The Inverse Method: A Collective Approach
All of absolute dating methods described in the preceding
sections have advantages and disadvantages. The counting
of layers and flow modeling are precise methods in terms of
the duration of events (at least for relatively recent periods)
because they are based on an assessment of the thickness of
the annual layers. However, errors accumulate and inaccuracy in the absolute age increases rapidly with depth.
Orbital tuning is generally applicable over the entire
length of a core, as long as the stratigraphy is maintained. In
addition, accuracy does not diminish with depth, and so far,
this is the most accurate method for dating the lower part of
polar cores. Unfortunately, this method is not very accurate
in terms of duration of events, and the accuracy in terms of
absolute ages is limited by the assumption that there is a
constant phase difference between the record being studied
and the insolation (and obviously does not allow the variations in this phase to be reconstructed). Research into indicators of local insolation paves the way for a significant
improvement in methods of orbital alignment, with a possibility of uncertainties below 1000 years. However, we still
lack sufficient distance to assess the real accuracy of these
methods. In addition, other methods will always be more
accurate for very recent times.
Volcanic eruptions provide important dated horizons.
This is particularly true for the last millennium, but beyond
this, only a few have an absolute age that is sufficiently
precise. Comparison with other dated records is particularly
useful for the dating of Dansgaard-Oeschger events, which
can be precisely localized using variations in the isotopic
composition of ice (in Greenland cores only), or variations in
the methane content in air bubbles. Most of these events
have been dated accurately for the last ice age, especially
from speleothems and the U/Th method (Chap. 6). Other
studies are underway to improve this accuracy and to study
previous ice ages. In any case, these dated horizons do not
provide continuous dating and are mainly relevant to recent
periods.
As these different sources of chronological information
are complementary, it is clear that to obtain an optimum
dating from the ice records, it is essential to combine them.
This is what glaciologists have attempted to do with
Antarctica ice cores. Initially, the poorly known parameters
in flow models (such as melting and sliding at the base of the
glacier) were adjusted by trial and error to obtain a good
agreement with dated horizons. But this approach quickly
becomes difficult when there are several free parameters and
when the different error bars for the dated horizons must be
taken into account.
In the early 2000s, an inverse method to formalize and
systematize this optimization of dating in a probabilistic
framework was developed (Parrenin et al. 2001). In the
context of dated horizons with a certain error bar, it facilitates the determination of the probability density called ‘a
posteriori’ both for the uncertain parameters in the flow
models and for the final dating. Concretely, this probability
density not only provides an optimum dating, but also a
confidence interval. This probabilistic method, based on the
Metropolis-Hastings algorithm, was applied to the Vostok
(Parrenin et al. 2004), Dome C (EDC3 dating, Parrenin et al
2007a, b) and Dome Fuji (Parrenin et al. 2007b) ice cores.
This inverse approach, however, has several limitations.
Firstly, it only takes into account the errors related to the
lack of understanding of poorly known parameters. In other
words, the model is considered perfect once an optimal
estimation of these parameters is achieved. This is the same
as saying that the model is able to describe all the relevant
flow mechanisms and is therefore, in agreement with all sets
of markers whose age errors are correctly estimated. In
reality, however, many physical phenomena influencing
flow are not taken into account in the model, either because
they are not properly understood, or because the level of
complexity necessary to describe them is incompatible with
inverse modeling (direct model is too costly in computing
time with too many parameters to inverse). This limitation of
the models appears clearly at the base of the EDC core
where the model, even after optimization of its parameters, is
unable to reproduce the age markers obtained by orbital
alignment (Dreyfus et al. 2007). A second limitation of this
inverse method is that it can be applied to only one core at a
time, and the optimal dating obtained is different for each
core, making it difficult to compare climate and environmental signals. In reality, these cores can be synchronized
more accurately in the ice phase (for example, by volcanic
horizons), as well as in the gas phase (for example, CH 4 and
d
18 O atm ).
As a result, a new method of optimization was developed
(Lemieux-Dudon et al. 2010; Parrenin et al. 2015). This
considers the information gleaned from modeling to be
weakly constrained (the model is not deemed perfect) and
applies to several cores simultaneously, taking into account
the stratigraphic links between these cores, both in the ice
and in the air. This method can thus provide an optimum
dating common to the different cores from Antarctica and
Greenland.
