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3D curvature attributes: a new approach
for seismic interpretation
Pascal Klein, Loic Richard, and Huw James* (Paradigm) present a new method to compute
volumetric curvatures and their application to structural closure and qualitative estimation
of basic fracture parameters.
W
e present a different approach to computing volumetric curvature and the application of volume curvature attributes to seismic interpretation. Volume
curvature attributes are geometric attributes computed at each sample of a 3D seismic volume from local surfaces fitted to the volume data in the region of the sample. The
curvature attributes respond to bends and breaks in seismic
reflectors. Because volume curvature focuses on changes of
shape rather than changes of amplitude, it is less affected by
changes in the seismic amplitude field caused by variations in
fluid and lithology and focuses more on variations caused by
faults and folding. Tight folds at seismic scale may indicate
sub-seismic faults. Interpretation of the tight folds can also
provide qualitative estimates of basic fracture parameters
such as fracture density, spacing, and orientation. This
knowledge of both faults and fractures is valuable for the
estimation of structural frameworks including closure and
also for the estimation of reservoir flow characteristics.
Seismic interpreters have used attribute volumes for fault
interpretation of 3D seismic data since they became available. Coherency (Bahorich and Farmer, 1995) is without
doubt the most popular attribute for this purpose. More
recently, curvature attributes have been found to be useful
in delineating faults and predicting fracture distribution and
orientation. Because curvature is sensitive to noise and is a
relatively intensive computational task, calculations of curvature were initially performed geometrically for seismic
horizon data. Very recently, algorithms of volumetric curvature were formulated that make the assumption that the
structure is locally defined by an iso-intensity surface. These
approaches suppose, moreover, that the orientation volumes
(dip and azimuth) are available.
Donias (Donias et al., 1998) propose an estimate of the
curvature based on the divergence formulation of the dip-azimuth vector field calculated in normal planes. Chopra and
Marfurt (2007) use the fractional derivatives of apparent dip
on each time slice to extract measurements of the curvature at
each sample of the 3D volume. West et al. (2003) give a method where individual curvatures are computed as horizontal
gradients of apparent dip for a given number of directions, and
are then combined to generate a combined curvature volume.
*
This paper proposes a method to compute volumetric
curvatures and their application to structural closure and
qualitative estimation of basic fracture parameters. The illustration and discussion use a data set from offshore Indonesia. The original seismic data is zero-phase and comprises
300 in-lines and 1300 cross lines with in-line spacing of 25
m, cross line spacing of 12.5 m, and a sample rate of 4 ms.
The regional basin geometry is made of pull-apart basins
due to tectonic extrusion of Southeast Asia in response to
the collision of India since the early Tertiary. The structural
framework of the basin consists of a number of extensional grabens, half-grabens, normal faults, horsts, and en-echelon faults (Figure 1). Part of sedimentation was syntectonic implying important thickness variation in the sedimentary series. Literature describes four tectonic periods occurring
in the study area: extension, quiescence, compression, and
another period of quiescence.
Surface curvature
Surface curvature is well described by Roberts (Roberts,
2001) In brief, surfaces of anticlines will yield positive
curvature, synclinal surfaces will yield negative curvature,
and saddles will yield both positive and negative curvature.
Ridges will yield positive curvature in the direction across the
ridge and zero curvature in the direction along the ridge line.
Troughs will yield negative curvature in the direction across
the trough and zero curvature along the trough line.
At any point of a surface, the curvature can be measured
as a bending number (positive or negative) at any azimuth.
One of these azimuths will yield the largest curvature. This
curvature is named the maximum curvature and the curvature in the orthogonal azimuth is named the minimum curvature. This set of curvatures can be used for defining other
curvature attributes. For example, the average of the minimum and maximum curvature or any other pair of curvatures measured on orthogonal azimuths is called the mean
curvature. The product of minimum and maximum curvature is called Gaussian curvature.
Surfaces that are initially flat will have a minimum and
maximum curvature of zero and consequently a Gaussian
curvature of zero. Folding such surfaces may increase the
Corresponding author, E-mail: [email protected].
