Sigismundi and Soldo 2003 TLE

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The concept of surface curvature dates from work by Gauss

in the 1820s but practical applications have only been pos-sible with the advent of powerful workstations in recentyears.

In order to explain the concept of curvature, let’s firstfocus on a two-dimensional curve on a x-y coordinate (Figure1). This curved line can be thought of being made up of manyarcs of a circle, with differing centers and radii. The curva-ture at any given point on this curve is the reciprocal of theradius of the particular arc at that point. It can also bedefined as the derivative of the curve’s tangent angle withrespect to position on the curve at that point. In other words,if θ(s) denotes the angle which the curve makes with somefixed reference axis as a function of the path length s alongthe curve, then k = dθ/ds. In terms of Cartesian coordinatesx-y, tan(θ) = dY/dX.

A plane curve can be represented as a function Y = f(X)where X and Y are Cartesian coordinates. Pythagoras givesus (ds)2 = (dX)2 + (dY)2, which can be rearranged as ds/dX= [1 + (dY/dX)2]1/2. The curvature k can then be determined

by directly evaluating the derivative dθ/ds as follows:

(1)

If the X axis is tangent to the curve at the point in ques-tion, then tan (θ) approaches dθ and dX approaches ds (i.e.,the zero dip situation) and from equation 1 the curvature

can be defined as simply the second derivative, κ

= d2

Y/dX2

.We can get a sense for both the sign and magnitude of cur-vature of a curve if we replace these radii by normal vec-tors (Figure 2). In this case, we are following the conventionproposed by Roberts (2000). As indicated in Figure 2, it isquite simple to understand how the arrangement of thesevectors indicates where the curve bends (i.e., the vectors arediverging, converging, or parallel). Planar surfaces havezero curvature, anticlines positive curvature, and synclinesnegative curvature. Another useful property of the curva-ture attribute is that it is independent of orientation of thesurface, which is not the case with first derivative attributessuch as dip or edge.

This concept is straightforwardly extended to threedimensions (Figure 3) because by cutting any surface with

a plane, we can make a curve from which the curvature can be calculated. If we cut this surface with an orthogonalplane, we obtain what is known as normal curvature(Roberts, 2001). From any two sets of orthogonal normal cur-vature values, we can calculate other curvature attributessuch us mean curvature, Euler’s curvature, Gaussian cur-vature, etc.

Equation 1 shows that curvature depends on the secondderivative of the function. This means that our data should

be filtered to reduce or remove any noise and tracking mis-picks, etc. It is necessary, therefore, to run spatial filtersusing varying grid densities and to use many algorithms(median filters, despike filters, distance weighted filters,

etc.). Sometimes it is necessary to iterate this filtering, butone should take care not to “over smooth” the data to thepoint that all surface irregularities are removed. A balanceshould be kept between removing noise but maintaining sur-face information.

Visualization is routinely used in seismic interpretation,

1122 T HE L EADING EDGE NOVEMBER 2003

Curvature attributes and seismic interpretation: Case studies from Argentina basins

M ARIO E. S IGISMONDI , Petrobras Energía SA, Neuquén, Argentina J UAN C. S OLDO , Heriot Watt University, Scotland

Figure 1.Definition of

curvature k and its rela-tionship toradii.

Figure 2. Sign convention adopted for values of curvature.

Figure 3. 3D surface intercepted by several orthogonal sets of planes anddefining a few types of curvature (modified from Roberts, 2001).



especially when the intention is to identify changes in thegeometry and shape of the interpreted surface (faulting orsubtle trends). A good color map with an appropriate rangeof values can enhance discontinuities, even if they are sub-tle.

Special care must be taken with the color ramp and thecolor spectrum range because if we use many base colors,we can lose some definition. On the other hand, by usingtoo few colors or strong contrasts such as black and white,we can obscure results.

NOVEMBER 2003 T HE L EADING E DGE 1

Figure 4. Smoothed version of the time structure map.

Figure 5. W-E inline over a graben feature.

Figure 6. Dip map, top of Piedra Clavada Formation structure map.

Figure 7. Maximum curvature attribute of the interpreted horizon. The graben is within the circle.

Figure 8. Most negative curvature attribute and related detailed features.

Figure 9. Smoothed version of the time structure map at base of Vaca Muerta Formation.



Data examples from Argentina. The first data example isfrom a 3D survey (bin size = 25 25 m, nominal fold = 36)from Magallanes Basin, Argentina. The area of interest isapproximately 40 km2. The interpreted time horizon is a seis-mic marker within Palermo Aike Formation, an UpperCretaceous shale and sandstone deltaic sequence (equiva-lent to Piedra Clavada Formation). It was interpreted alonga 10 10 line/crossline grid and then autotracked over theentire area. Next a series of filters was applied to obtain a


Figure 10. Seismic section showing graben in Rio Neuquén Field.

Figure 11. Dip map of the time structure map.

Figure 12. Maximum curvature, base of Vaca Muerta Formation.

Figure 13. Gaussian curvature, base of Vaca Muerta Formation.

Figure 14. Most negative curvature, base of Vaca Muerta Formation.

