Testing the predictive capability of curvature analyses

Testing the predictive capability of curvature analyses

S. BERGBAUER

BP Exploration, Farburn Industrial Estate, Dyce, Aberdeen AB21 7PB UK

(e-mail: [email protected])

Abstract: The curvature of geological surfaces has been used to predict areas of elevated strain,deformation and fracture density. The research presented here tests if and to what extent a geo-metric measure such as normal surface curvature can be used as a proxy for deformation by com-paring field observations with geometric modelling results. This is achieved by first quantifyingfracturing in outcrop and then by performing a curvature analysis of the deformed beddingsurface. This research suggests that curvature analysis by itself does not allow for the predictionof deformation: fracture density in the Emigrant Gap anticline is unrelated to horizon curvature,and synfolding fractures are aligned with prefolding fractures instead of the directions ofprincipal curvature.

Normal surface curvature by itself has only limited value in predicting strain or fracture density;however, surface curvature is a unique descriptor of shape. Describing the geometry of a horizonquantitatively is an essential first step when attempting to compare physical and numerical modelswith natural surfaces. The tools presented here allow for unique descriptions of three dimensionalfolded surfaces that are based on the normal surface curvature, and they provide the necessarymathematical rigour and flexibility to allow for descriptions of non-cylindrical folded surfaces.

For over a century, structural geologists haveworked on elucidating the mechanisms by whichsedimentary strata are deformed during folding(e.g. Gilpin 1883; Cloos 1936; Ramberg 1961;Ramsay 1967; Stearns 1968; Johnson 1977; Davis1979; Treagus & Treagus 1981; Suppe 1983;Dunne 1986; Ramsay & Huber 1987; Suppe &Medwedeff 1990; Johnson & Fletcher 1994;Bobillo-Ares et al. 2000). A key element of thesestructural investigations is the quantitative descrip-tion of the geometrical aspects of folded sedimen-tary strata. The geometry, taken from field orseismic data, is used to create idealized foldmodels based on continuum mechanics principles.Results of these model experiments, whether phys-ical or numerical, are compared to the descriptionsof natural folds to test hypotheses about the foldingmechanisms. This paper challenges various geo-metrical measures of folds in common use andfinds them inadequate to describe the uniquespatial variations in fold shape. We argue that anaccurate description of the geometry of folded sur-faces is a prerequisite to all model studies and pointout that modern technological innovations such asLiDAR, GPS, 3D seismic reflection and 3D scan-ning provide data that warrant a more rigorousapproach to the description and subsequent geome-chanical analysis of folds.

Differential geometry is a mathematical tool forthe quantitative description of curves and surfaces(e.g. Struik 1961; Lipschutz 1969; Stoker 1969).Geological structures such as slickenlines, meta-morphic lineations, surface intersections, plumose

structures and rib marks, as well as bedding,faults, fractures, metamorphic foliations, sedimen-tary boundaries and unconformities are inherentlythree-dimensional and curved, so differential geo-metry is the appropriate tool for their quantitativedescription (Chapter 3 in Pollard & Fletscher2005). Although these structures are commonlydescribed as ‘lineations’ and ‘planes’, and fieldmeasurements of trend and plunge or strike anddip imply rectilinear and planar forms, this oversim-plified approach to description and data gathering isnot sufficient for a rigorous quantitative description.Tools and concepts of differential geometry such asthe surface normal vector, tangent vector, andbinormal vector, as well as the spatial changes ofthese quantities as expressed by the concepts oftorsion and curvature provide the quantitativerigour. Despite the fact that differential geometrywas a well-developed discipline more than onehundred and fifty years ago (e.g. Gauss 1827)when structural geology was in its infancy, it hasfound only limited application in the quantitativedescription of geological curves and surfaces.

Apart from describing geological surfaces (e.g.Bengtson 1981; Lisle 1992; Lisle & Robinson1995; Roberts 2001; Bergbauer & Pollard 2003;Bergbauer et al. 2003; Pollard & Fletscher 2005;Mynatt et al. 2007), most publications focusedaround quantifying the degree of deformation orstrain in deformed strata (e.g. Bevis 1986; Ekman1988; Lisle 1994; Nothard et al. 1996; Samson &Mallet 1997) and predicting fracture orientationsand densities in bent strata (e.g. Murray 1968;

From: JOLLEY, S. J., BARR, D. WALSH, J. J. & KNIPE, R. J. (eds) Structurally Complex Reservoirs.Geological Society, London, Special Publications, 292, 185–202.DOI: 10.1144/SP292.11 0305-8719/07/$15.00 # The Geological Society of London 2007.

at University of St Andrews on December 6, 2014http://sp.lyellcollection.org/Downloaded from

http://sp.lyellcollection.org/

Thomas et al. 1974; Lisle 1994; Fischer &Wilkerson 2000; Hennings et al. 2000).

In most cases these investigators have appliedtheir analyses to seek answers regarding the stateof deformation, or the degree of fracturing, associ-ated with folded and domed strata. However,results of such analyses have been somewhatambiguous. Whereas some authors concluded cur-vature analysis is a valid tool (e.g. Ewy & Hood1984; Lisle 1994; Hennings et al. 2000) othersquestion its predictive capability (e.g. Schultz-Ela& Yeh 1992; Jamison 1997). For example,Bellahsen et al. (2006) identify four fracture setsfrom detailed mapping at Sheep Mountain Anti-cline, Wyoming, and point out that the oldest ofthese predates folding and strikes oblique to thefold trend; the next younger set correlates in strikewith the Laramide compression direction relatedto folding; the next younger set correlates in strikewith the fold trend and is localized in the hingeregion; and the youngest set strikes parallel tothe oldest set, but is vertical in the dipping bedsof the back limb rather than perpendicular tobedding. Of these four sets only the third may beunambiguously associated in terms of fracturedensity and orientation with the greater curvatureof beds in the fold hinge.