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Figure 1 General overview of data set from Indonesia. a) Time structure of a shallow horizon H1; b) Amplitude map for horizon H1; c) Maximum curvature
extracted along H1; d) Amplitude section.
maximum curvature but so long as the minimum curvature stays at zero the Gaussian curvature will also remain
at zero. This is an indication that the surface has not
been deformed. Naturally, if the unit bounded by the surface has thickness and the unit is not completely plastic,
there will be some fracturing as the unit is folded. Gaussian curvature may have some role to play as an indicator
of deformation.
Instead of choosing the azimuth of maximum curvature
the choice of azimuth can be made to select the most pos-
Figure 2 Elliptical paraboloid.
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itive or the most negative curvatures. These measures will
yield similar measures to that of maximum curvature but
will preserve the sign of curvature so that curvature images
will consistently represent ridges or troughs. The disadvantage is that the interpreter needs to view two separate images of positive and negative curvature and fuse them into one
interpretation. Patterns that include both minimum and
maximum curvature are separated and may become less
apparent. Alternatively, if the initial choice of azimuths is
the azimuth of maximum surface dip, then the curvature is
called the dip curvature and the curvature in the orthogonal azimuth is called the strike curvature.
Three very simple shapes illustrate the previous discussions about curvatures attributes. All the simple surfaces have been made as anti-form, but the conclusions are
the same for the syn-form, only the sign of the curvature
attribute will be changed.
The first one considered here is the elliptic paraboloid
surface (Figure 2) which has geologic analogues of diapir, basin, and karst dissolution. The distributions of maximum, minimum and dip curvature are radial. We notice
also that the azimuth of the dip curvature is equal to the
azimuth of the maximum curvature.
The second shape is the cylindrical surface (Figure 3)
with geological analogues of diapir, syncline, and anticline.
The minimum curvature is equal to zero. We also remark
that the azimuth of the dip curvature is equal to the azimuth of the maximum curvature. Lineaments of the maximum curvature and dip curvature are parallel and show the
apex of the antiform or the axis of the synform.
The last shape is the hyperbolic paraboloid surface (Figure 4) with geologic analogues of diaper and spill point.
Lineaments from maximum curvature and lineaments for
minimum curvature are orthogonal. The intersection of the
both lineaments corresponds to a possible spill point.
Volume curvature will be computed directly from the
volume data but the same measures as surface curvature
are available.
Volume curvature
In the examples above, curvature is computed directly from
surfaces and these same computations may be applied to
interpreted horizon and fault surfaces. Instead of computing curvature for surfaces, it is possible to compute curvatures at every point of the volume. These curvatures may
then be extracted along interpreted surfaces, time and depth
slices, or any kind of seismic section. Volume curvatures
may also be displayed directly in volume or voxel visualization displays. We have found that volume curvature
extracted along an interpreted horizon is less noisy than
Figure 3 Cylinder.
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Figure 4 Hyperbolic paraboloid or saddle.
surface curvature for the same horizon. This is because the
volume curvature is directly measuring the curvature of
the amplitude field while the surface curvature is usually
following a ‘snapped horizon’ which will be influenced
by the shape of a single trace or the shape of a manually
interpreted fault which we can expect to contain noise due
to manual picking.
Methodology
The proposed estimation of curvatures is performed in
three stages. First, for each volume sample, a small surface is propagated around the sample within the defined
horizontal range of analysis. The surface depths are found
by finding the maximum cross-correlation value over a
vertical analysis window between the central trace and
each surrounding trace within the defined range for analysis. The cross-correlations are back interpolated, using a
parabolic fit to determine the precise vertical shift of the
maximal cross-correlation. Then a least squares quadratic
surface z(x,y) of the form is fitted to the vertical shifts
within the analysis range. Finally, the set of curvature
attributes are computed from the coefficients of quadratic
surface using classic differential geometry (Roberts, 2001).
The curvature attributes most frequently used are the
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maximum and minimum curvatures which we designate κ1
and κ2 respectively.
Coherency and curvature
Both coherency (Bahorich and Farmer, 1995) and curvature are used to delineate faults and stratigraphic features
such as channels. Coherency accentuates parts of the
amplitude volume where there are discontinuities in the
amplitude field. These occur where there are faults and
the horizon amplitudes are discontinuous because the
rocks are broken. Discontinuities also occur where channel boundaries interrupt horizons and these too are well
imaged by coherency.
