Full-Waveform Inversion of Two-Parameter Ground-Penetrating Radar Based on Quadratic Wasserstein Distance

Full-waveform inversion (FWI) is one of the most promising techniques in current ground-penetrating radar (GPR) inversion methods. The least-squares method is usually used, minimizing the mismatch between the observed signal and the simulated signal. However, the cycle-skipping problem has become an urgent focus of this method because of the nonlinearity of the inversion problem. To mitigate the issue of local minima, the optimal transport problem has been introduced into full-waveform inversion in this study. The Wasserstein distance derived from the optimal transport problem is defined as the mismatch function in the FWI objective function, replacing the L2 norm. In this study, the Wasserstein distance is computed by using entropy regularization and the Sinkhorn algorithm to reduce computational complexity and improve efficiency. Additionally, this study presents the objective function for dual-parameter full-waveform inversion of ground-penetrating radar, with the Wasserstein distance as the mismatch function. By normalizing with the Softplus function, the electromagnetic wave signals are adjusted to meet the non-negativity and mass conservation assumptions of the Wasserstein distance, and the convexity of the method has been proven. A multi-scale frequency-domain Wasserstein distance full-waveform inversion method based on the Softplus normalization approach is proposed, enabling the simultaneous inversion of relative permittivity and conductivity from ground-penetrating radar data. Numerical simulation cases demonstrate that this method has low initial model dependency and low noise sensitivity, allowing for high-precision inversion of relative permittivity and conductivity. The inversion results show that it, in particular, significantly improves the accuracy of conductivity inversion.