U.S. Department of the Interior

Scientific Investigations Report 2010–5236

Prepared in cooperation with the Michigan Department of Natural Resources and Environment

Use of Instantaneous Streamflow Measurements 

to Improve Regression Estimates of Index Flow 

for the Summer Month of Lowest Streamflow in Michigan

Use of Instantaneous Streamflow Measurements 

to Improve Regression Estimates of Index Flow 

for the Summer Month of Lowest Streamflow 

in Michigan

By David J. Holtschlag

Prepared in cooperation with the Michigan Department of Natural Resources 

and Environment 

Scientific Investigations Report 2010–5236

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2011

For more information on the USGS—the Federal source for science about the Earth, its natural and living resources, 

natural hazards, and the environment, visit http://www.usgs.gov or call 1-888-ASK-USGS

For an overview of USGS information products, including maps, imagery, and publications, 

visit http://www.usgs.gov/pubprod

To order this and other USGS information products, visit http://store.usgs.gov

Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the 

U.S. Government.

Although this report is in the public domain, permission must be secured from the individual copyright owners to 

reproduce any copyrighted materials contained within this report.

Suggested citation:

Holtschlag, D.J., 2011, Use of instantaneous streamflow measurements to improve regression estimates of index 

flow for the summer month of lowest streamflow in Michigan: U.S. Geological Survey Scientific Investigations Report 

2010–5236, 86 p.
iii

Acknowledgments

The author gratefully acknowledges the valuable technical reviews provided by David A. Hamilton, 

Chief of the Water Management Section of the Michigan Department of Natural Resources and 

the Environment, Lansing, Michigan; Julie E. Kiang, Research Hydrologist, USGS Office of the 

Chief of Surface Water, Reston, Virginia; and Rachel A. Esralew, Hydrologist, USGS Oklahoma 

Water Science Center, Oklahoma. Also, Bonnie Fink, USGS Enterprise Publishing Network, edited 

the report, thus improving the readability and consistency of the text, figures, and tables. 
iv

Contents

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

Introduction.....................................................................................................................................................2

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

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

An Existing Regression Model for Estimating Index Flows ...........................................................2

Flow Data.........................................................................................................................................................5

Continuous-Flow Data at Streamgages ............................................................................................5

Instantaneous Flow Measurements at Miscellaneous Stream Sites..........................................8

Improving Regression Estimates of Index Flow by use of Instantaneous 

Flow Measurements ........................................................................................................................9

Development of the Discrete, Integrated, and Site Estimators using

 Monte Carlo Methods..........................................................................................................13

Discrete Estimator.....................................................................................................................14

Integrated Estimator..................................................................................................................15

Site Estimator..............................................................................................................................16

Results of the Monte Carlo Simulations for the Discrete, Integrated, 

and Site Estimators ...............................................................................................................18

Discrete Estimator.....................................................................................................................18

Integrated Estimator..................................................................................................................22

Site Estimator..............................................................................................................................24

Towards the Development of an Augmented Estimator from Base Flow 

Recession Characteristics...................................................................................................34

Base-Flow Recession ...............................................................................................................37

Recession Characteristics and Estimation of Index Water Yield ......................................43

Summary and Conclusions.........................................................................................................................46

References Cited..........................................................................................................................................47

Appendix 1. Tables of streamgage attributes and flow characteristics used to 

assess the utility of instantaneous flow measurements for improving regression 

estimates of index flow in Michigan ...........................................................................................49

Appendix 2. Example Matlab Code computations of index flow 

estimates with simulated instantaneous measurements .......................................................71

2–1. IndexFlow_MiscQ......................................................................................................................72

2–2. User required input ...................................................................................................................73

2–3. Parameters in index regression equation.............................................................................74

2–4. Parameters in augmented index regression equation .......................................................76

2–5. Specify run parameters and labels........................................................................................77

2–6. Load data for index flow simulation.......................................................................................77

2–7. Analysis of instantaneous flow measurements ...................................................................78

2–8. Compute water yields and flows with the Discrete Estimator...........................................82

2–9. Compute water yields and flows with the Integrated Estimator.......................................83

2–10. Compute water yields and flows with the Site Estimator.................................................83

2–11. List results for all estimators to console .............................................................................84
v

Figures

1–3. Maps showing:

1. Distribution of streamgages used to develop the regression model for 

index flows in the Upper Peninsula of Michigan............................................................3

2. Distribution of streamgages used in this analysis in the 

Lower Peninsula of Michigan............................................................................................4

3. Michigan hydrologic subregions used in the analysis of index flows........................6

4–6. Graphs showing:

4. Number of streamgages operated during water years 19612008 from 

the 147 index streamgages in Michigan ..........................................................................7

5. The empirical cumulative distribution function of the number of years 

of flow data used from the 147 index streamgages in Michigan.................................7

6. Relation between the index flow and the median daily flow in the 

sampling frame for the month of minimum flow.............................................................8

7. Map showing the distribution of sites with instantaneous streamflow 

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

8–11. Graphs showing:

8. Number of streamflow measurements per year at miscellaneous sites, 

1931 to 2006, in Michigan..................................................................................................11

9. Distribution of instantaneous streamflow measurements, per month, 

in Michigan .........................................................................................................................11

10. Average measured streamflow, by month, at miscellaneous sites in Michigan ....12

11. Typical relation between the square roots of daily mean water yields 

and index water yields, illustrated with data from active streamgages 

in Michigan hydrologic subregion 2 and a simulated instantaneous 

flow measurement at U.S. Geological Survey streamgage 04108600, 

Rabbit River near Hopkins, Mich., on July 11, 1967......................................................19

12–13. Box plots showing:

12. Distribution of coefficients of determination for regression relations 

between the square roots daily water yields and the square roots of 

index water yield, illustrated with results for index streamgages in 

Michigan hydrologic subregion 2 ...................................................................................20

13. Distribution of single measurement DiscrY50 estimates for index 

streamgages in Michigan hydrologic subregion 2.......................................................21

14–15. Graphs showing:

14. Percent decrease in the root mean square error of the DiscrY50 

estimator with respect to the IndxrY50 estimator with the number of 

simulated streamflow measurements by Michigan hydrologic subregion..............22

15. Percent decrease in the root mean square error of the IntgrY50 estimator 

with respect to the IndxrY50 estimator with the number of simulated 

streamflow measurements by Michigan hydrologic subregion................................23

16–22. Box plots showing:

16. Error for the index regression estimator IndxrY50 and the distribution 

of errors for the integrated water yield estimator IntgrY50 with one 

simulated measurement for index streamgages within Michigan 

hydrologic subregion 1 .....................................................................................................25
vi

17. Error for the index regression estimator 
vii

33. The cumulative distribution function of the common logarithm of the 

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2

) 259.0 hectare (ha)

square mile (mi2

) 2.590 square kilometer (km2) 

Volume

cubic foot (ft3

) 0.02832 cubic meter (m3) 

Flow rate

foot per second (ft/s) 0.3048 meter per second (m/s)

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]

inch per year (in/yr) 25.4 millimeter per year (mm/yr)

mile per hour (mi/h) 1.609 kilometer per hour (km/h) 

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.

Water year: The 12-month period from October 1 through September 30. The water year is 

designated by the calendar year in which it ends.
viii

Symbols Used in this Report

Symbol Name

Q50 The index flow defined as the minimum of median flows at a stream site for 

July, August, and September, in cubic feet per second.

rY50 The index water yield defined as the square root of water yield corresponding to the square root of the quantity Q50 divided by the upstream 

drainage area (DArea), in cubic feet per second per square mile.

GageQ50 The measured index flow defined as the minimum of median flows for 

the months of July, August, and September measured at a long-term 

streamgage. 

GagerY50 The measured index water yield defined as the square root of water yield 

corresponding to the square root of the quantity GageQ50 divided by 

the upstream drainage area, in cubic feet per second per square mile.

IndxQ50 The index regression estimate of index flow computed as the IndxrY50

estimate squared times the corresponding drainage area.

IndxrY50 The index regression estimate of water yield.

DArea Drainage Area, in square miles.

RMSE Root mean square error, which is a common measure of the standard 

deviation of the model residuals adjusted for the number of estimated 

parameters. 

The estimated coefficient of determination computed by squaring the 

Spearman (rank) correlation coefficient. 

Jo A target streamflow site or streamgage. 

 A set of auxiliary streamgages used to estimate index flow at J o

. 

IndxrY50 o J σ2

( ) ˆ The estimated prediction variance for site J o

 based on the IndxrY50

estimator. 

Qdv A daily value of flow, in cubic feet per second.

rYdv A daily value of the square root of water yield computed as Qdv DArea .

rYmm An estimate of the square root of water yield based on an instantaneous 

flow measurement.

SLR Simple linear regression, which implies a linear-regression equation with 

one explanatory variable and one response variable.

SiterY50 An estimate of index water yield based on the Site Estimator.

SiteQ50 An estimate of the index flow computed as SiterY502

·DArea.

DiscrY50 An estimate of index water yield based on the Discrete Estimator.

DiscQ50 An estimate of index flow computed as DiscrY502

·DArea. 

IntgrY50 An estimate of index water yield based on the Integrated Estimator.

IntgQ50 An estimate of the index flow computed as IntgrY502

·DArea.

AugmrY50 An estimate of index water yield based on the Augmented Estimator.

AugmQ50 An estimate of index flow computed as AugmrY502

·DArea. 

Iq A set of 12 flow-measurement intensities i i q q iq 1 2 12   , ,, where the number 

of measurements, # I q , equals 1, 2, ,  7,10,15, 20, 50,100 .

rs

2

{J'}
ix

Symbol Name

MHS Michigan Hydrologic Subregion, which is referred to as subregion for 

brevity where ambiguity can be avoided. 

  bo , b , ,b 1  p A set of regression parameters estimated by ordinary least squares. 

ε An error term indicating an unexpected difference between measured 

and estimated values.

DiscrY50t J, o A set of discrete estimates of flow indexed by t for site J o

.

σp t, J

2

( ) ˆ o Estimated prediction error variances indexed by t for site J o

.

σw

2 ˆ The within measurement-day estimated variance. 

σb

2 ˆ The between measurement-day estimated variance.

ciDiscrY50 A 95-percent confidence interval about the DiscrY50 estimate.

ciIntgrY50 A 95-percent confidence interval about the IntgrY50 estimate.

ciSiterY50 A 95-percent confidence interval about the SiterY50 estimate. b b dv 0 t J rY 1 rY , 50   · o t J, Disc o

For all, as in for all elements in a set. 

∈ Is an element of, as an element of a set or a matrix. 

RMSE ∆

κ,κ+1 Percent decrease in the root mean-square error of two estimators 

indexed by κ .

rYt The (square) root water yield estimate at time t. 

rYo A multiplicative root water yield (base flow) recession parameter 

corresponding approximately to the root water yield during 

recession periods beginning after direct runoff and early time 

ground-water response have passed. 

τ Time constant for a linear system. 

e The mathematical constant e is an irrational number approximately 

equal to 2.71828.

#{A} The cardinality (number of elements) of set A. 

Use of Instantaneous Streamflow Measurements 

to Improve Regression Estimates of Index Flow 

for the Summer Month of Lowest Streamflow in Michigan

By David J. Holtschlag

Abstract

In Michigan, index flow Q50 is a streamflow characteristic defined as the minimum of median flows for 

July, August, and September. The state of Michigan uses 

index flow estimates to help regulate large (greater than 

100,000 gallons per day) water withdrawals to prevent adverse 

effects on characteristic fish populations. At sites where 

long-term streamgages are located, index flows are computed 

directly from continuous streamflow records as GageQ50. In 

an earlier study, a multiple-regression equation was developed 

to estimate index flows IndxQ50 at ungaged sites. The index 

equation explains about 94 percent of the variability of index 

flows at 147 (index) streamgages by use of six explanatory 

variables describing soil type, aquifer transmissivity, land 

cover, and precipitation characteristics. This report extends 

the results of the previous study, by use of Monte Carlo 

simulations, to evaluate alternative flow estimators, DiscQ50, 

IntgQ50, SiteQ50, and AugmQ50. The Monte Carlo simulations treated each of the available index streamgages, in 

turn, as a miscellaneous site where streamflow conditions are 

described by one or more instantaneous measurements of flow. 

In the simulations, instantaneous flows were approximated 

by daily mean flows at the corresponding site. All estimators 

use information that can be obtained from instantaneous flow 

measurements and contemporaneous daily mean flow data 

from nearby long-term streamgages. The efficacy of these 

estimators was evaluated over a set of measurement intensities 

in which the number of simulated instantaneous flow measurements ranged from 1 to 100 at a site. 

The discrete measurement estimator DiscQ50 is based on 

a simple linear regression developed between information on 

daily mean flows at five or more streamgages near the miscellaneous site and their corresponding GageQ50 index flows. 

The regression relation then was used to compute a DiscQ50

estimate at the miscellaneous site by use of the simulated 

instantaneous flow measurement. This process was repeated to 

develop a set of DiscQ50 estimates for all simulated instantaneous measurements, a weighted DiscQ50 estimate was 

formed from this set. Results indicated that the expected value 

of this weighted estimate was more precise than the IndxQ50

estimate for all measurement intensities evaluated. 

The integrated index-flow estimator, IntgQ50, was 

formed by computing a weighted average of the index estimate IndxQ50 and the DiscQ50 estimate. Results indicated 

that the IntgQ50 estimator was more precise than the DiscQ50

estimator at low measurement intensities of one to two measurements. At greater measurement intensities, the precision 

of the IntgQ50 estimator converges to the DiscQ50 estimator. 

Neither the DiscQ50 nor the IntgQ50 estimators provided 

site-specific estimates. In particular, although expected values 

of DiscQ50 and IntgQ50 estimates converge with increasing 

measurement intensity, they do not necessarily converge to the 

site-specific value of Q50.

The site estimator of flow, SiteQ50, was developed to 

facilitate this convergence at higher measurement intensities. 

This is accomplished by use of the median of simulated instantaneous flow values for each measurement intensity level. 

A weighted estimate of the median and information associated with the IntgQ50 estimate was used to form the SiteQ50

estimate. Initial simulations indicate that the SiteQ50 estimator 

generally has greater precision than the IntgQ50 estimator at 

measurement intensities greater than 3, however, additional 

analysis is needed to identify streamflow conditions under 

which instantaneous measurements will produce estimates that 

generally converge to the index flows. 

A preliminary augmented index regression equation was 

developed, which contains the index regression estimate and 

two additional variables associated with base-flow recession 

characteristics. When these recession variables were estimated 

as the medians of recession parameters computed from daily 

values defining recession events at index streamgages, this 

augmented estimator (AugIndxrY50), had 66.3-percent lower 

root mean square error (RMSE) than the IndxrY50 estimator. 

For the AugmrY50 estimator, in which the recession parameters for the augmented regression were estimated by Monte 

Carlo analysis, error characteristics decreased monotonically 

between an upper limit defined by error characteristics of the 

IndxrY50 and AugIndxrY50 estimators, depending on sampling intensity. Characterizing the uncertainty of new predicted values from the AugIndxrY50 estimator is problematic, 

however, because two of the key explanatory variables are 

uncertain. 
2 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

Introduction

The Michigan Legislature (2006) passed Public Act 33 in 

2006 (PA33–2006); this 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 effect. This effect is 

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

flow such that the ability of the stream 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 U.S. Geological Survey (USGS) streamflow-gaging-station 

(hereafter termed streamgage) records in Michigan. 

In Michigan, the index flow Q50 is defined as the minimum of median flows during the months of July, August, or 

September. From this definition, index flow is computed from 

daily mean flows measured at selected (index) streamgages 

operated in Michigan for the periods of systematic record 

available for July, August, and September. Although this calculated index flow GageQ50 varies somewhat with the period 

of record, it is an unbiased estimator of Q50 (which is never 

observed), and, if Q50 is stationary, GageQ50 converges to 

Q50 as the period of record increases. The summer months of 

July, August, and September are considered the time of greatest stress on characteristic fish populations because of lower 

flow rates and higher water temperatures than other months. 

Purpose and Scope

The purpose of this report is to develop and evaluate 

four techniques for increasing the accuracy of index flow 

estimates in Michigan by integrating information from one 

or more instantaneous flow measurements with continuous 

flow information at streamgages. The utilities of the methods 

are assessed on the basis of the existing daily flow data and 

index flow characteristics computed at index streamgages 

that were used to develop the multiple-regression equation 

for estimating index flows. 

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. 

Ontario, Canada, lies to the north and east of Michigan. To the 

west and south, bordering states include Wisconsin, Indiana, 

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

a total land area of 57,000 mi2

, 38,600 mi2 of Great Lakes 

waters, and 1,190 mi2 of inland waters (Michigan’s State 

Facts, 2010). According to the U.S. Census Bureau (2008), 

the population of Michigan in 2008 was estimated to be 

slightly more than 10 million. 

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 tends to have slightly less precipitation, whereas July 

through September tends to have slightly 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 (figs. 1 and 2) 

separated by the Straits of Mackinac, a body of water that 

connects Lake Michigan with Lake Huron. 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.

An Existing Regression Model for Estimating 

Index Flows

Hamilton and others (2008) developed a multiple-linearregression model for use in estimating index flow Q50 at 

ungaged sites. Regression models are based on the assumption 

that the errors (residuals or differences between measured and 

estimated values) are normally distributed. This assumption is 

unlikely to be satisfied if the response variable is not normally 

distributed. To more nearly satisfy this normality assumption, 

the GageQ50 estimate computed from streamgage records was 

divided by the upstream drainage areas DArea contributing to 

flow to compute index water yields GageY50. For simplicity, 

this notation will be abbreviated to Y50, where ambiguity can 

be avoided. Hamilton and others (2008) found that the square 

roots of index water yields GagerY50 were approximately 

normally distributed, so this metric was used as the response 

variable in the index regression model. The regression estimate IndxrY50 is an estimator that generally is applicable 

within Michigan. Squaring IndxrY50 and multiplying by the 

drainage area of the upstream basin produces the index regression estimate of flow IndxQ50. 

The index regression was based on a set of 147 USGS 

streamgages (referred to as index streamgages in this report) 

in or near Michigan that had (1) a minimum of 10 years of 

continuous-record flow data; (2) no appreciable effect on daily 

mean flows from water withdrawal, diversion, or augmentation; and (3) no effects of regulation, either natural storage in lakes or retention in regulated surface-water bodies, 

that would substantially mask the daily flow response from 
04059500

04059000

04057510

04060993 04056500

04061500 04046000

0412

04049500

04033000 04031500

04058000 04058400

04031000

04032000 04040500 04062200 04045500

04035000

04040000 04041500

04041500

04043050

04001000

04055000

04057800

U.S. Geological Survey streamgage location

04031500 U.S. Geological Survey streamgage number 

Stream segment

0 04030201 50 100 liM 0 15 30 45 60 120 Kilometers

EXPLANATION

90° 85°

STRAITS OF MACKINAC

Base map from U.S. Geological Survey, 1:100,000, 1980

Figure 1. Distribution of streamgages used to develop the regression model for index flows in the Upper Peninsula of Michigan.
4 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

42 O

45 O

83 O 87 O

0 04030201 50 Miles

0 30 60 Kilometers

U.S. Geological Survey streamgage location

Stream segment

04096015 U.S. Geological Survey streamgage number

EXPLANATION

STRAITS OF MACKINAC

Base map from U.S. Geological Survey, 1:100,000, 1980

04127997

04135000 04128990 04133501

04136500

04135700

04140500

04140000

04135600

04135500

04121000 04139000 04139500

04123500

04125500

04124500

04126200

04123000

04124000

04122200 04121900

04122500

04125000

04122100

04155000

04154000

04135000

04152238

04121300

04141500

04141000

0 04118500

04116500

04158500

04158000

04142000

04138000

04138500

04137500

04166200

04160000

04163500

04164800

04175600

04176000

04097540 04176605

04096515 04099000

04184500

04175700

04173000

04173500

04166300

04174800

04102500 4174500

04102700

04096015

04101500

04101800

04104945

04117000

04108600 04117500

04108801 04118000

04103010

04105700

04097170 04096405

04096600 04096900

04105000

04151500

04150500 04150000

4115000

04147500 04147990

04159492

04159900

1500

04112500

04114498

04160600 04144500

0411 04170500

04112000

04161500

04143900

04144000 04148200

04148440

04148300

04161000

0

Figure 2. Distribution of streamgages used in this analysis in the Lower Peninsula of Michigan.
Flow Data 5

precipitation. For index streamgages, the average length of 

record was 40.2 years, with a range of 11 to 91 years. Drainage areas among selected streamgages ranged from 1.1 mi2

 at 

South Branch Shepards Creek near Selkirk, Mich., (04141000) 

to 3,666 mi2 at St. Joseph River at Niles, Mich. (04101500). 

The first year of record used in the regression analysis was 

1901; 88 streamgages included data from water year 2005, 

which was the last water year used in the analysis. 

The index regression model contains six explanatory 

variables including two soil properties associated with runoff 

potential, two measures of aquifer transmissivity, percent 

forest cover, and normal annual precipitation for the period 

19712000. The regression estimate IndxrY50 accounts for 

70.8 percent of the variability in GagerY50. Based on Spearman’s rank correlation coefficient squared rs

2 (Conover, 

1980), IndxQ50 accounts for 94.0 percent of the variability 

in Q50 (Hamilton and others, 2008). Owing to the nonlinear 

transformation (squaring ) of IndxrY50, associated with the 

computation of flow, IndxQ50 values are not strictly unbiased 

estimators of Q50, although the bias is thought to be small 

because the unexplained variability is small. 

The IndxrY50 estimate at site J o, Indx

J

rY50 o, is associated with an estimated prediction variance σIndx Jo rY50

2

( ) ˆ . 

This variance is a function of the mean square error (MSE) 

of the index regression and the particular set of explanatory 

variables. The MSE represents the variance of the regression 

for a site where all the explanatory variables were equal to the 

mean of corresponding variables used to develop the regression equation. This variance increases, however, with the 

magnitude of differences between site values of the explanatory variables and the average values of those explanatory 

variables used to develop the regression equation (Draper and 

Smith, 1966). 

In addition to predicting index water yields, the index 

regression model can be used to estimate the uncertainty of the 

response for any site. This uncertainty is sometimes expressed 

as a prediction interval that under repeated experiment is 

likely to contain the true index yield at a particular stream site 

with a specified probability. The 95-percent prediction interval 

is commonly used. 

Hamilton and others (2008) defined seven hydrologic 

subregions in Michigan to check for potential bias in the 

spatial distribution of IndxrY50 error (fig. 3). The seven 

subregions are composed of the same cataloging units shown 

on USGS hydrologic unit maps (Seaber and others, 1987), 

although the units in this report are grouped to form subregions somewhat differently than Seaber and others (1987) 

to conform to local understanding of hydrologic conditions. 

No spatial bias was detected in the distribution of IndxrY50

errors (Hamilton and others, 2008). In this report, the seven 

subregions are referred to as Michigan hydrologic subregions 

(MHS). 

Flow Data

The USGS operates a network of streamgages that 

provides continuousflow and water-level data at hourly or 

smaller time intervals. Continuousflow data are computed 

from direct measurements of water levels, which are applied 

to a water-level-flow (stage-discharge) relation that is updated 

periodically by use of paired direct measurements of water 

level and flow (Rantz and others, 1982). Daily mean flows 

for streamgages are computed from hourly or smaller timeinterval data. Direct measurements of flow also are obtained 

nonsystematically at miscellaneous stream sites. In this report, 

these measurements are referred to as instantaneous measurements, which describe streamflow conditions at the time 

of measurement, but are not commonly used to statistically 

characterize streamflow. 

Continuous-Flow Data at Streamgages

Continuous-flow data used in this analysis were obtained 

from the same 147 streamgages (figs. 1 and 2) that were used 

to develop the index regression report (Hamilton and others, 

2008). Daily mean flows from 1901 to 2005 were analyzed to 

compute the index flows GageQ50, which were used again in 

this report (appendix 1, table 1–1). A subset of this daily flow 

data, plus information from water years 2006 to 2008, was 

used to form a sampling frame for the analysis described in 

this report. The sampling frame provided a basis for relating 

daily mean flows to index flows. 

The sampling frame is a matrix of daily mean flows 

formed by 17,532 rows corresponding to consecutive days 

from October 1, 1960 to September 30, 2008 (the 48 water 

years from 19612008), and 147 columns corresponding to 

the 147 index streamgages. Of these index streamgages, a 

maximum of 126 streamgages were operated in 1970 and a 

minimum of 80 streamgages were operated in 1999 (fig. 4).

An average of 98 index streamgages per year was operated 

during this period. Daily flows during periods when index 

streamgages were not operated are represented by a missing 

value, indicated within the matrix by the arithmetic representative for Not-a-Number (NaN). About 33.8 percent of the 

elements in the sampling frame were missing values. 

The 48 years spanned by the sampling frame is considerably less than the 105 years of data used to compute index 

flows (fig. 5). Thus, the maximum period of record in the 

sampling frame is 48 years for the 47 index streamgages that 

were operated continuously from 1961 to 2008, compared 

with a maximum period of record of 91 years of record used 

to compute index flows. Also, the first year of daily mean 

flows represented by the sampling frame was 1961, which 

post-dated the last year of record used to compute index 

flows at five index streamgages. These index streamgages 

include (04056000) West Branch Manistique River near 

Manistique, Mich.; (04110000) Orchard Creek at Munith, 

Mich., (04145500) Bad River near Brant, Mich., (04157500) 
6 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

1

1

1

2

2

2

3

3

3

3

4

4

4

4

4

5

5

5

5

5

5

6

6

7

7 7

7

7

6

5

5

2

2

2

2

2

2

1

1

1

1

1

1

1

1

1

1 1

1

1

EXPLANATION

Michigan hydrologic subregion number

Hydrologic cataloging unit boundary 

U.S. Geological Survey streamgage location

1

0 4 0 8 0 2 0 Miles

0 5 0 1 00 2 5 Kilometers

83 O

48 O

42 O

90 O

Base map from U.S. Geological Survey, 1:100,000, 1980

Figure 3. Michigan hydrologic subregions used in the analysis of index flows.
Introduction 7

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

70

80

90

100

110

120

130

WATER YEAR

NUMBER OF STREAMGAGES OPERATING

Figure 4. Number of 

streamgages operated during 

water years 19612008 from 

the 147 index streamgages in 

Michigan.

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

YEARS OF DAILY FLOWS

FREQUENCY AMONG STREAMGAGES

Miscellaneous-flow analysis

Index-regression analysis Figure 5. The empirical 

cumulative distribution 

function of the number of 

years of flow data used from 

the 147 index streamgages 

in Michigan.
8 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

10 -210 -110 010 110 210 310 4 10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

INDEX FLOW, IN CUBIC FEET PER SECOND

MEDIAN DAILY FLOW FOR INDEX MONTH FROM

SAMPLING FRAME, IN CUBIC FEET PER SECOND

r

2

s = 0.9977

Line of agreement

Figure 6. Relation between the index flow and the median daily flow in the sampling frame for the month of 

minimum flow. The sampling frame is a matrix of daily mean flows formed by 17,532 rows, corresponding to 

consecutive days from October 1, 1960 to September 30, 2008 (the 48 water years from 1961 to 2008), and 147 

columns corresponding to the 147 index streamgages.

