Two general algorithms for solving constrained minimization problems are presented and discussed in the context of analysis and assimilation of meteorological observations. In both algorithms. the original constrained problem is transformed by appropriate modifications into one unconstrained problem, or into a sequence of unconstrained problems. The main advantage of proceeding in this way is that the new unconstrained problems can be solved by classical descent algorithms, thus avoiding the need of directly solving the Euler-Lagrange equations of the original constrained problem.The first algorithm presented in the uugmented lugrangiun algorithm. It generalizes the more classical penalty and duality algorithms. The second algorithm, inspired from optimal control techniques, is based on an appropriate use of an adjoint dynamical equation, and seems to be particularly well adapted to the assimilation of observations distributed in time. Simple numerical examples show the ability of these algorithms to solve non-linear minimization problems of the type encountered in meteorology. Their possible use in more complex situations is discussed, in particular in terms of their computational cost.

Two general algorithms for solving constrained minimization problems are presented and discussed in the context of analysis and assimilation of meteorological observations. In both algorithms, the original constrained problem is transformed by appropriate modifications into one unconstrained problem, or into a sequence of unconstrained problems. The main advantage of proceeding in this way is that the new unconstrained problems can be solved by classical descent algorithms, thus avoiding the need of directly solving the Euler‐Lagrange equations of the original constrained problem. The first algorithm presented in the augmented lagrangian algorithm. It generalizes the more classical penalty and duality algorithms. The second algorithm, inspired from optimal control techniques, is based on an appropriate use of an adjoint dynamical equation, and seems to be particularly well adapted to the assimilation of observations distributed in time. Simple numerical examples show the ability of these algorithms to solve non‐linear minimization problems of the type encountered in meteorology. Their possible use in more complex situations is discussed, in particular in terms of their computational cost.

In variational data assimilation (VDA) for meteorological and/or oceanic models, the assimilated fields are deduced by combining the model and the gradient of a cost functional measuring discrepancy between model solution and observation, via a first-order optimality system. However, existence and uniqueness of the VDA problem along with convergence of the algorithms for its implementation depend on the convexity of the cost function. Properties of local convexity can be deduced by studying the Hessian of the cost function in the vicinity of the optimum. This shows the necessity of second-order information to ensure a unique solution to the VDA problem.In this paper a comprehensive review of issues related to second-order analysis of the problem of VDA is presented along with many important issues closely connected to it. In particular issues of existence, uniqueness, and regularization through second-order properties are examined. The focus then shifts to second-order information related to statistical properties and to issues related to preconditioning and optimization methods and second-order VDA analysis. Predictability and its relation to the structure of the Hessian of the cost functional is then discussed along with issues of sensitivity analysis in the presence of data being assimilated. Computational complexity issues are also addressed and discussed.Automatic differentiation issues related to second-order information are also discussed along with the computational complexity of deriving the second-order adjoint.Finally an application aimed at illustrating the use of automatic differentiation for deriving the second-order adjoint as well as the Hessian/vector product applied to minimizing a cost functional of a meteorological problem using the truncated-Newton method is presented. Results verifying numerically the computational cost of deriving the second-order adjoint as well as results related to the spectrum of the Hessian of the cost functional are displayed and discussed.

International audienceWe present a method to use lagrangian data from remote sensing observation in a variational data assimilation process for a river hydraulics model based on the bidimensional shallow water equations. The trajectories of particles advected by the flow can be extracted from video images and are used in addition to classical eulerian observations. This lagrangian data brings information on the surface velocity thanks to an appropriate transport model. Numerical twin data assimilation experiments in an academic flow configuration demonstrate that this method makes it possible to significantly improve the identification of local bed elevation and initial conditions