Ignoring density in waveform inversion

Review of waveform inversion

Our implementation is based on the constant-density acoustic wave equation

$$\left(\nabla^2 - \frac{1}{v(\vec{r})^2}\frac{\partial^2}{\partial t^2}\right)\Psi(\vec{r},t)=0,$$ (1)

where $\psi$ is a pressure-field solution, $\vec{r}$ are the model coordinates ($x$ and $z$ for the two-dimensional case), and $t$ is time. Though the implementation uses time-domain finite differences, it is more convenient to express the equation in frequency $\omega$ and slowness $\sigma$:

$$\left(\nabla^2 + \omega^2\sigma(\vec{r})^2)\right)\Psi(\vec{r},t)=0.$$ (2)