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Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data

Marian Domanski, Robert R. Holmes, Elizabeth N. Heal, Travis M. Knight, E.H. Elias, A. Rango · U.S. Geological Survey

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Groundwater and Streamflow Information Program

Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data

Scientific Investigations Report 2024–5129

U.S. Department of the Interior U.S. Geological Survey

Cover. U.S. Geological Survey hydrologic technician, Peyton Hendrix, measures discharge with acoustic Doppler current profiler on the Comite River in Louisiana during the 2016 flood. Photograph by James Fountain, U.S. Geological Survey Surface Water Specialist.

Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data By Marian M. Domanski, Robert R. Holmes, Jr., Elizabeth N. Heal, and Travis M. Knight

Groundwater and Streamflow Information Program

Scientific Investigations Report 2024–5129

U.S. Department of the Interior U.S. Geological Survey

U.S. Geological Survey, Reston, Virginia: 2025

For more information on the USGS—the Federal source for science about the Earth, its natural and living resources, natural hazards, and the environment—visit https://www.usgs.gov or call 1–888–392–8545. For an overview of USGS information products, including maps, imagery, and publications, visit https://store.usgs.gov/ or contact the store at 1–888–275–8747. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. Although this information product, for the most part, is in the public domain, it also may contain copyrighted materials as noted in the text. Permission to reproduce copyrighted items must be secured from the copyright owner. Suggested citation: Domanski, M.M., Holmes, R.R., Jr., Heal, E.N., and Knight, T.M., 2025, Dynamic rating method for computing discharge and stage from time-series data: U.S. Geological Survey Scientific Investigations Report 2024–5129, 92 p., https://doi.org/10.3133/sir20245129. Associated data for this publication: Domanski, M.M., Holmes, R.R., and Heal, E.N., 2022, Dynamic rating method for computing discharge from time series stage data-site datasets: U.S. Geological Survey data release, https://doi.org/10.5066/P955QRPQ. Domanski, M.M., Holmes, R.R., and Heal, E.N., 2022, Dynamic stage to discharge rating model archive: U.S. Geological Survey data release, https://doi.org/10.5066/P9YUV9DG. Domanski, M.M., Holmes, R.R., Heal, E.N., and Knight, T.M., 2025, Dynamic rating model archive: U.S. Geological Survey data release, https://doi.org/10.5066/P9X7QZFG. Knight, T. M., Domanski, M.M., and Heal, E.N., 2025, DynRat: U.S. Geological Survey software release, https://doi.org/10.5066/P13FOADX. ISSN 2328-0328 (online)

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Contents Abstract��1 Introduction��1 Dynamic Rating Method Theory��4 Solution Method��6 Evaluation Using Model-Generated Test Scenarios��7 Dataset Development��8 Evaluation��10 Scenario 1��10 Scenario 2��10 Scenario 3��10 Scenario 4��10 Evaluation Using Field Data��23 Dataset Development��24 Cross-Section Geometry��24 Bed Slope��26 Evaluation��26 Meherrin River near Bryants Corner, Virginia��27 Tug Fork at Kermit, West Virginia��27 Tittabawassee River at Midland, Michigan��30 Red River of the North at Fargo, North Dakota��34 Papillion Creek at Fort Crook, Nebraska��36 Gasconade River at Jerome, Missouri��38 Mississippi River at St. Louis, Missouri��40 Calcasieu River near Kinder, Louisiana��46 Rio Grande Near Cerro, New Mexico��46 San Joaquin River Near Mendota, California��49 Dynamic Rating Application Guidelines��89 Summary��89 Acknowledgments��90 References Cited��90

Figures 1. Graph showing 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 for the inflow hydrograph at the upstream end��3 2. Graph showing a representative cross section for simulated test datasets��8 3. Graphs showing stage plotted against four variables��9 4. Graphs showing time series for simulated test scenario 1��11 5. Graphs showing time series for simulated test scenario 2��12 6. Graphs showing time series for simulated test scenario 3��13

7. Graphs showing time series for simulated test scenario 4��14 8. Graphs showing time series for simulated scenario 1��16 9. Graph showing the relation between stage and computed discharge and discharge and computed stage for simulated scenario 1��17 10. Graphs showing time series for simulated scenario 2��18 11. Graph showing the relation between stage and computed discharge and discharge and computed stage for simulated scenario 2��19 12. Graphs showing time series for simulated scenario 3��20 13. Graph showing the relation between stage and computed discharge and discharge and computed stage for simulated scenario 3��21 14. Graphs showing time series for simulated scenario 4��22 15. Graph showing the relation between stage and computed discharge and discharge and computed stage for simulated scenario 4��23 16. Map of U.S. Geological Survey streamgage sites used to evaluate discharge computed with the dynamic rating methods��24 17. Graph showing cross sections derived from an acoustic doppler profiler and a digital elevation model at Tittabawassee River at Midland, Michigan��25 18. Graph showing combined cross section at Tittabawassee River at Midland, Michigan��26 19. Graph showing the cross section used to compute the stage and discharge time series at Meherrin River near Bryants Corner, Virginia��28 20. Graph showing the discharge time series computed with the DYNPOUND method shown with the time series of WSC-computed discharge and field measurements made at Meherrin River near Bryants Corner, Virginia��31 21. Graph showing the stage time series computed with the DYNPOUND method shown with the time series of WSC-measured stage and field measurements made at Meherrin River near Bryants Corner, Virginia��32 22. Graph showing the stage-discharge relation at Meherrin River near Bryants Corner, Virginia, using discharge computed with the DYNPOUND method, WSC-computed discharge, and field measurements��33 23. Graph showing the cross section used to compute the stage and discharge time series at Tug Fork at Kermit, West Virginia��35 24. Graph showing the discharge time series computed with the DYNPOUND method shown with the time series of WSC-computed discharge and field measurements made at Tug Fork at Kermit, West Virginia��37 25. Graph showing the stage time series computed with the DYNPOUND method shown with the time series of WSC-measured stage and field measurements made at Tug Fork at Kermit, West Virginia��38 26. Graph showing the stage-discharge relation at Tug Fork at Kermit, West Virginia, using discharge computed with the DYNPOUND method, WSC-computed discharge, and field measurements��39 27. Graph showing the cross section used to compute the stage and discharge time series for the Tittabawassee River at Midland, Michigan��41 28. Graph showing the discharge time series computed with the DYNPOUND method shown with the time series of WSC-computed discharge and field measurements made at Tittabawassee River at Midland, Michigan��43 29. Graph showing the stage time series computed with the DYNPOUND method shown with the time series of WSC-measured and field measurements made at Tittabawassee River at Midland, Michigan��44

30. Graph showing the stage-discharge relation at Tittabawassee River at Midland, Michigan, using discharge computed with the DYNPOUND method, WSC-computed discharge, and field measurements��45 31. Graph showing the cross section used to compute the discharge time series at Red River of the North at Fargo, North Dakota��47 32. Graph showing the discharge time series computed with the DYNPOUND method shown with the time series of WSC-computed discharge and field measurements made at Red River of the North at Fargo, North Dakota��50 33. Graph showing the stage-discharge relation at Red River of the North at Fargo, North Dakota, using discharge computed with the DYNPOUND method, WSC-computed discharge, and field measurements��51 34. Graph showing the cross section used to compute the stage and discharge time series at Papillion Creek at Fort Crook, Nebraska��52 35. Graph showing the discharge time series computed with the DYNPOUND method shown with the time series of WSC-computed discharge and field measurements made at Papillion Creek at Fort Crook, Nebraska��54 36. Graph showing the stage time series computed with the DYNPOUND method shown with the time series of WSC-measured stage and field measurements made at Papillion Creek at Fort Crook, Nebraska��55 37. Graph showing the stage-discharge relation at Papillion Creek at Fort Crook, Nebraska, using discharge computed with the DYNPOUND method, WSC-computed discharge, and field measurements��56 38. Graph showing the cross section used to compute the discharge time series at Gasconade River at Jerome, Missouri��58 39. Graph showing the discharge time series computed with the DYNPOUND method shown with the time series of WSC-computed discharge and field measurements made at Gasconade River at Jerome, Missouri��60 40. Graph showing the stage time series computed with the DYNPOUND method shown with the time series of WSC-measured stage and field measurements made at Gasconade River at Jerome, Missouri��61 41. Graph showing stage-discharge relation at Gasconade River at Jerome, Missouri, using discharge computed with the DYNPOUND method, WSC-computed discharge, and field measurements��62 42. Graph showing the cross section used to compute the discharge time series at Mississippi River at St. Louis, Missouri��64 43. Graph showing the discharge time series computed with the DYNPOUND method shown with the WSC-computed discharge time series and field measurements made at Mississippi River at St. Louis, Missouri��66 44. Graph showing the stage time series computed with the DYNPOUND method shown with the time series of WSC-measured stage and field measurements made at Mississippi River at St. Louis, Missouri��67 45. Graph showing the stage-discharge relation at Mississippi River at St. Louis, Missouri, using discharge computed with the DYNPOUND method, WSC-computed discharge, and field measurements��68 46. Graph showing the cross section used to compute the stage and discharge time series at Calcasieu River near Kinder, Louisiana��73 47. Graph showing the discharge time series computed with the DYNPOUND method shown with the time series of WSC-computed discharge and field measurements made at Calcasieu River near Kinder, Louisiana��75

