New results in the classical travel-time inversion problem

Abstract

Travel-time inversion is a fundamental problem of mathematical geophysics: explosions and earthquakes occur on (or close to) the surface of the Earth, instruments record signals from them, and the problem is 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. In the classical formulation of the problem the wave velocity is assumed to depend only on the depth. This classical inverse problem has just recently seemed to be completely exhausted. However, it has turned out not to be the case. This paper relates the newest results in the problem, presents new key notions emerged in the last years research and explains-using duality principle-why discrete measures (or, in geophysical terms, waveguides with a finite number of layers) are so significant for describing ambiguity in travel-time inversion.

Keywords:

• Travel-time curve
• waveguide
• universal sequence
• majorant
• singular points
• duality principle

AMS:

• 86Al5
• 86A22
• 73D50