cpd 11 5605 2015

cpd 11 5605 2015
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Discussion Paper
Clim. Past Discuss., 11, 5605–5649, 2015
www.clim-past-discuss.net/11/5605/2015/
doi:10.5194/cpd-11-5605-2015
© Author(s) 2015. CC Attribution 3.0 License.
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, S. Kipfstuhl , J. Freitag , H. Meyer , and T. Laepple
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Correspondence to: T. Münch ([email protected])
Published by Copernicus Publications on behalf of the European Geosciences Union.
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Received: 13 October 2015 – Accepted: 5 November 2015 – Published: 26 November 2015
Regional signal vs.
local noise in
Antarctic δ18 O
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Alfred Wegener Institute Helmholtz Centre for Polar and Marine Research, Telegrafenberg
A43, 14473 Potsdam, Germany
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Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476
Potsdam, Germany
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Alfred Wegener Institute Helmholtz Centre for Polar and Marine Research, Am Alten Hafen
26, 27568 Bremerhaven, Germany
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T. Münch
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11, 5605–5649, 2015
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Discussion Paper
Regional climate signal vs. local noise:
a two-dimensional view of water isotopes
in Antarctic firn at Kohnen station,
Dronning Maud Land
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Regional signal vs.
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In low-accumulation regions, the reliability of δ 18 O-derived temperature signals from
ice cores within the Holocene is unclear, primarily due to small Holocene climate
changes relative to the intrinsic noise of the isotopic signal. In order to learn about
the representativity of single ice cores and to optimise future ice-core-based climate
reconstructions, we studied the stable-water isotope composition of firn at Kohnen station, Dronning Maud Land, Antarctica. Analysing δ 18 O in two 50 m long snow trenches
allowed us to create an unprecedented, two-dimensional image characterising the isotopic variations from the centimetre to the hundred-metre scale. Our results show
a clear seasonal layering of the isotopic composition, consistent with the accumulation
rate, as well as high lateral isotopic variability caused by local stratigraphic noise. Based
on the horizontal and vertical structure of the isotopic variations, we derive a statistical
model for the stratigraphic noise. Our model successfully explains the trench data and
allows to determine an upper bound of the reliability of climate reconstructions from
seasonal to inter-annual time scales, depending on the number and the spacing of the
cores taken. Implications for our study region include that reliably detecting a warming
trend (0.1 ◦ C decade−1 ) in 50 years of data would require ∼ 10–50 replicate cores with
a horizontal spacing of at least 10 m. More generally, our results suggest that in order
to obtain high-resolution records of Holocene temperature change, fast measurements,
thus allowing multiple cores, are more important than to minimise analytic uncertainty
as the latter only plays a minor role in the total uncertainty.
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Introduction
Ice cores obtained from continental ice sheets and glaciers are a key climate archive.
They store information on past changes in temperature in the form of stable water
isotopes (EPICA community members, 2006), in greenhouse gas concentrations via
trapped air (Raynaud et al., 1993) and in many other parameters such as accumulation
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11, 5605–5649, 2015
Regional signal vs.
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rates (e.g., Mosley-Thompson et al., 2001) or aerosols (e.g., Legrand and Mayewski,
1997).
The quantitative interpretation of stable water isotopes builds on the strong relationship between the isotopic ratios in precipitation and local air temperature (Dansgaard,
1964; Fujita and Abe, 2006). Analysis of the isotope ratios recorded in single deep ice
cores provided milestones in the palaeo-climate research, including the investigation
of glacial-interglacial climate changes (Petit et al., 1999) and the existence of rapid
climate variations within glacial periods (Dansgaard et al., 1993).
In contrast to this coherent view from polar ice cores on millennial and longer time
scales, the reliability of single ice cores as archives of the Holocene climate evolution is
less clear (Kobashi et al., 2011). The small amplitude of Holocene climate changes and
the aim to reconstruct them at a high temporal resolution poses a challenge to the interpretation of ice-core signals. This is especially true for low-accumulation sites as here
the non-climate noise – to which we refer in this manuscript as the part of the isotopic
record that cannot be interpreted in terms of large-scale temperature variations – may
often be too high to accurately extract a climatic signal (Fisher et al., 1985). Despite the
challenges, quantifying the Holocene polar climate variability is the key foundation to
determine the range of possible future climate changes (e.g., Huntingford et al., 2013,
and references therein) as well as to test the ability of climate models in simulating
natural climate variability (Laepple and Huybers, 2014).
The quantitative estimation of climate variability from proxy data requires an understanding of non-climate influences in order to separate them from the climate signal
(e.g., Laepple and Huybers, 2013). Several mechanisms influence the isotopic composition of snow prior to and after its deposition onto the ice sheet and thus cause
non-climate noise in ice-core signals. Irregular deposition caused by wind and surface
roughness along with spatial redistribution and erosion of snow is a major contribution to non-climate variance (“stratigraphic noise”) (Fisher et al., 1985). Wind scouring
can additionally remove entire seasons from the isotopic record (Fisher et al., 1983).
Non-climate variability may further be introduced by spatial as well as temporal precip-
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itation intermittency (Persson et al., 2011; Sime et al., 2009, 2011). After deposition,
vapour exchange with the atmosphere by sublimation-condensation processes (SteenLarsen et al., 2014) can influence the isotopic composition of the surface layers; diffusion of vapour into or out of the firn driven by forced ventilation (Waddington et al.,
2002; Neumann and Waddington, 2004; Town et al., 2008) may represent an additional
component of post-depositional change. Finally, diffusion of water vapour through the
porous firn smoothes isotopic variations from seasonal to inter-annual and possibly
longer time scales, depending on the accumulation rate (Johnsen, 1977; Whillans and
Grootes, 1985; Cuffey and Steig, 1998; Johnsen et al., 2000).
In the last two decades, a growing number of studies analysed to which extent single
ice cores record a representative climate signal on sub-millennial time scales. One wellstudied region is Dronning Maud Land (DML) on the East Antarctic Plateau. Comparing
16 annually resolved isotope records from DML spanning the last 200 years, Graf et al.
(2002) found low signal-to-noise variance ratios of 0.14 for oxygen isotope ratios and
0.04 for accumulation rates. Karlöf et al. (2006) analysed 200 year-long records of oxygen isotopes and electrical properties in five cores with inter-site spacings of 3.5–7 km
and detected no relationship between the cores except for volcanic imprints. This result is consistent with Sommer et al. (2000a, b) who studied high-resolution records
of chemical trace species from three DML shallow ice cores (inter-site distances of
∼ 100–200 km) and discovered a lack of inter-site correlation on decadal time scales.
Reconstructed accumulation rates showed a weak but significant correlation between
two cores only on time scales larger than 30 years (Sommer et al., 2000a). The low
representativity of single low-accumulation records was also supported by process
studies comparing observed and simulated snow pits, the latter modelled by combining backward trajectories with a Rayleigh-type distillation model (Helsen et al., 2006).
While the model-data comparison exercise was reasonably successful for coastal highaccumulation regions of DML, it largely failed on the dryer East Antarctic plateau. Such
a relationship between accumulation rate and the signal-to-noise ratio of ice cores was
further demonstrated in different studies across the Antarctic continent (Hoshina et al.,
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Near Kohnen station, close to the EPICA deep ice core drilling site on Dronning
Maud Land (EDML, −75.0◦ S, 0.1◦ E, altitude 2892 m a.s.l., mean annual temperature
−44.5 ◦ C, mean annual accumulation rate 64 kg m−2 yr−1 , EPICA community members,
2006), two 1.2 m deep, 1.2 m wide and approximately 45 m long trenches in the firn,
named T1 and T2, were excavated during the austral-summer field season 2012/2013
using a snow blower. Each trench was aligned perpendicularly to the local snow-dune
direction. The horizontal distance between the starting points of T1 and T2 was 415 m.
