Autocorrelation and Geostatistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 224

Box 6.8 (continued)
y k ¼ X ½ n 1
i

;

where I 2

n X ½ n 1
i

2 ˀ Ik

n

2 ˀ Ik

n

n S i exp

S i ¼

1

n y i exp

¼ 1 .The

¼ ½ ¼ ½

2 with 2 degrees
of freedom (see e.g. Jenkins 1961 ). The average of q consecutive periodogram
values provides an estimate of the power spectrum E ( P i ) that is distributed as

R i against i. P i is distributed as

periodogram is a plot of P i ¼ ˇ

2
with approximately 2 q degrees of freedom. This method to construct power
spectra commonly makes use of the Fast Fourier Transform (FFT) method
introduced by Cooley and Tukey ( 1965 ). The power spectrum of a continuous
series is: P f ¼

ˇ

s 2 ( x ) R 1 1

r h cos 2 ˀ fh dh where r h is the autocorrelation function.

cs 2 ðÞ=ˀ x f c

s 2 For example, if r h ¼ c exp( a j h j ), Pf ðÞ ¼ ð 1 c Þ ðÞþ x

2 .

1 þ ð f = f c Þ

Fig. 6.21 Periodogram of 118 zinc values with quadratic curve fitted by least squares (Logarithms
base 10). The flattening of the curve toward higher frequencies is believed to be due to the nugget
effect (Source: Agterberg 2012 , Fig. 25)

Figure 6.21 shows the power spectrum E ( P i ) estimated by averaging pairs of
consecutive values of the periodogram for the 118 zinc values together with a
quadratic curve fitted by least squares. A best-fitting straight line for the same
values results in

0.72, but by means of an F -test it can be shown that the
quadratic fit of Fig. 6.21 is significantly better than the linear fit (for level of
significance ʱ ¼0.01). The slope of the curve at the origin in Fig. 6.21 gives
ʲ ¼

ʲ ¼

1.18 with gradually decrease to 0.49 at maximum log wave number on the