U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2008–5096

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

U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2008–5096

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
U.S. Department of the Interior

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

Mark D. Myers, Director

U.S. Geological Survey, Reston, Virginia: 2008

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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. 
iii

Contents

Abstract...........................................................................................................................................................1

Introduction.....................................................................................................................................................1

Purpose and Scope ..............................................................................................................................2

Previous Investigations........................................................................................................................2

Description of the Study Area ............................................................................................................2

Regression Modeling ....................................................................................................................................4

Development of a Regression Model for Index Flow Estimation...........................................................5

Selection of Streamflow-Gaging Stations ........................................................................................5

Identification of the Hydrologic Response Variable .......................................................................6

Index Flow.....................................................................................................................................6

Index Water Yield.........................................................................................................................6

Compilation of Hydrologic Characteristics for Use as Explanatory Variables ...........................9

Selection of Hydrologic Characteristics for Use as Explanatory Variables..............................18

Estimation of the Hydrologic Response Variables ........................................................................18

Spatial Distribution of the Regression-Model Error......................................................................19

Computation of the Index Flow..................................................................................................................25

Index Water Yield and Flow ..............................................................................................................25

Comparison of Index Flows...............................................................................................................25

Example Computation ........................................................................................................................25

Summary........................................................................................................................................................26

Acknowledgments.......................................................................................................................................27

References Cited..........................................................................................................................................28

Figures

1–3. Maps showing:

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

and province.........................................................................................................................3

2. U.S. Geological Survey streamflow-gaging stations in Michigan’s 

Upper Peninsula included in the analyses .....................................................................7

 3. U.S. Geological Survey streamflow-gaging stations in Michigan’s Lower 

Peninsula included in the analyses ..................................................................................8

4–6. Graphs showing:

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

and drainage area ...............................................................................................................9

5. Empirical and fitted normal distributions for median-water-yield data 

from the month of lowest flow for selected streamflow-gaging stations 

in Michigan .........................................................................................................................10

6. Distribution of estimated aquifer transmissivity classes in Michigan......................11

7–11. Maps showing:

7. Distribution of aquifer transmissivity classes in Michigan ........................................12

 8. Distribution of forest cover in Michigan........................................................................14
iv

9. Distribution of hydrologic soil groups in Michigan......................................................15

10. Distribution of normal annual precipitation in Michigan for 1971–2000...................16

11. Distribution of normal annual snowfall depths in Michigan for 1971–2000 .............17

12–13. Graphs showing:

12. Relation between RÎY50 (the index of water yield estimated by regression) 

and R

~

IY50 (the index of water yield computed on the basis of the streamflowgaging station records).....................................................................................................20

13. Distribution of explanatory variables selected for the regression model................22

14. Map showing hydrologic subregions used in the analysis of the spatial 

distribution of regression-model error....................................................................................23

15–16. Graphs showing:

15. Regional distribution of regression model errors for estimating median 

water yield during the summer month of minimum flow.............................................24

16. Relation between measured and computed index flows for selected 

streamflow gaging stations in Michigan .......................................................................26

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 .........................................19

2. Regression model parameters for estimating the hydrologic response variable ...........20

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

 land areas within Michigan .....................................................................................................21

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............................21

5. The inverse of the X’X matrix needed to compute prediction limits ..................................27

Appendix 1. Tables of streamflow-gaging station attributes, flow characteristics, and 

explanatory variables used in the development of the regression equation 

for estimating the index flow at ungaged streams in Michigan ................................29

1–1. Flow, yield, and record characteristics for streamflow-gaging stations 

used in the regression analysis.......................................................................................30

1–2. Values of selected explanatory variables used in the development of the 

regression equation for estimating the index flow ......................................................38

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............................................................................................................43
v

Conversion Factors and Abbreviations

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

meter (m) 3.281 foot (ft) 

kilometer (km) 0.6214 mile (mi)

Area

square mile (mi2) 259.0 hectare (ha)

square mile (mi2) 2.590 square kilometer (km2) 

Flow rate

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

cubic foot per second per square

mile [(ft3/s)/mi2]

 0.01093 cubic meter per second per square

kilometer [(m3/s)/km2]

gallon per day (gal/d) 3.785 liters per day (liters per day)

inch per year (in/yr) 2.54 centimeter per year (cm/yr)

Transmissivity*

foot squared per day (ft2/d) 0.09290 meter squared per day (m2/d) 

Temperature in degrees Fahrenheit (°F) may be converted to degrees Celsius (°C) as follows:

°C=(°F-32)/1.8

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 

(NAVD 88).

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).

Altitude, as used in this report, refers to distance above the vertical datum.

*Transmissivity: The standard unit for transmissivity is cubic foot per day per square foot times 

foot of aquifer thickness [(ft3/d)/ft2]ft. In this report, the mathematically reduced form, foot 

squared per day (ft2/d), is used for convenience. 
vi

List of Symbols

Symbol Name

y A column vector containing the hydrologic response variable

X The “design matrix,” which, in general, is composed of p columns of basin 

and climatic characteristics augmented with a leading column of ones that 

serve as explanatory variables to estimate the hydrologic response

β A column vector of parameters, β0, β1, . . . ,βp , that relate the explanatory 

variables to the hydrologic response variable 

βols The ordinary least-square estimator of β is denoted and computed as 

βols=(X′·X)–1 ·X′· y , where the prime symbol implies a matrix transpose 

and the –1 power implies a matrix inverse operation

ε A vector of residuals that is assumed to be normally distributed and independent with mean zero and constant variance σ2

, commonly written ε ~ 

NI(0, σ2). 

Cov(ε,X ) The covariance matrix between the residual vector, ε , and the explanatory 

variables contained in the design matrix, X 

LPLα/2 , 

UPL1– α/2

The lower prediction limit and the upper prediction limit, respectively 

centered about the regression estimate of the hydrologic response, ŷ = x0 · 

βols . The interval [LPLα/2,UPL1–α/2 ] is likely to contain the true hydrologic 

response, y0, with a probability of 1–α 

x0 A row vector containing the basin and climatic characteristics at a specific 

site, augmented with a leading one, that serves as the explanatory variables 

to estimate the hydrologic response at that site

α The specified alpha level for the confidence interval. For example, if alpha 

was specified as 0.2, there would be a 10-percent chance that the hydrologic response would be less than LPLα/2 and a 10-percent chance that it 

would be greater than UPL1–α/2, providing a total probability of 20 percent 

that the true hydrologic response would be outside the prediction interval. 

t

n – p – 1,1 – α /2 The ordinate from the Student’s ‘t’ probability distribution for a specified 

degrees of freedom equal to the number of observations (n), minus the 

number of estimated parameter for explanatory variables (p), minus one for 

the intercept. 

s2 The sample estimate of the population error variance σ2 

SST The total sum of squares, computed as ( y–y

–)′(y – y–), where y– is the sample 

mean

SSE The sum of squared errors, computed as ( y–ŷ)′(y– ŷ)=ε′ ε 

dfE The degrees of freedom in the error term, computed as n–p–1

MSE The mean square error, computed as s2=SSE /dfE 

RMSE The square root of the mean square error

R2

p The Pearson multiple coefficient of determination, computed as 1 – SSE /SST 

R2

Adj The Pearson multiple coefficient of determination adjusted for the number of 

estimated parameters
vii

Symbol Name

rP The Pearson product moment correlation coefficient, which the square root of 

R2

p 

rS The Spearman correlation coefficient, which is equal to the Pearson’s correlation coefficient if it were computed on the ranks of the data

R2

S The Spearman coefficient of determination, which is the square of the Spearman correlation coefficient

cov (βols)

i, j The covariance matrix among ordinary least-square parameter estimates

cor(βols)i, j The correlation matrix, computed as cov cov cov , , , βols β β i j ols i i ols ( ) ( ) j j ⋅ ( )

A Regression Model for Computing Index Flows 

Describing the Median Flow for the Summer Month of 

Lowest Flow in Michigan

By David A. Hamilton1, Richard C. Sorrell1, and David J. Holtschlag2

Abstract

In 2006, Michigan enacted laws to prevent new largecapacity 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. Regressionmodel estimates of water yield explain about 70.8 percent of 

the variability in index water yields indicated by streamflowgaging 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 

1

 Michigan Department of Environmental Quality.

2

 U.S. Geological Survey.

index water yield and flow statewide. In addition, a technique 

is presented for computing prediction intervals about the index 

flow estimates. 

Introduction

The Michigan Legislature (2006) passed Public Act 33 in 

2006 (PA33–2006); it and related laws are the first state laws 

to regulate water withdrawals. The legislation seeks to prevent 

any new or increased large-capacity withdrawal (generally 

referring to withdrawals that average more than 100,000 gallons of water per day (0.1547 ft3

/s) in any consecutive 30-day 

period) from causing an adverse resource impact. This impact 

is defined as decreasing the flow of a stream by part of the 

index flow such that the stream’s ability to support characteristic fish populations is functionally impaired. PA33–2006 further defines index flow as the 50 percent exceedance (median) 

flow for the lowest flow month of the flow regime (year), as 

determined over the period of record or extrapolated from 

analyses of the U.S. Geological Survey (USGS) streamflowgaging-station records in Michigan. 

In this report, the index flow is characterized as the 

median flow during the lowest flow in July, August, and 

September. The lowest monthly median summer flow was 

calculated by ranking the daily average flows at each USGS 

streamflow-gaging station (station) for the period of record, 

grouped by month. The median exceedance flow for each 

month was determined, and the lowest monthly value in the 

summer was selected as the index flow for each station. Summer is the time of greatest stress on the ecosystem from low 

flows and high temperatures.

Multiple linear regression models (Draper and Smith, 

1966) are commonly used to transfer streamflow information 

from gaged to ungaged sites. The regression model includes 

an equation for estimating or predicting the index water yield, 

computed as the index flow divided by the drainage area 
2 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

contributing to flow, using basin and climatic characteristics as 

explanatory variables. In this report, “estimation” refers to the 

process of computing the square root of water yield or the corresponding index flow for a gaged site that was used in model 

development, whereas “prediction” refers to the process of 

computing the square root of water yield or the corresponding 

index flow for an ungaged site. Unless ambiguity would result, 

the term “computation” is used when the distinction between 

estimation and prediction is unimportant. 

In addition to an equation for predicting the hydrologic 

response, regression models provide a probability model that 

describes the uncertainties of predicted responses. This uncertainty is sometimes expressed as a range of responses with a 

specified probability that is likely to contain the true hydrologic response at a particular stream site. The lower limit of 

this range can be used to help avoid overestimating a response, 

such as the index flow. 

Purpose and Scope

This report documents the development of a multiple 

linear regression model for predicting the expected magnitude 

and uncertainty of the index water yield. The index water yield 

is the water yield associated with the index flow, which is 

the median flow for the month of lowest summer streamflow 

in Michigan. For ungaged sites, the predicted index water 

yield can be multiplied by the corresponding drainage area 

upstream from the site to compute the index flow. In addition 

to the expected magnitude of the index flow, the uncertainty 

characterized by the regression model provides a basis for 

computing a range of flows within which the true index flow is 

likely to occur. An example computation is given to illustrate 

application of the regression model for predicting water yield 

and computing magnitude and uncertainty of the index flow. 

The regression model is applicable to Michigan streams where 

index flows are not significantly affected by existing water 

withdrawals, diversions, or augmentations. 

Previous Investigations

Knutilla (1967) and Holtschlag and Croskey (1984) 

developed statistical models for predicting a variety of low-, 

average-, and peak-flow characteristics for Michigan streams. 

Neff and others (2005) developed multiple regression equations for predicting base flow throughout the Great Lakes. 

None of these studies, however, resulted in a method for 

estimating the median streamflow during the summer month 

of lowest flow in Michigan. Longer periods of record and 

additional streamflow-gaging sites, combined with improved 

methods for determining basin and climatic characteristics, 

created an opportunity to improve estimation of streamflow 

characteristics in Michigan and support implementation of the 

2006 water-withdrawal legislation. 

Description of the Study Area

Michigan is in the eastern north-central part of the United 

States and is surrounded by four of the five Great Lakes 

(fig. 1). Ontario, Canada lies to the north and east of Michigan. 

To the west and south, border states are Wisconsin, Indiana, 

and Ohio. Michigan is the 10th largest state in the Union with 

a total land area of 58,110 mi2, 38,575 mi2 of Great Lakes 

waters, and 1,305 mi2 of inland waters (Michigan Library, 

2006). According to the U.S. Census Bureau (2007), the population of Michigan in 2006 was estimated to be 10,095,643. 

Michigan has a humid continental climate in which the 

average precipitation (rainfall plus water-equivalent snowfall 

depths) varies from about 28 to 38 in/yr. December through 

March tend to have slightly less precipitation, whereas July 

through September tend to have slighter more precipitation, 

than is typical for the rest of the year. Greater evapotranspiration during the summer, however, generally causes summer 

streamflows to be lower than those at other times of the year. 

Michigan consists of two peninsulas separated by the 

Straits of Mackinac, a body of water that connects Lake 

Michigan with Lake Huron (fig. 1). The straits are spanned 

by the Mackinac Bridge, where the northern tip of the Lower 

Peninsula is within about 5 mi of the southern coast of the 

eastern Upper Peninsula. The Upper Peninsula is heavily forested and somewhat mountainous in the west. Bedrock is at or 

near the surface in much of the Upper Peninsula. 

The Lower Peninsula is covered by a thick layer of glacial drift. The northern part is characterized by sandy material 

and is heavily forested. Trout streams, sustained by plentiful 

base flow, are common in that area. Much of the southeastern part of the Lower Peninsula is flat lakebed plains that are 

extensively agricultural or urban; base flow is meager. The 

southwestern part of the Lower Peninsula has a wide mixture 

of landforms, soil types, land uses, and stream types. 
Introduction 3

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

LAKE 

ERIE

L A K E S U P E R I O R

L

A

K

E

M

I

C

H

I

G A

N

L A K E

H U R O N

0 40 80 120 Kilometers

C A N A D A

O N T A R I O

Lower Peninsula

Upper Peninsula

M I C H I G A N

Straits of Mackinac

W I S C O N S I N

I L L I N O I S

I N D I A N A O H I O

0 40 80 Miles

42o

48o

90o 83o

M I C H I G A N
4 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

Regression Modeling

A multiple linear regression model was developed to 

predict index water yield. The model consists of a linear equation that is a function of selected hydrologic characteristics 

and model parameters estimated from index flow divided by 

drainage areas at gaged sites. This equation, plus the probability model underlying the error distribution, form the regression 

model. The following paragraphs describe the mathematical 

procedures used to estimate the model parameters from available data and assumptions underlying the probability model. 

