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.