Quantification of true displacement using apparent displacement

Tectonophysics 467 (2009) 107–118

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Quantification of true displacement using apparent displacement along an arbitraryline on a fault plane

Shunshan Xu ⁎, A.F. Nieto-Samaniego, S.A. Alaniz-ÁlvarezUniversidad Nacional Autónoma de México, Centro de Geociencias, Apartado Postal 1-742, Querétaro, Qro., 76001, Mexico

⁎ Corresponding author. Tel.: +52 442 2381116.E-mail address: [email protected] (S

0040-1951/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.tecto.2008.12.004

a b s t r a c t

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Keywords:FaultTrue displacementApparent displacementQuantification

e approaches to determine the true displacement (St) using an apparentdisplacement (Sm) measured from an arbitrary line on a fault plane. The considered parameters are thepitch of slip lineation (γ), the pitch of a cutoff (β), the apparent displacement along the observation line(Sm), and the pitch of the observation line on the fault plane (φ). We analyzed the following cases. First, ifthe apparent displacement is taken as the true displacement, the degree of overestimation orunderestimation of the true displacement can be calculated. The displacement cannot be obtained alongthe null line because the pitch of the observation line (φ) is equal to the pitch of the cutoff of the marker(β). Second, the total true displacement can be obtained not only along the slip direction but also alonganother particular line depending on the values of γ and β. Third, if the apparent displacements from twonon-parallel markers can be measured, the slip direction can be estimated. We apply the methods tocalculate the extensions due to the normal faults of San Miguelito in Mesa Central, Mexico. The resultsindicate that the largest fault strain reaches ca. 0.50 and the smallest fault strain is ca. 0.08. Also, theisolated faults show more regular strain profiles along the fault strikes than the faults with overlapping orintersecting geometries.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

The term “displacement” is an ambiguous word in geology (Tear-pock and Bischke, 2003). According to Walsh and Watterson (1988),displacement refers to the displacement accumulated through thewhole active period of the fault. This definition indicates thatdisplacement is a total slip or total true displacement. Displacementalso represents the variation in position of a marker displaced by thefault movement (Tearpock and Bischke, 2003). In the light of thisconcept, displacement is an apparent displacement. Previous workdid not distinguish a true displacement from an apparent displace-ment (e.g. Dawers et al., 1993; Clark and Cox, 1996). In this paper, weuse the term “true displacement” (St) by following Walsh andWatterson's definition. Therefore, the true strike displacement(Sth) refers to the component of St along the fault strike. True dipdisplacement (Std) refers to the component of St along the fault dip(Fig. 1).

Three problems influence the obtainment of the true displace-ment. First, observed sections in outcrops may not be vertical at times,and the sample lines may not be perpendicular to the strikes of faults.Second, the beds are not horizontal or the strikes of the beds are notparallel to that of the fault. Third, faults are not absolute dip-slip or

. Xu).

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strike-slip faults. For all cases above, it is necessary to establish aquantitative relationship between the true displacement and appar-ent displacements.

Traditionally, the main parameters to determine the fault dis-placement are slickenside lineations and kinematic indicators on ornear the fault (e.g. Billings, 1972; Suppe, 1985; Doblas et al., 1997a, b).Recently, there has been somework for quantitatively determining thefault true displacement (e.g. Rouby et al., 2000, Xu et al., 2004a; Xuet al., 2007). Billi (2003) analyzed the components of fault slip andseparations generated by cleavage-controlled fault zone contraction,on the assumption that shortening occurs perpendicularly to solutioncleavages. The methods by Xu et al. (2004a) are appropriate only forthe faults on subsurface maps. The approaches by Xu et al. (2007)consider only data measured from cross-section perpendicular to thefault strike or frommap view. These methods need more assumptionsthan the approaches that we introduce here.

In this paper we quantify the magnitude of true displacement andthe direction of fault slip on faults. The approaches introduced herecan be applied to data measured along arbitrary lines on the faultplane, which are more general than methods proposed by Xu et al.(2007).

This paper consists of two parts. The first part is to establishequations for obtaining the magnitude of true displacement, thedirection of fault slip, or both, according to the available data. Thesecond part gives an example of how to calculate the strain due tofaulting. In most cases, the accurate strain is difficult to obtain if the

Fig. 1. Block diagram showing that the slickenside lineation and the marker cutoff haveopposite directions. St=CC′=true displacement, Sth=HC′=true strike displacement, Std=CH=true dip displacement, γ=pitch angle of true displacement, β=pitch angle of cutoff ofthemarker. KC′ and JC′ are two arbitrary lines alongwhich the apparent displacements aremeasured.φ1 andφ2 are pitch angles of KC′ and JC′, respectively. Apparent displacement isFC′ for lineKC′, whereas, LC′ for line JC′. Note that the apparent dip displacement (Smd=EC′)is not equal to Std.

