Prepared in cooperation with the Chicago District of the U.S. Army Corps of Engineers

Evaluation of the Potential for Hysteresis in Index-Velocity 

Ratings for the Chicago Sanitary and Ship Canal near

Lemont, Illinois 

Scientific Investigations Report 2013–5095

U.S. Department of the Interior

U.S. Geological Survey
Cover photo. Photograph of a survey boat making measurements in the Chicago Sanitary 

and Ship Canal near Lemont, Illinois. (Photo by K. Johnson, U.S. Geological Survey.)
Evaluation of the Potential for Hysteresis in 

Index-Velocity Ratings for the Chicago Sanitary 

and Ship Canal near Lemont, Illinois

By P. Ryan Jackson, Sumit Sinha, Som Dutta, Kevin K. Johnson, James J. Duncker, and 

Marcelo H. Garcia

Prepared in cooperation with the Chicago District of the U.S. Army Corps of Engineers

Scientific Investigations Report 2013–5095

U.S. Department of the Interior

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

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2013

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Suggested citation:

Jackson, P.R.; Sinha, Sumit; Dutta, Som; Johnson, K.K.; Duncker, J.J.; and Garcia, M.H., 2013, Evaluation of the 

potential for hysteresis in index-velocity ratings for the Chicago Sanitary and Ship Canal near Lemont, Illinois: 

U.S. Geological Survey Scientific Investigations Report 2013–5095, 33 p., http://pubs.usgs.gov/sir/2013/5095/
iii

Contents

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

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

Theoretical Analysis......................................................................................................................................3

Conceptual Model Overview ..............................................................................................................3

Details of the Conceptual Model........................................................................................................5

Pseudo-Synthetic Hydrograph ..................................................................................................5

Synthetic Transverse Velocity Profiles ....................................................................................7

Index-Velocity Rating Development .........................................................................................7

Simulations Summary .................................................................................................................8

Hysteresis Analysis ..............................................................................................................................8

Effect of Flow Distribution and Sampling Configuration .......................................................8

Observed Transverse Velocity Distribution and Potential for Hysteresis ........................13

Numerical Modeling Analysis....................................................................................................................19

About the Numerical Model..............................................................................................................19

Model Development ...........................................................................................................................20

Simulated Events .......................................................................................................................20

The Computational Mesh .........................................................................................................22

Model Calibration and Validation............................................................................................22

Model Rating Development...............................................................................................................25

Sampling the Model Output .....................................................................................................25

Building the Ratings ..................................................................................................................28

Discussion............................................................................................................................................28

Conclusions...................................................................................................................................................31

References Cited..........................................................................................................................................32

Figures

1. Map of the Chicago Area Waterway System (CAWS) showing the 

Chicago Sanitary and Ship Canal (CSSC), Calumet Sag Channel, 

North Branch Chicago River, Little Calumet River, Grand Calumet River, 

and the other waterways in and near Chicago, Illinois. Locations of 

USGS streamgages and the extent of the model domain also are 

shown for reference.....................................................................................................................2

2. Schematic showing the instrument configurations and the channel 

bathymetry at the U.S. Geological Survey streamgage on the Chicago 

Sanitary and Ship Canal near Lemont, Illinois (05536890)......................................................4

3. Workflow schematic for the conceptual model ......................................................................5

4. Compilation plot for a single simulation using a symmetric flow 

distribution and the sampling configuration for the horizontal acoustic 

Doppler current profiler...............................................................................................................6

5. Index-velocity rating curves for six simulations using the H-ADCP 

configuration and beta density function coefficients as shown in each figure.................9
iv

6. Six-panel plot showing the sensitivity of the horizontal acoustic Doppler 

current profiler (H-ADCP) sampling configuration to various distributions 

of the transverse velocity profile over a typical high-flow event.......................................10

7. Six-panel plot showing the sensitivity of the acoustic velocity meter (AVM) 

sampling configuration to various distributions of the transverse velocity 

profile over a typical high-flow event......................................................................................11

8. Six-panel plot showing the sensitivity of a single 15-foot, centered-cell 

sampling configuration to various distributions of the transverse velocity 

profile over a typical high-flow event......................................................................................12

9. Six-panel plot showing the sensitivity of a five-cell, right-half with blanking

zone sampling configuration to various distributions of the transverse velocity 

profile over a typical high-flow event......................................................................................14

10. Index-velocity rating curves for two cases with highly skewed flow and a 

five-cell, right-half with blanking zone sampling configuration..........................................15

11. Distribution of α and β for observations from 102 high-flow events in the 

Chicago Sanitary and Ship Canal near Lemont, Illinois .......................................................15

12. Histograms of α and β for observations from 102 high-flow events in the 

Chicago Sanitary and Ship Canal near Lemont, Illinois .......................................................16

13. Compilation plot for the observed median flow distribution and the 

sampling configuration for the horizontal acoustic Doppler current profiler...................18

14. Comparison of the field-derived index-velocity rating for the horizontal 

acoustic Doppler current profiler (H-ADCP) in Chicago Sanitary and Ship 

Canal (CSSC) near Lemont, Illinois, to the conceptual-model rating curve 

developed by using median values of the beta distribution coefficients from 

observed velocity profiles .........................................................................................................19

15. Synthetic hydrographs for the Chicago Sanitary and Ship Canal (CSSC) 

and Cal-Sag Channel upstream of Sag Junction and measured discharge 

in the CSSC at Lemont and free surface elevation in the CSSC at 

Romeoville Road corresponding to case 1 and case 2 ........................................................21

16. Measured hydrographs for the Chicago Sanitary and Ship Canal (CSSC) 

and Cal-Sag Channel and water-surface elevation in the CSSC at 

Romeoville Road corresponding to event 1, event 2, event 3, and event 4.......................23

17.Top view and three-dimensional view of the computational mesh 

for the modeled domain.............................................................................................................24

18. Comparison between simulated and observed water-surface elevation 

at the USGS streamgage on the Chicago Sanitary and Ship Canal near 

Lemont, Illinois (05536890), corresponding to all simulated cases and 

observed events..........................................................................................................................26

19. Comparison between simulated and observed discharge at the USGS 

streamgage on the Chicago Sanitary and Ship Canal near Lemont, 

Illinois (05536890), corresponding to all synthetic cases and 

measured events ........................................................................................................................27

20. Model-derived index-velocity rating curves corresponding to the acoustic 

velocity meter (AVM) sensor configuration for all simulated cases and events.............29

21. Model-derived index-velocity rating curves corresponding to the horizontal 

acoustic Doppler current profiler (H-ADCP) sensor configuration for all 

simulated cases and events .....................................................................................................30
v

Tables

1. Annual mean and peak discharge measured at USGS streamgage on 

the Chicago Sanitary and Ship Canal near Lemont, Illinois (05536890), 

water years 2006 to 2012 .............................................................................................................7

2. Simulated high-flow events in the Chicago Sanitary and Ship Canal 

near Lemont, Illinois ...................................................................................................................22

3. Error between predicted and observed velocities in the nine cells of 

the horizontal acoustic Doppler current profiler in the Chicago Sanitary 

and Ship Canal near Lemont, Illinois. Median values of the root mean 

squared error (RMSE) are reported independently for the two 

stage-discharge relation regimes............................................................................................31
vi

Conversion Factors

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

meter (m) 3.281 foot (ft)

mile (mi) 1.609 kilometer (km)

Area

square foot (ft2

) 0.09290 square meter (m2)

square meter (m2

) 10.76 square foot (ft2)

Velocity

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

Flow rate

cubic foot per second (ft3

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

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

(NAVD 88) and the National Geodetic Vertical Datum of 1929 (NGVD 29).

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

Acknowledgments

The authors wish to thank Jim Best and Dan Parsons for the use of the multibeam bathymetry 

used in the model. A special thanks goes to the USGS hydrologists who improved this report 

through their helpful reviews and the USGS personnel who assisted with data collection.
Evaluation of the Potential for Hysteresis in Index-Velocity 

Ratings for the Chicago Sanitary and Ship Canal near 

Lemont, Illinois

By P. Ryan Jackson1, Sumit Sinha2, Som Dutta2, Kevin K. Johnson1, James J. Duncker1, 

and Marcelo H. Garcia2

Abstract 

The U.S. Geological Survey (USGS) is responsible for 

monitoring flows in the Chicago Sanitary and Ship Canal 

(CSSC) near Lemont, Illinois, as a part of the Lake Michigan Diversion Accounting overseen by the U.S. Army Corps 

of Engineers, Chicago District. Lake Michigan Diversion 

Accounting is mandated by a U.S. Supreme Court decree in 

order to monitor, and limit, the State of Illinois’ annual diversion of Great Lakes water through the manmade CSSC. Every 

5 years, a technical review committee consisting of practicing 

engineers and academics reviews USGS streamgaging practices in the CSSC near Lemont, Illinois. The sixth technical 

review committee expressed concern that the index-velocity 

rating—the method used to estimate mean cross-sectional 

velocity from a measured index velocity—may be subject to 

hysteresis at this site because of the unique, unsteady hydraulics of the canal. Hysteresis in index-velocity ratings can occur 

at sites where the flow distribution in the channel varies significantly between the rising and falling limbs of the hydrograph 

for the same discharge. Presently, hysteresis in index-velocity 

ratings has been documented only in tidally affected sites. This 

report investigates whether hysteresis can occur at this nontidal site, and the conditions under which it is likely to occur, 

by using both a theoretical approach and a three-dimensional 

hydrodynamic model. The theoretical analysis investigated 

the conditions required for hysteresis in the index-velocity 

rating, and the modeling analysis focused on the effect of the 

timing of the inflows from the CSSC and the Cal-Sag Channel on the potential for hysteresis and whether highly resolved 

simulations of actual high-flow events show any evidence of 

hysteresis. 

Based on both a theoretical analysis using observed 

historical data and an analysis using a three-dimensional 

hydrodynamic model, there is no conclusive evidence for 

the existence of hysteresis in the index-velocity rating at 

the USGS streamgage on the CSSC near Lemont, Illinois. 

Although the theoretical analysis indicated the possibility of 

hysteresis at this site, the hydrodynamic conditions required 

to generate hysteresis are not present at this site based on 

historical data. Ongoing streamgaging practices at this site will 

use the information in this report and include periodic assessment of the index-velocity rating for any signs of hysteresis 

that might result from future changes to the operation of this 

manmade canal.

Introduction

The construction of the Chicago Sanitary and Ship Canal 

(CSSC) in the late 1800s allowed the reversal of the Chicago 

River in 1900, enabling Chicago to send diluted wastewater 

effluent downstream to the Mississippi River system and away 

from the city’s freshwater source, Lake Michigan. Illinois’ 

use of Lake Michigan water for consumption and dilution 

of wastewater resulted in a net diversion of water from the 

Great Lakes Basin, leading to a U.S. Supreme Court decree 

limiting Illinois’ diversion of Great Lakes water. This decree 

established the need for diversion accounting and continuous 

measurement of the discharge of water out of Lake Michigan 

through the CSSC. To ensure that the best engineering practices and technology are being used in diversion accounting, a 

technical review committee consisting of practicing engineers 

and academics is convened every 5 years to review U.S. Geological Survey (USGS) streamgaging practices on the CSSC 

near Lemont, Illinois (Ill.) (station 05536890). With locks and 

control works throughout the Chicago Area Waterway System 

(CAWS) and significant influences from industry, the CSSC 

is a highly regulated, unnatural waterway with continuous 

perturbations that propagate from the lakefront to Lockport 

Lock and Dam just upstream of the confluence with the Des 

Plaines River (fig. 1). Continuous and accurate measurement 

of discharge in the CSSC requires innovative technology, measurement redundancy, and extensive data analysis. The reader 

is referred to Johnson and others (2012) and Jackson and 

1

U.S. Geological Survey.