The most common reason for calculating surfacecurvature is to predict areas of high fracture densityas well as the orientations of fractures that formedduring deformation of strata. Murray (1968) founda relationship between fracture permeability andcurvature. He argued that fracture porosity isrelated to the product of bed thickness and curva-ture, and that fracture permeability is related tothe third power of this product. Thomas et al.(1974) proposed an approach, based on thin platetheory, to predict the location of structural disconti-nuities where maximum principal curvatures areelevated. Despite these early attempts to derivefracture predictions through the relationship ofcurvature and stress–strain, others have argued fora direct relationship between fracture density andsurface curvature (Schultz-Ela & Yeh 1992;Ericsson et al. 1998; Stewart & Podolski 1998).These authors argued that zones of high fracturedensity should coincide with areas of elevated cur-vature. Lisle (1994) presented a similar argumentrelating high Gaussian curvature with areas ofhigh fracture density. Whereas Schultz-Ela & Yeh(1992) concluded that only moderate curvaturescorrelate well with productive fracture zones, buthighest curvature magnitudes do not, Lisle (1994)and Ericsson et al. (1998) argued for the predictivecapability of their approaches. Trying to predict theorientation of joints, Fischer & Wilkerson (2000)modelled the temporal evolution of a fold andargued that if joints formed during deformation,they would parallel the direction of minimum

curvature. Finally, Hennings et al. (2000) tested avariety of surface attributes extracted fromseismic data in their ability to predict the locationand intensity of observed fracturing of an exposedLaramide fold. They concluded that the rate of dipchange and Gaussian curvature relate best withtheir fracture density observation. The techniquesdescribed above are now widely used in the hydro-carbon industry to predict subseismic fracturing.

The theory of pure bending of a thin plate(e.g. Timoshenko & Woinowsky-Krieger 1959)appears to lend some credibility to the idea of asimple relationship between surface curvature anddeformation as it relates the bending strain in athin plate with the product of curvature of theneutral surface and vertical distance from theneutral surface (Fig. 1). However, if applicable todescribing the state of deformation of folded strata,assumptions inherent to the thin plate theory needto be applied to the layered strata as well.

Tools from differential geometry for the

description of surfaces

Natural surfaces may be idealized either by continu-ous functions, or characterized by samples at pointson irregular (geological) or regular (seismic orscanned) grids. In either case, the surface can bedefined as:

r(x, y) ¼ xex þ yey þ z(x, y)ez, (1)

where r is the position vector of every point on thesurface, x and y are two independent parameters,z(x, y) is a continuous function or a set of elevationmeasurements with respect to the x–y parameter

n eu tra l su rface

ε

Mode I

cracks

y

bend = –k∗y

Fig. 1. Thin plate theory. The pure bending strain(1bend) is linearly related to the surface curvature (k) andthe distance from the neutral surface (y). If jointsformed due to the pure bending of a layer, they wouldparallel the axis of zero (minimum) curvature.

S. BERGBAUER186



plane. ex, ey, and ez are the unit vectors in the direc-tion of the Cartesian coordinate system axes, andequation (1) is the parametric form of surfacesconsidered here.

Two partial differential equations, the so-calledFirst and Second Fundamental Forms, uniquelydescribe the shape of a surface in 3D Euclideanspace (Lipschutz 1969, p. 171). The first fundamen-tal form, I, is calculated from the square of anarclength and provides the ‘yard-stick’ on acurved surface; it can be used for the calculationof angles, distances, and areas:

I ¼ dr † dr ¼ axxdx2 þ 2axydxdyþ ayydy2: (2a)

The coefficients of the first fundamental form (aij)are sometimes referred to as the metric coefficients.These coefficients are calculated from scalar pro-ducts of the first partial derivatives of the parametricequation (1) for the surface:

axx ¼@r

@x†@r

@x, axy ¼

@r

@x†@r

@y, and

ayy ¼@r

@y†@r

@y,

(2b)

which reduce to scalar quantities that are productsof the partial derivatives of the elevation z(x, y):

axx ¼ 1þ @z

@x

� �2

, axy ¼@z

@x

� �@z

@y

� �, and

ayy ¼ 1þ @z

@y

� �2

:(2c)

The second fundamental form, II, and its coeffi-cients quantify how the unit normal vector Nchanges orientation on a curved surface. The unitnormal vector N to the surface is given by thecross product of two tangent vectors (tx and ty),which describe the tangent plane to the surface ata point, divided by the magnitude of this product(Lipschutz 1969, p. 175). N is defined as:

N ;tx � ty

tx � ty

�� ¼@r

@x� @r

@y

@r

@x� @r

@y

��: (3a)

The components of the vector N are calculated fromthe first partial surface derivatives and the

coefficients of the first fundamental form:

N ¼ez � @z

@xex � @z

@yeyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

axxayy � a2xy

q : (3b)

Changes in orientation of N with position therebyprovide information on the local shape of thesurface:

II ¼ dr † dN

¼ bxxdx2 þ 2bxydxdyþ byydy2: (4a)

Using the parametric equation (1), the coefficientsof the second fundamental form are:

bxx ;@2r

@x2† N, bxy ;

@2r

@x@y† N, and

byy ;@2r

@y2† N,

(4b)

which reduce to scalar quantities that are pro-portional to the second partial derivatives of theelevation z(x, y):

bxx ¼@2z

@x2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaxxayy � a2

xy

q ,

bxy ¼

@2z

@x@yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaxxayy � a2

xy

q , and

byy ¼

@2z

@y2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaxxayy � a2

xy

q ,

(4c)

These coefficients are sometimes referred to as thecurvature coefficients.

Coefficients of the two fundamental forms varydepending on the direction and coordinate systemin which they are calculated on the surface.However, the fundamental forms (as opposed tothe individual coefficients) are invariant withrespect to the choice of parameterization and thecoordinate system (Lipschutz 1969, p. 172, 175).Here, the coefficients of the two fundamentalforms are calculated at every grid point using afinite difference scheme (e.g. Ames 1992;Bergbauer & Pollard 2003). The only limitationsare the precision of the data, the spacing of thesample grid, and the necessity that surface parame-terizations be single-valued. Examples of multiple

PREDICTIVE CAPABILITY OF CURVATURE 187



values for z(x, y) as in overturned folds or thrustfaults can be analysed as separate patches. Thetwo fundamental forms provide a mathematically-sufficient, quantitative description of the shape ofthe surface at every grid point. In principle, knowl-edge of the first and second fundamental forms is allthat is required to describe a folded surface.

