Multi-source Least-squares Migration of Gulf of Mexico Data

Transcription

1 Multi-source Least-squares Migration of Gulf of Mexico Data Xin Wang. King Abdullah University of Science and Technology, Thuwal , Kingdom of Saudi Arabia Corresponding author is Xin Wang. address: ABSTRACT Kirchhoff based multisource least-squares migration (MSLSM) is applied to marine data recorded in the Golf of Mexico. Results show that MSLSM images are of better quality than standard Kirchhoff migration, especially at depth level less than.8 km. The computational cost of MSLSM is about the same as conventional least-squares migration, but its IO cost is significantly decreased. MSLSM, which can be applied to marine streamer data without any restrictions on the acquisition geometry. This paper is organized into the following three sections. The first part presents the theory of MSLSM. The next section presents synthetic and field data results that demonstrate the efficiency and effectiveness of the MSLSM algorithm. Finally, a summary is provided. THEORY The phase-encoded multisource data (i.e., supergather) can be represented as (Dai et al., ) INTRODUCTION It has been shown that least-squares migration (LSM) (Nemeth et al., 999; Duquet et al., ) can improve the resolution of the migration images and suppress the migration artifacts. However, one of the drawbacks of LSM is its high computational cost. Romero et al. () proposed a blended source method by encoding and stacking different shot gathers into a supergather. The blended source data was formed by phase encoding each shot gather and stacking the shot gathers together to get a supergather. Dai et al. () adapted LSM to blended source data, which I now define as the multisource least-squares migration (MSLSM) procedure. This algorithm is applicable to Kirchhoff migration (Dai et al., ), wave-equation migration (Huang and Schuster, ) and reverse time migration (Dai et al., ). Schuster et al. () provide rigorous formulas for predicting the level of crosstalk noise as a function of the encoding parameters. Application of MSLSM to marine streamer data is hampered by the mismatch of receiver position and receiver number for different shots for wave-equation migration or reverse time migration. Huang and Schuster () proposed a frequency division multiplexing strategy with multisource least-squares wave-equation migration. However, this procedure is not required for Kirchhoff based 5 d = P i d i, () i= where d is an M vector defined as a supergather data, d i is the M vector for the ith shot gather, S is the number of shots, and the M matrix P i represents the phase-encoding function. It is typically a scaled identity matrix, where scaling function is proportional to e iωτ for the traces shifted by τ (Schuster et al., ). The P i can also account for random polarity value of ±. It is shown in Schuster et al. () that the combination of random polarity changes and random time shifts is more effective at reducing crosstalk noise than using any of them alone. Each trace of the supergather is a superposition of traces from a number of shots, which introduces the crosstalk noise. A receiver-side phase-encoding function is applied to each trace of the shot gathers. As shown in Figure, two shot gathers with different recording aperture (b and c) are encoded by a receiver-side encoding function (d-g) to form a supergather (h). Since the recording aperture is known, these two shot gathers can be decoded correctly (i and j). We assume that the ith CSG d i and the reflectivity model m are related by d i = L i m, ()

