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

📝 Abstract

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.

💡 Analysis

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.

📄 Content

1   A Dual EnKF for Estimating Water Level, Bottom Roughness, and Bathymetry in a 1-D Hydrodynamic Model   Milad Hooshyar Department of civil, Environmental, and construction Engineering, University of Central Florida. hooshyar.milad@knights.ucf.edu Stephen C. Medeiros Department of civil, Environmental, and construction Engineering, University of Central Florida. Dingbao Wang Department of civil, Environmental, and construction Engineering, University of Central Florida.
Scott C. Hagen Department of civil and environmental engineering, Louisiana State University. Abstract 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 Manning’s n and bathymetry simultaneously.
Key words: Parameter estimation; State estimation; Kalman Filter; Hydrodynamic model; Bathymetry estimation; Dual estimation

1. Introduction Hydrodynamic models typically utilize numerical techniques; i.e., finite element [Hagen et al., 2000; Tamura et al., 2014], finite difference [Sadourny, 1975], finite volume [Mingham and Causon, 1998; Bradford and Sanders, 2002], to solve a form of the Shallow Water Equations (SWEs). Given the initial water depths and velocities throughout the domain along with the boundary conditions, hydrodynamic models compute the deviation of the water surface and velocity for all predefined temporally and spatially discrete points. Hydrodynamic models, as with all numerical models, contain error due to both parametric and structural uncertainties [Tatang et al., 1997]. Parametric uncertainty is primarily due to insufficient knowledge about bottom roughness [Mayo et al., 2014], wind canopy [Teeter et al., 2001], bathymetry [Wilson et al., 2010; Wilson and Özkan-Haller, 2012], boundary and initial conditions, whereas structural uncertainty is mainly caused by poor representation of physical details, oftentimes due to insufficient model resolution [Hagen et al., 2000; Kim et al., 2014].
   Methods to counteract these uncertainties begin with improvements to the characterization of: the geometric description of the domain through increased model resolution [Hagen et al., 2000; 2   Kim et al., 2014], and the parameters [Schubert et al., 2008; Medeiros and Hagen, 2013]. When the best possible physical representation of the natural system is achieved, we are left with calibration and/or data assimilation to reduce model uncertainty. In calibration, the set of model parameters are adjusted to maximize the agreement of model results and measured data [Trucano et al., 2006]. In data assimilation, measured data are integrated into the numerical model as the simulation is performed in order to balance information from simulation and observation [Wang and Cai, 2008]. Filtering, a type of data assimilation, sequentially updates the state of the system based on available measurements. Among all available filtering methods, the Kalman Filter (KF) has been widely applied to deal with the uncertainty in numerical problems. The traditional KF updates the state of the system by minimizing the mean square error of the observed and measured state in linear systems. The KF is generalized for nonlinear systems by linearization of the state equation at each time step. This approach is called Extended Kalman Filter (EKF) [Ljung, 1979]. Although the EKF mitigates the linear model constraints of the traditional KF, it is not preferable for highly nonlinear systems due to the elimination of higher order moments of the error covariance equation [Madsen and Cañizares, 1999; Evensen, 2009]. Another approach for extending the KF to nonlinear systems is based on coupling KF and Monte Carlo simulation. This approach is called Ensemble Kalman Filter (EnKF) [Evensen, 2003]. Instead of one single state, an ensemble of states moves forward in

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