Wave equation inversion and diffraction tomography repaired

The problem with diffraction tomographies and inversions is that unless the sources and receivers surround the region of interest such as in medical imaging applications, the wavenumber spectrum of the image is incomplete. For sources and receivers along a horizontal line, the image is missing the low vertical wavenumbers. This is the case in seismic tomographies and inversions where the sources and receivers are placed along the Earth's surface. Why is this so considering both low- and high-wavenumber (interval velocity and reflector location) information is contained in seismic wavefields? Typical analyses of the inversion problem using the Born approximation and a constant background velocity indicate that the spectrum of the image is incomplete. However, when a non-constant and non-smoothly varying background velocity is used in the analysis, it is found that all wavenumbers can be resolved up to some maximum value related to the highest frequency contained in the seismic wavelet. In other words, the presence of deeper reflectors simulates a second source-receiver line below the region of interest. Hence, we have a situation equivalent to that when sources and receivers surround the region of interest except that the reflector depths are unknown. However, the reflector locations can be approximately found in an inversion for the high-wavenumber velocity model (a migration-like inversion). Therefore, an iterative inversion where both high- and low-wavenumber velocity models are allowed to vary at each iteration will resolve almost all wavenumbers in the velocity image. This corresponds to a wave equation based inversion to obtain reflector locations and interval velocities simultaneously.