Radon Transform

In this section I show how to exploit the difference in residual moveout between primaries and multiples in ADCIG's given by Equation 24 to design a Radon transform that focuses the primaries and multiples to separate regions of the Radon domain. The general expression for the Radon transform in the angle domain is (, )

$$z(q,\gamma)=z_0+q\;g(\gamma).$$ (25)

where $q$ is a measure of curvature and $g(\gamma)$ is the function that approximates the residual moveout of the multiples as a function of the aperture angle $\gamma$. () and () used the tangent-squared approximation of Biondi and Symes (2004)

$$g(\gamma)=\tan^2\gamma,$$ (26)

but for the focusing of the multiples a more accurate approximation is given by Equation 24:

$$g(\gamma)=\frac{1}{1+\rho}\left[\frac{\cos\gamma(\rho^2-(1-\rho^2)\tan^2\gamma)}{\sqrt{\rho^2-\sin^2\gamma}}-\rho\right].$$ (27)

This approximation is more accurate because it takes into account ray bending at the multiple-generating interface. This is illustrated in Figure 7 which shows a comparison of the Radon transforms defined be equations 27 and 26 applied to a synthetic ADCIG. Notice that the focusing of the primaries does not change since their moveout is zero. The multiples, on the other hand, are better focused with the new transform (panel (c)) which more closely follows their residual moveout in the ADCIGs. The better focusing of the multiples translates to a better estimation of the multiple model (compare panels (d) and (e) computed from panels (b) and (c), respectively). Notice, however, that this synthetic ADCIG has high aperture angles for which the difference between the two approximations is greater. As the angle coverage decreases, so does this difference. In any event, the better focusing of the multiples helps in separating them from the primaries.

synth1
Figure 7. Comparison of Radon transforms for a synthetic ADCIG. Panel (a) shows the ADCIG. Panels (b) and (c) correspond to the envelopes of the Radon transform of panel (a) computed with the straight-ray approximation and the ray-bending approximation respectively. Panels (d) and (e) are the multiple models computed from panels (b) and (c). The ovals highlight the improved accuracy afforded by the new transform for the multiple model at the large aperture angles.