INTRODUCTION
Sedimentary rocks especially shales show anisotropy if seismic waves pass through them at different directions. The presence of anisotropy has considerable influence on many aspects of seismic wave propagation and therefore has profound implications for both conventional processing schemes and interpretation. When the subsurface is anisotropic, the moveout of Pwaves in horizontally layered media as observed in common midpoint gathers, deviates from being hyperbolic at far offsets. Thus the nonhyperbolic component of the moveout is routinely used for estimating relevant seismic parameters describing such anisotropy (Grechka and Tsvankin, 1998). For Pwaves in Vertical Transverse Isotropic (VTI) media, these parameters are the anellipticity parameter (η) and the normal moveout velocity (V_{nmo}). Quantitative measurement of seismic anisotropy provides a valuable clue to lithology and degree of stratification, a tool for imaging and can as well provide important quantitative information about depth discrepancies observed from depths determined from seismic and well log data (Van der Baan and Kendall, 2002).
Anisotropic properties are estimated using a variety of ways. Laboratory tests
are the main techniques used in measuring anisotropic properties. Vertical Seismic
Profiling (VSP) and surface seismic data are the in situ techniques used
to determine anisotropic parameters (Li, 2006). Using surface seismic data,
the most popular approach is using modified Taylor series expressions (Alkhalifah,
1997) or taup (τρ) based methods (Van der Baan and Kendall, 2002).
Recently, Douma and Calvert (2006) used a semblance based rational interpolation
method to analyse nonhyperbolic moveout in the timeoffset (tx) domain to
estimate anisotropy parameters and moveout velocity. These workers pointed out
that the picking of traveltimes in the tx domain which is often laborious and
prone to error in the τρ technique of Van der Baan and Kendall (2002)
is a practical disadvantage. To overcome this caveat, Douma and Van der Baan
(2006) combined the τρ technique of Van der Baan and Kendall (2002)
and the rational interpolation method of Douma and Calvert (2006) to estimate
anisotropy parameters.
In this study, we use the approach of Douma and Van der Baan (2006) to determine the lateral variations of both effective and interval values of the anellipticity parameter (η), the NMO (v_{nmo}) and horizontal (v_{hor}) velocities at selected CDP locations in a marine 2D seismic dataset acquired in the Gulf of Mexico.
THEORETICAL BACKGROUND
The τp transform relates traveltime t and offset x to intercept time τ and horizontal slowness p by:
Hake (1986) observed that the total intercept time τ for pwave reflections in horizontally layered transverse isotropy with vertical axis of symmetry (VTI) media is a summation of the products of layer interval zerooffset time (Δt_{0,i}), the vertical Pwave velocity (v_{0,i}) and the vertical slowness (q_{i}) expressed as:
Alkhalifah (1998) expressed the vertical slowness q_{i} in terms of v_{nmo} and the anellipticity parameter η by assuming that the interval shearwave phase velocity V_{so} has negligible influence on the traveltimes of Pwaves in VTI media. Alkhalifah’s equation is expressed as:
Where:
p 
= 
The horizontal component of the slowness vector 
Grechka and Tsvankin (1998) however demonstrated that the horizontal velocity expressed as V_{hor} = V_{nmo} [1+2η]^{1/2} is better suited for semblancebased moveout analysis than η as used in Alkhalifah’s equation. Thus, expressing Eq. 3 in terms of V_{hor} and V_{nmo} and substituting into 2, gives:
Where:
V_{hor,i} 
= 
Horizontal velocity in layer i 
V_{nmo} 
= 
NMO velocity in layer i 
For semblance analysis to be carried out in the tx domain requires the traveltime for each recorded offset. From Eq. 1,
Substituting Eq. 4 into 5 and simplifying,
gives:
The traveltime associated with this offset was obtained by combining Eq.
1, 2 and 6 giving:
NONHYPERBOLIC MOVEOUT ANALYSIS USING RATIONAL INTERPOLATION
A rational approximation to a function T(x) is generally written as Stoer and Bulirsch (1993)
where, N_{L}(x) and D_{M}(x) are polynomials of degree L and M. Such a rational approximation can be written as [L/M]. For a polynomial of degree 2, L = M = 2. Thus, Eq. 8 can be rewritten as:
For a [2/2] rational interpolation for a single horizontal VTI layer, Douma and Calvert (2006) derived the squared traveltimes T as a function of squared offset X as:
where, T_{0} = t^{2}_{0} is the squared zerooffset
twoway traveltime and n_{1}, n_{1}, d_{1} and d_{2}
are the coefficients of the numerator and denominator of the rational approximant.
To determine the unknowns, four traveltimes (t_{i}) and four associated
offsets (x_{i}) are required. The traveltimes (t_{i}) and the
associated offset (x_{i}) are determined using Eq. 7
and 11, respectively.
Where:
k_{i} 
= 
The offsettodepth ratios. 
The interpolation traveltimes (t_{i}) and (x_{i}) are used in determining the coefficients of Eq. 10. When once the coefficients of Eq. 10 are known, the traveltimes (t) for offset (x) can be evaluated. Douma and Van der Baan (2006) extended this approach to horizontally layered VTI media.
DATA PROCESSING
The data used in this study is a 2D marine seismic Pwave data set acquired
in the Gulf of Mexico. The data was processed by applying a bandpass filter
of frequency range 153570 Hz to remove frequencies outside of the useful
signal bandwidth. Common Midpoint gathers were then collected into groups of
nine and stacked (vertically) to form supergathers. Frequencywavenumber (FK)
filtering was then applied to each supergather to suppress coherent noise and
semblance analysis performed in order to obtain stacking velocities for use
in normal moveout (NMO) correction. Common noise observed after FK filtering
is random noise. To reduce the random noise further, frequencyoffset (FX)
deconvolution (Canales, 1984) was applied. Finally, a coherency filter was applied
to improve signal character and continuity within the frequency range of the
data.
ANISOTROPY PARAMETER ESTIMATION
Ten supergathers (3855938640) between CDPs 37999 and 38941 were chosen for parameter estimation. These ten supergathers hereinafter are referred to as ensembles 110. This zone was chosen because the stack section shows horizons that are relatively flat. Thus, it was assumed that the presence of any nonhyperbolic moveout will be due to anisotropy. In each supergather, three events were chosen for analysis, these events occur at 1.14, 1.44 and 2.4 sec (Fig. 1). The algorithm SUVEL2DF used in this study determines values of the effective and interval anellipticity parameter (η), horizontal velocity (v_{hmax}) and NMO velocity (v_{max}) through semblance analysis. Semblance analysis using SUVEL2DF requires the computation of the traveltimes for each offset acquired in the field for a specific combination of zerooffset time (t_{0}), NMO (v_{nmo}) and horizontal (v_{hor}) velocities.
The most important input parameters are the zerooffset time (t_{0})
of the layer of interest determined from the supergather gather and user specified
effective NMO (v_{nmo}), horizontal (v_{hor}) velocities and
four offsettodepth ratios ranging from 14. For each particular v_{hor}
and v_{nmo} input into the algorithm, the four offsettodepth ratios are
converted to target offsets (x_{j}) using Eq. 11.

