Block Kriging

The kriging principle can be used for the estimation of any linear combination of the data. In particular, instead of the estimation of Z at a target point, we might be interested in computing the average value of Z over a volume v, called block. The block kriging performs this calculation; it is obtained by modifying the right-hand side of the point kriging system (see the paragraph Kriging of One Variable in the IRF-k Case):

• K_ɑ0 by K_ɑv which corresponds to the integral of the covariance function between the data point and a point which describes the volume v :

The integral must be expanded over the number of dimensions of the space in which v is defined.

• f^l₀ by f^l_v which correspond to the mean values of the drift functions over the volume:

We obtain the following block kriging system:

The block kriging variance is given by

It requires the calculation of the term K_vv instead K₀₀ of the term

For each block v, the K_vv integral needs to be calculated once, whereas K_ɑv needs to be calculated as many times as there are points in the block neighborhood. Therefore these integral calculations have to be optimized.

Formal expressions of these integrals exist for a few basic structures. Unfortunately, this is not true for most of them, and moreover these formal expressions sometimes lead to time consuming calculations. Furthermore, the same type of numerical integration MUST be used for the K_vv and the K_ɑv terms, otherwise we may end up with negative variances.

Numerical integration methods relying on the discretization of the target block are therefore preferred in Isatis. Two types of discretization are combined:

• the regular discretization,
• the random discretization.

In the regular discretization case, the block is partitioned into equal cells and the target is replaced by the union of the cell centers c_i.This allows the calculation of the K_ɑv terms:

where N is the number of cells in the blocks.

The double integral of the K_vv calculation is replaced by a double summation:

Applying in this case only the regular discretization sometimes lead to over-estimating the nugget effect. A random discretization is therefore substituted, where the first point of the discretization describes the centers of the previous regular cells whereas the second point is randomly located within its cell. In this case, there is almost no chance that a point c_i coincides with the point c_j and the function K(h) is never called for a zero-distance. The nugget effect of the structure therefore vanishes as soon as the covariance is integrated. This effect is recommended as soon as the dimension of the block is much larger than the dimension of the sample, which is usually the case.

Note: The drawback of this method is linked to its random aspect. For each calculation of a K_vv term the set of points requires a set of random values to be drawn which will vary from one trial to another. This is why it is recommended that the user exercises this calculation to determine the optimum as a trade-off between accuracy and stability of the result on the one hand, and computation time on the other : this possibility is provided in the Neighborhood procedure.