The Gaussian beam approximation

The starting point for the approximation of the acoustic field as a Gaussian beam is given by the analytical expression

$$P(s,\mbox{$\mathbf{n}$}) = \frac {1}{4\pi} \sqrt{ \frac {c_0... ...hbf{n}$}\cdot\mbox{$\mathbf{n}$} \right) \right] \right\} ,$$ (3.1)

where $c_0(s)$ stands for sound speed along a ray trajectory:

$$c_0(s) = c(s,\mbox{$\mathbf{0}$})$$

and $\mbox{$\mathbf{n}$}$ represents the normal to the ray, which can be considered as two-dimensional vector:

$$\mbox{$\mathbf{n}$}= \left[ \begin{array}{c} n_1 \\ n_2 \end{array} \right] ,$$

such that $\mbox{$\mathbf{n}$}\cdot\mbox{$\mathbf{e}$}_s = 0$, where $\mbox{$\mathbf{e}$}_s$ was already defined by Eq.(2.26). In what it follows the projection of sound slowness along $\mbox{$\mathbf{n}$}$ will be written as $\mbox{\boldmath$\sigma$}_n$.

While $\mbox{$\mathbf{n}$}$ is a real vector both matrices $\mbox{$\mathbf{M}$}$ and $\mbox{$\mathbf{Q}$}$ are complex. Therefore, the complex part of the product $\mbox{$\mathbf{M}$}\mbox{$\mathbf{n}$}\cdot\mbox{$\mathbf{n}$}$ induces a Gaussian decay of ray amplitude along the normal (see Fig.3.1), while the real part introduces phase corrections to the travel time along $\mbox{$\mathbf{n}$}$. Additionally, a proper choice of initial conditions for $\mbox{$\mathbf{Q}$}$ will ensure that det $\mbox{$\mathbf{Q}$}\neq0$, which frees the Gaussian beam approximation of singularities.

Figure 3.1: Gaussian beams: amplitude decay along the normal.

Both components of $\mbox{$\mathbf{n}$}$ and $\mbox{\boldmath$\sigma$}_n$ can be considered as being dependent of a particular set of local ray parameters, let's say, ray arclength $s$, plus angles $\alpha_1$ and $\alpha_2$. At any point of the ray one can introduce a set of three orthogonal unit vectors, known as the polarization vectors; naturally, the first polarization vector is $\mbox{$\mathbf{e}$}_s$; the other two polarization vectors, which are going to be represented as $\mbox{$\mathbf{e}$}_1$ and $\mbox{$\mathbf{e}$}_2$, are within the plane perpendicular to $\mbox{$\mathbf{e}$}_s$ (see Fig.3.2).

Figure 3.2:Gaussian beams: polarization vectors.

The vectors $\mbox{$\mathbf{e}$}_1$ and $\mbox{$\mathbf{e}$}_2$ define the possible orientations of $\mbox{$\mathbf{n}$}$ and $\mbox{\boldmath$\sigma$}_n$ at any coordinate $s$ of the ray:

$$\mbox{$\mathbf{n}$}= n_1\mbox{$\mathbf{e}$}_1 + n_2\mbox{$\ma... ... \sigma\mbox{$\mathbf{e}$}_1 + \sigma\mbox{$\mathbf{e}$}_2 .$$

Besides matrices $\mbox{$\mathbf{Q}$}$ and $\mbox{$\mathbf{M}$}$ the Gaussian beam approximation involves two more matrices, called $\mbox{$\mathbf{P}$}$ and $\mbox{$\mathbf{C}$}$; all four matrices are related through the following relationships:

$$\frac{d}{ds}\mbox{$\mathbf{Q}$}= c_0 \mbox{$\mathbf{P}$}\hsk... ...M}$}= \mbox{$\mathbf{P}$}\mbox{$\mathbf{Q}$}^{-1} \hskip5mm ,$$ (3.2)

where

$$\mbox{$\mathbf{P}$}= \left[ \begin{array}{cc} p_{11} & p_{12} \\ p_{21} & p_{22} \end{array} \right] ,$$ (3.3)

$$\mbox{$\mathbf{Q}$}= \left[ \begin{array}{cc} q_{11} & q_{12} \\ q_{21} & q_{22} \end{array} \right] ,$$ (3.4)

and

$$\mbox{$\mathbf{C}$}= \left[ \begin{array}{cc} C_{11} & C_{12} \\ C_{21} & C_{22} \end{array} \right] ,$$ (3.5)

where

$$C_{ij} = \frac{\partial{^2c}}{\partial{n_i\partial n_j}} .$$

The components of $\mbox{$\mathbf{P}$}$ and $\mbox{$\mathbf{Q}$}$ correspond to partial derivatives of the ray normal and normal slowness, along the auxiliary parameters $\alpha_1$ and $\alpha_2$[5]:

$$p_{ij} = \frac{\partial{\sigma_i}}{\partial{\alpha_j}} \hskip... ..._{ij} = \frac{\partial{n_i}}{\partial{\alpha_j}} \hskip5mm .$$