1.1.

Background

Observations of river dynamics from remote (satellites, aerial, and bankside) platforms involve the direct measurement of water-surface area, which when divided by reach length yields reach-averaged width, as well as the WSE (or stage) along the reach (relative to a known datum). These measurements are made either at a prescribed section (in the case of bankside or aerial mounted instruments, or satellite nadir tracks) or as a reach average (in the case of satellite or aerial mounted swath instruments). Reach measurements of water-surface height can also yield water-surface slope. Observations of width and height are currently being measured routinely from various imaging and nadir pointing altimetric satellites, although not concurrently. The NASA/CNES/CSA/UKSA SWOT wide-swath altimeter mission launched December 2022,⁴ will concurrently measure both water-surface width, height, and slope. Fujita et al.,²⁰ Kääb et al.,²¹ and Legleiter and Kinzel²² have demonstrated that surface velocity in some rivers can be measured from satellite platforms by tracking ice floes and turbulent irregularities in the flow, however, widespread routine measurements of velocity from satellites are not expected in the near future.

The tracking of river dynamics and discharge from remote platforms rely on width and stage observations, coupled with limited in-situ (ground-based) observations to develop time-series data with intervals between observations on the scale of days to weeks.^23–32 More recently, Refs. 9, 11, and 12 have developed discharge algorithms designed to apply observations of width, stage, and slope observations from the SWOT mission. Limited field observations provide ground truth information for calibration and accuracy assessment of remotely sensed estimates of discharge (RSQ). Alternatively, modeled river flows also may serve as points of calibration (for example from global hydrologic models as reported in the global reach-scale a-priori discharge estimates for SWOT (GRADES) website.³³

The selection of the river reach is an important consideration in the accuracy and uncertainty of remote sensing-based discharge estimates because of the role the geometry of the channel reach plays in the energy slope and in the flow resistance. The measurement data used in this study were all collected at individual cross-sections within a reach that is not defined. As such, the reach characteristics are not included as part of the evaluation. In many remote sensing-based applications to rivers, the river reach is also not considered because it is defined by the orbital crossings rather than by physical attributes of the reach. In any case, Frasson et al.¹⁶ have shown that over longer reaches (5 to 15 km) errors in satellite observed width, height, and slope are reduced, indicating that over these reach lengths, estimate accuracy is likely to be better than over shorter reaches. Additionally, channel morphology characteristics that reflect river hydraulic conditions^24,34 can better be defined over longer reaches.

The goal of the RSQ methodology presented in this paper is to estimate discharge using an approach that incorporates the use of in-situ information as much as practicable. Even small amounts of in-situ information can provide the means to constrain unknown parameters to minimize the degrees of freedom available for calibration. RSQ estimates presented herein are developed for in-bank flow conditions and strive to represent the mean hydraulic conditions in the channel (mean depth and velocity). Overbank flood flow accuracy is not addressed.

Inherent limitations in flow-law-based discharge estimation methods include the use of a continuous representation of flow resistance within a flow resistance equation such as the Manning equation. These methods rely on the assumption that flow resistance is a linear function of observed variables and that the flow is contained within a defined channel. Additionally, the remote discharge estimates are limited to in-bank flow, periods of no ice cover and obstructing clouds, and to river widths $>50\mathrm{m}$ if using satellite imagery. The observational record is also constrained by the revisit period and the distance between the ground paths traced out by the satellite orbit. These latter two limitations in most cases will limit repeat observations to several days and several kilometers (km) at best. These constraints require that the discharge estimates are independent of one another, and therefore do not assume continuity between observations in time or space and do not rely on transient or routing-based flow computations.

The remote sensing capabilities include observations of the river width and the river height. Validation studies for both observed variables have been conducted at many locations using different instruments mounted on various satellite platforms^{25–27,31,35–42} and these have provided first-order expected observations accuracies. Along with these studies, the spatial ($x$, $y$, and $z$ coordinates) accuracies are generally reported based on the specifications of the observing instruments, including discussion of factors that may reduce accuracy. Recently, observations of water-surface height in rivers (and other water bodies) derived from the multi-agency Jason-2 and Jason-3 satellite altimeter missions provide an estimate of observational error. These estimates are based on qualitative characteristics of the target as well as atmospheric and instrument conditions and responses. As such, they are not precise errors but rather are a first order of an expected error that is quantitative, based on characteristics of the observation.⁴³

1.2.

Ground Truth Data

The ground truth information used to evaluate the remote sensing estimates includes direct measurements of discharge, water-surface height, water velocity, and water depth in river cross-sections, and the rating curve produced by relating the measured water-surface height to the discharge. These direct measurements are considered to be the best available truth set with the following qualifiers and caveats.

Measurement of water-surface height at USGS streamgages adheres to a rigorous accuracy standard (0.3 cm).⁴⁴ When ice is covering the water-surface, stage may be affected and ice corrections applied. Water velocity and depth in the cross-section is measured using acoustic doppler current profiler (ADCP) technology in most cases. The accuracy of these measurements depends on the operation checks on the equipment, flow conditions during the measurement, and operator skill. The depth and velocity measurements are made across the channel in a more or less continuous flow field, with width measured perpendicular to the direction of flow. The measurement accounts for lateral and vertical flow to derive a mean flow in the downstream direction.

The discharge is computed from the velocity-depth flow field as the product of mean depth, velocity, and width. The USGS rates the discharge measured as poor, fair, good, or excellent based on the quality of the measurement conditions and equipment operation. In general, it is assumed that a poor measurement has an accuracy that is $>8\%$ of the “actual” discharge, a fair measurement within 8%, a good measurement within 5%, and an excellent measurement within 2%.⁴⁵ However, inherent errors in the measurements are not considered in this study as part of the uncertainty in the remote sensing estimates of discharge.

2.1.

River Discharge Algorithm Development and Uncertainty

The algorithm specifically evaluated here was developed by the USGS to estimate discharge remotely and is based on the Manning equation. ⁴⁸ There are similar algorithms that have been developed that do not require calibration but rather enforce mass conservation based on the Manning equation and on hydraulic geometry principles.^49,50 The Manning equation is an empirical adaptation of the Chezy equation, which describes a state of uniform flow that is neither accelerating nor decelerating but is maintained at a steady rate (mean velocity remains constant in the reach).

The Chezy equation is given as

Refer to the Appendix for the conceptual development of these equations.

Both the Manning and the Chezy equations provide estimates of the mean velocity in the channel ($V$), and when multiplied by the cross-sectional area of the channel (top width, $W$, times the mean depth, $Y$) yield the discharge, $Q$. The hydraulic radius, $R$, is approximately equal to $Y$ within $+/-10\%$ or less for rivers with width to depth ratios $>18$ for rectangular channels⁵¹ and for parabolic channels this can be demonstrated for channels with width to depth ratios $>10$. Many rivers⁵² have width to depth ratios $>20$, which for parabolic shapes means the difference between the hydraulic radius and the mean depth is $<5\%$. Similarly, Bjerklie et al.²⁶ have shown that from a practical perspective, the water-surface slope yields similar statistics when used in the Manning equation compared to the energy slope for a wide range of rivers. As such, for application purposes, the water-surface slope is equivalent to the energy slope (S) given that the water-surface slope can be directly measured.

The Chezy equation is derived from the principle that the source of flow resistance via shear stress on the channel boundary (the bed and banks) is viscous friction. However, in practice, flow resistance in rivers includes a number of interactions that expend energy and contribute to flow resistance. Additionally, energy loss is associated with irregular channel geometry and flow conditions, bedload transport, turbulent lateral and vertical motion in the flow field, form drag around large obstacles (e.g., boulders and fallen trees on the channel bottom), and shallow water wave resistance.^53,54 The modification of the Chezy equation by Manning, in effect, describes some of the energy loss as a function of the depth (hence the exponent 0.67 rather than 0.5). Bjerklie²⁴ also has shown that some portion of the flow resistance is a function of slope and can account for some flow variance in rivers assuming a slope exponent of 0.33 rather than 0.5. The result of the modification by Manning to include a depth exponent of 0.67 renders the Manning coefficient and thus the Manning equation itself a semi-empirical flow law. Thus, the Manning flow resistance coefficient becomes an empirical loss coefficient that accounts for any and all forms of energy loss in the flow, with the caveat that some of that energy loss is already accounted for by the modification of the depth exponent.

