Diffraction imaging using specularity gathers

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Abstract

Sturzu, I., Popovici, A.M. and Moser, T.J., 2014. Diffraction imaging using specularity gathers. Journal of Seismic Exploration, 23: 1-18. The separate imaging of subsurface diffractors is a key ingredient in the development of high-resolution imaging technologies. We here produce images of diffractors using depth migration algorithms modified to attenuate the energy from specular reflectors. The seismic events from a pre-stack seismic dataset are migrated to proper depth and location using the final velocity model obtained by the velocity model building process, but the output is assigned to separate bins according to the value of a specific parameter called specularity. The specularity gathers are post-processed using a plane wave destructor filter to attenuate the contribution coming from specular reflectors. The method is demonstrated on two synthetic models and on a field data target in the Teapot Dome reservoir.

Keywords

seismic migration

diffraction imaging

high-resolution imaging

plane wave destructor

common image gather

specularity gather

References

multiple-reverberation effects make the stratigraphic interpretation of the images
difficult and unreliable. Decreasing the wavelength of the seismic waves
reflected at the target is nearly impossible because of the dissipative nature of
the overburden that attenuates the high-frequencies in the seismic wave-field.
Furthermore, the high frequencies that are present in the data are often lost
during standard processing. High-resolution imaging is of value, for instance to
enable identification of small scale faults and to locate formation pinch-outs.
Standard approaches to obtain high-resolution information, such as coherency
analysis and structure-oriented filters, derive attributes from stacked, migrated
images. In comparison, diffraction imaging can act directly on the pre-stack
data, and has the potential to focus and image super-resolution structural
information as a consequence of the redundancy present in the pre-stack data.
THEORY AND METHOD
Diffractions are the seismic response of small elements (or diffractors) in
the subsurface of the earth, such as small scale faults, near surface scattering
objects, and, in general, all objects which are smaller than the seismic
wavelength. Diffraction imaging uses diffractions to focus and image the
structural elements that produced those diffraction events. Since diffractors are,
by definition, smaller than the wavelength of seismic waves, diffraction imaging
has the potential of providing super-resolution information, to image details that
are beyond the classical Rayleigh limit of half a seismic wavelength. The
importance of diffractions in high-resolution structural imaging has been
emphasized in many recent publications (Shtivelman and Keydar, 2004;
Khaidukov et al., 2004; Taner et al., 2006; Fomel et al., 2006; Moser and
Howard, 2008; Moser, 2009; Klokov et al., 2010; Klokov et al., 2011; Dell
and Gajewski, 2011; Koren and Ravve, 2011; Klokov and Fomel, 2012). Still,
diffraction imaging is not widely used tool. In fact, most algorithms that are
used to process seismic data explicitly enhance reflections and implicitly
suppress diffracted energy. The goal of diffraction imaging is not to replace
these traditional algorithms, but rather to provide an additional 3-D or 4-D
volume containing information about diffractors that would be able to fill in the
small, but potentially crucial, structural details.
DIFFRACTION IMAGING 3
A true diffraction image is not optimally obtained by post-processing a
traditional seismic image even if the seismic image is obtained by an algorithm
that does not suppress diffractions. While diffractors will appear in the image,
usually in the form of discontinuities, they have much lower amplitudes than
reflecting structures. On the other hand, by imaging diffractors using the
pre-stack data, the diffractor amplitude can be enhanced while the specular
reflections can be attenuated. Furthermore, apparent discontinuities in the
seismic image can have a variety of causes other than diffractions, including
small errors in the velocity model of the earth that was used to obtain the image,
so the post-stack/post-processing approach would not be able to discriminate
between discontinuities coming from diffractions and those coming from
processing errors.
Techniques for diffraction imaging fall into two categories. In the first
category are methods that separate the seismic data into two parts, one that
contains the wave energy from reflections (specular energy) and the other that
contains the wave energy from diffractions. Each component is used to provide
an image through traditional seismic imaging methods. In the second category
