Multiscale solution of a wave equation using stable finite differences
Roman Kazinnik, Vladimir Bashkardin · 2010
A multiscale solution to wave equation is useful in seismic modeling. Different-scale solutions are helpful when computing quick initial approximations. Multiplicity of scales is commonly achieved by accurate decimation of data in space and low-pass filtering in time. Another of its applications can be put to use in the area of tomography problems. The minimization functional in a tomography problem may have many local minima at the original high-resolution scale, whereas a global minimum becomes dominant at low-resolution scales. The multiscale approach helps in solving tomography minimization problems by ascertaining an initial approximation at the low-resolution scale. This work improves on the existing multiscale paradigm with the ability to produce multiscale solutions in the original nondecimated time and space domain. We therefore propose using stable finite differences, which require neither decimation in space nor low-pass filtering in time. Because processing of nondecimated data is more time consuming than processing of smaller decimated spaces, we present an efficient implementation capable of attaining computational speed comparable to that of implicit finite differences.