A Polygon-based Line Integral Method for Calculating Vorticity, Divergence, and Deformation from Non-uniform Observations

Traditional observational analysis of vorticity and divergence usually relies on interpolating observations and evaluating spatial derivatives on either a Cartesian or spherical grid. Great care must be taken in selecting the domain, to avoid excessive extrapolation, and the interpolation scheme, to avoid introducing artifacts into the data. A number of alternative methods of calculating vorticity and divergence have been proposed and are based on evaluating line integrals on triangular regions according to Green's Theorem. Since this method relies on only three observations to perform calculations, it is sensitive to observations dominated by local phenomena as well as instrument noise. A few studies have attempted to minimize the impact of non-representative or noisy observations by using polygons, but have, so far, been limited to fitting regular polygons to near-regularly gridded data.

The current study proposes 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 based off of a triangle tessellation, there is no requirement for the data to be uniformly spaced. Additionally, this method reduces the impact of noise associated with individual observations with only a minimal loss in the length of the resolved scale. These polygons can also be used to calculated fields derived from vorticity and divergence, such as vertical velocity, potential vorticity, and three-dimensional moisture transport.