Stress tensor determination from focal mechanisms: development of new methodology, especially aiming at solving the problem of the choice between the nodal planes

$\\$The reduced stress tensor includes the orientations of the three principal stress axes $\sigma _1$, $\sigma _2$, $\sigma _3$ and the ratio of principal stress differences, $\phi =(\sigma_2-\sigma_3)/(\sigma_1-\sigma_3)$. For a homogeneous subset of double couple mechanisms, this tensor can be determined through inversion of data, under the classical assumptions of mechanical consistency. This is a classical problem for geologists in brittle tectonics. In the case of focal mechanisms, however, there is an additional problem, that of the choice between nodal planes of double couple focal mechanisms. Classically, the answer either consists of avoiding the choice by using the right dihedra (P- and T- dihedra) method, or aims at performing this choice in various ways prior to numerical inversion of data. It is demonstrated that there is a third possibility, which allows one to carry out the mathematical inversion of data and to obtain the complete reduced stress tensor, without doing any choice between the nodal planes of each earthquake. Furthermore, the formulation is such that it is possible to carry out the inversion by pure analytical means, based on cancelling partial derivatives relative to the unknowns of the reduced stress tensor. The solution of the problem is thus straightforward and numerical problems are avoided. Using a simple direct inversion of double-couple mechanisms of earthquakes enables one to focus on a more important problem, that of the inhomogeneity of datasets in terms of stress states. Separation into subsets is made easier because in the new method calculation is extremely brief despite the complexity of algebraic derivations.