Accurate estimates of water isotope diffusion lengths are crucial when reconstructing and interpreting water isotope records from ice cores. This is especially true in the deepest, oldest sections of deep ice cores, where thermally enhanced diffusive processes have acted over millennia on extremely thinned ice. Previous statistical estimation methods, used with great success in shallower, younger ice cores, falter when applied to these deep sections, as they fail to account for the statistics of the climate on millennial timescales. Here, we present a new method to estimate the diffusion length from water isotope data and apply it to the Marine Isotope Stage 19 (MIS 19) interglacial at the bottom of the EPICA Dome C (EDC, Dome Concordia) ice core. In contrast to the conventional estimator, our method uses other interglacial periods taken from further up in the ice core to estimate the structure of the variability before diffusion. Through use of a Bayesian framework, we are able to constrain our fit while propagating the uncertainty in our assumptions. We estimate a diffusion length of

Large ice sheets from the polar regions offer unique insights into the climate up to hundreds of thousands of years ago. The drilling of deep ice cores in Greenland and Antarctica enables measurements of water isotopic ratios (

By comparing the power spectrum of a diffused water isotope record with that of the undiffused isotopic climate signal, it is possible to estimate the diffusion length. Current estimation methods assume that the isotopic variability before diffusion is constant across all frequencies, i.e. white noise

In this study we estimate the diffusion length for the Marine Isotope Stage 19 (MIS 19) section of the EPICA Dome C (EDC, Dome Concordia) ice core (763–795 ka). Previous estimations derived from water isotope data suggest a diffusion length between 40 and 60 cm for the MIS 19 interglacial

We use discrete

Full high-resolution EDC

Depth ranges of analysed data and the corresponding time periods. Also included are the depths and times at which the

In summary, the diffusion length in deep ice is estimated using a modification of the existing statistical model by representing the power spectrum of the water isotope record before diffusion as a power law. An appropriate power law is estimated from selected water isotope data from shallower sections of the same ice core, where diffusion will not have had time to affect the relevant frequencies, spanning time periods of a similar climate state and length to the deep-ice section.

The diffusion length and the properties of the isotopic variability are estimated in the frequency domain. For this, we use the raw periodogram on linearly detrended data with a split cosine bell taper of 10 % as this estimator is unbiased and we are only using it for parametric fits.

The isotope time series are irregular in time due to ice flow thinning, which is incompatible with classical power spectra analysis. Therefore, when working with data on the time domain, the respective records were linearly interpolated to equidistant time points. Although this introduced power loss at high frequencies in the power spectra

For comparison, we also calculated diffusion lengths using the conventional white-noise model for a running window over the entire EDC ice-core record. Gaps in the data were resolved through linear interpolation, and large or frequent linear interpolated gaps can significantly affect the power spectra. Therefore, the estimate was only performed over windows in which less than 25 % of the

The diffusion of water molecules averages out variability, with rapidly increasing effectiveness as frequency increases. While the lowest frequencies remain mostly unchanged, information at higher frequencies can be greatly affected. This relationship allows for the diffusion length of a water isotope time series to be estimated through analysis of its spectral properties combined with a representative model.

Mathematically, the effect of diffusion on a time series can be represented by a convolution with a Gaussian filter

Therefore it is possible to estimate the diffusion length of a water isotope record by taking its power spectrum and fitting Eq. (

The white-noise climate assumption of conventional methods relates to the

Discrete sampling will also introduce a block-averaging effect, further smoothing the data and biasing our diffusion length estimate if not taken into consideration. The magnitude of this smoothing depends on the sampling resolution and can be computed using the method described in

There are two common approaches for fitting a spectrum with Eq. (

The linear approach

Assuming

Another possible method of statistically estimating the diffusion length involves modelling

While previous studies assumed

To get a realistic alternative model for

To parameterise the variability across a large range of frequencies, we use a power law,

For the case of the MIS 19 record from EDC, an appropriate time period to estimate the climate signal before long-term ice diffusion is another, more recent interglacial. Using the same EDC ice core, we selected sections of water isotope data from the MIS 1 (the Holocene), MIS 5, and MIS 9 interglacials, which were retrieved from depths shallow enough to remain unaffected by lower-frequency diffusion. Large data gaps are present over the MIS 7 and MIS 11 interglacials, so a reliable power spectral estimate could not be acquired. Interglacial records from deeper in the ice core (MIS 13/15/17) were not suitable for our analysis as diffusion has attenuated the frequencies over which we are inferring the spectrum of

We used a Bayesian approach for all our power spectrum fitting as it has a several advantages over classical methods. Most importantly, rather than having to either set parameters to specific fixed values or leave them free to assume any value, the prior distributions of a Bayesian model allow us to put physically realistic restrictions on the values of parameters while also allowing uncertainty in their true values to be propagated through into uncertainty in the final estimate of diffusion length. Additionally, the Bayesian method allows us to specify a gamma distribution which is the true distribution of the errors in power spectral density estimated by Fourier methods

Using a reduced model without diffusion (Eq.

