Advances in Geophysics, Vol. 43 by Barry Saltzman, Renata Dmowska

By Barry Saltzman, Renata Dmowska

This sequence presents a venue for longer experiences of present advances in geophysics. Written at a degree available to graduate scholars, the articles serve to develop wisdom of assorted fields and should be priceless in classes and seminars.

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4. Finite Element The finite element method can be applied to models where the geological irregularities are placed between two flat layered media, separated by vertical interfaces (Fig. 6). ,,~,,~,,,,,,. . r . :;. ~ , r . ~. ':. ':. 'r162:. ':. ":. ~ . 6. Typical structural model for finite element method. SEISMIC WAVE PROPAGATION IN HETEROGENEOUS MEDIA 23 The classic procedure (Lysmer and Drake, 1972) implies a stationary analysis with finite elements of the irregular area. Special boundary conditions simulate the presence of the two layered halfspaces at each side.

Expression (63) does not contain any approximation since A is a generic function of x and to. To obtain the approximation of the classic ray theory we have to expand the amplitude vector in a series of inverse powers of to oo A(x, Xo, to) - S(to) Z Ai(x, Xo)to -i , (64) i=O where S(to) is the FT of the source time function. x0) u(x, to) = S(to)A0(x, x0) ~/J (x, x0) (65) and represents an approximation of the solution of the wave equation valid only at high frequencies, when the higher terms can be neglected in (65).

51) Umax The spatial derivatives that appear in Eq. , Chebyshev polynomials, Fourier series; the proper choice depends on the periodicity of the problem). For example, the simplified Eq. (46) become OUy at 1 OCrxv p(x) Ox O(rx,, Ou~. at = IX(X) Ox (52) where the notation ~(x, k, t) -- f OG g(x, y, t)e-ikYdy (53) OG is adopted, and k is the wavenumber for the y variable. For each value of k, Eq. (52) can be solved in the wavenumber domain. In the actual calculations, k has to be discretized and such an operation is correctly done only by assuming that the source-medium configuration is periodic along the y-axis.

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