State Drain near Sebewaing, Mich., and (04160050) Black 

River near Port Huron, Mich. Daily values from these five 

streamgages were not included in the sampling frame, thus 

streamgages were effectively eliminated from this analysis. 

For this analysis, it is not considered essential that the 

period of flow record represented by the sampling frame correspond exactly to the period of record used to compute index 

flows. This is because the index-flow statistics, which were 

computed on the full period-of-streamflow record used to 

develop the index regression report, are thought to be stationary, such that at least the means and variances of the series do 

not change with time. Thus, the expected values of index flows 

are assumed to be independent of the period of record used in 

their computation. In confirmation, the relation between index 

flows and the median flows computed by use of the daily values 

in the sampling frame for the month of minimum flow (appendix 1, table 1–1) were found to be in close agreement (fig. 6). 

Data in the sampling frame were used to relate daily flow 

values to index flows. This relation is not expected to have 

changed during the period of streamflow record used by Hamilton and others (2008) nor that used in the compilation of this 

report. The sampling frame is thought to span a sufficient period 

of time to characterize the variability in the relation between 

daily mean flows (or instantaneous flows) and index flows.

Instantaneous Flow Measurements at 

Miscellaneous Stream Sites

In contrast to continuous-flow data at streamgages, instantaneous measurements provide a set of direct-flow measurements, 

generally obtained at irregular intervals. Instantaneous measurements are not sufficient to directly compute the index flow of a 

site. These data do, however, provide site-specific information 
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 9

on current streamflow conditions. The utility of instantaneous 

flow measurements can be enhanced by comparison with daily 

mean flows at nearby active streamgages on the same day and 

their corresponding streamflow characteristics. In this report, 

miscellaneous sites are discussed as sites where existing or 

newly acquired flow data could be applied with the methods developed in this report to improve the local estimates 

of index flow from the regression-model estimate. For the 

purpose of method development, however, instantaneous flow 

measurements are approximated by use of daily average flow 

at index streamgages. 

Direct measurements of flow commonly are obtained by 

subdividing a stream cross section into about 25 sections of 

approximately equal flow (Rantz and others, 1982). One or 

more point measurements of flow velocity along a vertical line 

in the center of each section are used along with section width 

and depth measurements to compute the flow within each section. Section flows are added to compute the total flow. Direct 

measurements of flow usually can be completed in about 

1 hour. 

Although instantaneous measurements quantify flow 

only during the measurement interval, they often are used to 

approximate daily mean flow. Compilations of instantaneous 

flow measurements in Michigan (Knutilla, 1974; Holtschlag 

and Eagle, 1985), for example, specify the date but not the 

time interval of the measurement. This approximation to daily 

mean flow is more reliable during base-flow conditions associated with index flows, than during periods of rapidly varying 

flow conditions commonly associated with storm events. 

Holtschlag and Eagle (1985) document 17,607 flow 

measurements at 2,897 miscellaneous sites in Michigan. Since 

then, this dataset has expanded to 22,368 instantaneous flow 

measurements at 3,692 stream sites (fig. 7) (Marlio Lesmez, 

Michigan Department of Natural Resources and Environment, 

written commun., 2009). Thus, each miscellaneous site has an 

average of about six flow measurements, although the median 

is two and the mode is one measurement. A single measurement is available at 1,350 of the 3,692 miscellaneous sites; 

the maximum number of flow measurements available at a 

miscellaneous site is 213. 

The instantaneous flow set includes measurements from 

1931 to 2006 (fig. 8). During this period, the average number of measurements per year increased rapidly after 1945 

and remained steady at about 321 measurements per year 

until about 1961. From 1962 to 1982, the average number 

of measurements per year increased to about 558 per year. 

From 1983 to 2006, the average number of measurements per 

year decreased to about 214. August has the highest frequency 

of instantaneous flow measurements (fig. 9), and it is the 

month where the median flow was lowest at 92 of the 147 

streamgages used to define index flow (Hamilton and others, 

2008). The lowest average instantaneous flow magnitudes are 

in August (fig. 10) and the highest average discrete flows are 

in April. 

Improving Regression Estimates of 

Index Flow by use of Instantaneous 

Flow Measurements

Monte Carlo simulations were used to investigate the 

utility of information contained in instantaneous flow measurements for improving the accuracy of index flows estimated by the index regression equation. Repeated simulations 

were used to randomly select streamgages from the sampling 

frame, where the columns in the frame correspond to the set of 

streamgages used in the development of the index regression 

model. Rows in the sampling frame correspond to days within 

the period from October 1, 1960 to September 30, 2008. 

Where measured, elements in the frame are daily values (Qdv) 

of mean flow. 

Once a streamgage was selected for estimation as a 

target site, 1 or more days, depending on the measurement 

intensity being investigated, were randomly selected from the 

sampling frame during July, August, or September within the 

period of included record for that streamgage. In this report, 

measurement intensities represent the number of simulated 

instantaneous measurements used in the estimation process. 

The set of 12 measurement intensities was {1, 2,..., 7, 10, 

15, 20, 50, 100}. Selection also was restricted to those days 

when the daily mean flow represented low- and medium-flow 

conditions. The selected daily mean flow at the target site was 

treated as a simulated instantaneous flow measurement. 

For each initially selected day, other active index 

streamgages were identified in the same Michigan hydrologic 

subregion as the target site. Streamgages where daily mean 

flows represented low- and medium-flow conditions were chosen as possible auxiliary streamgages. If the set included five 

or more streamgages, the auxiliary streamgages were used to 

develop a simple linear regression (SLR) relation between the 

square root of water yields from daily mean flow values (rYdv) 

and index water yields rY50 for the selected day. Otherwise, a 

different day was randomly selected and re-evaluated so that 

a minimum of five auxiliary streamgages were available for 

relating daily mean flow to index flow. Index water yields at 

the target site were then estimated on the basis of this relation using the simulated instantaneous measurement as the 

explanatory variable in the simple linear-regression equation. 

In this report, index water yields estimated at target sites 

on the basis of these simple linear-regression equations and 

simulated (discrete) instantaneous measurements are the basis 

of the Discrete Estimator, which is symbolized as DiscrY50 

for root index water yields and DiscQ50 for index flows. The 

uncertainty of a discrete estimate of water yield is the prediction variance of the linear regression relating daily yields to 

index yields at the simulated instantaneous measurement. For 

measurement intensities greater than one, a weighted average of DiscrY50 estimates is computed. Weights are inversely 
10 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

91° 90° 89° 88° 87° 86° 85° 84° 83° 82°

49°

48°

47°

46°

45°

44°

43°

42°

41°

1

2–3

4–5

6–7

8–10

11–15

16–20

21–159

Number of symbol measurements 

EXPLANATION

Michigan hydrologic subregion 

1

2

3

4

5

6

7

0

0 30 60 120 Kilometers

20 40 60 100 Miles

Figure 7. Distribution of sites with instantaneous streamflow measurements in Michigan.
Flow Data 11

1930 1940 1950 1960 1970 1980 1990 2000 2010

CALENDAR YEAR

0

200

400

600

800

1,000

NUMBER OF INSTANTANEOUS FLOW 

MEASUREMENTS AT MISCELLANEOUS SITES

Figure 8. Number of streamflow measurements per year at miscellaneous sites, 1931 to 2006, in Michigan.

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

MONTH OF MEASUREMENT

NUMBER OF INSTANTANEOUS FLOW

MEASUREMENTS AT MISCELLANEOUS SITES

0

4,000

3,000

2,000

1,000

Figure 9. Distribution of instantaneous streamflow measurements, per month, in Michigan.
12 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

0

50

100

150

200

250

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

MONTH OF FLOW MEASUREMENT

AVERAGE FLOW MEASURED AT MISCELLANEOUS

SITES, IN CUBIC FEET PER SECOND

Figure 10. Average measured streamflow, by month, at miscellaneous sites in Michigan.

proportional to each estimate’s corresponding prediction 

variances from the linear regression between daily mean rYdv

and index water yields GagerY50. Estimates are assumed to 

be independent; an assumption that is increasingly plausible 

as the time between instantaneous measurements increases. 

A weighted average of the DiscrY50 estimate and the index 

regression estimate IndxrY50 forms the Integrated Estimator 

of index water yields symbolized as IntgrY50 and index flows 

as IntgQ50. 

Like IndxrY50, both DiscrY50 and IntgrY50, are generalized estimators, which refers to an estimator that is applicable 

to all sites that can be approximated by a regional norm. 

Generalized estimates, however, are not expected to converge1

to site-specific characteristics with increasing measurement 

intensity. Convergence, however, is desirable where sitespecific estimates of index water yield and flow are needed, 

and where measurement intensity can be varied to help 

achieve this objective. The third estimator, referred to as the 

Site Estimator, combines the Integrated Estimate IntgrY50

with estimates of median water yield from simulated instantaneous measurements to estimate index water yield. Site 

Estimates of index water yield are symbolized as SiterY50 and 

index flow as SiteQ50. Following is a more formal discussion 

of the Monte Carlo methods used to test estimators DiscrY50, 

IntgrY50, and SiterY50. 

1

 Converge (and convergence) is used in this report to indicate the tendency 

for a sample statistic to approach the corresponding population statistic with 

increasing sample size. 
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 13

Development of the Discrete, Integrated, and 

Site Estimators using Monte Carlo Methods

Monte Carlo methods were used to assess the likely 

utility of instantaneous flow measurements for improving 

IndxQ50 estimates of index flow Q50. Monte Carlo methods 

are a broadly defined set of numerical techniques that use 

repeated random sampling to characterize complex phenomena (Sobol ́, 1994), four elements are common in Monte Carlo 

methods. First, a sampling frame is defined as a set of possible 

inputs. Second, random samples are repeatedly drawn from 

the sampling frame. Third, the results of the sample are used 

to compute an individual result. Fourth, individual results 

are aggregated to characterize the phenomena. These four 

elements are discussed in detail, with respect to the analysis 

described, in the following section. 

1. As previously described, the sampling frame used in 

this analysis was based on a table of daily mean flow 

values Qdvt j , , with t corresponding to a day within the 

48-year period Oct. 1, 1960–Sept. 30, 2008 (water years 

1961–2008), and j corresponding to an index streamgage 

from 1 to 14 

a. Only 142 of the 147 index streamgages used in 

the development of the index flow regression 

model could be used in the Monte Carlo analysis, 

because the first day of flow values in the sampling 

frame postdated the last day of daily values at five 

index streamgages. Of these five streamgages, 

one streamgage was omitted in each of MHS 1, 

2, and 6, and two streamgages were omitted from 

MHS 5 (fig. 3). 

b. Daily mean flows, Qdvt j , , in cubic feet per second, 

were converted to daily mean water yields Ydvt j , , in 

cubic feet per second per square mile, by dividing 

the flows by the drainage areas, DAreaj , in square 

miles, at the corresponding index streamgage. For 

consistency with assumptions underlying linear 

regression models, a square root transformation was 

applied to the daily water yields to form a matrix of 

square roots of daily water yields indicated as rYdvt j , . 

2. A series of 84 random-sampling experiments were conducted as part of the Monte Carlo analysis. Each experiment was conducted over 1 of the 7 Michigan hydrologic 

subregions, and 1 of 12 levels of measurement intensity. 

The number of random trials for each measurement intensity was specified as the number of index streamgages 

within the subregion times 10,00 Thus, on the average, 

there were 10,000 trials for each streamgage, although this 

number varied slightly because of random sampling.

Within each hydrologic subregion, measurement-intensity 

levels correspond to the number of instantaneous measurements simulated by using rYdv as instantaneous measurements. Each simulated instantaneous measurement 

generated a set of paired rYdv and rY50 values from data 

at the auxiliary sites. These data were used to estimate 

the slope and intercept parameters for the SLR equation. 

The equation was used to estimate index water yield at the 

target site given the simulated instantaneous measurement 

of water yield at the target site. Measurement-intensity 

levels I i i i q q ={ } q q 1 2 12

, ,, were {1, 2, ..., 7, 10, 15, 20, 50, 

100}, corresponding to 1 to 100 simulated instantaneous 

measurements. For example, the number of simulated 

instantaneous measurements corresponding to i

q12

 is 100. 

a. Each sampling experiment consisted of an average 

of 10,000 trials per streamgage. In each trial, one of 

the index streamgages was randomly selected with 

replacement2 from among the index streamgages in 

the target Michigan hydrologic subregion (MHS). 

This selected streamgage would be temporarily 

treated as the target (miscellaneous) site where the 

index water yield rY50 (and flow Q50) could not 

be estimated directly from continuous-records. 

The remaining streamgages in the subregion were 

potential auxiliary streamgages for use in estimation. Selection of auxiliary streamgages was limited 

geographically to the same subregion as the target 

site so that auxiliary streamgages would likely have 

similar weather and flow conditions. 

b. Once the target site J o was selected, a set 

of one or more days, depending on the measurement 

intensity iq, was randomly selected with replacement from all days at J o

 meeting the water yield 

selection criterion. This criterion specified that 

rY rY rY dv ≤Gage t J J Indx o J , o

50 +1 9. 6 σ 50 o · ˆ . Otherwise, 

a different day was randomly selected. Such a 

one-side confidence interval would be expected 

to contain rY50 for the site with 95-percent probability. Note that the confidence interval is based 

on GagerY50, which is a more precise estimator 

of rY50 than IndxrY5 As an unbiased estimator of 

rY50, IndxrY50 could be substituted for sites where 

flow records are unavailable, although the index 

estimate is sensitive to this criterion. 

2

 Sampling with replacement is a sampling technique in which a randomly 

selected unit is returned to the sampled population for possible reselection in 

subsequent random samples. 
is the mean square error of the linear

regression for the simulated 

measurement, which is estimated as

 

is the number of axilary streamgages

in J , and

 is the estimated variance of 

the miscellaneous (instantaneous) 

measurement rYmm as an estimator of 

the daily mean rYdv. For these simulations, the uncertainty was estimated by 

use of a one percent coefficient of 

variation of the gaged daily mean yield, 

which results in a variance of 

0.01·GagerYdv . In future 

applications, the variance might be 

based on rYmm . 

 

where

2

≡ Var rY50 − rY50t J, Disc o t Jo , σp t, Jo

2

( ) ˆ

σSLR t, J'

2

( ) ˆ 1+

1

#{ } J' +

rYdvt, J − rYdvt, J

(rYdv ) t, j − rYdvt, j

+

∑

∀j∈J '

( ) Var  rY50 − rYmmt J, o t Jo ,

... 2

1

#{ } J' − 2 ∑

∀j∈J '

( ) rY50 − rY50t j , Disc

t j ,

2



+

σSLR t, J'

2

( ) ˆ

#{J '}

Var ( ) rY50 − rYmmt J, o t Jo ,

t Jo ( , )

'

t J o ,

( ) 

o ( '

)

2

, and

14 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

Discrete Estimator

3. Once a day was selected initially; all other operating 

index streamgages in the subregion were potential auxiliary streamgages. A set of auxiliary streamgages { } J′ was 

identified, meeting the same daily water-yield criterion at 

their respective locations as the target site. The minimum 

number of auxiliary streamgages for use in estimation 

was five ( ) #{ } J 5 ′ ≥ . Otherwise, an alternative day was 

randomly selected and evaluated. Additional days were 

selected for the target site to form a set where the number 

of elements in the set corresponded to the measurement 

intensity being evaluated. 

a. Using daily values of water yield measured at auxiliary 

streamgages, a set of simple linear regression {SLR} 

equations was developed as

b. This regression equation then was used to estimate the 

index water yield at target site J o for time t as

c. In future applications, an actual instantaneous 

measurement indicating water yield rYmmt J o , might 

be used to approximate rYdvt Jo , . These measurements, 

subject to the same selection criterion as the simulated 

measurements, would provide the explanatory variable 

needed to estimate rY50t J, Disc o , using an equation of 

the form [2].

d. rY50t J, Disc o estimates from all sample days t for the 

measurement intensity iq being investigated were 

weighted inversely proportionally to their corresponding prediction variances and averaged to estimate 

iq

rY50 , J Disc o at the target site (Taylor, 1982). 

The estimated prediction variance of the regression 

estimate of water yield at the target site for the selected 

day was computed as: 

e. Once all selected days for the specified measurement 

intensity level were evaluated, a weighted average 

index water yield estimate was computed as (Taylor, 

1982)

DiscrY50  i J, o

q

∀t∈

∑i { q}

DiscrY50

σ p t, J

2

( ) ˆ

t J, o

o ∀t∈

∑i { q } σ p t, J

2

( ) ˆ o

1

f. The estimate of the index flow based only on a set iq

of simulated discrete instantaneous measurements at 

target site Jo and the relation between daily mean and 

index water yields at the auxiliary streamgages is:

DiscQ   50 o i J, q= DiscrY50 o i J, q o J · DArea . 2

g. For a measurement intensity of one, the variance of the 

weighted average water yield estimate within each 

simulated measurement-day set of length # t i { } ∈ q was 

computed as (Taylor, 1982)

w1 o = 1 i J, q

σ2

( ) ˆ ∀t∈

∑i { q } σ p t, J

2

( ) ˆ o

1

b b dv 0 t J rY 1 rY , 50 = + · o t J, Disc o

is a vector of water yield estimates 

of rY50 based on flow records at 

streamgages used as auxiliary 

sites J on a day indexed by t,

is the corresponding vector of the 

square roots of daily mean water 

yields at the auxiliary 

sites J on day t,

are the intercept and slope 

parameters, respectively,of a simple 

linear regression estimated by 

ordinary least squares to relate daily 

yields to index yields, and

is a vector of regression errors 

(residuals) of length # for time t.

Gage b b dv J o t J t J rY50 ' 1 rY , , = + ε '

+ ' ·

Gage J rY50 '

dvt J rY , '

b b o 1 ,

t J, ε '

{J′}

′

′

where

(2)

(4)

(5)

(6)

(3)

(1)
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 15

h. For measurement intensities greater than one, this 

variance was adjusted to 

ν

2

 

where,

  −

 

w1 o = 1 i J, q

σ2

( ) ˆ ∀t∈

∑i { q } σ p t, J

2

( ) ˆ o

1 · max 1, χ

is a correction for possible underdispersion 

of the variance of the weighted mean, and 

is computed as

ν

2 χ

DiscrY50i J, o

q

∀t∈

∑i { q}

DiscrY50

σ p t, J

2

( ) ˆ

t J, o

o

1

# t i { } ∈ q 1

− 2

.

i. When measurement intensities greater than one were 

evaluated, a between-measurement variance was 

included as 

where,

b1 i J, o = q

σ2

( ) ˆ ∀t∈

∑i { q }

1

1–v o i J, q

o i J, q

u  Dis  crY50i J, o q

DiscrY50t J, o − 2

u o i J, q

is a vector of standardized weights computed as

σ p t, J

2

( ) ˆ o

1

∀t∈

∑i { q } σ p t, J

2

( ) ˆ o

1

v o i J, q

is the sum of squared standardized weights ∀t∈

∑i { q }

o i J, q

u

j. The estimated variance of the DiscrY50 for measurement intensity i

q

 at site J o is

  DiscrY50 i J, o = q

σ2

( ) ˆ w o i J, q

σ2

( ) ˆ o i J, ( ) q b1

σ

2

1

k. A 95-percent confidence interval was developed for 

each estimate of index water yield based on the set of 

simulated instantaneous measurements as

0.975,nReg+nMea t

o i J,q

= o i J,q 0.025,nReg+nMeasQ−1 +− t DiscrY50 o i J, q

σ2

( ) ciDiscrY50 DiscrY50 ˆ

where,

 is the critical value of the t distribution for 

a two-sided test at the 95-percent level of probability with nReg+nMea−1 degrees of freedom, 

where nReg is the minimum number of streamgages available in the regression relating index 

to daily water yields, and nMeasQ is the measurement intensity being evaluated. This random 

interval contains GagerY50 about 95 percent of 

the time.

Integrated Estimator

l. An integrated estimate of the index water yield was 

computed by combining information from both the 

index regression equation and the simulated instantaneous measurements as

DiscrY50i J, o

q

IndxrY50J o

IntgrY50 ,

where

 is the integrated estimate of water yield 

based on

 is the index regression estimate (Hamilton 

and others, 2008) of water yield at streamgage 

J , and

 is the discrete estimate of water yield 

based on all available miscellaneous measurements at site J ,

 is the estimated prediction variance of the 

index regression estimate of water yield at 

streamgage J , and

 is the estimated prediction variance of the 

discrete estimator with i instantaneous 

measurements at site J .

# t i { } ∈ q

IndxrY50J o

DiscrY50i J, o

q

o

o

o

o

q

DiscrY50 o i J, q

σ2

( ) ˆ IndxrY50 o J σ2

( ) ˆ +

IndxrY50 o J σ2

( ) ˆ DiscrY50 o i J, q

σ2

( ) 1 + 1 ˆ o i J, q

=

IntgrY50 o i J, q

IndxrY50 o J σ2

( ) ˆ

DiscrY50 o i J, q

σ2

( ) ˆ

m. The Integrated estimate of the index flow based on 

combining information from the Index Regression 

Estimator IndxrY50 with the Discrete Estimator 

DiscrY50 and drainage area at the target site is

2

  IntgQ50 IntgrY50 · DAreaJ o o i J, q

= o i J, q

n. The within estimator variability of the integrated 

estimate was estimated as

w o i J, q

σ2

( ) ˆ 2



IndxrY50Jo

IntgrY50

w o i J, q

σ2

2( )





 





–

IndxrY50J  o



 





IntgrY50 –

IndxrY50 IntgrY50 J o



  



 .

+

IndxrY50 o J σ2

( ) ˆ 1

o i J, q

1 1

+ 2

2  +











o i J, q

o i J, q

+ 2

2

(7)

(8)

(9)

(10)

(11)

(12)

(13)
16 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

o. Similarly, the between estimator variability of 

integrated estimate was computed as

IndxrY50 IntgrY50 J o b o i J, q

σ2

( ) ˆ 2

= 1

1–v

u1( )

DiscrY50

– 2

u IntgrY50 2( ) – 2

+...

w o i J, q

σ2

1( ) w o i J, q

σ2

1( )

,

,

w o i J, q

σ2

1( )

o i J, q

o i J, q o i J, q

where,

u is a 2-vector of standardized weights computed as:

IndxrY50 o J σ2

( ) ˆ

1 1 +

IndxrY50 o J σ2

( ) ˆ

1 1

IndxrY50 o J σ2

( ) ˆ is the prediction variance of the index 

regression at site J ,

is the variance defined in equation 6, and

is the sum of squared weights u + u .

o

2 2

1 2 v

,

p. The estimated variance of IntgrY50 at measurement 

intensity i

q at site J o

 is

 b o i J, q

σ2

( ) ˆ 2

 w o i J, q

σ2

2( ) o i J, ( ) q

ˆ +

IntgrY50 σ

2

q. A 95-percent confidence interval was developed for 

each integrated estimate of index water yield based on 

the index regression estimate and the set of simulated 

instantaneous measurements as

0.025,nReg+nMeasQ–1 t

o i J,q

= o i J,q 0.025,nReg+nMeasQ−1 +− t

IntgrY50 o i J, q

σ2

( ) ciIntgrY50 IntgrY50 ˆ

where, 

 is the Integrated estimate of rY50 for the

measurement intensity i at site J ,

 is the prediction variance associated 

with the Integrated estimate of rY50 for 

measurement intensity i at site J ,

 is the critical value of the t distribution for 

a two-sided 95-percent level of probability with 

nReg+nMeasQ−1 degrees of freedoms where 

nReg is the minimum number of streamgages 

available in the regression relating index to 

daily water yields, and nMeasQ is the measurement intensity being evaluated. This random 

interval is expected to contain the GagerY50 

about 95 percent of the time.

o i J,q

IntgrY50

IntgrY50 o i J, q

σ2

( ) ˆ

q

q

o

o

,

Site Estimator

r. Both the IndxrY50 and DiscrY50 estimators that form 

the IntgrY50 estimator likely would improve as generalized estimators of index water yield with increasing 

numbers of streamgages in the index regression estimator and increasing numbers of instantaneous flow 

measurements in the discrete estimator. Thus, over 

the population of stream sites that are applicable, both 

index streamgages and instantaneous flow measurements are helpful. Increases in streamgages or instantaneous measurements, however, would not be likely 

to result in site-specific estimates of water yield that 

converge to water yield values computed from longterm streamflow records at every site. Convergence is 

not expected because site-specific conditions result in 

unique index water yield and flow conditions. 

s. To better ensure convergence to site-specific conditions, a third estimator of index water yields SiterY50

and flows SiteQ50 was developed and evaluated. This 

estimator weighs estimates of IntgrY50 estimates with 

estimates of the median water yield computed from 

three or more simulated instantaneous flow measurements obtained during July, August, or September. 

Again, simulated instantaneous flow measurements 

rYdv that were associated with water yields that were 

less than t J IndxrY50( ) o , Jo J rYdv ≤ GagerY50 + 1.96 · σ o

2 ˆ were 

included; otherwise, a different day t was selected. 

This estimator requires an estimate of the variability 

of the median water yield as a function of the number 

of instantaneous measurements. The assumption that 

index water yields and flows are equally likely to occur 

in July, August, and September, and that median flows 

are similar to the minimum of median flows during 

these months is implicit in the simulation procedure 

described in this report. This procedure readily could 

be modified, however, to target the month more likely 

to contain the minimum flow. 

t. An estimate of the variance of the median flow σ x n2

( )

from a small set of n instantaneous flow measurements is needed to implement the Site Estimator. 

Rider (1960) reported that variances of the median for 

samples of size 1, 3, 5, and 7 from a standard normal 

distribution (with mean zero and variance 1) are 1, 

0.4487, 0.2868, and 0.2104, respectively. Rider (1960) 

did not compute variances for other odd-numbered 

sample sizes or for any even-numbered sample sizes. 

Medians of even numbered samples are not strictly 

defined, although they are conventionally computed as 

the average of the two middle values in an ordered list. 

In this report, Rider’s (1960) estimates of variances 

of medians for selected sample sizes were confirmed 

and extended to include all investigated measurement 

intensities by use of resampling techniques involving 

(14)

(15)

(16)
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 17

1 million trials for each measurement intensity (table 

1). By extension, the variances of sample medians 

from normal populations is: 

σ x n2

( ) σ x n · 2

( ) σ x

2

= ∈N(0,1) ,

where,

is the sample estimate of the variance of the 

median based on n measurements,

is the sample estimate of the variance of the 

mean, and

is the sample estimate of the variance of the 

median of a standard normal variate (table 1).

