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Reconstructingatmospherichistoriesfrom
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DOI:10.1029/2002JD002545
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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. D24, 4780, doi:10.1029/2002JD002545, 2002
Reconstructing atmospheric histories from
measurements of air composition in firn
C. M. Trudinger, D. M. Etheridge, P. J. Rayner, I. G. Enting,
G. A. Sturrock,1 and R. L. Langenfelds
Commonwealth Scientific and Industrial Research Organisation Atmospheric Research, Aspendale, Victoria, Australia
Received 20 May 2002; revised 31 July 2002; accepted 1 August 2002; published 21 December 2002.
[1] This paper investigates the use of a numerical model of firn diffusion and bubble
trapping in the reconstruction of atmospheric records from firn measurements. We
describe the concept of mean age and effective age of tracers in firn air and how the
growth rate of a tracer in the atmosphere can alter the effective age. We discuss an iterative
method to assign effective ages to firn measurements for tracers with fairly simple
atmospheric histories, taking into account atmospheric growth rate variations. We then
develop a Bayesian synthesis inversion calculation for inverting firn concentration
measurements. This calculation gives estimates of the atmospheric concentration record
with uncertainties. The dating and inversion techniques are demonstrated here with carbon
tetrachloride measurements from a long firn record from Law Dome, Antarctica. The
techniques are then applied to measurements of a range of halocarbons in a companion
INDEX TERMS: 0365 Atmospheric Composition and Structure:
paper by Sturrock et al. [2002].
Troposphere—composition and chemistry; 1610 Global Change: Atmosphere (0315, 0325); 3210
Mathematical Geophysics: Modeling; 3260 Mathematical Geophysics: Inverse theory; KEYWORDS: firn, ice
core, inversion, diffusion, air age, dating
Citation: Trudinger, C. M., D. M. Etheridge, P. J. Rayner, I. G. Enting, G. A. Sturrock, and R. L. Langenfelds, Reconstructing
atmospheric histories from measurements of air composition in firn, J. Geophys. Res., 107(D24), 4780, doi:10.1029/2002JD002545,
2002.
1. Introduction
[2] Air extracted from polar firn has been analyzed in a
number of studies to reconstruct past atmospheric histories
of trace gas mixing ratios [e.g., Butler et al., 1999; Sturges
et al., 2000, 2001], isotopic ratios [Francey et al., 1999] and
elemental ratios [Battle et al., 1996]. Processes acting in the
firn (mainly diffusion, gravity and bubble trapping) influence tracer concentrations in the firn relative to atmospheric
levels and need to be taken into account when firn measurements are interpreted. We describe the use of a numerical
model of firn processes [Trudinger et al., 1997] to extract
information about past atmospheric concentrations from
measurements of firn air. Although we focus here on
interpreting firn air measurements, many of the principles
are also applicable to measurements of air trapped in
bubbles in ice.
[3] Firn is a porous layer of compacted snow, typically
40– 100 m deep. Air in the open channels in firn is in contact
with the atmosphere, and variations in atmospheric composition diffuse slowly through the firn column. Diffusion rates
decrease with depth, going to zero at the top of the lock-in
1
Now at School of Environmental Sciences, University of East Anglia,
Norwich, UK.
Copyright 2002 by the American Geophysical Union.
0148-0227/02/2002JD002545$09.00
ACH
zone. Air near the bottom of the firn, which at some sites can
be up to 100 years older than air at the surface, is gradually
trapped into bubbles. Air can be pumped from the firn layer
and collected for analysis with a technique developed by
Schwander et al. [1993]. Firn air can give insight into
processes that influence the air trapped in bubbles in ice,
as well as providing old air. Firn air is particularly useful for
reconstructing the history of tracers that require large volumes of air for accurate measurement, as it is possible to
pump much larger amounts of air directly from the firn than
is currently practical to extract from bubbles trapped in ice.
Like air archived in tanks or ice cores, firn air corresponding
to a range of ages can be analyzed within a short period of
time, therefore avoiding the problems of drifting standards
or instrument changes that can hamper records of direct
measurements over long periods. On the other hand, possible chemical and physical modification of the trace gas
concentrations and isotopic ratios during transport and
storage in the firn must be understood.
[4] There are several ways that firn measurements can be
interpreted in terms of atmospheric changes. One option is
to relate the concentration of the tracer to that of another
tracer, preferably with a known history. This has been done
by some authors [e.g., Battle et al., 1996; Sturges et al.,
2000] because interpreting the concentration of one tracer
relative to another is less sensitive to uncertainty in the
variation of diffusivity with depth in the firn. Carbon
dioxide (CO2), chlorofluorocarbon-12 (CFC-12, CCl2F2)
15 - 1
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15 - 2
TRUDINGER ET AL.: DATING FIRN AIR
and sulphur hexafluoride (SF6) have been used as reference
tracers for this purpose. Alternatively, firn measurements
have been used to test different scenarios for the atmospheric history, by running these scenarios in a firn diffusion
model and comparing the results with firn measurements
[e.g., Butler et al., 1999; Sturges et al., 2001]. In this
approach, reference tracers are used to calibrate the firn
model. Another option is to use a more formal inversion
method such as that described by Rommelaere et al. [1997],
again with a model calibrated with reference tracers.
[5] In this paper we describe two approaches for interpreting firn measurements to reconstruct the atmospheric
history. Both methods give the results as time histories,
which is the most useful way to present firn data. The first
approach, which we call ‘‘iterative dating,’’ is a simple
method for assigning effective ages to firn measurements
to recreate the atmospheric history. We then develop a
Bayesian synthesis inversion calculation, that inverts firn
measurements for the atmospheric history, and gives estimates of the uncertainties on the results. The two techniques are applied here to carbon tetrachloride (CCl4) data
from the DSSW20K firn on Law Dome, and are also used
by Sturrock et al. [2002] in the analysis of halocarbon
measurements.
