Conclusion and Discussion

Comparing non-iterative linearized migration/inversion, iterative linearized migration/inversion, and nonlinear waveform inversion, linearized migration/inversion has been carefully reviewed. The following statements reflect my opinions about linearized migration/inversion.

(1) There exists a paradox: linearized migration/inversion requires that the background velocity is as close as possible to the true velocity distribution; however the Born approximation needs a very smooth background velocity, so that the incident wavefield only includes downward propagating waves. The Born approximation is not a good linearization method and it should be replaced by more accurate approximations, such as De Wolf approximation.

(2) In essence, non-iterative linearized migration/inversion, iterative linearized migration/inversion, and nonlinear waveform inversion, are all inverse scattering imaging methods.

(3) Relative true-amplitude imaging is possible, the condition is that the distribution of the Fourier transform of the object function on $\Omega _{\eta}\left( y\right)$ is even, and the range of the distribution is the same at every imaging point (or scattering point).

(4) The Hessian matrix is band-limited and its inverse is also band-limited, its inner structure has close relation with the complexity of the velocity and the acquisition geometry. It reflects the energy illuminating imaging points.