σ x n2

( )

σ x n2

( )

σ x

2

∈N(0,1)

ˆ ˆ ˆ

ˆ

ˆ

ˆ

 is the estimator of square root of water 

yield based on IntgrY50 and the median 

Med of water yields computed from a 

corresponding set of miscellaneous 

measurements and drainage area at the 

target site,

is the estimated variance of the Integrated 

estimate,

 is the estmated variance of the median 

water yield in Michigan hydrologic 

subregion Mhs containing J . This variance 

was computed as the average variance for 

all index streamgages (table 1–2) in the 

corresponding Mhs and is listed in table 1–3 

for each measurement intensity. 

o

IntgrY50 Med ( ) rYdvi J, q

o

+

SiterY50

+ i J, o

q

SiterY50i J, o

q

q

o i J, q

= IntgrY50 o i J, q

σ2

( ) ˆ MedrYdv o i ,Mhs J σ2 ˆ ( ) ,

IntgrY50 o i J, q

σ2

( )

1 ˆ 1 MedrYdv σ2 ˆ

where

o i J, q

IntgrY50 o i J, q

σ2

( ) ˆ

MedrYdv o i , Mhs J σ2 ˆ ( )

( )

q o

(i ,Mhs( ) J )

( q )

u. Estimated variances of the median rYdv values 

MedrYdv o i ,Mhs J σ2 ˆ ( ) ( q ) for each of the 142 index streamgages j

and each of the 12 measurement intensities 

investigated in this report were estimated through a set 

of resampling experiments. Each experiment included 

500,000 trials in which median values were computed 

from random samples of rYdv of size iq drawn with 

replacement from the sampling frame during the 

months of July, August, and September and low- and 

medium-flow conditions. 

v. Estimated standard deviations of median water yields 

are listed as a function of measurement intensity 

MedrYdv o i , Mhs J σ2

ˆ ( )

σ2 ˆ (q ) by streamgage in appendix 1, table 1–2, 

and by Michigan hydrologic subregion in 

appendix 1, table 1–3. 

w. The estimated variance of the SiterY50 estimator for 

measurement intensity i

q

 at site J o is:

+ 1

1

1 =

IntgrY50 o i J, q

σ2

( ) ˆ

o

SiterY50 i , J σ2 ˆ ( ) q o

MedrYdv o i ,Mhs J σ2 ˆ ( ) ( q )

x. An approximate 95-percent confidence interval is 

estimated as

where,

 is the quantile from the t distribution with 

nReg –1 degrees of freedom, where nReg is the 

minimum number of streamgages available in 

the regressions relating index to daily water 

yields. The random interval that is expected to 

contain rY50 at site J about 95 percent of the 

time.

o

o i J,q

= o i J,q − 0.025,nReg−1 t SiterY50 o i J, q

· σ ( ) ciSiterY50 SiterY50 ˆ

0.025,nReg−1 t

+

4. Estimator accuracies were determined from Monte Carlo 

simulations and compared on the basis of the root mean 

square error (RMSE) criterion. The RMSE is computed for each of the four estimators in the ordered set 

rY50 = {IndxrY50, DiscrY50, IntgrY50, SiterY50}. 

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

squared plus the variance of the errors, which can be 

written as

Table 1. Variances of the median from sampled measurement 

intensities for a standard normal probability distribution.

Sample size Variance of the median 

1 1.000

2 .5006

3 .4491

4 .2985

5 .2866

6 .2145

7 .2108

10 .1683

15 .1665

20 .1382

50 .1374

100 .1175

(17)

(18)

(19)

(20)
18 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

where

is a column vector of errors associated 

with κ estimator rY50 indexed 

by measurement intensity i and 

streamgage j during trial τ, 

κ   i , j q ,τ

q

is the average error, and

is the variance of the errors, and



i j ,τ κ q,   rY50 e

i j κ q,    , τ

RMSErY50

∑ i j κ q,   rY50 e

τ

i j , κ q,   rY50 e  #

i j , κ q rY50   , e

i j , κ q rY50   , ·e  

+

'

i j , κ q,   rY50 e  #

2





 









 



 



 





i j , κ q rY50   , e

i j , κ q rY50   , ·e  

'

i j , κ q,   rY50 e  #



 





к indexes the four estimators in the 

ordered set rY50.



∑ i j , κ q rY50   , e

i j , κ q,   rY50 e  #



 









 



 i i

An RMSE equal to zero would imply a perfect estimator; 

an RMSE equal to the standard deviation of rY50 would imply 

that only the mean is being estimated. The RMSE of the index 

regression, which equals 0.12377 ft1.5/s0.5-mi, will be used as a 

basis to compare alternative estimators. 

The RMSE was computed for each of the 84 simulations over the 10,000 trials used per simulation within all 

the Michigan Hydrologic Subregions, MHS, and over all 

levels of measurement intensities I. The change in accuracy of the index flow estimate by supplementing the index 

regression estimate with information from one or more 

simulated instantaneous measurements is described by 

the percent decrease in RMSE, which was computed as

RMSE   rY50

∆

κ+1, κ

i ,q ∀j∈ MHS =

   RMSErY50κ+1, κ i ,q MHS    RMSErY50 κ i ,q MHS – ( ( –

   RMSErY50 κ i ,q MHS 

·100

where,

is the negative percent change in the 

RMSE between estimator rY50 and 

rY50 for measurement intensity i for 

all j streamgages in Michigan hydrolic

subregion MHS. κ indexes the four 

estimators in {rY50}

RMSE   rY50

∆

κ+1, κ

i ,q ∀j∈ MHS 





к

к +1



q

Positive RMSE values imply greater accuracy of the 

rY50κ +1 estimator relative to the rY50κ estimator for ordered 

estimators indexed by κ∈  1,2,3 .

Results of the Monte Carlo Simulations for the 

Discrete, Integrated, and Site Estimators

A three-step estimation process was developed and 

evaluated by Monte Carlo simulations to evaluate the potential utility of instantaneous flow measurements for improving 

estimates of index water yields rY50 and index flows Q50 

in Michigan from those based on the index regression equation, IndxrY50 and IndxQ50, respectively. Each step in the 

estimation process was evaluated within each of the seven 

hydrologic subregions spanning Michigan and at simulated 

instantaneous flow measurement intensities that ranged from 

1 to 100 per site. 

Discrete Estimator

Although the index regression estimator applies to all 

hydrologic subregions, the estimator DiscrY50 is based on the 

relation between daily mean water yields during specified flow 

conditions in July, August, and September and index water 

yields at streamgages within the same hydrologic subregion 

(fig. 11). This linear relation commonly explained a substantial amount of the variability in index water yields (fig. 12). 

The coefficient of variation3

 of single measurement DiscrY50

estimates varied from 9.15 percent in Michigan hydrologic 

subregion 4 to 21.7 percent in subregion 5, with a weighted 

average of 15.1 percent overall. The distribution of single 

measurement DiscrY50 estimates for subregion 2, which had 

an average coefficient of variation of 15.5 is shown in fig. 13. 

Results indicate that expected DiscrY50 estimates averaged 32.6 percent lower RMSE than the IndxrY50 for one 

simulated instantaneous measurement and 66.5 percent lower 

RMSE for 100 simulated measurements (fig. 14), for the 

7 hydrologic subregions in Michigan For one simulated instantaneous measurement, subregion 5 had the least improvement 

3

 The coefficient of variation is computed as the standard deviation divided 

by the mean, and expressed as a percent. 

(22)

(21)
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 19

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ROOT DAILY WATER YIELD AT ACTIVE INDEX STREAMGAGES IN MICHIGAN

HYDROLOGIC SUBREGION 2, IN CUBIC FEET PER SECOND PER SQUARE MILE

ROOT INDEX WATER YIELD AT INDEX STREAMGAGES IN MICHIGAN HYDROLOGIC SUBREGION 2, IN CUBIC FEET PER SECOND PER SQUARE MILE

Simulated instantaneous measurement at

04108600 Rabbit River near Hopkins, Mich., 

on July 11, 1967. 

R2 = 0.8784

y = -0.0995 + 0.9878 x

Figure 11. Typical relation between the square roots of daily mean water yields and index water yields, illustrated with data from 

active streamgages in Michigan hydrologic subregion 2 and a simulated instantaneous flow measurement at U.S. Geological Survey 

streamgage 04108600, Rabbit River near Hopkins, Mich., on July 11, 1967.
20 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

04096015040964050409651504096600040969000409717004097540040990000410150004101800041025000410270004103010041049450410500004105700041086000410880104111500041120000411250004114498041150000411650004117000041175000411800004118500

U.S. GEOLOGICAL SURVEY STREAMGAGE NUMBER IN MICHIGAN HYDROLOGIC SUBREGION 2

DISTRIBUTION OF COEFFICIENTS OF DETERMINATION (R2

)

Lower hinge

Lower quartile

Median

Upper quartile

Upper hinge

Outlier

EXPLANATION

Figure 12. Distribution of coefficients of determination for regression relations between the square roots daily water yields 

and the square roots of index water yield, illustrated with results for index streamgages in Michigan hydrologic subregion 2.
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 21

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

04096015040964050409651504096600040969000409717004097540040990000410150004101800041025000410270004103010041049450410500004105700041086000410880104111500041120000411250004114498041150000411650004117000041175000411800004118500

U.S. GEOLOGICAL SURVEY STREAMGAGE NUMBER IN MICHIGAN HYDROLOGIC SUBREGION 2

SQUARE ROOT OF WATER YIELD, IN CUBIC

FEET PER SECOND PER SQUARE MILE

Lower hinge

Lower quartile

Median Square root of index water yield Upper quartile

Upper hinge

Outlier

EXPLANATION

Figure 13. Distribution of single measurement DiscrY50 estimates for index streamgages in Michigan hydrologic subregion 2.
22 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

1 10 100

NUMBER OF SIMULATED INSTANTANEOUS STREAMFLOW MEASUREMENTS

20

30

40

50

60

70

80

PERCENT DECREASE IN THE ROOT MEAN SQUARE ERROR 

FROM THE INDEX REGRESSION WATER YIELD ERROR

Michigan hydrologic subregion

Symbol Subregion

5

6

7

Average

1

2

3

4

Figure 14. Percent decrease in the root mean square error of the DiscrY50 estimator with respect to the IndxrY50 estimator with 

the number of simulated streamflow measurements by Michigan hydrologic subregion.

of 26.3 percent, and subregion 3 had the greatest improvement 

of 40.7 percent. Thus, even a single instantaneous flow measurement, obtained during July, August, or September under 

the appropriate flow conditions, is expected to provide a more 

precise estimate of index water yield and flow than the index 

regression equation. For 100 simulated measurements, subregion 6 had the least improvement of 54.3 percent, and subregion 4 had the greatest improvement of 77.6 percent. DiscQ50

estimates formed by squaring DiscrY50 estimates and multiplying by the corresponding drainage areas are expected to be 

consistent with the improvements in the DiscrY50 estimates. 

Variances were computed in addition to the magnitudes 

of the corresponding estimates. The variances were used to 

combine estimates from different days of simulated measurement and to generate nominal 95-percent confidence intervals 

about resulting weighted estimates. Over all measurement 

intensities and subregions, results indicate that the DiscrY50

variances are slightly overestimated in that 96.9 percent of 

the GagerY50 values are contained within a nominal 95-percent confidence interval about DiscrY50 estimates (table 2). 

At a measurement intensity of 1, however, average variances 

are slightly underestimated in that the nominal 95-percent 

confidence interval only contains 92.1 percent of GagerY50

values. There is a slight increase in the average effective confidence widths with increasing measurement intensities from 

2 to 100. In subregions 1, 2, and 4, at measurement intensity 100, the maximum of 100 percent of rY50 values were 

contained within the nominal 95-percent confidence interval 

formed about DiscrY50; in subregion 4 at measurement 

intensity 1, the minimum of 88.7 percent of rY50 values were 

contained within the nominal 95-percent confidence interval. 

Integrated Estimator

Integrated estimates are a weighted average of Index 

regression estimates and Discrete estimates. Integrated 

estimates of water yield IntgrY50 averaged 8.3 percent lower 

RMSE than expected Discrete values of estimates with one 

simulated instantaneous streamflow measurement (fig. 15) 

for the 7 hydrologic subregions. Among hydrologic subregions, improvements in RMSE ranged from 1.6 percent for 

subregion 5 to 19.1 percent for subregion 7. This difference, 

however, decreases with to zero after about five simulated 

measurements. Subregions 3 and 5 showed some increase 
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 23

1 10 100

NUMBER OF SIMULATED INSTANTANEOUS STREAMFLOW MEASUREMENTS

-10

0

10

20

PERCENT DECREASE IN THE ROOT MEAN SQUARE ERROR 

FROM THE DISCRETE WATER WATER YIELD ESTIMATOR ERROR

Michigan hydrologic subregion

Symbol Subregion

5

6

7

Average

Zero reference

1

2

3

4

Figure 15. Percent decrease in the root mean square error of the IntgrY50 estimator with respect to the IndxrY50 estimator with 

the number of simulated streamflow measurements by Michigan hydrologic subregion.

Table 2. Percent of GagerY50 values contained within a nominal 95-percent confidence interval 

about DiscrY50, IntgrY50, and SiterY50 estimators.

Number of simulated 

instantaneous streamflow 

Percent of GagerY50 values contained within a nominal 95-percent 

confidence interval about the corresponding estimator

measurements DiscrY50 IntgrY50 SiterY50

1 92.10 90.62 88.17

2 96.03 94.62 91.16

3 96.65 95.35 92.49

4 96.98 95.81 92.45

5 97.15 96.05 93.01

6 97.31 96.28 92.87

7 97.38 96.37 93.09

10 97.57 96.61 92.71

15 97.72 96.78 92.34

20 97.80 96.87 91.31

50 97.99 97.02 85.52

100 98.09 97.07 77.66

Average 96.90 95.79 90.23
24 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

in the RMSE of the Integrated estimates with two or more 

simulated measurements; this increase also tended to zero 

with increasing numbers of simulated measurements. Variances developed for IntgrY50 estimates were used to form 

nominal 95-percent confidence intervals for all subregions 

and measurement intensity levels. These confidence intervals 

effectively included about 95.8 percent of GagerY50 values 

for all measurement-intensity levels (table 2). The effective 

confidence intervals contained a smaller percentage of observations than the nominal confidence intervals would indicate 

at lower measurement frequencies, but a larger percentage 

than nominal confidence intervals would indicate at higher 

measurement frequencies. 

Figures (fig. 16-22) show the distribution of errors 

for estimates of index water yield computed by use of the 

IndxrY50 and IntgrY50 estimators for each index streamgage 

within the seven Michigan hydrologic subregions. In these 

figures, IntgrY50 estimates are based on one simulated instantaneous measurement. The IndxrY50 residual is displayed as 

an asterisk, which is at the midpoint of a vertical line spanning 

the middle 50 percent of the probability distribution from the 

75- to 25-percent points of the cumulative-distribution function. This interval is based on the standard error of prediction 

for the index regression equation. In comparison, the distribution of IntgrY50 residuals is displayed as a boxplot in which 

the upper and lower limits of the box represent the 75th and 

25th percentiles, respectively, of the error distribution. Thus, 

this interquartile range represents the middle 50-percent of 

the error distribution from the average of 10,000 Monte Carlo 

trials per streamgage used in the analysis. The mean and 

median errors for each boxplot show the central tendencies of 

the distribution. 

For each paired boxplot of IndxrY50 and IntgrY50

values at a particular streamgage, the method in which the 

absolute value of the mean (expected) error is closer to zero 

provides the more accurate estimate. For all subregions with 

one simulated instantaneous measurement, the IntgrY50

estimator outperformed the IndxrY50 estimator 105 times 

for the 142 streamgages evaluated, or about 74 percent of the 

time. For 100 simulated measurements, the IntgrY50 estimator outperformed the IndxrY50 estimator 108 times out of 

142 streamgages evaluated. 

With one simulated instantaneous streamflow measurement, variations in the ranks of IntgrY50 account for 94.9 

percent of the variability in the ranks of GagerY50, and 

variations in IntgQ50 account for 98.0 percent of the variability in GageQ50 (fig. 23). With 100 simulated instantaneous 

measurements, variations in IntgrY50 account for 96.7 percent 

of the variability in GagerY50, and variations in IntgQ50

account for 98.8 percent of the variations GageQ50. Relative 

to the 70.8 percent of the variability in rY50 associated with 

IndxrY50 estimator, the IntgrY50 estimator is more accurate, 

explaining 94.9-percent or more of the variability in rY50. 

The accuracy of the IntgrY50 estimator, however, shows limited improvement with increased measurement intensity. 

Site Estimator

The Site Estimator of index water yields SiterY50 and 

flows SiteQ50 was developed to facilitate the convergence of 

index water yield and flow estimates to site-specific values 

of GagerY50 and GageQ50 with increasing numbers of 

instantaneous flow measurements. The technique is based 

on the premise that as the number of instantaneous flow 

measurements obtained during July, August, and September 

approaches the number of daily mean flow values used to 

compute GagerY50 and GageQ50, the estimates should tend 

to converge to the computed values assuming the monthly 

medians are similar. Because median flows in July, August, 

and September differ somewhat, the median of instantaneous 

measurements taken with equal probability from these months 

likely will exceed the index flow, which is defined as the minimum of median flows during these 3 months. 

Application of the Site Estimator is limited to sites 

where three or more instantaneous measurements are available. Determination of the variance of the median estimator 

requires that at least two instantaneous flow estimates be 

available for estimation. The median of two samples, by convention however, is indistinguishable from the mean of two 

samples, even though the mean generally has different probability distribution than the median. With increasing sample 

size, the variance anomaly diminishes between even- and 

odd-numbered sample sizes. 

Simulation results indicate that the decrease in RMSE of 

the Site Estimator SiterY50 relative to the Integrated Estimator IntgrY50 generally increases with increasing numbers 

of simulated instantaneous measurements greater than or 

equal to 3 for all hydrologic subregions in Michigan (fig. 24). 

All Michigan hydrologic subregions except 2 have higher 

average precisions than the corresponding integrated estimator 

for all measurement intensities investigated. The reason for 

the anonymous results in subregion 2, which do not indicate 

increases in precision until more than 20 measurements are 

obtained, is not known. 

Unlike the results for the Discrete and Integrated estimators, the average effective percentage of GagerY50 values 

contained with the nominal 95 percent confidence interval did 

not increase monotonically with increased sampling intensity. 

Instead, the maximum average of 93.1 percent occurred for a 

7 simulated instantaneous measurements, while the minimum 

average percentage occurred for 100 simulated instantaneous 

measurements (table 2). 
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 25

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

04001000040310000403150004032000040330000403500004040000040405000404150004043050040455000404600004049500040550000405650004057510040578000405800004058400040590000405950004060993040615000406220004127918

ERROR IN INDEX WATER YIELD ESTIMATE, IN THE SQUARE

ROOT OF CUBIC FEET PER SECOND PER SQUARE MILE

U.S. GEOLOGICAL SURVEY STREAMGAGE NUMBER 

Upper whisker error1

75th-percentile error 

50th-percentile error 

Mean error 

25th-percentile error

Lower whisker error1

EXPLANATION

Error

Index regression IndxrY50 Integrated Estimate IntgrY50 

75-percentile

confidence limit

25-percentile

confidence limit

Zero reference

Michigan Hydrologic Subregion 1

1

The 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. 

Numerous outliers occurred in the errors 

of the Integrator estimate within the 

10,000 trials used in Monte Carlo 

simulations at each streamgage. To 

increase the clarity of the plot, outliers 

are not shown. 

Figure 16. Error for the index regression estimator IndxrY50 and the distribution of errors for the integrated water yield estimator 

IntgrY50 with one simulated measurement for index streamgages within Michigan hydrologic subregion 1.
26 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

-0.2

-0.1

0

0.1

0.2

0.3

04096015040964050409651504096600040969000409717004097540040990000410150004101800041025000410270004103010041049450410500004105700041086000410880104111500041120000411250004114498041150000411650004117000041175000411800004118500

ERROR IN INDEX WATER YIELD ESTIMATE, IN THE SQUARE

ROOT OF CUBIC FEET PER SECOND PER SQUARE MILE

U.S. GEOLOGICAL SURVEY STREAMGAGE NUMBER

Upper whisker error1

75th-percentile error

50th-percentile error 

Mean error 

25th-percentile error

Lower whisker error1

EXPLANATION

1

The 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. 

Numerous outliers occurred in the errors

of the Integrator estimate within the

10,000 trials used in Monte Carlo simulations

at each streamgage. To increase the clarity

of the plot, outliers are not shown. 

Error

Index regression IndxrY50 Integrated Estimate IntgrY50

75-percentile

confidence limit

25-percentile

confidence limit

Zero reference

Michigan Hydrologic Subregion 2

Figure 17. Error for the index regression estimator IndxrY50 and the distribution of errors for the integrated water yield estimator 

IntgrY50 with one simulated measurement for index streamgages within Michigan hydrologic subregion 2.
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 27

-0.2

-0.1

0

0.1

0.2

0.3

04121000041213000412190004122100041222000412250004123000041235000412400004124500041250000412550004126200

ERROR IN INDEX WATER YIELD ESTIMATE, IN THE SQUARE

ROOT OF CUBIC FEET PER SECOND PER SQUARE MILE

U.S. GEOLOGICAL SURVEY STREAMGAGE NUMBER

Upper whisker error1

75th-percentile error 

50th-percentile error 

Mean error 

25th-percentile error

Lower whisker error1

EXPLANATION

1

The 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. 

Numerous outliers occurred in the errors

of the Integrator estimate within the

10,000 trials used in Monte Carlo simulations

at each streamgage. To increase the clarity

of the plot, outliers are not shown. 

Error 

Index regression IndxrY50 Integrated Estimate IntgrY50

Zero reference

Michigan Hydrologic Subregion 3

75-percentile

confidence limit

25-percentile

confidence limit

Figure 18. Error for the index regression estimator IndxrY50 and the distribution of errors for the integrated water yield estimator 

IntgrY50 with one simulated measurement for index streamgages within Michigan hydrologic subregion 3.
28 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

041279970412899004133501041350000413550004135600041357000413650004137500

ERROR IN INDEX WATER YIELD ESTIMATE, IN THE SQUARE

ROOT OF CUBIC FEET PER SECOND PER SQUARE MILE

U.S. GEOLOGICAL SURVEY STREAMGAGE NUMBER

Upper whisker error1

75th-percentile error 

50th-percentile error 

Mean error

25th-percentile error

Lower whisker error1

EXPLANATION

1

The span of the whiskers is the largest

and smallest values within 1.5 times the

interquartile range, which is formed as

the dierence betweem tje 75th- and

25th-percentiles. 

Numerous outliers occurred in the errors

of the Integrator estimate within the

10,000 trials used in Monte Carlo simulations

at each streamgage. To increase the clarity

of the plot, outliers are not shown. 

Error 

Zero reference

Michigan Hydrologic Subregion 4

Index regression IndxrY50 Integrated Estimate IntgrY50 

75-percentile

confidence limit

25-percentile

confidence limit

Figure 19. Error for the index regression estimator IndxrY50 and the distribution of errors for the integrated water yield estimator 

IntgrY50 with one simulated measurement for index streamgages within Michigan hydrologic subregion 4.
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 29

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

04138000041385000413900004139500041400000414050004141000041415000414200004143900041440000414450004146000041460630414750004147990041481400414816004148200041483000414844004148500041500000415050004151500041522380415350004154000041550000415800004158500

ERROR IN INDEX WATER YIELD ESTIMATE, IN THE SQUARE

ROOT OF CUBIC FEET PER SECOND PER SQUARE MILE

U.S. GEOLOGICAL SURVEY STREAMGAGE NUMBER

EXPLANATION

Michigan Hydrologic Subregion 5

Upper whisker error1

75th-percentile error

50th-percentile error 

Mean error 

25th-percentile error

Lower whisker error1

Error

Index regression IndxrY50 Integrated Estimate IntgrY50

75-percentile

confidence limit

25-percentile

confidence limit

Zero reference

1

The 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. 

Numerous outliers occurred in the errors

of the Integrator estimate within the

10,000 trials used in Monte Carlo simulations

at each streamgage. To increase the clarity

of the plot, outliers are not shown. 

Figure 20. Error for the index regression estimator IndxrY50 and the distribution of errors for the integrated water yield estimator 

IntgrY50 with one simulated measurement for index streamgages within Michigan hydrologic subregion 5.
30 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

04159492041599000416000004160570041606000416080004160900041610000416110004161500041615400416158004161800041634000416350004164100041643000416450004164800041660000416620004166300

ERROR IN INDEX WATER YIELD ESTIMATE, IN THE SQUARE

ROOT OF CUBIC FEET PER SECOND PER SQUARE MILE

U.S. GEOLOGICAL SURVEY STREAMGAGE NUMBER

Michigan Hydrologic Subregion 6

Upper whisker error1

75th-percentile error

50th-percentile error 

Mean error 

25th-percentile error

Lower whisker error1

EXPLANATION

1

The 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. 

Numerous outliers occurred in the errors

of the Integrator estimate within the

10,000 trials used in Monte Carlo simulations

at each streamgage. To increase the clarity

of the plot, outliers are not shown. 

Error

Index regression IndxrY50 Integrated Estimate IntgrY50

75-percentile

confidence limit

25-percentile

confidence limit

Zero reference

Figure 21. Error for the index regression estimator IndxrY50 and the distribution of errors for the integrated water yield estimator 

IntgrY50 with one simulated measurement for index streamgages within Michigan hydrologic subregion 6.
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 31

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0416950004170000041705000417150004172000041730000417350004174500041748000417560004175700041760000417660504184500

ERROR IN INDEX WATER YIELD ESTIMATE, IN THE SQUARE

ROOT OF CUBIC FEET PER SECOND PER SQUARE MILE

U.S. GEOLOGICAL SURVEY STREAMGAGE NUMBER

Michigan Hydrologic Subregion 7

Upper whisker error1

75th-percentile error

50th-percentile error 

Mean error 

25th-percentile error

Lower whisker error1

EXPLANATION

1

The 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. 

Numerous outliers occurred in the errors

of the Integrator estimate within the

10,000 trials used in Monte Carlo simulations

at each streamgage. To increase the clarity

of the plot, outliers are not shown. 

