Progress Report on: Interferometric Interpolation of 3D SSP Data

Progress Report on: Interferometric Interpolation of 3D SSP Data Sherif M. Hanafy ABSTRACT We present the theory and numerical results for interferometrically interpolating and extrapolating 3D marine SSP data. For the interpolation of SSP data we use the combination of a natural Green s function (SSP shot gathers) and a model-based Green s function for the water-layer model. Synthetic results show that the aliased SSP data with large receiver intervals can be accurately interpolated to smaller intervals. The resulting virtual shot gather contains some artifacts, a 2D multi-channel local image matching filter is used after interferometric interpolation to remove these artifacts. SSP data extrapolation also employs a natural Green s function and a model-based Green s function. This suggests that a sparse marine SSP survey can yield much more information about the reflectors if data are interpolated and extrapolated by interferometry. This assumes that the sources are located both far outside and within the recording aperture. Limitations of this theory are: interpolated traces are limited to 4 virtual traces between each two real ones and with extrapolation, near offsets virtual traces show some artifacts. INTRODUCTION Typical marine surface seismic profiles (SSP) are ideally designed for a regular recording grid, but in reality it suffers from irregularities in the recording geometry. Coarse receiver spacing and narrow recording apertures are common problem in marine acquisition, especially in the crossline direction. The result can be inadequate subsurface illumination and distortions in the migration image. To alleviate this problem, various algorithms were suggested to fill in the missing traces in marine data (Abma and Kabir, 2006; van Dedem and Verschuur, 2005; Muijs et al., 2007). Most of them require certain assumptions 27 such as, the data have sparse representation in a certain domain, or rely on a priori information, such as knowledge of seismic velocities (Baumstein et al., 2005). In all cases, there is no use of the redundant information available in the free-surface multiples. In this paper we show how to interpolate marine SSP data by interferometry. This method transforms surface and seabed related multiples into primaries recorded at virtual receiver both inside and outside the receiver array. In this proposed technique, no velocity model for the sediments is needed. The interpolated data can be used for migration, velocity analysis, and tomography. The multiples also can be used to illuminate much wider areas of the subsurface. This work is the 3D extension of the 2D interferometric method of Dong and Hanafy (2008) This paper is divided into three parts. First, the theory of interferometric interpolation and extrapolation is presented. Then, tests are carried out on 3D synthetic data generated from a multi-layer model. Finally, a brief conclusion and discussion is given. THEORY Wang et al. (2009) applied the interferometric interpolation method to surface seismic profile (SSP) data. In that method, the surface related multiples are used to predict missing traces in gaps. Sparse SSP data can also be interpolated where an SSP SSP correlation transform is used. This transform is similar in spirit to the VSP VSP correlation transform (Schuster, 2009), except now we use a two-layer model-based Green s function with a band limited point source, as shown in Figure 1. The diagrams show how SSP traces (with both sources and receivers just below the free surface) are correlated with sparsely distributed SSP traces to generate a dense distribution of SSP traces. This correlation operation is required by the acoustic reciprocity equation of correlation type for a two-state system, where one state is the acoustic field associated with the multi-layered model shown in

