Elastic Dislocation Modelling using Metropolis MCMC

Fault Kinematics Along the San Andreas Fault from GPS Data Using Metropolis MCMC

Nicolás Castro-Perdomo

Indiana University, 2022

Main Goal:

• Implement a random walk Metropolis sampling algorithm to estimate fault kinematic parameters $(a, v_0, D_L, x_0)$ in an elastic half-space dislocation model (e.g., Weertman and Weertman, 1964; Savage and Burford, 1973).

• The model describes the theoretical horizontal velocity profile across a vertical fault as a function of the spatial variable $x$:

$$ v(x) = a + \frac{v_0}{\pi} tan^{-1} \Big( \frac{x-x_0}{D_L} \Big) $$

where $a$ is a constant vertical shift applied to the velocity profile, $v_0$ is the fault slip rate, $D_L$ is the fault locking depth, and $x_0$ is the fault location.

Model parameters

• The parameter domain is defined as follows:

  • x $\in$ [−150, 150] km
  • a $\in$ [-5, 5] mm/yr
  • $v_0$ $\in$ [0, 50] mm/yr
  • $D_L$ $\in$ [0, 50] km
  • $x_0$ $\in$ [−25, 25] km

• A Gaussian error model will be used, so that at any given location $x_j$, fault-parallel velocities satisfy:

$$ v_j = v(x_j) + \eta_j $$

where $\eta_j$ $\sim\mathcal{N}$(0, 1) and all $\eta_j$ are independently identically distributed

Metropolis MCMC:

Grid search inversion: