COMPUTATION OF ADJOINT OPERATORS

The adjoint operator ${\bf H^T}$ to the summation along a hyperbola in (t,x)-space is the spreading of each point in $(\tau,m)$-space onto a hyperbola in (t,x)-space. Similarly, the adjoint operator ${\bf P^T}$ to the summation along a parabola in $(\tau,m)$-space is the spreading of each point in (t,x)-space onto a parabola in $(\tau,m)$-space.

The application of ${\bf H^T}$ may be summarized as follows:

for all sloths m 
		 for all offsets x 
		 		 $\bold d_x=\bold d_x+{\bf NMO^T u}_m\;.$

 

${\bf NMO^T}$ is a transpose operation to ${\bf NMO}$ (Claerbout, 1985). In words, the process proceeds as follows:

The algorithm takes the first trace of the velocity analysis panel, applies ${\bf NMO^T}$ corresponding to the first offset and adds the result to the first offset. Then ${\bf NMO^T}$ is applied to the same trace for the second offset and the result is added to the second offset, and so on for other offsets. The same process is repeated with the rest of the traces of the velocity analysis panel.

For the operator ${\bf P^T}$ the process is the same; the only difference is that ${\bf NMO^T}$ is replaced by the transpose to the corresponding compressing operation.