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What can machine learning do for seismic data processing?
An interpolation application
Yongna Jia1and Jianwei Ma1
ABSTRACT
Machine learning (ML) systems can automatically mine data
sets for hidden features or relationships. Recently, ML methods
have become increasingly used within many scientific fields.
We have evaluated common applications of ML, and then we
developed a novel method based on the classic ML method of
support vector regression (SVR) for reconstructing seismic data
from under-sampled or missing traces. First, the SVR method
mines a continuous regression hyperplane from training data
that indicates the hidden relationship between input data with
missing traces and output completed data, and then it interpo-
lates missing seismic traces for other input data by using the
learned hyperplane. The key idea of our new ML method is sig-
nificantly different from that of many previous interpolation
methods. Our method depends on the characteristics of the
training data, rather than the assumptions of linear events, spar-
sity, or low rank. Therefore, it can break out the previous assump-
tions or constraints and show universality to different data sets. In
addition, our method dramatically reduces the manual workload;
for example, it allows users to avoid selecting the window size
parameters, as is required for methods based on the assumption of
linear events. The ML method facilitates intelligent interpolation
between data sets with similar geomorphological structures,
which can significantly reduce costs in engineering applications.
Furthermore, we combine a sparse transform called the data-
driven tight frame (so-called compressed learning) with the SVR
method to improve the training performance, in which the train-
ing is implemented in a sparse coefficient domain rather than in
the data domain. Numerical experiments show the competitive
performance of our method in comparison with the traditional
f-xinterpolation method.
INTRODUCTION
With the large amounts of data preserved by the Internet, machine
learning (ML) is emerging as a new kind of algorithm designed to
automatically learn the features and relationships hidden in large
data sets. This is often a very attractive alternative to performing the
same work manually. MLs have emerged as workhorses for many
applications, including (but not limited to) spam filters (Androut-
sopoulos et al., 2000;Guzella and Caminhas, 2009), recommender
systems (Bobadilla et al., 2013), credit scoring (Huang et al., 2007),
fraud detection (Ravisankar et al., 2011), and stock trading (Huang
et al., 2005). A vast array of previous research has established that
the primary tools of ML include linear/logistic regression, artificial
neural networks (Haykin, 2004), support vector machines (SVMs)
(Burges, 1998), decision trees (Murthy, 1998), and instance-based
learning (Dutton and Conroy, 1996). The primary functions
performed by ML include classification (for discrete outputs),
regression (for continuous outputs), cluster, association analysis
(Brijs et al., 1999), and anomaly detection (Hassan et al., 2015).
These techniques have a variety of applications. For example, ML
regression (Kwiatkowska and Fargion, 2003) is a promising tool for
improving data mergers, and SVM can facilitate satellite ocean
color sensor cross-calibrations. In addition, clustering can be used
to increase the efficiency and accuracy of regression. As another
example, detecting abnormal wafers can help workers to find faults
in the semiconductor domain (Hassan et al., 2015), and using SVM
to classify random-input sensor data can allow workers to recognize
motor fault (Banerjee and Das, 2012). Overall, due to its good per-
formance, ML has spread rapidly throughout multiple fields, sug-
gesting that it is likely to drive the next big wave of innovation.
What can ML do for seismic data processing? Despite the above-
listed advances, it is still unclear how ML can be used to improve
seismic exploration. ML-related techniques have preliminarily been
applied in the field of reservoir characterization to determine param-
Manuscript received by the Editor 7 June 2016; revised manuscript received 30 December 2016; published online 15 March 2017.
1Harbin Institute of Technology, Department of Mathematics, Harbin, China. E-mail: jiayongna123@163.com; jma@hit.edu.cn.
© 2017 Society of Exploration Geophysicists. All rights reserved.
V163
GEOPHYSICS, VOL. 82, NO. 3 (MAY-JUNE 2017); P. V163–V177, 19 FIGS., 6 TABLES.
10.1190/GEO2016-0300.1
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eters such as sand fraction, shale fraction, porosity, and permeabil-
ity. In previously published research, Lim (2005) is able to charac-
terize these reservoir properties using fuzzy logic and neural
networks from well data, and Helmy et al. (2010) propose the use
of hybrid computational models to characterize oil and gas reser-
voirs. Zhang et al. (2014) propose ML-based automated fault de-
tection in premigrated seismic data. This method generates a set of
seismic traces from velocity models containing faults with varying
locality and other properties, which then uses these known exam-
ples to train a ML model to identify the presence or locality of faults
in previously unseen traces.
Inspired by ML, seismic data interpolation can be regarded as
a regression problem for continuous output. In other words, ML
methods can be used to generate an approximate function (a con-
tinuous hyperplane for an interpolation projection). In this paper,
we attempt to interpolate seismic data using ML. The subsequent
seismic processing steps, including multiple suppression, migra-
tion, and imaging, among others, generally require a dense seismic
record. However, due to economic and physical constraints, seismic
records are often distributed sparsely, or they are often missing
traces. These missing traces can be reconstructed using interpola-
tion, which can greatly decrease this economic cost.