48. Graph showing the stage time series computed with the DYNPOUND method shown with the time series of WSC-measured stage and field measurements made at Calcasieu River near Kinder, Louisiana��76 49. Graph showing the stage-discharge relation at Calcasieu River near Kinder, Louisiana, using discharge computed with the DYNPOUND method, WCS-computed discharge, and field measurements��77 50. Graph showing the cross section used to compute the stage and discharge time series at Rio Grande near Cerro, New Mexico��78 51. Graph showing the discharge time series computed with the DYNPOUND method shown with the time series of WSC-computed discharge and field measurements made at Rio Grande near Cerro, New Mexico��80 52. Graph showing stage time series computed with the DYNPOUND method shown with the time series of WSC-measured stage and field measurements made at Rio Grande near Cerro, New Mexico��81 53. Graph showing the stage-discharge relation at Rio Grande near Cerro, New Mexico, using discharge computed with the DYNPOUND method, WSC-computed discharge, and field measurements��82 54. Graph showing the cross section used to compute the stage and discharge time series at San Joaquin River Near Mendota, California��84 55. Graph showing the discharge time series computed with the DYNPOUND method shown with the time series of WSC-computed discharge and field measurements made at San Joaquin River near Mendota, California��86 56. Graph showing the stage time series computed with the DYNPOUND method shown with the time series of WSC-measured stage and field measurements made at San Joaquin River near Mendota, California��87 57. Graph showing the stage-discharge relation at San Joaquin River near Mendota, California, using discharge computed with the DYNPOUND method, WSC-computed discharge, and field measurements��88

Tables 1. Bed slope and ratio of bed slope to average wave slope of simulated test data scenarios��9 2. Performance statistics for the DYNPOUND computation method��15 3. Streamgage number and name, drainage area, and slope of the field sites used to evaluate the DYNPOUND dynamic rating method��25 4. Stage and roughness coefficient values used to calibrate the DYNPOUND method at Meherrin River near Bryants Corner, Virginia��28 5. Discharge calibration results for the DYNPOUND ratings at Meherrin River near Bryants Corner, Virginia��29 6. Stage calibration results for the DYNPOUND ratings at Meherrin River near Bryants Corner, Virginia��30 7. Discharge computed for an event-based time series at Meherrin River near Bryants Corner, Virginia, with the DYNPOUND methods and the associated error��34 8. Stage computed for an event-based time series at Meherrin River near Bryants Corner, Virginia, with the DYNPOUND methods and the associated error��34 9. Stage and roughness coefficient values used to calibrate the DYNPOUND method at Tug Fork at Kermit, West Virginia��35

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10. Calibration results for the DYNPOUND discharge ratings at Tug Fork at Kermit, West Virginia��36 11. Calibration results for the DYNPOUND stage ratings at Tug Fork at Kermit, West Virginia��36 12. Discharge computed for an event-based time series at Tug Fork at Kermit, West Virginia, with the DYNPOUND methods and the associated error��40 13. Stage computed for an event-based time at the streamgage at Tug Fork at Kermit, West Virginia, with the DYNPOUND methods and the associated error��40 14. Stage and roughness coefficient values used to calibrate the DYNPOUND method at Tittabawassee River at Midland, Michigan��41 15. Discharge calibration results for the DYNPOUND ratings at Tittabawassee River at Midland, Michigan��42 16. Stage calibration results for the DYNPOUND ratings at Tittabawassee River at Midland, Michigan��42 17. Discharge computed with the DYNPOUND method and associated error for an event-based time series at Tittabawassee River at Midland, Michigan��46 18. Stage computed with the DYNPOUND method and associated error for an event-based time series at Tittabawassee River at Midland, Michigan��46 19. Stage and roughness coefficient values used to calibrate the DYNPOUND method at Red River of the North at Fargo, North Dakota��47 20. Discharge calibration results for the DYNPOUND ratings at Red River of the North at Fargo, North Dakota��48 21. Stage calibration results for the DYNPOUND ratings at Red River of the North at Fargo, North Dakota��48 22. Discharge computed with the DYNPOUND method and associated error for an event-based time series at Red River of the North at Fargo, North Dakota��49 23. Stage and roughness coefficient values used to calibrate the DYNPOUND method at Papillion Creek at Fort Crook, Nebraska��52 24. Discharge calibration results for the DYNPOUND ratings at Papillion Creek at Fort Crook, Nebraska��53 25. Stage calibration results for the DYNPOUND ratings at Papillion Creek at Fort Crook, Nebraska��53 26. Discharge computed for an event-based time series at Papillion Creek at Fort Crook, Nebraska, with the DYNPOUND methods and the associated error��57 27. Stage computed for an event-based time series at Papillion Creek at Fort Crook, Nebraska, with the DYNPOUND methods and the associated error��57 28. Stage and roughness coefficient values used to calibrate the DYNPOUND method at Gasconade River at Jerome, Missouri��58 29. Discharge calibration results for the DYNPOUND ratings at Gasconade River at Jerome, Missouri��59 30. Stage calibration results for the DYNPOUND ratings at Gasconade River at Jerome, Missouri ��59 31. Discharge computed for an event-based time series at Gasconade River at Jerome, Missouri, with the DYNPOUND methods and the associated error��63 32. Stage computed for an event-based time series at Gasconade River at Jerome, Missouri, with the DYNPOUND methods and the associated error��63 33. Stage and roughness coefficient values used to calibrate the DYNPOUND method at Mississippi River at St. Louis, Missouri��64

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34. Discharge calibration results for the DYNPOUND ratings at Mississippi River at St. Louis, Missouri��65 35. Stage calibration results for the DYNPOUND ratings at Mississippi River at St. Louis, Missouri��65 36. Discharge computed for an event-based time series at Mississippi River at St. Louis, Missouri, with the DYNPOUND method and the associated error��69 37. Stage computed for an event-based time series at Mississippi River at St. Louis, Missouri, with the DYNPOUND method and the associated error��71 38. Stage and roughness coefficient values used to calibrate the DYNPOUND method at Calcasieu River near Kinder, Louisiana��73 39. Discharge calibration results for the DYNPOUND ratings at Calcasieu River near Kinder, Louisiana��74 40. Stage calibration results for the DYNPOUND ratings at Calcasieu River near Kinder, Louisiana ��74 41. Discharge computed for an event-based time series at Calcasieu River near Kinder, Louisiana, with the DYNPOUND methods and the associated error��78 42. Stage computed for an event-based time series at Calcasieu River near Kinder, Louisiana, with the DYNPOUND methods and the associated error��78 43. Stage and roughness coefficient values used to calibrate the DYNPOUND method at Rio Grande near Cerro, New Mexico��79 44. Discharge calibration results for the DYNPOUND ratings at Rio Grande near Cerro, New Mexico��79 45. Stage calibration results for the DYNPOUND ratings at Rio Grande near Cerro, New Mexico ��79 46. Discharge computed with the DYNPOUND method and associated error for an event-based time series at Rio Grande near Cerro, New Mexico��83 47. Stage computed with the DYNPOUND method and associated error for an event-based time series at Rio Grande near Cerro, New Mexico��83 48. Stage and roughness coefficient values used to calibrate the DYNPOUND method at San Joaquin River near Mendota, California��84 49. Discharge calibration results for the DYNPOUND ratings at San Joaquin River near Mendota, California��85 50. Stage calibration results for the DYNPOUND ratings at San Joaquin River near Mendota, California��85 51. Discharge computed with the DYNPOUND method and associated error for an event-based time series at San Joaquin River near Mendota, California��89 52. Stage computed with the DYNPOUND method and associated error for an event-based time series at San Joaquin River near Mendota, California��89

