Parallel matrix factorization algorithm and its application to 5D seismic

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GEOPHYSICS, VOL. 80, NO. 6 (NOVEMBER-DECEMBER 2015); P. V173–V187, 23 FIGS., 1 TABLE. 10.1190/GEO2014-0594.1

Parallel matrix factorization algorithm and its application to 5D seismic reconstruction and denoising

Jianjun Gao1, Aaron Stanton2, and Mauricio D. Sacchi2

2005; Trad, 2009; Hunt et al., 2010; Chiu, 2014), projection onto convex sets (POCS) interpolation (Abma and Kabir, 2006; Gao et al., 2010, 2013b), matching pursuit regularization (Özemir et al., 2008; Vassallo et al., 2010), and antileakage Fourier transform interpolation (Xu et al., 2005; Schonewille et al., 2009; Xu et al., 2010). All the aforementioned techniques use Fourier kernels to solve the seismic data reconstruction problem. In short, Fourier reconstruction works as follows: The irregularly sampled data are used to estimate the wavenumber Fourier coefficients that synthesize the observed seismic data. A simplicity constraint is used to estimate these Fourier coefficients. Then, the estimated Fourier coefficients are used to synthesize data at unobserved spatial positions. Simplicity on the distribution of wavenumber domain coefficients is sought by restriction operators to impose band limitation (Duijndam et al., 1999; Schonewille et al., 2003) or by promoting sparsity (Sacchi et al., 1998; Zwartjes and Gisolf, 2007; Naghizadeh and Sacchi, 2013). Norms used to promote sparse or parsimonious solutions are the l1 -norm (Taylor et al., 1979); the Cauchy norm (Sacchi and Ulrych, 1995, 1996; Guspi and Introcaso, 2000); or in the case of MWNI (Liu and Sacchi, 2004), a weighted l2 -norm. One could also adopt a hybrid l1 –l2 -norm that mimics the behavior of the l1 -norm as proposed by Lee et al. (2006) and Li et al. (2012). Notice the strong connection of Fourier reconstruction methods to reconstruction solutions provided by the field of compressive sensing (Herrmann, 2010). The field of compressive sensing started in 2005 with contributions by Candes et al. (2006a, 2006b) and Donoho (2006). Parsimonious or sparse solutions for geophysical data inversion and reconstruction have been profusely adopted by our community prior to the inception of compressive sensing (Taylor et al., 1979; Levy et al., 1982; Thorson and Claerbout, 1985; Sacchi and Ulrych, 1995, 1996). We stress, however, that compressive sensing provides recovery conditions that were not discussed in early geophysical applications of sparsity driven inversion and reconstruction. In addition, the field of compressive sensing has become a production playground for the development of improved

ABSTRACT Tensors, also called multilinear arrays, have been receiving attention from the seismic processing community. Tensors permit us to generalize processing methodologies to multidimensional structures that depend on more than 2D. Recent studies on seismic data reconstruction via tensor completion have led to new and interesting results. For instance, fully sampled noise-free multidimensional seismic data can be represented by a low-rank tensor. Missing traces and random noise increase the rank of the tensor. Hence, multidimensional prestack seismic data denoising and reconstruction can be tackled with tools that have been studied in the field of tensor completion. We have investigated and applied the recently proposed parallel matrix factorization (PMF) method to solve the 5D seismic data reconstruction problem. We have evaluated the efficiency of the PMF method in comparison with our previously reported algorithms that used singular value decomposition to solve the tensor completion problem for prestack seismic data. We examined the performance of PMF with synthetic data sets and with a field data set from a heavy oil survey in the Western Canadian Sedimentary Basin.

INTRODUCTION In recent years, different methods for multidimensional seismic data reconstruction have been proposed. Most of the techniques currently used by industry are based on Fourier reconstruction methods. In this category of methods, simplicity constraints are adopted to estimate the wavenumber domain coefficients that model the observed data. Examples of the latter are minimum weighted norm interpolation (MWNI) (Liu and Sacchi, 2004; Sacchi and Liu,

Manuscript received by the Editor 16 December 2014; revised manuscript received 29 June 2015; published online 2 September 2015; corrected version published online 11 September 2015. 1 China University of Geosciences, Key Laboratory of Geo-detection, Ministry of Education, Beijing, China and University of Alberta, Department of Physics, Edmonton, Alberta, Canada. E-mail: [email protected]. 2 University of Alberta, Department of Physics, Edmonton, Alberta, Canada. E-mail: [email protected]; [email protected]. © 2015 Society of Exploration Geophysicists. All rights reserved. V173

