Geoscience Reference
In-Depth Information
Keywords Time series analysis Autocorrelation Power spectrum Spatial
statistics Semivariogram Nugget effect Geometric probability Extension
variance Pulacayo zinc values Whalesback copper deposit KTB copper KTB
geophysics
6.1 Time Series Analysis
Time series analysis is a well-established topic of mathematical statistics. It is
closely related to Fourier analysis. A good review of the history of this topic can be
found in Bloomfield (
2000
). It is of interest to mathematical geoscientists for
analysis of geoscientific time series and space series. The latter arise when a rock
unit is sampled at regular intervals along a line. Obviously, there is then in a 3-D
situation. Geostatistics was originally developed by Matheron (
1962
) for 3-D
domains. It was adopted by mathematical statisticians under the name “spatial
Suppose that a given series {
x
k
} is part of an infinite series which, in turn, is a
realization of an ordered set of random variables {
X
k
},
k
.
The theoretical set can be seen as a population that is doubly infinite, because any
point out of an infinite number of points along the line can assume any one value in an
infinite set of possible values. Suppose the random variables {
X
k
} all have the same
probability distribution with constant mean and variance. However, they can be
from panels that are (
h
¼
1
,
,
1, 0, 1,
,
1
...
...
) 30 ft. apart have approximately the same mean and
variance but are autocorrelated with
r
(
h
)
¼
0.59. The serial correlation coefficient
would become larger for shorter distance and less when
h
is increased. A series is
“weakly” stationary if the autocorrelation function for the {
X
k
} (with mean
¼
μ
and
2
) is constant and does not depend on location along the series. A property
of stationary series is that
r
(
h
)goestozerowhen
h
approaches infinity. Serial
correlation
r
(
h
) can be negative even if a series is stationary. Many series are not
stationary in that the mean of the {
X
k
} changes systematically along the series.
Discussions of the concept of “stationarity” of time or spatial series can be found
in Bloomfield (
2000
, Sect. 9.3) or Cressie (
1991
). “Strong” stationarity involves
time- or space-independence of frequency distributions and implies “weak”
stationarity also simply referred to as “stationarity”. Cressie (
1991
, p. 40) defines
“intrinsic” stationarity through first differences between successive values with the
properties
E
(
X
k
+
h
˃
variance
2
ʳ
h
is the
semivariogram. This assumption is weaker than “weak” stationarity because it
allows for infinitely large variance and non-existence of the mean
X
k
)
¼
0 and
˃
(
X
k
+
h
X
k
)
¼
2·
ʳ
h
where
, a topic to be
discussed in more detail in Sect.
6.2.1
. A time series is called “ergodic” if the time
average of a quantity is equal to its ensemble average (Brillinger
1981
, Sect. 2.11)
where “ensemble” comprises all possible realizations of the series. Ergodicity
implies stationarity. In this chapter, stationarity and ergodicity will be assumed to
μ
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