Sparse Regularization Based on Orthogonal Tensor Dictionary Learning for Inverse Problems

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Introduction
In exploration geophysics, seismic data processing is an essential task in processing various properties of the earth's subsurface.Due to the physical and budget constraint, only a subset of seismic data is acquired for this process.The acquired seismic data may be subjected to various noise contamination, and some important significant traces may also distort.To increase the resolution of the seismic records, random noise attenuation and interpolation are the two critical steps in seismic data processing.Improving the quality of seismic data by removing random noises and reconstructing the missing irregular seismic traces play a vital role in providing high-resolution processing, oil, and gas exploration, multiple suppression, and wave-equation migration [1].Seismic event detection, migration, and inversion demand a high quality of seismic data [2].Different methods have been proposed for seismic noise attenuation and interpolation [3][4][5].Most methods in seismic noise suppression are applied in the transform domain in which the signal and noise are distinguishable [6].Even though reconstructing the missing seismic traces (interpolation) and attenuating seismic noise (denoising) methods are different in their output, they are nearly identical operations in the process of obtaining clear seismic data at the final [7].
Based on the prediction method in the time-spatial t-x domain [8,9], the frequency-spatial f-x domain [10,11] based on the predictability of linear events without the prior knowledge of lateral coherence of the events has been proposed.In the frequency-wavenumber, f-x domain [7,12] has been proposed by utilizing a convolutional prediction filter computed from the low-frequency parts to predict the highfrequency components.These methods are only applied to the seismic data with linear events and are applied for only regularly sampled seismic data.
Based on the rank-reduction Cadzow method such as truncated singular value decomposition (SVD)-based matrix rank reduction of constant frequency slices for trace interpolation [13], multichannel singular spectrum analysis [3], adaptive rank-reduction based on the energy entropy [14] in solving rank reduction were proposed.Low-rank matrix completion [15,16], tensor higher order SVD [17], and nuclearnorm minimization-based matrix completion [18] have been proposed in seismic data restoration as an extension of Cadzow method.
Based on the wave-equation, seismic data restoration utilizes the inherent constraint of seismic data from wave equation to reconstruct seismic data [1].Different methods have been proposed based on the wave-equation method [19].This method attains good performance in the restoration of seismic data.However, it depends on only the known velocity model.
The fourth method is based on dictionary transform, promoting sparsity in seismic data processing.In seismic data processing, interpolation and denoising are the two inverse problems in which we can employ sparsity constraints [20].The sparse dictionary learning (DL) algorithm has been also proposed to attenuate random noise in a transform-based denoising framework [21].Sparse representation represents input data as the linear combination of basis elements such that most of the representation coefficients are zero [22].For a given signal x 2 R n , the sparsest representation is to find a sparse vector α 2 R m such that y ¼ Dα, where D is the dictionary and each of its columns is an atom.The idea of sparsity in seismic data processing has been used in two different ways: fixed (analytic) basis transform and learning (adaptive) dictionaries.
Fixed basis transforms have been applied in seismic data processing by using a known set of basis functions to estimate representation coefficients of the input data.Wavelet multiresolution analysis [23], physical wavelet transforms [24], Radon transforms [25,26], Fourier transform [27,28], curvelet transforms [29,30], seislet transforms [31,32], and shearlet transform [33] have been proposed.In the nonsubsampled contourlet transform domain, threshold shrink method [34] for denoising seismic data has been also proposed.In analytic DL approaches, the analytic construction is employed, and the mathematical model of the signal is formulated to represent the model.These methods are simple and have fast computation in creating data features for different applications in seismic data processing.
