  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, possibly disjoint, and only partially known, sampling
times. The potential of the proposed approach is illustrated using a spectral
estimation problem in paleoclimatology. In this problem, 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. In addition, sampling times are only approximately
known. The resulting joint estimate exploiting all available data is formulated
using a sparse reconstruction framework allowing for a reliable and robust
estimate of the underlying periodicities. The corresponding misspecified
Cram\'er-Rao lower bound, accounting for the expected sampling uncertainties,
is derived and the proposed method is shown to attain the resulting bound when
the signal to noise ratio is sufficiently high. The performance of the proposed
method is illustrated as compared to other commonly used approaches using both
simulated and measured ice core data sets.
