# Some identities on degenerate Bell polynomials and their related identities

Taekyun Kim  Department of Mathematics, Kwangwoon University, Seoul 139-701,
Republic of Korea<EMAIL_ADDRESS>and  DAE SAN KIM  Department of Mathematics,
Sogang University, Seoul 121-742, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

Recently, several types of degenerate Bell polynomials have been introduced as
degenerate versions of the ordinary Bell polynomials. The aim of this paper is
to study some identities for the degenerate Bell polynomials and their related
identities by making use of series transformation formula obtained by
Boyadzhiev. In particular, this answers to the natural question about the
relationship between two different types of degenerate Bell polynomials.

###### Key words and phrases:

degenerate Bell polynomial; degenerate geometric polynomial; Eulerian
polynomial; degenerate Stirling numbers of the first kind; degenerate Stirling
numbers of the second kind

###### 2010 Mathematics Subject Classification:

11B68; 11B83; 05A19

##  1\. Introduction

In [3], Carlitz initiated a study of degenerate Bernoulli and degenerate Euler
polynomials, which are degenerate versions of the ordinary Bernoulli and Euler
polynomials. In recent years, some mathematicians have explored degenerate
versions of many special polynomials and numbers which include the degenerate
Bernoulli numbers of the second kind, degenerate Stirling numbers of the first
and second kinds, degenerate Bell numbers and polynomials, and so on (see
[5–9,12] and the references therein).

The Bell number $B_{n}$ counts the number of partitions of a set with $n$
elements into disjoint nonempty subsets. The Bell polynomials are natural
extensions of Bell numbers and also called Touchard or exponential
polynomials. As degenerate versions of these Bell polynomials and numbers,
several different versions of degenerate Bell polynomials and numbers are
introduced and investigated (see [7–9]).

The aim of this paper is to study some identities for the degenerate Bell
polynomials and their related identities by using generating functions and the
series transformation formula obtained by Boyadzhiev (see [2]). In particular,
this answers to the natural question about the relationship between two
different types of degenerate Bell polynomials. In addition, it gives us an
expression for the generating function of the sums of $\lambda$-falling
factorials of consecutive nonnegative integers, which reduces to the
generating function of sums of powers of consecutive nonnegative integers.
This paper demonstrates that the series transformation formula can be used in
studying various types of the degenerate Bell polynomials as well.

This paper is outlined as follows. In Section 1, we will recall degenerate
exponential functions, degenerate Stirling numbers of the first kind and
degenerate Stirling numbers of the second kind which are respectively
degenerate versions of exponential functions, Stirling numbers of the first
kind and Stirling numbers of the second kind. Then we will go over the
definitions of some degenerate special polynomials, namely three types of
degenerate Bell polynomials and the degenerate Bernoulli polynomials. As to
two types degenerate Bell polynomials among the three types, we ask what their
relationship is. In addition, we recall the geometric polynomials in
connection with the ordinary Bell polynomials.

In Section 2, we will recall explicit expressions for the three types of
degenerate Bell polynomials. Then we will introduce the degenerate geometric
polynomilas in connection with the partially degenerate Bell polynomials.
Next, we will state Theorem 3 about the series transformation formula which is
proved in [2]. This is a key fact that we are going to apply in order to get
some interesting results. By choosing $g(x)=e_{\lambda}(x)$ as the degenerate
exponentials in Theorem 3, we get Theorem 4. And then, further taking
$f(x)=e^{xt}$ as the usual exponential, we will be able to answer to the
question regarding the relationship between two different types of degenerate
Bell polynomials. As is well known, the sums of powers of consecutive
nonnegative integers can be expressed in terms of Bernoulli polynomials. Here
we express the generating function of the sums of $\lambda$-falling factorials
of consecutive nonnegative integers in terms of the geometric polynomial and
the Stirling numbers of the first kind. By letting $\lambda\rightarrow 0$, the
generating function of the sums of powers of consecutive nonnegative integers
is expressed by the geometric polynomial. Then we obtain an identity involving
Stirling numbers of the second kind, the Carlitz degenerate Bernoulli numbers
and the degenerate geometirc polynomial. Finally, we conclude this paper in
Section 3.

For any nonzero $\lambda\in\mathbb{R}$, the degenerate exponential functions
are defined by

(1) $e_{\lambda}^{x}(t)=(1+\lambda t)^{\frac{x}{\lambda}},\quad
e_{\lambda}(t)=e_{\lambda}^{1}(t)=(1+\lambda
t)^{\frac{1}{\lambda}},\quad(\mathrm{see}\ [5]).$

Thus, by (1), we get

(2)
$e_{\lambda}^{x}(t)=\sum_{n=0}^{\infty}(x)_{n,\lambda}\frac{t^{n}}{n!},\quad(\mathrm{see}\
[6]),$

where the $\lambda$-falling factorials are given by

(3)
$(x)_{0,\lambda}=1,\quad(x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots(x-(n-1)\lambda),\
n\geq 1.$

It is well known that the Stirling numbers of the first kind are defined by

(4) $(x)_{n}=\sum_{l=0}^{n}S_{1}(n,l)x^{l},\quad(n\geq 0),\quad(\mathrm{see}\
[7,9,11]),$

where $(x)_{0}=1,\ (x)_{n}=x(x-1)\cdots(x-n+1),\ (n\geq 1)$.

As the inversion of (4), the Stirling numbers of the second kind are given by

(5) $x^{n}=\sum_{l=0}^{n}S_{2}(n,l)(x)_{l},\quad(n\geq 0),\quad(\mathrm{see}\
[7,11]).$

In [5], the degenerate Stirling numbers of the first kind are introduced as

(6) $(x)_{n}=\sum_{l=0}^{n}S_{1,\lambda}(n,l)(x)_{n,\lambda},\quad(n\geq 0).$

As the inversion of (6), the degenerate Stirling numbers of the second kind
are given by

(7) $(x)_{n,\lambda}=\sum_{l=0}^{n}S_{2,\lambda}(n,l)(x)_{l},\quad(n\geq
0),\quad(\mathrm{see}\ [5]).$

Recently, the degenerate Bell polynomials are defined by

(8)
$e_{\lambda}\big{(}x(e^{t}-1)\big{)}=\sum_{n=0}^{\infty}\mathrm{Bel}_{n,\lambda}(x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[9]).$

Note that $\displaystyle\lim_{\lambda\rightarrow
0}\mathrm{Bel}_{n,\lambda}(x)=\mathrm{Bel}_{n}(x),\ (n\geq 0)\displaystyle$,
where $\mathrm{Bel}_{n}(x)$ are the ordinary Bell polynomials given by

(9)
$e^{x(e^{t}-1)}=\sum_{n=0}^{\infty}\mathrm{Bel}_{n}(x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[1,11]).$

In [7], the degenerate Bell polynomials of the second kind are defined by

(10) $e_{\lambda}(xe^{t})\cdot
e_{\lambda}^{-1}(x)=\sum_{n=0}^{\infty}\mathrm{bel}_{n,\lambda}(x)\frac{t^{n}}{n!}.$

Observe that $\displaystyle\lim_{\lambda\rightarrow
0}\mathrm{bel}_{n,\lambda}(x)=\mathrm{Bel}_{n}(x)\ (n\geq 0)\displaystyle$.
The following natural question arises natually from (8) and (10).

Question. What is the relationship between $\mathrm{Bel}_{n,\lambda}(x)$ and
$\mathrm{bel}_{n,\lambda}(x)$?

This question will be answered in Theorem 6.

For $n\geq 0$, the geometric polynomials are defined by

(11) $W_{n}(x)=\sum_{k=0}^{n}S_{2}(n,k)k!x^{k},\quad(\mathrm{see}\ [2,12]).$

From (11), we note that

(12) $\sum_{n=0}^{\infty}W_{n}(x)\frac{t^{n}}{n!}=\frac{1}{1-x(e^{t}-1)}.$

By (12), we get

(13)
$W_{n}(x)=\int_{0}^{\infty}\mathrm{Bel}_{n}(xy)e^{-y}dy,\quad(\mathrm{see}\
[2]).$

In [8], the partially degenerate Bell polynomials are defined by

(14)
$e^{x(e_{\lambda}(t)-1)}=\sum_{n=0}^{\infty}\phi_{n,\lambda}(x)\frac{t^{n}}{n!}.$

Note that $\displaystyle\lim_{\lambda\rightarrow
0}\phi_{n,\lambda}(x)=\mathrm{Bel}_{n}(x),\ (n\geq 0)\displaystyle$.

