# appendix paleostress analytical methods appendix paleostress analytical methods the angelier (1984)

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Minor and others: Oblique Strain Transfer Middle Rio Grande Rift

1

APPENDIX

PALEOSTRESS ANALYTICAL METHODS The Angelier (1984) formulation utilized in this study for computing paleostress tensors is a least squares minimization scheme that aims at minimizing, for each fault measurement, the non-slip shear stress component (i.e., the computed stress component on the fault plane normal to the measured slip direction), leading to an over-determined set of linear equations. Measured strike and dip values of fault planes and associated slickenline rake angles (i.e., angles between slickenlines and fault strike) and slip senses are required as input. As with most other inversion methods, the Angelier algorithm finds a best-fit reduced stress tensor consisting of the orientation of the principal stresses σ1, σ2, and σ3 where σ1 > σ2 > σ3 (compressive stress positive), and the stress ratio φ = (σ2-σ3) / (σ1-σ3). The value of φ, which ranges from 0 to 1 and indicates the relative magnitudes of the three principal stresses, is discussed in more detail below. For the stress tensor computations in our study we used a coded version of the Angelier (1984) direct inversion algorithm within the commercially available fault analysis software MyFault (v. 1.03) (Pangaea Scientific, 2007). The MyFault routine requires that fault strikes are entered using the 0°-360° right-hand convention and entered rakes are measured in the footwall in a clockwise direction relative to the fault strike azimuth. The actual numeric rake value entered, which can range from 0° to 360°, depends on the sense of slip, or specifically the direction of movement of the hanging-wall block relative to the fault strike direction. The advantage of this numeric convention is that both the rake (or pitch) of the slickenlines and the slip sense are revealed in a single number, as illustrated in Figure A1. According to the convention, faults with rakes between 0° and 180° have components of normal slip, rakes between 180° and 360° indicate components of reverse slip, rakes between 270° and 090° indicate components of sinistral slip, and rakes between 090° and 270° indicate components of dextral slip (Figure A1). In addition to fault strike, dip, and rake, MyFault also requires as input the confidence ranking of each slip-sense determination (i.e., C, P, or I – see Methods section in main text). Basic assumptions of fault-slip inversion methods and their implications have been previously addressed (e.g., Angelier, 1984; Zoback, 1989; Pollard et al., 1993). All inversion methods assume that (1) a rock mass in which multiple fault slips are measured was subject to a homogeneous stress state during all measured slip events, and (2) the slip directions measured on the fault surfaces coincide with the maximum resolved shear stress directions on those surfaces. In reality, these assumptions can be violated to varying degrees mainly due to secondary mechanical effects (e.g., frictional anisotropy; fault-block obstructions; variations in slip-surface geometry), overprinted slip events resulting from changing stress regimes, observational errors, and inadequate data discrimination before analysis (Etchecopar et al., 1981; Angelier, 1984; Zoback, 1989; Pollard et al., 1993). In each stress tensor computed in this study, as many diversely oriented, representative faults were used as possible (mostly a minimum of eight slip

2013200 for GSA Special Paper 494, chap. 14

Minor and others: Oblique Strain Transfer Middle Rio Grande Rift

2

measurements) to optimize the tensor definition and to increase the likelihood that observational errors and secondary effects are cancelled out.

R RD

DR

D

DN

ND N

NS

SN

S

SR

RS

270O

0O (360O)

