A recent coherent chronology has been built for four Antarctic ice cores and the NorthGRIP (NGRIP) Greenland ice core (Antarctic Ice Core Chronology 2012, AICC2012) using a Bayesian approach for ice core dating (Datice). When building the AICC2012 chronology, and in order to prevent any confusion with official ice core chronology, the AICC2012 chronology for NGRIP was forced to fit exactly the GICC05 chronology based on layer counting. However, such a strong tuning did not satisfy the hypothesis of independence of background parameters and observations for the NGRIP core, as required by Datice. We present here the implementation in Datice of a new type of markers that is better suited for constraints deduced from layer counting: the duration constraints. Estimating the global error on chronology due to such markers is not straightforward and implies some assumption on the correlation between individual counting errors for each interval of duration. We validate this new methodological implementation by conducting twin experiments and a posteriori diagnostics on the NGRIP ice core. Several sensitivity tests on marker sampling and correlation between counting errors were performed to provide some guidelines when using such a method for future dating experiments. Finally, using these markers for NGRIP in a five-core dating exercise with Datice leads to new chronologies that do not differ by more than 410 years from AICC2012 for Antarctic ice cores and 150 years from GICC05 for NGRIP over the last 60 000 years.

The reference timescale for Greenland ice cores, GICC05, has been obtained by
layer counting back to 60 ka

This chronology has been used as a reference for many records of the North
Atlantic region

Still, the strong tie of AICC2012 to GICC05 has led to some technical problems when optimizing the chronology with Datice. Three glaciological parameters are indeed optimized during this process: accumulation rate, ice thinning and lock-in depth (i.e. the depth at which air is trapped when snow is sufficiently compacted). The Bayesian approach requires starting with first-guess (background) scenarios for the three parameters. They are then modified within their imposed variance range so that the final chronology fits the absolute and relative age constraints for each ice core within error bars.

In practice, to force the NGRIP AICC2012 chronology to fit the GICC05 age
scale, the modelled thinning function and accumulation rate of the GICC05
chronology

Even if the uncertainty of the GICC05 timescale is well constrained, this is
not true for the DJ–GICC05 scenarios of thinning and accumulation. The
thinning function is deduced from a simple Dansgaard–Johnsen (DJ) ice flow
model

Recently, studies combining air isotopic measurements (

It thus appears that the way NGRIP was implemented in the Datice tool for the AICC2012 chronology is not optimal. In addition to GICC05 chronological uncertainties that were not taken into account by construction, imposing the DJ–GICC05 accumulation rate and thinning scenarios with artificially reduced variances most probably led to incorrect output scenarios for these glaciological parameters.

In this paper, we propose an improvement of Datice to better implement the chronological uncertainties. Markers of duration are integrated in Datice with associated counting error. This allows for the strong constraints on thinning and accumulation rate to be relaxed. It also allows the NGRIP chronology to differ from GICC05 chronology within its error bars.

The outline of the manuscript is as follows. First, a methodological section presents and validates the improvements made to the Datice tool in order to integrate the duration constraints with their uncertainties. Then, we discuss different ways to implement the counting errors within the global chronological uncertainty. We also present some sensitivity experiments using the modified Datice tool for optimizing the sampling strategy and correlation between counting errors. Finally, we focus on how this new version of Datice modifies the NGRIP and the four Antarctic ice core chronologies compared to AICC2012.

The purpose of the following section is to describe the modifications
implemented in Datice

Datice aims at obtaining the best age model scenario by formulating an
optimization problem with a cost function that is accounted for by two main
types of constraint: the palaeo-observations

In practice, Datice is applied to several ice cores with a large set of
palaeo-observations in order to calculate coherent chronologies for both the ice and
gas phases. The chronologies are deduced from the scenarios of three
glaciological parameters for each core indexed with

The Datice cost function formulation (Eq.

Until now, observation

In this article, we wish to design Datice experiments with duration
constraints derived from the GICC05 counted layer chronology. In Sect.

In this section, twin experiments are performed to test the incorporation of duration constraints within the Datice tool. A twin experiment enables one to test any data assimilation system. It consists in the construction of some synthetic data and background by applying random perturbations of known statistical distribution to a given model scenario. The unperturbed model scenario is referred to as the “truth”. The aim of this validation method is to rebuild the truth by running the data assimilation system on the perturbed data and background.

In our case, we designed a twin experiment based on 51 simulations where the Datice system is run with the NGRIP ice core alone. The duration constraints are the only type of observation included. The GICC05 age scale is considered as the “truth”. We construct synthetic observations and backgrounds by applying random perturbations to the “true” GICC05 model scenario. The objective is to run the Datice system with the synthetic data as input and to rebuild GICC05 as accurately as possible.