Carlitz introduced the degenerate Bernoulli numbers given by

(15)
$\frac{t}{e_{\lambda}(t)-1}=\sum_{n=0}^{\infty}\beta_{n,\lambda}\frac{t^{n}}{n!},\quad(\mathrm{see}\
[3]).$

Note that $\displaystyle\lim_{\lambda\rightarrow 0}\beta_{n,\lambda}=B_{n},\
(n\geq 0)\displaystyle$, where $B_{n}$ are the ordinary Bernoulli numbers
given by

(16)
$\frac{t}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!},\quad(\mathrm{see}\
[1-12]).$

More generally, the Bernoulli polynomials $B_{n}(x)$ are defined by

(17) $\frac{t}{e^{t}-1}e^{xt}=\sum_{n=0}^{\infty}B_{n}(x)\frac{t^{n}}{n!}.$

##  2\. Some identities on degenerate Bell polynomials and their related
identities

From (8) and (10), it is immediate to see that

(18)
$\mathrm{Bel}_{n,\lambda}(x)=\sum_{k=0}^{n}(1)_{k,\lambda}S_{2}(n,k)x^{k},\quad(n\geq
0),$

and

(19)
$\mathrm{bel}_{n,\lambda}(x)=e_{\lambda}^{-1}(x)\sum_{k=0}^{\infty}\frac{(1)_{k,\lambda}}{k!}k^{n}x^{k},\quad(n\geq
0).$

By (14), we get

(20) $\phi_{n,\lambda}(x)=\sum_{k=0}^{n}S_{2,\lambda}(n,k)x^{k},\quad(n\geq
0).$

By inversion, from (18) and (20), we have

$\displaystyle
x^{n}=\frac{1}{(1)_{n,\lambda}}\sum_{k=0}^{n}S_{1}(n,k)\mathrm{Bel}_{k,\lambda}(x),\quad
x^{n}=\sum_{k=0}^{n}S_{1,\lambda}(n,k)\phi_{k,\lambda}(x).$

Now, we define the degenerate geometric polynomials by

(21) $W_{n,\lambda}(x)=\sum_{k=0}^{n}S_{2,\lambda}(n,k)k!x^{k},\quad(n\geq
0).$

Note that $\displaystyle\lim_{\lambda\rightarrow 0}W_{n,\lambda}(x)=W_{n}(x),\
(n\geq 0)\displaystyle$. Further, by inversion we also have

$x^{n}=\frac{1}{n!}\sum_{k=0}^{n}S_{1,\lambda}(n,k)W_{k,\lambda}(x).$

From (21), we have

(22) $\displaystyle\sum_{n=0}^{\infty}W_{n,\lambda}(x)\frac{t^{n}}{n!}\ $
$\displaystyle=\
\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}k!S_{2,\lambda}(n,k)x^{k}\bigg{)}\frac{t^{n}}{n!}$
$\displaystyle=\
\sum_{k=0}^{\infty}x^{k}k!\sum_{n=k}^{\infty}S_{2,\lambda}(n,k)\frac{t^{n}}{n!}$
$\displaystyle=\
\sum_{k=0}^{\infty}x^{k}\big{(}e_{\lambda}(t)-1\big{)}^{k}=\frac{1}{1-x(e_{\lambda}(t)-1)}.$

We observe that

(23) $\displaystyle\int_{0}^{\infty}e^{xy(e_{\lambda}(t)-1)}e^{-y}dy$
$\displaystyle=\int_{0}^{\infty}e^{-y(1-x(e_{\lambda}(t)-1))}dy$
$\displaystyle=\frac{1}{1-x(e_{\lambda}(t)-1)}=\sum_{n=0}^{\infty}W_{n,\lambda}(x)\frac{t^{n}}{n!}.$

On the other hand,

(24)
$\int_{0}^{\infty}e^{xy(e_{\lambda}(t)-1)}e^{-y}dy=\sum_{n=0}^{\infty}\int_{0}^{\infty}\phi_{n,\lambda}(xy)e^{-y}dy\frac{t^{n}}{n!}.$

Therefore, by (23) and (24), we obtain the following theorem.

######  Theorem 1.

For $n\geq 0$, we have

$\int_{0}^{\infty}\phi_{n,\lambda}(xy)e^{-y}dy=W_{n,\lambda}(x).$

Later, we will need the following lemma which is proved in [4, p.218] and also
mentioned in [2].

######  Lemma 2.

Let $D=\frac{d}{dx}$, and let $f$ be analytic on an open set $U$ in
$\mathbb{C}$. Then, for each $x\in U$, we have

$(xD)^{n}f(x)=\sum_{k=0}^{n}S_{2}(n,k)x^{k}D^{k}f(x),\quad(n\geq 0).$

In [2], K. N. Boyadzhiev presented a formula that turns power series into
series of functions. He also gave many valuable formulas related to
polynomials. The following series transformation formula is proved in [2,
Theorem 4.1]. Indeed, it can be shown by computing $f(xD)g(x)$ in two
different ways with the help of Lemma 2. We are going to apply this fromula
for various $f$ and $g$ in order to get a few interesting results.

######  Theorem 3.

Assume that $f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$ and
$g(x)=\sum_{k=0}^{\infty}c_{k}x^{k}$ are power series convergnt on some open
disks centered at the origin. Then we have

$\displaystyle\sum_{k=0}^{\infty}\frac{g^{(k)}(0)}{k!}f(k)x^{k}=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\sum_{k=0}^{n}S_{2}(n,k)g^{(k)}(x)x^{k}.$

First, we choose $g$ as $g(x)=e_{\lambda}(x)$ in Theorem 3. Then we see that
the following holds:

(25)
$g^{(k)}(x)=\bigg{(}\frac{d}{dx}\bigg{)}^{k}e_{\lambda}(x)=(1)_{k,\lambda}e_{\lambda}^{1-k\lambda}(x)=\frac{(1)_{k,\lambda}}{(1+\lambda
x)^{k}}e_{\lambda}(x).$

From Theorem 3 and (25), we note that

(26)
$\sum_{n=0}^{\infty}\frac{(1)_{n,\lambda}}{n!}f(n)x^{n}=e_{\lambda}(x)\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\sum_{k=0}^{n}S_{2}(n,k)(1)_{k,\lambda}\bigg{(}\frac{x}{1+\lambda
x}\bigg{)}^{k}.$

By (18) and (26), we get

(27)
$\sum_{n=0}^{\infty}\frac{(1)_{n,\lambda}}{n!}f(n)x^{n}=e_{\lambda}(x)\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\mathrm{Bel}_{n,\lambda}\bigg{(}\frac{x}{1+\lambda
x}\bigg{)}.$

Therefore, we obtain the following theorem.

######  Theorem 4.

Let $f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$ be a power series convergnt on some
open disk centered at the origin. For $n\geq 0$, we have

$\sum_{n=0}^{\infty}\frac{(1)_{n,\lambda}}{n!}f(n)x^{n}=e_{\lambda}(x)\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\mathrm{Bel}_{n,\lambda}\bigg{(}\frac{x}{1+\lambda
x}\bigg{)}.$

Now, by using (19) we observe that

(28) $\displaystyle(xD)^{n}e_{\lambda}(x)$
$\displaystyle=(xD)^{n}\sum_{k=0}^{\infty}\frac{(1)_{k,\lambda}}{k!}x^{k}=\sum_{k=0}^{\infty}\frac{(1)_{k,\lambda}}{k!}k^{n}x^{k}$
$\displaystyle=\bigg{(}e_{\lambda}^{-1}(x)\sum_{k=0}^{\infty}\frac{k^{n}}{k!}(1)_{k,\lambda}x^{k}\bigg{)}e_{\lambda}(x)=\mathrm{bel}_{n,\lambda}(x)e_{\lambda}(x).$

Hence, by (28), we get the next result.

######  Theorem 5.

For $n\geq 0$, we have

$\mathrm{bel}_{n,\lambda}(x)e_{\lambda}(x)=(xD)^{n}e_{\lambda}(x)=\sum_{k=0}^{\infty}\frac{(1)_{k,\lambda}}{k!}k^{n}x^{k}.$

Taking $f(x)=e^{xt}$ in (26), we have

(29) $\displaystyle\sum_{n=0}^{\infty}\frac{(1)_{n,\lambda}}{n!}e^{nt}x^{n}$
$\displaystyle=e_{\lambda}(x)\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\sum_{k=0}^{n}S_{2}(n,k)(1)_{k,\lambda}\bigg{(}\frac{x}{1+\lambda
x}\bigg{)}^{k}$
$\displaystyle=e_{\lambda}(x)\sum_{n=0}^{\infty}\mathrm{Bel}_{n,\lambda}\bigg{(}\frac{x}{1+\lambda
x}\bigg{)}\frac{t^{n}}{n!}.$

Thus, by (29), we get

(30)
$\displaystyle\sum_{n=0}^{\infty}\mathrm{Bel}_{n,\lambda}\bigg{(}\frac{x}{1+\lambda
x}\bigg{)}\frac{t^{n}}{n!}$
$\displaystyle=e_{\lambda}^{-1}(x)\sum_{n=0}^{\infty}\frac{(1)_{n,\lambda}}{n!}e^{nt}x^{n}$
$\displaystyle=e_{\lambda}^{-1}(x)e_{\lambda}\big{(}xe^{t}\big{)}=\sum_{n=0}^{\infty}\mathrm{bel}_{n,\lambda}(x)\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (30), we obtain the
following theorem. This answers to the question that is asked in the
Introduction.

######  Theorem 6.

For $n\geq 0$, we have

$\mathrm{bel}_{n,\lambda}(x)=\mathrm{Bel}_{n,\lambda}\bigg{(}\frac{x}{1+\lambda
x}\bigg{)}.$

From (8), (10) and (30), we note that

(31) $e_{\lambda}\bigg{(}\frac{x}{1+\lambda
x}\big{(}e^{t}-1\big{)}\bigg{)}=e_{\lambda}(xe^{t})\cdot e_{\lambda}^{-1}(x).$

For $g(x)=\frac{1}{1-x}$, we have
$g^{(k)}(x)=\big{(}\frac{d}{dx}\big{)}^{k}g(x)=k!\big{(}\frac{1}{1-x}\big{)}^{k+1}$,
$g^{(k)}(0)=k!$.

