In the medium composed of parallel sedimentary layers, model of elliptic anisotropy is often better approximation of real conditions than simple layers with constant velocity. In such model a velocity is described with 3 parameters: a, b (responsible for vertical velocity and its gradient) and χ - elliptic anisotropy coefficient, describing changes of velocity with direction of wave propagation. In this case, seismic rays are no longer straight lines but elliptic-shaped curves. In this contribution we present a tool for inversion of velocity parameters in multilayered medium considering model of elliptic anisotropy. Input data contains measured traveltimes of direct wave between all sources and receivers. The depth of boundaries and existence of elliptic anisotropy in specific layers are not subject of inversion. In order to obtain each synthetic traveltime, necessary for the target function, ray tracing must be carried out using Fermat’s principle directly. The optimization in both parameters inversion and ray tracing is local and performed with Simplex algorithm from GSL library. The tests of the tool was conducted on the synthetic data with various types of start model containing one or more layers with or without elliptic anisotropy.

Inversion of velocity parameters for the walkaway VSP data in a multilayered medium can be impeded by velocity gradients and anisotropy in some layers. A problem occurs if we compare velocities obtained from borehole seismic profiling which are equal to their vertical components with the velocities calculated with paths coming from far offsets where the horizontal component plays an important role, especially when the vertical gradient exists and the ray paths are curve-shaped. In this contribution we present the results of velocity model inversion for VSP data considering velocity gradient and elliptical anisotropy. The algorithm consists of two steps, optimization of velocity parameters and optimization of ray paths for the given model. Both procedures use the Nelder-Mead simplex method which finds local minima. Due to the character of optimization we performed also multistart analysis which can provide information about possible equivalences between parameters. Analysis was conducted for different parameterizations, in some cases allowing introduction of additional parameters: vertical gradient and elliptical anisotropy coefficient. The optimal model for a specific set of data is chosen with the help of Bayesian Information Criterion to balance complexity of model with quality of approximation of traveltimes.

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 in situ 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).

We present a strategy for selecting the values of model parameters by comparing walkaway vertical seismic profiling data with a multilayered model in the context of Bayesian information criterion. We consider P-wave traveltimes and assume elliptical polar velocity dependence. A model with different propagation speeds, depending on the angle of propagation, can be a good approximation for a medium composed of thin layers. While elliptical anisotropy in a one-layer model yields good results, an efficient tool for multilayer modelling would provide improved inversion results. To obtain the proper set of velocity values for specific parameterizations, we require two steps of optimization. In the first step, we find the signal trajectory; in the second step, we obtain parameter values by minimizing the misfit between the model and the data. By comparing models and data, we choose the best model in the sense of the Bayesian information criterion.

We present a strategy for selecting the values of elasticity parameters by comparing walkaway vertical seismic profiling data with a multilayered model in the context of Bayesian Information Criterion. We consider P -wave traveltimes and assume elliptical velocity dependence. The Bayesian Information Criterion approach requires two steps of optimization. In the first step, we find the signal trajectory and, in the second step, we find media parameters by minimizing the misfit between the model and data.