On the interaction of observation and prior error correlations in data assimilation

摘要

The importance of prior error correlations in data assimilation has long been known, however, observation error correlations have typically been neglected. Recent progress has been made in estimating and accounting for observation error correlations, allowing for the optimal use of denser observations. Given this progress, it is now timely to ask how prior and observation error correlations interact and how this affects the value of the observations in the analysis. Addressing this question is essential to understanding the optimal design of future observation networks for high-resolution numerical weather prediction. This paper presents new results which unify and advance upon previous studies on this topic. ududThe interaction of the prior and observation error correlations is illustrated with a series of 2-variable experiments in which the mapping between the state and observed variables (the observation operator) is allowed to vary. In an optimal system, the reduction in the analysis error variance and spread of information is shown to be greatest when the observation and prior errors have complementary statistics. For example, in the case of direct observations, when the correlations between the observation and prior errors have opposite signs. This can be explained in terms of the relative uncertainty of the observations and prior on different spatial scales. The results from these simple 2-variable experiments are used to inform the optimal observation density for observations of a circular domain (with 32 grid points). It is found that dense observations are most beneficial when they provide a more accurate estimate of the state at smaller scales than the prior estimate. In the case of second order auto-regressive correlation functions, this is achieved when the lengthscales of the observation error correlations are greater than those of the prior estimate and the observations are direct measurements of the state variables.