Seismic inversion is the process of using measurements of seismic waves on the surface of the earth to determine geophysical properties of the subsurface. This process is marginally ill-posed and naturally noise play an important role. We will analyze this role in the context of uncertainty quantification, Bayesian and frequency-based analysis. The particular setting we will consider is the recently popular, so-called Full Waveform Inversion, which in mathematical terms is PDE-constrained optimization. The objective or loss function in the optimization measures the mismatch between data and a forward wave equation. We will consider Sobolev norms and the Wasserstein metric from optimal transport to measure this mismatch.