Sampling of a continuous flow in the plane based on two-dimensional variogram models

Abstract

Theory of sampling from one-dimensional continuous material flows is well established in developments following the original path breaking work by P. Gy, based on an extensively employed tool named variogram and its modelling. These developments could be regarded as a continuous analogue of the discrete time series approach in one dimensional sampling due to W. G. Cochran, whose work was further unified with the variogram approach by G.H. Jowett. Extension of Cochran’s discrete time series approach to plane sampling, from equally spaced grid locations in two dimensions, was carried out, independently, by M. H. Quenouille and R. Das. Gy’s development is an off-shoot of a closely related geostatistical theory, pioneered by G. Matheron, for collecting and analysing data in three dimensional space. Geostatistical approach generally employed a variogram based on a Euclidean distance to provide a three dimensional modelling of spatial variation structures. Most of the successful variogram modelling approach is essentially based on geometrically isotropic one-dimensional variogram models, instead of variogram models based on generalised distance functions or distance functions in separate dimensions. Quenouille explicitly proposed a geometrically anisotropic two-dimensional autocorrelation function which could be readily reparameterised to obtain a two-dimensional elliptical variogram. Both plane sampling approach and geostatistical approach provide techniques of assessing the uncertainty of the sample average as an estimate of the overall population mean, by the respective techniques of two-dimensional discrete sampling and spatial variogram modelling. The present paper (i) outlines the connection between the precision formula based on plane sampling approach and that based on traditional geostatistical approach, via a unified framework in which the two methods can be compared on an equal footing, with or without equal spacing along the X-direction and/or the Y-direction (ii) proposes an elliptical empirical variogram model, as a two-dimensional exponential type variogram, obtained as a re-parameterised version of Quenouille’s elliptical auto-correlation function and (iii) provides a computationally robust algorithm for fitting a two-dimensional elliptical variogram model to an observed variogram in two dimensions by an approximate likelihood method, in addition to a demonstration of the developed methodology to a data set, available in the published literature.