\subsubsection{Elastic properties}
Elastic properties are usually computed through a rock-physics model. This model is a set of equations that transform petrophysical variables, typically porosity, mineralogy and fluid saturations, into elastic properties such as $P$- and $S$-wave velocity and density. For the purpose of this work, we first estimate the elastic properties of the solid phase, i.e.\ bulk ($K_s$) and shear ($G_s$) moduli, by applying the arithmetic average of the upper and lower Hashin-Shtrikman bounds \citep{Hashin1963}. Then we compute the elastic properties of the fluid phase, i.e.\ bulk ($K_f$) modulus and density ($\rho_f$), using fluid mixing laws. 

The new CO_2 saturated rock properties $K_{sat}$ is then calculated using Gassmann's relation \cite{Gassmann}:
\begin{equation}
K_{sat} = K_{dry} + \dfrac{\bigg(1-\dfrac{K_{dry}}{K_s}\bigg)^2}{\dfrac{\phi}{K_{fl}}+\dfrac{(1-\phi)}{K_s}-\dfrac{K_{dry}}{K^2_s}},
\label{eq:gassmann}
\end{equation}
where $\phi$ refers to porosity. The shear modulus of the saturated rock is simply the modulus of the dry rock ($G_{sat}$=$G_{dry}$). 
The saturated density $\rho_{sat}$ is computed as a linear combination of the solid density $\rho_s$ and fluid density $\rho_{fl}$ weighted by their respective volume fractions:
\begin{equation}
\rho_{sat} = \phi\ \rho_{fl} + (1 - \phi)\rho_s,
\label{eq:rho}
\end{equation}
and the new $P$- and $S$-wave velocity are calculated using the saturated elastic properties $K_{sat}$ and $G_{sat}$ and density $\rho$:
\begin{equation} 
\label{eq:Vp}
V_p = \sqrt{\Bigg(\dfrac{K_{sat}+\tfrac{4}{3}G_{sat}}{\rho_{sat}}\Bigg)},
\end{equation}
and
\begin{equation} 
\label{eq:Vs}
V_s = \sqrt{\dfrac{G_{sat}}{\rho_{sat}}}.
\end{equation}
Then, a synthetic full-wave seismic response $d_{synth}$ is computed  using an elastic finite-difference time-domain approach \cite{Bohlen_2002}.
The mismatch between $d_{synth}$ and $d_{obs}$ is evaluated using the same objective function of the previous step.
At the end of the stochastic inversion process, we obtain the fields of $V_p$, $V_s$, density ($\rho$) and porosity ($\phi$) that best honor static and dynamic data.
    
    
    
  