The double square root equation for laterally varying media in midpoint-offset coordinates provides a convenient framework for developing efficient 3-D prestack wave-equation depth migrations with screen propagators. Offset-domain pseudoscreen prestack depth migration downward continues the source and receiver wavefields simultaneously in midpoint-offset coordinates. Wavefield extrapolation is performed with a wavenumber-domain phase shift in a constant background medium followed by a phase correction in the space domain that accommodates smooth lateral velocity variations. An extra wide-angle compensation term is also applied to enhance steep dips in the presence of strong velocity contrasts. The algorithm is implemented using fast Fourier transforms and tri-diagonal matrix solvers, resulting in a computationally efficient implementation. Combined with the common-azimuth approximation, 3-D pseudoscreen migration provides a fast wavefield extrapolation for 3-D marine streamer data. Migration of the 2-D Marmousi model shows that offset domain pseudoscreen migration provides a significant improvement over first-arrival Kirchhoff migration for steeply dipping events in strong contrast heterogeneous media. For the 3-D SEG-EAGE C3 Narrow Angle synthetic dataset, image quality from offset-domain pseudoscreen migration is comparable to shot-record finite-difference migration results, but with computation times more than 100 times faster for full aperture imaging of the same data volume.