285O

315O

345O

015O

045O

075O090O105 O

135O

165O

180O

195O

225O

255O

fault strike direction

Figure A1

Figure A1. Diagram showing rake numeric convention used in this study and its relation to slip sense. For a given measurement, the assigned numeric rake value corresponds to the angle made by the movement vector of the hanging-wall block relative to the fault strike direction (based on “right-hand” rule); the rake angle can range from 0° to 360°. Typically the hanging-wall direction of movement is determined using slickenlines preserved on the fault slip surface and one or more slip-sense indicators, as described in the main text. Single- and two-letter acronyms within each 30° rake sector correspond to the primary (first or single letter) and secondary slip-sense components for that range of rake values. Slip components: R, reverse; S, sinistral; N, normal; D, dextral. Dark-, light-, and non-shaded regions in the diagram correspond to the same rake ranges as represented by similarly shaded histogram bars shown in the slip-rake frequency plots in Figures 8, 9, 10, and 17. The actual slip direction on a given fault surface may not coincide with the maximum resolved shear stress largely due to the factors mentioned above. Thus, an important measure of data dispersion and fitness of each stress tensor solution is the mean angular deviation (γ) between the measured slip directions and the calculated maximum shear stress directions resolved on the fault planes. The MyFault analysis package computes both the individual and mean γ values for each solution. The smaller the mean value of γ, the better the fit of the stress tensor to the data. In each stress tensor computation conducted for this study we strived for a mean γ value less than about 25°, and individual fault-slip data with γ values of generally more than about 60° were selectively removed

2013200 for GSA Special Paper 494, chap. 14

Minor and others: Oblique Strain Transfer Middle Rio Grande Rift

3

from the data subset and the remaining data were then reanalyzed. In a few cases several iterations of data removal and reanalysis were necessary before acceptable individual and mean angular deviations were obtained. The MyFault software estimates, using a bootstrap resampling method, the orientational error of each of the principal stress axes that is graphically displayed on an equal-area plot using contours representing a preset confidence or uncertainty limit. MyFault also calculates the mean “fault angle” (θ) between the best-fit σ1 axis and the measured fault planes, which is a separate indicator of the quality and reasonableness of each solution. In this study, stress solutions with mean θ values greater than 45° were generally rejected, as such values would suggest that a significant fraction of the analyzed faults slipped under relatively high normal and low shear stresses, which would be unlikely unless such faults had very low friction, were formed under significant fluid pressures, or were otherwise effectively weak. The general quality of the computed φ ratios can be qualitatively assessed by observing the degree of variability of the inverted fault-slip data using equal-area plots of the fault planes and slickenlines and frequency distribution plots of the fault strikes, dips, and rakes. Those φ values determined from fault data having diverse geometries and distributions are generally considered to be more robust than φ values computed from uniform data having little variation in strike, dip, and rake (Celerier, 1988). Robust φ ratios can be used, in turn, to estimate how well constrained or “stable” the stress axis orientations of each solution are. As φ approaches 0 or 1 the relative orientations of the σ2 and σ3 axes or σ1 and σ2 axes, respectively, become unconstrained and prone to permutations where the two axes “flip flop”. In this study we assume that all three computed stress axis directions are reasonably well constrained if: (1) the fault-slip data are numerous (at least 8 measurements) and geometrically diverse, (2) φ > 0.l0 and < 0.90, and (3) the mean and individual values of γ are within the tolerance levels specified above. Only one stress axis direction is well determined if the value of φ is < 0.l0 or > 0.90. Poor-quality stress tensor solutions with large γ values are commonly computed from geometrically and kinematically heterogeneous data sets that result from polyphase (i.e., temporally overprinted) fault slip (Angelier, 1984). In our study, data sets that yielded initial stress tensor solutions with large (>>25°) mean γ values, particularly those solutions resulting in numerous large (>>60°) individual angular deviations (γ), were subdivided and reanalyzed (see following explanation) to obtain multiple, better-fit solutions that were consistent with available field evidence of polyphase slip (cf., Marrett and Allmendinger, 1990). First, all fault-slip data that poorly fit the initial solution (i.e., γ >> 60°) were separated from the initial grouping and combined into a new grouping. The resulting two new subsets were then separately reanalyzed. Depending on the quality of the new solutions, any remaining poorly fit data associated with one of the solutions were reassigned to the other data grouping, and vice versa, and both modified subsets were analyzed again. In a few cases multiple iterations of data separation and recombination and (or) three data groupings were necessary before solutions with adequate fit were obtained. To help guide the data segregation process, consideration was given to any recorded relative-age observations (i.e., fau

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