Ratings are used for a variety of reasons in water-resources investigations. The simplest rating relates discharge to the stage of the river. From a pure hydrodynamics perspective, all rivers and streams have some form of hysteresis in the relation between stage and discharge because of unsteady flow as a flood wave passes. Simple ratings are unable to represent hysteresis in a stage/discharge relation. A dynamic rating method is capable of capturing hysteresis owing to the variable energy slope caused by unsteady momentum and pressure.

A dynamic rating method developed to compute discharge from stage for compact channel geometry, referred to as DYNMOD, previously has been developed through a simplification of the one-dimensional Saint-Venant equations. A dynamic rating method, which accommodates compound and compact channel geometry, referred to as DYNPOUND, has been developed through a similar simplification as a part of this study. The DYNMOD and DYNPOUND methods were implemented in the Python programming language. Discharge time series computed with the dynamic rating method implementations were then compared to simulated discharge time series and discrete discharge measurements made at U.S. Geological Survey streamgage sites.

Four sets of stage and discharge time series were created using one-dimensional unsteady simulation software with compound channel geometry to compare the results of both dynamic rating methods to results from the full one-dimensional shallow water equations. Discharge time series were computed from stage time series using DYNMOD and DYNPOUND. DYNPOUND outperformed DYNMOD in all four scenarios. The minimum and maximum mean squared logarithmic error (MSLE) for the DYNMOD results were 2.75×10^{−2} and 3.40×10^{−2}, respectively. The minimum and maximum MSLE for the DYNPOUND results were 2.51×10^{−7} and 1.91×10^{−4}, respectively.

The dynamic rating methods were calibrated for six U.S. Geological Survey streamgage sites using observed discharge data collected at the sites. The calibration objective for each site was to minimize the MSLE of the discharge computed with the rating method with respect to observed discharge. For each site, the calibration included all field measurements within a selected water year. The DYNMOD method failed to compute discharge for the full calibration time series for three sites. A method fails to compute when the implementation returns a nonfinite value at a time step. Because the values computed for following time steps are dependent on the previous time step, a nonfinite value results in nonfinite values that follow. For the three sites for which DYNMOD computed the complete discharge time series, the minimum MSLE for calibration was 2.19×10^{−3} and the maximum was 9.77×10^{−3}. The MSLE of the DYNPOUND computed discharge calibration time series for the six sites ranged from 3.70×10^{−3} to 1.25. For each site, an event-based time period was selected to compare the discharge time series computed with the dynamic rating methods to discrete discharge field measurements made at the streamgage sites. The DYNMOD-computed discharge time series for the three sites had an MSLE range of 2.76×10^{−3} to 3.14×10^{−2}. The range of MSLE for the six DYNPOUND sites was 3.64×10^{−3} to 7.23×10^{−2}. Although the DYNMOD method outperforms the DYNPOUND method when calibrated streamgage sites are under consideration, the DYNMOD method failed to compute a discharge time series at three of the six sites. The DYNPOUND method, therefore, was more robust than the DYNMOD method. Improvements to the implementation of the DYNPOUND method may improve the accuracy of the method.

Domanski, M.M., Holmes, R.R., and Heal, E.N., 2022a, Dynamic rating method for computing discharge from time series stage data—Site datasets: U.S. Geological Survey data release,

Domanski, M.M., Holmes, R.R., and Heal, E.N., 2022b, Dynamic stage to discharge rating model archive: U.S. Geological Survey data release,

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Multiply | By | To obtain |

Length | ||

foot (ft) | 0.3048 | meter (m) |

mile (mi) | 1.609 | kilometer (km) |

Area | ||

square mile (mi^{2}) |
259.0 | hectare (ha) |

square mile (mi^{2}) |
2.590 | square kilometer (km^{2}) |

Flow rate | ||

cubic foot per second (ft^{3}/s) |
0.02832 | cubic meter per second (m^{3}/s) |

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 (NAVD 88), National Geodetic Vertical Datum of 1929 (NGVD 1929), and World Geodetic System (WGS 1984).

A water year is defined as the 12-month period, October 1 through September 30, and is designated by calendar year in which it ends.

acoustic doppler current profiler

digital elevation model

Hydrologic Engineering Center River Analysis System

mean squared logarithmic error

U.S. Geological Survey

Universal Time Coordinated

A relation to estimate discharge using a continuous surrogate measure is termed a “rating.” Ratings are used for a variety of reasons in water-resources investigations, but a predominant use of ratings is at streamgages, where autonomously collected stage is converted to discharge by use of a rating. No widely accepted method for direct discrete continuous measurement of discharge is available. In the absence of direct discrete continuous discharge measurements, discharge typically is determined through continuous surrogate measures of one or more variables such as stage, water-surface slope, rate of change in stage, or index velocity, which are collected at a streamgage. The derivation of discharge through these surrogate variables utilizes various models that will be termed a “rating.” The rating is developed and calibrated using discharge measurements collected onsite by field staff.

The simplest rating relates discharge to stage of the river (simple rating). Hydrologists and engineers have long recognized hysteresis (loops) exist in relations between stage and discharge (

“The hysteretic relation of stage to discharge indicates that estimates of instantaneous dynamic discharge based on rating curves can be significantly in error. On the other hand, estimates of mean dynamic discharge based on rating curves may not be so severely affected by hysteresis because integration of the underestimated flow during the rising stages is frequently compensated for by a corresponding overestimate during falling stages.”

For both of these reasons, simple ratings are often adequate for discharge computation.

Simple ratings do not work as well for streamgages on low-gradient streams, streams with variable backwater, streams with large amounts of channel or overbank storage, streams with highly unsteady flow (rapid rises via flood wave movement), or streams with highly mobile beds (

This report documents the testing of the

Theoretical determination of the relation between stage and discharge for a 100-foot wide rectangular prismatic channel using a one-dimensional unsteady fully dynamic open-channel hydraulic model with varying bed slopes and rates of unsteadiness (rate of change in local velocity with respect to time) for the inflow hydrograph at the upstream end.

Figure 1. Graph showing theoretical determination of the relation between stage and discharge for a 100-foot wide rectangular prismatic channel using a one-dimensional unsteady fully dynamic open-channel hydraulic model with varying bed slopes and rates of unsteadiness for the inflow hydrograph at the upstream end.

The one-dimensional flow in a stream can be described by the Saint-Venant equations (

and an equation that represents the one-dimensional streamwise form of the conservation of momentum as

is the wetted cross-section area of the channel, in square feet;

is the mean velocity of flow, in feet per second;

is the streamwise distance along the channel, in feet;

is the channel width, in feet;

is the water-surface elevation above a datum plane, in feet;

is the time, in seconds;

is the acceleration of gravity, in feet per second squared;

is the flow depth, in feet;

is the friction slope, in feet per feet; and

is the bed slope, in feet per feet.

Flow resistance in a channel is often represented by the well-known Manning’s equation (

is the discharge, in cubic feet per second;

is the Manning’s roughness coefficient; and

is the hydraulic radius, in feet.

If the channel geometry, water-surface elevation, and Manning’s roughness coefficient (_{f}_{f}

lateral inflow and outflow are negligible;

the channel width is essentially constant (

energy losses from channel friction and turbulence are described by the Manning’s equation;

the geometry of the section is essentially permanent (scour and fill are negligible);

the bulk of the flood wave is moving approximately as a kinematic wave, which implies _{f}_{0} and the wave propagates only in the downstream direction; and

the flow at the section is controlled by the channel geometry, _{f}_{0}, and the shape of the flood wave.

Because the method utilizes data at a single streamgage, the above assumptions are used to adjust

The various components that make up _{f}_{0}; a pressure term,

Placing

This equation then becomes

At this point, the pressure term

is the flood wave velocity, in feet per second; and

is defined as the ratio of _{0} to the average wave slope (_{W}

The flood wave velocity can be represented (

Allowing

Hydraulic radius,

According to

Combining

The celerity coefficient (_{c}

It should be noted that this work does not follow

The second term in _{0}_{W}_{W}

is the stage at the peak of a typical flood, in feet;

is the stage prior to beginning of the typical flood, in feet;

is the velocity of the flood wave, in feet per second; and

is the elapsed time between the beginning of the typical flood to the peak of the flood, in seconds.

