Thanks to smooth intrinsic nature of normalized integral wavefield (NIW), full waveform inversion (FWI) using NIW has a potential to retrieve the long wavelength model parameters from seismic data even in the absence of low frequencies. However, if reflection events are dominant in the data, this strategy might fail in rendering a convergent result. To steer the focus of inversion more onto the matching of first arrivals' kinematics, hence the tomographic result, we construct objective functions using the logarithmic NIW and the NIW raised to the power of small positive number for FWI. We derive the gradient formulae of the new objective functions using the adjoint-state method, thus the computational cost of our approaches remains compatible to that of the conventional FWI. Synthetic data example on 2D SEG/EAGE salt-dome model, in which reflections are prevalent, indicates that our FWI approaches generate tomographic convergent results even without low frequency information.