the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 30 Oct 2025
An Innovative Equivalent River Channel Method for Integrated Hydrologic–Hydrodynamic Modeling
Abstract. To address the lack of river cross-sectional data in hilly regions, this study proposes a novel method that transforms Muskingum parameters (K and X) into Conceptual Equivalent River Channel (CERC). By integrating linear or nonlinear Muskingum parameters with characteristic discharge, roughness, and other relevant inputs, this approach derives simplified yet hydraulically representative cross-sections. Two types of CERC models are introduced: single-layer and dual-layer. The single-layer CERC model includes rectangular, parabolic and triangular cross-sections, while the double-layer CERC builds upon these with an exponential shape. The proposed method was applied to two river reaches in China: the Chenggouwan–Linqing reach in the Haihe River Basin and the Huayuankou–Jiahetan reach in the Yellow River Basin. Using previously calibrated and validated Muskingum parameters, the resulting channel geometries were incorporated into a one-dimensional (1-D) hydrodynamic model. Results indicated that CERCs accurately replicated observed hydrographs, and the dual-layer approach improved performance in reaches with strong nonlinear characteristics. Furthermore, the model effectively captured changes in water level and flow velocity, confirming the suitability of CERC for hydrodynamic modeling. A sensitivity analysis examined the impact of variations in roughness (n) affected the Conceptual Equivalent River Channel Cross-sections (CERCXs) and discharge outcomes, demonstrating the robustness of the proposed method. While CERCs simplify the natural complexity of river channels, their parametric framework represents the channel’s storage capacity and allows flexible shape selection, enabling accurate simulations of water levels and flow velocities when adjusted to match measured cross-sections. This research provides a practical solution that bridges traditional hydrological and hydrodynamic routing methods in regions with limited data availability, especially in hilly areas.
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Status: closed
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RC1: 'Comment on egusphere-2025-4059', Anonymous Referee #1, 06 Dec 2025
- AC1: 'Reply on RC1', Gang Chen, 03 Mar 2026
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RC2: 'Comment on egusphere-2025-4059', Anonymous Referee #2, 06 Jan 2026
Summary: This study links calibrated Muskingum parameters to effective channel cross section parameters. The paper uses the well-known equivalence of the Muskingum-Cunge lumped routing approach and the diffusive wave approximation of the de Saint-Venant equations. I see little novelty in the material presented here and also have some concerns regarding the methodology, pls see comments below. I do not recommend acceptance of this paper in its current form.
Review comments:
- The basic idea of this paper is that calibrated Muskingum parameters for a river reach can be translated into effective river cross sections. However, the equivalence between the diffusive wave approximation of the de Saint-Venant equations and the Muskingum-Cunge model is well understood and not new.
- The purpose of the effective river cross sections is unclear. Normally, the motivation for using hydraulic instead of lumped routing is interest in continuous water levels along the river (e.g. for flood risk assessment). However, the approach presented here will generate average, reach-scale cross sections, which will only reproduce large-scale behavior not detailed small-scale backwater effects etc. I believe the water levels produced with this approach will essentially be equivalent to water levels derived from reach-scale rating curves, thus making the hydraulic routing model obsolete.
- Methodology: International readership is used to Muskingum and Muskingum-Cunge algorithms. It would be useful to align your description with these standard algorithms. Is your nonlinear Muskingum model equivalent to Muskingum-Cunge? While that seems to be the case for the parameterization of X (eq 7), the parameterization of K (eq 6) is different. In Muskingum-Cunge, parameter K (i.e. reach length divided by wave speed) decreases with increasing discharge (wave speed increases), while in your approach (eg 6) it seems that you assume increasing K with increasing discharge. This seems counter-intuitive at first sight and needs more explanation/motivation. Any deviation from standard Muskingum-Cunge should be carefully explained and motivated.
- Methodology: Equation 12: Variable H is not explained. I assume that this is water surface elevation (WSE) or water level. The formulation f(H) then correctly describes that flow cross-sectional area depends on WSE, according to the chosen geometry assumption (rectangular, triangular, etc.). However, hydraulic radius R is written without WSE dependence. Is this just a mistake or do you assume constant hydraulic radius? In reality, R will definitely be R(H).