To provide an absolute height reference, vertically aligned bamboo poles were stuck
into the snow every 60 cm applying a spirit level. Additionally, a laser level device was
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Data and methods
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2014; Jones et al., 2014; McMorrow et al., 2002). A similar question of representativity
also arises for Artic and Greenlandic records, although the higher accumulation rates
generally lead to a higher signal content (Fisher and Koerner, 1994; Steen-Larsen
et al., 2011; Gfeller et al., 2014).
Despite this large body of literature, quantitative information about the signal-to-noise
ratios and the noise itself is mainly limited to correlation statistics of nearby cores. While
a relatively good understanding of stratigraphic noise exists in Arctic records (Fisher
et al., 1985), this does not apply to large parts of Antarctica where the environment is
markedly different with the accumulation being considerably reworked in and between
storms (Fisher et al., 1985).
Here we provide a direct visualisation and analysis of the signal and noise in an
Antarctic low-accumulation region by an extensive two-dimensional sampling of the
firn column in two 50 m long snow trenches. Our approach, for the first time, offers
a detailed quantitative analysis of the spatial structure of isotope variability on a centimetre to hundred-metre scale. This is a first step towards a signal and noise model
to enable quantitative reconstruction of the climate signal and its uncertainties from ice
cores.
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where Rsample is the isotopic ratio of the sample ( O/ O) and Rreference that of a reference. The isotopic ratios are calibrated to the international V-SMOW/SLAP scale by
means of a linear three-point regression analysis with different in-house standards. Additionally, a linear drift-correction scheme and a memory-correction scheme (adapted
from van Geldern and Barth, 2012) is applied. The memory correction allows the reduction of repeated measurements per sample; here, we have used three repeated
measurements instead of six suggested for Picarro instruments when no memory correction is applied, thereby approximately halving the measurement time. The analytical
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precision of the calibrated δ O measurements of all trench samples is on average
0.09 ‰.
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used to check the bamboo pole heights, yielding in each snow trench a vertical accuracy better than 2 cm. No absolute height reference between the two trenches could be
established, but, based on a stacked laser level measurement, the vertical difference
between the trenches was estimated to be less than 20 cm.
Both trenches were sampled for stable-water-isotope analysis with a vertical resolution of 3 cm. In T1, 38 profiles were taken at variable horizontal spacings between 0.1
and ∼ 2.5 m. In T2, due to time constraints during the field campaign, only four profiles
at positions of 0.3, 10, 30 and 40 m from the trench starting point were sampled. All firn
samples (a total number of N = 1507) were stored in plastic bags and transported to
Germany in frozen state. Stable isotope ratios were analysed using Cavity Ring-Down
Spectrometers (L2120i and L2130i, Picarro Inc.) in the isotope laboratories of the Alfred Wegener Institute in Potsdam and Bremerhaven. The isotope ratios are reported
in the usual delta notation in per mil (‰) as
!
Rsample
δ=
− 1 × 103 ,
(1)
Rreference
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3.1
Results
Trench isotope records
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11, 5605–5649, 2015
Regional signal vs.
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Antarctic δ18 O
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The firn samples obtained from trench T1 provide a two-dimensional image of the δ O
structure of the upper ∼ 1 m of firn on a horizontal scale of ∼ 50 m (Fig. 1a).
The surface height profile of the trench exhibits a snow topography which is typical for the sampling region. It is characterised by small-scale dunes with their main
ridges elongated parallel to the mean wind direction, significantly higher density than
the surrounding firn, and typical spatial dimensions of 4 m width, 8 m length and 20 cm
maximum height (Birnbaum et al., 2010). Trench T1 features one prominent dune located between 25–40 m, accompanied by a dune valley between ∼ 8–18 m, and some
smaller-scale height variations. The peak-to-peak amplitude of the large dune undulation is ∼ 10 cm, the entire height variations exhibit a standard deviation (SD) of 2.9 cm.
The trench surface height profile is adopted as a local coordinate system (surface coordinates), the mean surface height serves as reference for absolute coordinates.
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Overall, the trench δ O record shows a diverse picture. The delta values observed
in T1 span a range from −54 to −34 ‰ with a mean of −44.4 ‰. A similar range of
−50 to −38 ‰ is observed in T2 (Fig. 2) with a mean of −44.0 ‰. We can clearly identify alternating layers of high and low isotopic composition in the T1 record. Following
the surface undulations, the uppermost layer (first 6 cm) shows high but also strongly
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variable δ O values between −54 and −34 ‰ with a variance of 19 (‰) , thereby already spanning the range of the entire trench record. Less negative values tend to be
located in the valleys, however, the limited data do not allow to conclude whether this
is a general feature. Located below this first layer, a band of generally lower δ 18 O values is found in an absolute depth of 5–20 cm, likewise following the snow surface but
exhibiting less horizontal variability with a range of −53 to −46 ‰.
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To further analyse the layering, we track the pronounced maxima and minima of δ O
values along the trench by automatically determining the local extrema of each isotope
profile and visually assigning summer and winter to these extrema, resulting in lateral
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Regional signal vs.
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seasonal layer profiles as a function of depth (Fig. 1b). Implicitly this assumes that the
past isotopic isolines are also temporal isolines which is a rough approximation considering the highly variable isotopic composition of the surface layer (Fig. 1a). Our results
of the seasonal layer profiles will thus likely overestimate the respective surface height
profile of each past season. To analyse the similarity between the past seasonal layer
profiles and the present surface, we calculate the root-mean-square deviation (rmsd)
between the vertical anomalies, i.e., the mean-subtracted seasonal layer profiles, and
the horizontal reference as well as the present surface height profile. We find that the
first summer layer follows the present surface undulations (rmsd difference of 1.8 cm
between the comparison to the horizontal and the surface profile reference). The next
three layers show on average a much weaker link with the present surface (rmsd difference of 0.5 cm), and the layers below 40 cm are on average horizontally aligned (difference of −0.8 cm). Comparably to the present surface undulations, the vertical layer
anomalies feature peak-to-peak amplitudes of 6–24 cm (average SD of 3.7 cm). Supporting our above assumptions, the vertical separation of the observed lateral layer
profiles is approximately 20 cm, in accord with the local mean annual accumulation
rate of snow (64 kg m−2 yr−1 ) and the mean firn density of ρfirn = 340 kg m−3 measured
in trench T1. The layering is strongly perturbed primarily in the depth of ∼ 60–100 cm
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for profile positions < 30 m. Here, a broad and diffuse region of rather constant δ O
values around −40 ‰ is present together with a very prominent, 20 cm-thick feature of
high delta values between 18 and 28 m.
The four profiles obtained from trench T2 (Fig. 2) show similar results as trench T1.
Roughly five seasonal cycles can be identified, however, with remarkable inter-profile
deviations especially at depths of 50–90 cm. This coincides with the region of strong
perturbations identified in T1. As in trench T1, the T2 profiles suggest a direct relation
between the isotopic layering and the local snow height profile for the surface snow,
i.e., till a depth of 10–20 cm. Below that, the profiles diverge considerably (not shown)
but show a better alignment on absolute coordinates (Fig. 2).
