TY - JOUR

T1 - Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics

AU - Gejadze, I.Yu

AU - Copeland, G.J.M

AU - Le Dimet, F.X.

AU - Shutyaev, V.

PY - 2011/9/10

Y1 - 2011/9/10

N2 - 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 data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term.

AB - 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 data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term.

KW - posterior covariance

KW - computational physics

KW - nonlinear dynamics

UR - http://www.scopus.com/inward/record.url?scp=80052669410&partnerID=8YFLogxK

UR - http://www.sciencedirect.com/science/article/pii/S0021999111001902

U2 - 10.1016/j.jcp.2011.03.039

DO - 10.1016/j.jcp.2011.03.039

M3 - Article

VL - 230

SP - 7923

EP - 7943

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 22

ER -