NUMERICAL SOLUTION OF THE ACOUSTIC AND ELASTIC WAVE EQUATIONS BY A NEW RAPID EXPANSION METHOD¹

Abstract

We present a new rapid expansion method (REM) for the time integration of the acoustic wave equation and the equations of dynamic elasticity in two spatial dimensions. The method is applicable to spatial grid methods such as finite differences, finite elements or the Fourier method. It is based on a Chebyshev expansion of the formal solution to the appropriate wave equation written in operator form. The method yields machine accuracy yet it is faster than methods based on temporal differencing. Its disadvantages are that it does not apply to all types of material rheology, and it can also require much storage when many snapshots and time sections are desired. Comparisons between numerical and analytical solutions for simple acoustic and elastic problems demonstrate the high accuracy of the REM.

The pseudo-acoustic equations of the scalar wavefield in anisotropic media

“…temporal frequency ω a is expressed by an algebraic equation of the sixth order. This is a consequence of the fact that in elements a 11 , a 22 , a 33 of matrix (7) quadric frequency ω a remains.…”

“…Równania bazują na dyspersyjnych relacjach wyznaczających wartości własne -częstotliwości w funkcji liczb falowych i parametrów anizotropii na podstawie pełnego systemu równań akustycznych. Zaprezentowano kilka przypadków propagacji falowej określonej jednostronnym równaniem w formie migawkowych zdjęć dla różnych typów sygnału.Słowa kluczowe: anizotropia, poprzeczna izotropia (TI), modelowanie, jednostronne równanie falowe, relacja dyspersyjna.Forward modelling of seismic waves in anisotropic media should be executed by solving a full system of elastic wave equations using numerical techniques such as finite-difference, spectral and pseudo-spectral method [3,7,9,10,11,17,19,20,23]. Due to the very time-consuming calculation system of modelling three component acoustic field and insufficient data concerning elastic parameters, in practical seismic research the tendency to simplify the theory and to limit oneself to the considerations of a few models of anisotropy can be noted.…”

“…Forward modelling of seismic waves in anisotropic media should be executed by solving a full system of elastic wave equations using numerical techniques such as finite-difference, spectral and pseudo-spectral method [3,7,9,10,11,17,19,20,23]. Due to the very time-consuming calculation system of modelling three component acoustic field and insufficient data concerning elastic parameters, in practical seismic research the tendency to simplify the theory and to limit oneself to the considerations of a few models of anisotropy can be noted.…”

Pseudo-spectral full-waveform inversion

“…Ос-новной сложностью с вычислительной точки зрения здесь является моделирование волновых полей по из-вестной опорной модели m, которая представляется либо векторной функцией распределения в простран-стве m = m(x), либо просто вектором параметров m = (m 1 ,…m N ). Задача моделирования решается с по-мощью псевдоспектрального метода [Kosloff et al, 1989;Fornberg, 1987Fornberg, , 1998]. …”

Análise dos métodos de diferenças finitas e expansão rápida na migração reversa no tempo

“…Dentre os métodos explícitos, temos: diferenças finitas (finite difference -FD), que faz uso da expansão de Taylor (Etgen, 1986); o método de expansão rápida (rapid expansion method -REM), que faz uso da expansão de Chebyshev (Kosloff et al, 1989) e o método "two-step explicit marching" (Soubaras & Zhang, 2008), que faz uso duma expansão polinomial. Na indústria o FD explícitoé normalmente utilizado para implementar a RTM.…”