Basics

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

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

where for the data vector,

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

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

Construction with InSAR LOS data

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

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

• Note: The difference in sign for convergence and extension for dip-slip and tensile-slip is not idea but it seems consistent with Okada? Should I change this so that it is more intiutive (e.g., positive numbers are always convergence) or leave as is for consistency with Okada?