  Full waveform inversion (FWI) has recently become a favorite technique for
the inverse problem of finding properties in the earth from measurements of
vibrations of seismic waves on the surface. Mathematically, FWI is PDE
constrained optimization where model parameters in a wave equation are adjusted
such that the misfit between the computed and the measured dataset is
minimized. In a sequence of papers, we have shown that the quadratic
Wasserstein distance from optimal transport is to prefer as misfit functional
over the standard $L^2$ norm. Datasets need however first to be normalized
since seismic signals do not satisfy the requirements of optimal transport.
There has been a puzzling contradiction in the results. Normalization methods
that satisfy theorems pointing to ideal properties for FWI have not performed
well in practical computations, and other scaling methods that do not satisfy
these theorems have performed much better in practice. In this paper, we will
shed light on this issue and resolve this contradiction.