are methods that do not separate the input seismic data, but rather performs
filtering during migration. Moser and Howard (2008) and Moser (2009)
extracted the local direction of specularity from a previously obtained migration
stack, and used this information during a subsequent migration step in order to
filter the events that satisfied (to a given degree) Snell’s law. Koren and Ravve
(2011) pre-computed a directivity-dependent specularity attribute using
information from the velocity model and the acquisition geometry and used
angle domain gathers in order to suppress the specularity energy associated with
horizontal events in the angle domain gathers. In this paper we follow the
approach from Moser and Howard (2008). The parameters governing the
specularity filtering are rather arbitrary, if no further investigation is implied.
Sturzu et al. (2013) introduced a new concept - specularity gathers - that proved
to be very useful in the design of proper parameters for the specularity filter.
Below we show how we can selectively filter the specular energy within
specularity gathers to obtain the diffraction images after stacking along the
specularity dimension.
In Kirchhoff migration, energy is propagated to all possible reflection
points in the model space. After all events on all traces are propagated, an
image is generated by stacking (summing) all individual contributions. The
propagation of the events usually uses Green’s functions computed in the form
of travel-time tables (the time of propagation from the source defined by the
trace to the image point and further to the receiver defined by the trace.)
Stacking reinforces in-phase energy corresponding to true reflectors and cancels
out-of-phase energy that does not correspond to a true reflector. A conventional
Kirchhoff migration forms a seismic image as:
4 STURZU, POPOVICI & MOSER
V(x) = ) dtdsdrU'(t,s,r)é6[t 一 T(s,x,r)] , (1)
where 6 is the Dirac delta function, U'(t,s,r) the (second time derivative)
pre-stack data, depending on time t and shot/receiver positions s/r, T(s,x,r) is
the travel time from s to r via the subsurface image point x, computed by ray
tracing in a given reference velocity model, and V(x) the resulting migrated
image. The sum is carried out over the time samples and all source and receiver
pairs (s.r), in the seismic data.
In the earth’s subsurface the local discontinuities (reflectors) can be
modeled either as surfaces, edges, or isolated tips or points. For a smooth
surface reflector, the corresponding part of the image is a locally continuous,
planar surface that generates specular events, meaning that they strictly obey
Snell’s law. Events backscattered from all other types of discontinuities are
diffractive and do not obey Snell’s law. Diffraction imaging attenuates the
contribution of specular events in a migrated image.
In the final image stack, a specular element can be approximated locally
as a planar surface, while the isochrone surfaces (computed using the travel-time
tables) should be tangent to the planar surfaces corresponding to the specular
elements, as a direct consequence of Snell’s law. For diffractive events,
however, there is no such constraint. For a given trace in the data and a point
in the image (1), we define the specularity in terms of the specularity angle,
defined by the normal to a locally planar structure (dominant at that image
point) and the direction of the gradient of the total travel-time from the source
to the receiver via the image point. We choose the absolute value of the cosine
of the specularity angle as the actual value of specularity.. This can be expressed
mathematically as:
S(.xr) = [wT |/|T I, Q)
where T, denotes the gradient of T(s,x,r)with respect to x, and n is the unit
vector normal to the reflector surface, also depending on x, and the dot denotes
the scalar product. If x is located on a strong reflector, the value of the
specularity should be equal to unity, S = 1, because the two rays (coming from
the source and that going toward the receiver) then obey Snell’s Law with
respect to the normal to the reflector n, so the angle bisector of the two rays is
aligned with the normal to the surface. In diffraction imaging framework, this
is a pure specular reflection that has to be attenuated. If the angle bisector of the
rays and the normal to the surface are not perfectly aligned, S < 1 and the
energy is non-specularly scattered.
The concept of pure specularity as defined above has to be amended using
the concept of the Fresnel zone, which is a frequency-dependent volume around
DIFFRACTION IMAGING 5
the ray within which most of the wave energy is interfering constructively and
can be treated as a single arrival wave. All the points from a Fresnel zone have
to be considered together, even if they have slightly sub-unitary values for
specularity. That is why pure specularity has to be defined by a frequency-