2 5 Wang et. al where L i is the linear forward modeling operator associated with the ith shot. Plugging equation () into (), we get d = P i L i m = Lm, () i= where the supergather modeling operator is defined as L = P i L i. () i= Multisource least-squares migration To suppress crosstalk noise, we solve equation in the least-squares sense. That is, define the objective function as f(m) = d Lm + λ m m apr, (7) so that, an optimal m is sought to minimize the objective function in equation (8). The second term is the regularization (Tikhonov and Arsenin, 977), and λ is the damping parameter. The optimal model m can be found by a gradient type optimization method Multisource migration From equation (), the supergather migration operator is defined as the adjoint of the supergather modeling operator, L T = L T i P T i. (5) i= Figure depicts key steps for applying multisource Kirchhoff migration to marine streamer data. Two distinct shot gathers with different recording apertures (Figure a) are first phase-encoded (Figure b) then stacked to generate a supergather (Figure c). In Kirchhoff migration, each trace is decoded for the correct shot and receiver locations before migration. For example, if a trace is encoded by time shifting s, then that trace is decoded by a - s time shift (Figure d). In Figure e, the event in the blue solid circle is smeared to the model space (solid blue ellipse) with the traveltime from associated source and receiver of CSG, while the event in the red solid circle is smeared with the traveltime from associated source and receiver of CSG. Events with dashed line are smeared with dashed ellipse. Therefore, the solid lines indicate the migration artifacts and the dashed lines indicate the crosstalk noise. The supergather migration image is S m mig = L T d = L T P i L i m = = = L T j P T j j= i= j= i= i= P i L i m i= L T j P T j P i L i m standard mig crosstalk {}}{{}}{ L T i L i m + L T j P T j P i L i m, (6) j i i= consisting of two terms: the first term is the standard migration image and the second term is the crosstalk noise introduced by multisource blending of shot gathers. The magnitude of the crosstalk term for a variety of different encoding functions is derived in Schuster et al. (). m (k+) = m (k) αf(l T (Lm (k) d) + λm (k) ), (8) where L T (Lm (k) d) + λm (k) is the gradient, F is a preconditioning matrix and α is the step length. As both the forward modeling and migration operators are linear and adjoint to each other, the analytical step length formula can be used. Alternatively, in order to improve the robustness of the MSLSM algorithm, a quadratic line search method is carried out with the current model and two trial models. In this study, we use the conjugate gradient (CG) method for static encoding (keep the encoding function the same for all iterations), steepest descent (SD) method for dynamic encoding (change encoding function for each iteration), and SD+CG method (reset the conjugate direction at each re-encoding iteration) for hybrid encoding (change the encoding function every few iterations). NUMERICAL EXAMPLE The Kirchhoff-based MSLSM algorithm is tested on both synthetic and field data. The goal is to compare the quality of the MSLSM against that of the KM image. Synthetic Data Test Synthetic data are generated by a finite-difference solution to the acoustic wave equation for the D Marmousi model (Figure ) with the marine streamer acquisition geometry. The model size is, and the grid interval is m, with 9 shot gathers having a shot interval of 6 m, and each shot gather is recorded with a km long streamer with receivers. A Ricker wavelet with a peak frequency of Hz is used as the source wavelet. Figure shows the standard migration image and Figure 5 shows the least-squares migration image after 5 iterations. Three zoom views are shown in Figures 6-Figure 8. Compared with standard Kirchhoff migration, LSM decreases the migration artifacts (especially in the shallow depth), balances the amplitudes, and increases the resolution. For the MSLSM, 6 supergathers are generated with the receiver side encoding function (both polarity and time delay encoding), and each supergather consists of stacked shot gathers. Figure 9 shows the MSLSM result with static encoding after 5 iterations. Figure shows the MSLSM result with hybrid encoding (change the encoding

