Representation, representativity, representativeness error, forward interpolation error, forward model error, observation‐operator error, aggregation error and sampling error are all terms used to refer to components of observation error in the context of data assimilation. This article is an attempt to consolidate the terminology that has been used in the earth sciences literature and was suggested at a European Space Agency workshop held in Reading in April 2014. We review the state of the art and, through examples, motivate the terminology. In addition to a theoretical framework, examples from application areas of satellite data assimilation, ocean reanalysis and atmospheric chemistry data assimilation are provided. Diagnosing representation‐error statistics as well as their use in state‐of‐the‐art data assimilation systems is discussed within a consistent framework.

Data assimilation is used in atmospheric chemistry models to improve air quality forecasts, construct re-analyses of three-dimensional chemical (including aerosol) concentrations and perform inverse modeling of input variables or model parameters (e.g., emissions). Coupled chemistry meteorology models (CCMM) are atmospheric chemistry models that simulate meteorological processes and chemical transformations jointly. They offer the possibility to assimilate both meteorological and chemical data; however, because CCMM are fairly recent, data assimilation in CCMM has been limited to date. We review here the current status of data assimilation in atmospheric chemistry models with a particular focus on future prospects for data assimilation in CCMM. We first review the methods available for data assimilation in atmospheric models, including variational methods, ensemble Kalman filters, and hybrid methods. Next, we review past applications that have included chemical data assimilation in chemical transport models (CTM) and in CCMM. Observational data sets available for chemical data assimilation are described, including surface data, surface-based remote sensing, airborne data, and satellite data. Several case studies of chemical data assimilation in CCMM are presented to highlight the benefits obtained by assimilating chemical data in CCMM. A case study of data assimilation to constrain emissions is also presented. There are few examples to date of joint meteorological and chemical data assimilation in CCMM and potential difficulties associated with data assimilation in CCMM are discussed. As the number of variables being assimilated increases, it is essential to characterize correctly the errors; in particular, the specification of error cross-correlations may be problematic. In some cases, offline diagnostics are necessary to ensure that data assimilation can truly improve model performance. However, the main challenge is likely to be the paucity of chemical data available for assimilation in CCMM.

This review discusses recent advances in geophysical data assimilation beyond Gaussian statistical modeling, in the fields of meteorology, oceanography, as well as atmospheric chemistry. The non-Gaussian features are stressed rather than the nonlinearity of the dynamical models, although both aspects are entangled. Ideas recently proposed to deal with these non-Gaussian issues, in order to improve the state or parameter estimation, are emphasized.The general Bayesian solution to the estimation problem and the techniques to solve it are first presented, as well as the obstacles that hinder their use in high-dimensional and complex systems. Approximations to the Bayesian solution relying on Gaussian, or on second-order moment closure, have been wholly adopted in geophysical data assimilation (e.g., Kalman filters and quadratic variational solutions). Yet, nonlinear and non-Gaussian effects remain. They essentially originate in the nonlinear models and in the non-Gaussian priors. How these effects are handled within algorithms based on Gaussian assumptions is then described. Statistical tools that can diagnose them and measure deviations from Gaussianity are recalled.The following advanced techniques that seek to handle the estimation problem beyond Gaussianity are reviewed: maximum entropy filter, Gaussian anamorphosis, non-Gaussian priors, particle filter with an ensemble Kalman filter as a proposal distribution, maximum entropy on the mean, or strictly Bayesian inferences for large linear models, etc. Several ideas are illustrated with recent or original examples that possess some features of high-dimensional systems. Many of the new approaches are well understood only in special cases and have difficulties that remain to be circumvented. Some of the suggested approaches are quite promising, and sometimes already successful for moderately large though specific geophysical applications. Hints are given as to where progress might come from.