Conversion Factors U.S. customary units to International System of Units

Multiply

To obtain

Length foot (ft)

0.3048

meter (m)

mile (mi)

1.609

kilometer (km)

Area square foot (ft2)

929.0

square foot (ft2)

0.09290

square mile (mi2)

259.0

square mile (mi2)

2.590

square centimeter (cm2) square meter (m2) hectare (ha) square kilometer (km2)

Flow rate foot per second (ft/s)

0.3048

meter per second (m/s)

cubic foot per second (ft3/s)

0.02832

cubic meter per second (m3/s)

Datum 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).

Supplemental Information A water year is the 12-month period from October 1 through September 30 of the following year and is designated by the calendar year in which it ends.

Abbreviations ADCP

acoustic Doppler current profiler

DEM

digital elevation model

DYNMOD

Fread’s original method

DYNPOUND

dynamic rating method

HEC–RAS

Hydrologic Engineering Center River Analysis System

MSLE

mean squared logarithmic error

USGS

U.S. Geological Survey

UTC

coordinated universal time

WSC

U.S. Geological Survey Water Science Center

Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data By Marian M. Domanski,1 Robert R. Holmes, Jr.,2 Elizabeth N. Heal,3 and Travis M. Knight4

Abstract Ratings are used for several reasons in water-resources investigations. The simplest rating relates discharge to the stage of a river (the stage-discharge relation). From a pure hydrodynamics perspective, all rivers and streams have some form of hysteresis in the relation between stage and discharge because flow becomes unsteady as a flood wave passes. The stage-discharge relation is unable to represent hysteresis. However, a dynamic rating method can capture hysteresis, which is driven by the variable energy slope of a flood wave. A dynamic rating method called DYNPOUND, which accommodates compact and compound channel geometry, was developed by simplifying the one-dimensional Saint-Venant equations. The DYNPOUND method was developed in the Python programming language and computes discharge from stage and stage from discharge. Stage and discharge time series computed with this dynamic rating method were compared to the U.S. Geological Survey (USGS) published stage and discharge time series. The results from the DYNPOUND method were also compared to in-person field measurements of stage and discharge made at 10 USGS streamgages. DYNPOUND was calibrated for 10 USGS streamgages using published discharge time-series data computed with a simple rating method. The calibration objective was to minimize the mean squared logarithmic error (MSLE) of the DYNPOUND-computed discharge with respect to the discharge time series computed by a simple rating method. For each site, the calibration process also included comparing all field measurements within a selected water year to the corresponding DYNPOUND-computed discharge data points. The MSLE of the DYNPOUND-computed discharge time series for the 10 sites ranged from 8.51×10−4 to 1.36×10−1. For each site, an event-based period was selected to compare the discharge time series computed with the dynamic rating method to discharge field measurements 1Former U.S. Geological Survey. 2Retired U.S. Geological Survey. 3Lower Mississippi-Gulf Water Science Center. 4Hydrologic Networks Branch.

made at the streamgages; the range of MSLE for the 10 DYNPOUND-computed discharge sites was from 4.79×10−4 to 2.30×10−2.

Introduction A relation using a continuous surrogate measure to estimate discharge is termed a “rating.” Ratings are used for a variety of reasons in water-resources investigations, but they are predominantly used at streamgages, where autonomously measured stage is used to compute discharge by use of a rating (Kennedy, 1984). No widely accepted method for direct discrete continuous measurement of discharge in natural channels is available. Commonly then, the rating is developed and calibrated using discharge measurements made onsite by field staff. When direct discrete continuous discharge measurements are not available, discharge is typically determined by continuous surrogate measures of one or more variables such as stage, water-surface slope, rate of change in stage, or index velocity; all measurements of these surrogate variables are collected at a streamgage. The derivation of discharge through these surrogate variables uses various models to create and implement the rating (Rantz and others, 1982). The simplest rating relates discharge to stage of the river (simple rating). Hydrologists and engineers have long recognized hysteresis (loop effect) in relations between stage and discharge (Jones, 1915; Corbett, 1943; Fread, 1973; Faye and Cherry, 1980; Rantz and others, 1982; Kennedy, 1984). From a hydrodynamics perspective, disregarding channel-bed mobility, all rivers and streams have some form of hysteresis (loop effect) in the relation between stage and discharge. This also applies to prismatic channels without floodplains because flow is unsteady as the flood wave passes (fig. 1). The hysteresis is sometimes small enough to be hidden within the error of the measurements. Likewise, when the discharge event period is long enough, the hysteresis averages out. For example, a mean daily discharge value will often mitigate the effects of hysteresis, which are more evident in instantaneous hourly or 15-minute discharge values as explained in Faye and Cherry (1980, p. 19):

2 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data (The hysteretic relation of stage to discharge indicates that estimates of instantaneous dynamic discharge based on rating curves can be [substantially] in error. On the other hand, estimaes 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 reasons, simple ratings are often adequate to compute discharge for most streamgages. Simple ratings do not work when a unique relation between stage and discharge is lacking, such as 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 (Holmes, 2017). In these situations, a complex rating is often required. A complex rating relates discharge to stage and other variables because of the lack of a unique, univariate relation between stage and discharge. Complex rating methods vary from simply adding a second independent variable in the process of computing discharge to sophisticated computer models solving the Saint-Venant equations, which are conservation-of-momentum and conservation-of-mass partial differential equations (French, 1985). For the governing differential assumptions, Fread

(1973) developed what was termed a “dynamic loop” rating method for channels with compact geometry (no floodplain); this method computes discharge from a time series of stage measurements at a single streamgage. This rating method accounts for the variable energy slope defined by Fread (1975, p. 214) as being “associated with the dynamic inertia and pressure forces of the unsteady flood discharge” as opposed to rating loops imposed by alluvial bedform dynamics or scour and fill processes. This report documents the development and testing of an expansion of Fread’s (1973) original dynamic loop method that includes channels with noncompact channel geometry (channels with floodplains). Testing the expanded method consists of comparing DYNPOUND-computed discharge and stage to simulated and U.S. Geological Survey (USGS) Water Science Center (WSC)-computed discharge and WSC-measured stage. Simulated discharge and stage time series were generated using the modeling software, Hydrologic Engineering Center River Analysis System (HEC– RAS; U.S. Army Corps of Engineers, 2016), that computes results using the one-dimensional Saint-Venant equations. Discrete discharge measurements and the associated stage value provide the observed (field measurement) discharge at streamgage sites.

Introduction 3 20

Stage, in feet

0 0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

Discharge, in cubic feet per second EXPLANATION Bed slope (Domanski and others, 2022a) 5 feet per mile, mild unsteadiness for inflow hydrograph 1 foot per mile, mild unsteadiness for inflow hydrograph 5 feet per mile, moderate unsteadiness for inflow hydrograph

Figure 1. Graph showing 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.

4 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data

Dynamic Rating Method Theory By making simplifying assumptions, Fread (1973) used the conservation of mass and momentum equations to develop a method to estimate the friction slope from a single streamgage’s time series of stage and knowledge of how the flood wave moved through a short section of channel at the streamgage location. The simplifying assumptions made are as follows: 1. lateral inflow and outflow are negligible;

is the non-uniform velocity distribution coefficient;

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

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

is the friction slope, in feet per feet (dimensionless).

