Modeling Tangential Vector Fields on a Sphere

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 start with constructing parametric models for curl-free and divergence-free vector fields that are tangential to the unit sphere through applying the surface gradient or the surface curl operator to a scalar random potential field on the unit sphere. Using the Helmholtz–Hodge decomposition, we then construct a class of simple but flexible parametric models for general tangential vector fields, which are represented as a sum of a curl-free and a divergence-free components. We propose a likelihood-based parameter estimation procedure, and show that fast computation is possible even for large datasets when the observations are on a regular latitude–longitude grid. Characteristics and practical utility of the proposed methodology are illustrated through extensive simulation studies and an application to a dataset of ocean surface wind velocities collected by satellite-based scatterometers. We also compare our model with a bivariate Matérn model and a non-stationary bivariate global model. Supplementary materials for this article are available online.

KEYWORDS:

• Curl-free
• Divergence-free
• Helmholtz–Hodge decomposition
• Matérn model
• Ocean surface wind

Supplementary Materials

The online supplementary materials contain additional technical details, numerical results, and data example results.

Additional information

Funding

Division of Atmospheric and Geospace Sciences [1025089]; Division of Mathematical Sciences [1407530,1512945,1513484]; Division of Polar Programs [1443703].