Several seismic interpolation methods currently exist. Some re-
searchers have proposed methods assuming that the seismic record
of an original section comprises a limited number of linear events.
For example, Spitz (1991) assumes the existence of a series of linear
events and proposes one classical first-order f-x
interpolation method that interpolates missing
traces by using a set of linear equations. In this
case, one-step predictability may be seen as a
necessary but insufficient condition for signals
formed of linear events. Therefore, a suboptimal
solution can instead be generated for the cases of
curved events. Data with curved events are proc-
essed by using window techniques. However, the
quality of these reconstructions is significantly
influenced by window parameters. The method
proposed by Spitz has been extended to other do-
mains, including the time-spatial domain (Claerb-
out, 1992), the frequency-wavenumber domain
(Gülünay, 2003), and the curvelet domain (Naghi-
Figure 1. Example of transforming to a linear classification. (Left) The separated line in
R2is an ellipse (nonlinear). (Right) The separated hyperplane in R3is changed to a linear
hyperplane via a mapping Φ.
0 1 2 x* 5 6 7
−1.5
y*
0
1.5
Feature vector : x
Label : y
(x,y)
y = f(x)
(x*,y*)
a)
b)
Figure 3. Schematic diagram depicting the process of ML. In the
training stage, ML methods are used to mine an approximate func-
tion y¼fðxÞto fit training set (x,y). In the prediction stage, one
can input the feature vector xinto the trained function fðxÞto ob-
tain the unknown label y.
Figure 2. Explanation of the relationship between the classification
and regression. (a) The ε-insensitive loss function. (b) The regres-
sion hyperplane fðxÞ. The separating hyperplane of the convex hull
can guarantee the training data within the threshold of ε.
V164 Jia and Ma
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zadeh and Sacchi, 2010). Sparsity-promoting
methods, e.g., the Fourier transform (Sacchi et al.,
1998;Liu and Sacchi, 2004), curvelet transform
(Herrmann and Hennenfent, 2008), and dictionary
learning (Liang et al., 2014;Yu et a l . , 2 0 1 5 ), are
also popular in the field of seismic data interpo-
lation. These sparsity-promoting methods assume
that reconstructed data should be sparser than ob-
served data with missing traces in these transform
domains. Low-rank methods (which attempt to
achieve sparsity for singular values of seismic
data) have also attracted attention in recent years.
These methods assume that seismic data are of
low-rank structures after some pretransformation
ways, such as texture-patch mapping (Ma, 2013;
Yang et al., 2013), Hankel re-embedding (Trickett
et al., 2010;Oropeza and Sacchi, 2011;Naghiza-
deh and Sacchi, 2012;Jia et al., 2016), and
coordinate transformation (Kumar et al., 2013).
Under the assumption of low-rank conditions,
the interpolation problem can thus be translated
to become a rank-reduction matrix completion
problem.
In this paper, we use the support vector regres-
sion (SVR) method (Drucker et al., 1997), a
state-of-the-art ML regression tool, for learning
interpolation of seismic data. The use of SVR is
motivated by three factors: (1) SVR has a solid
theoretical foundation, and it can transform a
nonlinear classification/regression problem in
a low-dimensional space to a linear problem in a
high-dimensional space. Linear regression can be
done more easily. (2) SVR has a good generation
ability in predicting output labels for input data.
(3) SVR is effective at function approximation,
especially in cases with a high-dimensional input
space. It has been successfully used in other
fields, facilitating such programs as traveltime
prediction in intelligent transportation systems
(Wu et al., 2004), wind-speed forecasting
(Mohandes et al., 2004;Santamaría-Bonfil et al.,
2016), and image superresolution (Ni and
Nguyen, 2007). In this paper, the interpolation
project can be learned by a multivariate function
(a hyperplane) from the given example data sets.
Then, the function can be used for prediction and
interpolation of input data with missing traces,
without making any preassumptions about linear
events, sparsity, or low rank. Furthermore, a new
adaptive sparse transform, referred to as the data-
driven tight frame (DDTF) (Cai et al., 2014;Yu
et al., 2015), is combined with SVR to improve
its performance. The learning is implemented in
a sparse coefficient domain rather than in the
original data domain. The sparse transform is
helpful in increasing learning efficiency. Numeri-
cal experiments performed on varying data
demonstrate the applicability and competitive
performance of our method.
Figure 4. Schematic diagram of Gauss SVR. I: A local image patch centered the pixel at
ði; jÞ. II: The feature vector can be generated after the Gauss-weighted process.
Figure 5. Flow chart depicting the process of interpolation method based on SVR.
Algorithm 1. Interpolation method based on SVR.
