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1.3. QUANTITATIVE MIGRATION AND LINEARISED INVERSION 5
the wave equation have shown to be more robust since they account for later arrivals (Ehinger
et al., 1996). However, even when the macro-model explains correctly the kinematic aspect of
the imaging, an accurate migration was not originally expected to provide a quantitative estimation
of the reflectivity field. Therefore, the result of migration cannot be linked to physical
parameters that may more rigorously describe the model.
1.3 Quantitative migration and linearised inversion
Many techniques have been proposed to provide quantitative estimates of the model parameters.
Most of these techniques rely on: 1- the ray theory and 2- the Born approximation (Born, 1923)
or the Kirchhoff approximation (Schneider, 1978). They can all be defined within the context
of linearised inverse problem theory (Lailly, 1983; Tarantola, 1984b), where the inverse of the
linear forward operator must be estimated. In an early work, Cohen and Bleistein (1979) defined
a direct inverse operator for zero offset data. Thereafter, inversion procedures for prestack
data were proposed (Clayton and Stolt, 1981; Beylkin, 1985; Ikelle et al., 1986; Miller et al.,
1987). These developments initiated the concept of migration/inversion using kinematic Kirchhoff
migration where amplitude are estimated using the Born or the Kirchoff approximations
(Bleistein, 1987; Bleistein et al., 1987; Beydoun and Mendes, 1989). These types of migration
methods are commonly used because of their computational efficiency for manipulating very
large data sets (Thierry et al., 1999). As an alternative to the direct inverse method, iterative
methods are often considered more robust (Lambaré et al., 1992; Jin et al., 1992; Nemeth et al.,
1999). Iterative methods rely on the explicit minimisation of least-square misfit function and are
more efficient in handling incomplete data sets. As discussed in the previous section, one of the
difficulty of ray-based migration is its difficulty in accounting for multipathing. More advanced
ray tracing methods have therefore been proposed with the concept of waveform construction
(Vinje et al., 1993; Lambaré et al., 1996) and have shown to improve the quantitative estimation
of migration/inversion (Operto et al., 2000).
However, the accuracy of the estimation of the model parameter using linearised inversion
is limited by the validity of the Born or the Kirchoff approximation. For example, the
Born approximation only accounts for first order scattering and is only valid in the presence of
small velocity perturbation (Keller, 1969; Beylkin and Ostroglio, 1985; Beydoun and Tarantola,
1988). On the other hand, the Kirchoff approximation accounts for the reflectivity which cannot
directly be related to velocity. Amplitude versus offset (AVO) analysis are then required to