Cokriging

This time, we consider two random variables Z1 and Z2 characterized by:

  • the simple covariance/variogram of Z1 denoted C11/ γ11
  • the simple covariance/variogram of Z2 denoted C22/ γ22
  • the symetrical cross-covariance of Z1 and Z2 denoted C12 ( where C12= C21)
  • the cross-variogram of Z1 and Z2 denoted γ12

Note: It is because the cross covariance is supposed to be symetrical, which is a particular case, that the cokriging system can be easily translated from covariance to variograms.

We assume that the variables have unknown and unrelated means:

Let us now estimate the first variable at a target point denoted "0", as a linear combination of the neighboring information concerning both variables and using respectively the weights λ1 and λ2:

The first variable is also called the main variable.We still apply the unbiasedness condition

which leads to:

Let us consider the optimality condition and minimize the variance of the estimation error:

under the unbiasedness conditions.

This leads to the cokriging system:

In matrix notations:

with the estimation variance:

In the intrinsic case with symetrical cross-covariances, the cokriging system may be written using variograms:

with the estimation variance:

Note: If instead of Z1*, we want to estimate Z2*, the matrix is unchanged and only the right-hand side is modified:

and the corresponding estimation variance:

Let us first remark that both variables Z1 and Z2 do not have to be systematically defined at all the data points. The only constraint is that when estimating Z1, the number of data where Z2 is defined is strictly positive.

This system can easily be generalized to more than two variables. The only constraint lies in the "multivariate structure" which ensures that the system is regular if it comes from a linear coregionalization model.