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We provide new proofs based on the Myers-Steenrod theorem to confirm thattravel time data, travel time difference data and the broken scatteringrelations determine a simple Riemannian metric on a disc up to the naturalgauge of a boundary fixing diffeomorphism. Our method of the proof leads to aLipschitz-type stability estimate for the first two data sets in the class ofsimple metrics.

Three travel time inverse problems on simple Riemannian manifolds