# American Institute of Mathematical Sciences

June  2019, 13(3): 479-511. doi: 10.3934/ipi.2019024

## 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

* Corresponding author

Received  March 2018 Revised  November 2018 Published  March 2019

Fund Project: Results in this paper were obtained in part using a high-performance computing system acquired through NSF MRI grant DMS-1337943 to WPI.
The first author is supported by a Simons Foundation Collaboration Grant.

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.

Citation: Darko Volkov, Joan Calafell Sandiumenge. A stochastic approach to reconstruction of faults in elastic half space. Inverse Problems and Imaging, 2019, 13 (3) : 479-511. doi: 10.3934/ipi.2019024
##### 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.
The Guerrero gap region of Mexico. The subduction zone studied in this paper meets the sea floor of the Pacific ocean along a nearly linear course called the Middle American Trench: it appears on this figure as a dashed line. The large triangles mark the locations of GPS stations that were used to record the 2006 Guerrero SSE
Test case 1. Top left: the fault $\Gamma$ and the slip field $\tilde{ \mathit{\boldsymbol{h}}}$. The red line is $x_3 = -2$, the blue line is $x_3 = -40$, on the plane $x_3 = ax_1 + bx_2 + d$. The circles stand for the surface measurement points $P_j$. They appear on the map in Figure 1. Units for surface distances are kilometers. Color bar shows $|\tilde{ \mathit{\boldsymbol{h}}}|$, in meters. $\tilde{ \mathit{\boldsymbol{h}}}$ points in the direction of steepest ascent. The next two panels show the resulting surface displacements at the $P_j$s. The red line segment indicates the scale: 100 mm
Test case 1. Examples of plots of $C(i_1, i_2, i_3)$ against $\mathrm{{\bf Err}}/\|u^{(3N)}\|$
Test case 1. Computed marginal distributions for the geometry parameters $a$, $b$, and $d$. The blue star curve corresponds to the assumption $\sigma_{hor} = 1, \sigma_{ver} = 3$, the red circle curve corresponds to the assumption $\sigma_{hor} = 2, \sigma_{ver} = 6$, and the orange cross curve corresponds to the assumption $\sigma_{hor} = 3, \sigma_{ver} = 9$
Test case 1. Solid grid: the points with depth $x_3<0$ are on the plane containing the fault. Dashed grid: the points with depth $x_3<0$ are on the reconstructed plane. Red arrows: slip field. Blue arrows: reconstructed slip field. Top graph: maximum likelihood solution. Bottom graph: example of a reconstructed geometry two standard deviations away from the maximum likelihood solution for the case $\sigma_{hor} = 3$ and $\sigma_{ver} = 6$, see Figure 4. To improve legibility, the grids on these figures are coarser than the grids used in computations
Test case 1. Left: Computed local minima of the error functional (3.2) from [27]. These minima were obtained by using a grid of starting points. Right: corresponding values of $a, b, d$
Test case 2. Top left: the fault $\Gamma$ and the slip field $\tilde{ \mathit{\boldsymbol{h}}}$. The red line is $x_3 = -2$, the blue line is $x_3 = -40$, on the plane $x_3 = ax_1 + bx_2 + d$. The circles stand for the surface measurement points $P_j$. Units for surface distances are kilometers. Color bar shows $|\tilde{ \mathit{\boldsymbol{h}}}|$, in meters. $\tilde{ \mathit{\boldsymbol{h}}}$ points in the direction of steepest ascent. The next two panels show the resulting surface displacements at the $P_j$s. The red line segment indicates the scale: 100 mm
Test case 2: computed marginal distributions for the geometry parameters $a$, $b$, and $d$. The blue star curve corresponds to the assumption $\sigma_{hor} = 1, \sigma_{ver} = 3$, the red circle curve corresponds to the assumption $\sigma_{hor} = 2, \sigma_{ver} = 6$, and the orange cross curve corresponds to the assumption $\sigma_{hor} = 3, \sigma_{ver} = 9$
Test case 2. Reconstructed slip field
Test case 2. Solid grid: the points with depth $x_3<0$ are on the plane containing the fault. Dashed grid: the points with depth $x_3<0$ are on the reconstructed plane. Red arrows: slip field. Blue arrows: reconstructed slip field. Top graph: maximum likelihood solution. Bottom graph: example of a reconstructed geometry two standard deviations away from the maximum likelihood solution for the case $\sigma_{hor} = 3$ and $\sigma_{ver} = 6$, see Figure 8. To improve legibility, the grids on these figures are coarser than the grids used in computations
Test case 2. Left: Computed local minima of the error functional (3.2) from [27]. These minima were obtained by using a grid of starting points. Right: corresponding values of $a, b, d$
Test case 3: the fault $\Gamma$ and the slip field $\tilde{ \mathit{\boldsymbol{h}}}$. The red line is $x_3 = -2$, the blue line is $x_3 = -40$, on the plane $x_3 = ax_1 + bx_2 + d$. The circles stand for the surface measurement points $P_j$. Units for surface distances are kilometers. Color bar shows $|\tilde{ \mathit{\boldsymbol{h}}}|$, in meters. The red line segment indicates the scale: 100 mm. The direction of steepest ascent is indicated by the green arrow while the red arrow indicates the direction of slip
Test case 3: computed marginal distributions for the geometry parameters $a$, $b$, and $d$. The blue star curve corresponds to the assumption $\sigma_{hor} = 1, \sigma_{ver} = 3$, the red circle curve corresponds to the assumption $\sigma_{hor} = 2, \sigma_{ver} = 6$, and the orange cross curve corresponds to the assumption $\sigma_{hor} = 3, \sigma_{ver} = 9$
Solid grid: the points with depth $x_3<0$ are on the plane containing the fault. Dashed grid: the points with depth $x_3<0$ are on the reconstructed plane. Red arrows: slip field. Blue arrows: reconstructed slip field. Top graph: maximum likelihood solution. Bottom graph: example of a reconstructed geometry two standard deviations away from the maximum likelihood solution for the case $\sigma_{hor} = 3$ and $\sigma_{ver} = 6$, see Figure 13. To improve legibility, the grids on these figures are coarser than the grids used in computations
Test case 3. Left: Computed local minima of the error functional (3.2) from [27]. These minima were obtained by using a grid of starting points. Right: corresponding values of $a, b, d$
The 2007 Guerrero SSE. Computed marginal distributions for the geometry parameters $a$, $b$, and $d$. The blue star curve corresponds to the assumption that $\sigma_{hor} = .5, \sigma_{ver} = 1.5$, the red circle curve corresponds to the assumption that $\sigma_{hor} = 1, \sigma_{ver} = 3$, and the orange cross curve corresponds to the assumption that $\sigma_{hor} = 2, \sigma_{ver} = 6$
Computed average slip (left) and standard deviation (right) for the Guerrero 2007 SSE. Note the change of scale for the color bars between the two figures
Computed geometry profile and slip 2007 SSE. Dashed grid: the points with depth $x_3 < 0$ are on the expected plane containing the fault. The corresponding slip field is represented using blue arrows. Dotted grid: the points with depth $x_3 < 0$ are on a plane two standard deviations away from the expect plane containing the fault. The corresponding slip field is represented using green arrows
Computed geometry profile and slip 2007 SSE. Left: Computed local minima of the error functional (3.2) from [27]. These minima were obtained by using a grid of starting points. Right: corresponding values of $a, b, d$
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