A stochastic approach to reconstruction of faults in elastic half space

Figure 1.  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

Figure 2.  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

Figure 3.  Test case 1. Examples of plots of $ C(i_1, i_2, i_3) $ against $ \mathrm{{\bf Err}}/\|u^{(3N)}\| $

Figure 4.  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 $

Figure 5.  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

Figure 6.  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 $

Figure 7.  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

Figure 8.  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 $\sigma_{hor} = 3, \sigma_{ver} = 9$

Figure 9.  Test case 2. Reconstructed slip field

Figure 10.  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

Figure 11.  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 $

Figure 12.  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

Figure 13.  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 $

Figure 14.  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

Figure 15.  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 $

Figure 16.  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 $

Figure 17.  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

Figure 18.  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

Figure 19.  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 $