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c 1 ¼
0.019. The Hurst numbers for both
negative exponentials are equal to 0.5, more than twice the Hurst number of best
fit using the von K ´ rm ´ n equation.
Because the series considered in the preceding paragraphs is binary, it is possible
to interpret the scaling constants a i ( i
0.684, c 2 ¼
0.316, a 1 ¼
0.001 and a 2 ¼
1, 2) along the lines presented at the
beginning of this section. Suppose the two binary states along the borehole
are written as +1 and
¼
1. If the mean can be set equal to zero, the autocorrelation
ˁ
( h ) is equal to the sum of the probability that number of state changes over the
interval h is even minus the sum of the probability that it is odd. If P k represents
the Poisson-type probability that there exist k state changes over h :
1
k
ðÞ
ʻ
h
e 2 ʻh
e ʻh
ˁ
ðÞ ¼
h
ð
P 2 k
P 2 k 1
Þ ˁ
ðÞ ¼
h
P k ¼
;
k
!
k ¼0
where
is number of state changes per unit of distance. A similar result is obtained
when the mean is not equal to zero.
For the Goff-Hollinger KTB example, the fact that there are two separate negative
exponentials illustrates that, over short distances, there are rapid lithology changes or
a “nugget effect” for i
ʻ
¼
2, but changes at larger scale are controlled by the other
negative exponential ( i
1) function. Thus alternation between mostly metabasite
and mostly felsic gneisses in KTB is subject to two separate random processes. The
alternation either has high or low frequency with probabilities controlled by the c i
( i
¼
1, 2) coefficients. This type of modeling only applies to the binary residual V p
profile for KTB. For example, Marsan and Bean ( 1999 , 2003 ) have demonstrated that
the KTB sonic log can be modeled using a multifractal approach. Also, Goff and
Hollinger ( 2003 ) have developed a generic model for the so-called 1/ f nature of
seismic velocity fluctuations. In that paper, these authors modeled the autocovariance
function of KTB depth-detrended sonic log through the superposition of four von
K´rm´n autocovariances using negative exponentials with Hurst numbers
¼
ʽ ¼
0.5 for
large, medium, and intermediate scales but
ʽ ¼
0.99 for the small scale.
References
Agterberg FP (1965) The technique of serial correlation applied to continuous series of element
concentration values in homogeneous rocks. J Geol 73:142-162
Agterberg FP (1966) Trend surfaces with autocorrelated residuals. In: Proceedings of the sympo-
sium on computers and operations research in the mineral industries. Pennsylvania State
University, State College, Pennsylvania, pp 764-785
Agterberg FP (1967) Mathematical models in ore evaluation. J Can Oper Res Soc 5:144-158
Agterberg FP (1968) Application of trend analysis in the evaluation of the Whalesback Mine,
Newfoundland. Can Inst Min Metall 9:77-88
Agterberg FP (1974) Geomathematics. Elsevier, Amsterdam
Agterberg FP (1994) Fractals, multifractals, and change of support. In: Dimitrakopoulos R
(ed) Geostatistics for the next century. Kluwer, Dordrecht
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