3 MSLSM 55 function for every 5 iterations) after iterations. Figure shows the MSLSM result with dynamic encoding after iterations. Figures show the same zoom view for these three different encoding functions. Among the three images with different encoding function, the dynamic encoding result show the best quality image which is close to that of single source LSM. The second best is the one generated by MSLSM with hybrid encoding. If IO cost is considered, then static encoding is the most efficient, and second best is the hybrid encoding. Table compares the costs of these methods. LSM and MSLSM are compared at 5 iterations, the CPU, IO and memory cost for KM are defined as.. LSM I does not store data in the memory, but LSM II does. For MSLSM, supergather data are small enough to be stored in the memory. GOM Data Test A Gulf of Mexico dataset is tested with MSLSM. The reflectivity model size is, with a grid size of 6.5 m, and Figure 5 shows the P-velocity model estimated from full waveform inversion (Boonyasiriwat et al., ). There are 96 shots with a shot interval of.5 m, and each shot has a streamer length of 6 km with a receiver interval of.5 m. For each shot the nearest offset is m. A Hz bandpass filter is applied to the raw data, and the source wavelet is estimated by stacking the near-offset ocean-bottom reflections. Figure 6 shows the standard migration image and Figure 7 shows the least-squares migration image after iterations. Thirty two supergathers are generated from the total of 96 shot gathers, and each supergather consists of 5 or 6 shots. Figure 8 shows the multisource least-squares migration image with dynamic encoding after 5 iterations. Two detailed areas are shown in Figures 9 - Figure. Numerical results shows that both LSM and MSLSM can achieve better image quality compared to standard Kirchhoff migration. The difference image is shown in Figure and reveals the high-frequency details ignored by standard KM. CONCLUSIONS A multisource least-squares migration algorithm is proposed to efficiently produce high quality images. A receiverside and source-side phase-encoding function is used to generate the supergathers. This algorithm with Kirchhoff migration method can easily be applied to marine steamer data. Synthetic and field results show that both LSM and MSLSM can decrease migration artifacts, balance the amplitudes and increase the image resolution. MSLSM also has a anti-aliasing filter built in. Compared with single source LSM, MSLSM can significantly decrease the IO cost and memory cost, but not the CPU cost. Numerical results also suggest that dynamic encoding achieves the best image quality but at the highest IO cost. Static encoding incurs the least IO cost but suffers from more crosstalk noise. The performance of Hybrid encoding lies Table : Comparison of image quality and CPU, IO and memory cost among KM, LSM and MSLSM. Image quality CPU cost IO cost Memory cost KM... LSM I LSM II Static MSLSM Hybrid MSLSM Dynamic MSLSM in between, while the performance of hybrid encoding lies in between. Applying this method to D GOM data is now an active line ACKNOWLEDGEMENT We thank the CSIM members for supporting this research. REFERENCES Boonyasiriwat, C., G. T. Schuster, P. Valasek, and W. Cao,, Applications of multiscale waveform inversion to marine data using a flooding technique and dynamic early-arrival windows: Geophysics, 75, R9 R6. Dai, W., C. Boonyasiriwat, and G. T. Schuster,, Three dimensional multi-source least-squares reverse time migration: SEG Technical Program Expanded Abstracts, 9,. Dai, W., X. Wang, and G. T. Schuster,, Leastsquares migration of multisource data with a deblurring filter: Geophysics, 76, R5 R6. Duquet, B., K. J. Marfurt, and J. A. Dellinger,, Kirchhoff modeling, inversion for reflectivity, and subsurface illumination: Geophysics, 65, Huang, Y. and G. T. Schuster,, Multisource least-squares migration of marine streamer and land data with frequency-division encoding: Geophysical Prospecting, accepted. Nemeth, T., C. Wu, and G. T. Schuster, 999, Leastsquares migration of incomplete reflection data: Geophysics, 6, 8. Romero, L. A., D. C. Ghiglia, C. C. Ober, and S. A. Morton,, Phase encoding of shot records in prestack migration: Geophysics, 65, 6 6. Schuster, G. T., X. Wang, Y. Huang, W. Dai, and C. Boonyasiriwat,, Theory of multisource crosstalk reduction by phase-encoded statics: Geophysical Journal International, 8, 89. Tikhonov, A. and V. Arsenin, 977, Solutions of ill-posed problems: V.H. Winston and Sons.

4 56 Wang et. al Theory of Phase encoding (b) CSG (d) Polarity change (f) Time delay (i) Decoded CSG (a) Marine aperture CSG CSG (h) Supergather (c) CSG (e) Polarity change (g) Time delay (j) Decoded CSG Figure : Steps for creating a phase-encoded supergather and its decoding. (a) show the receiver and shot position associated with two shot gather, (b), (d) and (f) show the original, polarity encoded and time delay encoded CSG, while (c), (e) and (g) are for CSG; (h) is the supergather after summing of (f) and (g); and (i) and (j) are the decoded CSGs for migration. Shift back Figure : Illustration of steps for multisource Kirchhoff migration. (a) shows the single shot CSGs with a marine streamer acquisition aperture, receiver position are different for each shot, (b) shows phase-encoding which shifts CSG s later, (c) show the supergather data generated by stacking traces from CSG and CSG, (d) shows the decoded shot, which shift the supergather trace s back and (e) shows the migration of the decoded CSGs with the associated source and receiver positions.