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algorithms for the solution of linear problems with sparsity constrains (van den Berg and Friedlander, 2008; Beck and Teboulle, 2009; Donoho et al., 2009). Methods categorized under the matrix or tensor completion category have also been proposed to solve the seismic regularization problem. In these techniques, a matrix or a tensor with missing entries is observed, and the task is to find (to complete) the missing entries of the matrix or of the tensor by assuming that the reconstructed signal has low rank. The problem is often encountered in chemometrics (Tomasi and Bro, 2005), image processing (Liu et al., 2013), and more recently it was proposed to solve the 5D seismic reconstruction problem by Kreimer and Sacchi (2012), Kreimer et al. (2013), Ely et al. (2013), and Silva and Herrmann (2013a, 2013b). Two subcategories of reduced rank methods for seismic data reconstruction have been proposed. The first subcategory applies rank reduction to multilevel block Hankel or Toeplitz matrices formed from the entries of the tensor. In other words, multidimensional data are rearranged into a block Hankel or Toeplitz matrix, and a rank reduction algorithm is used to improve the signal-tonoise ratio (S/N) and to reconstruct the data. Methods in this subcategory are often named Cadzow reconstruction methods (Trickett et al., 2010) or multichannel singular spectrum analysis (MSSA) reconstruction (Oropeza and Sacchi, 2011; Gao et al., 2013a). The second subcategory of rank reduction methods encompasses techniques that are based on dimensionality reduction of multilinear arrays or tensors. In this case, the multichannel seismic data are viewed as multilinear arrays, and dimensionality reduction techniques are directly applied to the multilinear array. Methods in this category have first been proposed by Kreimer and Sacchi (2012) who adopt the high-order singular value decomposition (HOSVD) to solve the 5D seismic data reconstruction problem in the frequency-space domain. More recently, Kreimer et al. (2013) present a new algorithm that uses nuclear norm minimization to solve the tensor completion problem in exploration seismology. The nuclear norm plays a role similar to that played by the l1 -norm in reconstruction via Fourier methods. Nuclear norm minimization promotes solutions that are simple in terms of the rank of the structure one would like to reconstruct (e.g., a matrix or a tensor) (Recht et al., 2010; Gandy et al., 2011). Kreimer et al. (2013) show that the properly sampled multidimensional noise-free seismic data can be embedded into a low-rank tensor. The missing traces and noise increase the rank of the tensor. Several methods have been proposed to recover missing samples of prestack seismic data via tensor completion techniques. In this article, multidimensional seismic data are recovered via low-rank solutions that are estimated via the parallel matrix factorization (PMF) algorithm (Xu et al., 2015). The PMF method recovers a tensor with missing observations by simultaneously performing low-rank matrix factorizations to the different tensor unfoldings, which are the different ways, a tensor can be represented in matrix format (Kolda and Bader, 2009). The PMF algorithm is a singular value decomposition (SVD)-free approach, and therefore, it can make a significant impact on computational efficiency when processing largescale seismic data sets or when the data are broken down into many 5D patches to be processed independently (Trad, 2009). We show that the PMF algorithm described in this article is faster than existing reconstruction algorithms that were proposed by our group. For instance, our original research was focused on algorithms that relied on the application of the SVD to compute the HOSVD (Kreimer

et al., 2013) and to apply singular value thresholding for the minimization of the nuclear norm of the tensor (Kreimer et al., 2013). We avoid the computation of the SVD by a straightforward matrix factorization method (Wen et al., 2012; Xu et al., 2015) that reduces the overall cost of the iterative reconstruction. Comparisons of different methods for 5D seismic reconstruction were performed by Stanton et al. (2012). An interesting outcome of the comparison is that tensor reconstruction methods tend to better cope with the reconstruction of curved waveforms than do Fourier reconstruction techniques. However, the field of Fourier reconstruction has reached maturity, and nowadays, industry researchers depend on highly efficient methodologies to reconstruct seismic data via Fourier methods. On the other hand, tensor reconstruction techniques for multidimensional seismic data are still in their infancy. Research at this time mainly concerns computational efficiency and the development of practical algorithms for real data applications. This article contributes to the problem of tensor completion for 5D seismic data processing in several ways. First, we discuss and study the computational cost of PMF versus the cost of our previously reported algorithms that require the SVD. Second, we show that the PMF algorithm leads to a final recursion that resembles typical imputation methods used by the MSSA reconstruction method (Oropeza and Sacchi, 2011) and POCS seismic interpolation (Abma and Kabir, 2006). In essence, the final algorithm has a parameter that controls the reinsertion of existing observations into the final solution. Third, we discuss the problem of implementation of the algorithm in terms of a series of free parameters that control the quality of the reconstruction. We have conducted a Monte Carlo test with synthetic simulations that provide optimistic results in terms of the sensitivity of PMF to the determination of the unknown dimensionality of the data. Finally, we apply PMF to the reconstruction of a real 3D field data set with 85% of the traces missing.

THEORY Preliminaries In this article, the PMF algorithm (Xu et al., 2015) is implemented to reconstruct prestack seismic volumes in the midpointoffset frequency-space domain. We denote the seismic data in the frequency-space domain by Dðω; rx ; ry ; sx ; sy Þ, where rx and ry indicate receiver coordinates and sx and sy source coordinates. The seismic data can be transformed into midpoint-offset coordinates and expressed as Dðω; x; y; hx ; hy Þ, where x; y; hx and hy indicate the spatial coordinates of the inline midpoint, crossline midpoint, inline offset, and crossline offset, respectively. After binning the data in the midpoint-offset domain, the seismic volume for one particular frequency can be represented via a fourth-order tensor D, with elements Di1 ;i2 ;i3 ;i4 , where i1 ; i2 ; i3 ; and i4 are bins indices for the spatial coordinates x; y; hx and hy , respectively. Notice that we also have dropped the dependency on ω to simplify the notation, but bear in mind that the subsequent analysis is carried out for all frequencies ω (Kreimer, 2013). The proposed reconstruction method is only suitable for reconstructing irregularly missing seismic traces in a grid with uniform spatial sampling. Low-rank tensor reconstruction with irregular grids is still an open problem. Kumar et al. (2015) propose an improved matrix rank minimization method for regularization and

5D seismic reconstruction: PMF method interpolation for 2D seismic data sets on irregular grids. The method could be extended to cope with tensors on nonuniform grids.

