On The Impact of Wavefield Interpolation on Imaging Techniques and AVA Analysis Bin Liu and Mauricio D Sacchi* Department of Physics, University of Alberta, Edmonton, AB, T6G 2J1
[email protected] and Henning Kuehl Shell Canada, Calgary, AB ABSTRACT In 2D/3D survey, seismic data are often sparsely and irregularly sampled along spatial coordinates. This produces problems for multi-channel processing techniques such as migration which often requires regular and dense sampling. Wavefield interpolation can be used to produce densely and regularly sampled data, and therefore it is an effective mean to attenuate acquisition footprint. In this abstract, we study the impact of interpolation on the migration of angle gathers. The interpolation/resampling problem is posed as an inversion problem where from inadequate and incomplete data one attempts to recover the complete seismic wavefield. A minimum DFT-weighted norm interpolation (MWNI) method is used to interpolate prestack seismic data. The impact of the interpolation on migration and AVA analysis for 2D data has been tested with the Marmousi data set. The effectiveness of 3D MWNI scheme is illustrated on a real 3D common azimuth seismic data set. Introduction In 2D/3D survey, seismic data are often sparsely and irregularly sampled along spatial coordinates. This produces problems for multi-channel processing techniques such as suppression of coherent noise (multiple and ground roll) and migration which often require regular and dense sampling. Pre-stack interpolation of seismic traces is an effective mean to solve such problems. The interpolation/resampling problem can be posed as an inversion problem where from inadequate and incomplete data one attempts to recover the complete seismic wavefield. The problem, however, is often effectively under-determined which, as is well known, can be satisfied by many solutions. In this case, a regularized solution can be used where the regularizer serves to impose a particular feature on the solution (Cary, 1997; Duijndam et al., 1999; Hindriks and Duijndam, 2000; Liu and Sacchi, 2000, 2001; Sacchi and Ulrych, 1998). For example, a band-limited solution can be obtained using band-width constraints when we assume seismic data is band-limited in the spatial wavenumber domain (Duijndam et al., 1999; Hindriks and Duijndam, 2000); a regularizer imposed by
the Cauchy distribution can be used to obtain an estimation of the Fourier transform with sparse distribution of spectral amplitudes in the Fourier domain (Sacchi and Ulrych, 1998). In the method of weighted minimum norm interpolation (Liu and Sacchi, 2000, 2001), we have used an adaptive DFTweighted norm regularization term which incorporates a prior knowledge of energy distribution in wavenumber domain to regularize the solution. The technique is an optimal reconstruction method and is also time efficient since FFTs and an iterative solver like preconditioned conjugate gradient algorithm are used to accelerate the interpolation. In this abstract, we develop schemes to apply the adaptive weighted minimum norm algorithm to pre-stack interpolation in 2D and 3D. The impact of pre-stack interpolation on imaging techniques and AVA analysis is also examined. Adaptive weighted regularization The one-dimensional AWMNI algorithm (Liu and Sacchi, 2000, 2001) can be extended to higher dimensional algorithms using the properties of Kronecker product of matrices (Philip, 1971). We denote N-dimensional (ND) sampling problem as y = Tx , where T is the sampling matrix of the problem, y and x are the vectors of lexicographic ordering of the elements of ND observations and unknowns, respectively. The ND AWMNI solution can be obtained by minimizing following objective function 2 J ( x) = Tx − y 2 + µ 2 x T Qx where ⋅ stands l 2 norm, µ 2 is a specified weighting factor controls the trade off between the data norm and misfit of observations, and x T Qx defines the DFTT weighted norm. In specifics, Q = FND ΛFND , FND is a ND Fourier transform matrix and Λ is a diagonal matrix with entries 1 / p k 2 k ∈ Ω ND Λ kk = k ∉ Ω ND 0 where Ω ND denotes indexes for the pass-band of the data and Pk2 is a positive function which is similar in shape to the power spectrum of the ND unknown data in lexicographic form. The key to successful reconstruction is to define a DFTweighted norm that resembles the spectrum of the true solution. In particular, since the interpolation is always performed for each temporal frequency slice, we found that the DFT-weighted norm for the current interpolation can be derived from the wavenumber domain information obtained from previous frequency slices. Example: Marmousi shot reconstruction along two spatial coordinates The 2D AWMNI algorithm is demonstrated on Marmousi shot gathers. The original Marmousi shots and receiver sampling intervals are both 25 meters. We simulate a new survey with shot and receiver intervals increasing to 75 meters. A