108 S. Xu et al. / Tectonophysics 467 (2009) 107–118

true fault slips are not known. The example in this paper provides anexcellent application based on our methods.

2. Calculations of the true displacement

Inorder todefine thepitch angles, the following conventions areused.

(a) The angle of pitch is in the range from 0° to 90°. Starting fromthe strike line, the angle is measured in a sense which is downthe dip of the plane. This is used in conjunction with con-ventions b and c.

(b) Direction of pitch is the direction of the strike from which theangle of pitch is measured.

(c) Opposite direction of pitch refers to the direction of strikewhich isopposite to the strike fromwhich the angle of pitch is measured.

For calculation, four parameters should be known: the pitch of sliplineation (γ), the pitch of a cutoff (β) of a marker (bed, vein, etc.), thepitch of an observation line (φ), the apparent displacement along theobservation line (Sm). To calculate the magnitude of true displace-ment, the following cases can be considered (Table 1).

(1) Case of slickenside lineation with opposite pitch direction tothat of marker traces on the fault. There are two sub-cases:(a) the observation line has the same pitch direction as theslickenside lineation; (b) the observation line has the oppositepitch direction to the slickenside lineation.

(2) Case of slickenside lineation with the same pitch direction asmarker trace on the fault. Two sub-cases are considered: (a) theobservation line has the same pitch direction as the slickensidelineation; (b) the observation line has the opposite pitchdirection to the slickenside lineation. Two further situationsshould be included: sub-cases where βNγ and where βbγ.

Table 1Cases for calculation of true fault displacement

Case 1: slickenside lineation with pitch direction opposite tothat of marker traces

Case 2: slickenside lineatio

Case 1a: the observation line with the same pitch directionas the slickenside lineation.

Case 2a: the observation lias that of the slickenside l

Case 1b: the observation line with the opposite pitch directionto that of the slickenside lineation.

Case 2a-a: βNγCase 2a-b: βbγ

There are some principles for determining whether a slickensidelineation and a marker trace or an observation line on the fault hasthe same or opposite pitch direction (Xu et al., 2007). Sometimes,the pitch of intersection line of a bed with a fault plane could not bedirectly obtained. This problem can be resolved using the equations(Eqs. A1 to A18) established by Xu et al. (2007). For example, if afault with an angle of α intersects a marker with an arbitrary mark-er with an angle of θ, the pitch angle of the cutoff (β) will be β =arctan sinμ tanα tan θ

sinα tan θ cos μ− tanαð Þ� �

, where µ is the acute intersection angle ofthe fault strike with the marker strike.

2.1. Case of slickenside lineation with opposite pitch direction to markertrace on the fault

For this situation, we should consider two sub-cases (Table 1)according to the pitch direction of an observation line on the faultplane (Fig. 1). The first sub-case is that the observation line has thesame pitch direction as the slickenside lineation. Let the pitch angle ofthe line be φ1. In Fig. 1, for the triangle CC′E, we can obtain

BCECV= 90−β;BCCVE = 90−γ;thenBCVCE = β + γ;

ð1Þ

where ∠ indicates angle.For the triangle FCC′, FC′=Sm, CC′=St, because∠FC′E=90−φ1,∠CC′E=

90−γ, thereforewe can infer that∠CC′F=∠CC′E−∠FC′E=φ1−γ,∠CFC′=180−(φ1+β). By using the Law of Sines of triangle, the following equa-tion can be established

St =Sm sinð180− u1 + βð Þ

sin γ + βð Þ =Sm sin u1 + βð Þsin γ + βð Þ ð2Þ

The condition Sm=St holds for sin(φ1+β)=sin(γ+β) and sin[180−(φ1+β)]=sin(γ+β) reflecting periodicity of trigonometric function;then, we obtain

u1 = γ ð3Þ

and

u1 = 180−γ−2β ð4Þ

Eqs. (3) and (4) indicate that ifφ1=γ, orφ1=180−γ−2β, themeasureddisplacement is equal to the true displacement. For the given values of βand γ (marker and slickenside pitches), the true displacement can beobtained along two lines of observation. One of them is not theslickenside direction. A special case is β+γ=90°, in that case Sm and Stare equal to each other for only one value of φ1. For example, curve 4 inFig. 2a has two intersection points with true displacement (St). At point a,φ1 is equal to 40°, and at point b, φ1 is equal to 80°.

From Eq. (2), given two of the three angles β, γ, and φ1, therelationship between Sm and the third angle can be calculated (Fig. 2). Itcan be seen that the value of Sm (apparent displacement along theobservation line) could be larger, equal, or smaller than the total truedisplacement. From Fig. 2a, we can see that there are two curvetendencies between Sm andφ1. For small value of φ1, the value of Sm hasa negative relationship with the value of φ1, whereas for large value of

n with the same pitch direction as that of marker traces

ne with the same pitch directionineation.