The velocity of the flood wave is estimated from _{c}

is the peak discharge for a typical flood, in cubic feet per second;

is the discharge prior to the beginning of the typical flood, in cubic feet per second; and

is the wetted cross-section area associated with the average stage,

Utilizing

Given the developments shown through

is for the current time, and

is for time

_{f}_{j}_{j}

The conservation of momentum and mass equations (

The method developed for compound channel geometry is derived as follows. The equations for one-dimensional conservation of mass and momentum take the form of

The variable

the discharge for the total cross section is equal to the sum of discharges in each subsection, and

_{f}_{f}

In

The water-surface elevation slope, _{0}

Taking the partial derivative with respect to

Using the chain rule, the partial derivative of

Moving the partial derivative of cross-sectional area with respect to time to the right-hand side of

Using the chain rule in taking the partial derivative of

Substituting

Discharge is related to conveyance and _{f}_{f}

Substituting

As in the compact geometry method described previously, the compound geometry method uses

Under the kinematic wave assumption, _{f}_{0}

Because all variables in

Both methods described in this report were implemented in the Python programming language. DYNMOD is the dynamic rating method that computes discharge from stage for compact channel geometry, whereas DYNPOUND is the newly developed method that solves for discharge in compact and compound channels. The computation procedure for DYNMOD uses the Newton-Raphson numerical method (

To compute an unknown discharge for a time _{j}_{j-1}

known constants, which are _{0},

a known discharge value _{j-1}_{j-1}

a known stage value _{j-1}_{j-1}

a known stage value _{j}_{j}

The method to compute discharge at a site with compound channel geometry contains continuous derivatives that need to be discretized to determine discharge time-series values. Beginning with the derivative in the first term of

The second term of

is the cross-sectional area for stage _{j}

The pressure term

In the current implementation of the method,

The discrete form of

Substituting all discrete approximations of derivatives into

_{j}_{j}

These solution methods for the compact (DYNMOD) and compound (DYNPOUND) channel geometries were implemented in the Python programming language. The results shown in this report were computed using the Python implementation.

Simulated scenario test datasets were created from one-dimensional unsteady Hydrologic Engineering Center River Analysis System (HEC–RAS;

A prismatic channel geometry (_{0} and

Representative cross section for simulated test datasets.

Figure 2. Graph showing representative cross section for simulated test datasets.

Stage plotted against four variables.

Figure 3. Graphs showing stage plotted against four variables.

Different inflow hydrographs were developed for the evaluation to test a range of unsteadiness in the simulated responses from the three computation methods: HEC–RAS, DYNMOD, and DYNPOUND. A normal depth boundary condition was used at the downstream end in each scenario with the appropriate _{0} assigned to the normal depth relation.

All scenarios were simulated in HEC–RAS. The HEC–RAS computed stage and discharge time series at the cross-section midway (40 miles from the inflow point) between the most upstream and most downstream cross sections of the 80-mile reach were extracted and used for computation and comparison of the discharge with the dynamic rating methods. The midpoint cross section was selected to reduce the effects of the boundary conditions on the simulation results. The Manning’s _{0}, and

The width of the main channel of the simulation cross section was 300 ft, the flood plains have a total width of 600 ft, and the total width of the cross section was 900 ft. The bankfull depth of the main channel was 30 ft. The subsection stations coincide with the bank stations at 300 and 600 ft (

Table 1. Bed slope and ratio of bed slope to average wave slope of simulated test data scenarios.

[

Scenario | Bed slope | |

1 | 0.0001 | 10 |

2 | 0.0001 | 100 |

3 | 0.001 | 10 |

4 | 0.001 | 100 |

Four hydrographs, which were used as upstream boundary conditions for each scenario, were developed to simulate stage and discharge time series under varied channel slope and unsteadiness conditions in the test scenarios (

Simulated time series for simulated test scenario 1.

Figure 4. Graphs showing simulated time series for simulated test scenario 1.

Simulated time series for simulated test scenario 2.

Figure 5. Graphs showing simulated time series for simulated test scenario 2.

Simulated time series for simulated test scenario 3.

Figure 6. Graphs showing simulated time series for simulated test scenario 3.

Simulated time series for simulated test scenario 4.

Figure 7. Graphs showing simulated time series for simulated test scenario 4.

Both the DYNMOD and DYNPOUND methods performed well in comparison to the full one-dimensional unsteady flow equations within HEC–RAS because the mean percent error was within about 5.8 percent (

Table 2. Performance statistics for the DYNMOD and DYNPOUND discharge computation methods.

[DYNMOD, the dynamic rating method that computes discharge from stage for compact channel geometry; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; MSLE, mean squared logarithmic error]

Scenario | DYNMOD | DYNPOUND | ||||

Mean percent error | Maximum absolute percent error | MSLE | Mean percent error | Maximum absolute percent error | MSLE | |

1 | −3.91 | 49.3 | 3.40×10^{−2} |
0.444 | 10.4 | 1.91×10^{−4} |

2 | −5.81 | 45.7 | 3.35×10^{−2} |
−0.0100 | 0.723 | 8.24×10^{−7} |

3 | −1.16 | 41.2 | 2.75×10^{−2} |
0.0370 | 2.73 | 4.31×10^{−5} |

4 | −4.78 | 43.4 | 3.13×10^{−2} |
−0.00572 | 0.358 | 2.51×10^{−7} |

Scenarios 1 (

In all test scenarios, jumps in the discharge computed with the DYNMOD method are observed in the time series. The first jump from a higher to a lower discharge takes place when the stage rises from below to above the channel bank elevation, and the second jump from a lower to higher discharge takes place once the elevation falls below the bank elevation. This jump takes place because of abrupt changes in the relations of top width, wetted perimeter, and area with stage, as indicated in

Overall, the magnitude of the mean percent error is much greater in the results computed with the DYNMOD method when compared to the mean percent error in the results computed with the DYNPOUND method. The error is smaller in the time series computed with the DYNPOUND method because this method relies on the conveyance, as well as area and top width (see

In all scenarios, the DYNMOD computed discharge has a maximum percent error ranging from about 40 to almost 50 percent that takes place when the water surface rises above and falls below the channel bank elevations. The DYNMOD method performs about the same in all four test scenarios, whereas the DYNPOUND method performs better in scenarios 2 and 4 than it does in scenarios 1 and 3 (

The scenario 1 (^{−2} (^{−1} percent and a MSLE of 1.91×10^{−4} (

Time series for simulated scenario 1.

Figure 8. Graphs showing time series for simulated scenario 1.

Relation between stage and computed discharge for simulated scenario 1.

Figure 9. Graphs showing relation between stage and computed discharge for simulated scenario 1.

The results of scenario 2 indicate a lack of hysteresis of the computed hydrograph but substantial relative error in the DYNMOD solution compared to the HEC–RAS and DYNPOUND methods (^{−2} (^{−2} percent and a MSLE of 8.24×10^{−7}.

Time series for simulated scenario 2.

Figure 10. Graphs showing time series for simulated scenario 2.

Relation between stage and computed discharge for simulated scenario 2.

Figure 11. Graphs showing relation between stage and computed discharge for simulated scenario 2.

The stage/discharge relation of the HEC–RAS computed values for scenario 3 shows hysteresis (^{−2} (^{−5}.

Time series for simulated scenario 3.

Figure 12. Graphs showing time series for simulated scenario 3.

Relation between stage and computed discharge for simulated scenario 3.

Figure 13. Graphs showing relation between stage and computed discharge for simulated scenario 3.

The stage/discharge relation of the scenario 4 time series (^{−2} (^{−7}.

Time series for simulated scenario 4.

Figure 14. Graphs showing time series for simulated scenario 4.

Relation between stage and computed discharge for simulated scenario 4.

Figure 15. Graphs showing relation between stage and computed discharge for simulated scenario 4.

Discrete discharge measurements and stage time series from six USGS streamgages were used for evaluating the DYNMOD and DYNPOUND methods (

Site datasets consisted of discharge, stage, and field measurements; cross-section geometry; and bed slope. Discharge, stage, and field measurements were obtained from the National Water Information System (NWIS;

Locations of U.S. Geological Survey streamgage sites used in the evaluation of discharge computed with the dynamic rating methods.

Figure 16. Map showing locations of U.S. Geological Survey streamgage sites used in the evaluation of discharge computed with the dynamic rating methods.

Table 3. Station number and name, drainage area, and slope of the field sites used in the evaluation of the DYNMOD and DYNPOUND dynamic rating methods.