The Chezy equation may be considered to be based on physical principles provided it is accepted that the shear stress is a function of velocity squared per Eq. (1). Fundamental assumptions of the equation – or any flow resistance equation – are that the slope and resistance are primarily a function of the channel topography and boundary. The Manning equation indicates that the Chezy resistance coefficient is not constant in a river reach and is not purely a function of the boundary shear stress but is dependent on the depth to the one-sixth power, resulting in the exponent on the hydraulic radius being raised to the two-third power rather than a half. This improvement in flow prediction at varying depths using Manning as opposed to Chezy is demonstrated by Bjerklie et al.²⁵ As such, we can state that using the Chezy equation to compute flow at any given flow level can be assumed to be physically based and free of model error provided the width and mean depth of the channel are accurately known, and the mean velocity is computed accurately via a known Chezy resistance coefficient. However, the Manning equation has an implicit assumption that flow resistance is related to the hydraulic radius, which imparts non-dimensional homogeneity and therefore possible error due to the assumption.

Given the wider application of the Manning equation, we can assume that it also can be considered free of error for a given flow level provided the width, mean depth, and manning resistance coefficient are accurately known. However, due to the application of Manning at varying flow levels, even given the improvement rendered in its more general application due to relating the resistance to change in depth to the one-sixth power, we cannot argue that the Manning is a physically based formula for all flow levels due to the uncertainty and empirical nature of defining the flow resistance (the Manning $n$).

Remote observations provide measurements of the channel top width, water-surface height (and thus the change in depth), and the water-surface slope. As a consequence, the water-surface slope becomes the proxy for the friction slope that is the basis for the Manning and Chezy equation formulations, and some error is associated with this assumption. Bjerklie et al.²⁶ have shown that using an average slope over the range of flows in a wide range of rivers causes minimal change in discharge estimates compared to using a dynamic slope. Based on this, we might assume that using the water-surface slope rather than the friction slope would also have minimal impact under most natural conditions.

While remote observations provide river water-surface heights, without a way to measure the channel bottom height to translate the water-surface height to a mean channel depth, an assumption must be made regarding the shape of the channel cross-section. Bjerklie et al.⁵⁵ have shown that for thousands of rivers over a large range in discharge, the ratio of the maximum to the mean depth in a river is close to 1.5, which is the ratio that would result if the typical cross-section were parabolic. An assumed parabolic cross-section shape allows for a linear relationship between the width squared and the height observations to be extrapolated to a bottom height or elevation of zero flow (represented by the $y$-intercept of the height as a function of the width). Similarly, the assumption of a parabolic shape allows for the height change to be converted to a mean depth by dividing the height change by 1.5. The error associated with the assumption of a max-to-mean depth ratio of 1.5 can be inferred from the statistics elaborated by Bjerklie et al.⁵⁵

As previously mentioned, given the nature of flow in natural channels, the flow resistance coefficient is also associated with backwater effects due to the contraction and expansion of flow and changes in cross-section,^56–59 submerged obstructions, channel curvature,^{24,51,60–64} sediment load,⁵⁸ and changes in boundary roughness along the reach and with changing depth. Factors affecting changes in flow resistance with depth include sediment sorting, occurrence of bedrock, presence of sand and cobble bars, presence of vegetation, etc.⁵⁸

Because of the complexity of natural rivers, a constant value for the Manning flow resistance ($n$) under all flow conditions in a river reach would be an unrealistic assumption.^25,57,58,65 Numerous studies have shown that the Manning $n$ in most rivers varies with depth at a point and spatially within the reach. In addition, the Manning $n$ can vary significantly between rivers and in a downstream direction. Several studies have shown that the bankfull Manning $n$ is related to the channel slope and that the Manning $n$ can be modeled as a logarithmic or power function of the relative depth (diminishing with increasing depth^15,51,58,66). Given the empirical and highly variable nature of Manning $n$ in any given river, the independent assessment of accuracy in its estimation for all flow levels cannot be realistically accomplished. However, the bankfull or high flow estimate of its value may be compared to literature values and the many flood studies conducted by the USGS and others.

The model framework (algorithm) we present for computing discharge from remote sensing observations is based on a modified form of the Manning equation with a calibratable Manning $n$ function continuous over the range of flows (i.e., no discontinuities in channel configuration of roughness elements) coupled with the assumption of a regular cross-section geometry as indicated by Bjerklie et al.⁵⁵ The application of this equation is based on the following general assumptions—the slope and resistance are primarily a function of the channel topography and boundary, and flow resistance can be modeled as a continuous function of depth. It is also assumed that the stage has a monotonic relation to both width and discharge. The application of the parabolic assumption to the river cross-section also has an unknown independent impact on the discharge estimate but is assumed to be minor (see Sec. 2.2.1 following). The Manning equation model used for estimating discharge is parsed to be able to uniquely estimate the Manning roughness coefficient:¹⁵

The value of the Manning $n$ is calibrated based on a conceptual relative depth relation that is represented by an assumed power law relation.^51,58,66 The relative depth is formulated between the bankfull depth and the flow depth at the time of the observation multiplied by the baseline Manning $n$ at bankfull depth, assuming that the Manning $n$ at bankfull is at its lowest value.^57,64,65,67 This relation is given as¹⁵

The Manning equation can be calibrated to any continuous monotonic distribution of discharge values in a river provided the Manning $n$ can vary to match the discharge given the physical structure of the equation is fixed—meaning the depth and slope exponents are constant. However, given that the Manning $n$ will not (and cannot) be calibrated to each individual discharge, a continuous Manning $n$ function, as shown for Eq. (4) above, must be used. The form of the Manning $n$ function will constrain how well the Manning $n$ will match each discharge estimate and will not be able to account for variation beyond the limits of the function.

2.1.1.

Testing the discharge algorithm

Information from a set of flow measurements was assembled to test the application of the discharge algorithm derived in the previous section. A sample of USGS flow measurement data was downloaded from USGS National Water Information System (NWIS) database⁶⁸ from gaging stations on 30 rivers in the United States (Fig. 1). The rivers were selected to provide a range of geographic locales and sizes of rivers, with all widths greater than 30 m (i.e., a minimum size putatively observable by spaceborne instruments). The measurement data were supplemented with water-surface slope and other geomorphologic information and assumed to represent ground truth. The data set was composed of 37 to 160 USGS discharge measurements from each of 30 gaged rivers, chosen to be geomorphologically and geographically diverse across the United States, with eight rivers located in Alaska (Table 1). The measurements include the height of the water-surface stage for a specific location, the measured water-surface width, mean depth, mean velocity, and the discharge computed from width times depth times velocity. The width, depth, and velocity data were measured at cross-section locations that vary in the gaged reaches, and as such do not necessarily represent the reach averaged quantities that will be observed remotely. To address this, a quality control process was implemented that includes a rule that the width and stage must correlate positively and vary monotonically.

Fig. 1

Map showing location of 30 USGS streamgages used to develop the 30-river discharge data set.

For the purposes of this study, the maximum measured stage and depth are considered representative of the bankfull state. Information on river channel physical characteristics was also collected for each stream gage including the slope, meander length, sinuosity of the reach containing the gage, Landsat measured width, and modeled mean and 2-year discharge.⁶⁹ The characteristic data were obtained from Frasson et al.^18,19 or were independently measured by the USGS or obtained from available Geographic Information System (GIS) sources. These data and sources are reported in LeNoir et al.⁶⁹ The data were selected without consideration of the distribution of flow duration such that moderate to low flows are likely over-represented relative to high flows in frequency of measurement.

2.1.2.

Quality control of the discharge data

Quality control requirements were applied to the flow measurements to ensure the representativeness of the data. The primary screening criteria required that the reported discharge within 5% of the discharge as calculated by multiplying the reported width, depth, and velocity. Additional quality control requirements are itemized below; data that failed any single screening criterion were censored from the record:

• 1. The width-stage relation reasonably represents a relation where a unique width is associated with a unique stage (monotonic) and the width and stage both increase in the same direction (positive correlation) at different rates, often as a power law. This requirement considers some normal random scatter around a general monotonic trend that can be smoothed by fitting a line to the relation.

