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In cooperation with the Michigan Department of Environmental Quality and the Michigan Department of Natural Resources

A Regression Model for Computing Index Flows Describing the Median Flow for the Summer Month of Lowest Flow in Michigan

By David A. Hamilton, Richard C. Sorrell, and David J. Holtschlag

ABSTRACT In 2006, Michigan enacted laws to prevent new large capacity withdrawals from decreasing flows to the extent that they would functionally impair a stream’s ability to support characteristic fish populations. The median streamflow for the summer month of lowest flow was specified by state decision makers as the index flow on which likely impacts of withdrawals would be assessed. At sites near long-term streamflow-gaging stations, analysis of streamflow records during July, August, and September was used to determine the index flow. At ungaged sites, an alternate method for computing the index flow was needed. This report documents the development of a method for computing index flows at ungaged stream sites in Michigan. The method is based on a regression model that computes the index water yield, which is the index flow divided by the drainage area. To develop the regression model, index flows were determined on the basis of daily flows measured during July, August, and September at 147 streamflow-gaging stations having 10 or more years of record (considered long-term stations) in Michigan. The corresponding index water yields were statistically related to climatic and basin characteristics upstream from the stations in the regression model. Climatic and basin characteristics selected as explanatory variables in the regression model include two aquifer-transmissivity and hydrologic-soil groups, forest land cover, and normal annual precipitation. Regression model estimates of water yield explain about 70.8 percent of the variability in index water yields indicated by streamflow-gaging station records. Index flows computed on the basis of regression-model estimates of water yield and corresponding drainage areas explain about 94.0 percent of the variability in index flows indicated by streamflow-gaging station records. No regional bias was detected in the regression-based estimates of water yield within seven hydrologic subregions spanning Michigan. Thus, the single regression model developed in this report can be used to produce unbiased estimates of index water yield and flow statewide. In addition, a technique is presented for computing prediction intervals about the index flow estimates.

Revised February 25, 2011 First posted August 1, 2008 Report PDF (4.71 MB) 43 pages (8.5" by 11" paper) For additional information contact: U.S. Geological Survey Michigan Water Science Center 6520 Mercantile Way, Suite 5 Lansing, MI 48911–5991 GS-W-MIlns_DC@usgs.gov For more information about USGS activities in Michigan, visit the USGS Michigan Water Science Center home page. Part or all of this report is presented in Portable Document Format (PDF); the latest version of Adobe Reader or similar software is required to view it. Download the latest version of Adobe Reader, free of charge.

Suggested citation:

Hamilton, D.A., Sorrell, R.C., and Holtschlag, D.J., 2008, A regression model for computing index flows describing the median flow for the summer month of lowest flow in Michigan: U.S. Geological Survey Scientific Investigations Report 2008–5096, 43 p. Date Posted: August 1, 2008: available only at [http://pubs.water.usgs.gov/sir/2008/5096/].

Contents

Abstract

Introduction

Purpose and Scope

Previous Investigations

Description of the Study Area

Development of a Regression Model for Index Flow Estimation

Selection of Streamflow-Gaging Stations

Identification of the Hydrologic Response Variable

Index Flow

Index Water Yield

Compilation of Hydrologic Characteristics for Use as Explanatory Variables

Selection of Hydrologic Characteristics for Use as Explanatory Variables

Estimation of the Hydrologic Response Variables

Spatial Distribution of the Regression-Model Error

Computation of the Index Flow

Index Water Yield and Flow

Comparison of Index Flows

Example Computation

Summary

Acknowledgments

References Cited

Figures

     1–3. Maps showing:

     1. Michigan’s Upper and Lower Peninsulas and surrounding states and province.

     2. U.S. Geological Survey streamflow-gaging stations in Michigan’s Upper Peninsula included in the analyses.

     3. U.S. Geological Survey streamflow-gaging stations in Michigan’s Lower Peninsula included in the analyses.

     4–6. Graphs showing:

     4. Relation between estimates of index flow from gaging station records and drainage area.

     5. Empirical and fitted normal distributions for median-water-yield data from the month of lowest flow for
         selected streamflow-gaging stations in Michigan.

     6. Distribution of estimated aquifer transmissivity classes in Michigan.

   7–11. Maps showing:

     7. Distribution of aquifer transmissivity classes in Michigan.

     8. Distribution of forest cover in Michigan.

     9. Distribution of hydrologic soil groups in Michigan.

    10. Distribution of normal annual precipitation in Michigan for 1971–2000.

    11. Distribution of normal annual snowfall depths in Michigan for 1971–2000.

12–13. Graphs showing:

   12. Relation between RÎY₅₀ (the index of water yield estimated by regression) and RÎY₅₀
        (the index of water yield computed on the basis of the streamflow-gaging station records).

   13. Distribution of explanatory variables selected for the regression model.

      14. Map showing hydrologic subregions used in the analysis of the spatial distribution of regression-model error.

15–16. Graphs showing:

   15. Regional distribution of regression model errors for estimating median water yield during the summer month of
        minimum flow.

   16. Relation between measured and computed index flows for selected streamflow-gaging stations in Michigan.

Tables

   1. Lower triangular elements of the diagonally symmetric correlation matrix among candidate explanatory variables
       and the square root of median water yield for the summer month of lowest flow in Michigan.

   2. Regression model parameters for estimating the hydrologic response variable.

   3. Cross-tabulation of land use-land cover areas with hydrologic soil groups for land areas within Michigan.

   4. Lower triangular elements of the diagonally symmetric correlation matrix among parameters of selected
       explanatory variables and the square root of median water yield for the summer month of lowest flow in Michigan.

   5. The inverse of the X’X matrix needed to compute prediction limits.

1–1. Flow, yield, and record characteristics for streamflow-gaging stations used in the regression analysis.

1–2. Values of selected explanatory variables used in the development of the regression equation for estimating the
       index flow.

1–3. Cross-tabulation of cell counts and percentages for Michigan Resource Information System (MIRIS) 1978
       land use-land cover and hydrologic soil groups in Michigan.