28 Hanafy Figure 1 and the other state is associated with a sea-floor model. The reciprocity equation of correlation type can be described as (Appendix A): G(B A) G 0 (A B) = [G 0 (x B) G(x A) (1) S s n x G(x A) G 0(x B) n x ]d 2 x, where, G 0 (x B) is the Green s function for a water layer model and G(x A) is the Green s function for the actual earth model (Dong and Hanafy, 2008). Here, S s is the boundary just below the sea surface and the integration along the free surface vanishes because both Green s functions are zero there. The contributions from the vertical boundaries at infinity to the left and to the right of the boat will be ignored (Wapenaar and Fokkema, 2006). Here, A and B are just below the free surface and below the horizontal source line S o The above equation is a reciprocity equation of correlation type for two different states, which can be used for interpolation or extrapolation of traces. The far-field approximation to equation 1 yields the SSP SSP far-field transform SSP {}}{ G(B A) = 2ik S s SSP SSP SSP {}}{{}}{{}}{ G(x A) G 0 (x B) dx 2 + G 0 (A B), (2) To implement this equation, the SSP data are used to estimate G(x A) and a finite-difference solution to the wave equation is used to estimate G 0 (x B) for the ocean-layer model that only contains the free surface and ocean bottom interfaces. This FD calculation is possible because the sea floor topography is well known beneath any exploration survey. The key idea for interpolation is that the free surface acts as a perfectly reflecting mirror so that 2 nd and 3 rd views, i.e., free-surface related multiples, of the subsurface can be used to fill in the trace gaps, as indicated by Figure 1. NUMERICAL TEST Multi-layer Velocity Model To test the proposed approach, a 3D acoustic multi-layer velocity model (Figure 2) is generated and finite-difference (FD) solutions of the acoustic wave equation are used to simulate SSP traces in that model. The velocity model is 3000 x 3000 x 1400 m 3 with a 300 by 300 receiver points at receiver interval of 10 m in both X and Y directions. The water layer has a thickness of 750 m and the water bottom is a flat layer. At this stage, one common shot gather is generated, where the source is located at location (10,10,30) m for x, y, and z directions. In the following section we show the 2D test results and then the 3D results. Details for implementing the 2D interpolation and extrapolation method are given in Dong and Hanafy (2008) 2D Interpolation Results A 2D line is extracted from the CSG s to test the proposed approach. The extracted 2D line is located at X = 10 m and has 300 traces in the Y direction with a trace interval of 10 m. Figure 3a shows the original CSG and Figure 3b shows the sparse version of the extracted CSG, where only 75 traces are kept with a trace interval of 40 m. Equation 2 is used to interpolate the missing traces and Figure 3c shows the virtual CSG after one iteration before application of the matching filter. To eliminate the artifacts generated during interpolation process we use the following 2-steps; Use a 2D multi-channel local image matching filter. This matching filter can be repeated several times to minimize the artifacts. Repeat the interpolation - matching filter process several times, where the input CSG to each new iteration is the virtual CSG from the previous iteration. In this work we used 8 interpolation iterations, each interpolation iteration is followed by 10 iterations of matching filter. The final virtual CSG is shown in Figure 3d. Comparing Figures 3a and 3d shows a very good correlation between the virtual CSG and the original CSG. Artifacts at the far offsets are not completely eliminated due to the paucity of true traces at the far offsets that could be used for interpolation. 3D Interpolation Results The proposed approach is tested on a 3D CSG, where Figure 4a shows a part of the original CSG. The sparse CSG contains 60 parallel lines with 100 receivers per line (Figure 4b). The inline offset is 30 m and the crossline offset is 50 m. The sparse version of the CSG contains 6000 traces, and our target is to interpolate missing traces and get a virtual CSG with 10 m offsets in both inline and crossline directions. Equation 2 is used to interpolate the missing traces, where we used 3 interpolation iterations, each interpolation iteration is followed by 8 iterations of matching filter. Figure 4c shows the final virtual CSG with matching filter. Comparing the original (Figure 4a) and virtual (Figure 4c) CSGs shows a very good correlation. SEG/EAGE Velocity Model Another velocity model, the SEG/EAGE velocity model, is used to test the proposed approach. In this test we simulated a near azimuth marine survey with 8 streamers each one has 170 receiver. The crrossline interval is 30 m and the inline interval is 12 m. Near offset is at 0 m, while far offset is at 2028 m. Our target is to interpolate virtual traces in both the inline and crossline directions,

