The study of past climates is based on proxy measurements obtained from natural climate archives such as cave speleothems, lake and ocean sediments, and ice cores. Paleoclimate reconstructions derived from proxies suffer from 3-fold uncertainty. First, the proxy measurement itself involves the typical measurement uncertainties. Second, the interpretation of proxy variables, such as isotope ratios in terms of physical variables, such as temperature, is often ambiguous, and typically no one-to-one mapping can be established between the measured proxies and the climatic quantities of interest. Third, the age has to be measured alongside the proxy variable. In most cases an age model can be inferred that provides a quantitative relationship between the depth in the archive under consideration and the corresponding age. Such age models are also subject to uncertainties.

This study is exclusively concerned with the dating uncertainties of so-called layer counted archives, where the dating is performed based on counting periodic signals in the proxy archives such as annual layers arising from the impact of the seasonal cycle on the deposition process e.g.,. This type of archive comprises varved lake sediments, ice cores, banded corals, tree rings, and some speleothems . Using the example of the NGRIP ice core and its associated chronology, the GICC05 , we present here a statistical approach to generate ensembles of age models that may in turn be used to propagate the age uncertainties to any subsequent analysis of the time series derived from the NGRIP record. Our method can be directly adapted to other layer counted archives.

Layer counting assesses the age increments along the axis perpendicular to the layering, whose summation yields the total age. In turn, also the errors made in the counting process accumulate such that in chronologies obtained from counting annual layers, the absolute age uncertainty grows with increasing age e.g.,.

Most importantly, dating uncertainties make it challenging to establish an unambiguous temporal relation between signals recorded in different, and possibly remote, archives. Therefore, it is often not possible to decipher the exact temporal order of events and distinguish causes from consequences across past climate changes. For example, abrupt Greenland warmings known as Dansgaard–Oeschger (DO) events evidenced in ice core records from the last glacial are accompanied by changes in the east Asian monsoon system, which are apparent from Chinese speleothem records e.g.,. However, since the dating uncertainties exceed the relevant time scales of these abrupt climate shifts, a clear order of events cannot be determined. This prevents us from deducing if in the context of DO events, the abrupt Greenland warming triggered a hemispheric transition in the atmosphere, or vice versa, or if these changes happened simultaneously as part of a global abrupt climatic shift .

For the quantification of dating uncertainties in radiometrically dated archives, there exist well established generalized frameworks. One example is the Bayesian Accumulation Model which models the sediment accumulation rate as a first order autoregressive process with gamma distributed innovations. Other methods or software include OxCal and BChron . Contrarily, the uncertainties of layer counted archives are targeted systematically only by few studies. present a probabilistic model, where the number of missed and double-counted layers are expressed as counting processes shaped by corresponding error rates. However, this approach requires knowledge about these rates and further does not account for any uncertainty associated with them. An alternative Bayesian approach for quantifying the dating uncertainty of layer counted archives is presented in , where the uncertainty is shifted from the time axis to the proxy value. However, this approach does not allow for generation of ensembles of chronologies as required for uncertainty propagation.

Even though dating uncertainties are conveniently quantified for many archives, many studies ignore these uncertainties and instead draw inference from “average” or “most likely” age scales, as already highlighted by . This involves the risk of losing valuable information, as shown, for example, in . In some cases, rigorous propagation of uncertainty may yield results that qualitatively differ from results obtained by using the “average” or “best fit” age model. In this context, propose to apply the respective analysis to an ensemble of possible age scales to ensure the uncertainty propagation, in line with the strategy proposed by .

We focus on the layer counted part of the GICC05 chronology, a synchronized age scale for several Greenland ice cores . It was obtained by counting the layers of different Greenland ice cores and synchronizing the results using matchpoints. While the recent part of the chronology is compiled from multiple cores, the older part (older than 15 kyr b2k) is based exclusively on the layer counting in the NGRIP core. We introduce a new method to generate realistic age ensembles for the NGRIP core, which conveniently represent the uncertainty associated with the GICC05.

Originally, the dating uncertainty of the GICC05 is quantified in terms of the maximum counting error (MCE). The MCE increases by 0.5 years for every layer which is deemed uncertain by the investigators during the counting process: 1MCE(z)=0.5Nuz, where Nu is the number of uncertain layers down to depth z. In contrast to certain layers, which can be identified unambiguously in the records, uncertain layers are less pronounced and therefore it seems less certain that these signals truly correspond to physical layers. The accumulation of uncertain layers results in high values for the MCE for the older parts of the core (MCE =2.6 kyr at 60 kyr b2k estimated age). However, it seems highly unlikely that all uncertain layers are consistently either true layers or no layers, which is why we think that the MCE is an overly careful quantification of the age uncertainty, as already suggested by . One might, alternatively, be tempted to treat the uncertain layers as a Bernoulli experiment with Nu repetitions and a probability of one half for each uncertain layer to be a true layer. However, this would neglect any sort of bias in the assessment of the uncertain layers and would lead to unrealistically small uncertainties, since over- and undercounting practically cancel each other out in this Bernoulli type interpretation (see for instance ).

