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Journal of Structural Geology 33 (2011) 78e91 Contents lists available at ScienceDirect Journal of Structural Geology journal homepage: www.elsevier.com locate jsg Mechanical analysis of fault slip data: Implications for paleostress analysis J.O. Kaven, F. Maerten, D.D. Pollard Now at U.S.G.S. Menlo Park, Menlo Park, CA 94025, USA IGEOSS, A Schlumberger Company, 34790 Grabels, France c Department of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305, USA article info Article history: Received 21 September 2010 Received in revised form 12 November 2010 Accepted 6 December 2010 Available online 10 December 2010 Keywords: Fault slip data Paleostress Stress inversion Mechanical interaction abstract Stress inversions are a useful and popular tool for structural geologist and seismologist alike. These methods were first introduced by Wallace (1951) and Bott (1959) and subsequent studies continue to be based on their assumptions: the remote stress tensor is spatially uniform for the rock mass containing the faults and temporally constant over the history of faulting in that region, and the slip on each fault surface has the same direction and sense as the maximum shear stress resolved on that surface from the remote stress tensor. Furthermore, successful implementation requires that slip accumulates on faults of diverse orientation. Many studies employ these methods on isolated faults or on fault systems with limited ranges of orientations, which can lead to erroneous results. We propose a new method that incorporates the effects of mechanical interaction of the entire fault or fault system, and solves the complete mechanical problem rather than employing empirical relationships between slip and stress or strain (or strain rate). The method requires knowledge of the fault geometry and information on at least one slip vector component along portions of the known fault geometry. For example, if throw is known, the strike-slip component can be solved for. We test the method using a single synthetic fault with anisotropic roughness similar to that measured at fault outcrops. While the orientation of remote stress may be determined precisely, the lack of diverse fault orientations introduces a systematic error in the remote stress ratio. We further test the effect of diversity of fault orientations and find that WallaceeBott type inversions do not perform as well for limited ranges of orientations when compared to the proposed method. Finally, we use published data from the 1999 Chi-Chi, Taiwan, earthquake, and find that the method using surface data only, and surface data with subsurface focal mechanisms, produce similar results. The resulting stress orientations are in good agreement with results from WallaceeBott inversions. Furthermore, the slip distribution is in general agreement with kinematic slip inversions using coseismic surface deformation. Stress inversion methods using fault slip data can thus be improved upon, significantly in some cases, by solving a mechanical boundary value problem that takes into account the geometry of faults or fault systems. As a bonus, the solution provides the stress, strain, and displacement fields throughout the region and the slip distributions on the faults. ? 2010 Elsevier Ltd. All rights reserved. 1. Introduction Over the course of the 20th Century geologists sought to understand the origin and evolution of faults, and the tectonic history of faulted regions, by relating fault orientation and slip direction to the state of stress in Earth’s crust (e.g. Anderson, 1942; Price, 1966; Voight, 1966; Mandl, 1988). This relationship may be elucidated through both forward and inverse problem solving. In typical forward problems the equations of motion are solved with * Corresponding author. E-mail addresses: okaven@usgs.gov (J.O. Kaven), fmaerten@igeoss.com (F. Maerten), dpollard@standard.edu (D.D. Pollard). 0191-8141 $ e see front matter ? 2010 Elsevier Ltd. All rights reserved. doi:10.1016 j.jsg.2010.12.004 a prescribed remote stress state as boundary conditions, yielding the local stress, strain, and displacement fields, and the slip distributions over the model faults (e.g. Hafner, 1951; Sanford, 1959; Couples, 1977; B?rgmann et al., 1994; Willemse et al., 1996; Maerten et al., 1999). Assumptions about the constitutive behavior, the magnitudes of the strains, and the relative magnitudes of dynamic and static forces (Malvern, 1969, Chapters. 6, 4, and 8, respectively) enable one to reduce the underlying conservation laws to the relevant equations of motion (Pollard and Fletcher, 2005, Chapter 7). While the correspondence of such models to faulting in Earth’s crust depends upon the accuracy of the assumptions, each of which requires careful assessment, the efficacy of the methodology rests securely on the foundation of a complete mechanics (Fletcher and Pollard, 1990). J.O. Kaven et al. Journal of Structural Geology 33 (2011) 78e91 79 In typical inverse problems the directions of the remote principal stresses and a ratio of their magnitudes are constrained by analyzing field data on fault orientations and slip directions as inferred from striations such as slickenlines on exposed fault surfaces (e.g. Carey and Brunier, 1974; Etchecopar et al., 1981; Angelier et al., 1982; Gephart and Forsyth, 1990; Angelier, 1984; Michael, 1987; Reches, 1987; Fry, 1999; Shan et al., 2004). The adoption of this methodology is facilitated by an instructive exposition and computer codes in the textbook by Ramsay and Lisle (2000) and by the availability of other computer codes (e.g. Huang, 1988; Hardcastle and Hills, 1991; Orife et al., 2002). The enthusiastic implementation of the methodology by the structural geology community is witnessed by global compilations of paleostress results from 250 sites for the World Stress Map Project (Reinecker et al., 2004) and from 2791 independently chosen sites (Lisle et al., 2006) for a Special Issue of the Journal of Structural Geology on “New Dynamics in Palaeostress Analysis” (Blenkinsop et al., 2006). The equations of motion are not invoked for this inverse problem, and perturbations of the local stress field by fault slip are ignored. In other words, the mechanical role played by the faults in the tectonic deformation is not