Approximation of Harmonic Covariance Functions on the Sphere by non Harmonic Locally Supported Ones
Harmonic splines have been used in many branches of geodesy to interpolate and to predict data discretly given on the sphere. Spline technique permits the use of data of different kinds and with different noise-characteristics but has the drawback that one needs to solve a positive definite system as many equations as the number of data. The matrices of such systems are full matrices because the covariance functions even for very long distances are different from zero due to the harmonicity of the covariance function. However, there is an appropriate angle of separation beyond which the covariance function values are negligible small and thus permits the use of finite instead of full covariance function. The computational savings which might be gained if finite covariance functions could be used would be large. If we used a grid with N x N values, the full matrices would contain N 2 x (N 2—l)/2 elements. If finite covariance functions are used and if the observations are ordered in a reasonable manner, the matrices would contain non-zero elements of the order N 3. This fact made us propose three techniques to approximate harmonic covariance functions by finite supported positive definite functions. Because all covariance function related to the anomalous potential of the earth can be seen as the spherical convolution of a so-called original function with itself and because the convolution of a finite supported function with itself gives a finite positive definite function, we approximate the covariance function by the self spherical convolution of a finite approximation of the original function.
In this talk the theoretical background of our three methods is given and comparison between the so-called finite covariance function of Sansò and Schuh (1987) and our three techniques have been carried out for three types of covariance functions associated with the determination of the anomalous gravity potential from gravity anomalies. After that we compare the solution of the full linear system with the sparse one associated with our first method for the third covariance function.