The purpose of this report is to respond to the recommendations of the sixth and seventh TRC reports regarding uncertainty assessments for the velocity meters, discharge measurements, and computed annual total discharge at the Lemont streamgage. This report builds on prior velocity meter and discharge uncertainty analyses by the USGS for the LMDA project (Over and others, 2004; Duncker and others, 2006), which focused on a statistical approach, using properties of the index-velocity rating regression; this report continues that approach but also investigates incorporation of measurement uncertainty in stage, index velocity, and discharge. The analyses presented in this report consist of the following:

The different IVRs estimated in analysis step 3 are computed based on how the uncertainty estimates from steps 1 and 2 are used in the IVR regressions. For example, ordinary least squares (OLS) regression ignores such information, whereas weighted least squares (WLS) regression considers the uncertainties of the predictand (the y-axis variable; here, the velocity associated with the ADCP discharge). Additional types of regression considered here allow for consideration of uncertainties in the predictor (the x-axis variable; here, the index velocity) and for correlations among the uncertainties. The different IVRs arising from these additional types of regression can each be used for prediction; that is, to compute discharge using the continuous velocity data provided by the index-velocity meters and continuous cross-sectional area values computed from the continuous water-level data.

The results presented in this report are based on preliminary results presented by Over and others (2017). An updated version of the PowerPoint used in that presentation is attached to this report as appendix 2, with descriptions of the slides in appendix 1. Callouts to the slides by number have been inserted in the report text.

At the Lemont streamgage, the Chicago Sanitary and Ship Canal is an excavated rectangular channel about 162 feet (ft) wide, and the typical stage of 25 ft is 25–27 ft above a variable bottom elevation of 549–551.5 ft (above the North American Vertical Datum of 1988; Jackson and others, 2012; figs. 2A, B, and 3; appendix 2, slides 5 and 6). Just downstream (to the southwest) from the gage house is a sloughed bank, which reduces the cross-sectional area in that reach. A noncontact pressure transducer is used in conjunction with a gas-purge bubbler system to provide continuous stage (water-level) measurements (appendix 2, slide 7).

Two continuous velocity meters were present at the Lemont streamgage during this study: an ADVM and an AVM. The AVM was originally (2004) the primary velocity meter and the ADVM was the secondary or backup velocity meter; the order of priority was switched at the beginning of WY 2016. The AVM failed in April 2016 and has since been removed.

The ADVM is a 600-kilohertz Teledyne RD Instruments Channel Master, which has two beams that point 20 degrees upstream and downstream from the cross channel perpendicular, respectively. The ADVM is mounted on the northwest side of the channel next to the gage house, 13.2 ft above the gage datum, and during this study, it was set to return velocities in nine horizontal segments (cells) at 5 meters per cell, thus measuring the central 45 meters of the nominal 162-ft channel width. The ADVM velocity was computed from the mean of the values in the cells reporting valid data at a 1-minute time step (Jackson and others, 2012; fig. 4A and B; appendix 2, slide 8).

The AVM was an Accusonic O.R.E. 7510 with three paths, one each at 7.8, 13.2, and 18.4 ft above the gage datum, so its second path was at the same depth as the ADVM. Its transducers were placed so that the acoustic path between them had a length of 234.8 ft and was at a 44.5-degree angle relative to a perpendicular cross section. From each of the three AVM paths, 10-minute means of the velocities were averaged to obtain a composite AVM velocity for the cross section. When velocity data for all three paths were valid, a simple average was computed. If velocity data for one or two paths were missing or invalid, path coefficients were applied to the valid data before averaging, as described by Jackson and others (2012).

At the site, Qms are made using a Teldyne RD Instruments Rio Grande ADCP, mounted on a moving boat and tethered to a cross-channel tagline. From January 2014 to January 2018, an uplooking ADCP was at the site; its data were analyzed by Jackson (2018) but are not used in this study.

