Proc. of Conf. on Effects of Surface Geology, Odawara, 1992

Yoshiaki Hisada, Shunroku Yamamoto, and Sukenobu Tani
Waseda University, Ohkubo 3-4-1, Shinjuku, Tokyo 169, Japan

We propose a new Boundary Element Method (BEM), which is based on the exact Green's function of the layered half-space and the normal mode solution. As compared with the usual BEM based on the Green's function of the homogeneous full-space, this BEM drastically reduces both the computer memory and the CPU time for the seismic response problems of alluvial valleys or sedimentary basins, because the boundary elements along the basin bottom and the free surface are not required, in addition to the fast computation of the normal mode solution. We demonstrate the validity and the advantage of this BEM, by comparing results between this BEM and the usual BEM for 2D problems.

Since the tragedy of Mexico City during the 1985 Michoacan earthquake, the long-period strong ground motions in alluvial valleys or sedimentary basins has been widely investigated by a lot of researchers, by mainly 1D or 2D numerical codes. Recent observations, however, indicate that the long-period motions mainly consist of surface waves generated in the basins and are strongly affected by 3-D basin structures (e.g. Perez-Rocha et al. 1991). Although there have been several 3D numerical analyses for basin problems, all those studies restrict to very simple basin models, because of the limitations of computer memory and CPU time (Sanchez-Sesma 1983, Satoh et al. 1988, Mossessian and Dravinski 1990, Horike et al. 1990, Ohori et al. 1990). We, thus, need to develop an effective 3D numerical code for more quantitative study of the basin response problems.
In this study, we propose a new BEM for multi-layered 2D and 3D basin response problems. BEM is the one of the most effective methods for the basin problems, as well known. However, the usual BEM, which is based on the Green's function of the homogeneous half or full spaces, is still too heavy for 3D computations, because a lot of elements are necessary along the basin bottom and the free-surface. On the other hand, when we look at the actual structures of sedimentary basins or alluvial valleys, their thickness are usually very thin as compared with their width. For example, the thickness and the width of the Kanto sedimentary basin, in which Tokyo is located, are 2-3 km and 50-100 km, respectively. Therefore, we may assume that the sedimentary layers are approximately horizontally flat and surface waves are dominant for the propagation of long-period seismic waves in the basin. The proposed BEM is, thus, formulated under those assumptions, as described as follows.

As shown in Fig.1 for a 3D case, the basin model considered here consists of two flat-layered domains; one is an inside domain composed of sedimentary layers and underlying crustal layers, and the other is an outside domain of crustal layers. Wave fields of both domains are formulated by boundary integral equations based on two kinds of fundamental solutions: 1) the exact Green's function of layered half-space for the near field where body waves are dominant; 2) the normal mode solution for the non-near field where surface waves are dominant. We obtain both the exact Green's function and the normal mode solution by following Harkrider (1964), and consider complex Vs and Vp to avoid the singularities corresponding to the Rayleigh and the Love poles for the computations of the exact Green's function. The interface between both domains is discretized from the free surface to a certain depth by the BEM technique, using the fact that the amplitude of both fundamental solutions decreases exponentially with depth in the underlying half-space.

To check the validity and the advantage of the proposed BEM, we compute numerical seismograms considering 2D sedimentary basin models. As shown in Fig.2 and Fig.3 for anti-plane and in-plane cases, respectively, those models consist of a single sedimentary layer (Vs=1 km/s) with a rectangular edge, embedded in a homogeneous half-space bedrock (Vs=3 km/s) which includes a Haskell-type source. Although 43 and 120 boundary elements are required for the usual anti-plane and in-plane BEMs (Hisada et al. 1988), respectively (see Fig.2 (a) and Fig.3 (a)), only 6 elements are necessary for the proposed BEM (see Fig.2 (b) and Fig.3 (b)). This is because boundary elements along the basin bottom and the free-surface are not required for the proposed BEM. Thus, the computer memory is about one-fiftieth and one-four hundredth of those of the usual anti and in-plane BEMs, respectively.
Fig.4 shows synthetic seismograms using three kinds of BEMs. Solid lines are corresponding to the results by the usual BEMs considering the models of Fig.2 (a) and Fig.3 (a). Dotted lines are computed by the BEM based on the exact Green's function of layered half-space only, and dash-dotted lines are by the proposed BEM based on both the exact Green's function for the near field and the normal mode solution for the non-near field. The latter two kinds of lines are obtained using the models of Fig.2 (b) and Fig. 3 (b). As shown in the figures, the results of the proposed BEM show good agreements with those of the usual BEM and the BEM based on the exact Green's function. The computing time of the proposed BEM is much faster than those of the usual BEMs, because of the fast computations of the normal mode solutions.

The proposed BEM compute seismic responses of layered sedimentary basins with much less computer memory and CPU time than the usual BEM. Although we have demonstrated the advantage and the validity of this BEM by simple 2D models, we can easily extend this BEM to 3D basin problems considering realistic and complicated structures of actual sedimentary basins.

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