Similar to the concepts of stress or strain, whicheach require six independent quantities (the com-ponents) for a complete description of the state ata point, a complete description of a surface at apoint requires the six coefficients of the two funda-mental forms. Also, similar to the spatial variationsof stress or strain in a three-dimensional volume,these coefficients vary depending on the locationon the surface in three-dimensional space. Ratherthan providing renderings of all six stress com-ponents throughout a volume, a state of stress is pre-sented more conveniently by providing the threeprincipal magnitudes and their directions. In differ-ential geometry, because of their differential nature,displaying the two fundamental forms themselves isnot practical. One could provide maps of the sixcoefficients; however, for applications that aremainly concerned with describing the shape of asurface, a measure called the normal curvature isgenerally calculated, and its two principal valuesand their orientations are mapped onto the surfaceto convey information on the shape of that surface.

From the two fundamental forms, the normal cur-vature can be calculated (Lipschutz 1969, p. 180):

kn ¼II

I¼

bxxdx2 þ 2bxydxdyþ byydy2

axxdx2 þ 2axydxdyþ ayydy2: (5)

The magnitude of the normal curvature at a point ona surface depends on the direction in which it is cal-culated, and it varies smoothly with direction accord-ing to Euler’s equation from minimum to maximumvalues at 908 intervals (Lipschutz 1969, p. 196).Thus, a convenient way of representing the normalcurvature at any point is by providing the two princi-pal values, Kmin and Kmax, and their directions, whichare orthogonal (Lipschutz 1969, p. 181). From thetwo principal values, the Gaussian (G) and mean(M) curvatures can be calculated at every gridpoint (Lipschutz 1969, p. 184):

G ¼ Kmax † Kmin (6)

M ¼ 0:5ðKmax þ KminÞ: (7)

A combination of G and M, which we refer to asshape-curvature, has been used to describe localsurface geometries qualitatively (Roberts 2001;Bergbauer et al. 2003). They describe the localsurface shape in the context of second-order

surfaces (Fig. 2). Apart from discerning areas thatare shaped like these six surfaces (plane, antiform,synform, basin, dome, saddle), shape-curvature pro-vides a quantitative method for the discriminationof areas that resemble the cylindrical shapesshown in Figure 2 (G ¼ 0) from non-cylindricalshapes (G = 0). A cylindrical fold is a fold madeof a straight line that is moved parallel to itselfalong a curve in space (Marshak & Mitra 1988,p. 157), and therefore one of the two principal cur-vature magnitudes and subsequently the Gaussiancurvature G of a cylindrical fold is always zero.Although zero Gaussian curvature is a necessarycondition of cylindrical folds, not all folds thathave zero Gaussian curvature are cylindrical (suchas conical folds). In the context of second order sur-faces, conical folds would resemble domal orbasin-like structures.

Shape curvature can be used to calculate the cur-vature magnitudes that depart from a perfectlycylindrical fold shape, thus allowing for a quantifi-cation of ‘non-cylindricity’ of the structure. More-over, shape-curvature provides information on thelocal closure direction of the folded surface. Thisconcept in combination with locations of zero dipcan be used for the automatic evaluation ofhydrocarbon-trapping structures, their spillpoints,and trapped volumes.

Use of curvature to predict deformation

Pure bending of a thin plate

Thin plate theory (Fig. 1) relates bed curvature tostrain, and strain relates to stress via the elasticproperties. Whereas the curvature is the same forall bedding parallel surfaces in that plate, stressesrange from tension above to compression belowthe neutral surface. Since fracturing occurs as a

Fig. 2. Six fundamental geometries can be distinguishedbased on the signs of Gaussian (G) and mean (M)curvatures. Geometries with G ¼ 0 (plane, antiform,synform) are called cylindrical (modified from Roberts2001).

S. BERGBAUER188



response to stresses acting across a potential fractureplane, fracture densities are expected to vary signifi-cantly across the bent plate. Thin plate theoryassumes that the plate only experiences purebending deformation, that strains are infinitesimal,that the thickness of a mechanical unit is small com-pared to its lateral extent, that planar cross-sectionsremain planar during deformation, and that shearstrains are negligible. However, strains are generallyconsidered infinitesimal below 1%, thus renderingthe applicability of thin plate theory to describe thedeformation of most geological folds questionable.Moreover, geological dome structures are oftenformed by significantly stretching the strata as theyare pushed up. The bulk of the strain experiencedby strata involved in these folds is therefore notwell described by the pure bending of a thin plate.Despite these limitations, several investigators haveproposed a direct relationship between deformation(i.e. degree of deformation or strain, fracturedensity, permeability) and curvature (e.g. Murray1968; Thomas et al. 1974; Schultz-Ela & Yeh 1992;Ericsson et al. 1998; Stewart & Podolski 1998;Fischer & Wilkerson 2000; Hennings et al. 2000).

Fractures are initiated and propagated as a resultof stress concentrating at flaws and at fracture tips(Griffith 1924; Anderson 1951; Lawn & Wilshaw1975; Pollard & Aydin 1988; Willemse & Pollard1998), hence understanding the stress distributionin deforming strata in space and time is critical.Stresses that arise in strata during fold developmentare unlikely to be explained by one mechanismalone, such as pure bending. For example, stratathat are domed by a rising diapir experience con-siderable stretching, which is not included in thecalculation of pure bending (Jackson & Pollard

1988). Also, the interaction between deforminglayers of different lithologies has the potential toalter the local stress field (e.g. Treagus 1988; Bai& Pollard 2000a, b; McConaughy & Engelder2001; Wilkins et al. 2001; Young 2001). Moreover,the conditions of the layer boundaries affect thestresses within strata during folding (Cooke &Underwood 2001). If, for example, no bedding-parallel slip occurs at interfaces, the entirepackage acts as a single mechanical unit. If bedding-parallel slip develops, or if more ductile interbeddedshale or salt layers accommodate significant strain,the stress distribution within the package can bequite complicated. Finally, the stress distribution inthe strata changes over time as the structure evolves(Fischer & Wilkerson 2000). Early formed openingfractures, for example, may rotate along with thefolding strata, and may be reactivated in shear asthe deformation proceeds. These early fracturesmay perturb the stress field of the fold if they slipor open, and subsequent fracturing may be localizedalong their tiplines and at bends and steps along theirsurfaces. Because thin plate theory only accounts forpure bending stresses (thus ignoring plate stretching,layer interactions, or stress perturbations due to het-erogeneities such as pre-existing fractures) it is anunsuitable model for investigating fracturing duringfolding of layered strata. The following curvatureanalysis of an outcropping bedding surface providesan example of the deformation in a folded surface thatis ill described by thin plate theory.

