A resampling approach to test stress-ﬁeld uniformity from fault data

SUMMARY Several methods have been proposed to constrain the stress ﬁeld from fault plane orientations and slip directions within a crustal volume characterized by brittle deformation. All the methods are based on the assumption that the stress ﬁeld is uniform in the volume considered. If this hypothesis is not checked in advance, however, the methodology may lead to misleading conclusions. In this work, a procedure is deﬁned to check stress-ﬁeld uniformity by a statistical analysis of the available fault data. Since, in most cases, the statistical features of the uncertainties that a V ect such data are not well known, a distribution-free approach is proposed. It is based on a simple search algorithm, devoted to selecting stress conﬁgurations compatible with available data, combined with a bootstrap resampling approach. The test results are more conservative than the ones so far proposed in the literature. When the test allows stress heterogeneities to be safely excluded, approximate conﬁdence intervals for the principal stress directions can be obtained; otherwise, the level of stress heterogeneity present in the volume under study can be assessed. An application of the proposed procedure to a sample of fault data deduced from seismological data is presented.

INTRODUCTION the assumption that slip direction corresponds to the direction The definition of the tectonic stress field is of great importance of resolved shear stress on the fault plane seems to be physically for geodynamic and seismic hazard studies. In situations where plausible and reliable in most cases. On the other hand, as a direct measurements are lacking, fault data (i.e. fault plane result of the complexity of tectonic processes, the reliability of orientations and slip directions) can be used to constrain the the stress uniformity assumption is problematic in many cases. regional deviatoric stress field. However, since the unknown Thus, in order to avoid misleading conclusions, the reliability mechanical heterogeneities of crustal rocks are the most of this assumption should be carefully checked for each case important factor in controlling fault kinematics, the relationship studied. To this end, two procedures have been proposed, by between the fault geometry and stress responsible for faulting Wyss et al. (1992) and Yin and Ranalli (1993). In both cases, is not unique (McKenzie 1969). Thus, to deduce stress from assumptions are required about the statistical properties of fault geometries, a number of additional assumptions are the uncertainties that affect observed slip directions and fault necessary (see, e.g. Michael 1984;Gephart and Forsyth 1984; orientations: the average values of such uncertainties in the Caputo and Caputo 1988; Angelier 1990; Yin and Ranalli first approach; and the form of the relative parent probability 1993;Choi 1995;Yin 1996).
distributions in the second. However, in most cases, the experi-The first assumption is that, on each fault plane, slip occurs mental uncertainties that affect individual fault geometries and in the direction of the shear stress resolved on the same plane.
relative parent distributions are not well known. It is therefore Furthermore, it is assumed that it is possible to select a crustal difficult both to make a careful check of stress-field uniformity volume in which the unknown stress field is uniform. Faults and to assess reliable confidence intervals for the final results within this volume are thus analysed in order to identify the of stress-field inversion. The aim of the present work is to regional stress field. In particular, the 'best-fitting' stress con-develop a simple algorithm to overcome these difficulties by figuration is sought which minimizes any conventional 'loose' allowing a check on the stress-field uniformity hypothesis without restrictive assumptions about the experimental uncertainties. function representative of angular differences between resolved shear stress on the considered fault planes and observed slip.
In the following, only the principal stress directions will be of concern, and no attempt will be made to obtain information A discussion concerning the statistical and numerical properties of the various 'loose' functions and search procedures so far about other important parameters of the stress tensor (e.g. the stress ratio, etc.). proposed can be found in Yin and Ranalli (1993).
The case of negligible experimental uncertainties is con-the fault geometries within the volume of interest are compatible with these principal stress directions, they constitute a sidered first, and a simple distribution-free test is developed on the basis of geometrical and probabilistic considerations.
