  Full waveform inversion (FWI) is an important and popular technique in
subsurface earth property estimation. However, using the least-squares norm in
the misfit function often leads to the local minimum solution of the
optimization problem, and this phenomenon can be explained with the
cycle-skipping artifact. Several methods that apply optimal transport distances
to mitigate the cycle-skipping artifact have been proposed recently. The
optimal transport distance is designed to compare two probability measures. To
overcome the mass equality limit, we introduce an unbalanced optimal transport
(UOT) distance with KullbackLeibler divergence to balance the mass difference.
Also, a mixed L1/Wasserstein distance is constructed that can preserve the
convex properties with respect to shift, dilation, and amplitude change
operation. An entropy regularization approach and scaling algorithms are used
to compute the distance and the gradient efficiently. Two strategies of
normalization methods that transform the seismic signals into non-negative
functions are provided. Numerical examples are provided to demonstrate the
efficiency and effectiveness of the new method.
