  Calibrating for direction-dependent ionospheric distortions in visibility
data is one of the main technical challenges that must be overcome to advance
low-frequency radio astronomy. In this paper, we propose a novel probabilistic,
tomographic approach that utilises Gaussian processes to calibrate
direction-dependent ionospheric phase distortions in low-frequency
interferometric data. We suggest that the ionospheric free electron density can
be modelled to good approximation by a Gaussian process restricted to a thick
single layer, and show that under this assumption the differential total
electron content must also be a Gaussian process. We perform a comparison with
a number of other widely successful Gaussian processes on simulated
differential total electron contents over a wide range of experimental
conditions, and find that, in all experimental conditions, our model is better
able to represent observed data and generalise to unseen data. The mean
equivalent source shift imposed by our predictive errors are half as large as
the best competitor model's. We find that it is possible to partially constrain
the ionosphere's hyperparameters from sparse-and-noisy observed data. Our model
provides an alternative explanation for observed phase structure functions
deviating from Kolmogorov's 5/3 turbulence, turnover at high baselines, and
diffractive scale anisotropy. We show that our model implicitly cheaply
performs tomography of the free electron density. Moreover, we find that even a
fast, low-resolution approximation to our model yields better results than the
best alternative Gaussian process, implying that the geometric coupling between
directions and antennae is a powerful prior that should not be ignored.