'stress-field solution' representative of the actual stress field in the volume under study. This result is the basis of the 'right Then, to take experimental errors into account, the test is combined with a numerical resampling procedure which allows dihedra' approach proposed by Angelier and Melcher (1977) for the graphical inversion of the stress field from faulting data, the hypothesis of stress-field uniformity to be checked and the confidence intervals for principal stress directions to be and can be used as a simple test for stress-field uniformity: in fact, the lack of a stress-field solution implies that stress is not approximated.
uniform in the volume under study. However, this condition is necessary but not sufficient for stress-field uniformity. A A DISTRIBUTION-FREE TEST FOR finite probability exists that, even when the stress field is not STRESS-FIELD UNIFORMITY uniform, at least one stress field solution will be found by chance. To demonstrate this, we consider the case of N faults The case of negligible experimental uncertainties within a given crustal volume. In the general case that the stress field is not uniform, the volume can be considered to be In the assumption of shear faulting, fault geometry is fully composed of K subdomains, each characterized by a uniform described by two perpendicular vectors U and N ( boldface is stress field significantly different from those active in the other used in the following to indicate vectors), which represent, subdomains. respectively, the slip direction and the normal to the fault We assume that the stress field active in the ith subdomain plane. An alternative description can be supplied by the use of is responsible for n i of the available fault geometries. A tentative the two axes T and P related to U and N by the following direction for the unknown principal stress axis s 1 (or s 3 ) is vectorial relations: considered. In the case that it corresponds to the actual s 1 (or s 3 ) axis in any ith subdomain, the considered direction lies T= 1 √ 2 ( U+N ) , in the P (or T) dihedron corresponding to each of the n i faults in the ith subdomain. However, since the P (or T) dihedron of each fault includes 50 per cent of all possible directions, a P= 1 (1) probability of 0.5 exists that, by chance, the explored direction is also compatible with the N−n i faults that belong to different (see, for example, Jost and Herrmann 1989). In practice, the T stress domains. Thus, the probability that all the faults in the and P axes bisect the four solid angles (right dihedra) defined crustal volume under study are compatible by chance with the by two perpendicular planes: the 'principal' one, normal to N, considered direction despite the stress-field inhomogeneity is and the 'auxiliary' one normal to U. 0.5N−n i . The two opposite right dihedra including the T axis are said to be dilatational (or T) and the remaining ones are said to be This probability becomes vanishingly small as N approaches compressional (or P). Eq. (1) shows that interchanging U and reasonable sample sizes (say, >20). Thus, it appears that an N does not affect the directions of the T and P axes. This 'apparent' stress-field uniformity can be safely excluded for any makes the description of fault geometries in terms of the P practical purpose. However, if the search for possible stress and T axes less informative, but particularly useful in the case axes is performed by exploring a number M ( large) of possible of seismic fault plane solutions where U and N cannot be solutions, the probability P(H 0 ) that at least one direction discriminated by seismological data alone. Furthermore, the T compatible with the whole set of fault data will be found by and P directions are physically meaningful, being representative chance could become significant. In fact, it holds that of the principal strain axes (e 3 and e 1 , respectively) locally accommodated by the fault displacement (Marrett and (3) shows that P(H 0 ) could become significant for reasonable responsible for the slip may lie everywhere in the compressional sample sizes N given that M is sufficiently large. dihedra, while the principal direction s 3 is located somewhere Various inhomogeneity patterns can be explored with eq. (3). in the dilatational dihedra along a direction perpendicular to In the extreme case that no explored direction corresponds to s 1 . In particular, any of the active stresses (n i =0), eq. (3) becomes If an exhaustive search is performed over a sufficiently dense grid, it seems more realistic to assume that all stress configurations where the dot indicates the scalar product of the relevant that exist in the K subdomains are actually explored. In this vectors (eigenvalues of the stress tensor are not considered).
case, eq. (3) assumes the form When relationships (2) hold for a given fault geometry, we can say that this fault geometry is compatible with a stress P(H 0 (5) field represented by the principal stress axes s 1 and s 3 . If all As expected, in the case of a uniform stress field (K=1 in eq. 5), the probability of finding at least one stress-field solution is unity. The case of the highest level of stress-field complexity that can actually be detected corresponds to the presence of K different subdomains, one for each available fault geometry (K=N, n i =1 and K<M). In this case, eq. (5) becomes Some numerical results showing the dependence of P(H 0 ) on N and M are given in Fig. 1. These results indicate that strong stress-field heterogeneities can be easily detected when a reasonable data set (say, N>10) is available. This is not true, however, when low levels of heterogeneity in the stress field are of concern. To illustrate this, we note that a low level of stress-field complexity can be associated with the presence of only two stress subdomains in the volume under study. In this case (K=2 in eq. 5), it can be easily only, coexist in the volume under study with a main stress field responsible for N−n events. How rapidly this convergence occurs as N increases is shown in Fig. 2 for K=2 and n=1, which corresponds to the extreme case of 'least' detectable of stress heterogeneity: this probability is the significance level corresponding to the hypothesis H 0 that stress field is hetero-heterogeneity; that is, the lowest level of stress-field complexity.
