A review of the use of optimal transport distances for high resolution seismic imaging based on the full waveform

Métivier, Ludovic; Brossier, Romain; Kpadonou, Félix; Messud, Jérémie; Pladys, Arnaud

MathematicS In Action, Tome 11 (2022) no. 1, pp. 3-42.

This study is a review of recent applications to seismic imaging of optimal transport based numerical tools. Modern seismic imaging methods used in the industry rely on the interpretation of the full signal. The characterization of the subsurface mechanical properties is formulated as a PDE-constrained optimization problem, solved through local optimization strategies. The choice of the misfit function used to measure the distance between actual seismic recordings and those synthetized by the solution of wave propagation PDE is crucial. Indeed, the conventional least-squares distance function leads to a non-convex optimization problem whose solution through local optimization then strongly depends on the initial guess. Using an optimal transport distance is an interesting alternative from its convexity properties with respect to translation and dilation. Specific strategies need however to be implemented as seismic data are oscillatory, while the optimal transport theory has been developed for the comparison of positive measures. In this study we review two optimal transport based misfit functions, from their mathematical formulation to their application to field data through their numerical implementation. Advantages and drawbacks of both strategies are discussed. Numerical experiments show that they represent two interesting and complementary alternative to the classical least-squares misfit function, mitigating the dependency to the choice of the initial guess.

Classification : 35R30, 86A22, 86A15
Mots clés : Optimal transport, convexity, optimization, seismic imaging

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