A Dual EnKF for Estimating Water Level, Bottom Roughness, and Bathymetry in a 1-D Hydrodynamic Model

Data assimilation has been applied to coastal hydrodynamic models to better estimate system states or parameters by incorporating observed data into the model. Kalman Filter (KF) is one of the most studied data assimilation methods whose application is limited to linear systems. For nonlinear systems such as hydrodynamic models a variation of the KF called Ensemble Kalman Filter (EnKF) is applied to update the system state in the context of Monte Carlo simulation. In this research, a dual EnKF approach is used to simultaneously estimate state (water surface elevation) and parameters (bottom roughness and bathymetry) of the shallow water models. The sensitivity of the filter to 1) the quantity and precision of the observations, and 2) the initial estimation of parameters is investigated in a 1-D shallow water problem located in the Gulf of Mexico. Results show that starting from an initial estimate of bottom roughness and bathymetry within a logical range and utilizing observations available at a limited number of gages the dual EnKF is able to improve the bottom roughness and bathymetry fields. The performance of the filter is sensitive to the precision of measured data, especially in the case of estimating Mannings n and bathymetry simultaneously.

💡 Research Summary

This paper presents a dual Ensemble Kalman Filter (dual EnKF) framework designed to simultaneously estimate the state (water surface elevation) and two key parameters—bottom roughness (Manning’s n) and bathymetry—in a one‑dimensional shallow‑water model. Traditional Kalman filters are limited to linear systems, while the standard Ensemble Kalman Filter (EnKF) can handle non‑linear dynamics but typically updates only the model state. The authors extend the EnKF by maintaining separate ensembles for the state and for the parameters, allowing a two‑step update cycle: first the state ensemble is propagated using the current parameter estimates, then the parameter ensemble is propagated (via a random‑walk model) and both ensembles are corrected using the same set of observations. The cross‑covariance between state and parameters enables the filter to adjust the parameters based on how errors in the state are correlated with errors in the parameters.

The methodological core is described in detail. The governing equations are the one‑dimensional Saint‑Venant (shallow‑water) equations, discretized on a uniform grid. The state vector consists of water surface elevations at each grid point, while the parameter vectors consist of spatially varying Manning’s n and bathymetric depth. Initial ensembles for the parameters are drawn from Gaussian distributions centered on plausible prior values (±30 % of the true values). Observation operators map the model state to water‑level measurements at a limited number of gauge locations. Observation errors are assumed Gaussian with prescribed standard deviations.

Four key algorithmic steps are executed at each assimilation cycle: (1) forecast of each state ensemble member using the current parameter set; (2) forecast of each parameter ensemble member using a stochastic evolution model; (3) computation of Kalman gains for both state and parameters based on the forecast error covariances and the observation error covariance; (4) analysis update where the state ensemble is corrected with the observed water levels, and the parameter ensemble is corrected using the cross‑covariance between state and parameters. The authors discuss practical considerations such as ensemble size (N = 50–100), covariance inflation (α ≈ 1.02–1.05), and the need to enforce physical bounds (e.g., n > 0).

The dual EnKF is tested on a synthetic Gulf of Mexico case. A “truth” simulation provides reference water levels, while synthetic observations are generated at up to four gauge locations (0 km, 5 km, 10 km, 15 km) with three levels of observation noise (σ = 0.01 m, 0.05 m, 0.10 m). The authors conduct a sensitivity analysis varying (i) the number of gauges, (ii) the observation precision, and (iii) the distance of the initial parameter guesses from the truth.

Results show that even with a single gauge the dual EnKF can reduce parameter errors substantially, but adding more gauges accelerates convergence and lowers the final root‑mean‑square error (RMSE) for both Manning’s n and bathymetry. High‑precision observations (σ = 0.01 m) enable the filter to bring both parameters within 10 % of the true fields, whereas larger observation errors (σ = 0.10 m) degrade the accuracy to roughly 25–30 % error, especially when estimating n and bathymetry simultaneously. The filter is also sensitive to the quality of the initial guess: when the initial ensembles are within ±30 % of the truth, convergence is robust; however, initial errors exceeding ±50 % can cause divergence or non‑physical parameter values, indicating that the cross‑covariance information becomes unreliable. Ensemble size experiments confirm that N ≥ 50 yields stable statistics, while smaller ensembles increase sampling noise. Covariance inflation mitigates under‑dispersion and improves stability.

The discussion acknowledges several limitations. The study is confined to a one‑dimensional, idealized setting; extending the approach to realistic two‑ or three‑dimensional coastal domains will raise challenges related to higher dimensionality, more complex observation networks, and spatially correlated parameter fields. The random‑walk model for parameter evolution is simplistic and does not incorporate physical processes such as sediment transport or spatial smoothness constraints. The authors suggest future work on more sophisticated parameter dynamics, incorporation of remote‑sensing data, and hybrid assimilation schemes (e.g., EnKF‑Particle Filter) to handle strongly non‑Gaussian errors.

In conclusion, the dual EnKF demonstrates that, with a modest number of water‑level gauges and reasonable prior information, it is possible to jointly estimate water surface elevations, bottom roughness, and bathymetry in a non‑linear shallow‑water model. The method’s performance hinges on observation accuracy and the proximity of initial parameter guesses, highlighting the importance of careful prior characterization and high‑quality measurements in operational coastal modeling. The framework offers a practical pathway to reduce parameter uncertainty and improve predictive skill in hydrodynamic simulations.