[6] The outline of this paper is as follows: The firn model
is described briefly in section 2. In section 3 we discuss
different definitions of age, and implications of variations in
the atmospheric growth rate for dating firn measurements.
The iterative dating approach is described in section 4, and
the Bayesian synthesis inversion calculation is described in
section 5. Section 6 summarizes the results.
2. Numerical Model
[7] We use the one-dimensional finite difference model of
firn diffusion and bubble trapping of Trudinger et al.
[1997]. The ice sheet physical properties density and
diffusivity are modeled as constant with time. The model
uses a coordinate system that moves downward with the
accumulating ice. The model includes diffusion, gravitational settling and trapping of air into bubbles. It does not
include thermal diffusion [Severinghaus et al., 1998], an
upward flux of air due to compression [Schwander et al.,
1988; Schwander, 1989] or fractionation during closure of
bubbles [Battle et al., 1996] - these processes are expected
to be negligible for the applications described here. The
model has already been applied to a number of different firn
sites by Trudinger et al. [1997] and Trudinger [2000].
[8] The majority of the calculations in this paper relate to
the DSSW20K site on Law Dome, East Antarctica [Smith et
al., 2000; Sturrock et al., 2002]. Air was collected from the
DSSW20K firn in January 1998. Samples were pumped
from eight levels through the firn to the impermeable ice,
with the deepest sample at 52 m. The firn column at
DSSW20K is relatively short, with diffusion ceasing at
about 43 m. The accumulation rate is 150 kg m2 yr1.
Details about the site and collection of air are given by
Smith et al. [2000] and Sturrock et al. [2002].
[9] For density in the model we use a spline fit to density
measurements of the core drilled at the DSSW20K site.
Diffusivity varies greatly through the firn and is the main
physical property that determines the firn air composition
and age. Measurements of diffusivity in firn samples have
been made at other firn sites [Schwander et al., 1988;
Arnaud, 1997]. However, measured diffusivity may not
accurately represent the in situ diffusivity of the firn layer
as a whole [Etheridge, 1999; Fabre et al., 2000]. We tune
the diffusivity profile to obtain optimal fits to firn measurements with a minimization procedure. We relate diffusivity
in the firn to density, and use a diffusivity versus density
profile tuned to give the best fit to CO2, SF6 and hydrofluorocarbon HFC134a (CH2FCF3) concentration measurements. The model included a 4 m region near the surface
that was well-mixed by turbulence or convection, as suggested by the d15N2 measurements (where d15N2 is defined
as ((((14N15N/N2)sample)/((14N15N/N2)surface air)) 1) 1000). Previous studies of the Vostok site found a similar
well-mixed region extending 13 m into the firn [Bender et
al., 1994]. d15N2 in the firn may also have been affected
near the surface by thermal diffusion [Severinghaus et al.,
2001]. We have not explored this possibility here, but the
effect is expected to be small due to the small temperature
gradients at Law Dome, and the fact that the influence of
thermal diffusion does not extend very deep into the firn.
The diffusion of different tracers is related to the diffusion
of CO2 using the relative diffusion coefficients determined
with the method of Fuller et al. [1966] (also given by
Tucker and Nelken [1982]). To validate the tuned diffusivity, depth profiles are also calculated for 14C of CO2
[Smith et al., 2000] and methane (CH4) concentration with
the tuned diffusivity and compared to the firn measurements. All of our calculations except d15N2 are performed
without gravity in the model, because the firn measurements
can be corrected for gravity using d15N2, and running the
model without gravity greatly simplifies the iterative dating
calculations and use of the age distributions.
[10] The atmospheric record used to drive the model for
CO2 was a spline fit to DE08 and DE08-2 ice core CO2
measurements from Law Dome [Etheridge et al., 1996]. For
SF6 we used the quadratic equation from Maiss et al. [1996]
up to 1990 then the more recent equation from Maiss and
Brenninkmeijer [1998] after 1990. Both SF6 equations are
based on direct measurements and the Cape Grim Air
Archive [Langenfelds et al., 1996]. For HFC134a we use
annual averages from the Cape Grim Air Archive [Oram et
al., 1996; Oram, 1999]. A spline fit to Law Dome ice core
measurements was used for methane [Etheridge et al.,
1998], and tree ring data from Vogel and Marais [1971],
Manning and Melhuish [1994] and Stuiver and Braziunas
[1993] combined by Levchenko et al. [1997] for 14C.
[11] The model fit to the firn measurements for the tuned
diffusivity profile is shown in Figure 1. Overall, the model
is able to match the measured concentration profiles well for
a number of tracers with vastly different atmospheric
growth rates. There are some minor mismatches, such as
CO2 in the deepest samples, 14C at 47 m (the top of the
bomb pulse) and d15N2 at depth. There is a distinct flattening around the 1940s in the ice core CO2 record used to
drive the model. It is unclear what the actual atmospheric
CO2 variations may have been around this time [Trudinger,
2000] and this may explain the CO2 mismatch. For 14C, it
is possible to substantially alter the modeled depth of the
bomb pulse in the ice by varying the diffusivity profile, but
only possible to make very minor changes to the amplitude.
TRUDINGER ET AL.: DATING FIRN AIR
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15 - 3
Figure 1. Measurements and modeled trace gases in the DSSW20K firn. (a) CO2. (b) SF6. (c)
HFC134a. (d) d15N2 (see Langenfelds [2002] for details of measurement corrections). (e) CH4. (f ) 14C
of CO2, which is normalized for 13C and corrected for decay as described by Smith et al. [2000] and
Stuiver and Polach [1977]. 14C measurements are from Smith et al. [2000].
