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
References:
[1]

K. Aki and P. G. Richars, Quantitative Seismology 2nd edition, University Science Books, 2002.Google Scholar

[2]

H. AmmariE. BretinJ. Garnier and A. Wahab, Time reversal in attenuating acoustic media, Contemporary Mathematics, 548 (2011), 151-163. doi: 10.1090/conm/548. Google Scholar

[3]

H. Ammari, J. Garnier, W. Jing, H. Kang, M. Lim, K. Solna and H. Wang, Mathematical and Statistical Methods for Multistatic Imaging Springer, 2013. doi: 10.1007/978-3-319-02585-8. Google Scholar

[4]

C. Bardos and M. Fink, Mathematical foundations of the time reversal mirror, Asymptotic Analysis, 29 (2002), 157-182. Google Scholar

[5]

H. Brezis, Functional analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7. Google Scholar

[6]

N. W. CarlsonW. R. BabbittT. W. MossbergL. J. Rothberg and A. G. Yodh, Storage and time reversal of light pulses using photon echoes, Opt. Lett., 8 (1983), 483-485. doi: 10.1364/OL.8.000483. Google Scholar

[7]

H. ChenH. QiR. Long and M. Zhang, Research on 10-year tendency of China coal mine accidents and the characteristics of human factors, Safety Science, 50 (2012), 745-750. doi: 10.1016/j.ssci.2011.08.040. Google Scholar

[8]

Z. Dai-Ying and N. Bai-Sheng, Statistical analysis of China's coal mine particularly serious accidents, Procedia Engineering, 26 (2011), 2213-2221. doi: 10.1016/j.proeng.2011.11.2427. Google Scholar

[9]

F. Du, N. Hu, Y. Xie and G. Li, An undersea mining microseism source location algorithm considering wave velocity probability distribution Mathematical Problems in Engineering 2014 (2014), Article ID 805267, 7 pages. doi: 10.1155/2014/805267. Google Scholar

[10]

Y. V. Egorov and M. A. Shubin, Foundations of the Classical Theory of Partial Differential Equations Springer-Verlag, Berlin, 1998. Google Scholar

[11]

L. Evans, Partial Differential Equations 2nd edition, American Mathematical Society, Providence, 2010. Google Scholar

[12]

M. Fink, Time reversal of ultrasonic fields. Ⅰ. Basic principles, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 39 (1992), 555-566. doi: 10.1109/58.156174. Google Scholar

[13]

M. Fink, Time-reversed acoustics, Scientific American, 281 (1999), 91-97. doi: 10.1063/1.1373736. Google Scholar

[14]

G. C. GarciaA. Osses and M. Tapia, A heat source reconstruction formula from single internal measurements using a family of null controls, Journal of Inverse and Ill-posed Problems, 21 (2013), 755-779. doi: 10.1515/jip-2011-0001. Google Scholar

[15]

L. Geiger, Probability method for the determination of earthquake epicenters from the arrival time only (translated from Geiger's 1910 German article), Bulletin of St. Louis University, 8 (1912), 56-71. Google Scholar

[16]

S. J. Gibowicz and A. Kijko, An Introduction to Mining Seismology Academic Press, 1994.Google Scholar

[17]

Y. Hristova, Time reversal in thermoacoustic tomography–an error estimate, Inverse Problems, 25 (2009), 055008 (14 pp). doi: 10.1088/0266-5611/25/5/055008. Google Scholar

[18]

J. Joy, Occupational safety risk management in Australian mining, Occupational Medicine, 54 (2004), 311-315. doi: 10.1093/occmed/kqh074. Google Scholar

[19]

W. A. KupermanW. S. HodgkissH. C. SongT. AkalC. Ferla and D. R. Jackson, Phase conjugation in the ocean: Experimental demonstration of an acoustic time-reversal mirror, The Journal of the Acoustical Society of America, 103 (1998), 25-40. doi: 10.1121/1.423233. Google Scholar

