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direction produces the same result. The autocorrelation function and the autocor-
relation are the same because
ʓ 0 ¼
1.
6.3.1 KTB Geophysical Data Example
The following geophysical example of spatial covariance modeling of velocity and
lithology logs provides another example of application of the model of Fig. 6.25 .
An autocovariance function consisting of two superimposed negative exponentials
with different scaling constants originally was obtained by Goff and Holliger
(1999) for binary lithology values derived from velocity and lithology logs for
the main borehole of the German Continental deep Drilling Program (KTB).
In Fig. 6.19 , a 1 ¼
0.1892 for larger scale variability and a 2 ¼
2 was assumed for
the nugget effect. In Goff and Holliger's Fig. 7, a 1 ¼
0.001 for the “large scale” and
a 2 ¼
0.019 for the “small scale” model. The dimensionless ratio a 2 / a 1 for KTB
binary lithology is 19 and somewhat greater than the ratio of 11 in Fig. 6.19 .
Lithology in the main KTB borehole was determined at points that are 1 m apart
over a length of about 7 km. In general, significant pre-processing is required for the
analysis of long series of this type. Goff and Hollinger ( 1999 ) commenced this
process by plotting raw compressional velocity ( V p ) averaged within more or less
homogeneous lithological sections against depth. A deterministic component
derived from this plot was extracted for the purpose of detrending followed by
conversion of the lithology log into a binary residual V p profile for which the spatial
covariance in (km/s) 2 was estimated. The two rock types retained in the binary plot
are mainly metabasite ( V p ¼
0.2 km/s).
The von K ´ rm ´ n autocovariance model has been used extensively by geophys-
icists to characterize crustal heterogeneity properties not only for velocity log
properties (e.g., Wu and Aki 1985 ; Wu et al. 1994 ; Goff and Hollinger 1999 ,
2003 ) but also for geological maps of crustal exposures (e.g., Goff et al. 1994 ;
Goff and Levander 1996 ), seafloor morphology (Goff and Jordan 1988 ), and in field
simulations (Goff and Jennings 1999 ). This model was proposed by von K´rm´n
( 1948 ) (but independently by Mat´rn, in 1948; cf . Box 6.6 ) and can be written as:
+0.2 km/s) and mainly gneiss ( V p ¼
ð ʽ K ʽ ah
ah
ðÞ
ˁ
ðÞ ¼
h
2 ʽ 1
ʓðÞ
where
is the Hurst number ( cf ., Mandelbrot 1983; Chemingui 2001 ; Klimeˇ
2002 ), and K ʽ
ʽ
. Fitting of the
two-parameter von K´rm´n model to an estimated covariance function can be
performed using the inversion methodology of Goff and Jordan ( 1988 ). If
is the modified Bessel
function of order
ʽ
ʽ ¼
0.5,
the preceding equation reduces to
ah ). Goff and Hollinger's ( 1999 )
best von K´rm´n model fit for the KTB binary residual V p profile has
ˁ
( h )
¼
exp(
ʽ ¼
0.21 and
a
0.00072. However, a better fit for the autocovariance of this series was obtained
by these authors using the nested semi-exponential design model with c 0 ¼
¼
0,
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