Some Researches On The Approximation On Sphere

Approximation on the sphere is becoming more and more important in recent times.The obvious applications involve approximation of data from environmental sources,such as meteorology, oceanography, and pollution. Applications include representingfunctions which model temperature, pressure, ozone,gravitational and magnetic forces,elastic deformation etc. at all points on the surface of the earth based on a discretesample of values taken at arbitrary points. But there are also applications in biomedicalengineering, for example the electroencephalographic description of scalp potentials andscalp current densities from discretely placed electrodes.A major field of interest is gravity field and geoid determination in physical geodesywhich enjoys a renewed popularity in many positioning, mapping, and exploration ap-plications with the advent of satellite based techniques (like the Global PositioningSystem(GPS), satellite altimetry etc.). In former geophysical prospecting, which wasdominated by seismic reflection surveying, gravity methods have mostly been used ascomplements when difficulties with seismic methods have arisen. Nowadays, however,satellite gravity methods have brought a new concept into prospecting. High precisionand resolution of the gravity field, obtained with space-borne satellite techniques, willchange the ordinary routine in future prospecting. From being a secondary prospectingtool, the gravity field or the geoid, computed from (scattered) terrestrial and satellite,will be used to locate prospective regions as well as individual prospects.If the data are localized, approximation problems on the sphere can be solvedthrough application of methods designed for two-dimensional Euclidean space. How-ever, problems like gravity determination involve essentially the entire surface of thesphere, or a sufficiently large part that modeling the data as arising in two-space isno longer appropriate. Since there is no differential mapping of the entire sphere to abounded planar region, there is a need to developed approximation methods over thesphere itself.Many different methods have been given by the ones interested in approximation on the sphere. We will introduce several briefly in section 2. These include methodsbased on spherical harmonics, tensor-product spaces on a rectangular map of the sphere,functions defined over spherical rectangular map of the sphere, functions defined overspherical triangulations, spherical splines, spherical radial basis functions, and someassociated multi-resolution methods. In addition, a new method which can adapt thegiven data is given in section 3. A better result will be achieved through changing theshape of the support according to the distribution of the data points in the local region.