[20]

P. KyritsiG. PapanicolaouP. Eggers and A. Oprea, Time reversal techniques for wireless communications, Vehicular Technology Conference, 2004. VTC2004-Fall. 2004 IEEE 60th, 1 (2004), 47-51. doi: 10.1109/VETECF.2004.1399917. Google Scholar

[21]

C. Larmat, J. -P. Montagner, M. Fink, Y. Capdeville, A. Tourin and E. Clévédé, Time-reversal imaging of seismic sources and application to the great Sumatra earthquake Geophysical Research Letters 33 (2006), L19312. doi: 10.1029/2006GL026336. Google Scholar

[22]

P. D. Lax and R. S. Phillips, Scattering Theory 2nd edition, Academic Press, Boston, 1989. Google Scholar

[23]

S. Mallick and K. Mukherjee, An empirical study for mines safety management through analysis on potential for accident reduction, Omega, 24 (1996), 539-550. doi: 10.1016/0305-0483(96)00020-5. Google Scholar

[24]

M. Matsu'ura, Bayesian estimation of hypocenter with origin time eliminated, Journal of Physics of the Earth, 32 (1984), 469-483. doi: 10.4294/jpe1952.32.469. Google Scholar

[25]

H. T. NguyenJ. B. AndersenG. F. PedersenP. Kyritsi and P. Eggers, Time reversal in wireless communications: A measurement-based investigation, IEEE Transactions on Wireless Communications, 5 (2006), 2242-2252. doi: 10.1109/TWC.2006.1687740. Google Scholar

[26]

N. Ricker, The form and laws of propagation of seismic wavelets, Geophysics, 18 (1953), 10-40. doi: 10.1190/1.1437843. Google Scholar

[27]

L. SanmiquelJ. M. Rossell and C. Vintró, Study of Spanish mining accidents using data mining techniques, Safety Science, 75 (2015), 49-55. doi: 10.1016/j.ssci.2015.01.016. Google Scholar

[28]

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods 3rd edition, The Clarendon Press, Oxford University Press, New York, 1985. Google Scholar

[29]

H. C. SongW. A. KupermanW. S. HodgkissT. Akal and C. Ferla, Iterative time reversal in the ocean, The Journal of the Acoustical Society of America, 105 (1999), 3176-3184. doi: 10.1121/1.424648. Google Scholar

[30]

W. Spence, Relative epicenter determination using P-wave arrival-time differences, Bulletin of the Seismological Society of America, 70 (1980), 171-183. Google Scholar

[31]

P. Stefanov and G. Uhlmann, Multi-wave methods via ultrasound, Inverse Problems and Applications, Inside Out Ⅱ, MSRI Publications, 60 (2013), 271-323. Google Scholar

[32]

L. WuZ. JiangW. ChengX. ZuoD. Lv and Y. Yao, Major accident analysis and prevention of coal mines in China from the year of 1949 to 2009, Mining Science and Technology (China), 21 (2011), 693-699. doi: 10.1016/j.mstc.2011.03.006. Google Scholar

[33]

F. WuJ. L. Thomas and M. Fink, Time reversal of ultrasonic fields. Ⅱ. Experimental results, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 39 (1992), 567-578. doi: 10.1109/58.156175. Google Scholar

[34]

M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, 11 (1995), 481-496. doi: 10.1088/0266-5611/11/2/013. Google Scholar

[35]

M. F. Yanik and S. Fan, Time reversal of light with linear optics and modulators Phys. Rev. Lett. 93 (2004), 173903. doi: 10.1103/PhysRevLett.93.173903. Google Scholar

[36]

B. Y. Zel'Dovich, N. F. Pilipetsky and V. V. Shkunov, Principles of Phase Conjugation Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-540-38959-0. Google Scholar

show all references

References:
[1]

K. Aki and P. G. Richars, Quantitative Seismology 2nd edition, University Science Books, 2002.Google Scholar

[2]

