U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2009–5080

Prepared in cooperation with International Upper Great Lakes Study Board 

and U.S. Army Corps of Engineers

Detection of Conveyance Changes in St. Clair River Using 

Historical Water-Level and Flow Data with Inverse OneDimensional Hydrodynamic Modeling

Detection of Conveyance Changes 

in St. Clair River Using Historical 

Water-Level and Flow Data with Inverse 

One-Dimensional Hydrodynamic Modeling

By David J. Holtschlag and C.J. Hoard

Prepared in cooperation with International Upper Great Lakes Study Board 

and U.S. Army Corps of Engineers

Scientific Investigations Report 2009–5080

U.S. Department of the Interior

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

KEN SALAZAR, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2009

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reproduce any copyrighted materials contained within this report.

Suggested citation:

Holtschlag, D.J., and Hoard, C.J., 2009, Detection of conveyance changes in St. Clair River using historical water-level 

and flow data with inverse one-dimensional hydrodynamic modeling: U.S. Geological Survey Scientific Investigations 

Report 2009–5080, 39 p.

ISBN 978-1-4113-2435-0
iii

Contents

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

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

Study Area..............................................................................................................................................1

Problem...................................................................................................................................................3

Purpose and Scope ..............................................................................................................................4

Data and Models Provided by Other Agencies ...............................................................................5

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

Conventional Current-Meter Measurements.................................................................5

Acoustic Doppler Current Profiler Measurements .......................................................5

Water-Level Data.........................................................................................................................6

The Standard 1-D Hydrodynamic Model of the St. Clair River.............................................6

Model Schematization .......................................................................................................6

Model Geometry and Parameterization..........................................................................8

Approach.........................................................................................................................................................8

Water Levels at Gaging Stations........................................................................................................8

Water-Level Fall between Gages .............................................................................................8

Reach Conveyance......................................................................................................................9

Conveyance Ratios....................................................................................................................10

Inverse 1-D Hydrodynamic Modeling of the St. Clair River.........................................................10

1-D Hydrodynamic Model of the St. Clair River....................................................................10

Inverse Modeling.......................................................................................................................11

Nonlinear Regression ......................................................................................................11

Parameter Convergence .................................................................................................11

Parameter Magnitudes and Uncertainties...................................................................12

Parameterization Strategy ..............................................................................................12

Simulation of Flow under Annual Parameterizations Derived from Inverse Modeling...........12

Results of Conveyance Analyses..............................................................................................................13

Inferring Reach-Conveyance Changes from Water-Level Measurements ..............................13

Water-Surface Fall.....................................................................................................................13

Conveyance Ratios....................................................................................................................15

Inferring Changes in Reach Conveyance from 1-D Hydrodynamic-Model Parameters.........18

Time-Varying Parameter Estimation .......................................................................................22

Effective Channel Roughness Estimated from Conventional Measurements .......22

Effective Channel Roughness Estimated from Acoustic Doppler Current 

Profiler Measurements .....................................................................................24

Fixed-Parameter Estimation.....................................................................................................29

Scenario 1962 to 1968.......................................................................................................29

Scenario 1981 to 1985.......................................................................................................30

Scenario 1996 to 2007.......................................................................................................30

Integration of Time-Varying and Fixed-Parameter Estimation Results.............................32

Effective Channel Roughness Ratios and Conveyance Ratios ..........................................34

Inferring Changes in River Conveyance from 1-D Hydrodynamic Simulations........................37
iv

Summary and Conclusions.........................................................................................................................38

Acknowledgments.......................................................................................................................................38

References Cited..........................................................................................................................................39

Figures

1. Map showing St. Clair River study area in southeastern Michigan and southern 

Ontario, Canada ............................................................................................................................2

2–23. Graphs showing:

2. Monthly historical water-level characteristics for 1918–2007 and 2007 water levels 

on Lake Michigan–Huron ............................................................................................................3

3. Water levels and water-level differences between Lake Michigan–Huron and 

Lake St. Clair from 1962 to 2007 ..................................................................................................4

4. Schematic of the St. Clair River Hydrologic Engineering Center-River Analysis 

System (HEC-RAS) model showing channels and junctions defined in the 

standard model .............................................................................................................................7

5. Differences in average April 1 to November 30 water levels between adjacent 

reaches on the St. Clair River...................................................................................................14

6. Lower diagonal scatter-plot matrix of conveyance-ratios from 1962 to 2007 for 

paired reaches on the St. Clair River ......................................................................................16

7. Average flow of the St. Clair River for 1962–2006 based on coordinated monthly 

flow estimates .............................................................................................................................18

8. Lower diagonal scatter-plot matrix of conveyance-ratio innovations from 1962 to 

2006 for paired reaches on the St. Clair River........................................................................20

9. Relation between simulated flow and flow measured on the St. Clair River from 

1962 to 1985 using conventional current meters..................................................................... 23

10. Relation between simulated flow and flow measured on the St. Clair River from 

1996 to 2007 using acoustic Doppler current profilers. ........................................................25

11. Annual estimates of effective channel roughness with 95-percent confidence 

intervals for the St. Clair River reach from Fort Gratiot to Dunn Paper, Michigan ..........26

12. Annual estimates of effective channel roughness with 95-percent confidence 

intervals for the St. Clair River reach from Dunn Paper, Michigan, to Point Edward, 

Ontario ..........................................................................................................................................26

13. Annual estimates of effective channel roughness with 95-percent confidence 

intervals for the St. Clair River reach from Point Edward, Ontario, to the mouth of 

Black River, Michigan ................................................................................................................27

14. Annual estimates of effective channel roughness with 95-percent confidence 

intervals for the St. Clair River reach from the mouth of Black River to Dry Dock, 

Michigan ......................................................................................................................................27

15. Annual estimates of effective channel roughness with 95-percent confidence 

intervals for the St. Clair River reach from Dry Dock to St. Clair State Police, 

Michigan ......................................................................................................................................28

16. Annual estimates of effective channel roughness with 95-percent confidence 

intervals for the St. Clair River reach from St. Clair State Police, Michigan, to 

Port Lambton, Ontario ................................................................................................................28

17. Annual estimates of effective channel roughness with 95-percent confidence 

intervals for the St. Clair River reach from Port Lambton, Ontario, to Algonac, 

Michigan ......................................................................................................................................28
v

18. Relation between measured flow and flow simulated on the St. Clair River by use 

of a fixed, seven-parameter version of the 1-D HEC-RAS (One-dimensional 

Hydrologic Engineering Center-River Analysis System) model developed using 

data obtained from1962 and 1968.............................................................................................30

19. Relation between measured flow and flow simulated on the St. Clair River by use 

of a fixed, nine-parameter version of the hydrodynamic model for the period 

1996–2007 .....................................................................................................................................32

20. Fixed-parameter channel roughness estimates for nine St. Clair River reaches 

for the periods 1996–2001 and 2002–2007...............................................................................33

21. Relation between annual conveyance ratios estimated on the basis of effective 

channel roughness estimates and measured water-level data for the period 1996 

through 2007 ................................................................................................................................36

22. Relation between annual model-parameter estimates and simulated average flow 

from 1962 to 2007 for 1962 and 2007 water-level boundary conditions..............................37

23. Relation between annual parameter estimates and changes in average simulated 

flow from the 1962 parameterization .......................................................................................37

Tables

1. Selected water-level gaging stations along the St. Clair River ............................................6

2. St. Clair River reaches delimited by water-level gaging stations and used for 

this study ........................................................................................................................................9

3. Trends in water-level fall within St. Clair River reaches delimited by water-level 

gaging stations, 1962–2007........................................................................................................15

4. Trends in conveyance ratios among paired reaches on the St. Clair River..............................17

5. Relation among estimated flow from April 1 to November 30 and conveyance ratios 

of reaches on St. Clair River...........................................................................................................19

6. Trends in conveyance ratio innovations among paired reaches on the 

St. Clair River......................................................................................................................................21

7. Effective channel roughness estimated for reaches spanning the study area of 

the St. Clair River in years with one or more conventional flow measurements 

from 1962 to 1985 ...............................................................................................................................22

8. Estimates of effective channel-roughness values for reaches spanning the study 

area of the St. Clair River in years with one or more acoustic Doppler current 

profiler flow measurements from 1996 to 2007 ......................................................................24

9. Fixed-parameter estimates of effective channel roughness for St. Clair River 

reaches based on measurements from 1962 to 1968...............................................................29

10. Fixed-parameter estimates of effective channel roughness on reaches of 

St. Clair River for the period 1996–2007 .......................................................................................31

11. Lower diagonal elements of the symmetric correlation matrix for fixed-parameter 

estimates of effective channel roughness coefficients on St. Clair River for the 

period 1996–2007 ...............................................................................................................................31

12. Basis vectors selected in a stepwise-regression analysis of flow residuals from 

the 1996 to 2007 fixed, nine-parameter model of St. Clair River............................................34

13. Relation between ratios of effective channel roughness and conveyance ratios 

among reaches of the St. Clair River based on data from 1996 to 2007 ..............................35
vi

Conversion Factors and Abbreviations

Multiply By To obtain

Length

millimeter (mm) 0.03937 inch (in.)

meter (m) 3.281 foot (ft) 

kilometer (km) 0.6214 mile (mi)

kilometer (km) 0.5400 mile, nautical (nmi) 

meter (m) 1.094 yard (yd) 

Area

square meter (m2) 0.0002471 acre 

square kilometer (km2) 247.1 acre

square meter (m2) 10.76 square foot (ft2) 

square kilometer (km2) 0.3861 square mile (mi2)

Volume

liter (L) 0.2642 gallon (gal)

cubic meter (m3) 264.2 gallon (gal) 

cubic meter (m3) 0.0002642 million gallons (Mgal) 

cubic meter (m3) 35.31 cubic foot (ft3)

Velocity

centimeters per year (cm/yr) 2.54 inches per year (in.yr)

Flow rate

meter per second (m/s) 3.281 foot per second (ft/s) 

cubic meter per second (m3/s) 35.31 cubic foot per second (ft3/s)

cubic meter per second per square kilometer [(m3

/s)/km2] 91.49 cubic foot per second per square mile [(ft3/s)/mi2]

liter per second (L/s) 15.85 gallon per minute (gal/min) 

cubic meter per second (m3/s) 22.83 million gallons per day (Mgal/d) 

Mass

gram (g) 0.03527 ounce, avoirdupois (oz)

kilogram (kg) 2.205 pound avoirdupois (lb)

megagram (Mg) 1.102 ton, short (2,000 lb)

megagram (Mg) 0.9842 ton, long (2,240 lb)

Density

kilogram per cubic meter (kg/m3) 0.06242 pound per cubic foot (lb/ft3) 

gram per cubic centimeter (g/cm3) 62.4220 pound per cubic foot (lb/ft3) 
vii

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

°F=(1.8×°C)+32

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

°C=(°F-32)/1.8

Vertical coordinate information is referenced to the International Great Lakes Datum of 1985 

(IGLD 85).

Horizontal coordinate information is referenced to the Michigan South (zone 2113) State Plane 

Coordinate System of 1983 (SPCS 83).

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

Abbreviations

1-D one-dimensional

ADCP acoustic Doppler current profilers 

CHS Canadian Hydrographic Service 

GLERL Great Lakes Environmental Research Laboratory 

GPS global positioning system 

HEC-RAS Hydrologic Engineering Center’s River Analysis System 

IGLD85 International Great Lakes Datum of 1985 

IJC International Joint Commission 

IUGLS International Upper Great Lakes Study 

NOAA National Oceanic and Atmospheric Administration

SPCS State Plane Coordinate System

UCODE universal parameter estimate code

USACE U.S. Army Corps of Engineers 

USGS U.S. Geological Survey

Abstract

St. Clair River is a connecting channel that transports 

water from Lake Huron to the St. Clair River Delta and Lake 

St. Clair. A negative trend has been detected in differences 

between water levels on Lake Huron and Lake St. Clair. This 

trend may indicate a combination of flow and conveyance 

changes within St. Clair River. To identify where conveyance change may be taking place, eight water-level gaging 

stations along St. Clair River were selected to delimit seven 

reaches. Positive trends in water-level fall were detected in 

two reaches, and negative trends were detected in two other 

reaches. The presence of both positive and negative trends in 

water-level fall indicates that changes in conveyance are likely 

occurring among some reaches because all reaches transmit 

essentially the same flow. Annual water-level fall in reaches 

and reach lengths was used to compute conveyance ratios 

for all pairs of reaches by use of water-level data from 1962 

to 2007. Positive and negative trends in conveyance ratios 

indicate that relative conveyance is changing among some 

reaches. Inverse one-dimensional (1-D) hydrodynamic modeling was used to estimate a partial annual series of effective 

channel-roughness parameters in reaches forming the St. Clair 

River for 21 years when flow measurements were sufficient 

to support parameter estimation. Monotonic, persistent but 

non-monotonic, and irregular changes in estimated effective 

channel roughness with time were interpreted as systematic 

changes in conveyances in five reaches. Time-varying parameter estimates were used to simulate flow throughout the St. 

Clair River and compute changes in conveyance with time. 

Based on the partial annual series of parameters, conveyance 

in the St. Clair River increased about 10 percent from 1962 to 

2002. Conveyance decreased, however, about 4.1 percent from 

2003 to 2007, so that conveyance was about 5.9 percent higher 

in 2007 than in 1962. 

Introduction

In February 2007, the International Joint Commission 

(IJC), a binational organization whose main purpose is to 

prevent and resolve disputes between Canada and the United 

States concerning trans-boundary water-quality and -quantity 

issues, initiated an International Upper Great Lakes Study 

(IUGLS). The purposes of the study were to (1) determine 

whether the regulation of Lake Superior outflows could be 

improved to address the evolving needs of the upper Great 

Lakes, and (2) investigate whether physical changes in the 

St. Clair River have occurred and are causing the difference between Lake Huron and Lake St. Clair water levels to 

decrease. The upper Great Lakes include lakes Superior, Michigan, Huron, and Erie, and the interconnecting channels of St. 

Marys River, St. Clair River, Lake St. Clair, Detroit River, and 

Niagara River. The study component described in this report 

addresses the second purpose of the study and uses historical water-level and flow1

 data with inverse one-dimensional 

hydrodynamic modeling of the St. Clair River to infer possible 

temporal and spatial changes in conveyance characteristics. 

Study Area

St. Clair River forms part of the international border 

between the United States and Canada in southeastern Michigan and southern Ontario (fig. 1). As a connecting channel 

on the Great Lakes Waterway, St. Clair River receives water 

from Lake Huron and discharges water to Lake St. Clair. Near 

the outlet of Lake Huron, St. Clair River has a drainage area 

of 583,000 km2. Between Lake Huron and Lake St. Clair, the 

St. Clair River flows generally from north to south a distance 

of about 64 km; water-surface elevations along the St. Clair 

1 Flow is used synonymously with the term discharge or streamflow. 

Detection of Conveyance Changes in St. Clair River 

Using Historical Water-Level and Flow Data with 

Inverse One-Dimensional Hydrodynamic Modeling

By David J. Holtschlag and C.J. Hoard
2 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

LAKE HURON

MICHIGAN

ONTARIO

Lake St. Clair 

Anchor Bay 

St. Clair Cutoff

Corunna

Mooretown

Courtright

Marysville

St. Clair 

Point Edward

Sarnia

Port Huron 

Port Lambton 

Marine Sombra City

Algonac

Talford Creek

Black River 

MICHIGAN

St. Clair Flats 

Bassett Channel 

Chenal Ecarte 

Middle Channel 

North Channel 

UNITED STATES 

CANADA

Average water level: 176.2 meters 

Average water-level: 

174.7 meters 

South Channel 

CS-206

CS-212

CS-204

CS-214

CS-220

CS-216

CS-207

Mouth of Black 

River gage

Base from National Oceanic and Atmospheric Administration, 1999.

Lake elevations are shown in meters.

WATER-LEVEL GAGING STATION

FLOW-MEASUREMENT CROSS

SECTION AND NUMBER

CS-200

CS-202

MICHIGAN

43° 00’

82 82° 20’ ° 45’

42° 40’

Roberts

Landing

gage

Figure 1. St. Clair River study area in southeastern Michigan and southern Ontario, Canada. 
Introduction 3

River fall about 1.5 m. The St. Clair River has an average flow 

of 5,150 m3/s and an average water yield of 28.2 cm/yr from 

the contributing basin. 