Field observations

The Emigrant Gap anticline, located near Casper,Wyoming, is a doubly-plunging anticlinal fold

Fig. 3. Location of the Emigrant Gap anticline, and simplified geological map of parts of Natrona County, WY (fromLageson et al. 1980). Only relevant roads and waterways are shown.




that parallels the nearby hydrocarbon-producingPine Mountain – Oil Mountain trend (Stiteler1954; Cavanaugh 1976; Winn 1986; Henningset al. 2000; Bergbauer & Pollard 2004). The brea-ched anticline trends approximately NW–SE andis exposed for about 30 km (Fig. 3). The anticlineis asymmetrical, with the west flank dipping moresleepy than the east flank. The Emigrant Gap anti-cline is interpreted to be a forced fold related toLaramide-age movement along the NE-dippingEmigrant Gap thrust (Hennings et al. 2000;Bergbauer & Pollard 2004).

The exposed portion of the anticline consists ofthe Mesozoic Mowry Shale and the FrontierFormation (Fig. 4). These units are part of theWestern Interior Cretaceous Basin fill, deposited

in a foreland setting, during the subduction of theFarallon plate under the North American cratonand creation of the fold-and-thrust belt of westernNorth America (Kauffman 1984). The Cretaceousstrata in Wyoming consist of approximately1.5 km of mostly siliciclastic sediments derivedfrom both sides of the basin. The Upper Cretaceousstratigraphy for this part of the Casper Arch is sum-marized in Figure 4b. Mesozoic and older rockswere deformed and uplifted during the LaramideOrogeny (Schmidt et al. 1993).

The topographic expression of the Emigrant Gapanticline (Fig. 5a) is shaped by resistant sandstonebeds of the Frontier Formation, which consistof a cyclic package of sandstones and shales(Fig. 4b). According to Cavanaugh (1976), each

Fig. 4. Exposed Cretaceous stratigraphy. (a) Photograph along the fold axis looking approximately south showingthe exposed stratigraphy and fractures in the A2 sandstone bed. (b) General stratigraphy of the Big Horn BasinWyoming (modified from Sundell 1986) and detailed stratigraphy of the Frontier Formation (modified fromCavanaugh 1976).

S. BERGBAUER190



transgressive–regressive cycle of the FrontierFormation starts with a shale unit, which coarsensupward and is capped by a relatively prominentsandstone bed. These sandstone beds mark the endof each of the five cycles (A1, A2, B, C, D).Bedding-parallel slip probably occurred along theshale units during folding. However, the authorfound very little evidence for slip along bedding sur-faces within the competent sandstones beds. Inaddition to the five sandstone beds identified byCavanaugh (1976), two more resistant sandstonebeds can be traced around the northern end of theanticline; these represent minor regressive eventswithin one of the major cycles. These two beds arenamed B2 (B minus) and C2 (C minus) accordingto their location within the stratigraphy (Fig. 5a).

Near the northern termination of the anticline acreek cuts through the entire stratigraphy of theFrontier Formation leaving the lowest sandstonebed (A1) exposed continuously across the fold

hinge. This location is particularly useful forcollecting data in three dimensions (Fig. 5b) asthe sandstone bed is exposed in map view andcross section allowing for fractures to be tracedfrom the pavement into the cross section. Otherbeds of the Frontier formation also form extensivepavements on the fold limbs and are evensporadically-exposed around the fold hinge, allow-ing for extensive data collection (Fig. 4a).

From observations on these outcrops, Bergbauer& Pollard (2004) developed a conceptual modeldescribing the deformation of the exposed sand-stone layers, which considers the existence of twojoint sets prior to folding. This is based on the obser-vation that the same two fracture sets occur infolded and non-folded beds of the Frontier For-mation across the field area shown in Figure 3(see Bergbauer & Pollard (2004) for a morethorough discussion). These prefolding joints intro-duced anisotropy in the sandstone layers that

Fig. 5. (a) Oblique aerial photographs looking north showing exposes sandstone beds and fractures. (b) Close-up of A1

sandstone unit, showing mapped joints (blue) and faults (red).




influenced the local stress field during folding(Fig. 6). On the fold limbs, this deformationinduced further propagation of the existing jointsof both sets along strike and also led to infilling ofnew joints between older joints. This mechanismresulted in two systematic joint sets with fracturedensities of approximately four times thoseobserved in non-folded outcrops of the same sand-stone bed, regardless of the structural position onthe fold. The prefolding joints influenced the localstress field to the extent that new joints propagatedsubparallel to existing ones, despite the rotation ofbeds during folding. Because the beds on the foldlimbs are only gently curved, this tilting is bestdescribed as a nearly rigid body rotation withminor internal deformation.

In contrast to the limbs of the anticline, stratalocated near the fold hinge were significantly bentduring folding, leading to an increase in fracturecomplexity there (Fig. 6). Similar to the limbs,joint sets in the hinge region were also developedfurther by propagation of early-formed joints andby infilling. This led to a more dense set of sub-parallel fractures similar in density to joints

observed in the fold limbs. However, duringbending some of these joints were locally reacti-vated in shear, now seen as minor shear offsets.The observed shear fractures are absent on thefold limbs suggesting a different deformation mech-anism. Near the upper and lower tiplines of theseinclined shear fractures, tailcracking locallyincreased fracture intensity above levels observedin the fold limbs. Neither of the two sets of fracturesobserved on the fold (limbs and hinge) strike paral-lel or perpendicular to the fold axis.

Curvature analysis of the surface and

comparison to fracture model

To reconstruct the top of the lowest sandstone bedof the Frontier formation numerically the outcropswere sampled in the field (black dots in Fig. 7)using a TrimbleTM Pro XL GPS receiver at intervalsnot exceeding 100 m (where outcrop permitted).Vertical precisions of 0.5 m to 1.5 m, and horizontalprecision of less than 0.7 m were obtained afterrecording approximately 5 readings at eachlocation, and after real time differentially correctingeach position. 2529 point measurements of thenorthern part of the outcropping sandstone bedwere collected. Stereographic projections of 59poles to bedding measured along this sandstonebed indicate an approximate fold axis of 3458,048. When viewed along a calculated fold axis(down-plunge view), the top of the sandstone beddefines an anticlinal folded surface with approxi-mately planar limbs and a rounded hinge (Fig. 8).From the GPS data was created a digital model ofthe surface using gOcadTM and MatlabTM (Fig. 7),despite the limitation that only c. 25% of theintended area of the model is exposed in the field.However, a model of the bedding surface wascreated to illustrate geometric tools, with the under-standing that inferences from the geometric analysisare valid only in regions where data constrains thesurface model. Prior to the geometric descriptionthe created surface was smoothed using the filterdescribed by Bergbauer et al. (2003).

