Numerical comparison

The Amoco 2.5-D synthetic dataset Dellinger et al. (2000); Etgen and Regone (1998) provides an excellent test for the weighting functions discussed above.

The velocity model (Figure ) contains significant structural complexity in the upper 3.8 km, and a flat reflector of uniform amplitude at about 3.9 km depth. Since the entire velocity model (``Canadian foothills overthrusting onto the North Sea'') is somewhat pathological, I restricted my experiments to the North Sea section of the dataset (x>10 km). The data were generated by 3-D acoustic finite-difference modeling of the 2.5-D velocity model. However, making the test more difficult is the fact that the FFD one-way recursive extrapolators Ristow and Ruhl (1994) that I use for modeling and migration do not accurately predict the 3-D geometric spreading and multiple reflections that are present in this dataset. Figure  illustrates this by comparing a gather produced by full two-way 3-D acoustic finite-differences with a gather modeled with the one-way depth extrapolation algorithm.

 

amocovel
Figure 1 Velocity (in km/s) model used to generate the synthetic Amoco 2.5-D dataset.

 

amocogather18
Figure 2 Synthetic shot-gathers from the Amoco 2.5-D dataset (s_x=18.2 km). Panel (a) shows the gather generated by full two-way finite-difference modeling. Panel (b) shows the gather generated by linear one-way modeling. Panel (c) shows the same two-way equation gather as panel (a) but with a top-mute to enable easier comparison with panel (b).

Figure  compares the migrated image (${\bf m}_1$)with the results of remodeling and remigrating the three reference images described above. The imprint of the recording geometry is clearly visible on the three remigrations in Figures  (b-d).

 

amocomigs
Figure 3 Comparison of calibration images: (a) original migration, (b) original migration after modeling and migration, (c) random image after modeling and migration, and (d) flat event image after modeling and migration.

Figure  compares the illumination calculated from the three reference images with the shot illumination from section . Noticeably, the shot-only weighting function [panel (a)] does not take into account the off-end (as opposed to split-spread) receiver geometry. Panel (b), the weighting function derived from model ${\bf m}_1$, appears slightly noisy. However, in well-imaged areas (e.g. along the target reflector), the weighting function is well-behaved. Panel (c) shows the weighting function derived from the random reference image (${\bf m}_2$). Despite the smoothing, this weighting function clearly bears the stamp of the random number field. A feature of white noise is that no amount of smoothing will be able to completely remove the effect of the random numbers. The final panel (d) shows the flat-event illumination weighting function, derived from ${\bf m}_3$. This is noise-free and very well-behaved since it depends only on the velocity model and recording geometry, not the data.

 

amocowght
Figure 4 Comparison of weighting functions: (a) shot illumination as described in Chapter , (b) reference model was migrated image, (c) reference model was random image, and (d) reference model consisted of purely flat events.

For a quantitative comparison, I picked the maximum amplitude of the 3.9 s reflection event on the calibrated images. The normalized standard deviation (NSD) of these amplitudes is shown in Table , where

$$\mbox{NSD} = \sqrt{ \sum_{i_x} \left( \frac{a_{i_x}}{\bar a} - 1 \right )^2}.$$ (97)

Table , therefore, provides a measure of how well the various weighting function compensate for illumination difficulties. The amplitudes of the raw migration, and the migration after flat-event normalization are shown in Figure . This illustrates the numerical results from Table : for this model the normalization procedure improves amplitude reliability by almost a factor of two.

 

2||c|Weighting function: Normalized standard deviation:

2||c|No weighting function 0.229

2||c|Shot illumination 0.251

$${\bf m}_{\rm ref}={\bf m}_1$$ (migrated image) 0.145

$${\bf m}_{\rm ref}={\bf m}_2$$ (random image) 0.195

$${\bf m}_{\rm ref}={\bf m}_3$$ (flat events) 0.140

2||c|Four iterations of CG 0.157

 

eventampnm5 Figure 5 Normalized peak amplitude of 3.9 km reflector after migration (solid line), and then normalization by flat-event illumination (dashed-line). The ideal result would be a constant amplitude of 1.

To compare the results of a well-scaled adjoint with full L2 Fourier finite-difference migration, I ran 10 iterations of full conjugate gradients, using Paul Sava's out-of-core optimization library Sava (2001). Figure  shows images after four and ten iterations. I did not impose an explicit regularization (``model-styling'') term during the inversion, so as the solution evolves less well-constrained components of the model-space start to appear in the solution, including both low and high frequency noise and steeply-dipping energy. This causes the NSD to actually begin to increase after the fourth iteration (see Figure ).

 

amocoinv
Figure 6 Results of full L2 inversion of the Amoco 2.5-D dataset with FFD modeling/migration. Panel (a) shows results after four iterations, and panel (b) shows results after ten iterations.

 

invnorm Figure 7 Normalized standard deviation of flat reflector versus iteration number. After four iterations the noise-level causes degradation of amplitude reliability.