[mi^{2}, square mile; DYNMOD, the dynamic rating method that computes discharge from stage for compact channel geometry; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels]

Station number | Station name | Drainage area^{2}) |
Bed slope^{1} |

03214500 | Tug Fork at Kermit, West Virginia | 1,277 | 0.000352 |

04156000 | Tittabawassee River at Midland, Michigan | 2,400 | 0.000139 |

05054000 | Red River of the North at Fargo, North Dakota | 6,800 | 0.000145 |

07010000 | Mississippi River at St. Louis, Missouri | 697,000 | 0.000110 |

08263500 | Rio Grande near Cerro, New Mexico | 8,440 | 0.00360 |

11254000 | San Joaquin River near Mendota, California | 3,940 | 0.000248 |

Bed slope was calculated using elevation contour and topographical maps and the National Hydrography Dataset (

To obtain cross-section geometry, an ADCP discharge measurement was selected for each site that generally corresponded with a peak flow event. The ADCP measurement was then converted into a “station, depth” coordinate format and imported into AreaComp2. AreaComp2 was used to convert the “station, depth” coordinates to “station, elevation” (in stage datum) coordinates. Next, a digital elevation model (DEM;

Cross sections derived from acoustic doppler profiler (ADCP) and digital elevation model (DEM) for Tittabawassee River at Midland, Michigan (U.S. Geological Survey station 04156000).

Figure 17. Graph showing cross sections derived from acoustic doppler profiler and digital elevation model for Tittabawassee River at Midland, Michigan.

Combined cross section for Tittabawassee River at Midland, Michigan (U.S. Geological Survey station 04156000).

Figure 18. Graph showing combined cross section for Tittabawassee River at Midland, Michigan.

The cross section created from the ADCP measurement most accurately represented the channel, and the cross section created from the DEM was used to represent the larger flood plain. In some cases, a measurement was made at a high enough stage to capture some of the flood plain. If this was the case, then “station, elevation” (in stage datum) coordinates from the ADCP measurement superseded the overlapping coordinates from the DEM. Difficulties were encountered in geospatially locating ADCP transects at sites without geolocation data associated with ADCP measurements. If a cross section created from the DEM was used exclusively, it generally lacked accurate channel geometry. If a cross section created from the ADCP was used, it generally lacked flood-plain geometry. In some cases, cross sections with these limitations did not produce accurate results in either of the dynamic rating computation methods. When there was a lack of associated geolocation data for an ADCP measurement, and the location of the cross section was otherwise unable to be located, the cross-section geometry was taken from the DEM exclusively and properties of the channel geometry were estimated.

Bed slope (_{0}) was calculated using elevation contour and historical topographical maps and the National Hydrography Dataset flowline shapefiles (_{0}, the following equation was used:

is bed slope, in feet per feet;

is upstream elevation, in feet;

is downstream elevation, in feet; and

is length of reach, in feet.

Discharge time series were computed with the dynamic rating methods at gaged field sites where cross-section geometry was created and real-time stage and discharge data were being collected. Discharge time series were computed at six streamgage sites in California, Michigan, Missouri, New Mexico, North Dakota, and West Virginia.

For all sites, an event was chosen for computing the _{0}. The peak stage from this event is _{p}_{0} is

Subsections were added to the cross section for each site based on the need to (i) develop a smooth conveyance and stage relation, (ii) add regions where transitions in roughness can be made in the flood plain, (iii) remove those areas of the cross section that do not contribute to the momentum of the flow, and (iv) allow for computation of the non-uniform velocity distribution coefficient.

To determine the smoothness of the conveyance/stage relation, conveyance was plotted against stage and visually analyzed. If an abrupt change in slope of the relation was observed, the cross section was analyzed for sudden changes in geometry. Existing subsections that contained sudden changes in geometry were split into two subsections by inserting a split where the changes take place.

A full water year of stage and discharge time series was chosen for calibrating the method at each site (a water year is defined as the 12-month period, October 1 through September 30, and is designated by calendar year in which it ends). At least five field measurements were compared for calibration at each of the six streamgages for which results are discussed. To calibrate the computations for a site, the Manning’s

After the calibration for the site was completed, a different period of time in the record was selected to evaluate the method. For each site, the evaluation time period is after the calibration time period, which contains a substantial amount of field measurements and a wide range of observed discharge values. Discharge values from field measurements were then compared to dynamic rating computed discharge at the corresponding times.

In some cases, stage data were missing from the observed time series. As long as the period of missing data was short, the missing stage data were estimated through linear interpolation.

The USGS streamgage Tug Fork at Kermit, West Virginia (USGS station 03214500) represents an upstream basin of 1,277 square miles (mi^{2}). The computed bed slope for the site is 0.000352 (

Cross section used in the computation of the discharge time series for the Tug Fork at Kermit, West Virginia (U.S. Geological Survey station 03214500). Elevation is referenced to 574.07 feet above North American Vertical Datum of 1988.

Figure 19. Graph showing cross section used in the computation of the discharge time series for the Tug Fork at Kermit, West Virginia.

Six discharge measurements from the 2016 water year were used for calibration. The stage time series for the 2016 water year was used to compute discharge (

The MSLE for the DYNMOD calibration was 3.38×10^{-3}, and the MSLE for the DYNPOUND calibration was 3.70×10^{−3}. The mean percent error of the DYNMOD calibration was 2.13 percent, and the mean percent error of the DYNPOUND calibration was 0.0158 percent (

Table 4. Calibration results for the DYNMOD and DYNPOUND ratings at the Tug Fork at Kermit, West Virginia.

[Observed discharge data from ^{3}/s, cubic feet per second; DYNMOD, the dynamic rating method that computes discharge from stage for compact channel geometry; SLE, squared logarithmic error; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNMOD discharge^{3}/s) |
DYNMOD error |
DYNMOD SLE | DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

11/3/2015 | 19:48 | 344 | 394 | 14.6 | 1.84×10^{−2} |
390 | 13.6 | 1.58×10^{−2} |

1/13/2016 | 19:22 | 641 | 643 | 0.444 | 9.70×10^{−6} |
636 | −0.662 | 6.13×10^{−5} |

3/22/2016 | 17:48 | 1,090 | 1,053 | −3.34 | 1.19×10^{−3} |
1,036 | −4.91 | 2.58×10^{−3} |

5/12/2016 | 19:11 | 7,540 | 7,636 | 1.28 | 1.60×10^{−4} |
7,141 | −5.28 | 2.96×10^{−3} |

7/21/2016 | 18:53 | 805 | 817 | 1.53 | 2.19×10^{−4} |
806 | 0.167 | 1.54×10^{−6} |

9/19/2016 | 17:40 | 688 | 676 | −1.71 | 3.10×10^{−4} |
668 | −2.82 | 8.70×10^{−4} |

-- | -- | -- | 2.13 | 3.38×10^{−3} |
-- | 0.0158 | 3.70×10^{−3} |

For evaluation of the ratings for the USGS streamgage Tug Fork at Kermit, West Virginia, discharge was computed for the time period between February 1 and March 1, 2018, which also included five field measurements (^{−2} and 1.77×10^{−2} for the DYNMOD and DYNPOUND computed time series, respectively (

Table 5. Discharge computed for an event-based time series at the Tug Fork at Kermit, West Virginia, with the DYNMOD and DYNPOUND methods and the associated error.

[Observed discharge data from ^{3}/s, cubic feet per second; DYNMOD, the dynamic rating method that computes discharge from stage for compact channel geometry; SLE, squared logarithmic error; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNMOD discharge^{3}/s) |
DYNMOD error |
DYNMOD SLE | DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

2/5/2018 | 18:51 | 3,030 | 2,814 | −7.10 | 5.47×10^{−3} |
2,725 | −10.0 | 1.13×10^{−2} |

2/11/2018 | 19:24 | 27,600 | 30,932 | 12.1 | 1.30×10^{−2} |
28,975 | 4.98 | 2.36×10^{−3} |

2/11/2018 | 20:52 | 29,100 | 33,620 | 15.5 | 2.08×10^{−2} |
31,394 | 7.88 | 5.76×10^{−3} |

2/11/2018 | 22:21 | 34,400 | 36,496 | 6.09 | 3.50×10^{−3} |
34,040 | −1.04 | 1.11×10^{−4} |

2/12/2018 | 20:59 | 22,100 | 30,995 | 40.2 | 1.14×10^{−1} |
28,746 | 30.1 | 6.91×10^{−2} |

-- | -- | -- | 13.4 | 3.14×10^{−2} |
-- | 6.38 | 1.77×10^{−2} |

Discharge time series computed with the DYNMOD and DYNPOUND methods shown with the U.S. Geological Survey-computed discharge time series and field measurements made at Tug Fork at Kermit, West Virginia (U.S. Geological Survey station 03214500).

