Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data

Transcription

1 Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data Mostafa Naghizadeh and Mauricio D. Sacchi University of Alberta, Department of Physics Edmonton, Alberta, Canada Submitted to GEOPHYSICS on December 009 ABSTRACT We propose a robust interpolation scheme for aliased regularly sampled seismic data that uses the curvelet transform. In a first pass, the curvelet transform is used to compute the curvelet coefficients of the aliased seismic data. The aforementioned coefficients are divided into two groups of scales: alias-free and alias-contaminated scales. The alias-free curvelet coefficients are upscaled to estimate a mask function that is used to constrain the inversion of the alias-contaminated scale coefficients. The mask function is incorporated into the inversion via a minimum norm least squares algorithm that determines the curvelet coefficients of the desired alias-free data. Once the aliasfree coefficients are determined, the curvelet synthesis operator is used to reconstruct seismograms at new spatial positions. The proposed method can be used to reconstruct both regularly and irregularly sampled seismic data. We believe that our exposition leads to a clear unifying thread between f-x and f-k beyond-alias interpolation methods and curvelet reconstruction. Likewise in f-x and f-k interpolation, we stress the necessity of examining seismic data at different scales (frequency bands) in order to come up with viable and robust interpolation schemes. Synthetic and real data examples are used to illustrate the performance of the proposed curvelet interpolation method. INTRODUCTION Interpolation and reconstruction of seismic data has become an important topic for the seismic data processing community. It is often the case that logistic and economic constraints dictate the spatial sampling of seismic surveys. Wave-fields are continuous; in other words, seismic energy reaches the surface of the earth everywhere in our area of study. The process of acquisition records a finite number of spatial samples of the continuous wave field generated by a finite number of sources. This leads to a regular or irregular distribution of sources and receivers. Many important techniques for removing coherent noise and imaging the earth interior have stringent sampling requirements which are often not met in real surveys. In order to avoid information losses, the data should be sampled according to the Nyquist criterion (Vermeer, 990). When this criterion is not honored, reconstruction can be used to recover the data to a denser distribution of sources and receivers and mimic a properly sampled survey (Liu, 00). The final result of the reconstruction stage could have a significant impact on subsequent seismic processing steps such as noise removal (Soubaras, Curvelet Interpolation

2 00), AVO analysis (Sacchi and Liu, 00; Hunt et al., 008), and imaging (Liu and Sacchi, 00). Methods for seismic wave field reconstruction can be classified into two categories: waveequation based methods and signal processing methods. Wave-equation methods utilize the physics of wave propagation to reconstruct seismic volumes. In general, the idea can be summarized as follows. An operator is used to map seismic wave fields to a physical domain. Then, the modeled physical domain is transformed back to data space to obtain the data we would have acquired with an ideal experiment. It is basically a regression approach where the regressors are built based on wave equation principles (in general, approximations to kinematic ray theoretical solutions of the wave equation). The methods proposed by Ronen (987), Bagaini and Spagnolini (999), Stolt (00), Trad (00), Fomel (00), Malcolm et al. (00), Clapp (006) and Leggott et al. (007) fall under this category. These methods require the knowledge of some sort of velocity distribution in the earth s interior (migration velocities, root-mean-square velocities, stacking velocities). While reconstruction methods based on wave equation principles are very important, this paper will not investigate this category of reconstruction algorithms. Seismic data reconstruction via signal processing approaches is an ongoing research topic in exploration seismology. During the last decade, important advances have been made in this area. Currently, signal processing reconstruction algorithms based on Fourier synthesis operators can cope with multidimensional sampling as demonstrated by several authors (Duijndam et al., 999; Liu et al., 00; Zwartjes and Gisolf, 006; Schonewille et al., 009). These methods are based on classical signal processing principles and do not require information about the subsurface. However, they utilize specific properties of seismic data as apriori information for interpolation purposes. In addition, most of these methods are quite robust in situations where the optimality condition under which they were designed are not completely satisfied (Trad, 009). Signal processing methods for seismic data reconstruction often rely on transforming the data to other domains. The most commonly used transformations are the Fourier transform (Sacchi and Ulrych, 996; Sacchi et al., 998; Duijndam et al., 999; Liu et al., 00; Xu et al., 00; Zwartjes and Gisolf, 006), the Radon transform (Darche, 990; Verschuur and Kabir, 99; Trad et al., 00), the local Radon transform (Sacchi et al., 00; Wang et al., 009) and the curvelet transform (Hennenfent and Herrmann, 008; Herrmann and Hennenfent, 008). Another group of signal processing interpolation methods rely on prediction error filtering techniques (Wiggins and Miller, 97). Spitz (99) and Porsani (999) introduced beyond-alias seismic trace interpolation methods using prediction filters. These methods operate in the frequency-space (f-x) domain. In both cases, low frequency data components in a regular spatial grid are used to estimate the prediction filters needed to interpolate high frequency data components. An equivalent interpolation method in the frequencywavenumber (f-k) domain was introduced by Gulunay (00) and often referred as f-k interpolation. We finally point out that interpolation methods that rely on t-x domain prediction filters were investigated by Claerbout (99), Crawley (000) and Fomel (00). The main contribution of this paper is the introduction of a strategy that utilizes the curvelet transform to interpolate regularly sampled aliased seismic data. It is important to stress that the curvelet transform has been used by Hennenfent and Herrmann (007, 008) and Herrmann and Hennenfent (008) to interpolate seismic data. In their articles, they reported the difficulty of interpolating regularly sampled aliased data with the curvelet transform and therefore, proposed random sampling strategies to circumvent the