Fig. 1: 
Ensemble 38604 showing reflection events chosen for parameter
estimation in each supergather 

Fig. 2: 
Semblance scan for layer 1 (ensemble 6) showing optimal semblance
response 
The algorithm also computes the appropriate horizontal slowness (p_{j})
to target offsets (x_{j}) using Eq. 6 and subsequently
determine the corresponding traveltimes (t_{j}) by the use of Eq.
7. The four values of the target offsets (x_{j}) and traveltimes
(t_{j}) constitute the support points for the [2/2] rational interpolation.
The algorithm then determines the desired traveltimes t(x) at any offset x using
the rational approximant (Eq. 10). The resulting tx curves
are then used for the semblance analysis.
We determined both effective and interval values of η, v_{hor }and v_{nmo}. In determining effective values, we treated the overburden of the reflection event of interest as a single horizontal homogeneous VTI layer. As result, the effective values are average values. Interval values were obtained using a layerstripping approach. The accuracy of the inversion was confirmed by the high semblance maxima (S_{max}≈0.81.0) which coincided with the semblance contours nicely centred in the respective semblance scans (Fig. 2).
RESULTS AND DISCUSSION
Effective values of v_{nmo} and v_{hor } range between 18502250±111 and 19752475±122 m sec^{1} while interval values of v_{nmo} and v_{hor } range between 17502650±120 and 21003100±126 m sec^{1}, respectively. Values of the anellipticity parameter (η) as a function of ensemble number are shown in Fig. 3 and 4, respectively. The effective values are mainly between 0.030.16 ±0.01 in all the three layers investigated. Interval values range mainly between 0.030.2±0.03. Variation in anellipticity is observed in both effective and interval values suggesting lateral variation in anisotropy along the seismic line presumably due to varying shale content since the presence of shale is widely accepted as the cause of anisotropy (Wang, 2002; Li, 2006). The recovered values of v_{hor} and v_{nmo} exhibit a similar behaviour. In layer 2 in Fig. 3 (ensemble 1) and layer 2 in Fig. 4 (ensembles 4 and 8), we observe an abrupt increase in anellipticity values where the anellipticity is as high as 0.3 and 0.4, respectively. These values of η are high compared to values typically reported from nonhyperbolic moveout analysis (Toldi et al., 1999).
Without detailed geologic information, we can only speculate as to the cause
of these localised extreme values. The relatively high η values might reflect
some internal property of the medium, presumably increased shale content or
that these are inaccurate values due to the inability to obtain optimal semblance
response occasioned by the use of inaccurate input model parameters. If the
first assumption was correct, the nonhyperbolic moveout in the input CDP gathers
in these events should have been substantial.

Fig. 3: 
Effective
anellipticity parameter (η) against ensemble number for all three
layers showing lateral the lateral variation of anisotropy at ensemble
locations 

Fig. 4: 
Interval
anellipticity parameter (η) against ensemble number. Layer 1 is not
shown because in layer 1, effective and interval values are the same 
We observe that the nonhyperbolic moveout in these events was rather small
and therefore might not be able to cause such a high level of anellipticity,
since the strength of the anisotropy determines the amount of nonhyperbolic
moveout (Wookey et al., 2002). On the other hand, a careful look at the
semblance scans of these events show more than one semblance contour making
it difficult to obtain optimum semblance response. Since correct anellipticity
values are only obtained at optimum semblance response, we are convinced that
the anellipticity values of 0.3 observed in ensemble 1, layer 2 (Fig.
3) and ~0.4 in ensembles 4 and 8, layer 2 (Fig. 4) are
not correct and do not represent the actual subsurface anellipticity level at
these ensemble locations. The presence of more than one semblance contour might
be due to the presence of multiples.
CONCLUSION
Results from this study show that the three layers investigated have considerable variations in lithology. The effective and interval values of the anellipticity parameter vary mainly between 0.030.2±0.01. Effective and interval values of v_{nmo} and v_{hor} range between 19752825±111 and 17502650±126 m sec^{1}, respectively. We attribute the variation in anisotropy to varying shale content. These results can be employed as new constraints in processing algorithms that include anisotropy so as to improve the imaging of dipping reflectors, such as fault planes in the study area.