A numerical tour of wave propagation

Split PML (SPML) for acoustics

It is possible for us to split the wave field into two components: x-component $p_x$ and z-component $p_z$ (Carcione et al., 2002). Then the acoustic wave equation becomes

$$\left\{ \begin{split}p=&p_x +p_z\\ \frac{\partial p_x}{\partial... ...}{\partial t}=& \frac{\partial p}{\partial z} \end{split} \right.$$ (22)

To absorb the boundary reflection, by adding the decaying coefficients $d(u)$ the SPML governing equation can be specified as (Collino and Tsogka, 2001)

$$\left\{ \begin{split}p=p_x +p_z&\\ \frac{\partial p_x}{\partial... ...l t}+d(z)v_z &= \frac{\partial p}{\partial z} \end{split} \right.$$ (23)

where $d(x)$ and $d(z)$ are the ABC coefficients designed to attenuate the reflection in the boundary zone, see Figure 8. There exists many forms of ABC coefficients function. In the absorbing layers, we use the following model for the damping parameter $d(x)$ (Collino and Tsogka, 2001):

$$d(u)=d_0(\frac{u}{L})^2, d_0=-\frac{3v}{2L}\ln(R), u=x,z$$ (24)

where $L$ indicates the PML thinkness; $x$ represents the distance between current position (in PML) and PML inner boundary. $R$ is always chosen as $10^{-3}\sim 10^{-6}$ . It is important to note that the same idea can be applied to elastic wave equation (Collino and Tsogka, 2001). The split version of wave equation is very suitable for the construction of seismic Poynting vector. A straightforward application is the angle gather extration using Poynting vector, see Section .

A numerical example of SPML using 8th order staggered finite difference scheme is given in Figure 9.

snapspml
Figure 9. Wavefield snap of SPML with 8th order finite difference

A numerical tour of wave propagation