  Understanding centennial scale climate variability requires data sets that
are accurate, long, continuous and of broad spatial coverage. Since
instrumental measurements are generally only available after 1850, temperature
fields must be reconstructed using paleoclimate archives, known as proxies.
Various climate field reconstructions (CFR) methods have been proposed to
relate past temperature to such proxy networks. In this work, we propose a new
CFR method, called GraphEM, based on Gaussian Markov random fields embedded
within an EM algorithm. Gaussian Markov random fields provide a natural and
flexible framework for modeling high-dimensional spatial fields. At the same
time, they provide the parameter reduction necessary for obtaining precise and
well-conditioned estimates of the covariance structure, even in the
sample-starved setting common in paleoclimate applications. In this paper, we
propose and compare the performance of different methods to estimate the
graphical structure of climate fields, and demonstrate how the GraphEM
algorithm can be used to reconstruct past climate variations. The performance
of GraphEM is compared to the widely used CFR method RegEM with regularization
via truncated total least squares, using synthetic data. Our results show that
GraphEM can yield significant improvements, with uniform gains over space, and
far better risk properties. We demonstrate that the spatial structure of
temperature fields can be well estimated by graphs where each neighbor is only
connected to a few geographically close neighbors, and that the increase in
performance is directly related to recovering the underlying sparsity in the
covariance of the spatial field. Our work demonstrates how significant
improvements can be made in climate reconstruction methods by better modeling
the covariance structure of the climate field.
