Context. Helioseismic holography is a useful method for detecting active regions on the Sun's far side and improving space weather forecasts. Aims. We aim to improve helioseismic holography using a clear formulation of the problem, an accurate forward solver in the frequency domain, and a better understanding of the noise properties. Methods. Building on the work of Lindsey et al., we define the forward-and backward-propagated wave fields (ingression and egression) in terms of a Green's

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... . This Green's function is computed using an accurate forward solver in the frequency domain. We analyse overlapping segments of 31 h of SDO/HMI dopplergrams, with a cadence of 24 h. Phase shifts between the ingression and the egression are measured and averaged to detect active regions on the far side. Results. The phase maps are compared with direct extreme-ultraviolet (EUV) intensity maps from STEREO/EUVI. We confirm that medium-sized active regions can be detected on the far side with high confidence. Their evolution (and possible emergence) can be monitored on a daily time scale. Seismic maps averaged over 3 days provide an active-region detection rate as high as 75% and a falsediscovery rate as low as 7% for active regions with areas above one thousandth of a hemisphere. For a large part, these improvements can be attributed to the use of a complete Green's function (all skips) and the use of all available observations on the front side (full pupil). Conclusions. Improved helioseismic holography enables the study of the evolution of medium-sized active regions on the Sun's far side.