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Problem set up and notation We will designate tensors or multilinear arrays via calligraphic fonts. For instance, the Nth-order tensor X is the N-dimensional array with elements given by X i1 ;i2 ;i3 : : : iN , where in ¼ 1; : : : ; I n ; n ¼ 1; : : : ; N. An Nth-order tensor along each mode can be unfolded into N matrices. Similarly, a matrix can be folded into a tensor. We will designate the mode k unfolding and folding operators as follows:

XðkÞ ¼ unfoldk ½X ;

k ¼ 1; : : : ; N;

ΦC ¼

(1)

(2)

Figure 1 shows the folding and unfolding operator for a third-order tensor (N ¼ 3). The mode k unfolding reorganizes an Nth-order tensor into a matrix of dimension I k × ðI 1 I 2 : : : I k−1 I kþ1 : : : I N Þ (Kolda and Bader, 2009).

Cost function for tensor completion via parallel matrix factorization Consider a linear problem, where Z is a low rank Nth-order tensor. We also assume that we only have knowledge of some samples of the tensor. The observed data are represented by the Nth-order tensor D that contains missing entries. Furthermore, we will replace missing entries by zeros. This mathematical model for data reconstruction can be expressed via a linear system of equations:

P ∘ Z ¼ D;

(3)

where P is a sampling operator and “∘” indicates Hadamard’s product (elementwise product). The goal is to recover the tensor Z from observations D. In general, we also assume that the observations are noisy, and therefore the model P ∘ Z ≈ D is adopted. To estimate the complete data Z, we first define a cost function of the form

Φ ¼ ΦC þ μΦM ;

Cm×r and C ∈ Cr×n (Hogben, 2013). Then, the low-rank matrix factorization is applied to each mode unfolding of Z by seeking matrices XðkÞ ∈ CIk ×rk and YðkÞ ∈ Crk ×I1 : : : Ik−1 Ikþ1 : : : IN , such that ZðkÞ ≈ XðkÞ YðkÞ for k ¼ 1; : : : ; N, where rk is the rank of the mode k unfolding matrix ZðkÞ . In this article, we define the regularization term ΦC by the following expression:

ΦC ¼

X ¼ foldk ½XðkÞ :

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¼

N 1X kfoldk ½XðkÞ YðkÞ  − Zk2F ; 2 k¼1

(6)

N 1X kfoldk ½XðkÞ YðkÞ  − Zk2F ; 2 k¼1 N 1X kX Y − ZðkÞ k2F ; 2 k¼1 ðkÞ ðkÞ

(7)

where ZðkÞ ¼ unfoldk ½Z. The regularization constraint forces a low-rank representation of the unfolded matrices that are obtained from the unknown tensor Z. The cost function is nonlinear and it can be minimized via the alternating least-squares algorithm (Arrow et al., 1968; Glowinski and Marrocco, 1975; Gabay and Mercier, 1976; Wen et al., 2012; Xu et al., 2015). We rewrite the cost function Φ in equation 4 as the function of the variables X, Y, and Z:

ΦðX; Y;ZÞ ¼

N 1X μ kXðkÞ YðkÞ − ZðkÞ k2F þ kP ∘ Z − Dk2F ; 2 k¼1 2

(8) where X ¼ ðXð1Þ ; : : : ; XðNÞ Þ and Y ¼ ðYð1Þ ; : : : ; YðNÞ Þ. We minimize Φ by updating one variable X, Y, and Z at a time, while

(4)

where ΦC is the low-rank constraint term and μ is a regularization parameter. The term ΦM is the data misfit, which is given by the l2 -discrepancy functional (the Frobenius norm of the error):

1 ΦM ¼ kP ∘ Z − Dk2F : 2

(5)

Our problem is ill posed because there are an infinite number of solutions that satisfy equation 4. The ill-posed problem could be transformed into a family of well-posed problems by introducing a regularization constraint. It is well known that any matrix A ∈ Cm×n of rank r can be written into a matrix product form A ¼ BC, where B ∈

Figure 1. Schematic representation of the unfolding and folding operator for a thirdorder tensor X with dimensions I 1 × I 2 × I 3 .

