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The receiver function (RF) method is fundamental in assessing mantle seismic discontinuity depths and reflectivities. Most of the current approaches rely on phase equalization, though in many applications, high levels of incoherent noise, incomplete and irregular sampling customarily interfere with the analysis of weak secondary phases. In recent years, advancements in the field of multidimensional seismic data processing have triggered a shift in interest towards its application to RFs, specifically to single station gathers that depend on a single spatial dimension. Our work generalizes the application of singular spectrum analysis to RFs that rely on two and four spatial dimensions recorded by dense seismic arrays. We adopt a multidimensional signal processing approach known as multichannel singular spectrum analysis. We develop a strategy to assemble and enhance 3-D and 5-D seismic volumes via matrix rank reduction and a reinsertion algorithm to simultaneously suppress random noise, retrieve absent observations and boost identifiability of secondary conversions. We provide informative synthetic examples to gain insight into the effectiveness and limitations of our approach. In the real data example, we improve weak conversions from the mantle transition zone (MTZ) recorded by the USArray in the Yellowstone area. The reconstruction algorithm accurately recovers the timing and polarity of conversions associated with the 410-, 520- and 660-km seismic discontinuities. Our investigation shows that the simultaneous processing of several spatial variables expedites signal restoration, particularly in directions where large recording gaps exist due to a lack of earthquakes, which aids the mapping and identification of the MTZ interfaces. This study presents a theoretical/practical framework for the reconstruction of multidimensional RF data, and its full potential can be exploited with dense acquisition available from the emerging seismic nodal arrays to improve passive seismic imaging. [ABSTRACT FROM AUTHOR]