Chapter 8 Equal-Area Projections and Structural Analysis 155

Problem 8-4The equal-area plot of Figure 8-12a shows the same

data as that used in Problem 8-1. Determine theappropriate diameter for the counting circle, for the Kambmethod of contouring, and contour the data using thismethod.

Step 4: Using Equation 8-6, calculate a value for 0'(1.57). Contour the concentrations at 20', 40',60', and 80'(Fig. 8-12c). Notice that the contoured diagram issignificantly different from those obtained by the Schmidtand Kalsbeek methods.

Method 8-4Step 1: Determine the appropriate diameter of the

counting circle. Using Equation 8-8 for N = 72, r is foundto be 0.188 of the radius of the equal-area projection. For aIS-em-diameter net, the counting area is a circle of diameter2.82 cm.

Step 2: Construct a counting grid with a spacingequal to the radius of the counting circle, i.e., 1.41 em.

Step 3: Follow the same method as in Schmidtcontouring to obtain concentrations at the grid points (Fig.8-12b).

Additional methods of contouring equal-area plots andof statistical analysis of equal-area plots are described byVistelius (1966).

8-4 PATTERNS OF POINT DATAON EQUAL-AREA PROJECTIONS

The distribution of points on an equal-area projectiongraphically expresses the degree of preferred orientation (orlack thereof) of a particular structural element (such asfoliation or lineation). The key to interpreting the

'-

2-

l. 1 .. 1'- ..I! 19 U 4

.....,'. Il 20 19 8

b 7 3a72 poles to follati on

Figure 8-12. The Kamb method of con­touring described in Problem 8-4, for the samedata as Problem 8-1. Contours drawn at 20,40, 60, and 80, Contours at 2 o •• 0 I 60 and 80

156 Elementary Techniques Part I

projection lies in recognizing the pattern defined by thedistribution of points (where "points" refers to either theprojection of a line representing a linear structure or theprojection of a pole representing a planar structure).Recognition of patterns is often easier to do with acontoured diagram. There are four main patterns that canbe recognized:

Uniform Distribution: A statistically randomdistribution in the orientation of structural elements isexpressed by a scatter of points on an equal-area plot inwhich there are no obvious local concentrations. The lackof any concentration on a plot is called a uniformdistribution (Fig. 8-13a).

Point Maximum: A preferred orientation ofstructural elements is represented by a high concentration(significant cluster) of points symmetrically distributedaround a single mean orientation (Fig. 8-13b). The centerof the cluster is the point maximum. A single data setcan show more than one point maximum.

Great-Circle Girdle: A concentration of pointsalong an arc approximating a great circle is called agreat-circle girdle (Fig. 8-13c). The pole of the great-circlegirdle is called the girdle axis. A girdle may contain one or

more distinct maxima. In some cases, two girdles mayintersect, forming a crossed girdle pattern. A girdle patternfor linear elements indicates that the lineations all lie in asingle plane but are not parallel to one another. In such acase the girdle approximates the attitude of the planecontaining the lineations, and the girdle axis is the pole tothat plane. A girdle pattern for poles to planar elementsindicates that the planes could all intersect along the sameline. For example, a girdle pattern is obtained by plottingpoles to bedding taken around a cylindrical fold (see below).

Small-Circle Girdle: A small-circle girdle is aconcentration of points along an arc that approximates asmall circle (Fig. 8-13d). Such a girdle may contain one ormore distinct maxima. For both linear and planar elementssuch a girdle indicates a preferred orientation in a coneabout a single axis (the girdle axis).

We can describe the point-distribution pattern on anequal-area plot in terms of the type of symmetry displayed(e.g., the number of mirror planes that can be drawn on theplot, across which the point clusters are mirror images ofone another) by analogy with the description of pointgroups in crystallography. For example, a fold may be

+ +

a b

Figure 8-13. Patterns of pointdata on an equal-area projection.(a) Uniform distribution; (b) pointmaximum; (c) great-circle girdle; (d)small-circle girdle.d

. '". ',...-7'--. ".:,.:::".' '~~' .

. /::. :'\./. . . . ·.V.,'/ . . . . . \ "

.. ·f· • . . \.,

I" ~ .·:T... ~,' + Cone exi. ::./:.'\ . /:.\ :);.,

\ . I .

