Full-waveform inversion (FWI) is an iterative locally linearized data-fitting technique. The FWI method attempts to move from an initial low-wavenumber representation of earth parameters to a broader representation of the medium. An issue with the method is that FWI is an ill-posed problem, oversampled for the numerical forward discretization. The success of the model parameter reconstruction can often be greatly affected by external factors such as the presence of noise in the input field data or other artifacts arising from the imaging condition present in the FWI gradient computation. We have developed a strategy for mitigating against the influence of such external factors by preconditioning the discrete data gradient using an efficient differential approach instead of the often used integral formulation. Such an application of a smoothing correlation operator allows one to use prior information to locally filter along expected geological dips while being consistent with faults. The application of this preconditioning strategy to real and synthetic 2D data sets illustrates how this incremental additional step makes the FWI workflow less sensitive to noise and spatial aliasing artifacts. Nothing prevents a possible 3D acoustic extension, thanks to the small added computer cost for this local filter application. Three-dimensional elastic FWI may require this inexpensive filtering strategy due to the prohibitive forward-modeling costs that could be partially mitigated by using a coarse shot increment in conjunction with gradient preconditioning.