Sequential Indicator Simulations

The Sequential Indicator Simulations procedure enables you to perform several simulations of a categorical variable in an output file (not necessarily organized as a regular grid) using the Sequential Indicator method. Simulations can be conditional, that is to say they are constrained to adhere to all the active information.

This method does not correspond to any model in particular. The author even claims that it is a model-free method. The way it has been implemented in Isatis.neo is to produce a simulation of a categorical variable (i.e. facies).

The variation range of the input data Z is divided into a set of intervals. Each interval corresponds to a lithotype. For each interval, the corresponding global proportion (or frequency) is given.

In practice, the simulation domain often has too many target points and the procedure cannot apply directly. Instead, the simulation of each target point requires the selection of the neighboring relevant information. A standard ellipsoidal moving neighborhood is established, centered on the target point. Within this neighborhood, the procedure searches for data from initial data or from the already simulated sites (data and simulated sites are mixed together before applying the neighborhood search criteria).

Note: Main advantages of this algorithm are:

  • it does not require the migration of each input data point to its closest grid node, unlike the sequential process, which may introduce a bias in data location when the data is latter used for the conditioning,
  • the standard neighborhood takes anisotropies into account, whether in the spatial structure of the variable itself or in the sampling pattern. As such, the same neighborhood as for kriging can be used for instance if you wish to compare your simulation results vs. the kriging of the attribute.

The next step consists of a kriging phase taking into account the previous information. For sake of simplicity, the model is built starting from generic model (usually based on the one of the representative lithotype), tuning its sill according to the proportion or each lithotype. In fact, as we will not consider the calculation of estimation variances, the tuning is not even necessary.

The estimation does not necessarily lie between 0 and 100%. They are simply truncated to positive values and normalized a posteriori. At the target site, the vector of these estimations (for all the intervals) is considered as cumulative probability function, conditionally to the neighboring information. Then we simply draw a random value uniformly between 0 and 100% and compare it to the cumulative probability function. The simulated value corresponds to the rank of the interval to which the random value belongs.

An additional specific feature has been added in Isatis.neo which takes the theoretical proportions into account. The neighborhood search around a target site may end up with no data (neither hard data nor already simulated node). The simulation then becomes non conditional. The simulated lithotype is then simply drawn from proportions. These proportions correspond to the theoretical ones (entered by the user) modified accordingly to the values of the grid nodes already simulated.

For more information, please refer to Alabert F.G., Stochastic Imaging of Spatial Distributions Using Hard and Soft Data (M.Sc. Dissertation, Dept of Applied Earth Sciences, Stanford University, 1987, 198p).

Post-processing results can be computed from the facies simulations. You can store at each grid node location:

  • The probability of each facies,
  • The most probable facies,
  • The least probable facies,
  • The facies proportion difference,
  • The entropy,
  • The corrected most probable facies (after Soares correction).