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Regional signal vs.
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On the horizontal dimension of the trenches, the observed lateral variance (Fig. 3)
reflects processes that are not related to variations of atmospheric temperatures as
these are coherent on this spatial scale. According to the terminology adopted here,
the lateral variance is non-climate noise. The link between the lateral seasonal layer
profiles of the isotopic composition and the present snow surface decreases with depth
(Fig. 1b). This has direct consequences for the analysis of the lateral variability of δ 18 O
values (Fig. 3). For the first 20 cm, the lateral variance is significantly higher when
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evaluated on absolute coordinates than on surface coordinates (mean of 16.0 (‰)
2
vs. 7.8 (‰) , p = 0.1). For older firn layers (z < 20 cm) the situation seems to reverse
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2
with a mean of 3.6 (‰) in the former and 4.5 (‰) in the latter case. In both cases,
the lateral variance shows a pronounced drop from high values in the surface layer
to rather constant values deeper in the firn. The overall mean lateral variability of the
2
2
T1 record is σl,T1 ' 5.9 (‰) . Due to the rather horizontal stratigraphy of the isotopic
composition in the deeper trench parts all further plots and calculations will refer to the
horizontal reference and not to the actual snow surface.
The observation of such a considerable lateral variance or noise level poses the
questions on how representative single firn profiles from low-accumulation sites are,
and how much they reflect the original climate signal that is sought to be reconstructed.
One indicator for the similarity of profiles is the pairwise Pearson correlation coefficient.
The possible correlations (N = 152) between single profiles of T1 and single profiles of
T2 (Fig. 4) are substantially scattered around a mean of ∼ 0.50 (1 SD = 0.13). Each
single correlation mimics the potential result obtained from correlating two “classical”
snow pits taken at a distance of 500 m. Due to the lack of an absolute height reference
between the trenches, vertical shifting of the T2 profiles of up to ±12 cm is allowed to
maximise the correlations. The relative majority (∼ 43 %) of the profile pairs shows an
optimal shift of +3 cm which is well below the estimated upper vertical height difference of the trenches. Our results indicate that only by chance the classical snow-pit
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Single-profile representativity
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Regional signal vs.
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Antarctic δ18 O
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Here, var(ε)/var(S) =: F −1 is the inverse of the signal-to-noise variance ratio of the
profiles, |x −y| is the inter-profile spacing, and λ denotes the decorrelation length of the
autocorrelation. The variance ratio determines the limit of Eq. (2) for |x − y| → ∞. It is
estimated from the data using the mean inter-profile correlation for the profile spacings
−1
between 10–35 m, giving a value of F = 1.1 ± 0.1. An estimate of the decorrelation
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length is obtained from the lateral δ O variations of T1 by calculating the autocorrelation at a lag of ∆` = 1 m. To account for the irregular lateral sampling, we apply the
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To quantify the spatial noise structure in the trench isotope record, we investigate the
inter-profile correlation as a function of profile spacing (Fig. 5). To this end, all possible
profile pairs for a given spacing are selected, allowing a tolerance in the lateral position
of 5 %, and the mean inter-profile correlation of the pairs is calculated. The correlation
approaches one for nearest neighbours and rapidly drops with increasing inter-profile
distance before it stabilises around a value of ∼ 0.5 for spacings &10 m.
This spatial correlation structure can be described using a simple statistical model:
We assume that each profile consists of a common signal S and a noise component
ε independent of the signal. The noise component is modeled as a first-order autoregressive process (AR(1)) in the lateral direction. The inter-profile correlation coefficient
can then be expressed (see Appendix A) as
|x − y|
var(ε)
1
rX Y =
exp −
.
(2)
1+
var(ε)
λ
var(S)
1 + var(S)
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Spatial noise structure
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method can yield two profiles that share significant common features (half of the profile
pairs show a correlation ≤ 0.49, only two pairs (∼ 1.3 %) exhibit a correlation above
0.8). In general, due to the inherent noise, single firn profiles cannot be regarded as
representative recorders of isotopic proxy signals on the vertical scales analysed here.
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9.5 (‰) , T2:
2
σv,T2
2
' 7.3 (‰) ) reduced by the noise variance. For T1 we obtain var(S) '
10
= 3.6 (‰)2 . This gives a variance ratio of ∼ 1.6 which is of the same order of
magnitude as the estimate from the inter-profile correlation but slightly underestimates
the signal strength.
3.4
Regional signal vs.
local noise in
Antarctic δ18 O
T. Münch et al.
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The spatial mean of all T1 profiles (Fig. 6) is highly correlated with the spatial mean
of the T2 profiles (rT1,T2 = 0.81), indicating a common seasonal isotopic signal reproducible over a spatial scale of at least 500 m. It is interesting to note that this value
is above most of the single inter-profile correlations (Fig. 4). Due to the surface undulations, the number of existing observations evaluated on absolute coordinates varies
for the first three depth bins. To obtain non-biased mean profiles, only the depth range
covered by all profiles is used in the averaging process. A vertical shift of the mean T2
profile of ±12 cm was allowed to maximise the correlation and, consistent to the results
obtained for single profiles, an optimal shift of +3 cm was obtained. In both profiles, we
observe five seasonal cycles spanning a range of ∼ 6–7 ‰ at the surface, but being
attenuated further down and exhibiting no clear sinusoidal shape in the “fourth” year
(65–90 cm depth). Interestingly, this obscured part without any clear signal of depleted
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δ O “winter” values is found in both trenches, indicating that this feature persists over
several hundred of metres and is thus likely of climatic origin. Despite the statistically
significant correlation (p = 0.01, accounting for the full autocorrelation structure and
allowing for vertical shifting of ±12 cm), pronounced differences between the mean
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Trench mean profiles
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2
σv,T1
−σl,T1
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Gaussian kernel correlation discussed in Rehfeld et al. (2011) and find that the noise
correlation has decreased to 1/e at a distance of λ ' 1.5 m.
The signal-to-noise variance ratio can also be directly estimated from the data if
2
we identify the noise variance with the mean lateral trench variance, var(ε) = σl , and
assume that the noise is isotropic and independent of the signal. Then, the signal
2
variance var(S) can be estimated with the mean down-core variance σv2 (T1: σv,T1
'
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profiles are present, such as a significantly lower isotopic composition of the T2 mean
between 50–80 cm and a considerably higher one within depths up to ∼ 40 cm as well
as for the lowermost region of the trenches.
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In order to obtain annual-mean δ O time series we define annual bins through
the six local maxima determined from the averaged profile of the two mean trench
profiles. The mean peak-to-peak distance of these maxima is 19.8 cm, consistent with
the accumulation rate. Three alternative sets of annual bins are derived from the five
local minima as well as from the midpoints of the slopes flanking these minima. The
annual-mean time series derived from these four sets are averaged to obtain a single
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time series for each trench (Fig. 6). The correlation of the average annual-mean δ O
time series of 0.87+0.07
−0.20 (range represents the four binning methods) is comparable to
that of the mean seasonal profiles (0.81). However, five observations of annual means
are too short to reliably estimate the correlation and its significance.
CPD
11, 5605–5649, 2015
Regional signal vs.
