Simple kriging

When $\beta$ is known, simple kriging prediction is obtained. It only involves the prediction of $e(s_0)$:

$$\hat Z (s_0 ) = f(s_0)\beta + v_0 V^{-1} (z(s_0 ) - f(s_0)\beta)$$

having variance

$${\rm Var}(Z (s_0 ) - \hat Z (s_0 )) = \sigma^2_{Z (s_0)} - v_0 ' V^{-1} v_0$$