3 aliasing problem. In this paper, however, we propose a new methodology which successfully eliminates the requirement of randomization to avoid aliasing. We create a mask function from the alias-free curvelet scales (low frequencies) to constrain the interpolation of aliascontaminated scales (high frequencies). The proposed method is an attempt to utilize early principles of f-x and f-k domain beyond alias interpolation methods (Spitz, 99; Gulunay, 00) in the curvelet domain. In summary, by carefully understanding well-established beyond alias interpolation methods (Spitz, 99; Gulunay, 00) in conjunction with novel signal processing tools like the curvelet transform (Candes et al., 00), we were able to develop an algorithm capable of reconstructing aliased regularly sampled data. Alias-free (coarse) scales can drive the inversion of fine scale curvelet coefficients because we assume that the local dip of a seismic event is constant for all frequencies. In other words, we assume non-dispersive seismic events. In addition, the proposed algorithm can deal with the simpler problem of regularization of randomly sampled data where the alias footprint is annihilated by random sampling. In view of the fact that the curvelet transform is a local transformation, the proposed algorithm can easily cope with strong variations of dips. This is a clear advantage with respect to reconstruction via Fourier bases (Duijndam et al., 999; Liu, 00; Liu et al., 00; Zwartjes and Gisolf, 006; Trad, 009) and prediction filtering techniques (Spitz, 99; Naghizadeh and Sacchi, 00) that assume data composed of a superposition of a few plane waves. The latter requires spatial and temporal windowing to satisfy the plane wave assumption. One needs to stress, however, that efficient multidimensional algorithms based on Fourier bases and prediction filtering are already part of current industrial methodologies to reconstruct data that depend on and spatial dimensions (Schonewille et al., 00; Abma and Kabir, 006; Schonewille et al., 009; Trad, 009). Embarking on multidimensional reconstruction by means of curvelet basis functions could be extremely difficult as, to our knowledge, current practical implementations of multidimensional curvelet transforms are limited to dimensions (Ying et al., 00). Furthermore, curvelet transform reconstruction algorithms appear, at present time, not to be amenable of a fast implementation such as the one encountered in D spatial Fourier reconstruction algorithms that solely rely on the Fast Fourier Transform (Liu, 00; Trad, 009). We stress that we are not attempting to contrast curvelet based reconstruction with well-tested efficient multidimensional Fourier reconstruction algorithms. Our exposition mainly aims at presenting an interesting strategy for reconstructing aliased data using the curvelet transform. It is clear that an important amount of work is needed to implement curvelet regularization strategies that can be used to regularize D spatial data. THEORY The curvelet transform The curvelet transform is a local and directional decomposition of an image (dat into harmonic scales (Candes and Donoho, 00). The curvelet transform aims to find the contribution from each point of data in the t-x domain to isolated directional windows in the f-k domain. If we assume that m(t, x) represents seismic data in the t-x domain, we define a set of functions ϕ(s, θ, t, x), where s indicates scale (increasing from coarsest to finest), θ, is angle or dip and t 0, x 0 are the t-x location parameters. These functions, known as curvelets, are used to decompose the original data in local components of various scales