Input: Exemplified seismic data M¼fMtg;t¼1;2; :::, missing seismic data Y,
sample matrix F, patch size m.
1. Training stage
1) Down sample the exemplified seismic data M¼fMtgby F,t¼1;2;:::.
2) Calculate the initial preinterpolation data using the bicubic method.
3) Obtain the m2dimensional feature vectors xij and their corresponding labels yij .
4) Input all the pairs of ðxij;y
ijÞto the SVR system and build a continuous
regression function (hyperplane, fðxÞ).
2. Prediction stage
1) Apply the bicubic interpolation to the missing seismic data Y.
2) Extract feature vectors in the same manner as in the training stage.
3) Input all feature vectors into the regression function (hyperplane, fðxÞ) and
obtain the missing labels.
Output: Reconstructed seismic data ~
Y.
Figure 6. Schematic diagram of DDTF SVR. I: A local image patch centered the pixel at
ði; jÞ. II: The coefficients in the DDTF domain of each patch are equal to WTPijMBI . III:
Convert the coefficient matrix to the feature vector.
ML for seismic data interpolation V165
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THEORY
A review of SVR
In ML, SVMs are supervised learning models used for classifi-
cation, regression, and other learning tasks. Support vector classi-
fication is used for data classification, which produces discrete
outputs, and SVR is used for data fitting and regression, which pro-
duces continuous outputs. Here, first, we briefly explain the termi-
nology of regression. Suppose that we are given a training set with n
point pairs as
Ω¼fðxi;y
iÞ;i¼1;2; :::;nÞg;(1)
where xi∈Rdis the feature vector (e.g., the local neighbor infor-
mation of a sampling point) and yi∈Ris the corresponding label of
xi. Solving regression problems requires the construction of an
approximate function fðxÞextending from xto y(y¼fðxÞ)to
fit these point pairs. This function fðxÞis also used to predict other
labels yin point pairs (x,y), in which the feature vector is known
and the label is unknown. These unknown labels ycan thus be
output after the feature vectors xare input into function fðxÞ.
The function form fðxÞof SVR (regression) is generated from
support vector classification, a technique that assumes the nonli-
nearly distributed sample points can be separated linearly if they are
projected to a high-dimensional space through a mapping. Figure 1
presents a simple example (Vapnik, 1995) to explain this idea.
In considering the mapping Φ:R2→R3with Φðx1;x
2Þ¼
ð½x12;½x22;ffiffiffi
2
px1x2Þ, support vector classification can separate
the elliptically distributed points in R2into two categories with a
linear hyperplane in R3. This example reveals that support vector
classification can use a mapping to transform a nonlinear classifi-
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a)
b)
Figure 7. (a) Original synthetic data with linear events. (b) Deci-
mated data with 75% regular missing traces (1∕a, where a¼4).
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Figure 8. Four exemplified seismic data used to
support training point pairs.
V166 Jia and Ma
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cation problem in low-dimensional space to a linear problem in a
high-dimensional space. This linear classification can therefore be
made fairly easily.
The use of SVR to perform for data regression is expressed with
an ε-insensitive loss function Lε(Drucker et al., 1997) (also as
shown in Figure 2a):
Lε¼0jy−fðxÞj ≤ε;
jy−fðxÞj −εotherwise;(2)
in which εrepresents the ε-insensitive loss parameter and the loss
function ignores the error within the threshold of ε. In other words,
if the requirement of error jfðxiÞ−yij≤εis fulfilled, the prediction
value fðxiÞis equal to the label yi. Based on this loss function, the
approximate function should produce as many data pairs as possible
in the threshold. Therefore, the regression hyperplane fðxÞshould
exist as the separating hyperplane of the convex hull (Figure 2b):
Dþ¼fðxT
i;y
iþεÞT;i¼1; :::;ng;(3)
D−¼fðxT
i;y
i−εÞT;i¼1; :::;ng:(4)
By taking the example provided in Figure 1into account, the re-
gression hyperplane fðxÞcan be of the linear form in a high-dimen-
sional space with a mapping ϕ:
fðxÞ¼hw; ϕðxÞi þ b: (5)
The optimization problem to calculate the regression function
fðxÞis to minimize kwk2and guarantee the absolute error jfðxiÞ−
yijin the range of ε:
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a)
b)
c)
d)
Figure 9. Reconstruction results (1∕a,where
a¼4)by(a)bicubic(S∕N¼27.80 dB), (b) f-x
(S∕N¼38.97 dB), (c) Gauss SVR (S∕N¼
41.05 dB), and (d) DDTF SVR (S∕N¼
42.10 dB).
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b)
Figure 10. (a) Original synthetic data with curved events. (b) Deci-
mated data with sampling ratio 1∕a, where a¼3.