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the other two are fixed. The update is given by the following expressions:

Xiþ1 ¼ arg min ΦðX; Yi ; Z i Þ;

(9a)

equal to one. After a series of mathematical manipulations, Xu et al. (2015) show that only the products XðkÞ YðkÞ affect Z. Therefore, the algorithm can be simplified by replacing equation 11a by the following update:

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X i i H Xiþ1 ðkÞ ¼ ZðkÞ ðYðkÞ Þ ;

Yiþ1 ¼ arg min ΦðXiþ1 ; Y; Z i Þ;

Z

We now define the gradients

∂Φ ∂Φ ∂X , ∂Y ,

and

∂Φ ∂Z

Exact data fitting

(9c)

Let us consider the hypothetical case in which one desires an exact-fit solution to the original data. The problem entails finding the low-rank approximation for all tensor unfoldings subject to P ∘ Z ¼ D. The solution requires invoking the method of Lagrange multipliers (each linear equation now needs to be fully honoured). In this case, we minimize the following cost function:

and set them to zero

∂Φ ¼ XðkÞ YiðkÞ ðYiðkÞ ÞH − ZiðkÞ ðYiðkÞ ÞH ¼ 0; k ¼ 1; :::; N; ∂XðkÞ (10a)

Φ ¼ hW; P ∘ Z − Di þ ΦM ;

∂Φ iþ1 H i H iþ1 ¼ ðXiþ1 ðkÞ Þ XðkÞ YðkÞ − ðXðkÞ Þ ZðkÞ ¼ 0; k ¼ 1; :: :;N; ∂YðkÞ (10b)

N X ∂Φ iþ1 0 ¼− foldk ðXiþ1 ðkÞ YðkÞ Þ þ NZ þ μP ðPZ − DÞ ¼ 0: ∂Z k¼1

(10c) We can solve equations 10a, 10b, and 10c to obtain the following updates:

k ¼ 1; : : : ; N; (11a)

iþ1 H iþ1 † iþ1 H i Yiþ1 ðkÞ ¼ ððXðkÞ Þ XðkÞ Þ ðXðkÞ Þ ZðkÞ ; k ¼ 1; :::;N;

Z iþ1 ¼ ðI − αPÞ ∘ C þ αD;

(11b)

(11c)

with parameter α given by

μ ; α¼ Nþμ where the Nth-order tensor C is given by

C¼

N 1X iþ1 foldk ½Xiþ1 ðkÞ YðkÞ : N k¼1

(13)

The symbol “†” denotes the Moore-Penrose pseudoinverse. The tensor I is used to indicate an Nth-order tensor with all elements

(16)

Expression 16 is equivalent to expression 11c when the parameter α ¼ 1. We also notice that α ≈ 1 when μ ≫ N. In essence, the observed data D contribute to a larger proportion of the reconstructed data Z, when α ≈ 1. An algorithm with the form of equations 8 and 15 is also used by Oropeza and Sacchi (2011) for Cadzow-based denoising and reconstruction. In this case, however, rank reduction is applied to Hankel matrices constructed from spatial samples of seismic data. The POCS reconstruction algorithm can also be represented by equations 8 and 16 (Abma and Kabir, 2006; Gao et al., 2013b). However, in the POCS algorithm, the projection governing the reconstruction is frequency-wavenumber thresholding, whereas in the proposed methodology, factorization (rank reduction) of the unfolded tensor is the main driver of the reconstruction.

Parallel matrix factorization tensor completion algorithm The PMF method reduces to algorithm 1. The variable “rel” in Algorithm 1 represents the relative error that is used as stopping criterion. The parameter N iter represents the maximum number of iterations. The algorithm is initiated with random matrices Y0ðkÞ ; k ¼ 1; 2; : : : ; N. Similarly, Z 0 is initialized by an Nth-order random tensor. The elements of the aforementioned random matrices and tensor are normally distributed with unit variance. For noise-free data, we evaluate the scalar rel ¼

(12)

(15)

P P ¯ where hA; Bi ¼ Ii11¼1 : : : IiNN¼1 A i1 : : : iN Bi1 : : : iN . Without going into lengthy derivations, the resulting algorithm is given by

Z ¼ ðI − PÞ ∘ C þ D:

i i H i i H † Xiþ1 ðkÞ ¼ ZðkÞ ðYðkÞ Þ ðYðkÞ ðYðkÞ Þ Þ ;

(14)

(9b)

Y

Z iþ1 ¼ arg min ΦðXiþ1 ; Yiþ1 ; ZÞ:

k ¼ 1; : : : ; N:

kP∘Z iþ1 −Dk2F , kDk2F

which is

used as a stopping criterion. For noisy data, the algorithm adopts the following scalar rel ¼

kZ iþ1 −Z i k2F kZ i k2F

as a stopping criterion. The vari-

able “tol” is a small positive value that is provided by the user. Similarly, in the noisy case, we provide the parameter α rather than the parameter μ. This parameter has a more appealing meaning. It can be interpreted as an add-back data parameter that controls the amount of reinsertion of the original data into the final solution. The processor needs to define the rank r and α. In general, for practical application, we have found that it is simpler to define α ∈ ð0; 1Þ than the parameter μ ∈ ð0; ∞Þ.

