Point Kriging

Suppose that instead of trying to estimate a block one simply wants to estimate the grade at one particular point. Then all the coefficients of the krige system become very simple and involve only some distance computation. The matrix remains the same as before, it depends only on the samples at locations ${/boma x_i}$.

Let us take an example:

Example 6.1: (David, 1977[5]) Consider a stratiform deposit with a spherical variogram where $C = 20$; $C_0 = 2$; $a = 200'$ and one wants to estimate the grade $z_0$ of point ${/boma x}_0$ (Figure 6.2) from points ${/boma x}_1, {/boma x}_2, {/boma x}_3, {/boma x}_4$, where the grades are $z_1, z_2, z_3, z_4$. How do we weight these known grades for the estimation?

Figure 6.2: Point Kriging: Sketch of Sample Locations.

The definition of the variogram can now be given by the formula

and

Because the regions of the known values $v_{i}$ and also the region $v_{o}$ of the interesting value are at a point support, we denote $C(v_{i},v_{j})$ $(i,j=0,1,/ldots,4)$ simply $s_{ij}$, such that we obtain the following kriging system:

We thus only need the values of the covariances $s_{ij}$. The diagonal elements of the matrix are all equal the variances, i.e.

Furthermore, from the sketch of the sampling plan, it can be seen that

From this the solution of the system of equations follows:

Remark 1:

Although samples ${/boma x}_2$ and ${/boma x}_3$ have altogether little influence on the overall estimation, their relative influence is not linear in relation to their distance from ${/boma x}_o$. Sample ${/boma x}_3$ is further away than ${/boma x}_2$ and carries more influence (8.9%) than sample ${/boma x}_2$. This is due to the fact that ${/boma x}_o$ is directly under the influence of ${/boma x}_3$, while ${/boma x}_2$ is in fact in the ``shadow'' of ${/boma x}_1$. This means that introducing the covariances in the computation of weights will not give undue weight to clusters of samples as a simple distance method would do.

Remark 2:

Suppose that we now want to estimate the point ${/boma y}_o$ from ${/boma x}_1, {/boma x}_2, {/boma x}_3$ and ${/boma x}_4$. The only new effort is to recompute the $/sigma_{oi}$, since the matrix ${/boma /Sigma}$ remains unchanged. This should be kept in mind when writing a program of kriging since it saves a lot of computing time.