Integration of the variogram using spline functions for sampling error estimation

Abstract

The component of the sampling error caused by taking discrete samples from a continuous process is the integration error, IE. This error can be estimated using P.M. Gy's variographic technique. This method involves the integration of the variogram. The variogram can be calculated from a time series of discrete samples. If the variogram is simple, it can be modelled and integrated. This method has been generally used in, for example, geostatistics. Gy has pointed out that chemical processes often have variograms that are too complicated to be modelled with simple equations and has, therefore, proposed a numerical point-by-point integration method for the experimental variogram. Although this method is reliable, it underestimates the integration error for systematic sampling if a sampling interval close to that used in the variographic experiment is used. In this study, the integration of the variogram is carried out using a cubic smoothing spline function. The integration errors calculated with Gy's method and with the cubic smoothing spline function are compared with the best estimate of the integration error for the simulated data. The integration errors calculated from the bleaching process with the aforementioned methods are also compared. On average, the integration error calculated with this new method corresponds better with the best estimate of the integration error than that calculated by Gy's method for the few first multiples of the sampling interval used in the variographic experiment. The difference between the methods is not significant with longer sampling intervals, say, five times the interval of the variographic experiment or longer.