4 and dip. Curvelets can be considered as wavelets with the additional important property of directionality (dip). The continuous curvelet transform can be represented as the inner product of the data m(t, x) and the curvelet function c(s, θ, t 0, x 0 ) = C[m] = t x m(t, x) ϕ(s, θ, t 0 t, x 0 x)dtdx. () In our study we will adopt the discrete curvelet transform (Candes et al., 00). The latter is an efficient implementation of the above equation where the number of coefficients varies with scale and direction. This leads to a smaller number of curvelet coefficients in the coarser scales with respect to the finer scales. The mathematical basics of curvelets and discrete curvelet transforms are fully developed and discussed in Candes et al. (00) and Candes and Donoho (00). Our algorithm was developed utilizing the curvelet transform provided by the package CurveLab. In the rest of this article, we refer to the discrete curvelet transform as curvelet transform. Figure a shows the partitioning of the f-k domain adopted by the curvelet transform. This example contains 6 scales represented by the co-centric squares (in this case half of the f-k plane since we assume real t-x signals/images). Except for scale (coarsest scale), the rest of scales are divided into smaller windows each representing a specific direction. The coarsest scale of the curvelet transform does not have directional properties. Notice that the finest scale (scale 6 in this example) covers 7% of the f-k domain. For completeness, we have also indicated the 8 directions associated with scale. For the methodology discussed in this paper, the directional properties of the curvelets are of great importance. We will also display the curvelet coefficient as t 0 -x 0 patches positioned in a large matrix according to the patch scale and nominal angle or dip. The latter is shown in b. Using matrix notation, the curvelet transform C can be represented as follows c = Cm, () where the discrete set of coefficients computed by the curvelet transform is represented via the vector c. Similarly, the t-x discrete data are represented by the vector m and the transform via the matrix C. One needs to stress, however, that the coefficients and the data are not stored in vector format. Moreover, the curvelet transform is not implemented via an explicit matrix multiplication. The latter is just a notation that permits us to use the simple language of linear algebra to solve our reconstruction problem. The discrete curvelet transform is a tight frame and therefore, the adjoint operator C T is equal to the pseudo-inverse of C (Candes and Donoho, 00). Consequently, the inverse curvelet transform (our synthesis operator) is obtained using the adjoint transform m = C T c. () Seismic signal representation in the curvelet domain In what follows, we will use simple examples to illustrate the decomposition of seismic data by means of the curvelet transform. For this purpose, we will utilize the t-x data portrayed