dependent interval close to unity. In Fig. 1, we illustrate this situation by
comparing a pure specular reflection with a pure diffraction. The Fresnel zones
are depicted by hatched areas.
Fig. 1. Comparison between a pure specular reflection and a pure diffraction case. The Fresnel zone
for the reflection are depicted by the hatched areas.
A straightforward procedure for obtaining a diffraction image is outlined
in Moser and Howard (2008) and Moser (2009). First, using pre-stack Kirchhoff
migration, we obtain the seismic image; this image will include both reflections
and diffractions, but the reflections dominate the image. The second step is to
analyze the structures in the Kirchhoff image and determine the normal vector
to these structures at each image point. Using a migration stack to obtain this
information, rather than to extract it from the final velocity model, is critical in
cases when the stratigraphic non-conformity is important (Moser, 2009). The
second step has to be performed on an optimally focused image, obtained using
the best velocity model, so that the information extracted is related to the
geological geometry of the undersurface. In a subsequent migration run, the
migrated seismic events are stacked using a weight designed in order to
attenuate the contribution of the specular events (specularity value close to 1).
6 STURZU, POPOVICI & MOSER
An important challenge comes from the fact that there is no a priori
procedure to define the limits of the pure specularity region as a function of
specularity itself, so one cannot design a proper weighting function before the
last migration run. A simple trial-and-error method can be too computationally
demanding. Specularity gathers, a novel technique introduced in Sturzu et al.
(2013) can be used to increase the efficiency and accuracy of the diffraction
imaging technique. A specularity gather is similar to an offset or angle common
image gather, in which the migrated seismic events are separated according to
the value of specularity rather than of the offset or, respectively, reflection
angle. The events are migrated to the proper depth and are partially stacked
according to the specularity values in pre-defined specularity bins. The
specularity gather can be formally written as:
V,.(x,8) = J dtdsdrU'(t,s,r)6[t — T(s,x,n)]6(S — |n-T,|/T,|]) . @)
In a post-processing technique similar to the muting of the offset gathers,
the diffraction image can be obtained after a weighted stack over all the
specularity values:
Vux) = J dSwix,S)V, (8) . 4)
The use of specularity gathers has the advantage that the weighting
function is designed after migration and therefore is constructed, and updated,
very efficiently. In particular, the weighting function can be spatially variable
[w = w(x,S)] and adapted to the local Fresnel zone width, which is difficult to
estimate a priori, but becomes feasible using specularity gathers. Also, feedback
from interpretation can be easily included in the weighting function, and hence
in the final diffraction image.
As shown in Sturzu et al. (2013), for a correct velocity model and in the
ideal infinite-frequency limit, a specular reflection event appears in the
specularity gathers as a focused spot on the S=1-axis. Point diffractions appear
as flat events extending over 0 < S < 1. Edge diffractions in three dimensions
appear as dipping events, as they obey Snell’s law only along the edge, but not
transversely to it (Moser, 2011). For finite bandwidth seismic responses, the
situation is slightly different. Specular reflections also appear as dipping events,
as the non-specular part of reflected energy outside the Fresnel zone is not
related to the shortest reflection path following Fermat’s principle.
In a Common Image Point section of the specularity gather, the image is
obtained from different portions of the isochrones of different traces. For
example, in the case of a horizontal flat reflector with a constant velocity
DIFFRACTION IMAGING 7
overburden, the isochrones are ellipses. The specularity angle in each point of
the ellipse is monotonically changing from zero below the Common Image Point
to 90 degrees at zero depth. In a Common Image Point section of the specularity
gather, the bin for the maximum specularity (specularity angle close to zero) is
formed by pieces of the ellipses corresponding to maximum depth. Here, traces
with any offset should contribute. The very next specularity bin (in the same
Common Image Point section) is formed by the contributions of the traces with
neighboring mid-points, but having the right offset to yield the designed value
of the specularity. The location of the events should be shallower than the
location of the event in the previous bin. This interpretation pattern can be
applied to subsequent specularity bins, ending with the bin for zero specularity,