Case 2b: the observation line with the opposite pitchdirection to that of the slickenside lineation.Case 2b-a: βNγCase 2b-b: βbγ

Fig. 2. Changes of the values of Smwithφ1,β, andγ. For all curves, the total displacement Stis assumed to be equal to 100 with no unite. The results are based on Eq. (2) to the casewhen slickenside lineation has opposite pitch direction to marker trace on the fault. Thecurves in (a) show relationship Sm and φ1, given β=30° and γ=10° to 90°. The curves in(b) show relationship between Sm and γ, given β=35° and φ1=10° to 90°. The curves in(c) show relationship between Sm and β, given γ=20° and φ1=20° to 80°.

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φ1, the value of Sm has a positive relationship. There are curves in Fig. 2athat coincide completely, for example, the curves for γ=90° and γ=30°.This indicates that different combinations of β, γ, and φ1 can producethe same value of measured displacement Sm. In Fig. 2b, for small valueof γ, the value of Sm has a positive relationship with the value of γ,whereas for large value of γ, the value of Sm has a negative relationship.Also, there are curves that coincide in this figure. In Fig. 2c, there arethree types of curves.Whenβ is smaller thanφ1 (e.g., curve 1), the valueof β has a negative relations with the value of Sm. When β is equal to φ1

(e.g., curve 2), the value of Sm is equal to the value of St, independent upthe value of β. When β is larger than φ1 (e.g., curves 3, 4, 5, 6, 7), thevalue of β has a positive relations with the value of Sm.

Additionally, from Fig.1, we can also infer the relationship betweenthe total true displacement (St) and the apparent displacement (Smd)along the dip direction. For the triangle CC′E, EC′ is equal to Smd. Byusing the Law of Sines of triangle, we can obtain

St =Smd sin 90−βð Þsin γ + βð Þ =

Smd cosβsin γ + βð Þ ð5Þ

The second sub-case is that the measured line has opposite pitch/para direction to the slickenside lineation. In this case, JC′ is theapparent displacement (Sm) in Fig.1. For the triangle CC′J,∠CJC′=φ2−β,∠C′CJ=β+γ, then ∠CC′J=180−φ2−γ. By using the Law of Sines oftriangle, we obtain

St= sin u2−βð Þ = Sm= sin β + γð Þ

St =Sm sin u2−βð Þsin γ + βð Þ

ð6Þ

This equation is only for φ2Nβ. Similarly, for φ2bβ, we have

St =Sm sin β−u2ð Þsin γ + βð Þ ð7Þ

For Sm=St in Eq. (6), the following equation can be obtained

u2 = 2β + γ ð8ÞSimilarly, from Eq. (7), by assuming Sm=St, we have

u2 = γ + 2β−180 ð9ÞFrom Eqs. (8) and (9), giving different values of β and γ, we can

calculate the pitch (φ2) of the line in which we can measure the truedisplacement. Note that it is not the slip direction. This is useful,because when the true displacement along the slip direction cannotbe measured due to outcrop problem or any other problems, we canobtain the true displacement by measuring along a line obtained fromEqs. (8) and (9).

From Eqs. (6) and (7), by assuming a values of Sm and giving twoarbitrary values from the values ofβ,γ andφ2, the tendency of Sm can beshown in Fig. 3. From Fig. 3a, we can obtain the following results. First,whenφ2 is smaller thanβ, the value of Sm has positive relationshipwiththe value of φ2, whereas, when φ2 is larger than β, the value of Sm hasnegative relationship. Second, for φ2bβ the value of Sm is always largerthan the true displacement (St). For φ2Nβ, some measured values (Sm)are larger than the true displacement (St). When φ2N50°, the values ofSm approach the true displacement (St). Third, when the value of φ2

approaches the value of β, the value of Sm tend to have infinite value.This produces large difference between the values of Sm and St. Theobservation line with this value of φ2 is consistent with the null line ofRedmond (1972). Along the null line the fault displacement cannot beobserved. Fig. 3b shows that for β =60° and γb30°, the value of γ haspositive relationship with the value of Sm, whereas for γN30°, the valueof γ has negative relationship with the value of Sm. On the other hand,there are coincided curves in Fig. 3b. For example, the curves for γ=90°andγ=20° coincide completely. In Fig. 3c,whenβ is smaller thanφ2, thevalue of Sm has a positive relationship with the value of β, whereas,whenβ is larger thanφ2, the value of Sm has a negative relationship. Thenull line is β=50°. When β=φ2, the value of Sm tends to infinity.

