F020 Methods for Computing Angle Gathers Using RTM M. Vyas* (WesternGeco), X. Du (WesternGeco), E. Mobley (WesternGeco) & R. Fletcher (WesternGeco) SUMMARY Different techniques can be used to compute angle-domain common-image point gathers. They differ from one another not only from an algorithmic viewpoint but also in terms of cost and quality. The choice of method is thus dependent upon the purpose for which the gathers are being generated. In this paper we discuss some of the popular methods that have been proposed over the last few years along with a novel hybrid approach. We also allude to the relative advantages and dis-advantages of these techniques with the help of synthetic and real data examples.
Introduction Recently, there has been a renewed interest in common-image point (CIP) gathers and different methods have been proposed to compute angle-domain CIP gathers for reverse-time migration (RTM). These techniques can be categorized under two main classes: post-imaging and pre-imaging (Biondi, 2006). Generally speaking, the post-imaging algorithms are more attractive from a computational standpoint, whereas, the pre-imaging workflows are considered more accurate. There exist various types of methods belonging to both of these broad classes; they differ in terms of implementation, cost and accuracy. In this paper we discuss some of the popular ways of computing angle gathers both post- and pre-imaging and also their respective advantages and disadvantages. We also propose an approach that combines the benefits of some popular pre-imaging condition methods. Finally, we present results for synthetic as well as wide-azimuth field datasets. Post-imaging condition algorithms Post-imaging methods usually involve two steps: evaluating an extended imaging condition and transforming the extended gathers to angles and azimuths. We can choose to compute space-lag gathers (Biondi and Symes, 2004) or time-lag gathers (Sava and Fomel, 2006; Vyas et al., 2010) or a combination of both. The extended imaging condition in its most general form could be written as I(x,y,z,h x,h y,h z,τ) = U S (x + h x,y + h y,z + h z,t + τ) U R (x h x,y h y,z h z,t τ) (1) x, y and z represent the coordinates of the image point, t is time, τ is the time lag, h x, h y and h z are spatial lags, U S and U R are source and receiver wavefields and I is the image. Computing all the lags is computationally very expensive and the memory requirements to work with a 7D image volume are very high. Hence, it is common to only evaluate a few lags. Computing space lag in a single direction is usually not sufficient in 3D scenarios as it introduces a bias towards a particular coordinate axis and is commonly referred to as the 2D approximation. However, it is possible to compute accurate angle gathers using time lags alone as discussed below. Time-lag gathers The time-lag imaging condition does not have a bias towards any coordinate axes, and hence, no 2D assumption. However, it has an implicit integration over all the reflection azimuths, which implies time lags do give accurate angle information, but alone, they are not sufficient for computing both angles and azimuths. Vyas et al. (2010) suggest improvements to the time-lag gather algorithm presented by Sava and Fomel (2006) that make its application in the presence of conflicting dips, steeply dipping reflectors and anisotropy accurate. The transformation of time lags to angles can also be formulated in a least-squares sense to enhance the resolution and remove certain artifacts. Since the time-lag gathers provide accurate angle information they can be used for 3D applications where we do not expect strong azimuthal variations like shallow sediments or for narrow azimuth surveys. It should be mentioned that computing angle gathers using time lags is much cheaper than wavefield decomposition-type methods. Here we provide an example of WAZ data where angle gathers aided the interpretation of top salt. Figure 1(a) is the image obtained after a sediment flood migration and Figure 1(b) shows the angle gathers for the location marked with a red line in the image. Gathers above the salt look clean and free from any artifacts. Pre-imaging condition algorithms These refer to methods that estimate the opening angle and azimuth before or at the time of applying the imaging condition. To compute these quantities at the image point an estimate of the direction(s) of the incoming wavefield and the reflected wavefield is needed. The dot product between the two provides the opening angle (θ) whereas the cross product defines the reflection plane, hence the azimuth (φ). From an implementation standpoint, this can be accomplished by either carrying out the space-time