• 2. The relation between stage and width reasonably represents a consistent measurement location that can be assumed to be representative of a reach average. Data that were clearly collected at a different location in the river as evidenced by a defined stage-width relation that exhibited obvious discontinuity on either the stage, width, or both axes of a width versus stage graphical plot (Fig. 2) were removed.

• 3. Data points that represent an overbank flow condition as evidenced by a clear discontinuity or distinct inflection point in the graphical plot of width versus stage (Fig. 2), or are above-referenced flood stage was removed. No additional verification of overbank was conducted under the assumption that no clear discontinuity would necessarily imply that the flow remains in a contiguous and confined channel rather than spreading out beyond the channel confines.

• 4) Individual measures of stage, width, or both that were not screened by criteria 2 or 3 above that nevertheless plotted well above or below (by $>\sim 20\%$) the general relation between width and stage and violated the monotonic and/or positive correlation $c$ requirement (Fig. 2) were removed.

Fig. 2

Example of stage-width relation for the Susquehanna River near Waverley NY (station 01515000) with anomalous value outside of general trend (red point) that was removed from the data set, and values that represent overbank flow and/or measurement at a different cross-section (yellow points) were also removed from the data set.

The quality control measures described above resulted in the removal of 13% of the measurements. The quality-controlled data set is publicly available through a USGS data release at Ref. 62.

2.1.3.

Algorithm calibration procedure

The algorithm calibration proceeded by deriving a linear relation between width and stage assuming that the width varies with depth by a power function (e.g., parabolically—where the width squared varies linearly with stage). Using this relation, width was computed as a function of stage for each river. This step assures a uniform, monotonic linear increase in width with an increase in stage and assumes the calibrated width is representative of the average channel width for the reach. Figure 3 graphically displays the calibration process.

Fig. 3

Flow chart illustrating the procedure for calibrating the algorithm parameters – USGS National Water Information System (NWIS).

The maximum width in the derived time series and the maximum stage in the time series were assumed to represent bankfull channel dimensions. Bankfull values served as the reference point for determining relative depth in the channel. The bankfull depth in the channel for any given site was calculated from the following parabolic relation (bankfull stage – B)/1.5 where B (elevation of zero flow) is the no flow or bottom boundary of the assumed cross-section. The value 1.5 is the ratio of the maximum to mean depth in a cross-section that has a parabolic shape, with stage representing the change in depth at the center (deepest) part of the parabolic cross-section. The value of B was determined by matching the mean of the observed depths in the calibration data with the mean of the estimated depths determined from the observed stage minus B divided by 1.5 (see Fig. 3).

Calibration to all observed discharges proceeded by matching the mean and standard deviation of algorithm-estimated discharges to USGS observed discharges by adjusting the bankfull reference Manning $n$ (fitted parameter $n_b$) and the exponent $x$ of the Manning $n$ transform function that relates the $n_b$ to the relative depth as in Eq. (4). The parameter B was held constant.

An alternative calibration minimizes the normalized root mean square error (NRMSE; normalized to the mean flow) and maximizes the Kling–Gupta efficiency (KGE).^70,71 Comparing the two calibration methods shows very little difference, indicating that the algorithm optimizes to similar values using different calibration objectives.

2.1.5.

Calibration performance as a function of number of calibration points

In practice, it is assumed there would be a limited number of observed in-situ discharges available for calibration. A Monte Carlo style experiment was conducted to determine an optimally minimum number of observations (calibration points) needed to reliably calibrate $n_b$ and $x$ in Eq. (4). Random, unique, and independent subsamples were taken from the USGS measurement data set for each river producing 100 sets of calibration results for a range of subsample sizes. The subsample sizes represented a different number of observations available for calibration, starting with two observations up to 40 observations. The calibration points for each subsample were compared to the calibration results using the entire record used for the complete calibration described above. The results using the entire record represent the objective values, which are considered to be the best possible estimate. As such, the full calibration is assumed to represent algorithm uncertainty apart from uncertainty associated with the input data and uncertainty associated with differences in river flow regime and morphology.

For example, the results for Willamette River show that the parameter values will converge to within 10% of the optimal $n_b$ on average using seven randomly selected observations of discharge and within 10% of the optimum value for the parameter $x$ on average using 20 calibration points (Fig. 4; Table 3). Additionally, the optimum value for the KGE converges using three calibration points on average (Table 3). Based on average values from these experiments, it may be concluded that a satisfactory calibration may be achieved by calibrating with a minimum of approximately six in-situ discharge observations (as measured by the KGE), and optimum values for the algorithm parameters will be attained when 13 to 19 calibration points are available.

Fig. 4

Example Monte Carlo experiment $n_b$ and $x$ calibration results for the Willamette River at Salem (station 14191000) showing the mean Manning reference flow resistance ($n_b$) (a) and the exponent of the Manning $n$ function ($x$) (b) in the subsampled groups (points) and two standard deviations above and below the mean (bars) compared to the control $n_b$ and $x$ values calibrated from the entire record (horizontal line). As the number of calibration points increases, the mean values converge toward the control value and standard deviation decreases.

Table 3

Calibration performance as a function of number of calibration points: convergence characteristics for the two-parameter calibration results.

Station Number	River name	Number of calibration points to converge on stable value for nb	Number of calibration points to converge on stable value for x	Number of calibration points to converge on stable value for KGE
01184000	Connecticut River at Thompsonville, CT.	20	20	4
01434000	Delaware River at Port Jervis, NY	40	40	7
01515000	Susquehanna River near Waverly, NY	20	40	3
01576000	Susquehanna River at Marietta, PA	8	20	4
01638500	Potomac River at Point of Rocks, MD	40	40	8
02087500	Neuse River near Clayton, NC	20	30	8
02129000	Pee Dee R NR Rockingham, NC	40	40	8
03593500	Tennessee River at Savannah, TN	5	4	4
05082500	Red River of the North at Grand Forks, ND	20	30	7
06185500	Missouri River near Culbertson, MT	30	30	30
6820500	Platte River near Agency, MO	40	40	7
06934500	Missouri River at Hermann, MO	3	3	4
7020500	Mississippi River at Chester, IL	3	3	6
07022000	Mississippi River at Thebes, IL	3	3	5
9261000	Green River near Jensen, UT	5	30	4
11377100	Sacramento R AB Bend Bridge NR Red Bluff, CA	3	4	4
11389500	Sacramento R A Colusa, CA	6	20	5
12150800	Snohomish River near Monroe, WA	30	30	3
12399500	Columbia River at International Boundary	4	3	3
12422500	Spokane River at Spokane, WA	6	7	3
12462500	Wenatchee River at Monitor, WA	9	20	4
14191000	Willamette River at Salem, OR	7	20	3
15304000	Kuskokwim R at Crooked Creek, AK	3	20	7
15356000	Yukon R at Eagle, AK	3	8	4
15453500	Yukon R NR Stevens Village, AK	3	6	3
15485500	Tanana R at Fairbanks, AK	10	20	6
15515500	Tanana R at Nenana, AK	3	7	6
15565447	Yukon R at Pilot Station, AK	4	7	6
15875000	COLVILLE R AT UMIAT AK	5	4	10
15908000	Sagavanirktok R NR Pump STA 3, AK	3	20	4
	Mean	13	19	6
	Stdev	13	13	5
	Max	40	40	30
	Min	3	3	3
	Median	6	20	5
	75th	20	30	7
	25th	3	6	4
	Alaskan rivers
	Mean	4	12	6
	Stdev	2	7	2
	Max	10	20	10
	Min	3	4	3
	Median	3	8	6
	75th	4	20	6
	25th	3	7	4

nb, bankfull reference Manning n value; x, exponent on the relative depth Manning n function; number of calibration points to converge, the number of measurements needed to calibrate to a relatively stable value equal to the value of all observations points are used to calibrate; KGE, Kling–Gupta efficiency.

These tests also indicate that the calibrated parameters stabilize even over several years of discharge observations used as calibration points, showing that the calibration and the parameters are not subject to substantial fluctuations over time.

2.1.6.

Error distribution and algorithm improvements

Errors in the discharge estimates generated from the Manning algorithm applied in this study exhibit a pronounced positive bias for lower discharges in approximately one-third of the rivers (examples shown in Fig. 5). Note that the mean bias was near zero for the calibrated models because the calibration included matching the mean (Table 2).