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Antarctic δ18 O
T. Münch et al.
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Climate reconstructions based on proxy data rely on the assumption that at least part
of the measured signal is related to a climate parameter, such as temperature in case
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of ice/firn-core derived δ O (Dansgaard, 1964). However, proxy signals are inherently noisy with uncertainties arising prior to deposition of the proxy into the archive,
post-depositionally during archive storage, as well as later in the human sampling and
measurement process (Evans et al., 2013; Laepple and Huybers, 2013; Steig, 2009).
Our trench data confirm earlier results that individual firn records of δ 18 O from lowaccumulation regions are strongly influenced by local noise (Fisher et al., 1985; Karlöf
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et al., 2006). However, going beyond this finding, our two-dimensional δ O dataset
also allows to determine the spatial structure and to learn about the causes of the
noise. In this section, we discuss our findings in the context of the possible noise
sources and derive implications for inter-annual climate reconstructions based on firn
cores.
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Discussion
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Regional signal vs.
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Antarctic δ18 O
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A hypothetical, horizontally stratified trench with zero isotopic variance in lateral direction would yield perfectly correlated single profiles. However, in the actual trenches
we observe a high lateral variance (see Fig. 3 for T1) with a mean variance that is
comparable to the mean down-core variance (Table 1).
Several pre- and post-depositional effects induce lateral variance of the firn layer,
the relative importance of each varies on the spatial scales considered. Starting on the
m-scale, the principal contribution is induced by the surface roughness, closely related
to snow drift events including spatial redistribution, erosion, reworking and dune formation (“stratigraphic noise”, Fisher et al., 1985). Possibly, exchange of water vapour with
the atmosphere by sublimation-condensation processes (Steen-Larsen et al., 2014),
potentially accompanied by forced ventilation (Waddington et al., 2002; Neumann and
Waddington, 2004; Town et al., 2008), acts as a further noise source. Going to larger
spatial scales (&1 km), spatial precipitation intermittency (e.g., Richardson et al., 1997;
Persson et al., 2011; Sime et al., 2009, 2011) presents an additional component, influencing a certain snow layer via spatially varying precipitation weighting.
The down-core variance includes the isotopic signal from seasonal and longer climate variations. In addition, the vertical isotope record is also subject to modifications arising prior to and after the deposition of snow. Temporal precipitation intermit18
tency can bias the δ O record (Laepple et al., 2011) but also induces vertical variability caused by inter-annual variations of the timing of precipitation events (Persson
et al., 2011; Sime et al., 2009, 2011). Diffusion of water vapour through the porous
firn along seasonal isotopic gradients (Johnsen, 1977; Johnsen et al., 2000; Whillans
and Grootes, 1985; Cuffey and Steig, 1998) obscures seasonal and longer isotopic cycles, depending on the accumulation rate. Forced ventilation acts perpendicular to the
pressure isolines in the firn, generated by the steady wind flow across the undulating
surface (Waddington et al., 2002). Depending on the dune undulations, this may en-
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Trench δ18 O variance levels
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4.1
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hance the vertical diffusion in the first tens of centimetres of firn and shorten the time
for the snowpack to reach isotopic equilibration.
The pronounced drop in the lateral variance with depth (Fig. 3) can likely be explained by isotopic diffusion. This is suggested by a simple numerical estimate diffusing an artificial trench record that initially exhibits a rectangular isotope variation (25 %
summer precipitation) as well as a sinusoidal surface topography with a wavelength
of 10 m and a peak-to-peak amplitude of 10 cm (Fig. B1, see Appendix B for details).
While these are promising results, the theoretical estimate of Waddington et al. (2002)
as well as a numerical diffusion model including forced ventilation by Neumann and
Waddington (2004) showed that the true rate of diffusion in the first metre might be
higher. Furthermore, Town et al. (2008) demonstrated that forced ventilation also attenuates the seasonal cycle. In total, at the current stage of investigation we are not
able to clarify the importance of water vapour exchange and forced ventilation. For this,
more field measurements and a thorough numerical treatment are necessary.
CPD
11, 5605–5649, 2015
Regional signal vs.
local noise in
Antarctic δ18 O
T. Münch et al.
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Spatial structure of lateral variance
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In Sect. 3.3 we showed that the inter-profile correlation as a function of profile spacing
(Fig. 5) can be described by a common signal overlayed by lateral noise following an
AR(1) model.
This demonstrates firstly that each single trench profile features a local isotopic signal common only over a few metres which is induced by small-scale covarying noise.
The decorrelation length of ∼ 1.5 m of this noise is related to the intermittent deposition
of snow and, in particular, to the dune scale: A sinusoidal surface height variation with
a wavelength ν of .10 m would lead to zero autocorrelation for a shift of ν/4, similar to
our observations. While the real surface topography is more complicated, it suggests
18
that stratigraphic noise is an important noise component in our δ O records. In addition, vapour exchange with the atmosphere driven by forced ventilation might contribute
to the overall noise level since it is likewise related to the surface roughness. Secondly,
the remaining correlation of ∼ 0.5 for inter-profile spacings of &10 m, implying roughly
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11, 5605–5649, 2015
Regional signal vs.
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Antarctic δ18 O
T. Münch et al.
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Our statistical model of covarying stratigraphic noise allows to determine the seasonal
signal content depending on the number of profiles and the profile spacing. As the
model is entirely based on parameters estimated from the T1 data, we can use the
T2 data to validate the model. Therefore, we determine and predict the correlation
of an averaged set of T1 profiles with the T2 trench mean, the latter thus serving as
a reference isotopic signal.
To determine the correlation from the data for a given number of profiles and a profile
spacing, all possible unique sets of T1 profiles are selected that fulfill the given criteria.
Due to the uneven spacing of the T1 profiles, we allow an absolute uncertainty of the
spacing between the profiles in a set of 0.5 m. The correlation is given as the mean
correlation over all sets. Empirically, we find a steady increase in the correlation with
the T2 reference for increasing number of profiles used in the T1 set (Fig. 7). The
observed increase in correlation is expected since also for autocorrelated noise the
noise variance of the set decreases with the number of profiles. Additionally, as a direct
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Representativity of isotope signals on seasonal to inter-annual time scales
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the same amount of signal and noise variance in single profiles, is due to a regionally coherent (.1 km) isotope signal, supported by the fact that it is comparable to the
mean correlation between individual T1–T2 records (Fig. 4). However, this regional isotope signal does not directly translate into a regional climatic signal of local surface air
temperature as various effects can influence the isotopic composition of precipitation
(Jouzel et al., 1997). Further, there is the possibility of an additional noise component
with a spatial decorrelation length larger than the distance between both trenches, for
example caused by spatial precipitation intermittency.
The spatial autocorrelation structure and the inter-profile correlation provide estimates of an optimal sampling strategy for firn-coring efforts in the study region. To
ensure that the local noise is uncorrelated, single profiles should be spaced at distances several times the decorrelation length. Visually, we find a minimum spacing of
∼ 10 m to be optimal (Fig. 5).
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Regional signal vs.
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consequence of the autocorrelation structure, the correlation increases with a wider
spacing between the individual profiles of the T1 set (Fig. 7). A given number of profiles
at a spacing of 2.4 m share more common noise variance than the same number of
profiles at a larger spacing. Thus, when the two profile sets are averaged, the latter set
will show a higher correlation with the reference signal. This finding also explains the
comparable reduction of the noise levels of the trench mean profiles (for T1 the levels
drop by 46 % compared to the mean of the individual down-core variances, for T2 by
55 % (Table 1)): The 38 T1 profiles have varying inter-profile distances from 0.1–2.5 m,
whereas the four T2 profiles are already spaced at large, more optimal distances of
10–20 m.