Adaptive/learned DL uses the learned basis functions from the input data to estimate the missing data and restore the degraded data by a sparsity-promoting model.Based on adaptive/learned DL, different methods have been proposed in image processing.The adaptive DL algorithm has also been proposed to attenuate automatic coherent noise [35].This approach can learn the features of both signals and coherent noise and leave obvious morphological differences in the dictionary atoms.Principal component analysis (PCA) [36], generalized PCA [37], method of optimal directions (MOD) [38,39], and K-singular value decomposition (K-SVD) [40] are adaptive/learned dictionary proposed in image science.PCA is widely used in statistics for multivariate analysis.It reduces the dimensionality of the data by preserving the variability of data variables.K-SVD learns an overcomplete dictionary from the noisy image patches and uses the learned dictionary to sparsely represent model and image denoising [41].K-SVD has been proposed for signal processing applications [42] and seismic data denoising [38].However, the input data are considered as a vector in K-SVD, and then at each iteration step, SVD is used to transform the matrix to update each column of the dictionary [20].It leads to the redundancy of dictionaries in which most of the atoms are similar or contribute little role in the representation of input data.K-SVD-based sparse representation has also been used in seismic data denoising and compression [43].In this case, K-SVD is used to sparsely represent the 4D seismic dataset to find a representation with as few coefficients as possible while preserving the main characteristics of the data.In the learning dictionary for sparsity-based seismic data restoration, high redundancy dictionary leads to a poor sparse approximation of data.This approach is computationally infeasible when dealing with seismic data and requires high computational cost [44].Data-driven tight frame (DDTF) [45], which learns dictionary with a prescribed block Toeplitz structure and satisfies the perfect reconstruction property for denoising, has been proposed to reduce the computational complexity of K-SVD.The high performance of DDTF in interpolation and denoising of high dimensional seismic data was analyzed [46,47].
The hybrid method, which combines the advantage of fixed basis and learned-based DL, has been applied in seismic data processing.A double sparsity method which is based on seislets and DDTF [48] was proposed.However, most of these methods deal with vectors leading to loss of seismic data structure and poor sparse representation.To overcome this problem, tensor-based DL methods [49,50] have been proposed for various sparsity-based restoration.In the tensor DL method, the input data are treated as tensors instead of vectors and the dictionary is learned by tensor decomposition [44].The DDTF of Kronecker type (KronTF) [20], tensor decomposition [17], and a Kronecker-based dictionary in dynamic computed tomography [51] have been proposed to avoid the vectorization of input data and applying a learned dictionary based on tensor decomposition.
Many existing DL methods would be computationally infeasible when dealing with high-dimensional seismic data.Moreover, the vectorization method which likely destroys the input data's important information and reduces the discriminability and expressibility of the obtained representation are applied to those methods.In this paper, we have proposed the 2 Mathematical Problems in Engineering orthogonal tensor DL that learns a dictionary from the input data by employing orthogonality and separability.We use this method for seismic data denoising and interpolation.The separability of the dictionary makes the proposed method highly scalable, and the orthogonality among dictionary atoms leads to a very efficient sparse coding computation.The scalability and computational efficiency make the proposed method suitable for processing seismic data in the tensor domain [44].The rest part of this paper is organized as follows.In Section 2, we discussed the basic idea of orthogonal DL and its construction which greatly simplify the computation of both dictionary updating and sparse coding for the sparse representation of input data.In Section 3, orthogonal tensor-based DL and its property are verified.The orthogonal tensor DL is applied to seismic data denoising and interpolation problems.The numerical results are also presented in Section 4. Finally, the conclusion of this paper is addressed in Section 5.

