Interactive comment on “ Bimodal or quadrimodal ? Statistical tests for the shape of fault patterns

Dear editor and authors, This is a technical paper that present a new numeric tool for the analysis of fault/fracture sets. As the authors discuss, it not always easy to separate large (or small) orientation databases into sub-sets, and to identify the type of distribution. Therefore, the proposed method has the potential to be utilized by researchers in the field. In this respect, the paper can be a good contribution. The paper is clearly written and include multiple clear illustrations of synthetic and field data. It can be shorten as suggested below. The authors are strongly urged to revise the paper following the suggestions below. Major comments 1. The paper is written as a statistical manuscript and not as a tool for geologists. The methodology is presented as a black box without sufficient explanation on the rationale behind it and/or the statistical terminology. Below are few examples. a. The “eigenvalues of the 2nd and 4th rank orientation tensors”

This kinematic limitation is inconsistent with field and laboratory observations that 39 document the existence of polymodal or quadrimodal fault patterns, and which produce 40 triaxial strains in response to triaxial stresses (e.g. Aydin & Reches, 1982;Reches, 1978;41 Blenkinsop (stereographic) or equal-area projections of poles to fault planes or great circles. Azimuthal 52 projection methods (hereafter 'stereograms') provide a measure of the orientation 53 distribution, including the attitude and the shape of the overall pattern. However, these plots 54 can be unsatisfactory when they contain many data points, or the data are quite widely 55 dispersed. Woodcock (1977) developed the idea of the fabric shape, based on the fabric or 56 orientation tensor of Scheidegger (1965). The eigenvalues of this 2 nd rank tensor can be used 57 in a modified Flinn plot (Flinn, 1962;Ramsay, 1967) to discriminate between clusters and 58 girdles of poles. These plots can be useful for three of the five possible fabric symmetry 59 classes -spherical, axial and orthorhombic -because the three fabric eigenvectors coincide 60 with the three symmetry axes. However, there are issues with the interpretation of 61 distributions that are not uniaxial (Woodcock, 1977). We address these issues in this paper. 62 Reches (Reches, 1978;Aydin & Reches, 1982;Reches, 1983;Reches & Dieterich, 1983) has 63 exploited the orthorhombic symmetry of measured quadrimodal fault patterns to explore the 64 relationship between their geometric/ kinematic attributes and tectonic stress. More recently, 65 Yielding (2016) measured the branch lines of intersecting normal faults from seismic  66  reflection data and found they aligned with the bulk extension direction -a feature consistent  67 with their formation as polymodal patterns. Bimodal (i.e. conjugate) fault arrays have branch 68 lines aligned perpendicular to the bulk extension direction. 69

Rationale 70
The fundamental underlying differences in the symmetries of the two kinds of fault pattern -71 bimodal/bilateral and polymodal/orthorhombic -suggest that we should test for this 72 symmetry using the orientation distributions of measured fault planes. The results of such 73 tests may provide further insight into the kinematics and/or dynamics of the fault-forming 74 process. This paper describes new tests for fault pattern orientation data, and includes the 75 program code for each test written in the R language (R Core Team, 2017). The paper is 76 organised as follows: the next section (2) reviews the kinematic and mechanical issues raised 77 by conjugate and polymodal fault patterns, and in particular, the implications for their 78 orientation distributions. Section 3 describes the datasets used in this study, including 79 synthetic and natural fault orientation distributions. Section 4 presents tests for assessing 80 whether an orientation distribution has orthorhombic symmetry, including a description of 81 the mathematics and the R code. The examples used include synthetic orientation datasets of 82 known attributes (with and without added 'noise') and natural datasets of fault patterns 83 measured in a range of rock types. A Discussion of issues raised is provided in Section 5, and 84 is followed by a short Summary. The R code is available from http://www.mcs.st-85 andrews.ac.uk/~pej/2mode_tests_Rcode190418. 86  . This is the simplest non-uniform distribution for describing undirected lines, and has 118 probability density 119 where  is a measure of concentration (low  = dispersed, high  = concentrated) and  is the 121 mean direction. To obtain a synthetic conjugate fault pattern dataset of size n we combined 122 two datasets of size n/2, each from a Watson distribution, the two mean directions being 123 separated by 60°. We generated synthetic bimodal datasets with  = 10, 20, 50 and 100 and corresponding to the range of the discrete datasets shown in a). 199 We calculated the 2 nd rank orientation tensor (Woodcock, 1977) for each of the synthetic 200 datasets shown in Figures 3 and 4 (bimodal and quadrimodal, respectively). The eigenvalues 201 of this tensor (S1, S2 and S3, where S1 is the largest and S3 is the smallest) are used to plot the 202 data on a modified Flinn diagram (Figure 6), with loge(S2/S3) on the x-axis and loge(S1/S2) on 203 the y-axis. The points corresponding to the bimodal (shown in red) and quadrimodal (shown 204 in blue) datasets lie in distinct areas. Bimodal (conjugate) fault patterns lie below the 1:1 line, 205 on which S1/S2 = S2/S3. This is due to the S3 eigenvalue being very low (near 0) for these 206 distributions, which for high values of  begin to resemble girdle fabric patterns confined to 207 the plane of the eigenvectors corresponding to eigenvalues S1 and S2 (Woodcock, 1977). In 208 contrast, the quadrimodal patterns lie above the 1:1 line, as S3 for these distributions is large 209 relative to the equivalent bimodal pattern (i.e. for the same values of  and n). The modified 210 Flinn plot therefore provides a potentially rapid and simple way to discriminate between 211 bimodal (conjugate) and quadrimodal fault patterns. Note, however, that the spread of the 212 bimodal patterns in Figure 6a along the x-axis is a function of the  value of the underlying 213 Watson distribution, with low values of  -low concentration, highly dispersed -lying closer 214 to the origin. Dispersed or noisy bimodal (conjugate) patterns may therefore lie closer to 215 quadrimodal patterns (see Discussion below). 216