Our noise model allows to calculate the theoretical inter-trench correlation coefficient
(Eq. A13). Using the variance ratio of F −1 = 1.1 obtained in Sect. 3.3, the model prediction is in good agreement with the empirical data (Fig. 7). We can conclude that the
first-order autoregressive noise model captures the major noise component for isotopic
records on spatial scales of at least 500 m as well as on temporal scales of a few years.
With the noise model validated between the trenches, implications for climate reconstructions using firn-core isotope records can be deduced. We define the representativity of a set of trench profiles as the correlation of this set with a hypothetical, common
climate signal (Eq. A12). This representativity can be interpreted as the upper limit of
the correlation to a temperature times series obtained from a weather station located
in the study region. In the limit of independent noise terms (vanishing autocorrelation),
our definition of representativity yields the same expression as derived by Wigley et al.
(1984).
The representativity is time-scale-dependent since signal and noise variance are
both a function of the time scale. For example, the seasonal variability is much larger
than any year-to-year variations of the istopic signal. Analysing seasonal variability,
the representativity can be readily calculated with the variance ratio F −1 given above.
For reconstructions on inter-annual time scales, the isotope records are additionally
averaged in the vertical direction and thus, the results depend on the vertical noise
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Regional signal vs.
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The effective annual noise variance var(ε)σ annual (Eq. A11) for case I depends on the
autocorrelation parameter aannual which is estimated from the mean autocorrelation
function of the vertical δ 18 O data of T1 after subtracting the mean seasonal profile.
We obtain a value of aannual ≈ 0.61 for a lag of ∆` = 3 cm, equivalent to a decorrelation
length of λannual ≈ 6 cm. As the best-possible estimate, an annual signal variance of
2
18
var(S)annual ' 0.68 (‰) is obtained from the mean of the variances of the annual δ O
time series (Fig. 6) of the two trenches (Table 1). The seasonal noise variance var(ε) is
set to the observed mean lateral T1 variance (Table 1). Altogether, we obtain an annual
−1
−1
variance ratio of Fannual ' 1.8 for case I, and of Fannual ' 8.7 for case II. Note that using
the seasonal noise variance as calculated from the entire trench data might represent
a slight overestimation given the exceptionally high variability observed in the surface
layer (Fig. 3).
For single profiles, the representativity on the seasonal time scale is around 0.69
(Fig. 8). On the inter-annual time scale, single profiles have a representativity of 0.59
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covariance. Snow-pit studies around Vostok station have shown significant temporal
non-climatic oscillations of the isotopic composition (Ekaykin et al., 2002), indicating
a vertical spatial noise structure. The observed time scales of the oscillations range
0
1
2
from 10 –10 , possibly up to 10 years, and are linked to the movement of accumulation waves of various scales. Here, due to the limited data coverage in vertical direction,
we are only able to investigate two limiting cases. As the simplest best-case scenario
(case I), the vertical noise covariance is given by an AR(1) process as in the lateral
direction. In the worst case (case II), averaging one annual firn layer does not reduce
the noise level at all, assuming a complete interdependence of the noise on the subannual time scale. The variance ratio of noise over signal is for the inter-annual time
scale thus given by

 var(ε)σ ∗2annual case I
var(ε)
annual
−1
var(S)annual
Fannual =
=
(3)
var(S)annual  var(ε)
case II.
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Regional signal vs.
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in the best-case scenario (Fig. 8a) and a much lower one in the worst-case scenario
(0.32, Fig. 8b).
In general, the representativity increases with the number of profiles averaged, and
the increase is stronger for larger inter-profile spacings. However, spacings above 10 m
do not increase the representavities any further as the stratigraphic noise is practically
decorrelated (Fig. 5). To obtain a representativity of 0.8 for inter-annual signals with
profiles separated by 10 m, one needs to take a minimum of 4–16 cores (from best
to worst case). Demanding a representativity of 0.9, the number of cores required
increases to 8–37.
The low single-profile representativity on the inter-annual time scale is consistent
with previous findings from Dronning Maud Land. The 16 annually resolved δ 18 O
records of the study of Graf et al. (2002), taken in an area extending 500 km from
east to west and 200 km from north to south, showed a low signal-to-noise variance
ratio in the individual records of F = 0.14. Due to the large inter-profile spacing, the
stratigraphic noise covariance in the records is decorrelated. Then, the variance ratio F from the cross-correlations
directly translates into the representativity of a single
√
profile as rSX = 1/ 1 + F −1 ' 0.35, consistent with our findings for the worst-case scenario (case II). However, this accordance does not necessarily mean that our worstcase scenario is the more realistic one since the measured cross-correlations are also
subject to potential dating uncertainties and additional variability caused by spatially
varying precipitation-weighting and possibly other effects.
Stratigraphic noise does not only affect isotopic records but also other proxies derived from ice cores, such as aerosol-derived chemical constituents. Gfeller et al.
(2014) investigated the representativity of ion records from five Greenland firn cores
on seasonal and inter-annual time scales, taken at varying distances from 7–10 m in
the vicinity of the NEEM drilling site. With the definition of representativity based on
the theoretical work of Wigley et al. (1984), for inter-annual time scales Gfeller et al.
(2014) found representativities of ∼ 0.55–0.84 for single cores, and of ∼ 0.84–0.95 for
the average of all five cores, depending on the ions considered. These numbers are
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Regional signal vs.
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The noise level identified in our trench data poses a significant challenge for the interpretation of firn-core-based climate reconstructions on seasonal to inter-annual time
scales. In the following, we discuss examples of implications of the noise model concerning (1) the required measurement precision of water isotopes, (2) the potential
noise fraction in isotope signals of the EDML ice core and (3) the detectability of anthropogenic temperature trends in low-accumulation firn cores.
The noise of an isotopic signal consists of the stratigraphic noise discussed here as
well as the noise caused by the measurement process. Thus, obtaining the best signal
is a trade-off between measurement precision and the amount of analysed samples.
For seasonal as well as on inter-annual time scales, the measurement uncertainty of
18
the trench data of ∆δ O = 0.09‰ is much lower (∼ 4–8 %) than the standard deviation
of the stratigraphic noise (Table 2). This ratio is independent of the temporal resolution
if a lower temporal resolution is obtained by averaging annually resolved data as both,
the noise level and the measurement uncertainty, decrease by the same amount in
the averaging process, assuming independence between the samples. In such a case,
priority should be given to measuring and averaging across multiple cores in order to
reduce the (stratigraphic) noise levels instead of performing high-accuracy measure18
ments on single cores, given that we are only interested in δ O. As an example, for
Cavity Ring-Down Spectrometers as those that have been used for this work, much
faster measurements are possible by reducing the number of repeated measurements
down to one per sample, resulting only in a slight decrease in measurement precision
when a memory correction scheme as applied to our data is used.
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Implications
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slightly higher than our best-case-scenario results for δ O, a fact which is expected
as the accumulation rate at the NEEM site is about three times higher than at Kohnen
station (NEEM community members, 2013).
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5624
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Discussion Paper
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Regional signal vs.
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T. Münch et al.
