Recent developments in curvelet-based seismic processing

Transcription

1 0000 Recent developments in curvelet-based seismic processing Felix J. Herrmann Seismic Laboratory for Imaging and Modeling Department of Earth and Ocean Sciences The University of British Columbia SUMMARY In this paper, we present recent developments in nonlinear curvelet-based sparsity-promoting formulations of problems in the seismic data processing flow. We present our latest work on a parallel curvelet transform and recent work on a curvelet-regularized formulation for the focal transform, the prediction of multiples and the computation of the inverse data space. We show that the curvelet s wavefront detection capability and invariance under wave propagation lead to a formulation of these problems that is stable under noise and missing data.

2 OUTLINE During this presentation an overview will be given of the latest developments concerning the application of the curvelet transform to seismic data processing. The following topics will be discussed extensively: the definition of a windowed parallel curvelet transform, which allows for the application of the curvelet transform to large data volumes (Thomson et al., 2006); combination of the curvelet transform with the (de)focal transform (= a data adaptive transform defined by a cross-convolution/correlation with the primaries) for the purpose of improved interpolation and multiple prediction (Berkhout and Verschuur, 2006) a curvelet-based formulation for the computation of the data inverse (Berkhout, 2006). Parallel Windowed FDCT Seismic data volumes are often too large to fit into the memory of single processor. At the expense of a low-frequency cutoff, a parallel curvelet transform can be constructed by applying a domain decomposition (see Fig. 1), implementing a partitioning of unity (the squares of the windows sum to 1). Consider equally sized windows with uniform overlap ɛ. The overlapping regions are tapered such that the energy of the system remains unity when points in the data set are duplicated at the overlaps. The tapering ensures that edges of the data goes smoothly to zero, eliminating potential edge artifacts. Once the data is split into overlapping windows and tapered the fast discrete curvelet transform (FDCT, Candes et al., 2006; Hennenfent and Herrmann, 2006), is performed on each window separately. The shape of the taper function and the overlap of neighboring windows are both shown in Fig. 1. The data in this example is split into sixteen overlapping windows. The dashed lines in the image represent the edges of the overlapping windows, with the region between nearby parallel lines being shared amongst the two windows. The taper function is shown, where one can see the value going to zero at the window edges.

