Offset-to-angle transformation

In 2-D, the offset-to-angle transformation is done with the relation  

$$\tan \gamma=-\frac{k_{h}}{k_z},$$ (1)

where $\gamma$ is the aperture angle of the reflection, k_h is the offset wavenumber associated with the subsurface horizontal offset, and k_z is the vertical wavenumber. Tisserant and Biondi (2003) presented a 3-D generalization of Equation 1:

$$\begin{eqnarray} \tan \gamma &=&-\frac{k_{h_x}'} {\sqrt{k_z^2+k_{m_y}'^2}},\\ ... ...y} &=&-\frac{ k_{m_y}' k'_{m_x} k'_{h_x} } { k_z^2 + k_{m_y}'^2 }\end{eqnarray}$$ (2) (3)

where ${\bold k_m}$ and boldk_h are the midpoint and offset vector wavenumber, respectively, and where the reflection azimuth, $\beta$, is introduced through

$$\begin{eqnarray} k'_{m_x}&=&\cos \beta k_{m_x} - \sin \beta k_{m_y} \\ k'_{m_y}&... ...eta k_{h_y} \\ k'_{h_y}&=&\cos \beta k_{h_x} + \sin \beta k_{h_y}.\end{eqnarray}$$ (4) (5) (6) (7)

The offset-to-angle transforms a (k_z,k_mx,k_my,k_hx,k_hy) five dimensions cube into another 5-D one $(k_z,k_{m_x},k_{m_y},\gamma,\delta)$.Figure is the measured aperture-azimuth distribution for the configuration displayed in Figure obtained with ray-tracing. We set the lower boundary in $\gamma$ because of an increased incertitude in the estimation of $\beta$ as $\gamma$ gets close to . The upper boundary in $\gamma$ is reached when one of the two rays begins to overturn.

 

beta_f_gamma_v_correct Figure 2 $\gamma-\beta$ distribution corresponding to Figure

We now present a more complex 3-D extension: the one addressing the 3-D full prestack migration with a wrong migration velocity.