Filtering Model Components
Let us imagine that the target variable Z can be considered as a linear combination of two random variables Y1 and Y2, called scale components, in addition to the mean:
where Y1 is centered (mean is zero), characterized by the variogram γ1 and Y2 by γ2. If the two variables are independent, it is easy to see that the variogram of the variable Z is given by:
Instead of estimating Z, we may be interested in estimating one of the two components, the estimation of the mean has been covered in the previous paragraph. We are going to describe the estimation of one scale component (say the first one):
Here again, we will have to distinguish whether the mean is a known quantity or not. If the mean is a known constant, then it is obvious to see that the unbiasedness of the estimator is fulfilled automatically without implying additional constraints on the kriging weights. If the mean is constant but unknown, the unbiasedness condition leads to the equation:
Note that the formalism can be extended to the scope of IRF-k (i.e. defining the set of monomials fl(x) which compose the drift) and impose that:
Nevertheless the rest of this paragraph will be developed in the intrinsic case of order 0 and we can establish the optimality condition:
This leads to the system:
The estimation of the second scale component Y2*, will be obtained by simply changing 
into 
in the right-hand side of the kriging system, keeping the left-hand side unchanged.
Similarly, rather than extracting a scale component, we can also be interested in filtering a scale component. Usually this happens when the available data measure the variable together with an acquisition noise. This noise is considered as independent from the variable and characterized by its own scale component, the nugget effect. The technique is applied to produce an estimate of the variable, filtering out the effect of this noise, hence the name. In Isatis instead of selecting one scale component to be estimated, the user has to filter out components.
Because of the linearity of the kriging system, we can easily check that:
This technique is obviously not limited to two components per variable, nor to one single variable. We can even perform components filtering using the cokriging technique.
