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A source time reversal method for seismicity induced by mining

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  • 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.

    Mathematics Subject Classification: Primary: 35R30, 86A15, 86A22; Secondary: 65M06, 65M32, 65N21.

    Citation:

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  • Figure 1.  Diagram of STR method describing how to recover the source term $f(x)$

    Figure 2.  Functions selected as temporal source terms $g(t)$

    Figure 3.  Functions selected as spatial source terms $f(x)$

    Figure 4.  Spatial source term reconstruction for the different sources $f_i(x)g_j(t)$ $i,j\in\{1,2,3\}$

    Figure 5.  Functions selected as temporal source terms $g(t)$ to generate tremors

    Figure 6.  Spatial source term reconstruction using $g_\gamma(t)$ for the sources $f_i(x)g_a(t)$

    Figure 7.  Spatial source term reconstruction using $g_\gamma(t)$ for the sources $f_i(x)g_b(t)$

    Figure 8.  Relative error variation of the reconstruction with respect to the constant $c_0$

    Figure 9.  (a) Original function $f_4(x)$ and (b)-(j) Reconstructions $\widetilde{f}_4(x)$ for different sources and values of constant $c_0$

    Figure 10.  Space-and time-dependence in the synthetic seismic experiment

    Figure 11.  Spatial source term reconstruction in seismic experiments

    Table 1.  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%
     | Show Table
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    Table 2.  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 = 0.624.3%29.4%43.4%25.4%28.5%43.3%
    γ = 0.714.2%19.4%30.2%10.3%17.7%32.1%
    γ = 0.811.6%12.1%23.5%6.3%7.5%18.3%
    γ = 0.915.0%19.5%29.2%21.5%27.7%30.9%
    γ = 1.034.9%29.6%41.7%47.0%31.7%47.1%
     | Show Table
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    Table 3.  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; $c_0=0.1$)
     | Show Table
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