Different paleoclimate proxy records evidence repeated abrupt climate
transitions during previous glacial intervals. These transitions are thought
to comprise abrupt warming and increase in local precipitation over Greenland,
sudden reorganization of the Northern Hemisphere atmospheric circulation, and
retreat of sea ice in the North Atlantic. The physical mechanism underlying
these so-called Dansgaard–Oeschger (DO) events remains debated. A recent
analysis of Greenland ice core proxy records found that transitions in
In view of anthropogenic global warming, concerns have been raised that
several subsystems of the earth's climate system may undergo abrupt and
fundamental state transitions if temperatures exceed corresponding critical
thresholds
Time series of
Apart from
In the search for the mechanism(s) causing or triggering DO events, several
attempts have been made to deduce the relative temporal order of these abrupt
changes by analyzing the phasing of corresponding abrupt shifts detected in
multi-proxy time series from Greenland ice cores
These observation-based studies are complemented by numerous conceptual
theories and modeling studies that explore a variety of mechanisms to explain
the DO events. Many studies emphasize the role of the AMOC in the emergence of
DO events
Here, we refine the investigation of a potential pairwise lead–lag
relationship between the four climate proxies
While
We will show that, if the uncertainties are averaged out at the level of the
individual transition onset lags – thus ignoring the uncertainties in the
onset detection – all tests indicate statistical significance (at
5
In addition to the quantitative uncertainty discussed here, there is always
qualitative uncertainty about the interpretation of climate proxies. Clearly,
there is no one-to-one mapping between proxy variables and the climate
variables they are assumed to represent. To give an example, changes in the
atmospheric circulation will simultaneously impact the transport efficiency of
sea-salt aerosols to Greenland.
This article is structured as follows: first, the data used for the study are described. Second, we introduce our methodology in general terms, in order to
facilitate potential adaptation to structurally similar problems. Within this
section, we pay special attention to clarifying the differences between the
approaches chosen in this study and by
In conjunction with their study,
While
DO events (Greenland interstadial onsets) for which
For their analysis,
We first briefly review the probabilistic method that we adopted from
Consider a fluctuating time series
The key idea is to model the transition as a linear ramp
Given the probability densities for the transition onsets of two proxy
variables
For the sake of simplicity, we omit the difference between the posterior density
distribution and the empirical posterior density distribution induced by an
MCMC sample. It is shown in Appendix
In the following, all probability densities that represent uncertainties with
origin in the transition onset observation will be referred to as uncertainty
distributions or uncertainty densities. This helps to distinguish them from probability distributions that generically characterize
random experiments. The random variables described by uncertainty distributions will
be termed uncertain variables and will be marked with a hat. Generally, we
denote all random (uncertain) variables by capital letters
Despite their diversity in terms of temperature amplitude, duration, and
frequency across the last glacial, the reoccurring patterns and their common
manifestation in different proxies suggest that the DO events follow a common
physical mechanism. If this assumption holds true, this mechanism prescribes a
fixed pattern of causes and effects for all DO events – at least on the scale
of interactions between climatic subsystems represented by the proxies under
study. However, natural variability randomly delays or advances the individual
parts of the event chain of the DO mechanism in each single realization,
without violating the mechanistic causality. The observed pairwise transition
onset lags can thus be regarded as realizations of independent and identically
distributed (i.i.d.) random variables generated in a random experiment
According to the data selection by
Before we introduce the tests deployed for this study, we discuss the
particularity that the individual observations of the i.i.d. variables that
comprise our samples are themselves subject to uncertainty and hence are
represented by probability densities instead of scalar values. The common
literature on hypothesis tests assumes that an observation of a random
variable yields a scalar value. Given a sample of
In contrast to this setting, the DO transition onset lags
A simplistic approach to test hypotheses on an uncertain sample would be to
average over the uncertainty distribution and subsequently apply the test to
the resulting expected sample
The uncertainty propagation relies on the fact that applying a function The hypothesis shall be rejected at the significance level
The hypothesis shall be rejected at the significance level
While the
We have introduced the notion of uncertain samples and its consequences for
the application of hypothesis tests. Here, we briefly introduce the tests used
to test our null hypothesis that the observed tendency for delayed transition
onsets in Let
We identified three tests that are suited for this task, namely the
The
The
Compared to the
For a given sample
The generalization of the WSR test to the uncertain sample
Given an observed sample of differences
In the case where the sample of differences is uncertain, as for
The three tests are applied in combination in order to compensate for their
individual deficits. If the population
For the derivation of the transition lag uncertainty distributions
Given a pair of variables
This implicitly assumes that all DO events share the exact same time lag
Thinking of the DO transition onset lags as i.i.d. random variables of a
repeatedly executed random experiment takes into account the natural
variability between different DO events, and hence it removes the restricting
a priori assumption