5D seismic reconstruction: PMF method

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Analysis of computational cost We have created a 5D seismic data set having four spatial dimensions. The size of the tensor is given by I 1 × I 2 × I 3 × I 4. In our tests, we have varied I k from 5 to 14 with an increment of 1 for k ¼ 1; 2; 3; 4. Each data set is composed of 301 time samples per trace. We have modeled three linear events with S=N ¼ 106 . We randomly remove 50% of the traces and perform the reconstruction using the proposed PMF algorithm, the HOSVD reconstruction method (Kreimer and Sacchi, 2012), and the nuclear norm minimization method (Kreimer et al., 2013). For the PMF and HOSVD methods, we set the rank rk ¼ 3 for mode k, k ¼ 1; 2; 3; 4, the maximum number of iterations N iter ¼ 100, and iteration stopping error tol ¼ 10−4 for each frequency. For the nuclear norm method, we set N iter ¼ 100, tol ¼ 10−4 , and the parameters λ ¼ 2.5 and β ¼ 15. The selection of these parameters is discussed in Kreimer et al. (2013). Figure 2 shows a comparison of the computational cost of the proposed PMF method, the nuclear norm minimization method, and the HOSVD method. For each iteration, the computation cost of the PMF method in equation 14 is OðNmnrÞ, where r ¼ rk ¼ maxfr1 ; r2 ; : : : ; rN g, m ¼ I k ¼ maxfI 1 ; I 2 ; : : : ; I N g, n ¼ I 1 I 2 : : : I k−1 I kþ1 : : : I N (N represents the order of seismic data H iþ1 tensor). In equation 11b, the product ðXiþ1 ðkÞ Þ XðkÞ is a rk × rk square matrix. Hence, we apply LU decomposition with partial pivoting to solve Yiþ1 ðkÞ in equation 11b. The computation cost of equa-

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faster than the nuclear norm minimization and the HOSVD. Our analysis of number of operations is reflected by the results provided by simulations in Figure 2. Following with our analysis, we also choose a synthetic data model containing 12 × 12 × 12 × 12 traces in the spatial directions and 301 time samples per trace (also used in the example in Figure 2) to examine the reconstruction quality of the new proposed PMF algorithm, HOSVD reconstruction, and the nuclear norm minimization reconstruction method. We define the reconstruction quality Q by



Q½dB ¼ 10 log10

 kDtrue k2 ; kDtrue − Drecon k2

(17)

3

tion 11b is OðNðmr2 þ mnr þ 2nr 3 ÞÞ (Golub and Loan, 1996). The computation cost of equation 11c is OðNmnrÞ. The total cost of 3 PMF algorithm in one iteration is OðNð3mnr þ mr2 þ 2nr 3 ÞÞ. In the nuclear norm minimization algorithm, an SVD is applied in each iteration. Therefore, the computational cost is OðNð2m2 n þ 2m3 ÞÞ per iteration (Golub and Loan, 1996). For the case of the HOSVD algorithm, we adopted the tensor toolbox from Bader et al. (2012), where the cost is dominated by the higher order orthogonal iteration algorithm with a computational cost OðNð4m2 n þ 13m3 þ mr3ðN−1Þ ÞÞ (Golub and Loan, 1996; Ishteva et al., 2011; Kreimer, 2013). Because typically 1 ≤ r < minfm; ng, the PMF algorithm is

Figure 2. Computational time comparison of the proposed PMF reconstruction method, the HOSVD reconstruction method and the nuclear norm minimization method for different 5D volumes with size of 301 × I 1 × I 2 × I 3 × I 4 , I k ¼ 5; 6; : : : ; 14, k ¼ 1; 2; 3; 4.

Algorithm 1. Low-rank tensor completion via PMF. Inputs: D, P, μ, r1 ; : : : ; rN , N iter , rel, tol Initializations: Y0 , Z 0 for i ¼ 0; 1; :::; N iter do for k ¼ 1; :::; N do i i H Xiþ1 ðkÞ ¼ ZðkÞ ðYðkÞ Þ iþ1 iþ1 H iþ1 † H i YðkÞ ¼ ððXðkÞ Þ XðkÞ Þ ðXiþ1 ðkÞ Þ ZðkÞ end for P iþ1 C ¼ N1 Nk¼1 foldk ½Xiþ1 ðkÞ YðkÞ  Z iþ1 ¼ ðI − αPÞ ∘ C þ αD if rel < tol then break; end if end for Output: Z iþ1

Figure 3. Reconstruction quality Q versus percentage of missing traces for the PMF, HOSVD, and nuclear norm reconstruction methods.

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where Dtrue and Drecon represent the true noise-free complete data and reconstructed data in the time-spatial domain, respectively. Figure 3 shows the reconstruction quality Q varying with the percentage of missing traces. From Figure 3, we find that the reconstruction quality obtained by the proposed PMF method and HOSVD algorithms are similar. They both perform better than the nuclear norm minimization method.

EXAMPLES

of temporal frequencies for reconstruction is 1–70 Hz. Figure 4 shows the reconstruction result. From Figure 4c and 4d, we observe that the missing traces are effectively recovered. Figure 5 shows the normalized singular value curve corresponding to the mode k unfolded matrices k ¼ 1; 2; 3; 4 for the temporal frequency f ¼ 20 Hz for the original complete data and decimated data. The solid lines represent the original completed data. The dashed lines represent the decimated data. From the solid lines, we observe that the data are low rank at each mode k. Hence, we confirm that rank reduction

Synthetic example For the third example, we test the reconstruction performance of the PMF method on noise-free data containing three-plane waves. We create a 5D data model of size 301 × 12 × 12 × 12 × 12 and S=N ¼ 106 . We randomly decimate 90% of the traces and apply the proposed PMF method to recover the missing traces. We set N iter ¼ 100, tol ¼ 10−4 , rk ¼ 3, k ¼ 1; 2; 3; 4, α ¼ 1, and the band

Figure 5. Normalized singular value distribution for one frequency f ¼ 20 Hz. The solid lines marked with triangles, circles, squares, and downward triangles represent the normalized singular values for modes 1, 2, 3, and 4, in which the original complete data tensor was unfolded. The dashed lines with triangles, circles, squares, and downward triangles represent the normalized singular values for modes 1, 2, 3, and 4 obtained by unfolding the decimated data. This figure corresponds to data composed of linear events.