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

The variable β is the non-uniform velocity distribution coefficient and is defined by equation 3 for a cross section divided into N discrete subsections, where the subscript i is the ith discrete subsection (Cunge and others, 1980). Equation 3 is derived under the assumptions that

4. the geometry of the section is assumed permanent (scour and fill and bedform effects are negligible);

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

5. the bulk of the flood wave moves approximately as a kinematic wave, which implies the friction slope is approximately equal to the bed slope, and the wave propagates only in the downstream direction; and

2. Sf for the total cross section is equal to Sf for each subsection.

2. the channel width is assumed constant in the streamwise direction (direction of flow);

6. the flow at the section is controlled by the channel geometry, the friction slope, the bed slope, and the shape of the flood wave. The development of Fread’s original method and a discussion of that method (DYNMOD) are in two publications by Fread (1973, 1975). The same assumptions made by Fread are used to develop the method described in this report (DYNPOUND). The development of the method follows. The one-dimensional flow in a stream can be described by the Saint-Venant equations (Cunge and others, 1980), which consist of an equation that represents the one-dimensional streamwise form of the conservation of mass as

∂A ∂Q _ + _ = 0 ∂t ∂x

(1)

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

∂ Q ∂ (β Q 2/ A) ∂h _ + _ + gA _ + gA Sf  = 0 ∂t ∂x ∂x where

A is the wetted cross-section area of the channel, in square feet; t is the time, in seconds; Q

is the discharge, in cubic feet per second;

is the streamwise distance along the channel, in feet;

(2)

2 i    A N K _ β = 2 ∑ i  _ Ai  K

(3)

In equation 3, K is conveyance and is defined by equation 4 for the whole cross section and equation 5 for the ith subsection.

1.486 _ 2/3 K = n A R

(4)

1.486 2/3 K i = _ ni Ai  Ri   

(5)

where

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 (n) are known, then β and K are known. The roughness coefficient hereinafter within the narrative is termed the “n-value.” Because the method uses data from a single streamgage, the above assumptions are used to adjust equations 1 and 2 so that differential terms with respect ∂ Q ∂ (β Q 2/ A) ∂ h to the downstream distance, x in (_, _, _ ), are ∂x ∂x ∂x replaced with approximations that eliminate the need for these terms. The process starts by taking the partial derivative with respect to x in the second term in equation 2, which yields equation 6.

∂ (β Q 2/ A) Q 2∂ β 2Q ∂ Q Q 2∂ A _ = _ _ + β _ _ − β _2 _ A ∂x A ∂x ∂x A ∂ x

(6)

Dynamic Rating Method Theory 5

∂y ∂β ∂β ∂_β _ Using the chain rule, the partial derivative of β with respect to x yields _ = . If _, the change in nonuniform ∂x ∂y ∂x ∂y ∂y _ velocity distribution coefficient with respect to depth and , the change in depth with respect to streamwise distance are both ∂x assumed to be much less than one, then the product of the two is considered to be negligible with respect to the rest of the terms, ∂β Q 2_ so the term _ is dropped and equation 6 reduces to equation 7. A ∂x ∂ (β Q 2/ A) 2Q ∂ Q Q 2∂ A _ = β _ _ − β _2 _ A ∂x ∂x A ∂ x

(7)

Moving the partial derivative of cross-sectional area with respect to time to the right-hand side of equation 1 yields equation 8.

∂Q ∂A _ = − _ ∂x ∂t

(8)

∂A Using the chain rule in taking the partial derivative of A with respect to x and using the assumption of _ = B (Henderson, ∂y 1966) yields equation 9. ∂y ∂A ∂A ∂y _ = _ _ = B _ ∂x ∂y ∂x ∂x

(9)

Substituting equations 8 and 9 into equation 7 yields equation 10.

∂ (β Q 2/ A) 2Q ∂ A Q 2∂ y _ = − β _ _ − βB _2 _ A ∂t ∂x A ∂ x

(10)

∂h The water-surface elevation slope, _ , which is in the third term of equation 2, is equivalent to the slope of the water depth ∂x minus bed slope, S0, as shown in equation 11. ∂y ∂h _ = _ − S0  ∂x ∂x

(11)

Discharge is related to K and Sf by Q = KT  Sf  1/2   , where K T  is the sum of the conveyance of each sub-section. Solving this equation for S  fgives equation 12 Sf in terms of Q and K  T.

Q S f = (_ ) KT  2

(12)

Substituting equations 10, 11, and 12 into equation 2, dividing through by the product gA, and rearranging the result yields equation 13.

2Q ∂ A Q 2 ∂ y Q 1 ∂Q _ _ − β _2 _ + (1 − βB _3) _ + (_ ) − So  = 0 K   gA ∂ t g A ∂ t g A ∂ x T 2

(13)

∂y The pressure term, _ , is the remaining partial derivative with respect to x. Henderson (1966) shows that for a flood wave ∂x moving approximately as a kinematic wave (assumption 5), the pressure term can be represented as S0  ∂y ∂h 1 _ 2 _ _ = − (_ c) − (_ ) 2 3 r ∂x ∂t

(14)

6 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data where

c second; and r

QP  + Q 0 _ V K = 1.3 _ 2A

is the flood wave velocity, in feet per is defined as the dimensionless ratio of S0 to the average wave slope (SW).

The flood wave velocity can be represented (Henderson, 1966) as equation 15.

dQ _ c = (15) dA

where

(16)

(17)

(20)

∂y Equations 14, 16, and 20 are used to calculate _ . Finally, ∂x the development of a discretized version of equation 13 and a solution method are discussed in the next section.

Solution Method To compute an unknown stage or discharge for a time tj, which is sometime after a time tj-1, the method requires the following: • known constants, which are S0, r, and g; • a known discharge value Qj-1 observed at a time tj-1; • a known stage value hj-1 observed at a time tj-1;

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

is the stage before the 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.

• A, B, β, and K as known functions of stage. An additional known value is required depending on the unknown value to be computed. If an unknown discharge Qj at time tj is to be computed, then a known stage hj is required. If an unknown stage hj at time tj is to be computed, then a known discharge Qj is required. Equation 13 contains continuous derivatives that need to be discretized to compute time-series values. Beginning with ∂Q the derivative in the first term, _ can be discretized as ∂t

Qj  − Qj−1   ∂Q _ ≈ _ tj  − tj−1   ∂t

Vk is defined by

Q V k = K c _ A

Qp is the peak discharge for a typical flood, in cubic feet per second; Q0 is the discharge before the beginning of the typical flood, in cubic feet per second; and _ A is the wetted cross-section area associated with the average stage, ( hp  + h 0)⁄ 2

(Qp  + Q 0) _ _ τ S  r = 0.65 (hp  − h0  )A 0

For this method, K in equation 16 is computed as KT.The second term in equation 14 is a small correction to account for the fact that flood waves do not move as a kinematic wave. The second term is dependent on a value of r that requires not only determination of S0 but also the SW. Information from a typical flood wave at the streamgage is needed to estimate the SW as the height of a flood wave divided by the half length of the flood wave, which is represented by the following equation:

hp  − h0  S W = _ τ Vk 

where

Using equations 17 and 19, the following relation is determined as

Under the kinematic wave assumption, Sf is equal to S0, so discharge is related to conveyance by Q = K S 01/2   (Henderson, 1966). Taking the derivative of Q in this relation gives

dK c = S0  1/2   _ dA

(19)

(18)

where Kc is the celerity coefficient (Fread, 1973). The velocity of the flood wave is estimated from equation 18 with the assumption that Kc has a value of 1.3 (Corbett, 1943) and average values used for the flow and area such that

(21)

∂A The derivative in the second term, _ , becomes ∂t Aj  − Aj−1   ∂A _ ≈ _ tj  − tj−1   ∂t

(22)

Evaluation Using Model-Generated Test Scenarios 7 where

is the cross-sectional area for stage hj.