The mean of an uncertain sample
In the following we apply the above methodology to the different pairs of
proxies that
We first study the uncertain sample means. As already mentioned, the sample
mean is the best available point estimate for the population mean. Hence,
sample means different from 0 may be regarded as first indications for
potential systematic lead–lag relationships and thus motivate the application
of hypothesis tests. We compare the results obtained for the uncertain sample
means with corresponding results for the combined estimate. Both
quantities indicate a tendency towards a delayed transition in
Based on their assessment of the combined estimate,
Comparison between the uncertain sample means (this study) and
“combined estimates” according to
With the sample mean being the best point estimator of the population mean, it
serves as a suitable indicator for a potential population mean different from
0. The expectations
Both quantities, the uncertain sample mean and the combined estimate point
towards delayed transition onsets in
Above, we identified three tests for testing the hypothesis that the samples
Results from the application of the
Exemplary application of the analysis to the proxy pair
Figure
Results of the hypothesis tests applied to the uncertain samples
of transition onset lags
Figure
We argue that the variability across different DO events cannot be ignored in
the assessment of the data. Although the DO events are likely to be caused by
the same physical mechanism, changing boundary conditions and other natural
climate fluctuations will lead to deviations in the exact timings of the
different processes involved in triggering the individual DO
events. Figure
Our main purpose was the consistent treatment of observational uncertainties
and we have largely ignored the vibrant debate on the qualitative
interpretation of the proxies. Surprisingly, we could not find any literature
on the application of hypothesis tests to uncertain samples of the kind
discussed here. The theory of fuzzy
The potential of the availability of data from different sites has probably
not been fully leveraged in this study. Naively, one could think of the NEEM
and NGRIP
We have presented a statistical reinterpretation of the high-resolution proxy
records provided and analyzed by
Even though we find that the uncertainty of the transition onset detection combined with the small sample size prevents the deduction of statistically unambiguous statements on the temporal order of events, we think that multi-proxy analysis is a promising approach to investigate the sequential order at the beginning of DO events. In this study, we refrained from analyzing the lags between the different proxies in a combined approach and focused on the marginal populations. However, a combined statistical evaluation – that is, treating the transition onsets of all proxy variables as a four-dimensional random variable – merits further investigation. Also, we propose to statistically combine measurements from NEEM and NGRIP (and potentially further ice cores) of the same proxy pairs. Finally, hierarchical models may be invoked to avoid switching from a Bayesian perspective in the transition onset estimation to a frequentist perspective in the statistical interpretation of the uncertain samples. Finally, effort in conducting modeling studies should be sustained. Especially proxy-enabled modeling bears the potential to improve comparability between model results and paleoclimate records. Together, these lines of research are promising to further constrain the sequence of events that have caused the abrupt climate changes associated with DO events.
In Sect.
For a given proxy and a given DO event, in a first step the MCMC algorithm
samples from the joint posterior probability density for the models parameter
configuration
As in the main text, in the following we denote uncertain quantities with a
hat. For a given proxy pair the starting point for the statistical analysis
however, is the uncertain sample
Having found a numerically manageable expression for the empirical uncertainty
distribution of the sample
Recall the statistic of the
As explained in Sect.
Results obtained from the application of hypothesis tests to the
control group. Reported are the mean
The results obtained from the control runs show only minor deviations from the
results presented in the main text and thus confirm the validity of the
reduction in the corresponding sets. Table
In the main text, we stated that the uncertain sample mean is given by the pairwise convolution of the individual uncertainty distributions that describe the uncertain sample members. Here, we show how the uncertain sample mean can be computed if the individual uncertainty distributions are known.
Consider
Self-iteration of Eq. (
The 10-year resolution time series of
Please note that the
All data in preprocessed form, together with the software used to generate samples from the posterior distributions of the transition onsets for all proxies at all interstadial onsets under study, were directly obtained from
The numerical implementation of the analysis presented here, building upon the aforementioned samples from the posterior distributions of transition onsets, is publicly available under
The supplement related to this article is available online at:
KR and NB conceived the study. KR carried out the numerical analysis. KR and NB discussed and interpreted the results and wrote the paper.
The authors declare that they have no conflict of interest.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Norbert Marwan for very helpful comments and discussions. This is TiPES contribution no. 60; the TiPES (Tipping Points in the Earth System) project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement no. 820970. Niklas Boers acknowledges funding by the Volkswagen foundation.
This research has been supported by the Horizon 2020 (TiPES (grant no. 820970)) and the Volkswagen foundation.The article processing charges for this open-access publication were covered by the Potsdam Institute for Climate Impact Research (PIK).
This paper was edited by Laurie Menviel and reviewed by two anonymous referees.