Kriging with Inequalities

Note: Isatis.neo task: Kriging with Use uncertain data/Interval data options.

The aim of this technique is to deal with a variable which is defined in some locations by a value (hard data) and in some other locations by an interval (soft data). One of the bounds of an interval can be undefined. These soft data correspond to the inequalities.

To solve this problem, Isatis.neo proposes to replace the soft data by a new set of hard data. The way to replace the intervals is to calculate the conditional expectation of the target variable at each soft data location.

To calculate the conditional expectation, Isatis.neo uses a Gibbs Sampler technique which simulates for each soft data a given number of realizations of our target variable according to its variogram model and conditioned by the intervals and the hard data.

Note: The Gibbs Sampler is an iterative algorithm which consist in starting with an authorized vector of gaussian values consistent with the inequality constraints. Each value is then modified in turn using a kriging procedure and adding a random value. In Isatis.neo the parameters attached to this algorithm are fixed; a unique neighborhood is compulsory for the simple kriging step.

The simulations can only be performed in the gaussian space. The user has previously to transform the hard data into a gaussian variable and keep the anamorphosis attached to this transformation in a Geostatistical set. The intervals represented by 2 variables have also to be transformed in the gaussian space by the same anamorphosis function (this step is done by the program).

After the simulation, the program has just to calculate the average value of the realizations (after back transformation in the raw space) at each soft data point. These average values are called the conditional expectation. This conditional expectation is in fact the most probable value of the variable at the soft data locations. The standard deviation of these realizations is also calculated and stored.

Then, the final step is to krige the target variable using both the hard data and the conditional expectation values.