Conclusion
The dating of glacial archives is a complex problem which,
in the absence of the radioactive methods, is based on several complementary techniques. For the Holocene and in
high accumulation sites, dating by counting layers is
9 The Dating of Ice-Core Archives
133
All of absolute dating methods described in the preceding
sections have advantages and disadvantages. The counting
of layers and flow modeling are precise methods in terms of
the duration of events (at least for relatively recent periods)
because they are based on an assessment of the thickness of
the annual layers. However, errors accumulate and inaccuracy in the absolute age increases rapidly with depth.
Orbital tuning is generally applicable over the entire
length of a core, as long as the stratigraphy is maintained. In
addition, accuracy does not diminish with depth, and so far,
this is the most accurate method for dating the lower part of
polar cores. Unfortunately, this method is not very accurate
in terms of duration of events, and the accuracy in terms of
absolute ages is limited by the assumption that there is a
constant phase difference between the record being studied
and the insolation (and obviously does not allow the variations in this phase to be reconstructed). Research into indicators of local insolation paves the way for a significant
improvement in methods of orbital alignment, with a possibility of uncertainties below 1000 years. However, we still
lack sufficient distance to assess the real accuracy of these
methods. In addition, other methods will always be more
accurate for very recent times.
Volcanic eruptions provide important dated horizons.
This is particularly true for the last millennium, but beyond
this, only a few have an absolute age that is sufficiently
precise. Comparison with other dated records is particularly
useful for the dating of Dansgaard-Oeschger events, which
can be precisely localized using variations in the isotopic
composition of ice (in Greenland cores only), or variations in
the methane content in air bubbles. Most of these events
have been dated accurately for the last ice age, especially
from speleothems and the U/Th method (Chap. 6). Other
studies are underway to improve this accuracy and to study
previous ice ages. In any case, these dated horizons do not
provide continuous dating and are mainly relevant to recent
periods.
As these different sources of chronological information
are complementary, it is clear that to obtain an optimum
dating from the ice records, it is essential to combine them.
This is what glaciologists have attempted to do with
Antarctica ice cores. Initially, the poorly known parameters
in flow models (such as melting and sliding at the base of the
glacier) were adjusted by trial and error to obtain a good
agreement with dated horizons. But this approach quickly
becomes difficult when there are several free parameters and
when the different error bars for the dated horizons must be
taken into account.
In the early 2000s, an inverse method to formalize and
systematize this optimization of dating in a probabilistic
framework was developed (Parrenin et al. 2001). In the
context of dated horizons with a certain error bar, it facilitates the determination of the probability density called ‘a
posteriori’ both for the uncertain parameters in the flow
models and for the final dating. Concretely, this probability
density not only provides an optimum dating, but also a
confidence interval. This probabilistic method, based on the
Metropolis-Hastings algorithm, was applied to the Vostok
(Parrenin et al. 2004), Dome C (EDC3 dating, Parrenin et al
2007a, b) and Dome Fuji (Parrenin et al. 2007b) ice cores.
This inverse approach, however, has several limitations.
Firstly, it only takes into account the errors related to the
lack of understanding of poorly known parameters. In other
words, the model is considered perfect once an optimal
estimation of these parameters is achieved. This is the same
as saying that the model is able to describe all the relevant
flow mechanisms and is therefore, in agreement with all sets
of markers whose age errors are correctly estimated. In
reality, however, many physical phenomena influencing
flow are not taken into account in the model, either because
they are not properly understood, or because the level of
complexity necessary to describe them is incompatible with
inverse modeling (direct model is too costly in computing
time with too many parameters to inverse). This limitation of
the models appears clearly at the base of the EDC core
where the model, even after optimization of its parameters, is
unable to reproduce the age markers obtained by orbital
alignment (Dreyfus et al. 2007). A second limitation of this
inverse method is that it can be applied to only one core at a
time, and the optimal dating obtained is different for each
core, making it difficult to compare climate and environmental signals. In reality, these cores can be synchronized
more accurately in the ice phase (for example, by volcanic
horizons), as well as in the gas phase (for example, CH 4 and
d
18 O atm ).
As a result, a new method of optimization was developed
(Lemieux-Dudon et al. 2010; Parrenin et al. 2015). This
considers the information gleaned from modeling to be
weakly constrained (the model is not deemed perfect) and
applies to several cores simultaneously, taking into account
the stratigraphic links between these cores, both in the ice
and in the air. This method can thus provide an optimum
dating common to the different cores from Antarctica and
Greenland.
Conclusion
The dating of glacial archives is a complex problem which,
in the absence of the radioactive methods, is based on several complementary techniques. For the Holocene and in
high accumulation sites, dating by counting layers is
9 The Dating of Ice-Core Archives
133