Estimation of Anisotropic Viscosities for the Stochastic Primitive Equations
Igor Cialenco, Ruimeng Hu, Quyuan Lin
TL;DR
The paper tackles the problem of estimating anisotropic viscosities $ν_h$ and $ν_z$ in stochastic primitive equations using a finite set of Fourier modes from a single observed path on $[0,T]$. It develops several MLE-type estimators anchored in a barotropic/baroclinic decomposition, employing a novel hat-projection to make joint inference feasible for the near-highest-derivative parameters. Under suitable regularity ($\gamma>4$) and sampling power ($α>\gamma-2$), the authors prove weak consistency for the primary estimators and, further, joint asymptotic normality for $ν^N_{h1}$ and $\widehat{ν^N_{z1}}$ with rate $N^2$ and explicit covariance, enabling asymptotic confidence intervals. The methodological contribution includes a careful treatment of nonlinear terms, the use of Galerkin-type projections, and an explicit discussion of the optimal projection choice parameterized by a rational $q$, with implications for inference in other SPDEs having anisotropic parameters. The results provide a rigorous statistical foundation for anisotropic parameter estimation in high-dimensional stochastic geophysical models and suggest practical paths for CI construction in applied settings.
Abstract
The viscosity parameters play a fundamental role in applications involving stochastic primitive equations (SPE), such as accurate weather predictions, climate modeling, and ocean current simulations. In this paper, we develop several novel estimators for the anisotropic viscosities in the SPE, using a finite number of Fourier modes of a single sample path observed within a finite time interval. The focus is on analyzing the consistency and asymptotic normality of these estimators. We consider a torus domain and treat strong, pathwise solutions in the presence of additive white noise (in time). Notably, the analysis for estimating horizontal and vertical viscosities differs due to the unique structure of the SPE and the fact that both parameters of interest are adjacent to the highest-order derivative. To the best of our knowledge, this is the first work addressing the estimation of anisotropic viscosities, with the potential applicability of the developed methodology to other models.