2-D and 3-D Trend Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 250

Fig. 7.11 Contours from exponential cubic hypersurface based on copper data from 20 explor-
atory holes drilled downward from the topographic surface. Outline of orebody is approximately
1 % copper contour based on later underground information (Source: Agterberg 1968 , Fig. 6)

at n

¼

5 points (P 1 -P 5 ). These five values are to be used to estimate values at any
other point in the neighborhood; e.g., at P 0 in Fig. 7.12 . Suppose, firstly, that the five
known values are corrected for a hypothetical mean value, which may be a constant
estimated as the average of all values in a larger neighborhood. Then application of
the general linear model provides estimates of the coefficients (usually called
“weights”) to be assigned to the n ( ¼ 5 in Fig. 7.12 ) known values. The weights
can written as a column vector Wwith solution W ¼ R 1 1 R 0 where R 1 is an ( n n )

matrix consisting of the autocorrelation coefficients for all possible pairs of the
n known points in the neighborhood, and R 0 is a column vector of the autocorre-
lation coefficients for distances between the locations of the n known values and the
point at which the kriging value is to be estimated. In a slightly different version of
this simple kriging technique, it is required that the sum of the weights to be
assigned to the known values is unity. This constraint can be incorporated by
using a Lagrangian multiplier (Sect. 7.2.3 ). A useful generalization of simple
kriging is to correct all values for a trend value instead of for a constant regional
mean. The final estimated value at a point P 0 with arbitrary coordinates then is the
sum of the trend value at P 0 and the positive or negative value resulting from the
simple kriging.