Figure 20. Graph showing discharge time series computed with the DYNMOD and DYNPOUND methods shown with the U.S. Geological Survey-computed discharge time series and field measurements made at Tug Fork at Kermit, West Virginia.

Stage/discharge relation of the discharge computed with the DYNMOD and DYNPOUND methods shown with U.S. Geological Survey-computed discharge and field measurements made at Tug Fork at Kermit, West Virginia (U.S. Geological Survey station 03214500). Elevation is referenced to 574.07 feet above North American Vertical Datum of 1988.

Figure 21. Graph showing stage/discharge relation of the discharge computed with the DYNMOD and DYNPOUND methods shown with U.S. Geological Survey-computed discharge and field measurements made at Tug Fork at Kermit, West Virginia.

The USGS streamgage Tittabawassee River at Midland, Michigan (USGS station 04156000) represents an area of 2,400 mi^{2}. The computed bed slope for the site is 0.000139 (

Cross section used in the computation of the discharge time series for the Tittabawassee River at Midland, Michigan (U.S. Geological Survey station 04156000). Elevation is referenced to 579.47 feet above North American Vertical Datum of 1988.

Figure 22. Graph showing cross section used in the computation of the discharge time series for the Tittabawassee River at Midland, Michigan.

Nine discharge measurements collected during the 2017 water year were used for calibration (^{−2} and the mean percent error was −2.36 percent (

Table 6. Calibration results for the DYNPOUND rating at the Tittabawassee River at Midland, Michigan.

[Observed discharge data from ^{3}/s, cubic feet per second; SLE, squared logarithmic error; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND |

10/12/2016 | 15:22 | 905 | 1,097 | 21.3 | 3.70×10^{−2} |

12/1/2016 | 18:14 | 2,960 | 2,292 | −22.5 | 6.54×10^{−2} |

1/27/2017 | 19:03 | 6,020 | 4,502 | −25.2 | 8.44×10^{−2} |

3/16/2017 | 17:15 | 2,290 | 1,843 | −19.5 | 4.72×10^{−2} |

5/10/2017 | 15:51 | 2,040 | 1,774 | −13.0 | 1.95×10^{−2} |

6/24/2017 | 15:12 | 37,700 | 39,831 | 5.65 | 3.02×10^{−3} |

6/24/2017 | 16:58 | 38,800 | 40,035 | 3.19 | 9.82×10^{−4} |

6/26/2017 | 15:31 | 19,100 | 19,871 | 4.04 | 1.57×10^{−3} |

8/25/2017 | 11:40 | 765 | 954 | 24.8 | 4.87×10^{−2} |

-- | -- | -- | −2.36 | 3.42×10^{−2} |

Discharge measurements and discharge time series computed with DYNPOUND and the simple rating method were evaluated for the time period between May 1 and June 30, 2020 (

The discharge values computed with DYNPOUND were compared to three field measurements and indicated the DYNPOUND method had a mean percent error of 5.58 percent and an MSLE of 3.64×10^{−3} (

Table 7. Discharge computed with the DYNPOUND method and associated error for an event-based time series at Tittabawassee River at Midland, Michigan.

[Observed discharge data from ^{3}/s, cubic feet per second; SLE, squared logarithmic error; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

5/20/2020 | 17:48 | 51,500 | 53,537 | 3.96 | 1.50×10^{−3} |

5/20/2020 | 19:31 | 49,900 | 51,475 | 3.16 | 9.66×10^{−4} |

5/21/2020 | 19:18 | 30,100 | 32,996 | 9.62 | 8.44×10^{−3} |

-- | -- | -- | 5.58 | 3.64×10^{−3} |

Discharge time series computed with the DYNPOUND method shown with the rated discharge time series and field measurements made at Tittabawassee River at Midland, Michigan (U.S. Geological Survey station 04156000).

Figure 23. Graph showing discharge time series computed with the DYNPOUND method shown with the rated discharge time series and field measurements made at Tittabawassee River at Midland, Michigan.

Stage/discharge relation of the discharge computed with the DYNPOUND method shown with U.S. Geological Survey-computed discharge and field measurements made at the Tittabawassee River at Midland, Michigan (U.S. Geological Survey station 03214500).

Figure 24. Graph showing stage/discharge relation of the discharge computed with the DYNPOUND method shown with U.S. Geological Survey-computed discharge and field measurements made at the Tittabawassee River at Midland, Michigan.

The USGS streamgage Red River of the North at Fargo, North Dakota (U.S. Geological Survey station 05054000) represents a basin draining an area of 6,800 mi^{2}. The computed slope for the site is 0.000145 (

Cross section used in the computation of the discharge time series for the Red River of the North at Fargo, North Dakota (U.S. Geological Survey station 05054000). Elevation is referenced to 862.88 feet above North American Vertical Datum of 1988.

Figure 25. Graph showing cross section used in the computation of the discharge time series for the Red River of the North at Fargo, North Dakota.

Twelve field measurements collected during the 2019 water year were used for calibration. The stage time series from the 2019 water year was used to compute discharge. The DYNMOD method failed to compute discharge during the entire period of calibration, so the calibration results for the DYNMOD method are not quantified. The cross section was split into four subsections for the DYNPOUND computation. Subsection stations are 620, 875, and 1,090 feet. The calibrated Manning’s ^{−1} and the mean percent error was 61.9 percent (

Table 8. Calibration results for the DYNPOUND rating at the Red River of the North at Fargo, North Dakota.

[Observed discharge data from ^{3}/s, cubic feet per second; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; SLE, Squared logarithmic error; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

11/2/2018 | 16:13 | 434 | 1,586 | 266 | 1.68 |

1/23/2019 | 18:00 | 442 | 1,621 | 267 | 1.69 |

3/5/2019 | 23:15 | 432 | 1,621 | 275 | 1.75 |

4/2/2019 | 22:28 | 7,340 | 4,655 | −36.6 | 2.07×10^{−1} |

4/5/2019 | 17:42 | 15,500 | 14,683 | −5.27 | 2.93×10^{−3} |

4/6/2019 | 18:23 | 17,200 | 18,066 | 5.04 | 2.41×10^{−3} |

4/7/2019 | 18:50 | 19,500 | 19,759 | 1.33 | 1.74×10^{−4} |

4/8/2019 | 19:49 | 19,200 | 20,329 | 5.88 | 3.26×10^{−3} |

4/15/2019 | 23:04 | 11,400 | 11,952 | 4.84 | 2.24×10^{−3} |

4/23/2019 | 21:32 | 13,000 | 13,318 | 2.45 | 5.84×10^{−4} |

6/11/2019 | 17:27 | 3,390 | 2,332 | −31.2 | 1.40×10^{−1} |

7/23/2019 | 15:34 | 2,450 | 2,179 | −11.1 | 1.37×10^{−2} |

-- | -- | -- | 61.9 | 4.57×10^{−1} |

A discharge time series was computed with DYNPOUND for the time period between March 16 and May 5, 2020, and used to evaluate the generated rating with six field measurements collected during this period (^{−2}. As shown graphically in the time series and stage/discharge plots, and quantitatively by the mean percent error, the DYNPOUND computed discharge is substantially lower than the observed discharge during the time period.

Table 9. Discharge computed with the DYNPOUND method and associated error for an event-based time series at the Red River of the North at Fargo, North Dakota.

[Observed discharge data from ^{3}/s, cubic feet per second; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; SLE, Squared logarithmic error; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

3/26/2020 | 17:32 | 4,250 | 2,944 | −30.7 | 1.35×10^{−1} |

3/31/2020 | 18:03 | 11,700 | 9,692 | −17.2 | 3.55×10^{−2} |

4/4/2020 | 17:36 | 7,750 | 7,630 | −1.54 | 2.44×10^{−4} |

4/9/2020 | 14:51 | 10,300 | 9,208 | −10.6 | 1.26×10^{−2} |

4/15/2020 | 16:58 | 5,990 | 4,232 | −29.3 | 1.21×10^{−1} |

5/1/2020 | 15:44 | 3,400 | 2,370 | −30.3 | 1.30×10^{−1} |

-- | -- | -- | −19.9 | 7.23×10^{−2} |

Discharge time series computed with the DYNPOUND method shown with the rated discharge time series and field measurements made at the Red River of the North at Fargo, North Dakota (U.S. Geological Survey station 05054000).

