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\author{Chaiwoot Boonyasiriwat}
\title{Visco-acoustic Modeling}
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\section{Abstract}
This article explains how to model visco-acoustic wave propagation using frequency-domain finite-difference method. In this work, perfectly matched layer was used to prevent spurious reflections from the domain boundary. A MATLAB code was developed to simulate the wave propagation and has been tested with synthetic velocity and quality factor models. Numerical experiment shows the effect of attenuation to the amplitude and phase of waves.
\section{Introduction}
The anelastic attenuation factor or seismic quality factor denoted as $Q$ quantifies the effect of anelastic attenuation on the seismic wavelet caused by fluid movement and grain boundary friction \cite{b:sg95}. It is defined as
\begin{equation}
\label{eq:q_relation}
Q = 2 \pi \frac{E}{\Delta E},
\end{equation}
where $E/\Delta E$ is the fraction of energy lost per cycle. Seismic attnuation is a dispersive behavior because the rate of attenuation increases with frequency.
In addition to the quality factor $Q$, the attenuation coefficient $\alpha$ is another commonly used measure of attenuation \cite{b:tj81}. The quality factor $Q$ and attenuation coefficient $\alpha$ are related as follows.
\begin{equation}
\frac{1}{Q} = \frac{\alpha v}{\pi f},
\end{equation}
where $v$ is the velocity and $f$ is the frequency.
The attenuation coefficient $\alpha$ is defined as the exponential decay constant of the amplitude of a plane wave traveling in a homogeneous medium. For plane wave propagating in a homogeneous medium, the amplitude is given by
\begin{equation}
A(x,t) = A_0 e^{i(kx-\omega t)},
\end{equation}
where $\omega$ is the angular frequency and $k$ is the wavenumber. Attenuation may be introduced mathematically by allowing either the frequency or wavenumber to be complex. In the latter case,
\begin{equation}
k = k_r + i \alpha
\end{equation}
so that
\begin{equation}
A(x,t) = A_0 e^{-\alpha x} e^{i(k_r x-\omega t)},
\end{equation}
where the phase velocity is
\begin{equation}
\label{eq:v}
v = \frac{\omega}{k_r}.
\end{equation}
Now I show the derivation of complex velocity $v_c$ for visco-acoustic modeling.
\begin{eqnarray}
\label{eq:v1}
v_c & = & \frac{\omega}{k} \nonumber \\
& = & \frac{\omega}{k_r + i \alpha} \nonumber \\
& = & \frac{\omega}{k_r + i \alpha} \cdot
\frac{k_r-i\alpha}{k_r-i\alpha} \nonumber \\
& = & \frac{\omega \left( k_r-i\alpha \right)}{k_r^2+\alpha^2}
\end{eqnarray}
Rearranging equation \ref{eq:q_relation} yields
\begin{equation}
\label{eq:alpha}
\alpha = \frac{\pi f}{Qv} = \frac{\omega}{2Qv}.
\end{equation}
Substituting equation \ref{eq:alpha} into equation \ref{eq:v1}, we obtain
\begin{eqnarray}
\label{eq:v2}
v_c & = & \frac{\omega \left( k_r-i \frac{i\omega}{2Qv} \right)}
{k_r^2+ \left( \frac{\omega}{2Qv} \right)^2} \nonumber \\
& = & \frac{\omega k_r \left( 1 - \frac{i\omega}{2Qvk_r} \right)}
{k_r^2 \left( 1 + \left[ \frac{\omega}{2Qvk_r} \right]^2 \right)}
\end{eqnarray}
Using equation \ref{eq:v}, equation \ref{eq:v2} can be rewritten as
\begin{eqnarray}
\label{eq:vc}
v_c & = & v \frac{\left( 1 - \frac{i}{2Q} \right)}
{\left( 1 + \frac{1}{(2Q)^2} \right)} \nonumber \\
& = & v \left( 1 + \frac{i}{2Q} \right)^{-1}
\end{eqnarray}
Equation \ref{eq:vc} is then used to combine velocity and quality factor models to simulate visco-acoustic wave propagation which takes into account the effect of anelastic attenuation.
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