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Scientific Investigations Report 2008–5096

Pubs Warehouse

### 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 

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### 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/].

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### 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.

 

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