and its implications for the curvelet design Hervé Chauris, Ecole des Mines de Paris Summary This paper is a first attempt towards the migration of seismic data in the curvelet domain for heterogeneous background velocity models. We first explain how to build a simple curvelet decomposition/reconstruction code, based on the use of Fast Fourier Transforms (FFTs). It is directly related to ideas from Candès (Candès et al., 2005), but appears to be slightly more general. We then show how to derive simple operations (image shift, rotation and stretch) directly in the curvelet domain. The basic transforms are needed for subsequent seismic migration. We demonstrate how to interpolate in the curvelet domain, using the Shannon interpolation formula. It appears that the interpolation scheme imposes some specific conditions on the shapes of the filters used for the curvelet decomposition. In particular, the transform has to be redundant. Finally, we combine the basic operations to migrate seismic data in the curvelet domain. The 2-D common offset section is first decomposed into curvelet coefficients. All coefficients are map-migrated independently using the notion of direction inherent to curvelets. The final depth section is obtained by performing an inverse curvelet transform. The curvelet migration is applied on a very simple background velocity model but for a complex reflectivity structure. The results are as good as for the Kirchhoff migration, but are obtained by map-migrating the curvelet coefficients instead of smearing energy along isochrones. In previous papers (Candès and Demanet, 2005; Douma and de Hoop, 2005), the authors had the hope that it would be possible to only consider after mapmigration the nearest coefficient in the curvelet domain. We show that this approach unfortunately does not provide in practice the expected quality for the result. An interpolation in the curvelet domain as developed here is indeed necessary. Introduction Curvelets can be seen as an extension of wavelets (Mallat, 1989; Daubechies, 1992; Meyer, 1993) for multidimensional data. They recently appeared, keeping the multi-resolution and localization aspects of the wavelets (Candès and Donoho, 2002) and (Do, 2001; Do and Vertelli, 2003). They were initially designed for (nonseismic) image compression and denoising, whenever the data contains some geometrical structures (Candès and Donoho, 2002). Both wavelets and curvelets have the same following characteristics: Multi-resolution (from coarse to fine resolution); Localization in spatial and frequency domains; Critical sampling for the fast discrete transform (Candès et al., 2005; Do and Vetterli, 2005). The key difference between the two families is that only curvelets are really directional: the basis curvelet functions are elongated, the width being proportional to the square of the length at the fine resolution (Candès et al., 2005) (Fig. 1). Fig. 1: Example of a tiling of the Fourier space (left: for scale 3, right: for scale 4). In both cases, the number of directions is 64. Fig. 2: Example of a curvelet for direction #11 and scale #4 (left: in the spatial domain, right: in the frequency domain). In this particular case, the curvelet has a sharp support in the Fourier domain. From a geophysical point of view, curvelets can be seen as local plane waves (Fig. 2) characterized in 2-D by two positions, a direction and a central frequency (Candès et al., 2005). They seem to be very attractive for seismic imaging, at least at first glance, for two reasons: they provide a efficient decomposition of local seismic events and they are almost invariant under the migration operation (Candès and Donoho, 2002; Candès and Demanet, 2005; Herrmann, 2003; Douma and de Hoop, 2005). 2406
In order to process curvelets for seismic imaging, we first have to better understand how to build curvelets, and then how to perform some simple operations (shift, rotation and stretch) in the curvelet domain. In a second phase, the combination of these basic operations will indeed allow for migration, at least for simple background velocity models where the propagation of a local plane wave does not induce a significant curvature of the wavefront. The curvelet construction The idea of the curvelet construction proposed here is directly inspired from the work of (Candès et al., 2005). In our transformation, we have a greater flexibility for the design of the filters. The main differences will be underlined below. We deal with the 2-D space, where the vertical axis is denoted by z and the horizontal one by x. We first build a tiling of the 2-D Fourier space. Let us define Ŵi(j, k) some positive functions for i [1 : N]. They can be seen as 2-D filters. In the curvelet construction, the Ŵi functions have a local support ( wedge ) and a specific shape (see for example Fig. 1). Their definition is specified below. The only additional restriction is that the sum ŝ[j, k] = N i=1 Ŵi[j, k] 2 is non zero for all (j, k). In (Candès et al., 2005), ŝ = 1, which is not needed here. For