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The inversion of dispersive Rayleigh-wave data has been shown to be successful in providing reliable estimated shear-wave velocities within unconsolidated materials in the near surface. However, in a case where the multi-channel analysis of surface waves method was applied to a site consisting of clay residuum overlying basalt bedrock, inversion for the fundamental-mode Rayleigh wave resulted in shear-wave velocities within the rock that are less than half of expected values. Forward modeling reveals that the fundamental-mode dispersion curve is hardly sensitive to bedrock velocity perturbations over a practical range of wavelengths, leading to poorly constrained solutions. Standard surface-wave methods can fail because of a shortage of phase-velocity estimates at the low frequencies that are necessary to properly constrain shear-wave velocities at depth. The commonly used guideline that maximum investigation depth is roughly half of the largest recorded wavelength can be misleading. Data at much lower frequencies (i.e., longer wavelengths) than typically acquired might be required to obtain a meaningful shear-wave velocity profile, particularly for a site with a high-velocity half-space beneath a low-velocity layer. For such cases, layer geometry appears to have a large impact on inversion results. Consequently, Rayleigh-wave methods can be effective in determining depth to bedrock in simple, layered geologies (e.g., soft sediment over hard bedrock) when independent information of shear-wave velocity is available. Analysis techniques that address higher modes of Rayleigh-wave propagation may be useful for more accurately resolving depth and velocity of a high-velocity half-space. In the studied case, higher modes can theoretically reach the asymptotic high-velocity limits within the range of recorded frequencies.