\appendix

\section{Porous Media Model}
\label{app_biot}

\subsection*{The Model}

The equations of motion for a porous sediment layer may be written in
the form\cite{stern}
\begin{eqnarray*}
\mu \nabla^2 {\bf u} + (H - \mu) \nabla e - C
\nabla \zeta & = & \rho \ddot{\bf u} - \rho_f \ddot{\bf w}
\\
 C \nabla e - M \nabla \zeta & = & \rho_f
\ddot{\bf u} - \frac{c_m \rho_f}{\phi} \ddot{\bf w} - \frac{F
\eta}{\kappa} \dot{\bf w}
\end{eqnarray*}
where  {\bf u}  is the displacement vector associated with the frame
and  ${\bf w} = \phi ( {\bf u} - {\bf u_f} )$ is the weighted (by
porosity) relative displacement of the frame with respect to the pore
fluid.  Also  $e = {\rm div\,} {\bf u}$ and $\zeta = {\rm div\,}{\bf
w}$;
$\rho = \phi \rho_f + (1 - \phi )\rho_s$ is the average density of the
saturated sediment  and  $H$, $C$, and $M$ are (complex) elastic moduli
calculated from the prescribed moduli as
\begin{eqnarray*}
M & = & \frac{K_r}{1 - \frac{K}{K_r} + \phi ( \frac{K_r}{K_f} - 1 )} \\
C & = & \frac{1 - \frac{K}{K_r}}{M} \\
H & = & (1 - \frac{K}{K_r} ) C + K + \frac{4}{3}\mu
\end{eqnarray*}
Finally, $F$ is a frequency dependent drag coefficient which is taken
as\cite{biot}
\[
F = \frac{\xi T ( \xi )}{4 [ 1 + 2 i \frac{T ( \xi )}{\xi} ]}\]
where
\[\xi = a \sqrt{\frac{\omega \rho_f}{\eta}}{~~~\rm and ~~~}
T ( \xi ) = \frac{{\rm ber}'\,(\xi) + i \/ {\rm bei}'\,(\xi)}
{{\rm ber}\,(\xi) + i \/ {\rm bei}\,(\xi)}
\]
The constitutive equations describing the Biot porous sediment model
on which the above equations of motion are based may be written as
\begin{eqnarray*}
{\mbox{\boldmath $\sigma$}} & = & \mu [ \nabla {\bf u} + ( \nabla {\bf
u} )^{\rm T} ] + [(H - 2\mu ) e - c \zeta ] {\rm {\bf I}} \\
p & = & M \zeta - C e
\end{eqnarray*}
where {\mbox{\boldmath $\sigma$}} is the total stress in the saturated
sediment and
$p$  is the pore fluid pressure.