Figure 4. (a) Original data with S=N ¼ 106 . (b) Decimated data with 90% of the traces missing. (c) Reconstructed data. (d) Difference between panels (a and c).

Figure 6. Convergence of PMF versus iteration for frequencies f ¼ 10, 20, and 50 Hz.

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5D seismic reconstruction: PMF method via the PMF method is a reasonable approach for reconstruction. For completeness, Figure 6 shows the convergency of the proposed PMF method for frequencies f ¼ 10, 20, and 50 Hz. The PMF method can interpolate not only linear events but also curved events. For the fourth example, we synthesize a noise-free model data with four curved events. The spatial size of the model is 12 × 12 × 12 × 12 with 301 time samples per trace and S=N ¼ 106 . We have randomly decimated 90% of the traces, and set the rank to r1 ¼ r2 ¼ r3 ¼ 5 for modes 1; 2; 3 and r4 ¼ 4 for mode 4. We also set N iter ¼ 300, tol < 10−4 and α ¼ 1. Figure 7 shows the reconstruction result. From the difference section in Figure 7d, we conclude that the PMF method can also reconstruct curved events. Figure 8 shows the normalized singular values for the unfolding matrix at mode k, k ¼ 1; 2; 3; 4 for a temporal frequency of f ¼ 20 Hz. We find that even though the events are curved, they indeed preserve the low-rank property for each one of the associated unfolded matrices. The latter has also been discussed in Stanton et al. (2012). Figure 9

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shows the convergence curves for the PMF method for frequencies f ¼ 10, 20, and 50 Hz for this particular example. We also examine reconstruction results for data contaminated with noise. We add random noise to the noise-free data in Figures 4 and 7. We set S=N ¼ 1, N iter ¼ 300, tol < 10−4 , and μ ∈ ½0.001; 100 to estimate the best reconstruction quality factor Q. Figure 10 shows the reconstruction result for the noisy data with linear events using μ ¼ 4.175 (α ¼ 0.51) corresponding to the best Q ¼ 11.0. Figure 11 shows the reconstruction result for the noisy

Figure 8. Normalized singular values distribution for one frequency f ¼ 20 Hz. The solid lines marked with triangles, circles, squares, and downward triangles represent the normalized singular values for modes 1, 2, 3, and 4 that were obtained by unfolding the complete data. The dashed lines with triangles, circles, squares, and downward triangles represent the normalized singular values for modes 1, 2, 3, and 4 that were obtained by unfolding the decimated data. This figure corresponds to data composed of curved events.

Figure 7. (a) Original data with S=N ¼ 106 . (b) Decimated data with 90% of the traces missing. (c) Reconstructed data. (d) Difference between panels (a and c).

Figure 9. Convergency versus iteration for the PMF method for different frequencies at f ¼ 10, 20, and 50 Hz.

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data with curved events for μ ¼ 6.210 (α ¼ 0.61) corresponding to the best Q ¼ 8.1.

cmpx ∶N 1 ¼ 300; increment ¼ 5 m;

Field data example

cmpy ∶N 2 ¼ 220; increment ¼ 5 m;

ðmin; maxÞ ¼ ð80; 1575 mÞ; bin number ∈ ð1536; 1835Þ; ðmin; maxÞ ¼ ð180; 1275 mÞ; bin number ∈ ð656; 875Þ;

We also test the performance of the reconstruction method on a land data set that was acquired to monitor a heavy oil field in Alberta, Canada. Figure 12 shows the source and receiver distribution of the orthogonal survey. The data are binned on a 5 × 5 m common midpoint (CMP) grid, and a 100 × 100 m offset-x-y grid prior to interpolation. After the binning process, 91% of the original traces are preserved in the grid. NMO correction was applied to the data to avoid spectral wrapping in the frequency-wavenumber domain. We binned the traces sequentially, meaning the trace kept for any given bin was the last trace to fall in the bin. The reconstructed area includes 300 CMPx bins and 220 CMPy bins. The following parameters were chosen to examine the proposed algorithm:

We ran our algorithm for frequencies from 1 to 100 Hz and divided the whole survey data into 2640 overlapping blocks. Each block

Figure 10. Denoising and reconstruction using the proposed PMF method with μ ¼ 4.175. (a) Original noisy data with S=N ¼ 1. (b) Decimated data with 90% of the traces missing. (c) Reconstructed data. (d) Difference between Figures 4a and 10c.

