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A Bayesian Algorithm for Reconstructing Climate Anomalies




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A Bayesian Algorithm for Reconstructing Climate Anomalies in Space and Time. Part I: Development and Applications to Paleoclimate Reconstruction Problems MARTIN P. TINGLEY Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, North Carolina, and Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts PETER HUYBERS Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts (Manuscript received 6 January 2009, in final form 30 October 2009) ABSTRACT Reconstructing the spatial pattern of a climate field through time from a dataset of overlapping in- strumental and climate proxy time series is a nontrivial statistical problem. The need to transform the proxy observations into estimates of the climate field, and the fact that the observed time series are not uniformly distributed in space, further complicate the analysis. Current leading approaches to this problem are based on estimating the full covariance matrix between the proxy time series and instrumental time series over a ‘‘calibration’’ interval and then using this covariance matrix in the context of a linear regression to predict the missing instrumental values from the proxy observations for years prior to instrumental coverage. A fundamentally different approach to this problem is formulated by specifying parametric forms for the spatial covariance and temporal evolution of the climate field, as well as ‘‘observation equations’’ describing the relationship between the data types and the corresponding true values of the climate field. A hierarchical Bayesian model is used to assimilate both proxy and instrumental datasets and to estimate the probability distribution of all model parameters and the climate field through time on a regular spatial grid. The output from this approach includes an estimate of the full covariance structure of the climate field and model pa- rameters as well as diagnostics that estimate the utility of the different proxy time series. This methodology is demonstrated using an instrumental surface temperature dataset after corrupting a number of the time series to mimic proxy observations. The results are compared to those achieved using the regularized expectation–maximization algorithm, and in these experiments the Bayesian algorithm produces reconstructions with greater skill. The assumptions underlying these two methodologies and the results of applying each to simple surrogate datasets are explored in greater detail in Part II. 1. Introduction To put current and projected future changes of the climate system into context, it is imperative to under- stand the natural variability and past evolution of the climate system. Particular attention has been given in this regard to the time evolution of the surface temper- ature field over the last several thousand years, as this variable is of societal importance and features a relatively complete instrumental record extending back to about 1850. Given that a longer record is desirable for both investigating the dynamics of the system and testing the output of climate models, it becomes necessary to call upon paleoclimate observations, which are noisy and sparsely distributed in space, to extend reconstructions back in time. Information about surface temperatures over the last few millennia can be derived from historical documents, ...





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