The One-Dimensional Inverse Problem of Reflection Seismology

@article{citeulike:915842, abstract = {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 \&\#215; 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 solutions of the discrete schemes and of the differential equations is discussed. Finally the favored discrete downward continuation (DC) algorithm is demonstrated by means of numerical examples, which are illustrated graphically.}, author = {Bube, Kenneth P. and Burridge, Robert }, citeulike-article-id = {915842}, journal = {SIAM Review}, keywords = {acoustics, favorite\_papers, inverse, scattering}, number = {4}, pages = {497--559}, posted-at = {2006-10-28 13:59:37}, priority = {0}, title = {The One-Dimensional Inverse Problem of Reflection Seismology}, url = {http://www.jstor.org/stable/2029469}, volume = {25}, year = {1983} }

TY - JOUR ID - citeulike:915842 L3 - citeulike-article-id:915842 TI - The One-Dimensional Inverse Problem of Reflection Seismology JF - SIAM Review VL - 25 IS - 4 SP - 497 EP - 559 N2 - 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 × 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 solutions of the discrete schemes and of the differential equations is discussed. Finally the favored discrete downward continuation (DC) algorithm is demonstrated by means of numerical examples, which are illustrated graphically. KW - acoustics KW - favorite_papers KW - inverse KW - scattering AU - Bube, Kenneth AU - Burridge, Robert PY - 1983/// UR - http://www.jstor.org/stable/2029469 ER -