3 THEORY This entire process of windowing can be considered as a linear operator acting on a vector containing the reordered to-be-transformed data volumes. For the parallel transform, it is useful to distinguish between global vectors and transforms (=matrix-vector multiplication), which comprise the entire data volume, and local vectors and transforms that exist only on individual nodes. Our notation will reflect this distinction. For instance, the distributed transform and vector C and x are distinguished from the local transform and vector C i and x i that exist on a node indexed by i. Note that there will be cases when we want to apply a local transform C i to the data on every node in a global vector x. This involves a block diagonal global operator where each block consists of the particular C i corresponding to a given x i, which is simply a subsection of x. We now consider the entire process in the context of linear operators We denote the block diagonal acting data vectors. global In the paralleloperator context, it is useful tofor distinguish a local operator as [C i ]. nodes. Our notation will reflect this distinction. For instance, the distributed data operator and vector A and x are expanded easily distinguished from into the overlapping sections An arbitrary global local operator and vector A i and x i that exist on a node indexed by i. using the global windowing operator W. The tapering operator T is then Note that there will be cases when we want to apply a local operator A i to the data on every node in a global vector x. This involves a block diagonal global operator where each block consists of the particular applied across the system. A i corresponding It tois a given easy x i, which is tosimply see a subsection that of x. We the tapering operator is denote the block diagonal global operator for a local operator as [A i]. diagonal, and will have An arbitrary repeating global data vector ispatterns expanded into overlapping in sections blocks representing each using the global windowing operator W. The tapering operator T is then applied across the system. It is easy to see that the tapering opera- node. These tapering operators is diagonal, and will have T i repeating arepatterns local in blockson representing each node. Once W and each node. Alternatively, one can simply look at the tapering operator T have been applied, the as a local linear T operator C i can be applied on each node. i on each node. Once W and T have been applied, any We propose a method to parallelize an arbitrary linear operator that avoids problems related to edge artifacts and preserves overall energy. It requires relatively little communication between parallel nodes, making it highly scalable. Our particular interest here is focused on a scalable parallel generalization of the FDCT, but we stress the fact that any linear operator could take the place of the FDCT in this framework. Parallel Windowed FDCT The basic structures of this framework are overlapping and tapered windows. This scheme has previously been used in various parallel processing applications (see e.g. J. Fan and Rector, 1997). The details of these structures are quite flexible. In general, the sizes of the windows and the overlapping regions do not have to be uniform throughout the system. Here, though, we will only consider equally sized windows with uniform overlap throughout the system. One can express the overlap between windows by the value ε, which represents the depth to which one window receives adjacent data from its neighbours (Mallat, 1998), as illustrated in Figure 1. The overlapping regions of the windows are tapered such that the energy of the system remains constant when points in the data set are duplicated due to the overlaps. The tapering also ensures that edges of the data goes smoothly to zero at the edges, eliminating the potential of creating edge artifacts. The taper is applied across the outer region of the overlapping windows, affecting points that are within a distance of 2ε from an edge. The tapering function b must satisfy the relationship b 2 + b 2 = 1, (1) where b and b signify the tapers in adjacent windows that cover the same region. For consistency, we consider the application of an identical tapering function b to all windows. It follows from Eq. 1 that b 2 n + b 2 2ε n+1 = 1, (2) where n is an integer on the interval [1,2ε], and subject to the boundary conditions b 1 = 0 and b 2ε = 1. This condition ensures that the energy of the system is conserved when summing values from adjacent windows that represent the data at the same location. There are many examples of functions that satisfy this relationship (Mallat, 1998), the simplest being: ( ) (n 1)π b n = sin. (3) 2(2ε 1) To taper the end of the data set along each dimension, the function b is translated to the points n = {N,N 1,, N 2ε +1}, where N is the total size of the overlapping windows. The tapering function is then ( ) (N n)π b n = sin 2(2ε 1). (4) Once the data is split into overlapping windows and tapered, operations, like the FDCT in our particular case, can be performed on each window independently. The shape of the taper function and the overlap of neighbouring windows are both shown in Figure 1. The data in this case is split into sixteen overlapping windows, four horizontally and four vertically. The dashed lines in the image represent the edges of the overlapping windows, with the region between nearby parallel lines being shared between the two windows. The taper function is shown, where one can see the value going to zero at the window edges. Since the image and all windows are square in this case, the taper functions along the vertical axis are the same as the horizontal ones that are shown. Also, note that tapering is not done at the edges of the image, since this would remove energy from the system. This issue can be successfully dealt with in a number of ways, but we omit this discussion. between global vectors and operators, which comprise the entire parallel system, and local vectors and operators that exist only on individual arbitrary linear operator A i can be applied on each node. Figure 1: 2D synthetic seismic data with vertical traces. Figure 1: 2D synthetic seismic data with 30% missing vertical traces. Dotted lines Dotted lines represent borders of overlapping windows. Taper function is shown above for clarity. represent borders of overlapping windows. Taper function is shown above for clarity. Perhaps the most useful part of considering windowing and tapering as linear operators in matrix form is that looking at these matrices makes understanding the adjoints of these operations simple. Since T is diagonal, it is its own adjoint. For W, the adjoint operation involves summing together overlapping regions. Since the forward operation was a scatter, the adjoint becomes a gather, where data that correspond to the same point are summed (Claerbout, 1992). In a parallel computing realization, this means that a given node sends its outer band to the nodes to which they belong, then gathers data related to its inner band from those same neigbours, and sums them together. Importantly, the combination of these processes satisfies the relation Since T is diagonal, it is its own adjoint. For W, the adjoint operation involves summing together overlapping regions. Since the forward operation was a scatter, the adjoint becomes a gather, where data that correspond to the same point are summed (Claerbout, 1992). In a parallel computing realization, this means that a given node sends its outer W H T H TW = I, (5) band to the nodes to which they belong, then gathers data related to its

