Mondaic - Full Waveform Solutions

There exists a whole zoo of different parameterizations to describe the acoustic or elastic properties of a medium.

Salvus offers a lot of flexibility and supports various parameterizations. Because conversion can be a tedious and error-prone task, we list several parameter options and how the translate into each other below.

Internally, Salvus uses an abstract formulation of the acoustic wave equation, that allows to treat different parameterizations within the same implementation:

m_0 \partial_t^2 \phi - \nabla \cdot \big(m_1 \nabla \phi \big) = f.

Different choices of $m_0$ and $m_1$ lead to a different physical meaning of $\phi$ and $f$ .

parameter	symbol	unit	Salvus key
$m_0$	$m_0$	variable	`M0`
$m_1$	$m_1$	variable	`M1`
density	$\rho$	$kg \,m^{-3}$	`RHO`
velocity (sound speed)	$v_p$ or $c$	$m \,s^{-1}$	`VP`
impedance	$\mathcal{I}$	$kg \,m^{-2}\,s^{-1}$	coming soon
compressibility	$\beta$	$m\,s^2\,kg^{-1}$	coming soon

Below, we list conversion formulas for a displacement potential $\phi$ .

	$m_0 =$	$m_1=$	$\rho =$	$v_p =$	$\mathcal{I} =$	$\beta =$
$\left(\begin{array}{cc}m_0\\m_1\end{array}\right)$			$m_1^{-1}$	$\sqrt{\frac{m_1}{m_0}}$	$\frac{1}{\sqrt{ m_0 \, m_1 }}$	$m_0$
$\left(\begin{array}{cc}\rho\\v_p\end{array}\right)$	$\frac{1}{\rho\, v_p^2}$	$\frac{1}{\rho}$			$\rho\, v_p$	$\frac{1}{\rho \, v_p^2}$
$\left(\begin{array}{cc}\mathcal{I}\\v_p\end{array}\right)$	$\frac{1}{\mathcal{I}\, v_p}$	$\frac{v_p}{\mathcal{I}}$	$\frac{\mathcal{I}}{v_p}$			$\frac{1}{\mathcal{I}\, v_p}$
$\left(\begin{array}{cc}\mathcal{I}\\\rho\end{array}\right)$	$\frac{\rho}{\mathcal{I}^2}$	$\frac{1}{\rho}$		$\frac{\mathcal{I}}{\rho}$		$\frac{\rho}{\mathcal{I}^2}$
$\left(\begin{array}{cc}\beta\\\rho\end{array}\right)$	$\beta$	$\frac{1}{\rho}$		$\frac{1}{\sqrt{\rho\,\beta}}$	$\sqrt{\frac{\rho}{\beta}}$

Salvus can handle isotropic and anisotropic elastic media and supports different parameterizations.

In isotropic media, the elastic properties reduce to three parameters, which can either be expressed using the Lamé coefficients or by the velocities of compressional (P) and shear (S) waves.

parameter	symbol	unit	Salvus key
first Lamé coefficient	$\lambda$	$Pa$	`LAMBDA`
second Lamé coefficient, shear modulus	$\mu$	$Pa$	`MU`
density	$\rho$	$kg \,m^{-3}$	`RHO`
P-wave velocity	$v_p$	$m \,s^{-1}$	`VP`
S-wave velocity	$v_s$	$m \,s^{-1}$	`VS`

Here, the following conversion formula apply.

	$\lambda =$	$\mu =$	$v_p =$	$v_s =$
$\left(\begin{array}{c}\lambda\\\mu\\\rho\end{array}\right)$			$\sqrt{\frac{\lambda + 2\mu}{\rho}}$	$\sqrt{\frac{\mu}{\rho}}$
$\left(\begin{array}{c}\rho\\v_p\\v_s\end{array}\right)$	$\rho\,(v_p^2-2v_s^2)$	$\rho\,v_s^2$

Due to the symmtetry relations

\mathbf{C}_{ijkl} = \mathbf{C}_{klij} = \mathbf{C}_{jikl}

the fourth-order elastic tensor $\mathbf{C}$ reduces to at most 21 (in 3-D) or 6 (in 2-D) independent parameters. Using Voigt notation, these are fully specified by

parameter	symbol	unit	Salvus key
density	$\rho$	$kg \,m^{-3}$	`RHO`
Elastic tensor	$c_{ij}$		`Cij`

with 1 $\leq$ i $\leq$ j $\leq$ 6 in 3-D, and with 1 $\leq$ i $\leq$ j $\leq$ 3 in 2-D, respectively.

Tilted transversely isotropic material is an important special case of anisotropic media with are only two additional parameters compared to isotropic material that distinguish the wave speeds of horizontally and vertically traveling waves.

parameter	symbol	unit	Salvus key
first Lamé coefficient	$\lambda$	$Pa$	`LAMBDA`
second Lamé coefficient, shear modulus	$\mu$	$Pa$	`MU`
density	$\rho$	$kg \,m^{-3}$	`RHO`
horizontal P-wave velocity	$v_{ph}$	$m \,s^{-1}$	`VPH`
vertical P-wave velocity	$v_{pv}$	$m \,s^{-1}$	`VPV`
horizontal S-wave velocity	$v_{sh}$	$m \,s^{-1}$	`VSH`
vertical S-wave velocity	$v_{sv}$	$m \,s^{-1}$	`VSV`