deletions | additions       diff --git a/VSP methodology.tex b/VSP methodology.tex  new file mode 100644  index 0000000..690d8cd  --- /dev/null  +++ b/VSP methodology.tex 
...
\subsection{Methodology}  Seismic modeling is a technique for simulating wave propagation in the earth. The objective is to predict a seismogram given a composition and structure of the subsurface \citep{Carcione2010}.   Seismic wave propagation can be modeled by solving the wave equations for acoustic, elastic, viscoelastic, poroelastic or poro-viscoelastic media. \\  The elastic coefficients characterizing the porous media are:\\  The stiffness of the matrix ($E_m$)  \begin{equation}   E = K_m + \frac{4}{3}G ,  \end{equation}  The coupling modulus ($M$) between the solid and the fluid  \begin{equation}   M = \frac{K_{s}^{2}}{D-K_m} ,   \end{equation}  where  \begin{equation}   D = K_s \big[1 + \phi(K_s K_{f}^{-1} -1)\big],   \end{equation}  The poroelastic coefficient ($\alpha$) of effective stress.  \begin{equation}   \alpha = 1 - \frac{K_m}{K_s},   \end{equation}  where $K_m$, $K_s$, $K_f$ are the bulk moduli of the drained matrix, the solid and the fluid respectively; $\phi$ is the porosity, and $G$ is the shear modulus of the matrix (drained and saturated).  Poro-viscoelastic formulation represents perhaps the most effective tool to study the effect of the saturating fluid on seismic as fluid proeprties are directely taken into account in the equations \citep{Carcione2006}. \citet{Carcione1998} presented an aproach to introduce viscoelasticity into Biot's poroelastic equations, in which matrix-fluid mechanisms are modeled by generalizing the coupling modulus M to a time dependent relaxation function while the other elastic coefficients are independent of frequency. Detailed information on the implementation of the equations of motion could be found in \citet{Carcione1998} and \citet{Carcione1999}.\\  The seismic modeling in this study is directly inspired by the work of \citet{Carcione2006} that presents a seismic application of porviscoelastic modeling for monitoring underground CO$_2$ storage.   The algorithm used implement the convolutional perfectly matched-layer (PML) absorbing boundary method for solving the poroviscoelastic equations. \citep{Giroux2012}.