Physically Constrained Covariance Inflation from Location Uncertainty

Motivated by the concept of ``location uncertainty", initially introduced in citeMemin2013FluidFD, a scheme is sought to perturb the ``location" of a state variable at every forecast time step. Further considering Brenier's theorem citeBrenier1991, asserting that the difference of two positive density fields on the same domain can be represented by a transportation map, perturbations are demonstrated to consistently define a SPDE from the original PDE. It ensues that certain quantities, up to the user, are conserved at every time step. Remarkably, derivations following both the SALT citeHolm2015VariationalPF and LU citeMemin2013FluidFD, Resseguier2016GeophysicalFU settings, can be recovered from this perturbation scheme. Still, it opens broader applicability since it does not explicitly rely on Lagrangian mechanics or Newton's laws of force. For illustration, a stochastic version of the thermal shallow water equation is presented.