Minor tributaries contributing flow to the St. Clair River 

include Black River, Pine River, and Belle River. Streamflow 

records at the U.S. Geological Survey (USGS) gaging station (04160075) Black River near Port Huron, Mich., which 

is about 16 km upstream from the mouth of Black River at 

the confluence with St. Clair River, indicate an average flow 

of 8.2 m3/s when the station was operated continuously from 

1932–43. The drainage area at the Black River gaging station 

is about 1,620 km2. Streamflow records at the USGS gaging 

station (04160398) Pine River near Marysville, Mich., which 

is about 13 km upstream from the mouth of the Pine River, 

indicate an average flow of 4.7 m3/s during water year 2005, 

the only year of continuous record. The drainage area at the 

Pine River gaging station is 440 km2. Streamflow records at 

the USGS gaging station (04160625) Belle River near Marine 

City, Mich., indicate an average flow of 5.2 m3/s in 2005, the 

only year of continuous record data. The drainage area at the 

Belle River gaging station, which is about 10 km upstream 

from the mouth of the Belle River, is 552 km2. Flows from all 

tributaries along the St. Clair River are thought to contribute 

less than 1 percent of the total flow of St. Clair River. Chenal 

Ecarte is the most upstream distributary of the St. Clair River 

Delta near the mouth of the St. Clair River. Chenal Ecarte 

delivers about 3.8 percent of the flow from the St. Clair River 

through the Walpole Island First Nation Reserve to Lake 

St. Clair (Holtschlag and Koschik, 2001). 

St. Clair River is part of the navigational waterway on the 

upper Great Lakes. Dredging the navigational channel from 

minimum depths of 7.6 to 8.2 m in 1962 increased the conveyance in some reaches of the St. Clair River by increasing 

the cross-sectional area for flow. Quinn (1985) discusses the 

effects of St. Clair River dredging on lake levels and flows. 

Problem

Lakes Michigan and Huron (Michigan–Huron) are connected by the deep Straits of Mackinac and are considered to 

be one lake hydraulically with lake levels rising and falling 

together (U.S. Army Corps of Engineers and Great Lakes 

Commission, 1999). In 2007, monthly water levels on Lake 

Michigan–Huron averaged 0.5-m lower than average levels 

during 1918–2007 (fig. 2) (U.S. Army Corps of Engineers–

Detroit District, 2008). Low water levels and locally shallow 

depths reduce the effective cargo capacity of freighters, and 

increase the hazards of running aground for recreational boaters. During 1918–2007 on Lake Michigan–Huron, minimum 

water levels during the selected period occurred in 1964–65; 

maximum water-levels occurred in 1986–87. Hitt and Miller 

(1986) discuss effects of high water levels. 

Lake-wide average monthly water-level data from 1962 

to 2007 were retrieved for Lakes Michigan–Huron and Lake 

St. Clair (U.S. Army Corps of Engineers–Detroit District, 

2006). No trends were detected (2α = 0.05) in monthly water 

levels for Lake Michigan–Huron or Lake St. Clair based on 

the non-parametric Kendall’s tau test for trend (Gilbert, 1987). 

2

The Greek letter α indicates the pre-specified probability of rejecting the 

null hypothesis, known as type I error, when it is true.

JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

175.4

175.6

175.8

176.0

176.2

176.4

176.6

176.8

177.0

177.2

177.4

177.6

WATER LEVEL, IN METERS ABOVE IGLD 85

High

Low

Average

(1918-2007)

2007

EXPLANATION

Figure 2. Monthly historical water-level characteristics for 1918–2007 and 2007 water levels on Lake Michigan–Huron.
4 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Trends were detected, however, in water-level differences between Lake Michigan–Huron and Lake St. Clair 

(fig. 3). These trends indicate that decreases in average flow 

or increases in conveyance, or both, may be occurring. Such 

trends may indicate that lake levels may change in the future. 

Using Sen’s non-parametric slope estimator (Gilbert, 1987), 

the annual average of monthly water-level differences changed 

(p-value <0.0001) at an average rate of -6.17 mm/yr for the 

period 1962 to 2007. Similarly, significant decreases (p-value 

< 0.0001) also were detected in all months individually. A 

decrease in water-level differences between Lake Michigan–

Huron and Lake St. Clair indicates that the conveyance of St. 

Clair River is increasing if the flow rate is constant, or that that 

the outflow rate from Lake Michigan–Huron to Lake St. Clair 

is decreasing if the conveyance in St. Clair River is constant. 

There is insufficient information currently (2009) available 

on historical changes in conveyance or flow characteristics 

of St. Clair River to describe the possible contribution that 

changes in conveyance or changes in flow may be having on 

these water-level differences. 

Purpose and Scope

The purpose of this report is to ascertain possible conveyance changes in St. Clair River between Fort Gratiot and 

Algonac, Michigan, from 1962 to 2007 by use of hydraulic 

information. Interpretations are based on (1) analysis of reach 

conveyance investigated by use of water-level measurements 

at eight gaging stations on St. Clair River, (2) analysis of reach 

conveyance described by inverse modeling of flows and water 

levels with a 1‐D fixed-geometry hydrodynamic model, and 

(3) analysis of river conveyance investigated by flow simulations using water-level boundary conditions from 1962 to 

2007 and annual parameterizations based on inverse modeling 

results. 

Channel-roughness characteristics, which cannot be measured directly in the field, are only one component affecting 

conveyance that also includes channel cross-sectional area and 

hydraulic radius. Historical measurements of bathymetry and 

shoreline conditions, however, are not sufficient to fully characterize changes in cross-sectional area or hydraulic radius 

Figure 3. Water levels and water-level differences between Lake Michigan–Huron and Lake St. Clair from 1962 

to 2007.

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

YEAR

174

175

176

177

178

W

ATER LEVEL, IN MET

ERS A

BOVE IGLD 85

1.0

1.1

1.2

1.3

1.4

1.5

1.6

WAT

ER LEVEL DIFFER

ENC

E B

ETWE

EN LAKE MIC

HIGANHURON AND LAKE S

T. CLAIR, IN ME

T

E

R

S

Lake Michigan-Huron water levels

Lake St. Clair water levels

Water-level differences
Introduction 5

during the study period. Therefore, any systematic changes 

in estimated conveyances are associated with the combined 

effects of changes in channel-roughness characteristics, 

cross-sectional area, and hydraulic radius. Although nominally attributed to channel roughness, the methods used in this 

study do not provide a basis for identifying which component 

or components affecting conveyance may be changing with 

time or location. In this report, the term ‘effective channel 

roughness’ refers to the combined effect of physical channelroughness characteristics, changes in cross-sectional area, and 

wetted perimeter from those described based on a bathymetry 

survey in 2007. Effective channel roughness is expressed in 

units of Manning’s “n.”

This study was part of an integrated, multi-agency analysis of water levels and flows on the St. Clair River. In particular, the water-level and flow data were compiled and adjusted, 

as needed, by Nanette Noorbakhsh of the U.S. Army Corps 

of Engineer–Detroit District, Michigan. The data and models 

developed by other agencies were used without modification, 

except as indicated. 

Data and Models Provided by Other Agencies

A brief description of the flow and water-level data 

compiled, and the hydrodynamic model initially developed by 

other agencies is provided in the following paragraphs. 

Flow Data

The U.S. Army Corps of Engineers (USACE) has measured flow on the St. Clair River since 1856 (Coordinating 

Committee for Great Lakes Basic Hydraulic and Hydrologic 

Data, 1994). In Canada, Environment Canada also measures 

flow on the St. Clair River. Other U.S. Federal agencies, 

including the Great Lakes Environmental Research Laboratory 

(GLERL), and the USGS have occasionally contributed limited or experimental flow measurements to verify the accuracy 

of USACE measurements or for special studies. Consistent 

with the study area, only St. Clair River flow measurements 

from cross sections between water-level gaging stations at Fort 

Gratiot, and Algonac, Mich., were used. 

Flow measurements on the St. Clair River are only 

obtained during the ice-free season owing to special hazards associated with ice conditions. During the study period 

from 1962 to 2007, the earliest measurement in the year was 

obtained on April 26, and the latest measurement was obtained 

on December 5. Flow data were used to estimate the effective channel roughness by inverse hydrodynamic modeling. 

Flow measurements prior to 1995 generally were made with 

conventional current meters (Rantz and others, 1982), even 

though several moving-boat measurements are included. For 

ease of reference, all measurements prior to 1995 are referred 

to and included with the conventional measurements. After 

1995, all measurements used in this study were made with 

acoustic Doppler current profilers (Simpson, 2001). 

Conventional Current-Meter Measurements

Before 1995, flow measurements generally were made 

by use of the conventional current-meter method adapted for 

use on wide rivers (Rantz and others, 1982), although eight 

moving-boat measurements (Rantz and others, 1982) were 

made by the Water Survey of Canada (Environment Canada) 

in 1981. Applying current-meter methods involves subdividing 

a river cross section into a set of non-overlapping areas containing nearly equal amounts of flow. Within each area, horizontal velocities are profiled with depth to compute a weighted 

average that represents the depth-averaged horizontal velocity. 

Maintaining a stationary position long enough to measure an 

ensemble of horizontal velocities in deep, fast-moving water 

is complicated by the frequent need to yield to commercial 

shipping vessels. 

Flow is computed by integrating the average velocity 

with depth across the flow-measurement section. Measurement 

of flow by use of conventional current meters is difficult on St. 

Clair River because flow varies over short periods of time in 

response to continuous changes in wind, water-level, and wave 

conditions. Boat crews deploying current meters were supported by on-shore surveying crews to determine the locations 

of velocity profiles. From 1962 to 1985, 351 conventional flow 

measurements are documented on St. Clair River. All documented conventional current-meter measurements occurred 

on the main stem of St. Clair River; no conventional measurements were available to describe flow partitioning around Stag 

or Fawn Islands. 

Acoustic Doppler Current Profiler Measurements

After 1995, acoustic Doppler current profilers (ADCPs) 

generally have been used to measure flow on the St. Clair 

River (John A. Koschik, U.S. Army Corps of Engineers–

Detroit District, oral commun., 2001). ADCP uses sound to 

measure water velocity and channel-bottom depths. Positional 

information needed to complete the determination of crosssectional area corresponding to the velocity data generally 

are provided by a global positioning system (GPS) receiver, 

which is integrated with the ADCP unit. On the St. Clair River, 

the ADCP generally is attached to a boat that transects the 

selected measurement cross section. Measured flows are computed using the results from four or more transects of data in 

which similar flows are measured. ADCP provides more rapid 

measurements of flow than conventional current meters on the 

St. Clair River and can be obtained with a single boat crew. 

From 1996 to 2007, 353 ADCP measurements were made 

on the St. Clair River within the study area. Although most 

measurements are on the main stem of the river, paired measurement sections CS-208 and CS-210 (fig. 1) at Stag Island 

and paired measurement sections CS-216 and CS-218 at Fawn 

Island provide a basis for determining the partitioning of flows 

around the islands. 
6 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Water-Level Data

Water levels on the St. Clair River are monitored continually through a network of water-level gaging stations operated 

by the National Oceanic and Atmospheric Administration 

(NOAA) and the Canadian Hydrographic Service (CHS, a 

division of Fisheries and Oceans Canada). Although the active 

gaging stations operating in this network have changed with 

time, eight gaging stations were selected that were operated 

throughout the study period. Selected stations (table 1) have 

been operated continuously from 1962 to 2007 (except for 

periods of missing record) and currently (2009) can provide 

water-level data at hourly or smaller time intervals in near real 

time through publicly accessible Internet sites. Water levels at 

all selected gaging stations are referenced to the International 

Great Lakes Datum of 1985 (IGLD85). Water-level data were 

used to investigate water-level fall between gages as indicators of conveyance change and provided boundary-condition 

information for model simulation and calibration data for 

estimation of effective channel roughness.

The Standard 1-D Hydrodynamic Model of the 

St. Clair River

In this report, the geometry and initial parameterization 

of the one-dimensional (1-D) hydrodynamic model provided 

by Jason P. Giovannettone (U.S. Army Corps of Engineers, 

Hydrologic Engineering Center, Davis, Calif., written 

commun., 2008) is referred to as the standard model of the 

St. Clair River. The standard model is based on the generic 

hydraulic code HEC-RAS (Hydrologic Engineering Center’s 

River Analysis System) developed by the USACE (Brunner, 

2008). A brief description of the standard model follows. 

Model Schematization

The standard hydraulic model of the St. Clair River 

schematic defines seven channels and four junctions within the 

study reach from Fort Gratiot to Algonac, Mich. (fig. 4). Five 

channels are used to represent flow along the main stem, and 

two channels represent bifurcated flow around Stag and Fawn 

Islands. The four junctions provide a mechanism to rejoin 

flows and match water levels upstream and downstream from 

the two islands. Three-hundred and sixty-one cross sections 

represent the geometry of the St. Clair River within the study 

reach. 

The most upstream channel is referred to as Main Ch-1, 

which extends 13.1 km from the outlet of Lake Huron at the 

NOAA water-level gaging station (9014098) at Fort Gratiot, 

Mich., downstream to the first junction, which is north of 

Stag Island. Flow from Black River is added to the outflow 

from Lake Huron to represent the total flow of St. Clair River. 

Hourly water-level data from the Fort Gratiot gaging station 

provide the upstream boundary for model simulations. 

Flow splits at the junction around Stag Island into two 

channels: the western Main Ch-2, which is 5.2-km long, and 

the eastern Stag Island Ch, which is 5.0-km long. These two 

Table 1. Selected water-level gaging stations along the St. Clair River.

[SPCS, State Plane Coordinate System; IGLD85, International Great Lakes Datum of 1985; NOAA, U.S. National Oceanic and Atmospheric Administration]

Agency 

operating the 

gaging station

Gaging 

station 

number

Gaging station 

name assigned 

by operating 

agency

State 

or province 

where gaging 

station is 

operated

Gaging 

station 

identifier used 

in this report

Gaging station location, in 

Michigan SPCS (zone 2113) 

(meters)

Low water 

datum 

(meters 

above 

IGLD85) Easting Northing

NOAA1 9014098 Fort Gratiot Michigan FG 4,158,484 169,156 175.93

NOAA 9014096 Dunn Paper Michigan DP 4,158,514 168,817 175.78

CHS 11940 Point Edward Ontario PE 4,158,614 167,431 175.65

NOAA 9014090 Mouth of 

Black River Michigan BR 4,158,681 165,518 175.63

NOAA 9014087 Dry Dock Michigan DD 4,156,918 162,330 175.50

NOAA 9014080 St. Clair 

State Police Michigan SP 4,153,803 147,507 175.08

CHS 11950 Port Lambton Ontario PL 4,152,459 130,218 174.65

NOAA 9014070 Algonac Michigan AL 4,150,907 126,148 174.59

1

After 1970, NOAA assumed operation of all water-level gaging stations on the St. Clair River. These gaging stations had been built and operated by 

the U.S. Lake Survey, a division of the U.S. Army Corps of Engineers.
Introduction 7

channels rejoin at the second junction, which is south from 

Stag Island. Main Ch-3 extends 16.7-km downstream from the 

junction at the southern end of Stag Island to a third junction, 

which is upstream (north) of Fawn Island. Minor flows from 

Pine River and Belle River are added to the flow of St. Clair 

River in this reach. At the junction north of Fawn Island, flow 

splits into a western channel, referred to as Main Ch-4, which 

is 3.1-km long, and the eastern channel, referred to as Fawn 

Island Ch. Downstream of Fawn Island, these two channels 

rejoin at the fourth junction. 

Main Ch-5 extends 9.2-km downstream from the southern tip of Fawn Island to the northern tip of Russell Island 

(Michigan), where the St. Clair River branches into the North 

and South Channels. About 3.8 percent of the St. Clair River 

outflow flows into Chenal Ecarte upstream of Russell Island 

(Holtschlag and Koschik, 2001). Water-level data from the 

NOAA gaging station (9014070) at Algonac, Mich., provide 

the downstream boundary for model simulations. 

