Kriging prediction from a circular grid: application to wafer diffusion

In the production process of silicon wafers, which are crystalline slices used as substrate of electronic micro‐circuits, the thickness of the SiO2 deposition on their top is a main characteristic to be controlled during the process. The experimental design that is commonly used to monitor the thickness to the target value consists of a regular array of points lying on concentric circles, the silicon wafer itself being a disk. To speed up the control process, the engineers aim to use just only a limited subset of such points. To reconstruct the values on untried locations of the silicon wafer, the Kriging interpolation has been proposed because of its recognized ability in providing fairly good predictions. In this paper, we consider two methodological issues related to universal Kriging models. First, we discuss the modeling of the covariance structure among the measured points; in fact, spatial data usually show a strong correlation when they come from spatially near observed points. Second, we put forward an algebraic method to assess the identifiability of trend models, based both on the full experimental design and on special fractions of it. Our findings are illustrated by a data set from an industrial application. Copyright © 2013 John Wiley & Sons, Ltd.