Title: Background error covariance modeling using normal modes

Authors: Ruth Petrie (University of Reading)
Ross Bannister (University of Reading)

An important ingredient in operational variational data assimilation problems is the background error covariance matrix (B). This matrix describes statistically the degree of confidence in the prior state, and describes univariate and multivariate couplings. The dimensions of the P^f-matrix prohibits explicit calculation, storage or propagation with current resources. In conventional variational data assimilation this matrix is approximated by the background error covariance matrix (B). The B-matrix is typically modelled using a Control Variable Transform (CVT) which allows B to be represented in a compact and efficient way. The premise of the CVT is to simplify the problem by performing a series of transforms such that all the variables become decorrelated both spatially and from each other. Typically the CVT uses balance relationships (e.g. linear balance) to decorrelate multivariate relationships.

Leading edge forecast models for operational Numerical Weather Prediction (NWP) can operate at resolutions of 0(1 km) and at these resolutions it becomes possible to resolve small scale weather features such as thunderstorms. At the convective scale the larger aspect ratio implies that the use of balance relationships in the CVT may no longer be appropriate.

In this work an alternative approach to modelling B is proposed. A non-hydrostatic, 2D (longitude-height) toy-model is derived from the Euler equations. The control variable transform utilizes the normal modes of the linearized model, which are by definition uncorrelated (in a linear sense). The control variables are given as a function of physical mode (acoustic, gravity, rossby) and horiztonal and vertical wavenumber. This approach is more suited to convective scale data assimilation as no balance relationships are explicitly imposed. This approach is analyzed in terms of the implied covariances of the CVT.

GMAO Head: Michele Rienecker Global Modeling and Assimilation Office NASA Goddard Space Flight Center

Curator: Nikki Privé Last Updated: May 27 2011