From Theorem 3 , we have

(32) $\displaystyle\sum_{k=0}^{\infty}\frac{k!}{k!}f(k)x^{k}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\sum_{k=0}^{n}S_{2}(n,k)x^{k}k!\bigg{(}\frac{1}{1-x}\bigg{)}^{k+1}$
$\displaystyle=\frac{1}{1-x}\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\sum_{k=0}^{n}S_{2}(n,k)\bigg{(}\frac{x}{1-x}\bigg{)}^{k}k!$
$\displaystyle=\frac{1}{1-x}\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}W_{n}\bigg{(}\frac{x}{1-x}\bigg{)}.$

Thus, by (32), we get

(33)
$\sum_{k=0}^{\infty}f(k)x^{k}=\frac{1}{1-x}\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}W_{n}\bigg{(}\frac{x}{1-x}\bigg{)}.$

Applying (33) to $f(x)=(x)_{m,\lambda}$, and noting

$f(x)=(x)_{m,\lambda}=\lambda^{m}\bigg{(}\frac{x}{\lambda}\bigg{)}_{m}=\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}x^{l},$

and, for $n\leq m$,

(34)
$f^{(n)}(x)=\bigg{(}\frac{d}{dx}\bigg{)}^{n}f(x)=\sum_{l=n}^{m}S_{1}(m,l)\lambda^{m-l}(l)_{n}x^{l-n}.$

From (33) and (34), we have

(35)
$\sum_{k=0}^{\infty}(k)_{m,\lambda}x^{k}=\frac{1}{1-x}\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}W_{l}\bigg{(}\frac{x}{1-x}\bigg{)}.$

We observe that

(36) $\displaystyle
x^{m\lambda}\big{(}x^{1-\lambda}D\big{)}^{m}\bigg{(}\frac{1}{1-x}\bigg{)}$
$\displaystyle=x^{m\lambda}\big{(}x^{1-\lambda}D\big{)}^{m}\sum_{k=0}^{m}x^{k}$
$\displaystyle=\sum_{k=0}^{\infty}(k)_{m,\lambda}x^{k}.$

From (35) and (36), we have

(37) $\displaystyle
x^{m\lambda}\big{(}x^{1-\lambda}D\big{)}^{m}\bigg{(}\frac{1}{1-x}\bigg{)}$
$\displaystyle=\sum_{k=0}^{\infty}(k)_{m,\lambda}x^{k}$
$\displaystyle=\frac{1}{1-x}\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}W_{l}\bigg{(}\frac{x}{1-x}\bigg{)}.$

Note that

(38) $\displaystyle\frac{1}{1-x}\sum_{l=0}^{\infty}(l)_{m,\lambda}x^{l}$
$\displaystyle=\bigg{(}\sum_{j=0}^{\infty}x^{j}\bigg{)}\bigg{(}\sum_{l=0}^{\infty}(l)_{m,\lambda}x^{l}\bigg{)}=\sum_{k=0}^{\infty}\bigg{(}\sum_{l=0}^{k}(l)_{m,\lambda}\bigg{)}x^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\big{(}(0)_{m,\lambda}+(1)_{m,\lambda}+(2)_{m,\lambda}+\cdots+(k)_{m,\lambda}\big{)}x^{k}.$

By (38), we get

(39)
$\displaystyle\sum_{k=0}^{\infty}\big{(}(0)_{m,\lambda}+(1)_{m,\lambda}+(2)_{m,\lambda}+\cdots+(k)_{m,\lambda}\big{)}x^{k}$
$\displaystyle=\frac{1}{1-x}\sum_{l=0}^{\infty}(l)_{m,\lambda}x^{l}$
$\displaystyle=\frac{1}{(1-x)^{2}}\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}W_{l}\bigg{(}\frac{x}{1-x}\bigg{)}.$

Therefore, by (39), we obtain the following theorem.

######  Theorem 7.

For $m\geq 0$, we have

$\sum_{k=0}^{\infty}\big{(}(0)_{m,\lambda}+(1)_{m,\lambda}+(2)_{m,\lambda}+\cdots+(k)_{m,\lambda}\big{)}x^{k}=\frac{1}{(1-x)^{2}}\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}W_{l}\bigg{(}\frac{x}{1-x}\bigg{)}.$

######  Remark 8.

(a) The above theorem is equivalent to the following:

$\displaystyle\sum_{k=1}^{\infty}\big{(}(1)_{m,\lambda}+(2)_{m,\lambda}+\cdots+(k)_{m,\lambda}\big{)}x^{k}=\frac{1}{(1-x)^{2}}\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}W_{l}\bigg{(}\frac{x}{1-x}\bigg{)},\quad(m\geq
1),$
$\displaystyle\quad\quad\quad\quad\quad\sum_{k=0}^{\infty}(k+1)x^{k}=\frac{1}{(1-x)^{2}}.$

(b) It is well known that Faulhaber’s formula on the sums of powers of
consecutive nonnegative integers is equivalent to

(40)
$\displaystyle\sum_{i=0}^{k}i^{m}=\frac{1}{m+1}\big{(}B_{m+1}(k+1)-B_{m+1}\big{)}.$

By letting $\lambda\rightarrow 0$ in Theorem 7 and making use of (40), we have

$\displaystyle\sum_{k=0}^{\infty}\big{(}0^{m}+1^{m}+2^{m}+\cdots
k^{m}\big{)}x^{k}$
$\displaystyle=\frac{1}{m+1}\sum_{k=0}^{\infty}\big{(}B_{m+1}(k+1)-B_{m+1}\big{)}x^{k}$
$\displaystyle=\frac{1}{(1-x)^{2}}W_{m}\bigg{(}\frac{x}{1-x}\bigg{)}.$

For $n,m\in\mathbb{Z}$, with $n\geq 1,0\leq m\leq n$, let
$\genfrac{<}{>}{0.0pt}{}{n}{m}$ be the Eulerian numbers. Then we note that

(41)
$\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{l=0}^{k+1}(-1)^{l}\binom{k+1}{l}(k+1-l)^{n}\quad\mathrm{and}\quad\sum_{k=0}^{n}\genfrac{<}{>}{0.0pt}{}{n}{k}=n!.$

The Eulerian polynomials are defined by

(42) $A_{n}(x)=\sum_{m=0}^{n}\genfrac{<}{>}{0.0pt}{}{n}{m}x^{n-m},\quad(n\geq
0).$

From (37), we note that

(43) $\sum_{k=0}^{\infty}k^{m}x^{k}=\lim_{\lambda\rightarrow
0}\sum_{k=0}^{\infty}(k)_{m,\lambda}x^{k}=\frac{1}{1-x}W_{m}\bigg{(}\frac{x}{1-x}\bigg{)}.$

It is known that

(44)
$\sum_{k=0}^{\infty}k^{m}x^{k}=\frac{1}{(1-x)^{m+1}}A_{m}(x),\quad(\mathrm{see}\
[5]).$

By (43) and (44), we get

(45)
$\frac{1}{1-x}W_{m}\bigg{(}\frac{x}{1-x}\bigg{)}=\frac{1}{(1-x)^{m+1}}A_{m}(x).$

From (45), we note that

(46) $A_{m}(x)=(1-x)^{m}W_{m}\bigg{(}\frac{x}{1-x}\bigg{)}.$

By (37) and (46), we get

(47) $\displaystyle\sum_{k=0}^{\infty}(k)_{m,\lambda}x^{k}$
$\displaystyle=\frac{1}{1-x}\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}W_{l}\bigg{(}\frac{x}{1-x}\bigg{)}$
$\displaystyle=\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}\frac{1}{(1-x)^{l+1}}A_{l}(x).$

For $g(x)=(1-x)^{-r},\ (r\in\mathbb{N})$, we note that

(48)
$g^{(k)}(x)=\bigg{(}\frac{d}{dx}\bigg{)}^{k}g(x)=r(r+1)\cdots(r+k-1)(1-x)^{-r-k},$

and hence that

(49) $\displaystyle g^{(k)}(0)=r(r+1)\cdots(r+k-1)=\binom{-r}{k}k!(-1)^{k}.$

By Theorem 3, (48) and (49), we get

(50)
$\displaystyle\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}k!\binom{-r}{k}f(k)x^{k}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\sum_{k=0}^{n}S_{2}(n,k)r(r+1)\cdots(r+k-1)\frac{x^{k}}{(1-x)^{r+k}}$
$\displaystyle=\frac{1}{(1-x)^{r}}\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\sum_{k=0}^{n}S_{2}(n,k)\frac{\Gamma(r+k)}{\Gamma(r)}\bigg{(}\frac{x}{1-x}\bigg{)}^{k}$
$\displaystyle=\frac{1}{(1-x)^{r}}\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}W_{n}\bigg{(}\frac{x}{1-x}\
\bigg{|}\ r\bigg{)},$

where

(51) $W_{n}(x|r)=\sum_{k=0}^{n}S_{2}(n,k)\frac{\Gamma(r+k)}{\Gamma(k)}x^{k}.$

For $f(x)=(x)_{m,\lambda}$, we note that, for $n\leq m$,

(52) $\displaystyle\frac{1}{n!}\bigg{(}\frac{d}{dx}\bigg{)}^{n}f(x)$
$\displaystyle=\frac{1}{n!}f^{(n)}(x)=\frac{\lambda^{m}}{n!}\bigg{(}\frac{d}{dx}\bigg{)}^{n}\bigg{(}\frac{x}{\lambda}\bigg{)}_{m}$
$\displaystyle=\frac{1}{n!}\sum_{l=n}^{m}S_{1}(m,l)\lambda^{m-l}(l)_{n}x^{l-n}.$

By (50) and (52), we get

(53)
$\displaystyle\sum_{k=0}^{\infty}\binom{-r}{k}(-1)^{k}(k)_{m,\lambda}x^{k}$
$\displaystyle=\frac{1}{(1-x)^{r}}\sum_{n=0}^{m}\frac{1}{n!}\bigg{(}\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}(l)_{n}\bigg{)}0^{l-n}W_{n}\bigg{(}\frac{x}{1-x}\bigg{|}r\bigg{)}$
$\displaystyle=\frac{1}{(1-x)^{r}}\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}W_{l}\bigg{(}\frac{x}{1-x}\bigg{|}r\bigg{)}.$

From (53), we note that

(54)
$\sum_{k=0}^{\infty}\binom{-r}{k}(-1)^{k}(k)_{m,\lambda}x^{k}=\frac{1}{(1-x)^{r}}\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}W_{l}\bigg{(}\frac{1}{1-x}\bigg{|}r\bigg{)}.$

On the other hand,

(55) $\displaystyle x^{\lambda
m}\big{(}x^{1-\lambda}D\big{)}^{m}\frac{1}{(1-x)^{r}}$
$\displaystyle=x^{\lambda
m}\big{(}x^{1-\lambda}D\big{)}^{m}\sum_{k=0}^{\infty}\binom{-r}{k}(-1)^{k}x^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\binom{-r}{k}(-1)^{k}(k)_{m,\lambda}x^{k}.$

