Reflectivity matrix

The property of the angle-dependent reflectivity matrix R(z_n) in equation () becomes clearer if we look at it in the Fourier domain as de Bruin et al. (1990) does. In order to obtain an expression for $R(k_x,z_n,\omega)$,we start with the well-known angle-dependent reflection coefficient $R(z_n,\alpha)$ for two acoustic half-spaces separated by an interface at z_n:

 

$$R(z_n;\alpha) = {{\rho_2 c_2 \cos \alpha - \rho_1 \sqrt{ c_1... ...c_2 \cos \alpha + \rho_1 \sqrt{ c_1^2 - c_2^2 \sin^2\alpha} }},$$ (44)

where c₁ and c₂ are the velocities, $\rho_1$ and $\rho_2$ are the mass densities of the upper and lower half-space, respectively, and $\alpha$ is the angle of incidence. By substituting $k_1=\omega / c_1$, $k_2=\omega / c_2$, and $k_x = k_1 \sin \alpha$ into equation () we obtain

 

$$R(k_x,z_n,\omega) = {{\rho_2 \sqrt{k_1^2-k_x^2} - \rho_1 \sq... ...\over {\rho_2 \sqrt{k_1^2-k_x^2} + \rho_1 \sqrt{k_2^2-k_x^2}}}.$$ (45)

As an example, Figure  shows the angle-dependent reflection coefficients in the $(k_x,\omega)$ and $(x,\omega)$ domains when c₁=1500 m/s, c₂=3000 m/s, $\rho_1=1000 kg/m^3$, and $\rho_2=1000 kg/m^3$.Since the reflection is a convolution of the downgoing wave field with the reflection coefficient, the reflectivity matrix in equation () can be visualized by taking the reflection coefficient for a given frequency in Figure  and making a matrix whose columns are down-shifted reflection coefficients of each other. Figure  shows this reflectivity matrix when $\omega = \pi / 4$.

 

rc-wk-wx
Figure 1 Angle-dependent reflection coefficients in $(k_x,\omega)$ (left) and $(x,\omega)$ (right) domains when c₁=1500 m/s, c₂=3000 m/s,$\rho_1=1000 kg/m^3$, and $\rho_2=1000 kg/m^3$.

 

rc-mat Figure 2 Reflectivity matrix R for $\omega = \pi / 4$.