which has to have contributions only at zero depth, but from all traces.
Fig. 2 shows migration results for the case of a horizontal flat reflector
with a constant velocity overburden. The sub-figures depict: (a) a Common
Image Point (or vertical) section from the specularity gather (a partial image
obtained for a given vertical in the image space and all values of the
specularity), (b) the final stack over all values of specularity, equivalent to the
standard migrated image, and, in each of panels (c), (d) and (e) a specularity
section (a partial image obtained for a given value of the specularity for all
points in the image space) for S = 1.0, S = 0.9 and S = 0.8, respectively.
Sub-Figs. 2(d) and 2(e) contain out of Fresnel zone ghosts for the main specular
event depicted in sub-Figs. 2(b) and 2(c). Remarkably, stacking over all values
of specularity is able to fully cancel the contribution of all the ghosts in the final
Horizontal rane an
Fig. 2. Migration results for a horizontal flat reflector with a constant velocity overburden: (a) the
specularity gather for the horizontal position at x = 1250 m; (b) The stack over all values of
specularity, equivalent to a standard migrated image; (c) specularity section for S = 1.0; (d)
specularity section for S = 0.9; (e) specularity section for S = 0.8.
Depth [m]
8 STURZU, POPOVICI & MOSER
stack (b). Fig. 3 show similar results for a point diffractor in a constant velocity
medium. In the left panels is displayed the migration image, obtained by
stacking along all specularity values in the specularity gather. The specularity
gather in the exact location of the point diffractor is a flat horizontal event
(sub-figure 3(a) central panel); when moving slightly away from this point, the
horizontal event splits into two (sub-Fig. 3(b) central panel). The right panels
show two specularity sections, for S = 1.0 in sub-Fig. 3(a) and for S = 0.8 in
sub-Fig. 3(b).
Horizontal position [m] Specularity Horizontal position [m]
500 100015002000 0s 1.0 500 100015002000
500 os 10 500
Fig. 3. Migration results for a point diffractor in a constant velocity medium. Each panel displays
from left to right: the final stack, the specularity gather for a given horizontal position, x, and a
specularity section for given value of specularity, S. The horizontal location of the diffractor is at
1250 m. (a) x= 1250 m, S = 1.0 (b) x = 1150 m, S = 0.8.
a)
Depth fm1
b)
Depth [m]
DIFFRACTION IMAGING 9
Displaying common-image specularity gathers (vertical sections) may
become cumbersome when dealing with more complicated data. Fortunately,
using a common-depth display, i.e., showing sections along one of the
horizontal lines (corresponding to depth in the common-image gather) versus
specularity on the vertical axis is able to give a clearer image, especially for
cases with small lateral variations. For these cases, in the Common-Depth
Specularity Gathers (horizontal sections), the specular reflections are almost
horizontal events. An important issue is that in this display one can identify (out
of Fresnel zone-) ghosts of the specular events coming from deeper locations,
which are also almost horizontal. In this way, we can filter these ghosts together
with their primaries.
A workflow for diffraction imaging using common-depth specularity
gathers consists of:
I Standard pre-stack depth migration using formula (1) and associated
migration velocity analysis to obtain an optimally focused full-wave image
V(x);
II. Extraction of the unit vector normal to the reflector surface using V(x) in
each point;
Ill. Migrating using eq. (3) to obtain a specularity gather;
IV. Filtering the specular energy from the specularity gather;
V. Stacking over specularity dimension to obtain a diffraction image [eq.
(4)].
NUMERICAL RESULTS
The numerical results are obtained using the procedure outlined above
[eqs. (1)-(4)]. The filtering step IV can be done using any procedure able to
detect and attenuate laterally continuous seismic events. Here we used one of
them, the Plane Wave Destruction Filter (PWD) (Fomel, 2002). Before applying
the filter, we compute the dips in each section of the Common-Depth
Specularity Gather. Then, in each point from the gather, the filter is performing
a weighted stack along the dip in order to attenuate the seismic event from a
given vertical window, if a similar event is found along the dip. The first
numerical example has been designed for a proof of concept (Fig. 4) and
illustrates the functionality of specularity gathers on a simple diffraction ramp
model: a horizontal reflector at 900 m of depth, and a double ramp with the
base at 1400 m, as depicted in the perfect migration stack shown in the panel
(a) from Fig. 4. The synthetic data are generated using ray-Born approximation,
10 STURZU, POPOVICI & MOSER
a method proved to be very useful in forward modeling diffracted waves
(Moser, 2012). After pre-stack migration, the unit vector normal to the
