Common-angle processing using reflection angle computed by kinematic pre-stack time demigration

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1 Common-angle processing using reflection angle computed by kinematic pre-stack time demigration Didier Lecerf*, Philippe Herrmann, Gilles Lambaré, Jean-Paul Tourré and Sylvian Legleut, CGGVeritas Summary Recent works have shown the benefit of seismic data processing in common angle domain especially for AVA studies and stack resolution enhancement. When wide angle datasets are exploited, difficulties and reservations come with the genuine signification and the validation of the reflection angle value. The workflow presented includes two original methods providing accurate angle CDP gather from time migrated offset CDP gather. The dependencies offset, angle and time are computed through a kinematic pre-stack time demigration process taking into account local dips, the effective velocity and anelipticity model. The kinematic demigration takes advantage of the existing relationships between the reflection angle and the differential stretch in time of the wavelet for different offsets. This approach gives a better angle estimation than conventional methods especially for angles larger than 4 o and for dipping events. In addition to the mapping of offset, angle and time correspondences, a direct (and reverse) angle transform is defined in order to provide seismic traces regularly distributed in angle. The angle regularization is performed by an iterative Fourier reconstruction minimizing the spatial frequency leakage. Because the amount of stretch is nearly stationary in time (if there are no variations of dip in time) for a common angle time-migrated trace, a spectral harmonization along angles can be easily implemented by single spectral matching operators. This stretch compensation impacts on the stack quality and resolution. Introduction In reservoir characterization, wide angle seismic data give valuable information for improving lithology and fluids analysis. For example, AVO/AVA studies need incidence angle larger than 4 degrees for inverting density. Wide angle data may also provide significant improvements to the structural image by additional target illumination. Because imaging introduces about 5% wavelet stretch at 6 o incidence angle, conventional time processing sequences are not appropriate for such dataset. Data recorded beyond traditional 4 limit are generally muted to avoid distortion of the signal and large anisotropy effect. Using wide angle data for density inversion, Roy et al. (6) has shown the benefit of being in angle domain for analytically correcting the stretch of the signal. The combination of the Walden and the Dix equations are frequently used to estimate reflection angle up to Beyond this limit the estimation suffers from too many assumptions to be reliable. The incidence angle estimation is given by the relation (Walden, 1991): x. V int sin Θ = x V rms. t + V rms Where Θ is the incidence angle. x is the offset. t is the zero offset two-way time. V rms is the RMS velocity and V int is the interval velocity (estimated from Vrms using Dix equations). Therefore, Pèrez and Marfurt (6) have implemented an Angle-binned Kirchhoff time migration that delivers angle binned migrated traces using internally computed angle values. According to the authors, this makes the migration computationally more expensive than one whose bins over offset. This paper proposes to preserve the common offset prestack time migration algorithm (used in the industry and highly optimised) and presents an approach to computing common angle volume through a combination of a specific kinematic pre-stack time de-migration and a regularization process in angle domain. Incidence angle estimation with kinematic de-migration Our approach is based on kinematic pre-stack time demigration. For any position (time, offset) in a gather (at position x) and for a given associated dip (defined in the common offset time migrated image), time de-migration makes it possible to recover the associated shot and receiver positions (Figure 1). It also provides gradients of the one-way travel times in the time migrated image. Figure 1: Time de-migration of a localized reflector in a common offset time migrated image. 64