2

University of Illinois at Urbana-Champaign.
2 Evaluation of the Potential for Hysteresis in Index-Velocity Ratings for the Chicago Sanitary and Ship Canal near Lemont, Illinois

10 MILES

10 KILOMETERS

5

0 5

0

DUPAGE COUNTY

COOK COUNTY

WILL COUNTY

LAKE COUNTY

COOK COUNTY

42°15

41°45'

41°30'

42°

88°15' 88° 87°45' 87°30'

Des

R i v e r

P l a i n e s

Chicago Sanitary and Ship Canal

Calumet Sag

Channel

Salt

Creek

Little Calumet River

Calumet

R

Cr

Willow

Des

P l a i n e s

River

N o r t h

Branch

Chicago

River

South Branch

Chicago River

North Shore Channel

COOK COUNTY

SIONILLI

ANAIDNI

L A K E

M I C H I G A N

Grand

Calumet R

RNEHcM Y OC

Lockport Lock and Dam

Controlling works

O'Brien Lock

and Dam

Chicago Lock

Wilmette 

Gates

Illinois 

Map 

area

ILLINOIS

River

Base from U.S. Geological Survey 

1:100,000 Digital Data

Albers Equal-Area Conic Projection o o Standard Parallels 33 and 45 , o

central meridian -89

EXPLANATION

Chicago Sanitary and 

Ship Canal at Romeoville, 

Illinois (05536995)

Chicago Sanitary and Ship 

Canal near Lemont, Illinois 

(05536890)

Calumet Sag

Channel near Rt 83

at Sag Bridge

(05536700)

Inset

Modeling domain

U.S. Geological Survey streamgage 

 with location name and number

Calumet

Sag Channel

Des Plaines

River

Chicago

Sanitary

and

Ship

Canal

Lemont

Figure 1. Map of the Chicago Area Waterway System (CAWS) showing the Chicago Sanitary and Ship 

Canal (CSSC), Calumet Sag Channel, North Branch Chicago River, Little Calumet River, Grand Calumet 

River, and the other waterways in and near Chicago, Illinois. Locations of USGS streamgages and the 

extent of the model domain also are shown for reference.
Theoretical Analysis 3

others (2012) for detailed reviews of the streamgaging practices at this site and the unique hydraulics of this system. This 

report describes the investigation of the potential for hysteresis 

in index-velocity ratings for two types of acoustic velocity 

meters installed at the USGS streamgage on the CSSC near 

Lemont, Ill. (05536890).

The intense scrutiny of the Lake Michigan diversion 

accounting requires a high degree of accuracy and reliability 

in the USGS discharge measurements and the methods to 

compute continuous discharge in the CSSC. The current primary gage on the CSSC near Lemont, Ill., is an acoustic velocity meter (AVM) (fig. 2). This time-of-travel meter measures 

a bank-to-bank average velocity (index velocity) at specific 

elevations above the bed in the measurement reach, which is 

related to a mean cross-sectional velocity (discharge divided 

by the rated area) by an index-velocity rating. Redundancy 

is provided by a horizontal acoustic Doppler current profiler 

(H-ADCP), also called an acoustic Doppler velocity meter 

(ADVM), which serves as the backup gage at this site. The 

H-ADCP measures velocity in a series of “cells” at a specified 

distance in front of the right bank-mounted unit (fig. 2). Technical limitations prevent the unit from measuring near either 

bank (the unmeasured near-bank distance is a function of 

the unit configuration as well as channel geometry and water 

depth). Currently, the H-ADCP data are used only to estimate 

discharge during periods of missing record from the primary 

gage. The AVM and H-ADCP were compared in detail by 

Jackson and others (2012), who concluded that the H-ADCP 

was capable of replacing the AVM as the primary gage. In 

addition, Jackson and others (2012) reviewed and analyzed the 

complex hydraulics present in the system and the challenges 

this system presents to those applying standard continuous 

streamgaging techniques at this site. 

Of particular interest in the present analysis is the location of the streamgage relative to the confluence of the CSSC 

and the Calumet Sag Channel (Cal-Sag) and the possibility 

that the proximity of this gage to the confluence might result 

in hysteresis in the index-velocity rating curve due to variations in flow distribution downstream of this confluence during 

high-flow events. With regard to rating curves in open-channel 

hydraulics, hysteresis can be defined as the formation of 

looped rating curves during unsteady flows. Specifically, the 

rating curves for the rising and falling limbs of a flood wave 

are not coincident, resulting in a loop-shaped rating curve 

for the entire event. Hysteresis in index-velocity ratings has 

been documented at tidally affected sites (for example, Ruhl 

and Simpson, 2005) due to the large changes in flow distribution in the measurement section over the course of the tidal 

cycle. Changes in flow distribution in the section can result 

in two separate index velocities for the same discharge (or 

same mean cross-sectional velocity), leading to hysteresis. 

Although hysteresis in index-velocity ratings at nontidal sites 

is theoretically possible, documentation of this phenomenon 

is scarce. In contrast, hysteresis in stage-discharge ratings is 

well documented for backwater-affected channels and those 

with highly unsteady flow conditions. Fread (1975) explained 

that hysteresis in a stage-discharge rating curve is the manifestation of the fact that the discharge at a given section under 

unsteady flow conditions is the function not only of stage but 

also of the energy slope. Mander (1978) further explained 

that when a wave associated with a high-flow event passes 

through a given cross section, the effect of the wave front 

when upstream of the cross section is to increase the approach 

velocity at the cross section. As the flood peak passes downstream of the cross section, the rear of the wave enhances the 

backwater effect at the cross section and hence reduces the 

velocity at the cross section. The consequence of the floodpeak passage through the section is such that for the same 

stage, the discharge is higher during the rising stage than the 

falling stage. Consequently, one can obtain multiple values of 

the discharge for the same stage, depending on whether the 

stage measurement was made on the rising limb or the falling 

limb of the passing flood wave. To combat the problem of 

hysteresis in stage-discharge ratings, engineers and researchers 

have employed the index-velocity technique for measuring the 

discharge at a given section.

The objective of this report is to determine what conditions are required in the CSSC near Lemont, Ill., to generate hysteresis in the index-velocity ratings for the AVM and 

H-ADCP sampling configurations. Furthermore, this report 

examines whether those conditions are present at the site on 

the basis of observed data and simulations from conceptual 

and three-dimensional models. 

Theoretical Analysis

A theoretical analysis of the conditions under which 

hysteresis is present in the index-velocity rating curve for the 

Chicago Sanitary and Ship Canal near Lemont, Ill., allows 

the flexibility to test flow distributions that might not otherwise be present in field data or hydrodynamic models driven 

by observed boundary conditions. The following conceptual 

model tests more than 38,000 simulated flow events to determine the potential for hysteresis in the derived index-velocity 

rating.

Conceptual Model Overview

In order to determine the conditions required to produce 

hysteresis in the index-velocity ratings for the CSSC near 

Lemont, Ill., a conceptual model was developed. The model 

utilizes observed high-flow event data (stage and discharge) 

and the beta distribution to prescribe transverse velocity 

distributions (Seo and Baek, 2004) in order to investigate the 

sensitivity of the H-ADCP and AVM index velocity to flow 

distribution in the section (fig. 3). The observed hydrograph 

is divided into the rising limb and falling limbs (fig. 4A), and 

stage (G) is converted to area (A) by using the stage-area 

rating for this gage (A(ft2

)=139.83 G(ft)+670.39; Jackson and 

others, 2012) (figs. 4B and C). Mean velocity for the section 
4 Evaluation of the Potential for Hysteresis in Index-Velocity Ratings for the Chicago Sanitary and Ship Canal near Lemont, Illinois

EXPLANATION

Elevation, in feet above North American

 Vertical Datum of 1988

High: 568.14

Low: 548.98

N

Sloughed bank

Buried 

 cables

Cross-channel AVM

cables

Up-looking 

ADCP

(Installed near midchannel within

the acoustic velocity meter (AVM) 

measurement volume)

Horizontal acoustic Doppler 

current profiler (H-ADCP)

Gage

house

Measurement cross section

(see detail below)

162 feet

o 20

o

44.5

Flow

0 18 36 54 72 90 108 126 144 162

Distance, in feet

576.7, NAVD 88 (mean)

Channel cross section and bathymetry simplified for schematic

PATH 1 (570.2, NAVD 88)

PATH 2 (565.0, NAVD 88)

PATH 3 (559.6, NAVD 88)

551.5

0 18

15 30 45 60 75 90 105 120 135 150

36 54 72 90 108 126 144 162

555.0

560.0

565.0

570.0

575.0

576.7

Elevation above North American Vertical 

Datum of 1988 (NAVD 88), in feet

H-ADCP 

Elevation 565.1, NAVD 88

o

1.5

o

20

cell 1 cell 2 cell 3 cell 4 cell 5 cell 6 cell 7 cell 8 cell 9

Acoustic Velocity Meter (AVM) Section Horizontal Acoustic Doppler Current

Profiler (H-ADCP) Section

Figure 2. Schematic showing the instrument configurations and the channel bathymetry at the U.S. Geological Survey streamgage 

on the Chicago Sanitary and Ship Canal near Lemont, Illinois (05536890). The upper figure is in plan view.
Theoretical Analysis 5

(Vmean) is then computed by dividing the discharge (Q) by the 

rated area (A) and then plotted as the ordinate for the indexvelocity rating. The mean velocity is then used in conjunction 

with the coefficients of the beta density function (α, β, and ∆) 

to define the transverse velocity profile in the section (see 

section “Synthetic Transverse Velocity Profiles”). The coefficients of the beta density function are allowed to vary over the 

hydrograph, resulting in a variable shape of transverse velocity 

profile (fig. 4H). The index velocity (Vindx) is then computed 

by sampling the transverse velocity profile at each time step in 

the same configuration as the H-ADCP and AVM and plotted 

as the abscissa for the index-velocity rating. This workflow 

allows one to take observed data (stage and discharge), 

synthetically distribute the flow across the section, resample 

the flow using field-instrument configurations, and generate 

an index-velocity rating for each simulation. Replicating this 

process for a large range of flow distributions (the example in 

figure 4 is only one of 38,400 simulations run) allows one to 

test the conditions required to generate hysteresis in the indexvelocity rating in the CSSC near Lemont, Ill. The conceptual 

model was built and run in Matlab®.

Details of the Conceptual Model 

This section presents the details of the conceptual model 

diagramed in figure 3. Each step within the flowchart in 

figure 3 is discussed in detail, and key terms and methods are 

defined. In addition, the methods used to generate the large 

number of scenarios are presented. 

Pseudo-Synthetic Hydrograph

The hydrograph and stage data used in the conceptual 

model are based on observed data from the CSSC near Lemont 

during a large (13,200-ft3/s peak flow) event in November 

2010 (fig. 4A). For comparison, table 1 lists the annual mean 

and peak flows for the CSSC near Lemont, Ill. (05536890), for 

2006 to 2012. Ten-minute observations of stage and discharge 

during the November 2010 event were smoothed by using a 

110-minute moving average to remove noise and temporal 

variations due to lockages and other disturbances in the system 

(see Jackson and others, 2012). The derivative of the discharge 

(dQ/dt) is used to define the rising limb (dQ/dt > 0) and falling 

limb (dQ/dt < 0) of the hydrograph (fig. 4A).

Observed

data

Beta 

distribution

parameters

Input

Compute area (A)

using stage-area 

rating

Discharge (Q)

Stage

Define rising and

falling limbs

Compute mean velocity

V = Q/A mean

α β ∆

Compute α and β

event

time series over

Compute velocity distribution

for each time step

(using beta distribution) 

Sample velocity

distribution using 

virtual instrument 

Compute index velocity 

V indx

Build

indexvelocity

rating

Output

Figure 3. Workflow schematic for the conceptual model.
Discharge, in cubic 

feet per second

0

5,000

10,00015,000

Stage, in feet

23.5

24

24.5

25

Area, 

in square feet 

3,9004,0004,1004,200

Time, in minutes

Mean velocity, 

in feet per second

0 500 1,000 1,500 0

1

2

3

4

Index ve

Mean velocity, 

in feet per second

Discharge, in cubic 

feet per second

Stage, in feet

0 5,000 10,000 15,000

23.5

24

24.5

25

Green = rising limb Red = falling limb

0 20 40 60 80 100 

0

1

2

3

4

5

Velocity, in feet per second

Cell No. 1 2 3 4 5 6 

Time, in minutes

Coefficient value

0 500 1,00

1.41.2

1.6

1.8

2

α = β (symmetric)

∆ = 0.4

0 

0

1234

Loop 

in (ft

ABCD

E

F

Distance from right bank, in f

Figure 4. Compilation plot for a single simulation using a symmetric flow distribution and the sampling configuration for the horizontal acoustiprofiler. Plots on the left: 

A, the observed hydrograph; 

B, the observed stage; 

C, the computed cross-sectional area; and 

D, the computed mean 

the right: E, the time series of beta distribution coefficients; 

F, the stage-discharge loop; 

G, the index-velocity rating; and 

H, the temporal evolut

velocity profile over the event as defined by the beta distribution.
Theoretical Analysis 7

 

Table 1. Annual mean and peak discharge measured at USGS 

streamgage on the Chicago Sanitary and Ship Canal near 

Lemont, Illinois (05536890), water years 2006 to 2012.