These results can be used to check the hypothesis of stress-geneous. As an example, if at least one stress-field solution is found after a grid search carried out over 200 possible directions field uniformity when experimental errors can be considered negligible. The procedure as follows. Taking into account N by taking 20 fault geometries into account, the hypothesis that a high-level stress-field heterogeneity exists in the volume fault geometries, an exhaustive search is performed over a dense grid of M possible stress directions. If no stress-field under study (see Fig. 1 and eq. 6) can be safely excluded at a 10−3 significance level. solution is found, the hypothesis of stress-field homogeneity can be safely excluded. In the case that at least one solution is found, eqs (3) to (6) can be used to compute the probability General case: taking experimental uncertainties into that such a solution has been found by chance in the presence account In the previous discussion, the presence of at least one stressfield solution has been considered as a necessary condition for uniformity. On this basis, sufficiency conditions for stress-field uniformity have been analysed by estimating probabilities of 'apparent' uniformity arising by chance for different levels of actual heterogeneity in the stress field. These probabilities do not take into account random fluctuations induced in the data sample by random experimental errors, which result in the apparent 'displacement' of P and T directions from the actual ones. In this situation, it is possible that, by chance, 'apparent' stress homogeneity could arise from a heterogeneous stress field or 'apparent' uniformity could result from a heterogeneous stress field. Thus, the presence of at least one stress-field solution is neither a sufficient nor necessary condition for stress-field uniformity. As tentatively shown in Appendix B, the case of apparent uniformity induced by experimental errors seems to be less important, at least for a reasonable dimension of the data set, ability that at least one stress-field solution is found by chance under the relevant hypothesis of stress heterogeneity. This By using the bootstrap technique, eqs (3) to (7) can be used to check stress-field uniformity in cases in which unknown experi-reasonable conjecture allows a check to be made of stress-field uniformity when unknown experimental errors are present. mental errors affect the data. The starting set comprises N fault geometries, sampled from a parent multivariate population Let us assume that a number L of fault samples are available, each representative of the same active stress field. A grid search characterized by unknown statistical properties. By randomly resampling from the original data set, L (a large number) new of possible stress solutions is performed for each sample, and Q samples characterized by at least one stress-field solution samples are obtained, each characterized by the same parent population and representative of the same stress field. For each are found. By eqs (3) to (6) the 'upper bound' of probability P(H 0 ) that at least one stress solution is found by chance is sample, a grid search is performed and the eventual presence of at least one stress-field solution is checked. If Q such cases computed under any hypothesis H 0 about the stress-field heterogeneity level. The probability that, by chance, a number are actually found, eq. (7) can be used to check the significance level associated with any hypothesis H 0 of stress-field Q out of L examined samples are characterized by at least one stress-field solution under any hypothesis (H 0 ) about the heterogeneity. stress-field complexity level is simply found by the binomial equation Approximate confidence intervals for principal stress directions The bootstrap procedure described above can also be used to approximate possible confidence intervals for principal stress In the case that such a probability is lower than a threshold directions in cases when stress-field heterogeneity can be a, that specific level of stress-field complexity can be excluded excluded. at a significance level a. If this condition is not satisfied, stress-By following the approach described above, L sets, each field heterogeneity of the explored type cannot be excluded, with N fault geometries, are obtained by randomly resampling no matter whether it is 'true' or 'apparent'.
