The Full Waveform Inversion (FWI) gradient is composed of a low wavenumber tomographic component and a high wavenumber migration component. A successful application of FWI requires that the low wavenumber parts of the model be recovered before the high wavenumbers. This process becomes difficult in datasets dominated by pre-critical angle reflection energies. Reflection waveform inversion (RWI) has been proposed as an alternative to help bootstrap the FWI method for reflection data. In this thesis, I have made a novel contribution to RWI using Finite Di fference Explicit Wavefi eld Decomposition (FDEWD). This method improves the wavefi eld decomposition process by cleanly decomposing the wavefi elds into four components using fi nite diff erence method and Fourier transform. Four component wavefi elds travelling left, right, up and down are simultaneously derived in this method compared to just opposite directions possible with most other methods. FDEWD also lacks the evanescent energy present in traditional Fourier based separation. The extra layer of separation introduced by FDEWD ensures that the tomographic component of the gradient is formed by energies propagating within and close to the first Fresnel zone, hence yielding a cleaner tomographic update. The FDEWD method developed here was then used in an RWI context to successfully invert a synthetic dataset and a blind dataset. The scheme involved a migration update step with an exaggerated step length and a tomographic update step with true step length computation. The results obtained shows that the new method produces superior results compared to the method based on direct separation of the total wavefi elds. FDEWD also allows for transmission FWI to be performed without the need to mute the data in any way. We have implemented the scheme here in a 2-D constant density acoustic wave equation. It is, however, possible to extend this method to 3-D, anisotropic and elastic problems.