Electromagnetic Full Waveform Inversion Based on Quadratic Wasserstein Metric

Electromagnetic full waveform inversion (FWI) is a high-resolution method to reveal the distribution of dielectric parameters of the medium. Traditionally, the electromagnetic FWI is usually performed using the <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> norm misfit function, which suffers severely local minimum problems since the <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> norm misfit function is nonconvex in the model space. To mitigate the local minimum problem, the quadratic Wasserstein metric (<inline-formula> <tex-math notation="LaTeX">$W_{2}$ </tex-math></inline-formula>) derived from the optimal transport theory is proposed to replace the <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> norm to define the misfit function. Since the computation of the <inline-formula> <tex-math notation="LaTeX">$W_{2}$ </tex-math></inline-formula> metric requires the signal to be mass conservation and nonnegativity, the softplus function is applied to rescale the signal to meet these two requirements. Based on the softplus scaling method, a hierarchy inversion strategy that involves multiple scaling coefficients is proposed to mitigate the local minimum problem without sacrificing the inversion resolution. The numerical experiments prove that the FWI based on <inline-formula> <tex-math notation="LaTeX">$W_{2}$ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$W_{2}$ </tex-math></inline-formula>-FWI) is more independent on the initial model than the FWI based on the <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> norm (<inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-FWI), and the <inline-formula> <tex-math notation="LaTeX">$W_{2}$ </tex-math></inline-formula>-FWI could be more robust to zero-mean random noise than the <inline-formula> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula>-FWI. Finally, a comprehensive numerical experiment related to asteroid radar tomography is carried out to show the potential of <inline-formula> <tex-math notation="LaTeX">$W_{2}$ </tex-math></inline-formula>-FWI in realistic applications.