Comparison of Radon Transforms

The better fit of the ray bending approximation to the actual moveout of the multiple, as shown in Figure , suggests that a better focusing may be achieved for the multiples in the Radon domain by using equation 6 as the kernel of the Radon transform. To assess validity of this claim, I used the same synthetic data presented in Sava and Guitton 2003. Figure  is their figure 1 and shows a CMP and an ADCIG contaminated with multiples. Clearly, the primaries are flat in the ADCIGs (above about 350 m), whereas the multiples show the expected overmigrated residual moveout.

 

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Figure 3 A data CMP and an ADCIG from a simple synthetic model. Figure taken from Sava and Guitton 2003.

The general expression for the Radon transform in the angle domain is Sava and Guitton (2003)  

$$z(q,\gamma)=z_0+q\;g(\gamma).$$ (9)

The straight-ray approximation uses  

$$g(\gamma)=\tan^2\gamma.$$ (10)

The ray-bending approximation uses  

$$g(\gamma)=\frac{1}{1+\rho}\left[\frac{\cos\gamma(\rho^2-(1-\rho^2)\tan^2\gamma)}{\sqrt{\rho^2-\sin^2\gamma}}-\rho\right].$$ (11)

 

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Figure 4 Comparison of Radon transforms of the ADCIG shown in panel (a) of Figure . Panel (a) corresponds to the straight-ray approximation whereas panel (b) corresponds to the ray-bending approximation. Panel (c) and (d) are the envelopes of panels (a) and (b) respectively.

Figure  shows a comparison of both Radon transforms for the ADCIG shown in the left panel of Figure . 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 since the curvature more closely represents their residual moveout in the ADCIGs. In order to assess the improvement in focusing power of the new transform with real data, I applied both Radon transforms to an ADCIG from a real dataset Sava and Guitton (2003). Figure  shows the ADCIG and the transforms computed with the straight-ray and the ray-bending approximations. Again, the ray-bending approximation improves the focusing of the primaries. This may be better seen in the envelopes of the two transforms.

 

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Figure 5 ADCIG from a real dataset (a), the Radon transform corresponding to the straight-ray approximation (b), and the Radon transform with the ray-bending approximation (c). Panels (d) and (e) are the envelopes of panels (b) and (c).