[ft3/s, cubic feet per second]

Water 

year

Mean discharge 

(ft3

/s)

Peak discharge 

(ft3/s) Date of peak

2006 2,615 14,632 September 13, 2006

2007 3,167 16,572 August 23, 2007

2008 3,096 20,053 September 14, 2008

2009 3,267 18,649 December 27, 2008

2010 2,967 18,984 July 24, 2010

2011 2,838 16,862 July 23, 2011

2012 2,181 13,366 August 27, 2012

Synthetic Transverse Velocity Profiles 

The transverse velocity distribution across the measurement section is defined by using the beta distribution (Seo and 

Baek, 2004). The dimensionless transverse velocity distribution is defined by the beta density function 

u

V y y y

mean

= ( ) +

( ) ( )( ) − < < − − Γ

Γ Γ

α β

α β

α

' ' β ' ( )

1 1 1 0 1 (1)

where u is the depth-averaged velocity;

	 y′ is the normalized transverse coordinate 

defined as y′= y/W, where y is the 

transverse coordinate and W is the channel 

width; and

α is a positive coefficient of the beta density 

function; and

β is a positive coefficient of the beta density 

function; and

Vmean is the mean velocity for the cross section; and

	 Г	 is the gamma function defined as

 

 Γ α α α ( ) ∞

− − ∫

0

1 x e dx 0 x = > (2)

 

 Γ β β β ( ) = > ∞ − − ∫

0

1 x e dx 0 x (3)

The beta distribution has been found to be capable of representing the complete spectrum of properties of the transverse 

velocity distribution for natural streams (Seo and Baek, 2004). 

Although the CSSC is not a natural stream, the beta distribution is capable of replicating the transverse velocity distribution in the CSSC near Lemont (Jackson and others, 2012). 

The coefficients α and β in equation 1 can be varied over a 

large range of values to generate largely varying profiles. If α

and β are equal, the velocity distribution is symmetric about 

the channel centerline. Larger values of α and β generate a 

peaked profile with more flow in the center of the channel, 

whereas low values of α and β generate a more flat or uniform 

profile. If α and β are not equal, the flow is skewed to the left 

bank for α > β and skewed to the right bank for β > α. In this 

case, the banks are defined looking downstream with y′ = 0 on 

the right bank and y′ = W on the left bank. 

The beta distribution was utilized by specifying Vmean

(fig. 4D) and time series for α and β over the rising and falling 

limbs of the hydrograph (fig. 4E) to generate a velocity profile 

at each time step (fig. 4H). Change in the coefficients α and β

was driven by a normalized value of dQ/dt multiplied by Δ, 

a dimensionless parameter representing the total change in 

the values of the coefficients α and β over each limb. Physically, Δ can be thought of as a measure of the total prescribed 

change in the shape of transverse velocity profile over the 

high-flow event and is set by the user. Larger values of Δ will 

result in greater variation in the shape of the velocity profile over an event. Using the rate of change of Q to drive the 

change in α and β links the shape of the transverse profiles 

to the acceleration and deceleration of the flow. With this 

approach, the profiles become more peaked on the rising limb 

of the hydrograph (that is, α and β increase) and more flat on 

the falling limb of the hydrograph (lower α and β) (fig. 4H). 

This scenario allows the greatest variance in the flow profile 

between the rising and falling limbs of the hydrograph, thus 

producing the greatest likelihood of generating hysteresis. This 

synthetic variability in the flow distribution over an event will 

be compared with observed distributions later in this report. 

Index-Velocity Rating Development

Every simulation run by using the conceptual model 

resulted in a set of transverse velocity profiles that vary over 

the event hydrograph. In order to build an index-velocity 

rating from these data, one needs to compute the associated 

Vindx for every value of Vmean. Index velocities were computed 

for each time step by sampling each transverse velocity 

profile, using the field configuration for the AVM (single, fullwidth cell) and H-ADCP (nine 15-ft cells with unmeasured 

areas near both banks; see figs. 2 and 4H). In addition, several 

other configurations were tested including a single 15-ft cell 

in the center of the channel and five cells centered between the 

channel centerline and 15 ft from the right bank (measuring 

only the right half of the flow) to test the sensitivity of these 

other configurations to the variable flow distributions. The 

index velocity, Vindx, was then computed for each configuration, at each time step, as the average of the mean cell velocities (that is, for the H-ADCP, Vindx is the average of the mean 

velocities from each of the nine sampling cells). Therefore, for 

each simulation, Vmean was plotted against Vindx and points were 

color coded by the rising limb (green) or falling limb (red) of 

the hydrograph to determine whether hysteresis was present 

(fig. 4G). For every case in which a loop was generated, the 

area of the loop, computed as the total area between the two 

limbs, was computed as a measure of the magnitude of the 

hysteresis (a large loop area indicates substantial hysteresis).
8 Evaluation of the Potential for Hysteresis in Index-Velocity Ratings for the Chicago Sanitary and Ship Canal near Lemont, Illinois

Simulations Summary

In order to test a large range of possible flow distributions that could potentially lead to hysteresis, a large number 

of simulations were run for each instrument configuration. The 

initial beta distribution coefficients were varied from 1 to 5 at 

increments of 0.1, leading to 1,600 combinations of α and β. 

In addition, six different values of Δ were tested (0.0 to 0.5 at 

increments of 0.1) to examine the sensitivity of hysteresis to 

the total change in the coefficient value over the hydrograph. 

Therefore, 9,600 simulations were computed for each of the 

four instrument configurations (38,400 simulations total), 

with each simulation having an associated index-velocity 

rating curve and loop area. Figure 5 presents examples of 

index-velocity rating curves for six different simulations (all 

H-ADCP configuration) covering a large range of hysteresis 

magnitude (loop area). All computations were completed 

with the same observed hydrograph (fig. 4A) and stage record 

(fig. 4B). 

Hysteresis Analysis

The results of the conceptual model are discussed in this 

section and analyzed to understand the conditions required 

for hysteresis in the index-velocity rating in the Chicago 

Sanitary and Ship Canal near Lemont, Ill. In addition to the 

field configurations of the AVM and H-ADCP, several other 

common index-velocity sampling configurations are analyzed 

to determine their potential for hysteresis.

Effect of Flow Distribution and Sampling 

Configuration

Based on results of the conceptual model, hysteresis is 

most prominent in index-velocity ratings developed by using 

the H-ADCP sampling configuration for conditions in which 

the transverse flow profile is highly skewed to either bank 

and changes shape significantly between the rising and falling 

limbs of the hydrograph (fig. 6). With the exception of the 

Δ = 0 case, all simulations showed substantial hysteresis for 

flow profiles with extreme skewness to either bank. Hysteresis 

arises in this situation because of a combination of unmeasured areas near both banks and the magnitude and overall 

change in the flow profile within the unmeasured areas near 

the banks. Although subtle, the hysteresis is slightly larger 

for a flow skewed to the right bank relative to a flow skewed 

to the left bank. This is likely caused by the larger unmeasured zone near the right bank (15 ft) compared to the smaller 

unmeasured zone near the left bank (12 ft). 

For the Δ = 0 case (no change in the shape of the profile 

over the hydrograph, only changes in magnitude), no hysteresis was observed for any of the 1,600 cases simulated. In general, as Δ increases, there is greater overall change in the shape 

of the profile over the hydrograph, and hysteresis becomes 

more prominent (fig. 6). However, for the H-ADCP sampling 

configuration, hysteresis can be minimal even for large Δ if the 

flow is symmetric. In the case of a symmetric flow, hysteresis 

is greater for a flat flow profile than for a peaked flow profile. 

This condition likely arises because the peaked profile results 

in less overall flow in the unmeasured zones near the edges. 

When the entire cross section is measured and there are 

no unmeasured zones near the banks, as in the case of the 

AVM configuration, no hysteresis is observed in the conceptual model for any of the 9,600 cases simulated (fig. 7). Therefore, hysteresis is not caused solely by change in the profile 

over the hydrograph but rather by such changes that occur in 

unmeasured zones within the cross section and their overall 

influence on the computed index velocity. This result implies 

that instruments that measure the full width of a cross section 

with no unmeasured zones (such as the AVM) are not affected 

by hysteresis induced by variability in the transverse velocity 

profile. The AVM sampling configuration was modeled in this 

case as a single cell spanning the full width of the channel. 

Because the beta distribution represents the depth-averaged 

flow distribution, these results should hold for all three paths 

of the AVM, as well as for the index velocity built from their 

composite (see Jackson and others, 2012). 

The large dependency of hysteresis on the sampling configuration of the instrument used to measure the index velocity 

suggests that other potential sampling configurations for the 

CSSC near Lemont should be tested to determine their potential for hysteresis. As mentioned previously, the current (2013) 

configuration of the H-ADCP measures an index velocity by 

averaging nine 15-ft sampling cells evenly spaced across the 

channel with unmeasured zones near both banks (see fig. 2). 

Two additional sampling configurations were chosen on the 

basis of potential use of such a configuration at the site. The 

first alternative sampling configuration tested was a single, 

15-ft sampling cell along the channel centerline. Such a configuration might be used with both side-looking instruments as 

well as bottom-mounted up-looking ADCPs. The second alternative sampling configuration tested was a configuration with 

five evenly distributed sampling cells centered between the 

center of the channel and 15 ft from the right bank. This configuration approximates a common field situation when using 

instruments that do not have the capability to profile across 

the entire cross section. Neither of these two configurations 

have been field tested at this site. The remainder of this section 

presents the findings for these alternative configurations. 