Thus, by (54) and (55), we get

(56) $x^{\lambda
m}\big{(}x^{1-\lambda}D\big{)}^{m}\bigg{(}\frac{1}{1-x}\bigg{)}^{r}=\frac{1}{(1-x)^{r}}\sum_{l=0}^{m}S_{1}(m,l)\lambda^{m-l}W_{l}\bigg{(}\frac{x}{1-x}\bigg{|}r\bigg{)}.$

Now, we observe that

(57) $\displaystyle\frac{1}{e_{\lambda}(t)+1}$
$\displaystyle=\frac{1}{e_{\lambda}(t)-1}-\frac{2}{e_{\lambda}^{2}(t)-1}$
$\displaystyle=\frac{1}{t}\bigg{(}\frac{t}{e_{\lambda}(t)-1}-\frac{2t}{e_{\lambda}^{2}(t)-1}\bigg{)}=\frac{1}{t}\bigg{(}\frac{t}{e_{\lambda}(t)-1}-\frac{2t}{e_{\frac{\lambda}{2}}(2t)-1}\bigg{)}$
$\displaystyle=\frac{1}{t}\sum_{n=0}^{\infty}\big{(}\beta_{n,\lambda}-2^{n}\beta_{n,\frac{\lambda}{2}}\big{)}\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\bigg{(}\frac{\beta_{n+1,\lambda}-2^{n+1}\beta_{n+1,\frac{\lambda}{2}}}{n+1}\bigg{)}\frac{t^{n}}{n!}.$

Taking $x=-\frac{1}{2}$ in (21), we have

(58)
$\sum_{n=0}^{\infty}W_{n,\lambda}\bigg{(}-\frac{1}{2}\bigg{)}\frac{t^{n}}{n!}=\frac{2}{e_{\lambda}(t)+1}=\sum_{n=0}^{\infty}2\bigg{(}\frac{\beta_{n+1,\lambda}-2^{n+1}\beta_{n+1,\frac{\lambda}{2}}}{n+1}\bigg{)}\frac{t^{n}}{n!}.$

By comparing the coefficients on both sides of (58), we get

(59)
$W_{n,\lambda}\bigg{(}-\frac{1}{2}\bigg{)}=2\bigg{(}\frac{\beta_{n+1,\lambda}-2^{n+1}\beta_{n+1,\frac{\lambda}{2}}}{n+1}\bigg{)}.$

From (20), we have

(60)
$W_{n,\lambda}\bigg{(}-\frac{1}{2}\bigg{)}=\sum_{k=0}^{n}S_{2,\lambda}(n,k)k!(-1)^{k}2^{-k}.$

Therefore, by (59) and (60), we obtain the following theorem.

######  Theorem 9.

For $n\geq 0$, we have

$\sum_{k=0}^{n}S_{2,\lambda}(n,k)2^{-k}(-1)^{k}k!=W_{n,\lambda}\bigg{(}-\frac{1}{2}\bigg{)}=\frac{2}{n+1}\big{(}\beta_{n+1,\lambda}-2^{n+1}\beta_{n+1,\frac{\lambda}{2}}\big{)}.$

######  Remark 10.

We observe that

(61)
$\displaystyle\sum_{n=0}^{\infty}\bigg{(}\frac{\beta_{n+1,\lambda}-2^{n+1}\beta_{n+1,\frac{\lambda}{2}}}{n+1}\bigg{)}\frac{t^{n}}{n!}$
$\displaystyle=\frac{1}{e_{\lambda}(t)+1}=\sum_{k=0}^{\infty}(-1)^{k}e_{\lambda}^{k}(t)$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{\infty}(-1)^{k}(k)_{n,\lambda}\bigg{)}\frac{t^{n}}{n!}.$

By comparing the coefficients on both sides of (61), we get

$\sum_{k=0}^{\infty}(-1)^{k}(k)_{n,\lambda}=\frac{1}{n+1}\big{(}\beta_{n+1,\lambda}-2^{n+1}\beta_{n+1,\frac{\lambda}{2}}\big{)}.$

##  3\. Conclusion

It was Carlitz who initiated a study of the degenerate Bernoulli and
degenerate Euler polynomials. In recent years, intensive studies have been
done for degenerate versions of quite a few special polynomials and numbers.
Here we investigated three types of Bell polynomials by using the series
transformation formula proved by Boyadzhiev in [2]. We obtained some
interesting results by applying this key result to various choices of power
series $f$ and $g$. Specifically, we answered to the natural question
regarding the relationship between two different types of degenerate Bell
polynomials. In addition, we expressed the generating function of the sums of
$\lambda$-falling factorials of consecutive nonnegative integers in terms of
the geometric polynomial and the Stirling numbers of the first kind.

It is one of our future research projects to continue this line of research,
namely to study various degenerate versions of many special polynomials and
numbers and their applications to science, physics and engineering.

## References

  * [1] Bell, E. T. _Exponential numbers,_ Amer. Math. Monthly 41 (1934), no. 7, 411-419.
  * [2] Boyadzhiev, K. N. _A series transformation formula and related polynomials,_ Int. J. Math. Math. Sci. 2005, no. 23, 3849-3866.
  * [3] Carlitz, L. _Degenerate Stirling, Bernoulli and Eulerian numbers,_ Utilitas Math. 15 (1979), 51-88.
  * [4] Comtet, L. _Advanced combinatorics,_ D. Reidel, Dordrecht, 1974.
  * [5] Kim, D. S.; Kim, T. _A note on a new type of degenerate Bernoulli numbers,_ Russ. J. Math. Phys. 27 (2020), no. 2, 227-235.
  * [6] Kim, T.; Kim, D. S. _Degenerate zero-truncated Poisson random variables,_ Russ. J. Math. Phys. 28 (2021), no. 1, 66-72.
  * [7] Kim, T.; Kim, D. S. _Degenerate polyexponential functions and degenerate Bell polynomials,_ J. Math. Anal. Appl. 487 (2020), no. 2, 124017, 15 pp.
  * [8] Kim, T.; Kim, D. S. Dolgy, D. V. _On partially degenerate Bell numbers and polynomials,_ Proc. Jangjeon Math. Soc. 20 (2017), no. 3, 337-345.
  * [9] Kim, T.; Kim, D. S.; Lee, H.; Kwon, J. _A note on some identities of new type degenerate Bell polynomials,_ Mathematics 7 (2019), no. 11, 1086.
  * [10] Riordan, J. _Combinatorial identities,_ John Wiley & Sons, New York, 1968.
  * [11] Roman, S. _The umbral calculus,_ Pure and Applied Mathematics 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.
  * [12] Sharma, S. K.; Khan, W. A.; Araci, S.; Ahmed, S. S. _New construction of type 2 degenerate central Fubini polynomials with their certain properties,_ Adv. Difference Equ. 2020, Paper No. 587, 11 pp.


# Water isotopic ratios from a continuously melted ice core sample

V. Gkinis Correspondence to: V<EMAIL_ADDRESS>Centre for Ice and
Climate, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej
30,
2100 Copenhagen, Denmark  T. J. Popp  Centre for Ice and Climate, Niels Bohr
Institute, University of Copenhagen, Juliane Maries Vej 30,
2100 Copenhagen, Denmark  T. Blunier  Centre for Ice and Climate, Niels Bohr
Institute, University of Copenhagen, Juliane Maries Vej 30,
2100 Copenhagen, Denmark  M. Bigler  Physics Institute, Climate and
Environmental Physics and Oeschger Centre for Climate Change Research
University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland  S. Schüpbach
Physics Institute, Climate and Environmental Physics and Oeschger Centre for
Climate Change Research University of Bern, Sidlerstrasse 5, 3012 Bern,
Switzerland  E. Kettner  Centre for Ice and Climate, Niels Bohr Institute,
University of Copenhagen, Juliane Maries Vej 30,
2100 Copenhagen, Denmark  S. J. Johnsen  Centre for Ice and Climate, Niels
Bohr Institute, University of Copenhagen, Juliane Maries Vej 30,
2100 Copenhagen, Denmark  Science Institute, University of Iceland, Dunhaga 3,
107, Iceland

###### Abstract

A new technique for on-line high resolution isotopic analysis of liquid water,
tailored for ice core studies is presented. We built an interface between a
Wavelength Scanned Cavity Ring Down Spectrometer (WS-CRDS) purchased from
Picarro Inc. and a Continuous Flow Analysis (CFA) system. The system offers
the possibility to perform simultaneuous water isotopic analysis of
$\delta^{18}$O and $\delta$D on a continuous stream of liquid water as
generated from a continuously melted ice rod. Injection of sub ${\mu}$l
amounts of liquid water is achieved by pumping sample through a fused silica
capillary and instantaneously vaporizing it with 100 % efficiency in a home
made oven at a temperature of 170 ∘C. A calibration procedure allows for
proper reporting of the data on the VSMOW–SLAP scale. We apply the necessary
corrections based on the assessed performance of the system regarding
instrumental drifts and dependance on the water concentration in the optical
cavity. The melt rates are monitored in order to assign a depth scale to the
measured isotopic profiles. Application of spectral methods yields the
combined uncertainty of the system at below 0.1 ‰ and 0.5 ‰ for $\delta^{18}$O
and $\delta$D, respectively. This performance is comparable to that achieved
with mass spectrometry. Dispersion of the sample in the transfer lines limits
the temporal resolution of the technique. In this work we investigate and
assess these dispersion effects. By using an optimal filtering method we show
how the measured profiles can be corrected for the smoothing effects resulting
from the sample dispersion. Considering the significant advantages the
technique offers, i.e. simultaneuous measurement of $\delta^{18}$O and
$\delta$D, potentially in combination with chemical components that are
traditionally measured on CFA systems, notable reduction on analysis time and
power consumption, we consider it as an alternative to traditional isotope
ratio mass spectrometry with the possibility to be deployed for field ice core
studies. We present data acquired in the field during the 2010 season as part
of the NEEM deep ice core drilling project in North Greenland.

