Geology Reference
In-Depth Information
In the case of unequally spaced samples, the representation (2.134) of the time
sequence g j may be used if it is understood that the sample points t j are not at
uniform intervals. If the sample points are at equally spaced intervals, we shall see
that a least squares adjustment of this representation gives the DFT (2.148).
This suggests the following approach to the analysis of unequally spaced data. If
we have a record segment of length M , with 2 L +
1 samples, equally or unequally
spaced, we try to represent it in the frequency domain for N
L by 2 N
+
1 sinu-
soids, equally spaced in frequency, as
N
1
M
g j =
G k e i ( k / M ) t j
.
(2.178)
k =− N
The 2 N
1values, G k , of the DFT, once calculated, give a frequency domain
representation that is equally spaced in frequency with sample interval
+
1/ M .
The FFT algorithm can then be used to give an equivalent, equally spaced time
sequence.
To find the DFT sequence G k , we first construct the error sequence j as the
Δ
f
=
erence of the sequence g j and its representation g j expressed by the sum of
sinusoids (2.178),
di
ff
g j .
j
= g j
(2.179)
If each member of the sequence g j has an associated standard deviation σ j ,the G k
are chosen to minimise the objective function
j j
σ
L
I
=
j ,
(2.180)
2
j =− L
the superscript asterisk again indicating complex conjugation. Substitution for g j
from (2.178) in expression (2.180) for the objective function gives
g j g j
g j
M
L
N
N
1
σ
g j
M
G l e i 2π( l / M ) t j
G k e i 2π( k / M ) t j
I
=
2
j
j =− L
l
=−
N
k
=−
N
G k G l e i 2π(( k l )/ M ) t j .
N
N
1
M 2
+
(2.181)
k =− N
l =− N
 
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