Geostatistical simulation when the number of experimental data is small: an alternative paradigm

The usual paradigm in the application of geostatistical simulation has been to use a variogram model and a simulation algorithm to generate multiple realizations (conditional or non-conditional) of the random function model. In general the variogram model is inferred from the experimental data and then it has an uncertainty which can be large if the number of experimental data is small. However, this variogram uncertainty has usually been ignored, with the consequence that the simulated fields could be reproducing a spatial variability that does not mimic the underlying variability. Certainly there is an amount of variability in the local variogram of each simulated realization because of the ergodic fluctuations of the simulation algorithm that has been used. But we show in this paper that when the number of experimental data is small (which is not unusual in some disciplines of geosciences, such as groundwater hydrology or petroleum engineering) the description of the true underlying variability of the spatial variable may not be covered by the ergodic fluctuations of the random field realizations. Thus a change of paradigm is required. In the new paradigm, the uncertainty of the variogram parameters is taken into account and propagated into the simulations in a statistical way in order to cover up the underlaying variability. This alternative paradigm does not require new simulation algorithms; rather, it calls for choosing more carefully the range of variogram models (in opposition to using only the estimated model) that will be injected into the simulation algorithm. An example illustrates how the application of this approach is straightforward and can be important in reliability studies where geostatistical simulation is used to model the spatial uncertainty of a regionalized variable between the experimental locations.