Four steps were used in the analysis presented in this report (appendix 2, slide 9). The first step is the estimation of measurement uncertainty at the continuous sensors in use at the Lemont streamgage, consisting of the water-level sensor and the two index-velocity meters, using the first-order second moment (FOSM) method using a type B approach: AVM, ADVM, and stage. The second step is the estimation of the uncertainties of the Qms by ADCP used for calibration and fitting of the IVRs. The third step is the computation of two sets of IVRs, one each for each of the index-velocity meters. The different IVRs are distinguished by how the uncertainty estimates from the first two steps are used in the IVR regressions. In the fourth step, the different IVRs arising from the different IVR regressions are used to compute discharge using the continuous velocity data provided by the index-velocity meters and continuous cross-sectional areas computed from the continuous water-level data.

A dataset of 155 Qms made at the Lemont streamgage between January 12, 2005, and October 23, 2013, were processed for this study using QRev (Mueller, 2016; appendix 2, slide 10). These Qms have discharges ranging from −344 to 16,794 ft³/s with a median of 2,605 ft³/s, have durations ranging from 162 to 3,563 seconds with a median of 831 seconds, and consist of 1 to 18 transects with a median of 4. Of these Qms, 140 have corresponding AVM velocities, 130 have corresponding ADVM velocities, and 115 have both AVM and ADVM velocities. The 115 Qms having both AVM and ADVM velocities (termed here the “common” Qms) allow comparison of IVRs developed using the two index-velocity meters with the same Qms. The dataset is available as a USGS data release (Prater and others, 2021). Select data for the aforementioned Qms also are available in the USGS National Water Information System database (U.S. Geological Survey, 2021).

Error propagation is commonly done in one of two ways. One is FOSM analysis, where the second moment (the variance) is approximated using first-order (Taylor series) expansion (appendix 2, slide 12). This method makes two substantial assumptions: it reduces the specified probability distributions to their variances, and it reduces the DRE to a first-order (linear) form. As such, it is exact for Gaussian errors and a linear DRE but may give erroneous results for nonlinear DREs or non-Gaussian errors (for example, errors characterized by a rectangular distribution). To check for inaccuracies resulting from the assumptions inherent in the FOSM method, the second error propagation method can be used. This method is a Monte Carlo analysis, in which simulated errors that follow the specified error distributions are directly propagated through the DRE. In this study, both methods (FOSM and Monte Carlo) were implemented using the QMSys software package developed by Qualisyst Ltd. (2015), under the assumption of Gaussian elemental uncertainties. Even for the AVM DRE, which contains substantial nonlinearity, the FOSM and Monte Carlo methods produced sufficiently similar results that only FOSM results are presented. It is important to note that the approximate agreement of the FOSM and Monte Carlo results extended to the full distributions, not just the variance; that is, the Monte Carlo error distributions were approximately Gaussian.

Uncertainty in the FOSM method is computed as the variance, that is, the second moment, of some quantity Y, which is a function of one or more variables: Y=g (X₁, X₂, …, X_n),(1)where Y

is the quantity being computed;

AVMs operate by measuring the time of travel of sound waves between two transducers of known separation distance and at a known angle to the flow velocity; these waves are advected by the water, so the difference in the time of travel in the direction of water flow and that of the return can be used to estimate the flow velocity (appendix 2, slide 13; Laenen, 1985). The resulting DRE (eq. 5) expresses the “line velocity” (that is, the mean water velocity along the line between the transducers) as a function of the travel times, the flow velocity-acoustic path angle, and the path length. The DRE is nonlinear in the times of travel and angle between the acoustic waves and the flow velocity. VL= B 2cos A 1 TDC − 1 TCD ,(5)where

To the basic DRE that describes the AVM line velocity returned by the instrument (appendix 2, slide 13), error sources related to the field conditions, deployment practices, and data processing were added (fig. 5; appendix 2, slide 14). The elemental uncertainties for these error sources in terms of their standard uncertainties (standard deviations), which fully characterize an assumed Gaussian distribution assuming a zero mean, were estimated (table 1; appendix 2, slide 15) and entered into the QMSys software along with the DRE, and error propagation computations were carried out. The absolute uncertainty for the error source OC₃, which refers to the time synchronization between the AVM and Qms used in rating curve development, is indicated as “not applicable” in table 1 (appendix 2, slide 15) and does not contribute to the AVM uncertainty computation because it does not contribute to the AVM velocity uncertainty itself. The error sources with zero absolute uncertainties in table 1 (appendix 2, slide 15) were deemed to be negligible.