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If a lower temporal resolution is obtained by a coarser sampling of firn cores, the
measurement error to stratigraphic noise ratio will depend on the analysed resolution
(Table 2). For a resolution corresponding to ten years, our measurement uncertainty
might amount to up to 25 % of the stratigraphic noise level, assuming independence
of the stratigraphic noise between the years. For our data, the noise level of single
cores would become comparable to the measurement uncertainty for averages over
∼ 154 years (case I) or ∼ 728 years (case II).
The deep EPICA Dronning Maud Land ice core obtained in the vicinity of Kohnen
station shows the climate evolution in Antarctica over the last 150 000 years (EPICA
community members, 2006). Oerter et al. (2004) studied a section of the core covering
the last 6000 years with a resolution of ten years (their Fig. 2). We find a decadal
variance for this part of the core of ∼ 0.57 (‰)2 . If we assume that our estimates of
the stratigraphic noise variance hold over the last couple of thousand years, then ∼
20–100 % of the decadal variance seen in the EDML core over this time period might
be simply noise (Table 2). In order to reconstruct the Holocene climate variability of the
last millennium from low-accumulation regions, there is thus the clear need to either
average across multiple cores based on the results of the previous section, or, if only
the magnitude of variability is of interest, to correct the proxy variability for the noise
contribution (e.g., Laepple and Huybers, 2013).
As a final example of applying our noise model, we estimate the ability of firn cores
close to the Kohnen station to reconstruct a potential warming trend of the last decades.
In the last 50 years, the surface temperature over East Antarctica has warmed by about
half a degree (Steig et al., 2009). The probability to detect this trend or to reconstruct
5 18
its slope is estimated using a Monte Carlo approach creating 10 δ O time series
consisting of a signal (the linear temperature trend) and uncorrelated Gaussian noise
with variance equal to the annual trench noise variance for the best as well as the
worst case (Table 2). The trend is detected when the correlation of the time series with
the signal is positive at the significance level of p = 0.05. We define the probability for
determining the right slope as the fraction of cases where a linear regression yields
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a slope that lies in a range of 25 % around the true slope. To simplify matters, we assume a temperature-to-isotope gradient for δ 18 O of 1 ‰ K−1 , given the considerable
uncertainties associated with the spatial and temporal gradients discussed in the lit18
erature (e.g., Jouzel et al., 1997). We note that in general the δ O slope very likely
−1
−1
lies below 1 ‰ K (∼ 0.8 ‰ K for DML, EPICA community members, 2006) which
implies yet lower detection probabilities since the signal variance is then even smaller
compared to the noise variance. Finally, in the case of multiple cores it is assumed that
they are taken at distances on which the autocorrelation of the stratigraphic noise is
decorrelated (≥ 10 m).
The probability to detect the trend or to reconstruct its slope is below 20 % for single cores (Fig. 9). To reliably (> 80 % of the cases) detect the warming over the East
Antarctic plateau, our results suggest that averaging across at least ∼ 10–50 firn cores
taken at spacings of 10 m (Fig. 9) is needed, depending on the scenario for the annual
noise variance. Inferring the right slope would need three times that number of cores.
CPD
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Regional signal vs.
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T. Münch et al.
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Abstract
We presented extensive oxygen stable water isotope data derived from two snow
trenches excavated at Kohnen station in Dronning Maud Land, Antarctica. The twodimensional approach allowed a thorough investigation of the representativity of single
firn-core isotope profiles, as well as of the spatial structure of the signal and noise over
spatial scales of up to 500 m and a time span of approximately five years.
The trench data confirm previous studies that single isotope profiles obtained from
low-accumulation regions are poorly correlated and do not show a coherent signal, but
also demonstrated that the spatial average of a sufficient number of profiles provides
a representative isotopic signal. We further show that single profiles are strongly influenced by local, small-scale noise that exhibits a spatial covariance. A statistical model
describing this noise as a first-order autoregressive process successfully explains the
observed covariance structure and allows to reproduce the observed correlation statis-
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5626
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Discussion Paper
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Regional signal vs.
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Antarctic δ18 O
T. Münch et al.
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tics between the trenches. The autocorrelation of the noise occurs on spatial scales
that are of the same order of magnitude as the surface height variations introduced by
sastrugi and dunes and the intermittent deposition of snow, suggesting stratigraphic
noise as a major noise source. Extending the ordinary stacking of isotope records, our
results are used to infer appropriate sampling strategies. We derive the representativity
of isotope profiles for seasonal to annual resolution depending on the number of firn
cores and the inter-core spacing. For our low-accumulation (64 mm w.eq. yr−1 ) study
region, we find an optimal profile spacing of about 10 m where the noise covariance is
sufficiently decorrelated. The representativity depends on the time scale: For seasonal
resolution, five profiles taken with the optimal spacing are sufficient to obtain representative (R > 0.9) isotope signals; on inter-annual time scales, ∼ 2–8 times as many
profiles would be needed.
The low representativity of single firn profiles at our site hampers the quantitative
interpretation of isotope in terms of climate variations. The noise level observed in the
trench data suggests that large parts of the decadal variations seen in the EPICA DML
ice core over the last 6000 years might be noise. In addition, we show that faithfully
reconstructing the recent positive temperature trend observed over the East Antarctic
plateau is impossible by drilling only single cores; instead, averaging at least 10–50
firn cores would be necessary. This task is rendered easier by the fact that the annual
noise level is substantially larger than typical measurement uncertainties. Therefore, for
high-resolution single-proxy reconstructions it might be more advantageous to conduct
less precise measurements, e.g., by operating Cavity Ring-Down Spectrometers with
only one injection per sample, for the benefit of analysing many cores.
Since the stratigraphic noise is related to the intermittent deposition of snow and
the formation of surface dunes, it depends primarily on the local accumulation rates,
besides further factors such as wind strength, temperature, seasonal timing of the precipitation and snow properties. Therefore, to a first approximation we expect that our
representativity results improve (worsen) for regions with higher (lower) accumulation
rates. In effect, results similar to ours likely hold for large parts of the East Antarctic
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The Pearson pairwise correlation coefficient of two time series, or profiles, X and Y
reads
cov(X , Y )
,
σX σY
(A1)
cov(X , Y ) = hX Y i − hX i hY i .
11, 5605–5649, 2015
Regional signal vs.
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fn (t)
Xn (t) = S(t) + ε
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Here, h·i denotes the temporal average, thus the spatial average in vertical direction for
a trench profile.
We now assume that a trench isotope profile Xn (t) consists of a signal part S(t) and
fn (t) that is independent from the signal and following a standard
a noise component ε
normal distribution. In addition, to account for the spatial covariance of the noise in
lateral direction, we assume each noise term to be following an AR(1) autoregressive
proccess,
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(A2)
Discussion Paper
where σX and σY are the standard deviations of profile X and profile Y , respectively,
and cov(X , Y ) is the covariance of the profiles given by
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rX Y =
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Appendix A: Derivation of noise model
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plateau, but trench-like approaches in West Antarctica and Greenland – regions with
considerably higher accumulation rates – are needed. In addition, studies with deeper
trenches that cover longer times of isotopic variations are necessary to enhance our
knowledge about the vertical noise covariance structure which is crucial to determine
the representativity on longer time scales. Deeper trenches would also allow to link our
representativity results to actual correlations with temperature time series derived from
weather stations. The latter is part of ongoing work at Kohnen station.
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p
1 − a2 εn (t) .
(A3)
Here, a is the autocorrelation parameter with 0 ≤ a ≤ 1, and the square-root term in
front of εn (t) is a normalisation. If we consider P equidistant trench profiles numbered
1, 2, . . ., P , the noise term of profile n can be given recursively,
Xn (t) = S(t) + an−1 ε1 (t) +
1 − a2
n
X
an−i εi (t) .
