Unfaulting and unfolding 3D seismic images
Simon Luo & Dave Hale
Center for Wave Phenomena, Colorado School of Mines, Golden, CO 80401, USA
SUMMARY
One limitation of automatic interpretation methods such as
seismic image flattening is their inability to handle geologic
faults. To address this limitation, we propose to combine a
method for automatic image unfaulting with seismic image
flattening. First, using fault surfaces and fault throw vectors
estimated from an image, we interpolate throw vectors to produce a throw vector field, which we use to unfault the image.
Then, using seismic image flattening, we unfold the unfaulted
image to obtain a new image in which sedimentary layering is
horizontal and also aligned across faults. From this flattened
unfaulted image, we can automatically extract geologic horizons.
wf wh
t̃(wh )
Figure 1: A fault throw vector. At wf on the footwall side of
the fault, the fault throw is zero. At wh on the hanging wall side
of the fault, the fault throw vector t̃(wh ) specifies the location
of the image sample that, once shifted, aligns with the sample
on the footwall side.
INTRODUCTION
Extracting isochronal geologic surfaces—geologic horizons of
the same age—is a common problem in geophysics and geology. Such horizons are useful for interpretation of stratigraphic
features and analysis of structural deformation, as well as interpolation and correlation of subsurface properties. A geologic horizon is assumed to have been initially deposited as
a horizontal layer and subsequently subjected to faulting and
folding. So in order to extract such a horizon, it is necessary to
quantify faulting and folding apparent in a seismic image.
Perhaps the most straightforward way to extract a geologic
horizon is by manual picking. Manual picking is often used in
conjunction with autotracking methods (e.g., Howard, 1990),
which track seismic events by following local extrema or zero
crossings in amplitude in a seismic image. Alternatives to
horizon autotracking methods include volume interpretation
methods (e.g., Stark, 1996), which, rather than tracking single
events, process simultaneously an entire seismic volume. Automatic seismic image flattening (Stark, 2004; Lomask et al.,
2006; Parks, 2010) is an example of a volume interpretation
method. Automatic seismic image flattening could potentially
identify all horizons in an image, but the method is unable
to match horizons across faults unless additional information
(e.g., fault throw) is provided.
A fully automatic method for extracting geologic horizons is
ideal. Toward this end, we propose an automatic method that
can be used to extract all geologic horizons in an image, consisting of two steps: image unfaulting followed by image unfolding (i.e., image flattening). To unfault an image, we first
use the method described by Hale (2012) to estimate fault locations and fault throw vectors, displacement vectors along the
dip direction of a fault surface. To unfold an unfaulted image, we use non-vertical image flattening (e.g., Luo and Hale,
2011). By unfaulting and then unfolding an image, we obtain
an image in which a surface of constant relative geologic time
(i.e., a horizontal slice) maps to a geologic horizon.
IMAGE UNFAULTING
To unfault an image, we must first estimate fault locations and
fault slip. For the examples shown in this paper, we use the
method described by Hale (2012) to automatically compute
fault surfaces and fault throws from a 3D seismic image. Although we use Hale’s (2012) method, other methods (e.g., Borgos et al., 2003; Carrillat et al., 2004; Skov et al., 2004; Aurnhammer and Tönnies, 2005; Admasu, 2008; Liang et al., 2010)
could also be used to estimate fault locations and fault throw.
For an image f (x), where x = (x1 , x2 , x3 ) are coordinates in
the present-day space, the estimated fault throw vectors t̃(w),
where w = (w1 , w2 , w3 ) are coordinates in the unfaulted space,
can be used to compute an image
h̃(w) = f (w + t̃(w)) ,
(1)
in which seismic events are aligned across faults where t̃(w) is
specified. An example of a fault throw vector for a synthetic
2D seismic image is shown in Figure 1. In the figure, wf indicates the location of an image sample on the footwall side of
the fault, while wh indicates the location of the corresponding
sample on the hanging wall side of the fault. The fault throw
vector t̃(wh ) specifies the location of the image sample that,
once shifted to wh on the hanging wall side, aligns with the
image sample at wf on the footwall side. Note from equation
1 that events are shifted only at locations where the fault throw
t̃(w) is specified. Because we estimate fault throw only at locations where we have identified a fault surface, we must interpolate fault throw vectors at locations between faults to avoid
creating new discontinuities in an image when unfaulting. To
interpolate fault throw vectors at locations between faults, we
use blended neighbor interpolation (Hale, 2009).
Unfaulting and unfolding
a)
b)
c)
d)
Figure 2: A seismic image (a) overlaid with the vertical component of fault throw vectors, the unfaulted image (b), and the flattened
unfaulted image (c) computed using the composite shift vectors (d).
