Quantification of true displacement using apparent displacement along an arbitrary line on a fault plane Shunshan Xu ⁎, A.F. Nieto-Samaniego, S.A. Alaniz-Álvarez Universidad Nacional Autónoma de México, Centro de Geociencias, Apartado Postal 1-742, Querétaro, Qro., 76001, Mexico abstract article info Article history: Received 31 December 2007 Received in revised form 24 November 2008 Accepted 4 December 2008 Available online 10 December 2008 Keywords: Fault True displacement Apparent displacement Quantification This paper introduces some approaches to determine the true displacement (S t ) using an apparent displacement (S m ) measured from an arbitrary line on a fault plane. The considered parameters are the pitch of slip lineation (γ), the pitch of a cutoff (β), the apparent displacement along the observation line (S m ), and the pitch of the observation line on the fault plane (φ). We analyzed the following cases. First, if the apparent displacement is taken as the true displacement, the degree of overestimation or underestimation of the true displacement can be calculated. The displacement cannot be obtained along the 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 along another particular line depending on the values of γ and β. Third, if the apparent displacements from two non-parallel markers can be measured, the slip direction can be estimated. We apply the methods to calculate the extensions due to the normal faults of San Miguelito in Mesa Central, Mexico. The results indicate that the largest fault strain reaches ca. 0.50 and the smallest fault strain is ca. 0.08. Also, the isolated faults show more regular strain profiles along the fault strikes than the faults with overlapping or intersecting 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 the whole active period of the fault. This definition indicates that displacement is a total slip or total true displacement. Displacement also represents the variation in position of a marker displaced by the fault movement (Tearpock and Bischke, 2003). In the light of this concept, displacement is an apparent displacement. Previous work did not distinguish a true displacement from an apparent displace- ment (e.g. Dawers et al., 1993; Clark and Cox, 1996). In this paper, we use the term “true displacement” (S t ) by following Walsh and Watterson's definition. Therefore, the true strike displacement (S th ) refers to the component of S t along the fault strike. True dip displacement (S td ) refers to the component of S t 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 not parallel to that of the fault. Third, faults are not absolute dip-slip or strike-slip faults. For all cases above, it is necessary to establish a quantitative 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 or near the fault (e.g. Billings, 1972; Suppe, 1985; Doblas et al., 1997a, b). Recently, there has been some work for quantitatively determining the fault true displacement (e.g. Rouby et al., 2000, Xu et al., 2004a; Xu et al., 2007). Billi (2003) analyzed the components of fault slip and separations generated by cleavage-controlled fault zone contraction, on the assumption that shortening occurs perpendicularly to solution cleavages. The methods by Xu et al. (2004a) are appropriate only for the faults on subsurface maps. The approaches by Xu et al. (2007) consider only data measured from cross-section perpendicular to the fault strike or from map view. These methods need more assumptions than the approaches that we introduce here. In this paper we quantify the magnitude of true displacement and the direction of fault slip on faults. The approaches introduced here can be applied to data measured along arbitrary lines on the fault plane, which are more general than methods proposed by Xu et al. (2007). This paper consists of two parts. The first part is to establish equations for obtaining the magnitude of true displacement, the direction of fault slip, or both, according to the available data. The second part gives an example of how to calculate the strain due to faulting. In most cases, the accurate strain is difficult to obtain if the Tectonophysics 467 (2009) 107–118 ⁎ Corresponding author. Tel.: +52 442 2381116. E-mail address: sxu@dragon.geociencias.unam.mx (S. Xu). 0040-1951/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.tecto.2008.12.004 Contents lists available at ScienceDirect Tectonophysics journal homepage: www.elsevier.com/locate/tecto