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% |
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: |
Table 1.
Summary of the relative error
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
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% |
Table 3. Relative errors when reconstructing Phantom's source
(Fig. 9b; | (Fig. 9c; | (Fig. 9d; | |
(Fig. 9e; | (Fig. 9f; | (Fig. 9g; | |
(Fig. 9h; | (Fig. 9i; | (Fig. 9j; |
K. Aki and P. G. Richars,
Quantitative Seismology 2nd edition, University Science Books, 2002.
![]() |
|
H. Ammari
, E. Bretin
, J. Garnier
and A. Wahab
, Time reversal in attenuating acoustic media, Contemporary Mathematics, 548 (2011)
, 151-163.
doi: 10.1090/conm/548.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
C. Bardos
and M. Fink
, Mathematical foundations of the time reversal mirror, Asymptotic Analysis, 29 (2002)
, 157-182.
![]() ![]() |
|
H. Brezis,
Functional analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011.
doi: 10.1007/978-0-387-70914-7.![]() ![]() ![]() |
|
N. W. Carlson
, W. R. Babbitt
, T. W. Mossberg
, L. 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.![]() ![]() |
|
H. Chen
, H. Qi
, R. 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.![]() ![]() |
|
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.![]() ![]() |
|
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.![]() ![]() |
|
Y. V. Egorov and M. A. Shubin,
Foundations of the Classical Theory of Partial Differential Equations Springer-Verlag, Berlin, 1998.
![]() ![]() |
|
L. Evans,
Partial Differential Equations 2nd edition, American Mathematical Society, Providence, 2010.
![]() ![]() |
|
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.![]() ![]() |
|
M. Fink
, Time-reversed acoustics, Scientific American, 281 (1999)
, 91-97.
doi: 10.1063/1.1373736.![]() ![]() |
|
G. C. Garcia
, A. 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.![]() ![]() ![]() |
|
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.
![]() |
|
S. J. Gibowicz and A. Kijko,
An Introduction to Mining Seismology Academic Press, 1994.
![]() |
|
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.![]() ![]() ![]() |
|
J. Joy
, Occupational safety risk management in Australian mining, Occupational Medicine, 54 (2004)
, 311-315.
doi: 10.1093/occmed/kqh074.![]() ![]() |
|
W. A. Kuperman
, W. S. Hodgkiss
, H. C. Song
, T. Akal
, C. 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.![]() ![]() |
|
P. Kyritsi
, G. Papanicolaou
, P. 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.![]() ![]() |
|
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.![]() ![]() |
|
P. D. Lax and R. S. Phillips,
Scattering Theory 2nd edition, Academic Press, Boston, 1989.
![]() ![]() |
|
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.![]() ![]() |
|
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.![]() ![]() |
|
H. T. Nguyen
, J. B. Andersen
, G. F. Pedersen
, P. 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.![]() ![]() |
|
N. Ricker
, The form and laws of propagation of seismic wavelets, Geophysics, 18 (1953)
, 10-40.
doi: 10.1190/1.1437843.![]() ![]() |
|
L. Sanmiquel
, J. 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.![]() ![]() |
|
G. D. Smith,
Numerical Solution of Partial Differential Equations: Finite Difference Methods 3rd edition, The Clarendon Press, Oxford University Press, New York, 1985.
![]() ![]() |
|
H. C. Song
, W. A. Kuperman
, W. S. Hodgkiss
, T. 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.![]() ![]() |
|
W. Spence
, Relative epicenter determination using P-wave arrival-time differences, Bulletin of the Seismological Society of America, 70 (1980)
, 171-183.
![]() |
|
P. Stefanov
and G. Uhlmann
, Multi-wave methods via ultrasound, Inverse Problems and Applications, Inside Out Ⅱ, MSRI Publications, 60 (2013)
, 271-323.
![]() ![]() |
|
L. Wu
, Z. Jiang
, W. Cheng
, X. Zuo
, D. 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.![]() ![]() |
|
F. Wu
, J. 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.![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() |
|
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.![]() ![]() |
Diagram of STR method describing how to recover the source term
Functions selected as temporal source terms
Functions selected as spatial source terms
Spatial source term reconstruction for the different sources
Functions selected as temporal source terms
Spatial source term reconstruction using
Spatial source term reconstruction using
Relative error variation of the reconstruction with respect to the constant
(a) Original function
Space-and time-dependence in the synthetic seismic experiment
Spatial source term reconstruction in seismic experiments