We therefore could not fit both the highest value and the
sides of the peak. It is unlikely that the 14C measurement
at the top of the peak is low due to pumping air from a range
of depths above and below the peak in the ice [Sturrock et
al., 2002]. The modeled d15N2 does not appear to agree well
with the measurements below 40 m. The uncertainty in the
d15N2 measurements is estimated to be no more than 0.01
%. If we had used a diffusivity profile with zero diffusivity
below 40 m as might be suggested by the d15N2 measurements, we would not have been able to fit the concentration
profiles as well as we have. The reasons for these discrepancies are not known, but apart from them, the fit to the
measurements is good. We did not try to fit the deepest SF6
measurements, because the quadratic equation extrapolates
SF6 concentrations below about 0.5 ppt (i.e., before the start
of the Cape Grim Air Archive record).
[12] There is about 5% uncertainty in the diffusion
coefficient of each species in air relative to the diffusion
coefficient of CO2 in air [Tucker and Nelken, 1982]. For
example, values of the diffusion coefficient of methane
relative to CO2 ranging from 1.29 to 1.415 have been used
for firn air modeling in the past (e.g., 1.29 for DE08-2
[Trudinger et al., 1997], 1.35 for Summit [Schwander et al.,
1993], 1.35 for South Pole [Battle et al., 1996], 1.415 for
DE08-2 [Rommelaere et al., 1997]), based on different
equations or measurements. Variation in temperature from
one site to another will cause a small difference in the actual
relative diffusion, but the resulting effect is much smaller
than that due to the range of values used. For SF6 and
HFC134a, which were used to tune the diffusivity profile in
the model, we also tuned the relative diffusion coefficient
(this gave 0.628 for SF6 and 0.614 for HFC134a, compared
to the Fuller et al. [1966] values of 0.592 and 0.633). For
other tracers we use the values given by Fuller et al. [1966]
(i.e., 1.3, 0.992 and 0.53 for methane, 14CO2 and carbon
tetrachloride, respectively), but the uncertainty in the relative diffusion coefficient needs to be treated as part of the
model error and will be explored later.
[13] There is a limit to how well firn measurements can
be simulated with a one-dimensional model with a constant
accumulation rate and a diffusivity profile that is kept
constant with time. There may be transient variations in
the ice properties, even during relatively stable climatic
periods such as the Holocene, that could influence diffusion
in the firn and the measured concentration profile. To some
extent these variations would be averaged as mixing occurs
over the entire firn column. Tuning the diffusivity profile is
the best way to characterize the diffusion that actually
ACH
15 - 4
TRUDINGER ET AL.: DATING FIRN AIR
Figure 2. Response functions calculated with the firn
model for carbon tetrachloride in the DSSW20K firn at the
measurement depths. The curves correspond to collection of
air at the beginning of 1998.
occurs in situ [Etheridge, 1999; Fabre et al., 2000]. We
used a well-mixed region with a constant depth of 4 m. We
have no information as to whether this well-mixed region
always extends to this depth, but the modeled concentration
around and below the first firn measurement depth (29 m) is
not very sensitive to the inclusion of this layer. Despite the
limitations, the model certainly simulates the processes in
the firn well enough to be very valuable for interpreting firn
records of other trace gases.
3. Air Age
[14] In order to interpret firn air measurements in terms
of past atmospheric concentrations and isotopic ratios, we
need to know the age of the air in the firn. However, due to
the mixing processes in the firn, air at a particular depth in
the firn is a mix of different air parcels that left the
atmosphere over a range of times in the past. Thus firn
air is not characterized by a single age, but a distribution of
ages [Schwander et al., 1993]. The age distribution (or age
spectrum) can be determined with a numerical firn model.
As each tracer diffuses at a slightly different rate through
the air in the firn, the age distribution of each tracer differs
slightly. Figure 2 shows the age distributions for carbon
tetrachloride at the measurement depths in the DSSW20K
firn calculated with the firn model and corresponding to
sampling at the start of 1998.
[15] There are strong similarities between the characteristics of the age of air in the firn and the age of stratospheric
air relative to the troposphere, and recent work on determining the age of stratospheric air [Hall and Prather, 1993;
Hall and Plumb, 1994; Hall and Waugh, 1997; Andrews et
al., 1999, 2001] has parallels for firn air analysis. In fact, the
stratosphere has also been used as an ‘‘archive’’ of old air to
produce atmospheric records of CF4 and C2F6 [Harnisch et
al., 1996]. The age distributions for stratospheric air are of
similar form to those for firn air.
[16] Despite the fact that firn air is associated with a
distribution of ages, it is possible to use a single ‘‘age’’ to
characterize the distribution. However, there are different
definitions of age that can be used. One measure of age is
the mean of the age distribution, , (i.e., the first moment of
the distribution). Another is what is often referred to as the
‘‘effective age,’’ t, which is the elapsed time since the
surface was at the same concentration as is currently seen at
depth z, i.e., C(z, t) = C(0, t t). The effective age depends
on variations in the atmospheric growth rate of the tracer
[Hall and Waugh, 1997; Trudinger et al., 1997]. The
effective age and the mean age are the same for tracers
with a linear growth rate in the atmosphere [Enting and
Mansbridge, 1985; Hall and Prather, 1993]. In general, the
effective age will be less (i.e., younger) than the mean age
for tracers that increase more rapidly than linear, as a signal
propagates to the bottom of the firn (or to the stratosphere)
faster than the mean time lag of concentration for a tracer
with a linear growth rate [Hall and Plumb, 1994]. The
concept of effective age becomes ambiguous when the
atmospheric gradient disappears or changes sign, as there
is no longer a unique value satisfying C(z, t) = C(0, t t)
[Enting and Mansbridge, 1985]. The following two examples illustrate the variation of effective age in the firn with
the atmospheric growth rate.
[17] Figure 3a illustrates an example that was described
by Trudinger et al. [1997]. The solid line shows a hypothetical atmospheric concentration record that consists of a
sinusoidal pulse of 10 years duration with linear increase
before and after. The firn model was run for the DE08 site
[Etheridge et al., 1996; Trudinger et al., 1997] with this
atmospheric record, and the calculated concentration profile
with depth in the air bubbles is assigned the mean age of
CO2 at each depth. (This corresponds to a constant air
age—ice age difference of 30 years, as was used for the ice
core CO2 measurements from this site by Etheridge et al.
