Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Seismic Ray Theory

© 2001 by Chapman & Hall/CRC. Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems. Inverse Boundary Spectral Problems develop a rigorous theory for solving several types of inverse problems exactly. In it, the authors consider the following: “Can the unknown coefficients of an elliptic partial differential equation be determined from the eigenvalues and the boundary values of the eigenfunctions?” Along with this problem, many inverse problems for heat and wave equations are solved. The authors approach inverse problems in a coordinate invariant way, that is, by applying ideas drawn from differential geometry. To solve them, they apply methods of Riemannian geometry, modern control theory, and the theory of localized wave packets, also known as Gaussian beams. The treatment includes the relevant background of each of these areas. Although the theory of inverse boundary spectral problems has been in development for at least 10 years, until now the literature has been scattered throughout various journals. This self-contained monograph summarizes the relevant concepts and the techniques useful for dealing with them.

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Inverse Boundary Spectral Problems

Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.

Localized Light Waves: Paraxial and Exact Solutions of the Wave Equation (a Review)

An inverse problem for a semi-linear elliptic equation in Riemannian geometries

It is known that, using the Gaussian beam approximation, one can show that there exist solutions of the wave equation on a general globally hyperbolic Lorentzian manifold whose energy is localised along a given null geodesic for a finite, but arbitrarily long, time. We show that the energy of such a localised solution is determined by the energy of the underlying null geodesic. This result opens the door to various applications of Gaussian beams on Lorentzian manifolds that do not admit a globally timelike Killing vector field. In particular, we show that trapping in the exterior of Kerr or at the horizon of an extremal Reissner–Nordstrom black hole necessarily leads to a “loss of derivative” in a local energy decay statement. We also demonstrate the obstruction formed by the red-shift effect at the event horizon of a Schwarzschild black hole to scattering constructions from the future (where the red-shift turns into a blue-shift): we construct solutions to the backwards problem whose energies grow exponentially for a finite, but arbitrarily long, time. Finally, we give a simple mathematical realisation of the heuristics for the blue-shift effect near the Cauchy horizon of subextremal and extremal black holes: we construct a sequence of solutions to the wave equation whose initial energies are uniformly bounded, whereas the energy near the Cauchy horizon goes to infinity.

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Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes

In the paper formal asymptotic solutions of the wave equation are constructed which represent concentrated wave packets propagating along rays. The construction technique is a time generalization of the frequently used spatial complex ray method. Analogously to the method of complex germ of V. P. Maslov, it is based on considering complex solutions of the Hamilton-Jacobi equation.

V. V. Ulin

Papers

Complex space-time ray method and “quasiphotons”