A Polygon-Based Line-Integral Method for Calculating Vorticity, Divergence, and Deformation from Nonuniform Observations

Abstract

raditional observational analysis of derivative-based variables (e.g., vorticity) usually relies on interpolating observations and evaluating spatial derivatives either on a Cartesian grid or on a spherical grid. Great care must be taken in selecting the domain and the interpolation scheme to properly represent the features. There exist a number of alternative methods of calculating such variables by evaluating line integrals on triangular regions according to Green?s theorem. Since these methods rely on only three observations to perform calculations, they are overly sensitive to observations dominated by local phenomena as well as instrument noise. A few studies have attempted to minimize the impact of nonrepresentative or noisy observations by using higher-order polygons, but they have been limited to fitting regular polygons to near-regularly gridded data. The current study describes a new approach to calculating these fields by constructing higher-order polygons from a triangle tessellation and then applying Green?s theorem. Since the polygons are constructed using an automated triangle tessellation, the polygon construction process can proceed without the need for uniformly spaced data. The triangle tessellation employed here is unique for a given set of points, generating easily reproducible results. In addition, this method reduces the impact of noise associated with individual observations with only a minor loss in the length of the resolvable scale. An error analysis of the proposed method shows a large decrease in errors in comparison with purely triangle-based calculations. These improvements are present with a variety of data distributions (random and along research aircraft flight paths) and kinematic variables (vorticity, divergence, and deformation).