4 inner band from those same neigbours, and sums them together. The combination of these processes satisfies the relation W H T H TW = I, (1) which ensures perfect reconstruction of the data after applying the operators followed by their adjoints, i.e. our parallel curvelet transform is a tight frame. This implies that energy is preserved through the entire process. So, a new transform, that we will call the Parallel Windowed FDCT (PWFDCT), can thus be defined. The parallel curvelet transform corresponds to a block-diagonal global operator that is defined by applying the FDCT, C i, on each node independently after applying the global windowing and tapering operators T and W. In other words, C := [C i ]TW. (2) Since the local C is numerically tight, it follows that C H C = I. (3) Other properties of C are similarly shared by C. It is possible, then, to make use of the PWFDCT in existing algorithms that include the FDCT. It should be noted, though, that curvelets at very large scales (or, equivalently, low frequencies) are not represented in the same way they would be if a single FDCT were performed on the global data. Since the benefits of curvelets are mostly found at the finer scales (higher frequencies), the lack of large scale curvelet representation is not a problem. This makes a wide variety of algorithms capable of handling data sets much larger than otherwise possible. An example of a seismic data interpolation algorithm where the PWFDCT takes the place of the FDCT is included in Fig. 2. From the results presented in that figure one can conclude that algorithms, such as curvelet-based sparsity-promoting data recovery, can readily be scaled up by using our parallel windowed curvelet transform based on a partition of unity. EXTENSIONS OF THE CURVELET TRANSFORM So far, curvelets have mainly found their applications in the approximation of migration operators (Douma and de Hoop, 2006; Chauris, 2006), amplitude recovery (Herrmann et al., 2006) and seismic data interpolation (Herrmann and Hennenfent, 2007) and separation (Herrmann

