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An Objective Criterion for Selection of the Best Option in Hansen’s Method for Paleostress Estimation from Homogeneous Fault-Slip Data

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Abstract

Hansen’s direct linear inversion method is a rapid and robust technique for paleostress estimation from fault-slip observations. The method uses six- and nine-dimensional spaces and requires a user judgement of the best solution from four possible options, i.e., the tensors 6D, 6Df, 9D and 9Df. We propose an objective criterion for selection of the best solution. Using each of the four possible options, as the reduced stress tensor, 6D, 6Df, 9D and 9Df, the criterion computes the slip vector on each fault plane in a given population of homogeneous fault-slip data. The option that gives minimum average angle between the computed slip vectors and the true or observed slip vectors on the corresponding fault planes is the best solution. The criterion is successfully tested on 236 synthetic and 5 natural examples representing a variety of stress states in different tectonic settings. One example of the natural fault-slip observations contains a vorticity component, whereas all other test examples are devoid of vorticity. In the triaxial stress states, two correct solutions are obtained, one of which is either 6D or 6Df, and the other is 9D or 9Df. In the stress states that are characterized by axial compression or axial extension, the correct solution is either 6D or 6Df and the use of nine-dimensional space fails to invert the observations. In the natural example that contains a vorticity component, only 9D gave the correct result.

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(data from Table A1 in appendix V in Angelier 1990)

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Acknowledgements

This study was supported by a grant from the Department of Science and Technology, Government of India to D. C. Srivastava. We are grateful to Rahul Dixit for help and rigorous testing of the method during the revision of the manuscript and to Arun Ojha for the critical reading of the revised manuscript. We are also grateful to the two reviewers for their erudite and constructive suggestions. Carlos Liesa, in particular, provided very useful suggestions that helped improve the manuscript considerably.

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  1. Department of Earth Sciences, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247667, India

    Kartik Bhatnagar, D. C. Srivastava & P. K. Gupta

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  1. Kartik Bhatnagar
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Appendix 1

Appendix 1

Equal-area lower-hemisphere projections of 180 test data sets, simulated by using a known reduced tensor, are shown in Figs. 10, 11, 12, 13, 14, 15 and 16. These data sets are constrained by Byerlee’s law. In addition, 56 test data sets were simulated without imposing Byerlee’s law. Table 4 compares the 4 parameters in the best reduced stress tensors, selected by the minimum average angle criterion, with the parameters in the known reduced stress tensors used in the simulation of the 56 data sets.

Fig. 10
figure 10figure 10figure 10figure 10

1–48 Synthetic data sets for type-1 stress orientations and triaxial states, 0 < Φ < 1. Each data set, containing 15 fault-slip observations, is simulated for a true reduced stress tensor consisting of 4 parameters, the principal stress orientations (within matrix) and the stress ratio, Φ. One of the three principal stress axes is non-plunging in all the data sets

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Fig. 11
figure 11figure 11figure 11figure 11figure 11figure 11

1–64 Synthetic data sets for type-2 stress orientations and triaxial states, 0 < Φ < 1. Each data set, containing 15 fault-slip observations, is simulated for a true reduced stress tensor consisting of 4 parameters, the principal stress orientations (within matrix) and the stress ratio, Φ. All three principal stress axes are plunging in all the data sets

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Fig. 12
figure 12

1–12 Synthetic data sets with their respective stress orientations and stress ratio for Andersonian states. Stress states labelled 1–4, 5–8 and, 9–12 belong to extensional, compressional and strike-slip tectonic settings, respectively

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Fig. 13
figure 13

1–12 Synthetic data sets for type-1 stress orientations and prolate stress ellipsoids, axial compression (Φ = 0). Each data set, containing 15 fault-slip observations, is simulated for a true reduced stress tensor consisting of 4 parameters, the three principal stress orientations (within matrix) and the stress ratio, Φ = 0. One of the three principal stress axes is non-plunging in all the data sets

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Fig. 14
figure 14

1–12 Synthetic data sets for type-1 stress orientations and oblate stress ellipsoids, axial extension (Φ = 1). Each data set, containing 15 fault-slip observations, is simulated for a true reduced stress tensor consisting of 4 parameters, the three principal stress orientations (within matrix) and the stress ratio, Φ = 1. One of the three principal stress axes is non-plunging in all the data sets

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Fig. 15
figure 15figure 15

1–16 Synthetic data sets for type-2 stress orientations and prolate stress ellipsoids, Φ = 0. Each data set, containing 15 fault-slip observations, is simulated for a true reduced stress tensor consisting of 4 parameters, the three principal stress orientations (within matrix) and the stress ratio, Φ = 0. All three principal stress axes are plunging in all the data sets

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Fig. 16
figure 16figure 16

1–16 Synthetic data sets for type-2 stress orientations and oblate stress ellipsoids, Φ = 1. Each data set, containing 15 fault-slip observations, is simulated for a true reduced stress tensor consisting of 4 parameters, the three principal stress orientations (within matrix) and the stress ratio, Φ = 1. All three principal stresses axes are plunging in all the data sets

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Table 4 Results of tests of minimum average angle criterion on 56 such data sets that are unconstrained by Byerlee’s law
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Bhatnagar, K., Srivastava, D.C. & Gupta, P.K. An Objective Criterion for Selection of the Best Option in Hansen’s Method for Paleostress Estimation from Homogeneous Fault-Slip Data. Pure Appl. Geophys. 176, 1731–1766 (2019). https://doi.org/10.1007/s00024-018-2062-z

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