Techniques are described for using the model uncertainty and 

site-specific climatic and basin characteristics to bound model 

predictions with a specified level of certainty. 

The general form of a multiple linear regression equation 

is 

y=X· β+ε	 (1)

where

y is a column vector containing the hydrologic response 

variable; 

X is referred to as the “design matrix,” which, in general, 

is composed of p columns of basin and climatic 

characteristics augmented with a leading column of 

1’s that serve as explanatory variables to estimate the 

hydrologic response; 

β is a column vector of parameters, β0, β1, . . . ,βp, 

that relate the explanatory variables to the hydrologic 

response variable; the ordinary least-square estimator 

of β is denoted βols and computed as βols=(X′·X )–1

·X′· y , where the prime symbol implies a matrix 

transpose and the –1 power implies a matrix inverse 

operation; 

ε is a vector of residuals that is assumed to be normally 

distributed and independent with mean zero and constant variance σ2

, commonly written ε~NI(0,σ2). In 

addition, it is assumed in the regression model that the 

covariance between ε and X, Cov(ε,X ), equals zero. 

Along with the predicted value itself, the distributional 

characteristics of the regression model error and the hydrologic 

characteristics at the site of interest are a basis for assessing 

the uncertainty of the predicted value. Let [LPLα/2 , UPL1–α/2] 

be a prediction interval between the lower prediction limit 

LPLα/2 and the upper prediction limit UPL1–α/2 centered about 

the regression estimate that is likely to contain the hydrologic 

response, y0, with a probability of 1– α. For example, if α was 

specified as 0.2, there would be a 10-percent chance that the 

hydrologic response would be less than LPLα/2 and a 10-percent chance that it would be greater than UPL1–α/2 , providing a 

total probability of 20 percent that the true hydrologic response 

would be outside the prediction interval. 

Computationally,

LPL UPL x t s x X X x α 2 1 α 2 0 βols n p 1 1 α 2

2

0

1

0 , 1 / , /

' '

− − − −

−

 = ⋅ ± ( + ( ) ) , 

where x0 is a row vector of corresponding basin characteristics 

at the site of interest augmented by a leading 1, (X′ X )–1 is 

a function of the design matrix used to estimate the model 

parameters, s2=ε′· ε /(n–p–1) and t

n –p – 1,α /2 is the inverse 

of Student’s t cumulative distribution function with n–p–1 

degrees of freedom at the specified alpha level divided by 2. 

The assumption that the regression residuals are normally 

distributed is often difficult to satisfy with water yield or flow 

values. In particular, the density function of the normal distribution is symmetrical, whereas water yield and flow data tend 

to be positively skewed because these variables are bounded 

by zero on the left and unbounded on the right side of the 

distribution. Logarithmic and square-root transformations are 

commonly applied to water yield and flow values to produce 

a hydrologic response variable for which model residuals are 

likely to be normally distributed. Unlike the logarithmic transformation, the square-root transformation does not eliminate 

observations that have zero values. 

As a convenience to the interested reader, the following key statistics are defined. The total sum of squares 

is SST=(y– y

–)′(y – y–), where y– is the mean of the hydrologic response variable, the sum of squared errors is 

SSE =(y–ŷ)′(y – ŷ) ε′· ε, and the model sum of squares is SSM = 

SST – SSE. The mean square total is MST=SST /(n–1). Degrees 

of freedom for the error is dfE =n –p –1, which subtracts the 

number of model explanatory variables, p, plus 1 for the 

intercept term, to describe the effective number of observations associated with the model error. The mean square error is 

MSE ≡s2=SSE /dfE and the model mean square is MSM =SSM /p. 

The root mean square error is the square root of the mean 

square error, RMSE = . MSE In addition, an F statistic, computed by dividing the MSM by the MSE, characterizes the overall statistical significance of the model. On the basis of the F

probability distribution, a probability value (p-value) is computed with the F statistic, as well as the degrees of freedom in 

the model and error components, to assess the likelihood that 

the null hypothesis that all model parameters are zero is true. 

A small p-value, commonly (but not necessarily) less than 

0.05, is used to reject the null hypothesis, thereby accepting 

the alternative hypothesis that, overall, the regression model is 

statistically significant. 

The Pearson multiple coefficient of determination, here 

denoted as R2

p

, describes the fraction of the variability of 

y described by ŷ where R2

p=1 –SSE /SST .R2p

 is equal to the 

squared Pearson’s product moment correlation coefficient 

between hydrologic response variables computed on the basis 

of streamflow-gaging station records and values estimated by 

the regression equation, rp( y, ŷ), where 

r y ŷ

ŷ

ŷ

ŷ

ŷ

y y

y y

P

i i i

n

i

i

n

i

i

n ( , ) =

( ) − − ( )

( ) − ⋅ − ( )

=

= =

∑

∑ ∑

1

2

1

2

1

(2)
Development of a Regression Model for Index Flow Estimation 5

β1=βols,i

,

 is effectively accepted. Rejecting the null hypothesis 

implies that the parameter βols,i

, 

 and corresponding explanatory 

variable are needed in the regression model. 

Development of a Regression Model 

for Index Flow Estimation

Regression models are a statistical means of transferring 

flow information obtained at streamflow-gaging stations to 

ungaged sites in the same hydrologic region. The process of 

transferring flow information from gaged to ungaged basins is 

commonly referred to as “flow regionalization.” The transfer 

is facilitated by identifying climatic and basin (hydrologic) 

characteristics in the gaged basins that are statistically related 

to the flow statistics computed from gaging-station records. 

Once this statistical relation is identified and regression parameters in the model equation are estimated, only the selected 

climatic and basin characteristics upstream from the ungaged 

site are needed to estimate the flow statistic for that location 

by use of the regression equation. 

The regression model includes this equation and a set of 

assumptions pertaining to the model errors, which are the discrepancies between estimates of the flow statistics computed 

from gaging-station records and estimates computed by use 

of the regression model. It is often necessary to transform the 

streamflow statistics being estimated to satisfy assumptions 

associated with the model error. In regional flow analysis, a 

square-root or logarithmic transformation of the streamflow 

statistics is commonly applied prior to estimating regressionmodel parameters. The inverse transform is commonly applied 

to regression estimates to compute the flow statistics of interest. The spatial distribution of model error is investigated to 

assess whether any bias occurs among the hydrologic subregions forming the region. If no subregional bias is detected, 

the regression model is considered appropriate for estimating 

the flow characteristic of interest throughout the region. The 

regression model error characteristics also are a basis for computing an interval about the regression estimate in which the 

true, but unknown, value of the streamflow statistic is likely to 

occur. 

Selection of Streamflow-Gaging Stations

Development of the regression model for regional flow 

characterization includes selection of streamflow-gaging stations where (1) no trends occur in the mean and variance of 

flow, (2) the period of record is sufficiently long to accurately 

characterize flow conditions of interest through statistical 

analysis of station records, (3) flow characteristics of interest 

are not substantially affected by water withdrawals, diversions, or regulation, and (4) streamflow represents the natural 

hydrologic response to climatic conditions and basin characteristics that are typical of the area. 

Indicators of estimation accuracy, such as R2

p and 

RMSE, reflect the model fit to the dataset used in development of the regression model. These indicators tend to show 

model improvement with increasing numbers of explanatory 

variables because the model is increasing fit to the specific 

characteristics of the available observations. Prediction 

accuracy, which is associated with the accuracy of predicting the hydrologic response from a basin not used in model 

development, is more difficult to quantify with small datasets. 

Prediction accuracy, however, improves with the addition of 

explanatory variables only up to a point. Beyond this point, 

prediction accuracy may decrease with the further addition 

of explanatory variables because a model that too closely fits 

the specific characteristics of available observations may not 

generalize well. 

To lessen the inflation of the model fit sometimes indicated by the R2

p value, the adjusted coefficient of determination, R2

adj =1–(SSE /n– p–1)/(SST /(n – 1)), accounts for the 

number of parameters in the model. Spearman’s correlation 

coefficient is a more robust measure of association than Pearson’s correlation coefficient when the data distributions are 

skewed. The Spearman’s correlation coefficient is computed 

similarly to Pearson’s correlation coefficient except that the 

original data are replaced by their ranks, rs( y,ŷ)=rp(rank( y), 

rank( ŷ)), where the rank of the smallest value in the set is 

1 and the rank of the largest value is n. Finally, Spearman’s 

coefficient of determination, symbolized as R2

S , is the square of 

Spearman’s correlation coefficient. 

Like the response estimates, estimated parameters

βols associated with the individual explanatory variables 

are uncertain. As the sample size (n) becomes large, estimated parameters are unbiased and normally distributed 

about their true values, assuming that ε~NI(0,σ2). The 

covariance of the parameter estimates, cov(βols) is equal 

to σ2(X′·X )–1, which is commonly written βols ~ Np + 1 (β, 

cov(βols)). Diagonal elements of the covariance matrix 

describe the variance of the corresponding estimated parameters; off-diagonal elements describe the covariance among 

parameters. A large covariance among parameters indicates a 

coupling between one or more parameter estimates because 

of an approximate linear dependency among explanatory 

variables. Such a coupling complicates interpretation of 

parameter magnitudes associated with specific explanatory 

variables. The magnitude of parameter covariances is 

commonly evaluated on the basis of their correlations, 

computed as cor(βols)i, j =cov(βols)i, j cor c β β ov / cov c β β ov ols i j ols i j ols i i ols ( ) j j , = ( ), ( ), ⋅ ( ), ..

If the magnitudes of these correlations |cor(βols)

i, j| exceed 

0.95 (Poeter and others, 2005), the independence of the paired 

parameter estimates is uncertain. 

A t statistic computed from the data can be used to assess 

the significance of individual parameters as 

t= (βols)

i, j cor c β β ov / cov c β β ov ols i j ols i j ols i i ols ( ) j j , = ( ), ( ), ⋅ ( ), . This t statistic is used to compute the .

probability that the null hypothesis, β1=0, is true. If this computed p-value is small, say less than 0.05, the null hypothesis 

is commonly rejected, and the alternative hypothesis that 
6 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

each selected gaging station by month for the entire period of 

record available and selecting the 50th percentile. The median 

flow for each summer month (July, August, and September) 

was determined, and the summer month with the lowest 

median flow was used to estimate IQ50 at that gaging station. 

To distinguish the true index flow IQ50 from the value of the 

flow response computed by use of the finite period of gagingstation record, the gaging station statistic is symbolized as 

ĨQ50. The value of ĨQ50 is assumed to converge to IQ50 as the 

length of gaged record increases. This assumption requires that 

there is no trend in the streamflow data (the expected value 

of IQ50 does not vary with time) and that ĨQ50 is an unbiased 

estimator of IQ50. No trends were detected in streamflow data 

at the selected stations.

For the 147 stations selected for the analysis, the lowest median flow occurred in July at 5 stations, in August at 

92 stations, and in September at 50 stations. The index flow 

ranged from zero at stations 04157500, Sebewaing River State 

Drain near Sebewaing, Mich., and 04158000, Columbia Drain 

near Sebewaing, Mich., to 1,850 ft3/s at station 04101500, St. 

Joseph River at Niles, Mich. (Appendix A). The average index 

flow at selected stations was 116 ft3/s, the standard deviation 

of these flow was 228 ft3/s, and the (dimensionless) skewness 

was 4.5044. 

Index Water Yield

Much of the variability in index flow is related to drainage area (fig. 4)3

. In development of the predictive equation, 

there was concern that the dominant relation between index 

flow and drainage area indicated by the power equation 

ÎQ50=β0 ·DAβ1 could mask more subtle relations involving 

basin and climatic characteristics. Also, the estimated exponent in the power equation, β

ˆ

1 of 1.2301, implies a slightly 

nonlinear relation between drainage area and index flow 

(fig. 4). A nonlinear relation between index flow and drainage area is considered physically unlikely because much of 

the index flow is thought to be derived from ground-water 

sources, which would be approximately linearly related to 

drainage area (β1 1). To help identify the appropriate relations, the index water yield IY50 was selected as a preferred 

metric to the index flow. IY50 was estimated from station 

records by dividing ĨQ50 by the drainage area upstream from 

the corresponding gaged site, and it is symbolized by ĨY50 . 

For the 147 selected sites, ĨY50 ranged from zero at the 

two stations with zero index flows, to 1.3087 ft3/s-mi2 at station 04139000, Houghton Creek near Lupton, Mich. The average ĨY50 value was 0.3302 ft3

/s-mi2, the standard deviation was 

0.2600 ft3/s-mi2, and the skewness was 1.3422. The positive 

3

 Data for two of the selected streamflow gaging stations could not be 

included in this plot because they were zero and could not be represented on a 

logarithmic scale.

Streamflow data from the USGS network of continuousrecord streamflow-gaging stations operated in Michigan 

through water year 2005 were used for this analysis. A water 

year is the 12-month period from October 1 to September 30 

and is identified by the calendar year in which it ends. Stations 

were selected for the regression analysis with respect to the 

following criteria: 

1. A minimum of 10 years of continuous-record data was 

required to reduce the temporal sampling variability of the 

flow statistic. 

2. Estimates of daily flow were not thought to be appreciably 

affected by water withdrawal, diversion, or augmentation.

3. Effects of regulation, either from natural storage in lakes 

or retention in regulated surface-water bodies, were not 

thought to substantially mask the hydrologic response 

from precipitation. 

From these evaluations, 147 streamflow-gaging stations 

were selected for inclusion in the analyses (figs. 2 and 3). 

Among selected stations, the average length of record was 

40.2 years, and the range was from 11 to 91 years. The first 

water year of record used in the analysis was 1901, and 88 stations included data from water year 2005. 

Identification of the Hydrologic Response 

Variable

The regression equation described in this report is a 

basis for computing an estimate of the index flow, which is 

defined as the median streamflow for the summer month of 

lowest flow in Michigan. The statistical distribution of index 

flows, however, is not consistent with assumptions underlying the regression model. To find a metric of index flow that 

is consistent with these assumptions and one in which climatic and basin characteristics physically associated with the 

streamflow response are more readily identified, mathematical 

transformations of index flow values were investigated. As a 

result, the response variable used in the regression equation 

was formed as the square root of the quotient of index flow 

divided by the drainage area of its associated basin. In this 

report, the response variable is referred to as the “hydrologic 

response variable.” The inverse transformation is applied to 

the regression estimates of the hydrologic response to compute 

index flows. 

Index Flow

In accordance with PA33–2006, the median flow during the lowest summer flow month was the index flow and is 

represented symbolically as IQ50. A statistic was calculated 

to estimate IQ50 by ranking the daily mean flows measured at 
Figure 2. U.S. Geological Survey streamflow-gaging stations in Michigan’s Upper Peninsula included in the analyses.