ML for seismic data interpolation V167
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min
w;b;ξp;ξ
p
1
2kwk2þCX
n
p¼1ðξpþξ
pÞ
s:t:8
>
<
>
:
yi−ð<ω;ϕðxiÞ>þbÞ≤εþξp
ð<ω;ϕðxiÞ>þbÞ−yi≤εþξ
p
ξp;ξ
p≥0
;(6)
where ξpand ξ
prepresent the slack variables that can search for the
optimal solution in a larger feasible region.
Equation 6can be solved by using its convex dual problem, in
which the solution is expressed as
fðxÞ¼X
n
i¼1ðαi−α
iÞhϕðxiÞ;ϕðxÞi þ b;
¼X
n
i¼1ðαi−α
iÞKhxi;xiþb; (7)
where αand αrepresent the dual variables.
Equation 7shows that the mapping ϕcan be
an implicit mapping via the kernel function
Khxi;xi. This kernel function can be selected
as linear kernel, polynomial kernel, or Gauss ra-
dial basis function, among others. The inputs into
SVR are the known point pairs in equation 1, and
the output is a regression function (which is the
hyperplane fðxÞ). More details about SVR can
be found in Smola and Schölkopf (2004).
Interpolation method based on SVR
In this section, we first provide clarification by
briefly describing the interpolation method using
the ML-based regression as shown in Figure 3.
Some point pairs (x/feature vector and y/label)
such as the training set (hollow blue circles)
are known, and ML methods are used to mine
a regression function y¼fðxÞhidden in these
training point pairs. Other unknown labels y
in (x,y) can be output after the feature vectors
xare input to the trained function y¼fðxÞ.For
use in seismic data interpolation, it is necessary
to include a rule to transform the seismic data
into the form of point pairs (feature vector, label)
and to assign missing pixel values as the un-
known labels. Missing data should be first pre-
interpolated by using the bicubic method or
another method. A local patch in the preinterpo-
lated data is set as a feature vector, with the cor-
responding true pixel value set as a label. The
details of our SVR-based interpolation method
are described as follows.
Our method comprises two stages: the training
stage and the prediction stage. In the training
stage, we choose several examples without miss-
ing traces, whose geomorphological structures
are similar to those of the interpolated seismic
data. To distinguish between the two kinds of
seismic data, we categorize these examples as
either exemplified seismic data or missing seis-
mic data. As mentioned above, the use of SVR
requires that seismic data are transformed to a
new form (feature vector, label). To do this, we
first down-sample the exemplified data by using
the same sample matrix as the missing seismic
data. Second, we use bicubic interpolation to
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a) b)
c) d)
e) f)
g) h)
Figure 11. Reconstruction results (1∕a,wherea¼3)by(a)bicubic(S∕N¼28.64 dB),
(c) f-x(S∕N¼35.63 dB), (e) Gauss SVR (S∕N¼32.27 dB)and(g)DDTFSVR
(S∕N¼33.02 dB). (b), (d), (f) and (h) are their corresponding trace comparisons, respec-
tively. The dotted line represents the original trace, and the solid line represents the re-
constructed trace.
V168 Jia and Ma
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carry out an initial preinterpolation of the down-sampled exempli-
fied data. It is also possible to use other methods, such as f-x
method, to preinterpolate these down-sampled exemplified data
(see below). Next, for each pixel value at location ði; jÞin the pre-
interpolation matrix, we take a local image patch of size m×mcen-
tered at ði; jÞ. This patch is then weighted by a matrix, which is
constructed from a 2D Gaussian distribution. Finally, this weighted
patch is converted to a row vector (Figure 4):
xij ¼vecðWGðPijMBI ÞÞ;(8)
where MBI is the preinterpolation data obtained using bicubic in-
terpolation and Pij is the patch extraction operator. It is noted that
the patch notation is used to represent the local information of a
pixel, which is not the same as that used in patch-based seismic data
processing (Bonar and Sacchi, 2012). Here, WGrepresents the
weighted way through the Gaussian matrix. The function vec can
reshape a matrix of size m×mto a row vector of size 1×m2.
Therefore, an m2dimensional feature vector xij and its correspond-
ing label yij, the pixel value at position ði; jÞin the exemplified
seismic data, have been obtained.
Following this initial computation, we input all point pairs from
the exemplified data into the SVR system. Then, a continuous re-
gression function (hyperplane, fðxÞ) can be generated, and it is
saved for future use in the prediction stage.
In the prediction stage, the labels/pixel values of missing seismic
data are unknown, but they can be generated using their feature vec-
tors. Constructing feature vectors is done here in the same way as it
is done in the training stage. Note that the missing seismic data are
preinterpolated first with the bicubic method. All feature vectors are
input simultaneously into the regression function (hyperplane, fðxÞ)
trained in the previous stage, thus allowing us to obtain the labels
(missing pixel values). We call this interpolation method Gauss
SVR. The main steps in our interpolation algorithm are given in
Algorithm 1and further explained in Figure 5.
