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A variational gamma correction model for image contrast enhancement
A stochastic approach to reconstruction of faults in elastic half space
1. | Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA |
2. | Heat and Mass Technological Center (CTTC), Technical University of Catalonia (UPC), Colom 11, 08222 Terrassa (Barcelona), Spain |
We introduce in this study an algorithm for the imaging of faults and of slip fields on those faults. The physics of this problem are modeled using the equations of linear elasticity. We define a regularized functional to be minimized for building the image. We first prove that the minimum of that functional converges to the unique solution of the related fault inverse problem. Due to inherent uncertainties in measurements, rather than seeking a deterministic solution to the fault inverse problem, we then consider a Bayesian approach. The randomness involved in the unknown slip is teased out by assuming independence of the priors, and we show how the regularized error functional introduced earlier can be used to recover the probability density of the geometry parameter. The advantage of this Bayesian approach is that we obtain a way of quantifying uncertainties as part of our final answer. On the downside, this approach leads to a very large computation which we implemented on a parallel platform. After showing how this algorithm performs on simulated data, we apply it to measured data. The data was recorded during a slow slip event in Guerrero, Mexico.
References:
[1] |
H. Ammari, J. Garnier, H. Kang, W.-K. Park and K. Sølna,
Imaging schemes for perfectly conducting cracks, SIAM Journal on Applied Mathematics, 71 (2011), 68-91.
doi: 10.1137/100800130. |
[2] |
B. F. Atwater, A. R. Nelson, J. J. Clague, G. A. Carver, D. K. Yamaguchi, P. T. Bobrowsky, J. Bourgeois, M. E. Darienzo, W. C. Grant, E. Hemphill-Haley et al., Summary of coastal geologic evidence for past great earthquakes at the cascadia subduction zone, Earthquake Spectra, 11 (1995), 1–18.
doi: 10.1193/1.1585800. |
[3] |
E. Beretta, E. Francini, E. Kim and J.-Y. Lee, Algorithm for the determination of a linear crack in an elastic body from boundary measurements, Inverse Problems, 26 (2010), 085015, 13pp.
doi: 10.1088/0266-5611/26/8/085015. |
[4] |
E. Beretta, E. Francini and S. Vessella,
Determination of a linear crack in an elastic body from boundary measurements-lipschitz stability, SIAM Journal on Mathematical Analysis, 40 (2008), 984-1002.
doi: 10.1137/070698397. |
[5] |
L. Borcea, G. Papanicolaou and C. Tsogka, Theory and applications of time reversal and interferometric imaging, Inverse Problems, 19 (2003), S139–S164.
doi: 10.1088/0266-5611/19/6/058. |
[6] |
C. Dascalu, I. R. Ionescu and M. Campillo,
Fault finiteness and initiation of dynamic shear instability, Earth and Planetary Science Letters, 177 (2000), 163-176.
doi: 10.1016/S0012-821X(00)00055-8. |
[7] |
H. Dragert, K. Wang and G. Rogers,
Geodetic and seismic signatures of episodic tremor and slip in the northern Cascadia subduction zone, Earth Planets and Space, 56 (2004), 1143-1150.
doi: 10.1186/BF03353333. |
[8] |
H. Dragert, K. L. Wang and T. S. James,
A silent slip event on the deeper Cascadia subduction interface, Science, 292 (2001), 1525-1528.
doi: 10.1126/science.1060152. |
[9] |
A. Friedman and M. Vogelius, Determining cracks by boundary measurements, Indiana Univ. Math. J., 38 (1989), 527–556, http://conservancy.umn.edu/bitstream/handle/11299/4926/476.pdf.
doi: 10.1512/iumj.1989.38.38025. |
[10] |
I. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, Vol. 18 American Mathematical Society, Providence, R.I. 1969. |
[11] |
G. H. Golub, M. Heath and G. Wahba,
Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.
doi: 10.1080/00401706.1979.10489751. |
[12] |
P. C. Hansen,
Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, 34 (1992), 561-580.
doi: 10.1137/1034115. |
[13] |
I. R. Ionescu and D. Volkov,
Earth surface effects on active faults: An eigenvalue asymptotic analysis, Journal of Computational and Applied Mathematics, 220 (2008), 143-162.
doi: 10.1016/j.cam.2007.08.004. |
[14] |
M. E. Kilmer and D. P. O'Leary,
Choosing regularization parameters in iterative methods for ill-posed problems, SIAM Journal on Matrix Analysis and Applications, 22 (2001), 1204-1221.
doi: 10.1137/S0895479899345960. |
[15] |
V. Kostoglodov, W. Bandy, J. Dominguez and M. Mena,
Gravity and seismicity over the Guerrero seismic gap, Mexico, Geophys. Res. Lett., 23 (1996), 3385-3388.
