The temporal and spatial operators

The spatial operator was designed to be grid-centered of eighth order (9 points). It can be decomposed in three parts for each component:  

$$\begin{array} {cccc} {\bf A u} & = & \left[ \begin{array} {c... ...( \begin{array} {c} u_x \\ u_z \end{array} \right),\end{array}$$ (10)

where

$$\begin{eqnarray} {\bf a_x} & = & {(\hat{c}_{13} + \hat{c}_{55}) \over \rho \Delt... ..._{xx} + \delta_x \hat{c}_{55} \delta_x) \over \rho (\Delta x)^2}.\end{eqnarray}$$ (11)

${\bf a}$ and ${\bf b}$ are antisymmetric but ${\bf c}$ has no particular symmetry (except in homogeneous media); and their form is represented in Figure . The $\delta$ operators are grid-centered, normalized, bi-dimensional difference stars.

 

spaceop
Figure 4 Spatial difference operators described by equation (11).

The temporal updating uses the operator $\bf P^+$ described in Cunha (1991) in this report. To obtain the wavefield and its time derivative at time time $t + \Delta t$ only requires information from time t, that is,

$$\begin{array} {cccc} \left( \begin{array} {c} {\bf u} \\ {\... ...} {c} {\bf u} \\ \dot{\bf u} \end{array} \right)_t,\end{array}$$

where the forward time-propagation operator $\bf P_{+}$ has the form  

$${\bf P_{+}} = \left[ \begin{array} {ccc} {\bf I} + {{\bf A} ... ... \over 2} dt^2 + {{\bf A}^2 \over 24} dt^4 \end{array} \right],$$ (12)

which represents a fourth order approximation in time.