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This chapter discusses data assimilation in atmospheric chemistry and air quality. Atmospheric chemistry dynamics and its simulations are significantly forced by emissions, in addition to other parameters. Data assimilation typically aims at analysing the initial state of the system. Hence, for atmospheric chemistry simulations, data assimilation must be extended beyond initial-value identification to identify further parameters by estimation techniques. This chapter describes advanced spatio-temporal data assimilation techniques used in atmospheric chemistry. The objective is joint parameter-initial-state estimation for a coupled four-dimensional variational data assimilation/inversion system using the chemistry-transport model EURAD-IM and satisfying three criteria: both parameter families must have dominant influence on the dynamics, they must be poorly known, and they must impinge on the system on the same timescale. A coupled initial-value/emission-inversion system is described, along with assimilation of tropospheric satellite data from both gas-phase and aerosol retrievals. Examples of aerosol data assimilation are also given.

This chapter discusses some of the implementation details that are necessary to apply data assimilation in the context of numerical weather prediction (NWP). It is divided into three parts. The first part addresses the processing of observations, which includes the transformation of raw data into a form that can be processed by a data assimilation system, quality control, and data thinning. The second part discusses two important aspects of data assimilation for NWP: (i) filtering of the analysis to remove spurious inertia–gravity waves and (ii) methods to handle nonlinearities and non-Gaussian error statistics. The third part discusses the development of parallel algorithms for four-dimensional variational data assimilation (4D-VAR), in order to better exploit the parallel nature of the computers on which it is run and to maintain its status as an important and viable NWP data assimilation algorithm into the foreseeable future.

This chapter presents examples of variational data assimilation in coastal oceanography using the Regional Ocean Modeling System (ROMS). Realizing that satellite data is the only source of information in real time in most parts of the world ocean, the Ocean Modeling Group at Rutgers University has developed methodologies to exploit the information content in remotely sensed observations. This chapter evaluates the extent to which incremental, strong constraint, four-dimensional variational data assimilation (IS4DVAR) can improve prediction of mesoscale variability using ROMS. Examples of two applications of IS4DVAR in two very different dynamical regimes are presented: the East Australia Current (EAC) and the Middle Atlantic Bight (MAB). The two main sources of satellite information, namely sea surface temperature (SST) and sea surface height anomaly (SSHA), are found to be complementary, and therefore both need to be assimilated in order to approximate the three-dimensional structure of the ocean.

In this chapter, four-dimensional variational data assimilation (4D-VAR) is discussed in the context of numerical weather prediction (NWP). The analysis step in an NWP data assimilation cycle combines observations with a background forecast. Plausible models of error distributions involve transforms and statistics to describe the structure of errors at one time, plus a forecast model constraining the time evolution. They allow a Bayesian derivation of equations for the optimal analysis, by minimizing a 4D-Var penalty function using an adjoint model. Difficulties with the deterministic best fit of a nonlinear NWP model are discussed and a statistical approach to 4D-VAR based on the extended Kalman filter is presented. Advanced extensions to 4D-VAR can allow for nonlinearities and non-Gaussian distributions, arising from the physical limits to humidity, and from the possibility of erroneous observations. Ensembles provide useful information about likely background errors, which can be used in hybrid ensemble–variational data assimilation.

This chapter looks at the use of multigrid methods and local mesh refinement algorithms in the context of the variational data assimilation method. Firstly, the chapter looks back at basic properties of the traditional variational data assimilation method and considers on the role of the background error covariance matrix. The next section shows how multigrid algorithms can efficiently solve the resulting system. Then the chapter deals with local mesh refinements and the final part of the chapter gives some ideas on how to couple the two approaches in the view of local multigrid algorithms.

This chapter discusses the use of second-order methods for estimating error propagation in variational data assimilation. The basic variational approach to data assimilation exhibits the optimality system: it can be considered as a generalized model containing all the available information. To estimate the impact of errors due to the parameters of the model and/or to the observations, it is necessary to consider second-order properties. The variational approach can be used to estimate the propagation of uncertainties in the analysis. Two basic cases are considered. In the deterministic framework, the uncertainty is a virtual and deterministic perturbation on the model parameters, whose impact on some criterion is to be found. In the stochastic framework, the uncertainty is a random variable transported by the model as such. The output is a stochastic perturbation on the outputs of the analysis, for which it is necessary to determine its probabilistic characteristics.