Error

Index regression IndxrY50 Integrated Estimate IntgrY50

75-percentile

confidence limit

25-percentile

confidence limit

Zero reference

Figure 22. Error for the index regression estimator IndxrY50 and the distribution of errors for the integrated water yield estimator 

IntgrY50 with one simulated measurement for index streamgages within Michigan hydrologic subregion 7.
32 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

0.2 0.4 0.6 0.8 1 100 102

0.2 0.4 0.6 0.8 1 100 102

r2 

s = 0.9493 r

2 

s = 0.9801

r

2 r s = 0.9875 2 

s = 0.9667

100

101

102

103

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

GagerY50, IN SQUARE ROOT OF CUBIC GageQ50, IN CUBIC FEET PER SECOND FEET PER SECOND PER SQUARE MILE 

100

101

102

103

1 Simulated Instantaneous Streamflow Measurement

100 Simulated Instantaneous Streamflow Measurements

IntgrY50, IN SQUARE ROOT OF CUBIC

FEET PER SECOND PER SQUARE MILE 

IntgrY50, IN SQUARE ROOT OF CUBIC

FEET PER SECOND PER SQUARE MILE 

Intg

Q50, IN CUBIC FEET PER SECOND Intg

Q50, IN CUBIC FEET PER SECOND

1

2

3

4

5

6

7

Symbol Michigan Hydrologic Subregion

EXPLANATION

Line of agreement

Line of agreement Line of agreement

Line of agreement

Figure 23. Relation of IntgrY50 with GagerY50 and IntgQ50 with GageQ50 for 1 and 100 simulated instantaneous streamflow 

measurements.
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 33

1 10 100

- 2 0

- 1 0

0

1 0

2 0

3 0

4 0

5 0

6 0

1

2

3

4

5

6

7

Symbol Michigan Hydrologic Subregion

Average

EXPLANATION

NUMBER OF SIMULATED INSTANTANEOUS STREAMFLOW MEASUREMENTS

PERCENT DECREASE IN THE ROOT MEAN SQUARE ERROR

FROM THE INTEGRATED WATER YIELD ERROR ESTIMATOR

Figure 24. Decrease in root mean square error of the SiterY50 estimator with increased number of simulated instantaneous 

streamflow measurements.
34 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

The three estimators developed in this report tended to 

converge to their final values with increasing numbers of simulated instantaneous measurements. Only SiterY50 estimates, 

however, tended to converge to rY50 values with increasing 

measurement intensity. Rates of convergence varied widely 

among streamgages where convergence occurred (fig. 25), 

from about 6 simulated instantaneous flow measurements 

to more than 100. At other streamgages, however, SiterY50

estimates converged to values that were greater or less than 

GagerY50, at least within the range of simulated measurement 

intensities. Convergence of SiterY50 to GagerY50 values is 

thought to be sensitive to the month of flow measurement and 

the flow criterion under which instantaneous flow measurements are obtained. 

The tendency for convergence to GagerY50 values was 

compared by computing the RMSE between rY50 estimates 

and GagerY50 values for each estimator by measurement 

intensity (fig. 26). Results indicate that RMSE for DiscrY50

decreased with measurement intensity and converged with 

RMSE values for IntgrY50 at measurement intensities greater 

than 2. For measurement intensities greater than 10, the RMSE

for either DiscrY50 or IntgrY50 showed occasional substantial decreases. At measurement intensities greater than 3, the 

RMSE for SiterY50 estimates is consistently less than the 

RMSE for either DiscrY50 or IntgrY50. Moreover, the RMSE

for SiterY50 decreases throughout the range of measurement 

intensities investigated. This continued decrease indicates 

that SiterY50 estimates are converging to GagerY50 values. 

The RMSE for the IndxrY50 estimator is plotted for reference, 

although it is not a function of measurement intensity. 

To further improve convergence characteristics of the 

Site estimator, it may be appropriate to measure instantaneous 

flows during the month when minimum median flow is most 

likely to occur. Among the 147 index streamgages in Michigan, index flows occurred 62.9 percent of the time in August, 

34.0 percent of the time in September, and 3.4 percent of the 

time in July. Among Michigan hydrologic subregions, August 

was the most frequent month of minimum flow, expect for 

subregion 2, where minimum flows more commonly occurred 

in September (fig. 27). All index flows occurred in August 

within subregion 1, which corresponds to Michigan’s Upper 

Peninsula. July was the least frequent month of index flows 

among subregions. Convergence characteristics of the Site 

estimator also may be sensitive to the water-yield criterion 

used to restrict the selection of days for simulated instantaneous measurement. 

Further experimentation with the criterion for selecting daily values as estimators of rY50 values may facilitate 

convergence of the SiterY50 estimator. For example, it may 

be helpful if a threshold percentage of index streamgages in 

the target hydrologic subregion are above and below their 

index flows, thus forming a mixed index-flow condition for 

the subregion. Real-time monitoring of daily flows at index 

streamgages might be used to select a day for instantaneous 

measurement that would be expected to meet this criterion. 

For instance, an instantaneous measurement might be planned 

on a day when 10 percent or more of index streamgages in the 

hydrologic subregion are flowing at a rate less than their index 

flows, and 10 percent or more of the index streamgages are 

flowing at a rate greater than their index flows. Further analysis would help identify the likelihood of these mixed index 

flow conditions for a range of threshold percentages. Additional Monte Carlo simulations may clarify how the threshold 

percentages affect the rate of convergence of the SiterY50

estimates to rY50 values. 

Towards the Development of an Augmented 

Estimator from Base Flow Recession 

Characteristics

The augmented index regression estimator of water yield 

and flow (AugmrY50 and AugmQ50, respectively) is based on 

an augmented form of the index regression equation, which 

contains two additional variables characterizing base-flow 

recession. The augmented equation is developed by analysis 

of 9,724 base-flow recession events identified from daily flow 

values at index streamgages. Base flow recession characteristics at index streamgages are described, and recession parameters for a simplified recession equation are estimated. Monte 

Carlo simulations are used to describe the relation between the 

number of recession events measured and the accuracy of the 

augmented regression estimator. A set of two or more instantaneous flow measurements during one or more base-flow 

recession events can be used to estimate the base-flow recession variables needed to apply the augmented index regression 

equation. 

Eng and Milly (2007) proposed the use of base flow 

recession information to improve estimates of low-flow characteristics at partial-record stream sites. In their study, base 

flow recession segments were chosen only from the period 

April 1 to October 31, because low streamflow values generally occur during this period in their study region in the southeastern part of the United States. By use of daily flow data at 

streamgages, Eng and Milly (2007) identified sequences of 

eight or more daily values that (1) started with a peak daily 

streamflow that exceeded the median daily flow value, and 

(2) ended with the first day j for which both Qj+1 and Qj+2

exceed Qj. Eng and Milly (2007) concluded that (1) introduction of a base-flow recession time constant, estimated from a 

single pair of strategically timed streamflow measurements, 

approximately halves the root-mean-square estimation error 

relative to that of a conventional drainage area regression, and 

(2) additional streamflow measurements can be used to further 

reduce the error. A similar analysis was conducted as part of 

this report to assess the utility of base-flow recession information on the estimation index water yield and flow. 
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 35

0 1

0

0

2 3 4 5 6 7 10 15 20 50 100 0.40

0.45

0.50

0.55

0.60

0.65

04059500 Ford River near Hyde, Michigan

1 2 3 4 5 6 7 10 15 20 50 100

0.58

0.57

0.60

0.62

0.64

0.66

04138000 East Branch Au Gres River at McIvor, Michigan

WATER YIELD, IN SQUARE ROOT OF CUBIC FEET PER SECOND PER SQUARE MILE

1 2 3 4 5 6 7 10 15 20 50 100 0.35

0.40

0.45

0.50

04166300 Upper River Rouge at Farmington, Michigan

NUMBER OF SIMULATED INSTANTANEOUS STREAMFLOW MEASUREMENTS

IndxrY50

DiscrY50

IntgrY50

SiterY50

GagerY50

Figure 25. Convergence of DiscrY50, IntgrY50, and SiterY50 estimates with increasing number of simulated instantaneous 

streamflow measurements. 
36 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

1 2 3 5 7 10 20 50 100

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

SIMULATED NUMBER OF INSTANTANEOUS STREAMFLOW MEASUREMENTS

ROOT MEAN SQUARE ERROR

IndxrY50

DiscrY50

IntgrY50

SiterY50

ESTIMATOR

Figure 26. Relation between root mean square error and the number of simulated instantaneous streamflow measurements for 

estimators DiscrY50, IntgrY50, and SiterY50. 
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 37

1 2 3 4 5 6 7

MICHIGAN HYDROLOGIC SUBREGION 

0

5

10

15

20

25

30

FREQUENCY OF MEDIAN FLOW BY SUMMER MONTH

July

August

September

Figure 27. Frequency of summer month of minimum median flow by Michigan hydrologic subregion.

Base-Flow Recession

Consistent with the determination of index flows, recession segments were restricted to the period from July 1 to September 30. Like Eng and Milly (2007), flow on the first day 

of each base flow segment was required to exceed a minimum 

criterion, in this case the index flow Q50 for the corresponding 

streamgage. For consistency with the metric used by Hamilton 

and others (2008), recession events were defined on the basis 

of daily mean water yields rYdv rather than daily flows Qdv. 

Daily values contained in the sampling frame were the 

primary source of information for identifying base-flow recession events for index streamgages. Periods of daily record 

were added, however, for the five streamgages that were 

discontinued before first day of values in the sampling frame 

and for selected other index streamgages with short periods of 

record within the sampling frame. A total of 9,724 base-flow 

recession events were identified at 147 index streamgages 

(appendix 1, table 1-4). No recession events were identified, however, at streamgage 04135000 Thunder Bay River 

near Alpena, Mich. The lack of recession events meeting the 

specified criteria may be associated with flow regulation by a 

hydroelectric plant 1,000 ft upstream from this gage (Blumer 

and others, 1994). One to two base-flow recession events commonly occurred each year (fig. 28) at each streamgage. About 

one-half of the identified recession events were 8 or 9 days 

long (fig. 29). 

Following Eng and Milly (2007), parameters rYo and tau

for each recession event were estimated by least squares using 

all but the first five daily water yield values rYdv within the 

identified recession event as

rY = rY ·e , t o

t

tau -

where,

rYt is the square root of the estimated water yield at 

time t,

is the initial square root of water yield of the 

target base flow recession segment, and

a parameter associated with the recession time 

constant.

rYo

tau

(23)
38 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.00

0.05

0.10

0.15

0.20

0.25

NUMBER OF BASE-FLOW EVENTS PER SUMMER PER STREAMGAGE

RELATIVE FREQUENCY

Figure 28. Relative 

frequency of the number of 

base-flow events per summer 

per index streamgage.

8 10 12 14 16 18 20 22 24 26 28 30

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

DURATION OF BASE-FLOW RECESSION, TIME (t) IN DAYS

RELATIVE FREQUENCY

y(t)=0.30053 exp(-0.38167 (t-8) + 0.004382 (t-8)2 )+e(t)

Estimated y(t)

Empirical

Figure 29. Relative 

frequency of the durations 

of base flow recession 

segments identified at index 

streamgages.
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 39

Two examples are provided to illustrate the ideal match 

between measured and recession equation estimates of root 

water-yield recession. Figure 30 illustrates the importance of 

omitting the first five daily root water-yield values from the 

estimation process. In particular, the magnitude of the multiplicative parameter rYo would have been larger had the first 

five daily values been included in the estimation process. 

Also, the stepper gradient in the early period of the recession 

event would have been associated with a greater exponential 

term tau. Figure 31 shows that the fitted equation also can 

provide a close approximation to extended recession periods. 

The recession events selected as examples represent cases that 

are more consistent with the recession-equation estimates than 

perhaps are typical, but are not exceptional. 

20-Jul 22-Jul 24-Jul 26-Jul 28-Jul 30-Jul

0.34

0.36

0.38

0.40

0.42

0.44

0.46

0.48

0.50

0.52

SQUARE ROOT OF WATER YIELD, IN CUBIC FEET PER SECOND 

PER SQUARE MILE 

Water yield near the beginning

of the base-
ow recession event

Water yield used in the estimation

of recession parameters

Estimated recession curve

rYt = 0.3561 exp(-t/104.96 )

Extrapolated recession curve

showing lack of t to early

recession characteristics

1967

Figure 30. Base-flow recession starting July 20, 1967, at U.S. Geological Survey (USGS) streamgage 04112500 Red Cedar River at 

East Lansing, Mich. (t) is time in days since the beginning of the recession period used in parameter estimation.)
40 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

30-Jun 03-Jul 06-Jul 09-Jul 12-Jul 15-Jul 18-Jul 21-Jul 23-Jul

0.5

1.0

1.5

2.0

2.5

3.0

SQUARE ROOT OF WATER YIELD, IN CUBIC FEET PER SECOND

 PER SQUARE MILE 

1968

Water yield near the beginning

of the base-
ow recession event

Water yield used in the estimation

of recession parameters

Estimated recession curve

rYt = 0.496 exp(-t/15.3065 )

Extrapolated recession curve

showing lack of t to early

recession characteristics

Figure 31. Base flow recession starting July 1, 1968, at the U.S. Geological Survey (USGS) streamgage 04112500 Red Cedar River at 

East Lansing, Mich. (t is time in days since the beginning of the recession used in estimating the parameters).
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 41

The medians of the two parameters for all recession 

segments were used to estimate the corresponding recession 

characteristics at each index streamgage (appendix 1, table 

1–4). The distribution of median initial water yield recession 

mrYo is approximately normal with mean 0.5968 and variance 

0.05202 (fig. 32). The distribution of the common logarithms 

of the median tau value for each streamgage, designated as 

lmtau, is approximately normal with mean 1.612 and variance 

0.09872 (fig. 33). 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

RECESSION MODEL PARAMETER rY

CUMULATIVE PROBABILITY

Median rY data

Median rY fit 

N (0.5968,0.05202)

o

o

o

Figure 32. The cumulative distribution function of median recession parameter rYo at index streamgages.
42 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

CUMULATIVE PROBABILITY

N(1.612,0.0987)

COMMON LOGARITHM OF THE MEDIAN RECESSION MODEL TIME CONSTANT TAU 

FOR INDEX STREAMGAGES 

2.6

log10(median(tau))

Empirical Distribution

Normal Approximation

Figure 33. The cumulative distribution function of the common logarithm of the median recession parameter tau computed 

from base-flow recession events at index streamgages.
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 43

Recession Characteristics and Estimation 

of Index Water Yield

Hamilton and others (2008) developed an index regression model that contains six explanatory variables including 

measures of low and high aquifer transmissivities, L_Trans and 

H_Trans, respectively; percent forest covering the basin, 

Forest; normal annual precipitation for the period 1971–2000, 

Precip; and percent hydrologic soil groups A and B covering 

the basin, as Soil_A and Soil_B, respectively. The index 

regression estimate IndxrY50 accounts for 70.8 percent 

of the variability in rY50 and has an RMSE of 0.1238.

The utility of streamgage-specific recession parameters 

used as variables in an augmented form of the index regression equation was evaluated as an eight-variable regression 

model in this report. The variables were formed as the median 

of estimated rYo values (mrYo) and common logarithms of the 

median recession time constants tau (lmtau) from all recession 

events identified at index streamgages. Results indicate that 

mrYo and lmtau were highly significant (p-value <0.0001), and 

together with the original six variables, account for 97.4 percent 

of the variability in rY50 with an RMSE of 0.03684 (table 3). 

However, parameter estimates changed and their statistical 

significances were lower for some of the original variables in 

the index regression. 

To provide greater consistency with the index regression 

equation (Hamilton and others, 2008), a two-step least-squares 

form of the augmented equation is provided. The first step 

provides predicted values of root index water yield IndxrY50, 

which are used together with base-flow recession parameters 

mrYo

 and lmtau as explanatory variables computed from continuous streamflow data in the second step to estimate rY50. 

Results for this estimator, identified as AugIndxrY50, indicate 

that second-stage estimates account for 97.3 percent of the variability in rY50 with an RMSE of 0.03842. An analysis of variance and estimated parameters for the augmented index regression is shown in table 4. Parameters in the augmented index 

regression model may be biased because of the uncertainty 

in the explanatory variables, which are assumed to be known 

precisely under ordinary least squares regression. The large 

percentage of the total variability of rY50 explained by the 

regression, however, mitigates some of the concern about bias 

in the corresponding estimate of flow. The AugIndxrY50 has an 

RMSE that is 69 percent lower than the IndxrY50 estimator. 

An estimate of the variance of AugIndxrY50 at site Jo

 can 

be computed as

J o

1

2

 −

x

J o

( )−1

σ AugmrY50 J

2

( ) ˆ J o

2 = s · 1+ x o · ( ) X' · X · x ,

where,

s is the mean square error of the regression 

equation 0.03842 ,

is a column vector containing site-specific 

values of 1 IndxrY50 mrY Imtau , and

Inverse of the cross product matrix of the 

design matrix (table 5). 

2



o ] ]

X' · X 





Monte Carlo analysis was used to assess the effect of 

uncertainty in base flow recession variables on the uncertainty of augmented index regression estimates of root index 

water yields AugIndxrY50. For this estimator, identified as 

AugmrY50, mrYo and lmtau variables were estimated by 

random sampling with replacement from the set of recession 

parameters estimated for the identified base flow recession 

events at index streamgages. A set of 12 sampling intensities {1, 2,..., 7, 10, 15, 20, 50, 100} for recession events was 

evaluated. For each Monte Carlo trial, the medians of sampled 

recession parameters were used to estimate the m-o and lmtau

variables. 

Parameters estimated in the augmented index regression equation (table 4) were used to compute AugmrY50

estimates for each index streamgage in each of 10,000 trials 

per sampling intensity. Residuals were computed as the 

difference between GagerY50 values and AugIndxrY50

estimates. The interquartile range, the difference between the 

75th and 25th percentiles of residuals, was used to provide a 

robust measure of the spread of the residuals. 

The distribution of interquartile range for each sampling 

intensity was compared with the interquartile ranges of residuals for the index regression IndxrY50 and augmented index 

regression AugIndxrY50 estimates (fig. 34). Results show a 

monotonic decrease in the median interquartile range with 

increases in sampling intensity. The median interquartile range 

of residuals from AugmrY50 estimates based on 1 sampled 

recession event was slightly less than the interquartile range 

of residuals from IndxrY50 estimates, and the median interquartile range of residuals from AugmrY50 estimates based 

on 100 sampled recession events was slightly more than the 

interquartile range of residuals from AugIndxrY50 estimates. 

Daily values from streamgages operated less than 

10 years, but otherwise meeting the criteria for index 

streamgages, could be used instead of instantaneous flow 

measurements to estimate recession variables for the augmented index regression equation. Owing to the continuity of 

daily values throughout recession periods, data from shortterm streamgages would provide a more complete description 

of recession characteristics than limited sets of instantaneous 

measurements. 

Although the relation between the number of base-flow 

recession events used in the estimation of recession variables 

and estimation accuracy of AugmrY50 was evaluated, the 

relation between the number of instantaneous measurements 

per base-flow recession event and the accuracy of recession 

parameters was not. Eng and Milly (2007) argue that a single 

pair of strategically timed streamflow measurements approximately halves the RMSE of low-flow estimates relative to that 

of a conventional drainage area regression relation. Presumably, this timing would be informed by real-time streamflow 

data from streamgages near the site of interest and weather 

forecasts, which could help avoid the early (5-day) period 

following the start of the recession period and increases in 

streamflow from subsequent precipitation events. Owing 

to the autocorrelation of streamflow values, instantaneous 

(24)
44 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

Table 3. Analysis of variance table and parameter estimates for the index regression with base-flow recession parameters 

as variables.

{—, not applicable]

Source Degrees of freedom Sum of squares Mean square F value Probability of great 

F statistic

Model 8 7.45288 0.93161 624.42 <0.0001

Error 138 .20589 .00149 — —

Corrected Total 146 7.65877 — — —

Variable Degrees of freedom Parameter estimate Standard error t value Probability of a 

greater |t| statistic

Intercept 1 -0.17634 0.05945 -2.97 0.0036

L_Trans 1 .00019954 .00017088 1.17 .2449

H_Trans 1 -.00014948 .00016943 -.88 .3792

Forest 1 -.00002788 .00019767 -.14 .8880

Precip 1 -.00149 .00185 -.80 .4223

A_Soils 1 .00041971 .0002672 1.57 .1185

D_Soils 1 .00005736 .00034072 .17 .8665

mrYo 1 .82909 .02677 30.97 <.0001

lmtau 1 .15552 .01715 9.07 <.0001

Table 4. Analysis of variance table and parameter estimates for the augmented index regression AugIndxrY50 equation.

[—, not applicable]

Source Degrees of freedom Sum of squares Mean square F value Probability of great 

F statistic

Model 8 7.45510 0.93189 631.4 <0.0001

Error 138 .20367 .0014759 — —

Corrected Total 146 7.65877 — — —

Variable Degrees of freedom Parameter estimate Standard error t value Probability of a 

greater |t| statistic

Intercept 1 -0.22994 0.01698 -13.54 <0.0001

IndxrY50 6 .01082 .03145 .34 .7314

mrY 1 .81129 .02558 31.72 <.0001 o

lmtau 1 .16590 .01476 11.24 <.0001

Table 5. The inverse of the X’X matrix needed to compute prediction limits for the augmented index regression AugIndxrY50 equation.

Explanatory variables in the AugIndxrY50 regression equation

Intercept IndxrY 50 mrYo lmtau

0.202449 0.054157 0.023014 -0.147576

.054157 .694465 -.380471 -.119923

.023014 -.380471 .459334 -.059837

-.147576 -.119923 -.059837 .152903
Improving Regression Estimates of Index Flow by use of Instantaneous Flow Measurements 45

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

1 2 3 4 5 6 7 10 15 20 50 100

NUMBER OF BASE-FLOW EVENTS USED TO ESTIMATE RECESSION PARAMETERS

EXPLANATION

INTERQUARTILE RANGE

Interquartile range of

 residuals from the 

 Index Regression

Interquartile range of

 residuals from the 

 Augmented Index

 Regression

Lower quartile

Upper quartile

Median

Lower hinge

Upper hinge

Low outlier

High outlier Boxplots show the distribution of the interquartile 

ranges of residuals from Monte Carlo analysis. Each 

boxplot shows the results of 10,000 trials in which 

1 to 100 recession parameters, derived from 

measured base-ow events at each of the 147 

index streamgages, were randomly sampled with 

replacement. The median recession parameter for 

each trial was used to estimate the recession 

variable in the augmented index regression 

equation. Residuals were formed as the dierences 

between the augmented index regression estimate 

and the square root of water yield. 

Figure 34. Boxplots showing the distribution of interquartile ranges of residuals of augmented regression estimates as a 

function of the number of sampled base-flow recession events.
46 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

measurements separated by more than 3 days, for example, 

would be more informative than measurements separated by 

fewer than 3 days, assuming all measurements were in the 

later period of the recession event. A minimum of three measurements per recession event would be needed to provide an 

assessment of the recession model fit or parameter uncertainty. 

If a sufficient number of grouped instantaneous measurements 

are available to reliably estimate the recession variables mrYo

and lmtau for the target site, the experimental AugmrY50 estimator might be considered. 

In regression analysis, explanatory variables like mrYo

and lmtau are assumed to be measured without error. As 

described in this report, however, mrYo and lmtau are estimated from parameters simultaneously fit to a nonlinear 

equation from a limited set of baseflow recession events. The 

uncertainty associated with these variables will likely result in 

biased estimates of index flow. Additional analyses are needed 

to determine the error characteristics of the AugmrY50 estimator before application to issues affecting water withdrawal. 

Example calculations are provided (Appendix 2) to illustrate the application of the Discrete, Integrated, Site, and Augmented estimators to index streamgage 04112500 Red Cedar 

River at East Lansing, Mich. Based on 77 years of continuous 

streamflow data, the index flow (GageQ50) is 38.0 ft3/s and 

the root index water yield (GagerY50) is 0.332 ft 3

( )2 s·mi . 

The site is in hydrologic subregion 2 and has a drainage area 

of 344 mi2. The example calculation shows input data, results 

of intermediate and final computations. Matlab4 code is provided to document the computational approach in Appendix 2. 

Summary and Conclusions

In Michigan, index flow Q50 is a streamflow characteristic defined as the minimum of median daily flows during the 

months of July, August, or September. Index flow is used by 

the state of Michigan to regulate large water withdrawals so 

as to prevent adverse effects on characteristic fish populations. 

At U.S. Geological Survey (USGS) streamgages, daily flow 

values for the period of gage operation can be used to compute 

an index flow estimate GageQ50 that is expected to converge 

to Q50 with increasing length of record. Index flow commonly 

is expressed in units of cubic feet per second. 

Estimates of index flow also are needed at sites where 

streamgage records are not available or are not sufficiently 

long to compute a reliable estimate. Hamilton and others 

(2008) developed an equation to estimate index flow IndxQ50

as the product of the drainage area DArea contributing to 

flow upstream from a site, and the index (root) water yield 

across the contributing basin calculated by use of multiple linear regression. Water yield is expressed in units of 

cubic feet per second per square mile of drainage area.

4

 Matlab is a computer language for technical computation by the Mathworks, Inc. 

For consistency with statistical assumptions underlying regression, the index regression equation estimates the 

square root of water yield, which is indicated by the bold “r” 

for square root preceding “Y,” which is used to suggest yield, 

as IndxrY50. Daily streamflow data from water years 1901 

to 2005 for 147 selected streamgages in or near Michigan 

were used to compute GageQ50 and GagerY50 values for 

use in developing the index regression equation. Six explanatory variables were included in the equation, which included 

measures of aquifer transmissivity, land cover, annual precipitation, and hydrologic soil classification. Estimates from the 

index regression equation accounted for 70.8 percent of the 

variability in GagerY50, and 94.0 percent of the variability 

in GageQ50. Seven Michigan hydrologic subregions (MHS) 

were defined along watersheds to partition the state to more 

homogeneous units for analysis. 

At some ungaged stream sites, more accurate estimates 

of index flow are needed than IndxQ50. In this report, information obtainable from instantaneous measurements of flow 

was evaluated to determine the potential for increasing the 

accuracy of index flow estimates. Instantaneous measurements, which are direct measurements of flow that typically 

require about 1 hour to complete, are made at miscellaneous 

sites once or more often on a random or discontinuous basis to 

provide better areal coverage for describing hydrologic conditions than that provided by streamgages alone. In Michigan, a 

total of 22,368 flow measurements have been documented at 

3,692 miscellaneous sites (Marlio Lesmez, Michigan Department of Natural Resources and Environment, written commun., 2009). Although these measurements could not be used 

to develop the methods described in this report, many of them 

could be used to improve estimates of index flow at miscellaneous sites from those based on the index regression estimate. 

Monte Carlo simulations were used to assess a sequence 

of estimators for improving the estimates of index flow with 

information obtainable from instantaneous flow measurements. The Monte Carlo method is a generalized numerical 

technique that involves defining a sampling frame for analysis, 

repeated random sampling of the frame, evaluating alternative methods of estimation for each sample, and assessing and 

comparing the results. 

In this report, the sampling frame contained daily mean 

streamflow values for water years 1961 to 2008 for the 147 

streamgages used to develop the index regression equation. 

Each daily flow value during the months of July, August, and 

September was considered a possible simulated instantaneous 

flow measurement for use in estimating index flow. Each 

streamgage, in turn, served as a target site. Each target site 

was evaluated for a set of 12 measurement intensities corresponding to the total number of simulated instantaneous 

measurements, from 1 to 100, available for analysis. Each 

target site J o was evaluated for each measurement intensity by 

an average of 10,000 random samples of daily flow values; 

selected days were constrained to water yield conditions that 

were less than GagerY J IndxrY J 50 o+ 50 1.96 ·σˆ o . Otherwise, an 

alternate day was randomly selected to represent the simulated 
References Cited 47

instantaneous measurement. Other streamgages in the same 

hydrologic subregion, meeting the same site-specific flow 

criteria, were selected as auxiliary sites. 

