Data assimilation aims to correct a forecast of a physical system, such as the atmosphere or ocean, using observations of that system, in order to provide a best estimate of the current system state. Since it is not possible to observe the whole state it is important to know how errors in different variables of the forecast are related to each other, so that all fields may be corrected consistently. In the first part of this talk we consider how we may impose constraints between different physical variables in data assimilation. We examine how we can use our knowledge of atmospheric physics to pose the assimilation problem in variables that are assumed to be uncorrelated. Using a shallow-water model we demonstrate that this is best achieved by using potential vorticity rather than vorticity to capture the balanced part of the flow. The second part of the talk will consider a further transformation of variables to represent spatial correlations. We show how the accuracy and efficiency with which we can solve the transformed assimilation problem (as measured by the condition number) is affected by the observation distribution and accuracy and by the assumed correlation lengthscales. Theoretical results will be illustrated using the Met Office variational data assimilation scheme.