5 Normalized wavenumber Figure : Curvelet windows in the f-k domain. Representation of the curvelet coefficients which is adopted in this article. in Figure a. To complement the analysis, we first compute the f-k spectrum of the data shown in Figure b. The coefficients of the curvelet transform are displayed in c. At this point, an important comment is in order: this is not the traditional way of displaying the coefficients of the curvelet transform; because the number of coefficients at a given scale and direction is variable, each patch of coefficients for a given scale and direction was upscaled/downscaled to fit a matrix of size 0 0. In addition, each re-sampled patch was placed in a representation where the horizontal axis indicates angle or direction and, the vertical axis indicates scale. The reader can refer back to Figures a and b where we have presented our way of visualizing the coefficients of the curvelet transform. This representation allows us to make a clear analogy between the data representation in the curvelet domain and the more intuitive and classical f-k representation. In addition, we must stress that when analyzing t-x data one should represent directionality by means of dip rather than angle. However, without loss of generality, one can use angle to represent dip. For instance, the dip or slope is given by p = t x tan θ where t and x represent time and space sampling intervals, respectively. To gain understanding of the curvelet transform, our next step is to reconstruct the data by means of a limited number of curvelet coefficients. First, we set to zero all the curvelet coefficients except those belonging to scale. This is illustrated in Figure a. The data synthesized with the adjoint of the curvelet transform is portrayed in Figure b. Finally, Figure c shows the f-k spectrum of the synthesized data. In this example, we have preserved all the dips for only scale. The kinematics of the signal were preserved. One can clearly see reflections with distorted frequency content. To continue with our analysis, we now preserve only the coefficients representing one

6 Figure : Synthetic seismic data. Data in the t-x domain. Data in the f-k domain. Data in the curvelet domain. The curvelet coefficients were scaled to patches of size 0 0 for illustration purposes. scale and one direction. The latter is shown in Figure a. The synthesized data and its associated f-k spectrum are shown in Figures b and c, respectively. In this example, we have been able to synthesize the part of the data associated with one single dominant scale and dip. We finalize our exercise by setting to zero all curvelet coefficients but one. This is shown in Figure a. The synthesized data and its associated f-k spectrum are shown in Figures b and c, respectively. Minimum norm least squares curvelet interpolation We now turn our attention to the problem of reconstructing seismic data using the curvelet transform. For this purpose, we denote m our desired interpolated data in the t-x domain. In addition, the available traces are indicated by d. The available traces and the desired data are connected via a sampling operator G (Liu and Sacchi, 00) d = Gm + n, () where we have also incorporated the term n to include additive noise. The interpolation problem given by equation is an under-determined problem, therefore a priori information is needed for solving the problem. One way to solve the aforementioned problem is by introducing a change of variable and a regularization term to guarantee the stability and uniqueness of the solution. We posed the reconstruction problem in the curvelet domain (the change of variable) and, to attain this goal, we represent the data in terms of curvelet coefficients. The latter will allow us to incorporate a priori information about directionality and scale (frequency content) into our anti-alias interpolation scheme. The curvelet adjoint operator is used to represent the desired data. The alias-free desired data will be represented via

7 Figure : Data in Figure after eliminating all the curvelet coefficients except those belonging to scale. Curvelet coefficients. Data in the t-x domain obtained by applying the adjoint curvelet operator to the filtered coefficients in (. Data in the f-k domain Figure : Data in Figure after elimination all curvelet coefficients except those belonging to the patch associated to direction ( ) and scale. Curvelet coefficients. Data in the t-x domain obtained by applying the adjoint curvelet operator to the filtered coefficients in (. Data in the f-k domain.

8 Figure : Data in Figure after elimination all curvelet coefficients except a single coefficient in direction and scale (The position of the coefficient is indicated with an arrow). Curvelet coefficients. The arrow on the plot points to the location of the single curvelet coefficient. Data in the t-x domain obtained by applying the adjoint curvelet operator to the filtered coefficients in (. Data in the f-k domain. m = C T W c, () where we have introduced a mask function W that serves to preserve the subset of alias-free curvelet coefficients. Inserting equation into yields d = GC T W c + n. (6) The mask function W is a diagonal matrix with elements that are either 0 or so we are trying to prescribe with W the elements of the curvelet that are used to represent the desired data. Before describing how one can compute this important operator, let us assume for the moment that the operator W is known. The system of equations given by expression is under-determined (Menke, 989) and therefore, it admits an infinite number of solutions. A stable and unique solution can be found by minimizing the following cost function (Tikhonov and Goncharsky, 987) J = d G C T Wc + µ c. (7) The minimum of the cost function J can be computed using the method of conjugate gradients (Hestenes and Stiefel, 9). The advantage of using a semi-iterative solver like conjugate gradients is that there is no need of explicit knowledge of C in matrix form. The method of conjugate gradients requires the action of the operator C T and C on a vector in the coefficient and data spaces, respectively. The latter is a feature of conjugate gradients, and in general of many iterative and semi-iterative solvers, that is fully exploited in many seismic processing and inversion problems (Claerbout, 99). The goal of the proposed

