Conventional L2 norm based full waveform inversion usually has more than one local minima and is also sensitive to noise in the observed data. The Earth is at least attenuative, and thus, a pixel-to-pixel fitting of the waveform is not feasible in practice. We propose a normalized nonzero-lag crosscorrelation based elastic full waveform inversion algorithm, which intends to maximize the similarity of the predicted and observed data. The proposed objective function emphasizes that the matching the phases of the predicted and observed data is more immune to the simplified physics we often use to represent the medium. The normalization term can compensate the energy loss in the far offset and avoid the estimates being biased by extreme values in the observed data. We introduce a polynomial-type weighting function and suggest an approach to determine the optimal time lag. A modified elastic Marmousi model is used to verify the effectiveness of the proposed method.