[1996]). The parts of the concentration record that vary
linearly with time show good agreement between the ‘‘firnsmoothed’’ record and the atmospheric record that was used
to drive the model. However, the sinusoidal pulse is both
smoothed and time-shifted in the reconstructed record. Both
the leading edge and the maximum of the pulse appear in
the reconstructed record about 5 years earlier than in the
original record. Although the concentration at the bottom of
the firn at DE08 at time t is the same as the surface
concentration at time t-10 yr for a constant growth rate, a
change such as a pulse at the surface affects the concentrations at the bottom of the firn much earlier than 10 years
after it is seen in the atmosphere. This is a case in which the
atmospheric growth rate has changed sign twice, but
gradually enough that the feature is clear in the ice core
record.
[18] As another example, the firn model was run for SF6
in the DSSW20K firn with the Maiss et al. [1996] quadratic equation as the atmospheric input (solid line shown in
Figure 3b). The calculated depth profile was ‘‘sampled’’ at
5 m intervals. These ‘‘measurements’’ were assigned the
mean ages of SF6 for their depths (circles in Figure 3b),
and compared with the original atmospheric input. The
reconstructed record (i.e., the circles) differs from the
original one. A second step of redating the measurements
to take account of the growth rate changes, as will be
shown in the next section, allows almost perfect recovery
of the original record. This example demonstrates how a
systematic error could be introduced into the estimated ages
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TRUDINGER ET AL.: DATING FIRN AIR
15 - 5
Figure 3. (a) Hypothetical CO2 record (solid line) and this record as it would be reconstructed from
concentrations trapped in DE08 ice using a constant air age – ice age difference (dashed line). (b) Solid
line shows the SF6 quadratic equation from Maiss et al. [1996]. The circles show SF6 concentrations from
the firn model run for DSSW20K with the quadratic equation as atmospheric input, sampled at 5 m
intervals in the firn and dated with a linear tracer in the model (i.e., mean age).
of measurements of a tracer with an ‘‘SF6-like’’ atmospheric increase if the ages were determined by comparison
with a reference tracer with a fairly linear increase. It points
out the risks of using a reference tracer for dating if the
atmospheric histories of the reference and target species are
too different.
[19] Hall and Plumb [1994] investigated the difference
between the mean and effective ages in the stratosphere for
tracers with exponentially increasing concentrations in the
troposphere (i.e., conc = Aet/a + B). They found that the
growth time constant, a, must be greater than about 7 years,
which is true for all of the CFCs, to give less than 10%
difference between the mean and effective ages. We can
repeat this analysis for the firn, where the age distribution is
generally wider than that for stratospheric air. We first need
to quantify the width of the age distribution. As in the
stratospheric work, we can do this using the square root of
the second moment of the distribution, , also known as the
spectral width and defined as
2 ðzÞ ¼
1
2
Z
1
ðt Þ2 Gðz; t Þdt
ð1Þ
0
where z is depth, is the mean of the distribution (first
moment) and G is the distribution [Hall and Plumb, 1994;
Hall and Waugh, 1997; Andrews et al., 1999]. (The spectral
width for our asymmetric age distribution in the firn is the
analog of the standard deviation for the Gaussian distribution.) The spectral width for DSSW20K firn air after
diffusion stops is about 5 years, and for comparison the
spectral width for the South Pole firn air [Butler et al., 1999]
is about 18 years. Using this we can apply the equations
from Hall and Plumb [1994], which require
a > = 0:1 2
ð2Þ
for less than 10% difference between mean and effective
ages. The mean age of carbon tetrachloride at the bottom of
the DSSW20K firn is about 17 years, giving the requirement a > 15 years for less than 10% difference between
mean and effective ages, which is not the case for many of
the halocarbons. The absolute difference between the mean
and effective ages is given by
t¼
2
a
ð3Þ
where t is the effective age as defined earlier. Allowing up
to a 2 year difference between and t we would require
a > 12.5 years for DSSW20K and 160 years for the South
Pole firn. Even allowing 10 years difference between and
t would require a > 32 years at South Pole.
[20] These equations allow us to determine how well the
mean age will approximate the effective age for exponentially increasing tracers, given the growth time constant of
the tracer of interest and the spectral width at the firn site.
They can also be used for determining the likely error when
tracers with exponential-like atmospheric variation are
related to tracers with approximately linear atmospheric
history, such as CO2 or CH4 over recent decades, or tracers
with different exponential growth rates. When considering
the age of firn air, either mean or effective, it must be kept in
mind that the firn air profile is a smoothed representation of
the atmospheric record, and some features in the atmospheric record will either not be reflected at all, or be damped
in the firn.
[21] The age distribution discussed here applies in situ,
and does not take into account any effects of sampling air
from the firn. However, there is strong evidence that
collection of the air at the Law Dome firn sites does not
significantly widen the age spread compared to the in situ
effects of diffusion [Sturrock et al., 2002]. Briefly, as no
trend was measured in CO2 or CH4 in a series of flasks
collected successively from each depth, most importantly at
the deepest level, it is believed that the layered structure of
the firn causes air to be drawn from a narrow horizontal
layer, rather than from significantly above or below the
measurement depth.
