AUTHORS

Richard J. Lisle � Departamento deGeologıa, Universidad de Oviedo, Oviedo33005, Spain; present address: School ofEarth, Ocean and Planetary Sciences, CardiffUniversity, Park Place, Cardiff, Wales CF103YE, United Kingdom; lisle@cardiff.ac.uk

After his bachelor’s degree in geology (Birming-ham, 1969), Richard Lisle went on to spe-cialize in structural geology at the Royal Schoolof Mines, Imperial College (M.Sc., 1969; Ph.D.,1974). He has held posts at the Universitiesof Leiden, Utrecht, and Swansea and now isprofessor of structural geology at Cardiff Uni-versity, Wales, United Kingdom. His researchlies in the frontier area between geometryand geological structures.

Juan Luis Fernandez Martınez � Departa-mento de Matematicas, Universidad de Oviedo,Oviedo 3307, Spain; jlfm@orion.ciencias.uniovi.es

After graduating in mining engineering at theUniversity of Oviedo, Juan Luis trained as apetroleum engineer (Ecole Nationale duPetrole et des Moteurs, Paris, 1988; ImperialCollege, Royal School of Mines, London,1989). After some years working as a com-puting engineer in petroleum software inFrance, he gained a Ph.D. in mining engi-neering in Oviedo in 1994 and joined theMathematics Department. He currentlyworks on geomathematical methods: geo-statistics, inverse problems in geophysics,fold modeling, digital imaging processing,and finite-element methods.

ACKNOWLEDGEMENTS

Richard Lisle’s visiting professorship at theUniversidad de Oviedo, Spain, was funded bythe Secretarıa de Estado de Educacion yCiencia (Spain). This funding allowed collab-oration with the Folding Research Groupheaded by Fernando Bastida, project BTE02-00187. Useful reviews were provided byRichard Bischke, Richard H. Groshong, and ananonymous referee.

Structural analysis of seismicallymapped horizons using thedevelopable surface modelRichard J. Lisle and Juan Luis Fernandez Martınez

ABSTRACT

A new method for the analysis of folding of seismically mapped

horizons is described. Based on a model of developable surfaces, the

local geometrical properties are determined by analyzing the var-

iation of dip and strike along linear strips on the surface. By con-

sidering strips in different directions, a plunge line (the approxi-

mation to the generatrix of a developable fold) is identified as the

direction associated with the least variation of surface attitude. The

map pattern obtained by analyzing the plunge and trend of plunge

lines across an area allows the identification of domains where

folding accords with a developable geometry. Such domains are rec-

ognized from straight plunge lines, defining convergent or parallel

patterns. Deviations from these patterns correspond to regions of

structural complexity associated with ductile or brittle straining of

the horizon. We suggest that plunge-line analysis may offer a useful

technique for automatic fault recognition.

INTRODUCTION

The availability of three-dimensional (3-D) seismic data makes

possible the analysis of the 3-D form of folded geological surfaces.

A large number of techniques exist for extracting different types

of geometrical attributes of such surfaces (Roberts, 2001). For ex-

ample, structure contours maps and time structure maps provide

overall representations of the geometry, whereas other techniques

derive and display particular surface-derived attributes, e.g., maps

displaying variation of dip angle, dip azimuth, second derivative,

and curvature.

These attributes are calculated with the ultimate aim of pre-

dicting particular physical properties of the rocks in the subsurface.

GEOLOGIC NOTE

AAPG Bulletin, v. 89, no. 7 (July 2005), pp. 839–848 839

Copyright #2005. The American Association of Petroleum Geologists. All rights reserved.

Manuscript received June 26, 2004; provisional acceptance September 27, 2004; revised manuscriptreceived January 11, 2005; final acceptance January 30, 2005.

DOI:10.1306/01300504072

Geometrical attributes of structures have been used for

fracture prediction in the reservoir (Murray, 1968; Lisle,

1994; Stewart and Podolski, 1998; Fischer and Wilk-

erson 2000), understanding the relationship of folding

and faulting (Savage and Cooke, 2004), or automatic

fault detection (Lisle, 1994; Roberts, 2001; Bergbauer

et al., 2003). However, many of these attributes are of

little value for classifying the form of the surfaces con-

cerned and are therefore unable to characterize the fold-

ing style from a geometrical perspective. Furthermore,

these attributes do not allow the determination of ba-

sic geometrical features, such as the fold plunge, which

may relate to the direction of the strains associated with

folding.