- It should be clearly stated that the translation from Muskingum parameters to geometry (which is a direct consequence of the equivalence of Muskingum-Cunge and diffusive-wave approx. of de Saint-Venant) can only be done for given/known hydraulic roughness (n). However hydraulic roughness values are usually not available. Usually, Manning numbers are obtained from inverse modeling, i.e. fitting observed water levels for given geometry and discharge. This is a well-known parameter trade-off in hydraulic modeling, which stems from the fact that conveyance is a function of both geometry and hydraulic roughness. The impacts of both on conveyance cannot be separated based on water levels only.
Citation: https://doi.org/10.5194/egusphere-2025-4059-RC2 - AC2: 'Reply on RC2', Gang Chen, 03 Mar 2026
Status: closed
-
RC1: 'Comment on egusphere-2025-4059', Anonymous Referee #1, 06 Dec 2025
The manuscript proposes a workflow that takes reach-scale Muskingum routing parameters (K and X), translates them into an “equivalent” one-dimensional channel geometry (choosing among simple parametric shapes), and then uses those inferred cross-sections in a 1-D Saint-Venant solver to simulate flood wave propagation along long river reaches. The authors calibrate Muskingum parameters to reproduce observed hydrographs, present algebraic relations that map routing quantities to width, depth, slope and roughness, and report that the resulting hydrodynamic model reproduces the calibrated discharge time series at downstream gauges.
However, there are several limitations that undermine the central claim that routing parameters can be interpreted as physical channel geometry.
First, the inversion is underdetermined: the method attempts to recover multiple geometric and roughness unknowns from a small set of lumped routing observables, so the chosen shape family effectively forces one of many possible solutions rather than revealing a unique physical cross-section. The manuscript defines to mean that the cross-sectional area depends only on water depth , but Eqs. (17-19) explicitly contain additional variables (e.g. slope, roughness, geometric parameters) so the area is treated as a function of more than .
Second, the validation is circular and insufficient: Muskingum parameters are calibrated to match hydrographs, the calibrated values are converted to cross-sections, and the hydrodynamic model unsurprisingly reproduces the same hydrographs—this does not demonstrate that inferred widths, depths or slopes correspond to measured channel form.
Third, key algebraic approximations and a kinematic celerity assumption are introduced but not benchmarked against analytical solutions, synthetic channels with known geometry, or higher-fidelity (2D) models, so the error and applicability bound of the derivation are unknown.
Fourth, important practical issues are unaddressed: some reported weighting values fall outside conventional bounds, there is no propagation of calibration uncertainty into the inferred geometry, and no identifiability or equifinality analysis to show that the geometry estimates are robust.
Finally, representing long heterogeneous reaches as a single uniform cross-section ignores spatial variability (width, slope, tributaries, controls and backwater effects) that one-dimensional hydrodynamics are typically meant to capture.
To improve the manuscript the authors should clearly frame the product as an “equivalent storage” representation unless and until independent geometric validation is provided, perform synthetic and 2-D benchmark tests to quantify the limits of the algebraic approximations, propagate parameter uncertainty through to geometric and hydraulic outputs, and supply systematic comparisons of inferred widths/depths/slopes against surveyed or remote-sensing data at multiple locations and flow stages. Those steps will show whether the method offers a pragmatic engineering approximation or is simply a re-packaged calibration exercise.
Citation: https://doi.org/10.5194/egusphere-2025-4059-RC1 - AC1: 'Reply on RC1', Gang Chen, 03 Mar 2026
-
RC2: 'Comment on egusphere-2025-4059', Anonymous Referee #2, 06 Jan 2026
Summary: This study links calibrated Muskingum parameters to effective channel cross section parameters. The paper uses the well-known equivalence of the Muskingum-Cunge lumped routing approach and the diffusive wave approximation of the de Saint-Venant equations. I see little novelty in the material presented here and also have some concerns regarding the methodology, pls see comments below. I do not recommend acceptance of this paper in its current form.
Review comments:
- The basic idea of this paper is that calibrated Muskingum parameters for a river reach can be translated into effective river cross sections. However, the equivalence between the diffusive wave approximation of the de Saint-Venant equations and the Muskingum-Cunge model is well understood and not new.
- The purpose of the effective river cross sections is unclear. Normally, the motivation for using hydraulic instead of lumped routing is interest in continuous water levels along the river (e.g. for flood risk assessment). However, the approach presented here will generate average, reach-scale cross sections, which will only reproduce large-scale behavior not detailed small-scale backwater effects etc. I believe the water levels produced with this approach will essentially be equivalent to water levels derived from reach-scale rating curves, thus making the hydraulic routing model obsolete.
