A New NMO Correction Equation For HTI Medium
Samik Sil and Mrinal K. Sen
University of Texas at Austin, TX-78705, USA
ABSTRACT
Incorporating the effect of anisotropy during seismic data
processing and estimating anisotropic parameters is an active area
of research. In general five elastic coefficients are required to
describe a traveltime curve in a HTI medium. The problem of
estimating five elastic parameters by iterative fitting of travel
time data from a single azimuth recording is highly non-unique.
This can possibly be achieved by simultaneous fitting of multiple
azimuth travel time data. However that would require accurate
travel time picking and estimation, which involves numerical ray
tracing for multi-layered media. To circumvent these difficulties
we propose analysis of plane wave transformed azimuthal gathers
interactively using a single azimuth data at a time and a new delay
time equation which is a function of two parameters at each
azimuth. Results from independently estimated multi-azimuth gathers
can be combined to estimate stiffness or Thomsen coefficients.
Azimuthal -p analysis also avoids numerical ray tracing resulting
in a rapid algorithm. We demonstrate the applicability of our
method using a set of P wave synthetic seismograms from a
multi-layered medium consisting of isotropic and HTI layers.
Azimuth dependent anisotropy parameters are derived by delay time
fitting and NMO correction. The reflections from the bottom
interface of an isotropic layer with an anisotropic overburden show
apparent anisotropic travel time behavior which is easily accounted
for by our layer-stripping based azimuthal NMO analysis.
Introduction
A system of parallel vertical cracks embedded in an isotropic
medium exhibits horizontal transverse isotropy (HTI) (Tsvankin
1997). Unlike a VTI medium, seismic velocities in a HTI medium vary
with the azimuth () of the seismic survey line. An HTI medium can
be described by five Thomsen Style parametersand as defined by
Tsvankin (1997). A reflection traveltime curve obtained over a HTI
medium should be a function of the above 5 parameters. For a pure
mode P wave data in an HTI medium, the NMO correction in traveltime
offset (x-t) domain is generally performed by computing the
traveltime curve using a truncated three-term Taylor series
(Tsvankin and Thomsen 1994, Tsvankin, 1997, Al-Dajani and Tsvankin,
1998). The coefficients of the higher order terms are complicated
functions of the anisotropic parameters. By fitting a traveltime
curve with such a truncated Taylor series, NMO correction and
parameter estimation of the HTI medium is performed. Though this
method can sometime obtain satisfactory results, it has a few
limitations. The truncated Taylor series expression is approximate,
and such an approximation introduces uncertainty in the estimation
of the anisotropic parameters resulting in a mismatch with the
exact traveltime curve. For a multi-layer case with an isotropic
target layer below an anisotropic layer, the reflection from the
base of the isotropic layer show azimuthal variation in travel time
due to transmission through the upper anisotropic layer. If a
Taylor series curve fitting (using rms parameters) is applied to
compute moveout correction, false anisotropic parameters will be
obtained for the isotropic layer.
To overcome these problems, seismic gathers may be transformed
to delay-time slowness ( - p) domain. In this domain it is possible
to derive an exact equation for moveout correction in HTI (van der
Baan and Kendall, 2003) and general anisotropic media. The moveout
correction is performed here by a layer stripping approach. It is
therefore straightforward to apply NMO for each layer in succession
in depth allowing for exact correction of any overburden
effect.
Though an exact equation for NMO correction in - p domain exits,
it is difficult to implement it because of the presence of 5
anisotropic parameters in the equation. Five parameters make the
equation highly non-unique and difficult to implement (van der Baan
and Kendall, 2003) for practical purposes. To overcome this
limitation van der Baan and Kendall (2003) introduced a reduced
parameter equation for the HTI medium. This equation contains 4
parameters and performs better NMO correction than any Taylors
series based x-t domain method (Van der Baan and Kendall, 2003).
Note, however, that this approximate equation is still a function
of four parameters and therefore, may contain almost the same
degree of non-uniqueness as the exact equation. The approximate
equation for the VTI medium in - p domain is structurally different
from the approximate equation for the HTI medium. Motivated by the
search for a reduced parameter - p equation that can be easily
implemented, we develop an approximate equation in - p domain for P
wave reflection data, considering horizontally layered weak HTI
medium and demonstrate its effectiveness using full wave
anisotropic synthetic seismograms.
Theory:
The - p curve for a multilayered isotropic and transversely
isotropic earth model can be calculated using the following formula
(Stoffa et. al. 1982):
(1)
Where h is the thickness of the each layer and q is the vertical
slowness of that particular layer. The vertical slowness q can be
expressed as:
,
(2)
where v is the phase velocity which is a function of phase
angle.
For a weak homogeneous HTI medium, the phase velocity can be
written as (Sil and Sen, 2007):
, (3)
where and is the vertical P wave velocity in the HTI medium.
Putting equation 3 in equation 2 and with some mathematical
manipulations (Sil and Sen, 2007) we can obtain:
(4)
or
(5)
Where:
(6)
and
.
(7) are the HTI parameters defined by Tsvankin (1997).
