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.
Figure compares the migrated image ()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).
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 , 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 (). 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 . This is noise-free and very well-behaved since it depends only on the velocity model and recording geometry, not the data.
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
|2||c|Weighting function:||Normalized standard deviation:|
|2||c|No weighting function||0.229|
|2||c|Four iterations of CG||0.157|
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 ).
Figure 7 Normalized standard deviation of flat reflector versus iteration number. After four iterations the noise-level causes degradation of amplitude reliability.