04001000

USGS streamflow-gaging station

Stream segment

04031500 USGS station number

EXPLANATION

85O 90O

0 10 20 30 40 50 100 Miles

0 15 30 45 60 120 Kilometers

04046000

041279

04049500

04045500

04055000

04059500

04059000

04057510

04056500

04056000

04058400

04060993

04061500

04058000

04062200

04057800

04033000 04031500 04031000

04032000 04040500

04035000

04040000 04041500

04041500

04043050

#

# # #

#

#

#

#

# #

#

# # # #

#

#

# # # #

#

#

# #

#

#
8 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

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

# # #

# #

#

#

#

#

#

#

# # #

# # #

# #

#

# #

# # #

# #

# # # # # #

#

# # # # # # # # # # #

# # # #

#

# # # # # # # # # # # ## # # #

#

## # #

# # #

# # # # #

# #

#

#

##

#

#

###

# #

# # # #

#

#

# # #

#

#

# ## #

# ##

# #

#

# # #

# # #

#

#

#

42

45

87 83

Miles

30

EXPLANATION

USGS streamflow-gaging station

Stream segment

04096015 USGS station number

0 50

0 60 Kilometers

0 0 0 0

04127997

04135000 04128990 04133501

04123500 04136500

04135600 04135700

04135500

04121000 04139500

04137500 04124000

04152238

04122500 04125000

04125500

04142000

04124500

04123000

04121300

04138000

04138500 04141500

04141000

04140500

04140000

04139000

04145500 04151500 04155000

04122200 04121900 04150500 04150000

04154000 04157500

04158500

04158000

04122100

04147500 04147990

04159492

04159900 04160000 04115000

04117000

04163500

04111500

04112000 04117500 04161000 04108600

04160005

04164800

04112500

04114498

04161500 04108801 04118000

04143900

04144000 04148200

04160600

04118500

04148440 04144500 04148300

04173000

04173500

04110000 04104945 04166200

04166300

04170500

04102500

04174800

04174500

04103010

04116500

04105700 04102700

04097170

04105000

04175600

04096015

04176000

04176605

04101500

04097540

04096515

04099000

04101800

04175700

04096405

04096600 04096900

04184500

1 2 3 4
Development of a Regression Model for Index Flow Estimation 9

Figure 4. Relation between estimates of index flow from gaging station records, I

 ̃Q50 , and 

drainage area (DA) [R 2

p

, the Pearson coefficient of determination].

skewness indicates that index water yield values are spread 

out more to the right than to the left of the mean. Drainage 

areas range from 1.1 mi2 at station 04141000, South Branch 

Shepards Creek near Selkirk, Mich., to 3,670 mi2 at station 

04101500, St. Joseph River at Niles, Mich. 

A normal distribution fitted to the empirical ĨY50 data was 

inadequate to approximate the distribution of water-yield values (fig. 5), because the empirical distribution was frequently 

outside of the 95-percent confidence bounds of the fitted normal distribution. Formally, the Lilliefors test (Conover, 1980) 

rejected the null hypothesis that a normal distribution adequately approximated the distribution of ĨY50 at the 5-percent 

level (p<0.001) of significance. Similarly, the Lilliefors test 

rejected (p<0.001) the null hypothesis that a normal distribution adequately approximated the distribution of the common 

logarithm transform of the index yield (LĨY50 ). 

A square-root transformation was applied to the elements 

of ĨY50 to assess the effect on the empirical distribution of the 

resulting values. Based on a sample mean of 0.5274 (ft3/smi2

)1/2, variance of 0.0525 ft3/s-mi2, and a skewness of 0.1607, 

a normal distribution closely approximated the empirical 

distribution of square root (Root) transformed values symbolized as RĨY50 (fig. 5). A Lilliefors Test did not reject the null 

hypothesis that RĨY50 values were normally distributed at the 

5-percent level of significance (p=0.5). Therefore, the square 

root transformation RĨY50 was used as the hydrologic-response 

variable in the regression model. 

Compilation of Hydrologic Characteristics for 

Use as Explanatory Variables 

Hydrologic characteristics were compiled for the 147 

stations used in these analyses. Compiled hydrologic characteristics include basin and climatic characteristics that are 

considered physically and statistically related to the IY50. All 

hydrologic characteristics included as possible explanatory 

variables in the regression equation are available as Geographic Information System (GIS) files to facilitate computation of hydrologic characteristics. Basin characteristics 

included categories of aquifer transmissivity, forested area, 

and hydrologic soil group; climatic characteristics included 

normal (1971–2000) annual precipitation and annual snowfall 

amounts. The following paragraphs discuss the hydrologic 

1 10 100 1,000 10,000

DRAINAGE AREA, IN SQUARE MILES

0.01

0.1

1

10

100

1,000

10,000

INDEX FLO

W, IN CUBI

C FEET

P

E

R

S

E

C

O

N

D

Scatter of data

Power equation

1.2206

50 IQ = 0.078687⋅ DA

2 0.75284 RP =

Statistics of power equation:

EXPLANATION
10 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

characteristics evaluated as possible explanatory variables in 

the regression equation. 

Transmissivity is a measure of the capacity of an aquifer 

to transmit water. The transmissivity of an aquifer is equal to 

its hydraulic conductivity, commonly expressed in units of 

feet per day, multiplied by its saturated thickness, in feet. The 

Goundwater Mapping Project (http://gwmap.rsgis.msu.edu/), a 

multiagency study in Michigan, created a grid of the estimated 

transmissivities for the glacial deposits (Michigan Department 

of Information Technology, 2005a). The grid is composed of 

1-km (0.621-mi) square elements and is based on an interpolation of transmissivities assigned to 270,000 water wells on the 

basis of lithologic information described in well logs prepared 

by well drillers. In areas of thin glacial deposits (less than 30 

ft thick) the grid element was assigned a code of -1 to indicate 

that thin deposits prevented a reliable estimation of transmissivity at that element. Because of the uncertainty associated 

with interpolation over the highly heterogeneous aquifer 

transmissivity field, grid elements that were more than 2,000 

m (6,560 ft) from a well were assigned a code of -2 to indicate 

that interpolation uncertainties prevented reliable estimation of 

transmissivity at that element. Otherwise, grid elements were 

assigned an estimated transmissivity value that ranged from 0 

to 30,309 ft2/d. 

The Michigan Glacial Landsystems Coverage (Michigan 

Department of Information Technology, 2005b) classified the 

surface geologic deposits into 10 land systems. Each applicable land system was assigned to an aquifer transmissivity 

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

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-2 -1.5 -1.0 -0.5 0.0 0.5 1.0

INDEX YIELD FLOW METRIC

CUMULATIVE PROBABILITY

Empirical distribution

Upper 95-percent limit about normal fit

Lower 95-percent limit about normal fit

Normal ditribution fit to empirical distribution

Index flow computed from streamflowgaging station records, 

in cubic feet per second

Drainage area, in square miles

50 IQ

DA



L 50 10 I



Y = log

    50 

IQ

 DA 

50 R 50 IY

DA =  

50

50 IY

DA = 

IQ

IQ 

LIY50 

50 IY



R 50 I



Y
Development of a Regression Model for Index Flow Estimation 11

Figure 6. Distribution of estimated aquifer transmissivity within transmissivity classes in Michigan. 

class. Bedrock, lacustrine fine, and thin drift over bedrock 

land systems were assigned to the low-transmissivity class; 

lacustrine coarse, lodgement till or fine supraglacial drift, 

and ice-marginal till land systems were classified as medium 

transmissivity; and coastal dunes, ice-contact outwash, and 

proglacial outwash were assigned to the high-transmissivity 

class. Land systems designated as lakes were not assigned a 

transmissivity class. About 0.25 percent of the elements were 

assigned aquifer transmissivities of zero and could not be 

displayed by means of a common logarithm transformation 

(log10). The log10 transformed distribution of aquifer transmissivities that were estimated to be greater than zero are shown 

for low, medium, and high classes of transmissivities in figure 6. Median estimated aquifer transmissivities increased 

from 723 ft2/d in areas classified as low transmissivity, to 

2,020 ft2/d in areas classified as medium transmissivity, to 

3,780 ft2/d for areas classified as high transmissivity. The 

spatial distribution of estimated transmissivity classes in the 

glacial aquifers in shown in figure 7. 

Land-use and land-cover characteristics affect hydrologic response primarily by affecting the rate at which water 

infiltrates into the soil and subsequently either drains to the 

ground-water system or flows overland to a nearby stream. 

As indicated by Anderson and others (1976), land use refers 

to “man’s activities on the land that are directly related to 

the land” (Clawson and Stewart, 1965), whereas land cover 

describes “the vegetative and artificial construction covering 

the land” (Burley, 1961). 

 AQUIFER TRANSMISSIVITY,

Low Medium High

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

TRANSMISSIVITY CLASS

upper whisker1

75th percentile

median

25th percentile

lower whisker1

outlier2 Explanation

1The span of the whiskers is the largest 

and smallest values within 1.5 times the 

interquartile range, which is formed as the 

difference between the 75th and 25th percentiles.

2Outliers are defined as 

values outside of the 

whisker span.

LOG10 IN FEET SQUARED PER DAY
12 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

Figure 7. Distribution of aquifer transmissivity classes in Michigan.

Transmissivity

High (H_Trans)

Medium (M_Trans)

Low (L_Trans)

Open water

EXPLANATION

42o

48o

90o 83o

0

0

10

15 3045 60 120 Kilometers

20 30 40 50 120 Miles

Michigan aquifer transmissivity classes from Michigan Department of

Information Technology, 2005b
Development of a Regression Model for Index Flow Estimation 13

The State of Michigan uses the spatial data coverages in 

the Michigan Resource Information System (MIRIS) (1978) 

as the standard for hydrologic studies in Michigan. MIRIS 

contains land-use and land-cover data that had been compiled 

from county and regional planning commissions. The MIRIS 

data represent land-use and land-cover data in a grid that 

contains 26,319 rows and 25,247 columns of cells. Each cell 

represents a land area of 30 m square. The categories include 

Level I features (Anderson and others, 1976), which are coded 

in MIRIS as integers and are defined as follows: (1) urban or 

built-up land; (2) agricultural land; (3) rangeland; (4) forest 

land, which included Level II classification of deciduous, evergreen, and mixed forest lands; (5) water; (6) wetland; and 

(7) barren land. The code –9999 signifies no data or inapplicable, which occurs over areas such as the Great Lakes. The 

spatial distribution of forest land in the MIRIS coverage is 

shown in figure 8. 

Four hydrologic soil groups have been defined by the 

U.S. Department of Agriculture Natural Resources Conservation Service (NRCS) (2007): 

• Group A soils (basin characteristic variable A_Soils) 

have low runoff potential when thoroughly wet. Water 

is transmitted freely through the soil. Group A soils 

typically have less than 10 percent clay and more than 

90 percent sand or gravel and have gravel or sand 

textures. 

• Group B soils (B_Soils) have moderately low runoff 

potential when thoroughly wet. Water transmission 

through the soil is unimpeded. Group B soils typically 

have between 10 percent and 20 percent clay and 

50 percent to 90 percent sand and have loamy sand or 

sandy loam textures. 

• Group C soils (C_Soils) have moderately high runoff 

potential when thoroughly wet. Water transmission 

through the soil is somewhat restricted. Group C soils 

typically have between 20 percent and 40 percent 

clay and less than 50 percent sand and have loam, silt 

loam, sandy clay loam, clay loam, and silty clay loam 

textures. 

• Group D soils (D_Soils) have high runoff potential 

when thoroughly wet. Water movement through the 

soil is restricted or very restricted. Group D soils typically have greater than 40 percent clay, less than 

50 percent sand, and have clayey textures. 

The spatial distribution of hydrologic soil groups in Michigan 

is shown on figure 9 based on the MIRIS coverage. MIRIS 

represents soil data in a grid that contains 26,319 rows and 

25,247 columns of cells. Each cell represents a land area of 

30 m square. In MIRIS, hydrologic group A soils are coded as 

1, group B soils are coded as 2, group C soils are coded as 3, 

group D soils are coded as 4, and no data or inapplicable areas 

are coded as -9999. The hydrologic-soil-group grid is georeferenced the same as the MIRIS grid for land use and land 

cover. 

Runoff curve numbers (RCN) were developed by the 

U.S. Department of Agriculture National Resources Conservation Service (2004). Conceptually, RCN describes the direct 

runoff component of total flow that includes (1) the channel 

component representing precipitation falling directly on the 

stream channel, (2) the surface or overland flow component, 

which represents flow from precipitation that exceeds the infiltration rate on the land surface, and (3) the subsurface component, which represents infiltrated water that flows laterally 

underground to the stream without intercepting permanently 

saturated areas; this subsurface flow component is sometimes 

referred to as “interflow.” With reference to RCN, runoff does 

not include the base-flow component, which is likely the main 

component influencing IQ50. Large direct-runoff components, 

however, are likely to be associated with smaller median or 

base-flow components. In general, greater RCN values are 

associated with soils with greater peak runoff potential, such 

as areas underlain the hydrologic soil group D; within each 

soil group, RCN increases with percentages of impervious 

areas, land covers that are prone to produce runoff, and basins 

that are considered to be in poor hydrologic condition. The 

MDEQ has developed GIS processing techniques for computing RCN from land-use and soil GIS coverages. From 

a possible range of 0 (no direct runoff) to 100, RCN ranged 

from 48 to 85, with an average of 70 for the selected basins. 

No statewide coverage is available to display the geographic 

variation of RCN, although it is similar to the hydrologic soil 

groups and land-use characteristics from which it is derived. 

Normal annual precipitation for 1971–2000 ranged from 

about 28.5 in/yr in the northeastern part of the Lower Peninsula to about 38 in/yr in southeastern part of the Lower Peninsula (Michigan Climatological Resources Program, 2004). 

Precipitation in the far western part of the Upper Peninsula 

approaches 35 in/yr, whereas precipitation in the eastern part 

is about 32 in/yr (fig. 10). 

Normal annual snowfall depths (fig. 11) for 1971–2000 in 

Michigan generally trend from a minimum of 40 in. in southeastern Lower Peninsula to a maximum of 220 in. in the northwestern tip of the Upper Peninsula (Michigan Climatological 

Resources Program, 2004). Evidence of lake-effect snow is 

apparent along the western coast of the Lower Peninsula and 

in a trend of increasing snowfall depths from south to north in 

the Upper Peninsula. 
14 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

Figure 8. Distribution of forest cover in Michigan.

42o

48o

90o 83o

Land cover from Michigan Resource Information System, 1978a.

Forested areas (Forest)

Unforested areas

EXPLANATION

0 25 50 100 Miles

0 25 50 100 Kilometers
Development of a Regression Model for Index Flow Estimation 15

Figure 9. Distribution of hydrologic soil groups in Michigan.