(A4)
With the help of Eq. (A4), we can calculate the spatial mean of a set of N trench profiles,
(A5)
k∈{ν>1, ν≥i }
var(S) + cov(εX , εY )
var(S) + var(ε)
.
(A6)
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rX Y =
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|
∗
where we have defined ν := {n1 , n2 , . . ., nN } and ν := max(ν).
From Eq. (A1) the inter-profile correlation coefficient can be calculated for general
kinds of covarying noise terms, cov(εX , εY ) 6= 0. With cov(X , Y ) = var(S) + cov(εX , εY ),
var(εX ) = var(εY ) ≡ var(ε) and therefore var(X ) = var(S) + var(εX ) ≡ var(Y ) we obtain
15
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i =2
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n
o
X (t) := Xn1 (t), Xn2 (t), . . ., XnN (t)


nN

X
1

=S(t) +
ai −1  ε1 (t)
N
i =n1


n1
nN

p
X
X
an1 −i εi (t) + · · · +
anN −j εj (t)
+ 1 − a2 

i =2
j =2




!
ν∗


p
X
X
X
1
ν−1
k−i 
2

=S(t) +
a
ε1 (t) + 1 − a
a
εi (t)

N ν
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i =2
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5
p
Discussion Paper
= S(t) + aεg
n−1 (t) +
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cov(εX , εY ) = aξ var(ε) with ξ :=
Here, |x−y| is the distance between profile X and Y , and ∆` ist the spacing of adjacent
profiles. This can be seen ifD we set,E without loss of generality, X = X1 and Y = Xn and
cov(S, X )
;
σS σX
(A8)
correlating two profile sets yields the amount of variance shared by the sets,
15
cov(X , Y )
.
σX σY
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For statistically independent signal and noise terms we have cov(S, X ) = var(S). For
cov(X , Y ) we assume that one profile set is derived from T1, the other from T2. As
the trenches are separated by ∼ 500 m, the noise terms are to a good approximation
decorrelated, and therefore cov(X , Y ) ' var(S). What is left to calculate is the variance
σ 2 of a profile set. A straightforward calculation, again noting that only products of
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5629
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20
rX Y =
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rSX =
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calculate the spatial mean εX1 εXn , noting that only products of identical noise terms
have non-vanishing covariance. The parameter a is the value of the autocorrelation
function at lag one, a = exp −∆`/λ , where λ is the typical length scale on which the
autocorrelation decreases to the value of 1/e. Thus, the covariance of the noise terms
decreases exponentially with increasing inter-profile spacing |x − y|.
To obtain the representativity of a trench profile set, we correlate the profile set with
the signal S(t),
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(A7)
|
5
|x − y|
.
∆`
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Further, the idendity cov(εX , εY ) = hεX εY i, holds for noise. Thus, for AR(1)autocorrelated noise (Eq. A4) the covariance reads
X
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2
2
2
2
σ = hX i − hX i = hX i − hSi
2
X
= var(S) + var(ε)
∗2
.
is the effective noise variance of the profile set using the definition
∗
!2
5
σ
∗2
X
X
aν−1
:=
(A10)
+ (1 − a2 )
ν
X
X

i =2
ν
2

ak−i  .
(A11)
k∈{ν>1,ν≥i }
rX Y '
(A12)
∗2
1+
10
;
var(ε) σX
var(S) N 2
1
.
∗ 2 ∗ 2 1/2
σ
σ
var(ε)
var(ε)
1 + var(S) X2
1 + var(S) Y2
N
∗2
For vanishing autocorrelation, a → 0, Eq. (A11) gives σ → N. Thus, the representaX
tivity of a profile set, Eq. (A12), simplifies to the classical result
a→0
1
rSX −−−→ q
,
1 var(ε)
1 + N var(S)
(A14)
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X
11, 5605–5649, 2015
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N
(A13)
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1
rSX = r
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By combining Eqs. (A8) and (A9) with Eq. (A10), respectively, we finally obtain expressions for the representativity of a trench profile set as well as for the shared variance
of a T1 and a T2 profile set:
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X
X
N2
|
Here, var(ε)σ
σ
∗2
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identical noise terms do not vanish in the averaging process, yields
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5
where the noise variance scales with the number of profiles averaged.
For the full trench data, Eqs. (A12)–(A13) are referred to as the representativities on
var(ε)
the seasonal time scale with the corresponding seasonal variance ratio of var(S) . On
the inter-annual time scale, this variance ratio is replaced by the corresponding annual
var(ε)
ratio of var(S)annual , where for the annual noise variance, var(ε)annual , the two limiting
annual
cases discussed in the text are used.
|
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Regional signal vs.
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assuming zero vertical strain rate. The time t of burial since deposition is expressed in
terms of the depth of the respective snow parcel using the present accumulation rate ḃ
5631
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20
with a peak-to-peak amplitude of 2∆ = 10 cm and a wavelength of λ = 10 m.
For the numerical diffusion calculation, the diffusivity is taken approximately as a con18
−8
2 −1
stant over the first metre of firn with a value for δ O of D ≈ 2.9 × 10 cm s , which
has been calculated according to Johnsen et al. (2000) adopting the relevant parameters for Kohnen station. The diffusion length is modeled to vary with time as (Johnsen
et al., 2000)
p
(B2)
σdiff (t) ∼ 2Dt ,
11, 5605–5649, 2015
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15
To estimate the effect of isotopic diffusion through the porous firn on the lateral δ O
18
variance of the trenches, we apply a simple numerical approach. An artificial δ O
trench of 45 m length and 1.2 m depth is built by creating isotope profiles with a rect18
angular δ O variation (expressed as relative variation between −1 and 1) adopting
a summer fraction of 25 %. The lateral resolution is set to 0.6 m, resulting in 76 profiles; the vertical resolution is fixed at 0.5 cm. Each profile is vertically shifted to mimic
a surface height variation d of the form
2π
d (x) = ∆ · sin
x
(B1)
λ
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Appendix B: Estimate of the influence of isotopic diffusion
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5
−9
−1
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−1
of snow, t(z) = z/ḃ with ḃ = 0.2 m yr ≈ 6.3 × 10 m s . In the numerical approach,
for each depth z(t) the trench profiles are diffused with respect to the respective diffusion length σdiff by convoluting the original signal with a Gaussian with a standard
deviation of σdiff (t (z)).
18
The numerical lateral δ O trench variance after diffusion is in qualitatively good
agreement with the observational data of trench T1 (Fig. B1).
|
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10
Acknowledgements. We thank all the scientists, technicians and the logistic support who
worked at Kohnen station in the 2012/13 austral summer; especially Melanie Behrens, Tobias Binder, Andreas Frenzel, Katja Instenberg, Katharina Klein, Martin Schneebeli, Jan Tell
and Stefanie Weissbach, for assistance in creating the trench dataset. We further thank the
technicians of the isotope laboratories in Bremerhaven and Potsdam, especially York Schlomann and Christoph Manthey. All plots and numerical calculations were carried out using the
software R: A Language and Environment for Statistical Computing. This work was supported
by the Initiative and Networking Fund of the Helmholtz Association Grant VG-NH900.