Underlying distributions 218
To get a suitable general setting for our tests, we formalise the construction of the bimodal 219 and quadrimodal datasets considered in Section 3.1. Whereas the datasets considered in 220 Section 3.1 necessarily have equal numbers of points around each mode, for datasets arising 221 from the distributions here, this is true only on average. The very restrictive condition of 222 having a Watson distribution around each mode is relaxed here to that of having a circularly-223 symmetric distribution around each mode. 224 Suppose that axes x1, … xn are independent observations from some distribution of axes. If 225 the parent distribution is thought to be multi-modal then two appealing models are: 226 'pulling apart' a bimodal equal mixture distribution into two bimodal equal 234 mixture distributions with planes angle  apart, so that it has four equally strong modes. 235 More precisely, the probability density is: 236

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The problem of interest is to decide whether the parent distribution is (1) or (2). 242 243

The tests 244
Given axes x1, … xn we denote by ±̂1and ±̂, respectively, the largest and smallest 245 principal axes of the orientation tensor. S1 and S3 are the eigenvalues of this matrix. We can 246 also define 247 . 248 S1 and S2 are the 2 nd moments of x1, … xn along the 1 st and 3 rd principal axes, respectively, 249 whereas S11 and S33 are the 4 th moments along these principal axes. Therefore, both S1 -S3 250 and S11 -S33 are measures of anisotropy of x1, … xn. 251 Some algebra shows that 252 where 1 and 3 are the population versions of S1 and S3, respectively, and ± and ± are the 254 components of ± in the quadrimodal equal mixture model (2) along its 1 st and 3 rd principal 255 axes, respectively. Then (4) gives 256 and therefore, it is sensible to: 258 reject bimodality for small values of S1 -S3.
(7) 265 The significance of tests (5) or (7) Table 1 gives the p-values and corresponding decisions (at the 5% level) obtained by applying 274 the tests to some synthetic datasets simulated from the bimodal equal mixture model.  Table 3 gives the p-values and corresponding decisions (at the 5% level) obtained by applying 289 the tests to the natural datasets discussed in Section 3.2. For each dataset, the two tests come 290 to the same conclusion, which is plausible in view of Figure 5. Figure 7 shows the fabric 291 eigenvalue plot for these datasets. 292 (Gruinard) has the highest ratio of loge(S1/S2 samples deformed in the laboratory and then scanned by X-ray computerised tomography. 320 The Chimney Rock dataset is probably not orthorhombic according to the two tests, and lies 321 close to the line for k=1 on Figure 7. It is interesting to note that the Chimney Rock data, and 322 other fault patterns from the San Rafael area of Utah, are considered as displaying 323 orthorhombic symmetry by Krantz (1989) and Reches (1978). However, a subsequent re-324 interpretation by Davatzes et al. (2003) has ascribed the fault pattern to overprinting of 325 earlier deformation bands by later sheared joints. This may account for the inconsistent 326 results of our tests when compared to the position of the pattern on the eigenvalue plot. The 327 Central Italy dataset (taken from Roberts, 2007) is very large (n=1182) and the data were 328 measured over a wide geographical area. The dataset lies below the line for k=1 on the fabric 329 eigenvalue plot (Figure 7), which might suggest it is bimodal. However, for fault planes 330 measured over large areas there is a significant chance that regional stress variations may 331 have produced systematically varying orientations of fault planes. (conjugate) from quadrimodal fault patterns. However, we assert that this may not matter: a 342 noisy and disperse 'bimodal' conjugate fault pattern is in effect similar to a polymodal pattern 343 i.e. slip on these dispersed fault planes will produce a bulk 3D triaxial strain. 344 varying from 5 (dark blue) to 10 (yellow). c) Data from a) and b) merged onto the same plot 351 and enlarged to show the region close to the origin. Note the considerable overlap between 352 the conjugate (bimodal) data with the quadrimodal data, especially for  = 5 (dark blue).

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To assess the relative performance of the two tests presented in this paper, we generated 354 synthetic bimodal and quadrimodal distributions and compared the resulting p-values from 355 applying both the S1-S3 and S11-S33 tests to the same data. The results are shown in Figure 9, 356 displayed as cross-plots of p(S1-S3) versus p(S11-S33). While there is a slight tendency for the 357 p-values from the S11-S33 test to exceed those of the S1-S3 test (i.e. the points tend on average 358 to plot above the 1:1 line), neither of the tests can be said to 'better' or more 'accurate'. We 359 therefore recommend the S1-S3 test as simpler and sufficient. 360 361 Figure 9. Eigenvalue ratio plots comparing the relative performance of the two tests 362 proposed in this paper. The red lines denote p-values for either test at p=0.05, and the 363 diagonal black line is the locus of points where p(S1-S3) = p(S11-S33). a) For bimodal synthetic 364 datasets with size (N) varying from 32-360 and concentration () varying from 5-100, both 365 tests perform well and reject the majority of the datasets (p >> 0.05). The p-values for the S11-366 S33 test are, on average, slightly higher than those for the S1-S3 test across a range of dataset 367 sizes and concentrations. b) For quadrimodal synthetic datasets, many of the p-values are < 368 0.05, and this especially true for the larger datasets (higher N, green/yellow