83o 90o

42o

Hydrologic

soil group

Hydrologic soil groups from Michigan Resource Information System, 1978a.

EXPLANATION

A (A_Soils)

B (B_Soils)

C (C_Soils)

D (D_Soils)

0 25 50 100 Miles

0 30 60 120 Kilometers

47o
16 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

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

48o

90o 83o

42o

35.0

EXPLANATION

Line of equal normal 

annual precipitation,

in inches. Contour 

interval is 0.5 in.

0 25 50 100 Miles

0 25 50 100 Kilometers

Normal annual precipitation from Michigan Climatological Resources Program, 2004.

31.5

. 31 0

37.5

28.5

38.0

30.0

31.5

31.0

30.5

34.5

32.5 32.0

31.0

32.5

35.0

33.0

34.0

32.5

31.5

33.0

33.0

34.0

32.0

33.5

32.0

30

. 5

33.5

34.0

34.0

33.0

32.5

32.0

32.0

31.5

31.0

32.5

33.0

33.5

30.0

30.0

30.5

31.0

31.0

31.5

31.5

29.5

29.5

34.0

35.0

35.0

36.0

36.5 36.0

37.0

37.0

37.5

36.5

34.5

35.5

31.0

31.0

32.0
Development of a Regression Model for Index Flow Estimation 17

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

140

70

0 25 50 100 Miles

0 25 50 100 Kilometers

80

80

80

90

100

110

120

60

40

40

60

70

70

70

80

40

90

90

90

80

80

100

100

110

50

50

60

Line of equal normal 

annual snowfall depth, 

in inches (not water 

equalent). Contour 

interval is 10 in.

40

48o

42o

90o 83o

Snowfall depths from Michigan Climatological Resources Program, 2004.

EXPLANATION

150

140 150

150

170

180

170

70

70

200

210

220

230240

160

160

160

160

160

60

190

190

170

110 110

110

100

100 100

100

120

130

130

130 130

140

150

90

90

90

120

120 130

130

50

50

50
18 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

Selection of Hydrologic Characteristics for Use 

as Explanatory Variables

Explanatory variables used in the regression equation 

were selected on the basis of both their statistical and hydrologic significance. One of the initial screening devices for 

assessing statistical associations was the matrix of correlation 

coefficients (table 1). Here, the maximum positive correlation (0.63) was found between RĨY50 and forest (Forest); the 

maximum negative correlation (–0.72) was found between 

RĨY50 and runoff curve numbers (RCN). Among explanatory 

variables, large negative correlations were detected between 

RCN and A_Soils (–0.90). A large positive correlation also 

was found between Snowfall and Forest (0.83). 

Correlations between explanatory variables indicate some 

redundancy of information and result in some statistical ambiguity in identifying explanatory variables for inclusion in the 

regression equation. Percentages of land use classified within 

individual categories of both transmissivity and soil groups 

generally summed to 100 percent, except in some areas where 

soils or glacial drift were absent and the sum therefore was 

less than 100 percent. For these two sets of variables, intragroup categories were negatively correlated. Also, because 

the sums of all transmissivity and soil categories generally 

were 100 percent, all members of either the transmissivity or 

soil categories could not be included in the regression without 

special numerical constraints. 

Initial development of the regression equation proceeded 

in an automated, stepwise manner. In particular, the variable 

most highly correlated with RĨY50 was added to the equation 

first, followed by the variable that was most highly correlated 

given the presence of the first variable in the equation. The 

process continued until all the alternative explanatory variables were evaluated in turn. Introduction of new variables 

into the equation sometimes resulted in the elimination of 

variables previously included at an apparent significance level 

of 0.15. 

Final selection of the regression equation was based on 

the following criteria: 

• The model explained a significant amount of the variability in RĨY50 . 

• The estimation error of the overall model was low. 

• The number of selected explanatory variables was 

constrained so that model prediction error—the error 

applicable to sites not included in the development of 

the equation—would be similar to model estimation 

error.

• The signs and magnitudes of parameters associated 

with selected explanatory variables were generally 

consistent with the expected physical association 

between the individual explanatory variables and the 

hydrologic response.

• An apparent significance level of about 5 percent for 

individual parameters was generally maintained. 

Estimation of the Hydrologic Response 

Variables

The regression equation for estimating the hydrologic 

response variable, RĨY50 , contains six explanatory variables 

and an intercept term. Based on the computed R2

adj value, the 

regression model explains about 70.8 percent of the variability 

in RĨY50 (fig. 12). The RMSE was 0.12377, with corresponding 

MSE or s2 equal to 0.015320, and overall the p-value associated 

with the regression model was less than 0.0001 (p<0.0001). 

Based on the results of a Lilliefors test of normality, there was 

insufficient evidence to reject the normality of the residual 

distribution at the 0.01 level of significance (p=0.015). In this 

report, estimates of the index water yield, ÎY50 , were obtained 

by squaring estimates of RÎY50 . After squaring individual 

values of RĨY50 and RÎY50 to compute ĨY50 and ÎY50 values, 

respectively, the R2

p (RĨY50 , RÎY50 ) decreases from the 0.7080 

determined in the regression to an R2

p (ĨY50 , ÎY50 ) of 0.6128 

because of the skewed distribution of the squared values. The 

coefficient of determination based on the ranks of the squared 

values R2

S (ĨY50 , ÎY50 ), however, is 0.7498, which is slightly 

higher than the R2

p (RĨY50 , RÎY50 ) of the more normally distributed RĨY50 and RÎY50 values. Thus, the correlation between 

measured and estimated index water yield is preserved in the 

space appropriate to the distribution of the two variables. The 

mean and standard deviation of residuals between measured 

and estimated water-yield values are 0.0151, and 0.1622, 

respectively. 

Explanatory variables included in the regression model, 

parameter estimates, and associated statistics are listed in 

table 2. Only the parameter associated with low transmissivity 

(L_Trans) was negatively associated with RĨY50 . In apparent 

contradiction to the suspected physical relation, the parameter 

associated with D_Soils is positively associated with RĨY50 

and is similar in magnitude to the parameter associated with 

A_Soils. The anomalous sign associated with D_Soils may be 

related to an association between D_Soils and other land-use 

and land-cover characteristics. 

To investigate this possibility, a cross tabulation between 

the 1978 MIRIS land use-land cover areas with hydrologic 

soil groups was computed (table 1–3 of Appendix 1). The 

results of this tabulation indicate that 89.3 percent of the areas 

classified as water also were classified as group D soils and 

that 68.7 percent of the areas classified as wetlands also were 

classified as group D soils (table 3). Furthermore, 60.4 percent 

of the soils classified as group D also were classified as forest 

areas. Areas covered by water, wetlands, and forests would be 

expected to be associated with higher median flows than areas 

not associated with these land use-land cover characteristics. 

Thus, the positive sign of the parameter estimate for D_Soils is 

not considered physically anomalous. 
Selection of Hydrologic Characteristics for Use as Explanatory Variables 19

Table 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. 

[H_Trans, M_Trans, and L_Trans indicate the percentage of the land area underlain by high, medium, and low aquifer transmissivity classes, 

respectively; Forest indicates forest-covered lands; A_Soils, B_Soils, C_Soils, and D_Soils indicate the percent of land areas classified as 

hydrologic soil group A, B, C, and D, respectively; RCN indicates the runoff curve number; Precip indicates the normal annual precipitation for 

1971–2000; Snowfall indicates the snowfall depths (not water equivalent); and RĨY50 indicates the square root of the index water yield] 

H_Trans M_Trans L_Trans Forest A_Soils B_Soils C_Soils D_Soils RCN Precip Snowfall RĨY50 

H_Trans 1.00 . . . . . . . . . . .

M_Trans -0.69 1.00 . . . . . . . . . .

L_Trans -.57 -0.20 1.00 . . . . . . . . .

Forest .13 -.24 0.09 1.00 . . . . . . . .

A_Soils .57 -.53 -.18 0.53 1.00 . . . . . . .

B_Soils -.17 .29 -.09 -.57 -0.74 1.00 . . . . . .

C_Soils -.61 .41 .36 -.25 -.36 -0.19 1.00 . . . . .

D_Soils -.05 -.01 .08 .41 -.11 -.19 -0.26 1.00 . . . .

RCN -.52 .44 .21 -.74 -.90 .60 .48 0.01 1.00 . . .

Precip .16 -.02 -.18 -.18 -.23 .38 -.14 -.10 0.20 1.00 . .

Snowfall -.09 -.17 .31 .83 .26 -.37 -.08 .32 -.48 0.03 1.00 .

RĨY50 .59 -.40 -.35 .63 .63 -.43 -.47 .20 -.72 .12 0.43 1.00

Correlations among parameter estimates for explanatory 

variables (excluding the intercept term) ranged from -0.6398 

to 0.5075 (table 4), indicating no significant linear dependence 

among explanatory variables. Some ambiguity between the 

intercept term, which is associated with the leading column of 

1’s in the design matrix, and the parameter estimate associated 

with Precip is indicated by a correlation of –0.9881. 

Values of the selected explanatory variables for all 147 

observations used in regression model are in table 1–2 of 

Appendix 1. If a unit vector of equal length were appended 

before columns 3–8 in table 1–2, the table entries would be 

identical to the design matrix X used in the development of the 

regression model. Boxplots show the range and approximate 

distribution of the selected explanatory variables used in the 

regression equation (fig. 13).

Spatial Distribution of the Regression-Model 

Error

Taken over all streamflow-gaging stations in the analysis, 

the multiple linear regression equation developed in the report 

provides an unbiased estimator, RÎY50 , of RĨY50 . Estimation of 

spatially referenced quantities without corresponding spatially 

referenced gaging-station coordinates as explanatory variables, 

however, can result in spatial patterns in the regression error. 

A significant spatial pattern in the distribution of regression 

errors would indicate that estimates could be locally biased. 

To investigate the potential for local bias in regression estimates, each selected gaging station was assigned to a 

subregion within Michigan (fig. 14). The subregions used in 

this report are similar to subregions defined on USGS hydrologic unit maps (Seaber and others, 1987). So that similar 

numbers of streamflow-gaging stations would be included in 

each subregion, however, individual cataloging units shown on 

USGS hydrologic unit maps were grouped somewhat differently in this report than cataloging units grouped by the USGS 

to define subregions. In addition, the cataloging units forming the subregions in this report were clipped to the State’s 

boundaries. 

Notched boxplots show the distribution of model residuals by subregion (fig. 15). For each boxplot, the width of the 

notch is computed so that boxplots whose notches do not 

overlap would have different medians at the 5-percent level 

of significance. By examining the intervals spanned by the 

notches, however, the boxplots indicate no significant difference in median residual among hydrologic subregions. Similarly, a Kruskal-Wallis test (Conover, 1980), which compares 

the median residuals for each subregion, found no significant differences among subregions (p=0.3515). The lack of 

geographic bias among subregions implies that the regression 

equation is applicable for all hydrologic subreaches, which 

together span the State of Michigan. The median residual of 

-0.0438 in Michigan hydrologic subregion 7 is slightly less 

than zero. A bootstrap analysis of residuals in subregion 7 

alone, however, did not indicate that the median residual was 

biased at the 5-percent level of significance. 
20 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

Figure 12. Relation between RÎY50 (the index of water yield estimated by regression) and R

~

IY50 (the index 

of water yield computed on the basis of the streamflow-gaging station records) [R2

adj , the adjusted Pearson 

coefficient of determination].

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

[Intercept refers to a leading column of ones in the design matrix; L_Trans refers to the percentage of the basin classified as having low ground-water transmissivity; H_Trans refers to the percentage of the basin classified as having high 

ground-water transmissivity; Forest refers to the percentage of the basin where land cover is classified as forest; Precip

refers to the normal annual precipitation for the period 1971–2000, in inches; and A_Soils and D_Soils refer to the percentage of the basin classified in the A and D hydrologic soil groups, respectively]

Index i Hydrologic

characteristic

Parameter estimate

βi

Standard error of the 

parameter estimate

Student’s t

statistic p-value

0 Intercept -0.541982 0.1910 -2.838 0.0052

1 L_Trans -.00136258 .0005397 -2.524 .0127

2 H_Trans .00204796 .00051078 4.010 <.0001

3 Forest .00402190 .0005452 7.377 <.0001

4 Precip .0236424 .005778 4.092 <.0001

5 A_Soils .00225536 .0007683 2.935 .0039

6 D_Soils .00162107 .001136 1.427 .1557

Equation for predicting the hydrologic response variable:

RÎY50 = β0 + β1 · L_Trans + β2 · H_Trans + β3 · Forest + β4 · Precip + β5 · A_Soils + β6 · D_Soils

0.0 0.2 0.4 0.6 0.8 1.0 1.2

SQUARE ROOT OF INDEX WATER YIELD COMPUTED FROM STEAMFLOW-GAGING

STATION RECORDS, RIY50, IN CUBIC FEET PER SECOND PER SQUARE MILE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

S

Q

U

A

R

E

ROOT OF INDEX WAT

E

R

YIELD

E

S

TIM

ATE

D

B

Y

R

EGR

E

SSION (RIY50), IN

C

U

BIC

FEE

T

P

E

R

S

E

CON

D PE

R

S

Q

U

A

R

E

MILE

Scatter of paired values

Line of agreement

2 0.720 Radj =

~

^
Selection of Hydrologic Characteristics for Use as Explanatory Variables 21

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

within Michigan.

Land use/

land cover

Hydrologic soil group

A B C D No data1 Percent

Percentages of soil group by land use/land cover

Urban 41.4 42.3 10.8 5.3 0.1 100

Agriculture 16.7 54.5 24.7 4.1 0.0 100

Range land 39.4 31.7 10.0 18.9 0.0 100

Forest 38.9 24.5 7.7 28.4 0.6 100

Water 3.9 5.1 0.6 89.3 1.1 100

Wetland 16.8 10.1 3.6 68.7 0.7 100

Barren 48.6 4.2 4.5 22.7 20.0 100

No data1 0.0 0.0 0.0 0.1 99.9 100

Percentages of land use/land cover by soil group

Urban 8.5 7.9 5.4 1.4 0.0 --

Agriculture 15.9 47.3 57.1 5.2 0.0 --

Range land 10.3 7.5 6.3 6.6 0.0 --

Forest 61.9 35.4 29.6 60.4 0.1 --

Water 0.3 0.3 0.1 8.8 0.0 --

Wetland 2.8 1.5 1.5 15.5 0.0 --

Barren 0.2 0.0 0.0 0.1 0.0 --

No data 0.1 0.0 0.0 1.8 99.9 --

Percent 100 100 100 100 100 --

1

 “No data” indicates that a cell in the Michigan Resource Information System coverage was coded as -9999. The nodata codes typically represented areas outside the land areas in Michigan. Generally, 99.9 percent of the time, a no-data 

code for land use-land cover corresponded to a no-data code for hydrologic soil group. Occasionally, no-data codes did 

not match between coverages.