5 MSLSM 57 Velocity Model km/s Figure : Marmousi P-velocity model. Kirchhoff Migration Image 6 9 Figure : Standard Kirchhoff migration image. Three boxes indicates the areas for zoom view. Least squares Migration Image after 5 Iterations 6 9 Figure 5: Least-squares migration image after 5 iterations. Three boxes indicates the areas for zoom view.

6 58 Wang et. al (a) Velcocity (b) Reflectivity (c) Kirchhoff Migration (d) Least squares Migration Figure 6: Zoom in of red box in Figure and Figure 5. (a)-(d) shows the velocity, reflectivity, standard migration and least-squares migration. 5 (a) Velcocity 5 (b) Reflectivity (c) Kirchhoff Migration (d) Least squares Migration Figure 7: Zoom in of yellow box in Figure and Figure 5. (a)-(d) shows the velocity, reflectivity, standard migration image and least-squares migration image.. (a) Velcocity. (b) Reflectivity (c) Kirchhoff Migration (d) Least squares Migration Figure 8: Zoom in of blue box in Figure and Figure 5. (a)-(d) shows the velocity, reflectivity, standard migration image and least-squares migration image.

7 MSLSM 59 Multisource LSM Static Encoding after ites 6 9 Figure 9: Multisource least-squares migration image of 6 supergathers after iterations with static phase-encoding. Multisource LSM Hybrid Encoding after ites 6 9 Figure : Multisource least-squares migration image of 6 supergathers after iterations with hybrid phase-encoding. The encoding function changes every 5 iterations. Multisource LSM Dynamic Encoding after ites 6 9 Figure : Multisource least-squares migration image of 6 supergathers after iterations with dynamic phase-encoding.

8 6 Wang et. al (a) LSM after 5 ites (b) MSLSM Static after ites (c) MSLSM Hybrid after ites (d) MSLSM Dynamic after ites Figure : Zoom in of red box in Figure 5, Figure 9, Figure and Figure.(a)-(d) shows the LSM, static MSLSM, hybrid MSLSM and dynamic MSLSM images. 5 (a) LSM after 5 ites 5 (b) MSLSM Static after ites (c) MSLSM Hybrid after ites (d) MSLSM Dynamic after ites Figure : Zoom in of yellow box in Figure 5, Figure 9, Figure and Figure.(a)-(d) shows the LSM, static MSLSM, hybrid MSLSM and dynamic MSLSM images.. (a) LSM after 5 ites. (b) MSLSM Static after ites (c) MSLSM Hybrid after ites (d) MSLSM Dynamic after ites Figure : Zoom in of blue box in Figure 5, Figure 9, Figure and Figure.(a)-(d) shows the LSM, static MSLSM, hybrid MSLSM and dynamic MSLSM images.

9 MSLSM 6 Velocity Model km/s Figure 5: Velocity model of Gulf of Mexico data. Kirchhoff Migration Image Figure 6: Standard Kirchhoff migration image. Two boxes indicates the areas for zoom view.

10 6 Wang et. al Least squares Migration Image after ites Figure 7: Least-squares migration image after iterations. Three boxes indicates the areas for zoom view. MSLSM Dynamic after 5 ites Figure 8: Multisource least-squares migration image of supergathers after 5 iterations with dynamic phase-encoding.

11 MSLSM 6.9 (a) KM (b) LSM after ites (c) MSLSM Dynamic after 5 ites Figure 9: Zoom in of red box in Figure 6, Figure 7 and Figure 8. (a)-(c) shows the KM, LSM and dynamic MSLSM images.

12 6 Wang et. al (a) KM...8 (b) LSM after ites...8 (c) MSLSM Dynamic after 5 ites...8 Figure : Zoom in of yellow box in Figure 6, Figure 7 and Figure 8. (a)-(c) shows the KM, LSM and dynamic MSLSM images. (a) Difference image between KM and LSM. (b) Difference image between KM and MSLSM. Figure : Difference image between of zoom in of red box in Figure 6, Figure 7 and Figure 8. (a) shows the difference image between KM and LSM, and (b) shows the difference image between KM and MSLSM.