1 Introduction
Seismic data quality is vital to various geophysical processing. However, due to the financial and physical constraints, the real seismic survey data are usually incomplete. Seismic data reconstruction is a complex problem, and a large number of researchers have devoted themselves to the research of this field. Consequently, many approaches have been proposed to handle this problem. From the point of view of data organization, these methods can be divided into two categories.
The low dimensional based methods, such as transformbased methods which utilize the properties of the seismic trace in an auxiliary domain Hennenfent et al. (2010). Liner prediction theory use the predictability of the signal in the f‐x or t‐x domain Naghizadeh and Sacchi (2007). And methods which exploit the lowrank nature of seismic data embedded in Hankel matrices Oropeza and Sacchi (2011).
Those low dimensional based methods usually ignores the spatial structure of the seismic traces. While the spatial structure coherence is very important for the seismic data completion, hence recent developments in high dimensional tensor completion approaches exploit various tensor decomposition model are widely used in seismic data reconstruction Kreimer et al. (2013); Ely et al. (2015a)
. Those highorder tensor decomposition approaches have became a trend for seismic data completion, and the exist different definitions for tensor decomposition that lead to different tensor completion model, i.e. the higherorder singular value decomposition (HOSVD)
Kreimer and Sacchi (2011), the tuker decomposition Da Silva and Herrmann (2013), and tensor SVD (tSVD) decomposition Ely et al. (2015b).In this paper, we focus on a new seismic data reconstruction algorithm which based on lowtubalrank tensor decomposition model possesses extremely high precision seismic data recovery performance. Because of the high redundancy or coherence between one seismic trace to the others Ely et al. (2015a), we assume that the fulled sampled seismic volume has lowtubalrank property in the tSVD domain. Therefore, we can solve the seismic tensor completion problem through an alternating minimization algorithm for lowtubalrank tensor completion (TubalAltMin) Liu et al. (2016a, b). We have evaluated the performance of this approach on both synthetic and field seismic data.
Notations
The data in seismic survey is a natural highdimensional tensor, such as the 3D poststack seismic data which consists of one time or frequency dimension and two spatial dimensions corresponding to xline and inline directions.
Throughout the paper, we denote those 3D seismic tensor in time domain by uppercase calligraphic letter, , and denote the frequency domain 3D seismic tensor by correspondingly. Uppercase letter denotes matrix, and lowercase boldface letter
denotes vector. Let
denotes the set . In addition, we introduce an important tensor operator tproduct Kilmer et al. (2013).The tensorproduct of and is a tensor of size , , for and .
2 Problem Setup
From what we have stated above, we know that seismic data comprises many traces that provide a spatiotemporal sampling of the reflected wavefield. However, caused by various factors, such lost many important informations. Such as the exist of reservoir, residential or any other obstacle in the seismic data acquisition areas will lead to undersampled seismic record. The missing traces will complicate certain data processing steps such as the prediction accuracy of underground reservoirs. Hence, the completion step in seismic data processing is of grate significance.
In this paper, we explored the relationship between lowtubalrank and undersampled rate firstly, and found that lowtubal property is positively correlated with the sampling rate. Figure 1 shows the detail experiment result. Base on this work, we as
Alg. 1 Tubal Alternating Minimization: AM(, , , )
Input: Observation set and the corresponding elements
, number of iterations , target tubalrank .
1: Initialize(),
2: fft(, [], 3), fft(, [], 3),
fft(, [], 3),
3: for = 1 to do
LS_Y( ),
LS_X( ),
4: end for
5: ifft( , [], 3); ifft( , [], 3),
Output: Pair of tensors ( , ).
sume that the full sampled seismic data volume has lowtubalrank property and undersampled traces will increase the tubal rank. Therefore, the poststack 3D seismic data reconstruction can be tackled with tensor completion tools that using the low tubal rank property. The statement above transform to mathematical representation is that is a 3D seismic data with tubalrank equal to . Then, by observing a set of ’s elements, we get the undersampled seismic data . Then, our aim is to recover from . The tensor reconstruction problem can be formulated as following
optimization function:
(1)  
Here, denote the projection of a tensor onto the observed set , . The TubalAltMin algorithm proposed by Liu recently, which can complete the low tubal rank tensor with very high currency in several iterations is a perfect approach to solve this problem.
Solution
In the TubalAltMin algorithm, the target 3D seismic volume can be decomposed as , , , and is the target tubalrank. With this decomposition, the problem (1) reduces to
(2) 
This cost function can be solved by the alternating minimization algorithm for low tubal rank tensor completion designed by Liu. The main algorithm steps are showing as Alg 1.