Orthogonal Dictionary Learning
Sparse models in representing input data are an active research area in natural image processing.In a sparse model, the local input data patches are sparsely approximated by the linear combination of atoms in which the collection of these atoms forms a dictionary.The main challenge in sparse representation is the construction of a dictionary which is computationally feasible in seismic data representation.The orthogonal dictionary is the one which can greatly simplify the computation of both dictionary updating and sparse coding for the sparse representation of input data.In orthogonal DL, we aim at finding the orthogonal dictionary b D ¼ ½A; D 2 R n×n whose columns are dictionary atoms such that D T D ¼ I m and A T D ¼ 0 to sparsely approximate the given data Y ¼ fy 1 ; y 2 ; y 3 ; …; y m g 2 R n×m , where n and m are the size of orthogonal matrices.A 2 R n×n−m contains input orthogonal atoms and D 2 R n×m is the set of atoms to be learned from the input data.The corresponding minimization problem to learn orthogonal dictionary b D is formulated as follows [52]: , where V i; j are the vectors or entries of the matrix V and kVjj 0 denotes the number of nonzero coefficients of V. λ>0 is the parameter that balances the trade-off between the approximation term and the sparsity term.The solution of Equation ( 1) is disposed to increase the elements of b D such that V to become as small as possible.To solve Equation (1), we can take the alternating iterative scheme by decomposing the minimization problem into two steps.In this case, the K-SVD algorithm can solve Equation (1) by employing an iterative strategy that alternates between the two steps such that Equation ( 1) is reduced into only one unknown by fixing one as known.
Step 1 (sparse coding): For a given orthogonal dictionary b D, we need to find the sparse code V ðkÞ by solving the following minimization problem: To solve the minimization problem in step 1, we formulate the following remark which is equivalent to Equation (2).
Based on Remark 1 and for b has a unique solution which is given by V ðkÞ ¼ T λ ð b D T YÞ, where T λ is the hard threshold which is defined as follows: Step 2 (dictionary updating): For a given sparse code V, we want to update the dictionary b D by solving the minimization problem: such that D T D ¼ I m and A T D ¼ 0. The minimization problem in Equation ( 4) can be solved by employing the following remark which is equivalent to step 2.
Remark 2. The optimization problem in step 2 is equivalent to the following equation: This minimization problem has a unique solution which is given by the following equation: where R and H are the orthogonal matrices defined by the following SVD: where Σ is the singular value matrix of RH T , and Φ A defined from R n to the space spanned by the columns of the dictionary A is an orthogonal projection operator given by: Φ A l ¼ AðA T lÞ, for all l 2 R n .Finally, we adopt the following orthogonal DL algorithm.
Mathematical Problems in Engineering