(with replacement) from the original data set, also comprising The test becomes more effective as the sample L becomes N fault plane solutions. For each set, M possible directions larger. Thus, the application of such a procedure to check are evaluated as possible principal stress directions compatible stress-field uniformity requires a large number of data sets, with the relevant set of resampled fault plane solutions. Out each resulting from the same stress field and characterized by of the L runs, the mth direction resulted in a possible principal random experimental variations sampled from the same parent stress direction r(m) times. If L is large, the ratio r(m)/L can be population. Of course, with both the actual stress field and considered an estimate of the probability P that the direction statistical features of the experimental errors being unknown, m actually represents a principal stress direction compatible such data sets cannot be realistically drawn from experimental with the original data set. campaigns. However, as first proposed by Michael (1987) a Thus, the set of those directions m such that bootstrap resampling procedure can be used to obtain such r(m)/L >a (8) data samples from the only available data set.
The bootstrap method (see, e.g. Efron & Tibshirani 1986) is approximates the 1−a confidence interval for the inferred based on the assumption that realizations of a random variable principal stress axis. contain all the necessary information about relevant parent Confidence intervals defined in this way include a number probability distribution. This assumption is corroborated by a of possible stress-field solutions, each characterized by the same fundamental result of mathematical statistics which implies reliability. Since no 'best fit' criterion has been introduced, it that an empirical distribution function is the maximum likeliis not possible to select any particular solution as the 'best' one. hood estimator of the parent probability distribution of the sampled random variable. It can be shown that, if the experi-A CASE STUDY mental data set of n elements is sufficiently extended to be representative of the parent population, each new data sample The approach described above to check stress-field uniformity obtained by randomly extracting (with replacement) n values and to approximate confidence intervals for inverted principal from the original set is characterized by the same statistical stress directions has been applied to the data set reported properties as the original sample.
in Table 1 for the hypothesis that unknown experimental Several numerical procedures can be adopted to obtain errors affect the data. This sample comprises 25 seismic fault such new samples (Efron 1990). The simplest one requires plane solutions analysed by Cocina et al. (1997) to constrain the arrangement of the original data set (of n elements) in the the stress-field in the western sector of the Etna Volcano form of a sequential array. Then, a pseudo-random number (southern Italy) at crustal depths (h≥10 km). By following the generator (e.g. Press et al. 1992) is used to generate a sequence approach proposed by Gephart and Forsyth (1984) and Wyss of n integers in the range from 1 to n. Each number of the et al. (1992), which is a standard procedure for stress-field random series is used to pick up the corresponding element of inversion (see also Caccamo et al. 1996; Frepoli & Amato the array. The collection of all these elements will represent a 1997; Eva et al. 1997), Cocina et al. (1997 suggest that the 'randomly resampled' data set. This procedure can be iterated stress field in the explored crustal volume is uniform. In order in order to obtain an arbitrary number of data sets, each of n to check this conclusion, the data set in Table 1 has been elements. In a statistical sense, each new sample is a 'clone' analysed using the bootstrap procedure proposed here. of the original data set. The important point is that such a Up to 104 samples have been generated by randomly 'clone' can be realized without any knowledge about parent resampling from the data set in the Table. For each sample, a grid of 253 possible directions for principal stress axes has populations (Efron and Tibshirani 1986). been explored. These directions have been selected by following the procedure in Appendix C in order to ensure a uniform coverage of the equi-area stereonet with an average density of about 5°. For each run, the presence of at least one 'stressfield solution' (see eq. 2) has been checked. The relative frequency of such cases as a function of the number of samples considered is reported in Fig. 3 (pattern A). This frequency tends to stabilize around a value of 0.26 (2632 out of 104 trials) after a few thousand iterations. By using eqs (6) and (7) with N=25, M=253, L =104 and Q=2632, the hypothesis of maximum heterogeneity can be excluded at a high confidence level (P<10−6). However, a low level of heterogeneity cannot be excluded with the same high confidence level. This can be seen by assuming that only two stress subdomains exist, and that one of them is responsible for one fault geometry only ( least-stress heterogeneity). In this case, eqs (5) and (7) (with N=25, M=253, K=2, n=1, L =104 and Q=2632) indicate a probability P(H 0 ) near unity.   below 11 km depth. In this case, the bootstrap procedure (pattern B in Fig. 3) suggests a relative frequency of samples characterized by at least one stress solution higher than 0.5. By using eqs (5) and (7), it can be easily shown that, in this case, stress-field heterogeneity can be safely excluded also in the case of the lowest level of stress-field complexity.