Interpolation method combining the SVR and sparse
transforms
The feature vectors in classical SVR contain only their pixel val-
ues and Gauss-weighted local information. Including more informa-
tion in these feature vectors will likely improve the performance of
our method. For example, Chaplot et al. (2006) propose an idea to
combine wavelets and ML methods to classify magnetic resonance
images of the human brain. After using wavelets as the input to
SVM, they were able to achieve a good classification percentage
of more than 94%.
Similar to wavelet transforms, which have fixed and known basic
functions, the DDTF method (Cai et al., 2014;Liang et al., 2014;Yu
et al., 2015) is an adaptive sparse transform used for learning the
filters from the given data (see Appendix Afor more details). In-
spired by Chaplot et al.’s. (2006) work, we propose an interpolation
method by combining SVR and DDTF, referred to as DDTF SVR.
This new method differs only from Gauss SVR by the extraction
method of feature vectors. DDTF SVR is implemented in a sparse
domain under a DDTF tight frame, rather than in the data domain
(Figure 6). After the adaptive DDTF filter Wis generated, the co-
efficients of each patch are equal to WTPijMBI. Feature vectors are
then obtained by converting WTPijMBI to row vectors. Noted that
the corresponding labels in the training sets are the true pixel values
in the exemplified data, the same as Gauss SVR.
RESULTS AND DISCUSSION
We tested the Gauss SVR and DDTF SVR methods on synthetic
and field seismic data, and we compared the results with those of
bicubic interpolation (Keys, 1981)andSpitz’s(1991)f-xmethod.
To implement SVR, we used LibSVM (Chang and Lin, 2001)and
the Gauss radial basis function as the kernel function. We used a set
of 2D seismic data that includes one spatial dimension and one
temporal dimension. The interpolation addresses a regular sampling
problem, and the sampling ratio is 1∕a, which represents the sample
one trace out of atraces. The local patch size is set as 3×3; therefore,
the length of feature vector is nine. The reconstruction quality is
measured by using the signal-to-noise ratio (S/N), which is expressed
as
S∕NðdBÞ¼10 log10kIk2
F
kIn−Ik2
F;(9)
where Inand Irepresent the reconstructed data and the original data,
respectively. This numerical analysis was performed using MATLAB
on a PC with Windows 7, Intel core i-5, 3.2 GHz CPU, and
8GBRAM.
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a)
b)
Figure 12. (a) Original field data. (b) Decimated data with 50%
regular missing traces (1∕a, where a¼2).
ML for seismic data interpolation V169
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Synthetic example
Data with linear events
Synthetic seismic data with linear events are shown in Figure 7a.
In this section, we present a simple comparison of our results with
those of the bicubic method and the f-xmethod in a case of 25%
regular sampling ratio (1∕a, where a¼4) interpolation (Figure 7b).
Figure 8shows the four exemplified seismic data sets without miss-
ing data, from which 63,504 training point pairs of feature vectors
and labels can be extracted. The interpolation results are shown in
Figure 9. The S/N values of the four different methods (bicubic,
f-x, Gauss SVR, and DDTF SVR) are 27.80, 38.97, 41.05, and
42.10 dB, respectively. The fact that the S/N values obtained using
Gauss SVR and DDTF SVR are higher than those obtained by the
bicubic and f-xmethods indicates that our methods perform suc-
cessfully.
Data with curved events
Synthetic seismic data with curved events are depicted in Fig-
ure 10a. The size of these data is 1000 ×128, corresponding to the
total number of samples along the temporal and spatial directions,
respectively. As shown in Figure 10b, the sampling ratio is 1∕a,
where a¼3. Here, we classify four seismic data sets with three
different curved events as exemplified data. Because the only differ-
ence in the four exemplified data sets is the curvature, we do not
show the images here. Figure 11 shows reconstructed results and
their trace comparisons obtained using the bicubic, f-x,Gauss
SVR, and DDTF SVR methods. In this scenario, based on their
S/N values, f-xperforms better than SVR. Because optimal window
parameters must often be adjusted in the f-xmethod, its S/N values
are not always higher than those of SVR. As is common in ML tech-
niques, the effectiveness of SVR depends on the training set. Future
work should focus on the improvement of SVR in this respect.
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Figure 13. Eight exemplified field seismic data sets used to support the training point pairs.
V170 Jia and Ma
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Field data example
To better compare our new methods with the f-xmethod, we
tested a series of experiments on a field data set with size
128 ×128 (Figure 12a). The seismic data have been down-sampled
by a regular 50% (1∕a, where a¼2) sample matrix, which is
shown in Figure 12b. To construct the regression function, we select
eight field seismic data sets (Figure 13). These are actually eight
patches from a very large field data set, used as a simulation. These
exemplified data can provide 127,008 training point pairs. Figure 14
records the interpolation results and their trace comparisons ob-
tained using the f-x, Gauss SVR, and DDTF SVR methods. Our
proposed SVR-based methods yield slightly lower S/N values than
those of the f-xmethod. In this example, we test many experiments
to determine the best parameters (e.g., window size) for the f-x
method to achieve high S/N values.
