Sirgue, Laurent, 2003. Inversion de la forme d'onde dans le ...

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the larger the range of offsets, the fewer frequencies are required. Validation tests are carried
out on a 1-D velocity model that show the efficiency of frequency domain when a very limited
number of frequencies are adequately chosen. In these tests, the frequency domain performs
as well as time domain inversion at a much smaller computation cost. This strategy remains
efficient in 2D structures. This is demonstrated by the very good results that are obtained in the
2D Marmousi model from a wide-angle seismic acquisition survey, using only 3 frequencies.
Real seismic data do not contain very low frequencies and waveform inversion at higher frequencies
are likely to fail due to convergence into a local minimum. Preconditioning techniques
must hence be applied in order to enhance the efficacy of waveform inversion starting from realistic
frequencies. Because the high wavenumbers dominate the gradient image, the latter must
be smoothed to insure the proper reconstruction of lower wavenumbers. The determination
of the low wavenumbers is delicate as these correspond to the most non-linear components of
the model. These non-linearities may be mitigated by applying adequate preconditioning on the
data residuals that focus on the inversion of the early arrivals. A numerical test carried out on an
extended version of the 2D Marmousi model, in which a dense wide-angle survey is modelled,
shows that the proposed preconditioning strategy significantly improves the result of waveform
inversion. This test nevertheless also shows, that the starting model remains an important aspect
of waveform inversion.
The potential of waveform inversion in solving a given imaging problem may be evaluated
by defining the requirements of the starting model that ensure the success of the inversion. To
this end, I develop a linearity study that analyses the evolution of the misfit function with respect
to increasing degree of smoothness of the true model.
Such study is carried out on a 1D velocity model representing the sub-basalt imaging problem.
The analysis leads to the conclusion that waveform inversion may be used for the recovery
of the intra-basalt velocities, although the determination of the overburden sediments is more
difficult. Due to the presence of the basalt layer, only a migration-like image (i.e., the high
wavenumbers) can be obtained for the sub-basalt sediments. This study also emphasises the
importance of the low frequencies, as the requirements of the starting model are much more
demanding for higher frequencies.