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
where
= discharge;
and
= upstream and downstream water levels, respectively;
base = index of elevation at bottom of river; and
,
, and
= constants. Fay and Kerslake (
2009) noted that using the downstream gauge (
) for the St. Clair River and the upstream gauge (
) 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
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
,
, and
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,
, 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
where
= resistance coefficient;
= mean velocity at cross section;
= cross-sectional area;
= hydraulic radius;
= slope;
= fall measured between two gauges;
= reach length between the two gauges;
subscripts = reference conditions; and
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
, where
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
where
= measured discharge; and
= (normalized) rated discharge.
Next,
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
. Here,
represents the water level,
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
and
base are obtained from the least-squares regression. Rearranging this line of best fit, the curve-generated estimate of discharge,
, 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
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
. Using the line of best fit on Curve 2, an estimate of the curve-generated ratio
can be computed for any measured fall:
, where K is a constant determined using least-squares regression from the line of best fit. The estimated discharge,
, is then the product of these two curve-generated values:
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) [
and
, respectively]
Further details regarding the theoretical basis and application of SFQ-ISO are found in ISO 9123 (
ISO 2017).
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.
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 (
) 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.
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.
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.
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 (
) 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.
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
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
values greater than 0.9, root-mean-square error (RMSE) values lower than
, and bias values (
) lower than
(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
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
, and the average bias estimates range from
to
. 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
values range from 0.041 (MLR_FW_BP) to 0.93 (MLR_WP_WY), with four of the eight gauge pairs having
values above 0.89. The bias estimates are less than
(MLR_FW_BP) and the RMSE values range from 18 (MLR_WP_WY) to 54 (MLR_WP_BP)
. 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
values greater than 0.80 for both ISO calibration curves, as shown in Table
5. These seven gauge pairs typically had higher
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
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
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
, 17.0, 9.1, and
, 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
values for Calibration Curve 1 but relatively high
values for Calibration Curve 2. With the exception of ISO_WP_WY, all the gauge pairs had
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.