. (:. .' ;/............... _/

Small circle - --

c

Chapter 8 Equal-Area Projections and Structural Analysis 157

described as orthorhombic or monoclinic, depending on thepattern of clusters displayed on a plot of poles to bedding.For further discussion of this terminology see Turner andWeiss (1963).

8-5 ANALYSIS OF FOLDINGWITH AN EQUAL-AREA NET

Geometrically, a fold is merely a curved surface. There aretwo basic types of folds: (1) Cylindrical folds aregenerated by moving an imaginary straight line parallel toitself in space. The line that generates the fold is called thefold axis. (2) Noncylindrical folds are generated by a linethat moves in a nonparallel manner through space. If oneend of the generating line is fixed, the resulting fold formis called a conical fold. If the movement of the generatingline is nonsystematic, a complex fold results. Sometimescomplex folds can be subdivided into parts that areapproximately cylindrical. The geometry of cylindrical orconical surfaces can be analyzed with either B-diagrams or1t-diagrarns on an equal-area projection.

B-Dlagrams of Cylindrical Folds

Every segment of a cylindrically folded surface contains aline segment that is parallel to the fold axis. Any twotangential planes to the folded surface will intersect along aline that is parallel to the fold axis. On an equal-areaprojection, therefore, the great circles representing theattitudes of the folded surface at different points on the foldshould all intersect at a common point representing the foldaxis. This point is called the j3-axis. In practice, however,real folds do not have a perfectly cylindrical form, so strikeand dip measurements around the fold produce great circlesthat do not all intersect at a common point, although thepoints of intersection do show a point maximum that givesan average orientation for the B-axis. For n plotted planarattitudes (Fig. 8-14), the number of intersections (x) isgiven by the arithmetic progression

x =0 + 1 + 2 + ... + (n - 1) =n(n - 1)/2 (Eq. 8-9).

Thus, if there are 200 plotted planes, the number ofintersections is 19,900. Contouring of the intersection

a

+

Figure 8-14. ~-diagram of cylin­drical fold. The number of inter­sections of great circles increasesrapidly as the number of plottedplanes increases. (Adapted fromRamsay, 1967.)

2 planes. 1 p-point

4 planes. 1+2 ... 3 p-points

3 planes. 1+2 Il-points

n planes, n(n-1)/2 a-points

158

points will emphasize the maximum concentration ofintersections.

A plot of B-axes is not generally the best way torepresent attitude measurements on a fold, for severalreasons. First, the number of points on a B-axis plot is fargreater than the actual number of measurements; thus, sucha plot may make you think you have more data than youactually have. Second, if there isany scatter in the originaldata, there can be concentrations of B-axes away from themain concentration, leading to an erroneous interpretation.Such errors become unacceptably large if the interlimbangle is very small «400), as in tight folds, or very large(>140°), as in open folds. Finally, construction of aB-diagram is time consuming, because a large number ofgreat circles must be plotted, and the number ofintersections can become unmanageably large for even asmall data set.

x-Diagrams of Cylindrical Folds

Because of the disadvantages of the B-axis diagram, atr-diagram is the preferred method for representingmeasurements from a folded surface. A x-diagram is anequal-area plot of the poles to planes that are tangential tothe folded surface. Practically, this means that if we havestrike and dip measurements from many locations on afold, we plot the pole for each plane rather than thegreat-circle trace. On a cylindrical fold, each of the poles isperpendicular to the fold axis; thus, the poles are parallel toa plane perpendicular to the fold axis. On an equal-areaplot the poles approximate a great-circle girdle, which isvariously called the S-pole circle, the pole circle, or thetr-circle (Fig. 8-15). The pole to the x-circle is the x-axis(Fig. 8-15), and it represents the fold axis. The x-axisshould coincide with the B-axis on a plot. For a very openfold, with a very large interlimb angle, the x-diagram will

\ .'\. +~......... .......

.""'"- - . . /. - -_!...--.­S-pol e gi rdl e ~

or TT-girdle

Figure 8-15. n-diagram of cylindrical fold.Poles to planes lie on a great circle girdlewhose pole gives the fold axis (n z B).

Elementary Techniques Part I

show an elliptical point maximum. With progressivedecrease in the interlimb angle, the pole pattern changesfrom a point maximum, to an incomplete great-circlegirdle, and finally, to a complete great-cricle girdle (Fig.8-16; e.g., Ragan, 1985).

