Comparing the amplitudes of the methods

I can not compare impulse responses in constant offset sections because the MZO extracted from prestack migration includes an integration over the offset variable. The reason why I cannot just inverse Fourier transform in offset coordinates is explained in detail in the companion paper (Popovici, 1993). On short, the remapping of the field $p_h(\omega,k_h,k_y) \rightarrow p^*_h(\omega_0,k_h,k_y)$performs implicitly a summation over the offset. Therefore I compare the stacked output of the MZO in Fourier domain (applied to each constant offset section) and the output of the MZO extracted from prestack-migration applied to the whole prestack data cube. I will call the MZO defined in Fourier domain: FT-MZO, and the MZO extracted from prestack migration: Ps-MZO.

The first model is a diffractor situated at a depth of 600 meters. The velocity used is 2000 m/s and the number of offsets is 64. The maximum half-offset h is 1575 meters. The modeling was done by calculating the travel-time from source to diffractor and back to receiver, and assigning to each midpoint an amplitude equal to the product

$$Amp={1 \over {t_{src}}}{1 \over {t_{rec}}},$$

where t_src is the travel-time from source to the diffractor and t_rec is the travel-time from the diffractor to the receiver. The NMO applied to the constant-offset sections prior to processing via Hale's or Zhang's DMO has a Jacobian equal to

$$J_{NMO}={{t_{src}+t_{rec}} \over t_{NMO}}.$$

I compare the results obtained with and without applying the spherical divergence correction. Figure shows the first and last panel in the prestack data cube corresponding to the zero-offset and the maximum-offset common-midpoint (CMP) sections. Figure shows the output of the two MZO algorithms. Figure a shows the FT-MZO after stacking all independent offsets and Figure b shows the output of applying Ps-MZO to the entire prestack data cube. While in both cases the kinematics are the same, the maximum amplitudes along the zero-offset hyperbola shown in Figure a show there are significant differences between the two algorithms. The highest amplitude curve in Figure a belongs to the Ps-MZO section, followed by the original zero-offset model, Zhang's DMO, Hale's DMO and FT-MZO. I expected the FT-MZO to follow exactly Zhang's curve as it actually happens for the dipping reflector in Figure where the two curves practically coincide. Figure b shows results obtained when applying the spherical divergence correction. Figure compares the wavelets obtained after MZO. For each method I selected the trace passing through the CMP above the diffractor. Figure b has spherical divergence applied. As Hale's DMO curve always follows Zhang's DMO curve but at a lower amplitude, I removed this curve from the next comparisons, in an attempt to unclutter the figures.

The second model represents a dipping reflector through a medium with 2000 m/s velocity. Figure shows the first and last panel in the prestack data cube corresponding to the zero-offset and the maximum-offset common-midpoint (CMP) sections. The model was produced by considering the dipping reflector comprised of diffractors, each modeled separately. Figure a shows the output of the stacked FT-MZO, while Figure b shows the output of the Ps-MZO algorithm. Figure shows the amplitudes along the dipping reflector. This is the case where Zhang's amplitude curve is superimposed on the FT-MZO curve as I expected from theory. Figure b shows results obtained when applying the spherical divergence correction. The values at the sides of the figure should be discarded as the picking algorithm I used to collect the amplitudes along the reflector, chose the amplitudes along the artifacts generated by spatial wraparound. Figure compares the wavelets obtained after MZO. For each method I selected the trace passing through the middle of the section. Again Figure b has spherical divergence applied. Observe that in both cases the FT-DMO follows the zero-offset original curve, while the phase of the Ps-MZO wavelet is very different.

The third model represent a horizontal reflector through a medium with 2000 m/s velocity. Figure shows the first and last panel in the prestack data cube corresponding to the zero-offset and the maximum-offset common-midpoint (CMP) sections. Figure a shows the output of the stacked FT-MZO, while Figure b shows the output of the Ps-MZO algorithm. Figure shows the amplitudes along the horizontal reflector. Figure b shows results obtained when applying the spherical divergence correction. Again the maximum amplitude corresponds to the Ps-MZO, followed by Zhang's DMO, original zero-offset model and FT-MZO. When spherical divergence is applied all three methods show amplitudes higher than the original model. Figure compares the wavelets obtained after MZO. For each method I selected the trace passing through the middle of the section. Again Figure b has spherical divergence applied. While the two FT-MZO wavelets follow closely the zero-offset original model, the phase of the Ps-MZO wavelet is less similar.