Table 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.

[Intercept refers to a leading column of 1’s in the design matrix; L_Trans refers to the percentage of the basin classified 

as having low ground-water transmissivity; H_Trans refers to the percentage of the basin classified as having high 

ground-water transmissivity; Forest refers to the percentage of the basin where land cover is classified as forest; 

Precip refers to the normal annual precipitation for the period 1971–2000, in inches; and A_Soils and D_Soils refer to 

the percentage of the basin classified in the A and D hydrologic soil groups, respectively]

Parameter Intercept L_Trans H_Trans Forest Precip A_Soils D_Soils

Intercept 1 . . . . . .

L_Trans -0.1252 1 . . . . .

H_Trans .2255 0.5075 1 . . . .

Forest .1065 -.0545 0.2581 1 . . .

Precip -.9881 .0432 -.3134 -0.1215 1 . .

A_Soils -.3706 -.0988 -.6260 -.6398 0.3780 1 .

D_Soils -.2477 -.0231 -.2292 -.5779 .1899 0.4752 1
22 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

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

L_Trans, H_Trans, Forest, Precip, 

in inches per year

A_Soil, D_Soil,

in percent

0

10

20

30

40

50

60

70

80

90

100

MAGNITUDE OF EXPLANTORY VARIABLE

EXPLANATORY VARIABLE

in percent in percent in percent in percent

outlier2 Explanation

1The span of the whiskers is the largest and smallest 

values within 1.5 times the interquartile range, which is 

formed as the difference between the 75th and 25th percentiles.

2Outliers are defined as 

values outside of the 

whisker span.

upper whisker1

75th percentile

median

25th percentile

lower whisker1
Selection of Hydrologic Characteristics for Use as Explanatory Variables 23

Figure 14. Hydrologic subregions used in the analysis of the spatial distribution of regression-model error.

# #

#

# #

#

#

# #

#

# # # #

# # #

# #

#

# #

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# # # # # # # # # # # ## # # #

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## ## # # #

# # # ##

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#

#

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###

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#

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# # # #

#

#

#

# # #

# # # # # # # # # #

# # # #

#

### # # #

#

#

1

EXPLANATION

Streamflow-gaging station

Michigan hydrologic subregion

Hydrologic cataloging 

unit boundary

83O

48O

42O

90O

0 20 40 80 Miles

0 25 50 100 Kilometers

1

1

1

1

1

1

1

1

1

1

1

1 1

1

1

3

3

3

3

4

4

4

4

4

5

5

5

5

5

5

5

5 6

6

7

7

7

7

6

2

2

2

7

2

2

2

2

2

2

1
24 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

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

flow. RESIDUALS OF REGRESSION ESTIMATES OF THE SQUARE ROOT OF MEDIAN WATER YIELD DURING THE LOW FLOW SUMMER

MONTH, IN CUBIC FEET PER SECOND PER SQUARE MILE

MICHIGAN HYDROLOGIC SUBREGION 

95% CI3

upper whisker1

75th percentile

median

25th percentile

lower whisker1

outlier2

Explanation

1The span of the whiskers is

the largest and smmallest values

within 1.5 times the interquartile range, 

which is formed as the difference 

between the 75th and 25th percentiles. 

2Outliers are defined as 

values outside of the 

whisker span.

3The notch in the boxplot

spans the 95% confidence

interval (CI) about the median.

(26) (29) (13) (9) (33) (23) (14)

number of observations (26)

1 2 3 4 5 6 7

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5
Computation of the Index Flow 25

Computation of the Index Flow

The following sections describe computation of the 

index flow, ÎQ50 , which involves squaring the hydrologic 

response variable (the estimated square root of the index water 

yield), symbolized as RÎY50=ÎY50 , and multiplying by the 

corresponding drainage area. Using ÎY50 , the assumption of 

linearity between drainage area and index flow is evaluated. 

Then, the match between the index flows determined from the 

analysis of streamflow-gaging station records and index flows 

computed on the basis of regression estimates are compared. 

Finally, an example is provided for computing the index flow, 

ÎQ50 , given values for selected explanatory variables, and the 

upper and lower prediction limits, ÛPL1–α/2 and L

ˆPL1–α/2. 

Index Water Yield and Flow

In developing a regression equation for estimating 

the (square root of) index water yield, a linear relation was 

assumed between the index flow and corresponding drainage 

area. In particular, the drainage area raised to the first power 

was assumed to be proportional to flow. 

Two tests were done to evaluate the plausibility of this 

assumption. In the first test for unbiasedness, the estimated 

index flow was computed as ÎQ50=ÎY50 ·DA and a residual 

series as ξ1 =ĨQ50 – ÎQ50. Because ξ1 was not normally distributed, the nonparametric two-sided sign test (Conover, 1980) 

was applied under the null hypothesis that the median residual 

ξ1 did not differ significantly from zero. The resulting p-value 

was 0.4095, providing no statistical evidence to reject ÎQ50 as 

an unbiased estimator. 

Secondly, the unbiasedness of ÎQ50 and the linearity of relation between drainage area and the index flow were 

tested. In this case, the form of the model evaluated was 

ĨQ50=β0 + ÎY50 · DAβ1+ξ2

where it is assumed that the estimated value of β0 , β

ˆ

0

, was 

not significantly different from zero and that the estimated 

value of β1 , β

ˆ

1 , was not significantly different from 1. Nonlinear estimation of the above equation resulted in parameter 

estimates of β

ˆ

0=–2.2913 with an approximate standard error 

of šβ

ˆ

0

=5.8644 and β

ˆ

1=1.0093 with an approximate standard 

error of šβ

ˆ

1

=0.00322. Again, because ξ2 was not normally 

distributed, the conventional interpretation that rejection of the 

null hypothesis at a probability level α required that the interval [β

ˆ

0 –t1– α/2.147–9 ·šβ

ˆ

0

, β

ˆ

0+t1– α/2.147–9 ·šβ

ˆ

0

] not include zero and the 

interval [β

ˆ

1 –t1– α/2.147–9 ·šβ

ˆ

1

, β

ˆ

1+t1– α/2.147–9 ·šβ

ˆ

1

] not include 1 could 

not be strictly applied. The value of t

1–α/2,147–9 indicates the 

inverse of the Student’s t cumulative distribution function with 

a specified probability level α, commonly 0.05, and degrees 

of freedom 147-9, reflecting the total number of observations 

used to develop the regression equation and the total number 

of parameters used in estimating the square root of the yield 

and the relation between the yield and flow. These intervals 

provide no evidence, however, to indicate that β

ˆ

0 is statistically different from 0 or that β

ˆ

1 differs substantially (more 

than 1 percent) from its hypothesized value of 1. The approximate correlation between β

ˆ

0 and β

ˆ

1 was –0.4202, which does 

not indicate significant ambiguity between the two parameter 

estimates. Other nonlinear models investigated, including 

ĨQ50=β0 ·ÎY50 ·DAβ1+ξ3 and ĨQ50=β0+β1 ·ÎY50 ·DAβ2+ξ4 , 

resulted in one or more parameters having negative correlations less than –0.997, making the interpretations of individual 

parameter estimates unreliable. Therefore, ÎQ50 is considered 

an unbiased and physically plausible estimator of IQ50 .

Comparison of Index Flows

Index flows indicated by analysis of gaging-station 

records ĨQ50 and computed on the basis of the statewide 

regression equation ÎQ50 were compared for 147 streamflowgaging stations used in the development of the regression 

model. The resulting Spearman (rank) correlation was 0.97, 

and the corresponding coefficient of determination R2

S was 

0.9351. Although data for the two sites where index flows 

determined on the basis of streamflow-gaging station records 

equaled zero could not be displayed, a logarithmic plot of the 

measured and computed index flows shows a close match 

about the line of agreement (fig. 16). 

Example Computation

Following is an example computation to illustrate the 

procedure for estimating the index flow and computing the 

corresponding estimation interval. Station 04035000 is used 

to illustrate the computation. From table 1–2 Appendix 1, 

the explanatory variables for station 04035000 are L_Trans

=27.0 percent, H_Trans=23.9 percent, Forest=89.0 percent, 

Precip= 32.2 in., A_Soils=14.0 percent, and D_Soils=47.0 

percent. 

As an alternative to the matrix notation x0 · βols used previously, the regression equation for predicting the water yield 

response can be written as

RÎY50=β0+β1 ·L_Trans+β2 ·H_Trans+β3 ·Forest+β4 ·Precip

+β5 ·A_Soils+β6 ·D_Soils

Substituting the ordinary least square parameter estimates 

from table 4 for the beta coefficients and values of the explanatory variables for station 04035000, the regression equation 

can be written

RÎY50=–0.54198+(–0.0013626· 27.0)+(0.0020480· 23.9)+

(0.0040219 · 89.0)+ · · · +(0.023642· 32.2)+ 

(0.0022554· 14.0)+(0.0016211 · 47.0)

At station 04035000, the drainage area is 273 mi2, so the 

estimate of index flow, ÎQ50=RÎY2

50 ·DA=0.69722 · 273=

132.7 ft3/s, in this case compares closely with the measured 

value of ĨQ50=134 ft3/s. 
26 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

The interval formed by the range of the lower and upper 

prediction limits is a measure of the uncertainty of the hydrologic response estimate. In particular, the prediction interval is 

likely to contain IQ50 with probability 1 minus alpha (1 – α). 

The interval width will be smaller for a basin whose hydrologic characteristics are similar to those used to develop the 

regression than for basins whose characteristics are dissimilar. 

The computation of a lower estimation limit about RÎY50 

for α=0.2 will be shown with data from the site 04035000, as 

above. With this alpha value, the lower prediction limit will be 

less than IQ50 at a new site about 90 percent of the time. The 

lower prediction limit is computed as 

Figure 16. Relation between measured and computed index flows for selected streamflowgaging stations in Michigan [R2

S , the Spearman coefficient of determination].

LPL x t s x X X x α 2 0 βols 140 1 0 2 2

2

0

1

0 = ⋅ − ⋅ − (1+ ( ) ) −, . /

' '

where t

140,1 – 0.2/2=1.2876, s2=MSE =0.015320, and (X′X )–1

is from the entries in table 5, results in a lower 90-percent 

prediction limit of 0.5328. Similar computations resulted in 

an upper prediction limit of 0.8615 for RÎY50 . The 90-percent 

prediction interval about RÎY50 corresponds to a 90-percent 

prediction interval about ÎQ50 of [0.52382 ·273.2, 0.86152

·273.2]. Thus, the probability that IQ50 is contained within the 

estimation interval from [77.5,202.6] ft3/s is 80 percent, or 

Prob[77.5<IQ50 < 202.6]=0.8. 

Summary

In 2006, Michigan enacted legislation to prevent 

new large-capacity withdrawals from causing an adverse 

impact on a stream’s ability to support characteristic fish 

populations. The median streamflow for the summer month 

of lowest flow was selected as the index flow against which 

possible withdrawals would be assessed. This report describes 

a method to predict the index flow at ungaged stream sites in 

Michigan. This study was conducted by the U.S. Geological 

Survey (USGS) in cooperation with the Michigan Department 

of Environmental Quality and the Michigan Department of 

Natural Resources. 

A set of 147 USGS continuous streamflow-gaging 

stations were selected from among stations operated in Michi0.01 0.1 1 10 100 1,000 10,000

INDEX FLOW COMPUTED FROM STREAMFLOW-GAGING

STATION RECORDS, IQ50, IN CUBIC FEET PER SECOND

0.01

0.1

1

10

100

1,000

10,000

IND

E

X

FLO

W

COM

P

U

T

E

D

B

Y

U

S

E OF

T

H

E

R

E

G

R

ESSIO

N

M

O

D

EL, IQ50, IN

C

U

BIC

F

E

E

T

P

E

R

S

ECO

N

D

Scatter of paired values

Line of agreement

2 0.9351 RS =

^

~
Acknowledgments 27

gan for 10 or more years that were thought to represent the 

natural response of streamflow to precipitation. In particular, 

stations where median low flows were thought to have been 

appreciably affected by regulation or water withdrawals, augmentations, or diversions were excluded from the regression 

analysis. Of the 147 selected stations, minimum median flows 

occurred in July at 5 stations, in August at 92 stations, and in 

September at 50 stations. Index flows ranged from 0 to 

1,850 ft3/s. Index water yields, which were computed by dividing index flows by the corresponding drainage areas upstream 

from the stream measurement sites, ranged from 0 to 1.309 

ft3/s-mi2. A square-root transformation was applied to the 

index water yields so that the transformed values were 

approximately normally distributed. 

A multiple linear regression equation was developed 

to predict the square root of the index water yield at ungaged 

sites using selected basin and climatic characteristics as 

explanatory variables. Selected variables included percentages 

of land area underlain by low and high aquifer transmissivity, percentage of forest cover, normal annual precipitation, 

and percentages of land cover associated with hydrologic soil 

groups A and D (highly and poorly permeable soils, respectively). The regression model explains about 70.8 percent 

of the variability in the hydrologic response variable, which 

was the square root of the index water yield. No spatial bias 

in the regression estimates was detected among seven hydrologic subregions spanning Michigan. Therefore, the single 

regression equation developed in this report is appropriate for 

statewide application. 

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

[Intercept refers to a leading column of ones in the design matrix; L_Trans refers to the percentage of the basin classified as having low ground-water transmissivity; H_Trans refers to the percentage of the basin classified as having high ground-water transmissivity; Forest refers to the percentage of the basin where 

land cover is classified as forest; Precip refers to the normal annual precipitation for the period 1971–2000, in inches; and A_Soils and D_Soils refer to the 

percentage of the basin classified in the A and D hydrologic soil groups, respectively]

Explanatory variables in the regression model

Intercept L_Trans H_Trans Forest Precip A_Soils D_Soils

2.38035E+00 -8.42611E-04 1.43560E-03 7.23794E-04 -7.11625E-02 -3.54982E-03 -3.50692E-03

-8.42611E-04 1.90162E-05 9.13204E-06 -1.04607E-06 8.79202E-06 -2.67491E-06 -9.25476E-07

1.43560E-03 9.13204E-06 1.70298E-05 4.69128E-06 -6.03695E-05 -1.60371E-05 -8.67801E-06

7.23794E-04 -1.04607E-06 4.69128E-06 1.93996E-05 -2.49764E-05 -1.74926E-05 -2.33578E-05

-7.11625E-02 8.79202E-06 -6.03695E-05 -2.49764E-05 2.17903E-03 1.09528E-04 8.13390E-05

-3.54982E-03 -2.67491E-06 -1.60371E-05 -1.74926E-05 1.09528E-04 3.85354E-05 2.70690E-05

-3.50692E-03 -9.25476E-07 -8.67801E-06 -2.33578E-05 8.13390E-05 2.70690E-05 8.42089E-05

Index flows can be predicted at ungaged sites by 

squaring the predicted regression response and multiplying 

the result by the corresponding drainage area. The predicted 

index flow explains about 94.0 percent of the variability in 

index flows indicated by streamflow-gaging-station records. 