Based on the proposed DRE and estimated elemental errors, AVM velocity errors at a range of velocities from −1.5 to 6 ft/s were determined. For example, for “low flow” conditions in the Chicago Sanitary and Ship Canal, which are defined here as discharge of 1,300 ft³/s, stage of 24.92 ft, velocity of 0.322 ft/s, and cross-sectional area of 4,037 square feet), the AVM error expressed as a standard deviation (the “standard uncertainty”) was estimated by using the FOSM method as 0.00824 ft/s or 2.56 percent (table 1; appendix 2, slide 15) and as 0.00822 ft/s and approximately Gaussian by using the Monte Carlo method (fig. 6B; appendix 2, slide 16, lower left). The largest contributor was acoustic path deflection (OM₂), which contributed almost 0.002 ft/s of the error, followed by changes in spatial flow distribution (SI₂), sampling frequency (OC₁), travel time (I₁), wind-induced shear (OM₁), and temporary changes in flow distribution (OM₃), which together contributed about 0.006 ft/s, or most of the remainder (table 1; appendix 2, slide 15; fig. 6A; appendix 2, slide 16, upper right). Because changes in spatial flow distribution, sampling frequency, wind-induced shear, and temporary changes in flow distribution are expressed in percentages of the line velocity (each 1 percent), these phenomena scale directly with it, and thus, go to zero when the line velocity is zero and grow linearly with line velocity as it increases. The largest contributor to uncertainty at low flow conditions, acoustic path deflection, is expressed as a fraction of the acoustic path length (B) and the angle between the acoustic path and flow velocity (A), which remain constant regardless of flow velocity, so it also remains constant.

The basic DRE for ADVM velocity (appendix 2, slide 17) indicates the principle of operation of an ADVM (Morlock and others, 2002). Vi= Fd 2Fs C ,(6)where

Some fraction of the sound waves (“pings”) emitted by the instrument that strike scatterers, which are assumed to be carried along passively in the water, returns along the acoustic paths (fig. 4B; appendix 2, slide 8, right) to the transducer. According to the Doppler effect, the frequency of the returning echoes is shifted in proportion to the component of the velocity of the scatterers along the acoustic path such that the ratio of the Doppler shift to the transmit frequency (F_d/F_s) is proportional to the ratio of the velocity along the acoustic path to the speed of sound (V_i/C). Because there are two beams in the horizontal plane at an angle to each other (fig. 4B; appendix 2, slide 8, right), the components of velocity in that plane are resolvable. For general information on site selection, installation, and configuration of ADVMs in index-velocity applications, see Levesque and Oberg (2012). As deployed at the Lemont streamgage, the resulting velocities are returned in nine 5-meter bins covering most of the distance across the channel, which are averaged to obtain a single mean velocity for each time step. The uncertainty estimation was developed in terms of the mean velocity as indicated in the expanded DRE (fig. 7; appendix 2, slide 17).

As with the AVM, ADVM velocity errors at a range of velocities from −1.5 to 6 ft/s were estimated based on the adopted DRE and estimated elemental errors (table 2; appendix 2, slide 18). The errors from sources with absolute uncertainties listed as zero in table 2 (appendix 2, slide 18) were deemed to be negligible. The elemental uncertainty for the error source OM₂ (acoustic path deflection) is indicated as “not determined” in table 2 (appendix 2, slide 18); therefore, it does not contribute to the ADVM uncertainty computation. The actual value of this uncertainty is not zero; rather, it is unknown because the experiments needed to determine its magnitude are beyond the scope of this study. In addition, the effect of errors in the temperature sensor on the ADVM (which is used to compute the speed of sound in the velocity computation) was not considered in this study.