doi: 10.1029/96GL03159. |
[16] |
R. Kress, V. Maz'ya and V. Kozlov, Linear Integral Equations, vol. 17, Springer, 1989. |
[17] |
J. M. Lee and G. Uhlmann,
Determining anisotropic real-analytic conductivities by boundary measurements, Communications on Pure and Applied Mathematics, 42 (1989), 1097-1112.
doi: 10.1002/cpa.3160420804. |
[18] |
Y. Okada,
Internal deformation due to shear and tensile faults in a half-space, Bulletin of the Seismological Society of America, 82 (1992), 1018-1040.
|
[19] |
J. F. Pacheco and S. K. Singh, Seismicity and state of stress in Guerrero segment of the Mexican subduction zone, J. Geophys. Res., 115 (2010), 28PP.
doi: 10.1029/2009JB006453. |
[20] |
M. Radiguet, F. Cotton, M. Vergnolle, M. Campillo, B. Valette, V. Kostoglodov and N. Cotte,
Spatial and temporal evolution of a long term slow slip event: The 2006 Guerrero Slow Slip Event, Geophysical Journal International, 184 (2011), 816-828.
doi: 10.1111/j.1365-246X.2010.04866.x. |
[21] |
M. Radiguet, F. Cotton, M. Vergnolle, M. Campillo, A. Walpersdorf, N. Cotte and V. Kostoglodov, Slow slip events and strain accumulation in the Guerrero gap, Mexico, Journal of Geophysical Research, 117 (2012), 41PP.
doi: 10.1029/2011JB008801. |
[22] |
G. Suarez, T. Monfret, G. Wittlinger and C. David,
Geometry of subduction and depth of the seismogenic zone in the Guerrero gap, Mexico, Nature, 345 (1990), 336-338.
doi: 10.1038/345336a0. |
[23] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Annals of Mathematics, 125 (1987), 153-169.
doi: 10.2307/1971291. |
[24] |
D. Volkov,
A double layer surface traction free green's tensor, SIAM J. Appl. Math., 69 (2009), 1438-1456.
doi: 10.1137/080723697. |
[25] |
D. Volkov, C. Voisin and I. R. Ionescu,
Determining fault geometries from surface displacements, Pure and Applied Geophysics, 174 (2017), 1659-1678.
doi: 10.1007/s00024-017-1497-y. |
[26] |
D. Volkov,
An eigenvalue problem for elastic cracks in free space, Mathematical Methods in the Applied Sciences, 33 (2010), 607-622.
doi: 10.1002/mma.1182. |
[27] |
D. Volkov, C. Voisin and I. Ionescu, Reconstruction of faults in elastic half space from surface measurements, Inverse Problems, 33 (2017), 055018, 27PP.
doi: 10.1088/1361-6420/aa6360. |
show all references
References:
[1] |
H. Ammari, J. Garnier, H. Kang, W.-K. Park and K. Sølna,
Imaging schemes for perfectly conducting cracks, SIAM Journal on Applied Mathematics, 71 (2011), 68-91.
doi: 10.1137/100800130. |
[2] |
B. F. Atwater, A. R. Nelson, J. J. Clague, G. A. Carver, D. K. Yamaguchi, P. T. Bobrowsky, J. Bourgeois, M. E. Darienzo, W. C. Grant, E. Hemphill-Haley et al., Summary of coastal geologic evidence for past great earthquakes at the cascadia subduction zone, Earthquake Spectra, 11 (1995), 1–18.
doi: 10.1193/1.1585800. |
[3] |
E. Beretta, E. Francini, E. Kim and J.-Y. Lee, Algorithm for the determination of a linear crack in an elastic body from boundary measurements, Inverse Problems, 26 (2010), 085015, 13pp.
doi: 10.1088/0266-5611/26/8/085015. |
[4] |
E. Beretta, E. Francini and S. Vessella,
Determination of a linear crack in an elastic body from boundary measurements-lipschitz stability, SIAM Journal on Mathematical Analysis, 40 (2008), 984-1002.
doi: 10.1137/070698397. |
[5] |
L. Borcea, G. Papanicolaou and C. Tsogka, Theory and applications of time reversal and interferometric imaging, Inverse Problems, 19 (2003), S139–S164.
doi: 10.1088/0266-5611/19/6/058. |
[6] |
C. Dascalu, I. R. Ionescu and M. Campillo,
Fault finiteness and initiation of dynamic shear instability, Earth and Planetary Science Letters, 177 (2000), 163-176.