\subsection*{Implementation}

>From this point the development follows the lines given in the SAFARI
Users Guide\cite{hs:saf} and subsequent updates.  Scalar and
vector potentials are introduced so that
\[
{\bf u} = \nabla \Phi + {\rm curl}\,{\bf\Psi}~~~~~~~~
{\bf w} = \nabla \Phi^f + {\rm curl}\,{\bf\Psi}^f
\]
In terms of these potentials, and in the absence of sources, the
equations of motion become
\begin{eqnarray*}
H \nabla^2 \Phi - C \nabla^2 \Phi^f - \rho \frac{\partial^2
\Phi}{\partial t^2} - \rho_f \frac{\partial^2
\Phi^f}{\partial t^2} & = & 0 \\
C \nabla^2 \Phi - M \nabla^2 \Phi^f - \rho_f \frac{\partial^2
\Phi}{\partial t^2} + \frac{c_m \rho_f}{\phi} \frac{\partial^2
\Phi^f}{\partial t^2} - \frac{F \eta}{\kappa} \frac{\partial
\Phi^f}{\partial t} & = & 0 \\
\mu \nabla^2 {\bf\Psi} - \rho \frac{\partial^2 {\bf\Psi}}{\partial
t^2} + \rho_f \frac{\partial^2 {\bf\Psi}^f}{\partial t^2} & = & {\bf
0} \\
\rho_f \frac{\partial^2
{\bf\Psi}}{\partial t^2} + \frac{c_m \rho_f}{\phi} \frac{\partial^2
{\bf\Psi}^f}{\partial t^2} - \frac{F \eta}{\kappa} \frac{\partial
{\bf\Psi}^f}{\partial t} & = & {\bf 0}
\end{eqnarray*}
These equations are first subjected to Fourier transform in time and
either Fourier transform (for plane strain) or Hankel transform (for
axial symmetry) in the plane perpendicular to the depth coordinate
$z$.  We show details assuming plane strain with motion restricted to
the  $xz$-plane  (the axial symmetry case is similar and follows the
outline in the SAFARI Users Guide).  In this event the vector
potentials have only a single nontrivial component, ${\bf \Psi} =
\Psi {\bf j}$, so the displacement components take the form
\begin{eqnarray*}
u = {\Phi,}_x - {\Psi,}_z~~~ & ~~~~~~~~ & ~\, w = {\Phi,}_z +
{\Psi,}_x
\\
u^f = {\Phi^f,}_x - {\Psi^f,}_z & & w^f = {\Phi^f,}_z +
{\Psi^f,}_x
\end{eqnarray*}
Furthermore, the constitutive equations, in terms of the potentials,
become
\begin{eqnarray*}
\sigma_{xx} & = & H {\Phi,}_{xx} + ( H - 2\mu ) {\Phi,}_{zz} - 2\mu
{\Psi,}_{xz} - C ( {{\Phi^f},}_{xx} + {{\Phi^f},}_{zz} ) \\
\sigma_{zz} & = & H {\Phi,}_{zz} + ( H - 2\mu ) {\Phi,}_{xx} - 2\mu
{\Psi,}_{xz} - C ( {{\Phi^f},}_{xx} + {{\Phi^f},}_{zz} ) \\
\sigma_{xz} & = & \mu [ 2 {\Phi,}_{xz} + {\Psi,}_{xx} - {\Psi,}_{zz}
] \\
p & = & M [{{\Phi^f},}_{xx} + {{\Phi^f},}_{zz} ] - C [ {\Phi,}_{xx} +
{\Phi,}_{zz} ]
\end{eqnarray*}
The solution of the transformed equations take the form
\begin{eqnarray*}
\varphi & = & A^- e^{-\alpha z} + A^+ e^{\alpha z} + B^- e^{-\beta z} +
B^+ e^{\beta z} \\
\varphi^f & = & \delta_1 [ A^- e^{-\alpha z} + A^+ e^{\alpha z}] +
\delta_2 [ B^- e^{-\beta z} +
B^+ e^{\beta z} ] \\
\psi & = & i [ C^- e^{-\gamma z} + C^+ e^{\gamma z}] \\
\psi^f & = & \Gamma \psi
\end{eqnarray*}
where $\varphi$ and $\psi$ are the transformed potentials and
$A^{\pm}$, $B^{\pm}$,
and $C^{\pm}$ are functions of angular frequency $\omega$ and wave
number $s$.
Also
$$
\Gamma = \frac{\rho_f}{\frac{c \rho_f}{\phi} - i \frac{F
\eta}{\kappa \omega} }
$$
$$
\delta_1 = \frac{ H ( \alpha^2 - s^2 ) + \rho \omega^2}{C ( \alpha^2 -
s^2 ) + \rho_f \omega^2 } ~~~~~~~~~~
\delta_2 = \frac{ H ( \beta^2 - s^2 ) + \rho \omega^2}{C ( \beta^2 -
s^2 ) + \rho_f \omega^2 }
$$
and
$$
\alpha^2 = s^2 - {h_f}^2 ~~~~~~~~~~
\beta^2 = s^2 - {h_s}^2 ~~~~~~~~~~
\gamma^2 = s^2 - k^2
$$
where  $h_f$, $h_s$, and $k$  are respectively the complex wavenumbers
associated with fast, slow and shear waves in the Biot medium at
angular frequency  $\omega$.  For the shear wave we have
$$
k^2 = \frac{\rho - \Gamma \rho_f}{\mu} \/ \omega^2
$$
while the wave numbers associated with the fast and slow waves are
given by
$$
h^2 = b \pm \sqrt{ b^2 - g }
$$
where
\begin{eqnarray*}
b & = & \frac{M \rho + H [
\frac{c_m \rho_f}{\phi} - i \frac{ F \eta}{\kappa \omega} ]
- 2 C \rho_f}
{2(H M - C^2)}\/ \omega^2 \\
g & = & \frac{\rho [ \frac{c_m \rho_f}{\phi} - i \frac{F \eta
}{\kappa \omega }] - {\rho_f}^2}{H M - C^2} \/ \omega^4
\end{eqnarray*}
and the two compression wave numbers are ordered so that
$\Re (1/h_f) > \Re (1/h_s)$.

