Towards true-amplitude migration

If we know the subsurface reflectivity, simple linear wave propagation theory, such as that described in section , allows us to predict what seismic reflection data would be recorded in an experiment. This allows us to solve the forward problem of seismic imaging: given a reflectivity model, ${\bf m}$, we can construct a forward modeling operator, ${\bf A}$, such that the predicted data, ${\bf d}={\bf A} \; {\bf m}$.

Unfortunately, in a geophysical experiment we record ${\bf d}$, and would like to find ${\bf m}$. This is the so-called inverse problem, and is much more difficult to solve. As discussed in the previous section, rather than actually trying to invert the operator, ${\bf A}$, seismic migration amounts simply to applying its adjoint,

$$\hat{\bf m}= {\bf A}' \; {\bf d}.$$ (79)

Due to the symmetry of wave-propagation with respect to time-reversal, it turns out that migrating with the adjoint operator treats event kinematics correctly, and produces structurally correct images of the subsurface. Adjoint processing is also robust to the presence of noise, and missing or inconsistent data. However, the major shortcoming of migrating with the adjoint is that it does not treat seismic amplitudes correctly. Also, because processing places the emphasis on kinematics not amplitudes, amplitude terms are often completely ignored, or artificially constructed so that ${\bf A}' {\bf A} \approx {\bf I}$.

Interpreters, however, often try to extract more from seismic reflection images than kinematics: for example, rock physics studies show how important rock parameters such as porosity, lithology and fluid saturations may influence seismic amplitudes. The failure of migrating with the adjoint to correctly handle amplitudes has lead to the search for ``true-amplitude'', or ``amplitude-preserving'' migration operators. Rather applying the adjoint of a loosely-defined forward-modeling operator, true-amplitude schemes rigorously formulate the forward-modeling operator ${\bf A}_{\rm TA}$, and then approximate ${\bf A}_{\rm TA}^{-1}$ with a pseudo-inverse ${\bf A}_{\rm TA}^{\dagger}$. For example, Bleistein (1987) describes a Kirchhoff operator that becomes the inverse of the modeling operator in the high-frequency asymptotic limit.

If true-amplitude migration is the pseudo-inverse of the physically correct forward-modeling operator, then its adjoint, $({\bf A_{\rm TA}}')^{\dagger}$, is known as ``demigration''. Demigration is receiving increasing attention as part of amplitude-preserving processing flows Hubral et al. (1996). Figure  illustrates the relationships between forward modeling, migration, true-amplitude migration and demigration. Figure  also shows the relationship these operators have with what I refer to as ``industrial-strength'' migration - that is the pseudo-unitary operator, which is kinematically correct, but ignores the amplitude effects of wave-propagation in depth.

 

miginv4 Figure 2 Relationship between migration, modeling, their pseudo-inverses, and their (pseudo-unitary) industrial-strength counterparts.

 

unitary
Figure 3 Frequency (f-k_x) response of cascaded modeling and migration in a constant velocity medium. Panel (a) shows amplitude spectrum of dip-limited impulsive input. Panel (b) shows spectrum after exploding-reflector modeling and migration.