For the power of the noise we used a somewhat informative prior, as the uncertainty of the

The models were defined in the Stan language

In order to determine the significance of this new method, we first estimate the diffusion length using the conventional model, where the climate variability

The full power spectra of the selected interglacial time periods used to constrain

Using the Bayesian sampling method, the power-law fit is applied to each of the three more recent interglacials within these frequency bounds, with best fits using the mean

Best power-law fit for the power spectra of

Alpha and beta values estimated from each interglacial fit in Fig.

While prior distributions of

All parameters estimated from the MIS 19 spectrum, using different

Estimating diffusion length using an incorrectly parameterised best-fit model can significantly bias the result. To demonstrate this, we use the conventional method in Sect.

Best fits of a theoretical power spectrum of a deep-ice-core record simulated using Eq. (

Aware of this bias, we applied the conventional method to the MIS 19 spectrum to evaluate the difference between the two estimates, using the Bayesian approach with a very weak

To get an understanding about the evolution of diffusion length at Dome C, we applied the conventional method over a running window across the full Dome C record (Fig.

Applying the conventional method to a running window over EDC, the estimated diffusion lengths rapidly increase from

The downside of the proposed method is that it needs information about the isotope variability before diffusion (

Our implementation of a Bayesian fitting approach enables our prior knowledge of the physically realistic values of the parameters to be incorporated into the fit

Our estimate of

Possible improvements to the precision of our estimate could be made with a higher sampling resolution or data spanning longer time periods. Increasing the resolution would provide some opportunity to reduce the uncertainty, but the effect will be limited as the high frequencies do not contribute to the diffusion length estimate, with the main improvement arising from the effective measurement noise reduction at a given frequency. It would also lessen the smoothing effect of block averaging due to the rectangular sampling scheme, although in the future this could be directly incorporated into the fit. Alternatively, increasing the time span of our selected water isotope record would improve the reliability of our

While the focus of this new method was to improve the diffusion length estimate for deep ice, it should also be considered for application to shallower ice cores. Power-law behaviour is observable in the climate across multi-millennial to sub-annual timescales

Perhaps the main disadvantage of this new method compared to conventional estimation methods is the necessary inference of the undiffused isotopic profile from elsewhere in the ice core. It would be much more practical if it were possible to generalise the approach for an entire ice core, acquiring a full diffusion length profile with depth. This is potentially achievable using a different parameterisation for

We have described a new approach for water isotope diffusion length estimations in deep ice cores, resolving the biased assumption of a white-noise undiffused climate signal. Our method instead implements a power-law slope inferred from water isotope sections of a similar climate state in the shallower parts of the ice core, better representing the climate on millennial scales. Incorporating Bayesian statistics enables us to use priors, chosen based on our knowledge of the parameters and propagating our uncertainties into the fitting procedure. We applied our new method to the MIS 19 water isotope record from the bottom of the EDC ice core, estimating a diffusion length of 31

The data for the

To see the effect this might have on the resulting

Effect of interpolation on a simulated time series with spectral parameters from

We found the linear interpolation has a negligible effect in the frequency range the fit is applied over, as it only introduces error in much higher frequencies. For MIS 1 and MIS 9 there is not much data missing, so the interpolated fit matches almost perfectly with the “true” fit. This is not the case for MIS 5, as enough data are missing that it impacts the accuracy of the fit. However, the resulting fit still falls within the 90 % confidence interval, and given it only contributes partially to a suggestive prior, the MIS 19 fit is not strongly affected (less than 1 cm bias).

Priors for

MIS 19 fit priors and posteriors.

The

The measurement noise was estimated from the MIS 19 power spectrum, which tails off over the highest frequencies at a PSD of

The data used in this study were previously published and can be found at

TL and FS designed the study. TK and TL contributed to the spectral analysis and signal processing. AMD contributed to the Bayesian estimation. VG provided insights into diffusion and diffusion length estimation. FS performed the analysis and wrote the manuscript. All authors contributed to the interpretation and to the preparation of the final manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This article is part of the special issue “Oldest Ice: finding and interpreting climate proxies in ice older than 700 000 years (TC/CP/ESSD inter-journal SI)”. It is not associated with a conference.

This publication was generated in the frame of the DEEPICE project and received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 955750 and the ERC SPACE no. 716092. It contributes to Beyond EPICA, which received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 815384 (Oldest Ice Core). Vasileios Gkinis was supported by the Villum Foundation (“The whisper of ancient air bubbles in polar ice”, grant no. 00028061, and “Unraveling paleo-climate knots with lasers”, grant no. 00022995). Andrew M. Dolman was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, project no. 468685498, and SPP 2299, project no. 441832482). This is Beyond EPICA publication number 37.

This research has been supported by the European Research Council through the European Union Horizon 2020 research and innovation programme Marie Skłodowska-Curie Actions (grant no. 955750), the European Research Council through the European Union Horizon 2020 research and innovation programme (grant no. 716092), the Villum Foundation (grant nos. 00028061 and 00022995), Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, project no. 468685498, and SPP 2299, project no. 441832482), and European Union’s Horizon 2020 research and innovation programme under grant agreement no. 815384 (Oldest Ice Core).The article processing charges for this open-access publication were covered by the Alfred-Wegener-Institut Helmholtz-Zentrum für Polar- und Meeresforschung.

This paper was edited by Carlos Martin and reviewed by Christian Holme and one anonymous referee.