Additive and Multiplicative Incremental 4DVAR for Mixed Gaussian and Lognormal Distribution Errors

Dr. Steven Fletcher, Cooperative Institute for Research in the Atmosphere, Colorado State University

One of the advances that allowed 4DVAR to be operational for synoptic numerical weather prediction was the introduction of incremental 4DVAR. This method assumes that the errors are additive and Gaussian in nature. However, as work recently has shown, there are errors which are multiplicative. A full field version of the 4DVAR equations have been derived and tested in a toy problem for the situation where there is a mix of Gaussian and lognormal background and observational errors.

It is not straight-forward, however, to extend the incremental theory to multiplicative errors. One approach which has been suggested recently involves using a transform for the increment. It is shown here that the increment that is found is not the “incremental mode”, i.e. the most likely state for the increment, but rather a median state for the increment. To overcome the multiplicative nature of the errors we present a geometric tangent linear approximation which enables us to linearize the observation operator with respect to a consistent lognormal multiplicative increment.

In this talk we present an equivalent incremental version of the mixed lognormal-Gaussian which is based upon finding the most-likely state for additive increments for the Gaussian variables and lognormal for the multiplicative lognormal variables. We test this new approach with the Lorenz 1963 model under different size observational errors, observation window lengths, observation set sizes as well as a comparison against an assumed Gaussian fits all approach.

An interesting question about how would a user know when to assume a Gaussian or a lognormal error is also partially addressed in the second part of this talk where we present statistical tests designed to detect a lognormal signal in temporal sampling of a year of GFS moisture fields.