In this work we present a formalism that intends to solve the problem of modeling wave propagation in the context of seismic inversion. The formalism is based on the linear perturbation theory of Cauchy’s equations. Based on the foregoing, we derived an equivalent Helmholtz equation for the propagation of waves in a variable density media. Then, we defined a solution, by using the boundary conditions on a half plane. This solution is of an integral nature and resembles expansion in a Neumann series. We implemented the solution of the first terms in the series, considering only the incident wavefield and neglecting the reflections. We show how this approximation works in different media that include lateral in homogeneities in the velocity. The method presented hereunder is intended as a first step in the modelling process for the full wavefield, to be used in seismic inversion methods, Full Waveform Inversion, for example.

Full waveform inversion (FWI) is a tool for the inversion of seismic data. There are several sources of uncertainty in the results provided by FWI. The quantification of such uncertainties has been studied through the resolution matrix (Res), which rests on a quadratic approximation that interprets the Hessian matrix as the posterior covariance matrix. Despite efforts in the use of Res, there is no published analysis of the uncertainties contained in the full correlation matrix, (R). Our approach leads to build the full R matrix, which, at the end of the day, is the final quantity that includes all the information associated with uncertainties.We focused on uncertainties related to variation in the starting models of the FWI, and thus propose a method to study the full R matrix, which is-called the Density of Correlation Map, D. By using the D map, we found that the highest uncertainty zones in the FWI inverted model are near the sources, the model boundaries, and the interfaces. We argue that D can be a complement for the study and estimation of uncertainties in FWI.

En este trabajo se presenta una estrategia de trazado de rayos con aplicación a los problemas de tomografía sísmica. El trazador está basado en el método del camino más corto. La estrategia permite calcular trayectorias de rayos reflejados y refractados simultáneamente, generar rayos con varias fuentes y varios receptores, escoger puntos en interfaces de interés para generar reflexiones en dichos puntos, entre otras. Usando esta aproximación como estrategia de modelado, se implementó una tomografía de tiempos de propagación usando técnicas de reconstrucción algebraica, para estimar velocidades de propagación de ondas en el subsuelo. Se realizaron diferentes experimentos que muestran el comportamiento del trazador de rayos en diferentes escenarios con refracciones y reflexiones, así como los resultados del uso de la técnica de trazado de rayos en problemas de tomografía sísmica, obteniéndose resultados positivos en la identificación de estructuras en el modelo del subsuelo a un costo computacional relativamente bajo.