Our convention is that fault throw vectors t̃(w) specify throws
on the hanging wall side of a fault. Because we will interpolate
these throw vectors (e.g., Figure 2a) between faults, we must
also specify fault throw vectors on the footwall side of a fault
so that the relative throws on opposing sides of a fault do not
change after interpolation. Because fault throw vectors specify
throws on only the hanging wall side of a fault, the fault throws
on the footwall side must be zero (see Figure 1).
IMAGE FLATTENING
Figures 2a and 3a show subsections of a 3D seismic image
from offshore Netherlands with the vertical component of the
estimated fault throw vectors overlaid. Using the interpolated
throw vectors t(w) estimated from an input image f (x), the
unfaulted image h(w) is computed as
which has a corresponding Jacobian matrix J = ∂ w/∂ u:


1 − ∂ r1 /∂ u1
−∂ r1 /∂ u2
−∂ r1 /∂ u3
1 − ∂ r2 /∂ u2
−∂ r2 /∂ u3  . (4)
J =  −∂ r2 /∂ u1
−∂ r3 /∂ u1
−∂ r3 /∂ u2
1 − ∂ r3 /∂ u3
h(w) = f (w + t(w)) .
(2)
Figure 2b shows the unfaulted image computed according to
equation 2 from the input seismic image shown in Figure 2a
and the blended neighbor interpolation of the fault throw vectors whose vertical component is overlaid on the image in Figure 2a. Similarly, Figure 3b shows the unfaulted image computed from the input seismic image shown in Figure 3a and the
blended neighbor interpolation of the fault throw vectors also
shown in Figure 3a.
To flatten an (unfaulted) image h(w), we must find a mapping
w(u), where u = (u1 , u2 , u3 ) are coordinates in the flattened
space, such that the image g(u) = h(w(u)) is flat. We write
the mapping w(u) in terms of a shift vector field r(u):
w(u) = u − r(u) ,
(3)
Next, given normal vectors n = (n1 , n2 , n3 ), which we compute
from an image using structure tensors (van Vliet and Verbeek,
1995; Fehmers and Höcker, 2003), we can write the Jacobian
matrix for rotation:


n3 + n22 /(1 + n3 ) −n1 n2 /(1 + n3 ) n1
Jr =  −n1 n2 /(1 + n3 ) n3 + n21 /(1 + n3 ) n2  . (5)
−n1
−n2
n3
To flatten an image, we solve for an approximately isometric
mapping w(u) with Jacobian matrix J that satisfies
J� Jr = I,
(6)
Unfaulting and unfolding
a)
b)
c)
d)
Figure 3: A seismic image (a) overlaid with the vertical component of fault throw vectors, the unfaulted image (b), and the flattened
unfaulted image (c) computed using the composite shift vectors (d).
where I is the identity matrix. Isometric mappings are desirable because they preserve metric properties. Thus, if we could
isometrically map an image to a flattened image, then all metric properties (e.g., length, angle, area, and volume) of features
in the original image would be preserved in the flattened image. Isometric mappings, however, exist only in special cases
(Floater and Hormann, 2005), so in general, we solve for a
mapping w(u) that is only approximately isometric.
The columns of J contain vectors w1 (u), w2 (u), and w3 (u).
That is,
J = [ w1 (u) w2 (u) w3 (u) ] ,
(7)
where
�
��
∂ r1
∂ r2
∂ r3
∂ w(u)
= 1−
−
−
∂ u1
∂ u1
∂ u1
∂ u1
�
�
∂ w(u)
∂ r1
∂ r2
∂ r3 �
w2 (u) =
= −
1−
−
∂ u2
∂ u2
∂ u2
∂ u2
�
�
∂ w(u)
∂ r1
∂ r2
∂ r3 �
w3 (u) =
= −
−
1−
.
∂ u3
∂ u3
∂ u3
∂ u3
w1 (u) =
(8)
(9)
(10)
Vectors w1 (u) and w2 (u) are tangent to a surface (e.g., a horizon) at u and thus are orthogonal to a vector n(u) normal to the
surface at u. The vector w3 (u) is tangent to the line for which
the horizontal coordinates u1 and u2 in the flattened space are
constant, i.e., the line in coordinates w that maps to a vertical
line in coordinates u (Mallet, 2004). For an exactly isometric
mapping w(u), tangent vectors w1 (u), w2 (u), and w3 (u) are
orthonormal vectors, and the corresponding Jacobian matrix is
orthogonal.
Next, if we denote the columns of Jr as ŵ1 (u), ŵ2 (u), and
ŵ3 (u), then
Jr = [ ŵ1 (u) ŵ2 (u) ŵ3 (u) ] ,
and equation 6 states
 �
w1 ŵ1 w�
1 ŵ2
 w� ŵ1 w� ŵ2
2
2
w�
w�
3 ŵ1
3 ŵ2
 
w�
1
1 ŵ3
= 0
w�
2 ŵ3
0
w�
3 ŵ3
0
1
0
(11)

0
0 .