Devising more meaningful ways of characteriz-

ing the structure of a folded horizon requires an un-

derlying geometrical model for the folding. For folds

observed at outcrop, structural geologists have tradi-

tionally made great use of the cylindrical fold model

(Wegmann, 1929; Clark and McIntyre, 1951; Turner

and Weiss, 1963, p. 107), a model in which all lines

normal to the folded surface are coplanar, and all parts

of the surface contain a fixed line, referred to as the

fold axis. The notion of cylindrical folding provides the

conceptual basis for most methods of fold analysis and

underpins current fold terminology (Fleuty, 1964). For

cylindrical surfaces, the fold profile remains constant

in serial sections, so that folded surfaces that accord

with this model are characterized by successive struc-

ture contours of identical shape (Lisle, 2004). Three-

dimensional seismic mapping of reservoir structures,

however, reveals that folding of this type with a con-

stant direction of the fold axis is rather sparse. For this

reason, a conical model is sometimes considered an im-

provement on the cylindrical model (Bengston, 1980;

Groshong, 1999). In this article, we suggest that amore

versatile geometrical model for folding in hydrocar-

bon provinces is provided by the developable surface.

Developable surfaces are surfaces on which a series of

straight lines can be drawn, and where pairs of neigh-

boring straight lines, if extended far enough, will mu-

tually intersect. This model has two main advantages

over the cylindrical and conical models. First, it is more

general, i.e., it encompasses a broader range of fold

shapes and also subsumes the cylindrical and conical

models (Rech, 1977). Second, the developable surface

has special properties that relate to the deformation

of surfaces. Folds with developable geometry can, in

theory, be formed without any straining of the sur-

face that makes them the favored fold style for me-

chanically strong layers, especially in thinly bedded se-

quences that are strongly anisotropic in terms of shear

strength (Lisle, 1992). This property is important with

respect to structural restoration.

This article describes a new and conceptually sim-

ple technique for analyzing the variations of fold plunge

andof directions of plunge across any seismicallymapped

surface based on the geometrical model of the develop-

able surface. These variations of plunge can be used to

highlight patches on the folded horizon of particular

complexity and can assist in locating faults, zones of

abnormal fracturing, or localized ductile strain of the

folded beds.

The Developable Surface Model

In this article, we propose a new method for analyzing

the geometry of geological horizons by comparing them

to a class of surfaces that mathematicians refer to as

developable surfaces. In the context of geological struc-

tures, this name may appear strange because any real

folds must have developed (formed) in some manner.

In the language of mathematics, the term is derived

from a usage of the word ‘‘develop,’’ which means

‘‘unroll’’ (Hornby et al., 1963), because such surfaces

can be unrolled on to a flat plane.

Developable surfaces are of great use in other dis-

ciplines, e.g., manufacturing, shipbuilding (Clements,

1981), biomedical imaging (Hersch et al., 2000), etc.

Several algorithms exist for modeling and fitting de-

velopable surfaces (Chen et al., 1999; Pottmann and

Wallner, 1999).

Developable surfaces are described in texts on dif-

ferential geometry. The standard form of the equation

for a developable surface (Wardle, 1965, p. 65) can be

written as follows:

Xðt;uÞ ¼ rðtÞ þ u

:rðtÞ; u � r

whereP

(t,u) are the position vectors of points lying on

the surface, r(t) defines the position vectors of points

that lie on a nonplane curve called the edge of regres-

sion, and.rðtÞ are the tangent vectors of the edge of re-

gression curve at the corresponding t points. We there-

fore arrive at points lying on the surface by starting at

some point on the edge of regression determined by the

parameter t andmoving away by a distance equal to the

value of u k :rðtÞ k in the direction of the curve’s local

tangent. This implies that the surface is a ruled surface,

with the rulings or generators corresponding to the

straight tangent lines of the regression curve (Figure 1).