doi: 10.1016/S0012-821X(00)00055-8. |
[7] |
H. Dragert, K. Wang and G. Rogers,
Geodetic and seismic signatures of episodic tremor and slip in the northern Cascadia subduction zone, Earth Planets and Space, 56 (2004), 1143-1150.
doi: 10.1186/BF03353333. |
[8] |
H. Dragert, K. L. Wang and T. S. James,
A silent slip event on the deeper Cascadia subduction interface, Science, 292 (2001), 1525-1528.
doi: 10.1126/science.1060152. |
[9] |
A. Friedman and M. Vogelius, Determining cracks by boundary measurements, Indiana Univ. Math. J., 38 (1989), 527–556, http://conservancy.umn.edu/bitstream/handle/11299/4926/476.pdf.
doi: 10.1512/iumj.1989.38.38025. |
[10] |
I. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, Vol. 18 American Mathematical Society, Providence, R.I. 1969. |
[11] |
G. H. Golub, M. Heath and G. Wahba,
Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.
doi: 10.1080/00401706.1979.10489751. |
[12] |
P. C. Hansen,
Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, 34 (1992), 561-580.
doi: 10.1137/1034115. |
[13] |
I. R. Ionescu and D. Volkov,
Earth surface effects on active faults: An eigenvalue asymptotic analysis, Journal of Computational and Applied Mathematics, 220 (2008), 143-162.
doi: 10.1016/j.cam.2007.08.004. |
[14] |
M. E. Kilmer and D. P. O'Leary,
Choosing regularization parameters in iterative methods for ill-posed problems, SIAM Journal on Matrix Analysis and Applications, 22 (2001), 1204-1221.
doi: 10.1137/S0895479899345960. |
[15] |
V. Kostoglodov, W. Bandy, J. Dominguez and M. Mena,
Gravity and seismicity over the Guerrero seismic gap, Mexico, Geophys. Res. Lett., 23 (1996), 3385-3388.
doi: 10.1029/96GL03159. |
[16] |
R. Kress, V. Maz'ya and V. Kozlov, Linear Integral Equations, vol. 17, Springer, 1989. |
[17] |
J. M. Lee and G. Uhlmann,
Determining anisotropic real-analytic conductivities by boundary measurements, Communications on Pure and Applied Mathematics, 42 (1989), 1097-1112.
doi: 10.1002/cpa.3160420804. |
[18] |
Y. Okada,
Internal deformation due to shear and tensile faults in a half-space, Bulletin of the Seismological Society of America, 82 (1992), 1018-1040.
|
[19] |
J. F. Pacheco and S. K. Singh, Seismicity and state of stress in Guerrero segment of the Mexican subduction zone, J. Geophys. Res., 115 (2010), 28PP.
doi: 10.1029/2009JB006453. |
[20] |
M. Radiguet, F. Cotton, M. Vergnolle, M. Campillo, B. Valette, V. Kostoglodov and N. Cotte,
Spatial and temporal evolution of a long term slow slip event: The 2006 Guerrero Slow Slip Event, Geophysical Journal International, 184 (2011), 816-828.
doi: 10.1111/j.1365-246X.2010.04866.x. |
[21] |
M. Radiguet, F. Cotton, M. Vergnolle, M. Campillo, A. Walpersdorf, N. Cotte and V. Kostoglodov, Slow slip events and strain accumulation in the Guerrero gap, Mexico, Journal of Geophysical Research, 117 (2012), 41PP.
doi: 10.1029/2011JB008801. |
[22] |
G. Suarez, T. Monfret, G. Wittlinger and C. David,
Geometry of subduction and depth of the seismogenic zone in the Guerrero gap, Mexico, Nature, 345 (1990), 336-338.
doi: 10.1038/345336a0. |
[23] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Annals of Mathematics, 125 (1987), 153-169.
doi: 10.2307/1971291. |
[24] |
D. Volkov,
A double layer surface traction free green's tensor, SIAM J. Appl. Math., 69 (2009), 1438-1456.
doi: 10.1137/080723697. |
[25] |
D. Volkov, C. Voisin and I. R. Ionescu,
Determining fault geometries from surface displacements, Pure and Applied Geophysics, 174 (2017), 1659-1678.
doi: 10.1007/s00024-017-1497-y. |
[26] |
D. Volkov,
An eigenvalue problem for elastic cracks in free space, Mathematical Methods in the Applied Sciences, 33 (2010), 607-622.
doi: 10.1002/mma.1182. |
[27] |
D. Volkov, C. Voisin and I. Ionescu, Reconstruction of faults in elastic half space from surface measurements, Inverse Problems, 33 (2017), 055018, 27PP.
doi: 10.1088/1361-6420/aa6360. |


















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