  This paper is devoted to theoretical and numerical investigation of the local
minimum issue in seismic full waveform inversion (FWI).
  This paper provides a mathematical analysis of optimal transportation (OT)
type objective function's differentiability and proves that the gradient
obtained in the adjoint-state method does not depend on the particular choice
of the Kantorovich potentials. A novel approach using the softplus encoding
method is presented to generalize and impose the OT metric on FWI. This
approach improves the convexity of the objective function and mitigates the
cycle-skipping problem. The effectiveness of the proposed method is
demonstrated numerically on an inversion task with the benchmark Marmousi
model.
