Geology Reference
In-Depth Information
2.1.4 White noise and Wold decomposition
Time sequences may have specific statistical properties. Consider a time sequence,
...,
n
−
1
,
n
0
↑
,
n
1
,...,
n
j
,...,
(2.28)
which is completely uncorrelated. Its autocorrelation at lag
k
, time index
l
, is then
E
n
l
n
∗
l
−
k
k
E
n
l
n
∗
l
,
0
φ
nn
(
k
,
l
)
=
=
δ
(2.29)
0
where δ
k
is the Kronecker delta, if it is completely uncorrelated. Thus, the autocor-
relation vanishes for all lags, except zero lag where it is the power at the particular
time index
l
. Such a time sequence is called a
white noise sequence
.
If it is stationary and of unit power, its autocorrelation is
0
φ
nn
(
k
)
=
δ
k
.
(2.30)
The sequence
...,0,0,1
↑
,0,0,...
(2.31)
0
is called the
unit impulse sequence
. Its general term,
f
j
, is equal to δ
j
. Hence, in
the lag domain, the autocorrelation of a white noise sequence of unit power is the
unit impulse sequence.
Wold (1938) proved a very important theorem called the
Wold decomposition
theorem
(see Box
et al.
(1994)). The theorem shows that a stationary stochastic
sequence may be decomposed into the convolution of a deterministic energy signal
with a stationary white noise sequence of unit power.
Suppose we now represent a deterministic energy signal by the stationary stocha-
stic process that results from its convolution with a white noise sequence of unit
power. If we convolved the deterministic energy signal
f
j
with the unit impulse
sequence, we would get
∞
0
j
−
k
h
j
=
f
k
δ
=
f
j
,
(2.32)
k
=−∞
just the deterministic energy signal itself.
If we convolve it with white noise of unit power, we get the stationary sequence
∞
h
j
=
f
k
n
j
−
k
.
(2.33)
k
=−∞
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