A x-diagram not only gives information on theorientation of a fold axis but also contains clues to theform of the fold. For example, if a fold has a broad,rounded hinge, the density of points will be uniform withinthe x-circle girdle, and the two extreme points on the girdlewill define the interlimb angle (Fig. 8-17a). The x-circlegirdle for a fold with planar limbs and a narrow hinge zonewill contain maxima on the girdle corresponding to the twolimbs, and these maxima can be used to determine theinterlimb angle (Fig. 8-17b). For a chevron fold there isno well-defined girdle, and the x-circle on the projection isdefined by two point maxima corresponding to the twolimbs (Fig. 8-17c). Most natural folds show patterns thatare intermediate between the broad-hinge girdle and thetwo-maxima (limbs) girdle.

It is generally not possible to say anything conclusiveabout the symmetry of folds from just a x-diagram,because factors other than dips in the two limbs of thefolds determine fold symmetry (Hobbs, Means, andWilliams, 1976; Ramsay, 1967). A concentration ofpoints along a girdle may also be a consequence ofsampling bias. However, if the spatial distribution ofmeasurements in a train of folds is uniform, there will befewer readings from the short limbs of asymmetric folds,resulting in an asymmetry in the pole pattern on thex-diagram (Fig. 8-l7d). Generally, in order to determineasymmetry of folds, we need additional information such asvariation in thickness from limb to limb, orientation of theenveloping surface, and orientation of the axial surface (seeChapter 11).

The orientation of the axial surface (or the axial plane)can be determined if the x-axis is known and if theorientation of the axial trace at a locality can be determined;the great circle passing through these two points gives theorientation of the axial plane (Fig. 8-l8a). In the case ofchevron folds and kink folds, the axial plane may be definedas a plane containing the bisector of the interlimb angle.The bisector is represented by the point whose angulardistance from the two point maxima (measured along thex-circle girdle) is the same. The great circle passingthrough the bisector and x-axis represents the axial plane(Fig.8-18b).

The attitudes of the fold axis and the axial plane are, ofcourse, reflected in the position of the x-circle girdle on anequal-area projection. For example, if the fold axis ishorizontal, then the x-axis lies on the primitive and thegirdle passes through the center of the net, but if the foldaxis is plunging, then the x-axis lies inside the primitive,and the girdle follows a curve that does not pass through

Chapter 8 Equal-Area Projections and Structural Analysis 159

~IUnfolded inclined bed Gently plunging broad arch Plunging fold

a b c

Figure 8-16. Change in pole pattern on the n-diagram with decreasein fold interlimb angle. (Adapted from Ragan, 1985.)

the center of the net. If the axial plane of the fold isvertical, it is represented by a diameter of the equal-angleplot; if it is horizontal, it is represented by the primitive;and if it is inclined, it is represented by some intermediategreat circle. A fold with a plunging axis and an inclinedaxial plane may display a complex pattern on an equal-areanet. Figure 8-19 shows several examples of 1t-diagramsand the folds that they represent.

1t-Dlagrams of Noncylindrlcal Folds

If the folded surface is conical, with the cone having anapical angle JJ., each pole makes an angle of (900 - JJ.!2)with respect to the cone axis. In other words, the poles tobedding generate a coaxial cone with an apical angle of(180 - JJ.). Thus, the poles define a small circle, with itscenter representing the cone axis (Fig. 8-20). If anapproximate small-circle pattern is recognized, it may beworthwhile to replot the poles on a Wulff net, since asmall circle projects as a circle on the stereographicprojection. A small circle can then be fitted to the plottedpoints, and the center of the circle (representing the coneaxis) can be located. The cone axis can be rotated to theprimitive, and the small circles of the net can be used toanalyze the angular relationships within the fold.

In nonconical noncylindrical folds, both the axialsurface and the fold axis vary in attitude, and constructionof a 1t-diagram will generally give several possibleorientations for the 1t-axis. Commonly, areas ofsuperposed folding exhibit this kind of geometry. Toanalyze such folds, they must be subdivided, by trial anderror if necessary, into domains of plane cylindrical folding(see Problem 8-7). Each domain should have its ownconstant 1t-axis orientation. In plane noncylindrical foldsthe axial surface is planar and has a constant orientation,althought the orientation of the fold axis (1t-axis) mayvary. The mean orientation of the axial plane is defined asthe great circle passing through the axes of the differentcylindrical domains (Fig. 8-21).