Orthogonal Tensor Dictionary Learning
In this section, we extend the orthogonal DL that we have discussed in Section 2 by introducing tensor and learn a tensor dictionary with separability and orthogonality.In the conventional DL approach, the extracted image patches are first transferred into vectors to form training data and then the K-SVD method is used to learn a vector-based dictionary [48].In this strategy, the inherent spatial constraints of the input data structures can be lost or distorted when the dimension of input data is very high.To overcome this problem, the tensor-based method has been investigated for DL to achieve better performance in signal/image processing [53][54][55].Tensor is a multidimensional array [56].An nth order tensor is denoted as Let S M ¼ fD 2 R M×M g be the set of orthogonal square matrices.Then, we aim to learn the set of orthogonal dictionaries fD i 2 S M ; i ¼ 1; 2g to be learned from the input data simultaneously such that D T i D i ¼ I, for i = 1,2 and D T 2 D 1 ¼ 0 in the tensor domain.In this paper, Y 2 R n×m is considered as a tensor of two dimensions which can be written as follows: where D 1 and D 2 are orthogonal matrices from the set S M with the sparse coding tensor V and 2 are the tensor mode product operators.We formulate the following minimization problem to learn these dictionaries: argmin where k:jj 0 denotes the nonzero elements in the tensor domain, and k:jj 2 F is the Frobenius norm of the tensor which is defined as the square roots of the sum of the absolute squares of its elements.We decompose the problem in Equation ( 9) into two subproblems and solve them separately.
Step 1: For a given dictionaries D 1 and D 2 , we need to find the sparse tensor V ðkÞ by solving the minimization problem: To solve for the sparse coding V, we consider the vectorized version of the tensor representation in Equation ( 8) in terms of Kronecker dictionaries which is formulated as follows: where ⊗ represents the Kronecker product.As we discussed in Section 2, we can compute the corresponding sparse coding V 2 R M 1 ×M 2 ×N by applying a component-wise hard thresholding.Then, the solution for the sparse coding V is given by the following equation: where T λ is an operator that keeps the coefficients larger than λ and setting the other coefficients to be zero.
Step 2: For a given sparse coding V ðkÞ , we need to update the dictionaries D 1 and D 2 .The tensor representation Y in Equation ( 8) can be unfolded in order to update the two dictionaries D 1 and D 2 as follows: Then, D 1 and D 2 can be updated by solving the following minimization problem: The dictionary D n (n = 1, 2) can be updated by using an alternating least squares method while fixing the sparse coding V ðkÞ .To solve for the above two dictionaries D 1 and D 2 , we formulate the theorem which is related to the Input: Training dataset Y, Input orthogonal atoms A.
Output: Learned dictionary D. 1 Initialize the dictionary D ð0Þ 2 For k ¼ 0; 1; 2; …; K 3 Compute the sparse coding by using hard threshold V k D ¼ T λ ððD k Þ T VÞ, 4 Run the SVD for the following matrix Mathematical Problems in Engineering minimization problems in Equations ( 14) and ( (15).The proof of this theorem is given in the appendix section.

