ABSTRACT
This paper deals with the problem of smoothing noisy data. It is well known that a functional containing a term of smoothness of the smoothing function plus a term of fidelity of the smoothing function to the noisy data is to be minimized. This minimization is made for a given smoothing parameter which is a compromise between smoothness and fidelity. The optimum smoothing parameter can be obtained by minimizing the well-known Wahba’s generalized cross-validation function G C V (A). The minimization of the GCV function is generally done without knowing the variance of the noise. But, in some cases, it is reported (e.g. Wahba (1990), Utreras (1993)) that the minimization of the GCV function yielded an unsatisfactory smoothing parameter: 0 or oo. For these pathological cases, according to Wahba (1990), if even the order of magnitude of the error variance were known, it would be possible to find the good optimum parameter. In this paper, it is shown how to obtain a good estimate of the error variance from the gross variogram of the noisy data. In fact, the nugget effect of the variogram (the value at the origin, i.e. at 0+ ), is a good estimate of the error variance. With this estimate of the error variance, the optimum smoothing parameter is obtained by minimization of an unbiased estimate of the true mean square error averaged over the data.