included explicitly in the analysis. Instead, two basic assumptions are made: 1. The stress field is spatially homogeneous and temporally constant; and 2. The direction of slip and the direction of the maximum shear stress resolved on each would-be fault plane are coincident. These assumptions enable the inversion, which uses Cauchy’s Formula (Fung, 1977, p. 62) to relate the tangential tractions (maximum shear stresses) on planes with the measured fault orientations to the principal stresses in the corresponding homogeneous stress field. In a remarkably prescient paper, which to our knowledge is the earliest example of paleostress inversion, Anderson (1905) began, without comment or justification, by simply taking one principal stress direction as vertical at any point. This assumption was addressed explicitly 37 years later by Anderson (1942, p. 12 and Chapter VII). In his 1905 paper Anderson suggested that planes carrying the maximum tangential stress “will have much to do with determining the directions of faults in the rock”. He understood that there are two orientations of such planes at any point; that these planes intersect in the direction of the intermediate principal stress; and that they make equal angles of 45 x14 to the greatest principal compressive stress. He extended these relationships for stress at a point to rock volumes encompassing faults and conceived two conjugate sets of would-be faults corresponding to a single state of homogeneous stress. In calculating the resolved tangential stress on the conjugate planes Anderson used a variant of the Cauchy Tetrahedron (Malvern, 1969, p. 73) with one face corresponding to a would-be fault and made an interesting analogy: “This prism we suppose to exist in the rock, somewhat as the statue exists beforehand in the block of marble.” Apparently Anderson understood that slip on an actual fault would perturb the stress from its assumed homogeneous state. We appeal to his analogy of the would-be statue residing in the block of marble and refer to the entire class of inverse problems based on a homogeneous stress state as faultless paleostress analysis. The next stage in the development of faultless paleostress analysis was introduced in the middle of the last century when Wallace (1951) analyzed the maximum shear stress (tangential traction) on planes of arbitrary orientation for a homogeneous stress state using Cauchy’s Formula (e.g. Jaeger et al., 2007, p. 31). He illustrated the magnitude and orientation of this shear stress on stereonets and Mohr diagrams. Appealing to laboratory results and Mohr’s theory (N?dai, 1931, p. 61), Wallace proposed that “faults will tend to concentrate at orientations tangent to a cone, with apex angle less than 90 x14 (45 x14 radius), which has the axis of greatest compressive stress as its axis.” and that “Orientation of net slip on faults can be correlated almost directly with orientation of maximum shearing stress.”. In summary, he suggested that “If a complete picture of fault-plane orientations and net-slip orientations on several faults is available, it should be possible to determine with some degree of certainty the orientation and nature of the stress system producing the faults.” Taking a somewhat different approach conceptually, Bott (1959) contemplated the likely presence of strength inhomogeneity in the form of older faults, joints, and cleavage. Apparently supposing that whatever perturbation in the stress field due to the formation of these structures had relaxed, he suggested “These planes would remain unnoticed until the shearing stress within them should exceed the strength.”. Furthermore, Bott suggested “.frac Ключевые слова: pollard, eld stress, shear stress, atkinson, twiss, yamaji, discontinuity, aef, structural geology, error, cauchys formula, single fault, willemse, heuristic, strain, basic assumption, focal, material property, focal mechanism, mckenzie, test, stress ratio, blenkinsop, segall, plane, fault geometry, remote stress, resolved, triangular element, faultless inversion, angelier, mechanic, surface trace, seismology, carey, surface slip, range, ratio, taiwan, would-be fault, solution, distribution, trace, geometry, rupture, direction, paleostress, data, earthquake, gephart, stress, poissons ratio, rudnicki, fault set, orientation, equation solves, varying orientation, slip maximum, eld data, lin, society, applied, strain rate, mechanism, allmendinger, transactions, inversion, boundary condition, palaeostress analysis, surface data, grif?th, slip direction, slip data, traction, surface, variance bound, entire fault, faulting, principal stress, elsevier, chi-chi, paleostress inversion, condition, mechanics, model, fault, earthquake rupture, strike, diverse orientation, equation, principal strain, journal structural, heuristic model, dip varying, roughness, idealized fault, reches, tensor, slip, limited range, displacement elds, wallace, boundary, maerten, stress orientation, basic, geosciences, martel, shear, homogeneous stress, systematic error, journal, slip distribution, normal, displacement discontinuity, problem, geophysical, youngs modulus, faultless, inverse, lee, london, fault orientation, structural, strike-slip component, int, journal geophysical, seismological society, element, method, kaven journal, chi-chi earthquake, anderson, inversion technique, ue, standard deviation, proposed, eld, geology, displacement, slip magnitude, complete mechanic, maximum, shearing stress, inversion result, stress eld, measurement, fault trace, homogeneous isotropic, direct evaluation, magnitude, lisle, isolated fault, slip vector, journal structural geology, press, varying, model fault, inversion method, fracture, fry, rock, stress inversion, analysis, ?eld, assumption, varied percentage, fault surface, mechanical interaction, tectonophysics, kaven, earth, paleostress analysis, faultless method, rate, proposed method, component, relative vorticity, vector, varying percentage, fault tectonics, science, algorithm, result, principal, surface rupture, rst introduced, chelungpu fault, homogeneous, ha, subsurface geometry, local, active fault, fault slip, stress tensor, remote, complete, bulletin, horizontal, set, zoback, temporally constant, method proposed, vertical stress, inverse problem, fracture mechanics, special issue