Thus considering weak anisotropy, the - p curve for P-wave
reflection from horizontal HTI medium at a given azimuth can be
written as a function of 2 reduced parameters and , where the first
one is the elliptical velocity and the second one is an anelliptic
term, responsible for the non elliptic behavior of the - p curve
over anisotropic medium (Sen and Mukherjee 2003). We can use
equation 4 or 5 for a VTI medium as well, in that case for
different azimuth, the value of and will be constant and thus the
value of will be 0. When is 0, then equations 4 and 5 reduce to a
two-term approximate equation for the VTI medium, obtained by Sen
and Mukherjee (2003). When the medium is isotropic, all the
anisotropic parameters and will be zero. Thus the equation 4 or 5
can be used for NMO correction and parameter estimation in
isotropic, VTI and HTI media.
Example
It is possible to generate synthetic seismograms for a HTI
medium using exact p curve equation developed by van der Baan and
Kendal (2003). This is a forward problem, and as we have said, if
we use the 5 parameter exact equation for inversion, then the
result will be highly non-unique. We generate a seismogram for a
simple 3 layered case, where a target HTI layer is sandwiched
between two isotropic layers. The bottom layer is infinite and the
target HTI layer is weekly anisotropic (values of anisotropy
parameters are less than 1, a realistic assumption). The parameters
for the synthetic seismogram are shown in table 1:
Table 1: Parameter of the 2 layered case used in this study
.
Layer 1Layer 2Layer 3
System Type
ISOHTIISO
Thickness (km)
0.7101.300
Density (g/cm)
1.0002.5003.000
(km/sec)
2.9603.3303.500
(km/sec)
0.0002.1102.220
0.0000.2580.000
0.000-0.0780.000
0.0000.1820.000
We generate the synthetic seismogram based on the parameters in
table 1 for 3 different azimuths 0, 45 and 90 and converted them in
p domain (figure 1, figure 2 and figure 3). In p domain, for 0
azimuth seismogram, we also plot the exact solution (green curve),
van der Baan and Kendals approximation (magenta star), isotropic
solution (yellow curve) and our approximation (red dots). It is
clear that our approximation fits the exact curve better than any
other solutions and thus can perform better NMO correction.
Figure 2 shows the same type of seismogram for 45 azimuth. Here
we only plot our approximation (red line) and isotropic solution
(yellow line). As with increasing azimuth (in between 0 and 90) the
HTI medium becomes more isotropic, difference between red and
yellow line is getting less. In figure 3 we plot the seismograms
for 90 azimuth, at this azimuth isotropic medium behaves completely
isotropic. So the difference between our approximation (red line)
and isotropic solution is zero.
Now after fitting these 3 curves at 3 different azimuths,
solutions for all the anisotropic parameters can be resolved by
simple grid search method.
Conclusions
We have developed an azimuth dependent two parameter delay-time
equation in a horizontally layered weak HTI medium suitable for NMO
correction and parameter estimation. Since this equation contains
fewer parameters, a simple grid search method can be applied for
determining the value of the parameters and the extent of
non-uniqueness is less. The equation is developed in - p domain and
therefore, a layer stripping approach of data fitting can be used
to overcome the problem of an overlying anisotropic medium.
Figure 1: a) X-t domain seismogram of azimuth 0 based on the
parameters shown in table 1, b) -p domain seismogram with exact
solution (green line), previous approximation (magenta star),
isotropic solution (yellow line) and our approximation (red dot).
Our approximation feats the exact curve better than other
solutions.
Figure 2: a) X-t domain seismogram of azimuth 45 based on the
parameters shown in table 1, b) -p domain seismogram with isotropic
solution (yellow line) and our solution (red line). As the medium
is becoming more isotropic, the difference between two lines is
less here compared to 0 azimuth.
Figure 3: a) X-t domain seismogram of azimuth 90 based on the
parameters shown in table 1, b) -p domain seismogram with isotropic
solution (yellow line) and our solution (red line). As the medium
is now isotropic, the difference between two lines is zero.
References
Al-Dajani, A., and I. Tsvankin, 1998, Nonhyperbolic reflection
moveout for horizontal transverse isotropy: Geophysics, 63,
17381753
Sen. M. K., and A. Mukherjee, 2003, Tau-p analysis in
transversely isotropic media: Geophysical Journal International,
154,647658.
Sil. S. and M. K. Sen, 2007, Azimuthal -p analysis in HTI media:
SEG extended abstract, 183-186
Stoffa, P. L., J. B. Diebold, and P. Buhl, 1982, Velocity
analysis for wide aperture common midpoint data: Geophysical
Prospecting, 30, 2557.
Tsvankin, I., 1997, Reflection moveout and parameter estimation
for horizontal transverse isotropy: Geophysics, 62, 614629.
Tsvankin, I., and L. Thomsen, 1994, Nonhyperbolic reflection
moveout in anisotropic media: Geophysics, 59, 12901304.
Van der Baan M., and J-M. Kendall, 2003, Traveltime and
conversion-point computations and parameter estimation in layered,
anisotropic media by tau-p transform: Geophysics, 68, 210224.
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