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
$L_{2}$
norm misfit function, which suffers severely local minimum problems since the
$L_{2}$
norm misfit function is nonconvex in the model space. To mitigate the local minimum problem, the quadratic Wasserstein metric (
$W_{2}$
) derived from the optimal transport theory is proposed to replace the
$L_{2}$
norm to define the misfit function. Since the computation of the
$W_{2}$
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
$W_{2}$
(
$W_{2}$
-FWI) is more independent on the initial model than the FWI based on the
$L_{2}$
norm (
$L_{2}$
-FWI), and the
$W_{2}$
-FWI could be more robust to zero-mean random noise than the
$L_{2}$
-FWI. Finally, a comprehensive numerical experiment related to asteroid radar tomography is carried out to show the potential of
$W_{2}$
-FWI in realistic applications.