  A Bayesian approach is developed for the inference of an eddy-diffusivity
field from Lagrangian trajectory data. The motion of Lagrangian particles is
modelled by a stochastic differential equation associated with the
advection-diffusion equation. An inference scheme is constructed for the
unknown parameters that appear in this equation, namely the mean velocity,
velocity gradient, and diffusivity tensor. The scheme provides a posterior
probability distribution for these parameters, which is sampled using the
Metropolis-Hastings algorithm. The approach is applied first to a simple
periodic flow, for which the results are compared with the prediction from
homogenisation theory, and then to trajectories in a three-layer
quasigeostrophic double-gyre simulation. The statistics of the inferred
diffusivity tensor are examined for varying sampling interval and compared with
a standard diagnostic of ocean diffusivity. The Bayesian approach proves
capable of estimating spatially-variable anisotropic diffusivity fields from a
relatively modest amount of data while providing a measure of the uncertainty
of the estimates.
