On obtaining effective elasticity tensors with entries zeroing method

Abstract

The purpose of this paper is to propose a new method for obtaining tensors expressing certain symmetries, called effective elasticity tensors, and their optimal orientation. The generally anisotropic tensor being the result of <i>in situ</i> seismic measurements describes the elastic properties of a medium. It can be approximated with a tensor of a specific symmetry class. With a known symmetry class and orientation, one can better describe geological structure elements like layers and fissures. A method used to obtain effective tensor in the previous papers (i.e. Danek & Slawinski 2015) is based on minimizing the Frobenius norm between the measured and effective tensor of a chosen symmetry class in the same coordinate system. In this paper, we propose a new approach for obtaining the effective tensor with the assumption of a certain symmetry class. The entry zeroing method assumes the minimization of the target function, being the measure of similarity with the form of the effective tensor for the specific class. The optimization of orientation is made by means of the Particle Swarm Optimization (PSO) algorithm and transformations were parameterised with quaternions. To analyse the obtained results, the Monte-Carlo method was used. After thousands of runs of PSO optimization, values of quaternion parts and tensor entries were obtained. Then, thousands of realizations of generally anisotropic tensors described with normal distributions of entries were generated. Each of these tensors was the subject of separate PSO optimization, and the distributions of rotated tensor entries were obtained. The results obtained were compared with solutions of the method based on the Frobenius distances (Danek et al. 2013).