A simple sufficient condition for a variogram model to yield positive variances under restrictions

Estimation of linear combinations is accomplished by using the observed (available) data. Accordingly, to require the negative of a modeled variogram function to be positive definite for all possible data combinations is unnecessary when only the observed data are used in estimation. The requirement that the negative of a variogram model be conditionally positive semidefinite is then relaxed to apply at the observed spatial locations only. In this setting a simple, yet crude, sufficient condition is developed to ensure that a variogram model will yield nonnegative variances for the available data. It is seen that the condition is independent of the dimensionality of the data and applies to both isotropic and anisotropic models. An example of the application of the condition is also presented. The condition is harder to satisfy as the amount of data increases and must be adjusted as the variogram changes to accommodate new data.