8·6 ANALYSIS OF FABRICSWITH AN EQUAL-AREA NET

Types of Fabrics

The internal geometric and spatial configurations of thecomponents of a rock constitute its fabric. If a fabric isvisible in a rock regardless of the scale of observation, it issaid to be penetrative. Rocks that have penetrative fabric

160 Elementary Techniques Part I

Concentric fold Fold with narrow hinge

~~ ~.-- . --.../ ,

a b

Chevron fol d Asymmetric folds

Figure 8-17. Variation in1t-diagrams with change in foldform. (a) Fold with broad, roundedhinge; (b) fold with narrow hinge; (c)chevron fold; (d) asymmetric folds.d

~---.~ ~ ~

c

.--.-~-----~--.--..

resulting from deformation are referred to as tectonites.Three major classes of teetonites are recognized, based onwhether the fabric can be described as a foliation, alineation, or both. (1) S-tectonites have a strong foliationbut no lineation (Fig. 8-213). The foliation is defined byparallel alignment of platy minerals, lenticular mineral

aggregates, or flattened grains. The letter S is used becauseof the longstanding convention of referring to foliations asS-surfaces (Turner & Weiss, 1963). (2) L-tectonites have awell-developed lineation but no foliation (Fig. 8-22b). Thelineation in an L-tectonite is defined by alignment ofprismatic minerals or uniaxially elongated grains parallel to

Chapter 8 Equal-Area Projections and Structural Analysis 161

I Axial surface

1..11

Iiln-axis11

b

"-"-

"-"-

'" '\\' .

\ AXlsl surface

\\

~TT-aXiS

\\

Axial trace......

Figure 8-18. Determining att­itude of fold-axial surface from ax-diagram. a

c

\\

\

.\

\\

\\

\

\\

b

\\

\\

\\

~~•.....

....~::.: .....

\\

\\

\\

\

~~

aHorizontal normal fold Plunging normal fold Plunging inclined fold

f

......... -----/'

/"

//

/I

IfII

e

- - ----

d

Plunging overturned fold Reclined fold Recumbent fold

Figure 8-19. Variations in position of x-circle girdle with changes inattitude of the fold axis and axial plane. (a) Nonplunging upright(normal) fold; (b) plunging normal fold; (c) plunging inclined fold; (d)plunging overturned fold (note the presence of vertical beds indicatedby the plotting of some bedding poles on the primitive); (e) reclined fold;(f) recumbent fold; axial plane coincides with the primitive. (See Fig.11-17.)

162 Elementary Techniques Part I

"""-_'--:::::

:: .. ---. ...........

.. .".. .....

.::' .I

I

I\\\

\

" "--b

Ii) Fold axis

·i;·,F;i:j;7~~ -".. \-l·:l:·I.,

III\~

aSE-limb of Eureka syncline Sheba fault surface

c

Figure 8-20. Flexural-slip fold­ing of sedimentary rocks and a fault(at low angles to bedding). Thesoutheast limb of the Eurekasyncline shows cylindrical folding.The Sheba fault surface is conicallyfolded. (Adapted from Ramsay,1967.)

one another. (3) L-S-tectonites have both a foliation and alineation (Fig. 8-22c). In an L-S-tectonite either thelineation or the foliation may be more pronounced. TheL-S fabric may be defined by the alignment of elongatedplaty minerals or of ellipsoidal grains. It may also be theresult of crenulation of a foliation or the intersection oftwo foliations.

Traces on outcrop faces in a tectonite may representeither the lineation or the foliation or both. Planesperpendicular to, or at high angles to, the foliation willshow traces of the foliation. The best-developed traces inan L-S tectonite will be on a plane parallel to the lineationand perpendicular to the foliation. Lineation traces alonewill appear best developed on all planes parallel to, or atacute angles to, the lineation and are absent or, at most,poorly developed on planes perpendicular to, or at highangles to, the lineation. In an L-S-tectonite an outcropface parallel to the foliation itself shows the true attitude ofthe lineation.

