Comparing Discharge Computation Methods in Great Lakes Connecting Channels

Thompson, Aaron F.; Rodrigues, Sandrina N.; Fooks, Jeanette C.; Oberg, Kevin A.; Calappi, Timothy J.

doi:10.1061/(ASCE)HE.1943-5584.0001904

Open access

Case Studies

Mar 31, 2020

Comparing Discharge Computation Methods in Great Lakes Connecting Channels

This article has been corrected.

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Authors: Aaron F. Thompson https://orcid.org/0000-0001-7291-8476 [email protected], Sandrina N. Rodrigues, Ph.D. [email protected], Jeanette C. Fooks [email protected], Kevin A. Oberg [email protected], and Timothy J. Calappi, Ph.D. https://orcid.org/0000-0001-7087-0050 [email protected]Author Affiliations

Publication: Journal of Hydrologic Engineering

Volume 25, Issue 6

https://doi.org/10.1061/(ASCE)HE.1943-5584.0001904

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Abstract

Records of discharge for the connecting channels within the Great Lakes Basin are important to national governments of Canada and the United States and the various water management agencies and users in the basin. For more than 100 years, the official discharge records for the St. Clair and Detroit Rivers, two connecting channels within the Great Lakes Basin, have been computed using various stage-fall-discharge (SFQ) methods. However, as a result of technological advancements, newer methods have recently been considered for discharge computations. In this study, three discharge computation methods were compared: two SFQ methods and the index-velocity discharge (IVQ) method. Although the two SFQ methods have significantly different assumptions and use different data from the index-velocity method, the differences between the computed discharges derived from the methods are small, especially as the time step approaches monthly discharge values. Statistical analyses of discharge measurements and discharges computed using each of these methods indicate that there is no substantive difference in the discharges computed using the three methods. However, the IVQ method provides distinct advantages over the SFQ methods, including increased temporal resolution of computed discharge (minutes versus daily) and the ability to account for changes caused by aquatic vegetation and ice. Based on the results of the comparisons described herein, the IVQ discharge computation method is the most appropriate method for discharge computation in the St. Clair and Detroit Rivers. Updated SFQ equations for the St. Clair and Detroit Rivers, also presented herein, can be used to compute discharge during periods of missing or invalid IVQ record.

Introduction and Background

The Laurentian Great Lakes Basin (Fig. 1) consists of over

1,000,000 {km}^{2}

and is located along part of the international boundary between Canada and the United States. The basin extends from northern Minnesota and western Ontario to the Gulf of St. Lawrence. Within the Great Lakes Basin, a series of channels connect the Great Lakes, including the St. Clair River, which connects Lake Michigan-Huron to Lake St. Clair, and the Detroit River, which connects Lake St. Clair to Lake Erie. Accurate accounting of the discharge in these connecting channels is important to Canada and the United States because the international boundary bisects the Great Lakes and the connecting channels, thereby creating joint interest and responsibility for the stewardship of water levels of the Great Lakes and discharges in the connecting channels. Measurement of water levels and discharges is a shared responsibility of Canada and the US and involves staff from many different government agencies on both sides of the border.

Fig. 1. Great Lakes Basin. (Map by authors.)

The Coordinating Committee on Great Lakes Basic Hydraulic and Hydrologic Data (henceforth referred to as Coordinating Committee) was established in 1953 by the federal governments of the United States and Canada. Its mandate is to coordinate the collection and reporting of hydraulic, hydrologic, and vertical control data for the Great Lakes and St. Lawrence River Basin. Prior to 1953, Canada and the United States independently collected and compiled these data, with only superficial and informal coordination of some of the data. As a consequence, data sets that should have been identical were often discordant. The Coordinating Committee continues to meet semiannually to discuss and coordinate basic data for use by government agencies in Canada and the United States as well as for boards of the International Joint Commission (IJC), which helps prevent and resolve disputes over the use and quality of Canada and the United States boundary waters and to advise both governments on questions involving water resources.

Since the early 1900s, discharges in the St. Clair and Detroit Rivers have been computed using a variety of methods such as stage-fall-discharge (SFQ) ratings and unsteady hydraulic models. On several occasions the Coordinating Committee has reviewed the methods and published reports on single, deterministic estimates of monthly discharges for these rivers (Coordinating Committee 1982 and 1988). Flows have been coordinated approximately every 5 years by the Coordinating Committee and were most recently coordinated in 2009 using SFQ and unsteady hydraulic models calibrated to discharge measurements from 1996 to 2007 (which were collected using acoustic Doppler current profilers). In 2008, the USGS established new gauging stations on the St. Clair and Detroit Rivers, in response to recommendations made by the International Upper Great Lakes Study (IUGLS) to the IJC to install and maintain continuous-record stream gauging stations in the connecting channels (IUGLS 2009). Since that time, robust index-velocity (IVQ) ratings have been developed for the St. Clair and Detroit Rivers, presenting an alternative method for computing discharges in these connecting channels. The IVQ method is widely employed (Levesque and Oberg 2012; Rantz et al. 1982) and is increasingly common at stream gauges with rating complexity (Singh 2016). Given that the SFQ method has been the predominant method for computing discharges for more than 100 years, the Coordinating Committee decided to conduct a thorough analysis comparing SFQ methods to the IVQ method before implementing any changes.

Furthermore, the Coordinating Committee decided to evaluate the ISO stage-fall-discharge method (SFQ-ISO) as well as the previously employed multiple linear regression stage-fall-discharge method (SFQ-MLR). The rationale for comparing the SFQ-MLR to the ISO-SFQ method along with the IVQ method is twofold. First, the ISO-SFQ methods are widely used around the world for computing streamflow, and they are derived in a different manner than the one used by the Coordinating Committee (which has historically relied on MLR to develop SFQ equations). Second, the ISO-SFQ method was explored in the event that it proved to be a superior method for computing discharges for the St. Clair and Detroit Rivers.

The most widely used method for computing discharge in open channels uses a stage versus discharge relationship, commonly referred to as a stage-discharge rating (Leonard et al. 2000). A stage-discharge rating requires the installation of a water level monitoring gauge and repeated measurements of discharge over a range of concurrently observed water levels. The pairs of water level and discharge measurements are then used to create a numerical relationship between the stage and discharge. Subsequently, discharge can be computed continuously from the measurements of water level. The assumption of this method is that there is a unique relationship between the stage of the river and its discharge. This assumption is not valid during variable backwater conditions, when, depending on the backwater conditions, different discharges may be obtained for the same water level.

Methods for Computing Flows in Channels with Variable Backwater

Variable backwater can be caused by the impacts of downstream water bodies, control structures, ice, aquatic vegetation, and other obstructions. There are at least two ways to compute discharges for these types of rivers, including SFQ and IVQ methods. Both methods allow for continuous discharge computations, but in different ways. In the IVQ method, discharge is computed using both measured water velocities and water levels to determine discharge in the immediate reach impacted by variable backwater. In the SFQ method, a system of two water level gauges is used in the equation of discharge computations, so that the stage at a particular gauge and fall between a pair of water level gauges constantly accounts for any variable backwater. In this paper, two approaches to applying the SFQ method are analyzed and described in what follows: the multiple linear regression (MLR) and the ISO stage-fall discharge methods. These methods and their application to the St. Clair and Detroit Rivers will be evaluated in this paper.

Index-Velocity Method

IVQ ratings are commonly used to measure streamflow in highly controlled river systems with locks and dams, in tidal systems, in rivers with rapidly changing discharge, and in surface water systems with significant backwater effects (Jackson et al. 2012). Unlike more widely used stage-discharge ratings, index-velocity ratings can account for backwater effects and overcome hysteresis between stage and discharge, which can occur in stage-discharge ratings (Morlock et al. 2002). The IVQ method involves continuous measurement of velocity for a portion of a canal, stream, or river that serves as an indicator or index of the mean channel velocity. In addition, continuously measured stage data are used as an index of the channel cross-sectional area. Periodic measurements of discharge are obtained for the range of conditions experienced at the site. Index-velocity data are used in conjunction with these discharge measurements to develop a relation between index-velocity and measured-mean channel velocity known as the index-velocity rating. Similarly, the continuous stage data are used with previously obtained cross-sectional geometry data to develop a relation between the in situ stage and the cross-sectional area of the channel, and this relation is called the stage-area rating. These relations allow the computation of continuous mean velocity and cross-sectional area and are used to compute continuous records of discharge at a station.