The error sources denoted as SI₂ and OM₃ refer to changes in flow distribution (changes in spatial flow distribution and temporary changes in flow distribution, respectively). The first source of uncertainty (SI₂) refers to the effects of nonhomogeneous horizontal layers and a varying cross-sectional flow structure in the measurement volume over the range of measured velocities across seasons. The second source of uncertainty (OM₃) refers to temporary changes in flow distribution produced by barge passes or similar operations in the Chicago Sanitary and Ship Canal because of operations of the controlling structures.

At the same low flow conditions considered for the AVM (with a mean channel velocity of 0.322 ft/s) and for the estimated elemental uncertainties, the ADVM standard uncertainty was estimated as 0.0118 ft/s or 3.66 percent including consideration of salinity and 0.117 ft/s or 3.63 percent without consideration of salinity. Under these conditions, the largest contributors to the uncertainty were accuracy (I₁) and changes in spatial flow distribution (SI₂), which contributed about 0.0036 and 0.0035 ft/s, respectively, or about 60 percent of the total. Most of the remainder is estimated to come from resolution (I₂), factory settings (I₃), analytical methods (I₄), wind-induced shear (OM₁), and temporary changes in flow distribution (OM₃), which each contribute about 0.0009 ft/s and together contribute about 38 percent (table 2; appendix 2, slide 18). Of these sources, accuracy, resolution, factory settings, and analytical methods are estimated to have constant effects regardless of the flow velocity, whereas uncertainty arising from changes in spatial flow distribution, wind-induced shear, and temporary changes in flow distribution scale linearly with flow velocity and thus go to zero when the flow velocity does.

The results of applying FOSM analysis in a type B approach for index-velocity measurements at the Lemont streamgage indicate that AVM and ADVM standard uncertainties are generally similar and scale with flow velocity, V, approximately as 0.025V (fig. 8; appendix 2, slide 19). In addition, for both meters, the estimated standard uncertainties are mirrored around a flow velocity of zero. The ADVM uncertainties are somewhat higher, especially, in relative sense, near zero flow velocity, where the ADVM standard uncertainty is predicted as 0.0087 ft/s and the AVM standard uncertainty as 0.0031 ft/s. However, at a flow velocity of 6 ft/s, the ADVM and AVM standard uncertainties are predicted as 0.147 and 0.142 ft/s (2.45 and 2.37 percent), respectively. For comparison, Laenen (1985) indicated that single-path AVM systems may achieve accuracies of 3 percent and multipath systems may achieve accuracies of 1 percent. Duncker and others (2006) estimated the standard uncertainty of the AVM at the Romeoville streamgage as 0.0030 ft/s, but they did not consider operational error components. Le Coz and others (2008), testing the accuracy of a single horizontal ADVM, detected deviations from ADCP measurements within 5 percent within 60 meters of the ADVM.

The water-level sensor deployed at the Lemont streamgage is a gas-purge bubbler system, wherein nitrogen is bubbled through the orifice line (fig. 2A; appendix 2, slide 5, left) and out through a fixed orifice mounted at the bottom of the canal. The water pressure at the orifice is transmitted through the orifice and measured and converted to stage by a nonsubmersible pressure transducer in the gage house. General information on water-level measurement methods and associated error sources is available in Sauer and Turnipseed (2010). A part of the QMSys software input window showing the DRE used to estimate water-level uncertainty is shown in figure 9 (appendix 2, slide 20).

