Unique Neighborhood Case
We recall the principle of the kriging or cokriging, although only the kriging case will be addressed here for simplicity. We wish to estimate the variable Z at any target point (Z*) using the neighboring information Zɑ, as the linear combination:
where the kriging weights λɑ are the unknown.
The Kriging conditions of unbiasedness and optimality lead to the following linear Kriging System:
and the variance of the kriging estimation error is given by:
with the following notations:
| ɑ, β | Indices relative to data points belonging to the neighborhood of the target point |
| 0 | Index which refers to the target point |
| Cɑβ | The value of the covariance part of the structural model expressed for the distance between the data points |
| flɑ | The value of the drift function ranked "l" applied to the data point |
|
The value of the modified covariance part of the structural model expressed for the distance between the point and the target point. |
| fl0 | The value of the drift function ranked "l" applied to the target point |
|
The value of the modified covariance part of the structural model (iterated twice) expressed between the target point and itself. |
The terms 
and 
depend on the type of quantity to be estimated:
|
|
|
|---|---|---|
| punctual |
|
|
| drift |
|
|
| block average |
|
|
| first order partial derivative |
|
|
A second look at this kriging system allows us to write it as follows:
where:
| A | is the left-hand side kriging matrix |
| X | is the vector of kriging weights (including the possible Lagrange multipliers) |
| B | is the right-hand side kriging vector |
|
stands for the matrix product |
| * | will designate the scalar product |
It is essential to remark that, given the structural model:
- The left-hand side matrix depends on the mutual location of the data points present in the neighborhood of the target point.
- The right-hand side depends on the location of the data points of the neighborhood with regard to the location of the target point.
- The choice of the calculation option only influences the right-hand side and leaves the left-hand side matrix unchanged.
In the Moving Neighborhood case, the data points belonging to the neighborhood vary with the location of the target point. Then the left-hand matrix A, as well as the right-hand side vector B must be established each time and the vector of kriging weights X is obtained by solving the linear kriging system. The estimation is derived by calculating the product of the first part of the vector X (excluding the Lagrange multipliers) by the vector of the variable value measured at the neighboring data samples Z, that we can write in matrix notation as:
where 
is the vector of the variable value complemented by as many zero values as there are drift equations (and therefore Lagrange multipliers) and designates the scalar product.
Finally the variance of the estimation error is derived by calculating another scalar product:
In the Unique Neighborhood case, the neighboring data points remain the same whatever the target point. Therefore the right-hand side matrix is unchanged and it seems reasonable to invert it once for all A-1. For each target point, the right-hand side vector must be established, but this time the vector of kriging weights X is obtained by a simple scalar product:
Then, the rest of the procedure is similar to the Moving Neighborhood case:
If the variance of the estimation error is not required, the vector of kriging weights does not even have to be established. As a matter of fact, we can invert the following system:
The estimation is immediately obtained by calculating the scalar product (usually referred as the dual kriging system):
