Crosshole ground-penetrating radar full-waveform inversion by combining optimal-transport and least-squares distances

Gradient-based full-waveform inversion using the least-squares distance converges to a local minimum if the starting model is not close enough to the global minimum. In this article, we propose a full-waveform inversion algorithm for crosshole ground penetrating radar tomography that is independent of the starting model. This is achieved by using an optimal-transport distance, which features one broad global minimum and almost no local minima for common ground penetrating radar full-waveform inversion problems. As calculating an optimaltransport distance is computationally expensive and as the broad global minimum of the optimal-transport distance is impacted by numerical approximations in the vicinity of the optimal solution, we propose an algorithm that uses the optimal-transport distance only in the first few iterations of the inversion and switches to the least-squares distance once a solution close enough to the true solution has been found. An additional feature of the proposed algorithm is the sparse and explicit calculation of the gradient using random sets of master points and subsequent interpolation. This avoids the high sensitivity close to the antenna locations, which prohibits meaningful model updates in these regions. Furthermore, this approach smooths the gradient and, thus, the model without an explicit model-regularization term in the objective function. Finally, calculating the gradient explicitly allows to easily implement other distance measures as no update of the gradient calculation becomes necessary. As the explicit calculation of the gradient can be done in parallel and only a limited number of master points is necessary, the additional computational cost is limited. We demonstrate the capabilities of the algorithm on a simple synthetic dataset showing that the proposed algorithm works much better on our example dataset than if only least-squares or optimal-transport distances are used.