(a) (b) Figure 1 (a) Image after sediment flood migration and (b) time-shift angle gathers (0 to 60 ) for the marked location. The yellow line is the salt interpretation. domain decomposition, frequency-wavenumber ( f k) domain decomposition (Xu et al., 2010), or by estimation of the direction of the most energetic component (Yoon and Marfurt, 2006). 1. Space-time domain decomposition An obvious approach for computing opening angles and azimuths would be to apply the imaging condition after decomposing U S and U R into respective plane-wave components, I( x, p s, p r ) = U S D ( x, p s,t)u R D( x, p r,t)dt (2) Here, x is the spatial location vector, t is time, and p s and p r represent the plane-wave component of the decomposed source (U S D ) and the decomposed receiver wavefield (UR D ). The decomposition is carried out such that U S ( x,t) = U S D ( x, p s,t)d p s and U R ( x,t) = U R D ( x, p r,t)d p r. The imaging condition is evaluated for each pair of p s and p r and easily transformed to θ and φ. Although this approach for computing angle gathers is conceptually simple and intuitive, its implementation is extremely expensive and almost impractical with the current computational capacity. This leads us to other methods that try to achieve a similar objective, but in a more feasible fashion. It should be pointed out that if you choose to integrate equation (2) over different plane-waves, it becomes equivalent to the stereographic imaging condition proposed by Sava (2007). 2. Frequency-wavenumber domain approach Xu et al. (2010) suggest an f k equivalent of the approach presented in the previous section. p s and p r are now computed using wavenumbers from the source and receiver wavefields and opening angle and azimuth can be estimated thereafter. Like the previous approach, this method is accurate and correctly handles the conflicting energy in complex media. However, in 3D it involves 4D Fourier transformations of the wavefields and the proposed imaging condition involves evaluation of multidimensional convolutions. If we choose to work on the full image volume the computation will be prohibitively expensive even for a small 3D model. To reduce the cost, Xu et al. (2010) carry out the decomposition in local windows while taking care to control the boundary effects due to partial decomposition. They also employ the knowledge of the dispersion relationships to reduce the cost of the multidimensional convolutions even further. Nevertheless, the cost still remains high for its application on full volumes of 3D wide-azimuth surveys. To demonstrate the application of the f k domain approach, we choose an area above the salt in the 2D Sigsbee model. Figure 2(a) shows the raw zero-lag cross-correlation image from a single shot. Strong backscatter from the top of salt overrides the sediment energy. Figure 2(b) shows the angle gathers with angles ranging from 0 to 90, corresponding to CIP locations highlighted in Figure 2(a). We can see that the backscatter is correctly binned to high angles around 90, and that the sediment reflectors are
well separated (highlighted with red ellipses), due to the accuracy of the method. However, we would like to underscore that the cost, although substantially reduced, remains to be extremely high for its widespread application. (a) (b) (c) (d) Figure 2 (a) Image from migrating one shot close to top salt in the Sigsbee model and angle gathers using the (b) f k domain approach, (c) optical flow calculation and (d) our hybrid approach. 3. Direction vector based methods Another way to approach the problem is to estimate a single dominant direction of the propagating wavefield instead of a full or partial wavefield decomposition. Once the direction vector for the source and the receiver wavefield is estimated we can calculate the opening angle and azimuth. In spirit, the procedure is somewhat similar to the previous two methods, but is significantly cheaper. Yoon and Marfurt (2006) propose an algorithm for computing these direction vectors or Poynting vectors. Their method measures the energy flow in order to compute the direction of wave propagation using p = U t. U. Here, p is the direction vector, U is the energy or pressure, U t is the temporal gradient and U is the spatial gradient. Direct application of this equation results in Poynting vectors that are quite noisy and require some form of regularization or smoothing. Optical flow algorithms used in computer vision provide a potentially better solution for 3D motion vectors. The direction vector, p, is calculated as p. U = U