American Institute of Mathematical Sciences

January  2017, 11(1): 25-45. doi: 10.3934/ipi.2017002

A source time reversal method for seismicity induced by mining

 1 Departamento de Ingeniería Matemática, Universidad de Chile, Beauchef 851, Edificio Norte, Casilla 170-3, Correo 3, Santiago, Chile 2 Departamento de Ingeniería Matemática & Centro de Modelamiento Matemático, Universidad de Chile, Beauchef 851, Edificio Norte, Casilla 170-3, Correo 3, Santiago, Chile 3 Universidad del País Vasco (UPV/EHU), Leioa, Spain 4 Basque Center for Applied Mathematics (BCAM), Bilbao, Spain 5 Ikerbasque, Bilbao, Spain

Received  February 2016 Revised  July 2016 Published  January 2017

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.

Citation: Rodrigo I. Brevis, Jaime H. Ortega, David Pardo. A source time reversal method for seismicity induced by mining. Inverse Problems & Imaging, 2017, 11 (1) : 25-45. doi: 10.3934/ipi.2017002
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References:
Diagram of STR method describing how to recover the source term $f(x)$
Functions selected as temporal source terms $g(t)$
Functions selected as spatial source terms $f(x)$
Spatial source term reconstruction for the different sources $f_i(x)g_j(t)$ $i,j\in\{1,2,3\}$
Functions selected as temporal source terms $g(t)$ to generate tremors
Spatial source term reconstruction using $g_\gamma(t)$ for the sources $f_i(x)g_a(t)$
Spatial source term reconstruction using $g_\gamma(t)$ for the sources $f_i(x)g_b(t)$
Relative error variation of the reconstruction with respect to the constant $c_0$
(a) Original function $f_4(x)$ and (b)-(j) Reconstructions $\widetilde{f}_4(x)$ for different sources and values of constant $c_0$
Space-and time-dependence in the synthetic seismic experiment
Spatial source term reconstruction in seismic experiments
Summary of the relative error $\frac{\|\widetilde f_i - f_i\|_{L^2}}{\|f_i\|_{L^2}}$ in experiment smoothness of $f(x)$ and $g(t)$
 f1(x) f2(x) f3(x) g1(t) 0.7% 2.2% 8.7% g2(t) 1.3% 2.2% 8.2% g3(t) 0.9% 1.8% 4.1%
 f1(x) f2(x) f3(x) g1(t) 0.7% 2.2% 8.7% g2(t) 1.3% 2.2% 8.2% g3(t) 0.9% 1.8% 4.1%
Summary of the relative error $\frac{\|\widetilde f_i-f_i\|_{L^2}}{\|f_i\|_{L^2}}$ in experiment sensitivity with respect to $g(t)$
 f1(x)ga(t) f2(x)ga(t) f3(x)ga(t) f1(x)gb(t) f2(x)gb(t) f3(x)gb(t) γ = 0.6 24.3% 29.4% 43.4% 25.4% 28.5% 43.3% γ = 0.7 14.2% 19.4% 30.2% 10.3% 17.7% 32.1% γ = 0.8 11.6% 12.1% 23.5% 6.3% 7.5% 18.3% γ = 0.9 15.0% 19.5% 29.2% 21.5% 27.7% 30.9% γ = 1.0 34.9% 29.6% 41.7% 47.0% 31.7% 47.1%
 f1(x)ga(t) f2(x)ga(t) f3(x)ga(t) f1(x)gb(t) f2(x)gb(t) f3(x)gb(t) γ = 0.6 24.3% 29.4% 43.4% 25.4% 28.5% 43.3% γ = 0.7 14.2% 19.4% 30.2% 10.3% 17.7% 32.1% γ = 0.8 11.6% 12.1% 23.5% 6.3% 7.5% 18.3% γ = 0.9 15.0% 19.5% 29.2% 21.5% 27.7% 30.9% γ = 1.0 34.9% 29.6% 41.7% 47.0% 31.7% 47.1%
Relative errors when reconstructing Phantom's source
 $f_4(x)g_1(t)$ $35.8\%$ $\bf 13.2\%$ $30.9\%$ (Fig. 9b; $c_0=0$) (Fig. 9c; $c_0=2 \times {10^{ - 5}}$) (Fig. 9d; $c_0=0.01$) $f_4(x)g_2(t)$ $17.5\%$ $\bf 11.1\%$ $25.8\%$ (Fig. 9e; $c_0=0$) (Fig. 9f; $c_0=7 \times {10^{ - 4}}$) (Fig. 9g; $c_0=0.05$) $f_4(x)g_3(t)$ $8.0\%$ $\bf 5.7\%$ $8.2\%$ (Fig. 9h; $c_0=10^{-5}$) (Fig. 9i; $c_0=0.01$) (Fig. 9j; $c_0=0.1$)
 $f_4(x)g_1(t)$ $35.8\%$ $\bf 13.2\%$ $30.9\%$ (Fig. 9b; $c_0=0$) (Fig. 9c; $c_0=2 \times {10^{ - 5}}$) (Fig. 9d; $c_0=0.01$) $f_4(x)g_2(t)$ $17.5\%$ $\bf 11.1\%$ $25.8\%$ (Fig. 9e; $c_0=0$) (Fig. 9f; $c_0=7 \times {10^{ - 4}}$) (Fig. 9g; $c_0=0.05$) $f_4(x)g_3(t)$ $8.0\%$ $\bf 5.7\%$ $8.2\%$ (Fig. 9h; $c_0=10^{-5}$) (Fig. 9i; $c_0=0.01$) (Fig. 9j; $c_0=0.1$)
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