Computation of scaling factor $W_{\gamma,\phi}$

The last step is the computation of the scaling factor $W_{\gamma,\phi}$ from the partial derivatives ${\partial k_{x_h}}/{\partial \phi}$and ${\partial k_{y_h}}/{\partial \phi}$.

The unit vector ${\bf u}$tangent to the integration line at constant $\gamma$is given by

$${\bf u} = \left( \frac{ \frac{\partial k_{x_h}}{\partial \ph... ...+ \frac{\partial k_{y_h}}{\partial \phi}^2 } } {\bf y} \right).$$ (23)

The mapping along the azimuth axis of a vector of the same direction as u and length proportional to the sampling in the offset wavenumber domain $\left(\Delta k_{x_h},\Delta k_{y_h}\right)$is a segment of length

$$\delta \phi= \frac{2 \sqrt{\Delta k_{x_h}^2 + \Delta k_{y_h}... ...partial \phi}^2 + \frac{\partial k_{y_h}}{\partial \phi}^2 } }.$$ (24)

If $\Delta \phi$ is the azimuthal range of the stacking at $\gamma=0$,the scaling factor is set as in the following:  

$$W_{\gamma,\phi}= \left\{ \begin{array} {ll} 1 & {\rm if}\; ... ...hi} &{\rm if}\; \delta \phi< \Delta \phi. \\ \end{array}\right.$$ (25)