The development of cost-effective acoustic Doppler velocity meters (ADVMs) has increased the application of IVQ methods for computing discharge. At the USGS alone, Levesque and Oberg (2012) stated that ADVMs and the IVQ method were used to compute discharge records at more than 430 stream gauges in the USGS. An ADVM uses two velocity measurement beams oriented to measure horizontally across a section of the stream and another acoustic beam oriented vertically for measuring stage. On the St. Clair and Detroit Rivers, ADVMs are used to measure velocity and are colocated with water level gauges operated by Canadian Hydrographic Survey (CHS) and the National Oceanic and Atmospheric Administration (NOAA), respectively.

The stage-area rating is developed from one or more surveys of the standard cross section and stage data measured concurrently with the survey(s). The standard cross section is a cross section near the ADVM that is chosen for the development of the stage-area rating and that can be resurveyed at regular intervals. The index-velocity rating is developed from a set of discharge measurement data and concurrent velocities measured by an ADVM. For each discharge measurement, the mean cross-sectional velocity is computed by dividing the measured discharge by the cross-sectional area computed from the stage-area rating. An index-velocity rating is then derived by regression analysis in which the index-velocity for each discharge measurement is related to the measured mean cross-section velocity to establish an index-to-mean velocity rating. Once the index-velocity rating is developed, the mean channel velocity can be computed from the index-velocity rating using the index velocities (measured by the ADVM) and concurrent stage data as input. Flows determined using the IVQ method, are the product of computed mean channel velocity and computed area (the results of applying both ratings to measured values of stage and index-velocity).

In the application of the IVQ method, index-velocity ratings are developed using discharge measurements that span the range of stage and velocity conditions. In many cases, especially for nontidal rivers, 2–3 years of data collection are required before enough discharge measurements are available to develop an index-velocity rating. Subsequent discharge measurements are then used to validate the index-velocity ratings. On occasion, these validation measurements indicate that a change in the flow regime or channel conditions has taken place, and a new index-velocity rating is required. Similarly, stage-area ratings are validated by periodic cross-sectional surveys and changes made when indicated. Criteria for an index-velocity stream gauge and guidance for successfully applying the IVQ method are available in Levesque and Oberg (2012).

Multiple Linear Regression Stage-Fall-Discharge Method

The SFQ method that has been used by the Coordinating Committee is based on the derivation of Quinn (1964, 1979) and Moore (1933, 1946). The equation of discharge is derived from Manning’s equation and takes the form

Q_{e s t} = a {(H_{u} [or H_{d}] - b a s e)}^{b} {(H_{u} - H_{d})}^{c}

(1)

where

Q_{e s t}

= discharge;

H_{u}

and

H_{d}

= upstream and downstream water levels, respectively; base = index of elevation at bottom of river; and

a

,

b

, and

c

= constants. Fay and Kerslake (2009) noted that using the downstream gauge (

H_{d}

) for the St. Clair River and the upstream gauge (

H_{u}

) for the Detroit River in the middle factor in the SFQ equation typically resulted in a small improvement in the statistics of each river (higher

R^{2}

values, lower standard errors). In this analysis, the same approach as Fay and Kerslake (2009) was followed for each river. Individual equations are derived by fixing the base term to a representative elevation of the channel bottom in the reach. The constants

a

,

b

, and

c

are found by using measured discharge paired with water levels for the upstream and downstream stage gauges and log-transformed, least-squares regression. Once the coefficients of the equation are determined using discrete measurements of discharge and relating them to the concurrent water levels, the discharge,

Q_{e s t}

, may be calculated based on records of water levels. This method has been used extensively by the Coordinating Committee to compute discharges (Coordinating Committee 1982 and 1988).

ISO Stage-Fall-Discharge Method

The ISO-SFQ method is documented in the 2001 ISO Standard 9123 and utilizes two or more water level gauges to account for changes in the water-surface slope from the reference or typical slope. For additional information on the method, see Herschy (1995), Kennedy (1984), and Rantz et al. (1982). The SFQ-ISO method assumes that, for a fixed stage at the gauge where discharge is to be computed but there are different water-surface slopes, only the slope term will vary. The SFQ-ISO equation is derived as

\frac{Q_{m}}{Q_{r}} = \frac{V A}{V_{r} A} = \frac{C \sqrt{R S}}{C \sqrt{R S_{r}}} = \sqrt{\frac{S}{S_{r}}} = \sqrt{\frac{\frac{F_{m}}{L}}{\frac{F_{r}}{L}}} = \sqrt{\frac{F_{m}}{F_{r}}}

(2)

where

C

= resistance coefficient;

V

= mean velocity at cross section;

A

= cross-sectional area;

R

= hydraulic radius;

S

= slope;

F

= fall measured between two gauges;

L

= reach length between the two gauges;

r

subscripts = reference conditions; and

m

subscripts = conditions during any individual measurement. Reference conditions can represent the base relation between stage and discharge for uniform flow or for a constant backwater condition (constant-fall) developed from repeated observations of stage, discharge, and fall between two gauges. The cross-sectional area, hydraulic radius, and resistance coefficient remain unchanged between the reference and measurement condition for a given stage so that only the fall varies. Unlike the SFQ-MLR method, this procedure does not explicitly try to fit the discharge to a predefined equation.

The procedure to develop a SFQ-ISO equation is to use normalized discharges developed from measured discharges and corresponding water level data; the discharge measurements are normalized by

\sqrt{(F_{m} / F_{r})}

, where

F_{r}

is determined by taking the average of all measured falls for this reach (based on the available data set of measured discharge instances). The rated discharge is then calculated by

Q_{r} = \frac{Q_{m}}{\sqrt{(F_{m} / F_{r}})}

(3)

where

Q_{m}

= measured discharge; and

Q_{r}

= (normalized) rated discharge.

Next,

Q_{r}

is plotted against water level (which is called Curve 1 in the SFQ-ISO calibration; for an example, see Fig. 2), and least-squares regression is used to determine a line of best fit such that

W L = m_{1} \times Q_{C} + b a s e

. Here,

W L

represents the water level,

m_{1}

represents the slope of the regression line, and base is the intercept of the regression line and represents the lowest elevation at which these rating curves are applicable. The values of

m_{1}

and base are obtained from the least-squares regression. Rearranging this line of best fit, the curve-generated estimate of discharge,

{Q_{c}}^{*}

, can be computed for a given water level (the asterisk indicates that the variable is determined directly from a relationship curve). The magnitude of the

W L

must always be greater than the base constant in order to compute discharges greater than zero. Curve 2 is developed by plotting the measured fall against

{Q_{est} / Q_{c}}^{*}

. Using the line of best fit on Curve 2, an estimate of the curve-generated ratio

{(Q_{est} / Q_{c})}^{*}

can be computed for any measured fall:

{(Q_{est} / Q_{c})}^{*} = m_{2} \times F_{m} + K

, where K is a constant determined using least-squares regression from the line of best fit. The estimated discharge,

Q_{est}

, is then the product of these two curve-generated values:

Q_{est} = {Q_{C}}^{*} \times {(\frac{Q}{Q_{c}})}^{*}

(4)

or, when the lines of best fit from Curves 1 and 2 are substituted for the terms on the right-hand side of Eq. (3) [

Q_{c}

and

{(Q / Q_{C})}^{*}

, respectively]

Q_{est} = \frac{(W L - b a s e)}{m_{1}} * \times {(m_{2} \times F_{m} + K)}^{*}

(5)

Further details regarding the theoretical basis and application of SFQ-ISO are found in ISO 9123 (ISO 2017).

Fig. 2. Curves 1 and 2 for ISO method of deriving SFQ equations.

Application to St. Clair and Detroit Rivers

There are eight continuously recording stage gauges on the St. Clair River and six on the Detroit River. The data from these gauges are primarily used to aid in navigation through the St. Clair and Detroit Rivers but also may be used to compute time series of discharge using SFQ ratings. The gauges are listed in Table 1, and their locations are shown in Figs. 3 and 4. The NOAA gauges record stage at 6-min intervals and CHS gauges record stage at 3-min intervals. In 2008, ADVMs were installed by the USGS with assistance from Environment and Climate Change Canada (ECCC) at Port Huron in the St. Clair River and at Fort Wayne in the Detroit River. The ADVMs record 8-min averaged velocities (obtained every 12 min) for each river. The stage data for discharge computation are obtained from the NOAA and CHS gauges. For the St. Clair River index-velocity rating, the CHS Point Edward gauge is used, and for the Detroit River rating, the NOAA Fort Wayne gauge is used. Stage data collected by the USGS at these locations are used to fill any gaps in the record and for data review.

