Implications and impacts of transforming lognormal variables into normal variables in VAR

In this paper we compare two different approaches to deal with lognormal variables/observations, and hence their errors, in a 3-dimensional variational data assimilation framework. The first approach uses a transform to make the lognormal random variable into a normal random variable and hence use current data assimilation techniques. The second approach uses the correct distribution for a collection of normal and lognormal random variables through a hybrid distribution which gives a different cost function to minimise. The properties of these two different approaches is that the first finds an analysis median whilst the second find the analysis mode. The two are compared with the Lorenz 1963 model with different observational error variances and different times between the observations. The second approach out-performs the first here but the first is often used in operational centres with a form of bias correction and so we discuss this implication at the end of the paper.