To make the results in reconstruction quality more convincing,
the S/N values versus different sampling ratios (1∕a, where a¼
2, 3, 4) by using bicubic, f-x, Gauss SVR, and DDTF SVR are
recorded in Table 1. Of these four methods, DDTF SVR gets the
best quality with the sampling ratio 1∕a, where a¼3, 4, and it
yields a slightly smaller S/N value than f-xunder a¼2. In addi-
tion, our method can deal with interpolation problems of real field
data more intelligently, without needing to set complex parameters.
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Figure 14. First row: reconstruction results. Second row: trace comparison. Third row: magnified view of the rectangular area. The dotted line
represents the original trace, and the solid line represents the reconstructed trace. Methods used from the left to the right are as follows: f-x
(S∕N¼34.21 dB), Gauss SVR (S∕N¼32.20 dB), and DDTF SVR (S∕N¼32.54 dB).
Table 1. Comparison of S/N values (dB) obtained using differ-
ent methods.
a¼2a¼3a¼4
Bicubic 26.42 21.94 17.52
f-x34.21 21.58 18.04
Gauss SVR 32.20 24.47 20.83
DDTF SVR 32.54 25.24 22.12
ML for seismic data interpolation V171
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Table 3. Comparison of computational time (s) obtained using different percentages of training set.
Percentage 5% 15% 25% 35% 100%
Gauss SVR a¼2 2.91 0.02 23.77 0.24 63.75 0.24 245 74.02 2668.23
a¼3 2.90 0.01 23.82 0.19 63.62 0.28 207.44 9.58 3022.34
a¼4 2.92 0.01 24.17 0.10 65.20 0.34 252.41 53.38 3045.23
DDTF SVR a¼2 3.41 0.09 27.72 0.62 75.39 1.15 209.27 9.89 6120.54
a¼3 3.08 0.06 25.48 0.28 68.97 0.64 193.97 52.46 4478.94
a¼4 3.03 0.03 24.95 0.19 67.01 0.49 138.77 26.89 3343.21
Table 2. Comparison of S/N values (dB) obtained using different percentages of training set.
Percentage 5% 15% 25% 35% 100%
Gauss SVR a¼2 30.62 0.02 31.79 0.01 31.98 0.01 32.05 0.01 32.20
a¼3 23.05 0.01 23.98 0.01 24.14 0.01 24.23 0.01 24.47
a¼4 19.47 0.02 20.24 0.01 20.43 0.01 20.54 0.01 20.83
DDTF SVR a¼2 32.65 0.11 32.61 0.11 32.61 0.10 32.61 0.09 32.54
a¼3 24.81 0.14 24.99 0.12 25.07 0.10 25.12 0.08 25.24
a¼4 21.19 0.18 21.66 0.14 21.85 0.12 21.95 0.11 22.12
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Figure 15. Six exemplified field seismic data used
to support the training point pairs.
V172 Jia and Ma
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However, the ML training step is extremely time consuming. If
all of the training pairs (127,008) are used in the training stage, the
interpolation experiment takes more than 2500 s. To accelerate our
method, the skill of randomly selecting a certain subset (Reinartz,
2002) as the new training set is used for seismic data interpolation.
For each given sampling ratio, we test 10 times, and the resulting
mean values, along with their standard deviations in terms of S/N
and computational times, are listed in Tables 2and 3. The parameter
x%represents the used percentage of all training sets (out of a total
of 127,008). Table 2demonstrates that 15% of all the training sets
used in the training stage produce a very similar result in S/N with
that of all sets used. However, the computational time (Table 3) de-
creases significantly from more than 2500 s to almost 25 s. For in-
stance, as shown in Table 2, when a¼3and 15% of the training set
is used, the S/N of DDTF SVR is 24.99 0.12 dB. The value is
slightly less than that obtained when all training sets are used
(S∕N¼25.24 dB), whereas the computational time reduces from
4478.94 to 25.48 0.28 s.
Model training for a type of data
The biggest advantage of the ML method is that it can intelli-
gently accomplish tasks assigned by a human, thereby dramatically
reducing the manual workloads. Our method can decrease the com-
plexity in selecting parameters, such as window parameters. In ad-
dition, a trained regression function can be saved for future use in
analyzing a type of seismic data, not only for one particular seismic
data set. The details of this procedure are described as follows. Fig-
ure 15 demonstrates that six seismic data sets are used as exemplified
seismic data to construct a regression function with a 50% sampling
ratio (1∕a,wherea¼2). The trained regression function is then
saved and used to guide the interpolation of three different seismic
data sets (Figure 16). The reconstructed results of the first seismic
data set (shown in Figure 16) as obtained by the f-x, Gauss SVR,
and DDTF SVR are shown in Figure 17a,17d,and17g, respectively.
