Extracting physical parameters that cannot be directly measured from an observed data set remains a great challenge in several fields of science and physics. In many of these problems, the construction of a physical model from waveforms is hampered by the phase ambiguity of the recorded wave fronts. In this work, we present an approach for mitigating the effect of phase ambiguity in waveform-driven issues. Our proposal combines the optimal transport theory with the κ-statistical thermodynamics approach. We construct an energy function from the most probable state of a system described by a finite-variance κ-Gaussian distribution to introduce an optimal transport metric. We demonstrate that our proposal outperforms the classical frameworks by considering a nonlinear geophysical data-driven problem based on a wave equation numerical solution. The κ-generalized optimal transport metric is easily adapted to various inverse problems, from estimating power-law exponents to machine learning approaches in quantum mechanics.