Anisotropic parameter estimation from conversion point offsets
Jerry Yuan*, PGS Geophysical, 10550 Richmond Ave., Houston TX 77042, USA, and
Xiang-Yang Li, British Geological Survey, West Mains Road, Edinburgh EH9 3LA, Scotland
Summary
We present a new approach for parameter estimation in
VTI media, using conversion point offsets of converted
waves (C-waves). Traditionally, the analysis of anisotropy
is based on moveouts. However, conversion points of Cwaves are strongly influenced by VTI anisotropy. Unlike Pwaves, the ray-paths of C-waves are asymmetric. This
makes it possible to obtain conversion point information
from a correlation of positive and negative common-offset
gathers. A semblance analysis is used to estimate effective
velocity ratio eff and anisotropy. The validity of the
methodology is demonstrated through both synthetic and
field data examples. This analysis technique has an
advantage in the areas where P-wave velocities are not
reliable.
Introduction
One of the main problems in processing C-waves in 4C
marine seismic data is the wide occurrence of VTI
anisotropy (transverse isotropy with a vertical symmetry
axis). VTI affects both C-wave moveout and conversion
point. Traditionally, the analysis of VTI is performed
through moveout analysis either by P-waves or by Cwaves. These analysis techniques are either exclusively
dependent on P-wave velocity analysis (Alkhalifah 1997;
Lou et al 2002), or on combined P-wave and C-wave
velocity analysis (Yuan et al 2001). However, for the areas
where P-wave data have a low S/N ratio, such as seen in
gas clouds, it is difficult to estimate the strength of VTI
anisotropy without reliable P-wave velocity information.
We extend Yuan and Li (2001) and take a different
approach for the estimation of VTI anisotropy, using
conversion point instead of moveout. In fact, Yuan and Li
(2001) have demonstrated that VTI has a stronger influence
on the conversion point offsets than on the moveout. For
typical VTI materials, ignoring VTI will cause significant
error in calculating conversion points for offsets with
offset-depth ratios just above 0.5 (Yuan and Li 2001).
respectively, and Thomsen (1986) parameters i and i. The
conversion-point offset xc for a C-wave ray converted at the
bottom of the n-th layer and emerging at offset x can be
written as,

c x2 
xc  x c0  2 2 
(1)
1  c3 x 

where c3  c2 /(1  c0 ) (Thomsen 1999). Coefficients c0
and c2 are derived as (Yuan and Li 2001),
c0 
c2 
 eff
1   eff
, and
 eff 1   0 
2t c20Vc22 0
1   
3

0 eff
 
 1  8  eff  0 eff   eff

eff
(2)
where tc0 is the two-way vertical traveltime of C-wave, Vc2
is the C-wave short-spread hyperbolic moveout velocity, 0
is the ratio between the one-way vertical traveltime of Swave and that of P-wave, and eff is the short-spread
binning velocity ratio defined by Thomsen (1999). Two
effective anisotropic parameters eff and eff in equation (2)
are defined as
n 4
4 
 V p 2i t p 0i (1  8 i )  t p 0V p 2 ,
 i 1


1  n 4
4
 eff 
 V s 2i t s 0i (1  8 i)  t s 0V s 2  ,
8t s 0V s42  i 1

2
 i  ( i   i ) /(1  2 i ) , and  i   eff
i i ,
 eff 
1
8t p 0V p42
(3)
where eff was first introduced by Alkhalifah (1997), and
eff was an SV-wave anisotropic parameter introduced by
Yuan and Li (2001). In order to further simplify the
equation (2), we define
   eff  0  eff   eff
(4)
and c2 in equation (2) becomes
 eff 1   0 
c2  2 2
 0 eff  1  8 .
3
2tc 0Vc 2 0 1   eff