∂y ∂h The pressure term _ in equation 13 is computed from equation 14, which requires c and the derivative _ to be computed. ∂x ∂t dK _ Equation 16 is used to compute c and contains the derivative , which becomes dA Δ Kj  dK _ ≈ _ Δ Aj  dA

(23)

where Δh _ − K _ Δ Kj  = Khj  +Δh hj  − and

(24)

Δh _ − A _ Δ Aj  = Ahj  +Δh hj  −

(25)

For the implementation of this method, Δh=0.01 foot. After the derivative of K with respect to A is computed, equation 16 is used to compute the c. The partial derivative of stage with respect to time is computed as

hj  − hj−1   ∂h _ ≈ _ t   − t   ∂t j j−1

(26)

The discrete form of equation 14 becomes

  2 S0  ∂y j  − hj−1 Δh h 1 _ _ ≈ − _ _ − _2 1/2 t   − t   Δ K   ∂x S0    j j j−1 3 r

(27)

Substituting all discrete approximations of derivatives into equation 13 yields equation 28.

Q  − Q  2 Qj  Aj  − Aj−1   Qj  2  _   2 Q   S0  1 Δ Aj  hj  − hj−1 _ _ _j t  − t  − βj  _2 _ − 1 − β   B   1/2 _ _ j j   (   + 3 r 2) + (Kj  ) − S0  = 0 g Aj  j j−1 g Aj    tj  − tj−1 g Aj  3  ) (S0    Δ Kj  tj  − tj−1 j j−1 1 _ _

(28)

Equation 28 is a nonlinear function of the unknown variable (Qj or hj) because all other values of r are known. The root of equation 28, and thus the value of the unknown variable, is determined using the secant method (Dahlquist and Björck, 1974). The solution method to equation 28 was implemented (Domanski and others, 2025) in the Python programming language (Python Software Foundation, 2023). The results shown in this report were computed using the Python implementation (Domanski and others, 2025), and the software for its maintenance track is available through Knight and others (2025).

Evaluation Using Model-Generated Test Scenarios Simulated scenario test datasets were created from one-dimensional unsteady Hydrologic Engineering Center River Analysis System (HEC–RAS; U.S. Army Corps of Engineers, 2016) simulation results. The purpose of creating the simulated test datasets was to compare the results computed with the dynamic rating method (DYNPOUND) described in this report to results computed using the HEC–RAS model one-dimensional unsteady shallow water equations, of which the dynamic rating method is a simplification. The simulated test datasets were obtained from Domanski and others (2022a; 2025). The source code and calibration parameters for the stage-to-discharge DYNMOD and original DYNPOUND rating methods, along with the HEC–RAS project files, are available in Domanski and others (2022b). The source code and calibration parameters for the newly improved DYNPOUND rating method and updated HEC–RAS project files are available from Domanski and others (2025). The dynamic rating software, DynRat, was developed using the original source code and is available in Knight and others (2025). The improved DYNPOUND method has the functionality to specify stage and n-value pairs for a cross section or subsection,

8 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data with noteworthy shifts in flow patterns at specific stages but no obvious change in geometry. Additionally, the method computes both stage-to-discharge and discharge-to-stage time series. A prismatic channel geometry (fig. 2), with floodplains on each side and a main channel with a total length of 80 miles, was used for four different scenarios with different combinations of S0 and r (table 1). An n-value of 0.035 was used for all cross sections. The cross sections in the channels were split into three subsections to compute for β and to smooth out the stage-conveyance relation (fig. 3c). The subsection stationing includes the two bank stations so that two subsections contain the left and right overbank areas and one subsection contains the main channel. Different inflow hydrographs were developed for the evaluation to test the 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 of each scenario with the appropriate S0 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 (40 miles downstream from the inflow point, midway between the most upstream and most downstream cross sections of the 80-mile reach) were extracted and used to compute and compare the discharge with the dynamic rating methods. The midpoint of the cross section was selected to reduce the effects of the boundary conditions on the simulation results. The Manning’s n-value, S0, and r values used in the development of the HEC–RAS scenarios were assigned to the parameters in the dynamic rating discharge computations.

Dataset Development The width of the main channel of the simulated cross section was 300 feet (ft), the floodplains 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

100

Elevation, in feet, referenced to datum of gage

−10 –100

200

400

600

800

Station, in feet EXPLANATION Coordinate Subsection station

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

1,000

Evaluation Using Model-Generated Test Scenarios 9 A. Top width and wetted perimeter of cross section

100

EXPLANATION

Top width Wetted perimeter

60 40 20 0 0

200

400

600

800

1,000

1,200

Stage, in feet, referenced to datum of gage

Distance, in feet

B. Area of cross section

100 80 60 40 20 0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

Area, in square feet

C. Conveyance of cross section

100 80 60 40 20 0 0

10,000,000

20,000,000

30,000,000

40,000,000

50,000,000

Conveyance, in cubic feet per second

Figure 3. Graphs showing stage plotted against four variables. A, top width and wetted perimeter; B, area; and C, conveyance. Table 1. Bed slope and ratio of bed slope to average wave slope of simulated test data scenarios. [r, ratio of bed slope to average wave slope]

Scenario

Bed slope

0.0001

100

0.001

100

10 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data stations coincide with the bank stations at 300 and 600 ft (fig. 2). An n-value of 0.035 was used for all scenarios in the cross section. Four sets of 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 (figs. 4–7; Domanski and others, 2022b; Domanski and others, 2025). The value of unsteadiness can be characterized by r, which is the ratio of S0 to Sw (eq. 17). The larger the value of r for a particular S0, the lower the unsteadiness of the hydrograph; that is, the time from the onset of the flooding to the flood peak increases with increasing value of r. To determine the actual inflow hydrographs used for the scenarios, a flood wave slope was computed from a S0 and an assumed value of r (table 1). The rising and falling limbs of the stage hydrograph were computed using the constant value of the slope of the flood wave between the end points of 75 percent of the bankfull main channel depth (22.5 ft) to the peak stage (60 ft). The peak stage was chosen such that the total wetted area in the floodplain equaled the wetted area in the upstream channel location. Manning’s equation from the stage hydrograph was used to compute the discharge hydrographs for the upstream boundary condition.

Evaluation A preliminary analysis of the four test scenarios was performed using the DYNMOD method, in which jumps in discharge were observed in the time series (Domanski and others, 2022b). The first jump, from a higher to a lower discharge, took place when the stage rose from below to above the channel bank elevation, and the second jump, from a lower to higher discharge, took place once the elevation fell below the bank elevation. These jumps happened because of abrupt changes in the relations of top width, wetted perimeter, and area with stage (fig. 3). For more information about these test scenarios and the DYNMOD method, refer to Domanski and others (2022b). Overall, the magnitude of the mean percent error was much greater in the results computed with the DYNMOD method (Domanski and others, 2022b). 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 (refer to eq. 13). The function of conveyance with stage can be developed so that changes are less abrupt by creating subsections in the cross section, as was done for the simulated test scenarios.

The DYNPOUND method performed well compared to the full one-dimensional unsteady flow equations within HEC–RAS for all four scenarios. The mean percent error for the DYNPOUND-computed discharge was approximately 2.01×10−1 percent. The mean percent error for the DYNPOUND-computed stage was −1.05×10−1 percent. (table 2). Scenarios 1 (fig. 4) and 3 (fig. 6) have values of r equaling 10 (table 1) and, therefore, are highly unsteady and show pronounced hysteresis in the stage versus discharge curves. Scenarios 2 (fig. 5) and 4 (fig. 7), which have values of r equaling 100 (table 1), do not show hysteresis. Furthermore, the DYNPOUND method performs better for discharge and stage in scenarios 2 and 4 than it does in scenarios 1 and 3 (table 2). Scenarios 2 and 4 effectively have one-to-one stage-discharge relations.

Scenario 1 At approximately 30 ft, where flow begins to exceed the main channel, DYNPOUND-computed discharge and stage time series show a “jog” in the relation (fig. 8). This is likely due to the abrupt change in channel geometry and the application of a single n-value for the entire channel. The scenario 1 stage-discharge relation indicates hysteresis in the HEC–RAS results because of unsteady flow effects captured by the dynamic ratings simulations (fig. 9).

