Optimal solution error covariance in highly nonlinear problems of variational data assimilation

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem (see, e.g.[1]) to find the initial condition, boundary conditions or model parameters. The input data contain observation and background errors, hence there is an error in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can be approximated by the inverse Hessian of the cost functional of an auxiliary data assimilation problem ([2], [3]). The relationship between the optimal solution error covariance matrix and the Hessian of the auxiliary control problem is discussed for different degrees of validity of the tangent linear hypothesis. For problems with strongly nonlinear dynamics a new statistical method based on computation of a sample of inverse Hessians is suggested. This method relies on the efficient computation of the inverse Hessian by means of iterative methods (Lanczos and quasi-Newton BFGS) with preconditioning. The method allows us to get a sensible approximation of the posterior covariance matrix with a small sample size. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term. The first author acknowledges the funding through the project 09-01-00284 of the Russian Foundation for Basic Research, and the FCP program "Kadry"

Tidal straining, mixing and lagrangian flow residuals around headlands

Tidal flow past an idealized triangular headland was investigated using an explicit finite difference solver for the equations of shallow water flow. Flow separation and transient eddies developed downstream from the tip of the headland on each successive half tidal cycle. Details of the flow patterns were studied using Lagrangian particle tracks. Mixing and dispersion diagrams were created that showed the effect of the eddies on sets of colour-coded particles. These revealed both the inter-mixing of water masses originally located on each side of the headland and the straining of discrete volumes of water originating from locations seaward of the headland tip. The pattern of Eulerian tidal residuals was contrasted with the pattern of Lagrangian tidal residuals. The patterns were found to be very different from one another; the Lagrangian residuals revealed complex detail with some systematic features. It is only the Lagrangian residuals that can be used to make valid inferences about the mixing of dissolved or suspended material when the length-scale of the tidal excursion is similar to the size of the headland and of the eddies. In such cases, the use of the Eulerian residuals is invalid. The mixing rates were also quantified by evaluating the effective diffusion coefficient for the eddy system. Four distinct stages of development of dispersion were identified and elucidated by the use of mixing diagrams. The impact of transient eddies was shown to be confined to a distinct mixing zone within which high rates of strain and diffusion were observed

Open boundary control for Navier-Stokes equations including the free surface: adjoint sensitivity analysis

This paper develops the adjoint sensitivities to the free-surface barotropic Navier- Stokes equations in order to allow for the assimilation of measurements of currents and free-surface elevations into an unsteady flow solution by open-boundary control. To calculate a variation in a surface variable, a mapping is used in the vertical to shift the problem into a fixed domain. A variation is evaluated in the transformed space from the Jacobian matrix of the mapping. This variation is then mapped back into the original space where it completes a tangent linear model. The adjoint equations are derived using the scalar product formulas redefined for a domain with variable bounds. The method is demonstrated by application to an unsteady fluid flow in a one-dimensional open channel in which horizontal and vertical components of velocity are represented as well as the elevation of the free surface (a 2D vertical section model). This requires the proper treatment of open boundaries in both the forward and adjoint models. A particular application is to the construction of a fully three-dimensional coastal ocean model that allows assimilation of tidal elevation and current data. However, the results are general and can be applied in a wider context

Current data assimilation modelling for oil spill contingency planning

A method has been developed to assimilate, in near real-time (15 min updates), previous termcurrentnext term meter previous termdatanext term collected by Shell in its operations on the Brunei Shelf Sea. The results from the software are depth mean previous termcurrentnext term flow fields that take into account the requirements of mass and momentum conservation. The previous termdata assimilationnext term method minimises, in a least squares sense, a cost function that describes the differences between the measured previous termdatanext term and the model counterparts. The minimisation is subject to the weak constraint that the residuals of the shallow water equations are also minimised. Based on previous termcurrent datanext term collected over one year, the previous termdata assimilationnext term method was also used to calculate a set of 209 'typical' flow fields representing the most frequent flow conditions in that year. Continuous trajectories of hypothetical previous termspillsnext term were computed by integrating particle tracks forward through the year-long previous termcurrentnext term time series along with the corresponding wind previous termdata.next term At each time step, the most appropriate 'typical' flow that best represents the measured flow was selected and scaled for use in the integration. The trajectories were used to assess risks of previous termoilnext term landfall

Open boundary control problem for Navier-Stokes equations including a free surface: data assimilation

This paper develops the data-assimilation procedure in order to allow for the assimilation of measurements of currents and free-surface elevations into an unsteady flow solution governed by the free-surface barotropic Navier-Stokes equations. The flow is considered in a 2D vertical section in which horizontal and vertical components of velocity are represented as well as the elevation of the free surface. Since a possible application is to the construction of a coastal (limited area) circulation model, the open boundary control problem is the main scope of the paper. The assimilation algorithm is built on the limited memory quasi-Newton LBFGS method guided by the adjoint sensitivities. The analytical step search, which is based on the solution of the tangent linear model, is used. We process the gradients to regularize the solution. In numerical experiments we consider different wave patterns with a purpose to specify a set of incomplete measurements, which could be sufficient for boundary-control identification. As a result of these experiments we formulate some important practical conclusions

A numerical model for the propagation of short gravity waves and the resulting circulation around nearshore structures

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Water quality, Sepetiba Bay, Brazil

Sepetiba Bay is located at 23o S, 44o W in Rio de Janeiro State, Brazil. Water samples were taken at eight locations adjacent to the north shore of the Bay, near to villages and towns without sewage treatment provision. The samples were analysed and total and faecal coliform concentrations determined. A hydrodynamic model of the Bay was used together with a species dispersion model based on an adaptive quadtree mesh to predict faecal concentrations in the Bay. Effluent sources used in the model were defined using population data from census returns with flow and concentration values estimated using standard values recommended by the World Bank (WB) and the World Health Organisation (WHO). Sufficient agreement was obtained between the measured and predicted concentrations to support the use of WB and WHO summary statistics to estimate sources of sewage

Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term

Branch intensities and oscillator-strengths for the herzberg absorption systems in oxygen

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