Modeling Tangential Vector Fields on a Sphere

TL;DR

This paper introduces a new class of parametric models for tangential vector fields on a sphere, capturing physical constraints like curl-free and divergence-free properties, with efficient estimation and superior performance in wind data analysis.

Contribution

The paper develops a novel surface-based covariance model for curl-free and divergence-free vector fields on a sphere, with a likelihood-based estimation method and demonstrated advantages over existing models.

Findings

Models effectively capture physical properties of wind fields.

Proposed method outperforms bivariate Matérn models in estimation and prediction.

Efficient computation for large datasets on regular grids.

Abstract

Physical processes that manifest as tangential vector fields on a sphere are common in geophysical and environmental sciences. These naturally occurring vector fields are often subject to physical constraints, such as being curl-free or divergence-free. We construct a new class of parametric models for cross-covariance functions of curl-free and divergence-free vector fields that are tangential to the unit sphere. These models are constructed by applying the surface gradient or the surface curl operator to scalar random potential fields defined on the unit sphere. We propose a likelihood-based estimation procedure for the model parameters and show that fast computation is possible even for large data sets when the observations are on a regular latitude-longitude grid. Characteristics and utility of the proposed methodology are illustrated through simulation studies and by applying it to an ocean surface wind velocity data set collected through satellite-based scatterometry remote sensing. We also compare the performance of the proposed model with a class of bivariate Mat\'ern models in terms of estimation and prediction, and demonstrate that the proposed model is superior in capturing certain physical characteristics of the wind fields.