Table 1. Water level and discharge gauges

Gauge	Station number
St. Clair River
Fort Gratiot (FG)^a	9014098
Dunn Paper (DP)^a	9014096
Point Edward (PE)^b	11940
Mouth of Black River (MBR)^a	9014090
Dry dock (DD)^a	9014087
St. Clair state police (SCSP)^a	9014080
Port Lambton (PL)^b	11950
Algonac (AL)^a	9014070
St. Clair River at Port Huron^c^,^d	04159130^c
St. Clair River at Port Huron^c^,^d	02GG014^d
Detroit River
Windmill Point (WP)^a	9044049
Fort Wayne (FW)^a	9044036
Wyandotte (WY)^a	9044030
Amherstburg (AM)^b	11995
Gibraltar (GB)^a	9044020
Bar point (BP)^b	12005
Detroit River at Fort Wayne ^c^,^d	04165710^c
Detroit River at Fort Wayne ^c^,^d	02GH015^d

a

NOAA gauges.

b

CHS gauges.

c

ADVMs installed by USGS.

d

ADVMs installed by ECCC.

Fig. 3. Water level gauging stations along St. Clair River. [Base map by Esri, HERE, DeLorme, Intermap, Increment P Corp., GEBCO, USGS, FAO, NPS, NRCAN, Geobase, IGN, Kadaster NL, Ordnance Survey, Esri Japan, METI, Esri China (Hong Kong), swisstopo, MapmyIndia, © OpenStreetMap contributors and the GIS User Community.]

Fig. 4. Water level gauging stations along Detroit River. [Base map by Esri, HERE, DeLorme, Intermap, Increment P Corp., GEBCO, USGS, FAO, NPS, NRCAN, Geobase, IGN, Kadaster NL, Ordnance Survey, Esri Japan, METI, Esri China (Hong Kong), swisstopo, MapmyIndia, © OpenStreetMap contributors and the GIS User Community.]

Discharge measurements on the St. Clair River and the Detroit River are collected at regular intervals during open-water conditions (ice-free) by the USGS. Special measurements were obtained in response to various kinds of events (e.g., windstorms, seiches) to provide better definition to the index-velocity rating. The discharge measurements were made using the moving-boat acoustic Doppler current profiler (ADCP) method (Mueller et al. 2013). A total of 49 discharge measurements on the St. Clair River (at Port Huron) between December 2008 and September 2015 and a total of 66 discharge measurements on the Detroit River (at Fort Wayne) between September 2008 and November 2015 were used for this comparison analysis. It is often dangerous to navigate a boat during the winter because of floating ice in the St. Clair and Detroit Rivers. Moreover, boat launches are usually closed during the winter. As a result, there are no discharge measurements during ice conditions, so the comparison analysis covers only open-water conditions. Regardless, methods and considerations for calculation of discharge in the rivers will be discussed later in this paper.

A large number (

> 450

) of historic discharge measurements have been made on the St. Clair and Detroit Rivers. However, measurements made prior to the installation of the ADVMs could not be used in rating development. Therefore, for the purposes of this comparison, only those measurements used to develop the IVQ rating curves were used to develop SFQ equations. SFQ equations were derived with the SFQ-MLR method using Eq. (1) and the SFQ-ISO method using Eqs. (2) and (3). These measurements are hereafter referred to as calibration measurements. Comparison of the discharges computed using the different methods was accomplished by using available discharge measurements that were not used in rating development, referred to as validation measurements. The “Gauge” column in Tables 2–7 indicates three pieces of information, separated by an underscore: the first piece of information is the type of model used (MLR for multiple linear regression SFQ, ISO for the ISO-SFQ method, or IVQ for the IVQ method), and the second and third terms represent the upstream and downstream gauges, respectively, for which the model was run. The list of gauges and their abbreviations can be found in Table 1.

Table 2. St. Clair River stage-fall-discharge equations derived using multiple linear regression methods

Gauges	St. Clair River Equations	$R^{2}$	RMSE ( $m^{3} / s$ )	Bias ( $m^{3} / s$ )	Average \|%Diff\| (%)	Smearing coefficient
MLR_FG_PE	$Q = 122.40 {(PE - 167)}^{\land} 1.899 \times {(FG - PE)}^{\land} 0.232$	0.94	22	$- 0.41$	2.43	1.000398
MLR_FG_MBR	$Q = 464.01 {(MBR - 167)}^{\land} 1.422 \times {(FG - MBR)}^{\land} 0.423$	0.98	15	$- 0.25$	1.34	1.000155
MLR_FG_DD	$Q = 201.03 {(DD - 167)}^{\land} 1.732 \times {(FG - DD)}^{\land} 0.428$	0.97	15	$- 0.31$	1.55	1.000185
MLR_FG_SCSP	$Q = 262.84 {(SCSP - 167)}^{\land} 1.523 \times {(FG - SCSP)}^{\land} 0.554$	0.99	11	$- 0.02$	1.07	1.000088
MLR_FG_PL	$Q = 250.78 {(PL - 167)}^{\land} 1.463 \times {(FG - PL)}^{\land} 0.621$	0.98	12	0.11	1.11	1.000105
MLR_FG_AL	$Q = 270.22 {(AL - 167)}^{\land} 1.417 \times {(FG - AL)}^{\land} 0.635$	0.98	12	0.11	1.19	1.000117
MLR_DP_PE	$Q = 23.55 {(PE - 167)}^{\land} 2.429 \times {(DP - PE)}^{\land} 0.022$	0.64	216	0.17	3.20	1.000897
MLR_DP_MBR	$Q = 17.97 {(MBR - 167)}^{\land} 2.717 \times {(DP - MBR)}^{\land} 0.110$	0.88	32	$- 0.32$	2.75	1.000757
MLR_DP_DD	$Q = 29.64 {(DD - 167)}^{\land} 2.570 \times {(DP - DD)}^{\land} 0.250$	0.91	28	$- 0.46$	1.45	1.000563
MLR_DP_SCSP	$Q = 137.60 {(SCSP - 167)}^{\land} 1.876 \times {(DP - SCSP)}^{\land} 0.540$	0.97	15	$- 0.21$	1.40	1.000175
MLR_DP_PL	$Q = 175.09 {(PL - 167)}^{\land} 1.675 \times {(DP - PL)}^{\land} 0.631$	0.98	14	0.01	1.51	1.000148
MLR_DP_AL	$Q = 192.10 {(AL - 167)}^{\land} 1.619 \times {(DP - AL)}^{\land} 0.644$	0.97	15	0.01	2.44	1.000168
MLR_PE_DD	$Q = 45.42 {(DD - 167)}^{\land} 2.529 \times {(PE - DD)}^{\land} 0.376$	0.92	25	0.14	3.42	1.000478
MLR_PE_MBR	$Q = 12.18 {(MBR - 167)}^{\land} 2.795 \times {(PE - MBR)}^{\land} 0.019$	0.84	35	0.13	1.58	1.000966
MLR_PE_SCSP	$Q = 165.72 {(SCSP - 167)}^{\land} 1.820 \times {(PE - SCSP)}^{\land} 0.553$	0.97	16	0.24	1.58	1.000206
MLR_PE_PL	$Q = 165.95 {(PL - 167)}^{\land} 1.714 {(PE - PL)}^{\land} 0.594$	0.96	18	0.20	1.61	1.000267
MLR_PE_AL	$Q = 183.10 {(AL - 167)}^{\land} 1.656 \times {(PE - AL)}^{\land} 0.609$	0.95	18	0.20	2.43	1.000284
Index-velocity		0.95	22	1.18	1.66	—

Note: Average annual discharge of St. Clair $River = 5,190 m^{3} / s$ . Goodness-of-fit statistics based on calibration data. Bias computed as $Σ (modeled - measured) / (number of measurements)$ . Percent difference computed as $(modeled - measured) / measured$ .

