In 3D seismic data acquisition, sufficiently dense spatial sampling is most often not possible, because of physical, financial, and temporal constraints. The resulting aliased data is a major obstacle for accurate subsurface images. Therefore, methods are needed to reconstruct these sparse data to an adequate sampling. We develop a novel kinematic wavefront-based seismic data reconstruction method that uses the existing nonlinear beamforming (NLBF) framework, which allows for building a more detailed sampling from a sparse input. We present the theory and methodology of our NLBF reconstruction algorithm and test it on a synthetic Society of Exploration Geophysicists Advanced Modeling (SEAM) Arid dataset and a field dataset. We attempt to answer the following question: can the NLBF framework be used for seismic data reconstruction, and how do these results compare to some conventional reconstruction methods? Control parameter tests are performed to find the optimal NLBF reconstruction, and results are compared to those from several control methods including the convergent alternating projection onto convex sets (POCS) and bootstrap POCS methods, which were also developed for this study. Our NLBF reconstruction method successfully reconstructs high-quality data on both datasets. We find that the optimal NLBF control parameters are ultimately dataset-dependent as the concrete data acquisition geometry plays a central role. Specifically for our datasets, optimal parameters include a time window of 12∆ts, an operator aperture of 600 m x 600 m, and a parameter trace interval of ∆x = ∆y = 60 m. The synthetic SEAM Arid NLBF reconstruction results show high trace fidelity to the ground truth. The field data NLBF results show better-reconstructed gaps, and wavefield variations closer to those of physical propagating waves in comparison to our control methods. These results show the potential of the NLBF reconstruction method to become a common data reconstruction tool used in the seismic industry.