H. AmmariE. BretinJ. Garnier and A. Wahab, Time reversal in attenuating acoustic media, Contemporary Mathematics, 548 (2011), 151-163. doi: 10.1090/conm/548. Google Scholar

[3]

H. Ammari, J. Garnier, W. Jing, H. Kang, M. Lim, K. Solna and H. Wang, Mathematical and Statistical Methods for Multistatic Imaging Springer, 2013. doi: 10.1007/978-3-319-02585-8. Google Scholar

[4]

C. Bardos and M. Fink, Mathematical foundations of the time reversal mirror, Asymptotic Analysis, 29 (2002), 157-182. Google Scholar

[5]

H. Brezis, Functional analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7. Google Scholar

[6]

N. W. CarlsonW. R. BabbittT. W. MossbergL. J. Rothberg and A. G. Yodh, Storage and time reversal of light pulses using photon echoes, Opt. Lett., 8 (1983), 483-485. doi: 10.1364/OL.8.000483. Google Scholar

[7]

H. ChenH. QiR. Long and M. Zhang, Research on 10-year tendency of China coal mine accidents and the characteristics of human factors, Safety Science, 50 (2012), 745-750. doi: 10.1016/j.ssci.2011.08.040. Google Scholar

[8]

Z. Dai-Ying and N. Bai-Sheng, Statistical analysis of China's coal mine particularly serious accidents, Procedia Engineering, 26 (2011), 2213-2221. doi: 10.1016/j.proeng.2011.11.2427. Google Scholar

[9]

F. Du, N. Hu, Y. Xie and G. Li, An undersea mining microseism source location algorithm considering wave velocity probability distribution Mathematical Problems in Engineering 2014 (2014), Article ID 805267, 7 pages. doi: 10.1155/2014/805267. Google Scholar

[10]

Y. V. Egorov and M. A. Shubin, Foundations of the Classical Theory of Partial Differential Equations Springer-Verlag, Berlin, 1998. Google Scholar

[11]

L. Evans, Partial Differential Equations 2nd edition, American Mathematical Society, Providence, 2010. Google Scholar

[12]

M. Fink, Time reversal of ultrasonic fields. Ⅰ. Basic principles, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 39 (1992), 555-566. doi: 10.1109/58.156174. Google Scholar

[13]

M. Fink, Time-reversed acoustics, Scientific American, 281 (1999), 91-97. doi: 10.1063/1.1373736. Google Scholar

[14]

G. C. GarciaA. Osses and M. Tapia, A heat source reconstruction formula from single internal measurements using a family of null controls, Journal of Inverse and Ill-posed Problems, 21 (2013), 755-779. doi: 10.1515/jip-2011-0001. Google Scholar

[15]

L. Geiger, Probability method for the determination of earthquake epicenters from the arrival time only (translated from Geiger's 1910 German article), Bulletin of St. Louis University, 8 (1912), 56-71. Google Scholar

[16]

S. J. Gibowicz and A. Kijko, An Introduction to Mining Seismology Academic Press, 1994.Google Scholar

[17]

Y. Hristova, Time reversal in thermoacoustic tomography–an error estimate, Inverse Problems, 25 (2009), 055008 (14 pp). doi: 10.1088/0266-5611/25/5/055008. Google Scholar

[18]

J. Joy, Occupational safety risk management in Australian mining, Occupational Medicine, 54 (2004), 311-315. doi: 10.1093/occmed/kqh074. Google Scholar

[19]

W. A. KupermanW. S. HodgkissH. C. SongT. AkalC. Ferla and D. R. Jackson, Phase conjugation in the ocean: Experimental demonstration of an acoustic time-reversal mirror, The Journal of the Acoustical Society of America, 103 (1998), 25-40. doi: 10.1121/1.423233. Google Scholar

[20]