reflectors in the final stack (a) was computed and used to generate a specularity
gather [eq. (3)] in a subsequent migration run. The Common-Depth Specularity
Horizontal position [m] Horizontal position [m]
上
和
上
上
(b)
一
o
—
Depth{m]
S
Specularity
0 0.0 Os 10
=
一
=
区
Specularity
Os
0 0.0
(h)
&
Specularity
0s
0
Fig. 4. Diffraction ramp model: (a) Pre-stack migration image obtained by stacking over the values
of specularity in the specularity gather. (b) Diffraction image obtained by stacking over specularity
of the Plane Wave Destructor (PWD) filtered specularity gather. (c) Section of the specularity gather
in common-depth display for 1400 m of depth. (d) Section of the specularity gather in common-depth
display filtered with PWD for 1400 m of depth. (e) Section of the specularity gather in
common-depth display for 1140 m of depth. (f) Section of the specularity gather in common-depth
display filtered with PWD for 1140 m of depth. (g) Section of the specularity gather in
common-depth display for 900 m of depth. (h) Section of the specularity gather in common-depth
display filtered with PWD for 900 m of depth.
DIFFRACTION IMAGING 1
Gather of Fig. 4c shows two horizontal events close to S = 1 coming from the
specular reflections shown in the stack from Fig. 4a at 1400 m depth, while for
the diffractive events from the same depth at 750 m, 1500 m, and 2250 m along
the line, there are clearly defined peaks. Close to the central peak, we notice
also two dipping events from the specular reflections close to the edge of the
double-ramp. After applying the Plane Wave Destruction Filter the specular
energy is attenuated, and the result showing three diffraction peaks is displayed
in Fig. 4d. The specularity gather of Fig. 4e does not have horizontal events,
but displays, close to S = 1, dipping events corresponding to the specular
reflections at the top end of the ramps, visible in the stack from Fig. 4a at 1140
m in depth. For the diffractive events from the same depth at 750 m and 2250
m along the line, there are clearly defined peaks. In Fig. 4f is displayed the
result of applying the Plane Wave Destruction Filter on the section from Fig.
4e: the specular energy is almost completely attenuated and two diffraction
peaks are visible. The specularity gather of Fig. 4g shows a horizontal event
close to S = 1 coming from the specular reflections shown in the stack from
Fig. 4a at 900 m in depth, and two ghost dipping events coming from the
specular reflections on the ramps. After filtering with the Plane Wave
Destruction Filter, we obtain almost no energy - except for two very weak peaks
at the survey’s edges - as displayed in Fig. 4(h). Stacking the filtered specularity
gathers over the values of specularity gives the diffraction image shown in Fig.
4(b). Almost all of the specular energy was attenuated in the final image leaving
just the five points of discontinuity in the model.
Horizontal position [m] Specularity
4000 8000 12 000 0.0 0.5 1.0
(a) 8
(b)
Specularity
Depth [m]
Fig. 5. Mare di Cassis model: (a) Specularity gather in common-depth display (horizontal section)
for 2100 m. (b) Specularity gather in common-image display (vertical section) for the horizontal
position x = 4880 m.
12 STURZU, POPOVICI & MOSER
The second example is the Mare di Cassis data set, which is described in
Moser and Howard (2008). After regular pre-stack migration, the unit vector
normal to the reflectors in the final stack was computed and used to generate a
specularity gather in a subsequent migration run. Fig. 5 displays a comparison
between a Common Depth Specularity Gather (horizontal section) on the left,
and a vertical section from the specularity gather on the right. Visually it is
clear that the first one displays more information: the specular reflections are
identified as laterally continuous events, the out-of-Fresnel zone ghosts are
identified as similar events at smaller values of specularity, while diffractions
are identified by the numerous peaks. In the vertical display the diffractions are
identified by the horizontal peaks, while specular reflections by dipping events
that tend to align toward the zero-specularity point from the surface. The
specularity gathers were tapered along the specularity axis, and the plane
wave-destructor filter was applied in the common depth sections of the gather.
Horizontal position [m]
6 000 8 000 10 000 12 000
(a) 8
Specularity
(b)
Specularity
Fig. 6. Mare di Cassis model: (a) Common-Depth Specularity Gather for 1310 m. (b) PWD filtered
Common-Depth Specularity Gather for 1310 m.
DIFFRACTION IMAGING 13
In Figs. 6, 7, and 8, the result of applying the plane wave-destructor filter
on a common depth specularity gather is shown respectively for three values of
depth. Almost all the specular energy is attenuated, except for regions close to
