Inverting focal mechanisms for the stress tensor

We learned from the work of McKenzie [1969] that the crustal stress tensor cannot be estimated with confidence from the focal mechanism of one earthquake. Instead, we need an inversion mechanism that allows us to estimate the stress tensor from a larger number of earthquakes, and to formulate such a mechanism we will turn to structural geology and studies of faulting for help. The basic principles for slip on an arbitrarily oriented fault in a stress field were formulated by Wallace [1951] and Bott [1959]. Bott [1959] also proposed that slip on any fault plane will occur in the direction of maximum resolved shear stress (Bott's criterion) and showed, as in Equation 19, that the shear stress direction does depend on the orientation of the fault plane in the stress field and the relative size of the intermediate stress, $R$, but does not depend on the actual principal stress magnitudes. In his paper, Bott [1959] suggests that his equations could be used to determine the stress orientations and $R$, but it was not until Carey and Brunier [1974] utilized Bott's criterion that an inversion scheme for the causative state of stress was formulated. Carey and Brunier [1974] added the assumption that the motion shown by striae on the fault planes in their data set were all caused by a single common stress tensor. Their analysis was extended and improved on by a number of authors [e.g. Angelier et al., 1982; Reches, 1987; Etchecopar et al., 1981; Armijo et al., 1982; Michael, 1984; Angelier, 1979, 1990]. In common for most of these authors is the use of the geometric information of strike, dip and rake of the fault data for the estimation of the reduced stress tensor $\underline{T}$, i.e. the directions of the principal stresses and the ratio $R$. The reduced stress tensor is equivalent to finding four of the six independent components of the stress tensor. The last two, the magnitude of the maximum shear stress ( $\sigma_{1}$ - $\sigma_{3}$) and an isotropic component, cannot be determined without additional information. Estimating four parameters imply that we need at least four earthquake focal mechanisms as input data, see however the data discussion in Section 3.4.1 below. Inverting the fault plane measurements without other constraints than the ones of stress homogeneity and Bott's criterion discussed above, is a highly non-linear process. The papers quoted above use a wide variety of numerical techniques for the stress inversion and some add constraints to linearize the problem in their methods. Angelier [1979] proposed both an iterative inversion minimizing the angle between the calculated maximum shear direction and the slip direction in the fault plane, and a direct least-squares (LS) inversion minimizing the calculated shear stress perpendicular to the slip in the fault plane. The iterative LS-inversion scheme developed by Angelier et al. [1982] incorporates the errors of the fault plane measurements, Michael [1984] linearized the problem by requiring that the shear stress magnitudes on all failing faults are similar, Reches [1987] used the constraint that shear and normal stresses on the faults are related by a Coulomb failure criterion and Angelier [1990] minimized the angle between slip direction and shear stress while requiring large shear stress magnitudes to obtain a linearized inversion scheme. All these different approaches can yield a variety of results and, especially, very different error estimates.

Generalizing these inversion schemes from geological fault slip measurements to earthquake focal mechanisms [e.g. Michael, 1987; Angelier, 1984; Carey-Gailhardis and Mercier, 1987; Vasseur et al., 1983; Gephart and Forsyth, 1984] is rather straightforward but introduces the problem of the two nodal planes. We know from the double-couple discussion in section 3.1 that there are two indistinguishable nodal planes for each focal mechanism. Which is the ``correct'' plane, which plane did slip in the earthquake, which plane should be used to infer the causative state of stress? Using focal mechanisms in a stress inversion scheme requires us to either choose one of the nodal planes or to justify incorporating both. Angelier [1984] experimented with the inclusion of both nodal planes, inferring axisymmetric stresses ($R$ = 0 or 1) in which case both nodal planes are equally plausible. Gephart [1985] later showed that this is strictly admissible only if the B axis is coplanar with the two equal stress axes. Another common approach to the nodal plane problem is to choose the nodal plane which best fits surface geological evidence of fault orientations [e.g. Angelier, 1984; Gephart, 1990]. However, extrapolating surface fault orientations to earthquake focal depths is not always possible, an area can have complex surface faulting structures that are difficult to use to constrain the earthquake nodal planes. One might also wish to estimate subsurface fault structure using the stress inversion. In such cases the stress inversion technique itself must choose a preferred nodal plane. One method of picking the fault plane is to test both nodal planes in the stress field under consideration and choose the nodal plane which has the smallest angle between the shear stress and the slip direction (angular deviation) [e.g. Vasseur et al., 1983; Bergerat et al., 1998; Gephart and Forsyth, 1984]. I will refer to this method as the slip angle method. Tested by Michael [1987], the method works satisfactory when there are large differences in the angular deviations between the nodal planes. In more difficult situations, however, the reliability can become much degraded, see e.g. Magee [1997] and Paper II. Carey-Gailhardis and Mercier [1987] suggested an alternate fault picking mechanism; for each stress state they calculate an apparent value of $R$ for each nodal plane and since for non-axisymmetric stresses only one of the nodal planes can have $R$ in the allowed range $0 \leq R \leq 1$ [Gephart, 1985], the ``good'' nodal plane is picked. A method for choosing nodal plane based on a stability criterion is presented in Paper II of this thesis.

The grid search inversion method by Gephart and Forsyth [1984] introduced a novel approach to the angular deviation, or measure of misfit. Previously the misfit angle was usually defined as the angle between the tested shear stress direction and the observed slip direction in the fault plane. Gephart and Forsyth [1984] showed that this implicitly only considers errors in the slip direction but not in the orientation of the plane. They instead defined the misfit angle as the minimum rotation angle between the observed slip direction and the family of admissible fault geometries. Gephart and Forsyth [1984] also utilized a one-norm misfit criterion instead of an LS-criterion, based on observations that the angular misfit residuals were better fit by an exponential distribution than a normal distribution. Carey-Gailhardis and Mercier [1987] tested the fault planes obtained in the stress inversion against the first motion data from the focal mechanisms. This approach was used in conjunction with the Gephart and Forsyth [1984] inversion by Magee [1997] to better constrain the stress inversion and further utilizing the basic seismological data, Horiuchi et al. [1995] developed a completely integrated focal mechanism and stress tensor inversion based on polarities.

Paper II of this thesis introduces a stress tensor inversion technique based on the Gephart and Forsyth [1984] method but with considerable extensions in terms of allowing for the misfit of the earthquake focal mechanisms and in the choice of nodal plane.