Derivative Estimation from Marginally Sampled Vector Point Functions

Abstract

Several aspects of the problem of estimating derivatives from an irregular, discrete sample of vector observations are considered. It is shown that one must properly account for transformations from one vector representation to another. if one is to preserve the original properties of a vector point function during such a transformation (e.g., from u and v wind components to speed and direction). A simple technique for calculating the linear kinematic properties of a vector point function (translation, cud, divergence, and deformation) is derived for any noncolinear triad of points. This technique is equivalent to a calculation done using line integrals, but is much more efficient. It is shown that estimating derivatives by mapping the vector components onto a grid and taking finite differences is not equivalent to estimating the derivatives and mapping those estimates onto a grid, whenever the original observations are taken on a discrete, irregular network. This problem is particularly important whenever the data network is sparse relative to the wavelength of the phenomena. It is shown that conventional mapping/differencing fail to use all the information in the data, as well. Some suggesstions for minimizing the errors in derivative estimation for general (nonlinear) vector point functions are discussed.