Simulation

Simulation differs from kriging or any interpolation algorithm, in two major aspects:

1. In most interpolation algorithms, including kriging, the goal is to provide a “best”, hence unique, local estimate of the variable or any of its trend components without specific regard to the resulting spatial statistics of the estimates taken together. In simulation, reproduction of global features (texture) and statistics (histogram, covariance) take precedence over local accuracy. Kriging provides a set of local representations, say \(z^{*}(\mathbf{u}),\mathbf{u}\in A\), where local accuracy prevails. Simulation provides alternative global representations, \(z^{(l)}(u),u\in A\), where reproduction of patterns of spatial continuity prevails.

2. Except if a Gaussian model for errors is assumed, kriging provides only an incomplete measure of local accuracy, and no appreciation of joint accuracy when several locations are considered together. Simulations are designed specifically to provide such measures of accuracy, both local and involving several locations. These measures are given by the differences between \(L\) alternative simulated vlaues at any location (local accuracy) or the \(L\) alternative simulated fields (global or joint accuracy).

Different simulation algorithms impart different global statistics and spatial features on each realization, For example, simulated categorical values can be made to honor specific geometrical patterns as in object-based simulation or the covariance of simulated continuous values can be made to honor a prior covariance model as for Gaussian-related simulations. A hybrid approach could be considered to generate numerical models that reflect widely different types of features. For example, one may start with an object-based process or categorical indicator simulation to generate the geometric architecture of the various lithofacies, following with a Gaussian algorithm to simulate the distribution of continuous petrophysical properties within each sperate lithofacies, then a simulated annealing process could be used to modify locally the petrophysical properties to match, say, well test data.