The second and third rows represent the interpolated results of the
other two seismic data sets. These comparisons reveal that the DDTF
SVR method yields higher S/N values than the Gauss SVR method,
and it is competitive with the f-xmethod.
Discussion and extension
There are several factors that affect the performance of ML
methods, including the choice of preinterpolation method and
the training data set. In our ML methods, we typically use the bi-
cubic method to preinterpolate missing traces before extracting the
feature vectors. In some cases, the reconstructed quality of the f-x
method is slightly higher than that of our ML-based method. There-
fore, we try to use f-xas our chosen preinterpolation method. To
distinguish between the two preinterpolation methods, we refer to
our ML method as bicubic Gauss SVR, bicubic DDTF SVR, f-x
Gauss SVR, and f-xDDTF SVR. The original real field data used
in these test are presented in Figure 12a, and the S/N values obtained
using different sampling ratios are recorded in Table 4.Whena¼2,
the S/N value of the bicubic DDTF SVR (S∕N¼32.54 dB)is
slightly less than that of f-x(S∕N¼34.21 dB). However, f-x
DDTF SVR (S∕N¼34.31 dB) improves the reconstructed quality.
When a¼3, the S/N value of f-x(S∕N¼21.58 dB) is slightly
less than that of the bicubic DDTF SVR (S∕N¼25.24 dB).
Although the f-xDDTF SVR (S∕N¼23.85 dB) achieves a higher
S/N value than the f-x(S∕N¼21.58 dB), its S/N value is still lower
than that of the bicubic DDTF SVR. As mentioned above, the quality
of the reconstructed data is influenced by window parameters in the
f-xmethod, whereas the bicubic is generally easy to use. Therefore,
if one can ignore the slight change in S/N, the bicubic ML method is
likely a better choice.
ML methods work universally with different types of seismic data
and are able to greatly reduce manual workloads. However, the size
of today’s seismic databases often exceeds the size of data sets that
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b)
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Figure 16. Three original seismic data that need to be interpolated
under a same regression function.
ML for seismic data interpolation V173
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SVR can handle and such large quantities of data are not needed
to mine the regression hyperplane. Therefore, the processes of data
preparation, such as data selection and data cleaning, exert strong
controls on the efficiency of our SVR-based ML methods. Future
work should focus on determining how to support effective training
data sets (e.g., in assessing variance and deviation) and how
many training data sets should be used to avoid the over-fitting
problem.
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Figure 17. The reconstructed results (sampling ratio: 1∕a, where a¼2) of three different seismic data under a same regression function.
Methods used from the left to the right: f-x, Gauss SVR, and DDTF SVR. (Top to bottom) Three seismic data are shown in Figure 16. The S/N
values (dB) are (a) 59.51, (b) 62.09, (c) 66.79, (d) 59.15, (e) 61.73, (f) 64.77, (g) 59.40, (h) 62.53, and (i) 66.14.
Table 4. Comparison of S/N values (dB) obtained using differ-
ent preinterpolation methods.
a¼2a¼3a¼4
Bicubic Gauss SVR 32.20 24.47 20.83
DDTF SVR 32.54 25.24 22.12
f-x34.21 21.58 18.04
f-xGauss SVR 34.42 23.54 19.76
DDTF SVR 34.31 23.85 20.05
Table 5. Comparison of S/N values (dB) obtained using differ-
ent ratios with random sampling.
Sampling ratio 10% 30% 50%
Gauss SVR 13.47 19.52 26.61
DDTF SVR 13.77 19.93 26.82
V174 Jia and Ma
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In this section, we discuss some possible extensions of our
method in uses such as varying sampling strategy, simultaneous
denoising, and 5D interpolation. Regular sampling in the seismic
data interpolation problem has already been discussed in the paper.
But how about randomly sampling? Table 5records the S/N values
versus different ratios of the original real field data (Figure 12a)
as obtained by the interpolation problem. The data in this table
demonstrate that our ML methods can adequately solve interpola-
tion problems with random sampling. In this group of experiments,
the bicubic method is used for preinterpolation, and it is likely that
one could improve the S/N values with another more suitable
method for random preinterpolation.
The simultaneous denoising during interpolation is also an issue
in this field. The synthetic seismic data (Figure 7a) with a 50% regu-
lar sampling ratio (1∕a, where a¼2) are taken as an example, with
a partial close-up of these results shown in Figure 18. The interpo-
lation method based on SVR is limited in its ability to simultane-
ously suppress noise. Maybe this is because our regression function
is a continuous function. In the prediction stage, if the feature vector
contains significant noise, the calculated label/pixel value will yield
a certain deviation. Therefore, it is ideal to first suppress noise dur-
ing the process of constructing the feature vector.
