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- Author or Editor: Zhigang Xu x
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Abstract
A direct inverse method is presented for inferring numerical model open boundary conditions from interior observational data. The dynamical context of the method is the frequency-domain 3D linear shallow water equations. A set of weight matrices is derived via finite-element discretization of the dynamical equations. The weight matrices explicitly express any interior solution of elevations or velocities as a weighted sum of boundary elevations. The interior data assimilation is then cast as a regression problem.
The weight matrix may be singular, which implies there may be an infinite set of boundary conditions that fit the data equally well. With the singular value decomposition technique, a general solution is provided for this infinite set of minimum-squared-misfit boundary conditions. Among them, a particular boundary condition, which minimizes potential energy on the boundary (hence the whole domain), is studied in detail: its confidence interval is defined and a way to smooth it is discussed.
Green’s function maps for the weight matrix provide insights into the dynamics inherent to the model domain. Such maps should be useful for many purposes. One of their usages demonstrated in this paper is to provide a physical explanation for the singularity of the system. Also discussed are how to assess the compatibility between data and the model and how to design a null-space smoothing device for smoothing the potential energy minimum boundary condition. While maintaining the goodness of fit between data and model, smoothing of the boundary condition may improve the interior solutions at the locations where the data have poor control.
The method is tested in a realistic domain but with synthetic data. The test yields very satisfactory results. Application of the method to Lardner’s open bay tidal problem demonstrates its advantages in computational accuracy and inexpensiveness.
Abstract
A direct inverse method is presented for inferring numerical model open boundary conditions from interior observational data. The dynamical context of the method is the frequency-domain 3D linear shallow water equations. A set of weight matrices is derived via finite-element discretization of the dynamical equations. The weight matrices explicitly express any interior solution of elevations or velocities as a weighted sum of boundary elevations. The interior data assimilation is then cast as a regression problem.
The weight matrix may be singular, which implies there may be an infinite set of boundary conditions that fit the data equally well. With the singular value decomposition technique, a general solution is provided for this infinite set of minimum-squared-misfit boundary conditions. Among them, a particular boundary condition, which minimizes potential energy on the boundary (hence the whole domain), is studied in detail: its confidence interval is defined and a way to smooth it is discussed.
Green’s function maps for the weight matrix provide insights into the dynamics inherent to the model domain. Such maps should be useful for many purposes. One of their usages demonstrated in this paper is to provide a physical explanation for the singularity of the system. Also discussed are how to assess the compatibility between data and the model and how to design a null-space smoothing device for smoothing the potential energy minimum boundary condition. While maintaining the goodness of fit between data and model, smoothing of the boundary condition may improve the interior solutions at the locations where the data have poor control.
The method is tested in a realistic domain but with synthetic data. The test yields very satisfactory results. Application of the method to Lardner’s open bay tidal problem demonstrates its advantages in computational accuracy and inexpensiveness.
Abstract
This paper proposes an effective approach on how to predict tsunamis rapidly following a submarine earthquake by combining a real-time GPS-derived tsunami source function with a set of precalculated all-source Green's functions (ASGFs). The approach uses the data from both teleseismic and coastal GPS networks to constrain a tsunami source function consisting of both sea surface elevation and horizontal velocity field, and uses the ASGFs to instantaneously transfer the source function to the arrival time series at the destination points. The ASGF can take a tsunami source of arbitrary geographic origin and resolve it as fine as the native resolution of a tsunami propagation model from which the ASGF is derived. This new approach is verified by the 2011 Tohoku tsunami using data measured by the Deep-Ocean Assessment and Reporting of Tsunamis (DART) buoys.
Abstract
This paper proposes an effective approach on how to predict tsunamis rapidly following a submarine earthquake by combining a real-time GPS-derived tsunami source function with a set of precalculated all-source Green's functions (ASGFs). The approach uses the data from both teleseismic and coastal GPS networks to constrain a tsunami source function consisting of both sea surface elevation and horizontal velocity field, and uses the ASGFs to instantaneously transfer the source function to the arrival time series at the destination points. The ASGF can take a tsunami source of arbitrary geographic origin and resolve it as fine as the native resolution of a tsunami propagation model from which the ASGF is derived. This new approach is verified by the 2011 Tohoku tsunami using data measured by the Deep-Ocean Assessment and Reporting of Tsunamis (DART) buoys.