4. Iterative Dating
[22] The firn model can be used in a forward mode to
provide effective ages for firn measurements, taking into
ACH
15 - 6
TRUDINGER ET AL.: DATING FIRN AIR
Figure 4. The steps involved in the iterative dating of firn samples. The circles show the firn
‘‘measurements,’’ and the crosses show modeled concentrations at the measurement depths for
comparison with the driving atmospheric record. (a) Firn air concentration measurements to be dated. (b)
Linearly increasing atmospheric record used to drive firn model. (c) Calculated concentration depth
profile for the linear atmospheric record, crosses indicate modeled concentrations at the measurement
depths. (d) Mean age estimate determined by comparison of the modeled concentrations at the
measurement depths in Figure 4c with the driving atmospheric record in Figure 4b. (e) Curve created by
assigning the measurements in Figure 4a with the mean ages in Figure 4d. (f ) Modeled concentration
profile calculated with the atmospheric record shown in Figure 4e, crosses indicate modeled
concentrations at the measurement depths. (g) Second estimate of the effective age, determined by
comparison of the modeled concentrations at the measurement depths in Figure 4f with the driving
atmospheric record in Figure 4e. (h) Symbols show the measured concentrations in Figure 4a with the
first age estimates (grey circles) and the second age estimates (open circles). The line is a curve fitted
through the measurements with the second age estimates.
account the effect of variations in the atmospheric growth
rate described in the previous section. This technique
requires no prior knowledge of the atmospheric history,
and works best for tracers with reasonably simple atmospheric changes (e.g., monotonic). The mean age is used as a
first estimate for the effective age, then the model is used to
refine the age estimate. We refer to this procedure as
iterative dating. The procedure we use is as follows,
illustrated in Figure 4.
[23] Assume that we have some firn air concentration
measurements of tracer Z that we wish to date (Figure 4a).
The firn model must have been calibrated for the site, and we
need the diffusivity of tracer Z relative to that of CO2. We run
the model for tracer Z with a linearly increasing atmospheric
TRUDINGER ET AL.: DATING FIRN AIR
ACH
15 - 7
Figure 5. Carbon tetrachloride measurements from the DSSW20K firn [Sturrock et al., 2002]. (a) The
open circles show the firn measurements with the first age estimates (mean ages) and the solid symbols
show measurements with the second age estimates. The thick line shows the Cape Grim record [Prinn et
al., 2000]. The thin line shows a spline fit to measurements with the second age estimates. (b) The depth
profile for this atmospheric history. (c) Two possible atmospheric histories with different variation before
1950, assuming natural backgrounds of 5 ppt (solid line) and 0 ppt (dashed line). (d) Depth profiles for
the curves in Figure 5c.
concentration, C(z = 0, t) = At for 0 < t < Tf where A is an
arbitrary constant (Figure 4b), to generate a calculated
concentration depth profile (Figure 4c). The model starts
with zero concentration at all depths at t = 0, and must be run
long enough to establish nonzero concentration through the
entire depth range of interest. Because the effective age for a
linear atmospheric increase is equal to the mean age (see
previous section), this calculation allows us to determine the
mean age, (z), by comparing the calculated concentration at
the measurement depths, with the linear input atmospheric
function using (z) = Tf C(z, t)/A (Figures 4b, 4c and 4d).
(Mean ages can also be estimated by taking the mean of the
age distribution, but our method is simpler.)
[24] A first estimate of the atmospheric history of tracer Z
is created by assigning dates corresponding to these mean
ages to the concentration measurements (Figure 4e). The
dates used are Ts-(z), where Ts is the date of sampling (e.g.,
1998 for the DSSW20K firn). A smooth curve fitted
through these points (Figure 4e) is used in the firn model
to calculate a new concentration versus depth profile (Figure
4f ), which when compared with the input function (solid
line in Figure 4e) gives new estimates of the effective age
(Figure 4g). The atmospheric curve used at this stage needs
to be simple enough that comparison of the depth profile
with the input function is possible and gives unique ages
(i.e., a change in the sign of the growth rate is undesirable).
This second step can be repeated with the new age estimates, although generally this makes little difference to the
calculated ages.
[25] Figure 4h shows the concentration measurements
plotted against the first (solid grey circles) and second
(open circles) age estimates. The solid line shows the
function that was originally used to generate the synthetic
measurements used in this example. We have been able to
reconstruct the atmospheric variation very well without
prior knowledge of this variation. Although we have the
limitation that the atmospheric record must be relatively
simple, it is not too restrictive. Short time scale variations
(e.g., up to a few years) are not recorded in the firn, so they
do not affect the iterative dating procedure. Many anthropogenic tracers increase monotonically in the atmosphere,
and iterative dating allows us to correct dates for curvature
in the records.
[26] Figure 5 shows our iterative dating approach used to
reconstruct carbon tetrachloride in the atmosphere from
DSSW20K firn measurements [Sturrock et al., 2002]. The
model is run without gravity and the firn concentration data
are corrected first for the influence of gravity using measured d15N2 [Craig et al., 1988; Sowers et al., 1989] (the
gravitational correction for these measurements is a reduction of concentration by about 2%). The open circles in
Figure 5a show the firn measurements with the first age
estimates determined with a linearly increasing tracer in the
firn model (mean ages). The solid symbols show the
measurements with the second age estimates. The thin line
shows a spline fit to the measurements with the second age
estimates, and the depth profile generated when this record
is run in the firn model is shown in Figure 5b.
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15 - 8
TRUDINGER ET AL.: DATING FIRN AIR
[27] The iterative dating approach is used for a number of
halocarbons, including carbon tetrachloride, of Sturrock et
al. [2002] and the reconstructed records are discussed. It is
possible that carbon tetrachloride has a nonzero natural
background level, but the DSSW20K firn measurements
do not extend far back enough in time to show whether or
not this is the case. A range of atmospheric histories prior to
1950 are consistent with the DSSW20K measurements. The
history in Figure 5a increases from zero in 1910. Two other
possible histories consistent with the firn measurements are
shown in Figure 5c, one that increases from a steady level of
5 ppt in 1930, the other increasing from zero in 1930. The
corresponding depth profiles from the model are shown in
Figure 5d. Note that the reconstructed history will be
smoothed relative to the original record, as short time scale
variations are not recorded in firn and medium time scale
variations are damped. For example, the 14C bomb pulse,
an atmospheric feature lasting a few decades, is clearly
recorded in the DSSW20K firn, yet the amplitude is about
half that of the signal in the southern hemisphere atmosphere [Smith et al., 2000].
