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In media with strong vertical variations in elasto-dynamic properties, the characteristics of surface waves may deviate significantly from their typical appearance in media with small contrasts. Different numerical examples suggest that ignoring this behavior may lead to poor results in surface wave analysis. The cases analysed are: (1) Neighboring surface wave modes which exchange their major characteristics at osculation points of their dispersion curves; (2) Quasi-channel waves propagating inside a buried low velocity channel; (3) Surface waves in the presence of a low-velocity layer at the very surface; and (4) Rayleigh surface waves in the presence of a very high velocity contrast allowing normal quasi-compressional pseudo-modes. Amplitude-depth functions are used to analyze the physical cause of the unusual characteristics. One important result is a vanishing vertical displacement of the fundamental mode and dominance of the first higher Rayleigh mode at low frequency for cases with a large elastic contrast at shallow depth (increase in Vs approaching 300% at 5 m). If the displacement is treated as the fundamental mode, a systematic overestimate of deeper shear wave velocity will arise in the inversion, since higher modes have higher phase velocity. A second result is from a profile with a very thin, surficial low-velocity layer, where higher modes dominating the response might prevent fundamental mode detection and inversion. Finally, if shear wave velocity of the half-space exceeds the compressional wave velocity of an overlying layer, quasi-compressional waves (or guided P-waves) come into the velocity range of normal modes. This may lead to surface wave mode-misidentification.