2 From gradients of the one-way travel times we can obtain the gradient of the two-way travel time, whose vertical component is the local vertical stretching of the time migrated image for the considered offset (Figure ). Figure : The vertical stretching of the common offset time migrated image is defined by the vertical derivative of the two-way travel time. Finally, the reflection angle can be estimated at a given offset considering the ratio of the vertical stretching at the considered offset over the one at zero offset (Figure 3). process is adapted from an anti-leakage Fourier reconstruction algorithm (XU and PHAM, 4). The Fourier reconstruction aims to estimate the spatial frequency content of the seismic data on the irregularly sampled angle position. Using the Fourier coefficients, the seismic data can be reconstructed on any desired regular angle distribution. Unfortunately, spectral leakage appears from the non-orthogonality of the basis functions on an irregular grid. The Anti-leakage technique consists of orthogonalising the Fourier basis functions in an iterative way among Fourier coefficients. The process attenuates significantly boundary effects visible with conventional discrete Fourier transforms. In practice, the angle regularization is driven (at each time sample) using attribute traces collected in CDP gather and describing angle, offset and time dependencies, called Angle map CDPs. A reverse transform makes it possible to convert pre-stack angle CDP gather back to pre-stack offset CDP gather with a minimum lost in quality. The direct and inverse angle transforms allow any common angle processing (such as de-stretching) to be included into conventional offset processing sequence (such as residual normal move out). Time migrated stack Inline dip Crossline dip Models Time migrated gathers Kinematic De-migration Angle map gathers Figure 3: The cosine of the reflection angle, Θ, can be estimated as the ratio of the vertical stretching at the considered offset over the one at zero offset. Figure 4 summarizes the methodology providing angle map gathers. It shows the different inputs to the kinematic de-migration. Offset to Angle transform Once relations describing angle, offset and time dependencies have been established for each seismic trace, the next step consists of converting offset CDP gather into angle CDP gather. As an alternative to the partial angle stack, the transformation is defined as a regularization process. The angle transformation is done for each time sample converting an irregular angle distribution into regularly distributed angle traces. The regularization Figure 4: The diagram presents the different inputs to the kinematic de-migration algorithm in order to map accurately the offset, angle and time dependencies. Common angle processing Velocity Anelipticity Models (effectives) As pointed out by Lazaratos and Finn (4) and Roy et al. (5) the correction for the wavelet stretch in angle domain can be expressed analytically in frequency using: W(f) cos Θ W(f cos Θ) Consequently, the spectral operator Φ can be defined for each common angle volume by: Φ(f) 1/cos Θ W(f / cos Θ) 65

3 This operator can be considered stationary in time and consequently can be applied to the full angle traces as an analytical spectral reshaping. The mentioned authors have shown the benefit of the wavelet stretch correction in AVA studies inverting density with wide angle data. As an alternative to the analytical operator, we took a similar approach to Pèrez and Marfurt (6) estimating that a data dependent operator should be more robust. A statistical matching operator (time invariant) is then defined for each angle volume using a near angle stack (15 ) as reference. Then, a reverse transform (angle to offset) provides migrated offset CDP gather with a coherent spectral content between offset traces. In our data example, we have intentionally limited the application to the stack optimisation and we prefer to use the term spectral harmonization than de-stretch procedure (which supposes more constrains). Data Example We ve chosen a D long offset seismic survey in a complex area to illustrate the methodology. The geology covered is characterized by steep dip structures including a significant anelipticity effect. In this example, the assumption made in the Walden and Dix equations is clearly not valid for the wide angle component of the data. We start with migrated offset CDP gather using an effective velocity and anelipticity model. The presented workflow consists of six main stages: I. Determination of the dip within the initial migrated stack section. II. Kinematic pre-stack time de-migration to compute the Angle map CDP gathers (taking into account the dip, the velocity and the anelipticity model). Figure 5 shows the Angle map difference in degree between angles estimated through Walden relations and angles computed with the kinematic de-migration. Variation of about 5 can be observed in a range of 4-6 o. This represents significant variability for AVA studies. III. Offset to Angle transformation up to 6 incidence. IV. Common angle spectral harmonization for compensating the wavelet stretch. V. Angle to Offset reverse transformation Figure 6 summarizes the different stages in order to process initial migrated offset CDP gather into offset CDP gather with stretch compensation applied. Single spectral matching operators are defined for each common angle volume to match the amplitude spectrum of the reference angle volume 15. It can be noted, that we choose to be regularly sampled according to the tangent of the incidence angle. This angle domain helps to handle wide angles. VI. Stack Figure 7 presents the results on stack sections of the full process. The comparison (before and after) shows that the high frequency enhancement of the seismic section has a significant impact on the resolution improvement. a)kinematic de-migration Figure 5: Angle map CDP a) computed with the kinematic de-migration with mild dipping event. b) computed from Walden and Dix equations. c) Differences Discussion Beyond the presented application for common angle processing, pre-stack time migrated data are often provided for reservoir characterization studies. Partial angle stacks are standard inputs for sophisticated algorithm of AVA inversion. It is tempting to use advanced technique with fast-growing computing capability for extracting information from wide angle data. However, we would like to point out some key processes in the seismic data processing for such studies. An accurate offset-incidence angle dependency is a mandatory requirement for angles from 4 to 6. As other methodologies, the angle computation using the kinematic de-migration is highly sensitive to the correctness of the effective velocity and anelipticity models. In other words, effective velocity models have to be defined with care and are one of the most important prerequisites for an accurate angle determination. Whatever the complexity of the geological structures, the near angle time migrated image must be consistent with the image provided by wide angles. Success depends on the geophysicist s ability to define correct velocity and anisotropy values for wide angle data. Conclusions b)conventional deg c) Difference deg With increasing long offset/wide angle seismic surveys looking for new prospects, common angle processing offers a new prospective to extract structural information from wide angle data through full or partial stacks optimisation. 66