Scenario 2 Scenario 2’s computed hydrographs indicate a lack of hysteresis (figs. 10 and 11). The stage-discharge relation of the HEC–RAS results for scenario 2 is effectively one-to-one because the distance between the discharge values computed at a given stage is small (fig. 11).

Scenario 3 Scenario 3’s stage discharge relation computed by HEC– RAS shows hysteresis with a similar “jog” to scenario 1, at 30 ft, when the channel geometry changes (fig. 12 and fig. 13).

Scenario 4 The stage-discharge relation of the scenario 4 time series (fig. 14) as computed using HEC–RAS does not show hysteresis and can effectively be considered a one-to-one relation (fig. 15).

Evaluation Using Model-Generated Test Scenarios 11

180,000

A. Discharge time series for test scenario 1

160,000

EXPLANATION Discharge Upstream boundary condition Sampled cross section

Discharge, in cubic feet per second

140,000

120,000

100,000

80,000

60,000

40,000

20,000

B. Stage time series for test scenario 1

EXPLANATION

Stage, in feet, referenced to datum of gage

Stage Upstream boundary condition Sampled cross section

0:0 20 0 28 /

03 /

27 /

20 0

0:0

0 00 0:0 5/ 2 03 /2

03 /

23 /

20 0

0:0

:00 00 20 1/ /2 03

03 /

19 /

20 0

0:0

:00 00 03

20 5/ /1 03

:00

Simulation time, in month/day/year hour:minute (coordinated universal time [UTC]) format

Figure 4. Graphs showing time series for simulated test scenario 1 (Domanski and others, 2025). A, discharge and B, stage.

12 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data

180,000

A. Discharge time series for test scenario 2

EXPLANATION

160,000

Discharge Upstream boundary condition Sampled cross section

Discharge, in cubic feet per second

140,000

120,000

100,000

80,000

60,000

40,000

20,000

B. Stage time series for test scenario 2 EXPLANATION Stage Upstream boundary condition Sampled cross section

Stage, in feet, referenced to datum of gage

0 08 /

01 /

20 0

0:0

0 07 /

01 /

20 0

0:0

0 0:0 20 0 01 / 06 /

00 :00 05

20 1/ /0 04

00 :00

:00

Simulation time, in month/day/year hour:minute (coordinated universal time [UTC]) format

Figure 5. Graphs showing time series for simulated test scenario 2 (Domanski and others, 2025). A, discharge and B, stage.

Evaluation Using Model-Generated Test Scenarios 13

600,000

A. Discharge time series for test scenario 3

EXPLANATION

Discharge, in cubic feet per second

500,000

Discharge Upstream boundary condition Sampled cross section

400,000

300,000

200,000

100,000

300

B. Stage time series for test scenario 3

Stage, in feet, referenced to datum of gage

250

200

EXPLANATION Stage Upstream boundary condition Sampled cross section

150

100

/1 03

20 3/ /1 03

:00

:00 14

:00 20 3/ /1 03

08 03

20 3/ /1 03

:00

:00 06

:00 04 03

20 3/ /1 03

:00

Simulation time, in month/day/year hour:minute (coordinated universal time [UTC]) format

Figure 6. Graphs showing time series for simulated test scenario 3 (Domanski and others, 2025). A, discharge and B, stage.

14 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data

600,000

A. Discharge time series for test scenario 4

EXPLANATION

Discharge, in cubic feet per second

500,000

Discharge Upstream boundary condition Sampled cross section

400,000

300,000

200,000

100,000

B. Stage time series for test scenario 4 300

Stage, in feet, referenced to datum of gage

250

EXPLANATION

200

Stage Upstream boundary condition Sampled cross section

150

100

:00 03

:00 12 20 3/ /1 03

:00

Simulation time, in month/day/year hour:minute (coordinated universal time [UTC]) format

Figure 7. Graphs showing time series for simulated test scenario 4 (Domanski and others, 2025). A, discharge and B, stage.

Evaluation Using Model-Generated Test Scenarios 15 Table 2. Performance statistics for the DYNPOUND computation method. [DYNPOUND is the newly developed method that solves for stage and discharge in compact and compound channels. MSLE, mean squared logarithmic error]

DYNPOUND discharge

DYNPOUND stage

Mean percent error

Maximum absolute percent error

MSLE

Mean percent error

Maximum absolute percent error

MSLE

4.47×10−1

6.08

2.02×10−4

−2.36×10−1

1.77

2.60×10−5

−9.07×10−3

0.46

7.84×10−7

3.67×10−3

0.32

1.63×10−7

3.70×10−1

2.74

4.31×10−5

−1.89×10−1

0.65

1.09×10−5

−5.44×10−3

0.12

2.47×10−7

1.99×10−3

0.20

7.24×10−8

Mean

2.01×10−1

2.35

6.16×10−5

−1.05×10−1

0.74

9.28×10−6

Scenario

16 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data

180,000

A. Discharge time series for test scenario 1 EXPLANATION

160,000

Discharge DYNPOUND computed HEC–RAS computed

Discharge, in cubic feet per second

140,000

120,000

100,000

80,000

60,000

40,000

20,000

B. Stage time series for test scenario 1 EXPLANATION

Stage, in feet, referenced to datum of gage

Stage DYNPOUND computed HEC–RAS computed

00 :00

20 7/ /2 03

03 /2

5/ 2

00 0:0

00 :00

0 0:0 20 0 23 / 03 /

20 1/ /2 03

03 /1

9/ 2

00 0:0

00 :00

0 0:0 20 0 17 / 03 /

20 5/ /1 03

:00

Simulation time, in month/day/year hour:minute (coordinated universal time [UTC]) format DYNPOUND, A dynamic rating method which accommodates compound and compact channel geometry. HEC–RAS, Hydrologic Engineering Center River Analysis System.

Figure 8. Graphs showing time series for simulated scenario 1 (Domanski and others, 2025). A, Discharge computed with the DYNPOUND method; B, stage computed with the DYNPOUND method.

Evaluation Using Model-Generated Test Scenarios 17

Test scenario 1

Stage, in feet, referenced to datum of gage

EXPLANATION DYNPOUND-computed stage DYNPOUND-computed discharge HEC–RAS-computed discharge

20 20,000

40,000

60,000

80,000

100,000

120,000

140,000

160,000

Discharge, in cubic feet per second

DYNPOUND, A dynamic rating method which accommodates compound and compact channel geometry. HEC–RAS, Hydrologic Engineering Center River Analysis System.

Figure 9. Graph showing the relation between stage and computed discharge and discharge and computed stage for simulated scenario 1 (Domanski and others, 2025).

180,000

18 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data

180,000

A. Discharge time series for test scenario 2 EXPLANATION

160,000

Discharge DYNPOUND computed HEC–RAS computed

Discharge, in cubic feet per second

140,000

120,000

100,000

80,000

60,000

40,000

20,000

B. Stage time series for test scenario 2 EXPLANATION

Stage, in feet, referenced to datum of gage

Stage DYNPOUND computed HEC–RAS computed

08 /

01 /

20 0

0:0

0 00 0:0 1/ 2 07 /0

06 /

01 /

20 0

0:0

:00 00 05

20 1/ /0 04

:00

Figure 10. Graphs showing time series for simulated scenario 2 (Domanski and others, 2025). A, Discharge computed with the DYNPOUND method; B, Stage computed with the DYNPOUND method.

Evaluation Using Model-Generated Test Scenarios 19

Test scenario 2

Stage, in feet, referenced to datum of gage

EXPLANATION DYNPOUND-computed stage DYNPOUND-computed discharge HEC–RAS-computed discharge

20 20,000

40,000

60,000

80,000

100,000

120,000

140,000

160,000

Discharge, in cubic feet per second

DYNPOUND, A dynamic rating method which accommodates compound and compact channel geometry. HEC–RAS, Hydrologic Engineering Center River Analysis System.

Figure 11. Graph showing the relation between stage and computed discharge and discharge and computed stage for simulated scenario 2 (Domanski and others, 2025).

