Nonlinear least-squares full waveform inversion: SVD analysis

Nonlinear least-squares formulation for the inverse problem of seismic wave propagation is in the focus of computational geophysics community since 80th of the last century. Around the same time, the problem of trend component recovery becomes known. It is connected in the necessity to possess unrealistic low frequencies or extremely large source-receiver offsets in the data acquired to provide a reliable reconstruction of macrovelocity. At the same time, this component of the model determines the correct position of seismic images in space. Recently, thanks to the progress in the geophysical instrument industry, the substantive information becomes available for as low time frequencies as $5$ Hz; as a rule, however, this is not sufficient for the proper reconstruction of macrovelocity. This paper deals with the numerical singular value decomposition of the linearized forward map to bring to the light the mathematical roots of the troubles. On this basis, a modification of the cost function is proposed, discussed and compared with the standard least-square approach. The high reliability of this modification in macrovelocity determination is confirmed numerically.