Stepping in depth

If you are using finite differences on depth, or travel-time depth, then we have the relation (46):

$${ {\hat k}_z \, \Delta z \over 2 } \eq \tan \, { {k_z } \, \Delta z \over 2 }$$

which can be used in $\exp \, i k_z z$.Since dissipation may be present, the quantities above may be complex. Let N_z be the number of depth layers, generally equal or less than N_t. Adapting (49) to (59) gives  

$$e^{ - \, R ' \, N \, \Delta z / v } \ \ \ \approx \ \ \ \le... ...\ R ' \Delta z / 2 v \over 1\ +\ R ' \Delta z / 2 v} \right)^N$$ (60)