Fig. 5

Comparative RSQ estimate versus USGS ground-based gaged discharge (left panel) and error distribution as a function of the RSQ estimate (right panel) for three example rivers. (a) and (b) the Delaware, (c) and (d) Potomac, and (e) and (f) Green.

The low-end bias indicates that the Manning $n$ is underestimated at low flows relative to the middle and upper end of the hydrograph. This is likely because the hydraulic flow resistance at low flow becomes more influenced by individual flow obstructions on the bed and banks in the river.⁷³ Flow may become less turbulent in some portions of the reach and more turbulent in others because flow paths may diverge and become directed vertically and laterally, wave resistance becomes more significant as depth becomes shallower, and greater portions of the cross-section may not contribute to downstream conveyance (i.e., dead zones develop in the flow cross-section).

The largest error (and greatest uncertainty) in the discharge estimates occur at low flow for rivers that show the low-end bias (Fig. 5; Table 2; examples include the Pee Dee River, the Neuse River, the Potomac River). These rivers tend to be those with very low slopes and a wide range of discharge, and/or the flow regime at lower flow is dominated by obstructions. This same trend, although not nearly as prominent, is evident for many rivers that do not show the bias, indicating that the assumption of a linear function for transforming Manning $n$ flow resistance breaks down at low discharge as the relation between depth and flow resistance becomes non-linear. Examining the aggregate record of estimated discharge for all of the test rivers shows that the error (uncertainty) is clearly also related to the depth of flow and the discharge.

Three of the study rivers exhibit high errors, the Tennessee River, the Pee Dee River, and the Red River. The Tennessee River error is seen across the entire range of flow and not concentrated at the low end. This is attributed to flow, depth, and slope control by several dams along the channel both upstream and downstream of the measurement reach. The Pee Dee River shows the highest percent errors of any river, especially at the low end of the discharge and depth range of flow. The Pee Dee also has dams along the river reach that control the flow, depth, and slope, and exhibits very low Froude numbers at the low end of the discharge range. The Froude (F) number represents the ratio of inertial to gravitational forces in the flow,^48,51 and is computed directly from the USGS flow measurement data as the ratio of the mean velocity divided by the square root of the gravitation acceleration constant times the mean depth ($F=\sqrt{\frac{V}{gY}}$, where $V$ is the mean velocity, $Y$ is the mean depth, and $g$ is the gravitational acceleration constant).

The low Froude numbers may indicate that significant portions of the cross-section are not contributing to downstream flow due to stagnant water or pooling of water, with section control dominating the flow regime. The flow regime of both of these rivers reaches is considered to violate the assumptions inherent in the discharge algorithm that the channel control is the dominant control on slope, depth, velocity, and discharge. The Red River shows a majority of discharge estimates exhibiting relatively high error also at the low end of the discharge range with very low Froude numbers similar to the Pee Dee River. The Red River calibration is skewed to the low end because there are fewer midrange and high flow measurements that deviate from the stage-discharge trend defined by the lower discharges. This indicates that there are several flow regimes that are not adequately represented by a single flow resistance function. For this reason, the Red River also violates the general assumptions of the discharge algorithm. As such, these data were removed from the discharge database to facilitate further examination of the error trends among the remaining sites.

The percent fractional error for all of the estimated discharges in the discharge data set (excluding the Tennessee, Pee Dee, and Red River data) is largest at depths below 3 m [Fig. 6(a)], and discharge below $\sim 1000{\mathrm{m}}^3/\mathrm{s}$ [Fig. 6(b)]. The relation between error and discharge shows a distinct trend line [Fig. 6(b)] with decreasing error with discharge, although the trend line does not show a strong coefficient of determination ($R^2$).

Fig. 6

(a) and (b) Absolute value of percent error of discharge for the aggregate estimated discharges for the 30 study rivers plotted against the estimated depth of flow and for each discharge—note that this plot excludes the Tennessee, Pee Dee, and Red Rivers, as the depths of these rivers are considered to be artificially controlled or the flow does not meet assumptions of channel control due to excessive stagnant water in the channel. (b) The linear trend between discharge and error.

We tested a calibration procedure that allowed the parameter $B$ to be a freely adjusted parameter instead of assuming this to be a measured value. While this three-parameter calibration ($n_b$, $x$, and $B$) increased the number of degrees of freedom, the calibration was able to reduce low-end bias in many rivers. However, it was found that in almost all cases (with the exception of the Tanana River at Fairbanks, the Platte River, and the Columbia River) the calibrated value of $B$ was higher than observed resulting in shallower estimated depths, higher estimated velocities, and larger Froude numbers compared to the observed values with only marginal improvement in overall predictive power compared to using a measured depth.

The low-end bias has been observed in a general way when applying a Manning type equation to flows in many rivers,^25,74 which is hypothesized here to be due to a non-linear relationship between flow depth and flow resistance, although nonlinearity between flow resistance and slope and width may also be important. Dingman and Sharma⁷⁴ adjusted their statistically derived Manning type equation for non-linearity in slope. The roughness height/boundary layer theory⁴⁸ can be incorporated into the Manning $n$ transformation function by adding an additive term that includes a roughness height layer that does not effectively contribute to the uniform flow field. This results in a function that progressively offers more resistance to flow as the depth becomes shallower because the roughness height becomes more dominant relative to the full depth of flow. However, this approach can result in negative discharges when the roughness height exceeds the minimum height in the stage observations.

We adjusted for this observed nonlinearity by assuming the exponent x varies with relative depth to derive Eq. (5) recognizing the formulation presented in Eq. (4) results in over estimation of the low discharge. Therefore, a factor that would reduce the value of $x$ with relative depth would have the result of increasing flow resistance and decreasing discharge as depth decreases. We also used relative depth in the nonlinear function because it can be readily measured from satellites by tracking WSF. Future improvements may also include relative change in slope and width as nonlinear parameters. As such, we adopt Eq. (5) as the most robust Manning $n$ transform and adapt the overall strategy to measure depth ($B$) rather than let it be a freely calibrated parameter

Eq. (5)

$$n=n_b*\left(\frac{H-B}{h-B}\right)\exp (x*(\frac{H-B}{h-B}\left)\right).$$

Figures 7Fig. 8–9 show the results of the initial two-parameter calibration, the three-parameter calibration, and the modified Manning transformation function calibration for selected rivers. Comparative results for the three calibrations are shown in Table 4.

Fig. 7

Calibrated discharge estimates for the two-parameter (initial) calibration showing (a) the predicted discharge versus the observed discharge, (b) the predicted Froude number versus the observed Froude number, and (c) the error versus predicted discharge for the Susquehanna River at Marietta (station 01576000). USGS, U.S. Geological Survey; RSQ, remote sensing discharge.

Fig. 8

Calibrated discharge estimates for the three-parameter (initial) calibration showing (a) the predicted discharge versus the observed discharge, (b) the predicted Froude number versus the observed Froude number, and (c) the error versus predicted discharge (C) for the Susquehanna River at Marietta (station 01576000). USGS, U.S. Geological Survey; RSQ, remote sensing discharge.

Fig. 9

Calibrated discharge estimates for the two-parameter calibration with the modified Manning $n$ transform [Eq. (5)] showing (a) the predicted discharge versus the observed discharge, (b) the predicted Froude number versus the observed Froude number, and (c) the error versus predicted discharge for the Susquehanna River at Marietta (station 01576000). USGS, U.S. Geological Survey; RSQ, remote sensing discharge.

Table 4

Comparative calibration results for two- and three-parameter calibration of RSQ.