Hysteresis was pronounced in the case of a single 

sampling cell in the center of the channel for a large range 

of flow distribution scenarios (fig. 8). This configuration was 

found to be the most susceptible to hysteresis of the configurations tested. Total loop areas, representing the strength 

of the hysteresis, were found to be nearly twice that of the 

H-ADCP configuration case. Like the other configurations 

tested, the case of Δ = 0 resulted in no hysteresis. However, as 

Δ increased, hysteresis increased, with the greatest loop areas 

observed for flat, skewed profiles. The least hysteresis was 

observed for highly peaked, symmetric flow. This is likely 

because, for such a flow distribution, the center sampling cell 
Mean velocity, in feet per second Mean velocity, in feet per second 0 0.51 1.52 2.53

3.5

4

α = 2.0 β = 2.0∆ = 0.0Loop area

2 in (ft/s) = 0.0

α = 4.9β = 1.32∆ = 0.3Loop area

2 in (ft/s) = 0.6

α = 5.0 β = 1.51∆ = 0.5Loop area

2 in (ft/s) = 0.8

A

D E F

Index velocity, in feet per second

0 0.5 1 1.5 2 2.5 3 3.5 4

Index velocity, in feet per second

0 0.5 1 1.5 2 2.5 3 3.5 4

Index velocity, in fe

0 0.5 1 1.5 2 

0

0.5

1

1.5

2

2.5

3

3.5

4

α = 4.6β = 1.75∆ = 0.2Loop area

2 in (ft/s) = 0.2

B C

Figure 5. Index-velocity rating curves for six simulations using the H-ADCP configuration and beta density function coefficients as shown ipoints denote the rising limb of the hydrograph, and red points denote the falling limb. The loop area, a measure of the strength of the hystereach figure for reference.
β 1 2 3 4 5 β 1 2 3 4 5 β 1 2 3 4 

α 1 1.52 2.53 3.54 4.55 α 1 1.52 2.53 3.54 4.55 

∆ = 0.0 ∆ = 0.1 ∆ = 0.2 ∆ = 0.3 ∆ = 0.4 ∆ = 0.5 Line of symmetric flow 

Left bank skew Flat Peaked

Figure 6. Six-panel plot showing the sensitivity of the horizontal acoustic Doppler current profiler (H-ADCP) sampling configuration to variothe transverse velocity profile over a typical high-flow event. The flow profile is defined by the beta density function using coefficients α and overall change in α and β over the event, and hysteresis is quantified by using the loop area. Simulations in which either α and β fall below ufrom analysis because inverted profiles are not likely to occur in nature.
β 1 2 3 4 5 β 1 2 3 4 5 β 1 2 3 4 

α

1

1.5

2

2.5

3

3.5

4

4.5

5

α

1

1.5

2

2.5

3

3.5

4

4.5

5

∆ = 0.0 ∆ = 0.1 ∆ = 0.2

∆ = 0.3 ∆ = 0.4 ∆ = 0.5

Line of symmetric flow 

Left bank 

skew

Flat

Peaked

Figure 7. Six-panel plot showing the sensitivity of the acoustic velocity meter (AVM) sampling configuration to various distributions of the tprofile over a typical high-flow event. The flow profile is defined by the beta density function using coefficients α and β, Δ specifies the overaβ over the event, and hysteresis is quantified by using the loop area. Simulations in which either α and β fall below unity are omitted from aninverted profiles are not likely to occur in nature.
β 1 2 3 4 5 β 1 2 3 4 5 β 1 2 3 4 

α 1 1.52 2.53 3.54 4.55 α 1 1.52 2.53 3.54 4.55 ∆ = 0.0 ∆ = 0.1 ∆ = 0.2 ∆ = 0.3 ∆ = 0.4 ∆ = 0.5 Line of symmetric flow 

Left bank skew Flat Peaked 

Figure 8. Six-panel plot showing the sensitivity of a single 15-foot, centered-cell sampling configuration to various distributions of the tranprofile over a typical high-flow event. The flow profile is defined by the beta density function using coefficients α and β, Δ specifies the overover the event, and hysteresis is quantified by using the loop area. Simulations in which either α and β fall below unity are omitted from analprofiles are not likely to occur in nature.
Theoretical Analysis 13

was able to consistently sample the greatest amount of the 

flow. The widespread existence of hysteresis in this configuration suggests that caution should be used during analysis and 

rating development for any instrument used in the CSSC near 

Lemont that is set up in such a configuration. Only a small 

portion of the cross section is actually measured in this configuration, resulting in substantial hysteresis over a wide range 

of flow distributions. 

Perhaps contrary to intuition, for a sampling configuration in which only flows in the right half of the cross 

section are being measured (with the exception of the 

unmeasured zone within 15 ft of the right bank), the greatest hysteresis occurs when the flow is highly skewed to the 

right bank (fig. 9). In spite of having a large portion of the 

flow within the sampling volume when the flow is skewed 

to the right bank, the 15-ft unmeasured area near the right 

bank leads to significant hysteresis. The loop area for the case 

of α = 1.5, β = 5.0, Δ = 0.5 can be reduced by 84 percent by 

sampling the full right half of the flow and eliminating the 

unmeasured zone near the right bank. 

It should be noted that whereas loop area may adequately 

define the strength of hysteresis in a rating curve, the loop area 

may underestimate the potential uncertainty in the indexvelocity rating due to hysteresis in the case of a large, asymmetric, unmeasured area in the cross section. In this case, a 

true measure of the effect of hysteresis on index-velocity ratings is the difference between the predicted Vmean for the rising 

and falling limbs of a hydrograph for the same Vindx. Large differences in Vmean can lead to significant errors in the predicted 

discharge if the wrong limb of the rating is chosen. Large 

unmeasured areas in the cross section can create problems by 

shifting the slope of the rating curve due to a combination of 

skewed flow distributions and the relative proportion of the 

flow measured by the index-velocity instrument. For instance, 

for the configuration in which the right half of the flow is 

measured (less the 15 ft near the right bank), a flow skewed to 

the left bank (α = 5.0, β = 1.5) results in a steep rating curve 

(fig. 10) because only low velocities are being captured by 

the instrument. Although the loop area for such a case is only 

0.31 (ft/s)2, the orientation of the loop creates a large difference between the predicted Vmean values for the rising and 

falling limbs of the rating (fig. 10). When the flow is skewed 

to the right bank (α = 1.5, β = 5.0), the loop is 5 times larger 

in terms of area, but the slope is less, leading to a smaller 

difference in the predicted Vmean for the rising and falling limbs 

of the rating. 

Observed Transverse Velocity Distribution and 

Potential for Hysteresis

The previous section examined the conditions under 

which hysteresis is most prominent in the index-velocity 

rating for the CSSC near Lemont, Ill. Although this exercise 

provides useful information about the potential for hysteresis 

at this site, it does not necessarily dictate that hysteresis in the 

index-velocity rating exists for the range of observed flow conditions. To determine whether the hydraulics in the CSSC near 

Lemont support conditions for hysteresis, data from the from 

the H-ADCP can be utilized to examine the temporal variation 

in the shape of the transverse velocity profile over high-flow 

events. A total of 102 high-flow events were selected from 

available data in water years 2010 through 2012. Events were 

selected to represent a range of flow magnitudes. One-minute 

velocity observations from the H-ADCP in each of the nine 

cells of the sampling volume (fig. 2) were extracted and averaged over 10-minute intervals for each event. The peak in the 

discharge for each event was used to separate the rising and 

falling limbs of each event hydrograph; all data were categorized as to belonging to either a rising or falling limb and were 

later combined for statistical analysis. To understand the shape 

of the transverse velocity profile and its evolution in time, 

every 10-minute velocity profile (n = 18,094) was fit by using 

the beta density function, and α and β values for each best fit 

were recorded and analyzed.

The observed velocity profiles for 102 events in the 

CSSC near Lemont are flat and generally relatively symmetric, 

with a slight skew to the right bank (fig. 11). With the exception of several low flow profiles, the α and β values are tightly 

clustered between 1 and 1.75. There is no significant variation 

between the rising and falling limbs as the data clouds overlap. 

The rightward skew of the profiles is consistent with observations from November 2006 to January 2010 reported by 

Jackson and others (2012). 

Coefficients of the beta distribution determined by using 

observed data show almost no change between the median values for the rising and falling limbs of the hydrograph and have 

only a small variation about their median values for each event 

(fig. 12). Statistical analysis shows that for 8,163 observations 

on the rising limb and 9,931 observations on the falling limb, 

the median values for α for both limbs are 1.21 and the median 

values for β are 1.3 and 1.29 for the rising and falling limbs, 

respectively. Over each event, both α and β showed the greatest variance at low flows; and as the flow increased, the values 

of α and β decreased towards flatter profiles and coefficients 

just above unity (fig. 12 inset). Statistically, β showed slightly 

more variation over each event than α: the median values of 

the standard deviation of β were 0.07 and 0.083 for the rising 

and falling limbs, respectively, whereas the median values of 

the standard deviation of α were 0.056 and 0.067 for the rising 

and falling limbs, respectively. The slightly greater variability 

in β indicates the potential for increased asymmetry in the flow 

with the skew favoring the right bank. 

Several points concerning these observations are notable. 

First, the coefficients are nearly equal for the rising and falling 

limbs of the hydrograph, suggesting that there is little change 

in the shape of the transverse velocity profile between the 

rising and falling limbs of a high-flow event. In the previous 

section we examined a worst-case type scenario in which the 

coefficients varied greatly between the rising and falling limbs 

of the hydrograph in order to generate hysteresis. The second 

important point is that although α and β have similar median 
β 1 2 3 4 5 β 1 2 3 4 5 β 1 2 3 4 

α 1 1.52 2.53 3.54 4.55 α 1 1.52 2.53 3.54 4.55 

∆ = 0.0 ∆ = 0.1 ∆ = 0.2 ∆ = 0.3 ∆ = 0.4 ∆ = 0.5

Line of symmetric flow 

Left bank 

skew

Flat

Peaked

Figure 9. Six-panel plot showing the sensitivity of a five-cell, right-half with blanking zone sampling configuration to various distributions ovelocity profile over a typical high-flow event. The flow profile is defined by the beta density function using coefficients α and β, Δ specifies change in α and β over the event, and hysteresis is quantified by using the loop area. Simulations in which either α and β fall below unity are analysis because inverted profiles are not likely to occur in nature.
Theoretical Analysis 15

Index velocity, in feet per second

Mean velocity, in feet per second

0 1 2 3 4 5 6 7

0

0.5

1

1.5

2

2.5

3

3.5

4

α = 1.5 

β = 5.0

∆ = 0.5 2

Loop area in (ft/s) = 1.56

α = 5.0 

β = 1.5

∆ = 0.5 2 Loop area in (ft/s) = 0.31

δVmean = Vmean, Falling - Vmean, Rising

δVmean = 0.84 δVmean = 0.50

2.27 ft/s

1.77 ft/s

2.41 ft/s

1.57 ft/s

Figure 10. Index-velocity rating 

curves for two cases with highly 

skewed flow and a five-cell, 

right-half with blanking zone 

sampling configuration. Beta 

function parameters are shown 

along with the difference in mean 

velocity predicted by each rating 

curve for a given index velocity.

β

α

0 0.5 1 1.5 2 2.5 3 3.5

0

0.5

1

1.5

2

2.5

3

3.5

n = 18,094 (from 102 individual events) 

Line of symmetric flow 

Right bank 

skew

Left bank 

skew

Flat

Peaked

Figure 11. Distribution of α and β

for observations from 102 high-flow 

events in the Chicago Sanitary and 

Ship Canal near Lemont, Illinois. 

Points denote observations on the 

rising limb of the hydrograph, whereas 

open circles denote observations 

on the falling limb. Each event was 

assigned a different color to help 

distinguish events from one another.
α

Rising limb

n = 8,163

Number of

occurrences Median = 1.21

0 0.5 1 1.5 2 2.5 3 3.5 4 

0

200400600800

β

Rising limb

n = 8,163

Median = 1.3

0 0.5 1 1.5 2 2.5 3 3.5 4 

0

500

1,000

Number of

occurrences

α

Falling limb

n = 9,931

Median = 1.21

0 0.5 1 1.5 2 2.5 3 3.5 4 

0

500

1,0001,500

Number of

occurrences

β

 Falling limb

n = 9,931

Median = 1.29

0 0.5 1 1.5 2 2.5 3 3.5 4 

0

500

1,0001,500

Number of

occurrences

Standard deviation of 

α

Number of 

occurrences Median = 0.056

0 0.1 0.2 0.3 0.4 

0

5

10

n = 102 α

0.5 1 1.5 2 2.5 3 

024

Vindx, in ft/s 

Median = 0.067

0

5

10

Number of 

occurrences n = 102

α

0.5 1 1.5 2 2.5 3 

024

Vindx, in ft/s 

Standard deviation of 

α 0 0.1 0.2 0.3 0.4 

β

Median = 0.07

0

102030

Number of 

occurrences n = 102

β

0.5 1 1.5 2 2.5 3 

024

Vindx, in ft/s 

Standard deviation of 0 0.1 0.2 0.3 0.4 

Median = 0.083

0

102030

Number of 

occurrences n = 102

Vindx, in ft/s 

β

0.5 1 1.5 2 2.5 3 

024

Standard deviation of β 0 0.1 0.2 0.3 0.4 

Figure 12. Histograms of α and 

β for observations from 102 high-flow events in the Chicago Sanitary and Ship Canal near Lemont, Illinois. Inset 

histograms of the standard deviation of the coefficients over each event and the variation in the coefficient with velocity (color coded by event). for the rising and falling limbs separately.
Theoretical Analysis 17

values for the rising and falling limbs of the hydrograph, the 

coefficients do vary about these median values over each limb 

and cannot be considered to be constant. Therefore, Δ cannot be assumed to be equal to zero. The observed standard 

deviations in α and β equate to a Δ value of approximately 0.2. 

