Migration in the spectral domain Stolts method

The backward extrapolation (continuation) of the wavefield is especially easy to realize in the spectral domain (Stolt, 1978 Gazdag, 1978 Chun and Jacewitz, 1981). This approach is based on representation of the migration field as a sum of plane harmonic waves characterized by the most elementary law of propagation  

Substituting expression (15.211) into the wave equation, we obtain a simple second-order differential equation  

The boundary condition requires that the migration wavefield found in the lower half-space from relations (15.214) and (15.211) should be equal to the observed scattered field on the plane 2 = 0, i.e. 

From this it follows that, in accordance with the principles of migration transformation, only the second term should be retained on the right-hand side of equation (15.214), when o 0, i.e. coefficient 4( 3, ky,uj) should be made equal to zero. Then constant B kx,ky,uj) is found from relation (15.215), 

Conversely, for tc 0 the terms on the right-hand side of (15.214) will correspond to upgoing and downgoing waves respectively,. so that for w/c w 0 we have 

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