A source time reversal method for seismicity induced by mining

Abstract

In this work, we present a modified Time-Reversal Mirror (TRM) Method, called Source Time Reversal (STR), to find the spatial distribution of a seismic source induced by mining activity. This methodology is based on a known full description of the temporal dependence of the source, the Duhamel's principle, and the time-reverse property of the wave equation. We also provide an error estimate of the reconstruction when the measurements are acquired over the entire boundary, and we show experimentally the influence of measuring on a subdomain of the boundary. Numerical results indicate that the methodology is able to recover continuous and discontinuous sources, and it remains stable for partial boundary measurements.

Source time reversal (STR) method for linear elasticity

“…Here, we generate the elastic waves using three Figures 3b, 3c, and 3d show the reconstructions of the space-source term for waves generated with sources f ph (x)g i (t), i ∈ {1, 2, 3}. As it occurred with the results of the experiments perfomed in [19] for acoustic waves, the discontinuous time-source term g 3 (t) provides better reconstructions than the other terms.…”

“…STR [19] method modifies the classical Time-Reversal Mirrors (TRM) [28] for reconstructing a space-source term of the form s(x, t) = f (x)g(t), where g is a real-valued function with compact support. Taking advantage of the information provided by the time-source term g(t), the original source problem is related with a non-source wave problem in which the space-source term of the original problem appears as an initial condition and the boundaries information between the two problems are related by a Volterra integral equation of the first kind.…”

“…In [29], we can see the main limitations of considering the acoustic model for seismic events. Since STR method is oriented to solve problems in induced seismicity, in here we extend the original formulation for acoustic waves [19] to systems of elastic waves.…”

“…With the above regularization, high frequencies appear due to the division by F (g). This regularization method was considered in [19]. The addition of the constant c 0 makes the deconvolution stable and avoids divisions by zero but does not eliminate high frequencies.…”

“…In [19], a method called Source Time Reversal (STR) was proposed for reconstructing the space term of a source of the form f (x)g(t) for the acoustic wave equation. The work, with applications to mining seismicity, takes advantage of the information provided by a general time-source term g(t) to transform the original source problem into an synthetic with initial conditions and without source, where the unknown f (x) now appears as an initial condition for this new problem.…”