5 evident near window borders, as would be expected. In Figure 1, we show a subsection of the full output of interpolation runs using overlapping and non-overlapping windows. The same subsection is used for both, and contains one full window and parts of its nearest neighbours. In the example using non-overlapping windows (Figure 2(a)), the edges of the windows are obvious from the artifacts that appear. When overlapping windows are used (Figure 2(b)), the artifacts are no longer evident, and it is not clear at all where the window borders are. In essence, the interpolation performs just as well in the proximity of the window edges as it does in the middle of the window, which is clearly untrue for non-overlapping windows. The importance of overlapping and tapering is thus, as expected, clear from this example. We could potentially hold in memory. In order to run many algorithms in parallel, it is not sufficient to process data in separate pieces. At the same time, these same algorithms are often not scalable in their normal form due to exponential growth in the amount of communication between nodes that they require. For these reasons, we have developed a scheme that involves overlapping, tapered data windows that can be processed in parallel that is highly scalable since the communication costs do not grow with the number of processing nodes. We have used this method to define the PWFDCT, but we stress that this method is general and that the FDCT can be replaced by an arbitrary operator acting on each overlapping window independently as desired. We applied the PWFDCT to a seismic (a) data interpolation algorithm that is shown, in another presentation to this conference, to be successful using the FDCT. We have demonstrated that good results can be achieved with the PWFDCT in this algorithm, and shown the importance of the overlapping and tapering. Furthermore, we have demonstrated that it is possible to correct for the effect of tapering on threshold values in the transform domain. Much future work is expected to arise from the ideas described herein. In particular, we will apply the PWFDCT to other seismic data processing algorithms, enabling them to work on much larger data sets than is currently feasible. The parallelization method described here will also be applied to other operators besides the FDCT. Finally, we hope that this method will open up new possibilities in parallel signal processing in a variety of fields. ACKNOWLEDGMENTS The authors would like to thank the authors of the Fast Discrete Curvele Transform (FDCT) for making this code available at This work was in part financially supported by NSERC Discovery Grant 22R81254 of Felix J. Herrmann and was carried out as part of the SINBAD project with support, secured through ITF (the Industry Technology Facilitator), from the following organizations: BG Group, BP, Chevron, ExxonMobil and Shell. Much future work is expected to arise from the ideas described herein. In particular, we will apply the PWFDCT to other seismic data processing algorithms, enabling them to work on much larger data sets REFERENCES than is currently feasible. The parallelization method described here will also be applied to other operators besides the FDCT. Finally, we Claerbout, J. F., 1992, Earth soundings analysis: Processing versus hope that this method will open up new possibilities in parallel signal inversion: Blackwell Scientific publishing. processing in a variety of fields. Daubechies, I., M. Defrise, and C. de Mol, 2004, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint: Comm. Pure Appl. Math., ACKNOWLEDGMENTS Gupta, A. and V. Kumar, 1993, The scalability of FFT on parallel computers: IEEE Transaction on Parallel and Distributed Systems. (a) (b) The authors would like to thank the authors of the Fast Discrete Curvelet Hennenfent, G. and F. J. Herrmann, 2006, Application of stable seismic signal recovery to seismic interpolation. (submitted for pre- (a) Transform (FDCT) for making(b) this code available at Figure 2: Interpolation output for (a) non-overlapping windows and This work was in part financially supported by NSERC Discovery sentation at 76th SEG Conference & Exhibition). for (b) overlapping, tapered windows with threshold correction. Arrows indicate locations of artifacts in (a) and show the improvement in Grant 22R81254 of Felix J. Herrmann and was carried out as part of Herrmann, F. J. and G. Hennenfent, 2006, Non-parametric seismic the SINBAD project with support, secured through ITF (the Industry data recovery with curvelet frames. (to be submitted). Figure 2: Interpolation output for (a) non-overlapping (b). Technology Facilitator), windows from the following and organizations: for (b) BG Group, overlapping, J. Fan, K.T. Nihei, L. M. N. C. and J. Rector, 1997, Overlap domain BP, Chevron, ExxonMobil and Shell. decomposition method for wave propagation: Presented at the 67th also compared different methods for correcting the threshold value in SEG Conference & Exhibition. tapered windows with threshold correction. Arrows indicate locations of artifacts in (a) the curvelet domain to account for the taper. When the taper was not L. Ying, L. D. and E. Candès, 2005, 3D discrete curvelet transform. taken REFERENCES into account in thresholding, errors were evident in the overlapping region. When correcting threshold values by using a Monte Carlo submitted for publication. and show the improvement in (b). Mallat, S., 1998, A wavelet tour of signal processing: Academic Press. sampling Claerbout, or by J. F., evaluating 1992, Earth the taper soundings function analysis: at curvelet Processing centroids, versus artifacts inversion: related toblackwell erroneousscientific thresholding publishing. are eliminated. We omit the figures Daubechies, demonstrating I., M. Defrise, this dueand to space C. de limitations. Mol, 2004, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint: Comm. Pure Appl. Math., et al., 2007). During these applications, Gupta, A. andthe V. Kumar, invariance 1993, The scalability of FFTand on parallel computers: IEEE Transaction on Parallel and Distributed Systems. compressibility of curvelets is used. The curvelet mic signal properties recovery to seismic interpolation. can (submitted alsoforbe pre- used to (b) Hennenfent, G. and F. J. Herrmann, 2006, Application stable seis- Figure 2: Interpolation output for (a) non-overlapping windows and sentation at 76th SEG Conference & Exhibition). for (b) overlapping, tapered windows with threshold correction. Arrows seismic indicate locations of data artifacts in (a) recovery and show the improvement with in the data recovery focal with curvelet transform frames. (to be submitted). (Berkhout and Herrmann, F. J. and G. Hennenfent, 2006, Non-parametric seismic improve (b). J. Fan, K.T. Nihei, L. M. N. C. and J. Rector, 1997, Overlap domain Verschuur, 2006), multiple prediction decomposition method for wave propagation: Presented the 67th also compared different methods for correcting the threshold value in SEG and Conference the & Exhibition. computation of data inverse (Berkhout, taken into account in thresholding, 2006). errors werein evidenthis in the overlap- case, use is made of the property that submitted for publication. the curvelet domain to account for the taper. When the taper was not L. Ying, L. D. and E. Candès, 2005, 3D discrete curvelet transform. ping region. When correcting threshold values by using a Monte Carlo Mallat, S., 1998, A wavelet tour of signal processing: Academic Press. sampling or by evaluating the taper function curvelet centroids, ar- remain related to erroneous relatively thresholding are eliminated. sparse We omit theon data that is focused, defocused or curveletstifacts figures demonstrating this due to space limitations. inverted. Focused data in this context corresponds to data stripped of one propagation path interacting with the surface, while defocused data has a single propagation path added, mapping primaries into first-order multiples. The latter corresponds to SRME multiple predicted (Verschuur and Berkhout, 1997), while the former corresponds to the recently introduced focal transform (Berkhout and Verschuur, 2006). In all above described manifestations of seismic data, we are dealing with data that contains wavefronts whether these are primaries, multiples or the data inverse. Curvelets compress data with wavefronts and therefore serve as a regularization, where the primary operator, adjoint primary operator and data operator can stably be inverted with a curvelet sparsity norm. These different data manifestations can be calculated by solving of the following norm-one nonlinear program: P ɛ : { x = arg minx x 1 s.t. Ax y 2 ɛ f = ST x (4) in which y is the (incomplete) data, A the synthesis matrix and S T is the inverse sparsity transform and ɛ, a noise-dependent tolerance level. This constrained optimization problem is solved to within ɛ.