180,000

20 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data

450,000

A. Discharge time series for test scenario 3 EXPLANATION

400,000

Discharge DYNPOUND computed HEC–RAS computed

Discharge, in cubic feet per second

350,000

300,000

250,000

200,000

150,000

100,000

50,000

B. Stage time series for test scenario 3 EXPLANATION Stage DYNPOUND computed HEC–RAS computed

Stage, in feet, referenced to datum of gage

18 :00 03

16 :00 3/ /1 03

20 3/ /1 03

14 :00

0 01 2:0 3/ 2 03 /1

10 20 3/ /1 03

06 20 3/ /1 03

:00

08 :00

:00

0 4:0 20 0 13 / 03 /

20 3/ /1 03

:00

Figure 12. Graphs showing time series for simulated scenario 3 (Domanski and others, 2025). A, Discharge computed with the DYNPOUND method; B, Stage computed with the DYNPOUND method.

Evaluation Using Model-Generated Test Scenarios 21

Test scenario 3

Stage, in feet, referenced to datum of gage

EXPLANATION 25

DYNPOUND-computed stage DYNPOUND-computed discharge HEC–RAS-computed discharge

20 50,000

100,000

150,000

200,000

250,000

300,000

350,000

400,000

450,000

Discharge, in cubic feet per second

DYNPOUND, A dynamic rating method which accommodates compound and compact channel geometry. HEC–RAS, Hydrologic Engineering Center River Analysis System.

Figure 13. Graph showing the relation between stage and computed discharge and discharge and computed stage for simulated scenario 3 (Domanski and others, 2025).

22 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data

600,000

A. Discharge time series for test scenario 4 EXPLANATION Discharge DYNPOUND computed HEC–RAS computed

Discharge, in cubic feet per second

500,000

400,000

300,000

200,000

100,000

B. Stage time series for test scenario 4 EXPLANATION

Stage, in feet, referenced to datum of gage

Stage DYNPOUND computed HEC–RAS computed

00 :00

00 0:0 7/ 2 03 /1

00 20 6/ /1 03

03 /1

5/ 2

00 0:0

:00

0 00 0:0 4/ 2 03 /1

:00

Figure 14. Graphs showing time series for simulated scenario 4 (Domanski and others, 2025). A, Discharge computed with the DYNPOUND method; B, stage computed with the DYNPOUND method.

Evaluation Using Field Data 23

Test scenario 4

Stage, in feet, referenced to datum of gage

EXPLANATION DYNPOUND-computed stage DYNPOUND-computed discharge HEC–RAS-computed discharge

20 0

100,000

200,000

300,000

400,000

500,000

600,000

Discharge, in cubic feet per second

DYNPOUND, A dynamic rating method which accommodates compound and compact channel geometry. HEC–RAS, Hydrologic Engineering Center River Analysis System.

Figure 15. Graph showing the relation between stage and computed discharge and discharge and computed stage for simulated scenario 4 (Domanski and others, 2025).

Evaluation Using Field Data Field measurements of discharge and stage and WSC-computed discharge and WSC-measured stage from 10 USGS streamgages were used to evaluate the DYNPOUND method (U.S. Geological Survey, 2020). These streamgages

represent a variety of geographic locations and geomorphic conditions. The 10 streamgage sites chosen for evaluation were Meherrin River near Bryants Corner, Virginia (USGS streamgage 02052090); Tug Fork at Kermit, West Virginia (USGS streamgage 03214500); Tittabawassee River at Midland, Michigan (USGS streamgage 04156000); Red River of the North at Fargo, North Dakota (USGS streamgage

24 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data 05054000); Papillion Creek at Fort Crook, Nebraska (USGS streamgage 06610795); Gasconade River at Jerome, Missouri (USGS streamgage 06933500); Mississippi River at St. Louis, Missouri (USGS streamgage 07010000); Calcasieu River near Kinder, Louisiana (USGS streamgage 08015500); Rio Grande near Cerro, New Mexico (USGS streamgage 08263500); and San Joaquin River near Mendota, California (USGS streamgage 11254000) (fig. 16 and table 3).

Dataset Development Site datasets consisted of WSC-computed discharge and WSC-measured stage time series, field measurements of discharge and stage, cross-section geometry, and bed slope. Time series and field measurements were obtained from the National Water Information System (NWIS; U.S. Geological Survey, 2020). Cross-section geometry and bed slope were computed for each site using a combination of acoustic

Doppler profiler (ADCP) software, AreaComp2 USGS utility (U.S. Geological Survey, 2015), ArcGIS Pro (Esri, 2021), and HEC–RAS (U.S. Army Corps of Engineers, 2016).

Cross-Section Geometry To obtain cross-section geometry, an ADCP discharge measurement was selected for each site that generally corresponded to a high-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; U.S. Geological Survey, 2017; Michigan State University, 2020; U.S. Department of Agriculture, 2021) was imported into RAS Mapper within HEC–RAS, which was used to generate a “station, elevation” (in stage datum) cross section (fig. 17). This cross section created from the DEM

120°

105° 45°

90°

NORTH DAKOTA

75°

04156000

11254000

07010000

03214500

06933500

08263500

V W IR E G ST IN IA G IN IA

06610795

NEBRASKA

MIC

HIGA

05054000

02052090

NEW MEXIC

08015500

AN SI

30°

Base from U.S. Geological Survey (USGS) digital data Albers Equal-Area Conic, USGS contiguous United States projection North American Datum of 1983 0 USGS site number 02052090 03214500 04156000 05054000 06610795 06933500 07010000 08015500 08263500 11254000

Site name Meherrin River near Bryants Corner, Virginia Tug Fork at Kermit, West Virginia Tittabawassee River at Midland, Michigan Red River of the North at Fargo, North Dakota Papillion Creek at Fort Crook, Nebraska Gasconade River at Jerome, Missouri Mississippi River at St. Louis, Missouri Calcasieu River near Kinder, Louisiana Rio Grande near Cerro, New Mexico San Joaquin River Near Mendota, California

300 300

600 600

900 MILES

900 KILOMETERS

EXPLANATION 05054000

USGS streamgage and site number

Figure 16. Map of U.S. Geological Survey streamgage sites used to evaluate discharge computed with the dynamic rating methods (U.S. Geological Survey, 2020). USGS, U.S. Geological Survey.

Evaluation Using Field Data 25 Table 3. Streamgage number and name, drainage area, and slope of the field sites used to evaluate the DYNPOUND dynamic rating method. [Data from U.S. Geological Survey, 2020. DYNPOUND is the newly developed method that solves for stage and discharge in compact and compound channels. mi2, square mile]

Streamgage number

Streamgage name

Drainage area (mi2)

Bed slope1

807

0.00037809

02052090

Meherrin River near Bryants Corner, Virginia

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

06610795

Papillion Creek at Fort Crook, Nebraska

384

0.00025438

06933500

Gasconade River at Jerome, Missouri

2,840

0.00040953

07010000

Mississippi River at St. Louis, Missouri

697,000

0.000110

08015500

Calcasieu River near Kinder, Louisiana

1,700

0.00018992

08263500

Rio Grande near Cerro, New Mexico

8,440

0.00360

11254000

San Joaquin River near Mendota, California

3,940

0.000248

1Bed slope was calculated using elevation contour, topographical maps, and the National Hydrography Dataset (U.S. Geological Survey, 2017; Domanski and others, 2022a).

Stage, in feet, referenced to 579.47 feet above North American Vertical Datum of 1988

EXPLANATION Cross section DEM derived

ADCP derived 5

100

200

300

400

500

600

700

Station, in feet DEM, Digital elevation model. ADCP, Acoustic doppler current profiler.

Figure 17. Graph showing cross sections derived from an acoustic doppler profiler (ADCP) and a digital elevation model (DEM) at Tittabawassee River at Midland, Michigan (U.S. Geological Survey streamgage 04156000).