Abstract
The authors first derive both Coriolis-induced and viscosity-induced stresses for arbitrary water depth and arbitrary wave direction. Opportunity is taken here to succinctly and rigorously derive the Longuet-Higgins virtual tangential stress due to wave motion. It is shown that the virtual stress is a projection on the surface slope of two viscous normal stresses acting on the vertical and horizontal planes. Then a simple Eulerian model is presented for the steady flow driven by waves and by waves and winds This simple Eulerian model demonstrates that the wave forcing can he easily incorporated with other conventional forcing, rather than resorting to a complicated and lengthy perturbation analysis of the Lagrangian equations of motion. A further focus is given to the wave-driven flow when the various limits of the wave-driven steady flow are discussed. The wave-driven steady flow given by the model yields a unified formula between Ursell and Hasselmann's inviscid but rotational theory and the Longuet-Higgins viscid but nonrotational theory, and it becomes an Eulerian counterpart of Madsen's deep-water solution when the deep-water limit is taken. The model is further expanded for the case of unsteady wave forcing, yielding a general formula for any type of time variation in the wave field. Two examples are considered: a suddenly imposed wave field that is then maintained steady and a suddenly imposed wave field that is then subject to internal and bottom frictional decay. The extension of these results to the case of random waves is briefly discussed. Finally, an example is presented that suggests the need to add surface wave forcing in classical shelf dynamics.
Abstract
The authors first derive both Coriolis-induced and viscosity-induced stresses for arbitrary water depth and arbitrary wave direction. Opportunity is taken here to succinctly and rigorously derive the Longuet-Higgins virtual tangential stress due to wave motion. It is shown that the virtual stress is a projection on the surface slope of two viscous normal stresses acting on the vertical and horizontal planes. Then a simple Eulerian model is presented for the steady flow driven by waves and by waves and winds This simple Eulerian model demonstrates that the wave forcing can he easily incorporated with other conventional forcing, rather than resorting to a complicated and lengthy perturbation analysis of the Lagrangian equations of motion. A further focus is given to the wave-driven flow when the various limits of the wave-driven steady flow are discussed. The wave-driven steady flow given by the model yields a unified formula between Ursell and Hasselmann's inviscid but rotational theory and the Longuet-Higgins viscid but nonrotational theory, and it becomes an Eulerian counterpart of Madsen's deep-water solution when the deep-water limit is taken. The model is further expanded for the case of unsteady wave forcing, yielding a general formula for any type of time variation in the wave field. Two examples are considered: a suddenly imposed wave field that is then maintained steady and a suddenly imposed wave field that is then subject to internal and bottom frictional decay. The extension of these results to the case of random waves is briefly discussed. Finally, an example is presented that suggests the need to add surface wave forcing in classical shelf dynamics.
Abstract
The barotropic M2 tide over the Newfoundland and southern Labrador Shelves and adjacent deep ocean is studied using a linear harmonic finite-element model and a newly developed direct inverse method for data assimilation. The dataset includes harmonic tidal constituents from TOPEX/Poseidon altimetry, coastal tide gauges, bottom pressure gauges, and moored current meters. Three modeling approaches are taken: a conventional modeling approach with boundary conditions specified from along-boundary observations; a full interior data assimilative approach, which provides an optimal domain-wide solution; and a sensitivity study in which the roles of various data subsets and the frictional parameters are investigated.
The optimal solution from the full assimilative approach has rms misfits of 3.5 cm and 1.3 cm s−1 for elevation and current, respectively (in terms of distances on the complex plane), compared to overall rms amplitudes of 30 cm and 6 cm s−1. These misfits are reduced by more than 40% and 70% from those in the conventional solution. Formal confidence limits for the optimal solution can be estimated but depend on assumptions about the spatial covariance of the observational residuals. The sensitivity study provides quantitative indications of the importance of the quantity and location of the observational data and indicates little sensitivity to the specified frictional fields within a reasonable range. In particular, the inclusion of a fraction of the velocity data in the assimilation results in a significant improvement in the model fit to the velocity observations.
Abstract
The barotropic M2 tide over the Newfoundland and southern Labrador Shelves and adjacent deep ocean is studied using a linear harmonic finite-element model and a newly developed direct inverse method for data assimilation. The dataset includes harmonic tidal constituents from TOPEX/Poseidon altimetry, coastal tide gauges, bottom pressure gauges, and moored current meters. Three modeling approaches are taken: a conventional modeling approach with boundary conditions specified from along-boundary observations; a full interior data assimilative approach, which provides an optimal domain-wide solution; and a sensitivity study in which the roles of various data subsets and the frictional parameters are investigated.
The optimal solution from the full assimilative approach has rms misfits of 3.5 cm and 1.3 cm s−1 for elevation and current, respectively (in terms of distances on the complex plane), compared to overall rms amplitudes of 30 cm and 6 cm s−1. These misfits are reduced by more than 40% and 70% from those in the conventional solution. Formal confidence limits for the optimal solution can be estimated but depend on assumptions about the spatial covariance of the observational residuals. The sensitivity study provides quantitative indications of the importance of the quantity and location of the observational data and indicates little sensitivity to the specified frictional fields within a reasonable range. In particular, the inclusion of a fraction of the velocity data in the assimilation results in a significant improvement in the model fit to the velocity observations.
