Multifractals and Local Singularity Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 427

Fig. 11.13 Estimates of
semivariogram ʳ k .(a) Solid
line obtained by linear
regression after setting
˄ (2) ¼ 0.979. (b) Log-log
plot of A. (c) Logarithmic
approximation (Source:
Cheng and Agterberg 1996 ,
Fig. 5)

a

80

70

60

50

40

30

0 10 20 30 40 50 60 70

Lag

b

2

1.8

1.6

1.4

1.2

1

0 0.4 0.8

Log Lag

1.2 1.6 2

c

80

70

60

50

40

30
20
10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Log Lag

1) resulting from this equation are nearly the same as the
best-fitting semi-exponential (broken line in Fig. 10.12 ). It seems reasonable to use
the model for extrapolation toward the origin by replacing the second-order differ-
ence on the right side of this expression by the second derivative ( cf . Agterberg
2012a , Eq. 9):

The first 15 values ( k

h i

Þ ˄ ðÞþ 1 Þ ˄ ðÞþ 1 2 k ˄ ðÞþ 1 k ˄ ðÞ 1 ð k þ 1 þ ð k 1 ˄ f ðÞþ 2 1 g˄ ðÞ 2