[28] In the next sections we will explore estimation of the
uncertainty in the reconstructed record. There are three main
factors that contribute to the uncertainty: the first, and one
of the most important, is that there is a range of solutions
that are consistent with the measurements due to the
smoothing and loss of information caused by the firn
processes. The second is the uncertainty in the reconstructed
record due to uncertainty in the firn measurements (including analytical, calibration and sampling uncertainties). This
is expected to be very small for the halocarbon measurements of Sturrock et al. [2002]. The third cause is the
uncertainty due to model error, including uncertainty in the
relative diffusion coefficient. The Bayesian synthesis inversion described in the next section easily includes the first
two types of error, but model error is much more difficult to
include.
[29] We can try to estimate the effect of model error by
running the firn model for a range of model parameters,
including alternative diffusivity profiles that give reasonable
agreement for the tracers used for tuning (with and without
the well mixed region, and using different vertical resolution) and variation of ±5% in the diffusion coefficient of
carbon tetrachloride. Figure 6 shows the carbon tetrachloride depth profile for 10 different model runs with atmospheric variations given by the reconstructed history in
Figure 5. Of all the DSSW20K measurement depths, the
variation in the modeled concentration is greatest at 47 m,
with a standard deviation of 0.98 ppt (2% of the measured
concentration). This range is fairly small, and smaller in
some cases than the mismatch seen in Figure 1. We will use
these results later.
5. Bayesian Synthesis Inversion
5.1. Method
[30] In the previous section we used forward model
calculations to infer an atmospheric record by assigning
effective ages to the firn measurements. In this section, we
will use an inverse method known as Bayesian synthesis
inversion to invert the firn measurements for an atmospheric history. Bayesian synthesis inversion was used by
Figure 6. Modeled depth profiles of carbon tetrachloride
at DSSW20K using the atmospheric record shown in Figure
5b and 10 runs of the firn model with different model
parameters. The firn measurements [Sturrock et al., 2002]
are shown in grey.
Enting et al. [1995] and Rayner et al. [1999] to invert
atmospheric CO2 concentration measurements for CO2
sources and sinks. Our inversion is similar in some ways
to the inversion of firn data described by Rommelaere et al.
[1997].
[31] Like Waddington [1996] and Rommelaere et al.
[1997], we characterize the firn processes by using response
functions (also known as Green’s functions, transfer functions or age distributions) generated with the firn model.
The response function at a particular depth gives the
contribution to the concentration at that depth of the
atmospheric concentration for each year in the time interval
of interest. Because the model is linear, the response
functions can be used instead of the firn model to calculate
concentrations in the firn, ci, due to a time series of
atmospheric concentration over the past N years, (a1,
a2,. . ., aN), with
ci ¼
N
X
Rij aj
ð4Þ
j¼1
where Rij is the response function at depth zi for an
atmospheric concentration in the year j. The response
functions are calculated by running the firn model with an
atmospheric record that has concentrations of 1.0 for one
year then zero after that. The time series of concentration at
a particular depth from the model gives the response
function at that depth. We showed the response functions
calculated for carbon tetrachloride at the measurement
depths in the DSSW20K firn in Figure 2. The tuned
diffusivity for DSSW20K is zero below about 43 m, so the
response functions below this point have the same shape but
are translated in time. There is significant overlap of the
response functions at adjacent measurement depths.
[32] Equation (4) can also be written in terms of the yearto-year concentration change, sj = aj aj 1.
ci ¼
N
X
j¼1
Tij sj
ð5Þ
TRUDINGER ET AL.: DATING FIRN AIR
where
Tij ¼
N
X
ð6Þ
Rik
k¼j
In our inversion we will solve for sj rather than aj, for
reasons that will be explained shortly. The results will be
given in terms of concentrations. We will refer to sj as the
source, although it is the local (not global) year-to-year
change in concentration, and will include the effects of
sources, sinks and atmospheric transport.
[33] Synthesis inversion aims to estimate the source, sj, at
times tj for 0 j N to give the best fit to concentration
measurements in the firn, yi, for 0 i M. The calculation
is Bayesian, meaning that it uses prior estimates, sj, of the
sources, sj. The use of prior estimates stabilizes the inversion. The inversion minimizes the objective function, ¼
2
P
M
yi Nj¼1 Tij sj
X
i¼1
u2i
þ
2
N
X
sj sj
v2j
j¼1
ð7Þ
where ui is our estimate of the error standard deviation of
the firn concentration measurements and vi is the error
standard deviation of the prior estimate, si. Where atmospheric concentration measurements exist, such as from
Cape Grim, we can include them in the inversion as annual
means and with response functions of 1.0 in the relevant
year.
[34] An important aspect of our inversion is that uncertainties are tracked through the calculation to give uncertainties in the deduced atmospheric record. The uncertainties
estimated by the inversion include the first two causes of
uncertainty mentioned at the end of section 4, i.e., uncertainty due to firn smoothing and measurement uncertainty.
Firn model error is not included directly in this calculation,
but we can include it indirectly by increasing the measurement uncertainty to reflect the uncertainty in the model.
Because we are using both firn and atmospheric data, errors
in the firn model could cause a conflict between the two
types of data if model error is not included properly.
[35] The reasons why we invert for annual sources rather
than concentrations are as follows. If we inverted for annual
atmospheric concentrations, the uncertainties in the results
would be very large, because due to the smoothing effect of
the firn processes, firn measurements provide little information on annual atmospheric concentrations. We do know
that the atmospheric concentration in one year is unlikely to
be considerably different from the concentration in the
previous year. In other words, uncertainties on the prior
estimate of the annual concentrations are correlated. Inverting for atmospheric concentrations using correlated prior
estimates would require a priori specification of the correlations. A simpler solution is to invert for sources, and limit
the magnitude of the source with the prior uncertainty to
reflect realistic annual concentration changes. Another justification for inverting for sources is that other applications
using synthesis inversion of firn or ice core data may be
more interested in sources than concentrations, and could go
one step further to include a simple model of source (and
sink) evolution.