Application to Seismic Data Interpolation and Denoising.
To evaluate the effect of the proposed method on seismic data processing in terms of quality and computational efficiency, we applied for interpolation (reconstruction of missing traces) and denoising (attenuation of random noise).Let u be the complete data for recovery, f be the observed data, R be the trace sampling matrix which contains only 1 or 0, and ϵ denotes the amount of noise to be added on the input data.Let these data be related by the following model: When ϵ ¼ 0, the model in Equation ( 16) is the interpolation problem that can reconstruct the missing traces and if R is identity matrix I n and ϵ ≠ 0, it corresponds to the denoising problem.After the orthogonal tensor dictionary learned from the input seismic data simultaneously, we used it to propose sparse regularization which promotes sparsity in the seismic data restoration problem.Based on the learned orthogonal tensor dictionary model in Section 3, we formulate the seismic data restoration problem by employing a new sparse regularization.Let B i be an operator which can extract features of seismic data u, then the following minimization problem is formulated to reconstruct the missing traces and denoising seismic data: ð17Þ with the parameters λ 1 and λ 2 need to be determined.The first term in Equation ( 17) is the regularization term that corresponds to tensor-based orthogonal DL and the second term is the fidelity data in the projection domain with the sampling matrix R and the degraded data f.Basically, we can write 2 by Dv i and then Equation ( 17) becomes: We employed the alternating direction method of multipliers (ADMM) to solve Equation ( 18) based on the augmented Lagrangian and the separable paraboloid surrogate method.ADMM method has been widely applied to solve different minimization problems in seismic data processing [57].The surrogate method makes the cost function separable so that all traces of seismic data can be updated simultaneously.Let c ¼ B i u − Dv i and h be the two auxiliary variables used to transform the nonconvex optimization problem into a convex optimization problem, then we can rewrite Equation ( 18) as follows: We split the minimization problem in Equation ( 19) into three subproblems, where λ 1 , λ 2 , and β are the regularization parameters.Subproblem 1: updating DL and sparse coding: Subproblem 2: c-subproblem.The optimization problem for the auxiliary variable c is given by the following equation: Subproblem 3: u-subproblem.The corresponding minimization problem for u is as follows: We have discussed the solution for the one that corresponds to updating DL and sparse coding for subproblem 1 in Section 2. To solve the second subproblem, we apply the optimality condition for Equation ( 22): Then, After the dictionary D updated in the iteration process, the seismic data u should be updated with v i and D. To do so, the separable paraboloid surrogate method [58] is used to solve for u as follows: By using separable paraboloid surrogate method, u kþ1 is given by the following equation: where Then, The denominator of Equation ( 28) is the curvature of the paraboloid which can be obtained as follows:    Mathematical Problems in Engineering Therefore, Hence,  Finally, we formulate the following algorithm for the proposed method.