The factors for which the elemental errors were used to model the water-level uncertainty in this study consist of local disturbances (SI₂), accuracy of the meter (I₁), resolution (I₂), gage offset (OC₁), and periodic stage correction (OC₄); the other factors considered were deemed to be negligible and were set to zero (table 3; appendix 2, slide 21). Of the factors used in the uncertainty modeling, local disturbances had the largest effect, contributing about 54 percent of the total estimated standard uncertainty of 0.0271 ft. Resolution, gage offset, and periodic stage correction each contributed about 13.6 percent for a total contribution of about 41 percent. Only accuracy scales with the water level, but with an absolute uncertainty of 0.02 percent of stage that varies over a fairly small range, it does not cause the water-level uncertainty to change with stage, resulting in a constant standard uncertainty of 0.0271 ft for stage from 22 to 29 ft (the range of stage observed in this study). This uncertainty may be compared to the USGS policy regarding maximum stage uncertainty of at most 0.01 ft or 0.20 percent of stage, whichever is larger (U.S. Geological Survey, 1996), which was issued in the context of the use of stage in stage-discharge ratings. At a typical stage of 25 ft, a value of 0.20 percent translates to 0.05 ft; thus, a standard uncertainty of 0.0271 ft is deemed to satisfy the policy.

According to standard USGS practice (Levesque and Oberg, 2012), IVRs are developed by fitting a line (or, more generally, a curve) to the measured discharges converted to velocity by dividing by the stage-area rating corresponding to the index velocity (eq. 7) as a function of the index velocity and possibly other continuously measured quantities such as stage by OLS regression (appendix 2, slide 26). V_Qm=Qm/Area_index,(7)where V_Qm

is velocity from discharge measurement,

At the Lemont streamgage, the two index-velocity meters each have their own stage-area rating (appendix 2, slide 27); the ratings differ mainly because the AVM paths cross the part of the channel where the north bank has sloughed (fig. 3; appendix 2, slide 6). Because the Qm cross section is close to the ADVM measurement cross section (labeled as “Centerline of ADVM” on fig. 3 and appendix 2, slide 6), cross-sectional areas associated with the Qms and the ADVM velocities are similar, but as described, that condition is not required in the standard USGS practice (Levesque and Oberg, 2012).

For OLS regression to be the correct model for all purposes, several assumptions are required (table 5; appendix 2, slide 28; Helsel and others, 2020, p. 228). The two most fundamental are that the response variable (y) is indeed linearly related to the predictor variable (x) and that the data used for fitting are representative of the data of interest. The first was not formally tested in this study, but it is evident from the results that if deviations from linearity are present, they are small. Regarding the second, the distributions of the index-velocity values used for fitting the IVRs and those values used for prediction later in this report compare favorably except that the IVR fitting data are missing observations in the range of 1–3 ft/s (table 6 and fig. 11A, B; appendix 2, slide 29); further, the fitting data have somewhat narrower ranges, but the central tendencies measured by the medians are close. The next two assumptions, that the residuals are homoscedastic (that is, their variance of the residuals is constant) and that the residuals are independent of the x-axis variable, are required, along with the first two assumptions, to obtain a best (minimum variance) linear unbiased estimate of y given x, and homoscedasticity is required to get the correct variance (uncertainty estimate) of the predicted y value. However, the analysis of the index velocity and ADCP measurement uncertainties indicates that neither of these assumptions is true: the residuals depend on the x-axis variable through the uncertainty of the index velocities, and they vary among the ADCP velocities as a result of its variable measurement uncertainty. The final assumption, that the residuals are normally distributed (or Gaussian), is required for the validity of tests of statistical hypotheses related to the regressions. The distribution of the residuals and its implications for statistical testing are addressed in terms of the chi-squared (Χ²) test (appendix 2, slide 32).

Various generalized forms of linear regression have been developed to address situations where the assumptions of OLS regression are violated. This report focuses on three such generalizations that address the homoscedasticity and x-axis variable dependence of the residuals (table 7; appendix 2, slide 30): (1) WLS, in which the residuals are weighted inversely to the variance of the y values; (2) generalized least squares (GLS) regression (also called Gauss-Markov regression), in which the properties of the residuals are described by a covariance matrix with off-diagonal terms, which allows for correlations among the residuals; and (3) errors-in-variables (EIV) regression (also called generalized distance regression), in which variable uncertainties of the variables on both axes are considered. Regression models that include correlations among the errors on both the x and y axes (called generalized Gauss-Markov regression) also were considered, but results are not presented in this report because it was deemed unnecessarily complex for this application. For the alternative regressions, the errors were specified as predicted using the FOSM method, as described earlier. The computations were done using the R function lm (R Core Team, 2019) for OLS regression and using a translation into R of software developed by the National Physical Laboratory (2010) to implement the Technical Standard ISO/TS 28037 (International Organization for Standardization, 2010) (appendix 2, slide 31). Previously, González-Castro and Mohamed (2007), Zhang and others (2012), and Kim and others (2014) applied EIV, generalized Gauss-Markov regression, and EIV, respectively, to the problem of the fitting of stage-discharge rating curves, and they all used specified errors.

