E[Z°(y) ~ Z(v)] = 0 * E[Z (vy) - Z(v) ]* (unbiasedness) is a minimum. (minimum MSE) (7) The actual equations to which Equation 7 leads depends on the assumptions that have to be made on Z. There are basically two situations, according to whether or not the mean of Z(x) can be considered constant. In the simplest case, the variable Z(x) fluctuates about a constant value m, and it is natural to let E[Z(x)] = m for all x. In the scope of second-order stationary random functions, one could furthermore assume that Z(x) has a stationary covariance E[Z(x) - m] [Z(x + h) - mJ] = CCh) depending on the vector h only. Then Equation 7 could be made explicit in terms of C(h), leading to Wiener-Hopf type equations. But in practice, this approach has two drawbacks. One is that m is unknown and has to be replaced by an estimate; the other, noted for example by Matém (1960, p- 51), is that when the data are available over a restricted region, the covariance is in fact defined up to a constant. For these reasons, it is preferable to consider increments Z(X + h) - Z(x) which filter out the unknown mean m, and work with the variogram yh) = 3 E[ZGx +h) ~ 200) )? The assumption that the increments are stationary to the second order-- called the "intrinsic hypothesis"--is less restrictive than the classical stationarity assumptions on the process itself. very effective. 2A; Vx, - x5) +y = ¥(x,; v) 2,71 And it has proven to be Equation 7 then leads to the linear system: ; i= 1,2,...,N SIMPLE KRIGING SYSTEM ; (8) Where u stands for a Lagrange parameter and - 1 ¥ Ox, 5 v) = z J v(x, - x)dx is the average value of the variogram, when x; is the origin and the end point x sweeps throughout v. A more complicated situation is when Z(x) shows some systematic behavior, as is the case with Pu or Am concentrations that tend to decline with increasing distance from ground zero. Then it is more sensible to model Z(x) as the sum Z(x) = m(x) + Y(x) of a smooth deterministic function 377