Image-domain seismic inversion with subsurface offset extension may converge to kinematically accurate velocity models without the low-frequency data accuracy required for standard data-domain full waveform inversion. However, this robust alternative approach to waveform inversion suffers from very high computational cost, resulting from its use of nonlocal wave physics: the computation of strain from stress involves an integral over the subsurface offset axis, which must be performed at ev- ery space-time grid point. Additionally, under prototypical conditions of acquisition geometry, the existence of artefacts is very likely to deviate the velocity update from its path to the correct velocity. I show here three new approaches that significantly improve both efficiency and robustness of subsurface offset extended waveform in- version and migration velocity analysis (MVA). The global convergence property of the extended waveform inversion is achieved by adaptively determining the penalty weight. It is also shown that a combination of data-fit driven offset limits, grid coarsening, and low-pass data filtering can reduce the cost of extended inversion by one to two orders of magnitude. Lastly, a taper in angle domain depending on acquisition geometry and imaging point is introduced. The application of taper directly on extended image makes migration velocity analysis becomes more robust. I illustrate these new methods in the context of constant density acoustic waveform inversion, by recovering background model and perturbation fitting band-limited waveform data in the Born approximation.