In order to apply the methods of structural analysis,the fabric in a rock body must be homogeneous, meaningthat equal volumes of rock from different localities in the

body are structurally identical. True fabric homogeneitynever really occurs, but it is common to find rocks that arestatistically homogeneous, meaning that the sample overwhich the homogeneity is to be assessed is much largerthan the scale over which inhomogeneity occurs. A rockin which the degree of development (intensity) or theorientation of a fabric differs as a function of location isinhomogeneous. An inhomogeneous rock can usually besubdivided into homogeneous parts. Each of these parts isa three-dimensional portion of a rock body that isstatistically homogeneous and is called ajabric domain orsimply a domain.

If the fabric within a single domain has the sameproperties in all directions, then it is called isotropic. Inmost deformed rocks, however, the structural elementswithin any domain show some degree of preferredorientation, and the fabric is, therefore, said to beanisotropic. Rocks with anisotropic fabrics may be S-, L-,or L-S-tectonites. Equal-area nets are useful in analyzingfabric in two ways. First, they may be used to calculatethe true orientation of fabrics, given partial measurementson different planes; second, they may be used to describe

Chapter 8 Equal-Area Projections and Structural Analysis 163

II

.......

.------=-=--=-=-= -=--=- - - - - - ----

+ + +

II '"

Figure 8-21. Plane noncylin­drical folding. The area is sub­divided into domains of cylindricalfolding, each with its own fold axis,but all the fold axes lie on a commonaxial plane (as shown on thesynoptic equal-area plot). (Adaptedfrom Turner and Weiss, 1963.)

IV

+

V

+

Synoptic diagram

aS-tectonite

bL-tectonite

Grain shape Ellipsoidal markersElongated or

platy minerals

Figure 8-22. The penetrative Cfabric of a rock defined by overall LS-leclonite

grain shape. elongated or platy J-'<~~

minerals, and ellipsoidal markers.Three different classes are defined.(a) S-tectonite; (b) L-tectonite; (c)

LS-tectonite.

164

variations in the geometry of fabrics that occur betweendifferent domains. Finally, in rocks with multiple fabricsan equal-area projection may be the only way ofdistinguishing various fabric elements.

Calculation of Planar and Linear Attitudes

The trace of a planar structure on any surface is its apparentdip in that plane. If two or more such apparent dips canbe measured, the orientation of the planar structure can bedetermined on an equal-area projection (Method 5-8). Withmore than two traces, the plane is defined by thebest-fitting great circle that passes through the data points.

Problem 8-5The trace of a foliation (S) is seen on three nonparallel

faces. The attitudes of the faces were measured, and therake of the foliation trace on each face was measured, withthe following results:

AttitudeN44°E,60oNWS66°E,800SWS11°E,700NE

Elementary Techniques Part I

Determine the orientation of the foliation (S).

Method 8-5Step 1: On an equal-area projection, plot the

great-circle trace of each face (Fig. 8-23a, b, c). Measurethe rake of the lineation on each face, and plot the pointrepresenting the trace of the foliation for each face.

Step 2: Find the best-fitting great circle that passesthrough the three points representing the foliation traces(Fig. 8-23d). The great circle represents the plane of thefoliation, which has an attitude of N300E,400NW.

Often, lineations (such as minor fold axes or minerallineations) can be directly measured on an exposed surface.At some localities, however, linear structures do not lie ona plane of easy breakage and thus cannot be measureddirectly. The orientation or poorly exposed lineations canbe determined by measuring the apparent lineation on twoor more differently oriented faces and plotting the data onan equal-area projection. Imagine a rod-shaped fabricelement (e.g., a dowel with a circular cross section). Anysection of the rod oblique to its axis will be elliptical (Fig.8-24). The long ~is of the ellipse is an apparent lineationon the plane of exposure; the true linear structure is

NN

Figure 8-23. Procedure for det­ermining orientation of a foliationdescribed in Problem 8-6.

Chapter 8 Equal-Area Projections and Structural Analysis 165

Problem 8-6Traces of a lineation were measured on three

nonparallel faces. The orientations of the three faces andthe rake of the apparent lineation in each face are asfollows:

Determine the true attitude of the lineation.

OrientationS56°E,52°NWN82°E,300SN09°E,700W

contained in a plane perpendicular to the exposure plane andpassing through the long axis of the ellipse. Theorientation of the apparent lineation can be measured eitherby its rake on the plane of exposure or by its plunge andbearing. The measurements are plotted on a stereogram toobtain the true attitude of the lineation. The followingproblem illustrates the method.

