Seismic Inversion for the Calculation of Velocities Using the Generalized Inverse Linear Matrix, the Wave Equation, and a Non-Reflective-Boundaries Condition

Abstract

Objective: This work presents the results obtained in the development of a seismic velocity inversion model. The reference times recorded on the surface are taken and using the inversion model to obtain the initial reference model (hypocenters and velocities), starting from an unknown model. Methodology: A hypothetical reference model is proposed containing 64 blocks with interval velocity, 16 recording stations on the surface, and 64 earthquakes in the center of each block. With this model, the reference arrival times are generated for each earthquake registered in each station. The inversion model is made up of two parts: the direct model, which allows calculating the arrival times of the signal registered on the surface according to the hypo-central location of the earthquake and the velocity of the P and S wave of the medium; and the inverse model, which estimates a model of the velocity of the environment and hypo-central locations of the earthquakes that are the input variables of the direct model. The direct model was developed with the wave equation, while the inverse model was developed by modifying the generalized inverse matrix by introducing a factor called “damping.” Results: The discretization model is based on the finite difference method. When estimating the values of velocity and hypo-central location with the inverse algorithm, the propagation of the wave is simulated with the direct model, and then compared with the data of reference times measured on the surface. Depending on the mean square error, we proceed to modify the mean velocities and hypocenters of the earthquakes. This process repeates Seismic Inversion for the Calculation of Velocities Using the Generalized Inverse Linear Matrix González–Veloza., J.F. Duitama–leal ., a. Castillo–lópez., l.a. Gil – Gómez., J.H. y esquiVel., R.e. Tecnura • p-ISSN: 0123-921X • e-ISSN: 2248-7638 • Vol. 24 No. 66 • Octubre Diciembre de 2020 • pp. 13-26 [ 14 ] iteratively until the calculated error is less than a tolerance of 2x10-3s2. Conclusions: It was found that the estimated values of velocity and hypocentral locations coincide well for regions closer to the surface, while for deep regions the error more significant compared to the hypothetical reference model.