Average Length of Objects Defined from Binary Realizations: Effects of Discretization and Covariance Parameters

Abstract

There is often a misinterpretation of the range of the covariance as the only control on the average size of the “objects” that would be generated with pixel-based stochastic simulation techniques. Up to the point that it is not rare to read that the range of the covariance should be chosen equal to the size of the objects to be generated. (“Objects”, in this context, is interpreted as clustering of values below a given threshold.)

There are classical results from stochastic geometry that indicate that the controlling parameters of the average size are, the slope of the covariance at the origin and the object proportion. However, direct application of these theoretical results for the prediction of the average sizes of the objects generated over discrete grids do not yield satisfactory results. For instance, the mean length of the objects in a binary realization generated with a covariance with nugget effect, or by the truncation of a multiGaussian realization with underlying covariance of non-zero slope at the origin, is theoretically zero; while we observe that their experimental mean length is certainly non-null. The explanation resides in the discrepancy at short scales between the theoretical covariance and the discrete covariance of a realization generated over a discrete grid. This effect is specially noticeable for binary random functions with theoretical covariances that have a steep slope close to the origin.

A warning is sent to increase awareness about the average size of the objects that are generated by binary random functions and the real meaning of the parameters used to define the covariance in determining such size.