3D SSP Interpolation 29 so that the final results has 22 streamers and 805 receiver per streamer. The cross line interval is 10 m and the inline interval is 4 m. Figure 5 shows the SEG/EAGE velocity model we used in this test, Figure 6a shows a sample of the input CSG (1 st and 2 nd streamers), Figure 6b shows the virtual CSG (1 st to 4 th streamers), and Figure 6c shows the real CSG for comparison. Here, the artifacts are highly eliminated and the virtual CSG (Figure 6b) shows a very good correlation with the real CSG (Figure 6c). CONCLUSIONS An interferometric method for interpolating and extrapolating marine SSP data is tested on synthetic multi-layer and SEG/EAGE velocity models. The results show that this method can kinematically interpolate the sparse SSP data to a dense receiver distribution data and can widen the receiver aperture significantly. The least squares image matching filter is shown to suppress artifacts and can partly correct for amplitudes. No velocity model is required to apply this approach, however, the thickness of the water layer and the velocity of seismic wave in the water is required. The interpolated data can be used for migration, velocity analysis, and tomography. The multiples also can be used to illuminate much wider areas of the subsurface. The virtual CSGs are very good correlated with the real CSGs. With interpolation, the number of receivers can be increased 4 times. We do not recommend increase the number of receivers more than 4 times, since we got more artifacts with increasing number of virtual traces. The extrapolation is in principle similar to the interpolation, our extrapolated traces (not included in this work) show higher artifacts at the near offset zone. A future challenge is improve the extrapolated traces and eliminate the artifacts. ACKNOWLEDGMENTS We would like to thank the members of the 2008 University of Utah Tomography and Modeling/Migration (UTAM) Consortium (http://utam.gg.utah.edu). APPENDIX A ACOUSTIC RECIPROCITY EQUATION OF CORRELATION TYPE WITH TWO STATES Consider two states, one is the acoustic field associated with the multi-layered model shown in Figure 1 where G(x A) is interpreted as the acoustic wave field excited by an interior harmonic point source at A and recorded at x. And the other state is the acoustic field in the seafloor model, which only consists of a water layer, a free surface, and a sea floor below which is a homogeneous medium with the same velocity and density of layer 1. The Green s function associated with this new state is defined as G 0 (x B), and does not contain reflections from any interface below the sea floor. The Helmholtz equations satisfied by these two Green s functions are ( 2 + k 2 )G(x A) = δ(x A), (A-1) ( 2 + k 2 0)G 0 (x B) = δ(x B), (A-2) where k = ω/v(x) for the multi-layered model and k 0 = ω/v 0 (x) for the sea-floor model. The Laplacians can be weighted by Green s functions to give the following identities: G(x A) 2 G 0 (x B) = [G(x A) G 0 (x B)] G(x A) G 0 (x B), G 0 (x B) 2 G(x A) = [G 0 (x B) G(x A)] G 0 (x B) G(x A). (A-3) Instead of defining the integration volume over the entire multi-layered model the volume is restricted to the ocean layer where both the sea-floor model and multi-layer model agree. Subtracting and integrating the above equations over the ocean volume yields the reciprocity equation of correlation type for two different states: G(B A) G 0 (A B) = [G 0 (x B) G(x A) S s n x G(x A) G 0(x B) ]d 2 x, (A-4) n x where S s is the boundary along the sea floor and the integration along the free vanishes because both Green s functions are zero there. The contributions from the infinite vertical boundaries to the left and right of the boat will be ignored. Equation A-4 is a reciprocity equation of correlation type for two different states, which can be used for the interpolation or extrapolation of traces. REFERENCES Abma, R. and N. Kabir, 2006, 3d interpolation of irregular data with a pocs algorithm: Geophysics, 71, E91 E97. Baumstein, A., M. T. Hadidi, L. Hinkley, and W. S. Ross, 2005, A practical procedure for application of 3d srme to conventional marine data: TLE, 254 258. Dong, S. and S. M. Hanafy, 2008, Interferometric interpolation and extrapolation of sparse obs data: UTAM Annual Meeting, 7 21. Muijs, R., J. O. A. Robertsson, and K. Holliger, 2007, Prestack depth migration of primary and surfacerelated multiple reflections: Part 1 - imaging: Geophysics, 72, S59 S69. Schuster, G. T., 2009, Seismic interferometry: Cambridge Press, under press. van Dedem and D. Verschuur, 2005, 3d surface related multiple prediction: A sparse inversion approach: Geophysics, 70, V31 V43. Wang, Y., Y. Luo, and G. Schuster, 2009, Interferometric interpolation of missing seismic data: Geophysics, in press.

30 Hanafy Figure 1: Ray diagrams for transforming SSP data to SSP data. Here, the open geophones indicate the locations of virtual geophones where traces are created from the original SSP data recorded at the filled geophone positions. Both G(x A) and G o (x B) can be data based Green s functions, but in this case G o (x B) is model-based and so is computed for the two-layer sea-floor model. Figure 2: The 3D multi-layer velocity model used to test interpolation technique

3D SSP Interpolation 31 Figure 3: The 2D test results. a) The original common shot gather with a receiver interval of 10 m. b) The sparse SSP CSG used for the interpolation test, where the receiver interval is 40 m. c) The interpolation results before using the matching filter, where the interpolated data have a receiver interval of 10 m. d) The interpolation results after using the matching filter. Most artifacts are removed and the wavelet is consistent with the true data.

32 Hanafy Figure 4: The 3D test results. a) One of the original common shot gathers, here the receiver interval is 10 m. b) The sparse SSP CSG used for the interpolation test, where 60 parallel lines with 100 receivers per line are used. The inline receiver interval is 30 m, the crossline spacing is 50 m, and total number of receivers in the input data are 6000. c) The interpolation results after using the matching filter. Most artifacts are removed and the wavelet is consistent with the true data. The virtual CSGs contains 300 lines with 300 receiver per line. The inline receiver interval is 10 m, the crossline spacing is 10 m, and total number of receivers in the output data are 90,000.

3D SSP Interpolation 33 Figure 5: The SEG/EAGE velocity model used to test interpolation technique Wapenaar, K. and J. Fokkema, 2006, Green s function representations for seismic interferometry: Geophysics, 71, S133 S146.

34 Hanafy Figure 6: The 1 st and 2 nd streamers of the original common shot gather, here the receiver interval is 12 m, streamers offset is 30 m, and each streamer has 170 receivers.

3D SSP Interpolation 35 Figure 7: The interpolation results of the SEG/EAGE velocity model. a) The interpolation results after using the matching filter. Most of the artifacts are removed and the wavelet is consistent with the true data. The virtual CSGs contains 22 crosslines with 508 receiver per crossline. The inline receiver interval is 4 m, the crossline spacing is 10 m, and total number of receivers in the output data are 11,176. b) Synthetically generated CSG with the same geometry as in (a).