840 Geologic Note

These straight-line generators are curves of constant

value of the parameter t.Two additional properties of developable surfaces

are key to the method of analysis proposed here.

1. Immediately adjacent pairs of generators mutually

intersect at a point. This point lies on the regression

curve.

2. It follows fromproperty 1 that immediately adjacent

pairs of generators define a plane. For this reason, the

tangent planes of a developable surface are constant

in orientation at all points along a generator.

This second property is of crucial importance in

the present method of fold analysis because it allows

the identification of the local generators of the folded

surface. The first property allows us to test develop-

ability using the estimated local generators.

At all points on developable surfaces, one of the

principal curvatures is zero, which, in turn, means that

the Gaussian curvature, i.e., the product of the two

principal curvatures, is equal to zero everywhere.

Generators and Plunge Lines

In the special case of perfect cylindrical folds, the gen-

erators are constant in orientation across the surface

and are therefore mutually parallel; their orientation is

commonly referred to by structural geologists as the

‘‘fold axis.’’ It would be contrary to current practice to

use the term fold axis as a synonym for generator in

other developable folds because the latter has a defi-

nite location on the surface, whereas the former is a

direction and geometrical entity that applies to the

complete fold (Twiss and Moores, 1992; Davis and

Reynolds, 1996). Furthermore, for conical folds, the

term fold axis is used by some authors to refer to the

geometrical axis of the cone instead of a generator on

the cone’s surface. In their discussion of conical and

cylindrical folds, Wilson (1967) and DePaor (1988) re-

fer to the generators that vary across the surface as

plunge lines, and we suggest here that this usage could

be usefully extended to developable surfaces in general

without risk of confusion. Accordingly, the plunge of a

general developable fold could be constant at points

along a given plunge line but variable from one plunge

line to the next.

For describing the shape of cylindrical folds, fre-

quent use is made of the true profile plane of the fold,

i.e., a planar section through the structure perpendic-

ular to the fold axis (e.g., Wilson, 1967). Features such

as crest lines, trough lines, hinge lines, and inflection

lines are all defined with respect to points identified

in successive serial profiles (Fleuty, 1964). However,

the use of such terms with respect to general devel-

opable folds, which possess no natural section plane,

Figure 1. A developable surface, show-ing straight line generators, here referredto as plunge lines. Along any plungeline, the strike and dip of the surface re-mains constant. Adjacent plunge lines onthe surface mutually intersect, point p isan example of such an intersection point.The set of curves orthogonal to the plungelines consists of lines of maximum abso-lute curvature; these are referred to asprofile curves. Profile curves, which aregenerally not plane curves, can be usedwhen identifying hinge, inflection, andcrest lines of the fold.

Lisle and Fernandez Martınez 841

necessitates a broader definition of the term ‘‘fold pro-

file.’’ Therefore, we propose that the fold profile be

defined as a curve on a developable surface that is ev-

erywhere perpendicular to the plunge lines or gen-

erators. In the special case of cylindrical folds, suc-

cessive profiles are plane curves of identical shape and

size, but in other cases, these are nonplane curves and

vary in form across the surface. Defined in this way,

the successive profiles, together with the plunge lines,

form lines of curvature for the surface, i.e., two sets of

curves that are mutually orthogonal and track the

principal lines of curvature (Wardle, 1965, p. 75). The

developable surface has zero normal curvature in the

direction of its plunge lines and has maximum abso-

lute normal curvature in the direction of the profile

lines (Figure 1). Along any particular profile line, hinge

points and inflection points are located where the prin-

cipal curvature is greatest in absolute terms and zero,

respectively (Figure 2a). Likewise, crest and trough

points are the topographically highest and lowest points,

respectively, on the profile (Figure 2b). These types of

points, when joined to similar points across the

surface, define straight lines. These are plunge lines.

In map view, the relative trends of structure con-

tours and plunge lines allow the identification of crest

lines and trough lines and the distinction between

opposite limbs of a fold because the angle between

plunge line and strike is a function of the magnitude

of the plunge relative to the angle of dip of the sur-

face (Figure 2b).