A simple linear regression was developed between the 

square roots of daily values of water yields, rYdv, and index 

water yields, rY50, at the auxiliary sites. This relation was 

used to estimate index water yield at the target site based on 

the corresponding daily water yield at the target site. The 

first index water yield estimator in the sequence is based on 

simulated discrete water yields, and symbolized as DiscrY50

with corresponding index flow DiscQ50. For measurement 

intensities greater than 1, a weighted average of DiscrY50 

values was computed, where the weights are the reciprocals 

of their estimated variances. Although results varied among 

hydrologic subregions, DiscrY50 estimates with 1 simulated 

instantaneous measurement generally had a 32.6-percent 

lower root mean square error (RMSE) than IndxrY50 estimates, and DiscrY50 estimates computed with 100 simulated 

measurements had a 66.5- percent lower RMSE than IndxrY50

estimates. Thus, the expected discrete water yield returned by 

the discrete estimator is more precise than the index regression 

estimator for all measurement intensities investigated. 

The second estimator in the sequence was the integrated 

discrete estimator, which is formed by combining estimates 

based on the index regression estimator and the discrete estimator. In particular, a weighted integrated estimate IntgrY50

was formed as a weighted average of the IndxrY50 estimate 

and DiscrY50 estimates for all measurement intensities evaluated. Again, weights were the reciprocals of estimated variances. Results indicated that IntgrY50 had a lower RMSE than 

DiscrY50 for one or two simulated measurements. At greater 

measurement intensities, the RMSE of IntgrY50 converged to 

the DiscrY50 estimates. 

Neither the DiscrY50 nor the IntgrY50 estimates necessarily converge to site-specific values of GagerY50 with 

increasing measurement intensities. Thus, even if all daily 

flow values used to compute GagerY50 at a particular 

streamgage were used with the first two estimators, the resulting estimates would not match the corresponding index water 

yield or flow statistic at the streamgage. 

To facilitate this convergence, the third estimator in the 

sequence SiterY50 was developed. SiterY50 is a weighted 

average of IntgrY50 and the median daily water yield 

med(rYdv1, rYdv2, ...rYdvn ) on sampled days in July, August, 

or September at the target site for all simulated instantaneous 

measurements. Again, the weights were reciprocals of corresponding estimated variances, where the variances of the 

median (square root) water yields were computed as a function 

of measurement intensities from resampling experiments. 

Results indicate that the RMSE of the SiterY50 estimator with 

respect to IntgrY50 decreases with increasing measurement 

intensity. At some sites, convergence of SiterY50 estimates to 

site-specific GagerY50 values was evident. Convergence characteristics of SiterY50 estimates may be improved by selecting 

months for instantaneous measurements that are expected to 

have minimum median flows more than other months. Also, 

convergence characteristics may improve with an improved 

flow-selection criterion. It may be helpful, for example, if a 

threshold percentage of index streamgages in the target hydrologic subregion are above and below their index flows, thus 

forming a mixed index-flow condition for the subregion. Realtime monitoring of daily flows at index streamgages might be 

used to select a day for instantaneous measurement that would 

be expected to meet the criterion. Further analysis would help 

identify the likelihood of these mixed index flow conditions 

for a range of threshold percentages. Additional Monte Carlo 

simulations may clarify how the threshold percentages affect 

the rate of convergence of the SiterY50 estimates to rY50

values.

The accuracy of the index regression can be augmented 

with two additional variables that characterize base-flow 

recession by use of a two-parameter recession model. In this 

report, 9,724 recession events were identified at 147 index 

streamgages by use of daily streamflow values. Recession 

parameters estimated from these events provided a basis for 

the augmented index regression variables that provided a 

basis for the AugIndxrY50 estimates, which had a 66.3 percent 

lower RMSE than corresponding IndxrY50 estimates. Monte 

Carlo simulations were used to develop the relation between 

the number of recession events sampled and the precision 

of the estimator AugmrY50, which is based only on sampled 

events. In general, the uncertainty of the AugmrY50 estimates 

decreased monotonically with increased numbers of sampled 

recession events from slightly less than the IndxrY50 estimates 

with 1 recession event to slightly more than the AugIndxrY50

estimates with 100 events. Additional analysis of the Integrated estimator is needed before being used to estimate index 

flow to more fully assess the uncertainty of estimates where 

two of the critical explanatory variables are uncertain. 

This report develops empirical estimates of variances 

and variance components for all estimators. These variances 

may be helpful in forming the weighted estimators DiscrY50, 

IntgrY50, and SiterY50. All variance estimates assume 

independence between variance components. The utility of 

the estimated variances was confirmed by forming nominal 

95-percent confidence intervals that generally contained 

slightly more than 95 percent of the measurements. 

References Cited

Blumer, S.P, Behrendt, T.E., Larson, W.W., Minnerick, R.J., 

LeuVoy, R.L., and Whited, C.R., 1994, Water resources 

data, Michigan, water year 1993: U.S. Geological Survey 

Water-Data Report MI–93–1, 288 p.

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

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

Draper, N.R., and Smith, Harry, H1966, Applied regression 

analysis, New York, Wiley, 407 p.
48 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months 

Eng, Ken, and Milly, P.C.D., 2007, Relating low-flow 

characteristics to the base flow recession time constant at 

partial record stream gauges: Water Resources Research, 43, 

W01201, doi:10.1029/2006WR005293, 8 p.

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. (Also available at http://pubs.usgs.

gov/sir/2008/5096.) 

Holtschlag, D.J., and Eagle, D.V., 1985, Stream discharge in 

Michigan—Miscellaneous measurements: U.S. Geological 

Survey Open-File Report 85–350, 344 p. 

Knutilla, R.L., 1974, Compilation of miscellaneous streamflow measurements in Michigan through September 1970: 

Michigan Department of Natural Resources, Geological 

Survey Division, Water Information Series Report 5, 185 p.

Michigan Legislature, 2006, Public Acts of 2006, Act 

No. 33: 93rd Legislature, Regular Session of 2006, 

accessed Nov. 9, 2010, at http://www.legislature.mi.gov/

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

Michigan’s State Facts, 2010, accessed February 17, 2010, 

http://www.michigan.gov/som/0,1607,7-192-29938_30245-

67959--,00.html.

Rantz, S.E., and others, 1982, Measurement and computation of streamflow—v. 2. Computation of discharge: 

U.S. Geological Survey Water-Supply Paper 2175, 631 p.

Rider, P.R., 1960, Variance of the median of small samples 

from several special populations: Journal of the American 

Statistical Association, v. 55, no. 289, p. 148–150. 

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 June 21, 2010, at 

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

Sobol', I.M., 1994, A primer for the Monte Carlo method: 

Boca Raton, Fla., CRC Press, 107 p.

Taylor, J.R., 1982, An introduction to error analysis— 

The Study of Uncertainties in Physical Measurements: 

Mill Valley, Calif., University Science Books, 270 p. 

U.S. Census Bureau, 2008, State & county QuickFacts, 

accessed Feb. 17, 2010, at http://quickfacts.census.gov/qfd/

states/26000.html.
Appendix 1. Tables of streamgage attributes and 

flow characteristics used to assess the utility of 

instantaneous flow measurements for improving 

regression estimates of index flow in Michigan
Table 1–1. Streamgage and flow characteristics used to assess the utility of instantaneous flow measurements for improving index flow estimates. U.S. Geological Survey streamgage number Michigan hydrologic subregion U.S. Geological Survey Strreamgage name Latitude (decimal degrees) Longitude (decimal degrees) Drainage area (square miles) (DArea) Minimum monthly median flow (cubic feet per second) ( ) Q50 Minimum monthly median yield (cubic feet per second per square mile) (Y50) Month of minimum flow Year

of recor

04001000 1 Washington Creek at Windigo, Mich. 47.92306 89.14500 13.2 2.3 0.17424 August 39 04031000 1 Black River near Bessemer, Mich. 46.51134 90.07462 200 40.0 0.20000 August 33 04031500 1 Presque Isle River at Marenisco, Mich. 46.37217 89.69238 172 71.0 0.41183 August 38 04032000 1 Presque Isle River near Tula, Mich. 46.54689 89.77738 264 90.0 0.34078 August 29 04033000 1 Middle Branch Ontonagon River near Paulding, Mich. 46.35689 89.07736 162 100 0.61843 August 59 04035000 1 East Br Ontonagon River near Mass, Mich. 46.68994 89.07347 273 134 0.49048 August 38 04040000 1 Ontonagon River near Rockland, Mich 46.72077 89.20709 1,330 634 0.47530 August 64 04040500 1 Sturgeon River near Sidnaw, Mich. 46.58411 88.57597 169 44.0 0.26036 August 66 04041500 1 Sturgeon River near Alston, Mich. 46.72632 88.66208 343 184 0.53629 August 72 04043050 1 Trap Rock River near Lake Linden, Mich. 47.22854 88.38539 29.6 13.0 0.43919 August 39 04045500 1 Tahquamenon River near Paradise, Mich. 46.57501 85.26955 757 321 0.42410 August 52 04046000 1 Black River near Garnet, Mich. 46.11806 85.36537 33.5 10.0 0.29851 August 38 04049500 1 Manistique River at Germfask, Mich. 46.23331 85.92791 420 238 0.56721 August 33 04055000 1 Manistique River near Blaney, Mich. 46.08609 86.05930 716 352 0.49183 August 33 04056000 1 West Branch Manistique River near Manistique, Mich. 46.08886 86.16125 326 154 0.47312 August 19 04056500 1 Manistique River near Manistique, Mich. 46.03053 86.16125 1,130 605 0.53716 August 68 04057510 1 Sturgeon River near 

Nahma Junction, Mich.

45.94302 86.70570 184 82.0 0.44662 August 39 
Table 1–1. Streamgage and flow characteristics used to assess the utility of instantaneous flow measurements for improving index flow estimates.

U.S. 

Geological 

Survey 

streamgage 

number

Michigan 

hydrologic 

subregion

U.S. Geological Survey 

Strreamgage name

Latitude

(decimal

degrees)

Longitude

(decimal

degrees)

Drainage 

area 

(square 

miles)

(DArea)

Minimum 

monthly 

median flow 

(cubic feet 

per second)

( ) Q50

Minimum 

monthly median 

yield (cubic feet 

per second per 

square mile)

(Y50)

Month 

of 

minimum 

flow

Year

of 

recor

04057800 1 Middle Branch Escanaba 

River at Humboldt, Mich.

46.49910 87.88652 45.7 16.0 0.35011 August 47 

04058000 1 M Br Escanaba River near 

Ishpeming, Mich.

46.39438 87.75847 128 43.0 0.33515 August 22 

04058400 1 Goose Lake Outlet near 

Sands Station, Mich.

46.39300 87.49375 36.3 12.0 0.33058 August 17 

04059000 1 Escanaba River at 

Cornell, Mich.

45.90857 87.21375 871 330 0.37892 August 55 

04059500 1 Ford River near Hyde, Mich. 45.75552 87.20152 444 85.0 0.19140 August 51 04060993 1 Brule River near 

Florence, Wisc.

45.96079 88.31597 378 236 0.62269 August 62 

04061500 1 Paint River at 

Crystal Falls, Mich.

46.10578 88.33486 600 288 0.47976 August 52 

04062200 1 Peshekee River near 

Champion, Mich.

46.55688 88.00263 132 28.0 0.21244 August 23 

04096015 2 Galien River near 

Sawyer, Mich.

41.87365 86.57502 80.8 21.0 0.25990 September 11 

04096405 2 St. Joseph River at 

Burlington, Mich.

42.10282 85.04025 201 61.0 0.30348 September 43 

04096515 2 South Branch Hog Creek near 

Allen, Mich.

41.94866 84.82774 48.7 7.9 0.16222 September 36 

04096600 2 Coldwater River near 

Hodunk, Mich.

42.02921 85.10692 286 65.0 0.22759 September 27 

04096900 2 Nottawa Creek near 

Athens, Mich.

42.05560 85.30832 162 72.0 0.44444 September 31 

04097170 2 Portage River near 

Vicksburg, Mich.

42.11477 85.48555 68.2 30.0 0.43988 September 21 

04097540 2 Prairie River near 

Nottawa, Mich.

41.88838 85.40943 107 46.0 0.43152 September 43 

04099000 2 St. Joseph River at 

Mottville, Mich.

41.80088 85.75610 1,880 850 0.45227 September 82 

04101500 2 St. Joseph River at 

Niles, Mich.

41.82921 86.25973 3,670 1,850 0.50464 September 75 
Table 1–1. Streamgage and flow characteristics used to assess the utility of instantaneous flow measurements for improving index flow estimates. U.S. Geological Survey streamgage number Michigan hydrologic subregion U.S. Geological Survey Strreamgage name Latitude (decimal degrees) Longitude (decimal degrees) Drainage area (square miles) (DArea) Minimum monthly median flow (cubic feet per second) ( ) Q50 Minimum monthly median yield (cubic feet per second per square mile) (Y50) Month of minimum flow Year

of recor

04101800 2 Dowagiac River at Sumnerville, Mich. 41.91338 86.21307 252 177 0.70378 August 45 04102500 2 Paw Paw River at Riverside, Mich. 42.18615 86.36836 390 262 0.67197 August 54 04102700 2 South Branch Black River near Bangor, Mich. 42.35420 86.18753 83.5 35.0 0.41916 September 40 04103010 2 Kalamazoo River near Marengo, Mich. 42.26171 84.85581 270 147 0.54545 September 19 04104945 2 Wanadoga Creek near Battle Creek, Mich. 42.39643 85.13166 48.3 15.0 0.31056 August 11 04105000 2 Battle Creek at Battle Creek, Mich. 42.33199 85.15416 274 70.0 0.25547 September 72 04105700 2 Augusta Creek near Augusta, Mich. 42.35337 85.35389 36.8 31.0 0.84239 August 41 04108600 2 Rabbit River near Hopkins, Mich. 42.64225 85.72197 65.1 22.0 0.33794 September 40 04108801 2 Macatawa River near Zeeland, Mich. 42.77919 86.01837 66.9 4.3 0.06428 September 45 04110000 2 Orchard Creek at Munith, Mich. 42.39365 84.26496 47.3 5.3 0.11205 September 13 04111500 2 Deer Creek near Dansville, Mich. 42.60837 84.32080 16.3 0.9 0.05521 September 52 04112000 2 Sloan Creek near Williamston, Mich. 42.67587 84.36386 10.4 0.2 0.02308 September 52 04112500 2 Red Cedar River at East Lansing, Mich. 42.72781 84.47775 344 38.0 0.11047 September 77 04114498 2 Looking Glass River near Eagle, Mich. 42.82809 84.75943 284 40.0 0.14080 September 57 04115000 2 Maple River at Maple Rapids, Mich. 43.10975 84.69305 420 30.0 0.07141 September 62 04116500 2 Flat River at Smyrna, Mich. 43.05281 85.26474 516 222 0.42998 August 36 04117000 2 Quaker Brook near 

Nashville, Mich.

42.56587 85.09361 7.8 2.8 0.35897 September 33 
Table 1–1. Streamgage and flow characteristics used to assess the utility of instantaneous flow measurements for improving index flow estimates.

U.S. 

Geological 

Survey 

streamgage 

number

Michigan 

hydrologic 

subregion

U.S. Geological Survey 

Strreamgage name

Latitude

(decimal

degrees)

Longitude

(decimal

degrees)

Drainage 

area 

(square 

miles)

(DArea)

Minimum 

monthly 

median flow 

(cubic feet 

per second)

( ) Q50

Minimum 

monthly median 

yield (cubic feet 

per second per 

square mile)

(Y50)

Month 

of 

minimum 

flow

Year

of 

recor

04117500 2 Thornapple River near 

Hastings, Mich.

42.61587 85.23639 410 109 0.26553 September 61 

04118000 2 Thornapple River near 

Caledonia, Mich.

42.81114 85.48335 795 281 0.35337 September 41 

04118500 2 Rogue River near 

Rockford, Mich.

43.08225 85.59086 257 127 0.49378 September 50 

04121000 3 Muskegon River near 

Merritt, Mich.

44.33557 84.89003 352 115 0.32689 August 27 

04121300 3 Clam River at 

Vogel Center, Mich.

44.20057 85.05281 239 73.0 0.30506 August 40 

04121900 3 Little Muskegon River near 

Morley, Mich.

43.50253 85.34254 136 72.0 0.53137 July 30 

04122100 3 Bear Creek near 

Muskegon, Mich.

43.28863 86.22284 16.7 5.0 0.29940 August 40 

04122200 3 White River near 

Whitehall, Mich.

43.46418 86.23257 404 273 0.67491 August 49 

04122500 3 Pere Marquette River at 

Scottville, Mich.

43.94501 86.27869 689 455 0.65999 August 67 

04123000 3 Big Sable River near 

Freesoil, Mich.

44.12028 86.28008 115 101 0.87826 August 32 

04123500 3 Manistee River near 

Grayling, Mich.

44.69307 84.84726 132 170 1.28496 August 31 

04124000 3 Manistee River near 

Sherman, Mich.

44.43639 85.69868 865 856 0.98971 August 86 

04124500 3 East Branch Pine River near 

Tustin, Mich.

44.10251 85.51728 58.9 9.9 0.16808 August 26 

04125000 3 Pine River near Leroy, Mich. 44.06279 85.54894 130 51.0 0.39140 August 12 04125500 3 Pine River near 

Hoxeyville, Mich.

44.20306 85.79951 254 230 0.90658 August 31 

04126200 3 Little Manistee River near 

Freesoil, Mich.

44.18362 86.16758 185 141 0.76381 August 19 

04127918 1 Pine River near 

Rudyard, Mich.

46.18585 84.59783 202 77.0 0.38138 August 34 
Table 1–1. Streamgage and flow characteristics used to assess the utility of instantaneous flow measurements for improving index flow estimates. U.S. Geological Survey streamgage number Michigan hydrologic subregion U.S. Geological Survey Strreamgage name Latitude (decimal degrees) Longitude (decimal degrees) Drainage area (square miles) (DArea) Minimum monthly median flow (cubic feet per second) ( ) Q50 Minimum monthly median yield (cubic feet per second per square mile) (Y50) Month of minimum flow Year

of recor

04127997 4 Sturgeon River at Wolverine, Mich. 45.29890 84.61114 181 170 0.93975 August 64 04128990 4 Pigeon River near Vanderbilt, Mich. 45.15668 84.46669 57.7 58.0 1.00520 July 55 04133501 4 Thunder Bay River at Herron Road near Bolton, Mich. 45.12446 83.64721 586 309 0.52739 September 40 04135000 4 Thunder Bay River near Alpena, Mich. 45.09418 83.49970 1,240 474 0.38288 August 22 04135500 4 Au Sable River at Grayling, Mich. 44.65974 84.71253 96.6 61.0 0.63147 August 51 04135600 4 East Branch Au Sable River at Grayling, Mich. 44.66890 84.70558 71.2 34.0 0.47753 August 27 04135700 4 South Branch Au Sable River near Luzerne, Mich. 44.61474 84.45557 391 134 0.34236 August 38 04136500 4 Au Sable River at Mio, Mich. 44.66001 84.13112 1,360 788 0.57894 August 54 04137500 4 Au Sable R near Au Sable, Mich. 44.43640 83.43386 1,740 1,120 0.64383 September 19 04138000 5 East Branch Au Gres River at McIvor, Mich. 44.23252 83.70082 89.9 38.0 0.42269 August 23 04138500 5 Au Gres River near National City, Mich. 44.17613 83.74248 151 21.0 0.13871 August 31 04139000 5 Houghton Creek near Lupton, Mich. 44.39585 84.04722 29.8 39.0 1.30872 August 23 04139500 5 Rifle River at "The Ranch" near Lupton, Mich. 44.39335 84.03833 57.4 66.0 1.14983 August 22 04140000 5 Prior Creek near Selkirk, Mich. 44.33502 84.06833 21.0 6.7 0.31905 August 22 04140500 5 Rifle River at Selkirk, Mich. 44.31335 84.06944 116 87.0 0.74742 August 32 04141000 5 South Branch Shepards Creek near Selkirk, Mich. 44.30780 84.08694 1.1 0.0 0.03636 July 27 04141500 5 West Branch Rifle River near Selkirk, Mich. 44.26113 84.10916 65.4 32.0 0.48930 July 12 
Table 1–1. Streamgage and flow characteristics used to assess the utility of instantaneous flow measurements for improving index flow estimates.

U.S. 

Geological 

Survey 

streamgage 

number

Michigan 

hydrologic 

subregion

U.S. Geological Survey 

Strreamgage name

Latitude

(decimal

degrees)

Longitude

(decimal

degrees)

Drainage 

area 

(square 

miles)

(DArea)

Minimum 

monthly 

median flow 

(cubic feet 

per second)

( ) Q50

Minimum 

monthly median 

yield (cubic feet 

per second per 

square mile)

(Y50)

Month 

of 

minimum 

flow

Year

of 

recor

04142000 5 Rifle River near 

Sterling, Mich.

44.07252 84.01999 333 159 0.47791 August 69 

04143900 5 Shiawassee River at 

Linden, Mich.

42.81586 83.80190 81.9 22.0 0.26862 August 30 

04144000 5 Shiawassee River at 

Byron, Mich.

42.82364 83.94579 363 71.0 0.19543 September 36 

04144500 5 Shiawassee River at 

Owosso, Mich.

43.01503 84.18108 530 86.0 0.16214 September 75 

04145500 5 Bad River near Brant, Mich. 43.29669 84.22915 89.9 0.6 0.00667 August 11 04146000 5 Farmers Creek near 

Lapeer, Mich.

43.04475 83.33717 51.1 5.1 0.09980 August 73 

04146063 5 South Branch Flint River near 

Columbiaville, Mich.

43.15947 83.35078 211 48.5 0.23029 August 26 

04147500 5 Flint River near 

Otisville, Mich.

43.11114 83.51940 526 109 0.20715 August 52 

04147990 5 Butternut Creek near 

Genesee, Mich.

43.13586 83.59912 34.8 3.6 0.10345 August 14 

04148140 5 Kearsley Creek near 

Davison, Mich.

43.03364 83.58134 99.7 13.0 0.13039 August 40 

04148160 5 Gilkey Creek near Flint, Mich. 43.02419 83.62551 6.9 0.2 0.02319 August 14 04148200 5 Swartz Creek near 

Holly, Mich.

42.82753 83.62828 12.1 1.5 0.12397 September 20 

04148300 5 Swartz Creek at Flint, Mich. 42.98781 83.73246 114 5.6 0.04904 August 14 04148440 5 Thread Creek near 

Flint, Mich.

42.97503 83.63579 54.4 4.7 0.08640 August 14 

04148500 5 Flint River near Flint, Mich. 43.03892 83.77163 960 171 0.17805 August 73 04150000 5 South Branch Cass River near 

Cass City, Mich.

43.56696 83.11189 239 5.0 0.02090 September 32 

04150500 5 Cass River at Cass City, Mich. 43.58419 83.17606 363 11.0 0.03034 September 55 04151500 5 Cass River at 

Frankenmuth, Mich.

43.32780 83.74802 842 64.0 0.07597 September 69 
Table 1–1. Streamgage and flow characteristics used to assess the utility of instantaneous flow measurements for improving index flow estimates. U.S. Geological Survey streamgage number Michigan hydrologic subregion U.S. Geological Survey Strreamgage name Latitude (decimal degrees) Longitude (decimal degrees) Drainage area (square miles) (DArea) Minimum monthly median flow (cubic feet per second) ( ) Q50 Minimum monthly median yield (cubic feet per second per square mile) (Y50) Month of minimum flow Year

of recor

04152238 5 South Branch Tobacco River near Beaverton, Mich. 43.86697 84.54529 152 63.0 0.41366 September 19 04153500 5 Salt River near North Bradley, Mich. 43.70281 84.47056 145 8.5 0.05874 August 38 04154000 5 Chippewa River near Mount Pleasant, Mich. 43.62558 84.70779 409 150 0.36702 August 73 04155000 5 Pine River at Alma, Mich. 43.37948 84.65556 309 80.0 0.25882 August 75 04157500 5 State Drain near Sebewaing, Mich. 43.71196 83.42774 67.3 0 0.00000 August 15 04158000 5 Columbia Drain near Sebewaing, Mich. 43.72724 83.39607 33.9 0 0.00000 August 18 04158500 5 Pigeon River near Owendale, Mich. 43.76363 83.24606 53.3 3.0 0.05629 September 30 04159492 6 Black River near Jeddo, Mich. 43.15253 82.62409 479 22.0 0.04589 September 62 04159900 6 Mill Creek near Avoca, Mich. 43.05447 82.73465 169 6.8 0.04033 September 31 04160000 6 Mill Creek near Abbottsford, Mich. 43.04503 82.61381 184 8.5 0.04620 August 18 04160050 6 Black River near Port Huron, Mich. 42.99003 82.53770 683 19.0 0.02783 September 11 04160570 6 North Branch Belle River at Imlay City, Mich. 43.03031 83.06716 16.1 2.3 0.14286 August 36 04160600 6 Belle River at Memphis, Mich. 42.90086 82.76909 151 13.0 0.08587 September 43 04160800 6 Sashabaw Creek near Drayton Plains, Mich. 42.72003 83.35355 21.0 2.2 0.10476 September 46 04160900 6 Clinton River near Drayton Plains, Mich. 42.66031 83.39022 78.5 14.0 0.17834 August 46 04161000 6 Clinton River at Auburn Hills, Mich. 42.63337 83.22438 123 44 0.35685 August 34 
Table 1–1. Streamgage and flow characteristics used to assess the utility of instantaneous flow measurements for improving index flow estimates.

U.S. 

Geological 

Survey 

streamgage 

number

Michigan 

hydrologic 

subregion

U.S. Geological Survey 

Strreamgage name

Latitude

(decimal

degrees)

Longitude

(decimal

degrees)

Drainage 

area 

(square 

miles)

(DArea)

Minimum 

monthly 

median flow 

(cubic feet 

per second)

( ) Q50

Minimum 

monthly median 

yield (cubic feet 

per second per 

square mile)

(Y50)

Month 

of 

minimum 

flow

Year

of 

recor

04161100 6 Galloway Creek near 

Auburn Heights, Mich.

42.66725 83.20049 17.4 1.6 0.09195 August 32 

04161500 6 Paint Creek near 

Lake Orion, Mich. 

42.76753 83.21994 39.8 9.0 0.22613 August 23 

04161540 6 Paint Creek at 

Rochester, Mich.

42.68837 83.14299 71.8 21.0 0.29248 August 46 

04161580 6 Stony Creek near 

Romeo, Mich.

42.80086 83.09021 23.8 3.9 0.16387 August 41 

04161800 6 Stony Creek near 

Washington, Mich.

42.71531 83.09188 69.1 13.0 0.18813 August 48 

04163400 6 Plum Brook at Utica, Mich. 42.60142 83.07409 16.6 2.8 0.16867 August 40 04163500 6 Plum Brook near Utica, Mich. 42.58365 83.03048 23.8 0.7 0.02857 July 13 04164100 6 East Pond Creek at 

Romeo, Mich.

42.82253 83.02021 20.8 4.9 0.23558 August 47 

04164300 6 East Branch Coon Creek at 

Armada, Mich.

42.84586 82.88493 12.8 0.2 0.01641 August 47 

04164500 6 North Branch Clinton R near 

Mount Clemens, Mich.

