Geoscience Reference
In-Depth Information
right. A log-log plot of the two-point moving average of the periodogram of
Fig.
6.21
produces a pattern that is close to Lovejoy and Schertzer's (
2007
,
Fig. 3b) spectrum for the de Wijs data. This topic will be discussed in more detail
later (Fig.
12.35
). A straight line fit to the first 20 points of this two-point moving
average gives
ʲ
¼
ʲ
¼
1.03, which is close to
1.18 at the origin of Fig.
6.21
and close
ʲ
to
1.12. A possible explanation is that spectral analysis confirms validity of the
noise that tends to flatten the spectrum at higher frequencies.
For comparison, the preceding method also was applied in Agterberg (
2012
)toa
sequence of 132 titanium concentration values from the Black Cargo titaniferous
magnetite deposit, Los Angeles County, California (Benson et al.
1962
). This
sequence, previously analyzed in Agterberg (
1965
), is a composite of four
sub-sequences obtained from four different boreholes. All samples were 5 ft. in
length except for three 10 ft. samples at the subsequence meeting points. Mean and
standard deviation of the 132 numbers are 2.73 and 1.65 % TiO
2
, respectively.
The resulting periodogram (Agterberg
2012
, Fig. 26) is similar to Fig.
6.21
in that
the best-fitting quadratic trend line has a slope that decreases toward higher
frequencies. At the origin (
x
¼
0) its value is
1.088 and at maximum frequency
(
x
0.6186. Other results for this example also were similar
to those obtained for the 118 Pulacayo zinc values.
The curves in Figs.
6.20
and
6.21
indicate (1) the log-log plots of the three power
spectra are not straight lines but curves with slopes that decrease toward higher
frequencies; and (2) at their maximum frequency or highest position number the
curves are probably not horizontal indicating that the nugget effect is not white
noise with Dirac delta autocorrelation function. The sampling intervals of two data
sets used for example in this section are too wide to allow a better description of the
effect of the nugget effect on the power spectra.
¼
1.8195) the slope is
6.2.8 KTB Copper Example
The second example of detection of nugget effect is for a long series consisting of
2,796 copper (XRF) concentration values for cutting samples taken at 2 m intervals
along the Main KTB borehole of the German Continental deep Drilling Program
(abbreviated to KTB). These data are in the public domain (citation: KTB, WG
Geochemistry). Depths of first and last cuttings used for this series are 8 and
5,596 m, respectively. Locally, in the database, results are reported for a 1-m
sampling interval; then, alternate copper values at the standard 2 m interval were
included in the series used for example. Most values are shown in Fig.
6.22
together
with a 101-point average representing consecutive 202-m long segments of drill-
core. The data set was divided into three series (1, 2 and 3) with 1,000, 1,000 and
796 values, respectively. Mean copper values for these three series are 37.8, 33.7
and 39.9 ppm Cu, and corresponding standard deviations are 20.3, 11.0 and
20.6 ppm Cu, respectively. Figure
6.23
shows correlograms of the three series.
Each series shows a nugget effect that, for series 2 and 3, is accompanied by a
Search WWH ::
Custom Search