Moveout-corrected radial traces

Moveout correction may be regarded as a transformation from time to depth. When the moveout correction is properly done, all traces should show the same depth-dependent reflectivity. In principle, radial moveout correction proceeds by introducing z and eliminating t with the substitution $t v =\sqrt { z^2 + x^2 }$.In practice you would prefer a travel-time-depth axis to a depth axis. So the transformation equation becomes $t =\sqrt { \tau^2 + x^2 / v^2 }$.Eliminating x with rt we get  

$$t \eq {\tau \over \sqrt { 1 \ -\ {r^2} / {v^2} } }$$ (4)

Inspecting (4) we see that moveout correction in radial-trace coordinates is a uniform compression of the time t-axis into a $\tau$-axis. The amount of compression is fixed when r is fixed. The amount of compression does not change with time. The uniformity of the compression is an aid to modeling and removing the effects of shot waveforms and multiple reflections. It is curious to note that moveout correction of radial traces compresses time, while moveout correction of constant-offset data stretches time nonuniformly. Figure 4 shows a field profile before and after the radial-trace transformation.

 

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Figure 4 Field profile from Alberta (Western Geophysical) shown before and after deformation into radial traces.