Kriging with External Drift

We recall that when kriging the variable in the scope of the IRF-k, the expectation of Z(x) is expanded using a basis of polynomials: E[Z(x)] = a_lf^l(x) with unknown coefficients a_l

Here, the basic hypothesis is that the expectation of the variable can be written:

where S(x) is a known variable (background) and where a₀ and a₁ are unknown.

Once again, before applying the kriging conditions, we must make sure that the mean and the variance of the kriging error exist. We need this error to be a linear combination authorized for the drift to be filtered. This leads to the equations:

These existence equations ensure the unbiasedness of the system.

This optimality constraint leads to the traditional equations:

where K(h) is then a generalized covariance.

Introducing the Lagrange parameters μ₀ and μ₁, we must now minimize:

against the unknowns λ^ɑ, μ₀ and μ₁:

We finally obtain the kriging system with external drift:

In matrix notation:

and