IEEE Signal Processing - March 2018 - 88

y

t

Time (ms)

Texture Boundary

x

Texture Region 2

Texture Region 1

1,380
1,460
1,540
1,620
1,700
1,780
749

Center Point
First Cube

Second Cube

GoT in x-Direction

Gx = d (Wx- ,
Highest GoT is
obtained when
center point is
exactly on the
texture boundary.

(a)

799

849
899
Crosslines
(b)

949

999

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1

East
Time
North

Wx+)
At current position, GoT
drops as both cubes
have content from
the right texture
region.
x

(c)

Figure 5. (a) Computing GoT along the crossline, i.e., x-direction. (b) Salt-dome boundary detected by various delineation algorithms; magenta [15],
black [19], blue [24], green: reference salt-dome boundary. (c) A 3-D salt body detected from the F3 block in the North Sea using 3D-GoT [24].

Figure 5(a) illustrates synthetic seismic images in which a greendashed vertical line separates two textured regions depicted in
dotted and striped lines, respectively. To evaluate the GoT in the
x-direction (i.e., the crossline), as the center point and its two
neighboring cubes move along the blue line, texture dissimilarity
function d (·) yields a GoT profile shown at the bottom of Figure 5(a). Theoretically, the highest GoT value corresponds to the
highest dissimilarity and is obtained when the center point falls
exactly on the texture boundary. Similarly, GoT is also calculated
along t- (time or depth) and y- (inline) directions. To improve the
delineation efficiency and robustness, GoT employs a multiscale gradient expressed as follows:
G [t, x, y] = e

/

N

e/

1
2 2
n
n
~ n ·d ^ W i-, W i+ h o o ,

(3)

i ! {t, x, y} n = 1

where W ni- and W ni+ denote the neighboring cubes, n represents
the edge length of cubes, and ~ n represents the weight associated
with each cube size. To compute dissimilarity between cubes,
authors use a perceptual dissimilarity measure based on error
magnitude spectrum chaos, which is not only computationally
less expensive and performs better than nonperceptual dissimilarity measures, but also highlights texture variations in the most
effective manner. The perceptual dissimilarity measure is calculated as follows:
d ($) = E ` K U K U W ni- - W ni+ j,
i ! {t, x, y},

(4)

where U represents the tensor product, K is the Kronecker
matrix defined as K = D t 7 D x 7 D y, and D t, D x, and D y are
88

discrete Fourier transform matrices. The output of various saltdome delineation algorithms on a typical seismic inline and a salt
body detected using 3D-GoT [24] from the F3 block in the North
Sea are shown in Figure 5(b) and (c), respectively.

Fault and salt-dome tracking
The methods introduced in the previous two sections focus
mainly on the detection, or delineation, of faults and salt
domes in 2-D sections. To investigate the geological structures
of faults and salt domes, interpreters need to repeatedly apply
these methods on each section of a seismic volume. However, for
a seismic volume with a large size, the repeated detection in every
section may impair interpretation efficiency. Because of the slow
formation processes of subsurface structures, neighboring sections commonly have strong correlations. In recent years, fault
and salt-dome tracking methods have been proposed to utilize
correlations between sections to improve interpretation efficiency. Reference [13] borrows the concept of motion vectors in video
coding and grouped seismic sections into reference and predicted
sections. Faults in the predicted sections can be labeled by detection results in reference sections. Berthelot et al. [25] detected
the salt-dome boundary in one time section using the method
in [17] and tracked the boundary through adjacent sections by
minimizing a defined energy function that maintains boundaries' curvature and smoothness. Similarly, [26] takes advantage
of active contour to track salt-dome boundaries in neighboring
seismic sections. More recently, Wang et al. [27] have proposed a
salt-dome tracking method that extracts the features of salt-dome
boundaries in reference sections using tensor-based subspace
learning and delineates tracked boundaries by finding points
in predicted sections, which are the most similar to reference

IEEE Signal Processing Magazine

|

March 2018

|
Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018
