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_Journal of Structural Geology 32 (2010) 1960e1977_ Contents lists available at ScienceDirect Journal of Structural Geology journal homepage: www.elsevier.com locate jsg Determining brittle extension and shear strain using fault-length and displacement systematics: Part I: Theory Robert J. Twiss a,*, Randall Marrett b a Geology Department, University of California at Davis, One Shields Ave., Davis, CA 95616-8605, USA b Department of Geological Sciences, Jackson School of Geosciences, University of Texas at Austin, 1 University Station C1100, Austin, TX 78712-0254, USA Article info Article history: Received 25 April 2009 Received in revised form 2 April 2010 Accepted 12 April 2010 Available online 25 May 2010 Keywords: Brittle deformation Strain Extension Shear strain Fault systematics Fault scaling Abstract We derive the exact equations by which the continuum approximation to the extensional and shear strains can be determined from measurements of fault-lengths or fault-displacements in a faulted domain. We develop the theory by which we can infer the extensional and shear strain in a volume of brittlely deformed crust from an incomplete inventory of the faults. To that end, we use empirical power-law relationships between fault-length and fault-displacement, and the power-law cumulative frequency distribution for each of these variables, for sampling domains of one, two, and three dimensions. The theory 1) defines the relationships among the parameters in these power-laws, which allows the self-consistency of results from fault-length and fault-displacement studies in domains of one, two, and three dimensions to be evaluated; 2) defines constraints on the relative sizes of the sampling domain and the largest fault that can be included in an analysis using fault systematics; 3) shows that extensional and shear strains in faulted crust can be inferred knowing only an independent set of the parameters defining the population systematics plus the magnitude of either the displacement or the length for the largest fault in the domain; and 4) defines the constraints on three-dimensional strain imposed by sampling in one- and two-dimensional domains. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction The sizes of faults span a tremendous range so wide a range, in fact, that no single approach for observation could hope to achieve a complete inventory. It is observed empirically, however, that the abundance of faults as a function of fault-length is described by a power-law relationship, with the frequency of faults increasing exponentially as their size decreases (e.g., Scholz and Cowie, 1990; Walsh et al., 1991; Marrett and Allmendinger, 1992; Cladouhos and Marrett, 1996; Watterson et al., 1996; Marrett et al., 1999; Bonnet et al., 2001). Moreover, fault-displacement decreases systematically as fault-length decreases (e.g. Marrett and Allmendinger, 1991; Clark and Cox, 1996). As a consequence, although small faults individually contribute less than do large faults to strain and other bulk physical characteristics of the faulted volume of rock, their high abundance may compensate for their small individual contribution. Power-law distributions of fault sizes provide both obstacles and opportunities for addressing problems that inherently depend on populations of faults. For example, the magnitude of deformation due to faulting may depend not just on the largest faults, but rather on all faults of all sizes contained in a domain of interest, and available sampling of the faults is commonly insufficient to provide accurate estimates of the deformation. However, if data are lacking on the full range of fault sizes, the phenomenon of fault scaling provides a tool with which we may quantify statistically the contribution of the unobserved categories of faults. For current purposes, we limit attention to scaling of fault-lengths and displacements, as opposed to topological scaling of fault network geometry (e.g., via box counting; Walsh and Watterson, 1993). In this contribution we focus on fault-related strain, but analogous problems include permeability of fractured rock (recently reviewed by Molz et al., 2004), which can limit the rate of fluid flow; and fracture surface area, which can limit the rate of chemical interaction between fluids and rock (Marrett, 1996). A review of empirical results pertaining to fault scaling was published recently (Bonnet et al., 2001), but we have lacked a complete theory by which these empirical results can be compared and evaluated. Previous fragments of theoretical work have been intentionally narrow in order to simplify the analysis, but this also has limited the scope of applicability and in some cases R.J. Twiss, R. Marrett Journal of Structural Geology 32 (2010) 1960e1977 1961 has led to confusion. For example, Scholz and Cowie (1990) inferred that small faults contribute negligible strain, but their conclusions were potentially undercut by having applied fault data sampled in a two-dimensional domain to a three-dimensional problem without stereological correction. In this paper, we begin with first principles and derive a full and systematic theory for the application of fault scaling relations to the inference of fault-related extensional and shear strain for one-, two-, and three-dimensional domains that have been deformed by a large population of homogeneously distributed faults. A brief summary of part of this development for extension in three dimensions was published in Twiss and Moores (2007; Box 16-I, p. 440). We first derive the equations by which the continuum approximation of extensional and shear strain in a domain can be determined from displacements on a set of faults that deform the domain. We then adopt empirical power-law equations that relate fault-length to fault-displacement and that define the cumulative frequency distributions of faults as a function of each of these two variables. Different equations for cumulative frequency apply to data sampled in one-, two-, and three-dimensional domains, and heretofore, the relationships among these equations have not been clear. We show that the parameters in these equations are not all independent, and we derive the relationships among them. Implicit in the use of fault systematics to determine strain in a domain is an assumption that the faults are homogeneously distributed throughout the domain and that the domain is large relative to the size of the faults. We derive the size constraints that define, for a given size of domain, the largest fault that can be included in the analysis, or conversely, the minimum size of the domain that must be used to incorporate a given maximum-sized fault in the analysis. If we know a set of independent parameters that define the fault systematics in a faulted domain, the theory shows that both extensional and shear strains can be inferred from either the displacement on, or length of, the largest fault in the domain. Heretofore, it has not been clearly recognized that determinations of strain based on sampling in one- or two-dimensional domains do not necessarily define the three-dimensional strain exactly; nevertheless, they at least place constraints on it, and we derive equations that specify those constraints. The question, "Are small faults important?" has contradictory answers in the literature (e.g., Scholz and Cowie, 1990; Walsh et al., 1991; Marrett and Allmendinger, 1991, 1992), due in part to theoretical confusion and in part to semantics. In this paper, small faults are defined to be one order of magnitude or more smaller than the largest fault that can be included in the analysis for the domain under study, without regard to absolute scale (Walsh et al., 1991; Marrett and Allmendinger, 1992). We derive the relation between the size of the largest fault and the size of the domain in Section 3.4. In particular, we do not adopt the definition, used for example by Scholz and Cowie (1990), that small faults are those that do not span the brittle crust. Our results show that relatively small faults contribute significantly to the total strain in a brittlely deformed volume. In a companion paper (Twiss and Marrett, in this issue, referred to as Part II), we use the theoretical results to compare and evaluate multiple sets of data from the same domains, and we use empirical data to calculate the extensional strains for these areas and to test the predictions of the theory. Figure and equation references with numbers that begin with a "II:" refer to this companion paper. For ease of reference, all symbols used in the analysis are listed alphabetically in Table 1, along with a definition and the equation number of first use or occurrence that is relevant to the definition. 2. Continuum strains of faulted domains 2.1. Infinitesimal continuum extension from brittle faulting The first problem we address is how to find the continuum approximation to the extension in a specific direction across a faulted terrane. We begin with the result for the calculation of the average infinitesimal constant volume strain tensor that results from the slip on a set of non-rotating faults within a volume V of rock, where V is large relative to the dimension of the largest fault contained in the volume. The strain of a volume V that is contributed by a single fault completely contained within the volume, is (Kostrov (1974), referencing Riznichenko (1965); see also Twiss (2009)). ekl 1 2V M0ВЅhknl Гѕ hlnkВЉ; (2.1.1) where hk and nk are the components of unit vectors parallel, respectively, to the slip direction and the normal to the fault (Fig. 1). The slip direction is defined by t' _ Ключевые слова: e, r, o