Advection of the image point in probabilistically-reconstructed phase spaces

TL;DR

A probabilistic reconstruction method that enhances the hyper-parameterisation (HP) approach with ideas underlying the probabilistic-evolutionary approach that can be used as a fast reanalysis tool allowing the complex dynamics of a comprehensive ocean model to be approximated by the HP solution.

Abstract

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. The lack of data can significantly compromise the fidelity of results computed with data-driven methods or render them inapplicable. To challenge this problem, we propose a probabilistic reconstruction method that enhances the hyper-parameterisation (HP) approach with ideas underlying the probabilistic-evolutionary approach. We offer to use the HP method ``Advection of the image point'' on data sampled from the joint probability distribution of the reference dataset. The HP method has been tested regionally on the sea surface temperature and surface relative vorticity computed with the global 1/4-deg and 1/12-deg 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-deg resolution is more accurate than the 1/4-deg-solution computed with NEMO. Additionally, the HP solution is several orders of magnitude faster to compute than the 1/4-deg NEMO solution. 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. Furthermore, the method can be used as a fast reanalysis tool allowing the complex dynamics of a comprehensive ocean model to be approximated by the HP solution. It can also function as a dynamic interpolation method to fill gaps in observational data.

Figures (16)

• Figure 1: Schematic of the probabilistic reconstruction method.
• Figure 2: Shown is a schematic representation of the probabilistically-reconstructed phase space with the superimposed voids (grey regions). A new tendency (red vector) is computed as an average of reference tendencies (black vectors) in the neighborhood of sampled SST (red contour).
• Figure 3: Shown is a sketch of the probabilistically-reconstructed reference phase space (orange blob), vectors $\mathbf{F}(\mathbf{x}_i)$ (black arrows) pointing out from points $\mathbf{x}_i$. For the HP method all these points is a cloud of points which are not ordered in time. The HP solution $\mathbf{y}(t)$ is shown as a red trajectory. The filled red circle denotes the current state $\mathbf{y}(t)$ at time $t$, while the empty red circles represent other points along the HP trajectory. The neighborhood $\mathcal{U}(\mathbf{y}(t))$ is illustrated by the black contour surrounding the filled red circle. The red vectors (and also each point they are attached to) within the neighborhood are the $N$ reference fields $\mathbf{x}_i,\, i\in\mathcal{U}_I$ (and $M$ reference tendencies $\mathbf{F}(\mathbf{x}_i)$) nearest, in $l_2$-norm, to the HP solution $\mathbf{y}(t)$. We assume that $\mathcal{U}_I=\mathcal{U}_J$ and $M=N$, but the method is not bound to this particular choice.
• Figure 4: Shown is (a) the solution of Chua's system $\mathbf{x}(t)$ and (b) the corresponding vector field $\mathbf{F}(\mathbf{x}(t))$ for $t=[0,100]$.
• Figure 5: Shown is the marginal reference PDF (black) and reconstructed PDF (blue) for the solution of Chua's system (top row) and for the corresponding vector field (bottom row). Note that the reconstructed PDF is an accurate approximation to the reference PDF.