Each twin experiment inputs are prepared using the following method. To build
the 51 sets of synthetic markers of duration constraint

For this experiment, the markers

To construct the perturbed background thinning function

Twin experiments: 51 perturbed background chronologies (dashed blue lines) and the corresponding 51 analysed chronologies, i.e. output chronologies from Datice (orange lines). The GICC05 chronology (the “truth” in our twin experiments) is represented by the dashed black line for comparison.

Figures

Twin experiments: 51 analysed chronologies, i.e. output
chronologies from Datice. Top: comparison of the 51 analysed chronologies
with GICC05. Bottom: the corresponding 51 analysed errors (red). The dashed
black line represents the maximum counting error associated with GICC05 and
considered as equivalent to a 2

Twin experiments: (top) histograms of the 51 perturbed background (blue) and (bottom) the corresponding 51 analysed chronologies, i.e. output chronologies from Datice (red) at 1800 m depth for NGRIP ice core.

Finer diagnostics confirm the reliability of the Datice methodological
developments. As investigated in

One should note that we have applied perfectly calibrated background and
observation error covariance matrices. Indeed, the background and observation
errors specified in the cost function are exactly the

Layer counting consists in identifying annual cycles on the basis of annual
layer proxies recorded along the core. The identification of annual cycles is
subjected to errors. In order to deal with uncertain annual layers and to
derive a counting error estimate, GICC05 adopted the following statistical
approach. If the

Over which time window should we define our GICC05 markers of duration? Shall we apply markers of duration on 20 yr time window or choose another sampling rate (i.e. 20-, 40-, 60-year time window)?

How should we infer the associated error when applying different time windows?

The MCE can be expressed through formulation of the GICC05 counting process
with two normal probability density functions (pdf's): (i) the pdf of annual
cycles identified as certain with a 1-year mean and a variance that
tends to zero and (ii) the pdf of annual cycles identified as uncertain
with a mean and standard deviation both set to half a year. Under this
formalism (Appendix Sect.

Following this approach means that errors for duration constraints at 40, 60
or 80 years will be derived by summing up the GICC05 20-year-window MCE
2, 3 and 4 times, respectively, in the case of full correlation within the time
windows associated with the chosen sampling rate (Appendix Sects.

However, the final chronology error should not depend on the arbitrary choice of the sampling rate. The option has thus been included in the Datice approach to apply error correlation on a finite interval and avoid abrupt cut-off of error correlation between adjacent intervals. This development should permit sampling of the markers with a certain step and apply error correlations beyond this time interval. Indeed, in future chronologies constructions, the value of the error correlation may change along the core in relationship with changes of climatic periods.

For this formulation of error correlation on a finite range, the correlation
coefficients

With this new formulation of the error correlation, we can explore how both sampling and error correlation independently affect the final chronology and provide some guidelines for future constructions of ice core chronologies.

In this section, we extract several sets of duration constraints from GICC05,
with different sampling and/or different assumptions regarding their
associated errors. These inputs are used to conduct multiple Datice
experiments and thus to investigate the sensitivity of the solution to
sampling and error correlation assumptions. In the two following sections,
experiments are run on the NGRIP core alone with only duration constraints.
Details on the background settings are provided in Table

In these experiments, a classical 1 m depth grid resolution is imposed, as in
AICC2012. On such a depth grid, the annual layer thickness drops below 0.05 m yr

To study the influence of the sampling, we run three experiments with markers sampled at three uniform rates (100, 200 and 300 years) as well as one experiment with the adaptive sampling between 40 and 140 years.

In these four experiments, the associated errors are derived from the
20-year-window MCE data under the AddMCE assumption of full error
correlation between annual cycles over the length of the sampling interval.
As discussed in Sect.

Figures

Sensitivity of the age and error solution to sampling of duration
constraints and to the MCE assumptions. The difference between analysed
chronologies and GICC05 age scale are shown in the top panel. Analysed errors and
MCE are plotted in the bottom panel. The simulations settings are (i) three
different uniform sampling rates (300, 200 and 100 years) and (ii) three
adaptive samplings ranging from 40 to 140 years. The marker errors are derived
under the AddMCE assumption (full correlation between annual
cycles), except for (i) simulation 40–140 yr_SqrAddMCE, which is run
under the SqrAddMCE assumption (correlation cut-off above 20 years), and
(ii) simulation CorrCoeff_40–140 yr_SqrAddMCE, which is
run with a finite depth range correlation coefficient. Table

Summary of the simulation configurations for the experiments of
Sect.

