Analytical 1D transfer functions for layered soils

Transfer functions are constantly used in both Seismology and Geotechnical Earthquake Engineering to relate seismic ground motion at different depths within strata. In the context of diffusive theory, they also appear in the expression of the imaginary part of 1D Green's functions. In spite of their remarkable importance, their mathematical structure is not fully understood yet, except in the simplest cases of two or three layers at most. This incomplete understanding, in particular as to the effect of increasing number of layers, hinders progress in some areas, as researchers have to resort to expensive and less conclusive numerical parametric studies. This text presents the general form of transfer functions for any number of layers, overcoming the above issues. The mathematical structure of these transfer functions comes defined as a superposition of independent harmonics, whose number, amplitudes and periods we fully characterize in terms of the properties of the layers in closed-form. Owing to the formal connection between seismic wave propagation and other phenomena that, in essence, represent different instances of wave propagation in a linear-elastic medium, we have extended the results derived elsewhere, in the context of longitudinal wave propagation in modular rods, to seismic response of stratified sites. The ability to express the reciprocal of transfer functions as a superposition of independent harmonics enables new analytical approaches to assess the effect of each layer over the overall response. The knowledge of the general closed-form expression of the transfer functions allows to analytically characterize the long-wavelength asymptotics of the horizontal-to-vertical spectral ratio for any number of layers.