  Ensuring vertical separation is a key means of maintaining safe separation
between aircraft in congested airspace. Aircraft trajectories are modelled in
the presence of significant epistemic uncertainty, leading to discrepancies
between observed trajectories and the predictions of deterministic models,
hampering the task of planning to ensure safe separation. In this paper a
probabilistic model is presented, for the purpose of emulating the trajectories
of aircraft in climb and bounding the uncertainty of the predicted trajectory.
A monotonic, functional representation exploits the spatio-temporal
correlations in the radar observations. Through the use of Gaussian Process
Emulators, features that parameterise the climb are mapped directly to
functional outputs, providing a fast approximation, while ensuring that the
resulting trajectory is monotonic. The model was applied as a probabilistic
digital twin for aircraft in climb and baselined against BADA, a deterministic
model widely used in industry. When applied to an unseen test dataset, the
probabilistic model was found to provide a mean prediction that was 21% more
accurate, with a 34% sharper forecast.
