Reflection FWI in MBTT Formulation: Macro Velocity Reconstruction

Unfortunately we have to admit that after decades of development there are still no reliable techniques of full waveform inversion which guarantee reliable reconstruction of both macrovelocity model and reflectors reconstruction for reasonable acquisitions and frequency ranges. As reasonable we mean realistic offsets (about one-two depths of target objects) and temporal frequency above 5 – 7 Hz. The paper is devoted to the so-called Migration Based Travel Times (MBTT) formulation of the data misfit functional. This approach relies on the decomposition of a velocity model onto two subspaces – smooth propagator and rough depth reflectors. On this base the modified data misfit functional is introduced and compared with standard least squares formulation. Numerical Singular Value Decomposition proves that these two formulations produce functionals which have almost orthogonal stable subspaces. As is well known the classical formulation leads to stable subspaces mainly made of fast oscillating functions (reflectors). At the same time we prove that MBTT modification ensures appearance of the propagator in these stable subspaces. Numerical experiments prove the feasibility of full inversion for reflected waves in this modified reformulation for the well known Gullfaks velocity model.