Numerical Experiments
The numerical examples and the graphs used in this paper were simulated by MATLAB and the machine we used to run the code is DESKTOP-HKHFG0P: Processor: Intel(R) Core(TM) i7-7500U CPU @ 2.70 GHz 2.90 GHz and RAM: 8.00 GB.
We test the validity of the proposed method to noise free and noisy synthetic and real seismic datasets.The proposed method tested on both synthetic and real datasets by the different percentages of missing traces based on Jittered undersampled strategy which can help to obtain random properties as well as control the maximum gap between adjacent traces to meet the requirements of the compressive sensing theory.Throughout the numerical experiments, the patch size was 8 × 8 and the sparsity level was 4. The algorithm in Nazzal et al.'s [59] study is used to obtain the sparsity level.The regularization parameters are tuned based on the convergence condition.Accordingly, the choice of the parameters λ 2 ¼ 1 and λ 1 >0 is trivial.We empirically search for the optimal value of parameter λ 1 from the smallest set f0:0005; 0:0001; 0:005; 0:01g and search for the parameter β from the range of ½0:001; 0:2.To evaluate the quality of restored seismic data quantitatively,  Mathematical Problems in Engineering we introduced the signal-to-noise ratio (SNR) which is defined as follows: where u denotes the clean seismic data and v denotes the reconstructed seismic data.We simulate seismic data using synthetic datasets containing four layers with linear and curved events and synthetic datasets with six layers containing parabolic events.The 50% of missing traces is used based on the random sampling property as shown in Figures 1   and 2. We apply the proposed method to 2D synthetic data reconstruction with linear and curved events as shown in Figures 1 and 2, respectively.The result of the proposed method is compared with the results by K-SVD and orthogonal DL.In Figure 1, both K-SVD and orthogonal DL provide promising results in the reconstruction of missing traces when the seismic data are with the linear events.However, the reconstruction results by the K-SVD and orthogonal DL are not satisfactory when the seismic data are with the nonlinear events because of the elimination of some important features of the reconstructed data around the boundary, as shown in Figures 2(c  Mathematical Problems in Engineering missing seismic traces is presented in Figure 1(e).Furthermore, the residual which is defined as the difference between the interpolated and original seismic data is presented in Figure 3 to show the performance of K-SVD, orthogonal DL, and proposed methods in the reconstruction of missing traces.For the results by K-SVD and orthogonal DL, there are some important features of seismic signals left in the residual sections, as observed in Figures 3(a) and 3(b), respectively.There is no important information about seismic signals left in the residual sections as indicated in Figure 3(c).The minor residual indicates that the important features of the seismic signal are well preserved, and better interpolated seismic data can also be achieved.For further comparison, the SNR of the proposed method, K-SVD, and orthogonal DL are given in Table 1.The proposed method clearly shows much better signal preservation and SNR enhancement while reconstructing the missing seismic traces.
For additional comparison and to see the detail of the interpolated seismic data by the three methods, we plot a single seismic trace from the interpolated data in Figures 4 and 5. 14 Mathematical Problems in Engineering In the next example, we focused on the real seismic data to verify the performance of the proposed method in simultaneous interpolation and denoising and compare it with the K-SVD and orthogonal DL methods.In this case, we randomly remove 50% seismic traces and apply the same method with synthetic data.The real data with 50% missing trace and covered by strong real noise in which detecting the useful seismic signals are hard, and hard to see some events of seismic data are presented in Figure 6(a).The interpolated and denoised results are shown in Figure 6(c)-6(e).The proposed method shows promising performance in removing strong random noise and interpolating the missing seismic traces.Because of the special information contained in the tensor DL, the proposed method achieves better results in the interpolation and denoising of seismic data in terms of visual quality and preservation of seismic features.As shown in Figures 6(c) and 6(d), some important parts of the interpolated and denoised data are deteriorated.The noise sections from the original data (with strong noise) and the recovered data are presented in Figure 7.As presented in Figures 7(a The effectiveness of the proposed method is also verified by interpolating aliasing seismic data containing different features.The original seismic data, observed data with 50% missing traces, and the interpolated results are presented in Figure 8.The reconstructed results by K-SVD and orthogonal DL are unsatisfactory because of the insertion of artifacts on some parts of the interpolated results as shown in Figures 8(c) and 8(d), respectively.The performance of the proposed method in the reconstruction of missing seismic traces is presented in Figure 8(e).The single trace is extracted from the reconstructed data by K-SVD, orthogonal DL, and the proposed method as presented in Figure 9 to compare the details of the obtained results.
For the next example, we discussed the validity of the proposed method for removing random noise and compared it with K-SVD and orthogonal DL.Both synthetic and real seismic datasets are considered for random noise attenuation.Both synthetic and real seismic datasets are contaminated by band-limited Gaussian noise which is spatially uncorrelated random noise.The synthetic seismic data with random noise are presented in Figure 10.The denoised results by K-SVD and orthogonal DL are not satisfactory because of the elimination of some primary features of seismic data around the boundary and they introduce some artifacts as presented in Figures 10(c) and 10(d).Since K-SVD uses the redundancy of the seismic features over the datasets to attenuate random noise, it is computationally more expensive than orthogonal DL and proposed methods.Moreover, the denoising performance of K-SVD, orthogonal DL, and proposed methods are assessed by using the residuals obtained by subtraction of the denoised results from the noisy seismic data.In Figures 11(a For further comparison, a single seismic trace extracted from the denoised results in Figure 10 is displayed in Figure 12, and the comparison of SNR is presented in Table 2 with the different noise levels. Finally, we test the denoising of 2D real data with bandlimited Gaussian noise.The denoised result by the proposed method is shown in Figure 13(e), and the result is clear and the primary features of the data are well preserved.The other two methods cause some damage to the useful seismic signals and some part of the seismic data left in the residual parts as  Mathematical Problems in Engineering   Mathematical Problems in Engineering shown in Figures 14(a) and 14(b).In Table 2, the SNR values of the denoised seismic data for different noise levels are presented.Furthermore, a single seismic trace comparison is displayed in Figure 15 to demonstrate the performance and validity of the proposed method.

