Block Kriging

Example 6.2: (Journel and Huijbregts, 1978[11]) Consider a two-dimensional regionalization characterized by a point random function $Z(/boma{x})=Z(u,v)$ which, as a first approximation, is supposed to be intrinsic with an isotropic stationary semi-variogram: $/gamma (/boma{h}) = /gamma (/vert /boma{h} /vert)$. For example, $Z(u,v)$ may be the vertical thickness of a sedimentary seam.

The mean value $Z_V$ is to be estimated over a square panel $V=/overline{ABCD}$ of size $/ell/times /ell$ from the non-symmetric configuration of the four data of support $v$ each, located at $S_1$, (central sample) and $S_3, O_4, O_5$ (peripheral samples) as shown in the following Figure 6.3.

Figure 6.3: Block Kriging: Example.

For reasons of symmetry and for an isotropic semi-variogram $/gamma(/vert{/boma h}/vert)$ the two data $O_{4}$ and $O_{5}$ receive the same weight, and they can thus be grouped together to form the set $S_{2}=/{O_{4} /cup O_{5}/;/}$ of support $2v$. On the other hand, the two data $S_{1}$ and $S_{3}$ must be treated separately. The linear estimator thus considered contains three weights, namely

with

The kriging system is written as

and

with, for the various mean values $/bar{/gamma}$,

mean value over the data support,

(the dimensions of the support $v$ are neglectable with respect to the length $/ell$),

where $H(L,/ell)$ denotes an auxiliary function, which gives the mean of the semi-variogram $/gamma(/boma{h})$ if one end of the vector $/boma{h}$ is fixed at a corner of a rectangular of size $L /times /ell$ and the other end moves uniformly through the area of the rectangular. This function may be given analytically for simple variograms, values of more complicated ones may be read from graphical charts.

where $F(L,/ell)$ is a similar auxiliary function as $H$; it represents the mean of $/gamma(/boma{h})$ if both ends of $/boma{h}$ move uniformly through the area of the rectangular of size $L /times /ell$.

For further proceeding a linear model with nugget effect is considered:

The nugget effect $C_{o}(0)-C_{o}(r)$ is characterized on the regularization over the data support $v$ by a nugget constant $C_{ov}=A/v$. The nugget constant on the regularization of support $V$ is neglectable with respect to that of support $v$:

For the simple, linear standard model

the auxiliary functions $H$ and $F$ can be calculated analytically. We have

Our special cases are

and

The values of the mean variograms now are

where $/epsilon$ denotes the ``size'' of $v$; this term is neglectably small.

Three cases are considered:

(i)

the pure nugget effect with $C_{ov}=1,/; /omega /ell = 0/ / .$

(ii)

the partial nugget effect with $C_{ov}=.5, /; /omega /ell = .959/ / .$

(iii)

the absence of nugget effect, with $C_{ov}=0, /; /omega /ell = 1.918/ / .$

The previous values of the parameters $C_{ov}$, and $/omega/ell$ have been chosen so as to ensure, in three cases, a constant dispersion variance of data of support $v$ in the panel $V$:

For these special cases we perform the following computations.

(i)

Pure Nugget Effect

The Krige system follows:

After elimination of $C_{o}(0)$ we are left with

which gives

The Krige variance is

We can see that in this case of pure nugget effect, the weights $/lambda_{i}$ are proportional to the size of the support. The location is not important.

(ii)

Partial Nugget Effect

The Krige system follows as

We eliminate the constant $C_{o}(0)$ and the system solved gives

It is remarked that $S_{1}$ obtains the largest weight and that $/lambda_{2}$ is greater than $2 /times /lambda_{3}$. The Krige variance is $/sigma_K^2= .29.$

(iii)

No Nugget Effect

Elimination of $C_{o}(0))$ gives a Krige system

	1.918 $/lambda_{2}$ +	1.918 $/lambda_{3}$	+	$/mu$	=	.7338
1.918 $/lambda_{1}$	+	1.918 $/lambda_{2}$ +	2.712 $/lambda_{3}$	+	$/mu$	=	2.001
1.918 $/lambda_{1}$	+	2.712 $/lambda_{2}$ +			$/mu$	=	2.001
$/lambda_{1}$	+	$/lambda_{2}$ +	$/lambda_{3}$			=	1

with the solution

We can see that the central sample even gets a higher weight.

In the following summarizing table we also see results from other estimation methods. One is called POLY (polygon of influence) which uses positive weights only for samples which are in the region to be estimated. Therefore $/lambda_{1}= 1$ and $/lambda_{2} = /lambda_{3}= 0$. The methods of inverse distance and inverse-squared distance methods (ID and ID2) give the same weights to the three peripheral data: $/lambda_{3}=/lambda_{2}/2$. The central value $S_{1}$ obtains a greater weight by ID2 $(/lambda_{1}=.727)$ than by ID $(/lambda_{1}= .484)$. The mean distance from $S_{1}$ to the panel has been taken as a quarter of the diagonal, i.e., $/ell /sqrt{2}/4$.

In the stationary case, these standard methods ensure the unbiasedness of the estimation, as they satisfy the unique non-bias condition $/sum /lambda_i = 1$, but they do not, by themselves, provide the estimation variances $/sigma^2_E$. For this purpose, it is necessary to characterize in some form the spatial variability of the phenomenon under study. The geostatistical method is to use the structural function $/gamma(/boma{h})$ with which the estimation variance of any unbiased linear estimator can be calculated.

Nugget-
effect	Kriging	POLY	ID	ID2
	$/lambda_{1}=.25$	$/lambda_{1}= 1$	.484	.727
Pure	$/lambda_{2}=.50$	$/lambda_{2}= 0$	.344	.182
	$/lambda_{3}= .25$	$/lambda_{3}= 0$	.172	.091
	$/sigma^2_K= .25$	$/sigma_E^2= 1$	.323	.553
	$/lambda_{1}=.465$
Partial	$/lambda_{2}= .397$	as above	as above	as above
	$/lambda_{3}= .138$
	$/sigma_K^2= .29$	$/sigma^2_E= .734$	.296	.339
	$/lambda_{1}= .626$
Absent	$/lambda_{2}= .290$	as above	as above	as above
	$/lambda_{3}= .085$
	$/sigma_{K}^{2}= .224$	$/sigma^2_E= .468$	.27	.225