Since the stress field can be now assumed to be uniform, the bootstrap procedure can be used (eq. 8) to approximate 95 per cent confidence intervals for principal stress directions. Results of this analysis after 104 bootstrap trials are given in Fig. 4. Confidence intervals appear to be larger than the ones obtained by Cocina et al. (1997) (Fig. 4), and include the 'best-fitting' solutions obtained by these authors.

CONCLUSIONS
A distribution-free approach for testing stress-field uniformity with regard to fault geometry (plane orientation and slip direction) has been proposed. The test is based on a numerical resampling procedure (bootstrap), which is computationally simple and can easily be implemented in an efficient computer code. Since only fault geometries are considered, the test can be used for both structural and seismic data.
The basic assumption underlying the proposed approach is that fault slip occurs in the direction of shear stress resolved on the fault plane. No detailed knowledge of the uncertainties that affect fault geometries is required, but the data set must be known to be representative of the unknown parent population. All other assumptions, both concerning conventional 'bestfitting' criteria or statistical properties of parent populations of fault parameters, are unnecessary.
The test proposed here tends to be more conservative than other parametric tests (see, for example, Yin & Ranalli 1993), since it assumes stress heterogeneity as the hypothesis to be rejected by testing data: in practice, the stress field is supposed to be heterogeneous and data are used to falsify this hypothesis. This position seems to be more in line with geological information, which suggests strong variations in the crustal stress field as a result of strength heterogeneities at several scales (see, for example, Rebai et al. 1992). Thus, in cases in which insufficient information is available, stress heterogeneity is not excluded a priori, and possible misleading conclusions resulting by faulty assumptions of stress uniformity are avoided. The approach does not allow an 'optimal' stress-field performed over 253 directions (which corresponds to an average grid solution to be found from the available data. This is the result density of about 5°), and the data shown in ). An equal area lower emisphere projection is used. North is used to assess best-fitting solutions and stress ratios when stress upwards and east is to the right. uniformity has been safely assessed. Furthermore, since approximate confidence intervals for the principal stress directions can be obtained by the present approach, these can be used to select the initial positions required for non-linear inversion proposed by Wyss et al. (1992) for detecting stress-field heterogeneities. Furthermore, it has been shown how the present procedures (for example Gephart & Forsyth 1984).
An application of the present approach to real data has approach allows the level of stress complexity in the volume under study to be determined, and how this information can been given. It has been shown that, at least in the case considered here, this approach is more sensitive than the one be used to identify uniform-stress subvolumes. A comparison that M possible dihedra are explored, the probability that, due end, the grid has been defined over the equi-area stereonet projection (Wulff net) of the hemisphere. A unitary Wulff to random fluctuations, at least one of them can collect all P (or T) axes can be estimated by stereonet can be obtained by using the relationships y i =sin[(p/4)−(l i /2)] sin(w i ) (C2) which becomes rapidly small as N increases. As an example, for M=200 and N=15, the probability that 'apparent' (see, for example, Aki & Richards 1980) which reproduces the uniformity is induced by random errors is 0.6 per cent.
hemisphere on a plane circular surface of unitary radius: values x i and y i represent coordinates on the plane corresponding to the ith direction defined by the couple (w i , l i ). A uniform APPENDIX C: GRID FOR THE SEARCH OF coverage of the hemisphere can thus be obtained if x and y POSSIBLE STRESS-FIELD SOLUTIONS are allowed to vary in the intervals It is clear that a grid search performed over the lower x i µ[−1, +1] , hemisphere, allowing the trend (w) and plunge (l) angles to y i µ[−1, +1] , vary in the intervals with uniform step D along both the x and y axes. Directions 1µ[0, p/2] (C1) for the grid search can be easily obtained by converting these positions in space directions using the inverse of eqs (C2), with a uniform angular step d for both w and l does not result in a uniform grid. In fact, the grid elements tend to concen-which is given by trate around the direction normal to the horizontal plane.
w i =tan−1(x i /y i ) , In order to obtain an effective search for stress direction, the areal density of grid elements should be uniform. To this