An Efficient Approach to the Seismogram Synthesis for a Basin Structure Using Propagation Invariants

The AN-Lamer method is one of the cheapest methods for synthetic seismograms in irregularly layered media. In this article, we propose a new approach for a two-dimensional S H problem, solved originally by Aki and Lamer (1970). This new approach is not only based on the Rayleigh ansatz used in the original AkiLamer method but also uses further information on wave fields, i.e., the propagation invariants. We reduce two coupled integral equations formulated in the original AkiLarner method to a single integral equation. Applying the trapezoidal rule for numerical integration and collocation matching, this integral equation is discretized to yield a set of simultaneous linear equations. Throughout the derivation of these linear equations, we do not assume the periodicity of the interface, unlike the original AkiLarner method. But the final solution in the space domain implicitly includes it due to use of the same discretization of the horizontal wavenumber as the discrete wavenumber technique for the inverse Fourier transform from the wavenumber domain to the space domain. The scheme presented in this article is more efficient than the original Aki-Larner method. The computation time and memory required for our scheme are nearly half and one-fourth of those for the original Aki-Lamer method. We demonstrate that the band-reduction technique, approximation by considering only coupling between nearby wavenumbers, can accelerate the efficiency of our scheme, although it may degrade the accuracy. Introduction The Aki-Larner method (Aki and Lamer, 1970), based on the Rayleigh hypothesis (e.g., Millar, 1973) and the discrete wavenumber representation, is one of the widely used methods for synthetic seismograms in irregularly layered media. In the AN-Lamer method, the wave field in each layer is expressed as a superposition of plane harmonic waves, including inhomogeneous plane waves, and the boundary condition is matched in the horizontal wavenumber domain by taking advantage of the fast Fourier transform (FFr) in space. The method has been extended to the time domain by Bouchon (1973) and Bard and Bouchon (1980), to the case of multiple layers by Kohketsu (1987a) and Horike (1987), and to the case of vertically inhomogeneous layers by Bard and Gariel (1986). Recently, the Aki-Larner method has been extended to solve three-dimensional (3D) problems with two-dimensionally irregular interfaces by Horike et al. (1990), Ohori et al. (1990), and Uebayashi et al. (1992). Since the AN-Lamer method has the advantage that it requires less memory and less computation than most of the alternative numerical methods (e.g., Axilrod and Ferguson, 1990), it may be one of the cheapest methods. In this article, we further develop the Aki-Larner method by using the propagation invariants that were first introduced by Kennett (1984). A general form of the propagation invariants for time harmonic elastic fields in irregularly layered media was shown by Kennett (1984), which is closely related to Betti's theorem. The propagation invariants depend on the interaction of two distinct wave fields and vanish if both fields are identical. Haines (1988) extended the concept of propagation invariants to the spatial wavenumber domain for acoustic wave fields in laterally varying media. Subsequently, Kennett et al. (1990) used the propagation invariants to describe elastic wave fields in anisotropic, laterally varying media. Koketsu et al. (1991) exploited the propagation invariants in the coupled wavenumber domain to simplify the representation of the reflection and transmission of elastic waves in irregularly layered media. Recently, Takenaka et al. (1993) derived a number of new forms of propagation invariants for elastic waves in irregularly layered media, which are based on the transformation properties of integral operators represented in terms of displacement and traction components. Here we use a form of the propagation invariants presented by them.