Igor Yu. Gejadze

Adjoint to the Hessian derivative and error covariances in variational data assimilation

Abstract 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 optimal solution error is considered through the errors of input data (background and observation errors). The optimal solution error covariance operator is approximated by the inverse Hessian of the auxiliary (linearized) data assimilation problem, which involves the tangent linear model constraints. We show that the derivative of the inverse Hessian with respect to the exact solution may be treated as the measure of nonlinearity for analysis error covariances in variational data assimilation problems.

On optimal solution error covariances in variational data assimilation

Abstract 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.

On Variational Data Assimilation for 1D and 2D Fluvial Hydraulics

We address two problems related to variational data assimilation (VDA) as applied to river hydraulics (1D and 2D shallow water models). First, we seek to estimate accurately some parameters such as the inflow discharge, manning coefficients, the topography and/or the initial state. We develop a method which allow to assimilate lagrangian data (trajectory particles at the surface e.g. extracted from video images). Second, we develop a joint data assimilation - coupling method. We seek to couple accurately a 1D global net-model (rivers net) and a local 2D shallow water model (zoom into a flooded area), while we assimilate data. Numerical twin experiments are presented.

Hessian-based covariance approximations in variational data assimilation

Abstract The problem of variational data assimilation (estimation) for a nonlinear model is considered in general operator formulation. Hessian-based methods are presented to compute the estimation error covariances. The importance of dynamic formulation and the role of the Hessian and its inverse are discussed.

On model error in variational data assimilation

Abstract The 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 optimal solution error covariance can be constructed using the Hessian of the auxiliary data assimilation problem. The aim of this paper is to study the evolution of model errors via data assimilation. The optimal solution error covariances are derived in the case of imperfect model and for the weak constraint formulation, when the model euations determine the cost functional.

Reduced-space inverse Hessian for analysis error covariances in variational data assimilation

Abstract 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 considered in a reduced control space to show that the analysis error covariance operator can be approximated by the inverse Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints. The reduced-space Hessian is constructed in the explicit form, which allows an efficient computation of the analysis error covariance operator.