  Optimal transport has recently started to be successfully employed to define
misfit or loss functions in inverse problems. However, it is a problem
intrinsically defined for positive (probability) measures and therefore
strategies are needed for its applications in more general settings of
interest. In this paper we introduce an unbalanced optimal transport problem
for vector valued measures starting from the $L^1$ optimal transport. By
lifting data in a self-dual cone of a higher dimensional vector space, we show
that one can recover a meaningful transport problem. We show that the favorable
computational complexity of the $L^1$ problem, an advantage compared to other
formulations of optimal transport, is inherited by our vector extension. We
consider both a one-homogeneous and a two-homogeneous penalization for the
imbalance of mass, the latter being potentially relevant for applications to
physics based problems. In particular, we demonstrate the potential of our
strategy for full waveform inversion, an inverse problem for high resolution
seismic imaging.
