Mark Buehner
1. Preconditioning 3d/4d-var with Hessian Eigenvectors
Currently, the 3d/4d-var cost function is preconditioned with respect to
only the background term. In the first part of the talk I will describe
how the leading eigen-decomposition of the observation cost function
Hessian is used to improve the preconditioning of both 3d and 4d-var.
The approach is similar to the one recently implemented at ECMWF, however,
with additional simplifications. The result is a straightforward approach
that decreases the required number of iterations by between 35% and 50%
without degrading the quality of the analyses.
2. Singular Vectors: For Covariance Propagation and Model Diagnosis
Singular vectors provide an orthogonal set of perturbations which lead to
maximum growth in the forecast over a specified time period and with
respect to the specified norms. Other NWP centres are currently using SVs
for ensemble prediction and covariance propagation to improve the
background term of the 3d/4d-var cost function. Using examples with GEM,
I will show how SVs can effectively identify unwanted numerical
instabilities. I will also show how they can efficiently provide the
leading eigenvectors of the flow-dependent forecast error covariance
matrix which are then incorporated in the 3d-var cost function.