9 9 algorithm is to find the coefficients ĉ that minimize J and, use them to reconstruct the data via the curvelet adjoint operator ˆm = C T ĉ. Selection of W for regularly decimated aliased data This section describes a strategy to estimate W from the data, which is a key component of our algorithm. We first use the curvelet transform to find an initial vector of coefficients c (equation ). The coefficients are divided into two groups according to their scale: aliasfree and alias-contaminated scales. Let us define the indices j and l that indicate scale and angle, respectively. Furthermore, the parameter j a indicates the index of the maximum alias-free scale. With this definition in mind, the mask function for the alias-free scales can be computed as follows [W] j,l = { 0 [c]j,l < λ j [c] j,l λ j if j j a, (8) where [c] j,l is used to indicate all the coefficients for scale j and angle l. Similarly, [W] j,l represents all the elements of the diagonal mask function for scale j and angle l. The parameter λ j is a user defined threshold value for scale j. The remaining problem is to compute the mask function for alias-contaminated scales. For this purpose, we use the following algorithm [W] j,l = N [ W] j, l if j > j a, (9) where in the expression above N denotes the nearest neighbor operator that is needed to upscale the mask function from scale j to j and l indicates the directionality (angle) index closest to l. In essence, we use the mask function from a lower scale (alias-free) to constrain the curvelet coefficients of higher scales that are contaminated by aliasing. The mask behaves like a local and directional all-pass operator for the coefficients that are modeling the alias-free signal. In summary, equations 8 and 9 are used to estimate the mask function that will be used in the reconstruction algorithm outlined in the preceding section. SYNTHETIC EXAMPLES Regularly decimated aliased data In order to examine the performance of our beyond alias curvelet interpolation, we have created a synthetic seismic section with three hyperbolic events (Figure 6. Figure 6b shows the f-k spectrum of the data prior to decimation. Next, we decimate the original data by a factor of (eliminating traces between each pair of traces) to obtain the data in Figure 6c. The f-k spectrum of the decimated data is shown in Figure 6d. Figures 6e and 6f show the reconstructed data using the curvelet interpolation in the t-x and f-k domains, respectively. Figure 6g shows the difference between the original data in Figure 6a and the interpolated data in Figure 6e.

10 e) g) d) f) Figure 6: Synthetic example showing the curvelet interpolation method. Original data. Data decimated by a factor of. e) Interpolated data using curvelets. g) The difference between the original and the interpolated data., d), and f) are the f-k spectra of (, (, and (e), respectively.

