A Bayesian algorithm for reconstructing spatially averaged temperature - preliminary results

Martin Tingley

Abstract

The determination of spatially averaged temperature from point estimates is a non-trivial statistical problem. In the paleo-climate context, the additional need to convert proxy time-series into temperature estimates presents a serious challenge.

Most estimates of spatially averaged temperature at paleo-climate time-scales address these two issues sequentially: the proxy values are first averaged through space, and these estimates are then transformed onto the temperature scale via some form of regression. This two step approach distances the final estimate of temperature from the underlying data, complicating estimates of the associated uncertainty.

Our approach is to model the relationship between the true temperature field and the noisy, localized measurements of it using a hidden Markov model. We use a fully Bayesian algorithm to simultaneously estimate the coefficients linking the proxy values to temperature units, the parameters associated with both the temporal and spatial covariance structures, the observational error variances, the temperature values at a large number of uniformly distributed spatial locations, and the average of these estimated temperature values. A major benefit of this Bayesian approach is that, by drawing repeatedly from the full conditional posterior distributions, we obtain an estimate of the uncertainty covariance structure.

We present preliminary results of applying the Bayesian algorithm to 113 tree ring density series in the northern hemisphere high latitudes.