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 the present time, particularly on the synoptic and global scales. However, mesoscale and convective scale phenomena are considerably more chaotic and intermittent and it is clear that true 4-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 minimization of a 4-dimensional cost function with the forecast model as a weak constraint. This is a much more difficult problem than the widely discussed 4DVAR algorithm where the model is a strong constraint. Bennett and collaborators have considered a method of solution to the weak constraint problem, based on representer theory. However, their method is not suitable for the numerical weather prediction problem, because it does not cycle in time. In this paper, the representer method is modified to permit cycling in time, in a manner which is entirely internally consistent. The method was applied to a simple 1-dimensional constituent transport problem where the signal was sampled (perfectly and imperfectly) with various sparse observation network configurations. The cycling representer algorithm discussed here successfully extracted the signal from the noisy, sparse observations