4 In addition, advance reservoir characterization methodologies need accurate incidence angle estimation especially for angle values larger than 4 degrees. Conventional approaches are clearly not appropriate for such an angle range. Two original processes are presented in order to provide accurate common angle volume from a time-migrated offset CDP gather. Offset, angle and time dependencies are computed through a kinematic pre-stack time de-migration. In addition, a time-variant regularization by the Antileakage Fourier reconstruction is adapted in order to transform offset CDP gather into regularly distributed angle CDP traces. We choose a statistical matching operator for each common angle volume to harmonize the spectral content of the wavelet along angles. The reverse angle to offset transform allows returning to the offset domain providing as a result, offset CDP gather compensated for the stretch effect. The improvement of the stack resolution constitutes a part of the potential for common angle processing. Obviously, the benefit of the proposed workflow is not limited to the presented application. For example, residual move-out analysis will benefit significantly for such methodology. Acknowledgements The authors would like to thank TOTAL for the involvement in the project, the numerous discussions and the permission to show the data. Offset CDP Angle CDP tan( o ) tan(6 o ) Common Angle 15 Angle CDP tan( o ) tan(6 o ) Offset CDP Amplitude spectrum Hz 18 Common Angle 55 Amplitude spectrum Offset to Angle Transform Common angle gather Hz 18 Spectral angle harmonisation Angle to Offset Transform Figure 6: Left: Input offset CDP and Angle CDP regularly distributed in tangent of the angle (The stretch is time-invariant for each angle trace). Central: Common angle spectral harmonization: Single matching operators are defined for each common angle. (Angle 15 is taken as the reference). Right: The angle CDP and the offset CDP with the stretch compensation applied stack -6 o angle muted before spectral harmonization stack -6 o angle muted after spectral harmonization Figure 7: Stack before and after the spectral harmonization in angle domain. It can be noted the resolution improvement due to the stretch compensation. 67

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 7 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Lazaratos, S., and C. Finn, 4, Deterministic spectral balancing for high-fidelity AVO: 74th Annual International Meeting SEG, Expanded Abstracts, Pèrez, G., and K. Marfurt, 6, Correcting for wavelet stretch in common-angle migration improves vertical and lateral resolution: 76th Annual International Meeting SEG, Expanded Abstracts, Roy, B., P. Anno, R. Baumel, and J. Durani, 5, Analytic correction for wavelet stretch due to imaging: 75th Annual International Meeting SEG, Expanded Abstracts, Roy, B., P. Anno, and M. Gurch, 6, Wide-angle inversion for density: test for heavy-oil reservoir characterisation: 76th Annual International Meeting SEG, Expanded Abstracts, Walden, A. T., 1991, Making AVO sections more robust: Geophysical Prospecting, 39, Xu, S., and D. Pham, 4, Seismic data regularization with anti-leakage Fourier transform: 66th Conference and Exhibition, EAGE, Extended Abstract, D-3 68

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