(5)
Conversion point offset in layered VTI media
Consider an n-layered VTI medium. Each layer is
homogeneous with the following interval parameters for
the i-th layer (i=1,2,..,n): P- and S-wave vertical velocities
Vp0i and Vs0i, vertical one-way travel times tp0i and ts0i,
Hence  is the measurement of effects of anisotropy and
layering on conversion point at middle-far offsets. As
demonstrated by Yuan and Li (2001), the above equations
(1), (2) and (5) for calculating the conversion point offset
Anisotropic parameter estimation from conversion point
are very accurate up to an offset-depth ratio of 3.0 with
errors less than 0.5%.
Anisotropic parameter estimation from conversion
point
With the improved accuracy, equations (1), (2) and (5) can
be used for estimating the effective velocity ratio eff and
the anisotropic parameter , if xc, VC2 and 0 are known. Vc2
can be determined from short-spread C-wave velocity
analysis, and 0 from a coarse correlation of P- and C-wave
stacked sections. At certain locations with distinct
geological features, such as a fault point, the apex of an
anticline, or even amplitude anomaly, the conversion-point
offset xc can be obtained by cross-correlation of positive
with negative common-offset gathers. This method is
similar to the X-focusing analysis developed by Audebert
et al (1999), except that we also consider the influence of
anisotropy at middle-far offsets.
We first test the methodology on a synthetic dataset. Figure
1 shows a model with an anticline. The common-offset
gathers with an offset +/- 1000m are shown in Figures 2a
and 2b. It can be seen that there is a large spatial difference
of the displayed anticline between the positive and negative
offset gathers before any CCP sorting or migration. Ideally,
the spatial difference can be quantified by a lateral crosscorrelation of positive and negative offset gathers.
However, C-waves are notorious for diodic effects
(Thomsen 1999), which means that the events of positive
and negative offsets may have slightly different arrival
times. Therefore, we apply a 2D cross-correlation first
(Figure 3a), then find the maximum cross-correlation at
certain time shift. The 2D cross-correlation is performed
for all the +/- offset gathers, and the maximum crosscorrelations are plotted in Figure 3b.
Similar to the moveout velocity analysis technique used by
Alkhalifah (1997) for P-waves, and Yuan et al (2001) for
C-waves, we use a double-scan semblance analysis to
determine eff and  from the maximum cross-correlation
in Figure 3b. A contour plot of the semblance analysis is
shown in Figure 4. Vc2 and 0 are used as priori information.
The inversion result is very close to the theoretically
predicted value.
To further evaluate the methodology, the C-wave
conversion point analysis is applied to a dataset from the
North Sea. Figure 5 shows a small portion of the C-wave
image with a mild structure at 2.0s. The maximum crosscorrelation is shown in Figure 6a, and the semblance
analysis of eff and  is shown in Figure 6b. The vertical
velocity ratio is about 3.22 obtained from the correlation of
C-wave imaging with the P-wave image. The eff and are
determined as about 2.43 and 0.148, respectively, from
Figure 6b. The corresponding  is 0.011 about from
equations (3) and (4) assuming a single-layered VTI model.
Discussion and conclusions
We have tested the methodology on various synthetic and
field datasets. It works very well with mild structures.
However, sometimes it is difficult to apply the method to
events having relatively steep dips. Instead, one should
combine this method with the C-wave prestack migration
technique (Li et al, 2001). The conversion point analysis
should be applied after migration.
In the semblance analysis, we choose eff and  for
parameter inversion. An alternative way is to use c0(or eff)
and c2, then determine the anisotropic parameter later. The
reason we choose  over c2 is that direct inversion of c2
(whose value range is uncertain) may lead to some
unreasonable anisotropy estimation. However, because  is
coupled with parameters Vc2 and 0, errors in these
parameters will propagate into the  estimation, although
estimation of eff is robust.
In conclusion, anisotropy has a significant influence on the
conversion point. Based on an improved and accurate
conversion point approximation, we have developed an
alternative and robust way for C-wave parameter
estimation in VTI media. The methodology has been tested
on both synthetic and field data. The major advantage of
this methodology in C-wave processing is its weak
dependence on P-wave processing. The only information
needed from P-wave data is the vertical velocity ratio from
coarse correlation of P- and C-wave sections. Therefore,
the approach is particularly useful in areas where P-wave
velocities are unreliable, such as in the presence of gas
clouds.
Acknowledgements
We thank Shell Expro (UK) for providing field data to the
test. We thank Xianhuai Zhu and Don Pham for many
useful discussions and Min Lou for the synthetic modeling.
We thank PGS Geophysical for permission to show the
results. The work is carried out in collaboration with the
Edinburgh Anisotropy Project, and is presented with the
approval of all project partners.
References
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Anisotropic parameter estimation from conversion point
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Figure1: Synthetic model with an anticline. The VTI
material used is Dog Creek shale (Thomsen, 1986).
(a)
Yuan, J. and Li, X., 2001, PS-wave conversion-point
equation in layered anisotropic media, 71st Ann. Internat.
Mtg: Soc. of Expl. Geophys., 157-160.
(a)
(b)
Figure2: Positive (a) and negative (b) offset gathers showing the spatial
difference of the anticline before sorting and migration.
(b)
Figure3: (a) 2D cross-correlation of Figure 2 over spatial and time shift; (b)
maximum cross-correlation for all offset ranges.
Figure 4: Semblance analysis of Figure 2b for eff
and c using priori information for Vc2 and 0. The
circle is the expected value.
Anisotropic parameter estimation from conversion point
Figure 5: C-wave image of a dataset from the North Sea. The structure in the box (2.0s) is the analyzed area.
(a)
(b)
Figure 6: (a) Maximum +/- offset cross-correlation for all offset ranges; (b) semblance analysis of eff and 