Numerical investigation of the fundamental mode sensitivity to shear modulus inversion in soil deposits

The shear modulus ( $$G$$ ) profile of subsoil typically increases with depth. However, under specific geotechnical conditions or due to artificial surface stiffening, an inverted profile may occur, in which a stiffer layer overlies a softer one. This configuration challenges conventional assumptions in seismic site response analysis and motivates dedicated investigation. In this study, the governing equilibrium equations for a horizontally infinite soil deposit bounded by bedrock and free surface are derived assuming a shear modulus that decreases linearly with depth. The resulting equation of motion is spatially discretized and solved as an eigenvalue problem to estimate modal parameters (natural frequencies and mode shapes). A computationally efficient finite-difference model is developed and used to generate an extensive dataset through Monte Carlo simulations, which is then exploited to derive a predictive relationship for the fundamental frequency accounting for shear-wave velocity inversion. To better connect the idealised continuous inversion to common field configurations, an additional two-layer stepwise stiff-over-soft model is analysed using a conservative finite-difference discretization across the interface. A dedicated Monte Carlo study is carried out by varying the stiffness contrast and thickness ratio of the surficial stiff layer and comparing the first-mode frequency with the analytical homogeneous reference. Model results are validated against available analytical solutions from the literature and further compared with a 1D elastic site-response analysis in the frequency domain. Finally, the method is applied to a real geotechnical case study in the historical center of L’Aquila (Italy), an area characterized by complex subsurface conditions and significant velocity inversion, confirming the capability of the proposed approach to estimate modal parameters in realistic scenarios and providing additional insight into the dynamic behavior of inverted $$V_S$$ profiles.