11 The quality of the reconstruction in decibels (db) is evaluated by the following measure (Hennenfent and Herrmann, 006) Q = 0 log( m / m ˆm ), where m is the true desired synthetic data prior to decimation and ˆm is the reconstructed data. For this particular example Q =. db. Now we show, step by step, how to obtain the results provided by Figure 6. First, we need to guarantee that the desired signal (alias-free dat does not contain wrap-around energy in the f-k domain. The latter is achieved by interlacing zero traces and consequently, increasing the spatial Nyquist wavenumber. Figure 7a shows the original data (Figure 6 after interleaving zeros traces between each trace. The associated f-k spectrum is shown in Figure 7b. The spectrum of the desired signal does not show wrap-around artifacts. The solid box in Figure 7b shows the area covered by alias-free scales (scales ). These scales were used to derive the mask function to preserve the curvelet coefficients of the desired signal in scales 6. The interleaved data are the input to the curvelet interpolation algorithm. Figure 8a shows the curvelet domain representation of the data in Figure 7a. Scales - are free of alias while the alias energy emerges in scales and 6. Figure 8b shows the final mask function derived from the curvelet panel of the input data. Notice that the mask function for scales - is obtained via thresholding small coefficients. The mask function for scales and 6 was obtained using the procedure given by equation 9. Figure 8c shows the curvelet coefficients estimated by using the minimum norm least squares algorithm proposed by this paper. It is clear that the alias in scales and 6 were properly removed. The interpolated data (Figure 6e) were obtained by applying the adjoint curvelet transform to the coefficients obtained via the minimum norm least-squares algorithm. Irregularly sampled data The proposed method can also handle irregularly sampled data. However, the mask function W is estimated via a strategy that is different to that used for the regular sampling case. Irregular sampling attenuates alias and, at the same time, produces a noisy f-k spectrum (Zwartjes, 00). In this case, rather than bootstrapping scales like in the regularly sampled case, we design the mask function by applying thresholding (Hennenfent and Herrmann, 008). We then fit the curvelet coefficients that have survived the thresholding operator via the least-squares method. The algorithm is iterative and can be summarized as follows Initialization W 0 = I For k =,,... ĉ k = argmin{ d G C T W k c + µ c } c W k = T ĉ k End (0) where T indicates the operator given by equation 8 but now applied to all the scales and angles. The algorithm minimizes (0) using the method of conjugate gradients following by an update of the mask function. In general, we have found that only updates are required to converge to the solution. Figure 9a shows a synthetic seismic gather with three hyperbolic events. We have randomly eliminated 0% of the traces to obtain the section with missing traces (Figure

12 Figure 7: Pre-processed data for interpolation via the curvelet transform. Pre-processing is needed to guarantee that the desired signal does not have wrap-around energy in f-k spectrum. These data were obtained from Figure 6c by interlacing zero traces between each pair of existing traces. This is the input to the curvelet interpolation algorithm proposed in this article. The f-k spectrum of the pre-processed data. The solid box in ( represents the the f-k region where the alias-free coefficients can be found (scales ).

13 Figure 8: Curvelet representation of the data in Figure 7a. Mask function computed from ( using thresholding for scales to. The mask for scales -6 was bootstrapped from scale using the algorithm described in the text. Estimated curvelet coefficients via minimum norm least squares. The recovered data in Figure 6e were obtained by applying the adjoint curvelet operator to the inverted coefficients in (.

14 d) Figure 9: Synthetic example showing curvelet interpolation of irregularly sampled data. Original data. Data after randomly removing 0% of the traces. Interpolated data. d) The difference between the original and the interpolated data. 9. Figure 9c shows the reconstructed data using curvelet interpolation. Figure 9d shows the difference section between original data (Figure 9 and the interpolated data (Figure 9. Figures 0a-c show the f-k panel of Figures 9a-c, respectively. Figure a shows the curvelet representation of the data with missing traces (Figure 9. It is clear that the irregular sampling has created small amplitude artifacts in the curvelet domain. Figure b shows the curvelet panel of the interpolated data (Figure 9. This example yielded a quality of reconstruction Q =.9 db. We stress that similar results could be obtained using optimization techniques that promote sparse solutions via an l regularization term (Hennenfent and Herrmann, 008). It is also important to point out that sparsity promoting methods are not new to geophysical signal processing. They are a proven component of strategies to invert Radon operators for multiple removal and signal reconstruction (Thorson and Claerbout, 98; Sacchi and Ulrych, 99). On a same vein, sparse solutions for curvelet coefficients can be obtained with simple strategies based on iterative re-weighted least-squares. We have not found important differences between the algorithm outlined in this section and the classical iterative reweighted least-squares described, for instance, in Sacchi and Ulrych (99).

