Compilations of paleoceanographic observations for the deep sea now contain a few hundred points along the oceanic margins, mid-ocean ridges, and bathymetric highs, where seawater conditions are indirectly recorded in the chemistry of buried benthic foraminiferal shells. Here we design an idealized experiment to test our predictive ability to reconstruct modern-day seawater properties by considering paleoceanographic-like data. We attempt to reconstruct the known, modern-day global distributions by using a state estimation method that combines a kinematic tracer transport model with observations that have paleoceanographic characteristics. When a modern-like suite of observations (Θ, practical salinity, seawater δ18O, δ13CDIC, PO4, NO3, and O2) is used from the sparse paleolocations, the state estimate is consistent with the withheld data at all depths below 1500 m, suggesting that the observational sparsity can be overcome. Physical features, such as the interbasin gradients in deep δ13CDIC and the vertical structure of Atlantic δ13CDIC, are accurately reconstructed. The state estimation method extracts useful information from the pointwise observations to infer distributions at the largest oceanic scales (at least 10,000 km horizontally and 1500 m vertically) and outperforms a standard optimal interpolation technique even though neither dynamical constraints nor constraints from surface boundary fluxes are used. When the sparse observations are more realistically restricted to the paleoceanographic proxy observations of δ13C, δ18O, and Cd/Ca, however, the large-scale property distributions are no longer recovered coherently. At least three more water mass tracers are likely needed at the core sites in order to accurately reconstruct the large-scale property distributions of the Last Glacial Maximum.