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15 - 9
[36] The matrix of response functions for all sources at
the measurement depths constitutes the Jacobian of the
problem and contains all relevant information from the firn
model. We invert the Jacobian using the singular value
decomposition (SVD) technique [Press et al., 1992]. As the
inversion is linear, it will allow negative concentrations if
they seem to give the best fit to the measurements, but this
is not physically reasonable. We will discuss the use of
nonnegativity constraints in the inversion in section 5.4.
[37] Rommelaere et al. [1997] inverted firn and ice core
measurements for an atmospheric history using a similar
Green’s function formulation to that used here. Their
calculation is not Bayesian (i.e., they do not use a prior
estimate), instead they stabilize the calculation by controlling the ‘‘length’’ of the solution.
5.2. Data and Uncertainties
[38] We use the DSSW20K firn carbon tetrachloride
measurements from Sturrock et al. [2002] with uncertainties
(1s) that include the analytical uncertainty, calibration
uncertainty (1%), uncertainty due to sampling (0.1 ppt,
based on tests with the Firn Air Sampling Device), uncertainty in the gravitational correction (corresponding to 0.01
% uncertainty in the d15N2 measurements used for the
gravitational correction) and model error. For model error
we use twice the standard deviation determined by the set of
10 model runs shown in Figure 6. We use a range greater
than that determined with the model, as only one model was
used which probably underestimates the model error. The
different types of error are combined to give the uncertainty
for each measurement by taking the square root of the sum
of squares of the different errors. This assumes that the
errors are uncorrelated. In addition to the firn measurements, we use annual means based on in situ carbon
tetrachloride data from Cape Grim starting in 1979 [Prinn
et al., 2000]. We use uncertainties on the Cape Grim annual
means of 5% before 1993 and 2% after 1993 [Prinn et al.,
2001].
[39] The solution from iterative dating was used as the
prior estimate of the sources. For uncertainties on the prior
estimates we use the maximum source (i.e., year-to-year
change in concentration) for the whole record, so as not to
influence the solution too strongly. We calculate c2, which
compares the mismatch of the calculated and measured firn
concentrations with the uncertainties [Tarantola, 1987].
This confirms that the uncertainties are large enough that
the model can simultaneously match the two different types
of data (firn and Cape Grim) and the prior estimate.
5.3. Results
[40] Figure 7a shows the estimated time history of carbon
tetrachloride concentration (solid line), with (1s) uncertainties (dotted lines), calculated to give the best fit to the
DSSW20K firn measurements and annual means from Cape
Grim after 1979. We used a spline fit to the firn measurements with effective ages and the Cape Grim data as a
prior estimate for the annual concentrations (dashed line in
Figure 7a).
[41] Figure 7b shows the uncertainty in the estimated
concentration time history (solid line). This uncertainty
depends on the uncertainty in both the measurements and
the prior estimate, but not on the values of the prior estimate
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TRUDINGER ET AL.: DATING FIRN AIR
Figure 7. Results of the Bayesian synthesis inversion for carbon tetrachloride measurements from
Sturrock et al. [2002]. (a) The dashed line shows the prior estimate, and the solid line shows the estimated
history determined by the SVD inversion. The dotted lines indicate the uncertainty (1s) on the estimate.
(b) The concentration uncertainty is shown by the solid line. The asterisks show the mean age, and the
triangles the mode age, of carbon tetrachloride at the DSSW20K measurement depths. The prior
concentration uncertainty is shown by the dashed line. (c) The plus symbols show the DSSW20K firn
measurements, the diamonds show the concentration at the measurement depths for the prior estimate
from the previous section, and the circles show the concentration for the best estimate of the time history.
(d) The dashed line shows an alternative prior estimate, and the solid line shows the estimated history
calculated using this prior estimate, with uncertainties. (e) The constrained (solid line) and unconstrained
(short dashed line) solution with the linear prior estimate (long dashed line). (f ) Our best solution is the
solution subject to the constraint of nonnegative concentrations, with uncertainties from the
unconstrained (SVD) calculation and the iterative dating solution used for the prior estimate.
or the solution (as is the case for any linear least squares like
estimation problem). We also show the mean age of carbon
tetrachloride determined in the previous section (asterisks in
Figure 7b) and the year corresponding to the maximum of
the age distribution (i.e., the mode of the distribution) at the
four deepest measurement depths (triangles). The troughs in
the uncertainty (i.e., when the uncertainty in the reconstructed atmospheric record is lowest) do not correspond
exactly with either the means or the modes. The troughs,
means and modes all depend on the shape of the response
functions and not on the concentration history. However, the
means and modes correspond to each measurement depth
individually, while the overlap of the age distributions for
adjacent measurements can cause the troughs to be shifted
by a few years from where they would be for each measurement on its own.
[42] The concentration uncertainty before 1930 in Figure
7b increases as you go back in time, then decreases toward
TRUDINGER ET AL.: DATING FIRN AIR
1900. The reason for this is related to the prior concentration uncertainty. Recall that we use constant source
uncertainty for the entire period. This constant source
uncertainty corresponds to concentration uncertainty that
increases through the period, as shown by the dashed line in
Figure 7b that extends off the top of the graph. The firn
measurements reduce the uncertainty considerably after
about 1920, but they do not constrain the concentration
much at all in the first few decades of the 1900s, so the
calculated concentration uncertainty follows the prior concentration uncertainty.
