Kriging with uncertainties

We now assume that the observations $Y_i$ are affected by measurement errors $\epsilon_i$, and that:

1. the errors $\epsilon_i$ have zero mean:
   $$E[\epsilon_i] = 0 \qquad\qquad i=1,\ldots,n$$

2. the errors are uncorrelated:
      cov$$[\epsilon_i,\epsilon_j]=0 \qquad\qquad i\ne j$$

3. the errors are not correlated with the RF:
      cov$$[\epsilon_i,Y_i]=0$$

4. the variance $\sigma^2_i$ of the errors is a known quantity and can vary from point to point.

The new coefficient matrix $C$ of the linear system (2.5) is changed by adding to the main diagonal the quantity $-\sigma^2)i$:

$$C + \left[ \begin{array}{ccc} \sigma^2_1 & \cdot & 0 \\ 0 & \cdot & 0 \\ 0 & \cdot & \sigma^2_n \\ \end{array}\right]$$

and everything proceeds as in the standard case.