# 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 & Imaging, 2019, 13 (3) : 479-511. doi: 10.3934/ipi.2019024
##### References:

show all references

##### References:
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 $a, b, d$
 [1] Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1 [2] Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1 [3] Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1 [4] Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems & Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225 [5] Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059 [6] Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042 [7] Daniel Gerth, Andreas Hofinger, Ronny Ramlau. On the lifting of deterministic convergence rates for inverse problems with stochastic noise. Inverse Problems & Imaging, 2017, 11 (4) : 663-687. doi: 10.3934/ipi.2017031 [8] Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79. [9] Davide Guidetti. Some inverse problems of identification for integrodifferential parabolic systems with a boundary memory term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 749-756. doi: 10.3934/dcdss.2015.8.749 [10] Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19 [11] Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems & Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77 [12] Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems & Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183 [13] Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems & Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449 [14] Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002 [15] Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793 [16] François Monard, Guillaume Bal. Inverse diffusion problems with redundant internal information. Inverse Problems & Imaging, 2012, 6 (2) : 289-313. doi: 10.3934/ipi.2012.6.289 [17] Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511 [18] Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577 [19] Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040 [20] Johnathan M. Bardsley. Gaussian Markov random field priors for inverse problems. Inverse Problems & Imaging, 2013, 7 (2) : 397-416. doi: 10.3934/ipi.2013.7.397

2018 Impact Factor: 1.469