Option SqrAddMCE may therefore be a way to relax the dependence of
the analysed error to the sampling. However, as mentioned above, the abrupt
loss of correlation at the boundaries of sampling interval may be questioned.
At the junction of two duration constraints, neighbouring annual cycles from
either side do not share any error correlation, while each of them
correlates with much distant layers (as long as these layers are included in
the same sampling interval). We actually rather expect error correlations to
smoothly decrease with the distance between annual cycles. To circumvent this
problem, we have designed an experiment called CorrCoeff_40–140 yr_AddMCE (shown
in Figs.

We have demonstrated the sensitivity of the solution to the sampling and to the MCE assumptions applied to derive the observation error. However both issues are not fully decoupled in this first illustration. Hereafter, we investigate possible ways to study the error correlation independently of the sampling.

In this section, we apply different correlation coefficients between duration
constraints as implemented in Datice (Eqs.

In this set of experiments, due to large correlation coefficient values, the
level of observation error is largely increased compared to the experiments
of Sect.

Sensitivity of the age solution to sampling of duration constraints.
The simulations settings are as follows: (i) three different uniform sampling rates
(dark to light-blue lines: 300, 200 and 100 years) and (ii) one adaptive rate ranging
from 40 to 140 years (dashed coloured lines). The marker errors are derived
under the AddMCE assumption (full correlation between annual
cycles), except for (i) simulation 40–140 yr_SqrAddMCE, which is run
under the SqrAddMCE assumption (correlation cut-off above 20 years), and (ii) simulation CorrCoeff_40–140 yr_SqrAddMCE, which is
run with a finite depth range correlation coefficient. The curves represent
the difference between the different analysed and background chronologies.
The difference between GICC05 and background chronologies is displayed for comparison (dashed black
line). Table

In a first set of experiments, we investigate the InfiniteRangeCorr
option. The correlation coefficient

Figure

Sensitivity of the solution to correlation coefficient values applied between duration constraints (infinite depth range case). Top panel: GICC05 minus background chronology (black dashed line), Difference between analysed and background chronologies (blue to pink lines). Bottom panel: MCE (black dashed line) and analysed errors (blue to pink lines) The duration constraints are sampled every 100 years on GICC05, and correlation coefficients range from 0.8 to 0.2. The MCE assumption is AddMCE (full error correlation between cycles).

We test hereafter the FiniteRangeCorr experiment with the finite
depth range correlation coefficient. We ran simulations with five different
types of sampling: (i) four uniform sampling rates (300, 200, 100 and 80 years)
and (ii) the adaptive sampling between 40 and 140 years
(CorrCoeff_40–140 yr_AddMCE experiment). An error correlation is applied
between markers according to Eq. (

Sensitivity of the solution to correlation coefficient values
applied between duration constraints (finite depth range
case). Top panel: GICC05 minus background chronology (black dashed line) and
difference between analysed and background chronologies. Bottom panel: MCE
(black dashed line) and analysed errors (blue to pink lines). The duration
constraints are correlated through a correlation function (Gaussian times
triangle), the correlation length is 300 years. Uniform marker sampling at 80-,
100- and 200-year rates are shown with the blue to pink lines. The adaptive
sampling between 40 and 140 years is shown in brown. The MCE assumption is
AddMCE (full error correlation between cycles). Table

Summary of the simulation configurations for the experiments of
Sect.

Figure

In summary, the tests presented in the two previous sections suggest some guidelines for future constructions of chronology using the duration constraints. The central problem is the definition of the error associated with annual layer counting and how this error is correlated with other layers' error. We showed that making different assumptions on the error correlation leads to significant difference in the final chronology and associated error. For experiments with Datice applied to several ice cores including NGRIP, if the objective is to preserve the NGRIP age scale, our recommendations are (i) to sample the duration constraints over small time windows (e.g. 100 years or apply an adaptive sampling rate), (ii) to use a small uncertainty for the observations (this is directly linked with a large or short range of correlation between layer-counting errors), or (iii) to increase the NGRIP overall background error.

After having validated the new developments for the implementation of duration constraints and possible error correlation, we show a first application of the new Datice tool to a five-ice-core experiment (NGRIP, EDC, EDML, Vostok, TALDICE).

To use Datice properly, the age constraints and the background scenarios need to be independent from each other. This was not the
case when building AICC2012 for the NGRIP ice core. Here, the new development
of Datice allows one to use scenarios for background accumulation rate and
thinning function independently from the age constraints deduced from GICC05
for NGRIP. In this application, the thinning function is the same as for
AICC2012, obtained from the 1-D DJ glaciological model adapted to NGRIP

Concerning the age constraint, the absolute age markers deduced from GICC05
were replaced by the duration constraints. The markers of duration are
obtained from the GICC05 chronology with adaptive lengths of intervals
between 40 and 140 years under the AddMCE assumption (full
correlation between annual cycles) and a correlation length of 300 years. In
order to constrain the relative gas chronology vs. the ice chronology, we use
information derived from

New

Summary of the simulation configurations.

