Least squares collocation and regularization

Certain geodetic problems such as the downward continuation of gravity information from satellite or aerial altitudes to the surface of the earth or the inverse Stokes problem are improperly posed in the sense that the best approximate solution does not continuously depend on the given observations. In order to obtain a stable solution a technique of regularization is discussed which can be shown to be identical to the method of least squares collocation. The characteristic features of regularization are analysed transferring the problem through a singular value decomposition into the spectral domain which allows an easy interpretation as a special method of filtering. In practical applications the stability depends not only on the observational errors including the computer round-off but in the same way on the number of observations and their distribution. The regularized solution should achieve a proper trade-off between sufficient smoothness and highest possible resolution with a limit defined through the internal accuracy of the computer.