The key problem of Alg. 1 is the tensor least square minimization, which was solved by the providing methods in Liu’s paper. The main ideal is to decompose (2) into separate standard least squares minimization problem in the frequency domain. Then, we just need to solve a least square problem like the following form each step:
(3) 
Performance evaluation
To evaluate the performance of the algorithm we adopted, two commonly used evaluation criteria in seismic data completion filed have been used for comparison  the reconstruction error and the convergence speed.

Reconstruction error: here we adopted the relative square error as a scale standard which defined as RSE .

Convergence speed: we measure decreasing rate of the RSE across iterations by linearly fitting the measured RSEs.
In order to have an intuitive performance comparison, we compared with two other seismic data volume completion algorithm. The Parallel matrix factorization algorithm (PMF) Gao et al. (2015) and the tensor singular value decomposition based algorithm, also called tensor nuclear norm algorithm (TNN) Ely et al. (2015b) for seismic reconstruction. We applied those algorithm both on synthetic and real seismic data, the following subsections will demonstrate the detail performance comparison.
2.1 Synthetic data example
We use a Ricker wavelet with central frequency of 40 Hz to generate a simple 3D seismic model with two dipping planes. The seismic data corresponds to a spatial tensor of size , 256 time samples with the time sampling rate of 1 ms and 64 corresponding to inline and xline direction. Then, through tSVD decomposition we get , , and . According to the decomposition result, we choose the first 2 tubes of , and make other tubes elements equate to zeros, form a new tensor . Then we get the tensor which tubalrank is 2.
Firstly, we apply Algorithm 1 to decompose the lowtuablrank tensor of our data set at different sampling rate vary from 5% to 95% and set the tubalrank equate to 2. Using these decompositions, we generate our reconstructed data and compare the error between the lowtubalrank reconstruction and full sampled data. Applying two other algorithm we stated above to reconstruct the data at the same condition. Because of all of the three algorithms include random sampling operator, we averaged those algorithms’ performance under 20 above experiments. Figure 2 shows the relative square error (RSE) of the three algorithms. From these curves, we see that the performance of our algorithm is very outstanding. Even there are only 30% sampling traces, it can also get a relative perfect reconstruction data. As the increase of sampling rate, the reconstruction error of TubalAltMin algorithm decrease rapidly. Comparing with TNN, when the sampling rate over 50%, it improves the recovery error by orders of magnitude. It’s recovery error also better than the other algorithm almost in the whole sampling rate range.
Secondly, to evaluate the reconstruction error, we fixed the sa
mpling rate to be 40% and set the RSE tolerance of the three algorithms equal to 1e4. Then evaluate relative square error and convergence speed for the three algorithms. Figure 3 shows that TubalAltMin convergence at 9th iteration, TNNADMM terminates at the 60th iteration, PMF used 77th iterations to reach the preset threshold. The convergence speed of our algorithm is obviously better than other algorithms.
Then, we fixed the sampling rate at 40% to evaluate the completion performance on seismic data. The portion of reconstruction result shows in figure 4. From figure 4 (b), 4 (c) and 4 (d), we observe that the missing traces are effectively recovered.
2.2 Filed data example
We also test the performance of the reconstruction method on a land data set that was acquired to monitor a heavy oil field in Anyue mountain, China. Our test data is a tensor cut from the fullsampled area of the filed data. On this basis, we randomly sample 50% traces in the seismic tensor, then reconstruct it. To reconstruct the traces, we evaluated the lowtubalrank of the data manually. Determined by the tSVD, we found it’s first 16 eigentubes’s norm is much larger than the rest in almost 40 times. Consequently, we set the parameter of Alg. 1 . Figure 4 shows portion of the result cut along inline direction. From the comparison of figure 5 (a), 5 (c) and
5 (d), the reconstruction performance is such amazing that almost no residual between 5 (a) and 5 (c) Compared with the reconstruction result of TNN, it seems no difference. However, the RSE gap between them almost have a few orders of magnitude. Figure 5 shows the difference between the specific details of the seismic data traces restored by the two algorithms.
Conclusion
In this article, we formulate the problem of poststack seismic data reconstruction as an lowtubalrank decomposition problem. The adopted method is based on the alternating minimization approach for lowtubal rank tensor completion. The unknown lowtubalrank tensor is parameterized as the product of two much smaller tensors with the lowtubalrank property being automatically incorporated, and TubalAltMin alternates between estimating those two tensors using tensor least squares minimization. The biggest advantage of our algorithm is that it can achieve very high recovery accuracy in several it
erations. From the experimental result, it potently proved that compared with contrast algorithms our algorithm improves the recovery error by orders of magnitude with much better convergence speed for higher sampling rates.
3 Acknowledgments
We would like to acknowledge financial support from the National Natural Science Foundation of China (Grant No.U1562218).
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