The One-Dimensional Inverse Problem of Reflection Seismology

An impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth. The ensuing particle velocity at the surface is assumed to be measured. The problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one-dimensional inverse problem of reflection seismology.The problem is set up mathematically as an inverse problem for a first order $2 \times 2$ hyperbolic system. The role of propagation of singularities is explained and the relation between the solution of the inverse problem and the Cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem.It is found that recursive numerical techniques such as the Levinson algorithm and certain other fast methods for Cholesky factorization are intimately related to explicit finite-difference schemes for solving hyperbolic systems.The order of approximation between the ...