A solution of the following problem is presented: given a travel-time curve of a seismic wave, to determine the corresponding velocity distribution. This is a generalization of the Herglotz-Wiechert method to a medium with low-velocity zones.The velocity depends only on depth and is a piecewise double-smooth function with a finite number of waveguides. A complete mathematical description of this solution is presented. In the presence of wave-guides the solution is ambiguous. Necessary and sufficient conditions for a velocity to be a solution are formulated and the set formed by plots of solutions is obtained.The ambiguity arising from waveguides is reduced by a joint analysis of travel-time curves from surface and deep sources. In particular the following theorem is proved : If the travel times for a source between any adjacent waveguides as well as for a surface source are known, then the velocity between these waveguides can be determined uniquely.
The uniqueness of the determination of a velocity cross-section from the travel-time curves for surface and deep sources was investigated by Gerver & Markushevich (1966,1967). The Earth was assumed spherically symmetrical with a finite number of waveguides.The present report states the conditions when a solution of this inverse problem exists.
The problems of determination of velocity-depth functions from traveltime curves or from dispersion curves show that the solution of an inverse problem may not be unique.We study here, as a preliminary analogy of such problems, the derivation of the unknown density function for an inhomogeneous string capable of small transverse vibrations, with one end fixed and one free. A unit impulse is applied at the free end, and the subsequent motion of the free end is observed. We prove that the density as function of position on the string is uniquely determined by these observations, under certain conditions.If a more general disturbance is applied, and similar observations are made at an arbitrary point of the string, is the determination of density still unique? We show that it is, provided all modes of free oscillation of the string are excited when the string is symmetrical with free ends.Further, we examine the stability of our solution. Could very large variations of density correspond to small variations in the observed motion? If so, a solution from actual data, liable to error, would be useless. We show that the solution is stable for a wide class of strings provided the observation point does not coincide with a node of one of the first N modes, and that these modes are excited ' distinctly enough '. The choice of N and the meaning of ' distinctly enough ' are fully explained.The inverse problem of seismology is to determine the internal structure of the Earth from observations of the surface motion resulting from earthquakes and explosions. Some examples of inverse problems are the determination of velocitydepth functions from travel-time curves or from dispersion curves; and these examples show that the solution of the inverse problem may not be unique. The basic question arises: is it possible to determine the internal constitution of the Earth from observations on its surface?A priori, it is not excluded that differing models of the Earth may be indistinguishable when examined from the surface. The complete mathematical solution of this question is rather complicated. We will choose a simplified case: the inverse problem for the one-dimensional wave equation-' equation of the string ', Many aspects of this simplified problem will be important for the general problem. This simplified problem is also of independent interest, it is a problem of interpretation of normal incidence.
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