##  I introduction

Polar ice core records provide some of the most detailed views of past
environmental changes up to 800 000 yr before present, in large part via proxy
data such as the water isotopic composition and embedded chemical impurities.
One of the most important features of ice cores as climate archives, is their
continuity and the potential for high temporal resolution. Greenland ice cores
are particularly well suited for high resolution paleoclimatic studies,
because relatively high snow accumulation rates allow seasonal changes in
proxy data to be identified more than 50 000 yr in the past (Johnsen et al.,
1992; NGRIP members, 2004).

The isotopic signature of polar precipitation, commonly expressed through the
$\delta$ notation111Isotopic abundances are typically reported as deviations
of a sample’s isotopic ratio relative to that of a reference water (e.g.
VSMOW) expressed in per mil (‰) through the $\delta$ notation:
$\delta^{i}=\left(\displaystyle{\frac{{}^{i}{R}_{\textrm{sample}}}{{}^{i}{R}_{\textrm{{SMOW}}}}}-1\right)\times
1000$ where
${}^{2}{R}=\displaystyle{\frac{{}^{2}\textrm{H}}{{}^{1}\textrm{H}}}$ and
${}^{18}{R}=\displaystyle{\frac{{{}^{18}O}}{{{}^{16}O}}}$ (Epstein, 1953;
Mook, 2000) is related to the temperature gradient between the evaporation and
condensation site (Dansgaard, 1964) and has so far been used as a proxy for
the temperature of the cloud at the time of condensation (Jouzel and Merlivat,
1984; Jouzel et al., 1997; Johnsen et al., 2001). One step further, the
combined signal of $\delta\mathrm{D}$ and $\delta{{}^{18}\mathrm{O}}$ commonly
referred to as the deuterium excess (hereafter $\mathrm{D_{xs}}$ ),
constitutes a useful paleothermometer tool. Via its high correlation with the
temperature of the evaporation source (Johnsen et al., 1989), it has been used
to resolve issues related to changes in the location of the evaporation site
(Cuffey and Vimeux, 2001; Kavanaugh and Cuffey, 2002). A relatively recent
advance in the use of water isotope ratios as a direct proxy of firn
temperatures, has been introduced by Johnsen et al. (2000). Assessment of the
diffusivity of the water isotopologues in the porous medium of the firn column
can yield a temperature history, provided a dating model is available.

The measurement of water stable isotopic composition is typically performed
off-line via discrete sampling with traditional isotope ratio mass
spectrometry (hereafter IRMS). While high precision and accuracy can routinely
be achieved with IRMS systems, water isotope analysis remains an elaborate
process, which is demanding in terms of sample preparation, power consumption,
sample size, consumables, isotope standards and carrier gases. The analysis of
a deep ice core at its full length in high resolution (typically 2.5 to 5 cm
per sample) requires the processing of a vast amount of water samples and can
take years to complete. Additionally, these procedures often come at the
expense of not fully exploiting the temporal resolution available in the ice
core.

Laser spectroscopy in the near and mid infrared region has been demonstrated
as a potential alternative for water isotope analysis, presenting numerous
advantages over IRMS (Kerstel et al., 1999; Kerstel, 2004). A major advantage
of the technique is the ability to directly inject the sampled water vapour in
the optical cavity of the spectrometer where both isotopic ratios
${}^{18}\mathrm{O}/^{16}\mathrm{O}$ and ${}^{2}\mathrm{H}/^{1}\mathrm{H}$ are
measured simultaneuously. In contrast, in the most common IRMS techniques
water is not measured as such, but has to be converted to a different gas
prior to measurement. For $\delta{{}^{18}\mathrm{O}}$ analysis, the
$\mathrm{CO}_{2}$ equilibration method (Epstein, 1953) has been widely used,
whereas $\delta\mathrm{D}$ analysis commonly involves the reduction of water
to hydrogen gas over hot uranium (Bigeleisen et al., 1952; Vaughn et al.,
1998; Huber and Leuenberger, 2003), or chromium (Ghere et al., 1996). However,
the combined use of these two methods rules out simultaneous analysis of both
water isotopologues on a given sample. More recently, in combination with the
use of continuous flow mass spectrometers, conversion of water to CO and
$\text{H}_{2}$ is performed in a pyrolysis furnice (Begley and Scrimgeour,
1997) and allows simultaneous $\delta\mathrm{D}$ and
$\delta{{}^{18}\mathrm{O}}$ measurement, but still on a single discrete
sample. One of the drawbacks of this technique is the interference of
$\mathrm{NO}$ , formed at the ion source by the reaction of $\text{N}_{2}$ and
$\text{O}_{2}$ with the CO signal at m/z $=$ 30 (Accoe, 2008). Nowadays,
commercial IR spectrometers are available with a precision comparable to IRMS
systems (Lis et al., 2008; Brand et al., 2009). These units typically receive
a continuous stream of water vapor and offer ease of use and portability.

The analysis of another set of ice core proxies, that of chemical impurities,
has similarly been an elaborate process, traditionally performed with liquid
chromatography techniques. With the advent of Continuous Flow Analysis
(heareafter CFA) from continuously melted ice core segments, the measurement
of chemical impurities has reached the point of largely exploiting the high
resolution available in the core while it is often performed in the field
(Sigg et al., 1994; Röthlisberger et al., 2000; Kaufmann et al., 2008). The
continuous, on-line nature of the technique has resulted in a considerable
reduction in sample preparation and processing times. Recently, Schüpbach et
al. (2009) demonstrated the measurement of $\mathrm{CH}_{4}$ mixing ratios in
an on-line semi continuous mode with the use of a gas chromatograph combined
with a pulsed discharge and a thermal conductivity detector.

Here, we demonstrate the ability to perform continuous measurements of water
isotope ratios from a stream of water vapor derived from a continuously
melting ice rod by coupling a commercial IR spectrometer to a CFA system via a
passive, low volume flash evaporation module. In the following, we assess the
system’s precision, accuracy, and efficient calibration. We then comment on
issues related to sample dispersion in the sample transfer lines, the
evaporation module and the optical cavity of the spectrometer itself in order
to determine the expected smoothing imposed on the acquired data sets.
Finally, isotopic analysis of ice core samples from the NEEM deep ice core are
presented and compared to measurements performed in discrete mode.

##  II Experimental

###  II-A Continuous flow analysis

In the system described here, (Fig. 1) an ice rod measuring 3.2 $\times$ 3.2
$\times$ 110 cm (hereafter CFA run) is continuously melted on a copper, gold-
nickel coated melter at a regulated temperature of 20 ∘C. The concentric
arrangement of the melter’s surface facilitates the separation of the sample
that originates from the outer and inner part of the core. Approximately 90 %
of the sample from the inner part is transfered to the analytical system by
means of a peristaltic pump with a flow rate of 16 ml min-1. This
configuration provides an overflow of $\approx$10 % from the inner to the
outer part of the melter and ensures that the water sample that is introduced
into the analytical system is not contaminated.

A stainless steel weight sitting on top of the ice rod enhances the stability
and continuity of the melting process. An optical encoder connected to the
stainless steel weight, records the displacement of the rod. This information
is used to accurately define the depth scale of the produced water isotope
data. Breaks in the ice rod are logged prior to the melting process and
accounted for, during the data analysis procedure.

Gases included in the water stream originating from the air bubbles in the ice
core are extracted in a sealed debubbler, with a volume of $\approx$300
$\mu$l. The melt rate of the present system is approximately 3.2 cm min-1,
thus resulting in an analysis time of $\approx$35 min per CFA run. During the
intervals between CFA runs, mQ222filtered and deionized water with a
resistivity more than $18.2~{}\mathrm{M\Omega\,\,\,cm}$ and a total organic
content less than 10 ppb. water is pumped through the system. A 4-port
injection valve (V1 in Fig. 1) allows the selection between the mQ and sample
water. The mQ water is spiked with isotopically enriched water containing 99.8
atom % deuterium, Cortecnet Inc.) in a mixing ratio of $\approx$1 ppm. In this
way a distinction between sample and mQ water is possible, facilitating the
identification of the beginning and end times of a CFA run.

For further details on the analysis of chemical components or the extraction
of gases for greenhouse gas measurements the reader is refered to Kaufmann et
al. (2008) and Schüpbach et al. (2009).

###  II-B The water isotope measurement

We follow the same approach as previously presented in Gkinis et al. (2010) by
coupling a commercially available Cavity Ring Down IR spectrometer (hereafter
WS-CRDS) purchasecd from Picarro Inc. (Picarro L1102-i) (Crosson, 2008). The
spectrometer operates with a gas flow rate of 30 standard ml min-1. In the
optical cavity the pressure is regulated at 47 mbar with two proportional
valves in a feedback loop configuration up- and down-stream of the optical
cavity at a temperature of 80 ∘C. The high signal to noise ratio achieved with
the Cavity Ring Down configuration in combination with fine control of the
environmental parameters of the spectrometer, result in a performance
comperable to modern mass spectrometry systems taylored for water stable
isotope analysis.

A 6-port injection valve (V2 in Fig. 1) selects sample from the CFA line or a
set of local water standards. The isotopic composition of the local water
standards is determined with conventional IRMS and reported with respect to
VSMOW standard. A 6-port selection valve (V3 in Fig. 1) is used for the switch
between different water standards. A peristaltic pump (P3 in Fig. 1) in this
line with variable speeds, allows adjustment of the water vapor concentration
in the spectrometer’s optical cavity, by varying the pump speed. In that way,
the system’s sensitivity to levels of different water concentration can be
investigated and a calibration procedure can be implemented. We use high
purity Perfluoroalkoxy (PFA) tubing for all sample transfer lines.

