Sparse frequencies data inversion and the role of multi-scattered energy

Abstract

SummaryIn trying to extract a broad spectrum of model wavenumbers from the data, necessary to build a plausible model of the Earth, we are, in theory, bounded at the high end by the diffraction resolution limit, which is proportional to the highest usable frequency in the data. At the low end, and courtesy of our multi-dimensional acquisition, the principles behind diffraction tomography theoretically extend our range to zero-wavenumbers, mainly provided by transmissions like diving waves. Within certain regions of the subsurface (i.e. deep), we face the prospective of having a model wavenumber gap in representing the velocity. Here, I demonstrate that inverting for multi scattered energy, we can recover additional wavenumbers not provided by single scattering gradients, that may feed the high and low ends of the model wavenumber spectrum, as well as help us fill in the infamous intermediate wavenumber gap. Thus, I outline a scenario in which we acquire dedicated sparse frequency data, allowing for more time to inject more energy of those frequencies at a reduced cost. Such additional energy is necessary to the recording of more multi-scattered events. (Claerbout, 1985), as it helped us reduce the dimensionality of the wavefield. Even in reverse time migration, the reduction of dimensionality allows for practical storage of the source wavefield in 3D, needed for a fast implementation of the imaging condition. However, solving the Helmholtz wave equation requires finding the inverse of a large-sparse matrix, and in 3D, this is problematic especially for a large image space. Nevertheless, as soon as such an inverse is established (the Green's function), it can be used efficiently to obtain wavefield solutions for any source, including the residual wavefield, and more importantly secondary sources (like those from the Born series).