Stable isotope ratios

Here, we systematically analyse the properties and origins of

Our results explain a large set of observations discussed in the literature, providing a simple explanation for the interpretation of apparent cycles in shallow isotope records, without invoking complex mechanisms. Finally, the results underline previous suggestions that isotope signals in single ice cores from low-accumulation regions have a small signal-to-noise ratio and thus likely do not allow the reconstruction of interannual to decadal climate variations.

Stable isotope

High-resolution isotope data, thought to correspond to subannual variations, are now routinely measured at deep ice-core
sites

Example isotope profiles from EDML and EDC. Both profiles are visually similar despite the differing time periods covered,

For sites such as EDML and SP, their apparent cycle lengths match well with their annual snow layer thicknesses
and consequently their cycles have been explained as reflecting seasonal climate variation

Here we combine a statistical analysis of isotope profiles from eight Antarctic Plateau sites with theoretical
considerations and numerical simulations of the firn signal. We suggest that the presence of apparent cycles in the firn,
and their largely invariant length, can be explained by a combination of deposition-related noise in the surface isotope
signal and isotopic diffusion

Location of the sampling sites used in this study (solid squares). The 2500

Summary of the drilling and snow-pit sites used in this study. For each site we list latitude, longitude and elevation a.s.l. and for each corresponding record its name, the analysed depth, sampling resolution, measured proxy and original data reference. Some profiles have single missing measurements that are not included in the provided sampling resolution. The 35 missing

We first introduce the data set and the method used to compare the power spectral density of observed isotope profiles with those from a null model of diffused noise. We then provide an analytic solution for the expected distance between isotopic maxima (“cycle length”) and a method to estimate this cycle length from the observed firn profiles. Finally we provide a minimal numerical forward model for the isotopic variations.

We analyse data from eight sites on the Antarctic Plateau, for which vertical isotope profiles of

Spectra are estimated using Thomson's multitaper method with three windows

Significance testing of the power spectral density is performed against a null hypothesis of noise affected by firn
diffusion. More specifically, we assume the sum of white (temporally independent) noise subject to isotopic diffusion with
a depth-dependent diffusion length and additive white measurement noise. The diffusion length is calculated using the
site-specific accumulation rates as well as temperature and density profiles as described in
Sect.

The “wiggliness” of time series, assuming a stationary random process, is determined by the first moments of the
spectral density. This relationship, known as Rice's formula

A diffused white noise process has the power spectral density

To investigate the isotopic variations in a way similar to visual cycle counting

To improve robustness against measurement noise, we define a local maximum (or minimum) as that value of a profile which
is above (or below) all other values within a window of

We determine all local extrema for each observed or simulated profile in the described manner and record the distances
between subsequent extrema (i.e. the distances between two neighbouring maxima as well as between two neighbouring
minima) as a function of depth (midpoint of depth between the two extrema, Fig.

As a tool to understand the observations, we construct the following minimalistic model to simulate artificial isotope
profiles in the upper metres of Antarctic firn. We approximate the local climate conditions by the local near-surface air
temperatures,

Meteorological conditions and model parameters at the study sites. Listed are the annual mean temperature (

On monthly to multidecadal timescales, the local temperatures in Antarctica are dominated by the seasonal cycle. At our
studied sites, the seasonal cycle explains more than

Going from the isotope signal in the snow to the surface signal, the variability of the seasonal cycle is affected by
aliasing due to precipitation intermittency

Our model for the isotopic surface signal then is

The model also includes one implicit parameter, the resolution at which we evaluate the variance of the noise

Finally, the burial of the surface snow transfers the surface signal time series into the depth profile

Modelled diffusion lengths for the different sites. The diffusion lengths for

Power spectra of the firn

The effect of firn diffusion on the original isotope signal

The resulting effective diffusion lengths for our study sites are shown in Fig.

Despite originating from very different accumulation conditions, the power spectra of

The similarity of the power spectra between different sites, their similarity to the spectrum of diffused noise, and the
lack of evidence for periodic oscillations suggests that the apparent cycles might be independent of periodic variations
in the climate signal and instead represent a property of diffused noise. We note that “noise” here just describes
isotopic variations that are largely independent in time and thus exhibit a largely timescale-invariant (“white”) power
spectrum. This neither implies a non-climatic nor negates a climatic origin of these variations. The expected cycle
length for a diffused white noise signal is given by Rice's formula (Eq.