Figure 15. Time structure map of glauconitic marker.



time structure map or TSM (Figure 4). As can be seen, somefeatures need special consideration. In the lower center of the map is a graben structure that can also be observed onthe seismic section (Figure 5). Figure 6 shows the dip

attribute from the top of Piedra Clavada Formation TSM(Figure 4). The color map adopted for visualization is a

black and white scheme, which aids interpretation. Whilethere is a good discrimination of the fault system in the area,let us compare this to a curvature attribute called maximumcurvature or Kmax (Figure 7). This attribute can be explainedwith reference to Figure 3: of all the orthogonal planes thatintercept our 3D surface, the one that describes the maxi-mum absolute curvature is called maximum curvature. Thecolor map in Figure 7 was selected to bring out the main

faults.Faults are clearly delimited by the juxtaposition of red

against dark blue, where red represents the downthrownside and dark blue the upthrown side of the fault. Thisallows us to distinguish between faults (juxtaposition of color) and other linear features, such as ridges and valleys.This type of discrimination is not possible with the dipattribute. With this additional throw information in theKmax map in Figure 7, the graben (circled feature) can now

be easily delimited without the need to refer to a seismicsection. Another attribute called most negative curvature(Kmn) also helps identify very subtle surface lineaments con-tained within the surface. Clearly, the detailed interpretivequality has been improved by the use of curvature attrib-

utes (Figure 8).The second data example comes from a 3D survey overRio Neuquén Field, in Neuquén Basin. The TSM in Figure9 represents the base of Vaca Muerta Formation, an Upper

Jurassic (Titonian) transgressive marine sequence that is thesource rock for the majority of Neuquén Basin.

The area of study is approximately 40 km2 (bin = 30

NOVEMBER 2003 T HE L EADING E DGE 1

Figure 16. Most positive curvature, glauconitic marker.

Figure 17. Curvedness with 1, 2, 3, and 5 aperture ratio, glauconitic marker.



30 m, nominal fold = 30). This horizon was interpreted ona 5 5 inline/crossline grid and then autotracked. Thegraben structure feature in the center of Figure 9 is alsoclearly seen on the seismic section in Figure 10. The dip mapin Figure 11 also highlights this structural feature. Onceagain, in light colors (white to light gray), we can see themain graben faulting and the associated minor internalfaults. However, although this map adequately delineates

the fault system, curvature attribute maps reveal even moredetail and surface information. The attribute in Figure 12(maximum curvature) clearly shows the graben structuresand uses a similar color scheme to Figure 7. Figures 13 and14 illustrate other additional curvature attributes, respec-tively Gaussian curvature (Lisle, 1994) and most negativecurvature. These two attributes also help identify the faultsof the graben.

The third example also comes from a 3D survey inAustral Basin. The TSM (Figure 15) represents the top of theglauconitic marker in Chimén Aike Field. The study areacovers nearly 123 km2, with the follow acquisition parame-ters: BS = 25 25 m, fold = 30, and offset = 3089 m. Figure16 shows the most positive curvature, and 17 the curved-

ness with different “ratio aperture” and the sensitive fea-tures of the fault patterns.

An attribute comparison. During the course of this work,much discussion occurred with colleagues about curvatureattributes.

Can we compare both classical attributes and curvatureattributes? Is there a way to see both of them at once andmake a “difference” map? The answer is yes and no. In orderto compare curvature with dip values, they must be nor-malized to a common scale; that is why we answer “no” toour questions. In Figure 18, two sets of data attributes areplotted together with the values of TWT (two-way travel-time) from Austral Basin. Dark blue indicates values of two-way traveltime for an inline over the graben structure. Valuesof mean curvature are magenta and dip values are olivegreen. As we can see, there is a large contrast in values of mean curvature as compared to dip values. This is also truein Figure 19, where we compare the most negative curva-ture attribute with dip values. The curvature values showa much larger increase across the faulted areas (i.e., highercontrast). The greater the contrast (on an appropriate colormap), the better the visualization result and the easier to dis-criminate faults.

Conclusion. Curvature attribute maps are a powerful toolfor visualization and interpretation of structural features.Several curvature attributes were introduced, explained,and illustrated with two data sets. In both cases, they haveimproved the interpretation and structural understanding.

Finally, a comparison between common dip attributeand various curvature attributes has been made in order tohighlight the main differences between them. In both cases,a higher contrast in the curvature attribute helps to betteridentify the fault system.

Suggested reading. “Curvature attributes and their applicationto 3D interpreted horizons” by Roberts (First Break , 2001).“Detection of abnormal strain in structures using Gaussian cur-

vature analysis” by Lisle ( AAPG Bulletin, 1994). “Numericalmethods for computing interfacial mean curvature” by Bullardet al. (in Computational Materials Science 4, Elsevier, 1995).“Seismic attributes technology for reservoir forecasting andmonitoring” by Chen and Sidney (TLE, 1997). “Predicting nat-ural fracture distribution in reservoir from 3D seismic estimatesof structural curvature” by Belfield (2000 SPE Rocky MountainRegional/Low Permeability Reservoir Symposium andExhibition). T L E

Acknowledgments: We thank the management of the Development andProduction Division of Petrobras Energía SA for allowing the data setsto be presented. Special thanks to colleagues Peter (the highlander)

McGregor, Santiago Benotti, and Diego Lenge from Petrobras Energía

SA for general and helpful discussions. Also we thank Andy Roberts fromEnterprise Oil Norge for assistance with the software and his big contri-bution to applied reservoir geophysics. Bruce Hart developed the "Curv-Z software" to run the curvedness and other attributes.

Corresponding author: [email protected]; juan.soldo@ pet.hw.ac.uk


Figure 19. Comparison of seismic attributes (normalized values);magenta = most negative curvature and olive green = dip.

Figure 18. Comparison of seismic attributes (normalized values);magenta = mean curvature and olive green = dip.

Sigismundi and Soldo 2003 TLE

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