Estimation of average to maximum displacement ratio by using fault displacement-distance proﬁles S.-S. Xu ⁎, A.F. Nieto-Samaniego, S.A. Alaniz-Álvarez Centro de Geociencias, Universidad Nacional Autónoma de México (UNAM), Blvd. Juriquilla No. 3001, Querétaro 76230, Mexico abstract article info Article history: Received 11 March 2014 Received in revised form 22 August 2014 Accepted 23 August 2014 Available online 16 September 2014 Keywords: Average displacement Displacement proﬁle Simulation Fault linkage Fault displacement is an important factor in the study of discontinuous deformation. Considering that the values of average displacement (D av ) and maximum displacement (D mx ) are linearly related by D av = ρD mx , we calculate the values of ρ estimated from 205 published displacement-distance proﬁles. The following results are obtained: (a) the value of ρ is largest for the mesa-type or ﬂat-topped (M-type) proﬁles; (b) the value of ρ increases when ductile (continuous) deformation is added to the displacement proﬁle; (c) generally, the value of ρ for a linked fault array is smaller than that for segmented faults in the array, i.e., the value of ρ changes with fault evolution, and at the stage where linkage occurs, the value of ρ becomes smaller; (d) the simulation results indicate that for an ellipse function, the value of ρ varies from 0.667 to 0.785. For trapezoid (M-type) pro- ﬁles, the value of ρ is from 0.5 to 1, depending on the ratio of the upper base to the lower base. For best ﬁt poly- nomial curves, the value of ρ can be less than 0.5; (e) the values of ρ more frequently observed in the published proﬁles are between 0.6 and 0.7; the average value is 0.6023 and the standard deviation 0.1123. These data indicate that the displacement-distance proﬁles are hybrids from the triangular proﬁle to the elliptical or mesa proﬁle. The average value (0.6023) would be useful to determine the average displacement in cases where not enough displacement data can be obtained. Finally, the value change of ρ with fault evolution can be used to quantitatively evaluate the level of interaction between segmented faults. © 2014 Elsevier B.V. All rights reserved. 1. Introduction Displacement along faults is a major kinematic parameter of upper crustal deformation. Fault slip (displacement) data are necessary to infer both the orientations and relevant magnitudes of the principal stresses and the principal strain rates. Given a known average displace- ment, it is possible to quantify the fault-related strain (e.g., Molnar, 1983; Scholz and Cowie, 1990; Žalohar and Vrabec, 2008). Displacement-distance proﬁles along faults can provide insight into fault-growth history and information regarding fault interaction (e.g., Gupta and Scholz, 2000; Manighetti et al., 2001; Peacock and Sanderson, 1991). The ratio of the average displacement to the maxi- mum displacement is dependent on the shapes of displacement- distance proﬁles (Marrett and Allmendinger, 1990). When evaluating a seismic moment or a geometric moment, a linear relationship between the displacement averaged over fault surface (u av ) and the maximum displacement of a fault (u mx ) is given as u av = cu mx , with the value of c equal to 0.5 as a rough estimate (Marrett and Allmendinger, 1990). Based on the semi-elliptical displacement proﬁle, Olson (2003) presented a linear relationship of D av =(π/4) D mx , where D av is the average displacement and D mx is the maximum displacement. The coefﬁcient π/4 has been used to simulate the volumetric ﬂow per unit width that is normal to the direction of the ﬂow through fractures (Klimczak et al., 2010). Fault displacement is the change in position of a marker or horizon caused by fault movement. The displacement data can be commonly obtained from ﬁeld measurements or seismic reﬂection proﬁles. The ac- curacy of the observed displacement distributions on a fault strongly depends on the outcrop condition, the number of markers, and the measurement method. The aim of this paper is to determine the most common values of the ratio of the average-maximum fault displace- ment (ρ = D av /D mx ). To accomplish this, we investigate the effects of proﬁle shapes, ductile deformation, and fault linkage on the value of ρ, as calculated from the known, published displacement proﬁles. In addi- tion, the more common geometries of displacement-distance proﬁles are individually investigated. In this paper we estimated the values of ρ based on the published displacement-distance proﬁles (Section 2), and analyzed the values of ρ obtained from synthetic displacement proﬁles using trapezoidal, ellip- tical, and polynomial curves (Section 3). In the Section 4, the histogram of values of ρ from 205 representative proﬁles is obtained and the average value of ρ is calculated. Finally, in Section 5, we discuss the effects of fault interaction and off-fault damage on the value of ρ. Tectonophysics 636 (2014) 190–200 ⁎ Corresponding author. E-mail address: sxu@geociencias.unam.com (S.-S. Xu). http://dx.doi.org/10.1016/j.tecto.2014.08.023 0040-1951/© 2014 Elsevier B.V. All rights reserved. Contents lists available at ScienceDirect Tectonophysics journal homepage: www.elsevier.com/locate/tecto