A universal sequence in the classical travel-time inversion problem

Inversion of the travel-time curve is a fundamental problem of mathematical geophysics: explosions and earthquakes take place on (or close to) the surface of the Earth, instruments record signals from them, and it is required to find the velocity of elastic waves in the interior of the Earth from the times taken by the signals travelling from the sources to the receivers (the travel-time curve). After the pioneering work at the beginning of the century and the detailed research in the 1960s one would hardly have expected the appearance of fundamentally new results on this problem in its classical formulation, when the wave velocity is assumed to depend only on the depth. However, it has turned out to be premature to regard this formulation as settled. The theorems proved here on a universal sequence and extremal properties of discrete measures will probably surprise specialists in the inverse problem and will interest both experts and amateurs in extremal problems.