This was determined by using an empirical relation derived 

between the prescribed variation in α and β in the model (for 

example, fig. 4E), which is based on Δ, and the computed 

standard deviation of the α and β time series. These empirical relations are linear, with one relation for the rising limb 

( σ α β, .= 0 2661∆ ) and one for the falling limb (σα β, = 0 316 . ∆), 

where σ α β, is the standard deviation of α and β. Because Δ 

controls the variation in both α and β in the model, the relations are the same for α and β. The conceptual model can now 

be run using the median values of α and β and a Δ value of 

0.2 to determine whether hysteresis exists for these observed 

conditions. 

A simulation using a modified conceptual model with the 

observed median values of the beta density function coefficients shows no evidence of hysteresis in the index-velocity 

rating (fig. 13). The conceptual model was modified slightly 

to reflect the fact that the median values of α and β over the 

rising and falling limbs are equal but that they vary about the 

median values over each limb. Therefore, the absolute value 

of dQ/dt was used to drive the change in α and β, and the 

initial α and β values were chosen so that the overall mean 

values over each limb approximately matched the observed 

median values. The computed loop area for this configuration 

(H-ADCP, observed beta distribution, fig. 13H) is 0.0028 ft2

, 

suggesting that hysteresis is not significant for this configuration and that there is no substantial difference in the rating 

curve between the rising and falling limbs of the hydrograph 

(fig. 13G). Although it is possible that some high-flow events 

may deviate from this hydraulic configuration based on 

median values, the deviation should be relatively small based 

on the observed distributions of α and β, and the likelihood 

that such deviations will result in significant hysteresis is 

small.

The rating curve developed from the conceptual model 

using observed median values of the beta density function 

coefficients and the H-ADCP sampling configuration compares very closely with the index-velocity rating curve in 

use for the H-ADCP in the CSSC near Lemont, Ill. (fig. 14). 

Direct comparison of the two rating curves shows that the 

conceptual-model rating curve plots to the left of the fieldderived H-ADCP rating curve. This shift can be accounted for 

by considering the difference in velocity magnitude between 

the sampling elevation of the H-ADCP in the water column 

and the height above the bed of the depth-averaged velocity, as specified by the beta density function. The H-ADCP 

samples the flow higher in the water column compared to the 

location of the mean velocity, resulting in 4.96 percent higher 

index velocity if one assumes a logarithmic vertical velocity 

profile (Jackson and others, 2012). Therefore, if one modifies 

the field-derived rating curve by reducing Vindx by 5 percent 

over the full range of observations, the modified (shifted) 

rating describes the data from the conceptual model well and 

accounts for the left shift of the conceptual rating (fig. 14, red 

line). By accounting for the height of the sampling volume 

above the bed and using a logarithmic vertical velocity profile, 

we can conclude that the conceptual index-velocity rating 

developed by using the median observed values of the beta 

distribution is essentially equivalent to the field-derived rating 

in use for the H-ADCP in the CSSC near Lemont, Ill. It is 

important to note that the scatter in the rating points in the 

CSSC near Lemont cannot be accounted for by hysteresis in 

the rating curve; therefore, the scatter is likely due to uncertainties in the ADCP discharge measurements, unsteadiness in 

the flow over a measurement, and uncertainty in the indexvelocity measurement due to instrumentation noise and other 

sources of error, such as flow disturbance from vessel traffic 

and moored barges 0.7 mi upstream of the gage.
Discharge, in cubic 

feet per second

0

5,000

10,00015,000

Stage, in feet

23.5

24

24.5

25

Area, 

in square feet 

3,9004,0004,1004,200

Time, in minutes

Mean velocity, 

in feet per second

0 500 1,000 1,500 0

1

2

3

4

Index ve

Mean velocity, 

in feet per second

Discharge, in cubic 

feet per second

Stage, in feet

0 5,000 10,000 15,000

23.5

24

24.5

25

Green = Rising limb Red = Falling limb

0 20 40 60 80 100 

0

1

2

3

4

Distance from right bank, in f

Velocity, in feet per second

Cell No. 1 2 3 4 5 6 

Time, in minutes

Coefficient value

0 500 1,00

∆ = 0.2

0 

0

1234

Loop 

in (ft

ABCD

E

F

1.1

1.2

1.3

1.4

1.5

α

β

Figure 13. Compilation plot for the observed median flow distribution and the sampling configuration for the horizontal acoustic Doppler currenthe left: 

A, the observed hydrograph; B, the observed stage; 

C, the computed cross-sectional area; and 

D, the computed mean velocity. Plots on 

series of beta distribution coefficients; 

F, the stage-discharge loop; 

G, the index-velocity rating; and 

H, the temporal evolution of the transverse v

the event as defined by the beta distribution.
Numerical Modeling Analysis 19

Index velocity, in feet per second

Mean velocity, in feet per second

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

H-ADCP rating development 

points from Jackson and 

others (2012)

Theoretical rating points developed 

using November 2010 design storm

and observed median values of α and β

for 102 events (WY 2010-2012)

H-ADCP rating curve

Jackson and others (2012)

H-ADCP rating curve shifted 

to 5% lower Vindx to account for 

difference in sampling height

Velocity, in feet per second

Height above bottom, in feet

0 1 2 3 4

0

5

10

15

20

25

Depth-averaged velocity

(specified in the 

beta distribution)

H-ADCP sampling height

4.96%

δu change

Log law

(Jackson and others, 2012) 

EXPLANATION

Rising limb

Falling limb

Figure 14. Comparison of the 

field-derived index-velocity rating 

for the horizontal acoustic Doppler 

current profiler (H-ADCP) in Chicago 

Sanitary and Ship Canal (CSSC) near 

Lemont, Illinois, to the conceptualmodel rating curve developed by 

using median values of the beta 

distribution coefficients from 

observed velocity profiles. Inset 

figure shows the logarithmic profile 

of velocity for the CSSC near Lemont 

(see Jackson and others, 2012), 

including the sampling height of the 

H-ADCP relative to the height of the 

depth-averaged velocity as specified 

by the beta distribution.

Numerical Modeling Analysis

In order to further investigate the potential existence 

of hysteresis in the index-velocity rating for the CSSC near 

Lemont, a three-dimensional hydrodynamic model was 

applied. The computational domain for the model is shown 

in figure 1. A three-dimensional model was chosen for this 

analysis to capture flow physics in a realistic manner and to 

allow for inclusion of a virtual streamgage within the model, 

mimicking the configuration of sensors present near Lemont on the CSSC (fig. 2). The three-dimensional model was 

used to simulate actual high-flow events over a large range 

of measured boundary conditions. Additional simulations 

replicated the total discharge for a measured high-flow event 

but synthetically distributed the inflows from the CSSC and 

the Cal-Sag Channel over a range of scenarios in an attempt 

to elicit hysteresis in model-generated index-velocity ratings 

under a range of possible flow conditions. Index-velocity ratings were constructed by sampling the model flow field in the 

same manner as the field configuration of AVM and H-ADCP 

using virtual instruments (fig. 2). The flow events modeled as 

a part of this research were calibrated and validated with the 

help of observed water-surface-elevation and velocity data 

continuously measured at the USGS streamgage on the CSSC 

near Lemont, Ill.

About the Numerical Model 

The three-dimensional numerical modeling of the aforementioned domain (fig. 1) was carried out with TELEMAC-3D 

(Electricité de France, 2007). The TELEMAC suite of models has been in continuous development since 1987 when the 

Telemac project was launched by the Research and Development department of Electricité de France. TELEMAC-3D is 

based on finite-element methodology (Hervouet, 2007); the 

numerical model solves the three-dimensional form of the shallow water equations. These governing equations consist of the 

continuity and momentum equations in x and y directions:

 

∂

∂

+

∂

∂

+

∂

∂ = Ux

V

y

W

z 0

 

 (4)

∂

∂

+

∂

∂

+

∂

∂

+

∂

∂ = − ∂∂

+ ( ) +

U

t

U U

x

V U

y

W U

z g

Z

x

U F s

x υ∆ (5)

∂

∂

+

∂

∂

+

∂

∂

+

∂

∂ = − ∂∂

+ ( ) +

V

t

U V

x

V V

y

W V

z g

Z

y

V F s

y υ∆ (6)

 p p g Z z g dz (7) atm o s o

oz

Zs

= + ( ) − + ∫ ρ ρ ρ

ρ

∆

 
20 Evaluation of the Potential for Hysteresis in Index-Velocity Ratings for the Chicago Sanitary and Ship Canal near Lemont, Illinois

In the preceding equations, t is time in seconds, U, V and 

W are velocity in x, y and z directions, respectively, Zs denotes 

the water-surface elevation, p is the pressure, g is the acceleration due to gravity, ν is the turbulent eddy viscosity, ρo is the 

reference density, and Fx and Fy denote the source terms which 

represent the effect of wind, Coriolis force, and the bottom 

friction or any other process modeled in a similar manner. 

The governing equations (eqs 4–7) behind the computational 

kernel of the model are based on the following inherent 

assumptions:

1. Variation of density in the conservation of mass 

(continuity equation) is ignored, hence it represents 

incompressible flow.

2. Pressure is assumed to be hydrostatic along the water 

column, which implies that the pressure at any given 

depth is the sum of the air pressure at the fluid surface 

plus the weight of the overlying water body.

3. Boussinesq approximation for the momentum is 

assumed, which implies that the density variation is 

considered only in the gravity term.

The numerical solution of the governing equations 4–7 

is achieved by what is commonly known as the fractional 

step algorithm. The tremendous advantage and accuracy of 

the fractional step algorithm lies in the fact that the best and 

most highly accurate numerical schemes can be employed to 

solve different parts of the governing equations in a piecewise 

manner. 

The computational algorithm of TELEMAC-3D splits 

the governing equations into three distinct numerical steps. 

In the first step, only the advection component of the governing equations is solved, using method of characteristics 

(Hervouet, 2007). These advected velocities are then used in 

the second step, where both diffusion and the source component of equations are solved by using the finite element 

methodology. The water depth is computed from the vertical 

integration of the continuity and momentum equations and 

includes only the pressure-continuity terms. The new computed depth enables the computation of the new free surface. 

To account for a continually changing model domain due to 

variation of the free surface, TELEMAC-3D utilizes a sigma 

transformation (Hervouet, 2007). Finally, the vertical velocity 

(W) is also computed with the continuity equation. Conventional numerical modeling of riverine systems is accomplished 

by using a one-dimensional model (for example, HEC-RAS); 

but with advancements made in the field of parallel computing 

and easily accessible computational resources, three-dimensional modeling is becoming more common. 

One of the main advantages of TELEMAC-3D is the 

ability to scale up on cluster computers and minimize the 

computational time and resources spent on different simulations. TELEMAC-3D has been parallelized by using MPI 

(Message Passing Interface) paradigm (Hervouet, 2007). 

The TELEMAC suite of models are completely open 

source and freely available from www.opentelemac.org. 

The aforementioned three-dimensional numerical model was 

compiled and configured on Newton Cluster available at the 

Ven Te Chow Hydrosystems Lab at the University of Illinois 

at Urbana-Champaign (UIUC). The Newton Cluster is a 

7-node, 56-core Intel Xeon cluster. Each node has 8 cores 

and 32 GB of RAM. The nodes are connected to each other 

with an Ethernet interconnect and share a network file system 

(NFS). Running the simulations on the Newton Cluster using 

MPI parallelization dramatically reduced the computational 

time required for each simulation.

Model Development

This section is dedicated to a discussion of the methods 

used to define the boundary conditions for the simulations, 

the development of the computational mesh, and the calibration and validation of the model using observed water-surface 

elevation data and discharge in the CSSC near Lemont, Ill. 