P. KyritsiG. PapanicolaouP. Eggers and A. Oprea, Time reversal techniques for wireless communications, Vehicular Technology Conference, 2004. VTC2004-Fall. 2004 IEEE 60th, 1 (2004), 47-51. doi: 10.1109/VETECF.2004.1399917. Google Scholar

[21]

C. Larmat, J. -P. Montagner, M. Fink, Y. Capdeville, A. Tourin and E. Clévédé, Time-reversal imaging of seismic sources and application to the great Sumatra earthquake Geophysical Research Letters 33 (2006), L19312. doi: 10.1029/2006GL026336. Google Scholar

[22]

P. D. Lax and R. S. Phillips, Scattering Theory 2nd edition, Academic Press, Boston, 1989. Google Scholar

[23]

S. Mallick and K. Mukherjee, An empirical study for mines safety management through analysis on potential for accident reduction, Omega, 24 (1996), 539-550. doi: 10.1016/0305-0483(96)00020-5. Google Scholar

[24]

M. Matsu'ura, Bayesian estimation of hypocenter with origin time eliminated, Journal of Physics of the Earth, 32 (1984), 469-483. doi: 10.4294/jpe1952.32.469. Google Scholar

[25]

H. T. NguyenJ. B. AndersenG. F. PedersenP. Kyritsi and P. Eggers, Time reversal in wireless communications: A measurement-based investigation, IEEE Transactions on Wireless Communications, 5 (2006), 2242-2252. doi: 10.1109/TWC.2006.1687740. Google Scholar

[26]

N. Ricker, The form and laws of propagation of seismic wavelets, Geophysics, 18 (1953), 10-40. doi: 10.1190/1.1437843. Google Scholar

[27]

L. SanmiquelJ. M. Rossell and C. Vintró, Study of Spanish mining accidents using data mining techniques, Safety Science, 75 (2015), 49-55. doi: 10.1016/j.ssci.2015.01.016. Google Scholar

[28]

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods 3rd edition, The Clarendon Press, Oxford University Press, New York, 1985. Google Scholar

[29]

H. C. SongW. A. KupermanW. S. HodgkissT. Akal and C. Ferla, Iterative time reversal in the ocean, The Journal of the Acoustical Society of America, 105 (1999), 3176-3184. doi: 10.1121/1.424648. Google Scholar

[30]

W. Spence, Relative epicenter determination using P-wave arrival-time differences, Bulletin of the Seismological Society of America, 70 (1980), 171-183. Google Scholar

[31]

P. Stefanov and G. Uhlmann, Multi-wave methods via ultrasound, Inverse Problems and Applications, Inside Out Ⅱ, MSRI Publications, 60 (2013), 271-323. Google Scholar

[32]

L. WuZ. JiangW. ChengX. ZuoD. Lv and Y. Yao, Major accident analysis and prevention of coal mines in China from the year of 1949 to 2009, Mining Science and Technology (China), 21 (2011), 693-699. doi: 10.1016/j.mstc.2011.03.006. Google Scholar

[33]

F. WuJ. L. Thomas and M. Fink, Time reversal of ultrasonic fields. Ⅱ. Experimental results, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 39 (1992), 567-578. doi: 10.1109/58.156175. Google Scholar

[34]

M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, 11 (1995), 481-496. doi: 10.1088/0266-5611/11/2/013. Google Scholar

[35]

M. F. Yanik and S. Fan, Time reversal of light with linear optics and modulators Phys. Rev. Lett. 93 (2004), 173903. doi: 10.1103/PhysRevLett.93.173903. Google Scholar

[36]

B. Y. Zel'Dovich, N. F. Pilipetsky and V. V. Shkunov, Principles of Phase Conjugation Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-540-38959-0. Google Scholar

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%
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%
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%
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%
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\%$$8.2\%$
(Fig. 9h; $c_0=10^{-5}$)(Fig. 9i; $c_0=0.01$)(Fig. 9j; $c_0=0.1$)
$f_4(x)g_1(t)$$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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