Towards a true 4-dimensional data assimilation algorithm: application of a cycling representer algorithm to a simple transport problem

3-dimensional variational algorithms are widely used for atmospheric data assimilation at thepresent time, particularly on the synoptic and global scales. However, mesoscale and convectivescale phenomena are considerably more chaotic and intermittent and it is clear that true4-dimensional data assimilation algorithms will be required to properly analyze these phenomena.In its most general form, the data assimilation problem can be posed as the minimizationof a 4-dimensional cost function with the forecast model as a weak constraint. This is a muchmore difficult problem than the widely discussed 4DVAR algorithm where the model is a strongconstraint. Bennett and collaborators have considered a method of solution to the weak constraintproblem, based on representer theory. However, their method is not suitable for thenumerical weather prediction problem, because it does not cycle in time. In this paper, therepresenter method is modified to permit cycling in time, in a manner which is entirely internallyconsistent. The method was applied to a simple 1-dimensional constituent transport problemwhere the signal was sampled (perfectly and imperfectly) with various sparse observation networkconfigurations. The cycling representer algorithm discussed here successfully extractedthe signal from the noisy, sparse observations.