2.2. Case of slickenside lineation with the same pitch direction as themarker traces on the fault

This is a case in which the lineation and marker trace have thesame pitch direction is shown in Fig. 4. We consider two differentsub-cases according to pitches of slickenside lineations and markers:situation of βNγ and situation of βbγ.

Fig. 3. Relationships between the values of Sm and the values ofφ2, β, and γ, based on Eqs.(7) and (8) to the case when slickenside lineation has opposite pitch direction to markertrace on the fault. For all curves, the total displacement St is assumed to be 100 (horizontaldash line). In (a), β=20°, γ=10° to 90°. In (b), β=60°, φ2=10° to 90°. In (c), φ2=50°, γ=10°to 90°.

Fig. 4. Block diagrams showing that the pitches of the slickenside lineation and cutoff ofthe marker have the same directions. Marker 1 is parallel to marker 2. (a) The situationof βNγ. Lines JK and FD are two lines along which the apparent displacement aremeasured. (b) The situation of βbγ. Lines C′D and B′G are two lines along which theapparent displacement are measured.

110 S. Xu et al. / Tectonophysics 467 (2009) 107–118

2.2.1. Situation of βNγWhen pitch direction of the observation line (DF) is the same

as that of slickenside lineation (Fig. 4a), the pitch of this line is

defined as φ1 (Fig. 4a). First, we consider that the value of φ1 islarger than β. C′G is the apparent fault displacement (Sm) mea-sured along line DF (Fig. 4a). For the triangle CC′G, CC′=St, C′G=Sm,∠CGC′=φ1−β, ∠GCC′=β−γ, therefore by using the Law of Sines oftriangle

St= sin u1−βð Þ = Sm= sin β−γð Þ

St =Sm sin u1−βð Þsin β−γð Þ :

ð10Þ

Similarly, for φ1bβ, we have

St =Sm sin β−u1ð Þsin β−γð Þ : ð11Þ

From these two equations, by assuming the value of Sm and givingtwo of β, γ, and φ1, we can calculate the tendency of the value of Sm

111S. Xu et al. / Tectonophysics 467 (2009) 107–118

with the third angle. In Fig. 5a, for φ1bβ, the value of Sm has a positiverelationship with the value of φ1, whereas, for φ1Nβ, the value of Smhas a negative relationship. When the curves approaches the line ofφ1=β=50°, the value of Sm tends to infinite. This feature implies thatthe difference between the vales of Sm and St become larger when themeasurement line is more parallel to the cutoff of markers. In Fig. 5b,the values of γ has a negative relationship with the value of Sm. Thecurves coincide each other for values φ1=60° and φ1=90°, and forφ1=70° and φ1=80°. The gradients of the curves for the larger valuesof φ1 are greater. In Fig. 5c, for βbφ1, the value of Sm has a positive

Fig. 5.Maps showing relationships between the values of Sm and the values of φ1, β, andγ. The curves are based on Eqs. (10) and (11). For the curves in (a), β=50°, γ=5° to 45°.For the curves in (b), β=75°, φ1=10° to 90°. For the curves in (c), γ=10°, φ1=30° to 70°.

Fig. 6. Changes of the values of Smwithφ1,γ, andβ. For all curves, the total displacement St isassumed to be equal to 100 with no unite. The results are based on Eq. (14). (a) Relationshipbetween Sm and φ2, given β=60°, γ=10° to 50°. (b) Relationship between Sm and γ, givenβ=80°,φ2=10° to 90°. (c) Relationship between Sm and β, given γ=20°,φ2=10° to 90°.

relationship with the value of β, whereas, for βNφ1, the value of Smhas a negative relationship. The lines β=φ1 are null lines. When thecurves are close to the null lines, the value of Sm tends to show infinitevalue. This effect is similar to that in Fig. 5a.

Fig. 7. Changes of the values ofSmwithφ1,γ, andβ, based onEqs. (15) and (16). For (a),β=20°,γ=30° to 80°; For (b), β=15°,φ1=10° to 90°; For (c), γ=70°,φ1=20° to 90°.

112 S. Xu et al. / Tectonophysics 467 (2009) 107–118

For Eq. (10), if Sm=St, it can be deduced:

u1 = 2β−γ ð12Þ

Like in the previous cases we can obtain the total true displace-ment not only along the slip direction but also along other speciallines. On the other hand, from Eq. (11), we can infer that also in thecase of φ1=γ, the measured displacement is the true displacement.

