2-D and 3-D Trend Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 263

Fig. 7.24 Top diagram :
Artificial example of
sulphur variations along a
line to illustrate basic
concepts underlying Models
1 and 2. Bottom diagram :
Comparison of results in
westernmost parts of
Figs. 7.20 and 7.21 ;
triangles (for Model 1A)
represent predictions by
Model 1 applied to known
data south of 14,000 N only
(Source: Agterberg and
Chung 1973 , Fig. 4)

MODEL 2

MEAN

MODEL 1

HYPOTHETICAL
EXAMPLE

CONTROL

KNOWN DATA PREDICTIONS

DISTANCE

4.0

MODEL 2

MODEL 1A

3.0

MODEL 1

2.0

1.0
5,000N 10,000N 15,000N 20,000N 25,000N

DISTANCE IN FEET

single seam, changes from colliery to colliery and within the same colliery. Model
2 first attempts to capture this gradual change by means of trend surface analysis.
The trend replaces the constant mean (Fig. 7.24 , top). Next kriging was applied to
the residuals. Again, uncertainty increases with distance into the unknown area but
the predicted values converge toward the trend instead of to a constant mean. To
some extent, predictions by the two models can be tested against reality. A simple
test is illustrated in Fig. 7.24 (bottom). The five westernmost values in the rows of
numbers for known sulphur content printed in Figs. 7.20 and 7.21 were averaged to
give mean values for rectangular (2,000 ft. 10,000 ft.) blocks of coal located

between the W2,000 ft. and W12,000 ft. grid-lines. They represent averages of
known data in the area south of the N18,000 ft. line and predictions to the north of
this line. Models 1 and 2 yield different results: the known data in this part of the
Harbour seam show a rather abrupt decrease in sulphur content near the northern
edge of the worked area (between N15,000 and N17,000 ft.). This relatively
low sulphur content area influences the results of Model 1 more strongly than
those of Model 2. As an experiment, all known data north of N14,000 ft. were
omitted and statistical analysis by Model 1 was applied to the reduced data set.
The new (Model 1A) results also are shown in Fig. 7.24 (bottom). The decrease in
sulphur content at the northern edge of the No. 12 Colliery could not be predicted
in advance.