From modeling to full waveform inversion: A hands-on tour using Madagascar

Forward modeling

The wave equation we consider in this course material

$$(\frac{1}{v^2}\partial_t^2 - \nabla^2 ) p=f$$ (1)

Omitting the source, extrapolate your wavefield:

$$p^{k+1}=2p^{k}-p^{k-1}+\Delta t^2 v^2 \nabla^2 p^k$$ (2)

where

$$\nabla^2 p= \frac{p[ix][iz+1]-2p[ix][iz]+p[ix][iz-1]}{\Delta z^2} +\frac{p[ix-1][iz]-2p[ix][iz]+p[ix-1][iz]}{\Delta x^2}$$ (3)

The Clayton-Enquist absorbing boundary condition (ABC) (Clayton and Engquist, 1977)

$$:\frac{\partial^2 p}{\partial x\partial t}-\frac{1}{v}\frac{\partial^2 p}{\partial t^2}=\frac{v}{2}\frac{\partial^2 p}{\partial z^2}$$ (4)

The codes in every time step looks like