Station number	River name	Calibrated nb	Calibrated x	Calibrated B	NRMSE	NSE	Mean logRes (PCT)	Mean PctRes
		Calibrated best 2 parameter fit
01184000	Connecticut River at Thompsonville, CT	0.023	4.1	−1.9	0.19	0.96	−0.1	0.16
01434000	Delaware River at Port Jervis, NY	0.017	1.1	−0.7	0.27	0.98	−0.14	0.17
01515000	Susquehanna River Near Waverly, NY	0.021	0.5	−0.9	0.14	0.98	−0.17	0.26
01576000	Susquehanna River at Marietta, PA	0.037	1.4	7.1	0.11	0.99	−0.29	0.47
01638500	Potomac River at Point of Rocks, MD	0.021	0.4	−0.8	0.31	0.97	−0.23	0.53
02087500	Neuse River Near Clayton, NC	0.022	0.7	−0.7	0.23	0.97	−0.13	0.15
02129000	Pee Dee R NR Rockingham, NC	0.015	3.1	−3.5	0.33	0.93	−0.48	1.55
03593500	Tennessee River at Savannah, TN	0.023	8.3	−9.8	0.36	0.38	−0.1	0.44
05082500	Red River of the North at Grand Forks, ND	0.023	2.2	−0.6	0.35	0.86	−0.23	0.45
06185500	Missouri River near Culbertson, MT	0.015	1.2	−1.9	0.20	0.98	0.02	0
6820500	Platte River near Agency, MO	0.03	0.5	0.5	0.34	0.96	−0.52	0.92
06934500	Missouri River at Hermann, MO	0.029	1	−5.5	0.05	0.99	0	0
7020500	Mississippi River at Chester, IL	0.024	0.8	−7.8	0.04	0.99	0	0
07022000	Mississippi River at Thebes, IL	0.023	0.5	−6.3	0.08	0.98	0.02	−0.02
9261000	Green River near Jensen, UT	0.037	0.5	−0.5	0.09	0.99	−0.04	0.05
11377100	Sacramento R AB Bend Bridge NR Red Bluff, CA	0.031	−0.1	−1.5	0.04	1.00	−0.01	0.01
11389500	Sacramento R A Colusa, CA	0.038	0.2	8	0.07	0.99	−0.02	0.02
12150800	Snohomish River near Monroe, WA	0.012	1.9	−3.3	0.24	0.94	−0.2	0.28
12399500	Columbia River at International Boundary	0.019	0.5	22.6	0.04	0.99	0.02	−0.02
12422500	Spokane River at Spokane, WA	0.043	2.4	2.2	0.10	0.99	−0.16	0.26
12462500	Wenatchee River at Monitor, WA	0.029	1	3.9	0.12	0.99	−0.1	0.13
14191000	Willamette River at Salem, OR	0.022	0.5	−1.5	0.10	0.99	−0.04	0.05
15304000	Kuskokwim R at Crooked Creek, AK	0.018	0.9	−4.7	0.07	0.97	−0.03	0.04
15356000	Yukon R at Eagle, AK	0.022	0.4	−3.5	0.06	0.97	−0.01	0.04
15453500	Yukon R NR Stevens Village, AK	0.023	0.5	−0.9	0.04	0.99	0.02	−0.02
15485500	Tanana R at Fairbanks, AK	0.037	-0.3	2.1	0.14	0.92	0.06	−0.02
15515500	Tanana R at Nenana, AK	0.034	1.3	−4.4	0.07	0.98	0	0.01
15565447	Yukon R at Pilot Station, AK	0.019	2.5	−14.2	0.13	0.83	−0.01	0.03
15875000	Colville R at Umiat, AK	0.028	1.5	11.4	0.13	0.99	−0.06	0.64
15908000	Sagavanirktok R NR Pump STA 3, AK	0.031	1.7	3.3	0.26	0.86	−0.04	0.08
	Mean	0.026	1.4	−0.5	0.16	0.94	−0.10	0.22
	Stdev	0.008	1.6	6.7	0.11	0.12	0.14	0.34
	Max	0.043	8.3	22.6	0.36	1.00	0.06	1.55
	Min	0.012	−0.3	−14.2	0.04	0.38	−0.52	−0.02
	Median	0.023	1.0	−0.9	0.13	0.98	−0.04	0.07
	75th	0.031	1.7	1.7	0.24	0.99	0.00	0.28
	25th	0.021	0.5	−3.5	0.07	0.96	−0.16	0.01
	Alaskan rivers
	Mean	0.027	1.1	−1.4	0.11	0.94	−0.01	0.10
	stdev	0.007	0.9	7.4	0.07	0.06	0.04	0.22
	Max	0.037	2.5	11.4	0.26	0.99	0.06	0.64
	Min	0.018	−0.3	−14.2	0.04	0.83	−0.06	−0.02
	median	0.026	1.1	−2.2	0.10	0.97	−0.01	0.04
	75th	0.032	1.6	2.4	0.13	0.98	0.01	0.05
	25th	0.021	0.5	−4.5	0.07	0.91	−0.03	0.00
		Calibrated best two parameter fit using modified Manning $n$ function
01184000	Connecticut River at Thompsonville, CT	0.024	2.9	−1.9	0.12	0.98	−0.1	0.14
01434000	Delaware River at Port Jervis, NY	0.02	0.3	−0.7	0.09	1	0	0.01
01515000	Susquehanna River near Waverly, NY	0.022	0.2	−0.9	0.1	0.99	−0.1	0.13
01576000	Susquehanna River at Marietta, PA	0.039	0.7	7.1	0.03	1	0.01	0
01638500	Potomac River at Point of Rocks, MD	0.021	0.1	−0.8	0.25	0.98	−0.18	0.42
02087500	Neuse River near Clayton, NC	0.022	0.3	−0.7	0.21	0.97	−0.03	0.04
02129000	Pee Dee R NR Rockingham, NC	0.016	1.8	−3.5	0.22	0.97	−0.36	0.84
03593500	Tennessee River at Savannah, TN	0.024	7.2	−9.8	0.35	0.4	−0.07	0.41
05082500	Red River of the North at Grand Forks, ND	0.025	1.3	−0.6	0.3	0.9	−0.2	0.38
06185500	Missouri River near Culbertson, MT	0.015	0.6	−1.9	0.26	0.98	0.08	−0.04
6820500	Platte River near Agency, MO	0.035	0.1	0.5	0.29	0.97	−0.46	0.73
06934500	Missouri River at Hermann, MO	0.03	0.6	−5.5	0.05	0.99	0	0.01
7020500	Mississippi River at Chester, IL	0.024	0.5	−7.8	0.05	0.99	0	0
07022000	Mississippi River at Thebes, IL	0.023	0.3	−6.3	0.08	0.98	0.02	−0.02
9261000	Green River near Jensen, UT	0.039	0.2	−0.5	0.07	1	0.01	−0.01
11377100	Sacramento R AB Bend Bridge NR Red Bluff, CA	0.029	0	−1.5	0.06	0.99	0.01	−0.01
11389500	Sacramento R A Colusa, CA	0.038	0.1	8	0.07	0.99	0.01	−0.01
12150800	Snohomish River near Monroe, WA	0.013	1	−3.3	0.16	0.97	−0.15	0.18
12399500	Columbia River at International Boundary	0.019	0.3	22.6	0.04	0.99	0.03	−0.02
12422500	Spokane River at Spokane, WA	0.045	1.5	2.2	0.05	1	−0.07	0.08
12462500	Wenatchee River at Monitor, WA	0.032	0.4	3.9	0.06	1	−0.05	0.06
14191000	Willamette River at Salem, OR	0.023	0.2	−1.5	0.07	1	0	0
15304000	Kuskokwim R at Crooked Creek, AK	0.019	0.5	−4.7	0.07	0.98	−0.02	0.03
15356000	Yukon R at Eagle, AK	0.023	0.2	−3.5	0.06	0.97	0	0.03
15453500	Yukon R NR Stevens Village, AK	0.023	0.3	−0.9	0.02	1	0.01	−0.01
15485500	Tanana R at Fairbanks, AK	0.037	-0.3	2.1	0.14	0.92	0.02	0.02
15515500	Tanana R at Nenana, AK	0.035	0.9	−4.4	0.07	0.98	0.01	−0.01
15565447	Yukon R at Pilot Station, AK	0.0	2.5	−14.2	0.1	0.8	0.1	0.0
15875000	Colville R at Umiat, AK	0.03	0.8	11.4	0.16	0.98	0.22	−0.03
15908000	Sagavanirktok R NR Pump STA 3, AK	0.034	1	3.3	0.25	0.87	0.02	0.02
	Mean	0.027	0.9	−0.5	0.13	0.95	−0.04	0.11
	Stdev	0.008	1.4	6.7	0.09	0.11	0.13	0.22
	Max	0.045	7.2	22.6	0.35	1.00	0.22	0.84
	Min	0.013	-0.3	−14.2	0.02	0.40	−0.46	−0.04
	Median	0.024	0.5	−0.9	0.09	0.98	0.00	0.02
	75th	0.034	1.0	1.7	0.20	0.99	0.01	0.12
	25th	0.021	0.2	−3.5	0.06	0.97	−0.07	−0.01
	Alaskan rivers
	Mean	0.028	0.7	−1.4	0.11	0.94	0.04	0.00
	Stdev	0.007	0.8	7.4	0.07	0.07	0.08	0.03
	Max	0.037	2.5	11.4	0.25	1.00	0.22	0.03
	Min	0.019	−0.3	−14.2	0.02	0.80	−0.02	−0.04
	Median	0.027	0.7	−2.2	0.11	0.98	0.02	0.01
	75th	0.034	0.9	2.4	0.15	0.98	0.03	0.02
	25th	0.022	0.3	−4.5	0.07	0.91	0.01	−0.02
		Calibrated best 3 parameter fit
01184000	Connecticut River at Thompsonville, CT	0.005	−0.4	2.2	0.15	0.97	0.12	−0.12
01434000	Delaware River at Port Jervis, NY	0.015	0.3		0.11	1	−0.01	0.02
01515000	Susquehanna River Near Waverly, NY	0.013	−0.3	0.1	0.12	0.99	0.02	−0.02
01576000	Susquehanna River at Marietta, PA	0.024	0.11	9.1	0.03	1	−0.06	0.06
01638500	Potomac River at Point of Rocks, MD	0.018	−0.4	0.0	0.17	0.99	−0.15	0.37
02087500	Neuse River near Clayton, NC	0.015	−0.4	0.2	0.16	0.98	0.06	−0.05
02129000	Pee Dee R NR Rockingham, NC	0.004	0.7	−0.1	0.79	0.62	0.04	−0.01
03593500	Tennessee River at Savannah, TN	0.001	1.3	1.5	0.61	−0.77	0.58	−0.23
05082500	Red River of the North at Grand Forks, ND	0.007	−0.8	−2.0	0.19	0.96	−0.07	0.09
06185500	Missouri River near Culbertson, MT	0.016	1.2	0.9	0.2	0.98	0.02	0
6820500	Platte River near Agency, MO	0.022	0.8	−2.0	0.63	0.86	0.33	−0.25
06934500	Missouri River at Hermann, MO	0.018	0	−3.4	0.05	0.99	−0.02	0.02
7020500	Mississippi River at Chester, IL	0.015	−0.2	−0.9	0.07	0.98	−0.06	0.06
07022000	Mississippi River at Thebes, IL	0.013	−0.7		0.11	0.96	−0.04	0.05
9261000	Green River Near Jensen, UT	0.032	0.3	−0.2	0.09	0.99	0.03	−0.03
11377100	Sacramento R AB Bend Bridge NR Red Bluff, CA	0.027	−0.4	−0.8	0.06	0.99	0.02	−0.02
11389500	Sacramento R A Colusa, CA	0.031	−0.3	9.9	0.08	0.99	0.02	−0.02
12150800	Snohomish River Near Monroe, WA	0.006	−0.1	−0.2	0.15	0.98	−0.03	0.03
12399500	Columbia River at International Boundary	0.022	0.9	20.9	0.04	0.99	0	0
12422500	Spokane River at Spokane, WA	0.019	0.5	4.5	0.09	0.99	0.04	−0.04
12462500	Wenatchee River at Monitor, WA	0.019	0	4.8	0.1	0.99	−0.02	0.03
14191000	Willamette River at Salem, OR	0.016	−0.4	0.6	0.05	1	−0.03	0.03
15304000	Kuskokwim R at Crooked Creek, AK	0.012	0	−2.0	0.08	0.97	0.04	−0.04
15356000	Yukon R at Eagle, AK	0.008	−1	2.3	0.1	0.92	−0.06	0.07
15453500	Yukon R NR Stevens Village, AK	0.016	0	2.0	0.04	0.99	−0.02	0.02
15485500	Tanana R at Fairbanks, AK	0.1	0.8	−2.0	0.14	0.92	0.07	−0.02
15515500	Tanana R at Nenana, AK	0.014	−0.3	−0.8	0.07	0.98	−0.02	0.02
15565447	Yukon R at Pilot Station, AK	0.003	−1.00	1.9	0.15	0.80	−0.07	0.08
15875000	Colville R at Umiat, AK	0.200	1.20	12.5	0.16	0.98	0.34	−0.19
15908000	Sagavanirktok R NR Pump STA 3, AK	0.026	1.40	3.6	0.28	0.84	0.13	−0.09
	Mean	0.025	0.1	2.2	0.17	0.89	0.04	−0.01
	Stdev	0.037	0.7	5.2	0.18	0.32	0.14	0.11
	Max	0.200	1.4	20.9	0.79	1.00	0.58	0.37
	Min	0.001	−1.0	−3.4	0.03	−0.77	−0.15	−0.25
	Median	0.016	0.0	0.4	0.11	0.98	0.01	0.00
	75th	0.022	0.7	2.7	0.16	0.99	0.04	0.03
	25th	0.012	−0.4	−0.8	0.07	0.96	−0.03	−0.04
	Alaskan rivers
	Mean	0.047	0.1	2.2	0.13	0.93	0.05	−0.02
	Stdev	0.069	0.9	4.7	0.07	0.07	0.13	0.09
	Max	0.200	1.4	12.5	0.28	0.99	0.34	0.08
	Min	0.003	−1.0	−2.0	0.04	0.80	−0.07	−0.19
	Median	0.015	0.0	1.9	0.12	0.95	0.01	0.00
	75th	0.045	0.9	2.7	0.15	0.98	0.09	0.03
	25th	0.011	−0.5	−1.1	0.08	0.90	−0.03	−0.05