comparison of original and new sampling geometry is shown in Figure 1a where the x-mark indicates the positions of available traces, whereas the dots indicate the position of the traces to be interpolated (Note that only part of the full geometry are shown). The known seismic traces are then used as the input for pre-stack interpolation along shot and receiver dimensions. We first perform Fourier transform along the time axis. Reconstructions are then carried at temporal frequencies along two spatial (shot and receiver) coordinates simultaneously. All the missing traces have been reconstructed. The details of the reconstruction at shot positions 3075m, 3100m and 3125m are shown in Figure 1b-1d. Figure 1b portrays incomplete shots, Figure 1c shows the reconstructed shot records, and, finally Figure 1d shows the reconstruction error. Next, we migrate the complete, incomplete and reconstructed datasets with splitstep AVA DSR Migration. Figures 2a and 2c show both the stacked image and the migrated CIG at CMP location 7500m using the three different datasets. The migration with complete dataset yields a good stacked and CIG images (in Figure 2a). The coarse sampling of the wavefield results in severely aliased events in both the stacked image and the CIG (in Figure 2b). In Figure 2c, migration with the reconstructed wavefield yields an overall better stacked image without any aliasing. The continuity of events in the CIG is also improved compared to Figure 2b. To study the impact of wavefield interpolation on AVA analysis, a depth point located in the upper half of model is chosen for AVA inversion so that we have sufficient ray parameter/angle range coverage. Note that the picked target phase in the CIG is indicated with an arrow in Figure 2. In Figure 3, the reflection coefficient based on the acoustic approximation shows an increasing trend with the angle of incidence on the theoretical AVA curve. Despite its roughness, the AVA curve picked on the migrated CIGs from complete wavefield agrees with the theoretical AVA trend. The AVA curve picked on the migrated CIG from the reconstructed wavefield (in Figure 3a) is much closer to the original one comparing to the one picked on the migrated CIG from incomplete wavefield (Figure 3b). Example: Reconstruction of 3D seismic data with a common offset azimuth We have applied the adaptive weighted minimum norm pre-stack interpolation to a 3D real seismic dataset. The complete dataset contains 150 inlines and 40 crosslines with offsets align along a common azimuth. The offsets are originally irregularly sampled for individual CMPs (shown in Figure 4a). They have been binned to a regular grid before interpolation. Figure 4b shows the sparse sampling geometry in cross-line midpoint-offset for inline#6 (only parts of the crossline midpoints are shown) and Figure 4c shows the spatial positions after interpolation. The pre-stack interpolation is simultaneously applied along three dimensions, namely: inline, crossline and offset. Figure 5a is the original CMPs for inline#6 and crossline#15-18. The reconstructed result is shown in Figure 5b. Note all the gaps have been filled.
The 3D common azimuth AVA migration with split-step DSR propagator is applied to both original and interpolated datasets. Figure 6a-6b show the migrated image for crossline #36 and inline #71 where the original dataset are used as the migration input. Note that the irregular and sparse sampling result in poor quality images dominated by artifacts. With the interpolated dataset as input, the migration result shows an important reduction of artifacts (Figure 6c6d).The improvement on the stacked image can also be seen from depth slices. In Figure 7, we show a depth slice at 1470m of the migrated image. Note that the sparse and irregular acquisition geometry lefts a strong footprint on the depth slice (Figure 7a) when the original dataset are used as migration input. When migration takes interpolated data as input, we observe that the resulting depth slice (Figure 7b) shows uniform amplitude characteristic and reduced acquisition footprints. The impact of interpolation can also been seen in the ray parameter CIG domain. The CIG gathers for crossline#36 and inline#71 are shown in Figure 8. The migration with interpolated data as the input yields a CIG gather with reduced noise and better event continuity. Conclusions In this abstract, we applied AWMNI to pre-stack seismic data interpolation before amplitude versus angle wave equation migration. The method enables us to incorporate both band-width and spectrum shape into the interpolation problem and, therefore, it often yields optimal reconstructions. The method is extremely efficient since all computations are done in the flight using FFTs and a preconditioned conjugate gradients algorithm. It is important to mention that the method can be applied to seismic data in any domain with multiple spatial coordinates. Finally, examples with synthetic and real 3D data show an important reduction of sampling artifacts both in the structural image and in individual AVA gathers. Acknowledgements The Signal Analysis and Imaging Group at the University of Alberta would like acknowledge financial support from the following companies: Encana, Geo-X Ltd., and Veritas Geoservices. This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada and AOSTRA/Alberta Department of Energy and by the Schlumberger Foundation. Field data has been generously provided by Veritas Geoservices. References Cary, P., 1997, High-resolution "beyond Nyquist" stacking of irregularly sampled and spare 3D seismic data: Annual Meeting Abstracts, CSEG, 66-70. Davis, Philip J., 1979, Circulant matrices. New York: Wiley. Duijndam, A.J.W., Schonewille, M., and Hindriks, K., 1999, Reconstruction of seismic signals, irregularly sampled along on spatial coordinate: Geophysics, 64, 524-538.