Simulated Events

As shown in figure 1, the computational domain extends 

from USGS streamgage 05536995 on the downstream end to 

4.35 mi (7 km) upstream of Sag Junction (the confluence of 

the CSSC and Cal-Sag Channel) on both CSSC and Cal-Sag 

Channel. Boundary conditions were imposed at these three 

boundaries in order to drive the model. The choice of USGS 

streamgage 05536995 as the endpoint of the computational 

domain was facilitated by the fact that the USGS has records 

of water-surface elevation corresponding to various flow 

events modeled at this location. Boundary conditions for 

the numerical model consisted of inflow hydrographs at the 

upstream boundaries of the CSSC and Cal-Sag Channel and 

time series of the water-surface elevation at the downstream 

end at Romeoville (fig. 15).

In order to simulate a range of possible flow conditions 

for the CSSC near Lemont, Ill., the model simulations were 

separated into two groups based on the imposed boundary 

conditions (see table 2). In the first group of simulations, a 

large event (peak discharge of 18,984 ft3

/s) from July 2010 

was used as the design event (compare with other large events 

in table 1). To define the discharge time series in the CSSC 

and Cal-Sag Channel (upstream boundary conditions), the 

observed discharge in the CSSC near Lemont was divided 

several ways in order to generate a number of scenarios, but 

always ensuring that the contributions from the CSSC and 

Cal-Sag Channel summed to the observed discharge in the 

CSSC near Lemont. Estimating the inflow from the Cal-Sag 

Channel is necessary because there are no gage data for the 

Cal-Sag Channel during this time period. In all cases, the CalSag Channel contribution to the total discharge in the CSSC 

near Lemont during low flows was kept constant at 25 percent 

(estimated from discharges from the primary wastewater 

outfalls in the system, which make up most of the flow in the 

canal during low flows). During high flows, the discharge in 
Numerical Modeling Analysis 21

Date

B. Case 2–Start 00:00, July 22, 2010

Discharge in the Chicago Sanitary and Ship Canal near Lemont, Ill. (05536890)

Discharge in the Cal-Sag Channel near Rt 83

Discharge in the Chicago Sanitary and Ship Canal upstream of the Junction with the Cal-Sag Channel

Water-surface elevation in the Chicago Sanitary and Ship Canal at Romeoville, Ill. (05536995)

EXPLANATION

A. Case 1–Start 00:00, July 22, 2010

Discharge, in cubic feet per second

0

0

5,000

5,000

10,000

10,000

15,000

15,000

20,000

20,000

07/22 07/23 07/24 07/25 07/26 07/27

Free surface elevation, in feet above Chicago City Datum (CCD)

−8

−8

−6

−6

−4

−4

−2

−2

0

0

Figure 15. Synthetic hydrographs for the Chicago Sanitary and Ship Canal (CSSC) and 

Cal-Sag Channel upstream of Sag Junction and measured discharge in the CSSC at Lemont 

and free surface elevation in the CSSC at Romeoville Road corresponding to case 1 and case 

2. Water-surface elevations are referenced to the Chicago City Datum (CCD; 0.0 feet CCD 

corresponds to 579.48 feet NGVD 1929).
22 Evaluation of the Potential for Hysteresis in Index-Velocity Ratings for the Chicago Sanitary and Ship Canal near Lemont, Illinois

the Cal-Sag Channel was computed by modeling its percent 

contribution to the total flow in the CSSC near Lemont as 

a skew-normal distribution. Adjustment of the distribution 

parameters (peak, variance, skewness) allowed different CalSag Channel inflow scenarios to be defined. In case 1, the 

contribution from the Cal-Sag Channel was allowed to peak 

with a maximum contribution of 9,140 ft3/s before the peak of 

the hydrograph of the CSSC near Lemont (fig. 15A). Case 2 

was chosen to allow the Cal-Sag Channel to peak later than the 

CSSC near Lemont, with a maximum discharge of 7,980 ft3/s 

(fig. 15B). The reasoning behind these cases was to determine 

whether any hysteresis in the index-velocity rating for the 

CSSC near Lemont is sensitive to the timing of the peak of the 

hydrograph of the Cal-Sag Channel relative to the peak in the 

CSSC. For both cases, the peak discharge from the Cal-Sag 

Channel was defined as 60 percent of the total flow in the 

CSSC at Lemont, Ill. A peak contribution of 60 percent from 

the Cal-Sag Channel was chosen to represent an event that presented a potential for hysteresis because the Cal-Sag Channel 

and CSSC inflows would peak on opposite limbs of the hydrograph at Lemont and the Cal-Sag Channel inflow would briefly 

dominate the flow distribution, possibly leading to skewing of 

the velocity profile in the CSSC at Lemont and hysteresis. In 

addition, simulations with the peak Cal-Sag Channel contribution of 40 percent and 80 percent of the total flow in the CSSC 

at Lemont, Ill. were tested but are not presented in this report 

because they yielded no additional information.

In the second group of simulations, four different highflow events of varying peak discharge and time duration were 

simulated (fig. 16). All four events utilized measured discharge 

on the CSSC and Cal-Sag Channel for upstream boundary 

conditions, so no synthetic flow distributions were required 

due to the installation of the USGS streamgage on the Cal-Sag 

Channel near Route 83 (05536700) in 2011. The four different flow events simulated are hereafter called event 1, event 2, 

event 3, and event 4 (fig. 16). The timing and characteristics of 

each of the four events are given in table 2. Discharge from the 

CSSC and Cal-Sag Channel are shown for the four events in 

figure 16. 

The Computational Mesh

The computational mesh of the model was developed 

by using high-resolution bathymetric data provided by the 

U.S. Army Corps of Engineers, the USGS, and the University 

of Illinois. As shown in figure 17, the computational mesh 

consists of 21,000 nodal points and 36,000 triangular elements 

with 8 layers in vertical. TELEMAC-3D works in conjunction 

with an unstructured mesh. The use of an unstructured mesh 

tremendously reduces the preprocessing time required for 

the model setup and offers homogeneous mesh resolution in 

the entire domain. The advantage of using unstructured mesh 

becomes even more critical in the zone of the confluence. 

Because of the flexibility offered by the use of unstructured 

mesh, it was possible to keep the mesh resolution constant at a 

4-m edge length throughout the modeled domain. 

Model Calibration and Validation

The model was calibrated and the results were validated 

by using water-surface elevation records measured in the 

CSSC near Lemont (05536890). TELEMAC-3D provides the 

facility of splitting the turbulent eddy viscosity in horizontal 

and vertical components. Turbulence modeling can be accomplished through a variety of models. At the simplest level, 

eddy viscosity can be assigned a constant value throughout the 

domain and can be used as a calibration parameter. Such an 

assigned value has definite impact on size and shape of recirculation zones and eddy structures present in the flow being 

modeled. At a more complex level, the turbulence modeling is 

done through second-order model; for example, a κ-ε model 

(Wilcox, 2004). In the present simulations, the turbulence 

modeling in the horizontal direction was achieved by assigning a constant eddy viscosity of 0.001 m2

/s, which is reasonable compared with published values. Turbulence modeling 

in the vertical direction used a Prandtl’s mixing length model 

(Wilcox, 2004), where the vertical eddy viscosity is a function 

of mean velocity gradient. Prandtl’s mixing length model for 

specifying the vertical eddy viscosity is especially suitable for 

Table 2. Simulated high-flow events in the Chicago Sanitary and Ship Canal near Lemont, Illinois.

[CST, central standard time]

Group Case/

Event

Duration,

in hours Start time (CST)

Peak discharge, 

in cubic feet per 

second

Synthetic flow 

partitioning?

Group 1 Case 1 120 00:00 July 22, 2010 18,984 Yes

Case 2 120 00:00 July 22, 2010 18,984 Yes

Group 2 Event 1 72 12:00 July 17, 2012 12,697 No

Event 2 102 15:00 August 24, 2012 13,366 No

Event 3 23 22:00 July 23, 2012 8,706 No

Event 4 33 23:00 August 3, 2012 7,237 No
Discharge, in cubic feet per second

07/18 07/19 07/20 −2,000

0

2,0004,0006,0008,000

10,000

12,00014,000

08/25 08/26 08/27 08/28

Discharge, in cubic feet per second

00:00 06:00 12:00 18:00 −2,000

0

2,0004,0006,0008,000

10,00012,00014,000

00:00 06:00 12:00 18:00 00:00 

Date

Time

A. Event 1–Start 12:00, July 17, 2012C. Event 3–Start 22:00, July 23, 2012

B. Event 2–Start 15:00, August 24, 2012D. Event 4–Start 23:00, August 3, 2012

Discharge in the Chicago Sanitary and Ship Canal near Lemont, Ill. (05536890)Discharge in the Cal-Sag Channel near Rt 83 (05536700)Discharge in the Chicago Sanitary and Ship Canal upstream of the Junction with the Cal-Sag ChannelWater-surface elevation in the Chicago Sanitary and Ship Canal at Romeoville, Ill. (05536995)

EXPLANATION

Figure 16. Measured hydrographs for the Chicago Sanitary and Ship Canal (CSSC) and Cal-Sag Channel and water-surface elevation in the Road corresponding to event 1, event 2, event 3, and event 4.
-30.2 -23.5 -16.7 -10.0 -3.3 Bed elevation, in feet above Chicago City Datum (CCD) 21,000 points 36,000 elements 8 vertical layers 4 meter (13.12 ft) resolution Chicago Sanitary and Ship Canal at Romeoville, Illinois (05536995) 

Chicago Sanitary and Ship 

Chicago Sanitary and Ship Canal near Lemont, Illinois (05536890) Flow 0 2 MILES 0 2 KILOMETERS Sag Junction Calumet-Sag Ch

at Sag Bridge (0

Figure 17. Top view and three-dimensional view of the computational mesh for the modeled domain.
Numerical Modeling Analysis 25

flows with a high degree of free-surface variation, which is the 

case during high flow through the CSSC. The vertical eddy 

viscosity, ν, for the Prandtl’s mixing length model is given by

ν = L D m ij i Dj

2 2 (8)

where D U

x

U

x ij

i

j

j

i

= +

















1

2

δ

δ

δ

δ

and Lm is the mixing length.

The mixing length (Lm) in equation 8 is the proportionality constant between the eddy viscosity and mean velocity 

gradient. Physically, the mixing length is the distance traversed by a mass of fluid before it becomes blended in with 

neighboring masses (Prandtl, 1926). The version of Prandtl’s 

mixing length model presented in this simulation is a standard one, where Lm is determined by an empirical relationship between the water depth (H) and the distance from the 

bed (z). In this standard mixing length model, Lm is 0 at the 

bed (z = 0), increases linearly to Lm = 0.08H at z = 0.2H and 

is held constant at Lm = 0.08H for z > 0.2H (Hervouet, 2007). 

Another parameter that was adjusted during the calibration 

procedure was Manning’s roughness, which corresponds to the 

bottom roughness and is modeled as a part of the momentum 

equations. Both the calibration parameters (that is, Manning’s 

roughness and horizontal eddy viscosity) were calibrated to 

capture the water-surface elevation variation observed at the 

validation point near Lemont (USGS gage 05536890) (fig. 1). 

The best match between the observed and simulated water-surface elevation for the CSSC near Lemont was obtained with 

Manning’s n value of 0.0358 and horizontal eddy viscosity of 

0.001 m2/s.

The model was calibrated and validated with the watersurface elevation data recorded for the CSSC at the USGS 

streamgage near Lemont, Ill. (05536850). Cases 1 and 2 were 

primarily used for calibration of the model and events 1–4 and 

additional synthetic cases (not shown) served as validation 

data. The comparative plot between observed and simulated 

free-surface elevations corresponding to all synthetic cases 

and measured events is presented in figure 18. The model 

captures the free-surface variation in the CSSC near Lemont in 

a satisfactory manner. For example, during event 1 (fig. 18C) 

the match between the simulated and observed free surface 

is especially good after the drawdown starts, 42 hours after 

the start of the simulation. Before the drawdown begins, 

there are number of instances when the incoming flow from 

Cal-Sag Channel is negative; for instance, 8 hours after the 

start of the simulation the inflow from Cal-Sag Channel 

is −773 ft3/s (−21.88 m3/s), indicating an upstream flow 

(fig. 16A). In fact, upstream flow also occurred in the main 

channel of the CSSC near Lemont at this same time. Flow 

reversals on this system are relatively common and can occur 

because of control changes at Lockport Lock and Dam at 

Lockport, Ill., which is approximately 11.5 mi downstream of 

Lemont (Jackson and others, 2012). Each lockage through the 

system creates a disturbance that propagates upstream, causing 

much of the variation in discharge at low flows. In addition, 

periodic shutdowns of flow through the Lockport Powerhouse 

for maintenance and worker safety can cause flow reversals as 

the disturbance propagates upstream. 