Figure 26. Graph showing discharge time series computed with the DYNPOUND method shown with the rated discharge time series and field measurements made at the Red River of the North at Fargo, North Dakota.

Stage/discharge relation of the discharge computed with the DYNPOUND method shown with U.S. Geological Survey-computed discharge and field measurements made at the Red River of the North at Fargo, North Dakota (U.S. Geological Survey station 05054000). Stage is referenced to 862.88 feet above North American Vertical Datum of 1988.

Figure 27. Graph showing Stage/discharge relation of the discharge computed with the DYNPOUND method shown with U.S. Geological Survey-computed discharge and field measurements made at the Red River of the North at Fargo, North Dakota.

The USGS streamgage Mississippi River at St. Louis, Missouri (USGS station 07010000) represents a basin draining an area of 697,000 mi^{2}. The bed slope for the site is 0.000110 (

Cross section used in the computation of the discharge time series for the Mississippi River at St. Louis, Missouri (U.S. Geological Survey station 07010000). Elevation is referenced to 379.58 feet above North American Vertical Datum of 1988.

Figure 28. Graph showing cross section used in the computation of the discharge time series for the Mississippi River at St. Louis, Missouri.

Eleven discharge measurements collected during the 2014 water year were used for calibration of the dynamic ratings for the Mississippi River at St. Louis, Missouri, streamgage. The stage time series from the 2014 water year was used to compute discharge. The roughness coefficient used in the DYNMOD computations varied with stage (

Relation between stage and roughness coefficient (Manning’s

Figure 29. Relation between stage and roughness coefficient used in the computation of discharge with the DYNMOD method for the Mississippi River at St. Louis, Missouri.

For the DYNMOD calibration, the MSLE was 2.19×10^{−3} and the mean percent error was 1.82 percent. For the DYNPOUND calibration, the MSLE was 3.02×10^{−2} and the mean percent error was 1.42 percent (

Table 10. Calibration results for the DYNMOD and DYNPOUND ratings at the Mississippi River at St. Louis, Missouri.

[Observed discharge data from ^{3}/s, cubic feet per second; DYNMOD, the dynamic rating method that computes discharge from stage for compact channel geometry; SLE, squared logarithmic error; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNMOD discharge^{3}/s) |
DYNMOD error |
DYNMOD SLE | DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

10/31/2013 | 18:05 | 98,500 | 106,547 | 8.17 | 6.17×10^{−3} |
123,851 | 25.7 | 5.25×10^{−2} |

11/21/2013 | 17:57 | 106,000 | 105,171 | −0.782 | 6.16×10^{−5} |
122,172 | 15.3 | 2.02×10^{−2} |

1/15/2014 | 19:24 | 99,100 | 95,117 | 4.02 | 1.68×10^{−3} |
114,120 | 15.2 | 1.99×10^{−2} |

2/20/2014 | 16:50 | 92,500 | 93,874 | 1.49 | 2.17×10^{−4} |
114,548 | 23.8 | 4.57×10^{−2} |

3/13/2014 | 16:34 | 167,000 | 172,994 | 3.59 | 1.24×10^{−3} |
170,104 | 1.86 | 3.39×10^{−4} |

4/10/2014 | 18:13 | 245,000 | 257,606 | 5.15 | 2.52×10^{−3} |
226,891 | −7.39 | 5.90×10^{−3} |

5/21/2014 | 14:54 | 331,000 | 326,187 | 1.45 | 2.15×10^{−4} |
276,064 | −16.6 | 3.29×10^{−2} |

6/5/2014 | 17:19 | 291,000 | 296,933 | 2.04 | 4.07×10^{−4} |
257,341 | −11.6 | 1.51×10^{−2} |

7/10/2014 | 18:19 | 555,000 | 527,770 | 4.91 | 2.53×10^{−3} |
412,971 | −25.6 | 8.74×10^{−2} |

8/14/2014 | 18:37 | 137,000 | 150,668 | 9.98 | 9.04×10^{−3} |
154,331 | 12.7 | 1.42×10^{−2} |

9/18/2014 | 15:09 | 400,000 | 403,045 | 0.761 | 5.75×10^{−5} |
328,723 | −17.8 | 3.85×10^{−2} |

-- | -- | -- | 1.82 | 2.19×10^{−3} |
-- | 1.42 | 3.02×10^{−2} |

To evaluate the dynamic rating methods for this site, discharge for the time period between June 1 and August 15, 2015, was computed and observed, and computed discharge values were compared for 68 field discharge measurements. The mean percent error for the discharge computed with the DYNMOD method was 2.18 percent and the mean percent error for the DYNPOUND method was −22.1 percent (^{−3} and the 7.11×10^{−2} for the DYNPOUND results. The discharge computed with the DYNPOUND method was considerably lower than the observed discharge, whereas the DYNMOD computed discharge matches much better with the observed discharge values (

Although the mean percent error of the calibration for the DYNPOUND method shows low bias during the water year of calibration, the magnitude of discharge observed during the event time series was greater than the DYNPOUND computed discharge. It was possible to calibrate the DYNPOUND method by minimizing the MSLE for water year 2015, and the calibration resulted in a small mean percent error, but the magnitude of the largest percent error was 29.8 percent. The current implementation of the DYNPOUND does not allow for a stage/roughness relation to be specified in the computation of discharge. Because of this, the calibration of the DYNPOUND method resulted in low bias but large error during a full water year. Large discharge values are observed during the event time series used in the evaluation of the dynamic rating methods, so large errors occur in the discharge computed with the DYNPOUND method. Implementation of the feature that allows for a specification of a stage/roughness relation, as is possible with the DYNMOD method, may provide the ability to refine the DYNPOUND calibration and reduce the errors in the computed time series.

Table 11. Discharge computed for an event-based time series at the Mississippi River at St. Louis, Missouri, with the DYNMOD and DYNPOUND methods and the associated error.

[Observed discharge data from ^{3}/s, cubic feet per second; DYNMOD, the dynamic rating method that computes discharge from stage for compact channel geometry; SLE, squared logarithmic error; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNMOD discharge^{3}/s) |
DYNMOD error |
DYNMOD SLE | DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

6/10/2015 | 15:29 | 513,000 | 521,007 | 1.56 | 2.40×10^{−4} |
407,697 | −20.5 | 5.28×10^{−2} |

6/11/2015 | 15:09 | 490,000 | 506,108 | 3.29 | 1.05×10^{−3} |
396,416 | −19.1 | 4.49×10^{−2} |

6/17/2015 | 22:52 | 576,000 | 579,642 | 0.632 | 3.97×10^{−5} |
446,629 | −22.5 | 6.47×10^{−2} |

6/18/2015 | 23:06 | 605,000 | 604,613 | −0.0639 | 4.09×10^{−7} |
460,026 | −24.0 | 7.50×10^{−2} |

6/19/2015 | 17:09 | 633,000 | 648,606 | 2.47 | 5.93×10^{−4} |
490,626 | −22.5 | 6.49×10^{−2} |

6/20/2015 | 16:19 | 672,000 | 662,168 | −1.46 | 2.17×10^{−4} |
493,240 | −26.6 | 9.56×10^{−2} |

6/21/2015 | 16:49 | 677,000 | 670,074 | −1.02 | 1.06×10^{−4} |
496,378 | −26.7 | 9.63×10^{−2} |

6/21/2015 | 17:12 | 670,000 | 667,185 | −0.42 | 1.77×10^{−5} |
492,984 | −26.4 | 9.41×10^{−2} |

6/21/2015 | 17:32 | 691,000 | 669,176 | −3.16 | 1.03×10^{−3} |
495,147 | −28.3 | 1.11×10^{−1} |

6/21/2015 | 17:48 | 693,000 | 670,667 | −3.22 | 1.07×10^{−3} |
496,766 | −28.3 | 1.11×10^{−1} |

6/21/2015 | 18:02 | 685,000 | 671,277 | −2.00 | 4.10×10^{−4} |
497,512 | −27.4 | 1.02×10^{−1} |

6/21/2015 | 18:15 | 696,000 | 667,850 | −4.04 | 1.70×10^{−3} |
494,296 | −29.0 | 1.17×10^{−1} |

