de Bruin's imaging condition

In order to retrieve the full reflectivity matrix, de Bruin et al. (1990) proposes another imaging condition (Figure ). Instead of deconvolving in the spatial domain, they deconvolve in the Fourier domain $(k_x,\omega)$after Fourier transforming both downgoing and upcoming waves, as follows:  

$$X(k_x,z_n,\omega)={{{\bf g}(k_x,z_n,\omega){\bf s}^\ast(k_x,... ... s}(k_x,z_n,\omega){\bf s}^\ast(k_x,z_n,\omega) + \epsilon^2}}.$$ (52)

The next step is to transform $X(k_x,z_n,\omega)$ into $X(p,z_n,\omega)$by replacing the wavenumber k_x with the ray parameter $p = k_x / \omega$.Then imaging is carried out in the $p - \omega$domain along the lines of the constant p:  

$${\bf r}(p,z_n) = \sum_{\omega} X(p,z_n,\omega).$$ (53)

This imaging condition retrieve the full angle-dependent reflectivity. However, since the deconvolution is performed in the Fourier domain, it implies a spatially invariant reflectivity, which is the case of horizontal reflector in constant velocity medium. Thus it has the disadvantage that is of being difficult to implement a realistic model.

 

fk-img
Figure 5 Schematic representation of de Bruin's imaging condition