Particularly, when apparent displacement (Smd) is measured alongdip, we can obtain the relationship between St and Smd. In Fig. 4a, forthe triangle CC′E, CE′=Smd, ∠CEC′=90−β, ∠GCC′=β−γ, by using theLaw of Sines of triangle, the relationship between St and Smd can beshown that

St= sin 90−βð Þ = Smd= sin β−γð Þ

St =Smd cosβsin β−γð Þ :

ð13Þ

When pitch direction of the observation line (JK) is opposite to thatof the slickenside lineation, the pitch of this line is defined as φ2. InFig. 4a, C′H is the apparent displacement (Sm) measured along line JK.For the triangle CC′H, CC′=St,∠CHC′=180−(φ2+β),∠HCC′=β−γ, there-fore, by using the Law of Sines of triangle, we obtain

St= sin 180− u2 + βð Þ½ � = Sm= sin β−γð Þ

St =Sm sin u2 + βð Þ

sin β−γð Þ :

ð14Þ

For the given values ofβ, γ, andφ2, the tendency of the value of Sm isshown in Fig. 6. In Fig. 6a, givenβ=60°, for small value ofφ2, the value ofSm has a negative relationship with the value of φ2, whereas for largevalue of φ2, the value of Sm has a positive relationship. The inflectionpoints between the negative and positive relationship are located on theintersection between the line φ2=30° with curves. Two results can beobtained from Fig. 6b. First, for the given values of β andφ2, the value ofSm has a negative relationshipwith the value ofγ. Second, the curves forthe larger values ofφ2 have steeper gradients. The curves in Fig. 6c showthat for the given values of γ and φ2, the value of Sm has a positiverelationship with the value of β. The gradient of curves become largewhen the value of φ2 and β if close to 90°.

2.2.2. Situation of βbγFor this situation, two cases are considered (Fig. 4b). First, when the

pitch direction of the observation line (DC′) is the same as that of theslickenside lineation, the pitch of this line is defined asφ1. In Fig. 4b, C′F isthe fault apparent displacement (Sm) measured along the cutoff DC′. Forthe triangle CC′F, CC′=St, ∠FCC′=γ−β, ∠CC′F=∠CC′E−∠FC′E=(90−γ)−(90−φ1)=φ1−γ, then ∠CFC′=φ1−β, by using the Law of Sines of triangle,we obtain:

St= sin u1−βð Þ = Sm= sin γ−βð Þ

St =Sm sin u1−βð Þsin γ−βð Þ :

ð15Þ

This equation is for the case of φ1Nβ. Likewise, for the case ofφ1bβ, we have:

St =Sm sin β−u1ð Þsin γ−βð Þ : ð16Þ

Given values of β and γ and φ1, the tendencies of Sm are shown inFig. 7. It can be seen that for φ1bβ, the value of Sm has a positiverelationship with the value ofφ1, whereas, forφ1Nβ, the value of Sm hasa negative relationship (Fig. 7a). On the other hand,when the value ofφ1

is close to the value of β, the value of Sm tends to infinite. This impliesthat, forφ1=β (null line), the apparent displacement cannot beobtained.Fig. 7b shows that for given values of φ1 and β, the value of Sm has

positive relationshipwith the value ofγ. The curves for smaller values ofφ1 show larger gradients of curves. Fig. 7c presents that for βbφ1, thevalue of Sm has positive relationship with the value of φ1, whereas, forβNφ1, the value of Sm has negative relationship. Also, from the null lines,no displacement can be measured.

For Eq. (15), if Sm=St, we can be obtain

u1 = 2β−γ ð17Þ

Again, the total true displacement can also be measured along adifferent line from the slip direction. In addition, in Eq. (16), we can

Fig. 8. Curves of the value of Sm calculated from Eq. (19) by assuming St=100. In (a),γ=70°,β=10° to 60°; In (b), β=10°, φ2=10° to 90°; In (c), γ=85°, φ2=10° to 90°.

113S. Xu et al. / Tectonophysics 467 (2009) 107–118

infer that only in the case of φ1=γ, which means in the slip direction,the measured displacement is the true displacement (i.e. Sm=St).

In addition, in Fig. 4b, for the triangle CC′E, CE′=Smd, ∠CC′E=90−γ,∠ECC′=γ−β, then ∠CEC′=90+β, by using the Law of Sines of triangle,we establish the follow equations

St= sin 90 + βð Þ = Smd= sin γ−βð Þ

St =Smd cosβsin γ−βð Þ :

ð18Þ

When the pitch direction of the observation line (GB′) is opposite tothat of the slickenside lineation, the pitch of this line is defined as φ2

(Fig. 4b). HB′ is the fault apparent displacement (Sm)measured along thelineGB′. For the triangleBB′H,BB′=St,∠BB′H=γ−β,∠BB′H=180−(φ2+γ),then ∠BHB′=φ2+β, by using the Law of Sines of triangle, we obtain:

St= sin u2 + βð Þ = Sm= sin γ−βð Þ

St =Sm sin u2 + βð Þ

sin γ−βð Þ :

ð19Þ

For given values of β, γ, and φ2, the changes of Sm are shown in Fig. 8.From Fig. 8a, it can be seen that the value of Sm has a negative relationshipwith the value of φ2. Also, the curve gradient has a negative relationshipwith thevalueofφ2. Bycomparingdifferent curves, the curve gradients forsmaller valuesofφ2 are larger than those for larger valuesofφ2. The curvesin Fig. 8b indicate that the value of Sm has a positive relationship with thevalue of γ. On the other hand, the curve gradient has a negative rela-tionship with the value of γ. Fig. 8c shows that the value of Sm has anegative relationship with the value of β. For φ2N45°, the curves areconvex right-up; whereas, for φ2b45°, the curves are convex left-down.

