Discussion

Remark 1:

The existence and uniqueness of the solution. The kriging system has a unique solution if and only if the matrix of covariances $[/bar{C}(v_i, v_j)]$ is strictly positive definite and, thus, necessarily has a strictly positive determinant. For this purpose, it is enough that the point covariance model $C({/boma h})$ used is positive definite and that no data support $v_i$ coincides completely with another one. In fact, $v_i /equiv v_{i'}$ entails that $/bar{C}(v_i,v_j)=/bar{C}(v_{i'},v_j), /forall j$, and the determinant $/vert/bar{C}(v_i,v_j)/vert$ is, thus, zero.

This condition for the existence and the uniqueness of the solution of the kriging system thus entails that the kriging variance is non-negative.

Remark 2:

Kriging, which is an unbiased estimator, is also an exact interpolator, i.e., if the support $V$ to be estimated coincides with any of the supports $v_i$ of the available data, then the kriging system provides:

(i)

an estimator $/hat{Z}$ identical to the known grade $Z_i$, of the support $v_i = V$;

(ii)

a zero kriging variance, $/sigma_K^2=0$.

Remark 3:

The expressions of the systems and the kriging variances using the notions $/bar{C}$ and $/bar{/gamma}$ are completely general,

(i)

whatever the supports $v_i, v_j$ of the data and the support $V$ to be estimated, some data supports may partially overlap, $v_i /cap v_j /neq /emptyset$, but for $i/neq j$ it is imperative that $v_i /not/equiv v_j$-some data supports may be included in the volume $V$ to be estimated, $v_i /subset V$;

(ii)

whatever the underlying structure characterized by the model $C({/boma h})$ or $/gamma({/boma h})$, the structure may be isotropic or anisotropic, nested or not.

In cartography, it is said that ``the kriged surface passes through the experimental points''. Not every estimation procedure has this property, especially procedures using least square polynomials.

Remark 4:

The kriging system and the kriging variance depend only on the structural model $C({/boma h})$ or $/gamma({/boma h})$ and on the relative geometries of the various supports $v_i, v_j, V$, but not on the particular values of the data $Z_i$. Consequently, once the data configuration is known and before drilling is undertaken, the kriging system can be solved and the corresponding minimum estimation variance can be forecast. Thus, the kriging variance can be used to balance the cost of a new drilling campaign with its forecast utility (new data would decrease the estimation variance, thus providing narrower confidence intervals).

Remark 5:

The kriging matrix $K$ depends only on the relative geometries of the data supports $v_i, v_j$ and not at all on the support $V$ of the domain to be estimated.

Thus, two identical data configurations would provide the same kriging matrix $K$ and it is then enough to take the inverse matrix $K^{-1}$ only once. The two solution column vectors ${/boma /lambda}$ and ${/boma /lambda'}$ are then obtained by taking the products of the one inverse matrix $K^{-1}$ and the respective second-member vectors:

This, of course, suggests the very systematic and regular proceeding in data collecting.

Remark 6:

The kriging system and the kriging variance take into account the four essential and intuitive facts, which condition every estimation. These are as follows.

(i)

The geometry of the domain $V$ to be estimated, expressed in the term $/bar{/gamma}(V,V)$ in the expression of the kriging variance $/sigma_K^2$.

(ii)

The distances between $V$ and the supports $v_i$ of the information, expressed by the terms $/bar{/gamma}(v_i, V)$ of the vector ${/boma c}$.

(iii)

The geometry of the data configuration as expressed by the terms $/bar{/gamma}(v_i, v_j)$ of the kriging matrix $K$. The accuracy of an estimation depends not only on the number of data but also on their configuration in relation with the main features of the regionalization as characterized by the structural function $/gamma({/boma h})$ in the various terms $/bar{/gamma}(v_i, v_j)$.

(iv)

The main structural features of the variability of the phenomenon under study as characterized by the semi-variogram model $/gamma({/boma h})$.

Consider the estimation in two dimensions of the panel $V$ by the symmetric configuration of the four data points $A, B, C, D$ as shown on Figure 6.1. The underlying mineralization shows a preferential direction $u$ of continuity which appears in the anisotropic semi-variogram $/gamma(h_{u},h_{v})$ as a slower variability in the direction $u$. The kriging system thus gives a greater weight to the data $B$ and $D$, although they are at the same distance from $V$ as $A$ and $C$.

Figure 6.1: Influence of the Variability Structure on Kriging.