In seismic exploration, full-waveform inversion (FWI) relies on low-frequency data to succeed if a good initial model is unavailable. However, field seismic data excited by active sources are typically band-limited above 3 Hz. By extrapolated FWI, we can start inversion from computational low frequencies extrapolated from band-limited data. However, low-frequency extrapolation with deep learning is challenging for field data since a neural network trained on synthetic data usually generalizes poorly on real seismic data. Here we use a semi-supervised learning method to extrapolate low frequencies for field data by training with real data without real labels. Specifically, by training CycleGAN with unpaired images of field 4 - 10 Hz band-limited and synthetic 0 - 1 Hz low-frequency shot gathers, we can robustly extrapolate the 0 - 1 Hz low frequencies for the field data band-limited above 4 Hz. The source wavelet for the simulation of synthetic low-frequency data is used as the source in FWI using the extrapolated data. The inverted velocity model using only the extrapolated low frequencies is comparable to the tomography model, where tomography requires intense human efforts to build a good initial model for FWI. Our method strengthens the ability of FWI for mapping fine Earth structures by mitigating the cycle-skipping problem effectively.