15 Figure 0: - The f-k spectra of the data in Figures 9a-c. Regularly sampled aliased data with conflicting dips To continue with the synthetic data analysis, we have created a synthetic gather that consists of 8 events with conflicting dip and variable curvature. The data and their f-k spectrum are portrayed in Figures a and b, respectively. The decimated data and their f-k spectrum are portrayed in Figures c and d, respectively. The results of our interpolation algorithm are provided in Figures e and f. The input data to the algorithm, again, are prepared in order to secure that there is no wrap-around energy in the f-k domain. It was already mentioned that this is achieved by interleaving the data with zero traces (Figure a and. Figure a shows the curvelet panels of the input data in Figure a. It is easy to see that scales to are free of alias. On the other hand, scales and 6 are heavily contaminated by aliased energy. Figure b shows the mask function computed from Figure a using Equations 8 and 9 for j a =. Figure c shows the curvelet domain representation of the coefficients obtained via the minimum norm least squares method. These coefficients were used to synthesize the interpolated data portrayed in Figure e. Finally, the reconstruction quality for this example is given by Q =.9 db. REAL DATA EXAMPLES Figure a displays a shot record from a data set from the Gulf of Mexico. In this exercise we will try to decrease the spatial sampling from 0 meters to. meters. In other words, we want to insert new traces between each pair of existing traces. Figure c shows the result of curvelet interpolation with the algorithm proposed in this article. Figures b and d show the f-k panels of Figures a and c, respectively. The wrap around energy present in the original data (Figure is removed in the interpolated

16 Figure : Curvelet domain representation of the data in Figure 9b. Curvelet coefficients estimated via inversion with the minimum norm least squares method. The recovered data in Figure 9c were obtained by applying the adjoint curvelet operator to the inverted coefficients in (.

17 e) d) f) Figure : Synthetic data example with conflicting dips. Original data. Data after decimation by a factor of. Interpolated data. The panels, d) and f) display the f-k spectra of (, (, and (e), respectively.

18 Figure : Pre-processed data for interpolation via the curvelet transform. Pre-processing is needed to guarantee that the desired signal does not have wrap-around energy in f-k domain. These data were obtained from Figure c by interlacing zero traces between each pair of existing traces. This is the input to the curvelet interpolation algorithm proposed in this article. The f-k spectrum of the pre-processed data. The solid box in ( represent the region in f-k where alias-free coefficients are found (scales -).

19 Figure : Curvelet domain representation of the data in Figure a. Mask function computed from ( using thresholding for scales to. The mask for scales -6 was bootstrapped from scale using the algorithm described in the text. Curvelet patches of the interpolated data. The recovered data in Figure e were obtained by applying the adjoint curvelet operator to the inverted coefficients in (.

20 0 data (Figure d). The data used in this example do not contain frequencies above the normalized temporal frequency. Therefore, all the curvelet coefficients associated to scale 6 were excluded from the analysis. Figure 6a shows the curvelet panels of the input data. Figure 6b shows the mask function computed via thresholding and bootstrapping the mask from coarser to finer scales via equation 9. Figure 6c shows the curvelet coefficients computed via the minimum norm least squares method. Finally, we reiterate that the coefficients obtained via inversion were used to generate the interpolated data in Figure c. We also apply the curvelet reconstruction algorithm to interpolate a near offset section of data also from the Gulf of Mexico. Figure 7a shows the near offset section. The interpolated data using curvelets are presented in Figure 7b. To complete the analysis, the data was also interpolated using the adaptive f-x interpolation method proposed by Naghizadeh and Sacchi (009). We choose adaptive f-x interpolation because its capability to handle lateral variations of dips. In other words, f-x adaptive interpolation does not require processing the data in small windows to validate the linear event assumption. The comparison is shown in Figures 7b and c. Figure 8a, b and c show the f-k spectra of the original data and interpolated data using curvelets and adaptive f-x interpolation, respectively. Small artifacts are visible in both interpolation results. However, the curvelet interpolated data contain less evidence of the decimation footprint. In this example, the data do not contain frequencies above the normalized temporal frequency. Therefore, all the curvelet coefficients associated to scale 6 were excluded from the analysis. Figure 9a shows the curvelet representation of the input data. The mask function obtained via thresholding alias-free scales (scales= ) and by bootstrapping coarser to finer scales using equation 9 is provided by Figure 9b. The minimum norm least squares algorithm was used to obtained the curvelet coefficients displayed in Figure 9c. It is clear that these coefficients were used to generate the interpolated data showed in Figure 7b. DISCUSSIONS Curvelet coefficients have different sizes for different scales and directions. Therefore, it is not easy to provide a simple physical representation of the curvelet coefficients. In this paper, we use upscaling (or downscaling) methods to form image patches of size 0 0 samples for each individual scale and direction. This leads to an uncomplicated way of representing the curvelet transform that is amenable to seismic data processing tasks. We need to emphasize, however, that the computation of the mask function does not require us to form the 0 0 patches that we have used for visualization. The mask function is upscaled from a coarser scale to a finer scale via a simple nearest neighbor algorithm that directly operates on the curvelet coefficients at a given scale and direction. Other tricks could have been used to design the mask function, however, our tests indicate that our simple nearest neighbor upscaling suffices for our needs. We have also found that our algorithm works quite well when we set the regularization parameter µ = 0 and rely on the number of iterations as an equivalent trade-off parameter. This is also often used when solving large scale linear problems (Hansen, 987). It is also important to mention that a maximum of 0 iterations of conjugate gradients were used in all our examples. Again, the number of iterations does not seem to be a critical parameter