Hindriks, K. O. H., Duijndam, A.J.W., 2000, Reconstruction of 3-D seismic signals irregularly sampled along two spatial coordinates: Geophysics, 65 253256. Hugonnet, P., Herrmann, P. and Ribeiro, C., 2001, High Resolution Radon - a Review, 63rd Mtg.: Eur. Assn. of Expl. Geophys., Session: IM-2. Kuehl, H., 2002, Least-squares wave-equation migration/inversion: Ph.D. thesis, University of Alberta. Liu, B., and Sacchi, M.D., 2001, Minimum weighted norm interpolation of seismic data with adaptive weights: 71st Ann. Internat. Mtg., Soc. of Expl. Geophys., 1921-1924. Liu, B., and Sacchi, M.D., 2002, Minimum DFT-weighted norm interpolation of seismic data using FFT: Annual Meeting Abstract, CSEG. Sacchi, M.D., Ulrych, T.J., and Walker, C., 1998, Interpolation and extrapolation using a high-resolution discrete Fourier transform: IEEE Trans. Signal Processing, vol. 46, no. 1, 31-38. Zheng, Y., Gray, S., Cheadle, S., and Anderson, P., 2001, Factors affecting AVP analysis of prestack migrated gathers: 71st Ann. Internat. Mtg., Soc. of Expl. Geophys., 989-992. 0
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Fig. 1: (a) Source and receiver position map where the x-mark indicates the positions of available traces; the dots indicate the positions to be interpolated. (b) The incomplete shot gathers at 3075m, 3100m and 3125m. (c) The reconstructed shot gathers. (d) The reconstruction error.
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(c) Fig. 2: (a) Left: Migration of the Marmousi model. All wavefield data were used in the migration. Right: CIG at CMP location 7500m. (b) Left: Migration of the Marmousi model. The sparsely sampled wavefield (as shown in Fig. 1a) were used in the migration. Right: CIG at CMP location 7500m. (c) Left: Migration of the Marmousi model. The reconstructed wavefield data from the sampled wavefield were used in the migration. Right: CIG at CMP location 7500m.
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Fig. 3: (a) Picked AVA curve on migrated CIG using incomplete dataset is indicated with solid line, the dashed and dotted lines indict the picked AVA curve on migrated CIG using complete dataset and theoretical AVA curve respectively. (b) Picked AVA curve on migrated CIG using reconstructed dataset is indicated with solid line.
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Fig. 4: (a) The sparse and irregular sampling in midpoint and offset domain of the 3D common azimuth data. (b) Sampling in cross-line midpoint and offset for the original inline#6 (only part of the complete geometry is shown). (c) New sampling after interpolation.
(a) (b) Fig. 5: (a) The original traces in four CMPs corresponding to the inline No. 6 and crossline No. 15-18. (b) The reconstruction with 3D AWMNI.
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Fig. 8: (a) Ray parameter domain CIG at CMP location inline #71 crossline #36, the sparse dataset are used as the input for migration. (b) Ray parameter domain CIG at the same location where the interpolated dataset are used as the input for migration.