In recent years, the seismic industry has also been interested in
5D seismic data interpolation. The 5D data can be viewed as a five-
order tensor consisting of one time dimension and four spatial di-
mensions describing all locations of the sources as well as those of
the receivers on the surface. Different methods have been proposed
to solve this problem, including methods based on the Fourier trans-
form (Trad, 2009;Xu et al., 2010;Chiu, 2014), dictionary learning
(Yu et al., 2015), Hankel or Toeplitz matrix rearranged-based meth-
ods (Gao et al., 2013), and tensor completion-based methods
(Kreimer and Sacchi, 2012;Kreimer et al., 2013). Here, we also
attempt to test a 5D synthetic seismic data set based on ML. Fig-
ure 19a depicts a data set 32 ×16 ×16 ×16 ×16 in size that has
been modeled using the public MATLAB toolbox SeismicLab. The
decimated data with a regular sampling ratio (1∕a, where a¼3) are
shown in Figure 19b. Figure 19c depicts the reconstructed result by
the Gauss SVR, and Figure 19d displays the difference between the
original data (Figure 19a) and the result (Figure 19c). From these
data, it can be seen that our ML method can be used to solve a 5D
interpolation problem with satisfactory reconstructed results.
Currently, in most industries, the interpolation of down-sampled
regular data can be efficiently solved by f-x-ymethods and other
similar algorithms. Actually, not even Fourier techniques are
necessary for that particular problem because f-x-yis more efficient
and simpler. For irregular data, f-x-ytechniques are not suitable,
but Fourier techniques are. However, it is difficult to apply standard
techniques to very complex topographic scenarios with large gaps
or large separations between acquisition lines, on the order of hun-
dreds of meters. The objective of our ML method is to design a
database for interpolation that is suitable for multiple different types
of data, while simultaneously reducing manual labor. Our ML
method can still work with data sets featuring this kind of sparse
sampling, resulting from factors such as large gaps because it is able
to borrow information from training sets. For example, a typical 5D
interpolation of a wide azimuth land data set, will imply infilling a
4D grid (for each frequency), with a typical coverage of 3% (97%
empty cells). It is only through the use of very large multidimen-
sional windows on the order of 500–1000 m per side that we are
able to address these kinds of gaps. However, the application of ML
in seismic data processing is still in its infancy. Much work remains
to be done to make this technique efficient in those conditions and
then apply it to deal with complex structures, complex topography,
and noise. Finally, future work should also focus on using ML meth-
ods to accomplish reverse time migration and full-waveform in-
version.
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Figure 18. Simultaneous interpolation and denois-
ing for 2D synthetic data by Gauss SVR. (a) Origi-
nal clean data (Figure 7a); (b) decimated noisy data
with sampling ratio 1∕a,wherea¼2; (c) the re-
constructed result; and (d) difference between the
reconstructed result and the original clean data.
ML for seismic data interpolation V175
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CONCLUSION
In this paper, we propose an ML method for seismic data inter-
polation. A hidden relationship (called a continuous hyperplane
fðxÞ) can be mined from large amounts of exemplified training sets
and used to obtain missing data. We present the DDTF SVR method
to improve the performance of the Gauss SVR method. Our new
ML-based method allows us to break away from previous assump-
tions made in existing interpolation methods, and it is universally
applicable to varying data sets. Furthermore, the trained regression
function can be saved for future use to interpolate a type of seismic
data with similar geomorphological structure, which is useful in
production seismic processing. In future work, we will focus on
the improvement of SVR and try to investigate the use of deep
learning methods for seismic data processing. Deep learning is a
branch of ML based on a set of algorithms that attempt to model
high-level abstractions in data.
ACKNOWLEDGMENTS
The authors would like to thank the editors and reviewers for
their helpful comments and suggestions that improved this work,
as well as M. Sacchi for providing the SeismicLab toolbox. This
work is supported by National Natural Science Foundation of China
(grant numbers: NSFC 91330108, 41374121, 61327013, and
41625017), and the Fundamental Research Funds for the Central
Universities (grant number: HIT.PIRS.A201501).
APPENDIX A
DDTF
The DDTF can be briefly described as follows. The objective
function for the filters training in DDTF is
argmin
V;W
1
2kV−WPMBIk2
FþλkVk0s:t:WTW¼I; (A-1)
where MBI is the bicubic preinterpolation data, Pdenotes the patch
transform, Wis the matrix dictionary, Vis the coefficient matrix
after expanding PMBI on W, and Iis the identity matrix. Here,
kk2
Fand kk0indicate the Frobenius norm and L0norm (which de-
fines the number of nonzeros in a vector), respectively. The con-
straint WTW¼Iindicates that Wis a tight frame. The W
and Vcan be calculated alternatively. More details about DDTF
can be found in Cai et al. (2014) and Liang et al. (2014).
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