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This paper is devoted to the theoretical and numerical investigation of the local minimum problem in an inverse boundary value problem. We provide a mathematical analysis for the objective function of the optimal transportation type in the context of seismic full waveform inversion. In particular, we prove that the gradient obtained using the adjoint-state method does not depend on the specific choice of the Kantorovich potentials. Moreover, our frequency analysis results show a decreasing sensitivity of the reconstruction as the data misfit is concentrated in the high-frequency part. This confirms previous observations in many numerical experiments. We also propose a new method using the softplus encoding, which maps the seismic data into probability measures and, therefore, the Wasserstein metric can be applied. The softplus encoding retains the convexity of the data misfit with respect to translations and provides a parameter to tune the landscape of the objective function. The effectiveness of the proposed method is demonstrated numerically on an inversion task with the benchmark Marmousi model.

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