nb, bankfull reference Manning n; x, exponent of Manning n function; B, elevation of zero flow in channel (channel bottom); NRMSE, normalized root mean square error; NSE, Nash–Sutcliffe efficiency statistic; logRes, the log of the RSQ estimate minus the log of the observed discharge; PctRes, RSQ estimated discharge minus the observed discharge divided by the observed discharge; RSQ, remote sensing discharge.

The three-parameter calibration (optimized values for the value of $B$, $n_b$, and $x$) eliminated low-end bias and reduced error as compared to the initial two parameter calibration (Fig. 7), which used the observed depth to define the value of B (Fig. 9). However, to achieve the reduction in bias, the Froude number, as an index to hydraulic state of flow, is much larger due to shallower depth (increasing value of B) and predicted higher velocity. Thus, the three-parameter calibration improved on the low-end bias at the expense of realism in the hydraulics. Note also that the largest error is still concentrated at the low end of the discharge.

The calibration results for Eq. (5) (Fig. 8) are similar to the three-parameter calibration, successfully eliminating the low-end bias and reducing overall error with the advantage that it does not introduce an additional parameter that requires calibration, enabling the value of B to remain a measurable parameter.

The results of the three calibration options show that the two-parameter calibration with the modified (and nonlinear) Manning $n$ transform function (where $n_b$ and $x$ are calibrated holding $B$ constant) performs as well overall compared to the three-parameter calibration (Table 4). Increasing error is associated with a wider range of predicted Froude numbers (meaning wider range of flow conditions) as indexed by the ratio of the maximum to minimum Froude number (Fig. 10). This indicates that the preferred algorithm option for rivers with an expected wider range of flow would be the nonlinear Manning $n$ transform [Eq. (5)].

Fig. 10

Error as a function of Froude number index ($F_{\max }/F_{\min }$). RSQ, remote sensing discharge.

The overall NRMSE is also correlated with normalized stream power (Table 1) as defined and provided by Frasson et al.¹⁹ and illustrated in Fig. 11 (Stream power is defined in Frasson et al.,¹⁹ as the weight density of water times the flow rate times the slope, $\text{Power}=\mathrm{Q}\gamma \mathrm{S}$). Rivers with stream power $<200\mathrm{W}/\mathrm{m}$ are the rivers with the highest NRMSE. Froude number and stream power are strong indicators of algorithm performance.

