Geology Reference
In-Depth Information
PEP can be related to the cross-correlation coeffi-
cient R, a more traditional goodness of fit measure, by
PEP
PEP
¼
1
ð
energy in the residuals
=
trace energy
Þ
,
where the energy of a trace is the sum of the squares of
the time segment and the residuals are the difference
between a seismic trace and its matched synthetic.
R
2
. Thus, a cross-correlation of 0.7 effectively
means that just less than 50% of the energy is being
matched. Goodness of fit alone is not a definitive
measure of accuracy. The cross-correlation increases
with increasing wavelet length, but as the wavelet gets
longer there is more chance that noise is being
matched. A crude measure of accuracy is the normal-
ised mean square error (NMSE) which can be
approximately related to the phase error:
≈
Seismic segment
Wavelet length
1
bT
1
PEP
PEP
NMSE
¼
,
-0.1
and the approximate phase standard error in radians
will be
q
. Good well ties have PEP
NMSE
T=
0.5secs
>
0.7 and
NMSE
<
0.1.
L (secs)
0
4.4.1.2 Analysis of wavelet phase and timing
An appreciation of phase uncertainty can be derived
from an analysis of the phase variation with frequency
and the estimated standard errors (
Fig. 4.11
). The phase
and timing of the wavelet is calculated by making a
linear fit to the phase spectrum over the seismic band-
width between its standard errors. The intercept gives
the average phase and the slope of the line gives the
timing of the wavelet, where timing in ms
0.1
¼
slope
Figure 4.10
Seismic segment (
T
) and wavelet length (
L
).
a)
b)
High bandwidth data
Low bandwidth data
-40
Blue curve =
average phase
Estimated
standard errors
-20
-49º
-1.41ms
180
120
60
0
-60
-120
-180
180
120
60
0
-60
-120
-180
0
20
40
0
0.8
Amplitude
-40
-20
-40
17º
1.81ms
0
0
0
25
50
75
25
50
75
-20
20
40
-11º
2.67ms
Frequency
Frequency
0
20
40
1
0
Amplitude
0
0.8
Amplitude
45
Figure 4.11
Evaluating wavelet phase ambiguity through line fitting between standard errors; (a) high-bandwidth data with accurate
estimation of phase, (b) low-bandwidth data with appreciable phase ambiguity (modified after White and Simm,
2003
).
