Abstract

Geophysical or meteorological data collected over the surface of the earth via satellites or ground stations will invariably come from scattered sites. There are two extremes in the problems one faces in handling such data. The first is representing sparse data by fitting a surface to it. This arises in geodesy in conjunction with measurements of the gravitation field from satellites, or meteorological measurements – temperature, for example – made at ground stations. The second is analyzing dense data to extract features of interest. For example, one may wish to process satellite images for mapping purposes. Between these two extremes there are many other problems. We will review various aspects of fitting surfaces to scattered data, addressing problems involving interpolation and order of approximation, and quadratures. Analyzing data is a more recent problem that is currently being addressed via various spherical wavelet schemes, which we will review, along with multilevel schemes. We close by discussing quadrature methods, which arise in many of the wavelet schemes as well as some interpolation methods.

Introduction

Overview

In this survey, we discuss recent progress in the representation and analysis of scattered data on spheres. As is the case with ℝs, many practical problems have stimulated interest in this direction. More and more data is taken from satellites each year. This, in turn, requires for example, improved image processing techniques for fault detection and for generation of maps.