Molecular diffusion of stable water isotopes in polar firn as a proxy for past temperatures

Polar precipitation archived in ice caps contains information on past temperature conditions. Such information can be retrieved by measuring the water isotopic signals of $\delta{}^{18}\mathrm{O}$ and $\delta\mathrm{D}$ in ice cores. These signals have been attenuated during densification due to molecular diffusion in the firn column, where the magnitude of the diffusion is isotopologoue specific and temperature dependent. By utilizing the differential diffusion signal, dual isotope measurements of $\delta{}^{18}\mathrm{O}$ and $\delta\mathrm{D}$ enable multiple temperature reconstruction techniques. This study assesses how well six different methods can be used to reconstruct past surface temperatures from the diffusion-based temperature proxies. Two of the methods are based on the single diffusion lengths of $\delta{}^{18}\mathrm{O}$ and $\delta\mathrm{D}$, three of the methods employ the differential diffusion signal, while the last uses the ratio between the single diffusion lengths. All techniques are tested on synthetic data in order to evaluate their accuracy and precision. We perform a benchmark test to thirteen high resolution Holocene data sets from Greenland and Antarctica, which represent a broad range of mean annual surface temperatures and accumulation rates. Based on the benchmark test, we comment on the accuracy and precision of the methods. Both the benchmark test and the synthetic data test demonstrate that the most precise reconstructions are obtained when using the single isotope diffusion lengths, with precisions of approximately 1.0\,^\mathrm{o}\mathrm{C}. In the benchmark test, the single isotope diffusion lengths are also found to reconstruct consistent temperatures with a root-mean-square-deviation of 0.7\,^\mathrm{o}\mathrm{C}