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Abstract: Reverse time migration and full waveform inversion involve the crosscorrelation of two wavefields, propagated in the forward- and reverse-time directions, respectively. As a result, the forward-propagated wavefield needs to be stored, and then accessed to compute the correlation with the backward-propagated wavefield. Boundary-value methods reconstruct the source wavefield using saved boundary wavefields and can significantly reduce the storage requirements. However, the existing boundary-value methods are based on the explicit finite-difference (FD) approximations of the spatial derivatives. Implicit FD methods exhibit greater accuracy and thus allow for a smaller operator length. We develop two (an accuracy-preserving and a memory-efficient) wavefield reconstruction schemes based on an implicit staggered-grid FD (SFD) operator. The former uses boundary wavefields at M layers of grid points and the spatial derivatives of wavefields at one layer of grid points to reconstruct the source wavefield for a (2M?+?2)th-order implicit SFD operator. The latter applies boundary wavefields at N layers of grid points, a linear combination of wavefields at M–N layers of grid points, and the spatial derivatives of wavefields at one layer of grid points to reconstruct the source wavefield (0 ≤ N?