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Noise model

Consider a medium whose behavior is completely defined by the acoustic velocity, i.e. assume that the density $\rho \left (x,y,z \right)= \rho_0$ is constant and the velocity $v \left (x,y,z \right)$ fluctuates around a homogenized value $v_0 \left (x,y,z\right)$ according to the relation

\frac{1} {v^2 \left (x,y,z \right)} =
\frac{1 + \sigma m \left (x,y,z \right)}{v_0^2 \left (x,y,z \right)}\;,
\end{displaymath} (12)

where the parameter $m$ characterizes the type of random fluctuations present in the velocity model, and $\sigma$ denotes their strength.

Consider the covariance orientation vectors

$\displaystyle {\bf a}$ $\textstyle =$ $\displaystyle \left (a_x,a_y,a_z\right)^{\top} \in \mathbb{R}^3$ (13)
$\displaystyle {\bf b}$ $\textstyle =$ $\displaystyle \left (b_x,b_y,b_z\right)^{\top} \in \mathbb{R}^3$ (14)
$\displaystyle {\bf c}$ $\textstyle =$ $\displaystyle \left (c_x,c_y,c_z\right)^{\top} \in \mathbb{R}^3$ (15)

defining a coordinate system of arbitrary orientation in space. Let $r_a, r_b, r_c > 0$ be the covariance range parameters in the directions of ${\bf a}$,${\bf b}$ and ${\bf c}$, respectively.

We define a covariance function

\mathrm{cov} \left (x,y,z \right)= \exp \left [-l^{\alpha} \left (x,y,z \right)\right]\;,
\end{displaymath} (16)

where $\alpha \in [0,2]$ is a distribution shape parameter and
l \left (x,y,z \right)=
\sqrt{\left (\frac{{\bf a}\cdot {\b...
...ight)^2 +
\left (\frac{{\bf c}\cdot {\bf r}}{r_c} \right)^2}
\end{displaymath} (17)

is the distance from a point at coordinates ${\bf r}=\left (x,y,z \right)$ to the origin in the coordinate system defined by $\{r_a {\bf a}, r_b {\bf b}, r_c
{\bf c}\}$.

Given the IID Gaussian noise field $n \left (x,y,z \right)$, we obtain the random noise $m \left (x,y,z \right)$ according to the relation

m \left (x,y,z \right)=
\left [\sqrt{\wideh...
...ight)} \;
\widehat{ n} \left (k_x,k_y,k_z \right)
\end{displaymath} (18)

where $k_x,k_y,k_z$ are wavenumbers associated with the spatial coordinates $x,y,z$, respectively. Here,
$\displaystyle \widehat{\mathrm{cov}}$ $\textstyle =$ $\displaystyle \mathscr{F} \left [\mathrm{cov}\right]$ (19)
$\displaystyle \widehat{n}$ $\textstyle =$ $\displaystyle \mathscr{F} \left [n \right]$ (20)

are Fourier transforms of the covariance function $\mathrm{cov}$ and the noise $n$, $\mathscr{F}[\cdot]$ denotes Fourier transform, and $\mathscr{F}^{-1}[\cdot]$ denotes inverse Fourier transform. The parameter $\alpha$ controls the visual pattern of the field, and ${\bf a},{\bf b},{\bf c}, r_a,r_b,r_c$ control the size and orientation of a typical random inhomogeneity.

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