\\,," \,

Rod shaped fabric element

)

Plane containing linear element

Figure 8-24. Apparent lineation on a planarsurface obliquely intersecting a rod-shapedlinear structure.

Method 8-6Step 1: On an equal-area projection, plot each face,

its pole, and the trace of the lineation on the face (Fig.8-25a, b, c).

Figure 8-25. Procedure for det­ermining orientation of a lineationdescribed in Problem 8-7.

N

166

Step 2: For each face draw the great circle passingthrough the pole and the lineation trace (Fig. 8-25a, b, c).The three great circles intersect (ideally) in one pointdefining the aUitude of the lineation (Fig. 8-25c) as24°,52°W. This is referred to as Lowe's method (Lowe,1946).

Step 3: Alternatively, after step 1, find the poles(G1, Gz,G3) to the great circles that pass through the poleto the face and the lineation trace. All these "new" poleslie on a great circle that represents the plane perpendicularto the lineation (Fig. 8-25d). Thus, the pole to this greatcircle gives the aUitude of the lineation (Fig. 8-25d). Thismethod, which was devised by Cruden (1971), avoids theproblem of trying to find a single point of intersection todefine the attitude of the lineation by allowing a best-fitgreat circle to be drawn.

Analysis' of Fabric Geometry

Pauerns of variation in the altitude of fabrics around foldsmay help determine the chronology of fabric developmentwith respect to the fold. A number of pauerns arepossible, depending on the nature of the fabric, the timing

Elementary Techniques Part I

of fabric development with respect to folding, and themechanism of folding. The palterns are discussed nextindividually. A complete discussion of fabric types isbeyond the scope of this book; our goal here is solely toshow how fabric geometry can be practically analyzed withthe equal-area net.

Foliation Postdating Folding: If a planecylindrical fold that folds 51 and formed coevally with 52is cut by a later planar foliation (53)' the intersectionlineation between 53 and 51 will vary around the fold (Fig.8-26a). However, all the auitudes of the lineation lie on asingle plane (53) and fall along a great-circle girdle (Fig.8-26b). Also, the angle between the lineation and the F2fold axis varies as a function of 51 attitude (Fig. 8-26b).

Flexural-Slip Folding of a Lineation:Development of flexural-slip folds is accommodated bylayer-parallel slip with minimal internal distortion oflayers. Thus, to a first approximation, the movement of alayer during folding can be described as a rotation, and theangle (Il) between the fold axis and the preexistinglineation remains constant everywhere on the folded surface(Ramsay, 1967; Fig. 8-27a). On an equal-area projection,the points representing the lineation, therefore, lie on a

a

a

b

b

N

Figure 8-26. Intersection linea­tion produced by a later planarfoliation (83 ) cutting an earlierfolded foliation (81), (Adapted fromTurner and Weiss, 1963.)

Figure 8-27. Flexural-slip fold­ing of a preexisting lineation. linea­tion points lie on a small circlecentered on the fold axis. Lineationthat was perpendicular to the foldaxis (open circles on equal-areaplot) lies on a great circle afterfolding. (Adapted from Ramsay,1967.)

Chapter 8 Equal-Area Projections and Structural Analysis 167

small circle centered on the fold axis (3) (Fig. 8-27b),unless the original lineation is perpendicular to the foldaxis in which case the folded lineation lies on a great circle(Fig. 8-27b). Remember that the rotation of a line aroundan axis inscribes a small circle (see Chapter 7).

In reality, the discounting of material distortion oflayers is not correct. Because individual layers in aflexural-slip fold are boclded, each folded layer has a neutralsurface that shows ideal concentric geometry, but the outerarc is extended and the inner arc shortened (Fig. 8-28a).Therefore, on the outer arc, the angle between the lineationand the fold axis is slightly increased (j.J.'>~), and thelineation points lie on an arc that is broader than thesmall-circle arc but is still centered on the fold axis. (Fig.8-28b). Similarly, the angle between the fold axis and thelineation is slightly decreased (j.J."<~) on the inner arc, andthe lineation points lie on an arc that is narrower than thesmall-circle arc (Fig. 8-28b).