4,145,000

4,150,000

4,155,000

4,160,000

4,165,000

120,000

125,000

130,000

135,000

140,000

145,000

150,000

155,000

160,000

165,000

170,000

175,000

Fort Gratiot Gage

(Upstream waterlevel boundary)

Stag Island

Fawn Island

Main Ch-1

Chenal Ecarte

Black River

Pine River

Belle River

Main Ch-2

Main Ch-3

Main Ch-4

Main Ch-5

Stag Island Ch

Fawn Island Ch

Algonac 

Gage

(Downstream

water-level 

boundary)

NORTHING, IN MICHIGAN STATE PLANE COORDINATE 

SYSTEM OF 1983 (SOUTHERN ZONE) METERS

EASTING, IN MICHIGAN STATE PLANE COORDINATE 

SYSTEM OF 1983 (SOUTHERN ZONE) METERS

St. Clair River

Water-level 

 gaging station

Junction

LAKE HURON

LAKE ST. CLAIR

Main Ch-1 Channel

Channel names (Ch) are shown in italics

Figure 4. Schematic of the St. Clair River 

Hydrologic Engineering Center-River Analysis 

System (HEC-RAS) model showing channels and 

junctions defined in the standard model.
8 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

of flow and water levels to estimate annual reach-conveyance 

characteristics, and (3) simulation of flows using historical 

water-level boundary conditions with annual estimates of 

reach-conveyance characteristics. The approaches used for 

these investigations are described in the following paragraphs. 

Water Levels at Gaging Stations

Eight water-level gaging stations delimit seven reaches 

along the St. Clair River (table 2). Within each reach, waterlevel differences between upstream and downstream stations 

describe the fall of the water level over the reach. Trends in 

annual water-level fall, computed for the ice-free period April 

1 to November 30, were investigated to identify possible 

changes in conveyance and flow. Periods potentially affected 

by ice were not included because backwater caused by ice can 

increase water levels without corresponding increases in flow. 

Water-surface fall and reach lengths also were used to investigate relative conveyance changes among reaches. 

Water-Level Fall between Gages 

Conveyance is generally a monotonically increasing 

function of water level K(h) and is characteristically related to 

flow. For various flow magnitudes, longitudinal water-surface 

profiles tend to be parallel. Thus, water-level differences 

between gaging stations (the fall in water levels) are more 

nearly constant and less sensitive to changes in flow magnitudes than water levels themselves. In this report, trends in 

water-level fall between gages were used as an indicator of 

trends in K(t|h). 

The statistical significance and magnitudes of monotonic 

trends in water-level fall were investigated by use of two nonparametric procedures: Kendall’s tau rank correlation coefficient (Kendall’s tau) and Sen’s slope estimator. Kendall’s tau 

provides a measure of the degree of correspondence between 

year and water-level fall. Kendall’s tau (τ) is computed as: 

, (1)

where 

nc and nd are the number of concordant and discordant 

pairs, respectively, and 

n is the number of observations. 

A set of bivariate data pairs of length n.(n–1)/2 can be 

formed from n time values t and fall values f indexed by i and

j such that 1 ≤ i ≤ j ≤ n. Data pairs are considered concordant 

if sgn(tj–ti) = sgn(fj–fi) and discordant if sgn(tj–ti) = –sgn(fj–fi). 

The sgn function is the sign function such that sgn(x) = –1 if 

x < 0, 0 if x = 0, and 1 if x > 0. Kendall’s tau can be interpreted 

as the probability of a random pair being concordant minus 

the probability of the pair being discordant. In addition to the 

Kendall’s tau statistic, a probability (p-value) is computed for 

the likelihood that the null hypothesis that tau is equal to zero 

Model Geometry and Parameterization

The geometry of the standard model is defined based on a 

2007 multi-beam bathymetry survey obtained by the USACE–

Detroit District and interpreted by David Bennion (USGS 

Great Lakes Science Center, written commun., 2007). Information in the geometry tables was modified to remove areas 

of ineffective flow. Ineffective flow generally occurs near the 

shoreline where obstructions restrict flow or where channel 

curvature creates secondary flow patterns. Where available, 

identification of ineffective flow areas was guided by velocity 

data obtained during ADCP flow measurements. 

The standard model was parameterized to account for 

energy losses associated with channel roughness and losses 

associated with expansion and contraction of flow. In the standard model, channel cross sections were assigned time-invariant channel-roughness values for left over-bank, main channel, 

and right over-bank areas. Given the limited range in water 

levels on the St. Clair River, over-bank areas tend to carry 

only a small percentage of the flow. In the standard model, 

channel-roughness values assigned to left and right over-bank 

areas were the same and within the study area ranged from 

0.0263 to 0.0352, with an average value of 0.0287. Within the 

main channel, channel-roughness values ranged from 0.0240 

to 0.030, with an average value of 0.0258. Contraction and 

expansion coefficients were 0.1 and 0.3, respectively, for all 

cross sections in the study reach. 

Approach

Conveyance is a measure of a stream’s ability to transmit 

water with time and can be expressed in units of cubic meters 

per second. Conveyance is proportional to cross-sectional 

flow area and hydraulic radius of the channel (cross-sectional 

area divided by the wetted perimeter of the channel) and 

is inversely proportional to channel roughness. In general, 

conveyance K increases monotonically with increasing water 

level h, because increases in water level increase the crosssectional area and hydraulic radius. This generally monotonically increasing functional relation can be expressed as K(h). 

In this report, however, changes in conveyance generally refer 

to changes in conveyance with time t that are not attributable 

to changes in water level. This conditional relation of conveyance with time for a given water level K(t|h) may be constant 

or may contain deterministic or stochastic level or trend 

components, which appear monotonic or non-monotonic with 

time, and contain erratic (random) components. Trends may 

represent linear or nonlinear responses to continuous processes 

or discrete events that cause channel scour or fill or changes 

in actual channel-roughness characteristics. The generality of 

possible trend forms complicates the identification of trends in 

K(t|h). 

Changes in St. Clair River conveyance were investigated 

by (1) analysis of water-level data within reaches delimited 

at gaging stations, (2) inverse 1-D hydrodynamic modeling 

1 2 1 ( ) c d n n

n n

τ − = ⋅ ⋅ −
Approach 9

given the data is true. Small p-values (<0.05) are commonly 

used to reject the null hypothesis of no correlation. 

Should the hypothesis of no correlation be rejected, 

Sen’s slope estimator provides an estimate of the rate at which 

water-level fall is changing with time. Sen’s slope estimator is 

the median of n (n-1)/2 estimates of slope computed as

f f

t t

j i

j i

-

-

, again for all 1≤i≤j≤n . 

Reach Conveyance

Manning’s formula (Brunner, 2008) for computing flow 

(Q) is a function of conveyance (K) and the energy slope (Sf)

Q = K• S 1⁄2

f , (2)

where conveyance can be expressed in metric units as

. (3)

The reach-average cross-sectional area (A) and hydraulic 

radius (R) (cross-sectional area divided by the wetted perimeter) can be determined from field measurements. In contrast, 

effective channel roughness, which nominally describes channel roughness, can be ascertained quantitatively by inverse 

hydrodynamic modeling using flow and water-level data or 

estimated subjectively based on bed-material characteristics 

and channel form (Barnes, 1967, Arcement and Schneider, 

1989). 

Within a river reach of length L, the energy slope is 

S h f f = L . The energy head equals 

hf = ∆h + k • ( αu • V 2

u /2g – αd • V 2d /2g ), (4)

where 

∆h is the water-surface fall or the increase in water 

level from the downstream (d) limit of the 

reach to the upstream (u) limit; 

k is coefficient of contraction or expansion for 

gradual transitions, generally taken as 0.1 

and 0.3, respectively (Brunner, 2008); 

	αu and αd are velocity weighting coefficients, which vary 

with the velocity distribution in a reach; 

Vu and Vd are the average velocities at the upstream and 

downstream reach limits, respectively; and 

g is the constant of gravitational acceleration 

(9.806 m2/s). 

In reaches with steady uniform flow, the energy slope 

equals the change in water level divided by the reach length 

as Sf = S = ∆h/L. Although flow in the St. Clair River is not 

strictly uniform, the small bias introduced by assuming that 

the second term on the right-hand side of equation 4 is negligible would not significantly affect trend analyses, which are 

of primary interest in this study component. 

To identify local changes in the conditional conveyance 

K(t|h), the St. Clair River was subdivided into seven reaches 

at gaging stations between Fort Gratiot and Algonac, Mich. 

(table 2). In additional, two reaches around the east sides of 

Table 2. St. Clair River reaches delimited by water-level gaging stations and used for this study.

[ID, identification; FG, Fort Gratiot; DP, Dunn Paper; PE, Point Edward; BR, Black River; DD, Dry Dock; SP, State Police; EStag, East Stag Island; 

PL, Port Lambton; EFawn, East Fawn Island, and AL, Algonac]

Reach number Reach ID

Reach limits for the inverse model of 

St. Clair River (Gaging station name abbreviations 

used in italics where appropriate) 

Channel designation(s) 

for the standard hydraulic 

model of 

St. Clair River

Reach length 

(kilometers)

Upstream Downstream

1 FG_DP FG DP Main Ch-1 0.447

2 DP_PE DP PE Main Ch-1 1.47

3 PE_BR PE BR Main Ch-1 2.44

4 BR_DD BR DD Main Ch-1 3.89

5 DD_SP DD SP Main Ch-1,2,3 15.6

5.5 EStag Upstream limit 

of Stag Island

Downstream limit 

of Stag Island

Stag Island Ch 5.03

6 SP_PL SP PL Main Ch-3,4,5 18.5

6.5 EFawn Upstream limit 

of Fawn Island

Downstream limit 

of Fawn Island

Fawn Island Ch 3.15

7 PL_AL PL AL Main Ch-5 5.09

2

3 1 K A R

n = ⋅ ⋅
10 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Stag and Fawn Islands were included to describe the partitioning of flows measured around the islands with ADCP after 

1995. 

Conveyance Ratios

Within the St. Clair River, inflows and outflows from 

intervening tributaries and distributaries generally constitute a 

small percentage of the total flow of the river. Over extended 

periods of time, the unsteadiness of flow has little effect on the 

average flow, so that, for the i

th April 1 to November 30 period, 

the average flow in all reaches j are approximately equal Qi,j

≃ Q ̄i,.. By further assuming that the change in average velocities between reaches is small (thus that the second term on the 

right-hand side of equation 4 is negligible) with respect to the 

change in water-surface elevation over the reach, equation 2 

can be applied to pairs of reaches and solved for flow to compute conveyance ratios as: 

, i = {1962,1963,...2007}. (5)

A time series of conveyance ratios provides an indication 

of the relative change in conveyance among reaches based 

on direct measurements of water level over time. Monotonic 

trends in the annual series of conveyance ratios with time 

were evaluated based on the statistical significance of Kendall’s tau rank correlation coefficient (Conover, 1980). Where 

significant, the linear relation between conveyance ratios and 

time was estimated by use of Sen’s nonparametric estimator 

of slope ˆβSen1 (Gilbert, 1987). The slope was fitted through the 

point defined by the median conveyance ratio and median time 

to define the intercept (ˆβSen0

). 

The annual series of conveyance ratios K.,j /K.,j ́ may 

depend on estimates of average flow magnitudes in each 

year. Estimates of average flow magnitudes ( ˆQ ̄ i,.) for April 

through November were computed by use of monthly flows 

for the St. Clair River (Croley and Hunter, [2006]), which 

were developed by the Coordinating Committee for Great 

Lakes Basic Hydraulic and Hydrologic Data (Coordinating 

Committee). The statistical significance of Kendall’s tau correlation coefficient between conveyance ratios and annual-flow 

estimates was used to identify flow dependency. Where significant, a conditional estimate of conveyance ratios on flows was 

computed as:

=

 ˆQ1

 = ˆβSen0

( j,,j ́) + ˆ

βSen1

( j,,j ́)

•i + ei

( j,j ́), 

(6)

i = {1962,1963,...2006}.

Otherwise, the conditional estimate of the conveyance ratio 

was the median conveyance ratio of the time series. 

Innovations of the conveyance series were computed as 

the differences between the computed conveyance ratios and 

conditional conveyance ratios (equation 7). Innovations series 

were used to investigate the sensitivity of interpretations of 

changes in conveyance based on the conveyance ratios to 

estimates of the annual variations in flow. 

Qˆ

i

(7)

i = {1962,1963,...2006}

Inverse 1-D Hydrodynamic Modeling of the St. 

Clair River

Inverse modeling applies statistical techniques to estimate 

parameters of physically based models from sets of measurements. In this report, the physically based model is the HECRAS 1-D flow model of the St. Clair River. Water levels at the 

six intervening gaging stations and flow-measurement data 

are used to estimate model parameters that varied annually 

and parameters that were fixed over discrete intervals. Model 

parameters are values of the effective channel roughness 

within the main channel of reaches delimited by water-level 

gaging stations along the river. In this report, channel roughness is adjusted to improve the match among simulated and 

measured water levels and flows. 

Both water-level and flow data are needed to estimate 

unique values of effective channel roughness when water-level 

data are used as boundary conditions. When too few flow measurements are available, parameter estimates are highly correlated and depend on arbitrary, initial parameter values. This 

dependency is not a result of the inverse modeling approach, 

but is revealed in the output statistics. Highly correlated 

estimates of effective roughness parameters cannot reliably 

be used to infer reach conveyance. Although water-level data 

are generally available throughout the 46-year period of study 

from 1962–2007, one or more flow measurements are only 

available in 24 of these years. The availability of flow measurements necessarily limits the estimation of annual conveyance characteristics to these 24 years. 

1-D Hydrodynamic Model of the St. Clair River

Unsteady flow and water levels were simulated for 

selected years during periods from April 1 to November 30 

by use of the standard version of the HEC-RAS model of the 

St. Clair River. Hourly water-level data from the Fort Gratiot 

and Algonac, Michigan, gaging stations were used as boundary conditions. Assigned channel-roughness values in the main 

channel of the standard version were replaced with effective 

channel-roughness parameters estimated by inverse modeling 

techniques. Channel-roughness values in over-bank areas, and 

contraction and expansion energy-loss coefficients (specified 
Approach 11

in the standard model of 0.1 and 0.3, respectively), were used 

without modification. Flow augmentations for tributaries and 

diversions for distributaries described in the standard model 

were used. 

Inverse Modeling

In this report, inverse modeling applied a nonlinearregression technique referred to as universal parameter 

estimate code (UCODE) (Poeter and others, 2005) to iteratively modify parameters representing the effective channel 

roughness for selected reaches in a 1-D hydrodynamic model 

of the St. Clair River. The objective of the inverse modeling analysis was to minimize the sum of squared weighted 

residuals (differences) between simulated and measured water 

levels and flows. The iterative process was terminated when 

the parameter-estimation process converged or the specified 

number of maximum parameter iterations was exceeded. To 

assess whether parameters converged uniquely, two sets of 

starting values were used to initialize the parameter-estimation 

process, and converged pairs of estimates were compared. 

Annual-varying parameters were estimated to identify trends 

in the effective channel-roughness values that are needed to 

account for systematic differences between measured and 

simulated water levels and flows. A brief overview of inverse 

modeling is provided to aid in understanding the interpretations presented in this report. 

Nonlinear Regression

Nonlinear regression provides a generally applicable 

method for calibrating models in which simulated values vary 

nonlinearly with model parameters (Hill, 1998). In this study, 

the objective was to minimize the sum of squared weighted 

residuals as 

S b y y b i

N

i i i ( ) = −   ( )  ω1

2

S  , (8)

where 

S(b_) is the sum of squared weighted residuals 

between measured and simulated values;

b_ is a vector of model parameters of length p; 

N is the number of measurements in a particular 

year; 

yi is the i

th measurement of water level or flow; 

ỹi (b_) is the i

th simulated value, which is a function 

of the underlying model and associated 

parameters; and

ωi is the weight for the i

th measurement. 

The difference [yi– ỹi (b)]is referred to as a residual; the difference ωi 

1⁄2[yi

– ỹi (b_)] is a weighted residual. Measurement 

weights ω provide a mechanism for expressing the uncertainty 

of individual measurements and for including measurements 

in different scales (for example, flow, in cubic meters per 

second and water-surface elevation, in meters) within the same 

objective function. In this report, the weight matrix Ω was 

described as a diagonal matrix with elements ωii, which does 

not account for correlations among measurement errors. 

