The Annals of Applied Statistics

Reconstructing past temperatures from natural proxies and estimated climate forcings using short- and long-memory models

Luis Barboza, Bo Li, Martin P. Tingley, and Frederi G. Viens

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We produce new reconstructions of Northern Hemisphere annually averaged temperature anomalies back to 1000 AD, and explore the effects of including external climate forcings within the reconstruction and of accounting for short-memory and long-memory features. Our reconstructions are based on two linear models, with the first linking the latent temperature series to three main external forcings (solar irradiance, greenhouse gas concentration and volcanism), and the second linking the observed temperature proxy data (tree rings, sediment record, ice cores, etc.) to the unobserved temperature series. Uncertainty is captured with additive noise, and a rigorous statistical investigation of the correlation structure in the regression errors is conducted through systematic comparisons between reconstructions that assume no memory, short-memory autoregressive models, and long-memory fractional Gaussian noise models.

We use Bayesian estimation to fit the model parameters and to perform separate reconstructions of land-only and combined land-and-marine temperature anomalies. For model formulations that include forcings, both exploratory and Bayesian data analysis provide evidence against models with no memory. Model assessments indicate that models with no memory underestimate uncertainty. However, no single line of evidence is sufficient to favor short-memory models over long-memory ones, or to favor the opposite choice. When forcings are not included, the long-memory models appear to be necessary. While including external climate forcings substantially improves the reconstruction, accurate reconstructions that exclude these forcings are vital for testing the fidelity of climate models used for future projections.

Finally, we use posterior samples of model parameters to arrive at an estimate of the transient climate response to greenhouse gas forcings of 2.5°C (95% credible interval of [2.16, 2.92]°C), which is on the high end of, but consistent with, the expert-assessment-based uncertainties given in the recent Fifth Assessment Report of the IPCC.

Article information

Ann. Appl. Stat. Volume 8, Number 4 (2014), 1966-2001.

First available in Project Euclid: 19 December 2014

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Zentralblatt MATH identifier

External forcings long-memory proxies temperature reconstruction


Barboza, Luis; Li, Bo; Tingley, Martin P.; Viens, Frederi G. Reconstructing past temperatures from natural proxies and estimated climate forcings using short- and long-memory models. Ann. Appl. Stat. 8 (2014), no. 4, 1966--2001. doi:10.1214/14-AOAS785.

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Supplemental materials

  • Supplementary material: Supplement to: “Reconstructing past temperatures from natural proxies and estimated climate forcings using short- and long-memory models”. We provide a background on long-memory models, the multitaper estimator and scoring rules together with some calculations of our model’s posterior distributions. Finally, we include additional plots and tables.