IEEE Signal Processing - March 2018 - 94

u - WH
arg min X
W, H

2
F

+ m1 W

2
F

+ m2 H

2
F

+ c 1 HH T - B

s.t.W, H $ 0 and t (w i) = t w,

(9)

where matrix B ! R N f # N f contains random positive real numbers, and m 1, m 2, c 1, and t w are constants. To solve the problem
in (9), the following multiplicative update rules for W, and H are
derived, where
Wt + 1 =

and H t + 1 =

u H tT ) ij
W t 9 (X
t t tT
(W H H + m 1 W t) ij

(10)

u + c 1 (B + B T ) H t) ij
H t 9 (W t + 1 T X
. (11)
t + 1T
t+1 t
W H + c 1 H t H tT H t + m 2 H t) ij
(W

Here, 9 represents element-wise multiplication, and the superscript indicates the iteration number. Once W and H are initialized, the multiplicative update rules in (10) and (11) are applied
successively until both W and H converge.
Once W and H have converged, each column of H and h n,
indicates the features used to construct the nth image. Since every
feature in W should correspond to a single seismic structure, the
coefficients of each image can then be mapped into the seismic structures that make up that image. Thus, for every image
n ! [1, N s] we can obtain
L n = W final (Q 9 (h final
n 1 1 # N l))

6n ! [1, N s],

(12)

where Q ! {0, 1} N f # Nl is a cluster membership matrix such that
the element Q ij = 1 if the feature w i belongs to structure j, and
1 1 # N l is a vector of ones of size 1 # N l . The resulting matrix,
L n ! R +N p # Nl shows the likelihood of each seismic structure for
each pixel in the image. Then, the pixel-level label for each location i in image n corresponds to the seismic structure given by

A-A'

A
D

F-F'
F

B-B'

A'
B'

E-E'

North

E'
F'
C-C'

D-D'

Figure 10. An illustration of implementing CNNs for salt-body boundary
delineation from the poststack seismic amplitude. The detected boundaries
are clipped to six vertical sections for quality control. Note the good match
between the CNN detection and the poststack seismic images
94

yu n = arg max L n (:, j ) .

2
F

(13)

Emerging trends and open problems
From the recent advancement in seismic interpretation research
as we have previously reviewed, we observe two important factors
that contribute to the success of this endeavor. First, to address the
challenges rooted in the ever-increasing data size and complexity,
it becomes very critical to leverage the advanced machine-learning techniques, especially those based on deep learning. Second, being a unique type of visual signals, seismic volumes can
be interpreted effectively using image analysis algorithms that
utilize HVS characteristics and models. In this section, we will
briefly discuss the emerging trends and open problems regarding
these two aspects.

Deep subsurface learning
Deep learning is one of the most powerful learning techniques
available today. As a data-driven approach, it utilizes sophisticated neural networks with deep architectures to uncover complex hidden structures and characteristics directly from a large
amount of samples. When applied to subsurface data, deep
learning will allow geoscientists to make sense out of the massive data sets with many variables while avoiding the human
biases. Naturally, seismic interpretation based on deep learning
is emerging as a very promising trend. For example, Waldeland
and Solbergd [69] applied CNNs to classify salt bodies from
3-D seismic data sets. Huang et al. [70] provided an excellent
demonstration of the effectiveness of integrating CNNs and
multiattribute analysis from poststack amplitude in detecting
faults. An illustration of implementing CNNs for salt-body
boundary delineation is given in Figure 10. In this case, a good
match is observed between the detected boundaries and the
original seismic images. Besides the conventional poststack
data used for interpretation, geophysicists are turning their
attention to the prestack data. For example, Hami-Eddine et al.
[71] investigated a machine-learning approach to optimize the
use of both prestack and poststack seismic data. Araya-Polo
et al. [72] and Lin et al. [73] proposed using a deep neural
network to learn a mapping relationship between the raw seismic data and the subsurface geology so that the labor-intensive
processing stage could be avoided.
To fully explore the potential of deep learning, there are a few
open problems that need to be addressed carefully. First, the lack
of labeled data is a serious obstacle. Unlike natural image classification problems, public domain data sets with large sets of
labeled samples are rare for seismic interpretation. This severely
limits the application of supervised learning techniques. Alternatively, weakly supervised or unsupervised learning will be more
realistic choices in this case. Other techniques such as generative adversarial networks (GANs) and active learning, which will
be discussed shortly, can also provide an alternative approach to
supervised learning.
Second, more advanced network architectures need to be
explored for interpretation. Current works mainly use CNNs
as the core deep-learning structure. To account for the strong

IEEE Signal Processing Magazine

March 2018

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