Figure 11. Denoising and reconstruction using the proposed PMF method with μ ¼ 6.210. (a) Original noisy data with S=N ¼ 1. (b) Decimated data with 90% of the traces missing. (c) Reconstructed data. (d) Difference between Figures 7a and 11c.

hx ∶N 3 ¼ 7; increment ¼ 100 m; ðmin; maxÞ ¼ ð−300 ; 300 mÞ; bin number ∈ ð1; 7Þ; hy ∶N 4 ¼ 21; increment ¼ 100 m; ðmin; maxÞ ¼ ð−1000; 1000 mÞ; bin number ∈ ð1; 21Þ: (18)

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contains 20 × 20 CMP bins and 300 time samples. The overlap length for CMPx bin and CMPy bin are 10 bins. The following parameters were chosen for each block:

cmpx ∶n1 ¼ 20;

an overlap of 10 CMP bins;

cmpy ∶n2 ¼ 20;

an overlap of 10 CMP bins;

hx ∶n3 ¼ 7; hy ∶n4 ¼ 21;

Figure 12. Survey geometry for the real data example. Stars represent sources, and triangles represent receivers. The thick black rectangle represents the extent of the CMP bins that were reconstructed with the proposed algorithm. The horizontal green line AA′ represents the prestack slice for fixed CMPy bin 730 in Figure 14. The vertical green line BB′ represents the prestack slice for fixed CMPx bin 1565 in Figure 15.

Figure 13. Fold before reconstruction. The minimum fold number is 3 and the maximum is 122. The horizontal line CC′ represents the slice for fixed CMPy bin 705 in Figure 20. The vertical line DD′ represents the slice for fixed CMPx bin 1600 in Figure 21.

no overlap; no overlap:

(19)

On average, each patch of data contains approximately 85% missing traces. Figure 13 shows the fold number of each CMP bin before reconstruction. The minimum fold of the data is 3 and the maximum fold of the data is 122. We set μ ¼ 2.70 (α ¼ 0.40), rank rk ¼ 4; k ¼ 1; 2; 3; 4, and N iter ¼ 100 for the PMF reconstruction. The parameter α ¼ 0.40 was chosen by examining the quality of reconstruction in a visual manner. Figure 14 shows a zero-offset slice of the 5D volume with fixed CMPy (bin 730 and 550 m), hx (bin 4 and 0 m), and hy (bin 11 and 0 m). Figure 15 presents a zero-offset slice of the 5D volume with fixed CMPx (bin 1565 and 225 m), hx (bin 4 and 0 m), and hy (bin 11 and 0 m). Figure 16 displays a mid range offset slice of the 5D volume with fixed CMPy (bin 730 and 550 m), hx (bin 4 and 0 m), and hy (bin 6 and −500 m). Figure 17 shows a far-offset slice of the 5D volume with fixed CMPy (bin 763 and 715 m), hx (bin 4 and 0 m), and hy (bin 3 and −800 m). Figure 18 shows a CMP gather for a fixed CMPx bin 1550 and CMPy bin 765. The panels on top of (a and b) display the distribution of hx and hy for this gather in red and blue lines, respectively. Figure 19 shows the stacked data cube before and after interpolation. Figures 20 and 21 display a CMPx and CMPy slice of the stacked cube in Figure 19 before and after reconstruction. This example demonstrates the effectiveness of the PMF method to regularize noisy, irregularly sampled data. Although the stacked volume shows a modest improvement in its S/N, this could improve seismic attribute extraction and quantitative interpretation. Indeed, the regularized CMP gathers reveal subtle errors in the applied NMO correction velocity to be refined in subsequent processing. Different Figure 14. Zero-offset gathers for a constant CMPy bin 730. (a) Observed data. (b) Reconstructed data.

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blocks require different parameters. The real data processing experiment demonstrates that the reconstruction quality has a close relationship with the percentage of missing data in each block, the maximum number of iterations, the trade-off parameter (μ), and the S/N of each block. For instance, blocks having a higher number of missing traces and lower S/N require more iterations for the reconstruction. However, in our experiments with real data, we set the same parameters for all blocks. This leads to the amplitudes of the recovered data to appear dim, especially in near-offset sections, where gap sizes are typically large. This can be seen in Figures 14 (near offset section) and 16 (midrange offset section). A solution to this problem is to test for optimal reconstruction parameters to be Figure 15. Zero-offset gathers for a constant CMPx bin 1565. (a) Observed data. (b) Reconstructed data.

Figure 16. Mid-offset gathers for a constant CMPy bin 730, hx bin 0, hy bin 6. (a) Observed data. (b) Reconstructed data.

applied to different regions of the data set. A practical alternative is to scale the amplitudes of the reconstructed data by the ratio of the median root-mean-square (rms) amplitude of the original and reconstructed traces over a specific window. Considering that the rms amplitude should vary smoothly in the spatial dimensions provides a stable method to compensate for low-amplitude reconstructed traces.

