step 3

Backward time propagation of the recorded particle-displacement field and ${\bf w}$ and its time-derivative ${\bf \dot{w}}$, starting at the end time of the shot record and ending at time zero. These wavefields are introduced as time-dependent boundary conditions at z=0. Simultaneously with this backward propagation, this step also computes the lag zero of the correlation between the downgoing scalar field u(x,z) and the upcoming scalar field w(x,z). For each time step the following arrays are computed:

• Scalar upcoming particle-displacement field

  $$w(x,z,t) = \mbox{sign}(\dot{w}_z) \sqrt{\dot{w}_x^2 + \dot{w}_z^2}$$

• Unit vector in the group propagation direction of the upcoming wavefield

  $${\bf \hat{r}}(x,z,t).$$

Finally, the reflection coefficient image is computed as and the local Snell parameter image as

$$\check{p}(x,z) = {\int \mid u(x,z,t) w(x,z,t) \mid \, \tilde{p}(x,z,t) \, dt \over \int \mid u(x,z,t) w(x,z,t) \mid \, dt},$$

where $\tilde{p}(x,z,t)$ is computed from ${\bf \hat{i}}$ and ${\bf \hat{r}}$,as described in the next section.