Articles | Volume 19, issue 2
Nonlin. Processes Geophys., 19, 177–184, 2012
Nonlin. Processes Geophys., 19, 177–184, 2012

Research article 16 Mar 2012

Research article | 16 Mar 2012

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

V. Shutyaev1, I. Gejadze2, G. J. M. Copeland2, and F.-X. Le Dimet3 V. Shutyaev et al.
  • 1Institute of Numerical Mathematics, Russian Academy of Sciences, 119333 Gubkina 8, Moscow, Russia
  • 2Department of Civil Engineering, University of Strathclyde, 107 Rottenrow, Glasgow, G4 ONG, UK
  • 3MOISE project (CNRS, INRIA, UJF, INPG), LJK, Université de Grenoble, BP 53, 38041 Grenoble, France

Abstract. The problem of variational data assimilation (DA) for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition, boundary conditions and/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 function. For problems with strongly nonlinear dynamics, a new statistical method based on the 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. Numerical examples are presented for the model governed by the Burgers equation with a nonlinear viscous term.