Transfer matrix approach to earthquake amplification through layered soils

In this paper, the system transfer matrix for layered soil is obtained as the continued product of the transfer matrices for each individual layer. To determine the transfer function for the medium, the Laplace transformation is used to suppress the time variable in the governing equation, and the fact that the continuity of the shearing stress and the displacement (or equivalently of their transforms) must be maintained across each interface, naturally suggests their choice as the components of a state vector. After the problem has been recast into a transfer matrix setting, the continuity of the state vector is automatically assured by continued matrix multiplication.The transfer matrix for a single soil layer is considered first, for which the complex amplification is obtained. The influence of various levels of damping on the amplification is discussed, and it is shown that the response to a harmonic input is easily determined since the complex amplification is precisely the frequency response function. The multilayered medium is considered next, and it is shown that the amplification term is found through the multiplication of layer transfer matrices. Time response can then be obtained by means of a Fourier inversion which is easily accomplished through a fast Fourier transform algorithm. Finally, the problem of a soil layer on semi-infinite rock is considered in the absence of damping. Since the solution of this problem can be readily obtained in closed form, its discussion and interpretation are relatively simple. The fact that the transmission and reflection of waves at the soil-rock interface does not occur is analytically demonstrated, and explained on physical grounds.