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down-core σv2
5.9
5.3
9.5
7.3
seasonal
5.1
3.3
inter-annual
1.15
0.21
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T2
lateral σl2
2
σa
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2
σv
Discussion Paper
Table 1. The variance levels observed for the two trenches: The lateral variance is the mean
horizontal variance over all depth layers, the down-core variance gives the mean vertical variance over all respective trench profiles. The seasonal as well as the inter-annual variance levels
denote the variances of the respective mean seasonal and inter-annual δ 18 O time series of the
2
two trenches (Fig. 6). All numbers are in units of (‰) .
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18
SD in ‰
∆δ O/SD
5.9
1.25
5.9
0.13
0.59
2.43
1.12
2.43
0.36
0.77
4%
8%
4%
25 %
12 %
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seasonal
annual: case I
annual: case II
10 yr-avg.: case I
10 yr-avg.: case II
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11, 5605–5649, 2015
Regional signal vs.
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time scale
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Table 2. The noise variance and standard deviation (SD) of the trench data and the ratio of the
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measurement uncertainty (∆δ O = 0.09 ‰) and the respective noise SD, given for different
time scales and for the two scenarios of the annual noise variance.
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δ O [permil]
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Regional signal vs.
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Figure 1. (a) The two-dimensional δ 18 O profile of trench T1. The depth scale is relative to the
mean snow height (long-dashed black line); the solid black line shows the local snow height at
the sampling positions which are indicated by black dots above the snow profile. White gaps
indicate missing data. (b) The stratigraphy of trench T1 expressed as the seasonal layer profiles
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by tracking the local δ O extrema.
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depth [cm]
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depth [cm]
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δ18O [‰]
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Regional signal vs.
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T. Münch et al.
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0.3 m
10 m
29.8 m
40 m
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depth [cm]
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profile positions:
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Discussion Paper
Figure 2. The four isotope profiles from trench T2 as a function of depth below the mean snow
height.
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parallel to horizon
parallel to snow surface
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variance [(‰)^2]
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20
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Regional signal vs.
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0
20
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depth [cm]
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Figure 3. The lateral variance of T1 as a function of depth below the mean snow height. Blue
lines with circles give the lateral variance as calculated horizontally, red lines with circles display the variance computed for consecutive slices following the present snow surface. Dashed
horizontal lines show the mean variance of each variance profile for the depth ranges of 0–20
and 20– ∼ 110 cm where the shadings represent the 90 % confidence intervals of the respective
mean.
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frequency
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Regional signal vs.
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T. Münch et al.
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5
0
0.4
0.6
0.8
Figure 4. Histogram of all possible pairwise correlations (N = 152) between single profiles of
trench T1 and single profiles of trench T2. Displayed are the maximum correlations allowing
vertical shifts of the T2 profiles of up to ±12 cm. Shown as a red line is the correlation between
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the mean δ O profile of T1 and the mean δ O profile of T2 (Fig. 6).
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Trench 1
AR(1) noise model
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mean correlation
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Regional signal vs.
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0.2
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inter-profile spacing [m]
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0.3
Figure 5. The mean inter-profile correlation as a function of profile spacing for T1 (black line
with filled circles). Shadings denote the standard error of the mean (undefined if just one profile
pair is found for a given spacing), for each spacing calculated adopting an effective number of
profile pairs that is set to the lower value of the actually found number of pairs and the effective
degrees of freedom for the trench record in lateral direction. The dashed black line denotes the
theoretical inter-profile correlation calculated for first-order autoregressive noise (AR(1)).
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2012
2011
2010
2009
2008
Trench 1
Trench 2 - not shifted
Trench 2 - shifted by 3 cm
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δ18O [‰]
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Regional signal vs.
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T. Münch et al.
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Trench 1 annual mean
Trench 2 annual mean
0
20
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depth [cm]
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Figure 6. Comparison of the mean seasonal δ 18 O profiles as a function of depth below the
mean snow height obtained from trench T1 (black solid line) and T2 (red solid line). Vertical
shifting of ±12 cm was allowed to maximise the correlation, resulting in a shift of +3 cm of the
original T2 mean profile (black dashed line). The mean profiles are well correlated with rT1,T2 =
0.81. Additionally, red and black points with lines give the approximate annual-mean δ 18 O time
series for the trenches. Shadings represent the range due to different binning definitions. Note
that the first and last value of the annual-mean time series (years 2012 and 2008) are biased
since the trench data are incomplete here. The vertical dashed grey lines are the positions of
the six local maxima of the average profile obtained from the trench mean profiles.
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0.9
trench results
AR(1) noise model
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correlation
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0.8
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11, 5605–5649, 2015
Regional signal vs.
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Antarctic δ18 O
T. Münch et al.
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0.6
1
5
10
number of profiles
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Figure 7. The correlation between a set of averaged T1 profiles and the mean of all T2 profiles
depending on the number of profiles in the T1 set and their inter-profile spacing. Three different
spacings are investigated: 2.4 m (black), 4.8 m (red) and 9.6 m (blue). Solid lines show the
results for the actual trench data, dashed lines display the theoretical correlations calculated
for AR(1) autoregressive noise. The trench results are given as the mean of the correlations
obtained for all possible unique sets of profiles separated by the given spacing and are only
calculated when at least 15 sets are available.
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2.4 m
4.8 m
9.6 m
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(a)
case I
case II
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1.0
correlation
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T. Münch et al.
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Regional signal vs.
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Antarctic δ18 O
|
Figure 8. The representativity of an average set of δ O firn profiles expressed as the correlation with a hypothetical regional climate signal depending on the number of profiles averaged
as well as their inter-profile spacing. The dashed red line shows the representativity on the seasonal time scale for 10 m profile spacing. For the inter-annual time scale, the two limiting cases
discussed in the text are displayed (a: best-case scenario/case I, b: worst-case scenario/case
II), each for 2 m profile spacings (black) as well as 10 m profile spacings (blue).
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inter-annual signal, 2 m profile spacing
inter-annual signal, 10 m profile spacing
seasonal signal, 10 m profile spacing
11, 5605–5649, 2015
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inter-annual signal, 2 m profile spacing
inter-annual signal, 10 m profile spacing
seasonal signal, 10 m profile spacing
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probability
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0.8
0.6
0.4
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Regional signal vs.
local noise in
Antarctic δ18 O
T. Münch et al.
Title Page
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trend detection
slope determination
0.0
1
10
20
30
best case
worst case
40
50
number of cores
Figure 9. The probability of detecting a linear temperature trend of 0.5 C/50 yr (correlation > 0,
p ≥ 0.05) (solid lines) and of determining the strength of the trend with an accuracy of 25 %
(dashed lines), each as a function of the number of annually resolved firn cores averaged
and for the two scenarios of the annual noise variance discussed in the text (black lines: best
case/case I, blue lines: worst case/case II).
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0.75
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relative variance
variance [(‰)^2]
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parallel to horizon
parallel to snow surface
diffusion case study
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30
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11, 5605–5649, 2015
Regional signal vs.
local noise in
Antarctic δ18 O
T. Münch et al.
Title Page
Introduction
Conclusions
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0.25
0
0
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depth [cm]
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Figure B1. The lateral variance of T1 as a function of depth below the mean snow height. Blue
lines with circles give the lateral variance as calculated horizontally, red lines with circles display
the variance computed for consecutive slices following the present snow surface. Greyish-blue
dashed lines depict the numerical estimate of the vertical variance of a diffused artificial trench
record (see text for details).
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