The potential for temporal correlation in the errors of 

water-level residuals was reduced by comparing water-level 

measurements with simulated values at 8-hour sampling intervals. This 8-hour sampling interval provided 3 measurements 

per day for the 244-day periods from April 1 to November 30, 

or 732 measurements per year at each water-level gaging station. For the 6 water-level gages used in the calibrations, this 

nominally provided 4,390 water-levels annually for parameter estimation. Water-level values were assigned a weight ω

equal to (1/0.025)2 or 1,600, which is inversely proportional 

to a standard deviation of 0.025 m associated with water-level 

measurements. 

In contrast, 351 conventional and 353 ADCP flow measurements during selected years from 1962 to 2007 provided 

an average of 28 flow measurements per year for model 

calibration. Measured flows Qm were assigned a weight ω that 

varied with flow magnitude and was equal to (1/0.05•Qm)2. 

This weight is inversely proportional to a 5-percent coefficient 

of variation associated with measured flows. 

Parameter Convergence

Parameter convergence is based on three criteria that 

control the iterative procedure in the UCODE developed by 

Poeter and others (2005). First, the keyword TolPar specifies 

the maximum fractional change in parameter values between 

parameter-estimation iterations required for termination. If 

changes in all the parameters are less than the specified TolPar of 0.015, the criterion is satisfied. Second, the keyword 

TolSOSC specifies the fractional decline in the SOSC, here 

S(b_), over three parameter-estimation iterations. In this report, 

a TolSOSC of 1 percent was used. Third, the keyword MaxIter specifies the maximum number of parameter-estimation 

iterations, which was set to 50. If the TolPar and TolSOSC

criteria are both satisfied before the specified maximum 

number of parameter-estimation iterations are exceeded, the 

parameter estimates are considered to have converged for the 

specified initial values. Otherwise, parameter estimates are not 

considered to have converged. 

To reduce the possible dependence of final parameter 

estimates on the arbitrary initial values, two sets of effective 

channel-roughness values were used to initialize the parameter-estimation process. In the first set, effective channel-roughness values for all reaches were initialized at a value of 0.015, 

which is relatively low compared with the expected roughness 

characteristic of the channel. The corresponding parameter 

estimate for the yth year and rth reach resulting from this 

starting value is symbolized nˆy,r,0.015. In the second set, effective channel-roughness values were initialized at a relatively 

high value of 0.040. Similarly, the corresponding parameter 

estimate resulting from this starting value is symbolized 

nˆy,r,0.040. Percent relative differences (PRD) were computed for 
12 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

each year y and each reach r for which parameter estimation 

converged as 

PRD(nˆ y,r,0.040,nˆ y,r,0.015) = 

|nˆ y,r,0.040 – nˆ y,r,0.015|/max(|nˆ y,r,0.040|,|nˆ y,r,0.015|) •100. (9)

If the PRD between converged parameter estimates were 

less than 3 percent for all reaches, parameters were considered 

to have converged uniquely for that year. If the PRD for only 

one reach exceeded 3 percent, parameters were considered to 

have converged uniquely for the year except for the identified 

reach. If the PRD exceeded 3 percent for more than one reach, 

parameters were not considered uniquely converged for that 

year. 

Minimum variance estimates of the effective channelroughness parameters were computed as 

,

, , .

, , .

, , .

, , . var var

y r

y r

y r

y r

y r

      0 015

0 015

0 040

0 040

0 015 0 040

1 1

  

var y r, , . var  y r, , .

ˆ

n ˆn

ˆ

n ˆn

ˆ

n ˆn

ˆ

n (10)

where 

var(nˆ y,r,.) is the estimated variance of the corresponding 

parameter. 

Parameter Magnitudes and Uncertainties

For years of data in which parameters converged, several 

overall statistical measures of fit were computed. In particular, 

the estimated error variance is computed as 

s2 = S (b_f)/(N – p), (11)

where

S (b_f) is the weighted sum of squared residuals 

computed with the final estimate of the 

parameter vector.

This variance, together with the final estimate of the sensitivity matrix, is used to compute the symmetrical covariance 

matrix of the parameter vector as V(b_f) = s2

(XT

f •Ω •Xf)–1. The 

standard error of the regression is the square root of the estimated-error variance. The diagonal elements are the variances 

of the individual parameters s2

bj

 and the off-diagonal elements 

are covariances between parameter pairs. An (1–α)-level 

linear confidence interval is computed for each parameter as 

bj ± t

N – p,1 – α/2•sbj

 , where t is the Student-t statistic with N – p

degrees of freedom. Finally, the parameter-correlation matrix 

is computed from covariance elements of covij = V(b_f)ij and 

the correlation between the i

th and jth parameter corij = 

covij / . Parameter correlations can vary in the 

interval [–1, 1]. Parameter correlations with an absolute value 

greater than 0.95 indicate that some degradation in parameter 

estimates may have occurred because of the ambiguity of the 

effects caused by two model parameters on simulated results. 

Parameterization Strategy

In this report, the model-parameterization strategy refers 

to the flexibility of parameters to vary annually. Each period 

from April 1 to November 30 with adequate flow and waterlevel data provides a basis for independent model calibration 

by inverse modeling. Patterns of change in annual sequences 

of parameters were used to identify time-varying conveyance characteristics at individual reaches. Although persistent 

changes need not be monotonic to be significant, non-monotonic changes are difficult to confirm unambiguously unless 

the timings of likely changes in trends are known before they 

are revealed by the analysis. To help identify both monotonic 

and non-monotonic persistent changes, a minimum annual 

sequence of parameters representing five or more contiguous 

years was required. 

A fixed-parameter strategy follows a similar calibration approach as the time-varying strategy, but requires the 

additional assumption that reach conveyance is invariant 

over the time interval used in parameter estimation. For the 

fixed-parameter model, the validity of this assumption was 

investigated by analyzing the flow residuals formed as the differences between measured and simulated values. In particular, the annual sequence of effective channel-roughness values 

for reaches estimated in the time-varying models were used 

as basis functions for estimating trend components in the flow 

residuals. Those reaches whose basis function was significantly associated with flow residuals were considered to have 

significant time variability. 

Simulation of Flow under Annual 

Parameterizations Derived from Inverse 

Modeling

Each set of annual parameters for the 1-D hydrodynamic 

model of the St. Clair River characterized the conveyance 

during that year based on the available water-level and flow 

data. If conveyance were invariant with time, each annual 

parameterization would result in simulations that produce 

similar average flows for each set of water-level boundary 

conditions. Under conveyance invariance, variability among 

average simulated flows for specified water-level boundary conditions should only reflect parameter uncertainty. In 

contrast, persistence or trends in average simulated flow with 

annual parameterization would indicate conveyance change. 

Furthermore, because the water-level boundaries are invariant with the annual parameterization, the relative changes in 

flow with parameterization are identical to relative changes in 

conveyance. 

Following equation 2, average flow simulated under conveyance characteristics estimated for the i

th year using waterlevel boundary conditions from the jth year approximately 
Results of Conveyance Analyses 13

equals the conveyance described in the i

th year multiplied by 

the square root of the water-surface slope between the two 

gaging stations as Qi,j ≈ Ki

• Sj

1⁄2. The set Q•,j represents the set of 

average flows simulated under all annual parameterizations 

i Î {1962,1963,...2007} using boundary conditions for only 

the j

th year. 

For the parameterization set P indexed by p, say p = 

{1,2}, and an annual boundary-condition set B indexed by b, 

say b = {1}, we can write Qp,b = Q1,1 = K1

• S1

1⁄2 and Qp2,b = Q2,1 = 

K2

• S1

1⁄2. Solving for S11⁄2, we can write Q

Q

K

K

2 1

1 1

2

1

,

,

= .

Subtracting 1 from both sides allows us to write 

, or equivalently, 

Q Q

Q

K K

K

2 1 1 1

1 1

2 1

1

, ,

,

− = − . (12)

Thus, with respect to changes in parameterizations 

for fixed-boundary conditions, relative changes in flow are 

identical to relative changes in conveyance. The base year 

for parameterization can arbitrarily be taken as 1962, which 

provides a basis for describing the percent relative change in 

conveyance since 1962. 

Generally, the relative change in simulated flow for 

any parameterization set p ́>1ÎP with respect to p=1 for 

boundary conditions b can be written Δ  Q 

Q

p b

b

',

,

1

1

. 

The maximum relative changes in average flow for all sets of 

boundary conditions b Î B can be represented by the set 

, 

the minimum relative change as , 

and the average relative change as

 for all p ́>1 Î P. 

Results of Conveyance Analyses

Conveyance changes within reaches were evaluated 

based on water-level measurements at gaging stations and 1-D 

hydrodynamic-model simulations. Water-level measurements 

were used to compute water-surface fall and conveyance ratios 

among reaches. Analyses of these data provide a nearly continuous description of relative conveyance changes from 1962 

to 2007. Inverse 1-D hydrodynamic modeling was used to 

estimate a partial annual series of effective channel-roughness 

parameters from 1962 to 2007 for periods when both flow and 

water-level data were available to support estimation. Both 

analyses indicate systematic variations in conveyance within 

some reaches. Annual parameterizations of the 1-D hydrodynamic model also were used to evaluate changes in river 

conveyance from 1962 to 2007. 

Inferring Reach-Conveyance Changes from 

Water-Level Measurements

Synchronized water-level differences among eight gaging stations spanning the St. Clair River were used to characterize the annual fall in water levels within seven reaches 

forming the study area (table 2). Average water-surface falls 

between adjacent gages divided by their corresponding reach 

lengths were used to approximate annual energy slopes. The 

square root of the ratio of energy slopes along two reaches 

is inversely proportional to their corresponding conveyance 

ratios. The time series of conveyance ratios provide a basis for 

interpreting relative conveyance changes among reaches. 

Water-Surface Fall

Monthly average water-level data from the eight selected 

gaging stations along the St. Clair River were used to compute 

average annual water-surface falls through the corresponding 

reaches. When one or more months of data during this period 

were missing, the corresponding year of data also was considered missing and not used in the analysis. 

Some difficulties were encountered interpreting the 

reported monthly average water-level fall data. In the Point 

Edward to Black River reach (PE_BR), negative average 

monthly water-level falls were reported in 1964 for July, 

August, and September; in 1965 for April, July, August, 

October, and November; in 1966 for May through September; 

in 1967 for April through November (with July and August 

data missing); in 1968 for April through July, and October and 

November; in 1969 for April through July and November; in 

1970 for May and July; in 1971 for July and August; in 1973 

for April and August; in 1974 for June through August; and 

in 1982 for June. Negative monthly average water-level falls 

indicate that water would be flowing upstream within the 

reach for extended periods within the month, which is physically impossible. April to November periods when computed 

water-level falls were negative were not used to assess conveyance change. 

Opposite trends in water-level fall were detected in two 

adjacent reaches on St. Clair River. Water-level falls generally decreased from 1962 to 1992 in the Dunn Paper to 

Point Edward reach (DP_PE), and falls generally increased 

from 1962 to 1994 in the Point Edward to Black River reach 

(PE_BR) (fig. 5). Trends in water-surface fall may be associated with changes in either flow or conveyance or both. 

Opposite (decreasing and increasing) trends in water-surface 

fall through adjacent reaches, however, are inconsistent with 

the null hypothesis of no conveyance changes in either reach, 

because all reaches pass essentially the same annual flow. 
14 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Figure 5. Differences in average April 1 to November 30 water levels between adjacent reaches on the St. Clair River.

1960 1970 1980 1990 2000 2010

0.0

0.1

0.2

0.3

0.4

0.5

W

ATER

LEVEL DIF

F

E

RENCE

S

WITHI

N RE

ACH, IN ME

TERS

Reaches Delimited by Gaging Stations

Fort Gratiot to Dunn Paper (FG_DP)

Dunn Paper to Point Edward (DP_PE)

Point Edward to Black River (PE_BR)

Black River to Dry Dock (BR_DD)

Dry Dock to State Police (DD_SP)

State Police to Port Lambton (SP_PL)

Port Lambton to Algonac (PL_AL)
Results of Conveyance Analyses 15

Based on the available water-level data, a trend in conveyance 

is indicated at one or more of these adjacent reaches. Given 

the reflective symmetry of the trends, however, the possibility of an undocumented datum change at one or more of the 

water-level gaging stations cannot be definitively dismissed as 

a factor associated with the apparent trends in water-level fall 

and conveyance. 

Of the seven reaches investigated, negative monotonic 

trends in water-surface fall were detected (p-value < 0.05) at 

the reaches from Dunn Paper to Point Edward DP_PE and Dry 

Dock to State Police DD_SP, while positive monotonic trends 

in water-surface fall were detected at the reaches from Fort 

Gratiot to Dunn Paper FG_DP and Point Edward to Black 

River PE_BR (table 3). Even for reaches from State Police to 

Port Lambton SP_PL and Port Lambton to Algonac PL_AL

where the evidence of trends was less compelling (p-value < 

0.10), inspection of the convergence between water-surface 

fall profiles from 1962 to 1975 indicates that rates of changes 

in conveyance ratios may be time varying. 

Monthly estimates of flow through the St. Clair River 

have been computed for the period 1860 to 2006 by the 

Coordinating Committee, although values after 1990 are 

considered provisional. Use of the equation K Q S i j , , = i f . ,i j

would provide a basis for directly computing annual 

conveyance estimates for each reach. These estimates of 

flow, however, are based on stable relations assumed between 

measured water level and fall. The stability (time invariance) 

of conveyance (and hence the relation between flow, water 

level, and fall) is the subject of this investigation; therefore, 

interpretation of the resulting estimates would be problematic 

and are not used as a direct estimate of conveyance in this 

report. 

Conveyance Ratios

Conveyance ratios were computed by use of equation 5 

and water-level data from 1962–2007 (fig. 6). Results indicate 

that the conveyance in the FG_DP reach decreased significantly (α = 0.05) with time (table 4) relative to all reaches 

except PE_BR, where the relative conveyance may have 

increased (p-value = 0.0661). The relative decrease in conveyance at FG_DP with respect to DP_PE was greater than other 

reaches (-0.00949), because of the increase in conveyance 

at DP_PE relative to all downstream reaches. The tendency 

for an increased conveyance at FG_DP relative to PE_BR is 

thought to be related to the larger negative trend at PE_BR

relative to all downstream reaches. The conveyance at BR_DD

decreases significantly with respect to conveyance at DD_SP. 

Much of this relative decrease in conveyance is attributable 

to the sharp increase in conveyance of DD_SP relative to 

SP_PL, which is particularly evident in data prior to 1970. The 

increase in conveyance of DD_SP with respect to PL_AL in 

the pre-1970 data is consistent with increasing relative conveyance at DD_SP, although not statistically significant over 

the entire period. Furthermore, the decrease in conveyance of 

BR_DD relative to DD_SP is not evident in reaches further 

downstream. The patterns in relative conveyance at PL_AL are 

consistent with patterns in other reaches indicating that PL_AL

may be one of the more stable reaches in the St. Clair River. 

Possible relations between annual conveyance ratios and 

flow magnitudes were analyzed to determine whether systematic variations in flow might affect the interpretation of trends 

in conveyance ratios. In particular, greater flows likely would 

be associated with greater conveyance and water-level falls 

Table 3. Trends in water-level fall within St. Clair River reaches delimited by water-level gaging stations, 1962–2007.

[Reach ID, reach identification; FG_DP, Fort Gratiot to Dunn Paper; DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward to Black River; 

BR_DD, Black River to Dry Dock; DD_SP, Dry Dock to State Police; SP_PL, State Police to Port Lambton; PL_AL, Port Lambton to Algonac]

Reach ID Kendall’s tau p-value1 

Slope (change in 

average water-level 

fall, in meters, per year)

Median fall 

(meters per 

kilometer)

Median year 

of data

FG_DP 0.3169 0.0004 0.000708 0.106 1986.5

DP_PE -.4831 .0000 -.00253 .0942 1988.0

PE_BR .1652 .0084 .000353 .0219 1989.0

BR_DD -.1159 .2444 -.000107 .107 1984.0

DD_SP -.6570 .0000 -.00227 .379 1984.5

SP_PL -.1604 .0842 -.000732 .354 1985.0

PL_AL -.1778 .0555 -.000294 .0559 1985.0

1

p-value indicates the likelihood that the null hypothesis of no trend in water-level falls with time is true given the measured water level data. 