Table 3. Detroit River stage-fall-discharge equations (prior to March 11, 2011) derived using multiple linear regression methods

Gauges	Detroit River (1) Equations	$R^{2}$	RMSE ( $m^{3} / s$ )	Bias ( $m^{3} / s$ )	Average \|%Diff\| (%)	Smearing coefficient
MLR_WP_WY	$Q = 184.82 {(WP - 164)}^{\land} 1.594 \times {(WP - WY)}^{\land} 0.375$	0.911	141	0.04	2.24	1.000292
MLR_WP_AM	$Q = 368.89 {(WP - 164)}^{\land} 1.272 \times {(WP - AM)}^{\land} 0.392$	0.885	156	$- 0.34$	2.51	1.000390
MLR_WP_GB	$Q = 57.69 {(WP - 164)}^{\land} 1.965 \times {(WP - GB)}^{\land} 0.384$	0.874	166	0.55	2.44	1.000436
MLR_WP_BP	$Q = 338.59 {(WP - 164)}^{\land} 1.202 \times {(WP - BP)}^{\land} 0.365$	0.834	183	$- 0.14$	2.86	1.000556
MLR_FW_WY	$Q = 316.11 {(FW - 164)}^{\land} 1.565 \times {(FW - WY)}^{\land} 0.391$	0.857	174	$- 0.39$	2.85	1.0005898
MLR_FW_AM	$Q = 2281.3 {(FW - 164)}^{\land} 0.641 \times {(FW - AM)}^{\land} 0.403$	0.805	200	$- 0.21$	3.15	1.0006865
MLR_FW_GB	$Q = 103.47 {(FW - 164)}^{\land} 1.805 \times {(FW - GB)}^{\land} 0.409$	0.809	203	$- 0.49$	2.86	1.000669
MLR_FW_BP	$Q = 1617.12 {(FW - 164)}^{\land} 0.6 \times {(FW - BP)}^{\land} 0.353$	0.765	217	$- 0.36$	3.31	1.000796
	Detroit River index-velocity	0.907	179	$- 12.34$	2.49	—

Note: Average annual discharge of Detroit $River = 5,320 m^{3} / s$ . Goodness-of-fit statistics based on calibration data. Bias computed as $Σ (modeled - measured) / (number of measurements)$ . Percent difference computed as $(modeled - measured) / measured$ .

Table 4. Detroit River stage-fall-discharge equations (starting March 11, 2011) derived using multiple linear regression methods

Gauges	Detroit River (2) Equations	$R^{2}$	RMSE ( $m^{3} / s$ )	Bias ( $m^{3} / s$ )	Average \|%Diff\| (%)	Smearing coefficient
MLR_WP_FW	$Q = 72.29 {(WP - 164)}^{\land} 2.07 \times {(WP - FW)}^{\land} 0.414$	0.930	18	$- 0.07$	2.08	1.000332
MLR_WP_WY	$Q = 52.12 {(WP - 164)}^{\land} 2.139 \times {(WP - WY)}^{\land} 0.416$	0.934	18	$- 0.06$	2.06	1.000350
MLR_WP_AM	$Q = 40.15 {(WP - 164)}^{\land} 2.14 \times {(WP - AM)}^{\land} 0.436$	0.921	20	$- 0.03$	2.32	1.000400
MLR_WP_GB	$Q = 13.99 {(WP - 164)}^{\land} 2.554 \times {(WP - GB)}^{\land} 0.363$	0.910	22	0.00	2.29	1.000454
MLR_WP_BP	$Q = 5.778 {(WP - 164)}^{\land} 2.862 \times {(WP - BP)}^{\land} 0.001$	0.536	48	0.79	4.85	1.002406
MLR_FW_WY	$Q = 66.54 {(FW - 164)}^{\land} 2.245 \times {(FW - WY)}^{\land} 0.419$	0.887	24	$- 0.01$	2.78	1.000574
MLR_FW_AM	$Q = 42.73 {(FW - 164)}^{\land} 2.349 \times {(FW - AM)}^{\land} 0.446$	0.833	29	0.05	3.35	1.000842
MLR_FW_GB	$Q = 11.97 {(FW - 164)}^{\land} 2.702 \times {(FW - GB)}^{\land} 0.367$	0.858	27	0.12	2.83	1.000723
MLR_FW_BP	$Q = 14.60 {(FW - 164)}^{\land} 2.500 \times {(FW - BP)}^{\land} 0.003$	0.411	54	0.92	5.68	1.003051
	Detroit River index-velocity	0.759	197	21.55	3.27	—

Note: Average annual discharge of Detroit $River = 5,320 m^{3} / s$ . Goodness-of-fit statistics based on calibration data. Bias computed as $(modeled - measured) / (number of measurements)$ . Percent difference computed as $(modeled - measured) / measured$ .

Table 5. St. Clair River stage-fall-discharge equations derived using SFQ-ISO methods

Gauge	St. Clair River equation	$R^{2}$	RMSE ( $m^{3} / s$ )	Bias ( $m^{3} / s$ )	Average \|%Diff\| (%)	Curve 1 $R^{2}$	Curve 2 $R^{2}$
ISO_FG_PE	$[(FG - 175.87) / 0.00004684] \times [32.7878 (FG - PE) + (- 5.2498)]$	0.258	5,338	$- 1,814$	81.3	0.006	0.142
ISO_FG_MBR	$[(FG - 170.44) / 0.00109418] \times [1.9324 (FG - MBR) + (0.6238)]$	0.954	94	0.7	1.2	0.664	0.879
ISO_FG_DD	$[(FG - 171.19) / 0.00095115] \times [1.2861 (FG - DD) + (0.6174)]$	0.954	93	0.9	1.4	0.747	0.860
ISO_FG_SCSP	$[(FG - 170.95) / 0.00099812] \times [0.7078 (FG - SCSP) + (0.5704)]$	0.972	73	0.3	1.1	0.882	0.916
ISO_FG_PL	$[(FG - 171.13) / 0.00096842] \times [0.4547 (FG - PL) + (0.5815)]$	0.964	78	0.6	1.3	0.864	0.888
ISO_FG_AL	$[(FG - 170.78) / 0.00103022] \times [0.4309 (FG - AL) + (0.5772)]$	0.966	80	0.4	1.2	0.855	0.895
ISO_DP_PE	$[(DP - 175.91) / 0.00004208] \times [1152 (DP - PE) + (- 46.113)]$	0.056	26,203	10,731	358.0	0.014	0.067
ISO_DP_MBR	$[(DP - 174.81) / 0.00025598] \times [- 4.9269 (DP - MBR) + (1.4882)]$	0.506	686	105	9.6	0.130	0.140
ISO_DP_DD	$[(DP - 172.97) / 0.00060864] \times [1.3462 (DP - DD) + (0.7975)]$	0.855	183	7.8	2.6	0.533	0.371
ISO_DP_SCSP	$[(DP - 171.52) / 0.00088723] \times [0.8692 (DP - SCSP) + (0.6017)]$	0.950	98	1.0	1.4	0.837	0.844
ISO_DP_PL	$[(DP - 171.4) / 0.00091521] \times [0.5444 (DP - PL) + (0.5875)]$	0.949	93	1.1	1.5	0.845	0.849
ISO_DP_AL	$[(DP - 171.1) / 0.00096917] \times [0.5012 (DP - AL) + (0.5916)]$	0.952	95	0.8	1.5	0.830	0.856
ISO_PE_MBR	$[(PE - 175.51) / 0.00008204] \times [81.146 (PE - MBR) + (0.5793)]$	0.693	11,972	9,224	149.3	0.153	0.013
ISO_PE_DD	$[(PE - 173.2) / 0.00053396] \times [2.5452 (PE - DD) + (0.6721)]$	0.855	180	7.7	2.3	0.742	0.489
ISO_PE_SCSP	$[(PE - 172.09) / 0.00074857] \times [1.0435 (PE - SCSP) + (0.5466)]$	0.951	94	1.6	1.2	0.890	0.844
ISO_PE_PL	$[(PE - 172.22) / 0.00072451] \times [0.5049 (PE - PL) + (0.6185)]$	0.913	126	2.9	1.4	0.772	0.711
ISO_PE_AL	$[(PE - 172.16) / 0.00073617] \times [0.469 (PE - AL) + (0.6229)]$	0.902	133	3.0	1.6	0.743	0.683
	St. Clair River index-velocity	0.949	98	1.18	1.66	—	—

Note: Average annual discharge of St. Clair $River = 5,190 m^{3} / s$ . Goodness-of-fit statistics based on calibration data. Bias computed as $(modeled - measured) / (number of measurements)$ . Percent difference computed as $(modeled - measured) / measured$ .