6/21/2015 | 18:29 | 685,000 | 664,333 | −3.02 | 9.39×10^{−4} |
490,997 | −28.3 | 1.11×10^{−1} |

6/21/2015 | 18:43 | 688,000 | 660,654 | −3.97 | 1.64×10^{−3} |
487,546 | −29.1 | 1.19×10^{−1} |

6/21/2015 | 18:57 | 690,000 | 657,248 | −4.75 | 2.36×10^{−3} |
484,350 | −29.8 | 1.25×10^{−1} |

6/21/2015 | 19:10 | 691,000 | 658,895 | −4.65 | 2.26×10^{−3} |
486,067 | −29.7 | 1.24×10^{−1} |

6/21/2015 | 20:05 | 690,000 | 670,037 | −2.89 | 8.62×10^{−4} |
497,257 | −27.9 | 1.07×10^{−1} |

6/21/2015 | 20:18 | 687,000 | 668,785 | −2.65 | 7.22×10^{−4} |
495,843 | −27.8 | 1.06×10^{−1} |

6/21/2015 | 20:32 | 682,000 | 667,462 | −2.13 | 4.64×10^{−4} |
494,348 | −27.5 | 1.04×10^{−1} |

6/21/2015 | 20:46 | 690,000 | 666,155 | −3.46 | 1.24×10^{−3} |
492,872 | −28.6 | 1.13×10^{−1} |

6/21/2015 | 21:00 | 690,000 | 664,979 | −3.63 | 1.36×10^{−3} |
491,539 | −28.8 | 1.15×10^{−1} |

6/21/2015 | 21:13 | 689,000 | 666,017 | −3.34 | 1.15×10^{−3} |
492,603 | −28.5 | 1.13×10^{−1} |

6/21/2015 | 21:27 | 690,000 | 667,053 | −3.33 | 1.14×10^{−3} |
493,665 | −28.5 | 1.12×10^{−1} |

6/21/2015 | 22:47 | 686,000 | 655,874 | −4.39 | 2.02×10^{−3} |
483,251 | −29.6 | 1.23×10^{−1} |

6/21/2015 | 23:00 | 670,000 | 652,326 | −2.64 | 7.15×10^{−4} |
479,899 | −28.4 | 1.11×10^{−1} |

6/21/2015 | 23:14 | 664,000 | 653,527 | −1.58 | 2.53×10^{−4} |
481,333 | −27.5 | 1.04×10^{−1} |

6/21/2015 | 23:27 | 667,000 | 654,723 | −1.84 | 3.45×10^{−4} |
482,762 | −27.6 | 1.05×10^{−1} |

6/21/2015 | 23:43 | 668,000 | 656,184 | −1.77 | 3.19×10^{−4} |
484,505 | −27.5 | 1.03×10^{−1} |

6/22/2015 | 0:02 | 678,000 | 658,076 | −2.94 | 8.90×10^{−4} |
486,669 | −28.2 | 1.10×10^{−1} |

6/22/2015 | 0:18 | 668,000 | 660,762 | −1.08 | 1.19×10^{−4} |
489,298 | −26.8 | 9.69×10^{−2} |

6/22/2015 | 14:56 | 649,000 | 656,674 | 1.18 | 1.38×10^{−4} |
487,324 | −24.9 | 8.21×10^{−2} |

6/23/2015 | 21:48 | 649,000 | 670,035 | 3.24 | 1.02×10^{−3} |
499,177 | −23.1 | 6.89×10^{−2} |

6/24/2015 | 14:27 | 652,000 | 662,714 | 1.64 | 2.66×10^{−4} |
491,487 | −24.6 | 7.99×10^{−2} |

6/27/2015 | 0:52 | 642,000 | 645,670 | 0.572 | 3.25×10^{−5} |
481,318 | −25.0 | 8.30×10^{−2} |

6/27/2015 | 18:16 | 662,000 | 652,519 | −1.43 | 2.08×10^{−4} |
487,020 | −26.4 | 9.42×10^{−2} |

6/28/2015 | 21:13 | 700,000 | 672,513 | −3.93 | 1.60×10^{−3} |
498,927 | −28.7 | 1.15×10^{−1} |

7/1/2015 | 19:17 | 705,000 | 687,226 | −2.52 | 6.52×10^{−4} |
502,675 | −28.7 | 1.14×10^{−1} |

7/2/2015 | 18:04 | 642,000 | 675,390 | 5.20 | 2.57×10^{−3} |
496,564 | −22.7 | 6.60×10^{−2} |

7/3/2015 | 17:40 | 647,000 | 671,405 | 3.77 | 1.37×10^{−3} |
494,712 | −23.5 | 7.20×10^{−2} |

7/4/2015 | 23:57 | 632,000 | 660,803 | 4.56 | 1.99×10^{−3} |
488,807 | −22.7 | 6.60×10^{−2} |

7/5/2015 | 15:22 | 623,000 | 650,799 | 4.46 | 1.91×10^{−3} |
483,875 | −22.3 | 6.39×10^{−2} |

7/7/2015 | 16:07 | 555,000 | 581,397 | 4.76 | 2.16×10^{−3} |
442,827 | −20.2 | 5.10 ×10^{−2} |

7/7/2015 | 19:37 | 542,000 | 572,602 | 5.65 | 3.02×10^{−3} |
436,160 | −19.5 | 4.72×10^{−2} |

7/8/2015 | 16:07 | 500,000 | 550,114 | 10.0 | 9.12×10^{−3} |
425,120 | −15.0 | 2.63×10^{−2} |

7/9/2015 | 15:44 | 529,000 | 561,777 | 6.20 | 3.61×10^{−3} |
434,365 | −17.9 | 3.88×10^{−2} |

7/10/2015 | 14:47 | 583,000 | 609,643 | 4.57 | 2.00×10^{−3} |
463,880 | −20.4 | 5.22×10^{−2} |

7/11/2015 | 15:47 | 631,000 | 639,584 | 1.36 | 1.83×10^{−4} |
480,126 | −23.9 | 7.47×10^{−2} |

7/12/2015 | 23:45 | 592,000 | 607,837 | 2.68 | 6.97×10^{−4} |
457,480 | −22.7 | 6.64×10^{−2} |

7/13/2015 | 19:15 | 567,000 | 598,859 | 5.62 | 2.99×10^{−3} |
455,084 | −19.7 | 4.83×10^{−2} |

7/14/2015 | 15:14 | 565,000 | 593,089 | 4.97 | 2.35×10^{−3} |
450,621 | −20.2 | 5.12×10^{−2} |

7/15/2015 | 14:34 | 550,000 | 603,343 | 9.70 | 8.57×10^{−3} |
458,817 | −16.6 | 3.29×10^{−2} |

7/16/2015 | 14:08 | 561,000 | 587,649 | 4.75 | 2.15×10^{−3} |
446,697 | −20.4 | 5.19×10^{−2} |

7/17/2015 | 14:30 | 509,000 | 543,720 | 6.82 | 4.35×10^{−3} |
417,243 | −18.0 | 3.95×10^{−2} |

7/18/2015 | 16:31 | 467,000 | 506,263 | 8.41 | 6.52×10^{−3} |
396,197 | −15.2 | 2.70×10^{−2} |

7/19/2015 | 16:31 | 467,000 | 497,223 | 6.47 | 3.93×10^{−3} |
392,552 | −15.9 | 3.02×10^{−2} |

7/21/2015 | 19:36 | 537,000 | 555,475 | 3.44 | 1.14×10^{−3} |
428,400 | −20.2 | 5.10×10^{−2} |

7/24/2015 | 15:48 | 500,000 | 533,626 | 6.73 | 4.24×10^{−3} |
413,031 | −17.4 | 3.65×10^{−2} |

7/25/2015 | 15:21 | 454,000 | 494,058 | 8.82 | 7.15×10^{−3} |
387,429 | −14.7 | 2.51×10^{−2} |

7/26/2015 | 15:25 | 412,000 | 450,730 | 9.40 | 8.07×10^{−3} |
360,010 | −12.6 | 1.82×10^{−2} |

7/27/2015 | 14:24 | 412,000 | 447,976 | 8.73 | 7.01×10^{−3} |
361,637 | −12.2 | 1.70×10^{−2} |

7/28/2015 | 14:14 | 422,000 | 452,366 | 7.20 | 4.83×10^{−3} |
363,577 | −13.8 | 2.22×10^{−2} |

