Downward-continuation migration

In migration by downward-continuation, the wavefield at depth 127#127, 128#128, is obtained by phase-shift from the wavefield at depth z, 129#129. 

 130#130 (52)

This equation corresponds to the analytical solution of the ordinary differential equation  

 131#131 (53)

where the ' sign represents a derivative with respect to the depth z.

We can consider that the depth wavenumber (132#132) depends linearly, through a Taylor series expansion, on its value in the reference medium (133#133) and the laterally varying slowness in the depth interval from z to 127#127, 134#134

135#135 (54)

where s_r represents the constant slowness associated with the depth slab between the two depth intervals, and 136#136 represents the derivative of the depth wavenumber with respect to the reference slowness and which can be implemented in many different ways (). The wavefield downward-continued through the background slowness 137#137 can, therefore, be written as

138#138 (55)

from which we obtain that the full wavefield 128#128 depends on the background wavefield 139#139 through the relation

140#140 (56)

where 141#141 represents the difference between the true and background slownesses 142#142.