Effect of soil and bedrock anelasticity on the S-wave amplification function

We analyse how intrinsic attenuation and bedrock elasticity affect the amplitude and frequency of the resonance peaks of the S-wave amplification function. The Zener model (with a single relaxation peak) and the constant-Q model are used to describe attenuation. We consider two different cases, namely, the soil is softer than the bedrock (the usual situation, that is, a sediment overlying a stiff formation) and the upper layer is stiffer than the lower half-space (e.g. basalt over sediment). The presence of Zener loss in the upper layer causes a shift of the fundamental peak towards the low frequencies, while no shift is observed due to the non-rigid (viscoelastic) character of the half-space. In the constant-Q case, the shift to the low frequencies is not significant implying that it is difficult to estimate the attenuation parameters on the basis of the location of the resonance peaks. However, attenuation affects the amplitude of the higher modes, while these modes have the same amplitude of the fundamental mode no matter the degree of elasticity of the half-space. Attenuation of the layer and non-rigidity of the half-space affect the peaks, with the latter having a stronger effect. Examples are given, considering two real cases representing a glacier in Northern Italy and an ice stream in the Antarctic continent.