Fitting of Discrete Irregularly Spaced Data with Differentiable Functions: Application to Ray Tracing in the Ionosphere.

摘要

Frequently, data are collected at irregularly spaced grid points, and for a variety of reasons, there is usually a need for interpolation; for example, a two dimensional graphical representation in the form of maps showing contour lines of constant values (isobars in weather maps). In many cases, this is not a trivial task, especially when the grid points are irregular and large data gaps exist. This problem, as it exists for ionospheric data, was discussed by Rush et al. If the data can be fitted with a continuous function that has continuous partial derivatives, the computation of contour lines is relatively easy. The electron density in the ionosphere at a given time is a function of latitude, longitude, and height. Horizontal gradients of the electron density can cause significant deviation of the ray path distance and direction from that expected under the assumption of horizontal stratification of the ionosphere. To improve propagation predictions under such circumstances, ray tracing studies that require the knowledge of the density and its gradient for every point along each ray path are needed. Similar remarks apply to the vertical sounding of the ionosphere, where horizontal gradients cause deviation from vertical propagation and limit the accuracy of electron density profile calculations unless angle of arrival data are available and corresponding corrections can be obtained by means of ray tracing. This report introduces a process for fitting discrete data given, at irregular grid points with one function that is continuous and has finite continuous first- and higher-order derivatives everywhere.