A Matern based multivariate Gaussian random process for a consistent model of the horizontal wind components and related variables

TL;DR

This paper introduces a multivariate Gaussian random field model based on Matérn covariance functions that incorporates physical relationships in wind fields, enabling consistent and flexible modeling of related meteorological variables.

Contribution

It develops a novel bivariate Matérn covariance model for wind potentials and related variables, ensuring positive definiteness and physical consistency, with application to mesoscale weather forecast data.

Findings

The model accurately captures wind field correlations.

Parameter estimation shows negligible physical correlations between potentials.

The statistical estimator outperforms numerical methods in variance ratio estimation.

Abstract

The integration of physical relationships into stochastic models is of major interest e.g. in data assimilation. Here, a multivariate Gaussian random field formulation is introduced, which represents the differential relations of the two-dimensional wind field and related variables such as streamfunction, velocity potential, vorticity and divergence. The covariance model is based on a flexible bivariate Mat\'ern covariance function for streamfunction and velocity potential. It allows for different variances in the potentials, non-zero correlations between them, anisotropy and a flexible smoothness parameter. The joint covariance function of the related variables is derived analytically. Further, it is shown that a consistent model with non-zero correlations between the potentials and positive definite covariance function is possible. The statistical model is fitted to forecasts of the horizontal wind fields of a mesoscale numerical weather prediction system. Parameter uncertainty is assessed by a parametric bootstrap method. The estimates reveal only physically negligible correlations between the potentials. In contrast to the numerical estimator, the statistical estimator of the ratio between the variances of the rotational and divergent wind components is unbiased.