Observed and simulated discharges for the CSSC near 

Lemont, Ill., compare well for all the simulated events 

(fig. 19). Some simulated discharges vary significantly from 

the observed values at the start of the simulation but quickly 

tend to match well with the observed discharge in the CSSC 

near Lemont. Although the simulated discharges show slightly 

more variability than the observed discharges, lockageinduced spikes during dry-weather flow are captured. The 

model slightly underpredicts the peak discharge for case 1 

and case 2, but it otherwise simulates the observed discharge 

satisfactorily. 

Model Rating Development

For each simulation of the hydrodynamic model, the 

model output was sampled by using a virtual streamgage and 

employed in developing an index-velocity rating curve for 

each event. The remainder of this section discusses the details 

of this process. 

Sampling the Model Output

Modeled flow fields from each simulation were saved at 

10-minute time steps for rating development during postprocessing. Each flow field consisted of three-dimensional 

velocity data at the resolution of the computational mesh. 

For each of the six simulations and every 10-minute time step 

within the simulation, the model output was sampled with a 

virtual AVM and H-ADCP. Sampling volumes of each virtual 

instrument matched the sampling volumes of the physical 

instruments deployed in the field (see fig. 2). The virtual AVM 

velocity sensors were located at 3 different path elevations 

in the water column, and velocity magnitude was extracted 

at 40 equidistant points along the 3 AVM paths. The velocity 

magnitude corresponding to these 40 points were averaged at 

each AVM path elevation to get an average velocity magnitude for each path. Finally, the arithmetic mean of the average 

velocity magnitude at each AVM path elevation was computed 

to obtain the virtual AVM index velocity at each time step. 

The virtual H-ADCP sampled the flow field across the same 

nine-cell sampling volume of the field instrument, and the 

virtual H-ADCP index velocity was obtained by averaging the 

velocity magnitude within the H-ADCP sampling volume at 

each time step. These index velocities from the virtual AVM 

and the H-ADCP provided the abscissa values for the model 

index-velocity rating. 
A. Case 1 E. Event 3 F. Event 4 C. Event 1 B. Case 2 D. Event 2 0 20 40 60 80 100 120 −4−3−2−10 0 20 40 60 80 

−4−3−2−10 Water surface elevation, in feet above Chicago City Datum (CCD) 0 10 20 30 40 50 60 70 80 −4−3−2−10 0 20 40 60 80 −4−3−2−10 0 5 10 15 20 25 −4−3−2−10 Time, in hours 0 5 10 15 20 

−4−3−2−10 EXPLANATION Model water-surface elevation (rising limb) Model water-surface elevation (falling limb) Observed water-surface elevation in the Chicago Sanitary and Ship Canal near Lemont, Ill. (05536890) Figure 18. Comparison between simulated and observed water-surface elevation at the USGS streamgage on the Chicago Sanitary and 

Illinois (05536890), corresponding to all simulated cases and observed events.
A. Case 1

E. Event 3 F. Event 4

C. Event 1

B. Case 2

D. Event 2

0 20 40 60 80 100 120 0 20 40 60 80 

0 10 20 30 40 50 60 70 80 0 20 40 60 80 0 5 10 15 20 25

Time, in hours

0 5 10 15 20 

0

5,000

10,00015,00020,000

Discharge, in cubic feet per second 0 5,000 10,000 15,000 20,000

0

5,000

10,00015,00020,000

EXPLANATION

Model discharge (rising limb)Model discharge (falling limb)Observed discharge in the Chicago Sanitary and Ship Canal near Lemont, Ill. (05536890)

Figure 19. Comparison between simulated and observed discharge at the USGS streamgage on the Chicago Sanitary and Ship Canal ne(05536890), corresponding to all synthetic cases and measured events.
28 Evaluation of the Potential for Hysteresis in Index-Velocity Ratings for the Chicago Sanitary and Ship Canal near Lemont, Illinois

The ordinate values for the model index-velocity rating, 

Vmean (average velocity through the section), were computed by 

using the 10-minute flow fields from the model sampled at the 

measurement section identified in figure 2 and used in the field 

for boat-mounted ADCP (BM-ADCP) measurements. This 

procedure ensures that the discharge in the model is measured 

at the same section as the discharge is measured in the field. 

The total discharge through the section was computed by multiplying the width of the BM-ADCP cross section by the mean 

depth-averaged specific discharge of the measurement section. 

The total discharge was also computed by summing the fluxes 

through each grid node in the measurement section, and both 

methods were found to give approximately the same total 

discharge. The virtual Vmean was computed by dividing the total 

discharge at each time step by the rated cross-sectional area 

for that time step. The rated area was computed by using the 

stage-area rating for the USGS streamgage on the CSSC near 

Lemont (Jackson and others 2012). Differences between the 

area determined from the stage-area rating and the area from 

the model output were computed and found to be negligible. 

Building the Ratings

Sampling of the model output as described in the previous section allowed computation of virtual index velocities for 

both instruments and a virtual mean velocity for the channel 

at 10-minute intervals over the duration of each simulation. 

Index-velocity rating curves for each simulation were generated by plotting the mean velocity for the channel Vmean against 

the index velocity for each virtual instrument (figs. 20 and 21). 

To facilitate analysis, the rating points were color coded by 

points computed on the rising limb (green) and falling limb 

(red) of the hydrograph. Any differences in the rating points 

for the rising and falling limbs could introduce error into 

computations of discharge from an index-velocity rating, and 

large deviations suggest that hysteresis may occur at this site. 

For comparison, the field-derived rating curve and field measurements used to derive the rating are plotted on each figure, 

along with the residuals. 

Discussion

Model-derived index-velocity ratings for all six of the 

simulated high flow events are in good agreement with the 

field-derived index-velocity ratings for the Lemont gage given 

by Jackson and others (2012) (figs. 20 and 21). Six events 

were simulated with peak discharges ranging from approximately 7,200 to 19,000 ft3

/s, and all the model-derived indexvelocity ratings exhibit nearly identical behavior for the rising 

and falling limbs of the hydrograph. With the exception of the 

upper 10 percent of the rating for case 1 and case 2, the ratings 

are generally linear over the entirety of the event. Case 1 and 

case 2 show a slight break in the slope at high flows, and this 

deviation will be discussed in detail later in this section. The 

model-derived ratings for both instruments plot slightly to the 

left of the field-derived ratings for nearly all events, suggesting 

that index velocities in the model are slightly low compared to 

those computed in the field. In general, the model index velocities are lower than field values by less than 0.1 ft/s (with the 

exception of the upper range of the case 1 and case 2 ratings). 

However, the slight difference in the model and field ratings 

may be also attributed to the inability of the model to exactly 

replicate the observed stage and discharge time series over the 

full range of each event (figs. 18 and 19). Slight differences in 

discharge and stage between the model and the field observations can lead to changes in Vmean, thus affecting the rating. 

No substantial hysteresis was observed in the modelderived index-velocity ratings for any of the six simulated 

high-flow events (figs. 20 and 21). Slight deviation between 

the rising- and falling-limb rating points in case 1 and case 

2 at high flows can be attributed primarily to the inability of 

the model to accurately replicate the discharge and stage in 

the CSSC near Lemont during the peak of the event (figs. 18 

and 19). The nearly identical ratings developed for case 1 and 

case 2 suggest that the timing of the contribution from the CalSag Channel with respect to the flood wave in the CSSC has 

little to no effect on the rating curve in the CSSC near Lemont. 

(As discussed previously, case 1 and case 2 were the same 

event but with differing synthetic flow distributions between 

the CSSC and Cal-Sag Channel, case 1 being early Cal-Sag 

Channel contribution and case 2 being late Cal-Sag Channel 

contribution.) Finally, the scatter in the rating points derived 

from field measurements is significantly larger than any 

observed hysteresis in the model and therefore is likely not an 

artifact of hysteresis in the field measurements. 

The deviation in the linear trend of the ratings for case 

1 and case 2 (and to a lesser extent event 2) requires more 

discussion. The break in the slope of the rating occurs for case 

1 and case 2 at the point on the rising limb of the hydrograph 

when the stage-discharge relation at Romeoville (downstream 

boundary condition) transitions from an inversely proportional relation (higher discharge with lower stage) to a directly 

proportional relation (higher discharge with higher stage) 

(fig. 15). At this point, the stage in the canal increases in the 

middle of the drawdown period (a drawdown is a typical 

response due to anticipation of heavy precipitation to allow 

for storage in the canal) in response to a large influx of runoff 

to the canal. Stage increases because the influx of water to the 

canal exceeds the discharge of water through the generators 

and sluice gates at the Lockport Powerhouse and sluice gates 

at the Controlling Works upstream of Lockport. It is during 

this period that the stage-discharge relation transitions to one 

that is directly proportional (rather than inversely proportional), and the model-derived index-velocity rating changes 

slope. Once the stage-discharge relation transitions back to an 

inversely proportional relation, the model-derived rating transitions back to the original slope. This behavior can be seen 

in case 1 and case 2 and to a lesser extent in event 2 for both 

the AVM and H-ADCP. A direct comparison of the observed 

H-ADCP velocity data for each of the nine cells to the model 

output for each of the nine cells reveals that the median value 
0

1

2

3

4

Mean velocity, in feet per second

0 1 2 3 4 0 1 2 3 4 0

1

2

3

4

0 1 2 

EXPLANATION

Field measurement (Jackson and others, 2012)Model rating point (rising limb)Model rating point (falling limb)Field-derived rating (Jackson and others, 2012)

A. Case 1 B. Case 2 

D. Event 2 E. Event 3 

0 1 2 3

Residual, 

in feet per second

AVM index velocity, 

−0.2

−0.1

0

0.1

−0.3

in feet per second

0 1 2 3 4

Residual, 

in feet per second

AVM index velocity, 

−0.2

−0.1

0

0.1

−0.3

in feet per second

AVM index velocity, in feet per second

0 2 4 6

Residual, 

in feet per second

AVM index velocity, 

in feet per second

−0.2

−0.1

0

0.1

−0.3

AVM index velocity, 

in feet per second

0 2 4 6

Residual, 

in feet per second−0.2 −0.10

0.1

−0.3

0 1 2 

AVM index velocity, 

−0.2

−0.1

0

0.1

−0.3

in feet per second

0 1 2 3 

−0.2 −0.10

0.1

−0.3

AVM index velocity, 

in feet per second

Figure 20. Model-derived index-velocity rating curves corresponding to thmeter (AVM) sensor configuration for all simulated cases and events. The fivelocity rating for the AVM is shown for reference, along with residuals comfield rating curves (residual = Vmean (rating) − Vmean (model)).
0 2 4 6

Residual, 

in feet per second

H-ADCP index velocity, 

in feet per second

−0.2

−0.1

0

0.1

−0.3 0 2 4 6

Residual, 

in feet per second

H-ADCP index velocity, 

in feet per second

−0.2

−0.1

0

0.1

−0.3 0 1 2 3 

H-ADCP index velocity

in feet per second

−0.2

−0.1

0

0.1

−0.3

0 1 2 3 4

Residual, 

in feet per second

H-ADCP index velocity, 

in feet per second

−0.2

−0.1

0

0.1

−0.3

0 0.5 1 1.5 

H-ADCP index velocit

in feet per second

−0.2

−0.1

0

0.1

−0.3 0 1 2 3

Residual, 

in feet per second

H-ADCP index velocity, 

in feet per second

−0.2

−0.1

0

0.1

−0.3

0

1

2

3

4

Mean velocity, in feet per second

H-ADCP index velocity, in feet per second

0 1 2 3 4 0 1 2 3 4 0

1

2

3

4

0 1 2 

EXPLANATION

Field measurement (Jackson and others, 2012)Model rating point (rising limb)Model rating point (falling limb)Field-derived rating (Jackson and others, 2012)

A. Case 1 B. Case 2 

D. Event 2 E. Event 3 

Figure 21. Model-derived index-velocity rating curves corresponding to the Doppler current profiler (H-ADCP) sensor configuration for all simulated casederived index-velocity rating for the H-ADCP is shown for reference, along wthe model and field rating curves (residual = Vmean (rating) − Vmean (model)).
Conclusions 31

Table 3. Error between predicted and observed velocities in 

the nine cells of the horizontal acoustic Doppler current profiler 

in the Chicago Sanitary and Ship Canal near Lemont, Illinois. 