In addition, the report documents the technique and provides 

information needed to compute an interval about the predicted 

index flow. An example computation is provided. 

Acknowledgments

The authors appreciate the valuable technical contributions the following individuals made in developing the 

regression model: Jon W. Allan, Manager, and Scott F. DeBoe, 

Senior Environmental Planner, Consumers Energy Company; 

Paul Seelbach, Research Director, Michigan Department 

of Natural Resources; and Howard W. Reeves, Research 

Hydrologist, U.S. Geological Survey. Arthur Cooper, Research 

Assistant, University of Michigan, did the GIS analysis 

needed to characterize the explanatory variables at the selected 

streamflow-gaging stations. 
28 Regression Model for Computing Index Flows Describing the Median Flow, Summer Month of Lowest Flow, Michigan

References Cited

Anderson, J.R., Hardy, E.E., Roach, J.T., and Witmer, R.E., 

1976, A land use and land cover classification system 

for use with remote sensor data: U.S. Geological Survey 

Professional Paper 964, 41 p., accessed October 3, 2007, at 

http://landcover.usgs.gov/pdf/anderson.pdf.

Conover, W.J., 1980, Practical nonparametric statistics 

(2d ed.): New York, Wiley, 493 p.

Burley, T.M., 1961, Land use or land utilization?: Professional 

Geographer, v. 13, no. 6, p. 18–20.

Clawson, Marion, and Stewart, C.L., 1965, Land use information—A critical survey of U.S. statistics including possibilities for greater uniformity: Baltimore, Md., Johns Hopkins 

Press for Resources for the Future, Inc., 402 p.

Holtschlag, D.J., and Croskey, H.M., 1984, Statistical models 

for estimating flow characteristics of Michigan streams: 

U.S. Geological Survey Water-Resources Investigations 

Report 84–4207, 80 p. 

Draper, N.R., and Smith, Harry, 1966, Applied regression 

analysis: New York, Wiley, 407 p. 

Knutilla, R.L., 1967, Flow characteristics of Michigan 

streams: U.S. Geological Survey Open-File Report 67–135, 

55 p.

Michigan Climatological Resources Program, 2004, Michigan 

climatological normals, 1971–2000: East Lansing, Mich., 

State Climatologist’s Office, Michigan State University, 

digital media. 

Michigan Department of Environmental Quality [n.d.], Calculating curve numbers using GIS: Hydrologic Studies Unit, 

10 p., accessed Oct. 3, 2007, at http://www.michigan.gov/

documents/deq/lwm-cn-calc-using-gis_202628_7.pdf.

Michigan Department of Information Technology, 2005a, 

Glacial deposits estimated transmissivity geographic 

theme: Lansing, Mich., Center for Geographic Information, 

accessed December 6, 2007, at http://www.mcgi.state.mi.us/

mgdl/?rel=ext&action=sext.

Michigan Department of Information Technology, 2005b, 

Michigan glacial landsystems coverage by D.P. Lusch: East 

Lansing, Michigan, Michigan State University, accessed 

on December 6, 2007, at http://www.mcgi.state.mi.us/

mgdl/?rel=ext&action=sext.

Michigan Library, 2006, Michigan in Brief—Information 

about the state of Michigan: Lansing, Mich., accessed 

December 6, 2007, at www.michigan.gov/hal/0,1607,7-160-

15481_20826_20829-56001--,00.html.

Michigan Legislature, 2006, Public Acts of 2006, Act No. 

33: 93rd Legislature, Regular Session of 2006, accessed 

September 27, 2007, at http://www.legislature.mi.gov/

documents/2005-2006/publicact/pdf/2006-PA-0033.pdf.

Michigan Resource Information System, 1978, MIRIS Landuse/cover Polygon—Geographic data and metadata: Lansing, Michi., Michigan Department of Natural Resources, 

accessed December 6, 2007, at http://www.mcgi.state.mi.us/

mgdl/.

Natural Resources Conservation Service, 2004, Estimation of 

direct runoff from storm rainfall, Chapter 10 of Part 630, 

Hydrology, of National engineering handbook: Accessed 

December 7, 2007, at http://policy.nrcs.usda.gov/media/

pdf/H_210_630_10.pdf.

Natural Resources Conservation Service, 2007, Hydrologic 

soil groups, Chapter 7 of Part 630, Hydrology of National 

engineering handbook: Accessed December 6, 2007, at 

http://policy.nrcs.usda.gov/media/pdf/H_210_630_7.pdf.

Neff, B.P., Day, S.M., Piggott, A.R., and Fuller, L.M., Base 

flow in the Great Lakes Basin: U.S. Geological Survey 

Scientific Investigations Report 2005–5217, 23 p. 

Poeter, E.P., Hill, M.C., Banta, E.R., Mehl, Steffen, and Christensen, Steen, 2005, UCODE_2005 and six other computer 

codes for universal sensitivity analysis, calibration, and 

uncertainty evaluation: U.S. Geological Survey Techniques 

and Methods 6–A11, 283 p.

Seaber, P.R., Kapinos, F.P., and Knapp, G.L., 1987, Hydrologic unit maps: U.S. Geological Survey Water-Supply 

Paper 2294, 63 p., accessed December 7, 2007, at 

http://pubs.usgs.gov/wsp/wsp2294/.

U.S. Census Bureau, 2007, State & county QuickFacts, 

accessed December 6, 2007, at http://quickfacts.census.gov/

qfd/states/26000.html.
Appendix 1. Tables of streamflow-gaging 

station attributes, flow characteristics, and 

explanatory variables used in the development 

of the regression equation for estimating the 

index flow at ungaged streams in Michigan
Table 1–1. Flow, yield, and record characteristics for streamflow–gaging stations used in the regression analysis—Continued. [A water year is the 12–month period from October 1 to September 30. The water year is designated by the calendar year in which it ends] U.S. Geological Survey station number Station name Latitude (decimal degrees) Longitude (decimal degrees) Drainage area (square miles) Minimum monthly median flow (cubic feet per second) (IQ50) Minimum monthly median yield (cubic feet per second per square mile) (IY50) Month of minimum flow 

04001000 Washington Creek at Windigo, Mich. 47.92306 89.14500 13.2 2.3 0.17424 August 

04031000 Black River near Bessemer, Mich. 46.51134 90.07462 200 40.0 0.20000 August 04031500 Presque Isle River at Marenisco, Mich. 46.37217 89.69238 172 71.0 0.41183 August 04032000 Presque Isle River near Tula, Mich. 46.54689 89.77738 264 90.0 0.34078 August 04033000 Middle Branch Ontonagon River near Pauld- ing, Mich. 46.35689 89.07736 162 100 0.61843 August 04035000 East Br Ontonagon River near Mass, Mich. 46.68994 89.07347 273 134 0.49048 August 04040000 Ontonagon River near Rockland, Mich 46.72077 89.20709 1330 634 0.47530 August 04040500 Sturgeon River near Sidnaw, Mich. 46.58411 88.57597 169 44.0 0.26036 August 04041500 Sturgeon River near Alston, Mich. 46.72632 88.66208 343 184 0.53629 August 04043050 Trap Rock River near Lake Linden, Mich. 47.22854 88.38539 29.6 13.0 0.43919 August 04045500 Tahquamenon River near Paradise, Mich. 46.57501 85.26955 757 321 0.42410 August 04046000 Black River near Garnet, Mich. 46.11806 85.36537 33.5 10.0 0.29851 August 04049500 Manistique River at Germfask, Mich. 46.23331 85.92791 420 238 0.56721 August 04055000 Manistique River near Blaney, Mich. 46.08609 86.05930 716 352 0.49183 August 04056000 West Branch Manistique River near Manis- tique, Mich. 46.08886 86.16125 326 154 0.47312 August 04056500 Manistique River near Manistique, Mich. 46.03053 86.16125 1,130 605 0.53716 August 04057510 Sturgeon River near Nahma Junction, Mich. 45.94302 86.70570 184 82.0 0.44662 August 04057800 Middle Branch Escanaba River at Humboldt, Mich. 46.49910 87.88652 45.7 16.0 0.35011 August 04058000 M Br Escanaba River near Ishpeming, Mich. 46.39438 87.75847 128 43.0 0.33515 August 04058400 Goose Lake Outlet near Sands Station, Mich. 46.39300 87.49375 36.3 12.0 0.33058 August 

.
Table 1–1. Flow, yield, and record characteristics for streamflow–gaging stations used in the regression analysis—Continued.[A water year is the 12–month period from October 1 to September 30. The water year is designated by the calendar year in which it ends] 

U.S. 

Geological 

Survey 

station 

number Station name

Latitude 

(decimal 

degrees)

Longitude 

(decimal 

degrees)

Drainage 

area

(square 

miles)

Minimum 

monthly 

median flow 

(cubic feet 

per second) 

(IQ50)

Minimum 

monthly median 

yield (cubic feet 

per second per 

square mile) 

(IY50)

Month of

 minimum 

flow

04059000 Escanaba River at Cornell, Mich. 45.90857 87.21375 871 330 0.37892 August 

04059500 Ford River near Hyde, Mich. 45.75552 87.20152 444 85.0 0.19140 August 04060993 Brule River near Florence, Wisc. 45.96079 88.31597 378 236 0.62269 August 04061500 Paint River at Crystal Falls, Mich. 46.10578 88.33486 600 288 0.47976 August 04062200 Peshekee River near Champion, Mich. 46.55688 88.00263 132 28.0 0.21244 August 04096015 Galien River near Sawyer, Mich. 41.87365 86.57502 80.8 21.0 0.25990 September 04096405 St. Joseph River at Burlington, Mich. 42.10282 85.04025 201 61.0 0.30348 September 04096515 South Branch Hog Creek near Allen, Mich. 41.94866 84.82774 48.7 7.9 0.16222 September 04096600 Coldwater River near Hodunk, Mich. 42.02921 85.10692 286 65.0 0.22759 September 04096900 Nottawa Creek near Athens, Mich. 42.05560 85.30832 162 72.0 0.44444 September 04097170 Portage River near Vicksburg, Mich. 42.11477 85.48555 68.2 30.0 0.43988 September 04097540 Prairie River near Nottawa, Mich. 41.88838 85.40943 107 46.0 0.43152 September 04099000 St. Joseph River at Mottville, Mich. 41.80088 85.75610 1,880 850 0.45227 September 04101500 St. Joseph River at Niles, Mich. 41.82921 86.25973 3,670 1,850 0.50464 September 04101800 Dowagiac River at Sumnerville, Mich. 41.91338 86.21307 252 177 0.70378 August 04102500 Paw Paw River at Riverside, Mich. 42.18615 86.36836 390 262 0.67197 August 04102700 South Branch Black River near Bangor, 

Mich.

42.35420 86.18753 83.5 35.0 0.41916 September 

04103010 Kalamazoo River near Marengo, Mich. 42.26171 84.85581 270 147 0.54545 September 04104945 Wanadoga Creek near Battle Creek, Mich. 42.39643 85.13166 48.3 15.0 0.31056 August 04105000 Battle Creek at Battle Creek, Mich. 42.33199 85.15416 274 70.0 0.25547 September 04105700 Augusta Creek near Augusta, Mich. 42.35337 85.35389 36.8 31.0 0.84239 August 04108600 Rabbit River near Hopkins, Mich. 42.64225 85.72197 65.1 22.0 0.33794 September 04108801 Macatawa River near Zeeland, Mich. 42.77919 86.01837 66.9 4.3 0.06428 September 
Table 1–1. Flow, yield, and record characteristics for streamflow–gaging stations used in the regression analysis—Continued. [A water year is the 12–month period from October 1 to September 30. The water year is designated by the calendar year in which it ends] U.S. Geological Survey station number Station name Latitude (decimal degrees) Longitude (decimal degrees) Drainage area (square miles) Minimum monthly median flow (cubic feet per second) (IQ50) Minimum monthly median yield (cubic feet per second per square mile) (IY50) Month of minimum flow 

04110000 Orchard Creek at Munith, Mich. 42.39365 84.26496 47.3 5.3 0.11205 September 

04111500 Deer Creek near Dansville, Mich. 42.60837 84.32080 16.3 0.9 0.05521 September 

04112000 Sloan Creek near Williamston, Mich. 42.67587 84.36386 10.4 0.2 0.02308 September 04112500 Red Cedar River at East Lansing, Mich. 42.72781 84.47775 344 38.0 0.11047 September 04114498 Looking Glass River near Eagle, Mich. 42.82809 84.75943 284 40.0 0.14080 September 04115000 Maple River at Maple Rapids, Mich. 43.10975 84.69305 420 30.0 0.07141 September 04116500 Flat River at Smyrna, Mich. 43.05281 85.26474 516 222 0.42998 August 04117000 Quaker Brook near Nashville, Mich. 42.56587 85.09361 7.8 2.8 0.35897 September 04117500 Thornapple River near Hastings, Mich. 42.61587 85.23639 410 109 0.26553 September 04118000 Thornapple River near Caledonia, Mich. 42.81114 85.48335 795 281 0.35337 September 04118500 Rogue River near Rockford, Mich. 43.08225 85.59086 257 127 0.49378 September 

04121000 Muskegon River near Merritt, Mich. 44.33557 84.89003 352 115 0.32689 August 04121300 Clam River at Vogel Center, Mich. 44.20057 85.05281 239 73.0 0.30506 August 04121900 Little Muskegon River near Morley, Mich. 43.50253 85.34254 136 72.0 0.53137 July 04122100 Bear Creek near Muskegon, Mich. 43.28863 86.22284 16.7 5.0 0.29940 August 04122200 White River near Whitehall, Mich. 43.46418 86.23257 404 273 0.67491 August 04122500 Pere Marquette River at Scottville, Mich. 43.94501 86.27869 689 455 0.65999 August 04123000 Big Sable River near Freesoil, Mich. 44.12028 86.28008 115 101 0.87826 August 04123500 Manistee River near Grayling, Mich. 44.69307 84.84726 132 170 1.28496 August 04124000 Manistee River near Sherman, Mich. 44.43639 85.69868 865 856 0.98971 August 
Table 1–1. Flow, yield, and record characteristics for streamflow–gaging stations used in the regression analysis—Continued.[A water year is the 12–month period from October 1 to September 30. The water year is designated by the calendar year in which it ends] 

U.S. 