For regression models where the uncertainties are specified, the fit can be tested by applying a Χ² test (appendix 2, slide 32). This test is applied to the statistic SSR_norm, which is the sum of squares of the residuals normalized by their uncertainties (expressed as variances). Under the hypothesis that a linear model is correct and the errors are Gaussian with the specified variances, SSR_norm will be distributed as a Χ² random variable with n−p degrees of freedom (Press and others, 1992, p. 653–655), where n is the number of data points used and p is the number of parameters fitted. When SSR_norm is too large, then at least one of the assumptions (linearity or Gaussianity with specified variances) is wrong.

Because there are known to be errors in the x-axis variable, the true model (that is, the true relation between the index velocity measured without error and the true cross-sectional velocity) has a steeper slope and smaller intercept than that obtained by OLS regression. However, because the magnitudes of the errors in the x-axis variable are unknown, it is not known whether the alternative regressions presented in this report (tables 8–11; appendix 2, slides 37–40) are closer to the true model than is the OLS regression. The question of whether or not the true model needs to be estimated is a subtle one. Sometimes an estimate of the true effect of some predictor is desired for its own sake. In the present context, that of prediction, it is often the case, as is done here, that the prediction is done using uncertain measurements from the same population as was used for the regression fit, in which case, the true model is needed only if the fitting data and the prediction data are distributed differently (Fuller, 1987, section 1.6.3; Buonaccorsi, 2010, section 4.8). Here, as described previously and as listed in table 6 and shown in figure 11A, B (appendix 2, slide 29), the fitting and prediction data are broadly similar in distribution, although proportionally fewer high-velocity measurements are in the fitting data than in the prediction data. The other case where the true model may be needed for prediction is to determine the effect of changes in the predictor as opposed to its measured value, which is relevant if the predictor can be controlled.

As a first step in the exploration of the IVRs for the ADVM and AVM at the Lemont streamgage, IVRs were computed using OLS regression using the 115 measurements common to both index-velocity meters. These fits (fig. 12A, B; appendix 2, slide 33) look quite good overall, having intercepts near zero and slopes near one, and they provide no reason to doubt the linearity of the ADCP velocity–index-velocity relations nor to indicate another variable is needed to explain the data; however, they raise at least two issues regarding the residuals. One is that although the type B analysis indicated that the ADVM uncertainties are somewhat larger than those from the AVM, regressions with the same set of ADCP measurements indicate that the standard errors (SEs; that is, the standard deviation of the residuals) of the AVM residuals are about twice those of the ADVM residuals (about 0.104 ft/s compared to about 0.056 ft/s when considering all the residuals [fig. 12A, B; appendix 2, slide 33] or about 0.059 ft/s compared to about 0.033 ft/s when considering just the middle 50 percent of the values [fig. 13A, B; appendix 2, slide 34]). The second is that the residuals have non-Gaussian tails, as seen from the S-shaped fits to a theoretical Gaussian distribution for which the standard deviation is computed from the middle 50 percent of the values (fig. 13A, B; appendix 2, slide 34). Although evidence of heteroscedasticity (nonconstant variance) in the residuals is strong, according to the fitted lowess line (Helsel and others, 2020, section 10.3.2), their variance increases with velocity only modestly (fig. 14A, B; appendix 2, slide 35). Whether or not this increase is as fast as would be predicted by the combination of the predicted index-velocity and ADCP errors (figs. 8 and 10 [appendix 2, slides 19 and 24], respectively) has not been determined. The similar question of whether or not the overall magnitude of the predicted errors agrees with that of the empirical residuals is addressed by applying alternative regressions, as described later in this report. Approaches to inference regarding the structure of errors when they appear only on the y axis are given for example by Faraway (2005, sections 6.1 and 6.2) for the WLS and GLS models; apparently, the inference problem is much more complicated and subtle when there are errors in the variables on both axes.