efficiency reasons, the Fourier transforms are applied on restricted windows. This implies to define an appropriate phase correction. The decomposition is formulated as follows: Fourier transform of the image: ˆf = FFT2(f) For all i [1 : N], Filtering of the image (windowed for efficiency) ĝ i(j, k) = Ŵi(j, k) ˆf(j, k) Inverse Fourier transform of the filtered image: c i = IFFT2(ĝ i) The reconstruction scheme is defined as follows: For all i [1 : N] Fourier transform of the coefficients: ĉ i = FFT2(c i) Filtering of the image (still windowed) ĥ i(j, k) = Ŵi(j, k) ĉi(j, k) ĉ i are the curvelet coefficients associated to Ŵi. We have a perfect reconstruction (without any iteration scheme). In the example of the figure 1, the redundancy factor (ratio between the number of coefficients and the size of the input image) is equal to 3.55 for n x = n z = 256. In conclusion, we have defined a general scheme for data decomposition and reconstruction, with a very large choice for the shape of the filters Ŵ i. Among all these possibilities, curvelets have elongated shapes, but we still have some flexibility for their design. We will now see that some restrictions have to be imposed on the shape of the filters to process an image in the curvelet domain. Interpolation in the curvelet domain If some processing is applied in the curvelet domain, like a morphing of an image, the curvelet coefficients have to be interpolated. A linear interpolation scheme does not provide a good result. We test here the more general Shannon interpolation scheme. Basic operations for the curvelet migration are shift, rotation and stretch (Chauris et al., 2002; Douma and de Hoop, 2005). We give an example how to perform a rotation in the curvelet domain. After rotation, a single coefficient in the curvelet domain will be shifted to a new position and to a new direction. We thus need to interpolate between positions and between directions. As illustrated on figure 3, the Shannon interpolation between directions is not always possible due to the applicability of the Shannon theorem (large enough sampling rate). Sum of all intermediate results: ˆp(j, k) = N ĥi(j, k) i=1 Normalization by ŝ: ˆq(j, k) = ˆp(j, k)/ŝ(j, k) Fig. 3: Rotation of a curvelet. Top left: single curvelet, top right: rotation in the curvelet domain, bottom right: rotation in the spatial domain, bottom left: differences. The Shannon sampling theorem is violated outside the circle (bottom left). Inverse Fourier transform: q = IFFT2(ˆq) 2407
Migration in the curvelet domain Fig. 4: Same legend as for figure 3, but with different filters. The zone where the Shannon sampling theorem is violated is contained in the support of the curvelet. In an homogeneous background velocity model (v = 2000 m/s), we generated with the ray+born formalism and the original Marmousi reflectivity (Versteeg and Grau, 1991), a 2-D zero-offset seismic section. The data is decomposed into curvelet coefficients. Each coefficient is mapmigrated independently and interpolated in the curvelet domain. Using the direction information contained in the curvelet construction (Chauris et al., 2002; Douma and de Hoop, 2005), we do not have to smear energy along isochrones but we can directly use the specular conditions to migrate the data by shifting and interpolating the curvelet coefficients. After processing, an inverse curvelet transform is applied to get the final depth migrated image. The curvelet migration image (Fig. 6, middle) shows a very similar result compared to the classical Kirchhoff migration result (Fig. 6, bottom). There is a clear need for interpolation (Fig. 6, top). This illustrates the possibility to perform seismic migration directly in the curvelet domain, at least for an homogeneous background model, by simply combining shift, rotation, stretch and interpolation of the curvelet coefficients. Conclusions Fig. 5: Rotation of a seismic section (left) by an arbitrary angle of 23.5 deg (right). The rotation is performed in the curvelet domain. To solve this issue, some conditions are imposed on the shape of the filters Ŵi. After some formal computations, we derived a valid interpolation scheme between directions by ensuring that the zone where the Shannon formula is not applicable, is contained in the support of the curvelet in the spatial domain: in other words, either the Shannon formula is valid, or the energy of the signal is null. Thus, the shape of the filter cannot be arbitrarily chosen to perform a rotation in the curvelet domain. On figure 3, the Shannon interpolation scheme is only applicable inside the circle, showing large residuals outside. The residuals are removed on figure 4 as the zone outside the circle is contained in the support of the function. This guarantees a proper rotation for all coefficients (Fig. 4 and 5). Similar results are obtained for the shift and for the interpolation between scales in the curvelet domain. We have explained how to build a generic and efficient 2- D curvelet decomposition/reconstruction scheme. As in (Candès et al., 2005), it