26 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data was combined with the cross section created from the ADCP transect to create a cross section that contained the main channel and overbank sections (fig. 18). 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 floodplain. In some cases, a stage measurement was made during a peak-flow event that was high enough to capture some of the floodplain. If that happened, then “station, elevation” (in stage datum) coordinates from the ADCP measurement superseded the overlapping coordinates from the DEM. Geospatially locating ADCP transects at sites without geolocation data associated with ADCP measurements was difficult. 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 floodplain geometry. Sometimes cross sections with these limitations did not produce accurate results in either of the dynamic rating computation methods. When associated geolocation data for an ADCP measurement was lacking 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 Bed slope (S0) was calculated using elevation contours, historical topographical maps, and the National Hydrography Dataset flowline geospatial files (U.S. Geological Survey,

2017; Domanski and others, 2022a). Points were chosen upstream and downstream from each streamgage where an elevation contour line crossed the channel; reach lengths varied depending on available map contours, ranging from approximately 4,582 to 781,450 ft (Domanski and others, 2022a). To compute S0, the following equation was used:

Eus   − Eds   S 0 = _ Lr  where

(34)

is bed slope, in feet per feet (dimensionless);

Eus

is upstream elevation, in feet;

Eds

is downstream elevation, in feet; and

is length of reach, in feet.

Evaluation Discharge and stage time series were computed with the DYNPOUND method for streamgages where cross-section geometry was created and real-time stage and discharge data were being collected. These time series were computed at 10 streamgages in California, Louisiana, Michigan, Missouri, Nebraska, New Mexico, North Dakota, Virginia, and West Virginia.

Stage, in feet, referenced to 579.47 feet above North American Vertical Datum of 1988

100

200

300

400

500

600

700

Station, in feet

Figure 18. Graph showing combined cross section at Tittabawassee River at Midland, Michigan (U.S. Geological Survey streamgage 04156000). ft, foot.

Evaluation Using Field Data 27 For all sites, an event was chosen to compute the value of r (ratio of S0 to SW); the event was typically an isolated flood wave with a moderate peak stage and discharge. The stage at the time immediately before the onset of the flood was inserted into equation 20 as h0. The peak stage of the flood is hp in equation 20. The time of peak stage minus the time of occurrence of h0 is τ in equation 20. Subsections were added to the cross section of each site based on the need to (1) develop a smooth conveyance and stage relation, (2) add regions where transitions in roughness occur in the floodplain, (3) remove those areas of the cross section that do not contribute to the momentum of the flow (in other words, sections of the floodplain that were not inundated or sections of the channel higher in elevation than the peak stage during a high flow event), and (4) allow for computation of the non-uniform velocity distribution coefficient. To determine the smoothness of the stage-conveyance relation, conveyance was plotted against stage and visually analyzed. If an abrupt change in the 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. If there was a sudden change in the slope of the relation with no obvious change in channel geometry, the n-values of the cross section were modified according to the stage value at which the flow pattern shifted. The change was assumed to be because of unknown phenomena such as varying bed material or vegetation, obstruction, or channel meandering (Davidian, 1984; Arcement and Schneider, 1989). A full water year of stage and discharge time series was used to calibrate the method at each site. A water year is defined as the 12-month period, October 1 through September 30, and is designated by the calendar year in which it ends. Typically, at least 5 field measurements were used to calibrate each of the 10 streamgages for which results are discussed. Cross sections were also subsectioned so the DYNPOUND-computed measurement would match the field measurement. The S0, value of r, and cross-section geometry were considered fixed values for the calibration. After the site was calibrated, a different period in the record was selected to evaluate the method. For each site, the evaluation period typically follows the calibration period and contains at least five field measurements, and a wide range of WSC-computed discharge and WSC-measured stage time-series values. Discharge field measurements were then compared to DYNPOUND-computed discharge at the same times and vice versa for stage field measurements. Sometimes, WSC-computed discharge or WSC-measured stage data were missing from the time series, in which case the missing discharge or stage values of the DYNPOUND-computed time series were estimated through linear interpolation (Domanski and others, 2022b). If the period of missing data was longer than a few hours, a different period was chosen for calibration.

Meherrin River near Bryants Corner, Virginia The USGS streamgage at Meherrin River near Bryants Corner, Virginia (USGS streamgage 02052090), encompasses 870 square miles (mi2). The computed S0 for the site is 0.00037809 (table 3). The value of r computed for the event with a peak stage of 16.82 ft at 6:15 (coordinated universal time [UTC]) on December 28, 2015 (U.S. Geological Survey, 2020), is 80.63. The cross section used to compute the time series for this streamgage is shown in figure 19. The n-values chosen for the DYNPOUND computations vary from 0.045 to 0.45 (table 4). The cross section was split into five subsections; subsection stations are 250, 265, 350, and 375 ft from left bank. Twenty field measurements of stage and discharge from the 2017 water year were used for calibration (table 5 and table 6). The WSC-measured stage time series for the 2017 water year was used to compute discharge, and the WSC-computed discharge time series for the same water year was used to compute stage using the DYNPOUND method. (U.S. Geological Survey, 2020). The MSLE for the DYNPOUND discharge calibration was 1.30×10−2, and the mean percent error was 2.36 percent (table 5). The MSLE for the DYNPOUND stage calibration was 9.97×10−3, and the mean percent error was 0.37 (table 6). To evaluate the stage-discharge relation for the USGS streamgage at Meherrin River near Bryants Corner, Virginia, time series of stage and discharge were computed for the period between May 17 and 31, 2018, which included four field measurements used for error assessment (fig. 20 and fig. 21). A comparison of field measurement discharge and DYNPOUND-computed discharge for the period indicates the DYNPOUND values were biased high when compared to the field measurements and had a mean error of 6.25 percent (fig. 20). The MSLE was 6.40×10−3 (table 7). The mean error for the DYNPOUND-computed stage was −0.84 percent, and the MSLE was 2.48×10−3 (table 8). DYNPOUND captured hysteresis in the stage-discharge relation for the computed event, whereas the USGS-computed discharge is monotonic and did not capture hysteresis (fig. 22).

Tug Fork at Kermit, West Virginia The USGS streamgage Tug Fork at Kermit, West Virginia (USGS streamgage 03214500), encompasses 1,277 mi2. The computed S0 for the site is 0.000352 (table 3). The r value, computed from the event with the peak stage of 24.35 ft at 01:15 (coordinated universal time [UTC]) on April 25, 2017 (U.S. Geological Survey, 2020), is 37.19. The cross section used to compute the time series is shown in figure 23. However, it does not meet the criteria for subdivision (Dalrymple and Benson, 1967; Davidian, 1984). The n-values were chosen for the full cross section based on a plot of n-values, calculated with Manning’s equation and field

28 Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data

Elevation, in feet, referenced to 29.52 feet above North American Vertical Datum of 1988

USGS 02052090 Meherrin River near Bryants Corner, Virginia

EXPLANATION −5

Coordinate Subsection station

−10 0

100

200

300

400

500

600

Station, in feet

USGS, U.S. Geological Survey.

Figure 19. Graph showing the cross section used to compute the stage and discharge time series at Meherrin River near Bryants Corner, Virginia (U.S. Geological Survey streamgage 02052090; U.S. Geological Survey, 2020). Table 4. Stage and roughness coefficient values used to calibrate the DYNPOUND method at Meherrin River near Bryants Corner, Virginia (U.S. Geological Survey streamgage 02052090). [Data from Domanski and others, 2025. ft, foot]

Stage (ft)

Roughness coefficient (n-value)

3.00

0.45

4.00

0.3

7.00

0.2

11.00

0.18

17.00

0.045

Evaluation Using Field Data 29 Table 5. Discharge calibration results for the DYNPOUND ratings at Meherrin River near Bryants Corner, Virginia (U.S. Geological Survey streamgage 02052090). [Field measurement discharge data from U.S. Geological Survey, 2020. DYNPOUND is the newly developed method that solves for stage and discharge in compact and compound channels. MM, month; DD, day; YYYY, year; UTC, coordinated universal time; FM, field measurement; ft3/s, cubic foot per second; SLE, squared logarithmic error; NA, not applicable]

Measurement date (MM/DD/YYYY)

Measurement time (UTC)

FM discharge (ft3/s)

DYNPOUND discharge (ft3/s)

DYNPOUND discharge error (percent)

DYNPOUND discharge SLE

10/13/2016