Title: Mixed lognormal-Gaussian incremental variational data assimilation

Author: Steven J. Fletcher (Cooperative Institute for Research in the Atmosphere, Colorado State University)
Anton J. Kliewer (Cooperative Institute for Research in the Atmosphere, Colorado State University)
Andrew S. Jones (Cooperative Institute for Research in the Atmosphere, Colorado State University)
John M. Forsythe (Cooperative Institute for Research in the Atmosphere, Colorado State University)

The introduction of the incremental formulation enabled four dimensional variational (4DVAR) to be considered operationally viable. The additive increment enabled the conservation of the Bayesian problem through the property that the sum of two Gaussian independent variables is also a Gaussian random variable. However, this additive property is not guaranteed for lognormal random variables. The equivalent property for a geometric based distribution is that the product of two lognormal random variables is also a lognormal random variable. Given this property it has been possible to derive a geometric equivalent for the tangent linear model that then enables the linearization of the lognormal and mixed lognormal-Gaussian 3D and 4D VARS with respect to both additive and multiplicative increments. Therefore the resulting linearized formulation is probabilistically consistent. We present the theory to obtain the multiplicative and the mixed additive-multiplicative incremental formulation and present results with the Lorenz 63 model for different sizes of lognormal observation errors, frequency of obs, window lengths and show when the mixed approach produces the best analysis. We also compare a full Gaussian system to show that when the observation errors are more lognormal then the mixed approach remains stable while the Gaussian does not.

GMAO Head: Steven Pawson Global Modeling and Assimilation Office NASA Goddard Space Flight Center

Curator: Nikki Privé Last Updated: Feb 9 2015