6 Recovery from incomplete data by focusing Combination of the non-adaptive curvelet transform with the data-adaptive focal transform (Berkhout and Verschuur, 2006) gives rise to an interesting extension, where data is focused by inverting the primary operator (= a multidimensional convolution with the primaries). In the focal domain, data is sparser and hence the curvelet coefficients are sparser, which leads to an improved recovery. Recovery with focusing is possible by defining the synthesis matrix as A := R PC T and the inverse sparsity transform as S T := PC T. The operator P stands for a multidimensional convolution with the primaries. The data vector y contains the incomplete measurements, y = Rd with R the restriction matrix that selects the rows from the curvelet matrix that correspond to active traces. The solution yielded by P ɛ corresponds to a curvelet-regularized inverse of the primary operator. The estimated coefficients represent an estimate for the focused data. The synthesis operator defocuses the focused data back to data domain. The results as shown Fig 3 show as expected an improvement over interpolation without focusing. Multiple prediction with defocusing By defining the synthesis matrix by the adjoint of the primary operator, the solution of P ɛ, with the data y containing the data including multiples, yields an estimate for the predicted multiples, i.e. by setting S T = C T. This estimate corresponds to the inverse of the adjoint of the primary operator, i.e., A := P T C T. As can be observed from Fig. 4, the frequency content of the predicted multiples is increased by this prediction through curvelet-regularized inversion. Computation of the data inverse The previous two examples showed that a curvelet regularization for the inversion of the primary operator and its inverse improves the stability and yields better results for the multiple prediction. Similarly, the computation of the data inverse can benefit from the same regularization by solving P ɛ for the synthesis matrix, A := PC T with P the multidimensional convolution with the total data, and the data vector, y = δ, with δ the discrete delta-function. As the example in Fig. 5 illustrates, the

7 (a) (b) (c) Figure 3: Comparison between curvelet-based recovery by sparsity-promoting inversion with and without focusing. (a) Randomly subsampled data with 80 % of the traces missing. (b) Curvelet-based recovery. (c) Curvelet-based recovery with focusing. Notice the significant improvement from the focusing with the primary operator. SMRE predicted multiples CSRI predicted multiples (a) (b) Figure 4: Comparison between convolution-based multiple prediction (a) and sparsity-based multiple prediction (b).