42.62920 82.88881 198 12.0 0.06073 September 59 

04164800 6 Middle Branch Clinton River 

at Macomb, Mich.

42.70642 82.95909 41.2 3.5 0.08495 September 19 

04166000 6 River Rouge at 

Birmingham, Mich.

42.54587 83.22354 36.7 6.2 0.16894 September 56 

04166200 6 Evans Ditch at 

Southfield, Mich.

42.45781 83.26743 10.2 1.9 0.18627 September 48 

04166300 6 Upper River Rouge at 

Farmington, Mich.

42.46448 83.36966 17.6 4.1 0.23295 September 48 

04169500 7 Huron River at 

Commerce, Mich.

42.59031 83.48466 49.9 12.0 0.24048 August 30 

04170000 7 Huron River at Milford, Mich. 42.57892 83.62661 139 46.0 0.33141 August 57 04170500 7 Huron River near 

New Hudson, Mich.

42.51253 83.67633 155 54.0 0.34771 August 57 
Table 1–1. Streamgage and flow characteristics used to assess the utility of instantaneous flow measurements for improving index flow estimates. U.S. Geological Survey streamgage number Michigan hydrologic subregion U.S. Geological Survey Strreamgage name Latitude (decimal degrees) Longitude (decimal degrees) Drainage area (square miles) (DArea) Minimum monthly median flow (cubic feet per second) ( ) Q50 Minimum monthly median yield (cubic feet per second per square mile) (Y50) Month of minimum flow Years of recor

04171500 7 South Ore Creek near Brighton, Mich. 42.49781 83.80244 33.3 7.4 0.22222 September 18 04172000 7 Huron River near Hamburg, Mich. 42.46531 83.79994 320 103 0.32177 September 54 04173000 7 Huron River near Dexter, Mich. 42.38615 83.91106 538 120 0.22326 August 29 04173500 7 Mill Creek near Dexter, Mich. 42.30004 83.89856 131 23.0 0.17598 September 42 04174500 7 Huron River at Ann Arbor, Mich. 42.28615 83.73327 747 147 0.19684 August 91 04174800 7 Huron River at Ypsilanti, Mich. 42.24921 83.61244 817 235 0.28750 August 16 04175600 7 River Raisin near Manchester, Mich. 42.16809 84.07606 128 32.0 0.25059 August 33 04175700 7 River Raisin near Tecumseh, Mich. 41.94310 83.94578 266 62.5 0.23532 August 24 04176000 7 River Raisin near Adrian, Mich. 41.90421 83.98050 460 98.5 0.21422 September 46 04176605 7 Otter Creek at La Salle, Mich. 41.86699 83.45354 63.7 1.7 0.02669 September 18 04184500 7 Bean Creek at Powers, Ohio 41.67755 84.23217 205 19.0 0.09255 September 65 
Table 1–2. Standard deviation of the median water yield as a function of the number of simulated instantaneous measurements by streamgage

Michigan 

hydrologic 

subregion

U.S. 

Geological 

Survey 

Streamgage 

number

Number 

of 

eligible 

days

Number of simulated instantaneous measurements

1 2 3 4 5 6 7 10 15 20 

1 04001000 1122 0.16103 0.11380 0.11810 0.09615 0.09782 0.08441 0.08600 0.07107 0.06384 0.0551 04031000 816 0.14543 0.10254 0.09959 0.08120 0.07943 0.06842 0.06775 0.05471 0.04714 0.0401 04031500 463 0.14527 0.10287 0.10414 0.08493 0.08459 0.07323 0.07288 0.05909 0.05037 0.0421 04032000 307 0.14462 0.10238 0.10373 0.08467 0.08453 0.07285 0.07292 0.05936 0.05155 0.0441 04033000 1,191 0.09455 0.06682 0.06508 0.05307 0.05244 0.04550 0.04531 0.03717 0.03262 0.0281 04035000 551 0.08708 0.06162 0.05797 0.04696 0.04541 0.03915 0.03839 0.03076 0.02606 0.0221 04040000 1,110 0.11433 0.08085 0.07763 0.06335 0.06191 0.05354 0.05309 0.04300 0.03707 0.0311 04040500 909 0.18019 0.12743 0.13190 0.10746 0.10905 0.09417 0.09470 0.07752 0.06716 0.0571 04041500 991 0.19973 0.14136 0.12856 0.10389 0.09573 0.08214 0.07850 0.06242 0.05372 0.0461 04043050 1,146 0.10181 0.07195 0.06745 0.05483 0.05416 0.04665 0.04683 0.03811 0.03398 0.0291 04045500 1,275 0.11789 0.08339 0.08142 0.06617 0.06459 0.05580 0.05457 0.04349 0.03608 0.0291 04046000 910 0.11472 0.08131 0.07706 0.06226 0.05891 0.05042 0.04785 0.03673 0.02841 0.0221 04049500 297 0.08485 0.05987 0.05703 0.04643 0.04507 0.03890 0.03829 0.03079 0.02611 0.0221 04055000 281 0.09901 0.06993 0.06632 0.05374 0.05119 0.04413 0.04271 0.03376 0.02793 0.0231 04056500 1,302 0.09808 0.06939 0.06684 0.05439 0.05337 0.04613 0.04588 0.03749 0.03268 0.0281 04057510 914 0.12556 0.08892 0.09097 0.07415 0.07486 0.06468 0.06514 0.05335 0.04681 0.0401 04057800 973 0.15033 0.10631 0.10739 0.08735 0.08628 0.07449 0.07337 0.05880 0.04946 0.0411 04058000 332 0.14386 0.10171 0.10734 0.08729 0.08942 0.07722 0.07824 0.06445 0.05653 0.0481 04058400 462 0.12003 0.08489 0.08443 0.06863 0.06878 0.05947 0.05988 0.04908 0.04352 0.0371 04059000 1,234 0.12208 0.08633 0.08646 0.07054 0.07051 0.06093 0.06118 0.05003 0.04391 0.0371 04059500 1,282 0.14992 0.10614 0.10418 0.08479 0.08374 0.07237 0.07196 0.05847 0.05067 0.0431 04060993 1,378 0.09325 0.06585 0.06578 0.05368 0.05334 0.04613 0.04587 0.03737 0.03193 0.0261 04061500 997 0.11179 0.07910 0.07954 0.06476 0.06529 0.05640 0.05702 0.04697 0.04196 0.0361 04062200 571 0.18298 0.12927 0.13093 0.10661 0.10679 0.09235 0.09265 0.07578 0.06622 0.0562 04096015 396 0.09951 0.07031 0.06656 0.05417 0.05349 0.04621 0.04619 0.03777 0.03302 0.0272 04096405 1,055 0.14676 0.10377 0.09900 0.08041 0.07750 0.06697 0.06542 0.05264 0.04501 0.0382 04096515 1,020 0.15132 0.10711 0.10442 0.08479 0.08156 0.07008 0.06763 0.05317 0.04272 0.0342 04096600 653 0.15051 0.10642 0.10092 0.08153 0.07725 0.06625 0.06304 0.04840 0.03798 0.0312 04096900 770 0.11789 0.08339 0.08227 0.06709 0.06627 0.05735 0.05685 0.04606 0.03967 0.0332 04097170 379 0.08428 0.05955 0.05795 0.04714 0.04860 0.04195 0.04336 0.03619 0.03288 0.0282 04097540 1,079 0.12654 0.08958 0.08338 0.06774 0.06500 0.05610 0.05485 0.04402 0.03750 0.031
Table 1–2. Standard deviation of the median water yield as a function of the number of simulated instantaneous measurements by streamgage Michigan hydro- logic subregion U.S. Geological Survey Streamgage number Number of eligible days Number of simulated instantaneous measurements 1 2 3 4 5 6 7 10 15 20 

2 04099000 1,145 0.12197 0.08631 0.08240 0.06734 0.06556 0.05671 0.05574 0.04474 0.03752 0.0312 04101500 1,218 0.11560 0.08154 0.07918 0.06440 0.06292 0.05446 0.05357 0.04300 0.03645 0.0302 04101800 1,292 0.09871 0.06985 0.06865 0.05608 0.05501 0.04770 0.04701 0.03800 0.03234 0.0272 04102500 1,341 0.09024 0.06381 0.06094 0.04975 0.04841 0.04187 0.04123 0.03325 0.02801 0.0232 04102700 1,116 0.10376 0.07337 0.06785 0.05500 0.05218 0.04489 0.04348 0.03429 0.02847 0.0232 04103010 622 0.08884 0.06284 0.06157 0.05007 0.04891 0.04244 0.04171 0.03366 0.02887 0.0242 04104945 397 0.10355 0.07337 0.07222 0.05880 0.05871 0.05077 0.05097 0.04143 0.03627 0.0302 04105000 1,309 0.12277 0.08664 0.08411 0.06845 0.06627 0.05706 0.05578 0.04415 0.03642 0.0302 04105700 1,109 0.10968 0.07759 0.08089 0.06585 0.06720 0.05823 0.05905 0.04878 0.04316 0.0372 04108600 1,009 0.12366 0.08744 0.08472 0.06901 0.06734 0.05811 0.05705 0.04581 0.03873 0.0322 04108801 1,292 0.12244 0.08667 0.07468 0.05956 0.05337 0.04542 0.04234 0.03245 0.02632 0.0222 04111500 1,341 0.13989 0.09893 0.09334 0.07548 0.07210 0.06205 0.06016 0.04744 0.03936 0.0322 04112000 1,351 0.12837 0.09076 0.08382 0.06763 0.06380 0.05447 0.05233 0.04068 0.03279 0.0262 04112500 1,350 0.12967 0.09161 0.08857 0.07199 0.06981 0.06021 0.05915 0.04707 0.03983 0.0332 04114498 1,209 0.11465 0.08127 0.07478 0.06084 0.05741 0.04944 0.04765 0.03740 0.03065 0.0252 04115000 1,307 0.11841 0.08374 0.07521 0.06068 0.05802 0.05000 0.04893 0.03905 0.03305 0.0272 04116500 748 0.09718 0.06864 0.06542 0.05325 0.05217 0.04490 0.04445 0.03600 0.03086 0.0262 04117000 742 0.11578 0.08189 0.07841 0.06377 0.06211 0.05363 0.05288 0.04251 0.03632 0.0302 04117500 1,302 0.11602 0.08179 0.07868 0.06402 0.06238 0.05377 0.05289 0.04241 0.03568 0.0292 04118000 836 0.12052 0.08539 0.08415 0.06854 0.06899 0.05973 0.06026 0.04956 0.04365 0.0372 04118500 1,099 0.10066 0.07130 0.06946 0.05662 0.05526 0.04783 0.04697 0.03768 0.03175 0.0263 04121000 337 0.11171 0.07914 0.06802 0.05488 0.04837 0.04129 0.03759 0.02803 0.02153 0.0173 04121300 1,324 0.08149 0.05746 0.04975 0.04026 0.03691 0.03164 0.03009 0.02337 0.01896 0.0153 04121900 817 0.09505 0.06714 0.06365 0.05180 0.04984 0.04301 0.04185 0.03331 0.02791 0.0233 04122100 1,126 0.11879 0.08405 0.07623 0.06194 0.05816 0.05021 0.04838 0.03830 0.03179 0.0263 04122200 1,413 0.08082 0.05718 0.05586 0.04551 0.04426 0.03832 0.03769 0.03031 0.02546 0.0213 04122500 1,432 0.07383 0.05219 0.05283 0.04304 0.04304 0.03721 0.03720 0.03031 0.02629 0.0223 04123000 396 0.04437 0.03142 0.03046 0.02478 0.02446 0.02114 0.02097 0.01693 0.01436 0.0123 04123500 378 0.03323 0.02350 0.02401 0.01958 0.01982 0.01713 0.01732 0.01423 0.01252 0.0103 04124000 1,451 0.04759 0.03368 0.03135 0.02560 0.02440 0.02112 0.02052 0.01645 0.01390 0.0113 04124500 584 0.12003 0.08485 0.08478 0.06864 0.06762 0.05798 0.05740 0.04570 0.03873 0.032
Table 1–2. Standard deviation of the median water yield as a function of the number of simulated instantaneous measurements by streamgage

Michigan 

hydrologic 

subregion

U.S. 

Geological 

Survey 

Streamgage 

number

Number 

of 

eligible 

days

Number of simulated instantaneous measurements

1 2 3 4 5 6 7 10 15 20 

3 04125000 93 0.04566 0.03225 0.02324 0.01856 0.01558 0.01323 0.01214 0.00923 0.00764 0.0063 04125500 659 0.04542 0.03210 0.03084 0.02507 0.02451 0.02117 0.02086 0.01681 0.01427 0.0123 04126200 452 0.05010 0.03541 0.03451 0.02806 0.02764 0.02384 0.02369 0.01901 0.01609 0.0131 04127918 1,031 0.10449 0.07386 0.06980 0.05638 0.05414 0.04670 0.04551 0.03630 0.03041 0.0254 04127997 1,349 0.05926 0.04195 0.04198 0.03432 0.03430 0.02970 0.02971 0.02426 0.02099 0.0174 04128990 1,313 0.06907 0.04892 0.04912 0.04001 0.03980 0.03443 0.03422 0.02774 0.02364 0.0204 04133501 675 0.10989 0.07757 0.07566 0.06161 0.06072 0.05256 0.05187 0.04176 0.03515 0.0294 04135000 356 0.15431 0.10933 0.09876 0.07983 0.07457 0.06402 0.06194 0.04919 0.04170 0.0354 04135500 1,011 0.06888 0.04860 0.04407 0.03588 0.03419 0.02943 0.02865 0.02283 0.01904 0.0154 04135600 690 0.08536 0.06032 0.05415 0.04385 0.04122 0.03559 0.03438 0.02726 0.02299 0.0194 04135700 1,270 0.07035 0.04978 0.04107 0.03331 0.03090 0.02663 0.02570 0.02035 0.01698 0.0144 04136500 1,477 0.06465 0.04572 0.03820 0.03102 0.02907 0.02512 0.02430 0.01936 0.01622 0.0134 04137500 644 0.08005 0.05666 0.05496 0.04461 0.04376 0.03781 0.03720 0.03008 0.02557 0.0215 04138000 387 0.08192 0.05785 0.05045 0.04085 0.03745 0.03212 0.03052 0.02376 0.01959 0.0165 04138500 624 0.09124 0.06454 0.05456 0.04410 0.04097 0.03514 0.03406 0.02714 0.02315 0.0195 04139000 340 0.03282 0.02325 0.02218 0.01801 0.01772 0.01514 0.01532 0.01234 0.01134 0.0095 04139500 323 0.04205 0.02982 0.02906 0.02369 0.02328 0.02007 0.01992 0.01618 0.01429 0.0125 04140000 367 0.07156 0.05060 0.04385 0.03533 0.03264 0.02800 0.02674 0.02106 0.01722 0.0145 04140500 657 0.06574 0.04647 0.04427 0.03606 0.03483 0.03013 0.02946 0.02356 0.01996 0.0165 04141000 550 0.13429 0.09513 0.08820 0.07146 0.06709 0.05751 0.05492 0.04246 0.03438 0.0285 04141500 92 0.06246 0.04409 0.04104 0.03312 0.03237 0.02763 0.02728 0.02164 0.01833 0.0145 04142000 1,436 0.07367 0.05205 0.04711 0.03814 0.03639 0.03141 0.03065 0.02457 0.02102 0.0175 04143900 701 0.14420 0.10202 0.09542 0.07761 0.07482 0.06448 0.06312 0.05033 0.04216 0.0345 04144000 657 0.12899 0.09133 0.08959 0.07287 0.07208 0.06246 0.06222 0.05055 0.04370 0.0375 04144500 1,337 0.13964 0.09856 0.10060 0.08208 0.08235 0.07100 0.07139 0.05823 0.05072 0.0435 04146000 1,435 0.13954 0.09845 0.09693 0.07869 0.07759 0.06712 0.06637 0.05388 0.04655 0.0395 04146063 813 0.12602 0.08925 0.08801 0.07151 0.07075 0.06112 0.06069 0.04910 0.04203 0.0355 04147500 1,367 0.10391 0.07361 0.06042 0.04903 0.04548 0.03923 0.03805 0.03057 0.02626 0.0225 04147990 421 0.08399 0.05917 0.04745 0.03844 0.03577 0.03095 0.03030 0.02449 0.02116 0.0175 04148140 1,283 0.12471 0.08824 0.08269 0.06708 0.06457 0.05575 0.05460 0.04367 0.03701 0.0315 04148160 400 0.12332 0.08732 0.07962 0.06443 0.06116 0.05263 0.05077 0.04025 0.03324 0.027
Table 1–2. Standard deviation of the median water yield as a function of the number of simulated instantaneous measurements by streamgage Michigan hydro- logic subregion U.S. Geological Survey Streamgage number Number of eligible days Number of simulated instantaneous measurements 1 2 3 4 5 6 7 10 15 20 

5 04148200 432 0.14684 0.10370 0.09800 0.07954 0.07667 0.06610 0.06459 0.05203 0.04442 0.0375 04148300 418 0.11106 0.07826 0.06966 0.05667 0.05302 0.04563 0.04356 0.03408 0.02733 0.0225 04148440 422 0.11493 0.08129 0.07299 0.05949 0.05744 0.04972 0.04889 0.03945 0.03345 0.0285 04148500 1,411 0.10972 0.07752 0.07078 0.05758 0.05483 0.04715 0.04582 0.03683 0.03105 0.0265 04150000 577 0.08923 0.06317 0.04660 0.03708 0.03178 0.02719 0.02517 0.01942 0.01575 0.0135 04150500 1,225 0.11610 0.08215 0.07344 0.05927 0.05532 0.04754 0.04558 0.03586 0.02983 0.0245 04151500 1,326 0.11699 0.08258 0.07634 0.06200 0.05960 0.05114 0.04993 0.03999 0.03385 0.0285 04152238 592 0.09628 0.06802 0.06270 0.05091 0.04841 0.04180 0.04034 0.03226 0.02676 0.0225 04153500 341 0.07742 0.05480 0.04690 0.03805 0.03628 0.03128 0.03078 0.02474 0.02134 0.0185 04154000 1,411 0.09315 0.06571 0.06416 0.05238 0.05213 0.04509 0.04521 0.03711 0.03294 0.0285 04155000 1,387 0.11288 0.07988 0.07826 0.06372 0.06350 0.05488 0.05487 0.04481 0.03914 0.0335 04158000 80 0.15104 0.10694 0.12140 0.09870 0.10641 0.09148 0.09699 0.08153 0.07708 0.0675 04158500 621 0.11331 0.08015 0.06709 0.05414 0.04890 0.04179 0.03919 0.03010 0.02396 0.0196 04159492 1,342 0.10442 0.07380 0.06039 0.04858 0.04321 0.03683 0.03454 0.02681 0.02220 0.0186 04159900 991 0.09949 0.07036 0.06298 0.05101 0.04886 0.04192 0.04099 0.03276 0.02775 0.0236 04160000 124 0.10015 0.07079 0.06577 0.05292 0.04952 0.04211 0.03969 0.02962 0.02260 0.0176 04160570 1,078 0.14839 0.10480 0.09942 0.08099 0.07829 0.06762 0.06614 0.05331 0.04534 0.0386 04160600 1,319 0.12102 0.08554 0.07770 0.06282 0.05947 0.05102 0.04951 0.03922 0.03296 0.0276 04160800 1,273 0.15895 0.11250 0.10854 0.08832 0.08482 0.07326 0.07086 0.05582 0.04534 0.0376 04160900 1,355 0.15597 0.11038 0.11143 0.09041 0.08864 0.07611 0.07444 0.05849 0.04703 0.0386 04161000 664 0.13654 0.09673 0.09811 0.08015 0.07944 0.06894 0.06836 0.05548 0.04794 0.0406 04161100 886 0.16272 0.11509 0.11060 0.08995 0.08590 0.07387 0.07133 0.05591 0.04568 0.0376 04161500 493 0.13445 0.09513 0.09716 0.07918 0.07981 0.06912 0.06932 0.05688 0.04921 0.0416 04161540 1,313 0.11398 0.08056 0.07762 0.06333 0.06249 0.05403 0.05390 0.04373 0.03784 0.0326 04161580 1,313 0.12246 0.08637 0.08375 0.06816 0.06696 0.05783 0.05720 0.04601 0.03918 0.0336 04161800 1,367 0.14468 0.10251 0.10306 0.08403 0.08412 0.07278 0.07318 0.05970 0.05216 0.0446 04163400 1,054 0.17489 0.12383 0.12752 0.10406 0.10441 0.09029 0.09052 0.07399 0.06448 0.0556 04163500 185 0.09439 0.06652 0.05872 0.04771 0.04520 0.03908 0.03807 0.03030 0.02541 0.0216 04164100 1,304 0.14173 0.10000 0.09927 0.08069 0.07980 0.06911 0.06862 0.05561 0.04754 0.0406 04164300 1,436 0.09868 0.06986 0.05591 0.04492 0.04033 0.03457 0.03250 0.02511 0.02029 0.0166 04164500 1,354 0.13897 0.09846 0.09182 0.07449 0.07144 0.06163 0.06010 0.04781 0.03999 0.0336 04164800 550 0.13096 0.09256 0.08511 0.06862 0.06562 0.05632 0.05493 0.04398 0.03737 0.031
Table 1–2. Standard deviation of the median water yield as a function of the number of simulated instantaneous measurements by streamgage

Michigan 

hydrologic 

subregion

U.S. 

Geological 

Survey 

Streamgage 

number

Number 

of 

eligible 

days

Number of simulated instantaneous measurements

1 2 3 4 5 6 7 10 15 20 

6 04166000 1,242 0.15535 0.10994 0.10792 0.08769 0.08588 0.07401 0.07284 0.05829 0.04886 0.0416 04166200 1,188 0.13727 0.09685 0.09078 0.07380 0.07037 0.06072 0.05900 0.04671 0.03876 0.0326 04166300 1,047 0.15222 0.10784 0.11054 0.09015 0.09053 0.07831 0.07834 0.06408 0.05555 0.0477 04169500 409 0.12833 0.09086 0.09667 0.07878 0.08157 0.07051 0.07207 0.05960 0.05314 0.0457 04170000 1,244 0.12186 0.08612 0.08383 0.06846 0.06725 0.05803 0.05760 0.04669 0.04019 0.0347 04170500 1,189 0.12820 0.09065 0.09007 0.07362 0.07321 0.06330 0.06301 0.05120 0.04375 0.0377 04171500 222 0.15222 0.10742 0.10992 0.08928 0.08815 0.07579 0.07461 0.05920 0.05034 0.0427 04172000 1,166 0.13002 0.09176 0.09100 0.07405 0.07191 0.06201 0.06030 0.04810 0.03996 0.0337 04173000 377 0.10684 0.07545 0.06759 0.05489 0.05228 0.04517 0.04417 0.03555 0.03047 0.0257 04173500 1,037 0.10647 0.07552 0.06906 0.05598 0.05360 0.04621 0.04527 0.03634 0.03138 0.0267 04174500 1,395 0.12975 0.09177 0.08781 0.07138 0.06912 0.05961 0.05853 0.04698 0.03975 0.0337 04174800 449 0.12043 0.08508 0.08191 0.06657 0.06538 0.05652 0.05587 0.04536 0.03918 0.0337 04175600 923 0.15094 0.10662 0.10823 0.08798 0.08790 0.07610 0.07608 0.06189 0.05367 0.0457 04175700 537 0.12321 0.08703 0.07996 0.06483 0.06156 0.05304 0.05111 0.04055 0.03340 0.0277 04176000 1,202 0.12788 0.09052 0.09068 0.07384 0.07325 0.06307 0.06253 0.05035 0.04228 0.0357 04176605 614 0.14687 0.10403 0.10490 0.08538 0.08578 0.07418 0.07464 0.06132 0.05447 0.0477 04184500 810 0.11771 0.08319 0.07346 0.05958 0.05535 0.04782 0.04592 0.03621 0.02989 0.024
Table 1- 3. Average standard deviations of median water yields as a function of measurement intensity by Michigan hydrologic sub

Number of simulated instantaneous measurements Michigan Hydrologic Subregion Ave

ove

subre

1 2 3 4 5 6 7 1 0.131385 0.117701 0.078764 0.089280 0.108090 0.135032 0.128636 0.11

2 0.092913 0.083238 0.055697 0.063159 0.076425 0.095503 0.090942 0.083 0.092024 0.079558 0.052060 0.058399 0.071773 0.092317 0.089178 0.084 0.074860 0.064618 0.042246 0.047398 0.058270 0.075045 0.072544 0.065 0.074068 0.062688 0.040332 0.045451 0.056708 0.073181 0.071434 0.066 0.063932 0.054061 0.034719 0.039203 0.048862 0.063133 0.061662 0.057 0.063627 0.053040 0.033758 0.038324 0.048296 0.062120 0.061076 0.05

10 0.051684 0.042453 0.026778 0.030695 0.038970 0.049836 0.049355 0.0415 0.044771 0.035885 0.022411 0.025970 0.033650 0.042169 0.042388 0.0320 0.038164 0.030233 0.018839 0.021899 0.028545 0.035540 0.036042 0.0350 0.025028 0.019240 0.012000 0.013786 0.018419 0.022519 0.023479 0.02

100 0.018034 0.013741 0.008639 0.009710 0.012859 0.015910 0.016870 0.01
Table 1- 4. Base-flow recession parameters for index streamgages.

U.S.