Figure

Comparison of NGRIP

The Antarctic chronologies are not much modified compared to the AICC2012
chronologies. They all differ by less than 410 years from AICC2012 (Fig.

The Bayesian tool Datice used for the construction of coherent ice core chronology has been improved and now enables one to consider the duration of events as dating constraints. We validated this new methodological implementation by conducting twin experiments and a posteriori diagnostics.

In comparison to age markers, duration constraints are more coherent with the building of chronologies based on layer counting where the absolute error, defined as the maximum counting error for GICC05, increases with depth due to cumulative effects. To account for the fact that the counting errors on duration constraints are neither fully correlated nor uncorrelated, we have also introduced the possibility to adjust correlation between duration errors with a correlation coefficient that smoothly decreases with the distance between markers.

There is no objective way to choose the best representation of the
correlation, and future dating experiments may propose different correlation
coefficients for layer counting performed at different periods (glacial vs.
interglacial times). We have thus presented here some sensitivity tests for
the sampling and correlation of errors associated with duration constraints.
These tests lead to general guidelines for future dating experiments
including layer counting as absolute age constraints. For example, to best
respect an ice core chronology based on layer counting, we would favour a
high-frequency sampling of duration constraints with a correlation on a finite
depth range. Finally, the comparison of AICC2012 with the chronology obtained
over five polar sites using the improved Datice tool incorporating duration
constraints and associated correlation of errors shows differences of less
than 410 years over the last 60 kyr, well within the uncertainties associated
with the AICC2012 chronology. Huge efforts in annual layer counting were
produced in the recent years for ice core chronologies, in particular for the
Western Antarctic Ice Sheet (WAIS) ice core

The Datice age models are derived from three key ice core quantities: the
total thinning function

The gas chronology

Here we reiterate the formulations used to define the thinning function and the
LIDIE variances (Eqs.

The standard deviation of the thinning function is defined as

The standard deviation for the accumulation rate is

In order to avoid too small variances, a threshold value,

We have kept the same values for all Antarctic sites for

The formulation for the LIDIE standard deviation is

With the objective of better handling the MCE data, we make Gaussian
assumptions and reformulate the GICC05 layer counting with two probability
density functions (pdf's):

The duration of an annual cycle identified as certain is normally distributed with a 1-year average and a zero standard deviation.

The duration of an annual cycle identified as uncertain is normally distributed with a mean and a standard deviation both set to half a year.

The counting variables

It should be noted that such a formulation may be questioned: (i) the annual layer
counting has a discrete underlying nature, and one might rather prefer to
introduce discrete random variables to handle it; (ii) the Gaussian pdf
applies to continuous random variables ranging from

Sampling markers of duration from the GICC05 layer-counted chronology
may lead to a different amount of error correlation between individual
measures of annual cycles. Let us first sample the markers on a

Error covariance matrix

It is worth noting that Eq. (

When the correlation coefficients are identically null, we get the sum of the squared errors:

When the correlation coefficients are set to 1, we get the squared sum of the errors:

Below we briefly discuss the possible implications for choosing a sampling
rate of 20 or 40 years. While for the 20-year sampling, we may
straightforwardly implement the 20-year-window markers (

Option one: we believe that the full error correlation assessed over the 20-year time window between annual layers cuts-off. Then, no correlation exists
between the annual cycles included in the two separated but adjacent depth intervals

Option two: we believe that the full error correlation assessed over the 20-year time window between annual layers
extends over the 40-year time window (which means over the depth interval

From this simple illustration, it follows that markers of duration and errors sampled on GICC05 at different rates (i.e. 40–60–80–100 years), derived by summing up the GICC05 20-year-window MCE, must be understood as very different inputs and different simulation outputs must be expected.

In Sect. 2.3 of the main text, we refer to the AddMCE assumption (Eq. C9) when MCEs are added, while at the same time we refer to the AddSqrMCE assumption (Eq. C8) when the squared MCEs are added.

The background and observation errors

The analysed error

The ice age scale changes according to the correction function values

Datice calculates the components of

The variances of errors of the analysed chronology cumulate the error
covariances recorded in matrix

The age solution and its error are therefore largely determined by the
balance between observation and background errors (Eq.

If

Conversely, if

We thank Eric Wolff for his comments on a preliminary version of this article. We thank Paul Blackwell and the anonymous reviewers for their fruitful comments. This project was funded by the Fondation de France Ars Cuttoli. This work was supported by Labex L-IPSL, which is funded by the ANR (grant no. ANR-10-LABX-0018). This is LSCE contribution no. 5502. Edited by: N. Abram