrence of annual peak discharge by region, Montana.
3.    Graph showing example flood-frequency curve based on log-Pearson Type III probability distribution.
4.    Map showing regional skew coefficients for Montana.
5.    Graphs showing example flood-frequency curve based on mixed-population analysis.
6.    Map showing mean annual precipitation for the West and Northwest Regions, Montana, base period 1941-70 (PLATE 1).
7.    Diagrammatic sketch of a typical stream cross section showing active-channel and bankfull widths.
8.    Diagrammatic sketch of a typical alluvial stream showing best location for measuring channel width.
9.    Graphs showing comparison of new basin-characteristics equations with equations in previous U.S. Geological Survey report.
10.    Graphs showing maximum known floods, regional and national envelope lines, and regression lines relating 100-year flood to drainage area, West, Northwest, Northwest Foothills, and Northeast Plains Regions, Montana.
11.    Graphs showing maximum known floods, regional and national envelope lines, and regression lines relating 100-year flood to drainage area, East-Central Plains, Southeast Plains, Upper Yellowstone-Central Mountain, and Southwest Regions, Montana.
12.    Graphs showing average standard errors of prediction for various combinations of estimation methods in West, Northwest, Northwest Foothills, and Northeast Plains Regions, Montana.
13.    Graphs showing average standard errors of prediction for various combinations of estimation methods in East-Central Plains, Southeast Plains, Upper Yellowstone-Central Mountain, and Southwest Regions, Montana.    

Tables

1.    Hydrologic regions and flood characteristics in Montana.
2.    Basin-characteristics, channel-width, and flood-frequency data for selected streamflow-gaging stations, Montana and adjacent areas Idaho, Wyoming, and Canada.
3.    State Map Errors for various methods of determining regional skew.
4.    Regression equations based on basin characteristics.
5.    Student's t  for confidence interval of 90 percent for different regions.
6.    Range of values of basin characteristics used to develop regression equations.
7.    Regression equations based on active-channel width.
8.    Regression equations based on bankfull width.
9.    Range of channel widths used to develop regression equations.
10.  Comparison of results for estimation of 100-year flood from new basin-characteristics equations with those from previous equations.
11.    Cross-correlation coefficients between residuals for combinations of different estimation methods.
12.    Weights and average standard errors of prediction for various combinations of estimation methods.
13.     Regression coefficients for ordinary least squares regressions relating T-year flood to drainage area.
14.    (X^TX)^-1 and (X^TL^-1X)^-1 matrices for regression equations based on basin characteristics.
15.    (X^TX)^-1and (X^TL^-1X)^-1 matrices for regression equations based on active-channel width.
16.    (X^TX)^-1 and (X^TL^-1X)^-1 matrices for regression equations based on bankfull width.

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