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.