Conclusion
In this paper, we exploited sparse regularization which promotes sparsity for both seismic data interpolation and denoising.We have proposed a novel method to interpolate and attenuate random noise of seismic data based on orthogonal tensor DL.Due to the separability, orthogonality, and imposed tensor decomposition, the proposed method is computationally efficient and fast for learning the dictionary.The effectiveness of the proposed method is compared with both K-SVD and orthogonal DL methods in denoising and interpolation of seismic data.The numerical experiments demonstrate that the proposed method shows promising results in the denoising and interpolation of seismic data.Our approach uses the dictionary which can extract the local features and adapt to seismic data, with no introduced artifacts on the interpolated and denoised results.Compared with K-SVD and orthogonal DL methods, the proposed method is computationally effective.The experimental results and the performance of the methods in seismic data interpolation and denoising show the effective performance of the proposed method on synthetic and real seismic datasets.

Appendix
Theorem A.1.Let fD i : D i 2 S M i ; i ¼ 1; 2; 3; …; Ng be the set of orthogonal matrices and for the given Y; V 2 R M 1 ×M 2 ×M 3 …×M N ×R , the minimization problem: ðA:1Þ has a unique solution which is given by the following equation: with orthogonal matrices W and X such that where WΣX T is the SVD of E.
Proof.From Theorem 2.1 in Chrétien and Wei's [60] study, every tensor X 2 R n 1 ×n 2 ×n 3 …×n R can be written as follows: where each D R 2 R n r ×n r is an orthogonal matrix and VðXÞ 2 R n 1 ×n 2 ×n 3 …×n r is tensor of the same size with X.Let ⊗ be the standard Kronecker product for matrices, then: ðA:5Þ where V ðrÞ ðXÞ is the mode r matricization of VðXÞ.For convenience we can use U, V, and A to represent X ðrÞ , D r V ðrÞ ðXÞ, and ðD rþ1 ⊗ D rþ2 ⊗ … ⊗ D R ⊗ D 1 … ⊗ D r−1 Þ, respectively.Then, by using the r mode unfolding and length preservation property of orthogonal transform, we can reformulate the minimization problem of this theorem as follows: argmin ðA:6Þ such that E T E ¼ I.We can write the objective function in Equation (A.6) as follows:

□
Based on this theorem, we can solve for the dictionaries D 1 and D 2 in the same manner.Accordingly, for the minimization problem in Equation ( 14), the dictionary D 1 can be solved by fixing the dictionary D 2 and which is given by the following equation: ðA:17Þ with the SVD ðA:18Þ Similarly, for a fixed D 1 and D 2 is given by the following equation: ðA:19Þ with the SVD T : ðA:20Þ

FIGURE 1 :
FIGURE 1: Interpolated results: (a) original seismic data; (b) seismic data with 50% with missing traces; (c) interpolation by K-SVD method; (d) interpolation by orthogonal DL method; and (e) interpolation result by the proposed method.

FIGURE 2 :
FIGURE 2: (a) Original seismic data and (b) seismic data with 50% missing trace.Interpolation results using by (c) K-SVD method, (d) orthogonal DL method, and (e) the proposed method.

FIGURE 4 :
FIGURE 4: Single trace comparison of the reconstructed synthetic data: (a) trace from K-SVD result, (b) trace from orthogonal DL, and (c) trace from the proposed method.

FIGURE 5 :FIGURE 6 :
FIGURE 5: Single trace comparison of the reconstructed synthetic data in Figure 3 with the original trace from Figure 3(a).Trace from (a) K-SVD result, (b) orthogonal DL result, and (c) the proposed method.
) and 7(b), there are some useful seismic signals left in the noise sections.As we observe from Figure 7(c), there are no parts of seismic signals left in the noise sections.

FIGURE 9 : 8 FIGURE 10 :
FIGURE 9: Reconstructed single trace comparison by (a) K-SVD, (b) orthogonal DL, and (c) proposed method of the interpolated results in Figure 8.
) and 11(b), K-SVD and orthogonal DL residuals show some removed important seismic signals.In Figure 11(c), we observe that almost there were no important seismic signals left in the residual part.

FIGURE 11 :FIGURE 12 :
FIGURE 11: The noise sections from the denoised results by (a) K-SVD, (b) orthogonal DL, and (c) the proposed method corresponding to the one in Figure 10.

FIGURE 15 :
FIGURE 15: Trace comparison between original seismic data in Figure 13(a) and denoised results of Figure 13: (a) original trace, (b) noisy trace, trace from the denoised result by (c) K-SVD, (d) orthogonal DL algorithm, and (e) the proposed method.

TABLE 1 :
SNR values of three different methods applied to synthetic data in Figure2with different sampling ratio.

TABLE 2 :
SNR values of three different methods applied to real seismic dataset in Figure10.
16r U T VE T Since the first two terms in Equation (A.12) are constant, the minimization problem in Equation (A.12)By considering the SVD of U T V ¼ PΣQ T , we can rewrite Equation (A.13) as follows: Þ T ¼ I, since Q is orthogonal.Since Σ is diagonal matrix, then the maximization problem in Equation (A.16) is achieved when the diagonal of ðF * Þ T P is positive and22Mathematical Problems in Engineering maximum.By Cauchy-Schwartz inequality, this is true when F * ¼ P such that the diagonal elements are all 1.Therefore, E ¼ PQ T is the explicit solution for the minimization problem in Equation (A.6).
T E ¼ I. T E ¼ I. *