Comparison of expected and observed cycle length for all sites. The cycle length is evaluated
between 1 and 4

Illustrative examples of the effect of noise and firn diffusion on the cycle length and amplitude for three input
time series (

For a more quantitative comparison, we analyse the cycle lengths from measured

This similarity between observed cycle lengths and those predicted from diffused white noise is surprising, as we have not yet included any climate signal, such as the seasonal cycle, in our analysis. To better understand the combined influence on the firn signal of noise, the seasonal cycle and the diffusion process, we now analyse the extent to which simulated firn profiles depend on the input signal.

In contrast to the diffusion length, which is a function of snow depth, the climate signal should be largely invariant over time and thus, in first order, independent of the depth. Therefore, investigating the depth dependency of the cycle length in isotopic profiles should provide us with additional insights about the origin of the variations.

To understand the depth dependency of the cycle length, as well as of the signal amplitudes, we provide three examples of
simulated depth profiles (A–C) illustrating the effect of firn diffusion and noise (Fig.

(A) We assume a purely periodic surface isotope signal, such as the seasonal cycle (

Comparison of numerically predicted and observed

(B) We assume that the input signal is white noise (

(C) Finally we consider a mixture of cases (A) and (B), assuming that the input signal is equally partitioned (in
variance) between white noise and periodic signal (

These examples (Fig.

At all sites except SP, an increase in the estimated cycle length (grey bars) is observed with depth
(Fig.

For the very low-accumulation sites, Vostok and EDC, a small noise fraction already (

In summary, the presented evidence suggests that, with the exception of SP, diffused noise is the dominant source of the apparent cycles at the studied sites.

Visual comparison of measured and simulated profiles for EDML (left) and Vostok (right panel).
Upper row: simulations for

As a visual test of our finding, we compare the depth profiles and power spectra of simulated and observed example
profiles for the two representative sites EDML (72

We first analyse the low (

In contrast, the high (

These findings are confirmed by comparing the power spectra of the simulations and observations. As shown earlier
(Fig.

Stable isotope ratios in firn are usually interpreted as temperature proxy. Therefore, to a first approximation, vertical
isotopic variations in a snow pit should reflect the temperature variations. The naive expectation is thus that
a 3

To explain these findings, we constructed a simple forward model for isotope signals in firn cores, similar to the
ice-core proxy system model of

We applied Rice's formula to the problem of isotopic variations and showed that, assuming a white noise signal before
diffusion, the expected spacing between isotopic maxima or minima (“cycle length”) is

While in instrumental climate observations, a deterministic cycle (e.g. variations driven by the seasonal cycle) would be
clearly distinguishable from the realisation of a purely stochastic process, this is less clear for snow pits or firn
cores. Here, intra-seasonal and interannual changes in accumulation distort the seasonal cycle

Defining noise as deviations from the surface isotope signal that are unexplained by the local temperature time series,
the depth dependency of the cycle length and amplitude suggests a significant proportion of noise in the surface isotope
signal at all analysed sites (Fig.

This assertion may seem particularly troubling for the EDML site, as here the accumulation rate, as determined from snow
stakes or volcanic markers in firn cores, corresponds to an annual layer thickness of

For the three sites along the East Antarctic divide analysed here (DF, MP, DK) and later an extended set of sites,

At first sight, our results seem to contradict the finding that firn-core isotope profiles are significantly correlated
with impurities such as

Similar noise levels in the isotope and impurity signals at the surface, caused by common deposition and redistribution
processes, would also imply that little or no seasonality is preserved in the impurity records at those sites for which we
find a high noise level. This is consistent with the missing seasonality in the impurity signal at sites with very low
accumulation

Our result also has implications for estimating isotopic diffusion and for the usage of layer counting. Assuming a white
noise input signal, the observed cycle length is proportional to the diffusion length (Eq.