Fig. 11

Plot of the NRMSE against the Frasson et al.,^18,19 stream power.

Examining the aggregate error for all of the flows from the 30 rivers (Fig. 6) excluding the Tennessee and Pee Dee Rivers (due to flow controls by dams), as well as the Red and Pee Dee Rivers due to very low streampower (e.g., flow is more like flow through pools rather than flow controlled by slope with very low Froude numbers), we summarized threshold stats for NRMSE associated with depth and discharge (Table 5). Considering a depth threshold of $>2\mathrm{m}$ the data show that 52% have an error of 5% or less and 74% have an error of 10% or less compared to 39% and 58% for all flows. Using a threshold of $>3\mathrm{m}$, 63% are below 5% error and 87% below 10% error. Considering a discharge threshold of $>1000{\mathrm{m}}^3/\mathrm{s}$, the data show that 65% have an error of 5% or less and 88% have an error of 10% or less compared to 39% and 58% for all flows. Using a threshold of $>2000{\mathrm{m}}^3/\mathrm{s}$, 71% are below 5% error and 92% below 10% error.

Table 5

Error threshold statistics between observed and remote sensing estimated discharge (RSQ).

No. measurements	Fractional percent threshold	Fractional percent error of estimates below threshold	No. measurements	Fractional percent threshold	Fractional percent error of estimates below threshold
Fractional percent of measurements below error threshold for all data			Fractional percent of measurements below error threshold for all data
1177	<0.05	0.39	1177	<0.05	0.39
1745	<0.1	0.58	1745	<0.1	0.58
2078	<0.15	0.69	2078	<0.15	0.69
Fractional percent of measurements below error threshold for depth > 2 m			Fractional percent of measurements below error threshold for discharge $>1000{\mathrm{m}}^3/\mathrm{s}$
938	<0.05	0.52	650	<0.05	0.65
1341	<0.1	0.74	883	<0.1	0.88
1519	<0.15	0.84	947	<0.15	0.95
Fractional percent of measurements below error threshold for depth > 3 m			Fractional percent of measurements below error threshold for $\text{discharge}>2000{\mathrm{m}}^3/\mathrm{s}$
778	<0.05	0.63	559	<0.05	0.71
1071	<0.1	0.87	730	<0.1	0.92
1169	<0.15	0.95	765	<0.15	0.97

Fractional percent = Percent divided by 100; Fractional percent error = RSQ estimated discharge divided by measured discharge.

2.2.

Remote Sensing Observations Data Uncertainty

The remotely sensed river observations consist of the WSE, the width of the water-surface, and the water-surface slope. Although we anticipate limited ground-based observations of the discharge, velocity, and reach bathymetry to calibrate the RSQ estimates, we do not consider ground-based measurement error in this section, and it is generally assumed that uncertainty associated with ground-based observations is negligible. Validation studies for both observed variables have been conducted at many locations using different instruments mounted on various satellite platforms providing first-order expected observation accuracies. Along with these studies, the spatial ($x$, $y$, and $z$) accuracies are generally reported based on the specifications of the observing instruments, including discussion of factors that may reduce accuracy.

Recently, observations of water-surface height in rivers (and other water bodies) derived from the Jason-2 and Jason-3 satellite nadir altimeter missions provide an estimate of observational error. These estimates are based on qualitative characteristics of the target as well as atmospheric and instrument conditions and responses. As such, they are not precise errors but rather provide an expected error based on qualitative characteristics of the observation.⁴³

The accuracy of space-based altimeters that track water-surface heights globally in general have reported observational accuracies (when compared to a time series of in-situ observations) of a few to several tens of cm reported as the root mean square (rms) error. The accuracy is dependent on many factors,⁴³ including the surface roughness and width, and is typically $>10\mathrm{cm}$ rms for river reaches. For satellite-based laser altimeters, the accuracy is better and there is a greater potential to acquire stage for smaller tributaries.^75–77 There has been continuity of satellite radar altimeters over several decades but the latest radar altimeter missions, such as Sentinel-3 and Sentinel-6, use enhanced technology (Delay-Doppler SAR) in combination with innovative techniques such as fully-focused SAR (FF-SAR) with the aim of increasing the current spatial resolution of stage measurements along the ground, from 300 to 40–80 m.

Observations from the SWOT mission will be reported on a reach scale by averaging from a point cloud generated by wide swath altimetry (Biancamaria et al.³ SWOT | PO.DAAC/JPL/NASA). The accuracy of the SWOT mission water surface heights and widths has not been fully evaluated as observational data is only beginning to be received. The accuracy of averaged water surface heights over a 10 km reach and 100 m wide river is expected to be on the order of 10 cm.³ SWOT expected uncertainty in observations of water-surface width, expressed as the resolution of the image, is 15 m. Comparatively, Landsat resolution is 30 m,^39,40 however, the width accuracy is also a function of the classification algorithm that defines which pixels contain water. Although there are satellite-based imagers with much better resolution (10 m or better), these imagers do not routinely observe locations across the earth that can provide time series of observations. As such, it is expected that the best estimates of width observation uncertainty would be on the order of $+/-15\mathrm{m}$.

Bjerklie²⁶ showed that observational uncertainty—viewed as a fixed value for height and width (e.g., $+/-10\mathrm{cm}$ for height and $+/-30\mathrm{m}$ for width)—becomes more important in the overall accuracy of a discharge estimate as the river gets smaller. In other words, the effect of observation error becomes less important as rivers get larger because the uncertainty becomes a smaller percentage of the magnitude of the observed variable.

The reach lengths over which the averaging is accomplished may vary but fundamentally are constrained by the distances between satellite overpasses and the orientation of the river in relation to its intersection with the satellite crossing locations. Consequently, the impact of the reach on the average values is variable and somewhat unknown. Our assumption is that the reach averaging will not introduce substantial uncertainty provided the reach length is $>\sim 10$ rivers widths^78,79 and less than the distance between input tributaries. Uncertainty estimates of widths based on Landsat classification- were made through comparison with published ground-based values as well as those visually interpreted from finer-resolution satellite imagery collected coincidentally with the Landsat scenes being evaluated.^39,80 Estimates of uncertainty associated with satellite-based altimeters have been assessed by Birkett et al.⁴³ Bjerklie et al.¹⁵ summarize these data sources and compare observations with ground observations for the Yukon River in Alaska.

Here we look at how WSE error propagates into the estimates generated from the discharge algorithm as presented in Eq. (3). It is assumed that $r=2$ and the three parameters $B$, $n_b$, and $x$ are calibrated. The discharge uncertainty is calculated from the derivative of Eq. (3) with respect to the change in height ($dh$) to characterize the sensitivity of the equation to the magnitude of $h$. Using uncertainty quantification methods, the uncertainty in discharge due to the WSE observations is

Finally, the sensitivity of discharge to width is given as

Thus, discharge uncertainty is computed by combining Eqs. (6)–(9). Relative discharge uncertainty can be computed by dividing the result by Manning’s equation. One critical result that can be seen from the mathematics is that the derivatives are all a function of WSE themselves, and thus uncertainty changes with WSE; this is also true for the relative discharge uncertainty (not shown analytically). We explore the relative uncertainty by varying WSE and width values, along with width and WSE measurement uncertainty.

The resolution or uncertainty in the height observations for an example station on the Delaware River ranges from 1 to 15 cm, and the resolution or uncertainty in the width observations range from 5 to 15 m (Fig. 12). The uncertainty is far greater at low values of stage, and thus at low flow (e.g., for a 10 cm WSE uncertainty) the relative discharge uncertainty is 40% at a stage of 1 m, but only 5% at a stage of 5 m. Note that the discharge uncertainty due to width is significantly lower, and furthermore is far less responsive to the width itself. These results emphasize the importance of considering discharge uncertainty at a range of flows. The absolute error in the discharge resulting from absolute error in stage (i.e., depth) and width is proportional, as shown by Bjerklie et al.,²⁶ because both width and stage are multiplicative elements discharge continuity (e.g., $Q=W*Y*V$). As such, the contribution of fixed error in width and stage becomes a larger proportion of observational error as the width, stage, and discharge become smaller.