A small p-value (less than 0.05) indicates that the null hypothesis may be rejected.
16 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Figure 6. Lower diagonal scatter-plot matrix of conveyance-ratios from 1962 to 2007 for paired reaches on the St. Clair River. 

-0.20-0.10

0.000.100.200.30

0.080.120.160.200.240.280.280.320.360.400.44

0.240.280.320.360.400.44

0.240.280.320.36

1960 1980 2000

0.160.180.200.220.240.26

0.100.200.300.400.500.60

0.400.500.600.700.800.90

0.400.500.600.700.800.400.500.600.70

1960 1980 2000

0.200.300.400.500.60

1.002.003.004.005.001.002.003.004.005.001.002.003.004.00

1960 1980 2000

0.501.001.502.002.503.003.50

0.850.900.951.001.051.10

0.760.800.840.880.920.96

1960 1980 2000

0.500.550.600.650.700.750.80

0.800.840.880.920.961.00

1960 1980 2000

0.500.600.700.800.90

1960 1980 2000

0.600.650.700.750.800.850.90

B

R_

D

D

FG_D

P

PE_

B

R

DD_S

P

SP_P

L

P

L

_

AL

DP_P

E

DP_P

E

PE_

B

R

BR_D

D

DD_

S

P

SP_

P

L

FG_DP: Fort Gratiot to Dunn PaperDP_PE: Dunn Paper to Point EdwardPE_BR: Point Edward to Black RiverBR_DD: Black River to Dry DockDD_SP: Dry Dock to State PoliceSP_PL: State Police to Port LambtonPL_AL: Port Lambton to Algonac

EXPLANATION

Reach Identifier by Water-Level Gaging Station Trends based on a probability of less than 0.05 that 

the computed Kendall’s tau correlation coefficient would be obtained in random sampling from a population whose true correlation is zero:

Negative trendNo trendPositive trend

Note: In the lower diagonal scatter-plot matrix above, reaches are identified above each column  and to the left of each row. The column reach identifier is associated with the numerator and  the row reach identifier is associated with the denominator of the conveyance ratio. 
Results of Conveyance Analyses 17

Table 4. Trends in conveyance ratios among paired reaches on the St. Clair River.

[Reach ID, reach identification; FG_DP, Fort Gratiot to Dunn Paper; DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward to Black River; 

BR_DD, Black River to Dry Dock; DD_SP, Dry Dock to State Police; SP_PL, State Police to Port Lambton; PL_AL, Port Lambton to Algonac]

Upstream 

reach

Downstream 

reach Kendall’s tau p-value Intercept Median slope Median ratio Median year

FG_DP DP_PE -0.8050 <0.0001 19.4 -0.00949 0.490 1989.0

PE_BR .2344 .0661 -2.33 .00126 .184 1990.0

BR_DD -.4659 <.0001 3.46 -.00157 .338 1986.0

DD_SP -.6051 <.0001 4.34 -.00203 .308 1986.5

SP_PL -.5385 <.0001 3.15 -.00145 .278 1987.0

PL_AL -.4656 <.0001 2.49 -.00114 .214 1987.0

DP_PE PE_BR .6172 <.0001 -16.9 .00867 .383 1990.0

BR_DD .7916 <.0001 -15.1 .00794 .658 1988.0

DD_SP .8381 <.0001 -11.8 .00622 .604 1988.5

SP_PL .8674 <.0001 -12.5 .00657 .553 1989.0

PL_AL .7159 <.0001 -8.23 .00435 .423 1989.0

PE_BR BR_DD -.3826 .0015 32.9 -.01565 1.76 1989.0

DD_SP -.4394 .0002 38.7 -.01866 1.61 1989.0

SP_PL -.3609 .0047 24.5 -.01157 1.45 1990.0

PL_AL -.4069 .0013 23.6 -.01132 1.09 1990.0

BR_DD DD_SP -.5960 .0000 5.18 -.00214 .9234 1984.0

SP_PL -.0662 .5462 1.28 -.000222 .837 1984.5

PL_AL -.1684 .1191 2.95 -.00117 .634 1984.5

DD_SP SP_PL .6146 <.0001 -2.37 .00165 .903 1985.0

PL_AL .0964 .3765 -0.556 .000626 .687 1984.5

SP_PL PL_AL -.1042 .3304 1.88 -.000568 .756 1985.0
18 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

than would lesser flows. Monotonic relations between conveyance ratios and average April 1 to November 30 flows from 

monthly values for 1962–2006 published by the Coordinating 

Committee (fig. 7) were investigated by use of nonparametric 

Kendall’s tau correlation coefficient and Sen’s slope estimator 

(table 5). The five conveyance ratios for reaches downstream 

from DP_PE generally were negatively associated with flow. 

In contrast, conveyance ratios at the three reaches downstream 

from BR_DD generally were positively associated with flow, 

as were the conveyance ratios between DD_SP and PL_AL, 

and SP_PL and PL_AL. No other statistically significant correlations were detected. Relations between conveyance ratios 

and flow magnitudes were not consistent among reaches.

Innovations in the conveyance ratios (equation 7) also 

were analyzed for trends. Despite significant associations 

between conveyance ratios and estimates of average flow for 

some pairs of reaches, the lower diagonal scatter-plot matrix 

of conveyance-ratio innovations (fig. 8) showed similar trends 

(table 6) as the conveyance ratios themselves. Other than scaling changes associated with different means of the two ratios, 

the primary difference between the two sets of plots occurred 

for the reaches DD_SP and PL_AL. In particular, the trend 

in the conveyance ratios was not significant, while the trend 

in the conveyance-ratio innovations was positive (p-value = 

0.0017). Thus, available flow information provides only minor 

changes in the interpretation of conveyance change than evidence based on the ratios themselves. 

Inferring Changes in Reach Conveyance from 

1-D Hydrodynamic-Model Parameters

Two strategies were implemented to ascertain possible 

changes in reach conveyance from inverse 1-D hydrodynamic 

modeling of the St. Clair River. First, time-varying parameter 

estimation was used to describe possible continual changes 

in reach conveyance by use of annual estimates of model 

parameters describing effective channel roughness. Second, 

fixed-parameter estimation was used to describe possible 

discrete changes in reach conveyance from 1962 to 1968, 

1981 to 1985, and 1996 to 2007. Annual estimates of effective 

channel roughness from 1996 to 2007 in reaches were used as 

a basis for estimating variations in flow not accounted for by 

the fixed-parameter model. The results were used to identify 

reaches where conveyance changes are affecting river-flow 

estimates. 

Figure 7. Average flow of the St. Clair River for 1962–2006 based on coordinated monthly flow estimates. 

1960 1970 1980 1990 2000 2010

YEAR

4,000

4,500

5,000

5,500

6,000

6,500

7,000

ESTIM

A

T

E

D AVERAG

E

F

LOW

FROM APRI

L 1 TH

R

O

U

G

H

NO

VEMB

ER 30, IN 

CUBI

C M

ETER

S PE

R SE

C

O

N

D
Results of Conveyance Analyses 19

Table 5. Relation among estimated flow from April 1 to November 30 and conveyance ratios of reaches on St. Clair River.

[Reach ID, reach identification; FG_DP, Fort Gratiot to Dunn Paper; DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward to Black River; 

BR_DD, Black River to Dry Dock; DD_SP, Dry Dock to State Police; SP_PL, State Police to Port Lambton; PL_AL, Port Lambton to Algonac]

Upstream 

reach

Downstream 

reach Kendall’s tau p-value Intercept Median slope Median ratio Median year

FG_DP DP_PE 0.1872 0.1237 0.224 0.0000509 0.505 5,531.9

PE_BR -.2046 .1170 .320 -.0000246 .184 5,531.9

BR_DD -.1846 .0959 .434 -.0000175 .338 5,471.2

DD_SP -.0683 .5391 .345 -.0000068 .308 5,426.2

SP_PL -.0384 .7455 .296 -.0000033 .278 5,421.9

PL_AL .1778 .1195 .146 .0000128 .215 5,421.9

DP_PE PE_BR -.4069 .0013 .991 -.000110 .382 5,552.5

BR_DD -.3226 .0069 1.18 -.0000949 .657 5,552.5

DD_SP -.3479 .0030 1.03 -.0000782 .601 5,547.5

SP_PL -.3105 .0123 .986 -.0000791 .548 5,531.9

PL_AL -.2339 .0618 .671 -.0000448 .423 5,531.9

PE_BR BR_DD .1976 .1162 .834 .000169 1.77 5,552.5

DD_SP .2137 .0887 .548 .000191 1.61 5,552.5

SP_PL .1576 .2397 .753 .000126 1.45 5,547.5

PL_AL .2512 .0576 .256 .000151 1.09 5,547.5

BR_DD DD_SP .1966 .0610 .826 .0000174 .922 5,552.5

SP_PL .2683 .0132 .730 .0000188 .835 5,547.5

PL_AL .4293 .0000 .323 .0000561 .635 5,547.5

DD_SP SP_PL .0523 .6355 .874 .0000054 .904 5,531.9

PL_AL .3217 .0024 .458 .0000415 .687 5,531.9

SP_PL PL_AL .3393 .0016 .506 .0000453 .757 5,531.9
20 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Figure 8. Lower diagonal scatter-plot matrix of conveyance-ratio innovations from 1962 to 2006 for paired reaches on the St. Clair River. 

-0.20-0.10

0.000.100.200.30

-0.12-0.08-0.04

0.000.040.080.12

-0.08-0.04

0.000.040.080.12

-0.04

0.000.040.080.12

-0.04

0.000.040.08

1960 1980 2000

-0.06-0.04-0.02

0.000.020.040.06

-0.40-0.30-0.20-0.10

0.000.100.20

-0.30-0.20-0.10

0.000.100.20

-0.20-0.10

0.000.100.20

-0.20-0.10

0.000.10

1960 1980 2000

-0.20-0.10

0.000.100.20

-1.00

0.001.002.003.00

-1.00

0.001.002.003.00

-1.00

0.001.002.003.00

1960 1980 2000

-0.50

0.000.501.001.502.002.50

-0.10-0.05

0.000.050.100.15

-0.04

0.000.040.080.12

1960 1980 2000

-0.08-0.04

0.000.040.080.12

-0.12-0.08-0.04

0.000.040.08

1960 1980 2000

-0.15-0.10-0.05

0.000.050.100.15

1960 1980 2000

-0.12-0.08-0.04

0.000.040.080.12

B

R_

D

D

FG_D

P

PE_

B

R

DD_S

P

SP_P

L

P

L

_

AL

DP_P

E

DP_P

E

PE_

B

R

BR_D

D

DD_

S

P

SP_

P

L

FG_DP: Fort Gratiot to Dunn PaperDP_PE: Dunn Paper to Point EdwardPE_BR: Point Edward to Black RiverBR_DD: Black River to Dry DockDD_SP: Dry Dock to State PoliceSP_PL: State Police to Port LambtonPL_AL: Port Lambton to Algonac

EXPLANATION

Reach Identifier by Water-Level Gaging Station Trends based on a probability of less than 0.05 that 

the computed Kendall’s tau correlation coefficient would be obtained in random sampling from a population whose true correlation is zero:

Negative trendNo trendPositive trend

Note: In the lower diagonal scatter-plot matrix above, reaches are identified above each column  and to the left of each row. The column reach identifier is associated with the numerator and  the row reach identifier is associated with the denominator of the conveyance ratio. 
Results of Conveyance Analyses 21

Table 6. Trends in conveyance ratio innovations among paired reaches on the St. Clair River.

[Reach ID, reach identification; FG_DP, Fort Gratiot to Dunn Paper; DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward to Black River; 

BR_DD, Black River to Dry Dock; DD_SP, Dry Dock to State Police; SP_PL, State Police to Port Lambton; PL_AL, Port Lambton to Algonac]

Upstream 

reach

Downstream 

reach Kendall’s tau p-value Intercept Median slope Median ratio Median year

FG_DP DP_PE -0.8039 <0.0001 19.5 -0.00978 0.000004 1988.5

PE_BR .2506 .0534 -2.70 .00136 .000001 1989.5

BR_DD -.4564 <.0001 3.13 -.00158 .000000 1985.5

DD_SP -.6220 <.0001 4.21 -.00212 -.000004 1986.0

SP_PL -.5420 <.0001 2.93 -.00148 -.000001 1986.5

PL_AL -.4566 <.0001 2.28 -.00115 .000004 1986.5

DP_PE PE_BR .3977 .0017 -11.0 .00555 .0157 1989.5

BR_DD .6815 <.0001 -12.8 .00646 .0125 1987.5

DD_SP .6728 <.0001 -9.43 .00475 .0100 1988.0

SP_PL .6734 <.0001 -10.2 .00516 .0247 1988.5

PL_AL .7137 <.0001 -9.01 .00453 .000002 1988.5

PE_BR BR_DD -.3952 .0012 34.3 -.0173 .000005 1988.5

DD_SP -.4677 .0001 40.0 -.0201 .000002 1988.5

SP_PL -.3842 .0031 25.4 -.0128 -.000002 1989.0

PL_AL -.4187 .0012 25.1 -.0126 .000004 1989.0

BR_DD DD_SP -.6385 <.0001 4.41 -.00222 .000000 1983.5

SP_PL -.0707 .5244 0.356 -.00018 .00576 1984.0

PL_AL .0073 .9555 -0.120 .00006 -.00475 1984.0

DD_SP SP_PL .6446 <.0001 -3.44 .00173 .000003 1984.5

PL_AL .3370 .0017 -3.49 .00176 .00255 1984.5

SP_PL PL_AL .0964 .3765 -1.11 .00056 .000948 1984.5
22 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Time-Varying Parameter Estimation

Inverse modeling techniques were used to estimate the 

annual effective channel-roughness coefficients for years 

from 1962 to 2007 when water-level boundary and flow 

data were available to support estimation. Tabled results are 

presented separately for conventional measurements collected 

prior to 1986 and ADCP measurements obtained from 1996 

to 2007. No flow measurements were documented for 1986 

to 1995. Conventional flow measurements and water-level 

data supported estimation of roughness coefficients in seven 

reaches delimited by water-level gaging stations. ADCP flow 

measurements and water-level data supported estimation of 

the seven reaches estimated with conventional flow measurements, plus two additional reaches representing the east 

branches of Stag and Fawn Islands where flow was measured 

with ADCP instruments. 

Effective Channel Roughness Estimated from 

Conventional Measurements 

Annual parameter estimates for seven reaches associated with years of conventional flow measurements (prior to 

1986) are shown in table 7. Of the 13 years in which one or 

Table 7. Effective channel roughness estimated for reaches spanning the study area of the St. Clair River in years with one or more 

conventional flow measurements from 1962 to 1985.