It is, however, possible to gain some qualitative insight from the regression analysis regarding which uncertainties, those of the ADCP or those of the index-velocity meters, are larger by considering the correlation between the ADVM and the AVM residuals (fig. 15; appendix 2, slide 36). The Pearson and Spearman correlations (Helsel and others, 2020, sections 8.2 and 8.3) are positive but only of moderate magnitude, indicating that the ADCP errors are roughly of about the same magnitude as the index-velocity errors. If the residuals had been strongly correlated, that fact would have indicated that the ADCP errors, which the ADVM and AVM regressions have in common, made up most of the regression residuals, whereas if the residuals had been uncorrelated, that fact would have indicated that the index-velocity errors made up most of the residuals (assuming that the ADVM and AVM errors are independent).

The regression methods described in the “Alternative Types of Linear Regression” section were applied to the full ADVM and AVM datasets (as distinct from the common measurements used in figs. 12–15 [appendix 2, slides 33–36]). For those methods that allow for specification of errors, two sets of errors were considered: (1) the original predicted errors (figs. 8 and 10; appendix 2, slides 19 and 24) and (2) constant multiples of them (a multiple of 2 in the case of ADVM errors and 3 in the case of errors from the AVM). For the GLS method, which allows for correlations among y-axis (ADCP) errors, two assumptions were considered regarding those correlations: (1) that ADCP measurements made in immediate succession have correlations of 0.5 (indicated as “sel.corr.=0.5” in tables 8 and 9 [appendix 2, slides 37 and 38]) and (2) that all ADCP measurements are correlated at a level of 0.1 (indicated as “corr.=0.1” in tables 8 and 9 [appendix 2, slides 37 and 38]). These correlations among the ADCP errors were selected somewhat arbitrarily as a means of testing the sensitivity of the results to the presence of such correlations.

Relative to the standard method of OLS regression, all the alternative regression methods, whether they specified errors on only the y axis or both axes, except one version of GLS for the AVM data, resulted in steeper slopes and lower intercepts; thus, a counter-clockwise rotation of the regression line. That the slope of an OLS linear regression line is attenuated (biased toward zero) in the presence of errors on the x axis is well known (Fuller, 1987, chap. 1); thus, when these errors are accounted for, as indicated for the EIV regression, the line steepens and the intercept moves in the opposite direction because of its negative correlation with the slope. Based on these results and some exploratory simulations (not shown), accounting for variability in uncertainty on only the y axis using WLS regression generally has the same qualitative effect.

Other notable differences among the different regression methods include the following.

The predictions of OLS regression—for the values used to fit the regression—have a mean bias of zero and the smallest SE of prediction, but they have a reduced variance (tables 10 and 11; appendix 2, slides 39 and 40). This variance reduction in prediction with OLS regression is well known (Hirsch, 1982; Helsel and others, 2020, p. 228); its magnitude is 1−R², where R² is the coefficient of determination of the regression. Because the R² values of these OLS regressions are near 1 (0.996 and 0.988 for the ADVM and AVM regressions, respectively [tables 10 and 11, respectively]), the variance reduction effect is small. Nevertheless, one effect of the steeper regression lines of all but one of the other regressions is to increase the variance of the predictions; in these results, presumably because the specified errors do not match the true errors, the variance is usually overcorrected. In addition, the predictions of the alternative regressions have a nonzero mean bias, although it is less than 0.01 ft/s in absolute value for the ADVM regressions and less than 0.02 ft/s in absolute value for the AVM regressions, other than the case of GLS regression with hypothesized correlations among all the ADCP errors.