Geology Reference
In-Depth Information
X
=
realisations at time
t
0
t
0
Time
Figure 2.1 Successive realisations of a stochastic process. The expected value of
a random variable at a specific time
t
0
is found by averaging across an infinite
ensemble of such realisations.
Frequently, the stochastic process generating the time sequence will be
station-
ary
, or assumed to be approximately so. In this case, the statistical properties of the
process are independent of translations along the time axis. Then, the autocorrela-
tion is independent of the time index and we write it as
E
f
l
f
l
−
k
.
φ
ff
(
k
)
=
(2.18)
The autocorrelation of stationary sequences is a
Hermitian
function of lag, for
E
f
l
f
l
+
k
,
φ
ff
(
−
k
)
=
(2.19)
and, writing
m
for
l
+
k
,wehave
E
f
m
f
m
−
k
=
φ
∗
ff
(
k
).
φ
ff
(
−
k
)
=
(2.20)
For stationary sequences, averaging across an infinite ensemble of independent
realisations is equivalent to averaging along the time axis of a single record. This is
known as the
ergodic hypothesis
.For
stationary power signals
, the autocorrelation
can then be equivalently defined as
N
1
f
l
f
l
−
k
.
φ
ff
(
k
)
=
lim
N
→∞
(2.21)
2
N
+
1
l
=−
N
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