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

1.5. NON-LINEAR WAVEFORM INVERSION 7
misfit function using for example, Monte-carlo, genetic or simulated annealing algorithms. Although
these techniques are capable of handling non-linear behaviour by inverting for both low
and high wavenumbers, they require the computation of the same order of forward modellings
as there are model parameters involved. Another approach is to decouple the low and high
wavenumber information, inverting for each parameter set alternately (Snieder et al., 1989; Cao
et al., 1990; Hicks and Pratt, 2001). This latter approach achieves the required decoupling by
a re-parametrisation in time of the velocity model, thus ensuring the zero offset traveltimes
remain fixed during the reflectivity reconstruction. This method assumes a near-vertical propagation
of the near offset events, and is thus limited to situations in which the 1-D approximation
can be applied locally.
Because of the high computational cost of calculating synthetic seismic data, non-linear
waveform inversion is usually formulated as an iterative “descent” method, in which the minimisation
of residuals is achieved through the repeated calculation of a local gradient. The
gradient at each iteration provides the direction of minimisation of the “objective” functional,
usually the L 2 norm of the data residuals possibly combined with some form of regularising
constrains. Since 1983, it has been recognised that the calculation of the gradient is of the same
computational order as the forward modelling task, and that the gradient is closely related to
seismic migration (Lailly, 1983; Tarantola, 1984a). Success depends upon the topography of
the misfit function at the location of an initial guess of the velocity model (starting model). In
the presence of strong non-linearities, the inversion may fail and convergence into local minimum
may occur (Gauthier et al., 1986). The starting model for such scheme must therefore be
located in the neighbourhood of the global minimum, i.e., a descent path leading to the global
minimum must exist if the method is to succeed.
Iterative non-linear waveform inversion was first applied to near offset reflection data to
recover the high wavenumber components of the model (Tarantola, 1986; Mora, 1987; Pica et
al., 1990; Crase et al., 1990). The advantage of the non-linear approach over the linearised
approach is that it will locate more efficiently the global minimum since the misfit function is
not required to be locally quadratic. In other words, the recovery of the high wavenumbers is
not limited by the validity of the Born approximation, since the forward problem used does
take into account multiple scattering. However, the determination of the macro-model remains
a critical aspect of the non-linear inversion of near offset data as it will cause convergence into
a local minimum if the starting macro-model velocities are not sufficiently accurate. Non-linear
inversion of near offset data therefore suffers from the same limitation than migration/inversion