I don't have a good explanation yet for the difference in amplitudes between Zhang's DMO and FT-MZO in the case of the single diffractor and in the case of the horizontal reflector. A loss in amplitude in the FT-MZO methods, occurs due to stacking, as the amplitude of the zero-offset sections after DMO tend to decrease slightly with the offset. This could be due to my modeling or to the fact that the DMO Jacobian doesn't completely account for the loss in amplitude with offset. In think the spherical divergence correction should be part of the DMO process and not an independent process, as it should be dip dependent.

 

coFig1
Figure 3 Input to the DMO and MZO algorithms.
a. First panel in the input cube: zero-offset.
b. Last panel in the input cube: maximum-offset.

 

zoFig1
Figure 4 Output of the DMO and MZO algorithms.
a. DMO+NMO.
b. MZO from Prestack Migration.

 

ampDMO1
Figure 5 a. Amplitudes along the zero-offset hyperbola. H-line: DMO via Hale after NMO. Z-line: Zhang DMO. M-line: MZO (DMO+NMO) in Fourier domain. Dotted-line: MZO from Prestack Migration. Continuous line: amplitudes along original zero-offset hyperbola.
b. Amplitudes with spherical divergence applied. H-line: DMO via Hale after NMO. Z-line: Zhang DMO. M-line: MZO (DMO+NMO) in Fourier domain. Dotted-line: MZO from Prestack Migration. Continuous line: amplitudes along original zero-offset hyperbola.

 

wavDMO1
Figure 6 a. Zero-offset wavelets through top of hyperbola. H-line: Hale. Z-line: Zhang. M-line: MZO by FT. Dotted-line: MZO from PsM. Continuous line: original wavelet.
b. Zero-offset wavelets with spherical divergence applied. H-line: Hale. Z-line: Zhang. M-line: MZO by FT. Dotted-line: MZO from PsM. Continuous line: original wavelet.

 

coFig2
Figure 7 Input to the DMO and MZO algorithms.
a. First panel in the input cube: zero-offset.
b. Last panel in the input cube: maximum-offset.

 

zoFig2
Figure 8 Output of the DMO and MZO algorithms.
a. DMO+NMO.
b. MZO from Prestack Migration.

 

ampDMO2
Figure 9
a. Amplitudes along a dipping reflector. Z-line: DMO via Zhang after NMO. M-line: MZO (DMO+NMO) in Fourier domain. Dotted-line: MZO from Prestack Migration. Continuous line: amplitudes along original zero-offset reflector.
b. Amplitudes with spherical divergence applied. Z-line: DMO via Zhang after DMO. M-line: MZO (DMO+NMO) in Fourier domain. Dotted-line: MZO from Prestack Migration. Continuous line: amplitudes along original zero-offset reflector.

 

wavDMO2
Figure 10
a. Zero-offset wavelets. Z-line: Zhang. M-line: MZO by FT. Dotted-line: MZO from PsM. Continuous line: original wavelet.
b. Zero-offset wavelets with spherical divergence applied. Z-line: Zhang. M-line: MZO by FT. Dotted-line: MZO from PsM. Continuous line: original wavelet.

 

coFig3
Figure 11 Input to the DMO and MZO algorithms.
a. First panel in the input cube: zero-offset.
b. Last panel in the input cube: maximum-offset.

 

zoFig3
Figure 12 Output of the DMO and MZO algorithms.
a. DMO+NMO.
b. MZO from Prestack Migration.

 

ampDMO3
Figure 13
a. Amplitudes along a horizontal reflector. Z-line: DMO via Zhang after NMO. M-line: MZO (DMO+NMO) in Fourier domain. Dotted-line: MZO from Prestack Migration. Continuous line: amplitudes along original zero-offset reflector.
b. Amplitudes with spherical divergence applied. Z-line: DMO via Zhang after DMO. M-line: MZO (DMO+NMO) in Fourier domain. Dotted-line: MZO from Prestack Migration. Continuous line: amplitudes along original zero-offset reflector.

 

wavDMO3
Figure 14
a. Zero-offset wavelets. Z-line: Zhang. M-line: MZO by FT. Dotted-line: MZO from PsM. Continuous line: original wavelet.
b. Zero-offset wavelets with spherical divergence applied. Z-line: Zhang. M-line: MZO by FT. Dotted-line: MZO from PsM. Continuous line: original wavelet.