 Geological 

Survey streamgage 

number 

Base-flow 

events

Attained 

significance

Lower 

confidence limit 

of rYo

Median

rYo

Upper 

confidence interval 

rYo

Lower 

confidence limit 

of tau

Median

tau

04001000 98 3.34 0.483002 0.546629 0.629298 18.6389 20.6585 04031000 62 3.00 0.448095 0.522489 0.562420 17.5324 22.2514 04031500 51 4.89 0.679478 0.733736 0.816257 24.0804 32.7301 04032000 24 2.27 0.570470 0.658792 0.783379 18.6917 27.6530 04033000 108 4.28 0.820272 0.838981 0.857654 57.3214 64.4034 04035000 38 3.36 0.681058 0.711272 0.750226 46.5927 56.8473 04040000 22 1.69 0.651279 0.818127 0.918836 15.9055 29.5217 04040500 124 3.84 0.598121 0.634689 0.699791 16.8858 18.5635 04041500 33 3.51 0.743946 0.862618 0.934153 16.3910 26.2351 04043050 117 4.15 0.690063 0.695454 0.717553 55.9373 64.9492 04045500 109 3.46 0.701353 0.738398 0.797655 38.1566 42.0212 04046000 75 3.70 0.576702 0.618992 0.661284 49.5608 61.3142 04049500 19 1.92 0.726628 0.797301 0.956832 43.9437 74.6480 04055000 19 1.92 0.701841 0.770897 0.947378 42.1947 69.4234 04056000 51 4.89 0.743125 0.779994 0.839601 59.2561 71.0280 04056500 106 4.09 0.768538 0.802955 0.845176 63.2556 71.0039 04057510 94 4.95 0.671898 0.699816 0.766339 39.1462 42.0183 04057800 105 3.13 0.611122 0.650636 0.715771 21.9841 24.9776 04058000 33 3.51 0.568197 0.692847 0.726962 28.8438 35.8778 04058400 45 3.57 0.647821 0.696660 0.761067 37.7426 51.7492 04059000 67 4.98 0.686220 0.712366 0.781436 23.9468 29.4356 04059500 116 3.23 0.521148 0.562709 0.622614 21.2654 23.3582 04060993 95 3.96 0.799921 0.838379 0.855140 66.8279 76.9984 04061500 58 4.79 0.803269 0.843825 0.870507 31.8663 36.7301 04062200 70 4.14 0.529393 0.629267 0.680828 15.4421 16.2633 04096015 45 3.57 0.562585 0.588737 0.627385 47.2838 59.9665 04096405 125 4.87 0.649902 0.687439 0.727149 35.2120 37.5957 04096515 112 4.67 0.524752 0.554696 0.620815 23.1516 24.8803 04096600 59 3.63 0.527473 0.616143 0.734279 30.9879 37.3271 04096900 79 4.22 0.753160 0.781469 0.810631 42.6499 50.5221 04097170 27 1.92 0.730409 0.778235 0.866843 71.5850 95.9012 
Table 1- 4. Base-flow recession parameters for index streamgages. U.S. Geological Survey streamgage number Base-flow events Attained significance Lower confidence limit of rYo Median rYo Upper confidence interval rYo Lower confidence limit of tau Median tau 

04097540 128 4.16 0.726704 0.763332 0.807841 39.2234 42.3626 04099000 31 2.94 0.763038 0.892894 1.035445 20.0465 26.6289 04101500 25 4.33 0.839802 0.924211 1.038887 27.1450 32.3344 04101800 109 3.46 0.859170 0.880146 0.908345 57.9519 63.8285 04102500 94 4.95 0.866953 0.892286 0.916344 64.0849 69.2473 04102700 117 4.15 0.673646 0.697698 0.721626 52.2417 60.9805 04103010 61 3.96 0.785379 0.803085 0.854193 70.8714 80.4017 04104945 38 3.36 0.538382 0.568339 0.590319 32.8699 38.5286 04105000 100 3.52 0.557284 0.598272 0.620611 33.0650 40.3533 04105700 97 4.17 0.925024 0.947272 0.979495 45.7398 52.4309 04108600 112 4.67 0.594491 0.621802 0.644489 47.3755 50.2165 04108801 91 3.54 0.278517 0.297660 0.316535 26.4110 30.0517 04110000 39 2.37 0.369166 0.406994 0.452727 44.2779 56.7189 04111500 111 3.63 0.257629 0.281818 0.332400 21.5977 24.3187 04112000 108 4.28 0.211287 0.235244 0.261780 20.1172 21.7884 04112500 129 3.42 0.377884 0.397825 0.425668 30.6599 37.4381 04114498 111 3.63 0.433943 0.461787 0.485582 36.3960 41.1107 04115000 125 4.87 0.343672 0.363153 0.394382 24.2164 27.2011 04116500 25 4.33 0.633943 0.694560 0.861086 32.6507 41.4245 04117000 58 4.79 0.601151 0.626004 0.658613 60.6439 67.8450 04117500 125 4.87 0.567363 0.590223 0.608695 45.8925 51.0001 04118000 27 1.92 0.658775 0.754606 0.839950 19.7161 24.7076 04118500 105 3.13 0.716853 0.740981 0.769832 57.7796 64.1379 04121000 18 3.09 0.590125 0.646933 0.882204 61.1240 76.3426 04121300 100 3.52 0.560966 0.582136 0.601953 89.5641 105.6101 04121900 63 4.30 0.728472 0.752692 0.789414 49.8366 64.9913 04122100 62 3.00 0.564544 0.622980 0.675024 34.1220 38.7786 04122200 106 4.09 0.828181 0.843891 0.869350 98.6235 111.5726 04122500 103 4.82 0.856050 0.883647 0.892394 100.7904 123.5606 04123000 73 3.44 0.940891 0.955533 0.977004 166.4333 197.7745 04123500 59 3.63 1.133718 1.146177 1.159290 334.9381 352.3355 04124000 91 3.54 0.993975 1.003933 1.021019 162.3519 210.5108 
Table 1- 4. Base-flow recession parameters for index streamgages.

U.S.

 Geological 

Survey streamgage 

number 

Base-flow 

events

Attained 

significance

Lower 

confidence limit 

of rYo

Median

rYo

Upper 

confidence interval 

rYo

Lower 

confidence limit 

of tau

Median

tau

04124500 50 3.28 0.426252 0.433725 0.510510 33.8514 49.8196 04125000 24 2.27 0.620494 0.640645 0.672955 63.5432 117.9455 04125500 35 4.10 0.951765 0.966906 0.996644 176.8408 231.1826 04126200 35 4.10 0.874727 0.888904 0.914592 144.0475 249.0355 04127918 78 3.08 0.614226 0.652666 0.686708 47.4254 55.9419 04127997 67 4.98 0.962550 0.985580 1.015928 95.3136 112.4758 04128990 26 2.90 0.982259 0.997077 1.018910 89.9831 140.9850 04133501 52 3.65 0.715664 0.740176 0.805400 65.4767 84.1466 04135000 1 100.00 1.002797 1.002797 1.002797 6.4630 6.4630 04135500 78 3.08 0.800509 0.812771 0.832210 96.7878 131.7077 04135600 63 4.30 0.702798 0.719072 0.740230 86.0927 108.3531 04135700 98 3.34 0.593227 0.602980 0.615988 100.7027 118.6502 04136500 43 3.15 0.786930 0.806003 0.842991 66.5729 88.1422 04137500 12 3.86 0.758783 0.813092 0.907695 22.0867 43.8408 04138000 40 3.85 0.650186 0.670375 0.685881 74.7462 93.6872 04138500 39 2.37 0.353523 0.393135 0.437380 31.4091 42.6136 04139000 32 2.01 1.129300 1.140576 1.165089 157.5767 346.8487 04139500 33 3.51 1.061818 1.074644 1.096418 135.7355 197.4808 04140000 22 1.69 0.538125 0.554761 0.617382 46.3023 59.3169 04140500 47 4.00 0.868647 0.876180 0.897961 60.6363 80.8578 04141000 41 2.75 0.169938 0.201503 0.234199 15.8593 21.1600 04141500 13 2.25 0.704042 0.718135 0.772268 43.9723 61.2213 04142000 89 3.34 0.692363 0.713763 0.734539 64.6641 75.0414 04143900 56 4.40 0.554992 0.612372 0.681827 21.1937 33.9247 04144000 34 2.43 0.505303 0.577050 0.698170 24.4487 33.1772 04144500 89 3.34 0.478131 0.507430 0.559247 26.7874 32.7474 04145500 33 3.51 0.102605 0.150521 0.223135 5.0849 8.6671 04146000 127 3.28 0.419966 0.450410 0.480807 19.0131 21.5966 04146063 63 4.30 0.533099 0.608216 0.660450 21.2629 25.4957 04147500 104 3.90 0.523054 0.544147 0.571029 28.5675 37.6635 04147990 35 4.10 0.335628 0.357206 0.374552 34.9159 49.4544 
Table 1- 4. Base-flow recession parameters for index streamgages. U.S. Geological Survey streamgage number Base-flow events Attained significance Lower confidence limit of rYo Median rYo Upper confidence interval rYo Lower confidence limit of tau Median tau 

04148140 88 4.22 0.374868 0.415290 0.465033 21.4115 23.9547 04148160 27 1.92 0.067678 0.119109 0.169963 5.5394 10.5667 04148200 34 2.43 0.372348 0.411545 0.543801 18.6920 23.3498 04148300 24 2.27 0.206442 0.321323 0.451109 16.5152 20.4521 04148440 29 2.41 0.270691 0.359427 0.449336 11.0525 15.1952 04148500 53 2.70 0.505167 0.571274 0.645807 20.4988 25.3080 04150000 44 4.88 0.169150 0.192955 0.213191 26.7781 36.1641 04150500 119 4.33 0.219755 0.239991 0.272774 17.9604 21.1901 04151500 119 4.33 0.337913 0.361777 0.400801 26.0523 28.4006 04152238 34 2.43 0.642240 0.657580 0.680344 51.1028 66.9182 04153500 86 3.99 0.264423 0.279119 0.289738 39.8810 47.2435 04154000 114 4.87 0.661305 0.674792 0.702158 57.0459 62.6003 04155000 47 4.00 0.548834 0.581255 0.701086 22.4104 28.0769 04157500 46 2.59 0.038547 0.071546 0.129938 4.7286 6.9118 04158000 50 3.28 0.054313 0.073249 0.131280 4.5969 7.5590 04158500 41 2.75 0.236818 0.264524 0.308933 14.9540 21.6309 04159492 118 3.38 0.254759 0.270237 0.291754 22.2436 26.6894 04159900 81 4.48 0.263603 0.277679 0.315992 23.9942 26.8720 04160000 44 4.88 0.230021 0.258253 0.283705 25.5116 31.2106 04160050 13 2.25 0.155340 0.199311 0.287650 11.7208 14.5183 04160570 70 4.14 0.367134 0.406298 0.456220 16.8720 20.3360 04160600 101 4.60 0.321488 0.345863 0.360782 26.9577 31.8084 04160800 108 4.28 0.385942 0.423073 0.452196 24.1915 27.6115 04160900 80 3.30 0.480556 0.527135 0.651762 21.6157 25.6273 04161000 23 3.47 0.564640 0.656908 0.843072 15.3164 20.3012 04161100 45 3.57 0.336247 0.385676 0.467745 18.6941 24.1681 04161500 41 2.75 0.506681 0.614673 0.734003 23.5259 27.6929 04161540 83 4.75 0.538669 0.584676 0.632497 32.2311 39.1186 04161580 58 4.79 0.427571 0.474596 0.562940 20.3628 31.6945 04161800 86 3.99 0.518170 0.551143 0.617837 18.2987 20.7397 04163400 53 2.70 0.376712 0.434570 0.519663 11.4057 15.3426 
Table 1- 4. Base-flow recession parameters for index streamgages.

U.S.

 Geological 

Survey streamgage 

number 

Base-flow 

events

Attained 

significance

Lower 

confidence limit 

of rYo

Median

rYo

Upper 

confidence interval 

rYo

Lower 

confidence limit 

of tau

Median

tau

04163500 23 3.47 0.096705 0.163732 0.204890 8.1369 14.3396 04164100 103 4.82 0.526775 0.577908 0.655446 24.1752 28.3403 04164300 71 3.19 0.174865 0.192491 0.263539 11.5435 13.7107 04164500 106 4.09 0.274002 0.303158 0.325461 16.6803 18.2998 04164800 30 4.28 0.255525 0.318423 0.388076 20.2289 26.5707 04166000 80 3.30 0.443979 0.474117 0.521379 26.1733 31.7530 04166200 47 4.00 0.369327 0.395604 0.445477 27.3076 36.9693 04166300 62 3.00 0.497637 0.556876 0.600234 24.3636 28.7248 04169500 28 3.57 0.524439 0.561612 0.684833 28.8345 40.4468 04170000 83 4.75 0.655854 0.690095 0.752802 31.4510 35.4578 04170500 88 4.22 0.671783 0.713902 0.789242 29.9405 35.0958 04171500 37 4.70 0.481984 0.557099 0.640337 25.0973 30.7579 04172000 119 4.33 0.631822 0.674905 0.721384 37.6269 41.7419 04173000 17 4.90 0.460651 0.499871 0.698957 32.3792 49.7533 04173500 100 3.52 0.446603 0.478743 0.505545 59.3894 73.4198 04174500 28 3.57 0.507609 0.611135 0.717864 18.4807 24.7191 04174800 2 50.00 0.504461 0.577293 0.650124 16.3412 59.1641 04175600 85 2.95 0.589416 0.647665 0.692521 25.4977 29.2724 04175700 35 4.10 0.520763 0.575401 0.726165 34.3525 39.3302 04176000 96 3.15 0.518386 0.573385 0.608250 33.4850 36.9313 04176605 55 3.00 0.189307 0.221861 0.270466 14.0255 15.9844 04184500 84 3.75 0.349484 0.400590 0.455913 25.4209 31.2190 

Appendix 2. Example computations of index flow 

estimates with simulated instantaneous measurements 

IndexFlow_MiscQ

User required input

Parameters in index regression equation

Parameters in augmented index regression equation

Specify run parameters and labels

Load data for index flow simulation

Analysis of instantaneous flow measurements

Compute water yields and flows with the Discrete Estimator

Compute water yields and flows with the Integrated Estimator

Compute water yields and flows with the Site Estimator

List results for all estimators to console
72 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months

Matlab Code 2- 1. IndexFlow_MiscQ.

Index Flow Estimation with Index Regression and Instantaneous Flows. Index flow is a streamflow characteristic defined as the 

median flow during the summer month (July, August, September) of minimum flow in Michigan. Index flows are used to help 

regulate new or increased water withdrawals exceeding 100,000 gallons per day in any consecutive 30-day period from causing 

adverse impacts on characteristic fish populations in Michigan.

The techniques used in this code are based on the Index regression equation developed by Hamilon, 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. In addition, estimation using 

instantaneous flow measurements is based on techniques developed by Holtschlag, D.J., 2010, Improving regression estimates of 

index flow, describing the median streamflows for the summer month of lowest flow of streams in Michigan, by use of instantaneous streamflow measurements: U.S. Geological Survey Scientific Investigations Report 2010-5236, xx p. The following code 

is considered experimental and may not return correct results. User accepts responsibility for accuracy, applicability, and utility 

of all results as the terms of use.

clear; clc;
Matlab Code 2–2. User required input 73

Matlab Code 2–2. User required input

Enter site specific value for regression variables

Mhs = 2; % Michigan hydrologic subregion.

SiteNo = ‘04112500’; % Given streamgage number.

SiteName = ‘Red Cedar River at East Lansing, Mich.’; % Given streamgage name.

DArea0 = 344.0 ;

L_Trans = 0.0 ;

H_Trans = 13.7 ;

Forest = 14.16626 ;

Precip = 32.6 ;

A_Soils = 11.0 ;

D_Soils = 14.0 ;

%

% Enter instantaneous flow for Discrete, Integrated, and Site Estimators

nMeasQ = 3;

% Preallocate variables

DateNo = NaN(nMeasQ,1); InstQmea = NaN(nMeasQ,1);

% Enter nMeasQ measurements of date and flow magnitude

DateNo(1) = datenum(‘08-Sep-2001’,’dd-mmm-yyyy’); InstQmea(1) = 37.0;

DateNo(2) = datenum(‘25-Sep-1979’,’dd-mmm-yyyy’); InstQmea(2) = 18.0;

DateNo(3) = datenum(‘20-Sep-2001’,’dd-mmm-yyyy’); InstQmea(3) = 70.0;

%

% Augmented Index regression explanatory variables

% For each baseflow recession period 1 through nRecess analyzed

% enter the corresponding parameter estimates of the rY0vec is a vector...

% of one or more initial root water yields

START SAMPLE DATA (Expected format shown with sample data.).

rY0vec = [0.38275 0.596658 0.39783]; % Vector of initial yields during baseflow 

recession.

tauvec = [25.2610 64.5668 37.4378]; % Vector of time constants during baseflow 

recession.

% END SAMPLE DATA.
74 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months

Matlab Code 2–3. Parameters in index regression equation

Explanatory vector

x0 = [1 L_Trans H_Trans Forest Precip A_Soils D_Soils]’;

% Parameter estimates

bIntcp = -0.541982;

bL_Trans = -0.00136258;

bH_Trans = 0.00204796;

bForest = 0.00402190;

bPrecip = 0.0236424;

bA_Soils = 0.00225536;

bD_Soils = 0.00162107;

% Parameter vector

bvec = [bIntcp bL_Trans bH_Trans bForest bPrecip bA_Soils bD_Soils];

%

% Inverse of the X’X matrx needed to compute prediction uncertainty

InvXpX = [ 2.38035e+0 -8.42611e-4 1.43560e-3 7.23794e-4 -7.11625e-2 -3.54982e-3 

-3.50692e-3; ...

 -8.42611e-4 1.90162e-5 9.13024e-6 -1.04607e-6 8.79202e-6 -2.67491e-6 

-9.25476e-7; ...

 1.43560e-3 9.13204e-6 1.70298e-5 4.69128e-6 -6.03695e-5 -1.60371e-5 

-8.67804e-6; ...

 7.23794e-4 -1.04607e-6 4.69128e-6 1.93996e-5 -2.49764e-5 -1.74926e-5 

-2.33578e-5; ...

 -7.11625e-2 8.79202e-6 -6.03695e-5 -2.49764e-5 2.17903e-3 1.09528e-4 

8.13390e-5; ...

 -3.54982e-3 -2.67491e-6 -1.60371e-5 -1.74926e-5 1.09528e-4 3.85354e-5 

2.70690e-5; ...

 -3.50692e-3 -9.25476e-7 -8.67801e-6 -2.33578e-5 8.13390e-5 2.70690e-5 

8.42089e-5];

%

IndxrY50_mse = 0.015320;

IndxrY50_rmse = sqrt(IndxrY50_mse);

% Index root yields

IndxrY50_hat = bvec * x0;

IndxrY50_var = IndxrY50_mse * (1 + x0’*InvXpX*x0);

IndxrY50_sep = sqrt(IndxrY50_var);

IndxrY50_l95 = IndxrY50_hat + tinv(.025,147-7)*IndxrY50_sep;

IndxrY50_u95 = IndxrY50_hat + tinv(.975,147-7)*IndxrY50_sep;

% Index flows

IndxQ50_hat = IndxrY50_hat^2 * DArea0;

IndxQ50_l95 = IndxrY50_l95^2 * DArea0;

IndxQ50_u95 = IndxrY50_u95^2 * DArea0;

%

% Index regression residuals

IndxrY50_res = [-0.007576813 -0.193160541 0.000288932 -0.076300339 0.084963459 ...

0.002973825 0.046814793 -0.124385304 0.061876491 0.201011933 -0.003608545 ...

-0.023682698 0.097397392 0.107360353 0.019729621 0.135140746 -0.169003059 ...

-0.031989724 -0.008403485 -0.096016247 -0.093484089 -0.203537087 0.212716133 ...

-0.021016441 -0.057917802 0.055962715 0.00068285 -0.070625215 -0.045399968 ...

0.008741081 0.049519021 0.050241785 0.082269415 0.104325843 0.137093468 ...

0.115758648 0.000997975 0.247519925 -0.009711837 -0.011725471 0.243007414 ...

-0.001314207 -0.180446343 -0.141835772 -0.116096304 -0.14654114 -0.028932584 ...

-0.016960894 -0.036515331 -0.007714564 0.123701013 0.065270498 0.109738346 ...
Matlab Code 2–3. Parameters in index regression equation 75

0.113171529 -0.194725929 -0.171603729 -0.041337003 0.057331834 -0.004784252 ...

-0.116128492 0.001748437 0.200130288 0.082559296 -0.195076717 -0.080462014 ...

0.164080737 -0.120009651 -0.017182485 0.094138057 0.184926391 0.070738178 ...

0.00068102 -0.103002066 -0.19917193 -0.309125987 -0.127259386 -0.047153008 ...

0.078919278 -0.038588799 0.508387742 0.436465315 0.006212689 0.241435973 ...

-0.051198964 0.09215994 0.099394298 0.107441485 0.041780081 0.026585722 ...

-0.123052071 -0.02621801 0.122248708 0.081317958 -0.065194777 -0.029654524 ...

0.001412433 -0.126592344 -0.093203984 -0.083599741 0.067769137 -0.184076643 ...

-0.177106108 -0.099015733 0.008299128 -0.035193265 -0.038619136 -0.071096618 ...

-0.11227454 -0.092503758 0.00888174 -0.023826084 -0.08363744 -0.068947764 ...

-0.089168557 0.142600675 0.094726854 -0.265190342 -0.116936969 0.103301189 ...

-0.122586974 -0.024504426 0.103048046 0.012930496 -0.000667989 0.173564612 ...

-0.073877409 0.113991476 -0.045810706 0.044647904 0.066259253 0.05151493 ...

0.255807281 0.131717401 -0.106459947 0.014539439 0.028374572 -0.047578852 ...

0.0303895 -0.063527979 -0.039972854 -0.064027292 0.036581922 -0.077301413 ...

-0.034568378 -0.013906751 -0.128537752 -0.078167275 ];

%

% Compute the Smearing estimate of flow

IndxQ50_sl95 = 0;

IndxQ50_shat = 0;

IndxQ50_su95 = 0;

for i = 1:147,

 IndxQ50_sl95 = IndxQ50_sl95 + (IndxrY50_l95 + IndxrY50_res(i)).^2;

 IndxQ50_shat = IndxQ50_shat + (IndxrY50_hat + IndxrY50_res(i)).^2;

 IndxQ50_su95 = IndxQ50_su95 + (IndxrY50_u95 + IndxrY50_res(i)).^2;

end

IndxQ50_shat = IndxQ50_shat/147 * DArea0;

IndxQ50_sl95 = IndxQ50_sl95/147 * DArea0;

IndxQ50_su95 = IndxQ50_su95/147 * DArea0;

%

ndash = 82;

fprintf(1,’Index water yield and flow for site %s %s \n’,SiteNo ,SiteName);

fprintf(1,[repmat(‘-’,1,ndash) ‘\n’]);

fprintf(1,’ Lower Upper 

StdErr \n’);

fprintf(1,’ESTIMATOR 95 Percent Expected 95 Percent 

Predict \n’);

fprintf(1,[repmat(‘-’,1,ndash) ‘\n’]);

fprintf(1,’Index Water Yield (CFSM) %10.3f %10.3f %10.3f %8.5f\n’,...

 IndxrY50_l95,IndxrY50_hat,IndxrY50_u95,IndxrY50_sep);

fprintf(1,’Conventional Index Flow (CFS) %10.1f %10.1f %10.1f \n’,IndxQ50_l95, 

IndxQ50_hat, IndxQ50_u95);

fprintf(1,’Smearing Index Flow (CFS) %10.1f %10.1f %10.1f \n’,IndxQ50_

sl95,IndxQ50_shat,IndxQ50_su95);

fprintf(1,[repmat(‘-’,1,ndash) ‘\n’]);

%

Index water yield and flow for site 04112500 Red Cedar River at East Lansing, Mich. 

----------------------------------------------------------------------------------

 Lower Upper StdErr 

ESTIMATOR 95 Percent Expected 95 Percent Predict 

----------------------------------------------------------------------------------

Index Water Yield (CFSM) 0.113 0.361 0.610 0.12569

Conventional Index Flow (CFS) 4.4 44.9 127.9 

Smearing Index Flow (CFS) 9.4 49.9 132.9 

----------------------------------------------------------------------------------
76 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months

Matlab Code 2–4. Parameters in augmented index regression equation 

Explanatory vector QA sample recession parameter to see if they differ from the default.

QARecess = 0; % QA recession parameter on by default

if median(rY0vec) == median([0.38275 0.596658 0.39783]) && ...

 log10(median(tauvec)) == log10(median([25.2610 64.5668 37.4378])) && ...

 QARecess == 1

 fprintf(1,’ERROR! Enter site specific recession parameters. \n’);

 nRecess = 0;

 mrY0 = []; lmtau = [];

else

 nRecess = length(rY0vec);

 mrY0 = median(rY0vec); lmtau = log10(median(tauvec));

end

%

AugmX0 = [1 IndxrY50_hat mrY0 lmtau]’;

% Parameter estimates

bAugmIntcp = -0.22994;

bAugmIndxrY50 = 0.01082;

bAugmrY0 = 0.81126;

bAugmlmtau = 0.16590;

% Parameter vector

bAugmvec = [bAugmIntcp bAugmIndxrY50 bAugmrY0 bAugmlmtau];

%

% Inverse of the X’X matrx needed to compute prediction uncertainty

AugInvXpX = [ 0.202449 0.054157 0.023014 -0.147576;...

 0.054157 0.694465 -0.380471 -0.119923;...

 0.023014 -0.380471 0.459334 -0.059837;...

 -0.147576 -0.119923 -0.059837 0.152903];

%

AugmrY50_mse = 0.001424;

AugmrY50_rmse = sqrt(AugmrY50_mse);

%

if nRecess >= 1

 AugmrY50_hat = bAugmvec * AugmX0;

 AugmrY50_var = AugmrY50_mse * (1 + AugmX0’*AugInvXpX*AugmX0);

 AugmrY50_sep = sqrt(AugmrY50_var);

 AugmrY50_l95 = AugmrY50_hat + tinv(.025,147-7)*AugmrY50_sep;

 AugmrY50_u95 = AugmrY50_hat + tinv(.975,147-7)*AugmrY50_sep;

 % Index flows

 AugmQ50_hat = AugmrY50_hat^2 * DArea0;

 AugmQ50_l95 = AugmrY50_l95^2 * DArea0;

 AugmQ50_u95 = AugmrY50_u95^2 * DArea0;

end

%
Matlab Code 2–5. Specify run parameters and labels 77

Matlab Code 2–5. Specify run parameters and labels

MinStaEst = 5; % Minimum number of active streamgages required for estimation

DV_cv = 0.01; % Coefficient of variation for approximating daily flows

% by a short-interval flow measurement

% Indices of months to use in estimation, Jan = 1;

MonthLab = {‘Jan’,’Feb’,’Mar’,’Apr’,’May’,’Jun’,’Jul’,’Aug’,’Sep’,’Oct’,’Nov’,’Dec’};

MhsName = {‘Upper Peninsula’,’Mhs0405’,’Mhs0406’,’Mhs0407’,’Mhs0408’,...

 ‘Mhs 04090001-0004’,’Mhs 04090005 0410’}; % Michigan subregion names

nmJndxP = NaN(nMeasQ,1); NumObs = NaN(nMeasQ,1);

Matlab Code 2–6. Load data for index flow simulation

Spreadsheet Qdv1960_2009.xlsx!Numeric has the daily values MLDate = XLDate + 693960; 

save(‘Qdv1960_2009’,’MLDate’,’Qdv’);

whos -file Qdv1960_2009;

load Qdv1960_2009 % Matlab version of spreadsheet

% whos -file StdMedrYdv;

% StdMedrYdv was computed w/o lower bound and 1.96s upper bound

% REMEMBER: StdMedrYdv has 15 fields:

% StaSeq Mhs StaNum Stdev(1:7,10,15,20,50,100)

load StdMedrYdv

% Eliminate streamgages w/o daily values

MeanQdv = nanmean(Qdv);

xBarNdx = find(~isnan(MeanQdv));

% [~, mo, ~, ~, ~, ~] = datevec(MLDate);

% %GMo is short for Good Month

% GMo = find(ismember(mo,MonthInd)==1 & ...