1
(12)
The matrix J� Jr is a metric tensor characterizing local metric properties such as length, angle, area, and volume (Mallet,
2002, 2004), so by setting this matrix equal to the identity, we
constrain the type of deformation parameterized by the mapping w(u). Equation 12 gives nine equations for the partial
derivatives of the shift vector field r(u):
�
�
∂ r1
∂ r2
∂ r3
n1 1 −
− n2
− n3
= 0,
∂ u1
∂ u1
∂ u1
(13)
�
�
∂ r1
∂ r2
∂ r3
−n1
+ n2 1 −
− n3
= 0,
∂ u2
∂ u2
∂ u2
Unfaulting and unfolding
a)
b)
1.5
1.6
1.7
1.8
s
1.1
s
1.2
s
1.3
s
s
s
s
1k
1k
m
m
Figure 4: Geologic horizons extracted using the composite shift vector fields shown in Figure 2d (a) and Figure 3d (b).
and
and
�
∂ r1
∂ r2
∂ r3
α 1−
−γ
+ n1
= 1,
∂ u1
∂ u1
∂ u1
�
�
∂ r1
∂ r2
∂ r3
γ 1−
−β
+ n2
= 0,
∂ u1
∂ u11.1 s ∂ u1
�
�
∂ r1
∂ r2 1.2 s ∂ r3
−α
+γ 1−
+ n1
= 0,
∂ u2
∂ u2 1.3 s ∂ u2
�
�
∂ r1
∂ r2
∂ r3
−γ
+β 1−
+ n2
= 1,
∂ u2
∂ u2
∂ u2
vector n, then the thickness of sedimentary layers will be preserved in the flattening process. For most images, however,
we cannot preserve thickness while flattening, so we give the
corresponding equations least weight.
�
�
�
∂ r1
∂ r2
∂ r3
−γ
− n1 1 −
= 0,
∂ u3
∂ u3
∂ u3
�
�
∂ r1
∂ r2
∂ r3
−γ
−β
− n2 1 −
= 0,
∂ u3
∂ u3
∂ u3
�
�
∂ r1
∂ r2
∂ r3
−n1
− n2
+ n3 1 −
= 1,
∂ u3
∂ u3
∂ u3
1.5 s
(14)
1.6 s
HORIZON EXTRACTION
1.7 s
We extract a horizon by first selecting a horizontal slice of con1.8 s
stant u3 in a flattened unfaulted image g(u). Then, we form a
composite shift vector field s(u) by combining the interpolated
throw vectors t(w) and flattening shift vectors r(u) as
s(u) = r(u) − t(u − r(u)) ,
−α
(15)
where α = n3 + n22 /(1 + n3 ), β = n3 + n21 /(1 + n3 ), and γ =
−n1 n2 /(1 + n3 ). We solve equations 13, 14, and 15 for the
components of the shift vector field r(u) by weighted leastsquares using conjugate gradient iterations.
Equation 6 describes an isometric mapping of an image to a
flattened image, but in general, we cannot expect to find an exactly isometric mapping for all images. In practice, this means
equations 13, 14, and 15 cannot be satisfied exactly, and we
must decide which equations to emphasize. For image flattening, we give most weight to equations 13, which determine the
angle between the surface tangent vectors w1 (u) and w2 (u)
and the normal vector n(u). If these equations are satisfied,
then the image g(u) obtained by applying the shifts r(u) will
be flat. We give less weight to equations 14. The four corresponding entries in the metric tensor on the left side of equation 12 form what is referred to as the first fundamental form
(Floater and Hormann, 2005), which characterizes lengths, areas, and angles measured on a surface. Finally, we give least
weight to equations 15, which determine the length of the tangent vector w3 (u) and the angles it forms with the surface tangent vectors. If w3 (u) is a unit vector parallel to the normal
(16)
The composite shift vector field allows for a direct mapping
from an image f (x) to a flattened unfaulted image g(u) with
g(u) = f (u − s(u)) .
(17)
Using the composite shift vector field s(u), we map a surface
of constant u3 , which corresponds to constant geologic time
or constant depositional time, to a geologic horizon in presentday coordinates. For example, for ũ = (u1 , u2 , k3 ) where k3 is
constant, the coordinates x̃ of the horizon in present-day space
are simply x̃ = x(ũ) = ũ − s(ũ).
Figures 4a and 4b show geologic horizon surfaces extracted
from the composite shift vector fields shown in Figures 2d and
3d, respectively. In the horizon in Figure 4a, notice the en
échelon faults that can be clearly seen in the seismic image
in Figure 2a. In the horizon in Figure 4b, notice the roughly
circular fault polygons, which correspond to the conical fault
surfaces described by Hale (2012).
ACKNOWLEDGEMENTS
Thanks to dGB Earth Sciences for providing the 3D seismic
images shown in this report. This research was supported by
the sponsors of the Center for Wave Phenomena at the Colorado School of Mines.
Unfaulting and unfolding
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