[PRD, is the percent relative difference. An “n” prepended to the reach identifier indicates the effective channel roughness coefficient of the corresponding 

reach. Reach identifiers include: FG_DP, Fort Gratiot to Dunn Paper; DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward to Black River; 

BR_DD, Black River to Dry Dock; DD_SP, Dry Dock to State Police; SP_PL, State Police to Port Lambton; PL_AL, Port Lambton to Algonac]

Year

Number 

of flow 

measurements

Sum

 of squared 

weighted 

residuals 

(PRD)

Annual estimate of effective channel roughness for corresponding reaches 

on the St. Clair River

nFG_DP

(PRD)

nDP_PE

(PRD)

nPE_BR

(PRD)

nBR_DD

(PRD)

nDD_SP

(PRD)

nSP_PL

(PRD)

nPL_AL

(PRD)

1962 67 1987

(0.15)

0.0016

(73.2)

0.0414

(0.07)

0.0205

(0.06)

0.0254

(0.40)

0.0309

(0.34)

0.0268

(0.34)

0.0255

(0.27)

1963 30 3053

(.01)

.0413

(.46)

.0337

(.01)

.0243

(.18)

.0238

(.12)

.0301

(.11)

.0266

(.12)

.0230

(.09)

1964 69 3577

(.01)

.0013

(20.8)

.0425

(.04)

.0209

(.33)

.0244

(.05)

.0299

(.02)

.0265

(.02)

.0225

(.02)

1966 30 2302

(.00)

.0128

(4.99)

.0406

(.08)

.0207

(.14)

.0247

(.22)

.0285

(.21)

.0262

(.21)

.0240

(.18)

1968 66 2502

(.00)

.0145

(.82)

.0434

(.00)

.0202

(.08)

.0251

(.05)

.0291

(.03)

.0267

(.03)

.0269

(.03)

1973 43 NB NB NB NB NB NB NB NB

1977 11 8645

(.04)

.0135

(24.9)

.0393

(1.00)

.0249

(.43)

.0242

(1.91)

.0282

(1.62)

.0267

(1.62)

.0266

(1.31)

1979 1 NB NB NB1 NB NB NB NB NB

1981 14 1922

(.01)

.0125

(5.9)

.0394

(.15)

.0271

(.15)

.0236

(.34)

.0286

(.31)

.0272

(.31)

.0254

(.22)

1982 1 2939

(.04)

.0379

(39.9)

.0410

(8.02)

.0286

(6.02)

.0269

(13.6)

.0311

(12.3)

.0297

(12.4)

.0271

(9.73)

1983 8 2291

(.02)

.0363

(8.18)

.0379

(.89)

.0261

(.63)

.0232

(1.88)

.0280

(1.62)

.0268

(1.62)

.0254

(1.25)

1984 10 1633

(.01)

.0363

(.95)

.0363

(.07)

.0266

(.01)

.0236

(.26)

.0276

(.21)

.0262

(.22)

.0263

(.10)

1985 1 2809

(.67)

.0075

(24.9)

.0352

(9.21)

.0256

(5.28)

.0207

(19.4)

.0246

(16.6)

.0236

(16.6)

.0237

(12.1)

1

 NB—No boundaries: Indicates that insufficient water-level boundary data were available at Fort Gratiot or Algonac for flow simulation and parameter 

estimation.
Results of Conveyance Analyses 23

more flow measurements were available to support parameter 

estimation, 11 years had sufficient hourly water-level data at 

the Fort Gratiot, Mich., and Algonac, Mich., gaging stations 

to provide boundary conditions needed for simulation and 

inverse modeling. In addition to the parameter estimate, the 

percent relative difference (PRD) is shown for each parameter. The PRD was computed between converged-parameter 

estimates based on starting values of 0.015 and 0.040. If the 

PRD was greater than 3, the corresponding parameter estimates were considered un-converged and un-interpretable 

with respect to conveyance. Based on this criterion, parameter 

estimates for all reaches in 1982 and 1985, when only one 

flow measurement was available for estimation, are considered 

un-interpretable. For the remaining years, parameter estimates 

for all reaches downstream from Fort Gratiot to Dunn Paper 

(FG_DP) converged uniquely. Among these years, however, 

parameter estimates for 1977 and 1983 had the greatest relative differences. Parameter estimates for the FG_DP reach 

converged uniquely for 1963, 1968, and 1984. The high 

variability in parameter estimates for these 3 years may be 

4,000 5,000 6,000 7,000

MEASURED FLOW, IN CUBIC METERS PER SECOND

4,000

5,000

6,000

7,000

SIMULATED FLOW, IN CUBIC METERS PER SECOND

Measurement Sites

Blue Water Bridge (450 meters downstream from CS-200)

GLERL (400 meters downstream from CS-202)

Bay Point (600 meters downstream from CS-202)

Roberts Landing (600 meters downstream from CS-220)

Line of agreement (n = 218, r2 = 0.845)

GLERL Great Lakes Environmental Research Laboratory

 r2 Sample coefficient of determination

n Number of measurements

Figure 9. Relation between simulated flow and flow measured on the St. Clair River from 1962 to 1985 using conventional current 

meters. 
24 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Table 8. Estimates of effective channel-roughness values for reaches spanning the study area of the St. Clair River in years with one or 

more acoustic Doppler current profiler flow measurements from 1996 to 2007.

[PRD, is the percent relative difference. An “n” prepended to the reach identifier indicates the effective channel roughness coefficient of the corresponding reach. 

Reach identifiers include: FG_DP, Fort Gratiot to Dunn Paper; DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward to Black River; BR_DD, Black 

River to Dry Dock; DD_SP, Dry Dock to State Police; SP_PL, State Police to Port Lambton; PL_AL, Port Lambton to Algonac]

Year

Number of

flow

measurements

Sum of 

squared 

weighted

 residuals 

(PRD)

Effective channel-roughness values for corresponding St. Clair River reaches

nFG_DP

(PRD)

nDP_PE

(PRD)

nPE_BR

(PRD)

nBR_DD

(PRD)

nDD_SP

(PRD)

nEStag

(PRD)

nSP_PL

(PRD)

nEFawn

(PRD)

nPL_AL

(PRD)

1996 20 1918

(0.00)

0.0381

(1.29)

0.0341

(0.13)

0.0253

(0.04)

0.0232

(0.42)

0.0269

(0.18)

0.0265

(1.56)

0.0266

(0.33)

0.0240

(0.29)

0.0248

(0.25)

1997 59 2186

(.01)

.0435

(.48)

.0347

(.03)

.0255

(.04)

.0229

(.14)

.0276

(.12)

.0282

(.11)

.0262

(.12)

.0226

(.31)

.0242

(.09)

1998 46 2167

(.01)

.0427

(.16)

.0339

(.02)

.0262

(.09)

.0236

(.03)

.0273

(.04)

.0283

(.14)

.0270

(.02)

.0242

(2.49)

.0251

(.04)

1999 44 2447

(.00)

.0395

(.20)

.0332

(.03)

.0250

(.10)

.0257

(.11)

.0271

(.05)

.0290

(.69)

.0270

(.13)

.0281

(.85)

.0228

(.07)

2000 8 2010

(.00)

.0356

(2.83)

.0324

(.24)

.0278

(.40)

.0244

(.75)

.0269

(.79)

.0277

(.40)

.0265

(.67)

.0248

(.92)

.0251

(.53)

2001 32 1640

(.01)

.0314

(.02)

.0323

(.00)

.0265

(.02)

.0229

(.02)

.0262

(.08)

.0278

(.78)

.0258

(.00)

.0256

(.19)

.0242

(.01)

2002 9 1364

(.00)

.0327

(.35)

.0317

(.01)

.0271

(.06)

.0224

(.10)

.0258

(.07)

.0277

(.11)

.0259

(.06)

.0265

(1.25)

.0225

(.06)

2003 39 1662

(.02)

.0335

(1.10)

.0311

(.17)

.0264

(.11)

.0231

(.34)

.0258

(.15)

.0265

(1.51)

.0257

(.26)

.0267

(1.76)

.0241

(.23)

2004 14 3688

(.01)

.0314

(1.28)

.0320

(.09)

.0251

(.06)

.0232

(.29)

.0259

(.31)

.0282

(.40)

.0261

(.27)

.0238

(1.21)

.0241

(.13)

2005 43 1843

(.00)

.0380

(.05)

.0308

(.02)

.0261

(.02)

.0232

(.00)

.0266

(.02)

.0288

(.02)

.0261

(.02)

.0274

(.16)

.0251

(.03)

2006 19 2046

(.00)

.0412

(.76)

.0293

(.02)

.0275

(.14)

.0237

(.20)

.0267

(.02)

.0286

(2.28)

.0261

(.22)

.0290

(.27)

.0256

(.15)

2007 20 2412

(.00)

.0403

(1.13)

.0322

(.17)

.0252

(.02)

.0229

(.47)

.0273

(.04)

.0305

(3.55)

.0266

(.30)

.0296

(2.11)

.0248

(.29)

associated with possible datum instability at some water-level 

gaging stations in upper St. Clair River, as discussed in the 

analysis of water-level falls. In particular, for 1968, the estimated parameter of 0.0145 is considered physically implausible and therefore un-interpretable with respect to conveyance 

change. 

In general, conventionally measured flows closely match 

flows simulated with time-varying parameterization from 1962 

and 1985 (fig. 9). Differences between average flows among 

measurement sites are attributed to long-term variations in 

flow. In particular, flow measurements at the Bay Point and 

Roberts Landing sections generally occurred from 1962 to 

1968 when average flows were less than average flows from 

1981 to 1984, when measurements at the Blue Water Bridge 

and GLERL sections were obtained. The standard deviation of 

flow residuals is 144 m3/s.

Effective Channel Roughness Estimated from Acoustic 

Doppler Current Profiler Measurements 

Annual parameter estimates for nine reaches associated 

with years of ADCP flow measurements (1996 to 2007) are 

shown in table 8. The nine reaches include the seven previously estimated, with the two additional reaches representing 

flows in the east branches around Stag and Fawn Islands. In 
Results of Conveyance Analyses 25

Figure 10. Relation between simulated flow and flow measured on the St. Clair River from 1996 to 2007 using acoustic Doppler 

current profilers.

0 1,000 2,000 3,000 4,000 5,000 6,000 7,000

MEASURED FLOW, IN CUBIC METERS PER SECOND

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

SIM

U

L

A

T

ED F

L

O

W, IN 

CUBI

C METER

S PE

R SE

C

O

N

D

Flow Measurement Cross Section

Line of agreement (n = 353, r2

= 0.996)

CS-200

CS-202

CS-204

CS-206

CS-207

CS-208

CS-210

CS-212

CS-214

CS-216

CS-218

CS-220

CS-230

r2

 Sample coefficient of determination

n Number of measurements

particular, the EStag reach represents a subdivision of the 

DD_SP reach, and the EFawn reach represents a subdivision 

of the SP_PL reach. Estimation of flow-resistance characteristics in the EStag and EFawn reaches was not possible before 

flows in the east branches were measured systematically with 

ADCP instruments. Parameter estimates converged uniquely 

for all years based on the PRD for SSWR of 1 percent and for 

parameters of 3 percent. 

In general, ADCP-measured flows closely match flows 

simulated with time-varying parameterization from 1996 and 

2007 (fig. 10). Lower-magnitude flows measured in the east 

branches around Stag and Fawn Islands contributed significantly to the increase in the estimated coefficient of determination (r2

) from the conventional measurements of 0.845 to 

0.996 for the ADCP measurements. The standard deviation of 

ADCP-flow residuals of 105 m3/s is somewhat lower, however, than the standard deviation of conventional flow-measurement residuals of 144 m3

/s.

Converged estimates of effective annual channel-roughness characteristics in the reach from Fort Gratiot to Dunn 
26 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

YEAR

0.01

0.02

0.03

0.04

0.05

ESTIMATED EFFE

CTIVE CHANNEL R

OUGHNESS

Upper limit of the 95-percent confidence interval

Expected parameter value

Lower limit of the 95-percent confidence interval

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

YEAR

0.01

0.02

0.03

0.04

0.05

ESTIMATED EFFE

CTIVE CHANNEL R

OUGHNESS

Upper limit of the 95-percent confidence interval

Expected parameter value

Lower limit of the 95-percent confidence interval

Figure 11. Annual estimates of effective 

channel roughness with 95-percent 

confidence intervals for the St. Clair River 

reach from Fort Gratiot to Dunn Paper, 

Michigan.

Figure 12. Annual estimates of effective 

channel roughness with 95-percent 

confidence intervals for the St. Clair River 

reach from Dunn Paper, Michigan, to Point 

Edward, Ontario.

Paper (FG_DP) are shown on fig. 11. Limited years with flow 

measurements prior to 1996, limited the years in which annual 

estimates of effective channel roughness could be developed. 

The physically improbable estimate of effective channel 

roughness of 0.0145 for 1968 and its departure from effective 

channel-roughness values of 0.04125 and 0.03634 estimated 

for 1963 and 1984, respectively, may have been affected by 

local datum irregularities associated with a negative fall of 

0.0145 m from Point Edward to Black River (PE_BR) from 

April 1 to November 30, 1968. A pattern of continual change, 

however, is evident as a decreasing trend in channel roughness 

(increasing conveyance) from 1996 to 2001 and an increasing trend in channel roughness (decreasing conveyance) from 

2002 to 2007. The non-monotonic pattern of change from 

1996 to 2007 would be difficult to associate with dredging 

prior to 1962. As indicated by the width of the 95-percent 

confidence interval about the expected parameter value, the 

median standard deviation for parameters in this reach is 

0.001157. 

For the St. Clair River reach from Dunn Paper to Point 

Edward (DP_PE), the effective channel roughness generally 

decreased with time (fig. 12), indicating generally increasing 

channel conveyance since 1962. The 1963 estimate of channel 

roughness of 0.0337 is less than other estimates for the 1960s. 

Coincidently, the average water-level fall indicated by gagingstation records for April 1 to November 30, 1963, of 0.072 m 

was less than water-level falls of 0.124, 0.121, 0.144, 0.141, 

0.148, 0.157, and 0.172 m for the corresponding period in 

1962, and 1964–1969, respectively. The lower channel-rough-
Results of Conveyance Analyses 27

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

YEAR

0.01

0.02

0.03

0.04

0.05

ESTIMATED EFFE

CTIVE CHANNEL R

OUGHNESS

Upper limit of the 95-percent confidence interval

Expected parameter value

Lower limit of the 95-percent confidence interval

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

YEAR

0.01

0.02

0.03

0.04

0.05

ESTIMATED EFFE

CTIVE CHANNEL R

OUGHNESS

Upper limit of the 95-percent confidence interval

Expected parameter value

Lower limit of the 95-percent confidence interval

Figure 13. Annual estimates of effective 

channel roughness with 95-percent 

confidence intervals for the St. Clair River 

reach from Point Edward, Ontario, to the 

mouth of Black River, Michigan.

Figure 14. Annual estimates of effective 

channel roughness with 95-percent 

confidence intervals for the St. Clair River 

reach from the mouth of Black River to 

Dry Dock, Michigan.

ness value would facilitate matching measured with simulated 

water levels at Dunn Paper. 

In contrast, the effective channel roughness for the St. 

Clair River reach from Point Edward to Black River (PE_BR) 

was a substantially higher value of 0.0243 in 1963 than values 

of 0.0205, 0.0209, 0.0207, and 0.0202 estimated for the years 

1962, 1964, 1966, and 1968, respectively (fig. 13). Similarly, 

the water-level fall for April 1–November 30, 1963, of 

0.0157 m was substantially greater than reported water-level 

falls of 0.0110, 0.0046, –0.0160, –0.0139, –0.0105, –0.0145, 

and –0.0026 m in 1962, and 1964–1969, respectively. Given 

the greater consistency of the estimated values for 1963 

conditions with post-1960s conditions on both reaches and 

with expected physical characteristics of channel roughness, 

it is possible that datum anomalies at the Dunn Paper or Point 

Edward water-level gaging stations degraded the estimates of 

effective channel roughness in other years during the 1960s. 

The median standard deviation of parameters in the 

DP_PE and PE_BR reaches were 0.000213 and 0.000246, 

which are substantially less than the standard deviation of 

parameters in the FG_DP reach, which was 0.00116. The 

generally greater parameter uncertainty in the FG_DP reach 

may be attributable to higher variability in water levels near 

Fort Gratiot, where water levels are affected by wind-induced 

waves on Lake Huron.

The effective channel roughness in the St. Clair River 

reach from Black River to Dry Dock (BR_DD) generally 

decreased (fig. 14) from 1962 to 2007, indicating slightly 
28 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

YEAR

0.01

0.02

0.03

0.04

0.05

ESTIMATED EFFE

CTIVE CHANNEL R

OUGHNESS

Upper limit of the 95-percent confidence interval

Expected parameter value

Lower limit of the 95-percent confidence interval Figure 15. Annual estimates of effective 

channel roughness with 95-percent confidence 

intervals for the St. Clair River reach from Dry 

Dock to St. Clair State Police, Michigan.

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

YEAR

0.01

0.02

0.03

0.04

0.05

ESTIMATED EFFE

CTIVE CHANNEL R

OUGHNESS

Upper limit of the 95-percent confidence interval

Expected parameter value

Lower limit of the 95-percent confidence interval

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

YEAR

0.01

0.02

0.03

0.04

0.05

ESTIMATED EFFE

CTIVE CHANNEL R

OUGHNESS

Upper limit of the 95-percent confidence interval

Expected parameter value

Lower limit of the 95-percent confidence interval

Figure 16. Annual estimates of effective 

channel roughness with 95-percent confidence 

intervals for the St. Clair River reach from St. 

