Wavefield injection in finite-difference (FD) grids can be described by the method of multiple point sources. The method teaches how synthetically generated wavefields and wavefield constituents can be reconstructed from surface recordings using a combination of monopole and dipole sources on an injection surface surrounding the model. We show how to properly record surface wavefields and inject point sources in staggered FD grids, in a way that is consistent with the order of spatial accuracy of the FD scheme. The description is general and can be used for schemes of any order. Only one or two surface wavefields are required to reconstruct the original wavefields or wavefield constituents to numerical precision, independent of the order of spatial accuracy of the FD stencil. We have applied the method for the separation of up- and downgoing wavefields and for source wavefield reconstruction for reverse time migration. Our implementation enables accurate source wavefield reconstruction with optimally minimal memory requirements.