In order to test if normal surface curvature mightpredict fracture density and fracture orientation, theprincipal curvatures of the modelled anticlinalbedding surface were calculated and then comparedwith the observed fracture style and density.Figure 9 shows minimum and maximum curvaturesrespectively. Minimum principal curvatures rangefrom –1e23 to 1e3 m21 and are roughly twoorders of magnitude smaller in value than themaximum principal curvatures, which range from–1.7e24 to 3.8e3 m21. Colours depicting elevatedmagnitudes of minimum curvature trend obliquelyacross the fold axis, which are caused by gentle

Fig. 6. Conceptual fracture-fold model for the EmigrantGap anticline acknowledging the existence of twoprefolding joint sets. (a) Prefolding stage: two, almostorthogonal joint sets (J1 and J2) form in flat-lying strataprior to folding. (b) Synfolding stage on the limbs: jointsform parallel to the two prefolding joint sets.(c) Synfolding stage on the fold hinge: fracture styledepends on the relative orientation of hinge lineand prefolding joints. Whereas shear fractures (SF1) andtailcracks form strike-parallel to the J1 joints, mostlyjoints form subparallel to the J2 set.

S. BERGBAUER192



surface undulations observed in outcrop. Thedirections of minimum curvature, shown as whiteticks, generally trend subparallel to the trend ofthe hinge area, and only change directions signi-ficantly across the gentle surface undulations. Themaximum principal curvature, which is mostlypositive and considerably larger in value than theminimum curvature, is greatest near the crest ofthe structure. Directions of principal maximumcurvature (white ticks) trend approximately perpen-dicular to the hinge region.

This curvature analysis predicts that fracturespacing should be smallest over the hinge of theanticline (where curvature is greatest), and fracturesshould trend approximately parallel to the fold axis(parallel to the direction of minimum curvature) asindicated in Figure 9c. However, we found thatfracture density was approximately constant onthe fold. Moreover, fractures strike oblique to the

directions of principal curvature (black lines inFig. 9c indicate fracture strikes), not following thedirection of minimum curvature as predicted bythin plate theory (Fig. 1). Thus, a direct relationshipbetween fracturing and normal surface curvaturecannot be established for the bedding surface mod-elled here. This is not surprising considering thatfractures here did not form in a systematic stressfield, as predicted by thin plate theory, but in acomplex stress field perturbed by the reactivationof prefolding joints and mechanical interactionsbetween beds. This observation should hold truefor most geological folds and domes, not onlybecause of the layered nature of geological strata,but also because of rocks generally exhibit hetero-geneities on all scales that influence fracture for-mation (e.g. prefolding fractures, fractures formedearly during folding that subsequently reactivate,sedimentological heterogeneities, spatial variations

–500 0 500 1000

–100

0

100

200

Fig. 8. Downplunge view of 2529 GPS points collected on one bedding surface. The fold axis was determined bystereographic projection of 59 poles to bedding (fold axis strike and plunge: 3458/048).

Fig. 7. Oblique view of smoothed model bedding surface. Black dots are GPS data used to create the model. Note thatsome of the GPS data is located below the surface.




Fig. 9. See p. 195 for caption.

S. BERGBAUER194



in rock properties). Thus, to predict fracturessuccessfully a model that considers heterogeneitiesof the appropriate scale (i.e. those that have thepotential to alter the stress field in which thesefractures form) is required.

Use of curvature as a tool for a unique

description of folded surfaces

Methods

In the pursuit of quantitative descriptions of foldedsurfaces concepts discussed, for example by Turner& Weiss (1963), Fleuty (1964, 1987), Ramsay(1967) and Hudleston (1973) are commonlyemployed. These authors invoke classificationschemes based on stereographic projections ofbedding attitudes and cylindrical fold profiles asseen in down-plunge views. From observationslargely in metamorphic rocks, Turner & Weiss(1963) provide abundant examples and techniqueson the usage of the stereographic projection forfold description. Fleuty (1964, 1987) and Ramsay(1967) focus on establishing ‘unambiguous’ classi-fication schemes for the accurate description offolds using concepts such as crest line, troughline, hinge line, fold axis, dip isogons, and foldtightness. Ramsay (1967) uses the concept of curva-ture, measured along cross-sections of cylindrically-folded surfaces, to establish a classification schemefor the relationship of adjacent folded surfaces.Hudleston (1973) establishes a visual approach toclassifying fold shapes by providing 30 idealizedcross-sectional views of cylindrical and symmetri-cal folds. These approaches are reviewed inmodern textbooks for students of structuralgeology (e.g. Davis 1984; Suppe 1985; Ramsay &Huber 1987; Marshak & Mitra 1988; Twiss &Moores 1992).

A general problem associated with these folddescriptions is the assumption of cylindricity ofthe folded surfaces. Although a fold might locallycontain cylindrical patches, a folded geologicalhorizon is never globally cylindrical. For example,folds do not extend forever, as required by theassumption of cylindricity. This fact is recognizedby most structural geologists, and often p-diagramsare used to discriminate patches of local cylindricity

along the three dimensional fold. A cylindrical foldwould have identical cross-sections regardless ofwhere the section is constructed, implying thatfold hinge lines, inflections lines, trough lines, andcrest lines are straight. Terms such as ‘trueprofile’, p-diagrams and b–axis all assume thefold to be cylindrical (Lisle & Robinson 1995).Similarly, fold classification schemes that usemeasures such as the fold tightness (Fleuty 1964),the shape of a fold profile (Hudleston 1973), andthe hinge: limb ratio (Ramsay 1967) assume thatthe fold maintains its shape along trend. Whencylindricity of the structure is assumed, it is suffi-cient to provide one fold profile, to give onemeasure for bluntness, and to measure one directionfor the fold hinge line. These aspects of the foldwould then be sufficiently described everywhere,because those measures are constant throughoutthe fold.

