Measures of observation impact in non-Gaussian data assimilation

Abstract

A B S T R A C T Non-Gaussian/non-linear data assimilation is becoming an increasingly important area of research in the Geosciences as the resolution and non-linearity of models are increased and more and more non-linear observation operators are being used. In this study, we look at the effect of relaxing the assumption of a Gaussian prior on the impact of observations within the data assimilation system. Three different measures of observation impact are studied: the sensitivity of the posterior mean to the observations, mutual information and relative entropy. The sensitivity of the posterior mean is derived analytically when the prior is modelled by a simplified Gaussian mixture and the observation errors are Gaussian. It is found that the sensitivity is a strong function of the value of the observation and proportional to the posterior variance. Similarly, relative entropy is found to be a strong function of the value of the observation. However, the errors in estimating these two measures using a Gaussian approximation to the prior can differ significantly. This hampers conclusions about the effect of the non-Gaussian prior on observation impact. Mutual information does not depend on the value of the observation and is seen to be close to its Gaussian approximation. These findings are illustrated with the particle filter applied to the Lorenz '63 system. This article is concluded with a discussion of the appropriateness of these measures of observation impact for different situations.

Observation impact in data assimilation: the effect of non-Gaussian observation error

“…(6)] and so as a measure of the influence of a non-Gaussian prior it provides a more consistent result. Mutual information was also seen in Fowler and van Leeuwen (2012) to be affected a relatively small amount by a non-Gaussian prior.…”

“…This work follows on from Fowler and van Leeuwen (2012), in which the effect of a non-Gaussian prior on the impact of observations was studied when the likelihood was restricted to a Gaussian distribution. The main conclusions from that paper are summarised below.…”

“…This is not a desirable property for a measure of observation impact. Fowler and van Leeuwen (2012) concluded that comparing the analysis sensitivity to the sensitivity when a Gaussian prior was assumed eq. (4), showed that the Gaussian assumption could lead to both a large overestimation and a large underestimation depending on the value of the observation relative to the background and the structure of the prior.…”

“…In Fowler and van Leeuwen (2012) a Gaussian mixture was used to describe the prior distribution to allow for a wide range of non-Gaussian distributions. It was shown that, in the scalar case, the sensitivity of the analysis to observations was still given by the analysis error variance divided by the observation error variance as in the Gaussian case [see eq.…”

Estimation and calibration of observation impact signals using the Lanczos method in NOAA/NCEP data assimilation system

“…In the following, we will show the square root of error variance reduction, i.e., diag(C) (referred to as observation impact signal or OIS in the following) for different variables at different levels. Similar quantities, such as information content, relative entropy, degrees of freedom for signal, and mutual information, for quantifying the impacts of observations in a DA system have been introduced and studied by Rodgers (2000), Xu (2007), Zupanski et al (2007) and Fowler and van Leeuwen (2012).…”