IEEE Signal Processing - March 2018 - 89

Reference Sections with
Labeled Boundaries

Predicted Section
v

Tracked
Boundary

~
INc,k

Maually
Labeled
Boundary

Centric Reference
Section

Extracting Texture Features

Estimating Tracked Positions

Np × Np

Tensor-Based
Subspace Learning
(MPCA, TLPP, ...)

Tensor
Generation

Postprocessing

Tracked
Salt-Dome
Boundaries

~
Boundary Features Gk

Tensor Group Gk

Nr
Manually Labeled
Salt-Dome Boundaries
in Reference Sections

Figure 6. The diagram of salt-dome tracking, in which features around salt-dome boundaries in reference sections are extracted by tensor-based subspace learning.

ones. Figure 6 illustrates the diagram of feature extraction process using tensor-based subspace learning adopted for salt-dome
tracking. Authors manually label the boundaries of salt domes in
N r reference sections, in which N c = ^N r /2h is the central one.
For each point in the labeled boundary of N c, its corresponding points in neighboring (N r - 1) reference boundaries are
identified. Since every point corresponds to a patch with the
size of N p # N p, stacking corresponding patches constructs
third-order texture tensors from reference sections, denoted
" A k ! R N p # N p # N r, k = 1, 2, f, K N c ,, where K N c represents
the number of points in the N c th reference boundary. Because
of strong correlations between neighbors, every point in the
N c th reference boundary corresponds to a tensor group, denoted G k = " A k - N s, f, A k, f, A k + N s ,, which contains 2N s + 1
tensors. Using multilinear dimensionality reduction methods
such as multilinear principal component analysis and tensorbased linearity projection preserving, one can obtain transfor(1)
(2)
(3)
mation matrices U k , U k , and U k and map tensor group G k
to its subspace as follows:
(1)
( 2)
( 3)
Au m = A m # 1 U k # 2 U k # 3 U k ,
m ! " k - N s, f, k + N s ,,

(5)

where # i, i = 1, 2, 3 denotes the i-mode product of a tensor by
a matrix, and tensor Au m with fewer dimensions in each mode

contains the features of reference boundaries. These extracted features can help identify tracked boundary points in predicted sections. The local and global comparison of the tracked boundary
(green) with the manually labeled one (red) shows the high similarity and accuracy of salt-dome tracking within seismic volumes.

Channel detection
The variety of rocks that are deposited in a channel system change
the acquired seismic signals, and, thereby, a channel system can
be detected by applying the common edge detection methods on
horizontal slices or well-picked horizon surfaces. For example,
the coherence attribute [28], [29] was first proposed for identifying the morphology of a channel system, and then a suite of new
attributes and algorithms have been developed for enhancing both
detection accuracy and noise robustness. However, while successfully highlighting the channel boundaries, such discontinuity analysis fails to depict the rock properties within the channel system,
as no such information is considered by the detection operators.
To overcome this limitation, the GLCM analysis was introduced
from the field of image processing and used for seismic facies
analysis [30]. Eichkitz et al. [31] presented a suite of algorithms
for generating ten different GLCM attributes from a seismic
cube. Di and Gao [32] made it possible to decompose a channel system by integrating the logistic/exponent gray-level transformations with the GLCM attributes. Machine-learning-based

IEEE Signal Processing Magazine

March 2018

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