Integral Geometry and Inverse Problems for Hyperbolic Equations

Abstract

I. Some Problems in Integral Geometry.- 1. Problem of Finding a Function from its Integrals over Ellipsoids of Revolution.- 2. Generalization to the Case of Analytic Curves.- 3. Existence Theorem for the Case of Ellipses.- 4. Determination of a Function from its Integrals over a Family of Curves Invariant to Displacement.- 5. The Integral-Geometric Problem for m Functions.- 6. Determination of a Function in a Circle from its Integrals over a Family of Curves Invariant to Rotation about Center of the Circle.- 7. Integral-Geometric Problem for Surfaces Invariant to Displacement.- 8. Integral-Geometric Problems for a Family of Curves Generated by a Riemannian Metric.- II. Inverse Problems for Hyperbolic Linear Differential Equations.- 1. General Information Concerning the Solution of the Cauchy Problem for Linear Hyperbolic Equations.- 2. One-Dimensional Inverse Problem for the Telegraph Equation in Three-Dimensional Space.- 3. Linearized Inverse Problem for the Telegraph Equation.- 4. The Problem of Finding the Coefficients of the Lower Order Derivatives in a Second-Order Equation.- 5. Linearized Inverse Kinematic Problem for the Wave Equation in Variable Isotropic Media.- 6. One-Dimensional Inverse Kinematic Problem for the Wave Equation in Anisotropic Media.- 7. Multidimensional Linearized Inverse Kinematic Problem for the Wave Equation in Anisotropic Media.- III. Application of the Linearized Inverse Kinematic Problem to Geophysics.- 1. The Earth's Structure from a Geophysical Standpoint and the Problem of Determining the Velocity Structure of the Earth's Mantle.- 2. Numerical Solution of the Linearized Inverse Kinematic Problem.- 3. Some Numerical Results.