Injection of water sample into the evaporation oven takes place via a
$\varnothing$ 40 $\mu$m fused silica capillary where immediate and 100 %
evaporation takes place avoiding any fractionation effects. The setpoint of
the evaporation temperature is set to 170 ∘C and is regulated with a PID
controller. The amount of the injected water to the oven can be adjusted by
the pressure gradient maintained between the inlet and waste ports of the T1
tee-split (Fig. 1). The latter depends on the ratio of the inner diameters of
the tubes connected to the two ports as well as the length of the waste line.
The total water sample consumption is $\approx$0.1 ml min-1 maintained by the
peristaltic pump P2 (Fig. 1). For a detailed description of the sample
preparation and evaporation module the reader may reffer to Gkinis et al.
(2010). A smooth and undisturbed sample delivery to the spectrometer at the
level of $\approx$20 000 ppm results in optimum performance of the system.
Fluctuations of the sample flow caused by air bubbles or impurities are likely
to result in a deteriorated performance of the measurement and are
occasionally observed as extreme outliers on both $\delta{{}^{18}\mathrm{O}}$
and $\delta\mathrm{D}$ measurements. The processes that control the occurence
of these events are still not well understood.

##  III Results and discussion – from raw data to isotope records

In this study we present data collected in the framework of the NEEM ice core
drilling project. Measurements were carried out in the field during the 2010
field season and span 919.05 m of ice core (depth interval 1281.5–2200.55 m).
Here we exemplify the performance of the system over a section of ice from the
depth interval 1382.152–1398.607. The age of this section spans $\approx$411
yr with a mean age of 10.9 ka b2k 333thousand years before 2000 AD (ka b2k).
The reported age is based on a preliminary time scale constructed by
stratigraphic transfer of the GICC05 time scale (Rasmussen et al., 2006) from
the NGRIP to the NEEM ice core.

In Fig. 2 we present an example of raw data as acquired by the system. This
data set covers 7 CFA runs (7.70 m of ice). A clear baseline of the
isotopically heavier mQ water can be seen in between CFA runs. At $t=1.9\times
10{{}^{4}}$ s one can observe a sudden drop in the signal of the water
concentration due to a scheduled change of the mQ water tank. Adjacent to
this, both $\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ signals present
a clear spike, characteristic of the sensitivity of the system to the
stability of the sample flow rates.

###  III-A VSMOW – water concentration calibrations

Before any further processing we correct the acquired data for fluctuations of
the water concentration in the optical cavity. To a good approximation the
$\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ signals show a linear
response to differences in water concentrations around 20 000 ppmv (Brand et
al., 2009; Gkinis et al., 2010). A correction is performed as:

$\Delta\delta=\alpha(R_{20}-1){}$ (1)

Here $R_{20}=\frac{[\mathrm{H}_{2}\mathrm{O}]}{20\,000}$, $\alpha_{18}=1.94$ ‰
and $\alpha_{\rm D}=3.77$ ‰ as estimated in Gkinis et al. (2010). The
estimation of these values has been performed several times during the period
July 2009–October 2011. Based on compiled data from 6 calibrations we can
report that these values do not appear to be drifting in the course of the two
years. The values reported here are the ones estimated chronologically closer
to the measurements we present here. These values are most likely instrument
specific and should be used with caution for other analyzers. We typically
operate the system in the area of 17 000–22 000 ppmv in which we observe no
impact of the water concentration level on the precision of the isotopic
signal. The mean and standard deviation of the water concentration signal in
the course of aprroximately 7 h as seen in Fig. 2 is $19\;939\pm 306$ ppmv.
The water concentration correction is applied to the raw data, scaling all the
isotopic values to the level of 20 000 ppmv.

Raw data are expressed in per mil values for both $\delta{{}^{18}\mathrm{O}}$
and $\delta\mathrm{D}$ and ppmv for the water vapour concentration. These
values are based on the slope and intercept values of the instrument’s stored
internal calibration line. Due to apparent instrumental drifts though, the
latter are expected to deviate with time. To overcome this problem we perform
frequent VSMOW calibrations by using 3 local water standards with well known
$\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ values measured by
conventional Isotope Ratio Mass Spectrometry combined with a pyrolysis glassy
carbon reactor (Thermo DeltaV–TC/EA). The water standards are transported and
stored in the field using the necessary precautions to avoid evaporation. We
used amber glass bottles with silicon sealed caps. Two of the water standards
are used to calculate the slope and the intercept of the calibration line and
the third is used for a check of the linearity and the accuracy. The frequency
of the VSMOW calibrations depends on the work flow of the overall CFA system.
For the particular section we present here a VSMOW calibration was performed
the same day.

In Fig. 3 we illustrate this calibration. Based on the measured and real
values of the standards “$-$22” and “$-$40” the measurement of the “NEEM”
standard can be used as a test for accuracy and linearity. In Table I we
present the VSMOW calibrated values of the water standards as estimated with
the IRMS system. The slope and intercept of the calibration lines are [1.002,
$-$0.007] for $\delta{{}^{18}\mathrm{O}}$ and [0.963, $-$8.214] for
$\delta\mathrm{D}$ . Based on these values one can calculate the isotopic
composition of the “NEEM” standard. The results we obtain are $-33$.4 ‰ and
$-256$.99 ‰ for $\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$
respectively. This results in a difference of $0.04$ ‰ ($0.$31 ‰) for
$\delta{{}^{18}\mathrm{O}}$ ($\delta\mathrm{D}$ ), giving an indication about
the accuracy obtained by the system.

TABLE I: Isotopic composition of the standards used for the VSMOW calibrations
in ‰.
|  $-$22 | NEEM |  $-$40
---|---|---|---
$\delta{{}^{18}\mathrm{O}}$ |  $-$21.9 |  $-$33.44 |  $-$39.97
$\delta\mathrm{D}$ |  $-$168.4 |  $-$257.3 |  $-$310

###  III-B The depth scale

The melting process is recorded by an optical encoder connected to the top of
the stainless steel weight that lies on top of the ice rod. The data acquired
by the optical encoder allow for a conversion of the measurement time scale to
a depth scale. In order to locate the beginning and end of every run we take
advantage of the isotopic step observed during the transition between mQ
baseline and sample water. A smoothed version of the discrete derivative of
the acquired isotope data for both $\delta{{}^{18}\mathrm{O}}$ and
$\delta\mathrm{D}$ reveals a local minimum (maximum) for the beginning (end)
of the measurement (Fig. 4). The logged depth of the top and the bottom of the
CFA run is assigned to these points. Data that lie in the transition interval
between mQ and sample water are manually removed from the series. Additional
breaks within a CFA run that can possibly be created during the drilling or
processing phase of the ice core, are taken into account at the last stage of
the data analysis. If necessary and depending on their size, the gaps can be
filled by means of some interpolation technique. Here, due to the small size
of the gaps we use a linear interpolation scheme. The use of more advanced
methods is also possible but is out of the scope of this work. The processed
profiles presented in Fig. 4 are reported with a nominal resolution of 5 mm.
The interpolated sections are highlighted with gray bars. Their width
indicates the length of the gaps.

###  III-C Noise level – comparison with discrete data

An estimate of the noise level of the measurements can be obtained from the
appropriately normalized power spectral density of the time series. Using the
$\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ data of the section under
consideration, we implement an autoregressive spectral estimation method
developed by Burg (1975) by the use of the algorithm introduced by Andersen
(1974). The order of the autoregressive model is $M$ $=$ 300\. The standard
deviation of the time series will be defined as:

$\sigma^{2}=\int_{-f_{c}}^{f_{c}}|\hat{\eta}(f)|^{2}df{}$ (2)

where the Nyquist frequency is $f_{c}=100$ cycles m-1 and
$|\hat{\eta}(f)|^{2}$ can be obtained by a linear fit on the flat high
frequency part of the spectrum (Fig. 5). By performing this analysis we obtain
$\sigma_{18}=0.055$ ‰ and $\sigma_{\rm D}=0.21$ ‰.

In order to validate the quality of the calibrations as well as the estimated
depth scale we compare the CFA data with measurements performed in a discrete
fashion using the same WS-CRDS spectrometer in combination with a sample
preparation evaporator system (Gupta et al., 2009) and an autosampler. The
discrete samples are cut in a resolution of 5 cm. The sample injection
sequence takes into account apparent memory effects and results are reported
on the VSMOW scale by appropriate calibration using local water standards. The
results are illustrated in Fig. 6 for $\delta{{}^{18}\mathrm{O}}$ and
$\delta\mathrm{D}$ . The comparison of the data sets demonstrates validity of
the followed calibration procedures. The benefits of the technique in terms of
achieved resolution can be seen when one compares the two datasets over
isotopic cycles with relatively small amplitude and higher frequency. Such an
example can be seen at the depth of 1390.5 m where a sequence of 4 cycles is
sampled relatively poorly with the discrete method when compared to the on-
line system. This performance can benefit studies that look into the spectral
properties of the signals by providing better statistics for the obtained
measurements.

###  III-D Obtained resolution – diffusive sample mixing

One of the advantages of the combined CFA–CRDS technique for water isotopic
analysis of ice cores lies in the potential for higher resolution measurements
relative to discrete sampling. However, diffusion effects in both the liquid
and the vapor phase are expected to attenuate the obtained resolution.

Attenuation of the initial signal of the precipitation occurs also via a
combination of in situ processes that take place after deposition. The porous
medium of the firn column allows for an exchange of water molecules in the gas
phase along the isotopic gradients of the profile. For the case of polar
sites, this process has been studied extensively (Johnsen, 1977; Whillans and
Grootes, 1985) and can be well described and quantified provided that a good
estimate of the diffusivity coefficient and a strain rate history of the ice
core site are available (Johnsen et al., 2000). The process ceases when the
porous medium is closed-off and the diffusivity of air reaches zero at a
density of $\approx$804 kg m-3. Deeper in the ice, diffusion within the ice
crystals takes place via a process that is considerably slower when compared
with the firn diffusion. At a temperature of $-$30 ∘C the diffusivity
coefficients of these two processes differ by 4 orders of magnitude (Johnsen
et al., 2000).

Assuming an isotopic signal ${\delta}_{\rm pr}$ for the precipitation, the
total effect of the diffusive processes, in-situ and experimental, can be seen
as the convolution of ${\delta}_{\rm pr}(z)$ with a smoothing filter
${\mathcal{G}}_{\rm tot}$.