Table 6. Detroit River Equations derived using SFQ-ISO methods (prior to March 11, 2011)

Gauge	Detroit River (1) equation	$R^{2}$	RMSE ( $m^{3} / s$ )	Bias ( $m^{3} / s$ )	Average \|%Diff\| (%)	Curve 1 $R^{2}$	Curve 2 $R^{2}$
ISO_WP_WY	$[(WP - 173.18) / 0.00032315] \times [1.3477 (WP - WY) + (0.5525)]$	0.748	280	$- 16.7$	5.0	0.408	0.799
ISO_WP_AM	$[(WP - 173.11) / 0.00033755] \times [1.1257 (WP - AM) + (0.5293)]$	0.670	332	20.2	4.9	0.342	0.758
ISO_WP_GB	$[(WP - 173.13) / 0.00033268] \times [0.6782 (WP - GB) + (0.535)]$	0.723	297	17.0	4.2	0.501	0.808
ISO_WP_BP	$[(WP - 173.56) / 0.00025605] \times [0.6146 (WP - BP) + (0.5207)]$	0.516	458	37.9	6.6	0.282	0.655
ISO_FW_WY	$[(FW - 172.51) / 0.00040649] \times [3.2772 (FW - WY) + (0.5407)]$	0.794	235	9.1	3.3	0.575	0.853
ISO_FW_AM	$[(FW - 172.99) / 0.00032394] \times [2.135 (FW - AM) + (0.4678)]$	0.613	350	19.3	5.0	0.240	0.720
ISO_FW_GB	$[(FW - 172.64) / 0.00038289] \times [0.8375 (FW - GB) + (0.5116)]$	0.777	242	8.3	3.3	0.553	0.849
ISO_FW_BP	$[(FW - 173.19) / 0.00028424] \times [0.6787 (FW - BP) + (0.4923)]$	0.562	387	23.6	5.5	0.288	0.692
	Detroit River index-velocity	0.907	179	$- 12.34$	2.49	—	—

Note: Average annual discharge of Detroit $River = 5,320 m^{3} / s$ . Goodness-of-fit statistics based on calibration data. Bias computed as $(modeled - measured) / (number of measurements)$ . Percent difference computed as $(modeled - measured) / measured$ .

Table 7. Detroit River equations derived using SFQ-ISO methods (starting March 11, 2011)

Gauge	Detroit River (2) equation	$R^{2}$	RMSE ( $m^{3} / s$ )	Bias ( $m^{3} / s$ )	Average \|%Diff\| (%)	Curve 1 $R^{2}$	Curve 2 $R^{2}$
ISO_WP_WY	$[(WP - 172.54) / 0.00043412] \times [1.1016 (WP - WY) + (0.6082)]$	0.811	279	18.2	5.5	0.312	0.758
ISO_WP_AM	$[(WP - 171.65) / 0.00059668] \times [0.9183 (WP - AM) + (0.5922)]$	0.890	210	9.0	4.5	0.409	0.857
ISO_WP_GB	$[(WP - 172.15) / 0.00049989] \times [0.5075 (WP - GB) + (0.6219)]$	0.884	220	11.3	4.6	0.518	0.844
ISO_WP_BP	$[(WP - 172.18) / 0.00049501] \times [0.4462 (WP - BP) + (0.6187)]$	0.862	243	13.6	5.5	0.414	0.813
ISO_FW_WY	$[(FW - 172.14) / 0.00046867] \times [2.91 (FW - WY) + (0.5695)]$	0.868	222	11.1	4.0	0.497	0.841
ISO_FW_AM	$[(FW - 171.81) / 0.00052437] \times [1.797 (FW - AM) + (0.5399)]$	0.878	221	10.1	5.2	0.453	0.843
ISO_FW_GB	$[(FW - 172.12) / 0.00046296] \times [0.6531 (FW - GB) + (0.5833)]$	0.882	217	9.7	4.0	0.650	0.850
ISO_FW_BP	$[(FW - 172.2) / 0.00044985] \times [0.552 (FW - BP) + (0.5739)]$	0.857	245	13.8	5.6	0.475	0.819
	Detroit River index-velocity	0.759	196	21.55	3.27	—	—

Note: Average annual discharge of Detroit $River = 5,320 m^{3} / s$ . Goodness-of-fit statistics based on calibration data. Average bias computed as $(modeled - measured) / (number of measurements)$ . Percent difference computed as $(modeled - measured) / measured$ .

Application of Index-Velocity Method

The IVQ method was used to compute discharges on the St. Clair River (at Port Huron) and the Detroit River (at Fort Wayne) at the interval of available stage data. This interval is 3 min for the St. Clair River and 6 min for the Detroit River. Stage-area and index-velocity ratings were developed for each location. Stage-area ratings were developed in accordance with methods presented by Levesque and Oberg (2012).

Index-velocity ratings were developed for each gauge after discharge measurements covering the expected range of stage and velocity were available. For the St. Clair River gauge (at Port Huron), an index-to-mean velocity rating was developed and remained valid for the entire 2008–2015 time period. The initial rating was made effective on November 8, 2008, using the concurrent discharge and index-velocity measurements then available. Subsequent concurrent discharge and index-velocity measurements indicate that the rating continues to be valid for computing discharge. The St. Clair River stage-area rating and index-velocity rating are shown in Figs. 5(a and b), respectively.

Fig. 5. (a) St. Clair River stage-area rating for index-velocity method; and (b) Detroit River stage-area rating for index-velocity method.

For the Detroit River gauge (at Fort Wayne), two index-velocity ratings have been used. The first rating for the Detroit River gauge was developed in October 2008 based on the concurrent discharge and index-velocity measurements then available for analysis. Subsequent concurrent discharge and index-velocity measurements validated the rating until March 2011. Measurements made after March 11, 2011, indicated that a new rating was needed. Therefore, a second index-to-mean velocity rating was developed based on measurements made from October 7, 2009, to November 21, 2012, and put into use starting on March 11, 2011. Subsequent concurrent discharge and index-velocity measurements have confirmed the second rating. Although the first index-to-mean velocity rating at the Detroit River gauge had very few validation measurements owing to the changeover to the second index-to-mean velocity rating, the second rating had 26 validation measurements at the time of this comparison, and validation measurements continue to be added to this list to confirm that the rating is appropriate. A single stage-area rating has been used for the entire period for the Detroit River. Graphs illustrating the stage-area rating for the Detroit River and the two index-velocity ratings are shown in Figs. 6 and 7(a and b), respectively.

Fig. 6. St. Clair River index-velocity rating.

Fig. 7. (a) Detroit River index-velocity Rating 1; and (b) Detroit River index-velocity Rating 2.

The discharges measured at the St. Clair and Detroit River gauge locations were compared to the computed IVQ discharges averaged over the length of each measurement (measurements may last 1 h or longer). The average of the absolute value of the percent differences between the IVQ discharges and the measured discharges for the calibration data was less than 1.7% for the St. Clair River gauge and less than 3.3% for the Detroit River gauge. The residuals (

Q_{m} - Q_{est}

) of the discharges computed using the IVQ method and the measured discharges for the calibration and validation data sets are shown in Fig. 8 for the St. Clair River (left) and two Detroit River IVQ ratings (center and right, respectively). Statistics describing the flows computed from the index-velocity methods against the measurements are included in the last lines in Tables 2–7 to allow direct comparison with the other flow computation methods.

Fig. 8. Residual plots comparing IVQ, SFQ-MLR, SFQ-ISO, and measured discharges for calibration and validation data. Detroit 1 = Detroit Rating 1, ending March 10, 2011; Detroit 2 = Detroit Rating 2, starting March 11, 2011.

Application of Multiple Linear Regression Stage-Fall-Discharge Method

To develop SFQ-MLR equations, gauge pairs were selected such that one gauge is upstream of the index-velocity meter and one is downstream, with all possible gauge pairs analyzed. This approach resulted in the creation of 17 gauge pairs for the St. Clair River and 8 for the Detroit River. Equations for all possible reaches were developed and the comparison of computed to measured discharge was used to select the optimal reaches. Gauge pairs were identified by the upstream and downstream gauge abbreviation (as listed in parentheses in Table 1) as well as an abbreviation for each method, MLR or ISO. The value of the base term in Eq. (1) was estimated as 167 m for the St. Clair River and 164 m for the Detroit River, based on local knowledge and previous work. These two values for the base term are reasonable estimates of the average thalweg elevations of the two rivers.