METHOD

Folds with developable geometry have been considered

previously in relation to layers that resist stretching

during folding (Lisle, 1992), and several qualitative

criteria have been suggested for the recognition of

folds with this type of 3-D form. A possible strategy

for the identification of developable folds would be to

determine the Gaussian curvature at points across the

surface to check whether it equals zero. In practice,

problems arise in the mapping of curvature because

the presence of noise in the data produces small-scale

complicated variations of curvature that can obscure

those of geological significance. These problems are

discussed in detail by Stewart and Podolski (1998).

In the present method, a quantitative approach is

used, which is based on the property that the surface

should contain straight generators along which the ori-

entation of the surface remains constant. The advantage

of this approach is that the existence of this property

can be tested by examining the data from strips of finite

size on the surface, and this helps reduce the problemof

noise.

The following algorithm is used to compute gener-

ator lines for a seismically mapped horizon (Figure 3):

1. Primary sampling points, P, on the map are selected;

these do not need to coincide with the data points.

The sampling pattern can be irregular or a regular grid.

2. At each primary point, a variety of radial sample

lines are considered with different directions in the

horizontal (xy) plane (labeled scan lines in Figure 3).

Figure 2. (a) Developable folds are bounded by inflectionlines and possess hinge lines, lines joining points of maximumcurvature on the profile curves. (b) Crest and trough lines arespecial plunge lines joining crest and trough points, respectively.Crest and trough points can be identified structure contourmaps because they are located where plunge lines intersectstructure contour lines at right angles.

842 Geologic Note

3. Along each sample line, the coordinates (x,y) of even-ly spaced secondary sampling points are calculated.

4. The original data points (x,y,z) on the mapped sur-

face in close proximity to each secondary point are

identified, and these neighbors are used to compute

the local orientation of the horizon associated with

each secondary point, in terms of the direction co-

sines of the normal to the surface.

5. Using a resultant vector method, the variation in

orientation of surface normals for each secondary

point along each sample line is determined, as well

as the average surface normal. The variation of ori-

entations of the surface normals is caused by both

curvature and torsion of the surface in the direction

of the sample line, as well as to measurement errors

in the data.

6. Of all the sample lines radiating from a primary sam-

ple point, the one producing the least variation of

the local surface normals is identified, and its direc-

tion in the xy plane is taken as the direction of the

best-fit generator at this primary point (local plunge

line).

7. The plunge (inclination) of the generator is calcu-

lated as the apparent dip of the average surface de-

termined in step 5, in the direction found in step 6.

8. Steps 1–7 are repeated at the other primary sample

points that describe the geometry of the horizon.

This procedure requires two decisions to be made

by the structural geologist. The first is to choose the

length of the sample line. This determines the size of

the circular patch of data that is used in the calculation

of the generator at each primary sample point on the

surface and should relate to the scale of local develop-

ability. For perfectly developable surfaces, this would

be unimportant because, in such cases, the generators

extend as straight lines across the whole surface. How-

ever, for surfaces that only approximate to a develop-

able form, the length of the sample line determines the

minimum scale of features identifiable in the analysis

and the degree of smoothing of the structural surface.

In the absence of a priori information about this sam-

pling scale, the magnitude of the resultant vector ob-

tained at step 5 above can be used to assess, by trial and

error, the constancy of dip associated with the best-fit

plunge line and can assist in deciding an optimal length

for the sample line. If the chosen sample line is too

short, the results may be adversely affected by noise. In

the example below, the sample line length was about

1800 m (5906 ft), and the resultant vector magnitude

typically exceeded 0.998. The second choice concerns

the number of secondary points defined along the sam-

ple line; this was 16 in our analysis.

Once generators have been calculated across a sur-

face, the pattern of variation in their orientations al-

lows the identification of surface patches that accord

or otherwise with the developable model. A develop-

able portion of a surface fold must show plunge lines

aligned in rectilinear trends with or without conver-

gence. However, these requirements apply in three di-

mensions, and judgements based on two-dimensional

(2-D) maps of plunge-line symbols can be difficult. An

Figure 3. The calculation of plunge lines.Through every primary point P on thesurface, scan lines in different directionsare used to sample the x,y,z data froman elongate strip on the surface. Alongeach scan line, the poles to best-fitplanes show a range of orientation.The scan line that produces the lowestvariety of orientations of the best-fitplanes is defined as the best-fit plungeline (best generator) at point P.

