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

Open-File Report 2022-1031
Groundwater and Streamflow Information Program
By: , and 



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.

Suggested Citation

Domanski, M., Holmes, R.R., Jr., and Heal, E.N., 2022, Dynamic rating method for computing discharge from time-series stage data: U.S. Geological Survey Open-File Report 2022–1031, 48 p., https://doi.org/10.3133/ofr20221031.

ISSN: 2331-1258 (online)

Table of Contents

  • Abstract
  • Introduction
  • Dynamic Rating Method Theory
  • Solution Method
  • Evaluation Using Model-Generated Test Scenarios
  • Evaluation Using Field Data
  • Dynamic Rating Application Recommendations
  • Summary
  • Acknowledgments
  • References Cited
Publication type Report
Publication Subtype USGS Numbered Series
Title Dynamic rating method for computing discharge from time-series stage data
Series title Open-File Report
Series number 2022-1031
DOI 10.3133/ofr20221031
Year Published 2022
Language English
Publisher U.S. Geological Survey
Publisher location Reston, VA
Contributing office(s) Central Midwest Water Science Center
Description Report: vii, 48 p.; 2 Data Releases; Dataset
Online Only (Y/N) Y
Google Analytic Metrics Metrics page
Additional publication details