Two-dimensional Stochastic Processes in Astronomy

In various astronomical applications, the impracticability of describing the phenomena under study with a relative simple model leads to the failure of any data fitting adjustment. Our inability to relate the observable quantities to the unknown physical parameters means that the residuals appear larger than expected and that they are strongly correlated. A powerful tool to attack this problem is the method of least-square collocation, borrowed from geodesy, combined with a technique for the estimation of the covariance function of the residuals. In this paper we describe the principles of this method, study its properties using simulations, and develop a partially new algorithm for the efficient estimation of the empirical covariance function. Several astronomical applications are presented and briefly discussed.