In terms of the six degrees of freedom $A^{\pm}$, $B^{\pm}$, and
$C^{\pm}$  for a porous sediment layer, the transformed displacement
and stress components are
\begin{eqnarray*}
\hat{u} & = & i s [ A^- e^{- \alpha z} + A^+ e^{\alpha z} + B^-
e^{-\beta z} + B^+ e^{\beta z}] + i \gamma [ C^- e^{- \gamma z} - C^+
e^{\gamma z}] \\
\hat{w} & = & \alpha [- A^- e^{- \alpha z} + A^+ e^{\alpha z}] +
\beta [ - B^- e^{-\beta z} + B^+ e^{\beta z}] + s [ C^- e^{-
\gamma z} + C^+ e^{\gamma z}] \\
\hat{w}^f & = & \alpha \delta_1 [- A^- e^{- \alpha z} + A^+ e^{\alpha
z}] +
\beta \delta_2 [ - B^- e^{-\beta z} + B^+ e^{\beta z}] + s \Gamma [ C^-
e^{-
\gamma z} + C^+ e^{\gamma z}] \\
\hat{\sigma}_{xx} & = & [( C \delta_1 - H)(s^2 - \alpha^2) - 2 \mu
\alpha^2] [ A^- e^{- \alpha z} + A^+ e^{\alpha z} ] \\
 & & + [( C \delta_2 - H)(s^2 - \beta^2) - 2 \mu
\beta^2] [ B^- e^{- \beta z} + B^+ e^{\beta z} ] + 2 \mu \gamma s [ C^-
e^{-
\gamma z} - C^+ e^{\gamma z}] \\
\hat{\sigma}_{zz} & = & [( C \delta_1 - H)(s^2 - \alpha^2) + 2 \mu
s^2] [ A^- e^{- \alpha z} + A^+ e^{\alpha z} ] \\
 & & + [( C \delta_2 - H)(s^2 - \beta^2) + 2 \mu
s^2] [ B^- e^{- \beta z} + B^+ e^{\beta z} ] - 2 \mu \gamma s [ C^-
e^{-\gamma z} - C^+ e^{\gamma z}] \\
\hat{\sigma}_{xz} & = & i \mu
\left[
2 \alpha s [- A^- e^{- \alpha z} - A^+ e^{\alpha z}] + 2 \beta s
[ B^- e^{- \beta z} + B^+ e^{\beta z} ] \right. \\
 & &  \left. - (s^2 + \gamma^2 ) [ C^- e^{- \gamma z} + C^+
e^{\gamma z}] \right] \\
\hat{p} & = & [ ( C - M \delta_1 )( s^2 - \alpha^2 )] [ A^- e^{-
\alpha z} + A^+ e^{\alpha z} ] \\
 & &  + [ ( C - M \delta_2 )( s^2 - \beta^2
)] [ B^- e^{- \beta z} + B^+ e^{\beta z} ]
\end{eqnarray*}

Boundary conditions to be satisfied at each interface involve
continuity of pore fluid normal  displacement and pressure, and
continuity of displacement and traction components for the frame.  To
complete table 1 of the SAFARI Users guide we add the following table
entries where the symbol $\sim$ means the parameter is not involved,
0 means it should be set to zero, and = means continuity is
enforced at the interface.
In the following table the first four columns correspond to the same
entries in the Users Guide table, while the last column corresponds to
the negative of the traction component $\sigma_{zz}$ in adjacent fluid
layers.

\begin{center}
\begin{tabular}{|l|cccccc|} \hline
\multicolumn{1}{|c|}{Type} & \multicolumn{6}{c|}{Field parameter} \\
\cline{2-7}
 & $w - w^f$ & $u$ & $\sigma_{zz}$ & $\sigma_{xz}$ & $w^f$ & $p$ \\
\hline
Biot/vacuum & $\sim$ & $\sim$ & 0 & 0 & $\sim$ & 0\\
Biot/fluid & = & $\sim$ & = & 0 & $\sim$ & = \\
Biot/solid & = & = & = & = & 0 & $\sim$ \\
Biot/Biot & ~~~=~~~ & ~~~=~~~ & ~~~=~~~ & ~~~=~~~ & ~~~=~~~
& ~~~=~~~ \\ \hline
\end{tabular}
\end{center}