The method presented here abandons the notion of certain and uncertain layers. Instead, we separate the GICC05 chronology into contributions that can be captured by deterministic model equations and corresponding residuals. We construct a new age–depth model by complementing the deterministic part with a stochastic component designed in accordance with the statistics of the residuals. This model can be used to generate age–depth ensembles in a computationally efficient manner. In turn, these ensembles facilitate uncertainty propagation to subsequent analysis. The model parameters are tuned with respect to the GICC05 chronology that includes every uncertain layer as half a layer.

The outline of this paper is as follows: Section  gives a description of the data used for this study. Section  introduces our statistical model for the dating uncertainties, and details about how we incorporate physical processes and how we deduce the noise of the model from the statistics of the residuals. In Sect.  we show how we can formulate our model in terms of a hierarchical Bayesian modeling framework that allows for the physical and noise components to be estimated simultaneously. That section also details how one can use the resulting posterior distributions of the model parameters to obtain a full description of the posterior distributions of the dating uncertainties using a sample-based approach. We finally demonstrate how a potential counting bias could be incorporated by the model, and how that would affect the results. In Sect.  we show how our model can be used to obtain a full description of the dating uncertainties of abrupt warming events, which takes into account the dating uncertainties as well as the uncertainties in determining their exact position in the noisy data. Further discussion and conclusions are provided in Sect. .

We use the Greenland Ice Core Chronology 2005 (GICC05) as defined for the NGRIP ice core together with the corresponding δ18O proxy record . An analogous analysis for the Ca2+ proxy record is presented in Appendix . The final age of the layer counted part of the GICC05 is 59 944 yr b2k, and we consider data up to 11 703 yr b2k. The following Holocene part of the record is excluded since it is governed by a substantially different climate than the last glacial interval . For the considered period, the NGRIP record is available at 5 cm resolution and thus equidistant in depth, but not in time. In total, the data comprises n=18672 data points of the form (zk,yk,xk),k∈{0,1,…,n-1}, where zk denotes the kth depth, yk the corresponding age as indicated by the GICC05, and xk the measured proxy value.

The GICC05 is based on counting annual layers which are evident in multi-proxy continuous flow measurements from the NGRIP, DYE3, and the GRIP ice cores. While the measurements from DYE3 and GRIP only facilitate layer counting up to ages of 8.2 and 14.9 kyr b2k, respectively, the NGRIP core allowed the identification of annual layers up to an age of 60 kyr b2k. The uncertainty of the GICC05 has been quantified as follows: whenever the investigators were uncertain about whether or not a signal in the data should be considered an annual layer, half a year was added to the cumulative number of layers while simultaneously adding ±0.5 yr to the age uncertainty. The total age uncertainty determined by the number of all uncertain layers up to a given depth is termed the maximum counting error (MCE). The MCE amounts to a relative age uncertainty of 0.84 % at the onset of the Holocene and 4.34 % at the end of the layer counted section of the core.

δ18O values from Greenland ice cores are interpreted as a qualitative measure of the site temperature at the time of precipitation . We include these data in our study since our modeling approach will make use of the relation between atmospheric temperatures and the amount of precipitation, which in turn affects the thickness of the annual layers. In addition we use the division of the record into Greenland stadial and interstadial phases as presented in Table 2 of . We label the depths at which stadial–interstadial transitions occur by z1∗,…,zp∗ and the corresponding ages by y1∗,…,yp∗. Figure  shows the measured δ18O values as a function of the GICC05 time scale, together with the Greenland stadial and interstadial onsets.

Motivated by the relationship between air temperature on the one hand and the water-holding capacity and thus precipitation on the other hand, we assumed a linear dependency of the number of incremental layers per 5 cm on respective values of δ18O, which serves as an air temperature proxy. The only other climate proxy variable which is available from the NGRIP ice core at the same resolution over the same time period is the Ca2+ particle concentration . From visual inspection one can already see that the negative logarithm of the Ca2+ mass concentration record shows high covariability with the δ18O record (see, for example, Fig. 1 of ). Changes in the Ca2+ concentrations are interpreted as changes in the local and hemispheric atmospheric circulation e.g.,, which would also affect the amount of precipitation over Greenland. Above, we have focused on the δ18O; and analogous analysis for -log⁡Ca2+ is presented in Appendix . The results of the -log⁡Ca2+-based chronology modeling are in very good agreement with the ones based on δ18, which corroborates our methodology.