The assumption of cylindricity is used for classi-fication schemes because it is a powerful and effec-tive idealization, and because three dimensionaldata that might challenge such a simplificationused to be rare. Moreover, the computational andvisualization tools for the analysis and presentationof three-dimensional data were not developed, andsubsequently the assumption of cylindricitybecame a standard for structural analysis.

Over the past twenty years this standard hasbeen questioned. Bengtson (1981) suggested thatvarious broad geological structures, such as domesand folds, could be distinguished based on theirdifferent ‘bulk’ curvatures. He argued that theoverall surface geometry can be extracted from dip-meter data collected at different structural positions.Lisle (1992, 1994), Lisle & Robinson (1995) andStewart & Podolski (1998) use the sign of Gaussiancurvatures calculated over variously curved sur-faces to distinguish between elliptical, hyperbolical,and cylindrical surfaces. Such analysis allows forthe distinction between surfaces that are shapedlike domes or basins (elliptical) and surfaces thatare either saddle-like (hyperbolic) or shaped likecylindrical planes, synforms or antiforms. Lisle &Robinson (1995) point out that most existingmethods to describe folds are based on theassumption that structures are cylindrically shaped(e.g. Ramsay 1967), and suggest that surfacecurvature might be a useful tool in describing the

Fig. 9. Principal curvature vectors derived from the normal surface curvature, kn. (a) Magnitude of minimum principalcurvature, Kmin, and directions of vector (white ticks). (b) Magnitude and directions (white ticks) of maximumcurvature vector (Kmax). White lines are structure contours of the model surface shown in Figure 5. (c) Magnitude ofmaximum principal curvature, Kmax, directions of minimum curvature (white ticks), and observed fracture directions inoutcrop (black lines). If the normal surface curvature were a valid proxy for fracture density and orientation, thenfracture spacing should be smallest over the hinge of the anticline, and observed fractures should trend approximatelyparallel to the white ticks.




non-cylindrical aspects of folds. Roberts (2001)expands on these concepts by using not only thesign of the Gaussian curvature (G), but also thesign of the mean curvature (M) to distinguishlocal surface shapes (Fig. 2) as either domes(G . 0, M . 0), bowls (G . 0, M , 0), saddles(G , 0), ridges (G ¼ 0, M . 0), planes (G ¼ 0,M ¼ 0), and synforms (G ¼ 0, M , 0). Bergbaueret al. (2003) refer to this measure as shape-curvature, and suggest this might be useful in thedetection of large-scale structural domains (asimplied by Reches 1976; Cruikshank & Aydin1995; Rawnsley et al. 1998).

It is because the means of measuring, analysing,and displaying 3D spatial data are now readilyavailable, that it is proposed that the assumptionthat a folded surface is shaped cylindrically, or sym-metrically is now obsolete. All natural folds arenon-cylindrical: they do not extend indefinitelyalong the fold axis without any changes in shape.Researchers who analyse or describe digitalmodels of asymmetrical and non-cylindricalfolded surfaces (e.g. Grujic et al. 2002) need toolsthat go beyond the stereonet and idealized cross-sections. Here we explore the usage of differentialgeometry in providing quantitative measures thatavoid assumptions of cylindrical fold shapes, andthus preserve spatial changes in these measuresover the entire folded surface.

Whereas it is possible to construct a stereo-graphic projection of poles to bedding that isunique (from a given data set of attitudes or mapsof strike and dip), the reverse transform (a construc-tion of a surface from the poles to bedding) is notunique, because the spatial information has beenlost. Thus, although the bedding surface from theEmigrant Gap anticline, which was constrained by2529 geographic coordinates (UTM), allows foran infinite number of very similar surfaces thatcan be constructed from that data, an infinitenumber of very different surfaces can be con-structed from a set of attitude measurements. Poss-ible 3D surfaces that only honour attitude data (andomit the spatial information) might have similarshapes but can significantly differ in their foldamplitudes, their closure direction, their lengthscales, and even the number of folds in a foldtrain. Other possible surfaces might be constructed,honouring the same data, by any spatial arrange-ment of the attitude information, which, in contrastto the bedding surface created from the UTMcoordinates, could result in significantly differentfold shapes. Thus, representing a folded surfaceby a stereographic projection of poles to beddinginevitably leads to subjective interpretations ofthe geometry.

A surface description that is based on the firstand second fundamental forms of differential

geometry is not only unique, but it is inherentlytied to the spatial variations of the surface anddoes not leave room for a subjective interpretationof the geometry. Thus, particularly for the quantitat-ive comparison of surfaces, in addition to collectingstrike and dip measurements in the field, geologistsshould record and use the geographic coordinatesfor every sampled point on the structures they aremapping. Meaningful interpretations of geologicalsurfaces can then be presented quantitatively, inlight of the sampling theorem (surface coverage),data precision, statistics, and motivation ofthe analysis.

Despite the fact that the first and second funda-mental forms fully describe the geometry of asurface, other descriptive measures are normallyused for surface descriptions (e.g. strike, dip,hinge and inflection line). As with kn, thesemeasures can be readily derived from the two fun-damental forms, but contrary to kn, they do notprovide a full description of the shape of asurface. Some of these concepts, such as strikeand dip, prove useful for the description of afolded surface, because bedding attitudes constitutea large part of available surface data (e.g. Grujicet al. 2002). The unit normal at a point on asurface is the pole to bedding commonly used fordescribing surfaces using stereographic projections.The components of the unit normal vector, N, areused to determine the dip and the strike of thesurface (Aki & Richards 1980, p. 115):

dip ¼ cos�1ðNzÞ (8)

and

strike ¼ tan�1ð�Ny=NxÞ; (9)

where Nx, Ny, and Nz are the components of N in theex (north), ey (east), and ez (vertical) directions.

Another descriptive term in use that can bederived from the two fundamental forms is theconcept of a hinge line. Hinge lines of cylindricalfolds, which are straight lines parallel to the foldaxis, are defined as lines of maximum curvature(e.g. Marshak & Mitra 1988, p. 214). For three-dimensional folded surfaces the concept of hingelines can be generalized to curves that passthrough the loci of greatest values of normal curva-ture. Thus, they can be calculated analytically andtheir geometry may depart from that of a straightline. On planar surface patches, the concept of ahinge line has to be replaced by that of a hinge area.

