Diffracted multiple

Figure  shows the raypath of a diffracted multiple from a dipping reflector. The image-space coordinates of the diffracted multiple are given by the same equations as the water-bottom multiple, i.e. equations 32-34. The main difference is that now $\alpha_r\ne\alpha_s+4\varphi$. In fact, $\alpha_r$depends exclusively oh the position of the diffractor with respect to the receiver and is given by (Appendix E)  

$$\alpha_r=\tan^{-1}\left[\frac{h_D+m_D-X_{diff}}{Z_{diff}}\right].$$ (47)

 

mul_sktch6 Figure 14 Diffracted multiple from a dipping water-bottom. Note that the receiver ray does not satisfy Snell's law at the diffractor.

The depth of the diffractor Z_diff can be computed as (Appendix E):  

$$Z_{diff}=\tilde{Z}_D\cos\varphi+(X_{diff}-m_D)\tan\varphi,$$ (48)

where $\tilde{Z}_D$, as before, is the perpendicular distance between the CMP and the reflector. It can be computed from the traveltime of the diffracted multiple of the zero surface-offset trace as shown in Appendix E. The traveltime segments from the source to the diffractor are the same as before and given by equations 39-41, while the traveltime from the diffractor to the receiver is simply  

$$t_{r_1}=\frac{1}{V_1}\sqrt{(h_D+(m_D-X_{diff}))^2+Z_{diff}^2}.$$ (49)

In order to have the image coordinates entirely in terms of the data space coordinates all that is left is to compute $\alpha_s$ (Appendix E):  

$$\alpha_s=\sin^{-1}\left[\frac{2\tilde{Z}_D\sin\varphi+(h_D+X... ...V_1t_m-\sqrt{(h_D+m_D-X_{diff})^2+Z_{diff}^2}}\right]-2\varphi.$$ (50)

 

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Figure 15 image sections at 0, -200 and 200 m subsurface offset for a diffracted multiple from a dipping water-bottom.

Figure  shows three image sections at subsurface offsets of 0, -200 and 200 m. These sections are a poor representation of either the reflector or the diffractor since the diffracted multiple is not imaged as a primary.

 

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Figure 16 SODCIGs at three different CMP locations: 1,800, 2,000 and 2,200 m for a diffracted multiple from a dipping water-bottom.

Figure  shows three SODCIGs at CMP locations 1800, 2200 and 2600 m. Again, we see that the SODCIGs are very different depending on their relative position to the diffractor, unlike the situation with the non-diffracted multiple which maps to negative subsurface offsets (for $h_D\ge 0$) for all SODCIGs.

 

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Figure 17 ADCIGs corresponding to the three SODCIGs of Figure .

The aperture angle and the image depth of the diffracted multiple in ADCIGs can also be computed with equations 11 and 12 with $\beta_r$ and $\beta_s$ given by equation 46. Figure  shows the ADCIG corresponding to the same ODCIG in Figure . Again, notice that the apex is shifted away from zero aperture angle.