Fourier pseudo spectral method for attenuative simulation with fractional Laplacian

Absorbing boundary condition

. One of the easiest absorbing boundary condtion (ABC) is sponge (or referred to as Gaussian taper) boundary condition proposed by [Cerjan et al.(1985)Cerjan, Kosloff, Kosloff, and Reshef]. The principle is very simple: Attenuating the refections exponentially in the extended artificial boundary (Figure 2) area by multiplying a factor less $d(u)$ than 1, i.e., $d(u)=\mathrm{exp}(-[\alpha(nb-i)]^2), u=x,z (i\Delta x \; \mathrm{or} \; i\Delta z)$ where $nb$ is the thickness of the artificial boundary on each side of the model. A usual choice is $\alpha=0.015$ for $nb=20\sim30$ absorbing layers. The sponge ABC can be easily applied to a wide range of wave propagation problems, including some governing wave equations for complicated medium.

extbndrwidth=0.6A schematic diagram of extended artificial boundary area. $A_1A_2A_3A_4$ is the original model zone, which is extended to be $B_1B_2B_3B_4$ with artificial boundary. In the extended bounary area, the attenuation coeffcient $d(u)\neq 0$ ; In the model zone $A_1A_2A_3A_4$ , $d(u)= 0$ , $u=x,z$ .

Fourier pseudo spectral method for attenuative simulation with fractional Laplacian