Similarly, inflection lines can be derived fromthe two fundamental forms. Inflection lines onfolded cylindrical surfaces are normally thoughtof as straight lines that pass through points of zero

S. BERGBAUER196



curvature measured along the fold profile (e.g.Ramsay 1967, p. 347). This concept also can begeneralized for three-dimensional surfaces tocurves passing through points across which atleast one of the principal curvature values changessign. Thus, the principal curvatures accuratelydefine the locations of inflection and hinge lineson folded surfaces and therefore a quantitativedescription of folded surfaces should start with aquantification of the two fundamental forms, fromwhich other measures in common use can readilybe derived.

Example of a geometrical description

Figure 10 summarizes some of the traditional geo-metrical measures of a bedding surface, includinga structure contour map (white lines) and colour-coded maps of the strike and dip of bedding, tocharacterize the surface shape. Bedding dips(Fig. 10a) as large as 358 occur on the west limbof the model surface, and up to 258 on the eastlimb. The points on the surface at which the dip isclose to zero mark the crest of the structure. Thisis a narrow strip that runs approximately parallelto the y-axis. Figure 10b provides a map of thesurface strikes: using the right hand rule (Marshak& Mitra 1988, p. 7), the east limb strikes about3508, whereas the west limb strikes about 1708.The crest is identified by the approximately 1808switch of the strike angle. The line of greatestmaximum curvatures is defined as the fold hinge(Marshak & Mitra 1988, p. 214), which is shownas the solid black line in Figure 9b. The hinge lineof our modelled bedding surface curves in space,and is located to the west of the crest line (dashedblack line). Thus, the surface describes a non-symmetrical fold, and the amount of asymmetry isquantified by the spatial departure between crestand hinge curves. The asymmetry changes alongthe trend of the anticline, almost vanishing nearthe northern extremity of the anticline.

A second application of the normal curvature,kn, assesses if a folded surface is cylindrical, andprovides a measure of the surface’s departurefrom a cylindrical shape. For a surface to be cylind-rical the minimum principal curvature must be zeroeverywhere, thus forcing the maximum principalcurvature to have the same distribution on everycross-section taken perpendicular to the fold axis.Not only is the minimum principal curvaturenon-zero across the modelled bedding surface(Fig. 9a), the maximum curvature distributionsdiffer from one cross section to another (Fig. 9b).Thus, the modelled surface is not cylindrical, andthe magnitudes of the principal curvatures thatdepart from a cylindrical shape thus quantifythat property.

Fig. 10. Surface attitudes derived from the unit normalvector, N, to the model surface. (a) Bedding dips, and(b) bedding strikes (right hand rule) were calculatedat every grid point. White lines are structure contours ofthe model surface shown in Figure 7.




Discrimination of cylindrical and non-cylindrical areas across the modelled beddingsurface, and quantification of the non-cylindricalcurvatures, can also be achieved using the conceptof shape-curvature. Using the colour code ofFigure 2 the modelled bedding surface is comprisedof domal and saddle-like areas, which reflect thegentle, hinge oblique surface undulations, superim-posed on the broader fold shape (Fig. 11a). Thus,the modelled bedding surface does not containany cylindrically-shaped areas (where G ¼ 0) ofsignificant extent.

The quantification of the surface’s departurefrom cylindrical can be achieved by subsequentlysetting the smallest values of principal curvaturesto zero. Doing so turns areas that minimallydepart from one of the cylindrical shapes (shownin Fig. 2), into cylindrically-shaped geometries.This might be helpful in distinguishing areas on sur-faces that are significantly non-cylindrical fromareas that are quite cylindrical. Thus, a surfacecan be ‘smoothed’ until the underlying, cylindrical

shape is extracted, and the difference in themagnitudes of the principal curvatures prior toand after extraction of the cylindrical shapes canbe used as a measure of the non-cylindrical geome-tries of the surface.

The smallest curvature magnitude across themodelled bedding surface equals 1.5e29 m21

(Fig. 9). Small principal curvatures are locatednear lines separating different surface shapes, andwhen principal curvature jkj , 1e25 are set tozero, these areas become shaped like cylindricalantiforms (Fig. 11a). Larger areas of equal cylindri-cal shapes emerge when principal curvatures ofjkj , 5e24 m21 are set to zero, and the surfaceassumes a roughly antiformal shape (Fig. 11b) withremnants of non-cylindrical areas. Upon settingthe threshold value to jkj , 1.5e23 m21 ¼ 0, theentire model surface becomes cylindrical, with twoplanar limbs and a cylindrical anticline along thefold (Fig. 11c). The surface thus contains approxi-mately five orders of magnitude (from jkj ¼1.5e29 to jkj ¼ 5e24 m21) of principal curvatures

Fig. 11. Shape-curvature derived from the normal curvature, kn. Colours indicate the local shape of the surfacebased on the background colours used in Figure 1. (a) Principal curvature magnitudes below jkj , 1e25 m21,(b) jkj , 5e24 m21, and (c) jkj , 1.5e23 m21 are zeroed out. Five orders of principal curvatures are cancelled beforethe horizon is approximated by mostly cylindrical surface shapes.

S. BERGBAUER198



that depart from a perfectly cylindrical surfaceshape. If the surface had been sampled on atighter grid, or smoothed less prior to curvature cal-culation, the magnitude of smallest curvature con-tained in the data would have been less than thatreported here. However, this would have notchanged the curvature threshold value at whichthe surface turns into cylindrical shapes.

The concept of shape-curvature also allows forthe determination of the locations of inflection‘lines’ across the fold. Lines defining areas ofequal shape in Figure 11 are inflection lines,because one of the principal curvatures changes itssign across them (in this case it is the minimumcurvature, Fig. 9a). Several hinge-line-obliqueinflection lines, defined by the gentle surface undu-lations, are present in Figure 11a. After extracting apurely cylindrical fold shape (Fig. 11c), two inflec-tion lines, trending along the long axis of the modelsurface, define the transition from the approximatelyplanar limbs to the anticlinal hinge. Moreover,closure direction of the structures can be inferredbased on the local shape of the surface: antiformsand domes provide upward closure, synforms andbasins are closed downward, and the closure direc-tion for saddle areas depends on the surface shapeon either side of the saddle. The three shape-curvature maps indicate that model beddingsurface is comprised of either saddle areas anddomes (Fig. 11a), or antiforms, domes and planes(Fig. 11b), or planes and antiforms (Fig. 11c).Thus, all three shape-curvature maps confirm theexpected upward closure of an anticline.