Lisle and Fernandez Martınez 843

alternative, more objective method of detecting devia-

tions from a developable surface has therefore been

developed. A pair of immediately adjacent generators

on a developable surface either are parallel or mutually

intersect whereas a pair of generators on a nondevel-

opable surfacewill form a pair of skew lines. The closest

3-D separation distance, d, between pairs of genera-

tors, computed using the method of Green (1963,

p. 32), is used here as an index of deviation from a de-

velopable geometry. The plunge-line skewness of a gen-

erator drawn at point P (xP,yP,zP) on the surface with

respect to a neighboring generator at pointN (xN, yN,zN) is defined as d/D, where D is the component, mea-

sured in a direction perpendicular to the plunge line

of the generator, of the straight line distance between

P and N. Skewness values range from 0 to 1.

This method does not require a regular grid of sur-

face heights. This is advantageous because the inter-

polation required to derive a regular grid from irreg-

ularly spaced data may introduce undesirable geomet-

rical artifacts.

The algorithm has been tested on synthetic exam-

ples of developable surfaces and gives consistent and

accurate results. The program has been developed in

MATLAB1 because of its capabilities for handling ma-

trix calculations and postprocessing. Details of this pro-

gram will be described in a future article.

RESULTS

The plunge-line method is applied to a 3-D data set

for a single folded horizon. For reasons of confiden-

tiality, details of the data set and geology are withheld

here, and only parts of the overall structure are de-

picted as examples of the method. A structure con-

tour map (Figure 4) shows the gross structure, which

consists of a north-south–trending periclinal anticline

Figure 4. Structure contour map (depths)of the area investigated, with locations ofdetailed maps in Figures 5–8. A periclinalantiform is the dominant structure.

844 Geologic Note

with an eastern limb that dips steeply (20j). However,

the application of the plunge-line method reveals that

the fold plunge is highly variable and also allows the

identification of the following structural features that

are not obvious from the structure contour pattern:

1. In parts of the area, plunge lines that intersect struc-

ture contours at high angles signify that the local

fold plunge has an angle approaching that of the

dip of the horizon. The crest lines and trough lines of

the fold can be determined by joining points where

the plunge lines trend in a direction at right an-

gles to the structure contours (Figure 5). The sense

of obliquity between the structure contours and the

plunge lines serves to distinguish the two limbs of

the fold. In other parts of the region, the trends of

the calculated plunge lines are nearly parallel to the

structure contours. These correspond to parts of the

structure that are nonplunging or at least where

the angle of plunge of the plunge lines is slight in

relation to the dip of the beds.

2. Plunge lines are lines of constant dip and are spaced

according to the structural curvature. A fan pattern

of plunge lines indicates a change of structural curva-

ture. Curvature increases in the direction of plunge-

line convergence. In Figure 5, the general pattern is

of plunge lines diverging southward; the anticline is

therefore decreasing in curvature in that direction.

3. The pattern of parallel plunge lines, similar with

respect to both trend and plunge, characterizes a

cylindrically folded region. This is illustrated in the

central area and the southern part of the area in

Figure 6, where the plunge lines are consistent in

trend, and their angles of plunge do not vary signif-

icantly (Figure 6b).

4. Structural domains with nondevelopable geometry

are recognized by two features. First, the trajectories

defined by tracking along plunge lines define curved

or kinked lines. This is observed in the northwest

corner and the eastern edge of the area in Figure 6,

where map trends are defected by about 40j. A sec-

ond feature is a significant change in the angle of

plunge of the plunge lines along their trajectories.

This is highlighted in Figure 6b, where contours of

the constant plunge angle run across the trends of

the plunge lines and are closely spaced. The presence

of one or both of these features warrants the conclu-

sion that the evolution of the structure involved strain-

ing of the surface by brittle or ductile mechanisms.

5. Abrupt changes of direction or plunge angle of

the calculated plunge lines are attributed to faults.