For the St. Clair River, SFQ-MLR models were developed for 17 gauge pairs using the same calibration discharge measurements used to develop index-velocity ratings. The discharge measurements were divided into calibration and validation data sets with 33 and 16 measurements, respectively (the same measurements were used to develop the index-velocity ratings). The equations derived using the calibration data and the associated goodness-of-fit statistics are shown in Table 2. For the Detroit River, two sets of equations were developed with the calibration data to provide a consistent comparison with the discharges computed using the IVQ method. The index-velocity rating for the Detroit River gauge was changed by the USGS, effective March 11, 2011, as indicated by discharge measurements. Therefore, two sets of SFQ-MLR models were developed for the eight gauge pairs, one set for the period from September 26, 2008–March 10, 2011, and another set for the period March 11, 2011–December 31, 2016. The resulting equations for the two periods are shown in Tables 3 and 4 along

20 m^{3} / s

ciated with the model development. As with the St. Clair data, discharge measurements made on the Detroit River were divided into calibration and validation data sets for both periods. The calibration data sets consisted of 20 and 18 measurements for the first and second periods, respectively. The validation data sets consisted of 4 and 26 measurements.

The SFQ-MLR method worked well for approximately half of the gauge pairs on the St. Clair River, which have

R^{2}

values greater than 0.9, root-mean-square error (RMSE) values lower than

20 m^{3} / s

, and bias values (

Q_{est} - Q_{m}

) lower than

1 m^{3} / s

(Table 2). The average absolute percent differences between the discharges computed for the St. Clair River using the SFQ-MLR method and the measured discharges ranged from 2.1% to 3.4%. Regressions using a log-transformed model tend to generate negative bias when transformed back to the original variable space (Helsel and Hirsch 2002). A smearing coefficient described by Duan (1983) was used to correct for this bias and is shown in Tables 2–4.

Statistics describing the SFQ-MLR model results for the calibration data during the period of September 2008 to March 10, 2011, on the Detroit River are given in Table 3. Average absolute percent differences from the measured discharges range from 2.2% (WP_WY) to 3.3% (MLR_FW_BP). The

R^{2}

values range from 0.765 (MLR_FW_BP) to 0.911 (MLR_WP_WY). The RMSE value for this data set range from 141 to

217 m^{3} / s

, and the average bias estimates range from

- 1.6

to

- 4.7 m^{3} / s

. Overall, the gauge pairs that performed best based on statistical and residual analyses were the MLR_WP_WY, MLR_WP_GB, and MLR_WP_AM.

The SFQ-MLR models applicable to the Detroit River for May 11, 2011, onwards have average absolute percent differences for the calibration data set ranging from 2.1% (MLR_WP_WY) to 5.7% (MLR_FW_BP), as shown in Table 4. The

R^{2}

values range from 0.041 (MLR_FW_BP) to 0.93 (MLR_WP_WY), with four of the eight gauge pairs having

R^{2}

values above 0.89. The bias estimates are less than

1 m^{3} / s

(MLR_FW_BP) and the RMSE values range from 18 (MLR_WP_WY) to 54 (MLR_WP_BP)

m^{3} / s

. Fig. 8 shows plots of the residuals over time for the calibration and validation data using the best SFQ-MLR method to compute discharges for the St. Clair River (left) and the two Detroit River ratings.

Application of ISO Stage-Fall-Discharge Method

Seventeen SFQ-ISO curves were developed for the St. Clair River, eight for the Detroit River for data from October 1, 2008, to March 10, 2011, and eight for the Detroit River starting from March 11, 2011. The SFQ-ISO curves are summarized in Tables 5–7, respectively.

The base term is a statistically determined value in the SFQ-ISO method [Eq. (5)], and as such the values render these equations invalid when water levels fall below the value of base. In reality, water levels could become much lower than the value of base and the discharge would still not become zero. The expected ranges of the upstream water levels are greater than the base term in most equations. The FG, DP, and PE average daily water levels from January 1, 2008, to December 31, 2016, range from 175.3 to 176.9 m, from 175.2 to 176.7 m, and from 175.2 to 176.6 m, respectively. The WP and FW average daily water levels from January 1, 2008, to December 31, 2016, range from 174.1 to 175.6 m and from 174.1 to 175.4 m, respectively. The SFQ-ISO equations for the St. Clair River in Table 5 that would not meet the criteria for the expected ranges of the upstream water level gauges are therefore discarded as possible methods for computing St. Clair River discharges are ISO_FG_PE, ISO_DP_PE, ISO_DP_MBR, and ISO_PE_MBR because the value of base for each of these equations is 175.87, 175.91, 174.81, and 175.51, respectively. The statistical metrics for these gauge pairs in Table 5 are noticeably poorer than the statistics for the remaining gauge pairs. As a result, they were eliminated from consideration as a method for computing St. Clair River discharges. All of the ISO-derived equations for the Detroit River in Tables 6 and 7 meet the criteria required to compute positive, nonzero values of discharge given the typical range of upstream values for each gauge pair, with the exception of ISO_WP_WY for the Detroit River from the data set spanning September 2008 to March 10, 2011.

Along the St. Clair River, 7 of the 17 gauge pairs used to compute discharges had

R^{2}

values greater than 0.80 for both ISO calibration curves, as shown in Table 5. These seven gauge pairs typically had higher

R^{2}

values and lower RMSE, bias, and average departure values. Overall, the ISO equations that are most likely to compute reliable discharges for the St. Clair River are ISO_FG_SCSP, ISO_FG_PL, and ISO_FG_AL.

When the ISO method was employed to develop equations for data on the Detroit River (for the data set prior to March 11, 2011) (Table 6); none of the gauge pairs had

R^{2}

values greater than 0.80 for both calibration curves. Despite this, when the method was used to estimate discharges, four of the eight gauge pairs had overall model

R^{2}

values greater than 0.7. These four gauge pairs, ISO_WP_WY, ISO_WP_GB, ISO_FW_WY, and ISO_FW_GB, had bias values of

- 16.7

, 17.0, 9.1, and

8.3 m^{3} / s

, respectively, and average departures of 1.1%, 4.2%, 3.3%, and 2.6%, respectively. The residual plots for the validation data consisted of only four data points for each gauge pair and, therefore, do not reliably assess the performance of these models. Overall, the three best ISO equations for this Detroit River data set are ISO_FW_WY, ISO_WP_GB, and ISO_WP_WY.

The ISO method performed well for the second set of Detroit River data (Table 7) for each gauge pair, with relatively low

R^{2}

values for Calibration Curve 1 but relatively high

R^{2}

values for Calibration Curve 2. With the exception of ISO_WP_WY, all the gauge pairs had

R^{2}

values greater than 0.85. Only half of the residual plots for the validation data show data that are normally distributed and approximately centered about zero: ISO_WP_WY, ISO_WP_AM, ISO_WP_GB, ISO_WP_BP. Overall, the three best ISO equations for this second set of Detroit River data are ISO_WP_AM, ISO_WP_GB, and ISO_WP_BP. In general, the high positive average bias values shown in the ISO tables (Tables 6 and 7) indicate that the ISO method tends to overestimate the computed discharges for both sets of Detroit River data.

Fig. 8 shows plots of the residuals of the calibration and validation data for the best-performing gauge pair along the St. Clair River and the two Detroit River ratings using the SFQ-ISO method. The residuals for the St. Clair River show relatively normally distributed data centered about zero; whereas, the residuals for the Detroit River appear to show somewhat of a bias over time.

Comparison of Discharge Computation Methods and Discussion

Comparisons of the computation methods were made using the best MLR and ISO-SFQ in Tables 2–7. Recall that the relationships were fitted using the same paired-discharge measurements and water level observations collected concurrently in time. Box plots illustrating the percent difference between the computation record and the measurements are shown in Fig. 9. The plots show that, typically, the percent differences are smallest when using the SFQ-MLR method, followed by the SFQ-ISO method. Student’s

t

-tests and Wilcoxon signed-rank tests failed to reject the null hypothesis at the 5% significance level that the mean of the differences between computed and measured discharges for the calibration data were zero for all three discharge computation methods. Therefore, there is no statistically significant difference between the three methods.

Fig. 9. Box plot of percent difference between discharge record computation method and discharge measurements for calibration and validation data sets for two Detroit River ratings and one St. Clair River rating.