A third application of kn involves the use ofshape curvature to distinguish structural domainson surfaces. A structural domain is defined as anarea of a rock mass that exhibits a common defor-mation style. In the absence of significant hetero-geneities within the same rock mass, the boundarybetween two structural domains is often markedby a change in horizon shape. For example, theshape curvature of Figure 11c distinguishesbetween an antiformal hinge (blue) and two planarlimbs (yellow). Our conceptual model (Fig. 6)shows that different deformation styles have beenobserved between the hinge and the limbs, andthat the change in style occurs approximatelywhere the surface changes its shape from an anticli-nal hinge to roughly planar limbs. If the shape of ahorizon is indeed responsible for producing differ-ent structural domains, then a predictive approachcan be employed for the non-exposed parts of thefold hinge using the concept of shape curvature:within an area of equal shape, the same fracturestyle is expected. This implies that for the non-exposed part of the fold hinge, likely a fracturestyle similar to that observed in outcrops of thehinge would be. Similarly, fracturing along the

non-exposed fold limbs is expected to be roughlysimilar to the fractures observed on parts of theexposed fold limbs.

Discussion

This analysis assumes that areas of similar surfaceshape deformed in a similar fashion, and assumingsimilar rock properties, that this resulted in similardeformation features. For the described beddingsurface, the roughly planar areas should havedeformed mainly by a rigid body rotation,whereas the areas shaped like an anticline experi-enced some bending, which resulted in shear reacti-vation of joints. This similarity analysis is invalidfor beds that exhibit significant variability in rockproperties. The observed fracture styles in onebed are often significantly different from fracturestyles observed in adjacent beds even if thenormal curvature is similar. Thus predicting frac-tures in a bed from observations of fractures inanother bed based on this similarity argument isnot recommended. Finally, faults that cut throughmost of the vertical or lateral extent of the fold(such as tear faults or the underlying thrust faultitself) locally create deformation that is not predict-able based on this analysis (Hennings et al. 2000).

Fitting a surface to the limited GPS coverageshown in Figure 7 is not a unique problem: aninfinite number of surfaces can be constructedthat more or less honour the 2529 data points.However, normal surface curvatures depend onthe surface they are calculated from, so a compari-son of the modelled surface with data and obser-vations from the bedding surface is necessary.Furthermore, conclusions about geometry inpoorly constrained areas should be viewed withappropriate scepticism.

A factor influencing the outcome of the modelbedding surface from the Emigrant Gap anticlineis the amount of seeds used for surface triangulationin gOcadTM. Triangles were seeded every 70 malong the border of the surface. This distance waschosen based on the sampling distance applied inthe field. The target of the sampling procedure inthe field was to resolve the broad scale surfaceshape, as well as the gentle, hinge line obliquesurface undulations. The maximum spatial wave-length resolved by a sampling procedure equalstwice the sampling spacing (Bracewell 2000). Ourchoice of maximum sampling distance across theaccessible patches of the surface was aimed at resol-ving surface undulations with wavelengths ofapproximately 200 m and larger, which requires aminimum sampling interval of 100 m. To avoidfurther surface aliasing, a slightly smaller seedspacing was chosen for the triangulation than the




100 m sampling interval employed in the field.Once the surface was triangulated, the choices ofre-sampling grid spacing and data interpolationtechnique (performed with MatlabTM) only hadminor effects on the final surface geometry. Theseparameters were chosen to maximize the correlationcoefficient R2.

A final factor influencing the outcome of themodelled surface is the amount of surface smoothingperformed prior to curvature calculations. Smooth-ing is necessary prior to any curvature calculation(Bergbauer & Pollard 2003; Bergbauer et al. 2003).Surface triangulation using gOcadTM produces arugged surface, full of artificial kinks where adjacenttriangles meet. The rugged gOcadTM surface isexported and smoothed using a technique based onthe fast Fourier transform (Bergbauer et al. 2003).The surface is smoothed by removing surface wave-lengths of 360 m and smaller (frequency filters ¼ 0.0028 m21). Larger wavelengths were kept,because they apparently describe the surface undula-tions targeted with the GPS sampling (Fig. 7). Thus,the maximum sampling interval of approximately100 m in the field successfully sampled these gentlesurface undulations.

Conclusions

1. At the Emigrant Gap anticline, there is nodirect link between surface curvature andstrike or density of synfolding fractures. Frac-tures there strike obliquely to either directionof the principal curvatures, and fracturedensity is roughly constant everywhere on thefold even where the magnitudes of the princi-pal curvatures are large.

2. At the Emigrant Gap anticline, the shape of thehorizon controls the deformation style: wherethe surface is significantly curved and theshape assumes that of an anticline (the hingeof the fold), shear fracturing is widespread.This is in contrast to roughly planar-shapedareas of the horizon (the limbs of the fold)where shear fractures are absent and mostfractures formed as joints. A measure calledshape-curvature can delineate these differ-ently-shaped areas. Shape curvature can there-fore be used to map out structural domains onthe anticline. This method for mapping outstructural domains should be generally appli-cable for structures where the shape of thehorizon is the dominant factor in controllingdeformation.

3. When describing folded surfaces, the first andsecond fundamental forms provide a tool thatuniquely describes their geometry. This is incontrast to common measures such as fold

hinge lines, inflections lines, true profile,tightness, hinge/limb ration, p-diagrams andb-axis etc, which all assume the fold to becylindrical, and more importantly, are inher-ently non-unique. These common measurescan be derived readily from the two fundamen-tal forms, but not vice versa.

4. It is proposed that where possible any folddescription should start with a quantificationof the surface in three dimensions. Thus,although measurements of strike and dip of asurface are useful, unambiguous representationof a surface geometry can only be achievedwhen the geographic coordinates of the datapoints are also recorded and included inthe analysis.

This research was supported by Phillips PetroleumCompany, the Stanford Rock Fracture Project, the Stan-ford McGee foundation, and by NSF EAR-0125935‘Strain Accommodation by Fracturing During Folding ofSedimentary Rock’. I would like to thank D. Pollard forhis countless contributions to this research. Reviews byR. Lisle, T. Couzens and S. Jolley have improved thismanuscript significantly.

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Documents

Testing the predictive capability of curvature analyses