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 1

Figure 2

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 3

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.

Figure 4

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

(97) |

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

2||c|No weighting function | 0.229 | |

2||c|Shot illumination | 0.251 | |

(migrated image) | 0.145 | |

(random image) | 0.195 | |

(flat events) | 0.140 | |

2||c|Four iterations of CG | 0.157 |

eventampnm5
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.
Figure 5 |

To compare the results of a well-scaled adjoint with full *L*2
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 6

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

5/27/2001