Comparisons of the methods were also completed using time series of discharges computed using each of the methods. The use of SFQ equations has been limited to computing discharges on daily and monthly time scales. The temporal limitation in computing discharges with SFQ equations is related to the underlying assumptions of the equations used to compute them. The fall term approximates the energy slope in the reach bounded by the stage gauge at the upstream and downstream limits of the reach. The difference in water-surface slope is a valid approximation of the energy slope only over sufficiently long intervals of time such that short-term disturbances in the water levels at either end of the reach due to seiches, winds, or other meteorological phenomena can be ignored. This temporal resolution is sufficient for the primary applications of the SFQ discharge data that are used for lake-level regulation and Great Lakes water balance calculations that are performed on daily and monthly time steps. However, it is known that discharges in the St. Clair and Detroit River vary on time scales much shorter than daily. Jackson (2016) demonstrated that oscillations of St. Clair River discharge in a repeatable 40-min period exist due to Lake Huron seiches. The period of oscillations in discharge on the Detroit River is longer, with the most dominant effects being about every 12 h, again caused by seiches on Lake Erie. These subdaily variations in discharge can only be resolved by sampling of discharge at higher frequencies than is possible with the SFQ method.

For each river and each of the SFQ methods, some gauge pairs performed better than others, based on statistical and residual analysis. The best SFQ-MLR and SFQ-ISO equations are shown in Table 8, which also presents the absolute average differences of computed daily and monthly flows compared to the index-velocity-based flows. Further discussion on comparisons of daily and monthly flows for the October 1, 2008–December 31, 2016 time period is described in the following section.

Table 8. Best SFQ-MLR and SFQ-ISO equations for each data set (one for St. Clair River and two for Detroit River) versus IVQ

Gauges	Dataset	Flow computation equations	Average \|Diff\| Daily IVQ (%)	Average \|Diff\| Monthly IVQ (%)
MLR_FG_SCSP	SCR	$Q = 229.64 {(SCSP - 167)}^{\land} 1.583 \times {(FG - SCSP)}^{\land} 0.545$	4.4	3.5
ISO_FG_SCSP	SCR	$Q = [(FG - 170.95) / 0.00099812] \times [0.7078 (FG - SCSP) + (0.5704)]$	7.8	3.5
MLR_WP_WY	DET1	$Q = 184.82 {(WP - 167)}^{\land} 1.594 \times {(WP - WY)}^{\land} 0.375$	4.5	1
ISO_FW_WY	DET1	$Q = [(FW - 172.51) / 0.00040649] \times [3.2772 (FW - WY) + (0.5407)]$	13.3	9.5
MLR_WP_WY	DET2	$Q = 52.12 {(WP - 164)}^{\land} 2.139 \times {(WP - WY)}^{\land} 0.416$	5.2	2
ISO_WP_AM	DET2	$Q = [(WP - 171.65) / 0.00059668] \times [0.9183 (WP - AM) + (0.5922)]$	5.9	1

Note: Average of absolute value of differences over 2008–2016 time period for daily and monthly records.

Comparison on a Daily Time Scale

Fig. 10 shows the daily discharges computed using the equations in Table 8, alongside the IVQ discharges, for the period from January 1 to December 31, 2016, for the St. Clair River. The corresponding plot for the Detroit River is shown in Fig. 11 using the best SFQ-MLR and SFQ-ISO methods and the IVQ discharges for the Detroit River data. Annual plots of daily discharge, comparable to Figs. 10 and 11, were created for each year from 2008 to 2016 but for brevity’s sake are not included. For the St. Clair River, the SFQ-ISO methods typically compute discharges that are lower than those computed using the IVQ method on a daily scale, whereas (at the same time scale) the SFQ-MLR method computes discharges that are closer in magnitude to those computed using the IVQ method. For the Detroit River, the SFQ-ISO methods and SFQ-MLR methods compute discharges that are close in magnitude to the IVQ discharges during some parts of the year (November and December 2016, for example), while at other times of the year they compute discharges that are very similar to each other in magnitude but that differ noticeably from the IVQ computed discharges (e.g., January–March 2016).

Fig. 10. Daily discharges computed using SFQ-MLR, SFQ-ISO, and IVQ methods, St. Clair River.

Fig. 11. Daily discharges computed using SFQ-MLR, SFQ-ISO, and IVQ methods for Detroit River.

The daily St. Clair River discharges computed from January 1, 2009, through December 31, 2016, using the best equations from the SFQ-MLR and SFQ-ISO methods differ from the corresponding IVQ discharges by 4.4% and 7.8%, respectively. The daily discharges computed from January 1, 2009, through December 31, 2016, for the Detroit River using the best equations of the SFQ-MLR and SFQ-ISO methods differ from the corresponding IVQ discharges by 4.5% and 13.3%, respectively, for the data set ending March 10, 2011, and 5.2% and 5.9%, respectively, for the data set starting on March 11, 2011. Figs. 10 and 11 show that the SFQ-MLR and SFQ-ISO methods agree when computing daily discharges for the St. Clair River; they do not agree as well when computing discharges for the Detroit River. In general, the SFQ-MLR method computes discharges that are more similar to the IVQ discharges than does the SFQ-ISO method; however, specifically and only for the St. Clair River, the two SFQ methods compute very similar daily discharges.

Comparison on a Monthly Time Scale

The daily discharges were averaged to compute monthly discharges from October 1, 2010, to December 31, 2016, as shown in Fig. 12 for the St. Clair River and Fig. 13 for the Detroit River (tagged in the legend as DET1 for the data set ending March 10, 2011, and DET2 for the remaining Detroit River data). The monthly discharges computed using the SFQ-MLR and SFQ-ISO methods both differ from the IVQ method by 3.5% for the St. Clair River. Fig. 12 shows that at a monthly time scale, the SFQ-MLR typically computes discharges that are similar in magnitude to the IVQ method for the St. Clair River except during ice conditions, which will be discussed in the next section. The SFQ-ISO method typically underestimates the discharges compared to the IVQ method. For the Detroit River, the SFQ-MLR differs from the IVQ method by approximately 1.0% and 2.0%, for the first and second data sets, respectively, and the SFQ-ISO computations differ by approximately 9.5% and 1.0% for the first and second Detroit River data sets, respectively. Fig. 13 shows Detroit River discharges on a monthly time scale. The SFQ-MLR typically computes discharges that are similar in magnitude to those computed using the IVQ method, whereas the SFQ-ISO methods seem to differ noticeably from the IVQ discharges, particularly for data prior to March 10, 2011. The Detroit River monthly discharge computations for March 11, 2011, onwards have noticeably less variation between the three discharge computation methods. Figs. 12 and 13 also show that the periods with the largest differences in the discharges are typically when there is ice cover, e.g., February 2010 and 2015.

Fig. 12. Monthly discharges computed using MLR, ISO, and IVQ methods on St. Clair River.

Fig. 13. Monthly discharges computed using MLR, ISO, and IVQ methods, on Detroit River.

Ice Considerations

The presence of ice and its impact on discharge is an important consideration for the St. Clair and Detroit Rivers. Ice forms most years on Lake Huron and enters the St. Clair River when there are northerly winds and flows downstream until it reaches or forms a jam at Lake St. Clair. As ice continues to flow into the jam, it extends upstream. The presence of a jam creates variable backwater downstream of the stage gauges, during which the stage tends to be higher than normal for the same open-water discharge. If the jam extends upstream of a gauge, the stage will be lower than normal for the same open-water discharge. Under these conditions, the relationship between discharge and stage and fall is no longer the same owing to the varying backwater effect caused by the ice. In the Detroit River, an ice bridge generally forms at the outlet of Lake St. Clair each year. The bridge remains stable in the lake and often extends downstream into the Detroit River. During periods of above-freezing temperatures, the ice bridge erodes back to Lake St. Clair and large sheets of ice break off and flow downstream. Lake Erie ice can also cause river ice jams at the lower end of the Detroit River, and then ice will build first in the lower river and later into the upper Detroit River. If there is a prolonged warm spell or break-up of Lake St. Clair ice, the entire Detroit River can fill with ice.

Past practice of the Coordinating Committee in computing discharges during ice conditions for the St. Clair River has been to select equations for reaches that are not ice covered when the typical reaches used for calculating discharges are covered or partially covered in ice. The location of the head of the jam must be determined using water level gauges and other information such as satellite imagery, ice cameras, and field observations. Stage readings of pairs of gauges are inspected to determine whether ice is affecting the water level measurements, and discharges are then calculated using equations for reaches that do not have ice cover. This process has inherent subjectivity because it is not always possible to precisely know the extent of the ice cover and determine which equations to use. Calculating discharges in the Detroit River during ice conditions is done using equations of reaches that are ice-free, but if no equations are available, discharges for the Detroit River are estimated using the St. Clair River discharges plus the net local supply from the Lake St. Clair Basin. The local supply computation considers runoff from the basin, precipitation to and evaporation from Lake St. Clair, and the change in storage of Lake St. Clair over the computation time period. More information on the calculation of the net local supply from Lake St. Clair or the St. Clair transfer factor can be found in Quinn (1976).