- Methodology: International readership is used to Muskingum and Muskingum-Cunge algorithms. It would be useful to align your description with these standard algorithms. Is your nonlinear Muskingum model equivalent to Muskingum-Cunge? While that seems to be the case for the parameterization of X (eq 7), the parameterization of K (eq 6) is different. In Muskingum-Cunge, parameter K (i.e. reach length divided by wave speed) decreases with increasing discharge (wave speed increases), while in your approach (eg 6) it seems that you assume increasing K with increasing discharge. This seems counter-intuitive at first sight and needs more explanation/motivation. Any deviation from standard Muskingum-Cunge should be carefully explained and motivated.
- Methodology: Equation 12: Variable H is not explained. I assume that this is water surface elevation (WSE) or water level. The formulation f(H) then correctly describes that flow cross-sectional area depends on WSE, according to the chosen geometry assumption (rectangular, triangular, etc.). However, hydraulic radius R is written without WSE dependence. Is this just a mistake or do you assume constant hydraulic radius? In reality, R will definitely be R(H).
- It should be clearly stated that the translation from Muskingum parameters to geometry (which is a direct consequence of the equivalence of Muskingum-Cunge and diffusive-wave approx. of de Saint-Venant) can only be done for given/known hydraulic roughness (n). However hydraulic roughness values are usually not available. Usually, Manning numbers are obtained from inverse modeling, i.e. fitting observed water levels for given geometry and discharge. This is a well-known parameter trade-off in hydraulic modeling, which stems from the fact that conveyance is a function of both geometry and hydraulic roughness. The impacts of both on conveyance cannot be separated based on water levels only.
Citation: https://doi.org/10.5194/egusphere-2025-4059-RC2 - AC2: 'Reply on RC2', Gang Chen, 03 Mar 2026
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The manuscript proposes a workflow that takes reach-scale Muskingum routing parameters (K and X), translates them into an “equivalent” one-dimensional channel geometry (choosing among simple parametric shapes), and then uses those inferred cross-sections in a 1-D Saint-Venant solver to simulate flood wave propagation along long river reaches. The authors calibrate Muskingum parameters to reproduce observed hydrographs, present algebraic relations that map routing quantities to width, depth, slope and roughness, and report that the resulting hydrodynamic model reproduces the calibrated discharge time series at downstream gauges.
However, there are several limitations that undermine the central claim that routing parameters can be interpreted as physical channel geometry.
First, the inversion is underdetermined: the method attempts to recover multiple geometric and roughness unknowns from a small set of lumped routing observables, so the chosen shape family effectively forces one of many possible solutions rather than revealing a unique physical cross-section. The manuscript defines to mean that the cross-sectional area depends only on water depth , but Eqs. (17-19) explicitly contain additional variables (e.g. slope, roughness, geometric parameters) so the area is treated as a function of more than .
Second, the validation is circular and insufficient: Muskingum parameters are calibrated to match hydrographs, the calibrated values are converted to cross-sections, and the hydrodynamic model unsurprisingly reproduces the same hydrographs—this does not demonstrate that inferred widths, depths or slopes correspond to measured channel form.
Third, key algebraic approximations and a kinematic celerity assumption are introduced but not benchmarked against analytical solutions, synthetic channels with known geometry, or higher-fidelity (2D) models, so the error and applicability bound of the derivation are unknown.
Fourth, important practical issues are unaddressed: some reported weighting values fall outside conventional bounds, there is no propagation of calibration uncertainty into the inferred geometry, and no identifiability or equifinality analysis to show that the geometry estimates are robust.
Finally, representing long heterogeneous reaches as a single uniform cross-section ignores spatial variability (width, slope, tributaries, controls and backwater effects) that one-dimensional hydrodynamics are typically meant to capture.
To improve the manuscript the authors should clearly frame the product as an “equivalent storage” representation unless and until independent geometric validation is provided, perform synthetic and 2-D benchmark tests to quantify the limits of the algebraic approximations, propagate parameter uncertainty through to geometric and hydraulic outputs, and supply systematic comparisons of inferred widths/depths/slopes against surveyed or remote-sensing data at multiple locations and flow stages. Those steps will show whether the method offers a pragmatic engineering approximation or is simply a re-packaged calibration exercise.