In addition, assuming the value of Sm is equal to the value of St, wehave:

u2 = γ−2β ð20ÞThis equation shows that there is a line, different from the slick-

enside, along which the total true displacement could be measured.

3. Calculation of pitch of the fault slip lineation

If there are two non-parallel markers, two sets of measured datafor these twomarkers can be obtained. Then, the pitch value (γ) of thetrue displacement can be estimated employing the two data sets. Theattitudes of twomarker horizons havemany combinations that can beused to evaluate the pitch of true displacement. For example, if thetwo marker horizons are consistent with the prerequisite of Eq. (2) inthe case of slickenside lineation with opposite pitch direction tomarker traces on the fault, then the Eq. (2) can be used to calculate thevalue of γ. Therefore, the following relationship can be established:

Sm1 sin u1 + β1ð Þsin γ + β1ð Þ =

Sm2 sin u1 + β2ð Þsin γ + β2ð Þ ð21Þ

where Sm1 and Sm2 are the apparent displacements measured alongline 1 for the marker 1 and marker 2, respectively, and β1 and β2 arethe pitch of marker 1 and marker 2, respectively.

By rearranging and simplifying, we obtain:

tanγ =Sm2 sin u1 + β2ð Þsinβ1−Sm1 sin u1 + β1ð ÞSm1sin u1 + β1ð Þ cosβ2−Sm2 sin u1 + β2ð Þ ð22Þ

Other combinations between two markers can be formulated,however, to avoid similarly inferring, they are not discussed here.

4. Case study: Extensional strain in Sierra de San Miguelito, Mexico

The fault strain is due to continuous fault slip along the dipdirection (Jamison, 1989; Peacock and Sanderson, 1993; Schultz and

Fossen, 2002). Fault strain provides important information about faultlinkage, fault formation, and lithologic influence on faulting process. Alarger fault strain corresponds to a larger fault slip (Westaway andKusznir, 1993). Therefore, the strain profiles should be similar to thedisplacement profiles, if the faults growth after rocks were deposited.For an isolated fault, the displacement profile has been consideredwith the form of cone or ellipse (Gupta and Scholz, 2000, Fig. 12c,d).Multipeak curves in the profiles suggest that the faults in the studyarea experienced complicated evolution and fault linkage (e.g. Segalland Pollard, 1983; Gudmundsson, 1987; Peacock, 1991; Trudgill and

Fig. 10. Sketch to illustrate the relationships among parameters for domino faultsassuming vertical shear supposed by Westaway and Kusznir (1993).

114 S. Xu et al. / Tectonophysics 467 (2009) 107–118

Cartwright, 1994; Xu et al., 2006). Ferrill et al. (1999) reported twomechanisms of linkage: breakthrough by curved lateral propagationand breakthrough by connecting fault formation. Both linkages maycause irregular displacement distribution of fault dip along faultstrike.

To apply our methods, we selected the Sierra de San Miguelito as adetailed study area (Fig. 9). The study area is within an elevatedplateau in central Mexico located in the south of theMesa Central. Thefault system in San Miguelito has been regarded as having a “dominostyle” because it consists of sub-parallel faults that systematically tiltthe volcanic beds to the NE (Labarthe-Hernández and Jiménez-López,1992; Nieto-Samaniego et al., 1997, Xu et al., 2004b). On the whole theindividual faults have a strike direction of N20°W–S60°E and dips tothe SW. The fault slickenside lineations are observed to plunge SW(occasionally SE) with a pitch of 75°–85°.There are some faults withN–S or NE–SW strike direction. Their main activity was dated byNieto-Samaniego et al. (1999) between 30.0 and 26.8 Ma, coeval withthe emission of the Cantera Ignimbrite and ended with theemplacement of the upper Panalillo Rhyolite (Labarthe-Hernándezand Jiménez-López, 1992).

In the studied area, the Cenozoic units are: Portezuelo Latite, SanMiguelito Rhyolite, Cantera Ignimbrite, Panalillo Rhyolite, and Hal-cones Conglomerate. A unwelded member of the Cantera Ignimbrite isselected as a marker for measurement.