Our findings further support the assumption of an initially white spectrum as it is used in isotopic diffusion studies

We showed that the level of noise in the input signal also determines the depth dependency of the amplitude of the
variations. The boundary case of a diffused pure seasonal cycle leads to an exponential decrease of amplitude with depth

The combination of isotopic diffusion with strong variability at the surface that is not directly related to temperature
also limits the effective resolution of climate signals that can be obtained by analysing firn-core isotopic
records. While the problem of diffusion could be overcome by undiffusing the signal

We provide an explanation of why snow pits across different sites in East Antarctica show visually similar variations in
stable isotope ratios

Our hypothesis does not exclude the existence of a climatic signal in the isotope time series, as any low-frequency surface signal would still be preserved in the diffusion process, and thus does not question the relevance of stable isotope ratios as a palaeo-temperature proxy. However, in particular for low-accumulation areas we show that the typical spacing of extrema in isotope profiles can be explained without invoking multidecadal climate changes or other climate-related hypotheses.

Our results underline previous findings that

The snow trench (T15) isotope data are available in the PANGAEA repository

Our previous calculations assumed an isotope surface signal that is a mixture of a seasonal cycle and uncorrelated (white) noise. While uncorrelated noise is the simplest hypothesis, it is likely that the surface signal exhibits more structure. Potential processes that lead to autocorrelation include precipitation events that deposit several centimetres of snow with similar isotopic composition, as well as mixing and redistribution by wind drift that might vertically homogenise the snow surface.

Unfortunately, the surface isotope signal before diffusion is largely unknown. To obtain a reasonable surrogate for the power spectrum of the surface isotope signal, we therefore resort to observed major ion profiles. This is motivated by the fact that, assuming an atmospheric source, wet deposited impurities are also influenced by precipitation intermittency and snow redistribution and might therefore show a similar variability structure as the isotope signal at the surface. However, in contrast to the isotopic composition, impurities are not affected by diffusion and therefore the variability in measured impurity firn profiles should in a first approximation reflect the temporal surface variability.

Interestingly, major ion profiles in snow pits

To test the effect of autocorrelated noise in the input signal on the resulting cycle length, we simulate profiles
assuming three different input signals: (1) white noise, (2) white noise subject to a 5

Sensitivity of the cycle length to the temporal correlation structure of the assumed input signal. Thin lines show the undiffused signal, while thick coloured lines the signal after diffusion. The correlation between the undiffused and the diffused signals is provided in the panels. For the more structured signals (mixed white noise and variability mimicking impurities), a larger fraction of the signal is preserved, leading to higher correlations. The resulting cycle length (right panel, coloured lines) is only weakly dependent on the input signal and is close to the theoretical result for white noise (Rice's formula, dashed line).

The results (Fig.

We note that while the cycle length is similar, the correlation between the input and the diffused signal is larger for the more structured input signals as a larger fraction of low-frequency variability is preserved after diffusion.

As Fig.

In the main text, we showed the observed and simulated cycle length statistics for the sites with multiple profiles
(Fig.

The depth dependency of the cycle length is less clear, which is likely caused by the large estimation uncertainty. In addition, MP shows a systematically smaller observed cycle length than the simulations. Potential reasons could be either uncertainties in the isotopic data set (independent noise leads to more minima and maxima and thus a smaller cycle length) or our choice of climatic parameters (accumulation rate, firn temperature).

Demonstration of spurious peaks when testing against a white noise null hypothesis. Coloured lines show power spectra for three realisations of purely random firn profiles. The grey horizontal line shows the critical level (

To demonstrate the effect of a white noise null hypothesis on the spectral analysis of oxygen / hydrogen isotope ratios in
snow and firn, we simulate random

The authors declare that they have no conflict of interest.

We thank Y. Hoshina and A. Ekaykin for sharing their data and J. Freitag and A. M. Dolman for discussions and detailed editing. We are grateful to the reviewers B. Markle and the two anonymous reviewers, for their detailed comments on the manuscript. T. Laepple and T. Münch were supported by the Initiative and Networking Fund of the Helmholtz Association grant VG-NH900. M. Casado and A. Landais have received funding from the European Research Council (ERC) under the European Union's Seventh Framework Programme (FP7/2007–2013)/RC grant agreement number 306045 and T. Laepple has received funding from the ERC under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 716092). The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Joel Savarino Reviewed by: Bradley Markle and two anonymous referees