Geological 

Survey 

station 

number Station name

Latitude 

(decimal 

degrees)

Longitude 

(decimal 

degrees)

Drainage 

area

(square 

miles)

Minimum 

monthly 

median flow 

(cubic feet 

per second) 

(IQ50)

Minimum 

monthly median 

yield (cubic feet 

per second per 

square mile) 

(IY50)

Month of

 minimum 

flow

04124500 East Branch Pine River near Tustin, Mich. 44.10251 85.51728 58.9 9.9 0.16808 August 

04125000 Pine River near Leroy, Mich. 44.06279 85.54894 130 51.0 0.39140 August 04125500 Pine River near Hoxeyville, Mich. 44.20306 85.79951 254 230 0.90658 August 04126200 Little Manistee River near Freesoil, Mich. 44.18362 86.16758 185 141 0.76381 August 04127918 Pine River near Rudyard, Mich. 46.18585 84.59783 202 77.0 0.38138 August 04127997 Sturgeon River at Wolverine, Mich. 45.29890 84.61114 181 170 0.93975 August 04128990 Pigeon River near Vanderbilt, Mich. 45.15668 84.46669 57.7 58.0 1.00520 July 04133501 Thunder Bay River at Herron Road near 

Bolton, Mich.

45.12446 83.64721 586 309 0.52739 September 

04135000 Thunder Bay River near Alpena, Mich. 45.09418 83.49970 1,240 474 0.38288 August 04135500 Au Sable River at Grayling, Mich. 44.65974 84.71253 96.6 61.0 0.63147 August 04135600 East Branch Au Sable River at Grayling, 

Mich.

44.66890 84.70558 71.2 34.0 0.47753 August 

04135700 South Branch Au Sable River near Luzerne, 

Mich.

44.61474 84.45557 391 134 0.34236 August 

04136500 Au Sable River at Mio, Mich. 44.66001 84.13112 1,360 788 0.57894 August 04137500 Au Sable R near Au Sable, Mich. 44.43640 83.43386 1,740 1,120 0.64383 September 04138000 East Branch Au Gres River at McIvor, Mich. 44.23252 83.70082 89.9 38.0 0.42269 August 04138500 Au Gres River near National City, Mich. 44.17613 83.74248 151 21.0 0.13871 August 04139000 Houghton Creek near Lupton, Mich. 44.39585 84.04722 29.8 39.0 1.30872 August 04139500 Rifle River at “The Ranch” near Lupton, 

Mich.

44.39335 84.03833 57.4 66.0 1.14983 August 

04140000 Prior Creek near Selkirk, Mich. 44.33502 84.06833 21.0 6.7 0.31905 August 04140500 Rifle River at Selkirk, Mich. 44.31335 84.06944 116 87.0 0.74742 August 04141000 South Branch Shepards Creek near Selkirk, 

Mich.

44.30780 84.08694 1.1 0.0 0.03636 July 
Table 1–1. Flow, yield, and record characteristics for streamflow–gaging stations used in the regression analysis—Continued. [A water year is the 12–month period from October 1 to September 30. The water year is designated by the calendar year in which it ends] U.S. Geological Survey station number Station name Latitude (decimal degrees) Longitude (decimal degrees) Drainage area (square miles) Minimum monthly median flow (cubic feet per second) (IQ50) Minimum monthly median yield (cubic feet per second per square mile) (IY50) Month of minimum flow 

04141500 West Branch Rifle River near Selkirk, Mich. 44.26113 84.10916 65.4 32.0 0.48930 July 

04142000 Rifle River near Sterling, Mich. 44.07252 84.01999 333 159 0.47791 August 04143900 Shiawassee River at Linden, Mich. 42.81586 83.80190 81.9 22.0 0.26862 August 04144000 Shiawassee River at Byron, Mich. 42.82364 83.94579 363 71.0 0.19543 September 04144500 Shiawassee River at Owosso, Mich. 43.01503 84.18108 530 86.0 0.16214 September 04145500 Bad River near Brant, Mich. 43.29669 84.22915 89.9 0.6 0.00667 August 04146000 Farmers Creek near Lapeer, Mich. 43.04475 83.33717 51.1 5.1 0.09980 August 04146063 South Branch Flint River near Columbiaville, Mich. 43.15947 83.35078 211 48.5 0.23029 August 04147500 Flint River near Otisville, Mich. 43.11114 83.51940 526 109 0.20715 August 04147990 Butternut Creek near Genesee, Mich. 43.13586 83.59912 34.8 3.6 0.10345 August 04148140 Kearsley Creek near Davison, Mich. 43.03364 83.58134 99.7 13.0 0.13039 August 04148160 Gilkey Creek near Flint, Mich. 43.02419 83.62551 6.9 0.2 0.02319 August 04148200 Swartz Creek near Holly, Mich. 42.82753 83.62828 12.1 1.5 0.12397 September 04148300 Swartz Creek at Flint, Mich. 42.98781 83.73246 114 5.6 0.04904 August 04148440 Thread Creek near Flint, Mich. 42.97503 83.63579 54.4 4.7 0.08640 August 04148500 Flint River near Flint, Mich. 43.03892 83.77163 960 171 0.17805 August 04150000 South Branch Cass River near Cass City, Mich. 43.56696 83.11189 239 5.0 0.02090 September 04150500 Cass River at Cass City, Mich. 43.58419 83.17606 363 11.0 0.03034 September 04151500 Cass River at Frankenmuth, Mich. 43.32780 83.74802 842 64.0 0.07597 September 04152238 South Branch Tobacco River near Beaverton, Mich. 43.86697 84.54529 152 63.0 0.41366 September 
Table 1–1. Flow, yield, and record characteristics for streamflow–gaging stations used in the regression analysis—Continued.[A water year is the 12–month period from October 1 to September 30. The water year is designated by the calendar year in which it ends] 

U.S. 

Geological 

Survey 

station 

number Station name

Latitude 

(decimal 

degrees)

Longitude 

(decimal 

degrees)

Drainage 

area

(square 

miles)

Minimum 

monthly 

median flow 

(cubic feet 

per second) 

(IQ50)

Minimum 

monthly median 

yield (cubic feet 

per second per 

square mile) 

(IY50)

Month of

 minimum 

flow

04153500 Salt River near North Bradley, Mich. 43.70281 84.47056 145 8.5 0.05874 August 

04154000 Chippewa River near Mount Pleasant, Mich. 43.62558 84.70779 409 150 0.36702 August 04155000 Pine River at Alma, Mich. 43.37948 84.65556 309 80.0 0.25882 August 04157500 State Drain near Sebewaing, Mich. 43.71196 83.42774 67.3 0 0.00000 August 04158000 Columbia Drain near Sebewaing, Mich. 43.72724 83.39607 33.9 0 0.00000 August 04158500 Pigeon River near Owendale, Mich. 43.76363 83.24606 53.3 3.0 0.05629 September 04159492 Black River near Jeddo, Mich. 43.15253 82.62409 479 22.0 0.04589 September 04159900 Mill Creek near Avoca, Mich. 43.05447 82.73465 169 6.8 0.04033 September 04160000 Mill Creek near Abbottsford, Mich. 43.04503 82.61381 184 8.5 0.04620 August 04160050 Black River near Port Huron, Mich. 42.99003 82.53770 683 19.0 0.02783 September 04160570 North Branch Belle River at Imlay City, 

Mich.

43.03031 83.06716 16.1 2.3 0.14286 August 

04160600 Belle River at Memphis, Mich. 42.90086 82.76909 151 13.0 0.08587 September 04160800 Sashabaw Creek near Drayton Plains, Mich. 42.72003 83.35355 21.0 2.2 0.10476 September 04160900 Clinton River near Drayton Plains, Mich. 42.66031 83.39022 78.5 14.0 0.17834 August 

04161000 Clinton River at Auburn Hills, Mich. 42.63337 83.22438 123 44 0.35685 August 

04161100 Galloway Creek near Auburn Heights, Mich. 42.66725 83.20049 17.4 1.6 0.09195 August 

04161500 Paint Creek near Lake Orion, Mich. 42.76753 83.21994 39.8 9.0 0.22613 August 04161540 Paint Creek at Rochester, Mich. 42.68837 83.14299 71.8 21.0 0.29248 August 04161580 Stony Creek near Romeo, Mich. 42.80086 83.09021 23.8 3.9 0.16387 August 04161800 Stony Creek near Washington, Mich. 42.71531 83.09188 69.1 13.0 0.18813 August 
Table 1–1. Flow, yield, and record characteristics for streamflow–gaging stations used in the regression analysis—Continued. [A water year is the 12–month period from October 1 to September 30. The water year is designated by the calendar year in which it ends] U.S. Geological Survey station number Station name Latitude (decimal degrees) Longitude (decimal degrees) Drainage area (square miles) Minimum monthly median flow (cubic feet per second) (IQ50) Minimum monthly median yield (cubic feet per second per square mile) (IY50) Month of minimum flow 

04163400 Plum Brook at Utica, Mich. 42.60142 83.07409 16.6 2.8 0.16867 August 

04163500 Plum Brook near Utica, Mich. 42.58365 83.03048 23.8 0.7 0.02857 July 04164100 East Pond Creek at Romeo, Mich. 42.82253 83.02021 20.8 4.9 0.23558 August 04164300 East Branch Coon Creek at Armada, Mich. 42.84586 82.88493 12.8 0.2 0.01641 August 04164500 North Branch Clinton R near Mount Clemens, Mich. 42.62920 82.88881 198 12.0 0.06073 September 04164800 Middle Branch Clinton River at Macomb, Mich. 42.70642 82.95909 41.2 3.5 0.08495 September 04166000 River Rouge at Birmingham, Mich. 42.54587 83.22354 36.7 6.2 0.16894 September 04166200 Evans Ditch at Southfield, Mich. 42.45781 83.26743 10.2 1.9 0.18627 September 04166300 Upper River Rouge at Farmington, Mich. 42.46448 83.36966 17.6 4.1 0.23295 September 04169500 Huron River at Commerce, Mich. 42.59031 83.48466 49.9 12.0 0.24048 August 04170000 Huron River at Milford, Mich. 42.57892 83.62661 139 46.0 0.33141 August 04170500 Huron River near New Hudson, Mich. 42.51253 83.67633 155 54.0 0.34771 August 04171500 South Ore Creek near Brighton, Mich. 42.49781 83.80244 33.3 7.4 0.22222 September 04172000 Huron River near Hamburg, Mich. 42.46531 83.79994 320 103 0.32177 September 04173000 Huron River near Dexter, Mich. 42.38615 83.91106 538 120 0.22326 August 04173500 Mill Creek near Dexter, Mich. 42.30004 83.89856 131 23.0 0.17598 September 04174500 Huron River at Ann Arbor, Mich. 42.28615 83.73327 747 147 0.19684 August 04174800 Huron River at Ypsilanti, Mich. 42.24921 83.61244 817 235 0.28750 August 04175600 River Raisin near Manchester, Mich. 42.16809 84.07606 128 32.0 0.25059 August 
Table 1–1. Flow, yield, and record characteristics for streamflow–gaging stations used in the regression analysis—Continued.[A water year is the 12–month period from October 1 to September 30. The water year is designated by the calendar year in which it ends] 

U.S. 

Geological 

Survey 

station 

number Station name

Latitude 

(decimal 

degrees)

Longitude 

(decimal 

degrees)

Drainage 

area

(square 

miles)

Minimum 

monthly 

median flow 

(cubic feet 

per second) 

(IQ50)

Minimum 

monthly median 

yield (cubic feet 

per second per 

square mile) 

(IY50)

Month of

 minimum 

flow

04175700 River Raisin near Tecumseh, Mich. 41.94310 83.94578 266 62.5 0.23532 August 

04176000 River Raisin near Adrian, Mich. 41.90421 83.98050 460 98.5 0.21422 September 04176605 Otter Creek at Lasalle, Mich. 41.86699 83.45354 63.7 1.7 0.02669 September 04184500 Bean Creek at Powers, Ohio 41.67755 84.23217 205 19.0 0.09255 September 
38 Regression Model for Computing Index Flows Describing the Median Flow , Summer Month of Lowest Flow, Michigan

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

the index flow.—Continued.

U.S. 

Geological 

Survey 

station 

number

Drainage 

area 

(square 

miles)

Percent of 

basin with low 

ground-water

 transmissivity 

(L_Trans)

Percent of 

basin with 

high groundwater 

transmissivity 

(H_Trans)

Percent of 

basin with 

forest cover 

(Forest)

Normal 

annual 

precipitation 

for 1971–2000 

(inches) 

(Precip)

Percent of 

basin with A 

hydrologic 

soil group 

(A_Soils)

Percent of 

basin with D 

hydrologic 

soil group 

(D_Soils)

Michigan 

hydrologic 

subregion

04001000 13.2 98.0 0.0 91.4 31.0 0.0 0.0 1

04031000 200 7.7 6.8 85.5 34.7 .0 9.0 1

04031500 172 .0 9.7 86.6 33.8 .0 10.0 1

04032000 264 0.8 11.3 90.3 34.2 .0 5.0 1

04033000 162 .0 21.0 86.6 31.7 1.0 62.0 1

04035000 273 27.0 23.9 89.0 32.2 14.0 47.0 1

04040000 1330 34.1 8.9 84.2 32.6 25.0 29.0 1

04040500 169 32.4 14.1 84.7 33.0 7.0 34.0 1

04041500 343 17.0 20.2 85.1 32.7 9.0 36.0 1

04043050 29.6 36.2 3.6 59.5 31.4 19.0 13.0 1

04045500 757 55.0 30.9 79.0 31.2 32.0 50.0 1

04046000 33.5 27.7 .0 75.4 30.7 32.0 30.0 1

04049500 420 31.5 48.9 68.4 30.5 54.0 14.0 1

04055000 716 46.6 36.4 60.8 30.5 62.0 12.0 1

04056000 326 60.3 29.9 77.9 31.6 72.0 5.0 1

04056500 1,130 53.4 32.1 66.4 30.8 32.0 49.0 1

04057510 184 12.7 85.5 79.8 31.4 45.0 35.0 1

04057800 45.7 56.5 24.3 80.9 33.0 5.0 47.0 1

04058000 128 68.3 20.7 77.1 33.0 6.0 47.0 1

04058400 36.3 66.6 33.4 78.7 32.5 53.0 19.0 1

04059000 871 22.4 29.7 81.7 32.4 30.0 36.0 1

04059500 444 .1 5.4 86.1 31.5 10.0 36.0 1

04060993 378 .0 29.7 72.6 31.2 3.0 13.0 1

04061500 600 1.8 19.7 84.5 31.5 3.0 78.0 1

04062200 132 97.9 .0 84.9 33.1 1.0 42.0 1

04096015 80.8 15.4 8.3 17.0 37.6 11.0 11.0 2

04096405 201 .0 69.6 18.5 35.3 8.0 14.0 2

04096515 48.7 .0 32.5 15.4 35.7 11.0 11.0 2

04096600 286 .0 57.2 16.0 36.0 4.0 14.0 2

04096900 162 .0 96.7 24.5 36.1 7.0 21.0 2

04097170 68.2 .0 86.8 19.0 37.0 1.0 15.0 2

04097540 107 .0 80.3 18.5 36.5 10.0 15.0 2

04099000 1,880 .0 76.0 18.2 36.4 9.0 14.0 2

04101500 3,670 .0 77.7 18.3 36.9 9.0 14.0 2

04101800 252 .0 94.5 20.8 37.8 23.0 13.0 2
Appendix 1 39

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

the index flow.—Continued.

