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Time-reversal algorithms in viscoelastic media. (English) Zbl 1326.74068

Summary: We consider the problem of reconstructing sources in a homogeneous viscoelastic medium from wavefield measurements. We first present a modified time-reversal imaging algorithm based on a weighted Helmholtz decomposition and justify mathematically that it provides a better approximation than by simply time reversing the displacement field, where artifacts appear due to the coupling between the pressure and shear waves. Then we investigate the source inverse problem in an elastic attenuating medium. We provide a regularized time-reversal imaging which corrects the attenuation effect at the first order. The results of this paper yield the fundamental tools for solving imaging problems in elastic media using cross-correlation techniques.

MSC:

74J25 Inverse problems for waves in solid mechanics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q74 PDEs in connection with mechanics of deformable solids
74J05 Linear waves in solid mechanics
35R30 Inverse problems for PDEs
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References:

[1] Mathematical Modeling in Biomedical Imaging II pp 57– (2011)
[2] DOI: 10.1017/S0956792509007888 · Zbl 1187.92058 · doi:10.1017/S0956792509007888
[3] DOI: 10.1137/11083191X · Zbl 1239.35181 · doi:10.1137/11083191X
[4] DOI: 10.1002/mma.1344 · Zbl 1381.76325 · doi:10.1002/mma.1344
[5] Mathematical and Statistical Methods for Imaging pp 151– (2011)
[6] Mathematical Modeling in Biomedical Imaging II pp 85– (2011)
[7] DOI: 10.1137/090748494 · Zbl 1257.74091 · doi:10.1137/090748494
[8] DOI: 10.1137/100786174 · Zbl 1230.35143 · doi:10.1137/100786174
[9] The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis (2003)
[10] Mathematics & Applications, Vol. 62 (2008)
[11] DOI: 10.1121/1.417118 · doi:10.1121/1.417118
[12] Quantitative Seismology, Vol. 1 (1980)
[13] DOI: 10.1146/annurev.bioeng.5.040202.121623 · doi:10.1146/annurev.bioeng.5.040202.121623
[14] An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I, Linearized Steady Problems (1994)
[15] DOI: 10.1016/j.wavemoti.2006.06.002 · Zbl 1231.76270 · doi:10.1016/j.wavemoti.2006.06.002
[16] Wave Propagation and Time Reversal in Randomly Layered Media (2007) · Zbl 1386.74001
[17] DOI: 10.1016/j.wavemoti.2005.03.001 · Zbl 1189.76459 · doi:10.1016/j.wavemoti.2005.03.001
[18] DOI: 10.1137/S0036139903422371 · Zbl 1126.76011 · doi:10.1137/S0036139903422371
[19] DOI: 10.1088/0266-5611/17/1/301 · doi:10.1088/0266-5611/17/1/301
[20] Phys. Today 50 pp 34– (1997)
[21] Lecture Notes in Computer Science and Engineering, Vol. 59 (2007)
[22] DOI: 10.1121/1.1815075 · doi:10.1121/1.1815075
[23] DOI: 10.1103/PhysRevLett.92.033902 · doi:10.1103/PhysRevLett.92.033902
[24] Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic Solvers on Rectangular Parallelepipeds (1999)
[25] DOI: 10.1103/PhysRevLett.100.064301 · doi:10.1103/PhysRevLett.100.064301
[26] Spectral Methods in Fluid Dynamics (1987) · Zbl 0636.76009
[27] DOI: 10.1002/mma.1404 · Zbl 1213.35400 · doi:10.1002/mma.1404
[28] DOI: 10.1103/PhysRevLett.93.254301 · doi:10.1103/PhysRevLett.93.254301
[29] DOI: 10.1088/0266-5611/18/5/303 · Zbl 1047.74032 · doi:10.1088/0266-5611/18/5/303
[30] DOI: 10.1007/s10483-007-0503-x · Zbl 1231.35264 · doi:10.1007/s10483-007-0503-x
[31] DOI: 10.1088/0266-5611/21/4/015 · doi:10.1088/0266-5611/21/4/015
[32] Mathematical Modeling in Biomedical Imaging I pp 73– (2009)
[33] Inverse Probl. 19 pp 134– (2003)
[34] DOI: 10.1121/1.428630 · doi:10.1121/1.428630
[35] DOI: 10.1109/TUFFC.2004.1367494 · doi:10.1109/TUFFC.2004.1367494
[36] DOI: 10.1137/0705041 · Zbl 0184.38503 · doi:10.1137/0705041
[37] Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory (2007) · Zbl 1220.35001
[38] DOI: 10.1016/S0301-5629(98)00110-0 · doi:10.1016/S0301-5629(98)00110-0
[39] DOI: 10.1017/S030821051000020X · Zbl 1448.74016 · doi:10.1017/S030821051000020X
[40] Elastic Wave Propagation and Generation in Seismology (2003)
[41] Quart. Appl. Math. 66 pp 139– (2008)
[42] DOI: 10.1088/0266-5611/18/6/320 · Zbl 1092.70522 · doi:10.1088/0266-5611/18/6/320
[43] DOI: 10.1137/070684823 · Zbl 1156.35103 · doi:10.1137/070684823
[44] Geophysics 71 pp I33– (2006) · Zbl 1109.85300
[45] DOI: 10.1121/1.1945468 · doi:10.1121/1.1945468
[46] DOI: 10.2478/s13540-013-0003-1 · Zbl 1312.35185 · doi:10.2478/s13540-013-0003-1
[47] DOI: 10.1016/j.jde.2010.07.012 · Zbl 1197.35309 · doi:10.1016/j.jde.2010.07.012
[48] Radio Sci. 40 pp RS6S12– (2005)
[49] DOI: 10.1029/2006GL026336 · doi:10.1029/2006GL026336
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