  Insufficient reference data is ubiquitous in data-driven computational fluid
dynamics, as it is usually too expensive to compute or impossible to observe
over long enough times needed for data-driven methods to properly operate out
of the sampled range. Ultimately, the lack of data can significantly compromise
the fidelity of results computed with data-driven methods or make the
data-driven approach inapplicable. In order to challenge this problem, we
propose a probabilistic reconstruction method which enriches the
hyper-parameterisation approach with ideas underlying the
probabilistic-evolutionary approach. We offer to use Advection of the image
point (a hyper-parameterisation method) on data sampled from the joint
probability distribution of the reference dataset. The idea is to blend
together the simplicity of the hyper-parameterisation (HP) method with an extra
source of reference data provided through sampling from the joint probability
distribution.
  The HP method has been tested on the sea surface temperature and surface
relative vorticity computed with the global 1/4-degree and 1/12-degree
resolution NEMO model. Our results show that the HP solution (the solution
computed with the HP method) in the probabilistically-reconstructed and reduced
(in terms of dimensionality) phase space at 1/4-degree resolution is more
accurate compared to the 1/4-degree solution computed with NEMO and it is also
several orders of magnitude faster to compute than the 1/4-degree run. The
proposed method shows encouraging results for the NEMO model and the potential
for the use in other operational ocean and ocean-atmospheric models for both
deterministic and probabilistic predictions.