PARAMETER SELECTION The PMF reconstruction algorithm requires that the user provides two parameters: the rank r and the trade-off parameter μ or α. It is

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5D seismic reconstruction: PMF method important to understand the performance of the algorithm when the true rank of the underlying fully sampled data is not known. To test the recovery properties of the PMF method, we have carried out two experiments. In the first experiment, we have created a 5D data set that is composed of three linear events. In the second experiment, we have created a 5D data set composed of four curved events. Both data sets contain 301 samples in time and 12 × 12 × 12 × 12 spatial samples. We have not added noise to the data, and therefore we have used equation 16 with α ¼ 1. The performance of the reconstruction as a function of rank is examined by the following procedure. For a given rank r and a level of random decimation of the data, we run 50 reconstructions. Each reconstruction uses a different pattern of random sampling with the same level of decimation. We count how many times the reconstruction is successful by defining that a successful run is a solution where the reconstruction quality Q ≥ 13 (relative error of 0.05). We repeat the procedure for the rank parameter varying in the interval r ∈ ½1; 10 and for decimation levels of 10%; 20%; : : : ; 90%. The results are shown in Figures 22 (linear events) and 23 (curved events). The color scale indicates the probability of success. For instance, white means that in 50 times out of 50, the recovery yields a solution with Q ≥ 13. Black, on the other hand, signifies that none of the realizations succeeded in recovering the data. From these experiments, we can draw important conclusions. First, we see that for moderate levels of decimations (e.g., 50%), a wide range of ranks yields accurate reconstructions. On the other hand, for decimations of the order of 80%–90%, a more precise knowledge of the rank is needed. These results are encouraging because they stress that precise knowledge of the rank parameter is not a stringent requirement for the recovery of the unknown data when approximately 50% of the data were decimated. In the case of real data with noise, the parameters α and r can be examined by carrying out the analysis on one patch of 5D data. For instance, one can run a reconstruction with different values of α and r and examine the quality of the reconstruction by, for instance, visualizing the stacked or the reconstruction along different slices of the patch. In general, if the reconstruction yields results that are overly smooth, then we recommend increasing α. In our real data

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Figure 18. CMP gather for a fixed CMPx bin 1560 and CMPy bin 765. (a) Before reconstruction. (b) After reconstruction.

Figure 17. Far-offset gathers for a constant CMPy bin 763, hx bin 0, and hy bin 3. (a) Observed data. (b) Reconstructed data.

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Figure 19. The stacked cube before and after reconstruction. (a) Stacked cube of the observed data. (b) Stacked cube of the reconstructed data. (c) Stacked cube using only the recovered missing traces. From the thick red rectangular area in panels (b and c), we can see that the diffraction curvatures and the discontinue points are more clear than in panel (a).

Figure 20. A CMPx slice view of the stacked cube at fixed CMPy bin 705. (a) Before reconstruction. (b) After reconstruction. (c) Stacked obtained by only using the recovered missing traces. We find that panel (c) has a similar geologic structure to panels (b and a). This similarity indicates that the proposed PMF reconstruction method is credible. Ths can also be seen in Figure 20.

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examples, we used α ¼ 0.4 and rk ¼ 4, k ¼ 1; 2; 3; 4. One can use a test to determine the optimal parameters (Ely et al., 2013). However, for our real data examples, we preferred to use a more practical ap-

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proach, and we have considered the visual inspection of residuals and reconstruction results, as it is often done in seismic data processing when one needs to select a trade-off parameter.

Figure 21. The CMPy slice view of the stacked cube at fixed CMPx bin 1600. (a) Before reconstruction. (b) After reconstruction. (c) Stacked obtained by only using the recovered missing traces.

Figure 22. Phase transition plot for the proposed PMF method for 5D data model with three linear events. The white color represents the case when the data are successfully recovered. The black color indicates that the reconstruction has failed. The decimated data ranging from 10% to 90% are successfully recovered when the rank r ∈ ½3; 6.

Figure 23. Phase transition plot for the proposed PMF method for a 5D data model with four curve events. The white color represents the case when the data are successfully recovered. The black color indicates that the reconstruction has failed. The decimated data ranging from 10% to 90% are successfully recovered when the rank r ∈ ½4; 6.

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CONCLUSIONS In this article, we have presented a method for multidimensional seismic data reconstruction via low rank tensor completion. The proposed algorithm (PMF) applies low-rank matrix factorization to mode unfoldings of the tensor and apply an alternating minimization algorithm to estimate the complete data tensor. Contrary to other low-rank reconstruction techniques based on the SVD algorithm, such as the nuclear norm minimization method or the HOSVD method, the proposed method is an SVD-free approach. The main computational cost of the method resides in the solving the pseudoinverse of small matrices. One of the main obstacles that might prevent industrial applications of tensor completion reconstruction is the computational cost of classical factorization implemented via the SVD. By adopting the PMF method, we have improved the computational efficiency of our previously developed algorithms (HOSVD and minimum nuclear norm reconstruction). However, we are still not clear on the benefits of tensor completion methods in comparison with well-tested and robust methods that are based on Fourier reconstruction techniques, such as MWNI, POCS, or the antileakage Fourier transform. Nevertheless, tensor (multilinear) algebra continues to be a buoyant field for the exploration of new methods to process seismic data, such as seismic denoising, seismic coding and decoding, seismic attribute analysis, and so on.

ACKNOWLEDGMENTS This research was supported by the sponsors of the Signal Analysis and Imaging Group at the University of Alberta and by an Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant (Fundamental and Applied Studies in Seismic Data Preconditioning and Inversion). J. Gao appreciates the financial support from the National Natural Science Foundation of China (no. 41304102), the Fundamental Research Fund for Central Universities (no. 2652013048), and the Fundamental Research Fund (no. GDL1206) for the Key Laboratory of Geo-detection (China University of Geosciences, Beijing), Ministry of Education. We also value the comments from N. Kreimer and the other anonymous reviewers, which have improved the quality of our paper.

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