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 article presents new results, which unify and advance upon previous studies on this topic.The interaction of the prior and observation-error correlations is illustrated with a series of two-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 two-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 length-scales of the observation-error correlations are greater than those of the prior estimate and the observations are direct measurements of the state variables.

Operational forecasting centres are currently developing data assimilation systems for coupled atmosphereÁ ocean models. Strongly coupled assimilation, in which a single assimilation system is applied to a coupled model, presents significant technical and scientific challenges. Hence weakly coupled assimilation systems are being developed as a first step, in which the coupled model is used to compare the current state estimate with observations, but corrections to the atmosphere and ocean initial conditions are then calculated independently. In this paper, we provide a comprehensive description of the different coupled assimilation methodologies in the context of four-dimensional variational assimilation (4D-Var) and use an idealised framework to assess the expected benefits of moving towards coupled data assimilation. We implement an incremental 4D-Var system within an idealised single-column atmosphereÁocean model. The system has the capability to run both strongly and weakly coupled assimilations as well as uncoupled atmosphere-or ocean-only assimilations, thus allowing a systematic comparison of the different strategies for treating the coupled data assimilation problem. We present results from a series of identical twin experiments devised to investigate the behaviour and sensitivities of the different approaches. Overall, our study demonstrates the potential benefits that may be expected from coupled data assimilation. When compared to uncoupled initialisation, coupled assimilation is able to produce more balanced initial analysis fields, thus reducing initialisation shock and its impact on the subsequent forecast. Single observation experiments demonstrate how coupled assimilation systems are able to pass information between the atmosphere and ocean and therefore use near-surface data to greater effect. We show that much of this benefit may also be gained from a weakly coupled assimilation system, but that this can be sensitive to the parameters used in the assimilation.

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.

The strong-constraint formulation of four-dimensional variational data assimilation (4D-Var) assumes that the model used in the process perfectly describes the true dynamics of the system. However, this assumption often does not hold and the use of an erroneous model in strong-constraint 4D-Var can lead to a sub-optimal estimation of the initial conditions. We show how the presence of model error can be correctly accounted for in strong constraint 4D-Var by allowing for errors in both the observations and the model when considering the statistics of the innovation vector. We demonstrate that, when these combined model error and observation-error statistics are used in place of the standard observation error statistics in the strong-constraint formulation of 4D-Var, a statistically more accurate estimate of the initial state is obtained.The calculation of the combined model error and observation-error statistics requires the specification of model error covariances, which in practice are often unknown. We present a method to estimate the combined statistics from innovation data that does not require explicit specification of the model error covariances. Numerical experiments using the linear advection equation and a simple nonlinear coupled model demonstrate the success of the new methods in reducing the error in the estimate of the initial state, even in the case when only the uncorrelated part of the model error is accounted for.