I am confused about the presentation of using variogram in simple kriging in the book of Multivariate geostatistics by H. Wackernagel (1995).
I understand the process of derivation of simple kriging equations which follows from determining the minimum of the variance of the estimation error. One challenge with the simple kriging equations is that they require the computation of covariances between locations ($cov[Z(x_{\alpha}), Z(x_{\beta})]=C(x_{\alpha}-x_{\beta})$). This equivalence follows from the assumption that the covariance between "random vectors" (I used "" because there are in fact no random vectors) in locations depends only on the distance between these locations). However, the covariances can only be calculated between random vectors and in the locations we don't have random vectors but only single numbers (because only one measurement in a single location was conducted). This frustrating problem is referred sometimes to as the "dead end" problem (Why do you have to provide a variogram model when you are kriging?) because we have the equations but we cannot solve them.
To circumvent the "dead end" problem, the use of the variogram is proposed in the next chapters of the book. The variogram function $\gamma(h)$ is defined as a half of the expected value of squared difference between locations separated by distance $h$: $\gamma(h)=0.5 \mathbb{E}[(Z(x+h)-Z(x))^2]$. The following equation can be then used to estimate the covariances $C(h)=C(0)-\gamma(h)$, where $C(h)$ is the estimated covariance between locations separated by distance $h$, $C(0)=\gamma(\infty)$ is variance/variogram sill. So far so good.
The problem is that the author writes (after some calculations) that the use of variogram is only authorized if the $n+1$ kriging weights sum up to zero. But because in simple kriging no restriction is imposed on kriging weights, the author concludes that the use of variogram is not authorized for simple kriging ("In particular, the variogram cannot in general be used in a simple kriging", p. 40, "Comment 11.1 The variogram is authorized for ordinary kriging, but not for simple kriging, because the latter does not include a constraint on the weight", p. 75).
So my understanding is that the author doesn't give the chance to circumvent the "dead end" problem in simple kriging: just because we are deprived of tools that should help us to solve the simple kriging equations. I assume that the problem must be somehow solved but I cannot find an explanation in the book how.
