Title: Accounting for linearisation error in the Extended Kalman Filter and 4D-Var

Authors: Tim Payne (Met Office)

Both the Extended Kalman Filter and incremental 4D-Var make use of a linear model, to propagate covariances explicitly in the first case and implicitly in the second. Experience with incremental 4D-Var at major operational centres suggests that the relative error in the linear model can approach 100% after 12 hours in several fields and model levels; this may be contrasted with the full model which shows significant skill even at 5 days. It would appear that linear model error can be far larger than the error in the full model. However, while much work has been done on the inclusion of full model error in data assimilation, no one has yet attempted to account for error in the linear model. This is ironic, as while full model error is largely unknown (even unknowable), we have in principle unlimited knowledge of every aspect of the errors in the linear model.

The sources of error in the linear model, on top of the error in truncating the Taylor expansion of the nonlinear model, are several. In operational DA probably the largest source of error comes from physical processes which are either not included at all, or only very approximately. This may be due to the large number of thresholds in their formulation, the complexity of the code, or their perceived cost. Another source of error is the lower resolution of the linear compared with full model, typically by a factor of between two and four.

We present an as far as possible rigorous analysis of linear model error, and how it may be accounted for in the Extended Kalman Filter and incremental 4D-Var.

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

Curator: Nikki Privé Last Updated: May 27 2011