Adopting Gaussian noise models in principle allows for a negative modeled number of incremental layers in a 5 cm incremental core segment, which does not seem plausible from a physical perspective. To avoid this, for comparison one could log-transform the age increments and then apply the modeling procedure. The model output would then have to be transferred back by taking the exponential. However, it turns out that the log transformation induces systematic deviations of the mean model output age from the reported GICC05 age. With the chances of negative layer increments being fairly small (less than 5 %), overall the original model outperforms the log-transformed model. A more detailed discussion on the log-transformed modeling approach is given in Appendix .

From Fig. b we observe that the variance of the residuals increases slightly with increasing depth down in the core. This heteroskedasticity can be incorporated into our latent Gaussian model by assuming that the variance depends on the core depth in some predefined way. As this implementation is rather technical we consider it beyond the scope of the current paper. Regardless, assuming constant variance appears to be a good first order approach.

We have developed a general statistical framework for quantifying the age–depth uncertainty of layer counted proxy archives. In these records the age can be determined by counting annual layers that result from seasonal variations, which in turn impact the deposition process. By counting these layers one can assign time stamps to the individual proxy measurements. However, there is a non-negligible uncertainty associated with this counting process. Proper quantification of this uncertainty is important since it carries valuable information and the error propagates to further analyses, e.g., dating of climatic events, determining cause and effect between such events, and model-data comparisons. Originally, the uncertainty of the GICC05 is quantified in terms of the maximum counting error (MCE), defined as half the number of uncertain layers. However, since this method assumes that uncertain layers are either true or false, we believe this to be an overly conservative estimate, giving too high uncertainty for deeper layers.

In our approach we express the number of layers per depth increment as the sum of a structured component and a stochastic component. The structured component represents physical layer thinning, a positive temperature–precipitation feedback, and persisting trends over individual stadials and interstadials. The stochastic component takes into account the natural variability of the layer thickness and the errors made in the counting process. After fitting the structured component in a least squares manner, we find the residuals to be approximately stationary, Gaussian distributed, and to exhibit short-range autocorrelation. These summary statistics motivate us to employ Gaussian white noise, or an autoregressive process of first or second order, as the stochastic part of the age–depth model.

After defining the structure of the model, we estimate all model parameters simultaneously in a hierarchical Bayesian framework. The resulting joint posterior distribution on the one hand serves as a quantification of the parameter uncertainty in the model and on the other hand allows to generate chronology ensembles that reflect the uncertainty in the age–depth relationship of the NGRIP ice core. The dating uncertainties obtained from this approach are significantly smaller than those from the MCE. We also find that our estimates do not deviate much from the GICC05 in terms of best estimates for the dating.

Additional information that may help to further constrain the uncertainties, such as tie points obtained via cosmogenic radionuclides , will be fed into the model in future research.

One of the biggest concerns regarding the layer counting is that of a potential counting bias. Such a systematic error cannot be corrected after the counting and, therefore, we investigate how a potential unknown counting bias increases the uncertainty of the presented age–depth model. If such a counting bias is restricted to ±4% we obtain total age uncertainties comparable to the estimates based on the MCE. Finally, we apply our method to the dating of DO events. Using a Bayesian transition onset detection we are able to combine the uncertainty of the onset depth with the corresponding age uncertainty, and to give a posterior distribution that entails the complete dating uncertainty of each transition onset. We find that previous estimates of the DO onsets reported in are well within our estimated uncertainty ranges both in terms of depth and age. However, the dating uncertainties of the abrupt warming event onsets are all considerably smaller than the MCE, even when accounting for the additional uncertainty associated with the onset depth.

In theory, it should be possible to apply this approach to other layered proxy records as well. However, there are some requirements that need to be fulfilled for this approach to be applicable. The first condition is that a potential layer thinning can be adequately expressed by a regression model. In our results we find the residuals to follow a Gaussian process, but it should be possible for the model to be adapted such that it supports other distributions for the residuals as well. However, depending on the model, if the residuals exhibit too long memory then this could lead to the simulation procedure having an infeasibly high computational cost. Moreover, if there are many effects in the regression model there needs to be sufficient data to achieve proper inference.