% MLDate<datenum([2008 10 1 0 0 0]));

% MLGMoDate = MLDate(GMo);

% nGMo = length(GMo);

% QdvGMo = Qdv(GMo,:);

% StaNumber is the streamgage number

% StaName is the streamgage name

% Q50 is the Index flow

% DArea is the drainage area at each of the streamgages

% rY50 is the sqrt of the index yield

% IndxrY50_h is the regression estimate of rY50

% IndxrY50_r is the residual rY50 - IndxrY50_h

% IndxrY50_s is the standard error of prediction from the index regression

% YrsDV is the number of years of daily values at the streamgage

% whos -file IndexReg147

load IndexReg147

Q50147 = Q50;

DArea147 = DArea;

 MhsJndx = intersect(xBarNdx,find(MiSubRegion == Mhs))’;

nMhsJndx = length(MhsJndx);

MhsQdv = Qdv(:,MhsJndx);

% Eliminate streamgages w/o daily values or outside Mhs

MhsStaNum = StaNumber(MhsJndx);

MhsStaNam = StaName(MhsJndx);

MhsQ50 = Q50( MhsJndx); % Index flow
78 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months

MhsDArea = DArea( MhsJndx); % Drainage area

MhsrY50 = SqrtY50( MhsJndx); % Square root of index water yield

MhsIndxrY50_h = SqrtY50hat(MhsJndx); % Index regression estimate of water yield

MhsIndxrY50_r = SqrtY50res(MhsJndx); % Index regression residual of water yield

MhsIndxrY50_s = SqrtY50SEP(MhsJndx); % Standard error of prediction

MhsYrsDV = YrsDV( MhsJndx);

% Specify flow max and min to be considered an Index flow estimator

MhsIndxQ50_h = MhsIndxrY50_h.^2.*MhsDArea;

%

Find if site (SiteNo ) is an index streamgage

 Jndx0 = strmatch(SiteNo ,MhsStaNum);

% if Jndx0 is a valid index, then drop the target site from the aux

% JndxP = setdiff(MhsJndx,MhsJndx(Jndx0));

% nJndxP = length(JndxP);

% MhsQdv = Qdv(:,JndxP);

%

ry50m_h = NaN(nMeasQ,1);

ry50m_v = NaN(nMeasQ,1);

Matlab Code 2–7. Analysis of instantaneous flow measurements

for m = 1:nMeasQ

 % Find the index of the measurement date

 Indx0 = find(DateNo(m)==MLDate);

 % Find days with valid flow data at Mhs streamgages

 mJndx = find(~isnan(Qdv(Indx0,MhsJndx))==1);

 mJndxP = setdiff(mJndx,Jndx0);

 mMhsrY50 = MhsrY50( mJndxP);

 mMhsQdv = Qdv(Indx0, MhsJndx( mJndxP))’;

 mMhsrYdv = sqrt(mMhsQdv ./ MhsDArea(mJndxP));

 %

 if(~isempty(Jndx0))

 mInstQmea = Qdv(Indx0, MhsJndx( Jndx0));

 mInstrYmea = sqrt(mInstQmea ./ MhsDArea( Jndx0));

 else

 mInstQmea = InstQmea(m);

 mInstrYmea = sqrt(mInstQmea ./ DArea0);

 end

 %

 % List daily flows at auxiliary streamgages

 ndash = 72;

 fprintf(1,[‘\n’ repmat(‘-’,1,ndash) ‘\n’]);

 fprintf(1,’Relation between daily mean and index flow on %s in Mhs %u \n’,...

 datestr(DateNo(m)),Mhs);

 fprintf(1,’ at %s %s for measurement%3u. \n’,SiteNo ,SiteName,m);

 fprintf(1,’ DArea Q50 rY50 Qdv 

rYdv\n’);

 fprintf(1,’ jSta iJndxP Streamgage (mi^2) (cfs) (cfsm) (cfs) 

(cfsm)\n’);
Matlab Code 2–7. Analysis of instantaneous flow measurements 79

 fprintf(1,[repmat(‘-’,1,ndash) ‘\n’]);

 for jSta = 1:length(mJndxP)

 fprintf(1,’%4u %4u %s %10.1f %10.1f %10.4f %8.1f %8.4f \n’,...

 jSta, mJndxP(jSta),...

 MhsStaNum{mJndxP(jSta)},...

 MhsDArea( mJndxP(jSta)), ...

 MhsDArea( mJndxP(jSta)) * mMhsrY50(jSta)^2, ...

 mMhsrY50( jSta )^2,...

 mMhsQdv( jSta ), ...

 mMhsrYdv( jSta ));

 end

 fprintf(1,[repmat(‘-’,1,ndash) ‘\n\n’]);

 % Compute Index regression estimates

 nmJndxP(m) = length(mJndxP);

 NumObs(m) = nmJndxP(m);

 xbar = mean(mMhsrYdv);

 X = [ones(size(mMhsrYdv)),mMhsrYdv];

 [b,bint,r,rint,stats] = regress(mMhsrY50,X);

 fprintf(1,’Regression results for measurement %u for %s \n’,...

 m,datestr(DateNo(m)));

 fprintf(1,’ nReg R2-value F-Stat p-value RMSE b_Intercept b_Slope \n’);

 fprintf(1,’ %3u %6.4f %6.2f %6.5f %7.4f %8.4f %8.4f \n’,...

 mJndxP(m),stats(1:3),sqrt(stats(4)),b);

 Sxx = sum((mMhsrYdv - xbar).^2);

 x2 = (mInstrYmea - xbar).^2 ;

 ry50m_h(m) = b(1)+b(2)*mInstrYmea;

 rYRes = mMhsrY50 - (b(1)+b(2)*mMhsrYdv);

 ry50m_v(m) = var(rYRes)*(1 + 1/length(mMhsrYdv)+x2/Sxx)+ ...

 (DV_cv*mInstrYmea)^2;

 %

 % Plot daily and index flows for each instantaneous flow measurement

 figure(5); clf(5);

 plot(mMhsrYdv,mMhsrY50,’bs’,...

 ‘MarkerFaceColor’,’b’,’MarkerSize’,4);

 hold on

 minx = min(mMhsrYdv);

 maxx = max(mMhsrYdv);

 plot([minx,maxx],b(1)+b(2)*[minx,maxx],’k-’);

 xlabel(‘ROOT DAILY WATER YIELD, IN CFSM’);

 ylabel(‘ROOT INDEX WATER YIELD, IN CFSM’);

 title([SiteNo ,’ ‘, SiteName,...

 ‘ on ‘,datestr(DateNo(m))]);

 text(minx+0.05*(maxx-minx),minx+.4,[‘R^2= ‘,num2str(stats(1),’%6.4f’)]);

 xmin = min(get(gca,’XLim’)); ymin = min(get(gca,’YLim’));

 plot([mInstrYmea,mInstrYmea],[ymin,ry50m_h(m)],’r-’);

 plot([mInstrYmea,xmin],[ry50m_h(m),ry50m_h(m)],’r-’);

 plot(mInstrYmea,ry50m_h(m),’r*’,’MarkerSize’,5);

 legend(‘Auxilary streamgages’,’Daily and Index flow regression’,...

 [SiteNo ‘ instantaneous flow’],’Location’,’NorthWest’);

 hold off

 pause

end
80 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months

------------------------------------------------------------------------

Relation between daily mean and index flow on 08-Sep-2001 in Mhs 2 

 at 04112500 Red Cedar River at East Lansing, Mich. for measurement 1. 

 DArea Q50 rY50 Qdv rYdv

 jSta iJndxP Streamgage (mi^2) (cfs) (cfsm) (cfs) (cfsm)

------------------------------------------------------------------------

 1 1 04096015 80.8 21.0 0.2599 20.0 0.4975 

 2 2 04096405 201.0 61.0 0.3035 56.0 0.5278 

 3 3 04096515 48.7 7.9 0.1622 6.5 0.3653 

 4 7 04097540 106.6 46.0 0.4315 50.0 0.6849 

 5 8 04099000 1879.4 850.0 0.4523 966.0 0.7169 

 6 9 04101500 3666.0 1850.0 0.5046 1980.0 0.7349 

 7 10 04101800 251.5 177.0 0.7038 157.0 0.7901 

 8 11 04102500 389.9 262.0 0.6720 229.0 0.7664 

 9 12 04102700 83.5 35.0 0.4192 34.0 0.6381 

 10 13 04103010 269.5 147.0 0.5455 159.0 0.7681 

 11 14 04104945 48.3 15.0 0.3106 16.0 0.5756 

 12 15 04105000 274.0 70.0 0.2555 70.0 0.5054 

 13 16 04105700 36.8 31.0 0.8424 27.0 0.8566 

 14 17 04108600 65.1 22.0 0.3379 33.0 0.7120 

 15 18 04108801 66.9 4.3 0.0643 26.0 0.6234 

 16 19 04111500 16.3 0.9 0.0552 0.6 0.1837 

 17 20 04112000 10.4 0.2 0.0231 0.1 0.1201 

 18 23 04115000 420.1 30.0 0.0714 31.0 0.2716 

 19 25 04117000 7.8 2.8 0.3590 3.5 0.6699 

 20 26 04117500 410.5 109.0 0.2655 123.0 0.5474 

 21 28 04118500 257.2 127.0 0.4938 185.0 0.8481 

------------------------------------------------------------------------

Regression results for measurement 1 for 08-Sep-2001 

 nReg R2-value F-Stat p-value RMSE b_Intercept b_Slope 

 1 0.7980 75.08 0.00000 0.0958 0.0380 0.8898 

------------------------------------------------------------------------

Relation between daily mean and index flow on 25-Sep-1979 in Mhs 2 

 at 04112500 Red Cedar River at East Lansing, Mich. for measurement 2. 

 DArea Q50 rY50 Qdv rYdv

 jSta iJndxP Streamgage (mi^2) (cfs) (cfsm) (cfs) (cfsm)

------------------------------------------------------------------------

 1 2 04096405 201.0 61.0 0.3035 45.0 0.4732 

 2 3 04096515 48.7 7.9 0.1622 4.3 0.2971 

 3 4 04096600 285.6 65.0 0.2276 48.0 0.4100 

 4 5 04096900 162.0 72.0 0.4444 42.0 0.5092 

 5 6 04097170 68.2 30.0 0.4399 30.0 0.6632 

 6 7 04097540 106.6 46.0 0.4315 32.0 0.5479 

 7 8 04099000 1879.4 850.0 0.4523 815.0 0.6585 

 8 9 04101500 3666.0 1850.0 0.5046 1650.0 0.6709 

 9 10 04101800 251.5 177.0 0.7038 173.0 0.8294 

 10 11 04102500 389.9 262.0 0.6720 295.0 0.8698 

 11 12 04102700 83.5 35.0 0.4192 30.0 0.5994 

 12 15 04105000 274.0 70.0 0.2555 54.0 0.4439 

 13 16 04105700 36.8 31.0 0.8424 26.0 0.8405 

 14 17 04108600 65.1 22.0 0.3379 20.0 0.5543 

 15 18 04108801 66.9 4.3 0.0643 5.6 0.2893 

 16 19 04111500 16.3 0.9 0.0552 0.1 0.0893 

 17 20 04112000 10.4 0.2 0.0231 0.1 0.0760 
Matlab Code 2–7. Analysis of instantaneous flow measurements 81

 18 22 04114498 284.1 40.0 0.1408 25.0 0.2966 

 19 23 04115000 420.1 30.0 0.0714 5.3 0.1123 

 20 24 04116500 516.3 222.0 0.4300 148.0 0.5354 

 21 26 04117500 410.5 109.0 0.2655 86.0 0.4577 

 22 27 04118000 795.2 281.0 0.3534 178.0 0.4731 

 23 28 04118500 257.2 127.0 0.4938 90.0 0.5915 

------------------------------------------------------------------------

Regression results for measurement 2 for 25-Sep-1979 

 nReg R2-value F-Stat p-value RMSE b_Intercept b_Slope 

 3 0.9400 328.94 0.00000 0.0508 0.1295 0.8756 

------------------------------------------------------------------------

Relation between daily mean and index flow on 20-Sep-2001 in Mhs 2 

 at 04112500 Red Cedar River at East Lansing, Mich. for measurement 3. 

 DArea Q50 rY50 Qdv rYdv

 jSta iJndxP Streamgage (mi^2) (cfs) (cfsm) (cfs) (cfsm)

------------------------------------------------------------------------

 1 1 04096015 80.8 21.0 0.2599 28.0 0.5887 

 2 2 04096405 201.0 61.0 0.3035 71.0 0.5943 

 3 3 04096515 48.7 7.9 0.1622 13.0 0.5167 

 4 7 04097540 106.6 46.0 0.4315 63.0 0.7688 

 5 8 04099000 1879.4 850.0 0.4523 1200.0 0.7991 

 6 9 04101500 3666.0 1850.0 0.5046 2520.0 0.8291 

 7 10 04101800 251.5 177.0 0.7038 209.0 0.9116 

 8 11 04102500 389.9 262.0 0.6720 261.0 0.8182 

 9 12 04102700 83.5 35.0 0.4192 49.0 0.7660 

 10 13 04103010 269.5 147.0 0.5455 168.0 0.7895 

 11 14 04104945 48.3 15.0 0.3106 25.0 0.7194 

 12 15 04105000 274.0 70.0 0.2555 92.0 0.5795 

 13 16 04105700 36.8 31.0 0.8424 34.0 0.9612 

 14 17 04108600 65.1 22.0 0.3379 33.0 0.7120 

 15 18 04108801 66.9 4.3 0.0643 41.0 0.7829 

 16 19 04111500 16.3 0.9 0.0552 3.3 0.4499 

 17 20 04112000 10.4 0.2 0.0231 0.6 0.2402 

 18 23 04115000 420.1 30.0 0.0714 35.0 0.2886 

 19 25 04117000 7.8 2.8 0.3590 4.8 0.7845 

 20 26 04117500 410.5 109.0 0.2655 130.0 0.5627 

 21 28 04118500 257.2 127.0 0.4938 212.0 0.9079 

------------------------------------------------------------------------

Regression results for measurement 3 for 20-Sep-2001 

 nReg R2-value F-Stat p-value RMSE b_Intercept b_Slope 

 3 0.7253 50.16 0.00000 0.1117 -0.0590 0.9098 
82 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ROOT DAILY WATER YIELD, IN CFSM

ROOT INDEX WATER YIELD, IN CFSM

04112500 Red Cedar River at East Lansing, Mich. on 08-Sep-2001

R2

= 0.7980

Auxilary streamgages

Daily and Index flow regression

04112500 instantaneous flow

Matlab Code 2–8. Compute water yields and flows with the Discrete Estimator

Standard definition of the weighted estimate

DiscrY50_hat = (sum(ry50m_h./ry50m_v))/sum(1./ry50m_v);

DiscrY50_vw = 1/sum(1./ry50m_v);

% Effective number of observations in the regression equation.

nReg = min(NumObs);

% Correcting for under dispersion of a weighted variance

DiscrY50_vb = 0;

if nMeasQ > 1

 cf = (1/(nMeasQ-1))*sum(((ry50m_h-mean(ry50m_h)).^2)./ry50m_v);

 DiscrY50_vw = DiscrY50_vw * max(1,cf);

 wstd = (1./ry50m_v)./sum(1./ry50m_v);

 V2 = sum(wstd.^2);

 DiscrY50_vb = 1/(1-V2)*sum(wstd.*(ry50m_h-DiscrY50_hat).^2);

end

%

DiscrY50_l95 = DiscrY50_hat-tinv(0.975,nReg+nMeasQ-1)*sqrt(DiscrY50_vb+DiscrY50_vw);

DiscrY50_u95 = DiscrY50_hat+tinv(0.975,nReg+nMeasQ-1)*sqrt(DiscrY50_vb+DiscrY50_vw);

% Test whether the index flow is within the CI

%
Matlab 2–8. Compute water yields and flows with the Integrated Estimator 83

Matlab Code 2–9. Compute water yields and flows with the Integrated Estimator

IntgrY50_hat = ( IndxrY50_hat/IndxrY50_var + DiscrY50_hat/DiscrY50_vw )/...

 (1/IndxrY50_var + 1/(DiscrY50_vw));

% Weighted variance with correction for underdispersion

IntgrY50_vw = 1/(1/IndxrY50_var + 1/(DiscrY50_vw))* ...

 ( ((IndxrY50_hat - mean([IndxrY50_hat,IntgrY50_hat])).^2) ...

 ./ IndxrY50_var + ...

 ((DiscrY50_hat - mean([IndxrY50_hat,IntgrY50_hat])).^2) ...

 ./ (DiscrY50_vw));

%

wstd = [1/IndxrY50_var,1/DiscrY50_vw]./(1/IndxrY50_var + 1/DiscrY50_vw);

V2 = sum(wstd.^2);

IntgrY50_vb = 1/(1-V2)*(wstd(1)*(IndxrY50_hat-IntgrY50_hat)^2+...

 wstd(2)*(DiscrY50_hat-IntgrY50_hat)^2);

%

IntgrY50_hat = (IndxrY50_hat/IndxrY50_var + 

DiscrY50_hat/(DiscrY50_vb+DiscrY50_vw))/...

 (1/IndxrY50_var + 

1/(DiscrY50_vb+DiscrY50_vw));

IntgrY50_var = 1 / (1/IndxrY50_var + 

1/(DiscrY50_vb+DiscrY50_vw));

%

IntgrY50_l95 = IntgrY50_hat+tinv(0.025,nReg+nMeasQ-1)*sqrt(IntgrY50_var);

IntgrY50_u95 = IntgrY50_hat+tinv(0.975,nReg+nMeasQ-1)*sqrt(IntgrY50_var);

%

Matlab Code 2–10. Compute water yields and flows with the Site Estimator

Compute root water yields from instantaneous flow measurements

InstrYmea = sqrt(InstQmea ./ DArea0) ;

if length(InstrYmea)>1

 % whos -file MhsStdMedrYdv

 load MhsStdMedrYdv

 % Average standard deviation of median rY50s as a function of nMeasQ

 % as a function of measurement intensity [1,...,100] in rows and

 % Mhs in corresponding columns.

 % Interpolate SdMednMeasQMhs from standard intensities

 StdnMeasQ = [1:7 10 15 20 50 100];

 MhsStdMedrYdv_hat = interp1(StdnMeasQ,MhsStdMedrYdv(:,Mhs)’,nMeasQ);

 % Look up the standard deviation of the median rY50 values for the

 % number of measurements and the Mhs.

 SiterY50_hat =(median(InstrYmea) / MhsStdMedrYdv_hat^2 + ...

 IntgrY50_hat /IntgrY50_var) ./ ...

 (1/(MhsStdMedrYdv_hat^2)+1/(IntgrY50_var));

 %

 SiterY50_var = 1 /( 1/MhsStdMedrYdv_hat^2 + 1/ IntgrY50_var);

 SiterY50_l95 = SiterY50_hat+tinv(0.025,nReg-1)*sqrt(SiterY50_var);

 SiterY50_u95 = SiterY50_hat+tinv(0.975,nReg-1)*sqrt(SiterY50_var);

 %

end
84 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months

Matlab Code 2–11. List results for all estimators to console

ndash = 77;

fprintf(1,[‘\n’,repmat(‘-’,1,ndash),’\n’ ]);

fprintf(1,’Results for all Estimators at %s %s \n’,SiteNo,SiteName);

fprintf(1,[repmat(‘-’,1,ndash),’\n’ ]);

-----------------------------------------------------------------------------

Results for all Estimators at 04112500 Red Cedar River at East Lansing, Mich. 

-----------------------------------------------------------------------------

fprintf(1,’INDEX REGRESSION ESTIMATOR: \n’);

fprintf(1,’Charact- Number of Index Standard_Deviation Confidence_Limits\n’);

fprintf(1,’teristic measurements estimate mean prediction 95-lower 95-upper 

\n’);

fprintf(1,[repmat(‘-’,1,ndash),’\n’ ]);

fprintf(1,’rYield (rcfsm):%3u %10.4f %9.6f %9.6f %8.4f %8.4f \n’,...

 0,IndxrY50_hat,IndxrY50_rmse,IndxrY50_sep,IndxrY50_l95,IndxrY50_u95);

fprintf(1,’Flow (cfs): %10.2f ----- ----- %8.2f %8.2f \n’,...

 IndxQ50_hat,IndxQ50_l95,IndxQ50_u95);

fprintf(1,[repmat(‘-’,1,ndash),’\n\n’ ]);

INDEX REGRESSION ESTIMATOR: 

Charact- Number of Index Standard_Deviation Confidence_Limits

teristic measurements estimate mean prediction 95-lower 95-upper 

-----------------------------------------------------------------------------

rYield (rcfsm): 0 0.3613 0.123774 0.125692 0.1128 0.6098 

Flow (cfs): 44.90 ----- ----- 4.38 127.92 

-----------------------------------------------------------------------------

fprintf(1,’DISCRETE ESTIMATOR: \n’);

fprintf(1,[repmat(‘-’,1,ndash),’\n’ ]);

fprintf(1,’Charact- Number of Discrete Standard_Deviation Confidence_Limits\n’);

fprintf(1,’teristic measurements estimate within between 95-lower 95-upper 

\n’);

fprintf(1,[repmat(‘-’,1,ndash),’\n’ ]);

fprintf(1,’rYield (rcfsm):%3u %10.4f %9.6f %9.6f %8.4f %8.4f \n’,...

nMeasQ,DiscrY50_hat,sqrt(DiscrY50_vw),sqrt(DiscrY50_vb),DiscrY50_l95,DiscrY50_u95);

fprintf(1,’Flow (cfs): %10.2f ----- ----- %8.2f %8.2f \n’,...

 [DiscrY50_hat^2,DiscrY50_l95^2,DiscrY50_u95^2].*DArea0);

fprintf(1,[repmat(‘-’,1,ndash),’\n\n’ ]);

DISCRETE ESTIMATOR: 

-----------------------------------------------------------------------------

Charact- Number of Discrete Standard_Deviation Confidence_Limits

teristic measurements estimate within between 95-lower 95-upper 

-----------------------------------------------------------------------------

rYield (rcfsm): 3 0.3328 0.042876 0.010639 0.2414 0.4241 

Flow (cfs): 38.09 ----- ----- 20.04 61.89 

-----------------------------------------------------------------------------

fprintf(1,’INTEGRATED ESTIMATOR: \n’);

fprintf(1,[repmat(‘-’,1,ndash),’\n’ ]);

fprintf(1,’Charact- Number of Discrete Standard_Deviation Confidence_Limits\n’);

fprintf(1,’teristic measurements estimate IndxrY50 DiscrY50 95-lower 95-upper 

\n’);
Matlab Codw 2–11. List results for all estimators to console 85

fprintf(1,[repmat(‘-’,1,ndash),’\n’ ]);

fprintf(1,’rYield (rcfsm):%3u %10.4f %9.6f %9.6f %8.4f %8.4f \n’,...

nMeasQ,IntgrY50_hat,sqrt(IndxrY50_var),sqrt(DiscrY50_vb+DiscrY50_vw),IntgrY50_

l95,IntgrY50_u95);

fprintf(1,’Flow (cfs): %10.2f ----- ----- %8.2f %8.2f \n’,...

 [IntgrY50_hat^2,IntgrY50_l95^2,IntgrY50_u95^2].*DArea0);

fprintf(1,[repmat(‘-’,1,ndash),’\n\n’ ]);

INTEGRATED ESTIMATOR: 

-----------------------------------------------------------------------------

Charact- Number of Discrete Standard_Deviation Confidence_Limits

teristic measurements estimate IndxrY50 DiscrY50 95-lower 95-upper 

-----------------------------------------------------------------------------

rYield (rcfsm): 3 0.3359 0.125692 0.044176 0.2497 0.4221 

Flow (cfs): 38.81 ----- ----- 21.45 61.29 

-----------------------------------------------------------------------------

if nMeasQ >= 3

 fprintf(1,’SITE ESTIMATOR: \n’);

 fprintf(1,[repmat(‘-’,1,ndash),’\n’ ]);

 fprintf(1,’Charact- Number of Site Standard_Deviation Confidence_

Limits\n’);

 fprintf(1,’teristic measurements estimate IntgrY50 MhsStdMedrY 95-lower 

95-upper \n’);

 fprintf(1,[repmat(‘-’,1,ndash),’\n’ ]);

 fprintf(1,’rYield (rcfsm):%3u %10.4f %9.6f %9.6f %8.4f %8.4f \n’,...

 nMeasQ,SiterY50_hat,sqrt(SiterY50_var),MhsStdMedrYdv_hat,SiterY50_l95,SiterY50_u95);

 fprintf(1,’Flow (cfs): %10.2f ----- ----- %8.2f %8.2f \n’,...

 [SiterY50_hat^2,SiterY50_l95^2,SiterY50_u95^2].*DArea0);

 fprintf(1,[repmat(‘-’,1,ndash),’\n\n’ ]);

else

 fprintf(1,’Too few instantaneous measurements to compute the site estimator.\n’);

end

%

if nRecess >= 1

 fprintf(1,’AUGMENTED ESTIMATOR with user data mrY0= %8.6f and lmtau= 

%8.6f\n’,mrY0,lmtau);

 fprintf(1,’*** Inaccurate results are expected if mrY0 or lmtau are poorly 

estimated. *** \n’);

 fprintf(1,’Charact- Number of Augmented Standard_Deviation Confidence_

Limits\n’);

 fprintf(1,’teristic measurements estimate mean prediction 95-lower 

95-upper \n’);

 fprintf(1,[repmat(‘-’,1,ndash),’\n’ ]);

 fprintf(1,’rYield (rcfsm):%3u %10.4f %9.6f %9.6f %8.4f %8.4f \n’,...

 nRecess,AugmrY50_hat,AugmrY50_rmse,AugmrY50_sep,AugmrY50_l95,AugmrY50_u95);

 fprintf(1,’Flow (cfs): %10.2f ----- ----- %8.2f %8.2f \n’,...

 AugmQ50_hat,AugmQ50_l95,AugmQ50_u95);

 fprintf(1,[repmat(‘-’,1,ndash),’\n\n’ ]);

else

 fprintf(1,’Baseflow recession variables not defined.\n’);

end

%
86 Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for Summer Months

SITE ESTIMATOR: 

-----------------------------------------------------------------------------

Charact- Number of Site Standard_Deviation Confidence_Limits

teristic measurements estimate IntgrY50 MhsStdMedrY 95-lower 95-upper 

-----------------------------------------------------------------------------

rYield (rcfsm): 3 0.3342 0.036918 0.079558 0.2572 0.4112 

Flow (cfs): 38.42 ----- ----- 22.75 58.17 

-----------------------------------------------------------------------------

AUGMENTED ESTIMATOR with user data mrY0= 0.397830 and lmtau= 1.573310

*** Inaccurate results are expected if mrY0 or lmtau are poorly estimated. *** 

Charact- Number of Augmented Standard_Deviation Confidence_Limits

teristic measurements estimate mean prediction 95-lower 95-upper 

-----------------------------------------------------------------------------

rYield (rcfsm): 3 0.3577 0.037736 0.038051 0.2825 0.4330 

Flow (cfs): 44.02 ----- ----- 27.45 64.48 

-----------------------------------------------------------------------------

Published with MATLAB® 7.10

Holtschlag—Use of Instantaneous Streamflow Measurements to Improve Regression Estimates of Index Flow for the Summer Month—SIR 5010–5236

Printed on recycled paper