Physically constrained covariance inflation from location uncertainty

Abstract. Motivated by the concept of “location uncertainty”, initially introduced in Mémin (2014), a scheme is sought to perturb the “location” of a state variable at every forecast time step. Further considering Brenier's theorem (Brenier, 1991), asserting that the difference of two positive density fields on the same domain can be represented by a transportation map, we demonstrate that the perturbations consistently define a stochastic partial differential equation (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 (stochastic advection by Lie transport; Holm, 2015) and LU (location uncertainty; Mémin, 2014; Resseguier et al., 2017a) settings can be recovered from this perturbation scheme. Still, it offers 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.