8 (a) (b) Figure 5: Computation of the data inverse. (a) shot record in the data space. (b) shot record in the inverse data space computed according to P ɛ. solution of P ɛ yields a stable estimate for the data inverse. DISCUSSION AND CONCLUSIONS Applications of the curvelet transform to seismic data processing bank on two favorable properties of curvelets, namely their ability to detect wavefronts and their approximate invariance under wave propagation. In this paper, we showed some exciting recent developments of this transform towards a large-scale parallelization and a combination with the physics of wave propagation. The successful application of curvelets, juxtaposed by sparsity-promoting inversion, opens a range of new perspectives on seismic data processing, wavefield extrapolation and imaging. Because of their singular wavefront detection capability, curvelets represent in our vision an ideal domain for future developments in exploration seismology. ACKNOWLEDGMENTS: The author would like to thank the authors of CurveLab for making their codes available, Darren Thomson for helping to implement the parallel curvelet transform and Eric Verschuur for his input. The examples presented in this paper were prepared with Madagascar (rsf.sourceforge. net/). SAGA and ExxonMobil are thanked for making test datasets available. This work was in part financially supported by the Natural Sciences and Engineering Research Council of Canada Discovery

9 Grant (22R81254) and Collaborative Research and Development Grant DNOISE ( ) of Felix J. Herrmann and was carried out as part of the SINBAD project with support, secured through ITF (the Industry Technology Facilitator), from the following organizations: BG Group, BP, Chevron, ExxonMobil and Shell. REFERENCES Berkhout, A. J., 2006, Seismic processing in the inverse data space: Geophysics, 71. Berkhout, A. J. and D. J. Verschuur, 2006, Focal transformation, an imaging concept for signal restoration and noise removal: Geophysics, 71. Candes, E. J., L. Demanet, D. L. Donoho, and L. Ying, 2006, Fast discrete curvelet transforms: SIAM Multiscale Model. Simul., 5, Chauris, H., 2006, Seismic imaging in the curvelet domain and its implications for the curvelet design: Presented at the 76th Ann. Internat. Mtg., SEG, Soc. Expl. Geophys., Expanded abstracts. Claerbout, J. F., 1992, Earth soundings analysis: Processing versus inversion: Blackwell Scientific publishing. Douma, H. and M. de Hoop, 2006, Leading-order seismic imaging using curvelets: Presented at the 76th Ann. Internat. Mtg., SEG, Soc. Expl. Geophys., Expanded abstracts. Hennenfent, G. and F. J. Herrmann, 2006, Seismic denoising with nonuniformly sampled curvelets: IEEE Comp. in Sci. and Eng., 8, Herrmann, F. J., U. Boeniger, and D.-J. E. Verschuur, 2007, Nonlinear primary-multiple separation with directional curvelet frames: Geoph. J. Int. To appear. Herrmann, F. J. and G. Hennenfent, 2007, Non-parametric seismic data recovery with curvelet frames. Submitted for publication. Herrmann, F. J., P. P. Moghaddam, and C. Stolk, 2006, Sparsety- and continuity-promoting seismic imaging with curvelet frames. In revision. Thomson, D., G. Hennenfent, H. Modzelewski, and F. Herrmann, 2006, A parallel windowed fast discrete curvelet transform applied to seismic processing: Presented at the SEG International Exposition and 76th Annual Meeting. Verschuur, D. J. and A. J. Berkhout, 1997, Estimation of multiple scat-

10 tering by iterative inversion, part II: practical aspects and examples: Geophysics, 62,