21 d) Figure : Shot gather from a data set from the Gulf of Mexico. Interpolated data using curvelets. and d) are the f-k spectra of ( and (, respectively.

22 Figure 6: Curvelet domain representation of the data from Figure a after pre-processing (interlacing 7 zero traces between each available traces). Mask function computed from ( using thresholding for scales to. The mask function for scales and were bootstrapped from scale following the algorithm discussed in the text. Curvelet coefficients of the interpolated data.

23 Figure 7: Near offset section from the Gulf of Mexico. Interpolated data using curvelets. Interpolated data using adaptive f-x interpolation method (Naghizadeh and Sacchi, 009).

24 Figure 8: - are the f-k panels of the data in Figures 7a-c, respectively. for our interpolation scheme. The algorithm converges quite fast (approximately at 8 to 0 conjugate gradient iterations) and we have never seen any evidence of numerical instabilities. The thresholding constant λ j was set with the following criteria. For synthetic data, we keep 0% of the largest amplitude coefficients at a given scale. For the real data examples, we keep about 0% of the largest coefficients at a given scale. These parameters were obtained with numerical tests where we visually examined the residual data after interpolation to decide for an optimal threshold constant. When working with real data, the thresholding constant was increased because a larger number of coefficients were needed to properly model the data. In addition, we would like to mention that existing curvelet reconstruction methods often use sparsity promoting algorithms to find a parsimonious data representation in terms of a small number of curvelet coefficients (Herrmann and Hennenfent, 008). In this work, we have avoided sparsity-promoting algorithms and we solely relied on a minimum norm least squares algorithm implemented via the method of conjugate gradients. Finally, we stress that other methods could be used to interpolate regularly sampled aliased data with strong variation of dips. As an example we cite f-x adaptive interpolation (Naghizadeh and Sacchi, 009); a fast alternative for the type of interpolation introduced here. CONCLUSIONS The curvelet transform is an effective tool for the decomposition of seismic data based on their local dip and frequency content. In this paper, we propose a novel method for interpolation of aliased seismic data using the curvelet transform. We have shown that spatially aliased data can be represented in the curvelet domain by two types of coefficients. Those that belong to coarser scales and that have been minimally affected by spatial sampling and those at finer scales that have been contaminated by alias. We assume that the seismic signal is expected to have similar local dips in lower and higher scales. This assumption is used

25 Figure 9: Curvelet domain representation of the data in Figure 7a after interlacing 7 zero traces between each pair of traces. Mask function computed from ( using thresholding for scales to. The mask function for scales and were bootstrapped from scale following the algorithm discussed in the text. Curvelet coefficients of the interpolated data.

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