Basics - brendanjmeade/celeri GitHub Wiki

The structure of the linear operator for the block model problem in `celeri`,

\begin{bmatrix} 
    \mathbf{v} \\
    \omega_\mathrm{c} \\
    \mathbf{s}_\mathrm{c} \\ 
    \mathbf{0} \\ 
    \mathbf{t}_\mathrm{c}(\mathrm{m}_1) \\ 
    \vdots \\ 
    \mathbf{0} \\ 
    \mathbf{t}_\mathrm{c}(\mathrm{m}_n) 
\end{bmatrix} 
=
\begin{bmatrix} 
    \mathbf{R}            & \mathbf{T}(\mathrm{m}_1) & \cdots & \mathbf{T}(\mathrm{m}_n) & \mathbf{E} & \mathbf{M} \\ 
    \mathbf{I}            & \mathbf{0}               & \cdots & \mathbf{0}               & \mathbf{0} & \mathbf{0} \\ 
    \mathbf{R}_\mathrm{s} & \mathbf{0}               & \cdots & \mathbf{0}               & \mathbf{0} & \mathbf{0} \\ 
    \mathbf{0}            & \mathbf{S}(\mathrm{m}_1) & \cdots & \mathbf{0}               & \mathbf{0} & \mathbf{0} \\ 
    \mathbf{0}            & \mathbf{I}               & \cdots & \mathbf{0}               & \mathbf{0} & \mathbf{0} \\ 
    \vdots                & \vdots                   & \ddots & \vdots                   & \vdots     & \vdots \\ 
    \mathbf{0}            & \mathbf{0}               & \cdots & \mathbf{S}(\mathrm{m}_n) & \mathbf{0} & \mathbf{0} \\ 
    \mathbf{0}            & \mathbf{0}               & \cdots & \mathbf{I}               & \mathbf{0} & \mathbf{0} 
\end{bmatrix} 
\begin{bmatrix} 
    \omega \\ 
    \mathbf{t}(\mathrm{m}_1) \\ 
    \vdots \\ 
    \mathbf{t}(\mathrm{m}_n) \\ 
    \epsilon \\ 
    \mathbf{m} 
\end{bmatrix}

where for the data vector,

$\mathbf{v}$ is a vector of geodetic velocities,

$\omega_\mathrm{c}$ are a priori constraints on block motion,

$\mathbf{s}_\mathrm{c}$ are a priori constraints on fault slip rates,

$\mathbf{0}$ is a data constraint for the TDE smoothing matrix,

$\mathbf{t}^1_\mathrm{c}$ are a priori constraints on TDE slip rates for mesh $i$.

For the design matrix, $\mathbf{R}$ is an operator relating block motion and elastic deformation around fully coupled segments to geodetic velocities, $\mathbf{T}^i$ relates TDE slip rates to geodetic velocities for mesh $i$, $\mathbf{E}$ relates homogeneous intrablock strain rates to geodetic velocities, and $\mathbf{P}$ relates volume change rates at Mogi sources to geodetic velocities. For constraints, $\mathbf{R}_\mathrm{s}$ projects block motion into fault slip rates and $\mathbf{S}^i$ smooths TDE slip rates.

Finally, we estimate $\omega$ as block motion vectors, $\mathbf{t}^i$ as TDE slip rates for mesh $i$, $\epsilon$ as intrablock strain rates, and $\mathbf{p}$ as volume change rates at Mogi sources.

Construction with InSAR LOS data

\begin{bmatrix} 
    \mathbf{v} \\
    \mathbf{LOS} \\
    \omega_\mathrm{c} \\
    \mathbf{s}_\mathrm{c} \\ 
    \mathbf{0} \\ 
    \mathbf{t}_\mathrm{c}(\mathrm{m}_1) \\ 
    \vdots \\ 
    \mathbf{0} \\ 
    \mathbf{t}_\mathrm{c}(\mathrm{m}_n) 
\end{bmatrix} 
=
\begin{bmatrix} 
    \mathbf{R}_\mathbf{v}            & \mathbf{T}_\mathbf{v}(\mathrm{m}_1) & \cdots & \mathbf{T}_\mathbf{v}(\mathrm{m}_n) & \mathbf{E}_\mathbf{v} & \mathbf{M}_\mathbf{v} \\ 
    \mathbf{R}_\mathbf{LOS}            & \mathbf{T}_\mathbf{LOS}(\mathrm{m}_1) & \cdots & \mathbf{T}_\mathbf{LOS}(\mathrm{m}_n) & \mathbf{E}_\mathbf{LOS} & \mathbf{M}_\mathbf{LOS} \\ 
    \mathbf{I}            & \mathbf{0}               & \cdots & \mathbf{0}               & \mathbf{0} & \mathbf{0} \\ 
    \mathbf{R}_\mathrm{s} & \mathbf{0}               & \cdots & \mathbf{0}               & \mathbf{0} & \mathbf{0} \\ 
    \mathbf{0}            & \mathbf{S}(\mathrm{m}_1) & \cdots & \mathbf{0}               & \mathbf{0} & \mathbf{0} \\ 
    \mathbf{0}            & \mathbf{I}               & \cdots & \mathbf{0}               & \mathbf{0} & \mathbf{0} \\ 
    \vdots                & \vdots                   & \ddots & \vdots                   & \vdots     & \vdots \\ 
    \mathbf{0}            & \mathbf{0}               & \cdots & \mathbf{S}(\mathrm{m}_n) & \mathbf{0} & \mathbf{0} \\ 
    \mathbf{0}            & \mathbf{0}               & \cdots & \mathbf{I}               & \mathbf{0} & \mathbf{0} 
\end{bmatrix} 
\begin{bmatrix} 
    \omega \\ 
    \mathbf{t}(\mathrm{m}_1) \\ 
    \vdots \\ 
    \mathbf{t}(\mathrm{m}_n) \\ 
    \epsilon \\ 
    \mathbf{m} 
\end{bmatrix}

Locking depth is positive down

Summary of Okada slip rate conventions:

type	sign	interpretation
strike-slip	positive	left-lateral
strike-slip	negative	right-lateral
dip-slip	positive	convergence
dip-slip	negative	extension
tensile-slip	positive	extension
tensile-slip	negative	convergence