On the aliasing problem in the spherical harmonic analysis

To apply the least squares method for the interpolation of harmonic functions is a common practice in Geodesy. Since the method of least squares can be applied only to overdetermined problem, the interpolation problem which is always under-determined, is often reduced to an overdetermined form by truncating a series of spherical harmonics. When the data points are the knots of a regular grid it is easy to see that the estimated harmonic coefficients converge to the correct theoretical values, but when the observation density is not constant a significant bias is introduced. The result is obtained by assuming that the number of observations tends to infinity with points sampled from a given distribution. Under the same conditions it is shown that quadrature and “collocation-like” formulas displays a statistically consistent behaviour.