Volume curvature will show high values where horizons are bent rather than broken. Volume curvature at discontinuities need not yield predictable results, but typically horizons are bent prior to breaking at faults so volume
curvature may well pick out a fault. For example, volume
curvature calculated in the region of a low throw normal
fault will show high positive curvature at the edge of the
footwall coupled with high negative curvature at the edge
of the hanging wall. This characteristic pair of high positive and negative curvatures can be used to interpret low
throw faults. At channel boundaries volume curvature may
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have high positive values at the levees and negative values
in the thalweg.
So both attributes can detect faults and channels.
Coherency can be calculated over relatively long time gates
to create very precise images of faults in plan view. When
used in this fashion, coherency becomes a detailed qualitative indicator of faults and their position. This allows
interpreters to quickly pick them without anguishing over
the precise position as they may do when using amplitude
data alone.
Volume curvature produces quantitative measures of
folds and is typically calculated over the interval of a single wavelet. The value of volume curvature may more reliably be used in further numerical calculations and volume
curvature is more likely to usefully indicate regions of folding or sub-seismic faulting.
These two attributes illuminate different features of
faults, folding, and stratigraphic features. So it is wise to
use both of them for detailed interpretation.
Filtering curvature lineament
Curvature attributes are mainly analyzed using the lineament concept, introduced by Hobbs (Hobbs, 1904). A lineament is a mappable, simple, or composite linear feature of
a surface, whose parts are aligned in a rectilinear or slightly
curvilinear relationship and which differs distinctly from
the patterns of adjacent features and presumably reflects
a subsurface phenomenon. Two dimensional analysis of
curvature attribute shows that lineaments do not necessarily indicate a geological structure, such as a deformation
zone or a sedimentary pattern. The general question is
how to identify features that are only related to geological
feature. The best answer is to reverse the question and try
to exclude non-geological pattern.
Filtering noise lineaments from anthropogenic sources,
such as surface installations, when interpreting for shallow hazards, can be easily managed. On the other hand,
acquisition footprint reduction is not an easy task. It is
recommended to perform this process within the seismic
data processing sequence. Regardless, volume curvature
allows us to significantly reduce the noise that results from
the acquisition footprint still remaining in the post stack
amplitude volume. Rose diagrams for azimuth and dip may
be plotted for lineaments interpreted from maximum curvature. These Rose diagrams can in turn be interpreted to
identify lineaments due to geology versus lineaments due to
surface noise or acquisition footprint.
Structural closure
The structural hydrocarbon traps are frequently composed
of three way dip closures occurring against faults. The
trapping efficiency of this kind in the tectonic regime of the
study area depends, among other factors, on the reservoir
© 2008 EAGE www.firstbreak.org
juxtaposition on the up thrown block against the downthrown block. For this reason, lateral continuity of the
fault and vertical displacement of the hanging wall from
the foot wall need to be carefully analyzed.
Curvature attributes allow quantifying and qualifying most of these aspects and illuminate the analysis of
each structural trap. Vertical throw in sub-vertical faulting is generally best seen on vertical seismic sections, while
strike-slip faults (lateral displacement) are better seen in
horizontal sections (slices). Horizontal sections extracted from the three-dimensional curvature cube enable the
interpreter to qualify vertical and strike-slip faulting displacement.
Minimum curvature and maximum curvature attributes
are highly sensitive to brittle deformation especially in the
fault nose areas. High values of major curvature correlate
directly with high values of brittle deformation. High values of minimum curvature and maximum curvature will be
spatially arranged in such a way that they will define geological lineaments corresponding to faults.(Figure 5c) Lateral continuity, length, orientation, spacing between faults
are defined from the analysis of lineaments on horizontal
sections (slices) extracted from the minimum and maximum curvature 3D attribute cubes. The result of this analysis will help to appraise the possible connectivity between
both blocks. In the present case study, lineament analysis
shows en-echelon patterns with an average length of the
fault equal to 400 m (Figure 6c).