Clair State Police, Michigan, to Port Lambton, 

Ontario.

Figure 17. Annual estimates of effective 

channel roughness with 95-percent confidence 

intervals for the St. Clair River reach from Port 

Lambton, Ontario, to Algonac, Michigan.
Results of Conveyance Analyses 29

increasing conveyance with time. Some short-term instability 

in effective roughness may be apparent from 1998 to 2003. 

The median standard deviation of effective channel roughness 

is 0.0002612. 

In the St. Clair River reach from Dry Dock to St. Police 

(DD_SP), the effective channel roughness generally decreased 

from 1962 to 2003 and then increased to 2007 (fig. 15). From 

1996 to 2007, the pattern in effective channel roughness in the 

reach from DD_SP was similar to the pattern of variability in 

effective channel roughness at the FG_DP reach (fig. 11). The 

median standard deviation in the roughness parameters for the 

DD_SP reach was 0.000241. 

There is limited evidence of trends or patterns in the 

effective channel roughness in the reach from St. Police to 

Port Lambton (SP_PL) (fig. 16) or in the reach from Port 

Lambton to Algonac (PL_AL) (fig. 17). The median standard 

deviations of parameter estimates in the SP_PL and PL_AL 

reaches were 0.000215 and 0.000194, respectively. 

Fixed-Parameter Estimation

Three scenarios were analyzed to estimate the average 

effective channel-roughness values for three periods: 1962–68, 

1981–85, and 1996–2007. In particular, the effective channelroughness value for each specified reach was considered fixed 

or constant during the period. The results of fixed-parameter 

estimation were compared with those of time-varying parameter estimation to better understand temporal variations in 

conveyance. 

Scenario 1962 to 1968

Seven parameters for the 1-D HEC-RAS model of the St. 

Clair River were estimated (table 9) based on water-level and 

conventional flow measurements obtained in 1962–64, 1966, 

and 1968 at the Bay Point (600 m downstream from CS-202) 

and Roberts Landing (600 m downstream from CS-220) sections (fig. 1). Intervening years without flow measurements or 

water-level data for boundary-condition specification were not 

used in parameter estimation. The sum of squared weighted 

residuals for the 19,221 water-level and 259 flow measurements used in the estimation was 16,605. Parameter estimates 

converged uniquely, with the largest magnitude correlation 

of 0.8967 between the effective channel-roughness parameter 

for the reach DD_SP (nDD_SP) and the parameter for the 

reach SP_PL (nSP_PL). The magnitude of this correlation is 

not thought to substantially degrade the estimation results. 

The match between simulated and measured flow (fig. 18) 

describes 83.9 percent of the variability in the flow measurements. The standard deviations of differences between 

measured and simulated water levels at the intervening gaging 

stations averaged 0.023 m.

Table 9. Fixed-parameter estimates of effective channel roughness for St. Clair River reaches based on 

measurements from 1962 to 1968.

[An “n” prepended to the reach identifier indicates the effective channel roughness coefficient of the corresponding reach. 

Reach identifiers include: FG_DP, Fort Gratiot to Dunn Paper; DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward 

to Black River; BR_DD, Black River to Dry Dock; DD_SP, Dry Dock to State Police; SP_PL, State Police to Port Lambton; 

PL_AL, Port Lambton to Algonac]

Name of effective 

channel-roughness 

parameter

Parameter 

estimate

95-percent confidence limits 

about the parameter estimate

Standard 

deviation of 

parameter

estimate

Coefficient 

of variation 

of parameter 

estimate Lower limit Upper limit

nFG_DP 0.0177 0.0164 0.0191 0.000698 0.0394

nDP_PE .0403 .0401 .0405 .000110 .00272

nPE_BR .0228 .0226 .0233 .000163 .00709

nBR_DD .0241 .0239 .0243 .000116 .00481

nDD_SP .0298 .0295 .0298 .0000867 .00292

nSP_PL .0265 .0264 .0267 .0000786 .00296

nPL_AL .0246 .0244 .0248 .0000924 .00376
30 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Scenario 1981 to 1985

The inverse model used to estimate seven fixed parameters associated with channel roughness in the HEC-RAS 

model of the St. Clair River did not converge using waterlevel and flow data available from 1981 to 1985. The estimation scenario that initialized effective channel-roughness 

values to 0.040 did not converge after 40 iterations, although 

the weighted sum of squared residuals gradually declined 

to 63,464. Failure to converge was caused by changes in 

parameters between iterations that did not decrease below the 

convergence criteria of 3 percent. High parameter correlation 

was detected among several pairs of parameters including 

(1) nFG_DP and nSP_PL, which had a correlation of 0.97; 

(2) nBR_DD and nSP_PL, which had a correlation of 0.94; 

(3) nBR_DD, and nSP_PL, which had a correlation of 0.92; 

(4) nDP_PE and nSP_PL, which had a correlation of 0.91; and 

(5) nFG_DP and nPL_AL, which had a correlation of 0.90. 

The availability of only a single flow measurement in 1982 

and 1985 may have contributed to the convergence failure. 

Scenario 1996 to 2007

The fixed, nine-parameter version of the 1-D HEC-RAS 

model of the St. Clair River was estimated based on waterlevel and ADCP flow measurements obtained from 1996 to 

2007. The sum of squared weighted residuals for the 50,546 

water-level and 346 flow measurements used in the estimation 

was 29,595. Parameters converged uniquely (table 10), with 

the largest magnitude correlation of 0.8983 (table 11) between 

the effective channel-roughness parameter for the reach 

FG_DP (nFG_DP) and the parameter for the reach SP_PL

(nSP_PL). The magnitude of this correlation is not thought to 

substantially degrade the estimation results. 

Simulated and measured flows match closely (r2 = 0.992) 

(fig. 19). Also, simulated and measured water levels match 

closely at all intervening gaging stations with coefficients 

of determination at all gaging stations exceeding 0.996. The 

standard deviations of the water-level residuals for all gaging 

stations average less than 0.019 m. 

Fixed-parameter estimates were computed for two 

subintervals from 1996–2001 and 2001–07, which spanned 

the period of available ADCP flow-measurement data. Results 

indicate that parameter estimates for individual reaches generally were similar for the two subintervals (fig. 20), except 

for the reach from DP_PE, where the channel roughness 

decreased (conveyance increased) with time (fig. 12). At FG_

DP (fig. 11) and DD_SP (fig. 15), negative trends in annual 

estimates of effective channel roughness during the first half 

of this interval were averaged with positive trends during the 

second half to produce similar average values for the two 

4,000 4,500 5,000 5,500 6,000

MEASURED FLOW, IN CUBIC METERS PER SECOND

4,000

4,500

5,000

5,500

6,000

SIMULATED FLOW, IN CUBIC METERS PER SECOND

Bay Point pairs (n = 224, r2 = 0.82)

Roberts Landing pairs (n=34, r2 = 0.61)

Line of agreement

n Number of measure-

 ment pairs

r2 Coefficient of determination

Figure 18. Relation between 

measured flow and flow 

simulated on the St. Clair 

River by use of a fixed, sevenparameter version of the 1-D 

HEC-RAS (One-dimensional 

Hydrologic Engineering CenterRiver Analysis System) model 

developed using data obtained 

from1962 and 1968. 
Results of Conveyance Analyses 31

Table 10. Fixed-parameter estimates of effective channel roughness on reaches of St. Clair River for the 

period 1996–2007.

[An “n” prepended to the reach identifier indicates the effective channel roughness coefficient of the corresponding reach. Reach 

identifiers include: FG_DP, Fort Gratiot to Dunn Paper; DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward to Black 

River; BR_DD, Black River to Dry Dock; DD_SP, Dry Dock to State Police; EStag, East Stag Island; SP_PL, State Police to Port 

Lambton; EFawn, East Fawn Island; PL_AL, Port Lambton to Algonac]

Name of effective 

channel-roughness 

parameter

Parameter 

estimate

95-percent confidence limits 

about the parameter estimate

Standard 

deviation of 

parameter

 estimate

Coefficient 

of variation 

of parameter

Lower limit Upper limit estimate

nFG_DP 0.0367 0.0360 0.0373 0.000320 0.00871

nDP_PE .0325 .0324 .0326 .0000539 .00166

nPE_BR .0260 .0259 .0262 .0000659 .00253

nBR_DD .0232 .0231 .0233 .0000696 .00300

nDD_SP .0265 .0264 .0266 .0000653 .00246

nEStag .0282 .0276 .0287 .000270 .00958

nSP_PL .0261 .0260 .0262 .0000557 .00213

nEFawn .0263 .0256 .0269 .000336 .0128

nPL_AL .0242 .0241 .0243 .0000538 .00222

Table 11. Lower diagonal elements of the symmetric correlation matrix for fixed-parameter estimates of effective channel roughness 

coefficients on St. Clair River for the period 1996–2007.

[An “n” prepended to the reach identifier indicates the effective channel roughness coefficient of the corresponding reach. Reach identifiers include: 

FG_DP, Fort Gratiot to Dunn Paper; DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward to Black River; BR_DD, Black River to Dry Dock; 

DD_SP, Dry Dock to State Police; EStag, East Stag Island; SP_PL, State Police to Port Lambton; EFawn, East Fawn Island; PL_AL, Port Lambton to Algonac]

Name of effective 

channel-roughness 

parameter

nFG_DP nDP_PE nPD_BR nBR_DD nDD_SP nEStag nSP_PL nEFawn nPL_AL

nFG_DP 1.0000 -- -- -- -- -- -- -- --

nDP_PE .2303 1.0000 -- -- -- -- -- -- --

nPD_BR .2860 -.2172 1.0000 -- -- -- -- -- --

nBR_DD .7359 .4423 -.0776 1.0000 -- -- -- -- --

nDD_SP .8177 .4641 .2783 .6519 1.0000 -- -- -- --

nEStag .0732 .0415 .0257 .0490 -.3422 1.0000 -- -- --

nSP_PL .8983 .5102 .3024 .7794 .8458 .0759 1.0000 -- --

nEFawn .0646 .0367 .0224 .0561 .0564 .0054 -.0079 1.0000 --

nPL_AL .6224 .3534 .2093 .5397 .5998 .0541 .5587 .0192 1.0000
32 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

subintervals (fig. 20). The standard deviations of flow residuals for the periods 1996–2001 and 2002–07 were 122, and 141 

m3/s, respectively.

Integration of Time-Varying and Fixed-Parameter 

Estimation Results

Although patterns are evident in some of the annual 

series of parameter estimates, the significance of any pattern 

for flow simulation was difficult to establish directly. In this 

analysis, the annual series of parameter estimates from 1996 to 

2007 are used as a basis for vectors to help explain the difference between measured and simulated flows in the fixed, nineparameter model also developed with data from 1996 to 2007. 

In particular, the regression design matrix is composed of nine 

columns corresponding to the nine reaches for which annual 

variations in effective channel roughness were estimated, plus 

a vector of ones for the intercept term. The vectors have a 

length equal to the number of flow measurements. The value 

in each row of the basis vectors corresponds to the estimated 

0 1,000 2,000 3,000 4,000 5,000 6,000 7,000

MEASURED FLOW, IN CUBIC METERS PER SECOND

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

SIM

U

L

A

T

ED F

L

O

W, IN 

CUBI

C METER

S PE

R SE

C

O

N

D

Flow Measurement Cross Section

Line of agreement (n = 353, r2 = 0.996)

CS-200

CS-202

CS-204

CS-206

CS-207

CS-208

CS-210

CS-212

CS-214

CS-216

CS-218

CS-220

CS-230

r2 Sample coefficient of determination

n Number of measurement pairs 

Figure 19. Relation between measured flow and flow simulated on the St. Clair River by use of a fixed, nine-parameter version of the 

hydrodynamic model for the period 1996–2007. 
Results of Conveyance Analyses 33

Figure 20. Fixed-parameter channel 

roughness estimates for nine St. Clair 

River reaches for the periods 1996–2001 

and 2002–2007.

FG_D

P

DP_PE

P

E_

B

R

BR_DD

DD_SP

EStag

S

P_

PL

EFawn

PL_

AL

REACH IDENTIFIER

0.01

0.02

0.03

0.04

0.05

ES

TIMA

TED EFFE

C

TIVE CHA

NNEL R

O

U

G

HNE

S

S

1996 - 2001

2002 - 2007

Fort Gration to Dunn Paper

Dunn Paper to Point Edward

Point Edward to Black River

Black River to Dry Dock

Dry Dock to State Police

East Stag Island

State Police to Port Lambton

East Fawn Island

Port Lambton to Algonac

FG_DP

DP_PE

PE_BR

BR_DD

DD_SP

EStag

SP_PL

EFawn

PL_AL

Reach

Identifier Reach Description

Upper limit of the 95-percent confidence interval Data Interval

Expected parameter value

Lower limit of the 95-percent confidence interval

effective channel-roughness value in the year of the flow measurement. A stepwise variable-selection process in a multiple-regression analysis was used to identify reaches where 

patterns in annual effective channel-roughness estimates were 

related to flow residuals. The basis vectors corresponding to 

the annual series of effective channel-roughness values are 

distinguished by a leading b rather than an n preceding the 

reach designator. For example, bFG_DP is the basis vector 

created from the annual series of effective channel-roughness 

estimates in nFG_DP. 

Results indicate that four basis vectors—bFG_DP, 

bDP_PE, bPE_BR, and bSP_PL—explain 38.1 percent 

of the variability in the flow residuals (table 12). Analysis 

of variance indicates that the overall model is statistically 

significant (p-value < 0.0001) based on an F statistic of 54.15 

with 341 degrees of freedom in the model error. Of the four 

included basis vectors, the non-monotonic variations in the 

annual series nFG_DP are associated with the basis vector 

with the largest significance. Although differing in scale, the 

annual series nDD_SP shows a similar pattern of variation as 

nFG_DP. The corresponding basis vectors have a correlation 

coefficient of 0.9201. This similarity (co-linearity) between 

the two basis vectors may mask the contribution of nDD_SP

in explaining the variability of flow residuals.

To assess the possible significance of annual variations in 

effective channel roughness at DD_SP, the stepwise multiple 

regression was re-run excluding the basis vector bFG_DP. 

Again, four basis vectors were selected with an α = 0.10 such 

that bDD_SP had the greatest significance of the eight available basis vectors (table 12). Furthermore, the same three basis 

vectors bDP_PE, bPE_BR, and bSP_PL are included in both 

models, and the estimated parameters associated with these 

variables are similar between the two regression analyses. 

The results of the regression analyses are used to ascertain that annual variations in effective channel-roughness values at reaches FG_DP, DP_PE, PE_BR, DD_SP, and SP_PL

contribute to the understanding of variability in flows measured from 1996 to 2007, which are not accounted for by flows 

simulated using the fixed, nine-parameter HEC-RAS model of 

the St. Clair River. Patterns in the annual series nFG_DP and 

n_DD_SP are similar, showing a gradual decrease in effective 

channel-roughness values (increase in conveyance) from 1996 

to about 2002, followed by an increase in effective channel 

roughness (decrease in conveyance) from 2003 to 2007. The 

annual series nDP_PE indicates a general decrease in effective channel roughness (increase in conveyance) from 1996 to 

2007, which is consistent with the pattern indicated by earlier 

estimates of nDP_PE. Annual variations in effective channel roughness in the series nPE_BR and nSP_PL show less 

persistence than variations at other selected reaches. Variations 

in effective channel roughness at non-selected reaches may 

contribute to unexplained variability in water levels, but have 

no demonstrable relation to flows. 

Stepwise regression tends to over indicate the statistical 

significance of individual selected variables, so the effective 

alpha value is likely less than 0.1. Based on the significance 

of the overall models, however, annual variations in effective 

channel roughness account for flow variability not explained 

by the fixed, nine-parameter HEC-RAS model of the St. Clair 

River. 
34 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Effective Channel Roughness Ratios and 

Conveyance Ratios

Equation 2 was used to compute conveyance ratios as a 

function of measured water levels (and reach lengths). Alternatively, conveyance ratios can be computed by use of equation 3 and the effective channel roughness values estimated by 

use of inverse modeling as 

K

K n A

n A

R

R

c

K

K

1

2

1 1 1

2

3

2 2 2

2

3 1 2

2

1

1

2

1

1

 ,

ˆ

ˆ

nˆ

nˆ • •

• •

(13)

provided the ratios of cross-sectional areas (A1, A2) and 

hydraulic radii (R1, R2) in any two reaches 1 and 2, given nˆ , 

are approximately constant c1,2 with time. 