Median values of the root mean squared error (RMSE) are 

reported independently for the two stage-discharge relation 

regimes.

[ft/s, feet per second; --, not computed; n/a, not applicable]

Case/Event

Velocity RMSE, in ft/s

Inversely proportional 

stage-discharge relation

Directly proportional 

stage-discharge relation

Case 1 0.18 0.56

Case 2 -- --

Event 1 0.17 n/a

Event 2 0.16 0.29

Event 3 0.16 n/a

Event 4 0.16 n/a

of the root mean squared error (RMSE) of the predicted velocities along the H-ADCP path was up to 3 times higher during 

the directly proportional stage-discharge period compared to 

the inversely proportional stage-discharge period (table 3). 

On average, the model underpredicted velocity magnitude in 

the H-ADCP section, and the underprediction significantly 

increases when the stage-discharge relation becomes directly 

proportional. The magnitude of this underprediction during 

the peak of the event for case 1 and case 2 (median = 0.56 ft/s) 

is more than enough to account for the deviation of the 

model rating from the field rating at the highest velocities 

(deviation ≈ 0.33 ft/s). 

The underprediction of velocity magnitude in the 

measurement cross section can be attributed to the inability 

of the model to accurately capture the stage and discharge 

during the peak of the events for case 1, case 2, and event 2 

(figs. 18 and 19). Modeled discharges for case 1 and case 2 

are underestimated by as much as 2,000 ft3/s on the rising 

limb near the peak of the event (fig. 19); and although the 

discharge on the falling limb is much closer to the observed 

discharge (slightly higher), the modeled stage on the falling 

limb is overpredicted (fig. 18). Both a low discharge and a 

high stage (high cross-sectional area) lead to lower velocities 

in the section. The inability of the model to accurately capture 

the peak of these flows will affect both the index velocity, 

Vindx, and the mean velocity, Vmean, and it appears to change the 

relation between the two velocities enough to cause a change 

in the slope of the index-velocity rating. In addition, because 

the change is induced by a low-biased discharge on the rising 

limb and a high-biased stage on the falling limb, the slope of 

the upper part of the rating curve for these events varies, causing slight hysteresis. This hysteresis, however, and the overall 

deviation in the linear trend of the rating curve at high flows 

for case 1 and case 2 (also for event 2) appear to be entirely 

due to inaccuracies of the model and are not supported by field 

measurements, which do not show any break in the linear ratings (figs. 20 and 21).

The cause of the uncertainty of the model during the 

directly proportional stage-discharge relation portion of the 

hydrograph (large event peaks) is not entirely clear, but may 

be related to the synthetic partitioning of the flow to generate 

the upstream boundary conditions for case 1 and case 2 combined with the choice of Romeoville as a downstream boundary condition. The control for this system is at the Lockport 

Powerhouse and Lockport Lock and Dam about 6 mi downstream of Romeoville. Although the water-surface elevation 

at Romeoville may be adequate as a downstream boundary 

condition for most flow conditions, it might be inadequate for 

the peaks of large events that display a directly proportional 

stage-discharge relation. Moving the downstream boundary 

condition to Lockport might result in better agreement over 

the full range of flows observed at Lemont, but doing so would 

require modeling of the complex reach between Romeoville 

and Lockport, which includes a powerplant with intakes and 

outfall, the Controlling Works with its seven sluice gates, 

and the Lockport Lock and Powerhouse (two turbines and 

sluice gates). Not only is this reach complicated to model and 

beyond the scope of this project, but much of the data required 

to properly model this reach is nonexistent. 

Conclusions

The U.S. Geological Survey is responsible for monitoring flows in the Chicago Sanitary and Ship Canal (CSSC) 

near Lemont, Illinois (Ill.), as a part of the Lake Michigan 

Diversion Accounting overseen by the U.S. Army Corps 

of Engineers, Chicago District. Lake Michigan Diversion 

Accounting is mandated by a U.S. Supreme Court decree in 

order to monitor, and limit, the State of Illinois’ annual diversion of Great Lakes’ water through the manmade Chicago 

Sanitary and Ship Canal. Every 5 years, a technical review 

committee consisting of practicing engineers and academics 

reviews the U.S. Geological Survey’s streamgaging practices 

in the CSSC near Lemont, Ill. The sixth technical review committee expressed concern that the index-velocity rating at this 

gage may be subject to hysteresis due to the unique, unsteady 

hydraulics of the canal. Hysteresis in index-velocity ratings 

can occur at sites where the flow distribution in the channel 

varies significantly between the rising and falling limbs of the 

hydrograph for the same discharge (Ruhl and Simpson, 2005). 

Presently, hysteresis in index-velocity ratings has been documented only at tidally affected sites. 

This report investigates the possibility of hysteresis at 

this nontidal site and the conditions under which it would be 

likely to occur. Study methods involved both a theoretical 

approach and use of a three-dimensional hydrodynamic model. 

The theoretical analysis investigated the conditions required 

for hysteresis in the index-velocity rating, whereas the modeling analysis focused on the effect of the timing of the inflows 

from the CSSC and the Cal-Sag Channel on the potential for 

hysteresis and whether highly resolved simulations of actual 

high-flow events show any evidence of hysteresis.
32 Evaluation of the Potential for Hysteresis in Index-Velocity Ratings for the Chicago Sanitary and Ship Canal near Lemont, Illinois

Theoretical analysis using a 13,200-ft3/s high-flow 

event hydrograph combined with theoretical transverse flow 

distributions in the CSSC near Lemont, Ill., showed that 

hysteresis in the index-velocity rating curve can be introduced 

by varying the transverse flow distribution in the measurement 

section between the rising and falling limbs of the hydrograph. 

Sensitivity analysis shows that hysteresis is most pronounced 

when the transverse flow distribution is peaked and skewed to 

one bank rather than symmetric in the cross section and when 

the distribution varies greatly between rising and falling limbs 

of the hydrograph. Although the theoretical analysis shows 

hysteresis can exist for the horizontal acoustic Doppler current profiler (H-ADCP) in the CSSC near Lemont and other 

configurations with unmeasured zones in the cross section, this 

theory does not produce hysteresis for the acoustic velocity 

meter (AVM), which samples the full cross section. And, 

although theory allows for the possibility of hysteresis, the 

observed transverse velocity profiles in the CSSC near Lemont 

do not exhibit the spatial variation over high-flow events 

required to generate hysteresis for either instrument configuration (H-ADCP or AVM). 

A three-dimensional hydrodynamic model was used 

to simulate six high-flow events of varying peak discharge 

between 7,200 to 19,000 ft3/s. The predicted flow field in the 

CSSC near Lemont from the model was then sampled over the 

duration of each event in a manner that replicates the gaging 

practices in the field. Data extracted from the simulations were 

used to develop index-velocity rating curves for the H-ADCP 

and AVM for each event. Model-derived index-velocity ratings for all six of the simulated high-flow events are in close 

agreement with the field-derived index-velocity rating for the 

Lemont gage given by Jackson and others (2012). No significant hysteresis was observed in the model-derived indexvelocity ratings for any of the six simulated high-flow events. 

Slight deviation from the field rating and small hysteresis in 

synthetic case 1 and case 2 at high flows can be attributed to 

the inability of the model to accurately replicate the discharge 

and stage in the CSSC near Lemont during the peak of the 

event and the associated underprediction of velocity magnitudes for the highest flows. This limitation of the model 

is likely due to a combination of synthetic flow partitioning 

used to define the upstream boundary conditions (given a lack 

of field observations for these cases) and the inability of the 

downstream boundary condition at Romeoville to fully capture 

the effect of the control at Lockport at peak flows. For synthetic case 1 and case 2, the ratings appeared to be insensitive 

to the timing of the discharge from the Cal-Sag Channel relative to the flow in the CSSC. The scatter in the rating points 

derived from field measurements is significantly larger than 

any observed hysteresis in the model and therefore is likely 

not an artifact of hysteresis in the field measurements. 

Based on both the theoretical analysis using observed 

historical data and the analysis using a three-dimensional 

hydrodynamic model, there is no definitive evidence for the 

existence of hysteresis in the index-velocity rating at the 

USGS streamgage on the CSSC near Lemont, Ill. Whereas the 

theoretical analysis allowed for the possibility of hysteresis, 

the hydrodynamic conditions required to generate hysteresis 

do not appear to be present at this site, based on historical data. 

Though the hydrodynamic model shows some limitations at 

peak flows, leading to some uncertainty in the model results, 

we can conclude that any hysteresis in the index-velocity rating 

at this site, if it occurs, is small and likely to be well within the 

range of uncertainty due to other sources in field measurements 

of discharge and index-velocity used to develop the rating. 

Ongoing streamgaging practices at this site will rely on the 

information in this report and include periodic assessment of 

the index-velocity rating for any signs of hysteresis that might 

result from future changes to the operation of this manmade 

canal. 

References Cited

Electricité de France, EDF-DER, 2007, Telemac Modelling 

System, 3-D Hydrodynamics TELEMAC-3D Code, 

release 5.8 operating manual, accessed March 28, 2013, at 

www.opentelemac.org.

Fread, D.L., 1975, Computations of stage-discharge relationships affected by unsteady flow: Water Resources Bulletin, 

v. 11, p. 213–218.

Hervouet, J.M., 2007, Hydrodynamics of free surface flows—

Modelling with the finite element method: Hoboken, N.J., 

John Wiley & Sons, 341 p., ISBN: 978-0-470-03558-0.

Jackson, P.R., Johnson, K.K., and Duncker, J.J., 2012, Comparison of index velocity measurements made with a horizontal 

acoustic Doppler current profiler and a three-path acoustic 

velocity meter for computation of discharge in the Chicago 

Sanitary and Ship Canal near Lemont, Illinois: U.S. Geological Survey Scientific Investigations Report 2011–5205, 42 p.

Johnson, K.K., Duncker, J.J., and Jackson, P.R., 2012, The role 

of the U.S. Geological Survey in Lake Michigan Diversion 

Accounting in Illinois, 1984–2010: U.S. Geological Survey 

Open-File Report 2012–1243, 73 p. 

Mander, R.J., 1978, Aspects of unsteady flow and variable 

backwater, in Herschy, R.W., ed., Hydrometry—Principles 

and practice (1st ed.): Chichester, UK, John Wiley & Sons, 

p. 205–246.
References Cited 33

Prandtl, L., 1926, Über die ausgebildete Turbulenz, in

Proceedings of the Second International Congress on 

Applied Mechanics, Zurich, Switzerland: p. 62–75.

 Ruhl, C.A., and Simpson, M.R., 2005, Computation of 

discharge using the index-velocity method in tidally 

affected areas: U.S. Geological Survey Scientific 

Investigations Report 2005–5004, 31 p.

Seo, I.W., and Baek, K.O., 2004, Estimation of the longitudinal dispersion coefficient using the velocity profile in 

natural streams: Journal of Hydraulic Engineering, v. 130, 

no. 3, p. 227–236.

Wilcox, D.C., 2004, Turbulence modeling for CFD 

(2d ed.): La Cañada, Calif., DCW Industries, Inc., 540 p., 

ISBN 1-928729-10-X. 

Jackson and others—Evaluation of the Potential for Hysteresis in Index-Velocity Ratings for the Chicago Sanitary and Ship Canal—Scientific Investigations Report 2013–5095