Stochastic fluids with transport noise: Approximating diffusion from data using SVD and ensemble forecast back-propagation

We introduce and test methods for the calibration of the diffusion term in Stochastic Partial Differential Equations (SPDEs) describing fluids. We take two approaches, one uses ideas from the singular value decomposition and the Biot-Savart law. The other backpropagates through an ensemble forecast, with respect to diffusion parameters, to minimise a probabilistic ensemble forecasting metric. We describe the approaches in the specific context of solutions to SPDEs describing the evolution of fluid particles, sometimes called inviscid vortex methods. The methods are tested in an idealised setting in which the reference data is a known realisation of the parameterised SPDE, and also using a forecast verification metric known as the Continuous Rank Probability Score (CRPS).