7/29/2015 | 13:53 | 428,000 | 453,158 | 5.88 | 3.26×10^{−3} |
363,829 | −15.0 | 2.64×10^{−2} |

7/31/2015 | 18:33 | 407,000 | 443,599 | 8.99 | 7.41×10^{−3} |
357,040 | −12.3 | 1.72×10^{−2} |

8/3/2015 | 18:11 | 375,000 | 401,511 | 7.07 | 4.67×10^{−3} |
329,327 | −12.2 | 1.69×10^{−2} |

8/5/2015 | 17:14 | 346,000 | 369,670 | 6.84 | 4.38×10^{−3} |
307,679 | −11.1 | 1.38×10^{−2} |

8/6/2015 | 15:09 | 320,000 | 365,868 | 14.3 | 1.79×10^{−2} |
305,039 | −4.68 | 2.29×10^{−3} |

8/7/2015 | 16:36 | 318,000 | 336,926 | 5.95 | 3.34×10^{−3} |
284,208 | −10.6 | 1.26×10^{−2} |

8/11/2015 | 17:21 | 233,000 | 273,525 | 17.4 | 2.57×10^{−2} |
239,576 | 2.82 | 7.75×10^{−4} |

-- | -- | -- | 2.18 | 2.76×10^{−3} |
-- | −22.1 | 7.11×10^{−2} |

Discharge time series computed with the DYNMOD and DYNPOUND methods shown with the U.S. Geological Survey-computed discharge time series and field measurements made at the Mississippi River at St. Louis, Missouri (U.S. Geological Survey station 07010000).

Figure 30. Graph showing discharge time series computed with the DYNMOD and DYNPOUND methods shown with the U.S. Geological Survey-computed discharge time series and field measurements made at the Mississippi River at St. Louis, Missouri.

Stage/discharge relation of the discharge computed with the DYNMOD and DYNPOUND methods shown with U.S. Geological Survey-computed discharge and field measurements made at the Mississippi River at St. Louis, Missouri (U.S. Geological Survey station 07010000). Stage is referenced to 379.58 feet above North American Vertical Datum of 1988.

Figure 31. Graph showing stage/discharge relation of the discharge computed with the DYNMOD and DYNPOUND methods shown with U.S. Geological Survey-computed discharge and field measurements made at the Mississippi River at St. Louis, Missouri.

The USGS streamgage Rio Grande near Cerro, New Mexico (USGS station 08263500) represents a basin draining an area of 8,440 mi^{2}. The bed slope for the site is computed as 0.00360 (

Cross section used in the computation of the discharge time series for the Rio Grande near Cerro, New Mexico (U.S. Geological Survey station 08263500). Elevation is referenced to 7,110 feet above National Geodetic Vertical Datum of 1929.

Figure 32. Graph showing cross section used in the computation of the discharge time series for the Rio Grande near Cerro, New Mexico.

Seven field measurements of discharge collected during the 2015 water year were used for calibration. The stage time series from the 2015 water year was used to compute discharge. No subdivision of the cross section (

The MSLE and mean percent error for the DYNMOD method were 9.77×10^{−3} and −2.95 percent, respectively. The calibration results for the DYNPOUND method were 1.02×10^{−2} and −3.86 percent, respectively (

Table 12. Calibration results for the DYNMOD and DYNPOUND ratings at the Rio Grande Near Cerro, New Mexico.

^{3}/s, cubic feet per second; DYNMOD, the dynamic rating method that computes discharge from stage for compact channel geometry; SLE, squared logarithmic error; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNMOD discharge^{3}/s) |
DYNMOD error |
DYNMOD SLE | DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

10/9/2014 | 15:02 | 306 | 299 | −2.05 | 5.36×10^{−4} |
296 | −2.99 | 1.10×10^{−3} |

12/4/2014 | 17:25 | 451 | 465 | 3.29 | 9.35×10^{−4} |
460 | 2.19 | 3.90×10^{−4} |

1/26/2015 | 17:16 | 273 | 274 | 0.381 | 1.34×10^{−5} |
271 | −0.558 | 5.41×10^{−5} |

2/26/2015 | 16:16 | 349 | 353 | 1.15 | 1.30×10^{−4} |
349 | 0 | 0 |

4/14/2015 | 17:59 | 189 | 166 | −11.7 | 1.68×10^{−2} |
165 | −12.4 | 1.84×10^{−2} |

7/30/2015 | 16:18 | 381 | 408 | 7.11 | 4.69×10^{−3} |
403 | 6.00 | 3.15×10^{−3} |

8/20/2015 | 14:50 | 167 | 135 | −18.8 | 4.52×10^{−2} |
134 | −19.4 | 4.85×10^{−2} |

-- | -- | -- | −2.95 | 9.77×10^{−3} |
-- | −3.86 | 1.02×10^{−2} |

The time period between April 1 and August 5, 2019, was used to evaluate the discharge computed with the dynamic rating methods for this site. Although only two measurements were made during the event time period, both dynamic rating methods were biased high (^{−2} and the MSLE for the DYNMOD results is 7.96×10^{−3} (

Discharge time series computed with the DYNMOD and DYNPOUND methods shown with the U.S. Geological Survey-computed discharge time series and field measurements made at the Rio Grande near Cerro, New Mexico (U.S. Geological Survey station 08263500).

Figure 33. Graph showing discharge time series computed with the DYNMOD and DYNPOUND methods shown with the U.S. Geological Survey-computed discharge time series and field measurements made at the Rio Grande near Cerro, New Mexico.

Stage/discharge relation of the discharge computed with the DYNMOD and DYNPOUND methods shown with U.S. Geological Survey-computed discharge and field measurements made at Rio Grande near Cerro, New Mexico (U.S. Geological Survey station 08263500). Elevation is referenced to 7,110 feet above National Geodetic Vertical Datum of 1929.

Figure 34. Graph showing stage/discharge relation of the discharge computed with the DYNMOD and DYNPOUND methods shown with U.S. Geological Survey-computed discharge and field measurements made at Rio Grande near Cerro, New Mexico.

Table 13. Discharge computed with the DYNMOD and DYNPOUND methods and associated error for an event-based time series at the Rio Grande near Cerro, New Mexico.

^{3}/s, cubic feet per second; DYNMOD, the dynamic rating method that computes discharge from stage for compact channel geometry; SLE, squared logarithmic error; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNMOD discharge^{3}/s) |
DYNMOD error |
DYNMOD SLE | DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

4/26/2019 | 16:20 | 776 | 861 | 11.0 | 1.08×10^{−2} |
850 | 9.62 | 8.30×10^{−3} |

5/29/2019 | 16:00 | 1,370 | 1,521 | 11.1 | 1.09×10^{−2} |
1,495 | 9.18 | 7.62×10^{−3} |

-- | -- | -- | 11.05 | 1.09×10^{−2} |
-- | 9.40 | 7.96×10^{−3} |

The USGS streamgage San Joaquin River near Mendota, California (U.S. Geological Survey station 11254000) represents a basin draining an area of 3,940 mi^{2}. The computed slope for the site is 0.000248 (

Cross section used in the computation of the discharge time series for the San Joaquin River Near Mendota, California (U.S. Geological Survey station 11254000). Elevation is referenced to 133.35 feet above North American Vertical Datum of 1988.

Figure 35. Graph showing Cross section used in the computation of the discharge time series for the San Joaquin River Near Mendota, California.

Nine discharge measurements collected during the 2017 water year were used for calibration. The stage time series from the 2017 water year was used to compute discharge. The DYNMOD method failed to compute discharge the entire period of calibration, so the calibration results for the DYNMOD method are not quantified. The cross section for the DYNPOUND analyses was subdivided into three subsections, with the subsection between 540 and 700 feet being the only subsection that is used in the computation of discharge. A Manning’s ^{3}/s, and the DYNPOUND computed discharge was 17 ft^{3}/s.

Table 14. Calibration results for the DYNPOUND rating at the San Joaquin River near Mendota, California.