t. To provide a stable estimate of p i.e. the gradient of energy, we use a regularized L2 inversion following Lucas and Kanade (1981). The scheme is relatively inexpensive and easy to apply. It should be remembered that these direction vector methods are substantially cheaper when compared with wavefield decomposition methods but they only estimate a single direction at any one point in space and time. This makes the direction vector estimates inaccurate when wavefields are very complex (in the presence of conflicting energy). Figure 2(c) shows the corresponding angle gathers obtained using the optical flow algorithm. Because undesirable upgoing reflections from salt overwhelm the desired but weaker downgoing reflections from shallow sediments, the reflected energy from sediments is not well separated from backscattered energy and is mapped to the wrong angle bin. However, we can improve the optical flow calculation if we successfully separate the source and the receiver wavefield into upgoing and downgoing components. Liu et al. (2007) proposed an imaging condition to eliminate low-frequency backscatter artifacts from the stacked RTM image. The technique relies upon separating different components of the wavefield and we use a similar concept here. We suggest that one should carry out an appropriate wavefield separation in the challenging areas (with complex wavefields) and then do the optical flow calculation on separated wavefields independently. This can be
visualized as a hybrid approach between full wavefield decomposition and direction vector estimates. Here we use wavefield decomposition but only to separate the upgoing wavefield from the downgoing wavefield and the direction vector calculation is done on these separated (simpler) wavefields. We not only avoid the multidimensional convolution in the f k domain but the direction vector estimates become more reliable as well. Figure 2(d) shows improved angle gathers from applying this approach. The result is close to that of the f k method but is significantly less expensive to compute. We now show RTM angle gathers for 3D wide-azimuth WAZ Gulf of Mexico data (image in Figure 3(a)) computed using the optical flow algorithm. We use both a slower velocity model and the correct velocity model to generate 3D angle gathers that contain 6 azimuths with 30 increments and reflection angles from 0 to 60 (gathers provided for the location marked with a red line). Notice the moveout in Figure 3(b) due to under migration, especially in the shallow sedimentary reflections. The events are almost flat in Figure 3(c), validating the correct migration velocity model. (a) (b) (c) Figure 3 (a) Image (in-line) and angle-azimuth gathers with (b) slower velocity and (c) correct velocity. Conclusions We have discussed different methods to compute angle and azimuth gathers for RTM indicating the advantages and disadvantages of each. We also introduced a method to compute angle gathers using a hybrid approach between wavefield decomposition and direction-vector based techniques. Choosing an appropriate method to compute gathers in our view is strongly dependent on the final purpose. For certain applications like creating image stacks, gathers must be computed everywhere so cheaper alternatives like time-shift imaging or direction vector methods may be used; whereas, for targeted AVO/AVA inversion, the wavefield decomposition techniques can be employed. Acknowledgments The authors would like to thank WesternGeco for the permission to publish this work. References Biondi, B. [2006] 3D Seismic Imaging. SEG. Biondi, B. and Symes, W. [2004] Angle-domain common-image gathers for migration velocity analysis by wavefield-continuation imaging. Geophysics, 69, 1283 1298. Liu, F., Zhang, G., Scott, M. and Levielle, J. [2007] Reverse-time migration using one-way wavefield imaging condition. SEG Expanded Abstracts, 26, 2170 2173. Lucas, B. and Kanade, T. [1981] An iterative image registration technique with an application to stereo vision. Proceedings of the DARPV IV workshop. Sava, P.C. [2007] Stereographic imaging condition for wave-equation migration. Geophysics, 72(6), A87 A91. Sava, P. and Fomel, S. [2006] Time-shift imaging condition in seismic migration. Geophysics, 71(6). Vyas, M., Mobley, E., Nichols, D. and Perdomo, J. [2010] Angle gathers from RTM using extended imaging conditions. SEG Expanded Abstracts, 29, 3252 3256. Xu, S., Zhang, Y. and Tang, B. [2010] 3D common image gathers from Reverse time migration. SEG Expanded Abstracts, 29, 3257 3262. Yoon, K. and Marfurt, K.J. [2006] Reverse-time migration using the Poynting vector. Exploration Geophysics, 37.