Origin error in estimation of 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. The optimal solution (analysis) error arises due to the errors of the input data (background and observation errors). For random and normally distributed input data 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. The so-called 'origin error' occurs [5] in estimating the analysis error covariance. An exact equation is derived for the origin error, and algorithms to estimate this error for computing the variances are presented.