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In this work, we introduce a novel approach for determining a joint sparse spectrum from several non-uniformly sampled data sets, where each data set is assumed to have its own, and only partially known, sampling times. The problem originates in paleoclimatology, where each data point derives from a separate ice core measurement, resulting in that even though all measurements reflect the same periodicities, the sampling times and phases differ among the data sets, with the sampling times being only approximately known. The proposed estimator exploits all available data using a sparse reconstruction framework allowing for a reliable and robust estimation of the underlying periodicities. The performance of the method is illustrated using both simulated and measured ice core data sets.