**C. Granier**, and C. Reeves, 2004: Data assimilation and inverse methods. In

*Emissions of Atmospheric Trace Compounds*, C. Granier, P. Artaxo, and C. E. Reeves (Eds.), Kluwer Academic Publishers, 477-515.

**INTRODUCTION**

In the previous chapter we reviewed the principal methods of observations of atmospheric chemical constituents and showed how they can be used in relatively simple models, often 2-dimensional, to provide constraints on emission estimates. We will now discuss how these observations can be used in conjunction with more complex 3-dimensional chemical transport models to yield useful knowledge about surface emissions and the chemical state of the atmosphere by employing methods based on estimation theory, called inverse method and data assimilation. Although there is a rich history of application of this theory in other fields, such as meteorology, seismology, and remote sensing, it is only recently that data assimilation and inverse techniques have been developed to address atmospheric constituent problems.

The objective of chemical data assimilation is to estimate the chemical state of the atmosphere given prescribed sources, whereas the objective of inverse methods is to estimate the chemical sources or to improve emission inventories. These methods have remarkable similarities, for instance they share the same mathematical apparatus of estimation theory, but their application is quite different. Data assimilation uses a short-term chemical forecast, typically of 6 hours of less, and combines this information with observations to get the best estimate of the chemical state of the atmosphere in space and evolving in time. Data assimilation is particularly useful to optimally fill in the gaps in the observational network, to maximize the use of information content of satellite observations which often provide only column amounts, and to infer information about unobserved species important, for example fast reacting compounds. In contrast, inverse methods use much longer integration times, typically of a month, and were developed primarily to infer surface emissions of long-lived gas species using surface observations. Inverse modelling tools will have important implications to monitor the application of international treaties on greenhouse gas emissions. However, inverse methods have important limitations. Long time integrations of models representing chemical tracers distributions, which are used in order to avoid estimating the spatial structure of the initial condition, result in a loss of spatial information due to diffusion and mixing, so that only the large-scale, slowly time-varying sources can be estimated.

In this chapter we will continue the basic formulation of both data assimilation and inverse methods, and draw attention to their similarities and differences. We will then proceed in extending these methods to a simultaneous state-source estimation scheme. The following section gives an overview on chemical transport, with emphasis on the numerical properties that transport schemes should have for chemical modelling. In section 3, we discuss the issue of data assimilation and inverse modelling by drawing into the analogy that those two methods have. Section 4 provides examples of application of data assimilation, inverse modelling, and dual state-source estimation. We conclude with a few remarks on future challenges.