Geology Reference
In-Depth Information
extra stability in generating supergathers with attend-
ant benefits in the extraction of AVO attributes.
is
β
,
then the residuals calculated by subtracting
sin
2
β
from each starting amplitude can be subjected
to the same median fit calculation. This process could
be iterated further, but instead Walden suggests
making a maximum-likelihood fit to the residuals at
this stage. This differs from a standard least-squares
fit by giving less weight to points with high residuals,
and zero weight to points with residuals beyond
a threshold. An example is shown in
Fig. 6.16
,
where the robust fit has been effective in removing
the influence of the high amplitudes on the last few
traces.
θ
6.3.5 Gradient estimation and noise
reduction
Notionally, it is a simple matter to calculate a volume
of intercept R
0
and of gradient G traces by making a
least-squares fit of the amplitude R (for each TWT
sample of each angle gather) to the equation R
¼
R
0
+
G sin
2
is the angle of incidence. It is
important that the angle range used to calculate the
gradient has good signal-to-noise ratio and that the
linearity assumption is valid. This can be tested by
analysing the angle behaviour of amplitudes in the
gathers and comparing it to well synthetics. As has
been discussed in previous sections a key issue is the
role of noise in the variability of the gradient (e.g.
Hendrickson,
1999
; Cambois,
1998
) and various
methods have been proposed to improve the robust-
ness of the gradient estimate in the presence of noise.
θ
, where
θ
6.3.5.2 Whitcombe
'
s noise reduction
Another method for reducing noise in the gradient
has been proposed by Whitcombe et al. (
2004
). It
makes use of the fact that the noise is well imaged as
a coordinate rotation on the intercept vs gradient
crossplot. Filtering of the rotated data is carried out
by a process rooted in the hodogram concept (Keho
et al.,
2001
). First, a coordinate rotation is applied to
the data in the intercept
gradient domain to align the
axes parallel and perpendicular to the noise trend
(
Fig. 6.17a
). Rotation of the axes is achieved by:
R
0
0
¼
-
6.3.5.1 Walden
'
s robust fitting
Walden (
1991
) proposed a two-stage approach as
follows. In the first step the data are divided into
two groups: nears and fars, separated at the median
value of sin
2
χ
χ
ð
Þ
R
0
cos
+G sin
,
6
:
7
. For each group, the median amplitude
and value of sin
2
θ
G
0
¼
χ þ
χ
:
ð
Þ
R
0
sin
G cos
6
:
8
are calculated. A straight line
drawn between the near-group point and the
far-group point gives an estimate of the gradient that
is robust against the noise, because the median is
insensitive to outliers. If this estimate for the gradient
θ
Within a sliding window the rotated values of inter-
cept and gradient for adjacent datapoints have a
regression line fit through them. The filtered output
for the central datapoint is determined directly from
a)
b)
c)
G
G'
G
R(0)
R(0)
R(0)'
Figure 6.17
(
2004
); (a) intercept vs gradient crossplot showing
the noise ellipse, (b) determining new values for the rotated gradient based on a regression of rotated intercept and rotated gradient points in
a sliding window (five samples in this case), (c) noise ellipse reduced in magnitude after rotation back into intercept gradient space (after
Whitcombe et al.,
2004
).
Filtering process to improve AVO gradient estimation, after Whitcombe
et al.
124
