Abstract The problem of predicting the dispersion of neutral material in the atmosphere is considered when rather more information than mean wind speed, turbulence and source position is available. Wind records on the site of a release of material could be used to indicate the initial velocity or direction of the cloud. In some cases a monitoring system might pick up the cloud some distance downstream. It is important that this extra information be used in the best possible way and this paper explores in part at least how it may be done. It turns out that a very simple theory is reasonably adequate. If the monitoring system can pick up the cloud several kilometres (or more) downstream, then the new position should be used as a new virtual source to predict, in the simple conventional way, the cloud's future position. An outcome of the theory, not immediately related to the above problem, is a method of estimating the Lagrangian time-scale from diffusion experiments over short distances of travel. This could be a real advantage, since hitherto such estimates have had to be made from long-range diffusion experiments both expensive and often difficult to run and interpret.