Moment tensor inversion typically involves minimization of a misfit function between observed and model predicted waveforms. In both the deterministic and stochastic inversion frameworks, a widely used misfit function is the ℓ2 distance. In this work we argue that ℓp distances are not well suited to comparing waveforms or, more generally, functional data. In fact, ℓp distances ignore the “horizontal” time structure of the signal and measure similarities in amplitude only, at each time point. This can have a detrimental effect on the quality of moment tensor estimates, particularly when the accuracy of the model used for inversion (e.g., the velocity model) is poor. Recently, optimal transport distances have been demonstrated as useful alternatives to ℓp distances in the context of full waveform inversion. In this paper we apply a specific type of Wasserstein distance, called the TLp distance, to the problem of moment tensor estimation, and we evaluate its robustness to misspecification of the velocity model. We cast our approach in a Bayesian statistical framework. Advantages of this distance over the classic ℓ2 and other Wasserstein-type distances are discussed theoretically and through some numerical examples.