Dip curvature is an attribute which often highlights the
areas where the layer is broken. In an extensive regime,
positive values of this attribute correspond to ‘bottom-up’
shapes such as fault noses; negative values correspond to
synform shapes such as erosional scours. High values of
this attribute indicate the deformation is brittle, relatively low values indicate ductile deformation or no deformation at all. Limits between ductile and brittle deformation may be highlighted on maps by colour coding. Lateral misalignment of these limits between the foot wall and
the hanging wall will reflect strike-slip movement. Qualification and quantification of the strike-slip displacement
is then possible. In the current case study, sinistral movement was evidenced with a horizontal average throw equal
to 150 m (Figure 6a).
Separation between strong negative and strong positive
values of the dip curvature attribute (red and blue colours on
Figure 6) measures the vertical displacement. In the present
case study, the vertical displacement was varying from 35
to 110 milliseconds (Figure 6b). Using the above-mentioned
attributes, it has been inferred that hydrocarbon trapping in
the study area is controlled by a series of normal north to south
trending en-echelon faults The maximum curvature and dip
curvature attributes suggest that the regime of constraint is a
trans-tensional stress with northeast-southwest sinistral shear.
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Figure 5 a) Structural slice with coherency; b) Structural slice with dip curvature; c) Structural slice with maximum curvature.
Figure 6 a) Lateral throw from dip curvature; b) Vertical displacement from dip curvature; c) Length from dip curvature.
Reservoir characterisation and fracture analysis
Naturally-fractured reservoirs are an important component of global hydrocarbon reserves. It is important for the
prediction of future reservoir performance to detect zones
of fracturing and, at least qualitatively, estimate their basic
parameters, for example, the density and orientation of
the fractures. Fractures are usually difficult to resolve from
seismic amplitude data due to the seismic frequency content which limits seismic resolution. In our example data
set, despite the fact that the fractures are poorly illuminated, the curvature attribute detected the fractured areas.
Fracture signatures derived from curvature attributes are
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indicated by a relatively medium to high value of the
minimum curvature. Most of the lineaments defined by the
spatial arrangement of the minimum curvature attribute
correspond to fractures.
In the present case study, zones of fracturing are mainly
detected close to the major brittle fault events (Figure 7).
Conclusions
The new technique proposed here to compute volumetric
curvature attributes performs calculations in a single step,
without requiring any pre-computation of intermediate
volumes such as dip and azimuth.
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Figure 7 Structural slice with minimum curvature. Fractured areas indicated close to faults.
Curvature attributes allow quantifying and qualifying lateral continuity of the fault and its vertical displacement. They support the analysis of structural traps occurring against faults.
Geological model properties benefit from the qualitative
and quantitative information extracted from the curvature
attributes, such as fracture density and orientation.
As a future perspective, a post processing of the curvature attributes may be implemented in order to sort out singular geological lineament orientations. This approach could also
be used to remove non-geological lineaments such as acquisition footprints
The curvature attributes can augment the coherency
attribute in the analysis of the geological scheme.
References
Acknowledgements
West, B. P., May, S. R., Gillard, D., Eastwood, J. E., Gross, M. D., and
Al-Dossary, S. and Marfurt, K. [2006] 3D Volumetric multispectral estimates of reflector curvature and rotation. Geophysics, 71(5).
Bahorich, M. and Farmer, S [1995] 3D seismic coherency for faults and
stratigraphic features. The Leading Edge, 14(10).
Chopra, S. and Marfurt, K. [2007] Curvature attribute applications to
3D surface seismic data. The Leading Edge, 26(4).
Donias, M., Baylou P., and Keskes, N. [1998] Curvature of oriented patterns: 2-D and 3-D Estimation from Differential Geometry. IEEE
International Conference on Image Processing, 1, 236-40.
Hobbs, W. H. [1904] Lineaments of the Atlantic border region. Geological Society of America Bulletin, 15.
Roberts, A. (2001) Curvature attributes and their application to 3D
interpreted horizons. First Break, 19(2)
We thank Paradigm for permission to publish this work
and Clyde Petroleum for the use of its seismic data.
© 2008 EAGE www.firstbreak.org
Frantes T. J. [2003] Method for analyzing reflection curvature in
seismic data volumes. US Patent No 6662111.
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