Kinematics in iterated correlations of a passive acoustic experiment

Green's function in the Born Approximation

We are interested in the Green's function in an inhomogeneous medium. We assume the velocity can be split into a background velocity $c_0$ and a perturbation $\alpha(\mathbf{x})$ as $c^{-2}(\mathbf{x}) = c^{-2}_0\left[ 1+ \alpha(\mathbf{x})\right]$. Assuming the perturbation is confined inside some finite domain $\mathbf{D}_s$, the Green's function in the Born approximation can now be computed in terms of a Green's function computed in the background, $G_0$, medium as

$$G(\mathbf{x},\mathbf{x}_s,\omega) = G_0(\mathbf{x},\mathbf{x}_s,\omega) + G_1(\mathbf{x},\mathbf{x}_s,\omega),\hspace{.5cm} \mathrm{with}$$ (70)

$$G_1(\mathbf{x},\mathbf{x}_s,\omega) = \oint_ {\mathbf{D}_s} G_0(\mathbf{x},\mathbf{x}',\omega) \frac{\o... ...\alpha(\mathbf{x}') G_0(\mathbf{x}',\mathbf{x}_s,\omega) \mathrm{d}\mathbf{x}'.$$ (71)

The Green's function in the background medium is computed using equation A-5 with $c=c_0$. When the medium consists of a homogeneous background with a series of $N$ scatters positioned at $\mathbf{x}_{c,1}, \mathbf{x}_{c,2},\mathbf{x}_{c,3} ... \mathbf{x}_{c,N}$ with strength $\alpha_1, \alpha_2, \alpha_3, ... \alpha_N$, then $\alpha(\mathbf{x}) = \displaystyle\sum_{i=1}^{N}\delta(\mathbf{x}-\mathbf{x}_{c,i})\alpha_i$. Hence the Green's function $G_1$ in equation A-10 can be written as

$$G_1(\mathbf{x},\mathbf{x}_s,\omega) = \displaystyle\sum_{i=1}^{N}... ...mega) \frac{\omega^2}{c_0^2}\alpha_i G_0(\mathbf{x}_{c,i},\mathbf{x}_s,\omega).$$ (72)

Kinematics in iterated correlations of a passive acoustic experiment