On analysis error covariances in variational data assimilation

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The equation for the analysis error is derived through the errors of the input data (background and observation errors). This equation is used to show that in a nonlinear case the analysis error covariance operator can be approximated by the inverse Hessian of an auxiliary data assimilation problem which involves the tangent linear model constraints. The inverse Hessian is constructed by the quasi-Newton BFGS algorithm when solving the auxiliary data assimilation problem. A fully nonlinear ensemble procedure is developed to verify the accuracy of the proposed algorithm. Numerical examples are presented

On optimal solution error covariances in variational data assimilation problems

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters such as distributed model coefficients or boundary conditions. The equation for the optimal solution error is derived through the errors of the input data (background and observation errors), and the optimal solution error covariance operator through the input data error covariance operators, respectively. The quasi-Newton BFGS algorithm is adapted to construct the covariance matrix of the optimal solution error using the inverse Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints. Preconditioning is applied to reduce the number of iterations required by the BFGS algorithm to build a quasi-Newton approximation of the inverse Hessian. Numerical examples are presented for the one-dimensional convection-diffusion model

On gauss-verifiability of optimal solutions in variational data assimilation problems with nonlinear dynamics

International audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. The optimal solution (analysis) error arises due to the errors in the input data (background and observation errors). Under the gaussian assumption the confidence region for the optimal solution error can be constructed using the analysis error covariance. Due to nonlinearity of the model equations the analysis pdf deviates from the gaussian. To a certain extent the gaussian confidence region built on a basis of a non-gaussian analysis pdf remains useful. In this case we say that the optimal solution is "gauss-verifiable". When the deviation from the gaussian further extends, the optimal solutions may still be partially (locally) gauss-verifiable. The aim of this paper is to develop a diagnostics to check gauss-verifiability of the optimal solution. We introduce a relevant measure and propose a method for computing decomposition of this measure into the sum of components associated to the corresponding elements of the control vector. This approach has the potential for implementation in realistic high-dimensional cases. Numerical experiments for the 1D Burgers equation illustrate and justify the presented theory

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

Design of the control set in the framework of variational data assimilation

International audienceSolving data assimilation problems under uncertainty in basic model parameters and in source terms may require a careful design of the control set. The task is to avoid such combinations of the control variables which may either lead to ill-posedness of the control problem formulation or compromise the robustness of the solution procedure. We suggest a method for quantifying the performance of a control set which is formed as a subset of the full set of uncertainty-bearing model inputs. Based on this quantity one can decide if the chosen 'safe' control set is sufficient in terms of the prediction accuracy. Technically, the method presents a certain generalization of the 'variational' uncertainty quantification method for observed systems. It is implemented as a matrix-free method, thus allowing high-dimensional applications. Moreover, if the Automatic Differentiation is utilized for computing the tangent linear and adjoint mappings, then it could be applied to any multi-input 'black-box' system. As application example we consider the full Saint-Venant hydraulic network model SIC2, which describes the flow dynamics in river and canal networks. The developed methodology seem useful in the context of the future SWOT satellite mission, which will provide observations of river systems the properties of which are known with quite a limited precision

On a 2D 'zoom' for the 1D shallow water model: coupling and data assimilation

In the context of river hydraulics we elaborate the idea of a 'zoom' model locally superposed on an open-channel network global model. The zoom model (2D shallow water equations) describes additional physical phenomena, which are not represented by the global model (1D shallow water equations with storage areas). Both models are coupled using the optimal control approach when the zoom model is used to assimilate local observations into the global model (variational data assimilation) by playing the part of a mapping operator. The global model benefits from using zooms, while no substantial modification to it is required. Numerical results on a toy test case show the feasibility of the suggested method

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

On error covariances in variational data assimilation

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 equation for the error of the optimal solution (analysis) is derived through the statistical errors of the input data (background and observation errors). The numerical algorithm is developed to construct the covariance operator of the analysis error using the covariance operators of the input errors. Numerical examples are presented

On optimal solution error covariances in variational data assimilation

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find some unknown parameters of the model. The equation for the error of the optimal solution is derived through the statistical errors of the input data. The covariance operator of the optimal solution error is obtained using the Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints