We perform two tests to investigate to which degree viscoelastic modeling is relevant during a full waveform inversion of shallow seismic surface waves. Firstly, we compare field data with synthetic elastic and viscoelastic data. We show that the optimized source time function acts as a low pass filter in the case of elastic wavefields and can compensate a significant fraction of the residuals between elasticly and viscoelasticly modeled data. However, the viscoelastic data can explain the recorded data better in some aspects like the amplitude decay with offset of the fundamental mode and the near offset traces. Secondly, we run inversion tests for simulated viscoelastic observations (Q=20) using both elastic as well as viscoelastic forward modeling with Q=20, 25, and 10 during the inversion. The results show that it is not possible to infer the steep gradient in the shear wave velocity model in the topmost meter using an elastic inversion. Using a slightly wrong Q factor in the inversion produces very similar results compared to the results obtained by an inversion using the correct Q factor. If we use Q factors that are too far away from the Q factor of the observed data the inversion result becomes worse.