is based on filtering the data in the Fourier domain. In order to perform simple operations in the curvelet domain (shift, rotation and stretch), the curvelet coefficients have to be interpolated. We propose to use here the Shannon interpolation scheme. Its applicability imposes some constraints on the shape of the filters, leading to redundant curvelet transforms. Based on these results, we combined shift, rotation and stretch to perform migration in the curvelet domain, showing nice results on an homogeneous background velocity model and a complex reflectivity. Despite an easier implementation, it is not possible to only use the nearest curvelet coefficient for the map-migration. A proper interpolation is crucial for the quality of the final result. It becomes now possible to study the effect of the curvature that may be introduced during the wave propagation in heterogeneous background velocity models. Acknowledgments We would like to thank Emmanuel Candès (Caltech) for providing us with his curvelet transform used for comparison. We would like to thank Shell E&P for partly funding the project and for permission to publish this work. We would also like to thank Fons ten Kroode (Shell 2408
E&P), Mark Noble and Podvin Podvin (Ecole des Mines de Paris) for fruitful discussions. References Candès, E., and Demanet, L., 2005, The curvelet representation of wave propagators is optimally sparse: Commnications on Pure and Applied Mathematics, pages 1 55. Candès, E., and Donoho, D., New tight frames of curvelets and optimal representations of objects with C 2 singularities:, Technical report, Caltech, 2002. Candès, E., Demanet, L., Donoho, D., and Ying, L., Fast discrete curvelet transform:, Technical report, Caltech, 2005. Chauris, H., Noble, M. S., Lambare, G., and Podvin, P., 2002, Migration velocity analysis from locally coherent events in 2-D laterally heterogeneous media, Part I: Theoretical aspects: Geophysics, 67, no. 04, 1202 1212. Daubechies, I., 1992, Ten lectures on wavelets: SIAM. Do, M. N., and Vertelli, M., 2003, Contourlet in Beyond wavelets: G. V. Welland, Academic Press. Do, M. N., and Vetterli, M., 2005, The contourlet transform: an efficient multiscale directional multiresolution image representation: To appear in IEEE Transactions on Image Processing. Do, M. N., 2001, Directional multiresolution image representations: Ph.D. thesis, Swiss Federal Institute of Technology Lausanne. Douma, H., and de Hoop, M. V., 2005, On common-offset prestack time migration with curvelets: 72 nd Annual SEG Meeting and Exposition, Soc. Expl. Geophys., Expanded Abstracts. Herrmann, F., 2003, Optimal imaging with curvelets: 70 th Annual SEG Meeting and Exposition, Soc. Expl. Geophys., Expanded Abstracts. Mallat, S., 1989, A theory for multiresolution signal decomposition: the wavelet representation: IEEE Pattern Anal. and Machine Intell., 11, 674 693. Meyer, Y., 1993, Wavelets: algorithms and applications: SIAM. Versteeg, R. J., and Grau, G., Eds., 1991, The Marmousi experience, Proceedings of the 1990 EAEG workshop on Pratical Aspects of Seismic Data Inversion, Eur. Ass. Expl. Geophys. Fig. 6: Migration results of the zero-offset ray+born Marmousi data set, generated in an homogeneous model and with the original Marmousi reflectivity. Top: migration in the curvelet domain without interpolation, middle: migration in the curvelet domain with interpolation, bottom: Kirchhoff migration. 2409
EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2006 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 Candes, E., and L. Demanet, 2005, The curvelet representation of wave propagators is optimally sparse: Communications on Pure and Applied Mathematics, 1 55. Candes, E., L. Demanet, D. Donoho, and L. Ying, 2005, Fast discrete curvelet transform: Caltech. Candes, E., and D. Donoho, 2002, New tight frames of curvelets and optimal representations of objects with C2 singularities: Caltech. Chauris, H., M. S. Noble, G. Lambare, and P. Podvin, 2002, Migration velocity analysis from locally coherent events in 2-D laterally heterogeneous media, Part I: Geophysics, 67, 1202 1212. Daubechies, I., 1992, Ten lectures on wavelets: SIAM. Do, M. N., 2001, Directional multiresolution image representations: Ph.D. thesis, Swiss Federal Institute of Technology Lausanne. Do, M. N, and M. Vertelli, 2003, Contourlet in G. V. Welland, Beyond wavelets: Academic Press. Douma, H., and M. V. de Hoop, 2005, On common-offset prestack time migration with curvelets: 72nd Annual International Meeting, SEG, Expanded Abstracts, 2009 2012. Herrmann, F., 2003, Optimal seismic imaging with curvelets: 70th Annual International Meeting, SEG, Expanded Abstracts, 997 1000. Mallat, S., 1989, A theory for multiresolution signal decomposition: the wavelet representation: IEEE Pattern Analysis and Machine Intelligence, 11, 674 693. Meyer, Y., 1993, Wavelets: Algorithms and applications: Versteeg, R. J., and G. Grau, 1991, The Marmousi experience: Proceedings of the EAEG workshop on Pratical Aspects of Seismic Data Inversion, ESAG. 2410