$\delta_{\rm m}(z)=\int_{-\infty}^{\infty}{\delta}_{\rm
pr}(\tau){\mathcal{G}}_{\rm tot}(z-\tau)d\tau=[\delta_{\rm
pr}\ast{\mathcal{G}}_{\rm tot}](z){}$ (3)

where $\delta_{\rm m}(z)$ is the measured signal and ($\ast$) denotes the
convolution operation. Since instrumental and in-situ firn-ice diffusion are
statistically independent, the variance of the total smoothing filter is the
sum of the variances of the in-situ and experimental smoothing filters
(hereafter $\mathcal{G}_{\rm firn}$, $\sigma_{\rm firn}$, $\mathcal{G}_{\rm
cfa}$, $\sigma_{\rm cfa}$).

$\sigma_{\rm tot}^{2}=\sigma_{\rm firn}^{2}+\sigma_{\rm cfa}^{2}{}$ (4)

It can be seen that any attempt to study firn and ice diffusion by means of
ice core data obtained with an on-line method similar to the one we present
here, requires a good assesment of the diffusive properties of the
experimental system. The latter is possible if one is able to estimate the
variance of the smoothing filter $\mathcal{G}_{\rm cfa}$ expressed by the
variance $\sigma_{\rm cfa}^{2}$ (hereafter diffusion length).

One way to approach this problem is to measure the response of the system to a
step function. Ideally, in the case of zero diffusion, a switch between two
isotopic levels would be described by a scaled and shifted version of the the
Heaviside unit step function as:

$\delta_{\rm H}(z)=\left\\{\begin{array}[]{ll}C_{2}&\qquad z<0\\\
C_{1}H(z)+C_{2}&\qquad z\geq 0\end{array}\right.{}$ (5)

where the isotopic shift takes place at $z=0$, $H(z)$ is the Heaviside unit
step function and $C_{1}$ and $C_{2}$ refer to the amplitude and base line
level of the isotopic step. Convolution of the signal of Eq. (5) with
$\mathcal{G}_{\rm cfa}$ and subsequent calculation of the derivative yields,

$\frac{d\delta_{\rm m}}{dz}=\frac{d\delta_{\rm H}}{dz}\ast\mathcal{G}_{\rm
cfa}=C_{1}\frac{dH}{dt}\ast\mathcal{G}_{\rm cfa}=C_{1}\delta_{\rm
Dirac}\ast\mathcal{G}_{\rm cfa}{}$ (6)

Thus the derivative of the measured signal, properly normalized, equals the
impulse respone of the system. Applying the Fourier transform, denoted by the
overhead hat symbol, in Eq. (6), and by using the convolution theorem, we
deduce the transfer function $\hat{\mathcal{G}}_{\rm cfa}$ of the system:

$\widehat{\frac{d\delta_{\rm m}}{dz}}=C_{1}\hat{\delta}_{\rm
Dirac}\cdot\hat{\mathcal{G}}_{\rm cfa}=C_{1}\cdot\hat{\mathcal{G}}_{\rm
cfa}{}$ (7)

In the case of the system presented here, an isotopic transition can be
observed when the main CFA valve (V1 in Fig. 1) switches between mQ water and
sample at the beginning and the end of each CFA run as shown in Fig. 4. By
using these transitions we are able to construct isotopic steps and estimate
the impulse response of the system. Such an isotopic step is illustrated in
Fig. 7a. We fit the data of Fig. 7a with a scaled version of the cumulative
distribution function of a normal distribution described as

$\delta_{\rm model}(z)=\frac{C_{1}^{\prime}}{2}\left[1+{\rm
erf}\left(\frac{z-z_{0}}{\sigma_{\rm
step}\sqrt{2}}\right)\right]+C_{2}^{\prime}{}$ (8)

The values of $C_{1}^{\prime}$, $C_{2}^{\prime}$, $z_{0}$ and $\sigma_{\rm
step}$ are estimated by means of a least square optimization and used
accordingly to normalize the length scale and the isotopic values of the step.
A nominal melt rate of 3.2 cm min-1 is used for all the calculations presented
here. We focus our analysis on the $\delta\mathrm{D}$ signal. The same
approach can be followed for $\delta{{}^{18}\mathrm{O}}$ . In Fig. 7b we
present the calculated impulse response of the system. The latter can be well
approximated by a Gaussian type filter described as:

$\mathcal{G}_{\rm cfa}(z)=\frac{1}{\sigma_{\rm
cfa}\sqrt{2\pi}}{e}^{-\frac{z^{2}}{2\sigma_{\rm cfa}^{2}}}{}$ (9)

The diffusion length term $\sigma_{\rm cfa}$ is equal to $13.4\pm 0.17$ mm
($1\sigma$) as calculated with the least squares optimization. The transfer
function for this filter will be given by its Fourier transform, which is
itself a Gaussian and is equal to (Abramowitz and Stegun, 1964):

$\displaystyle\mathfrak{F}[\mathcal{G}_{\rm cfa}(z)]=\hat{\mathcal{G}}_{\rm
cfa}$ $\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{1}{\sigma_{\rm
cfa}\sqrt{2\pi}}{e}^{-\frac{z^{2}}{2\sigma_{\rm
cfa}^{2}}}{e}^{-ikz}dk={e}^{\frac{-k^{2}\sigma_{\rm cfa}^{2}}{2}}$ (10)

where $k=\frac{2\pi}{\lambda}$ and $\lambda$ is the wavelength444Here the term
wavelength refers to the isotopic signal in the ice and should not be confused
with the wavelength of the light emitted by the laser diode of the
spectrometer. of a harmonic of the isotopic signal. Harmonics with an initial
amplitude $A_{0}$ and wavenumber $k$ will be attenuated to a final amplitude
equal to:

$A=A_{0}{e}^{\frac{-k^{2}\sigma_{\rm cfa}^{2}}{2}}{}$ (11)

An estimate of the transfer function based on the data and the cumulative
distribution model is presented in Fig. 8 (blue and pink curve, respectively).
As seen in this plot, cycles with wavelengths longer than 25 cm experience
negligible attenuation, whereas cycles with a wavelength of 7 cm are
attenuated by $\approx 50$ %.

The step response approach has been followed in the past for on-line chemistry
data. In some studies such as Sigg et al. (1994) and Rasmussen et al. (2005),
the resolution of the experimental system was assessed via the estimation of
the transfer function. In other studies (Röthlisberger et al., 2000; Kaufmann
et al., 2008), the characteristic time in which a step reaches a certain level
(typically $1/e$) with respect to its final value, is used as a measure of the
obtained resolution of the system. A common weakness of this approach as
applied in the current, as well as previous studies, is that it is based on
the analysis of a step that is introduced in the analytical system by
switching a valve that is typically situated downstream of the melting and the
debubbling system. Consequently, the impact of these last two elements on the
smoothing of the obtained signals is neglected. In this study, this is the
valve V1 in Fig. 1.

To overcome this problem we will present here an alternative way, based on the
comparison of the spectral properties of the on-line CFA data and the off-line
discrete data in 5 cm sampling resolution, presented in Sect. 3.2. In this
approach the diffusion length of the total smothing filter for the off-line
discrete analysis will be:

$\sigma_{\rm off}^{2}=\sigma_{\rm firn}^{2}+\sigma_{\rm 5\,cm}^{2}{}$ (12)

where $\sigma_{\rm 5\,cm}^{2}$ is the diffusion length of the smoothing
imposed by the sample cutting scheme at a 5 cm resolution. If one averages the
on-line CFA data at a 5 cm resolution by means of a running mean filter, the
diffusion length of the total smoothing filter for the on-line CFA
measurements averaged on a 5 cm resolution will be:

$\sigma_{\rm on}^{2}=\sigma_{\rm firn}^{2}+\sigma_{\rm 5\,cm}^{2}+\sigma_{\rm
cfa}^{2}{}$ (13)

From Eqs. (12) and (13) we get:

$\sigma_{\rm cfa}^{2}=\sigma_{\rm on}^{2}-\sigma_{\rm off}^{2}{}$ (14)

As a result, the term $\sigma_{\rm on}^{2}-\sigma_{\rm off}^{2}$ is directly
related to the diffusion length of the smoothing filter of the whole CFA-water
isotope system including the melting and debubbling sections. Based on Eq.
(11), the power spectral density of the signals will be:

$P=P_{0}{e}^{-k^{2}\sigma^{2}}{}$ (15)

where $\sigma^{2}$ refers in this case to $\sigma_{\rm on}^{2}$ or
$\sigma_{\rm off}^{2}$. Combining the power spectral densities of the on-line
and off-line time series we finally get:

$\ln\left(\frac{P_{\rm off}}{P_{\rm on}}\right)=\ln\left(\frac{P_{\rm
0off}}{P_{\rm 0on}}\right)+\sigma_{\rm cfa}^{2}k^{2}{}$ (16)

Hence, the logarithm of the ratio $P_{\rm off}/P_{\rm on}$ is linearly related
to $k^{2}$ with a slope equal to $\sigma_{\rm cfa}^{2}$. In Fig. 9 we perform
this analysis for $\delta\mathrm{D}$ and by applying a linear fit we calculate
the $\sigma_{\rm cfa}[D]$ to be equal to $16.4\pm 2.4$ mm. In a similar manner
$\sigma_{\rm cfa}[O18]$ is found to be equal to $16.8\pm 2.3$ mm.

The higher value calculated with the spectral method points to the additional
diffusion of the sample at the melter and debubbler system that could not be
considered in the analysis based on the step response. The impulse response of
the system based on the updated value of $\sigma_{\rm cfa}^{2}$ is presented
in Fig. 6.