In the estimation of effective channel roughness values 

by use of inverse modeling techniques, the bathymetry data, 

used to compute cross-sectional areas and hydraulic radii, did 

not vary with time. Therefore, the assumption of a constant 

cross sectional properties is reflected in the estimated effective 

roughness values. Although the reach-pair specific values of 

the proportionality constant c cannot be determined directly, 

c can be assumed to be positive because it is a function of 

channel cross section values that only take on positive values. 

Thus, conveyance ratios are assumed to be positively associated with the inverse of estimated effective roughness ratios 

(equation 13). 

To test this assumption, ratios of annual estimates of 

channel roughness values were computed for all possible 

reach pairs using inverse modeling results for the period from 

1996 through 2007. For 20 of the 21 pairs of these reaches, the 

ratios of channel roughness values were inversely proportional 

to conveyance ratios computed by use of water-level data 

(table 13). Thus, conveyance ratios computed on the basis of 

water level data (equation 2) are directly proportional (fig. 21) 

to conveyance ratios based on effective channel roughness 

estimated (equation 3). 

This consistency between estimators of conveyance ratios 

is thought to apply to data prior to 1996, when limited flow 

measurement data were available to support continual annual 

estimation of effective channel roughness values by use of 

inverse modeling analysis. Given the time-invariance of this 

relation, the conveyance ratios computed from water-surface 

data provide a basis for understanding and extending trends in 

conveyance ratios that are consistent with conveyance ratios 

computed by use of inverse modeling techniques.

Table 12. Basis vectors selected in a stepwise-regression analysis of flow residuals from the 1996 to 2007 fixed, nine-parameter 

model of St. Clair River.

[An “b” prepended to the reach identifier indicates a basis vector for the corresponding reach. Reach identifiers include: FG_DP, Fort Gratiot to Dunn Paper; 

DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward to Black River; BR_DD, Black River to Dry Dock; DD_SP, Dry Dock to State Police; SP_PL, 

State Police to Port Lambton; PL_AL, Port Lambton to Algonac]

Variable Degrees 

of freedom

Parameter

 estimate

Standard 

error

t-value Pr > abs(t)

Results of a stepwise-regression analysis using all nine basis vectors

Intercept 1 3,142 606 5.19 <0.0001

bFG_DP 1 -12,602 1,926 -6.54 <.0001

bDP_PE 1 -23,973 5,374 -4.46 <.0001

bPE_BR 1 -18,760 10,396 -1.80 .072

bSP_PL 1 -53,923 17,334 -3.11 .002

Results of stepwise-regression analysis using eight basis vectors, excluding bFG_DP

Intercept 1 5,177 573 9.04 <.0001

bDP_PE 1 -15,583 6,045 -2.58 .0104

bPE_BR 1 -23,279 10,370 -2.24 .0254

bDD_SP 1 -102,757 16,213 -6.34 <.0001

bSP_PL 1 -50,895 17,705 -2.87 .004
Results of Conveyance Analyses 35

Table 13. Relation between ratios of effective channel roughness and conveyance ratios among reaches of the St. Clair River based 

on data from 1996 to 2007..

[Reach ID, reach identification; FG_DP, Fort Gratiot to Dunn Paper; DP_PE, Dunn Paper to Point Edward; PE_BR, Point Edward to Black River; BR_DD, 

Black River to Dry Dock; DD_SP, Dry Dock to State Police; SP_PL, State Police to Port Lambton; PL_AL, Port Lambton to Algonac]

Upstream 

reach

Downstream 

reach

Kendall’s 

tau p-value1 Intercept Median slope

Median 

ratio

Median 

year

N2

FG_DP DP_PE -0.0182 0.560 0.432 -0.00455 0.428 0.895 11

PE_BR 0.4667 .0363 -0.0116 .267 .182 .722 10

BR_DD 0.7091 .0008 0.222 .157 .319 .612 11

DD_SP 0.5758 .0044 .221 .0970 .290 .704 12

SP_PL 0.697 .0005 .198 .103 .269 .691 12

PL_AL 0.4545 .0224 .122 .113 .196 .655 12

DP_PE PE_BR 0.9556 <.0001 -.829 1.62 .441 .783 10

BR_DD 0.7333 .0011 -.543 1.84 .762 .710 10

DD_SP 0.5273 .0132 -.767 1.80 .696 .815 11

SP_PL 0.7455 .0004 -.515 1.42 .647 .816 11

PL_AL 0.6727 .0016 -.191 .880 .470 .751 11

PE_BR BR_DD 0.6000 .0083 -1.98 4.14 1.74 .899 10

DD_SP 0.6889 .0023 -2.14 3.63 1.62 1.04 10

SP_PL 0.6444 .0046 -2.76 4.14 1.49 1.03 10

PL_AL 0.6000 .0083 -2.26 3.58 1.07 .931 10

BR_DD DD_SP 0.8545 <.0001 .184 .632 .907 1.14 11

SP_PL 0.7455 .0004 .084 .667 .835 1.13 11

PL_AL 0.6727 .0016 -.0298 .602 .606 1.06 11

DD_SP SP_PL 0.697 .0005 -.0379 .979 .926 .985 12

PL_AL 0.7576 .0001 -.0919 .827 .670 .922 12

SP_PL PL_AL 0.7176 .0008 -.205 .997 .725 .932 12

1

p-value is the probability that the null hypothesis that the correlation coefficient is not greater than zero, given the sample data pairs, is true.

2

N is the number of sample data pairs.
36 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

Figure 21. Relation between annual conveyance ratios estimated on the basis of effective channel roughness estimates and measured water-level data for the period 1996 through 2007.

0.100.200.300.400.500.60

0.100.200.300.400.500.60

0.100.200.300.400.500.60

DATA LE EV L R- ETA W ED UR S MEA ON D BASE IOS T RA E NC A CONVEY

0.100.200.300.400.500.60

0.100.200.300.400.500.60

0.5 0.7 0.9 1.1

0.100.200.300.400.500.60

0.200.400.600.801.00

0.200.400.600.801.00

0.200.400.600.801.00

0.200.400.600.801.00

0.6 0.7 0.8 0.9

0.200.400.600.801.00

1.201.401.601.802.002.20

1.201.401.601.80

1.001.201.401.601.80

0.8 0.9 1.0 1.1

CONVEY

ANC

E

R

ATIOS BASED ON

E

STIMATED 

E

FFE

C

TIVE

CHA

NNEL ROUGHNESS

VALU

E

S

0.600.801.001.201.40

0.840.860.880.900.920.940.96

0.780.800.820.840.860.880.90

0.8 0.9 1.0 1.1 1.2

0.520.560.600.640.680.72

0.900.920.940.960.98

0.80 0.88 0.96 1.04

0.600.640.680.720.760.80

0.8 0.9 1.0

0.640.680.720.760.80

BR_

D

D

FG_

D

P

PE_

B

R

DD_

S

P

SP_PL

P

L_AL

DP_PE

DP_PE

PE_

B

R

BR_D

D

DD_

S

P

SP_

P

L

FG_DP: Fort Gratiot to Dunn PaperDP_PE: Dunn Paper to Point EdwardPE_BR: Point Edward to Black RiverBR_DD: Black River to Dry DockDD_SP: Dry Dock to State PoliceSP_PL: State Police to Port LambtonPL_AL: Port Lambton to Algonac

EXPLANATION

Reach Identifier by Water-Level Gaging Station Trends based on a probability of less than 0.05 that 

the computed Kendall’s tau correlation coefficient would be obtained in random sampling from a population whose true correlation is zero:

No trendPositive trend

Note: In the lower diagonal scatter-plot matrix above, reaches are identified above each column  and to the left of each row. The column reach identifier is associated with the numerator and  the row reach identifier is associated with the denominator of the conveyance ratio. 
Results of Conveyance Analyses 37

Inferring Changes in River Conveyance from 1-D 

Hydrodynamic Simulations

The previous section described annual estimates of 

effective channel-roughness parameters that were consistently determined for each reach from water-level and flow 

measurements by statistical inference. These annual parameter estimates are used with historical water-level boundary 

conditions to show how the change in parameters from 1962 

to 2007 would affect corresponding estimates of average flow. 

Changes in flow with parameterization indicate changes in 

conveyance with time, because the boundary conditions are 

consistent across annual parameterizations. 

Hourly flows were simulated for all years from 1962 to 

2007 in which parameter estimates converged under all sets 

of converged parameters. Average annual flows from April 1 

to November 30 simulated near the water-level gaging station 

at Dry Dock, Mich., (HEC-RAS cross-section 125) were 

computed to provide a basis of comparison. Results indicate 

that simulated flows for 2007 water-level boundary conditions 

averaged 675 m3/s (12.5 percent) less than simulated flows for 

1962 water-level boundary conditions for all annual sets of 

parameters estimated (fig. 22). Simulated flows for both 1962 

and 2007 water-level boundary conditions, however, generally increased with annual parameter estimates of hydraulic 

conditions for 1962 to 2003, but decreased monotonically for 

conditions from 2004 to 2007. Likewise, simulated average 

flows generally increased for boundary conditions from 1963 

to 2003 with annual estimates of parameters relative to values 

simulated for parameters associated with 1962 hydraulic 

conditions. 

The relative increase in simulated flows from 1962 

hydraulic conditions varied from about 0.87 to 10 percent 

(fig. 23) from 1963 to 2003 and decreased to about 5.8 percent 

Figure 23. Relation between annual 

parameter estimates and changes in 

average simulated flow from the 1962 

parameterization.

Figure 22. Relation between annual 

model-parameter estimates and simulated 

average flow from 1962 to 2007 for 1962 

and 2007 water-level boundary conditions. 1960 1970 1980 1990 2000 2010

YEAR OF PARAMETER ESTIMATE

4,400

4,800

5,200

5,600

6,000

A

VERAGE SIMUL

ATED FLOW, 

IN CUBIC METERS PER SEC

OND

Flow Conditions

1962

2007

1960 1970 1980 1990 2000 2010

YEAR OF PARAMETER ESTIMATE

0

2

4

6

8

10

12

CHANGE IN AVER

A

GE SIMULATED FLOW

FROM 1962 PARAMETERIZATION, IN PERCENT

Maximum

Average

Minimum
38 Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data

by 2007. Changes in average and relative flows with year of 

parameterization are associated with changes in conveyance 

because water-level boundary conditions were consistent. As 

demonstrated by use of equation 12, the relative changes in 

simulated flows indicate the relative annual changes in conveyance (fig. 23).

Summary and Conclusions

Water levels on Lakes Michigan and Huron (Michigan–

Huron) are controlled by flow through the St. Clair River. In 

2007, water levels on Lake Michigan–Huron were 0.5-meters 

below historical averages for 1918–2007. Although no trends 

were detected in Lake Michigan–Huron or Lake St. Clair 

water levels, annual differences between these water levels 

were decreasing from 1962 to 2007. This decrease indicates 

that conveyance is increasing if average annual flow is constant or that flow is decreasing if conveyance is constant. This 

study was done to investigate possible conveyance changes 

from 1962 to 2007 by use of water-level and flow data with an 

inverse hydrodynamic model of the St. Clair River. 

A 47.3-kilometer reach of St. Clair River was investigated from the northern limit near Fort Gratiot, Michigan, 

to the southern limit near Algonac, Michigan. The river was 

subdivided into seven reaches delimited by the locations of 

eight water-level gaging stations. Monotonic trends in average 

water-level differences (water-surface fall) within reaches 

were analyzed. Positive trends in water-surface fall were 

detected in two reaches, and negative trends were detected in 

two other reaches. The presence of both positive and negative 

trends in fall is inconsistent with the hypothesis of no conveyance change regardless of possible trends in flow. 

Flow within a reach approximately equals the product 

of conveyance and the square root of the water-surface slope. 

Given that the annual flow through all reaches in the St. Clair 

River is approximately the same, annual conveyance ratios 

between reaches can be computed without knowing the magnitude of this flow. Analysis of trends in annual conveyance 

ratios indicates that conveyance in the reach from Fort Gratiot 

to Dunn Paper is decreasing with respect to all other reaches 

except Point Edwards to Black River. In contrast, conveyance 

in the reach from Dunn Paper to Point Edward is increasing 

with respect to all downstream reaches. Conveyance in the 

reach from Point Edward to Black River is decreasing with 

respect to all downstream reaches. If conveyances in the reach 

from Fort Gratiot to Dunn Paper and in the reach from Point 

Edward to Black River are both decreasing with time, the lack 

of significant change in conveyance between the two reaches 

is not unexpected. Conveyance in the reach from Black River 

to Dry Dock is decreasing with respect to conveyance from 

State Police to Port Lambton; conveyance from Dry Dock to 

State Police is increasing with respect to conveyance from 

State Police to Port Lambton. Although conveyance ratios 

were sometimes positively or negatively related to estimated 

flow in different reaches, adjusting conveyance ratios for flow 

did not substantially alter the interpretation of conveyance 

ratios that were not adjusted for flow. 

Inverse one-dimensional (1-D) hydrodynamic modeling of the St. Clair River was used to investigate conveyance 

change from 1962 to 2007, although missing water-level or 

limited flow-measurement data prior to 1996 limited annual 

estimation. In particular, inverse modeling techniques were 

used to estimate the magnitude and uncertainty of parameters 

associated with the effective channel roughness in each reach 

for selected time intervals under a fixed geometry model. 

Patterns in the time series of estimated parameters were 

interpreted as evidence of conveyance change by all factors that affect conveyance, including cross-sectional area, 

hydraulic radius, and channel roughness. Hourly water-level 

measurements were used as boundary conditions for unsteady 

flow simulations from April 1 to November 30 of each year, 

a period selected to avoid variable ice-induced backwater 

effects. 

Conveyance ratios can be computed for reach pairs on the 

basis of water-level measurements or effective channel roughness values estimated by use of inverse modeling. Comparison 

of annual conveyance ratios computed by use of both techniques for the period 1996 through 2007 showed that the two 

estimates are, in general, directly proportional. The consistency between estimates for the 1996-2007 periods supports 

the interpretation of trends in conveyance ratios computed by 

use of water-level data from 1962-2007. 

Possible changes in river conveyance cannot be ascertained directly from positive and negative changes in reach 

conveyances. To assess changes in river conveyance, annual 

flows were simulated under all available annual parameterizations for each corresponding water-level boundary-condition 

set. Over all boundary-condition sets, results show the relative simulated flows increased significantly with the year of 

parameterization from 1962 to 2002 by as much as 10 percent 

and decreased about 4.2 percent from 2002 to 2007. These 

non-monotonic trends in conveyance are difficult to associate 

with a causative event. 

Acknowledgments

This study was supported by the International Upper 

Great Lakes Study Board and U.S. Army Corps of Engineers. The authors are grateful to several individuals from the 

USACE Hydrologic Engineering Center in Davis, California. 

In particular, Jason Giovannettone developed the standard 1-D 

HEC-RAS model of the St. Clair River and responded to several requests for modifications from the authors. Mark Jensen 

developed a Visual Basic script that enabled the parameterestimation application—UCODE—to run HEC-RAS without 

user intervention. Gary Brunner, the developer of HEC-RAS, 

provided technical assistance to the authors in applying HECRAS to the St. Clair River. Nanette Noorbakhsh, USACE–

Detroit District compiled flow and water-level measurement 
References Cited 39

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Aaron Thompson, Environment Canada at the Canada Center 

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individuals from various local, State, and Federal agencies, 

including the Great Lakes Commission, who were instrumental in preserving NOAA’s water-level gaging-station network 

when elimination of numerous gages on the St. Clair–Detroit 

River Waterway was proposed in 2000. 

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Holtschlag and Hoard—Detection of Conveyance Changes in St. Clair River Using Historical Water-Level and Flow Data—Scientific Investigations Report 2009–5080 

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