Figure 7 shows an example of a plunge-line pattern

that is interpreted as a structure dissected by several

faults. The plunge lines in this example are calcu-

lated using scan lines of length 15 times the grid spac-

ing. Nevertheless, distinct discontinuities are visible

on the plunge-line map. These abrupt changes in the

trend of plunge correlate well with changes in the an-

gle of plunge shown by contours in Figure 7. By con-

sidering both components together, it seems likely

Figure 5. Structure contour and plunge-line map of the southern part of the anticline in Figure 4. The arrows are the calculatedplunge lines. The fold crest line is marked by the perpendicular relationship between plunge line and structure contour line trends.The divergence of the plunge lines toward the south indicates that the structure is decreasing in curvature in that direction. Thecurved pattern of the plunge lines in the west does not accord with the developable fold model.

Lisle and Fernandez Martınez 845

that a northeast-southwest–trending fault cuts the

central part, whereas an east-west fault exists in the

southeast corner of the area. It is significant that these

features are not evident from the structure contour

map of the surface, which implies that these faults

have little vertical separation. In other words, it ap-

pears that discontinuities of structural trend need

not always coincide with pronounced vertical offset.

6. More subtle deviations from developable fold form

are recognized on maps showing values of plunge-

line skewness. The calculated closest approach dis-

tance between neighboring generators provides a

more objective check on developability. Figure 8 il-

lustrates how skewness identifies linear zones of

nondevelopable folding in the eastern part of the

area. These zones, which are poorly marked by the

pattern of plunge-line directions alone, are proba-

bly an expression of faulting.

7. Any periclinal structure has nonzero Gaussian cur-

vatures and is therefore geometrically incompatible

Figure 6. (a) Structure contours andplunge lines (arrows). A parallel arrange-ment of plunge lines indicates a devel-opable fold of cylindrical type. In thenorthwest part of the area, a rapid changeof direction of plunge lines marks thelimit of the area of east-west cylindricalfolding. See Figure 4 for location. (b) Samearea as in (a). Plunge lines (arrows) andcontour lines of angle of plunge of plungelines. A rapid change in plunge angle ac-companies the change in trend of plungelines and delineates structural disconti-nuity in this part of the area. The in-terpreted faults seem to decouple adja-cent folding domains, allowing thedevelopment of different structural trends.

846 Geologic Note

with the developable fold model. In the structure

analyzed in the article, it appears that the overall

periclinal geometry is achieved by a complex patch-

work of domains with near-developable geometries

and mutually bounded by faults. Figure 7 illustrates

this phenomenon.

CONCLUSIONS

The newmethod is able to detect subtle fold structures

from 3-D seismic data. The plunge-line approach al-

lows the character of those folds to be assessed in de-

tail in terms of their directions of minimum surface

Figure 7. Different domains ofdevelopable folding are bound-ed by discontinuities (probablefaults). These discontinuitiesare recognized from a suddenchange in trend (arrows) andsteep plunge gradient (con-tours). One such discontinuityruns northeast-southwestacross the center of the map;another important structuralbreak has an east-west trend inthe southeastern part of the map.

Figure 8. Regions of nondevelopable surface geometry detected by plunge-line skewness. See Figure 4 for location. The shadedareas are those with values of skewness greater than 0.1 and probably represent zones close to faults.

Lisle and Fernandez Martınez 847

curvature. The nature of the variation of the plunge

lines across surface indicates the extent to which the

folding fits a developable fold model. Patches of the

structure that are found to deviate from a developable

geometry are likely to be zones of enhanced strain-

ing of the beds, either by ductile strain or by brittle

deformation.

In the analysis of a periclinal structure described in

this article, the method reveals that domains with ge-

ometry approaching developable folds are of limited

extent. These domains are sharply bounded, and these

boundaries are related to numerous faults. These faults

are commonly not apparent from the structure con-

tour map because these structures are commonly not

associated with major vertical offsets. Such faults are

defined by spatial variations in the 3-D structural geom-

etry and may not be easily recognizable on 2-D struc-

tural sections. We therefore suggest that plunge-line

analysis may offer a useful strategy for automatic fault

detection.

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848 Geologic Note