###  III-E Optimal filtering

In the ideal case of a noise-free measured signal ${\delta^{\prime}_{\rm
m}}(z)$ and provided that the transfer function $\hat{\mathcal{G}}_{\rm cfa}$
is known, one can reconstruct the initial isotopic signal $\delta_{\rm i}(z)$
from Eq. (3) as:

$\delta_{\rm
i}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\hat{\delta}^{\prime}_{\rm
m}(k)}{\hat{\mathcal{G}}(k)}e^{ikz}dk{}$ (17)

where the integral operation denotes the inverse Fourier transform and
$k=\frac{2\pi}{\lambda}$ with $\lambda$ being the wavelength of the isotopic
signals. In the presence of measurement noise $\eta(z)$, this approach will
fail due to excess amplification of the high frequency noise channels in the
spectrum of the signal.

Hereby we use the Wiener approach in deconvoluting the acquired isotopic
signals for the diffusion that takes place during the measurement. Considering
a measured isotopic signal

$\delta_{\rm m}(z)={\delta^{\prime}_{\rm m}}(z)+\eta(z){}$ (18)

an optimal filter $\varphi(z)$ can be constructed that when used at the
deconvolution step, it results in an estimate of the initial isotopic signal
described as:

$\tilde{\delta}_{i}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\hat{\delta}_{\rm
m}(k)}{\hat{\mathcal{G}}(k)}\hat{\varphi}(k)e^{ikz}dk{}$ (19)

Assuming that ${\delta^{\prime}_{\rm m}}(z)$ and $\eta(z)$ are uncorrelated
signals, the optimal filter is given by:

$\hat{\varphi}(k)=\frac{|\hat{\delta}_{\rm
m}^{\prime}(k)|^{2}}{|\hat{\delta}_{\rm
m}^{\prime}(k)|^{2}+|\hat{\eta}(k)|^{2}}{}$ (20)

(Wiener, 1949); where $|\hat{\delta}_{\rm m}^{\prime}(k)|^{2}$ and
$|\hat{\eta}(k)|^{2}$ are the power spectral densities of the signals
$\delta_{\rm m}^{\prime}(z)$ and $\eta(z)$.

In the same fashion as in the previous section we assume that the spectrum of
the noise free measured signal $|\hat{\delta}_{\rm m}^{\prime}(k)|^{2}$, is
described by Eq. (15) where $\sigma^{2}=\sigma_{\rm tot}^{2}$. Regarding the
noise, we assume red noise described by an AR1 process. The spectrum of the
noise signal will then be described by (Kay and Marple, 1981):

$|\hat{\eta}(k)|^{2}=\frac{\sigma_{\eta}^{2}\Delta
z}{\left|1+a_{1}\exp{\left(-ik\Delta z\right)}\right|^{2}}{}$ (21)

where $\sigma_{\eta}^{2}$ is the variance of the noise and $a_{1}$ is the
coefficient of the AR1 process. We vary the parameters $\sigma_{\rm tot}^{2}$,
$P_{0}$, $\sigma_{\eta}^{2}$ and $a_{1}$ so that the sum $|\hat{\delta}_{\rm
m}(k)|^{2}=|\hat{\delta}_{\rm m}^{\prime}(k)|^{2}+|\hat{\eta}(k)|^{2}$ fits
the spectrum of the measured signal. The set of parameters that results in the
optimum fit is used to calculate the optimal filter.

The constructed filters together with the transfer functions that were
calculated based on the two different techniques outlined in Sect. III-D are
illustrated in Fig. 8. One can observe how the restoration filters work by
amplifying cycles with wavelengths as low as 7 mm. Beyond that point, the
shape of the optimal filter attenuates cycles with higher frequency, which lie
in the area of noise. An example of deconvoluted $\delta\mathrm{D}$ data
section is given in Fig. 10. It can be seen that the effect of the optimal
filtering results in both the amplification of the signals that are damped due
to the instrumental diffusion, as well as in the filtering of the measurement
noise.

###  III-F Information on deuterium excess

Combining $\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ gives the
deuterium excess as $\mathrm{D_{xs}}$ $=$ $\delta\mathrm{D}$ $-$
8$\delta{{}^{18}\mathrm{O}}$ (Craig et al., 1963; Mook, 2000). The noise level
of the $\mathrm{D_{xs}}$ signal can be calculated by the estimated noise
levels of $\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ as:

${\sigma}_{\mathrm{D_{xs}}}=\sqrt{\sigma_{\rm
D}^{2}+64\cdot{\sigma}_{18}^{2}}=0.48\,\,\,\mbox{\permil}$ (22)

As seen in Fig. 11, the $\mathrm{D_{xs}}$ signal presents a low signal to
noise ratio. In this case, the technique of optimal filtering can effectively
attenuate unwanted high frequency noise components, thus reveiling a “clean”
$\mathrm{D_{xs}}$ signal.

The latter offers the possibility for the study of abrubt transitions as they
have previously been investigated in $\delta{{}^{18}\mathrm{O}}$ ,
$\delta\mathrm{D}$ and $\mathrm{D_{xs}}$ time series from discrete high
resolution samples (Steffensen et al., 2008). The on-line fashion in which
these measurements are performed has the potential to yield not only higher
temporal resolution but also better statistics for those climatic transitions.

##  IV Conclusions

[Summary and conclusions]

We have succesfully demonstrated the possibility for on-line water isotopic
analysis on a continuously melted ice core sample. We used an infrared laser
spectrometer in a cavity ring down configuration in combination with a
continuous flow melter system. A custom made continuous stream flash
evaporator served as the sample preparation unit, interfacing the laser
spectrometer to the melter system.

Local water standards have been used in order to calibrate the measurements to
the VSMOW scale. Additionally, dependencies related to the sample size in the
optical cavity have been accounted for. The melting procedure is recorded by
an optical encoder that provides the necessary information for assigning a
depth scale to the isotope measurements. We verified the validity of the
applied calibrations and the calculated depth scale by comparing the CFA
measurements with measurements performed on discrete samples in 5 cm
resolution.

By means of spectral methods we provide an estimate of the noise level of the
measurements. The combined uncertainty of the measurement is estimated at
$\approx$0.06, 0.2, and 0.5 ‰ for $\delta{{}^{18}\mathrm{O}}$ ,
$\delta\mathrm{D}$ and $\mathrm{D_{xs}}$ , respectively. This performance is
comparable to, or better than the performance typically achieved with
conventional IRMS systems in a discrete mode.

Based on the isotopic step at the beginning of each CFA run, the impulse
response, as well as the transfer function of the system can be estimated. We
show how this method does not take into account the whole CFA system, thus
underestimating the sample diffusion that takes place from the melter until
the optical cavity of the spectrometer. We proposed a different method that
considers the power spectrum of the CFA data in combination with the spectrum
of a data set over the same depth interval measured in a discrete off-line
fashion. The use of the optimal filtering deconvolution technique, provides a
way to deconvolute the measured isotopic profiles for apparent sample
dispersion effects.

The combination of infrared spectroscopy on gaseuous samples with continuous
flow melter systems provides new possibilities for ice core science.The non
destructive, continuous, and on-line technique offers the possibility for
analysis of multiple species on the same sample in high resolution and
precision and can potentially be performed in the ﬁeld.

## Acknowledgements

We would like to thank Dorthe Dahl Jensen for supporting our research.
Numerous drillers, core processors and general field assistants have
contributed to the NEEM ice core drilling project with weeks of intensive
field work. Withought this collective effort, the measurements we present here
would not be possible. Bruce Vaughn and James White have contributed to this
project with valuable comments and ideas. This project was partly funded by
the Marie Curie Research Training Network for Ice Sheet and Climate Evolution
(MRTN-CT-2006-036127).

Edited by: P. Werle

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Figure 1: Block diagram of the CFA-CRDS system. Figure 2: Raw signals spanning
7 CFA runs on 29 May 2010. Figure 3: VSMOW calibrations with three water
standards “$-$22”, “$-$40” and “NEEM”; Left – $\delta\mathrm{D}$ , right –
$\delta{{}^{18}\mathrm{O}}$ . Figure 4: The beginning and end of each CFA run
is determined by the extrema of the 1st derivative of the isotopic signal,
presented on the top graph. With gray bars we indicate the position and width
of sections with data that are missing due to breaks in the ice, or removed in
order to account for the transition from mQ water to sample and vice cersa.
Figure 5: $\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ power spectral
density based on the data of the interval 1382.152–1398.607 m. Figure 6:
Comparison CRDS-CFA with 5 cm discrete samples for $\delta\mathrm{D}$ (top)
and $\delta{{}^{18}\mathrm{O}}$ (bottom). Bars indicate the position and width
of sections with missing/removed data. Figure 7: (a) Isotopic
$\delta\mathrm{D}$ step. The length scale is normalized so normalized length
$=$ 0 when the normalized $\delta\mathrm{D}$ value equals 0.5 ‰. (b) Impulse
Response of the system for $\delta\mathrm{D}$ based on the step response (red)
and the spectral analysis (blue) with $\sigma_{\rm cfa}=13.4$ and 16.4 mm,
respectively. Figure 8: (a) Power spectral density of $\delta\mathrm{D}$ . (b)
Transfer function calculated based on the step response with $\sigma_{\rm
cfa}=13.4$ mm (pink – squares) and the comparison between discrete and CFA
analysis with $\sigma_{\rm cfa}=16.4$ mm (green – triangles). Restoration
filters built considering the two different transfer functions are illustrated
with orange circles ($\sigma_{\rm cfa}=16.4$ mm) and black diamonds
($\sigma_{\rm cfa}=13.4$ mm). Figure 9: Calculation of the diffusion length
for the transfer function of the CFA system. The dashed and solid lines
represent, respectively the power spectral density of the offline discrete
data and the CFA data averaged on a 5 cm resolution. Figure 10:
$\delta\mathrm{D}$ signal before and after optimal filtering. Figure 11:
$\delta{{}^{18}\mathrm{O}}$ , $\delta\mathrm{D}$ and $\mathrm{D_{xs}}$ signals
after the optimal filtering. For the $\mathrm{D_{xs}}$ we present the signal
before (light green) and after (black) the filtering. Gray bars indicate the
position and width of sections with missing/removed data.