Xu et al. (2004b) documented a vertical shear mechanism offaulting and bed tilting. For vertical shear, the extension produced byany planar normal fault is equal to its heave. Thus, the horizontal

Fig. 9. Geological map of the

distance between the footwall cutoff of one fault and the hanging-wallcutoff in the next fault will remain constant as deformation proceeds,maintaining its initial value (Lo in Fig. 10). So, we have DC=BC′=Lo andthe heave is h=D′B=Lbcot(α)sin(θ). Length before deformation is

Lo = Lb cos θð Þ; ð23Þ

Therefore, the extension is

e = h=Lo = cot αð Þ tan θð Þ ð24ÞWhether the extension calculated from this equation is true

depending on the orientation of the slickenside lineation and bedazimuth. For the faults with pure dip-slip (γ=90°) and the bed withstrike parallel to the fault strike (β=0°), the calculated extension istrue. For the oblique faults (γ≠90°), and the bed with strike notparallel to the fault strike (β≠0°), a correction factor should be addedfor calculation of true extension. For example, if the fault slip and

Sierra de San Miguelito.

115S. Xu et al. / Tectonophysics 467 (2009) 107–118

marker cutoff are consistent with Eq. (2), thus the equation to obtainthe true extension in an oblique fault will be

e =sin γ + βð Þsin u1 + βð Þ cot αð Þ tan θð Þ ð25Þ

where sin γ + βð Þsin u1 + βð Þ is a correction factor.

If the apparent displacement is measured along fault dip, the valueof φ1 is 90°, then, Eq. (25) will be

e =sin γ + βð Þcos βð Þ cot αð Þ tan θð Þ ð26Þ

In order to avoid confusion in terminology, now we will use theterm “fault strain” for the extension calculated as explained above. The

Fig.11. Fault strain profiles along the strike of faults with overlapping or intersecting geometrisolated faults in Fig. 12. Distance is measured from the northwestern end of each fault.

calculated results from Sierra de San Miguelito are shown in Figs. 11and 12, where the horizontal coordinate is normalized distance(distance/length). The characteristics of the fault strain profiles showvery irregular and multipeak curves for the faults that are intersectedor overlapped by other faults (Fig. 11). The largest fault strain reachesca. 50% along fault 1. The smallest fault strain is ca. 8% along faults 14and 16. For isolated faults shown in Fig. 12, more regular profiles wereobtained, but they are not very consistent with the displacementprofiles of other published isolated faults (e.g. Dawers et al., 1993;Dawers and Anders, 1995; Fossen and Hesthammer, 1997). Thecorrection factors from the section AA′ are shown in Table 2. It canbe seen that the correction factors can be either larger or smaller than1. The largest value is 1.15, and the smallest value is 0.97. These resultsimply that the differences between the true extensions and apparent

ies in the Sierra de SanMiguelito, Mexico. These profiles aremore irregular than those of

Fig. 12. Fault strain profiles along the strike of isolated faults in the Sierra de San Miguelito. These profiles are more regular than those of faults with overlapping or intersectinggeometries in Fig. 11, but still deviate from the theoretical displacement profile. Distance is measured from the northwestern end of each fault. In (e) and (f), the ideal displacementprofiles are present according to Gupta and Scholz (2000).

116 S. Xu et al. / Tectonophysics 467 (2009) 107–118

extensions are not very large. This is because the values of γ do notdeviate much from 90° (72°–78°).

5. Application limitations

When a fault is reactivated (e.g., Celerier, 1988; Alaniz-Alvarezet al., 1998), the observed slickenside lineation may be due to the last

Table 2Correction factors to calculate the extension due to faulting along the section in Fig. 9

No. 3 4 5 6 7 8

A1/α (°) 220/63 239/54 210/64 250/45 200/75 215/55A2/θ (°) 16/25 30/23 20/25 31/28 42/30 48/24γ (°)/A3 75SE 78SE 70SE 76SE 80NW 76NWβ (°) 12.1 17.7 5.5 36.3 14.5 10ξ 1.02 1.04 0.97 1.15 1.03 1.01

A1 is the dip direction of the fault. A2 is the dip direction of the bed. A3 is the pitch directionpitch of cutoff. ξ is correction factor.

movement of the fault. In this case, the calculated total displacementwill not represent the amount of total fault movement. This effect isshown in Fig. 13. Two slip generations of the fault (CC′ and C′C″) areassumed. If only last slickenside lineation (C′C″) is observed, theobtained displacement according to Eq. (5) is FC″. This value isneither equal to total displacement (CC″=St) nor the last displace-ment C′C″.