Passive Folding of a Lineation: Developmentof a passive fold is geometrically analogous to the passivereorientation of a marker layer by shearing on a set ofclose-spaced planes that are oblique to the foliation. In

reality, discrete slip planes need not exist. The axial planeof the fold is parallel to the hypothetical shear planes, andthe fold axis is parallel to the shear plane-marker layerintersection lineation. Points along an original linearfeature on the marker layer are transported variable distancesalong parallel lines (in the slip direction) and are positionedon the surface of the folded layer so that the folded lineationis contained in a plane defined by the original lineation andthe slip direction (Fig. 8-29a). Thus, on an equal-areaprojection the points representing the folded lineation lieon a great circle that is oblique to the fold axis (Fig.8-29b). This geometry is the same as that for anintersection lineation due to a foliation superposed on apreexisting fold, except that in this case, there may be nofoliation developed parallel to the plane containing thelineation.

Complex Refolding of Lineations: Manynatural folds show complex patterns for refolded lineation;the lineations often lie in arcs intermediate between a smallcircle and a great circle. Such modifications may resultfrom layer-parallel shortening prior to folding,homogeneous flattening after folding, or some form of

Figure 8-28. Effect of bucklingof individual layers during flexural­slip folding. The small-circle arcpattern of lineations is modified inthe outer and inner arcs of the fold.(Adapted from Ramsay, 1967.)

Figure 8-29. Passive folding ofalineation. Lineation points lie on agreat circle oblique to the fold axis.(Adapted from Ramsay, 1967.)

1\Inner arc

a

a

Outer arc

b

\ \ \ '\

\ \\ ., \.\ \ \ . .­

\ \

\

b

168 Elementary Techniques Part I

longitudinal layer-parallel strain accompanying folding.Details of these various possibilities are discussed inTurner and Weiss (1963) and Ramsay (1967).

Flexural-Slip Folding of Obliquely InclinedSurfaces: Folding of rocks containing two preexistingfoliations (SI and S2) that are oblique to one anotherresults in simultaneous folding of both foliations (Ramsay,1967). The foliations could, for example, be bedding andcleavage, or two preexisting cleavages, or even crossbedding and its enclosing master bedding. The geometricpatterns resulting from such folding are readily analyzed onan equal-area net. During flexural-slip folding, if theS I/S2 intersection lineation is parallel to the fold axis,then both surfaces are folded into cylindrical folds that arecoaxial (Fig. 8-30a). If, on the other hand, the SI/S2intersection lineation is perpendicular to the fold axis, andone surface (S2) is folded into a cylindrical fold, the othersurface (S2) maintains its dihedral angle with respect to S1and is folded in conical form, with the cone axis parallel tothe fold axis for SI (Fig. 8-30b). Oblique intersectionsbetween SI and S2 give rise to more complex patterns,with the S1/S 2 intersection lineation falling on a

small-circle arc centered on the fold axis for SI' and thedihedral angle between S1 and S2 varying continuouslyaround the fold (Ramsay, 1967) (Fig. 8-3Oc).

Passive Folding of Obliquely InclinedSurfaces: During passive folding the SI/S2 intersectionlineation is folded (Fig. 8-31a) but still lies in a planedefmed by its original orientation and the slip direction onthe hypothetical shear planes. Thus, after folding, theintersection lineations lie on a great circle (Fig. 8-31b).Both SI and S2 are folded into cylindrical folds with acommon axial plane (S3) parallel to the shear planes. Thetwo folded surfaces have different fold axes (81 and 82)determined by their lines of intersection with the shearplanes (Fig. 8-31b, c). The dihedral angle between SI andS2 generally varies across the fold (Ramsay, 1967) (Fig.8-31c).

n-Dlagram Analysis of Superposed Folds

Superposed folding refers to the overprint of a latergeneration of folds over an earlier one. Depending on theirorientation, the later generation of folding can cause

----Cone axis = ~

Greal circle (51)Great circle (81)

Small circle (52)

Greal circle (51,52)

a b cFigure 8-30. Flexural-slip folding of obliquely inclined surfaces. (a)$ 1/$2 intersection lineation parallel to the fold axis; (b) $1/$2intersection lineation perpendicular to the fold axis; (c) $1/$2 inter­section lineation oblique to the fold axis. (Adapted from Ramsay, 1967.)

Chapter 8 Equal-Area Projections and Structural Analysis

Axial Plane]

(83)

a82

169

Figure 8-31. Passive folding ofobliquely inclined surfaces. Sl/S2intersection lineation is folded butlies on a plane (great circle) afterfolding.

