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_Journal of Structural Geology 32 (2010) 1978e1995_ _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 II: Data evaluation and test of the 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 use the theoretical relations developed in Part I of this work to evaluate the self-consistency of fault-length and fault-displacement data gathered in domains of one and two dimensions from the Yucca Mountain area and from the coal?elds in south Yorkshire, U.K. These data sets are not all self-consistent. For the Yucca Mt. area, the theory shows that the volume over which the sampling of the faults must occur should have a horizontal width no smaller than 2.4 times the horizontal length of the largest fault and a depth no smaller than 1.6 times the vertical extent of the largest vertical-equivalent-fault. It also shows that the volumetric extension must be !95_ of the extension of a two-dimensional domain and !80_ of the extension of a one-dimensional domain. The theory successfully accounts for the observed cumulative extensional strain derived from fault-displacement data from a one-dimensional sampling domain at Yucca Mt., Nevada, U.S.A. Faults up to about four orders of magnitude smaller than the largest fault make a significant contribution to the strain. The most robust calculation of cumulative fractional strain requires the parameters inferred from sampling displacement in a one-dimensional domain. This sampling procedure therefore provides the most reliable results._ _? 2010 Elsevier Ltd. All rights reserved._ _1. Introduction_ _In the companion to this paper (Twiss and Marrett, in this issue, referred to as Part I), we develop the theory by which we can infer the extensional and shear strain in one-, two-, and three-dimensional domains that have been deformed by brittle faulting. The inference requires a knowledge of an independent set of parameters that define the fault systematics and the measurements of either the length of, or displacement on, the largest fault in the domain._ _In this paper (Part II), we apply the results of the theory to examine the fault-length and fault-displacement distributions from detailed one- and two-dimensional sampling of the Yucca Mountain area of southern Nevada, U.S.A. and from the coal?elds of southern Yorkshire, U.K. (Watterson et al., 1996). We review the results of these studies in Section 2._ _From the theory, we can calculate constraints on how large the sampling domain must be relative to the largest fault in the sample in order that we may include all the measured faults in this type of strain analysis. Conversely, we can calculate constraints on the size of the largest fault that can be included in a strain analysis of this type for a given size of sampling domain. We illustrate these constraints for the Yucca Mountain and south Yorkshire areas in Section 3._ _We then use the results of the theory to test the self-consistency of the different data sets from these two areas (Section 4). The theory provides equations that relate the parameters in the equations that define the fault-length vs. fault-displacement systematics, and that define the cumulative frequencies of faults as a function of length and displacement for one-, two-, and three-dimensional domains. These equations permit quantitative tests for consistency among different data sets from the same area. We show that the different data sets from the same area commonly are not self-consistent, and that at least for the Yucca Mountain area, the most reliable measurements derive from one-dimensional sampling of fault-displacement. We conclude that some sampling bias distorts the results from two-dimensional sampling._ _Determination of strain in one- and two-dimensional domains does not permit an exact determination of the three-dimensional strain, and the theory lets us calculate the lower bounds that such measurements impose on the volumetric strain. In Section 5 we illustrate these constraints for the Yucca Mountain area._ _We then test the theory against a data set from a detailed one-dimensional sampling of the Yucca Mountain area and show that it successfully predicts the fractional cumulative extensional strain as a function of fault-displacement (Section 6)._ _R.J. Twiss, R. Marrett Journal of Structural Geology 32 (2010) 1978e1995_ _In Section 7, we calculate for both these areas the strains contributed by the largest fault and compare that with the total strains. For the Yucca Mountain area, we calculate the cumulative strain as a function of displacement and show that faults up to several orders of magnitude smaller than the largest fault in the area contribute significant amounts to the total strain._ _For ease of reference, we collect in Table 1 the equations from Part I that are referenced in Part II._ _2. Empirical determination of parameters_ _The equations we have derived in Part I provide specific relations among the parameters that characterize the power-law relationship between fault-length and fault-displacement, and the frequency distributions for both of these fault attributes. We also derived specific relations among the parameters in the equations describing the frequency distributions for sampling domains having different numbers of dimensions. These equations can be applied to infer strain accurately only if we know the values for these parameters and if the measurement techniques used to evaluate the parameters do not include any systematic biases that would limit the accuracy of the sampling. In this section we review two studies that provide some of the best available data for determining the parameters in the equations for fault systematics and for testing the self-consistency of measurements made on fault-length and fault-displacement in one- and two-dimensional domains._ _2.1. Evaluation of the parameter p_ _Fig. 1 shows a log-log plot of fault-length vs. fault-displacement for nine different data sets from Clark and Cox (1996); it also includes one set from the Yucca Mountain area of southern Nevada, U.S.A. (compiled by Simonds et al. (1995); symbols labeled ‘Y’ in legend), and one set for the northeast-striking faults in the East Pennine coal?elds of south Yorkshire, U.K. from Watterson et al. (1996, their fig. 13f; symbols labeled ‘W’ in legend). We discuss the latter two data sets separately below._ _Clark and Cox (1996) performed a statistical evaluation of the parameters in Eq. (I:3.1.2) relating fault-length to fault-displacement for nine different data sets from different areas. They concluded that individually, the data sets have slopes p in Eq. (I:3.1.2)2 that, at the 95% confidence level, are statistically indistinguishable from_ _p ? 1:0_ _(2.1)_ _(Fig. 1, Table 2, Clark and Cox, 1996, their table 1). Taken all together, the data span a range of more than six orders of magnitude in both length and displacement, although individually, the sets each span a range of at most only about 2 orders of magnitude. The best-fits to the individual sets of data actually give different values of p ranging from 0.63 to 1.4 (Clark and Cox, 1996, their table 1). Yield et al. (1996), in their review of subsurface data, find values between 1.0 and about 2.0, with most results lying between 1.0 and 1.4 (their table 1 and Fig. 10C). For most of the individual data sets reviewed by Clark and Cox (1996), however, the span of values and the scatter of the data points are such that the slope is poorly defined, and the fact that a slope of p ? 1.0 provides a good fit to each of the data sets individually, and to all of the data collectively suggests that this is a reasonable value to use in a general application of Eq. (I:3.1.2) to faults. According to Cowie and Scholz (1992), the value of p in Eq. (2.1) is expected theoretically for faults that originate by a fracture mechanics mechanism with residual friction._ _Analysis of the nine data sets combined, also shows that the data cannot be fit well with a common intercept (?log B); the range of_ _Table 1 Equations from Part I referenced in Part II._ _Equations from PartI A?i? ? ?l?i?L?i??L?i? ? l?i??L?i??2_ _l?i? h?LA?i??i??2_ _l?i?, ?, 1Ll??ii??_ _or, l?i?, ?, p l?i? 4 L?i?,_ _Lp ? Bd_ _p log L ? log B ? logd d_ _, 1 LpB log d ? ?log B ? p logL N?z;v??L? ? G?z;v?L?mz_ _log N?z;v??L? ? log G?z;v? ? mz log L f?z;v??L? ? g?z;v?L?mz_ _log f?z;v??L? ? log g?z;v? ? mz logL f?z;v? ?L?hN?z;v? ?L?_D?z;v?_ _g?z;v? hG?z;v? _D?z;v?_ _D?3?hV ? T WH_ _D?2;th?hAh ? T W_ _&_, "'_ _D?1;v? hT W_ for v ? t for v ? w_ _N?z;v??d? ? G?z;v??Bd??mz_p_ _logN?z;v? ?d?, _, log?G?z;v?, B?mz_, _p_, _, mz p, log d_ _f?z;v??d? ? g?z;v??Bd??mz_p_ _log f?z;v??d?, _, log?g?z;v?, B?mz_, _p_, _, mz p, log d_ _r?1;t?, _, r?3? lB2_p, cos q ?s3, s3 ? 2_p?, 1 s2 ? s3 ? p_ _r?2;th?, _, r?3? mB1_p, sin r, s3, s3 ? 1_p_ _s1 ?, m1 p_, _m2 ? 1 p_, _m3 ? 2 p_ Ключевые слова: values, calculated relationship, yucca mountain, good, fault systematics, equation, log, cumulative-frequency measured, data set, slope, south yorkshire, marrett, mat, strain determined, one-dimensional, sin, frequency distribution, test, scaling systematics, applied, one-dimensional sampling, displacement data, satisfy, normal fault, largest vertical-equivalent-fault, orthogonal distance, displacement distribution, journal structural, systematics, down-dip width, three-dimensional strain, fault-displacement, plot, gray, fault intersected, marrett journal, elsevier, mutually consistent, gure number, clark cox, fault, self-consistency test, displacement measured, excellent, total extension, three-dimensional sampling, small fault, theoretical, cumulative, equality hold, cumulative-number measured, fault displacement, inset legend, size, fault-length, ?p b?, twiss, curve, order, extension, magnitude smaller, data digitized, gray dashedotedot, reasonable, one-two-, southern yorkshire, measurement, frequency, fault strike, mountain area, equation describing, dimensional sampling, largest fault, length data, eqs, clark, two-dimensional, iii, yielding, intercept, theoretical relation, b? th?, constraint, distribution, calculated distribution, yucca, displacement, relation, sampling domain, close, calculated, sanderson, observed, journal structural geology, journal, vertical extent, horizontal length, parameter, volumetric extension, linear, observed displacement, table, dimensional, cumulative strain, southern nevada, theoretical prediction, strain symbol, volume, data, smaller fault, self-consistent, empirical, th?, graphical construction, observed data, displacement systematics, variable, horizontal width, point, strain analysis, dimensionality, determined, self-consistency, calculate, consistency test, large range, structural geology, transect length, area, ?nd, solid, sciences, result, strain, consistency, fault-length distribution, two-dimensional domain, maximum displacement, structural, dimension, ? ?p, p b, fault-displacement data, yorkshire, length, twiss marrett, applies, fault-displacement systematics, magnitude, watterson, one-dimensional domain, acceptable, fault length, strike-slip fault, ii, fault-displacement distribution, walsh, eq, log-linear, dimensional domain, ne-striking fault, systematic bias, geology, largest, volumetric strain, set, total, faulted domain, parameter describing, total strain, logelog plot, type, maximum throw, smaller, extensional strain, measured, parameter inferred, shear strain, two-dimensional sampling, consistent, ne-striking set, strain contributed, plotted, cox, theory, best-?t, data distribution, south, twodimensional domain, values minimize, original data, normal, domain, mountain, nevada, parameter dening, analysis, sampling, shear