The influence of diffusion and associated errors on the adjoint data assimilation technique

We investigate the influence of diffusion, and errors associated with its representation, on the adjoint data assimilation technique. In order to determine how diffusion influences the retrieval of the initial state given a set of observations at later times, we perform a linear analysis of the one-dimensional diffusion equation and show that the retrieved initial state will be amplified (smoothed) if the diffusion in the prediction model is larger (smaller) than that present within the observations. This amplification (smoothing) not only increases dramatically as the length scale of the feature under consideration decreases, but also plays a rôle in suggesting an appropriate time period within which to assimilate observed data. These results are verified numerically for the simple case of a rising thermal in a neutrally-stratified environment using simulated pseudo-observations from a dry, three-dimensional Boussinesq model and its adjoint.