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- _ / \81~ 'r~-.Great circles

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reorientation of the earlier folds. Typically, in areas ofsuperposed folding there are multiple generations of foldsand multiple sets of foliations. Sometimes, each foliationset can be shown to be in an axial-planar orientation withrespect to a particular generation of folds. In analyzing anarea of superposed folding, the frrst step is to recognize anddefme domains of plane cylindrical folding of any foliation.The foliation that is analyzed may be different in differentdomains. The earliest foliation possible is bedding and isusually labeled SO, Successive later foliations are labeledS1,S2' ..., etc. Next, we illustrate how an area ofsuperposed folding can be analyzed with the aid of anequal-area net. Additional examples are provided in Chapter15.

Problem 8-7The example shown in Figure 8-32 is taken from

Turner and Weiss (1963). The map shows folded foliation(S 1)' There are two kinds of axial traces: frrst, brokenlines (F2), which are folded; and second, solid lines (F3),which do not show any consistent folding. With the aid ofan equal-area net, analyze the structures shown in the mapof Figure 8-32. Identify structural domains, and determinethe generation of fold responsible for the orientation offoliation in each domain.

Method 8-7Step 1: Divide the map into domains of plane

cylindrical folding by choosing areas with straight axialtraces. Plot poles to S1 foliation within each domain on aseparate equal-area plot. By trial and error, adjust domainboundaries so that the plot from each domain displays asingle 1t-axis; 1t-axes for different domains will be different.The axial trace, determined from the map, and 1t-axispermit calculation of the orientation of the axial plane foreach domain.

Step 2: Group the domains based on orientation ofthe axial plane defined by S1 foliation attitudes within thedomain. In this example, we can do this by inspection:Domains I to VII have S3 as the axial plane, whiledomains VIII to XIV have S2 as the axial plane.

Step 3: Draw synoptic diagrams for each group ofdomains: (a) For domains I to VII the fold axes(determined from S1 poles) lie on a great circle and arealmost coplanar to S3 planes; (b) for domains VIII to XIVthe axes (from S1 poles) also lie on a great circle. The S2planes intersect to define an axis that lies on the same greatcircle as the axes from domains I to VII. Generally, innatural examples the older folds do not show such a regularpattern because of inhomogeneities that develop duringrefolding.

170 Elementary Techniques Part I

5, - pole diagrams for domains of plane cylindrical folding

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SYNOPTIC PS, (B~:I SYNOPTIC S. SYNOPTIC JlS, (B~:) . SYNOPTIC S.

Figure 8-32. 1t-diagram analysisof superposed folding described inProblem 8-5, (Adapted from Turnerand Weiss, 1963,)

Chapter 8 Equal-Area Projections and Structural Analysis 171

Step 4: If S2 and S3 cannot be distinguished byinspection, then we can attempt to group the domainseither by trial and error or by using minor structures (folds,axial-plane foliations, intersection lineations) to establishage relationships among structures. The latter is usually

the more fruitful approach, since any macroscopic analysisof complex folding is unsatisfactory without theinformation provided by relationships among variousminor structures, and among minor and major structures.

EXERCISES

1. The following series of attitude measurements were obtained from the limbs of afold:

106°,36°SW150°,45°SW079°,400SE

N400E,600SES03°E,65°SWN53°E,500SE

(a) Plot a 13-diagram using these data.(b) Plot a 1t-diagram using these data.(c) Describe the structural significance of the 13-axis, the 1t-axis, and the 1t-circlegirdle. Give the orientation of each. Which diagram (1t or 3) is easier for you tointerpret?

2. A geologist measured a prominent mineral lineation on foliation surfaces of theHightower Gneiss exposed on a horizontal pediment near Tarantula Gulch,Arizona. A simplified version of the map of this area is presented as Figure8-Ml. The geologist measured the pitch of the lineations. This problememphasizes the fact that geometric calculations can readily be done with anequal-area net.

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Figure 8-M1. Map of the Tarantula Gulcharea for exercise 2.

(a) Complete the following table.

Attitude of foliation

A N37°E,300SEB N-S,400EC N23°W,6QONED N55°W,700SWE N85°W,400S

Rake of lineation

42°SWl1°S04°NWnOSE75°E

Plunge and bearing of lineation