U.S. 

Geological 

Survey 

station 

number

Drainage 

area 

(square 

miles)

Percent of 

basin with low 

ground-water

 transmissivity 

(L_Trans)

Percent of 

basin with 

high groundwater 

transmissivity 

(H_Trans)

Percent of 

basin with 

forest cover 

(Forest)

Normal 

annual 

precipitation 

for 1971–2000 

(inches) 

(Precip)

Percent of 

basin with A 

hydrologic 

soil group 

(A_Soils)

Percent of 

basin with D 

hydrologic 

soil group 

(D_Soils)

Michigan 

hydrologic 

subregion

04102500 390 1.9 71.5 25.8 37.7 36.0 16.0 2

04102700 83.5 1.1 36.1 32.5 37.7 23.0 26.0 2

04103010 270 .0 54.7 16.0 34.9 6.0 11.0 2

04104945 48.3 .0 55.4 24.9 35.5 12.0 18.0 2

04105000 274 .0 44.7 22.4 34.8 12.0 17.0 2

04105700 36.8 .0 99.1 28.2 36.6 4.0 16.0 2

04108600 65.1 .0 48.4 21.1 36.7 23.0 13.0 2

04108801 66.9 1.7 25.0 8.3 36.4 11.0 5.0 2

04110000 47.3 .0 39.5 19.6 32.7 25.0 18.0 2

04111500 16.3 .0 4.4 15.1 32.6 14.0 13.0 2

04112000 10.4 .0 .0 13.6 32.5 2.0 8.0 2

04112500 344 .0 13.7 14.2 32.6 11.0 14.0 2

04114498 284 .0 28.9 14.8 32.4 9.0 18.0 2

04115000 420 29.3 20.1 10.7 32.4 5.0 15.0 2

04116500 516 .0 79.0 26.7 34.0 45.0 19.0 2

04117000 7.8 .0 30.8 21.9 35.3 4.0 14.0 2

04117500 410 .0 30.3 18.4 34.9 5.0 12.0 2

04118000 795 .0 31.3 21.8 35.2 11.0 11.0 2

04118500 257 .0 46.6 29.8 34.5 36.0 12.0 2

04121000 352 .0 85.2 63.1 30.7 57.0 16.0 3

04121300 239 3.1 77.2 53.0 32.1 62.0 .0 3

04121900 136 .0 96.0 41.4 33.4 60.0 15.0 3

04122100 16.7 70.6 25.8 43.5 34.0 32.0 15.0 3

04122200 404 .0 81.7 57.9 34.0 62.0 15.0 3

04122500 689 .0 91.3 74.7 33.8 70.0 16.0 3

04123000 115 .0 91.1 79.7 33.5 64.0 21.0 3

04123500 132 .0 100.0 73.0 32.2 92.0 5.0 3

04124000 865 .0 94.4 76.3 31.9 80.0 12.0 3

04124500 58.9 .0 31.2 40.1 32.6 54.0 18.0 3

04125000 130 .0 54.5 51.7 32.7 61.0 11.0 3

04125500 254 .0 72.1 61.6 32.9 63.0 9.0 3

04126200 185 .0 99.3 83.3 33.5 84.0 10.0 3

04127918 202 44.5 26.6 71.3 32.1 30.0 43.0 1

04127997 181 .0 98.6 69.2 31.4 77.0 13.0 4

04128990 57.7 .0 90.6 64.4 30.9 66.0 22.0 4
40 Regression Model for Computing Index Flows Describing the Median Flow , Summer Month of Lowest Flow, Michigan

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

the index flow.—Continued.

U.S. 

Geological 

Survey 

station 

number

Drainage 

area 

(square 

miles)

Percent of 

basin with low 

ground-water

 transmissivity 

(L_Trans)

Percent of 

basin with 

high groundwater 

transmissivity 

(H_Trans)

Percent of 

basin with 

forest cover 

(Forest)

Normal 

annual 

precipitation 

for 1971–2000 

(inches) 

(Precip)

Percent of 

basin with A 

hydrologic 

soil group 

(A_Soils)

Percent of 

basin with D 

hydrologic 

soil group 

(D_Soils)

Michigan 

hydrologic 

subregion

04133501 586 1.1 55.4 71.4 28.5 38.0 24.0 4

04135000 1,240 2.8 54.6 67.5 28.4 29.0 27.0 4

04135500 96.6 .0 97.8 69.4 31.8 88.0 6.0 4

04135600 71.2 .0 100.0 70.9 31.2 85.0 8.0 4

04135700 391 .0 97.8 81.1 30.2 76.0 15.0 4

04136500 1,360 .5 94.7 80.1 30.2 81.0 11.0 4

04137500 1,740 2.0 89.5 80.9 29.1 79.0 12.0 4

04138000 89.9 .9 21.4 62.4 29.7 42.0 14.0 4

04138500 151 15.2 4.4 41.1 29.9 18.0 32.0 5

04139000 29.8 .0 48.7 58.1 29.6 54.0 14.0 5

04139500 57.4 .0 53.5 58.2 29.5 47.0 19.0 5

04140000 21.0 .0 46.7 45.7 29.8 41.0 15.0 5

04140500 116 .0 53.4 55.8 29.6 44.0 20.0 5

04141000 1.1 .0 .0 13.1 29.9 5.0 8.0 5

04141500 65.4 5.0 50.8 50.8 30.0 52.0 13.0 5

04142000 333 3.6 36.2 56.2 30.0 43.0 20.0 5

04143900 81.9 .0 49.8 14.6 31.5 8.0 18.0 5

04144000 363 .0 34.5 19.4 31.8 7.0 16.0 5

04144500 530 .0 32.0 16.6 31.7 6.0 14.0 5

04145500 89.9 56.4 .0 11.4 32.3 2.0 6.0 5

04146000 51.1 12.3 15.1 20.5 31.4 10.0 14.0 5

04146063 211 19.8 26.1 22.0 31.3 9.0 15.0 5

04147500 526 14.9 33.0 20.1 31.3 9.0 17.0 5

04147990 34.8 12.3 28.4 23.9 31.5 12.0 12.0 5

04148140 99.7 5.6 32.9 19.3 31.5 11.0 16.0 5

04148160 6.9 67.2 .0 6.5 31.6 2.0 4.0 5

04148200 12.1 .0 52.9 21.7 31.5 17.0 26.0 5

04148300 114 6.1 13.1 14.7 31.6 4.0 14.0 5

04148440 54.4 .5 26.9 18.1 31.5 11.0 14.0 5

04148500 960 15.3 27.5 17.7 31.5 9.0 15.0 5

04150000 239 .0 26.6 10.1 31.0 11.0 11.0 5

04150500 363 6.2 31.1 12.4 31.1 12.0 16.0 5

04151500 842 9.8 25.9 20.7 31.0 14.0 18.0 5

04152238 152 1.3 64.3 42.1 31.5 46.0 18.0 5

04153500 145 39.3 4.2 19.6 31.7 6.0 14.0 5
Appendix 1 41

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

the index flow.—Continued.

U.S. 

Geological 

Survey 

station 

number

Drainage 

area 

(square 

miles)

Percent of 

basin with low 

ground-water

 transmissivity 

(L_Trans)

Percent of 

basin with 

high groundwater 

transmissivity 

(H_Trans)

Percent of 

basin with 

forest cover 

(Forest)

Normal 

annual 

precipitation 

for 1971–2000 

(inches) 

(Precip)

Percent of 

basin with A 

hydrologic 

soil group 

(A_Soils)

Percent of 

basin with D 

hydrologic 

soil group 

(D_Soils)

Michigan 

hydrologic 

subregion

04154000 409 .7 76.7 34.6 32.5 43.0 16.0 5

04155000 309 .7 69.4 24.2 32.9 36.0 15.0 5

04157500 67.3 77.5 .0 4.2 30.9 2.0 5.0 5

04158000 33.9 89.0 .0 3.4 30.9 3.0 3.0 5

04158500 53.3 33.6 1.0 12.2 31.2 5.0 10.0 5

041594920 479 36.3 14.4 7.0 31.0 10.0 10.0 5

04159900 169 32.3 25.6 11.9 30.9 9.0 12.0 6

04160000 184 31.4 23.5 12.5 30.9 9.0 12.0 6

04160050 683 35.0 16.4 9.6 31.0 10.0 11.0 6

04160570 16.1 46.5 .0 12.3 31.0 15.0 15.0 6

04160600 151 53.6 .0 11.4 31.0 6.0 13.0 6

04160800 21.0 .0 100.0 18.7 31.3 34.0 21.0 6

04160900 78.5 .0 95.8 15.4 31.3 26.0 15.0 6

04161000 123 .0 81.0 11.9 31.4 24.0 16.0 6

04161100 17.4 .0 52.5 11.1 31.3 12.0 30.0 6

04161500 39.8 .0 87.3 14.5 31.2 19.0 15.0 6

04161540 71.8 .0 61.9 15.0 31.0 15.0 16.0 6

04161580 23.8 .0 33.6 22.6 31.2 4.0 17.0 6

04161800 69.1 .0 56.7 20.0 31.2 8.0 15.0 6

04163400 16.6 49.3 2.5 8.6 31.4 17.0 16.0 6

04163500 23.8 43.5 1.7 9.3 31.4 14.0 18.0 6

04164100 20.8 4.2 35.4 19.8 31.1 4.0 14.0 6

04164300 12.8 46.2 .0 8.0 31.1 .0 7.0 6

04164500 198 64.4 6.3 11.4 31.2 2.0 19.0 6

04164800 41.2 53.1 3.5 11.0 31.2 13.0 13.0 6

04166000 36.7 4.3 35.2 7.7 31.4 13.0 20.0 6

04166200 10.2 58.0 .0 3.7 31.6 9.0 9.0 6

04166300 17.6 .2 17.2 15.9 31.6 10.0 15.0 6

04169500 49.9 .0 98.5 22.5 31.5 25.0 28.0 6

04170000 139 .0 90.1 18.0 31.7 25.0 25.0 7

04170500 155 .0 90.6 18.3 31.7 24.0 25.0 7

04171500 33.3 .0 76.5 24.6 32.1 7.0 19.0 7

04172000 320 .0 84.4 19.2 32.0 17.0 21.0 7

04173000 538 .0 79.2 20.6 32.2 16.0 22.0 7

04173500 131 .0 50.3 15.4 33.0 12.0 18.0 7
42 Regression Model for Computing Index Flows Describing the Median Flow , Summer Month of Lowest Flow, Michigan

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

the index flow.—Continued.

U.S. 

Geological 

Survey 

station 

number

Drainage 

area 

(square 

miles)

Percent of 

basin with low 

ground-water

 transmissivity 

(L_Trans)

Percent of 

basin with 

high groundwater 

transmissivity 

(H_Trans)

Percent of 

basin with 

forest cover 

(Forest)

Normal 

annual 

precipitation 

for 1971–2000 

(inches) 

(Precip)

Percent of 

basin with A 

hydrologic 

soil group 

(A_Soils)

Percent of 

basin with D 

hydrologic 

soil group 

(D_Soils)

Michigan 

hydrologic 

subregion

04174500 747 .0 70.8 19.2 32.3 14.0 20.0 7

04174800 817 .0 67.9 18.8 32.3 13.0 21.0 7

04175600 128 .0 90.5 20.0 34.1 9.0 17.0 7

04175700 266 .1 71.6 17.7 34.0 7.0 15.0 7

04176000 460 .0 53.8 15.9 34.2 6.0 14.0 7

04176605 63.7 54.0 0.0 13.0 33.5 23.0 7.0 7

04184500 205 1.4 9.6 14.4 35.0 3.0 9.0 7
Appendix 1 43

Table 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 Michigan1.

Hydrologic soil group

Land use-land cover A B C D

Outside of 

Michigan2 Percent

Adjusted 

percent3

Urban 4,343,953 4,432,438 1,137,269 555,570 14,563 1.6 6.3

Agriculture 8,163,149 26,685,700 1,2077,467 2,000,247 2,392 7.4 29.3

Range land 5,292,626 4,255,525 1,338,514 2,544,935 5,416 2.0 8.0

Forest 31,740,852 19,970,933 6,257,886 2,3151,323 453,893 12.3 48.8

Water 148,888 193,023 24,015 3,391,954 42,373 0.6 2.3

Wetland 1,457,774 873,403 314,675 5,950,392 59,920 1.3 5.2

Barren 105,535 9,180 9,802 49,231 43,448 0.0 0.1

Outside of Michigan 28,474 7,383 1,510 693,402 496,646,760 74.9 --

Percent 7.7 8.5 3.2 5.8 74.8 100 100

Adjusted percent 30.7 33.7 12.7 22.9 -- 100 100

1

The Michigan Resource Information System represents the 1978 land use-land cover and hydrologic soil groups in Michigan as a rectangular grid of integers 

that contain 26,319 rows and 25,247 columns. Each grid is identically referenced geographically. Each cell in the grid represents a land area of 30 meters square 

(900 square meters). Numeric codes for the land use-land cover grid are as follows: (1) urban or built-up land, (2) agricultural land, (3) rangeland, (4) forest 

land, which included Level II classification of deciduous, evergreen, and mixed forest lands, (5) water, (6) wetland, and (7) barren land. For hydrologic soil 

groups numeric codes are as follows: (1) group A soils, (2) group B soils, (3) group C soils, and (4) group D soils. For both coverages, the code -9999 signifies 

no data or inapplicable, which occurs over extensive areas of adjacent states, the Province of Ontario, Canada, and the Great Lakes.

2 “Outside of Michigan” refers to land areas of adjacent states and the Province of Ontario, Canada, and water areas over the Great Lakes, both within and 

outside of Michigan. 

3

 Adjusted percentage accounts only for the land areas within Michigan. 


Hamilton and others—Regression Model for Computing Index Flows Describing Median Flow...Michigan—Scientific Investigations Report 2008–5096

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