Iterative direct inversion: An exact complementary solution for inverting fault-slip data to obtain palaeostresses

Abstract

The present study shows that reconstructing the reduced stress tensor (RST) from the measurable fault-slip data (FSD) and the immeasurable shear stress magnitudes (SSM) is a typical iteration problem. The result of direct inversion of FSD presented by Angelier [1990. Geophysical Journal International 103, 363–376] is considered as a starting point (zero step iteration) where all SSM are assigned constant value ($\lambda =\sqrt{3}/2$). By iteration, the SSM and RST update each other until they converge to fixed values. Angelier [1990. Geophysical Journal International 103, 363–376] designed the function upsilon ($\upsilon$) and the two estimators: relative upsilon (RUP) and (ANG) to express the divergence between the measured and calculated shear stresses.

Plotting individual faults’ RUP at successive iteration steps shows that they tend to zero (simulated data) or to fixed values (real data) at a rate depending on the orientation and homogeneity of the data. FSD of related origin tend to aggregate in clusters. Plots of the estimators ANG versus RUP show that by iteration, labeled data points are disposed in clusters about a straight line. These two new plots form the basis of a technique for separating FSD into homogeneous clusters.

Introduction

Recent literature clearly demonstrates the utility of palaeostress estimation to a wide range of fields, e.g. structural geology (Guiraud et al., 1989; Hwang et al., 1991), geodynamical studies (Pollard, 2000), foundation engineering (Amadei and Stephansson, 1997) and to study subsurface fluid flow (Nelson, 1985; Mandl, 2000).

The reconstruction of palaeostresses from microstructures has been discussed by Dunne and Hancock (1994). There is a wide range of such techniques: those which are mainly based on the orientation of faults and their associated slip indicators are by far the most popular. Also, there are several methods for inverting measured fault-slip data (FSD) to reconstruct the incomplete tensor which consists of the orientation and relative magnitudes of the principal stresses (PSA). Those methods are either analytical or based on a searching strategy.

The 4-D exploration method adopted by Angelier, 1975, Angelier, 1979, Angelier, 1984, Angelier, 1991 is an example of a grid search technique and consists of considering numerous tensors and selected the one that minimizes a simple function of the angles between the computed shear stresses on the fault planes and the actual slip vectors. Hardcastle and Hills (1991) also present an objective and non-elegant method for such an inversion. Thousands of tensor configurations are tested against a collection of data in the light of Mohr–Coulomb yield criteria.

The analytical methods are due to the work of Angelier, 1990, Angelier, 1994, Huang and Angelier (1989), Angelier et al. (1982) and Fry (1999).

Several recent works question the validity of inversion methods. Yamaji (2003) questions the correctness of the solutions of stress inversion. He displays graphically, using a special stereogram, the two object functions of Angelier, 1979, Angelier, 1990. These functions measure the angular misfit between the observed and calculated striations on the fault planes. The object function (F) is evaluated in terms of the stress ratio $\varphi$ and ${\sigma }_1$ at different points of the net which also represent the location of ${\sigma }_3$. $\varphi$ varies from zero to one in steps of 0.1, whereas ${\sigma }_1$ rotates about ${\sigma }_3$ from 0–180° in steps of 18°. Finally, these three values F, $\varphi$ and ${\sigma }_1$ are displayed on a stereogram using a color code and different degrees of color saturation and hue. The location of the extreme value of F allows the estimation of the stress tensor. Yamaji (2003) adds that heterogeneous FSD display several extrema. Accordingly, the method can be used for the analysis of polyphase data. Needless to say the discussed method is highly complicated, difficult to apply and the analysis of the data set KAM (Angelier, 1990) seems to be a rather qualitative.

Tobore and Lisle (2003) numerically process palaeostress results to overcome the incomplete information about the stress tensor. Pollard et al. (1993) criticize the assumptions on which stress inversion is based.

The direct inversion of FSD presented by Angelier (1990) is a reliable solution, and will be considered in the present study and briefly discussed in the next section. For details of the inversion technique, refer to Angelier (1990, Appendices I and II).