[43] Figure 7c shows the firn measurements (indicated by
the plus symbols), concentration at the firn measurement
depths for the prior concentration history (diamonds) and
for the best estimate of the time history (circles). The
agreement with the firn measurements was already very
good with the prior estimate, but it is better with the
synthesis solution. Figure 7d shows the time history estimated using a prior estimate that has a linear increase from
1920 to join the Cape Grim record in 1979 (long dashes).
This solution is very similar to the one obtained with the
prior estimate from iterative dating, but differs most in that
it is negative around 1920 and has a sharper increase. As
mentioned earlier, because the model is linear, it allows
solutions with negative concentrations. We can avoid this to
some degree by using a prior estimate obtained from other
methods that gives a reasonable fit to the measurements.
5.4. Constraints
[44] We know a priori that the concentration of carbon
tetrachloride cannot be less than zero at any time. In
addition, for very long-lived, synthetic tracers (e.g., SF6)
it is unlikely that the atmospheric concentration decreased
since (significant) production began. Because the synthesis
inversion is linear, it does not exclude negative concentrations or sources from the range of solutions. Including
nonnegativity constraints in the inversion is difficult. The
use of prior sources can help keep the solution stable to
some extent, but the inversion can still allow negative
concentrations in some cases, as we saw in the previous
section for the linear prior estimate. We can use additional
information to help the inversion. For example, if a tracer
has zero concentration in the firn at and below a particular
depth, z, we can reasonably assume that there was no
natural background concentration, and determine with the
firn model the earliest date that concentrations are likely to
have increased from zero. We then use the synthesis
inversion to invert for concentrations only after this date.
This is done by Sturrock et al. [2002]. A combination of the
results from forward modeling with the inversion calculation improves the results.
[45] The constraint of requiring the source to be nonnegative is a very strong constraint, and should lead to much
smaller uncertainties on the atmospheric concentration history than a solution without this constraint. The constraint
of nonnegative concentrations is important near the beginning of the record, and should also lead to smaller uncertainties. However, with these constraints our problem is
nonlinear, and cannot be solved by the linear synthesis
inversion method described above. Without the nonnegative
source constraint, the SVD solution has strong anticorrelations in the errors of the estimated concentrations in
ACH
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adjacent years, due to the strong smoothing of annual
concentrations by the firn processes. This means that
histories consistent with the synthesis solution and the error
covariance matrix have high frequency variations that are
unlikely to be realistic. We limit this variability to some
extent with the prior estimates of the sources, but the
uncertainty in the prior source estimates should not be too
small or they will have too much influence over the
solution.
[46] We can estimate the atmospheric record subject to the
nonnegativity constraints using a nonlinear minimization
routine such as CONSTRAINED_MIN in IDL (Research
Systems Inc., Boulder, Colorado, USA). Figure 7e shows the
solution using the CONSTRAINED_MIN routine to solve
the inversion with the constraints of nonnegative concentrations (solid line) with the linear prior estimate (long
dashes). The SVD solution is shown by the short dashed
line for comparison. A major disadvantage of this solution is
that it can not include the uncertainty analysis that was an
important part of the SVD calculation.
[47] As the uncertainties from the SVD calculation should
include as a subset the uncertainties for the constrained
problem, we take as our best solution the time history
calculated with the constraint of nonnegative concentrations, with the uncertainties from the SVD calculation
(Figure 7f ). The prior estimate is the solution from iterative
dating. The constrained solution in Figure 7f looks very
similar to the unconstrained solution in Figure 7a (the
uncertainties are the same, but the solutions do differ
slightly). If a different prior estimate had been used, such
as the linear prior in Figure 7d, these two solutions could
have been much more different. Here and in the work by
Sturrock et al. [2002] we have imposed the nonnegativity
constraint only on concentrations, not sources. The constraint of nonnegative sources is computationally just as
easy to include, but if used would have to be justified by
considering the lifetime of the particular tracer and whether
the atmospheric concentration is unlikely to have decreased
at any time in the past. Nonlinear Bayesian inversion
calculations, such as Markov chain Monte Carlo methods
[e.g., Tamminen and Kyrölä, 2001] may prove useful for
solving for the uncertainties with the nonnegativity constraints, but are beyond the scope of this paper.
6. Summary and Further Work
[48] We have investigated techniques for reconstructing
time histories from firn (or ice core) measurements. We
showed how effective ages can be assigned to firn measurements, taking into account variations in the atmospheric
growth rate that influence the effective age. We described a
synthesis inversion calculation that inverts the firn measurements to generate a concentration versus time history with
uncertainties. The uncertainties estimated by the synthesis
inversion include the effects of firn smoothing, measurement/sampling error and model error. Our aim here was to
discuss methods for producing time histories of tracers,
giving the results as variations with time. This is often a
more useful way to present the firn data than relating
measurements to other tracers as has often been done in
the past. Because firn smoothing causes a significant loss of
information, meaning that a range of possible atmospheric
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TRUDINGER ET AL.: DATING FIRN AIR
histories is consistent with the firn measurements, estimation of the uncertainty in the reconstructed history is far
superior to determining a single history that is consistent
with the measurements.
[49] The Bayesian synthesis inversion leads on to a
number of possibilities. We have used Bayesian synthesis
inversion with measurements from just one firn site. A
major advantage of the method is that the inversion could be
performed with measurements from a number of firn (or ice
core) sites with different characteristics, provided Green’s
functions are available for each site, to determine a concentration (or source) history consistent with all measurements,
given their uncertainties. Inversion of firn or ice core
measurements in this way could be combined with a process
model of the source or sink, perhaps inverting for model
parameters (e.g., atmospheric lifetime [Bloomfield et al.,
1994]) rather than the concentration history.
[50] Acknowledgments. The authors wish to thank Paul Steele and
Roger Francey for ongoing support of this work including GASLAB trace
gas measurements. We also thank the Glaciology Program of the Australian
Antarctic Division for drilling and logistic support of the firn air sampling
expeditions, and Ingeborg Levin (University of Heidelberg, Institut für
Umweltphysik) for SF6 measurements.
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