The Whole Story on the Hole Effect

The spatial continuity of the variables we model in geostatistics is dependent on the modelling and reproduction of the variogram. The variogram defines the relationship between variability (or geologic distance) and the lag distance (or Euclidian distance). As the magnitude of the lag separation vector increases, we typically expect the variogram to also increase. This is generally observed. The majority of variogram model structures are monotonic increasing. Nevertheless, non-monotonic structures may have a physical interpretation, provide valuable information and may be modeled with positive definitive models for more accurate geologic models. Non-monotonic variograms that show cyclic patterns are called ”hole effect” variograms. The theory and application of hole effect structures is explored in this report. Efforts are made to illustrate and provide examples of the hole effect structure. The generation of simulated realizations in presence of hole effect structures is investigated, with special attention paid to the reproduction of the histogram and variogram and their ergodic fluctuations. Hole effect variograms are commonly encountered and may be used in geostatistical simulation. The variogram structures are reproduced together with the histogram and the local conditioning data.