S=1 where the dip calculation is affected by either multiple dips concurrently
in the same image point or by vertical dips; consequently, the plane
wave-destructor filter is not able to clean all the specular energy. Fig. 9 displays
in panel (a) the standard migrated image, in panel (b) the diffraction image
obtained by stacking over specularities smaller than 0.97 of the plane
wave-destructor filtered specularity gather, and - for reference - in panel (c), the
diffraction image obtained using a cubic taper to filter out all the events from
the specularity gather corresponding to values of specularity larger than 0.92
(Sturzu et al., 2013). A visual comparison shows that the current procedure
gives better results than that obtained using the uniform taper.
Horizontal position [m]
0 2000 4000 6000 8000 10 000 12 000
Fig. 7. Mare di Cassis model: (a) Common-Depth Specularity Gather for 1960 m. (b) PWD filtered
Common-Depth Specularity Gather for 1960 m.
14 STURZU, POPOVICI & MOSER
Horizontal position [m]
6 000 8000 10 000
Specularity
(b)
Speculerity
Fig. 8. Mare di Cassis model: (a) Common-Depth Specularity Gather for 2080 m. (b) PWD filtered
Common-Depth Specularity Gather for 2080 m.
The third example contains a field dataset from Teapot Dome (Powder
River Basin, Wyoming). Here, diffraction imaging has been carried out using
the same steps as above, but in the framework of a full 3D depth imaging
process. Figs. 10 and 11 show the results for a region that contains the target
known as the Tensleep formation. In Fig. 10 we focus on the crossline 118 of
the survey: panel (a) is the standard migration result (the top of the Tensleep
formation is depicted in the figure), while panel (c) displays corresponding
section from the diffraction image. The vertical section of the specularity gather
at the location given by the vertical thin white line on the stack is shown in Fig.
10b, while the corresponding section from the plane wave-destructor filtered
specularity gather is displayed in Fig. 10d. The filtering procedure was applied
in each horizontal section (common depth specularity gather) separately, so Fig.
10d was not obtained by applying directly the plane wave-destructor in the
vertical section.
DIFFRACTION IMAGING 15
Horizontal position [m]
a 4000 6 000 8 000 12 000
10 000
Depth [m]
(b)
Depth [m]
(c)
Depth [m]
Fig. 9. Mare di Cassis model: (a) Standard migrated image; (b) Diffraction image obtained by
stacking over specularity of the PWD filtered specularity gather; (c) Diffraction image obtained using
a uniform taper filter above S = 0.92.
16 STURZU, POPOVICI & MOSER
In Fig. 11, we focus on a depth section at 2020 m (depicted in Fig. 10a with a
thin white horizontal line). Fig. 11a displays the depth section through the
standard migration result, while in Fig. 11b is shown the corresponding depth
section in the diffraction image. The common depth specularity gather for the
crossline 118 at the same depth is shown in Fig. 11c, while the corresponding
result filtered with plane wave-destructor method is depicted in Fig. 11d. In this
case, due to the 3D geometry, the diffraction peaks from the specularity gather
are not as clear as in the synthetic examples. However, applying the procedure
described above is able to delineate in the migrated image (Fig. 11b)
high-resolution diffractive elements related to the transition between different
stratigraphic formations.
(a) (b)
0 200
000
1600 2 400
人
全
已
N
(©) (d)
Fig. 10. Teapot Dome dataset, crossline 118: (a) Standard migrated image; (b) Vertical section of
the specularity gather at inline 138; (c) Diffraction image; (d) Vertical section of the specularity
gather at inline 138 filtered with PWD.
CONCLUSIONS
Specularity gather analysis proves to be a very useful instrument in
obtaining and/or optimizing diffraction images. The energy corresponding to
higher values of specularity can be attenuated by using tapers (uniform or based
on interpretation input). An automatic algorithm can be alternatively constructed
by using a filter, such as plane-wave-destructor, to attenuate the specular energy
at any location in the specularity gather. Further development of this method
will help in advancing diffraction imaging technology.
DIFFRACTION IMAGING 17
(b)
Crosslines
0 150 100 50
Specularity
5
加 — 全
Fig. 11. Teapot Dome dataset, depth section at 2020 m: (a) Standard migrated image; (b) Diffraction
image; (c) Common-Depth Specularity Gather, display at crossline 118; (d) Common-Depth
Specularity Gather filtered with PWD, displayed at crossline 118.
ACKNOWLEDGEMENTS
We thank Rocky Mountain Oilfield Testing Center and the U.S.
Department of Energy for providing the Teapot Dome dataset, and Opera (Pau)
for providing the Mare di Cassis dataset. Special thanks are due to our
anonymous reviewer, for comments that greatly improved the manuscript.
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