The IVQ method explicitly accounts for variable backwater effects caused by the formation of ice downstream in that the change in velocity (and discharge) can be sensed by the ADVM. Therefore, no additional intervention is required with the IVQ method except to verify that the meter is collecting data. The results in Figs. 10–13 of this paper compare SFQ-MLR and SFQ-ISO equations with index-velocity equations using the equations that best fit the river during ice-free conditions, which are the Fort Gratiot to St. Clair State Police reach in the St. Clair River and Windmill Point to Amherstburg reach in the Detroit River. Comparisons between shorter reach SFQ equations that are not affected by ice, e.g., Fort Gratiot to the mouth of the Black River in the St. Clair River, and index-velocity computed discharges would result in smaller differences during winter months with a strong ice impact.

Conclusions

Three river channel discharge computation methods suitable for variable backwater conditions, two SFQ methods and the IVQ method were compared using data from the Lake Huron-Lake Erie connecting channels. Although the two stage-fall-discharge methods have significantly different assumptions and use different data from the index-velocity method, the differences between the computed discharges derived from the methods are small, especially as the time step approaches monthly discharge values. The IVQ method offers an array of benefits compared to the SFQ methods. In particular, the IVQ discharges are based on real-time physical measurements of the observed, 8-min averaged velocities (obtained every 12 min) in the river, which can be used to determine discharges at the same resolution. In comparison, the stage-fall-discharge methods are based on a daily average water level, which may not accurately account for discharge variations throughout the day. The existence of frequent, significant, subdaily oscillations in the discharges of both the St. Clair and Detroit Rivers make the IVQ the preferred method for discharge computation. The IVQ method also inherently accounts for changes in backwater due to ice and weed conditions by continuously measuring the velocities that are affected by these conditions. The velocities are used to determine the discharge without the need to select gauge pairs that are not affected by ice, as was necessary with the stage-discharge methods.

For the St. Clair and Detroit Rivers, the results indicate that the SFQ-MLR method would typically be a more robust method for computing discharges compared to the SFQ-ISO method. Additionally, the SFQ-MLR method offers more gauge pairs with acceptable statistics in case of outage at one of the gauging stations. When the same data set is used to derive the SFQ-MLR and SFQ-ISO equations on daily and monthly time scales using the gauge pair with the best statistics for each method, both compute discharges that are comparable to the IVQ method for the St. Clair River. However, the same is not true of the Detroit River discharge computations. The daily SFQ-MLR and SFQ-ISO discharge computations for the Detroit River seem to diverge at various times over the October 2008–December 2016 period, with no discernable pattern.

Given the binational importance of the records of discharge for the St. Clair River and Detroit River, it is necessary to have redundancy in the methods used to compute discharges in case the IVQ instruments experience outages or communications failure, for example. The Coordinating Committee’s SFQ-MLR equations will be used during IVQ outages greater than a 24-h period for computing discharges in the St. Clair and Detroit Rivers. From this analysis, the discharges computed using the best SFQ-MLR equation for each river compare well to the SFQ-MLR method, which has been used by the Coordinating Committee since 2009. The SFQ-MLR as derived in this analysis would be a good back-up method to the IVQ method for computing discharges along the St. Clair River and Detroit River. Selecting the SFQ-MLR method over the SFQ-ISO method as the backup is justified because the SFQ-MLR method typically provides more statistically sound gauge pair options than the ISO method for the two rivers and, additionally, enables the Coordinating Committee to maintain some continuity with the previously employed methods used to compute past discharges allowing for long-term-trend and other analyses of discharge.

Data Availability Statement

The following data used during the study are available online:

•

USGS/Environment and Climate Change discharge measurements and discharge records for the St. Clair River: https://waterdata.usgs.gov/mi/nwis/uv/?site_no=04159130&agency_cd=USGS;

•

USGS/Environment and Climate Change Canada discharge measurements and discharge records for the Detroit River: https://waterdata.usgs.gov/nwis/uv/?site_no=04165710&agency_cd=USGS;

•

Water levels for the St. Clair and Detroit Rivers from the US NOAA for the gauges identified in Table 1: https://tidesandcurrents.noaa.gov/stations.html?type=Water+Levels; and

•

Water levels for the St. Clair and Detroit Rivers from the Department of Fisheries and Oceans Canada for the gauges identified in Table 1: http://www.isdm-gdsi.gc.ca/isdm-gdsi/twl-mne/inventory-inventaire/list-liste-eng.asp?user=isdm-gdsi&region=CA&tst=1.

Acknowledgments

The authors would like to thank the field crews of the USGS and Environment and Climate Change Canada, especially Don James and Jeanette Fooks, who installed the acoustic velocity meters and collected the discharge measurements used to derive the equations of discharge. The authors also thank Ms. Nanette Noorbakhsh of the USACE, Mr. David Fay of the International Joint Commission, and Dr. Frank Quinn (a NOAA retiree) who explained the previous work, history, and limitations of the stage-fall-discharge equations in the St. Clair and Detroit Rivers. The assistance of Justin Boldt from the USGS is also acknowledged for his statistical analysis. Finally, the authors thank Thomas Over from the USGS for his constructive comments on the paper.

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Information & Authors

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Published In

Journal of Hydrologic Engineering

Volume 25 • Issue 6 • June 2020

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.

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Received: Dec 20, 2018

Accepted: Oct 31, 2019

Published online: Mar 31, 2020

Published in print: Jun 1, 2020

Discussion open until: Aug 31, 2020

Authors

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Aaron F. Thompson https://orcid.org/0000-0001-7291-8476 [email protected]

P.Eng.

Manager in Engineering, Technical and Data Services, Environment and Climate Change Canada, 867 Lakeshore Rd., Burlington, ON, Canada L7S 1A1 (corresponding author). ORCID: https://orcid.org/0000-0001-7291-8476. Email: [email protected]

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Sandrina N. Rodrigues, Ph.D. [email protected]

P.Eng.

Water Resources Engineer, Environment and Climate Change Canada, 123 Main St., Winnipeg, MB, Canada R3C 1A3. Email: [email protected]

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Jeanette C. Fooks [email protected]

P.Eng.

Manager, Hydrometric Operations Ontario, Environment and Climate Change Canada, 867 Lakeshore Rd., Burlington, ON, Canada L7S 1A1. Email: [email protected]

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Kevin A. Oberg [email protected]

Scientist Emeritus, US Geological Survey, 405 N. Goodwin Ave., Urbana, IL 61801. Email: [email protected]

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Timothy J. Calappi, Ph.D. https://orcid.org/0000-0001-7087-0050 [email protected]

Adjunct Professor, Civil and Environmental Engineering, Wayne State Univ., 5050 Anthony Wayne Dr., Detroit, MI 48202; Hydraulic Engineer, Detroit District, US Army Corps of Engineers, 477 Michigan Ave., Detroit, MI 48226. ORCID: https://orcid.org/0000-0001-7087-0050. Email: [email protected]

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Comparing Discharge Computation Methods in Great Lakes Connecting Channels

Abstract

Introduction and Background

Methods for Computing Flows in Channels with Variable Backwater

Index-Velocity Method

Multiple Linear Regression Stage-Fall-Discharge Method

ISO Stage-Fall-Discharge Method

Application to St. Clair and Detroit Rivers

Application of Index-Velocity Method

Application of Multiple Linear Regression Stage-Fall-Discharge Method

Application of ISO Stage-Fall-Discharge Method

Comparison of Discharge Computation Methods and Discussion

Comparison on a Daily Time Scale

Comparison on a Monthly Time Scale

Ice Considerations

Conclusions

Data Availability Statement

Acknowledgments

References

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Abstract

Introduction and Background

Methods for Computing Flows in Channels with Variable Backwater

Index-Velocity Method

Multiple Linear Regression Stage-Fall-Discharge Method

ISO Stage-Fall-Discharge Method

Application to St. Clair and Detroit Rivers

Application of Index-Velocity Method

Application of Multiple Linear Regression Stage-Fall-Discharge Method

Application of ISO Stage-Fall-Discharge Method

Comparison of Discharge Computation Methods and Discussion

Comparison on a Daily Time Scale

Comparison on a Monthly Time Scale

Ice Considerations

Conclusions

Data Availability Statement

Acknowledgments

References

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