A technique for estimating the age–depth relationship in an ice core and
evaluating its uncertainty is presented. The age–depth relationship is
determined by the accumulation of snow at the site of the ice core and the
thinning process as a result of the deformation of ice layers. However, since
neither the accumulation rate nor the thinning process is fully known, it is
essential to incorporate observational information into a model that
describes the accumulation and thinning processes. In the proposed technique,
the age as a function of depth is estimated by making use of age markers and

Ice cores provide vital information on the climatic and environmental changes
over the past hundreds of thousands of years. In order to make use of the
chronological records from each slice of an ice core, it is crucial to
accurately determine the relationship between age and depth in the ice cores.
Many of the dating methods for determining the age–depth relationship rely
on glaciological modeling. However, since the glaciological processes
controlling the age–depth relationship are not fully known, it is essential
to reduce uncertainty by incorporating various types of observational
information into the glaciological model. In particular, it is important to
effectively make use of the information of age markers, which provides
significant constraints on the age–depth relationship. The Bayesian approach
is a powerful way to combine a variety of observational information with a
model, and it has been applied to the dating of ice cores in a number of
studies.

The remainder of the present paper is organized as follows. In
Sect.

The age–depth relationship is determined by two processes. One is
accumulation of snow at the site of the ice core and the other is the
thinning process due to long-term deformations within the ice sheet (e.g.,

Denoting the annual rate of snow accumulation by

In order to model the thinning factor

In order to obtain the age

At several depths, we can also use reliable age values given by age markers.
We used such age values as tie points when estimating the age–depth
relationship. The age, depth, and uncertainty (

In this section, the age–depth relationship is formulated in a framework of a state space model on the basis of the model described in the previous section. The state space model represents the evolution of variables by a recurrence equation. The state space model provides a platform for the sequential Bayesian estimation using PMCMC, which will be explained in the next section.

Discretizing the vertical coordinate

Equation (

Depth, age, and uncertainty of age at each tie point.

The accumulation rate

Based on Eqs. (

Estimates of

The

Like Eqs. (

We hereinafter combine

Our aim is to estimate

We can also estimate the parameter

Since the present accumulation

In order to approximate the conditional distributions

The SMC method, which is sometimes referred to as the particle
filter/smoother in time-series analysis

Equation (

An approximation of the marginal likelihood

Using the Monte Carlo approximation of the marginal likelihood

In the above algorithm, an approximated value of the marginal likelihood

As mentioned in Sect.

We applied the PMCMC method to the Dome Fuji ice core. In this study, the
thickness of the ice sheet

Figure

Estimated marginal distributions of the posterior distributions for the nine parameters.

Two-dimensional histograms of the marginal posterior
distribution of

In the posterior distribution, some of the parameters are correlated with
each other. Figure

Figure

Estimated age as a function of depth. The solid line indicates
the median of the posterior distribution. The 10th and 90th percentiles of
the posterior are indicated by red dotted lines. The black crosses indicate
the tie points. The result obtained by

Difference of the 10th and 90th percentiles
of the posterior distribution from the median of the posterior
(red dotted lines), difference of each tie point from the median
of the posterior (black crosses), and difference of the estimate
by

Estimated thinning factor

Estimated accumulation rate as a function of depth.
The median of the posterior is indicated by a red solid line,
the 10th and 90th percentiles are indicated by red dotted
lines, the difference between the 10th and 90th percentiles
is indicated by a blue dotted line, and the estimate by

Figure

In order to evaluate the robustness, we obtained the estimate without using
the last five tie points at below 2400 m depth. We estimated the parameters
and the age–depth relationship from the other 20 tie points and the

Estimated accumulation rate as a function of age.
The median of the posterior is indicated by a red solid line, the 10th and
90th percentiles are indicated by red dotted lines, the difference between
the 10th and 90th percentiles is indicated by a blue dotted line, and the
estimate by

Estimated marginal distributions of each of the nine parameters: without using the last five tie points (blue) and with all the tie points (red).

Difference of the 10th and 90th percentiles of the posterior distribution from the median of the posterior (red dotted lines) for the result without using the last five age markers. The difference of each tie point from the median of the posterior (black crosses) and the difference between the estimate with all the tie points and the estimate without using the last five tie points (grey line) are also shown.

Figure

The accumulation rate as a function of age was also estimated without using
the five tie points at the bottom of the ice core. Figure

The proposed technique requires a high computational cost because the SMC
sampling is performed at each iteration of the Metropolis method. At present,
it takes about 43 h to complete 250 000 iterations of the Metropolis
sampling with 5000 particles for the SMC on a workstation with two Intel Xeon
processors (12 cores for each processor; 2.7 GHz). The efficiency could be
improved by using a better proposal distribution used in SMC (e.g.,

Estimated accumulation rate as a function of age: without using the last five tie points (red) and with all the tie points (grey). The solid lines indicate the median of the posterior distribution. The 10th and 90th percentiles of the posterior are indicated by dotted lines.

There may be room for improvement in the model for the accumulation rate
described by Eq. (

This study used the

We have developed a technique for the dating of an ice core by combining
information obtained from age markers at various depths with a model
describing the accumulation of snow and glaciological dynamics. This
technique provides estimates of unspecified parameters in the model from the
posterior distributions calculated with the PMCMC method. In the PMCMC
method, the marginal posterior distributions of the parameters are obtained
using the Metropolis method; this is similar to other existing techniques

The main advantage of the proposed technique is that it can be applied to general nonlinear non-Gaussian situations. Since the relationship between accumulation rate and a temperature proxy is typically nonlinear, it is not necessarily justified to assume linearity and Gaussianity when using a temperature proxy to date an ice core. The PMCMC method allows us to use various kinds of data that are expected to have a nonlinear relationship with the model variables. Another advantage is that the PMCMC method estimates the model parameters simultaneously with the age as a function of depth. The uncertainty of age is therefore evaluated after taking into account the uncertainties in the model parameters.

This work was conducted under project “Exploration for Seeds of Integrated Research”, supported by the Transdisciplinary Research Integration Center, Research Organization of Information and Systems. Edited by: A. M. Mancho Reviewed by: M. Winstrup and two anonymous referees