Elastic wavefield directionality vectors

Elastic wavefield modeling

The elastic isotropic wave equation in index notation reads:

$$\frac {\partial \sigma_{ii}}{\partial x_i} + \frac {\partial \sigma_{ij}}{\partial x_j} + f_i (\bold X , t) = \rho \ddot U_i,$$ (1)

where $\sigma_{ii}$ are the normal stresses, $\sigma_{ij}$ are the transverse stresses, $f_i$ is the source function in direction $i$ , $X$ is the spatial source location operating at time $t$ , $\rho$ is density and $U$ is the displacement. The stresses are propagated using the stress-displacement relation:

$$\sigma_{ii} = (\lambda + 2 \mu) \frac {\partial U_i}{\partial x_i} + \lambda \frac {\partial U_j}{\partial x_j},$$ (2)

$$\sigma_{ij} = \mu \left( \frac {\partial U_i}{\partial x_j} + \frac {\partial U_j}{\partial x_i} \right),$$ (3)

where $\lambda$ and $\mu$ are the Lame elastic constants.

The finite-difference implementation follows the staggered grid methodology of Virieux (1986). The code I am using implements the variable grid size finite difference method, developed by Wu and Harris (2002), although I have not used the grid size variability so far. The code performs 3D elastic wavefield propagation, and was ported to Fortran90 by Robert Clapp of SEP.

Elastic wavefield directionality vectors