Ordinary Kriging (Intrinsic Case)

We designate by Z the random variable. We define the kriging estimate, denoted Z*, as a linear combination of the neighboring information Zɑ, introducing the corresponding weights λɑ.

For a better legibility, we will omit the summation symbol when possible using the Einstein notation. We consider the estimation error, i.e. the difference between the estimation and the true value Z* - Z0.

We impose the estimator at the target (denoted "0") to be:

  • unbiased:

    • (which assumes that the expectation of the linear combination exists).

  • minimum variance (optimal):

    • (which assumes that the variance of the linear combination exists).

We will develop the equations assuming that the random variable Z has a constant unknown mean value:

Then equation can be expanded:

This is usually called "the Universality Condition".

Introducing the equation is expanded using the covariance C

which should be minimum under the constraints given in the previous equation.

Introducing the Lagrange multiplier μ, we must then minimize the quantity:

against the unknown λɑ and μ.

We finally obtain the (Ordinary) kriging system:

Using matrix notation:

In the intrinsic case, we know that we can use the variogram γ instead of the covariance C and that:

We can then rewrite the kriging system:

In the intrinsic case, there are two ways of expressing kriging equations: either in covariance terms or in variogram terms. In view of the numerical solution of these equations, the formulation in covariance terms should be preferred because it endows the kriging matrix with the virtues of definite positiveness and involves an easier practical inversion.