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Tissue elastography measures tissue mechanical properties, which contain important physiological information and help medical diagnosis. Instead of tracking shear wave propagation inside tissue as do magnetic resonance elastography and ultrasound based techniques, this study focuses on monitoring the propagation of surface Raleigh waves stimulated by short impulses. The method is noncontact, noninvasive, and low cost and has a potential for clinical applications. A customized device designed to measure surface wave propagation is constructed based on a laser displacement sensor (LDS). Experiments are carried out on two porcine skin gelatin phantoms of different concentrations. For each phantom, the phase velocities of specific frequencies are extracted using a cross-spectrum method and then the material elasticity and viscosity are found by fitting the phase velocities with the Voigt's model. The results suggest that measuring viscoelasticity by monitoring the response to a surface impulse is an efficient method because of the richness of frequency content of impulse responses. The results are validated with a standard continuous wave (CW) method.