9 10 11 12 13 14 15

238/58 226/75 246/65 220/67 245/64 230/65 235/6625/22 30/15 30/22 10/13 46/18 35/19 38/1275SE 77SE 76SE 76SE 75SE 76SE 72SE15.8 4.4 16.1 5.8 7.9 6.4 4.21.04 0.99 1.04 0.99 1.01 0.99 0.97

of slickenside lineation on fault. α is fault angle. γ is pitch of slickenside lineation. β is

Fig. 13. Diagram showing twomovements of a fault. The trajectory of fault movement isC→C′→C″. Total true displacement is St =CC″. The last displacement is C′C″, which isnot equal to the total true displacement (CC″).

117S. Xu et al. / Tectonophysics 467 (2009) 107–118

Sometimes, normal faults show opening or pull-apart structures(e.g. Acocella et al., 2003; Gasperini et al., 2003). In this case, the totaldisplacement cannot be calculated, because the cutoffs of the samemarker cannot be observed on one fault surface (Fig. 14). For example,in Fig. 14b, if the fault has opening A′A″ or B′B″, the amount of the faultmovement is CC″. However, the total displacement on fault is CC′.The line CC″ is not on the fault surface. Evidently, the amount of CC″ is

Fig. 14. (a) Diagram showing a fault with no opening perpendicular to the fault strike.The total displacement on fault is St=CC′. (b) If the same fault in (a) has a furtheropening A′A″, the directly measured offset for the marker is CC″. This value is not equalto CC′. In this case, the value of St (CC′) cannot be directly measured.

not equal to CC′. To calculate CC' in this case, the value of C′C″, CC″, and∠C″CC′ are needed. However, the value of ∠C″CC′ is difficult to obtainin practice.

Our methods assume that faults are planar and have uniform slipacross the portion of the fault surface analyzed. These conditions arenot met in some cases.

For the scissor faults, the slip is not uniform along both the strikeand dip in the ranges of analysis (Fig. 15a). Also the slickensidelineations, commonly, are not a straight line. This produces the changeof the value of β across the portion of analysis. For the listric faults, thefault dip varies with depth (Fig. 15b). As a result, the value of βwill bedifferent for the same marker. Thus, the established equations cannotbe used to calculate the true displacement. Finally, if the beds alongthe fault zone experienced ductile deformation, the calculated truedisplacement will have error. In Fig. 15c, the true displacement is ABfor the bottom of bed b. The calculated true displacement is AC. Theerror of the true displacement is BC.

In general, it is dependent upon the scale of analysis to fulfill allrequirements of the geometric models. If the fault at the scale ofobservation is plane, the slickenside lineations are straight, and the hostrock is considered as rigid body, then the models can be well applied.This is similar to the other analysis of deformation for which it isnecessary to consider that the obtained results are valid only at thescales where the deformation are analyzed. For example, the deforma-tion is heterogeneous or homogeneous, depending on the scale ofobservation.

Fig. 15. (a). On a scissor fault, the magnitude and direction of fault are changeable.(b). For a listric fault, the fault dip along the dip is different. (c). For a ductile fault, thefault displacement is not the true slip due to ductile deformation.

118 S. Xu et al. / Tectonophysics 467 (2009) 107–118

6. Conclusions

The true and apparent displacements of a fault are related in ageometrical way. When the fault apparent displacement from onlyonemarker can bemeasured and slickenside lineation on the fault canbe observed, we established a series of equations for determining therelationship between the true displacement and apparent displace-ment according to the different combinations among pitch directionsof slickenside lineation, cutoffs of markers and observation lines.Three results are obtained from this scenario. First, if the apparentdisplacement is taken as the true displacement, the latter may beeither overestimated or underestimated. Second, for given values of βand γ, the measured displacement Sm (apparent displacement) isdependent on the pitch of the observation line (φ). When theobservation line is parallel to the cutoffs (φ=β), the displacementcannot be observed. The value of Sm is either positively or negativelyrelated to the pitch of observation line (φ), the pitch of cutoffs (β), andthe pitch of slickenside lineation (γ). Third, the total true displacementcan be obtained along one particular line except for the slip directionon the fault plane. The pitch of the particular lines can be calculated bythe values of γ and β. On the other hand, if the fault apparentdisplacements from two non-parallel markers can be measured, weinferred some equations to determine the direction of fault slip.

The methods are applied to calculate the extension due to thenormal faults of San Miguelito in Mesa Central, Mexico. The resultsindicate that the largest fault strain reaches ca. 0.508 and the smallestfault strain is ca. 0.08. For the isolated faults, the strain profiles alongthe fault strikes are similar to the displacement profiles and show thelargest strain near to center. For the faults with overlapping orintersecting geometries, the strain profiles showmultipeaks andmoreirregular than for the isolated faults.

Acknowledgements

This work was supported by the research projects D.01003 of theInstituto Mexicano del Petróleo, and 89867 and 049049 of Conacyt.We wish to thank A. Billi and an anonymous reviewer for theirpertinent comments.

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