Determination of stress fields in the elastic lithosphere by methods based on stress orientations

Abstract

Theoretical reconstruction of the stress fields in tectonic plates or particular tectonic regions is an important geophysical problem. Conventional approaches for solving this vital issue are based on classical formulations of boundary value problems of elasticity. In these approaches, stress fields are calculated for diverse boundary conditions defined on the margins of the region in order to fit the experimentally observed orientations of principal stresses inside the region. The present article identifies the major flaw in the conventional approach, which is the impossibility of obtaining a unique and reliable stress field, and suggests alternative methods based on the analysis of orientations of principal stresses. Three methods for determining the elastic state of stress in relatively stable blocks of the lithosphere are described and applied to particular tectonic domains. All of the methods are based on the direct use of experimental data on the stress orientations as input information. The first method exploits direct integration of the equations of elasticity when the field of principal stress trajectories is prescribed within a region. The second one utilizes the non-classical boundary value problem of elasticity, which uses experimentally obtained stress orientations at the region margins as boundary conditions. The third method is aimed at the numerical determination of the stress field from a given set of spatially discrete principal stress orientations. In contrast to the conventional approach, the methods suggested here do not require knowledge of the boundary stress magnitudes. As a consequence, the general solution of the problem becomes non-unique. However, in the case of an elastic medium, only a certain (finite) number of arbitrary parameters control the general solutions. These parameters can be determined from in situ stress measurements within the region under study. Therefore, for the selected spatial scale, the unique stress field can be singled out. In the second method, the number of parameters (and, thus, the minimum number of field measurements) is determined from an analysis of boundary stress orientations alone. In other methods, this number depends upon the harmonicity or non-harmonicity of inclination of the prescribed stress trajectories (for the second method) and the calculated stress trajectories (for the third method). To illustrate the essence of the proposed methods, they are applied to the determination of first-order stress fields in the West European and Australian platforms. These platforms represent two basically different types of stress domains. The stress field for the West European platform reflects nearly homogeneous stress orientations throughout the extent of the region, whereas the Australian platform is characterized by rotation of the principal stress axes while traversing the region margins. In the case of Australia, an important result is the existence (at the chosen spatial scale) of a singular point inside the Australian continent at which the curvature of the stress trajectories is infinite. The local state of stress near such a point has important geophysical and engineering implications. The proposed approaches can be applied not only for the determination of regional stress fields but also at other spatial scales, depending on the scale at which the stress indicators have been characterized.