Fig. 12

Example of the effect of observation uncertainty on discharge uncertainty for the Delaware River data. (a) and (b) Discharge performance expressed as uncertainty as a function of the uncertainty in height and width observations, respectively.

Across all 30 rivers in the test data set, for 10 cm WSE error the 67th percentile of error yields relative discharge uncertainty of 12% at median flow (Fig. 13). At low flow, most rivers have uncertainty $<25\%$. A minority of rivers (90th percentile) are at a discharge uncertainty of 20% or greater at median flow. Increasing the WSE error to 20 cm makes a large change, with most rivers (67th percentile) now at 23% discharge error. Error in width shows much less impact across the 30 test rivers, as the assumed uncertainty in width observations does not constitute a large portion of the total width of the water-surface. Additionally, the width is smoothed to reduce random measurement error as previously discussed.

Fig. 13

Cumulative number of rivers (CDF) for the 30-river data set as a function of relative discharge uncertainty assuming (a) a 10 cm and (b) 20 cm uncertainty in WSE observations.

Assuming the error associated with the algorithm (model error) is on the order of 10%, for approximately half of the rivers the observation error is larger than the model error. For nearly all (85%) of the rivers, the error associated with the proxy satellite observations is a major or dominant component of overall error. Depth controls most of the variability and observation error dominates for shallower rivers. For some rivers, observation error is far greater than model error.

In summary, observation error will almost certainly remain a large part of the total discharge uncertainty in remote sensing of discharge. Even when error in the observation of WSE is $<10\mathrm{cm}$ (which is a challenging goal) observation error will likely be greater than model error in most rivers. WSE uncertainty varies with river stage, increasing at low flow.

2.2.1.

Effect of slope and assumed regular cross-section (width) on discharge estimates

Dynamic water-surface slope data were collected for the Tanana River near Fairbanks and used to evaluate the effect of dynamic slope compared to static slope on algorithm-derived discharge estimates. Unfortunately, there are few data sets available to provide a broader range of evaluation. The Tanana River water-surface height was monitored at three locations along the river reach that flows past the USGS streamgage at Fairbanks, AK (collected June 9–October 12, 2016; https://doi.org/10.5066/P9ZUUVPS). The height data was coupled with continuous discharge data at the gage to derive a data set of reach averaged change in height, slope, and discharge that spanned a several month period in 2016. Discharge estimates were derived from the height and slope observations and compared with the USGS estimates based on the stage-discharge rating curve at the gage (Fig. 14). The dynamic reach averaged width was not directly measured and as such it was estimated based on a relation between stage and width derived from observations of the maximum width and estimates of the bankfull depth, assuming that the river channel is parabolic in shape (see Bjerklie et al.¹⁵). This was accomplished by fitting a parabola with the width and depth defining the parabolic dimensions. The parabola is given as $Y={\mathrm{aW}}^2$ with $Y$ being the depth and $W$ the width. The coefficient $a$ is set as the ratio of the bankfull depth, $Y_b$ divided by the bankfull width, ${W_b}^2$, and the bankfull dimensions obtained from the discharge measurement record as previously described. Riggs et al.¹³ have shown that reach averaged widths improve overall accuracy of discharge estimates compared to using a cross-section width, further supporting the value of width averaging. The assumption of a regular width derived from the stage also provides a means to test the importance of the accuracy of width in the discharge estimate algorithm.

Fig. 14

(a) The estimated discharge versus the observed discharge over the summer of 2016 for the Tanana River at Fairbanks using the dynamic slope and assumed width and maximum depth. (b) The estimated discharge using a constant slope versus the discharge estimated using a dynamic slope only. USGS, U.S. Geological Survey.

Bjerklie et al.²⁶ have previously shown that assuming a constant slope does not degrade the overall accuracy of discharge estimates using a Manning flow resistance law compared to using measured dynamic slope. Similarly, the discharge estimates for the Tanana River derived from assuming a constant slope compared to a dynamic slope show only a small reduction in accuracy (on the order of 1% difference in overall mean). Additionally, the assumed width could be used to derive accurate discharge estimates through calibration, indicating that width can be viewed more or less as a scalar and need not necessarily be highly accurate compared to ground-measured width provided the estimates can be calibrated. These results indicate that initial values of discharge can be derived from appropriately scaled width and constant slope as a precursor to full dynamic observations of width and slope without significant degradation of estimate accuracy.

Additionally, the impact of using the measured width as opposed to a smoothed width that enforces a monotonic relation between stage and width is explored. In particular, the rivers that showed a relative nonlinear but regular pattern of stage versus width were recalibrated using the observed width rather than the smoothed width, and the results were compared. These rivers include the Susquehanna River at Marietta and Waverley, Delaware River, Connecticut River, Potomac, Wenatchee, Platte and the Tanana at Fairbanks.

The Tanana at Fairbanks is the only river that shows a stage width relation where the width increases more rapidly at a higher stage – interestingly this river produced a negative value of $x$ for the two-parameter calibration, indicating that the rapid increase in width resulted in increasing flow resistance with increasing depth. Rerunning the calibration for these rivers using the observed width showed improvement in the mean log residual for several of the rivers including the Susquehanna at Marietta and Waverley, and the Platte with the NRMSE and the NSE remaining the same. The low-end bias for both Susquehanna stations also diminished indicating the value of a reliable observed reach averaged width where possible. However, given the present data set with width not measured consistently at the same location, the width smoothing is necessary. Additionally, width smoothing is necessary if stage and width are not measured at the same time.

2.3.

Parameter Uncertainty

The Manning equation described by Eqs. (3) and (4) ingests measured variables of water-surface slope, water-surface width, and water-surface height. The Manning equation parameters that must be estimated for each river reach where the equation is applied include the bankfull and bottom reference heights ($H$ and $B$) from which the water depth is referenced; the bankfull reference Manning $n$ flow resistance ($n_b$), which is the reference point from which Manning $n$ values at lower water-surface heights are referenced; the exponent $x$ controls the variation of Manning $n$ with respect to the relative depth; and the channel cross-section geometry coefficient $r$. As previously mentioned, we assume that the geometry coefficient is constant for all channels with a power of 2 (parabolic). Parameter errors are offset by averaging and calibration—minimizing the number of parameters to calibrate improves the overall accuracy of ground truth hydraulics and provides more reasonably accurate values for the mean velocity and depth.

3.1.

Overall Uncertainty

Considering the uncertainty associated with the algorithm, calibrated parameters, and measurement error, it is possible to develop a general understanding of expected uncertainty (accuracy) in the discharge estimates depending on flow regime. Combining the expected algorithm and parameter estimation accuracy for a well calibrated model, we would expect accuracy on the order of 10% of USGS ground-based values of discharge and a general input variable measurement accuracy on the order of 10% (primarily measurement errors associated with WSE). With these assumptions, the overall expected uncertainty would be $\sim 14\%$ for most rivers using

For rivers with a wide range of estimated Froude numbers and shallow slopes, as well as rivers with high BFI and impacts from dams, the accuracy would decrease due to algorithm and parameter uncertainty such that the expected algorithm uncertainty would increase up to 20% or more even as measurement accuracy remains the same at 10%. Thus, according to Eq. (10) the general expected accuracy would be on the order of 22%.

However, as has been demonstrated, for most rivers the uncertainty associated with the algorithm and parameter estimation as well as input data measurements increases significantly for shallower depths ($<3\mathrm{m}$) and lower discharge ($<1000{\mathrm{m}}^3/\mathrm{s}$). Thus, it can be said that as discharge and depth decrease, uncertainty increases especially for those rivers that have shallow slopes, are impacted by dams, and have a high BFI. The low discharge and shallow depth uncertainty may increase to 50% or more and at higher flows and depths uncertainty is significantly lower, which is key to understanding and applying the RSQ flow data.

In the context of an accuracy classification system the expectation would be that an RSQ gage on a large river that is well calibrated (10 to 20 calibration points) and meets the assumptions of the flow law algorithm that is free of dams has a low BFI range of Froude numbers, and a relatively steep slope would have good to fair discharge ratings (15% or less uncertainty). Smaller rivers that do not meet the assumptions of the flow law algorithm, show a wide range in Froude numbers, have relatively shallow slopes, have a relatively high BFI, and are impacted by dams, would be expected to have fair to poor discharge ratings (15% or greater uncertainty).