[Observed discharge data from ^{3}/s, cubic feet per second; SLE, squared logarithmic error; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

10/6/2016 | 15:49 | 347 | 350 | 0.989 | 7.41×10^{−5} |

12/1/2016 | 17:22 | 22 | 34 | 58.4 | 1.90×10^{−1} |

1/10/2017 | 20:48 | 0.67 | 17 | 2,530 | 1.05×10^{1} |

1/26/2017 | 0:21 | 767 | 783 | 2.19 | 4.26×10^{−4} |

2/22/2017 | 22:45 | 3,840 | 6,362 | 65.7 | 2.55×10^{−1} |

4/4/2017 | 16:40 | 3,320 | 5,354 | 61.3 | 2.28×10^{−1} |

5/26/2017 | 17:58 | 273 | 359 | 31.7 | 7.50×10^{−2} |

7/21/2017 | 15:05 | 567 | 621 | 9.62 | 8.28×10^{−3} |

9/21/2017 | 22:51 | 421 | 434 | 3.27 | 9.25×10^{−4} |

-- | -- | -- | 307 | 1.25 |

Discharge was computed for the time period between May 20 and July 10, 2019, to evaluate the DYNPOUND method at San Joaquin River near Mendota, California. Two field measurements are available for comparison during this time period. The DYNPOUND computed discharge is much higher than the observed and rated discharges at higher stage (^{−2} (

Discharge time series computed with the DYNPOUND method shown with the U.S. Geological Survey-computed time series and field measurements made at the San Joaquin River near Mendota, California (U.S. Geological Survey station 11254000).

Figure 36. Graph showing discharge time series computed with the DYNPOUND method shown with the U.S. Geological Survey-computed time series and field measurements made at the San Joaquin River near Mendota, California.

Stage/discharge relation of the discharge computed with the DYNPOUND method shown with U.S. Geological Survey-computed discharge and field measurements made at the San Joaquin River near Mendota, California (U.S. Geological Survey station 11254000). Stage is referenced to 133.35 feet above North American Vertical Datum of 1988.

Figure 37. Graph showing stage/discharge relation of the discharge computed with the DYNPOUND method shown with U.S. Geological Survey-computed discharge and field measurements made at the San Joaquin River near Mendota, California.

Table 15. Discharge computed with the DYNPOUND method and associated error for an event-based time series at San Joaquin River near Mendota, California.

[Observed discharge data from ^{3}/s, cubic feet per second; DYNPOUND, the newly developed method that solves for discharge in compact and compound channels; SLE, squared logarithmic error; --, not applicable]

Measurement date |
Measurement time |
Observed discharge^{3}/s) |
DYNPOUND discharge^{3}/s) |
DYNPOUND error |
DYNPOUND SLE |

6/9/2019 | 18:32 | 1,610 | 2,129 | 32.3 | 7.81×10^{−2} |

6/24/2019 | 18:40 | 627 | 655 | 4.50 | 1.91×10^{−3} |

-- | -- | -- | 18.4 | 4.00×10^{−2} |

Through the course of developing and testing the dynamic rating method, the following are suggested best practices for using this method.

Select an appropriate cross section for characterization of the channel geometry described as follows:

Select a reach where the flow is approximately one-dimensional (flow is orthogonal to the banks).

The flow direction should be well established in a one-dimensional nature. A rule of thumb for an ideal cross section would be one that is straight at least 100 times the bank full depth upstream and 100 times the bank full depth downstream.

Avoid cross sections within a river reach with abrupt changes in cross-sectional geometry.

If possible, select multiple flood events to compute and assess the value of

Create the channel cross-section geometry properties and ensure that the stage/conveyance curve is smooth by subdividing the cross section.

Choose a series of high-flow events to use as calibration for the Manning’s roughness coefficient by the DYNMOD and DYNPOUND methods. Evaluate the methods using a different set of high-flow events.

For compact channels (those without flood plains), although DYNPOUND was written for complex channels (those with flood plains), DYNPOUND may perform better than DYNMOD. As such, try both methods for all sites.

Ratings are used for a variety of reasons in water-resources investigations, but a predominant use of ratings is at streamgages, where autonomously collected stage is converted to discharge by use of a rating. In the absence of direct discrete continuous discharge measurements, discharge typically is determined through surrogate measures of one or more variables such as stage, water-surface slope, rate of change in stage, or index velocity collected at a streamgage. The rating is developed and calibrated using discharge measurements collected onsite by field staff. The simplest rating relates discharge to stage of the river (simple rating). Hydrologists and engineers have long recognized hysteresis (loops) exist in relations between stage and discharge. The hysteresis is sometimes small enough that it is hidden within the error of the measurements. Likewise, when the time of reporting of the discharge is large enough, the hysteresis averages out. For some sites, simple ratings work well. Simple ratings do not work as well for streamgages on low-gradient streams, streams with variable backwater, streams with large amounts of channel or overbank storage, streams with highly unsteady flow, or streams with highly mobile beds. In these cases, a complex rating is often needed. A complex rating relates discharge to stage and other variables because of the lack of a unique, univariate relation between stage and discharge. A dynamic rating is a rating that accounts for a variable energy slope owing to unsteady flow accelerations.

A previously developed dynamic rating method (DYNMOD) to compute discharge from a stage time series, which was developed for compact channel geometry, was described. A new dynamic rating method (DYNPOUND), which has been developed for compact and compound channel geometry, was introduced in this report. The mathematical formulation for DYNPOUND was derived and a numerical solution method was then formulated and described.

Discharge time series computed with the DYNMOD and DYNPOUND rating methods were compared to results computed from the one-dimensional unsteady shallow water equations. The simulated discharge time series, and corresponding simulated stage time series, were generated using one-dimensional hydraulic modeling software. The time series were simulated using a prismatic channel created from a compound cross section. Four scenarios were created using two different bed slopes and four different hydrographs that serve as the upstream boundary conditions. The hydrographs were created to capture a range of unsteadiness in the flow conditions. In the comparison, the discharge computed with the DYNPOUND method had a lower mean squared logarithmic error (MSLE) than discharge computed with the DYNMOD method in all scenarios. The MSLE for the DYNPOUND computed discharge ranges from 2.51×10^{−7} to 1.91×10^{−4}, and the MSLE computed from the DYNMOD computed discharge ranges from 2.75×10^{−2} to 3.40×10^{−2}. In all cases, the DYNPOUND method outperformed the DYNMOD method.

The results computed with the dynamic rating methods were then compared to field data previously collected at U.S. Geological Survey streamgage sites. Six streamgage sites were chosen for comparison. Cross-section geometry for the streamgage sites was created by combining “station, elevation” coordinates from acoustic doppler current profiler discharge measurements with digital elevation data. Coordinate data were extracted from previously collected discharge measurements. Bed slopes for the sites were estimated from topographic maps. Continuous stage, needed to compute discharge with the dynamic rating methods, were obtained from the U.S. Geological Survey National Water Information System database. Field measurements, which were used to calibrate and evaluate the performance of the dynamic rating methods, were also obtained from the National Water Information System database, along with discharge time series, which were computed with more traditional simple rating methods.

Dynamic ratings were developed and calibrated for each site. Calibration was accomplished by adjusting ^{−3} to 9.77×10^{−3}, and the DYNPOUND calibration has a range of 3.7×10^{−3} to 1.25.

One event-based time period was chosen for each site to evaluate the calibration of the dynamic rating methods. For each dynamic rating method, the calibrated rating was used, along with the stage time series from the time period, to compute a discharge time series. The DYNMOD method was not used to compute discharge for sites at which the method failed to compute the full calibration time period. The range of MSLE for the DYNMOD time series is 2.73×10^{−3} to 3.14×10^{−2}, and the range of MSLE for the DYNPOUND method is 3.64×10^{−3} to 7.23×10^{−2}. For sites that DYNMOD successfully computed the full calibration time series, DYNMOD performs better than DYNPOUND when using MSLE as a performance metric. However, DYNMOD failed to compute calibration time series at 3 out of 6 field sites, which indicates DYNPOUND is a more robust method. Additionally, improvements, including variable roughness with stage, can be made to the implementation of the DYNPOUND method, which may improve the computed results.

The field offices of the California Water Science Center, Caribbean-Florida Water Science Center, Central Midwest Water Science Center, Dakota Water Science Center, Lower Mississippi-Gulf Water Science Center, Nebraska Water Science Center, New England Water Science Center, New Mexico Water Science Center, Ohio-Kentucky-Indiana Water Science Center, Oregon Water Science Center, Upper Midwest Water Science Center, Utah Water Science Center, Virginia and West Virginia Water Science Center, and Washington Water Science Center were instrumental in providing site data for this project.

For more information about this publication, contact

Director, USGS Central Midwest Water Science Center

405 North Goodwin

Urbana, IL 61801

217–328–8747

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