Analytical 1D transfer functions for layered soils

Transfer functions are constantly used in both Seismology and Geotechnical Earthquake Engineering to relate seismic displacement 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 its 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. 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, one can extend the results derived elsewhere [Garcia-Suarez, Joaquin. 2021. “Trace Spectrum of 1D Transfer Matrices for Wave Propagation in Layered Media.” engrXiv. June 24. doi:10.31224/osf.io/ygt8z] in the context of longitudinal wave propagation in modular rods to seismic response of stratified sites. 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.