The firn samples obtained from trench T1 provide a two-dimensional image of the δ18O structure of the upper ∼1m of firn on a horizontal scale of ∼ 50m (Fig. a).

The surface height profile of the trench reflects the typical snow topography of the sampling region characterised by small-scale dunes with their main ridges elongated parallel to the mean wind direction . Trench T1 features one prominent dune located between 25 and 40 m, accompanied by a dune valley between 8 and 18 m, and some smaller-scale height variations. The peak-to-peak amplitude of the large dune undulation is ∼10cm; the entire height variations exhibit a standard deviation (SD) of 2.9cm.

Overall, the trench δ18O record shows a diverse picture. The delta values in T1 (Fig. a) span a range from -54 to -34‰ with a mean of -44.4‰ (SD 3.1‰). A similar range of -50 to -38‰ is observed in T2 (Fig. ) with a mean of -44.0‰ (SD 2.7‰). We can identify 8 to 10 alternating layers of enriched and depleted isotopic composition in the T1 record. The uppermost layer (first 6cm relative to the surface) essentially shows enriched (mean of -42.7‰) but also strongly variable δ18O values between -54 and -34‰ (SD 4.4‰), thus already covering the range of the entire trench record. Stronger enrichment tends to be found in the valleys; however, the limited data do not allow one to conclude whether this is a general feature. In an absolute depth of 5–20cm, a band of generally more depleted δ18O values is found exhibiting less horizontal variability compared to the first layer with a range of -54 to -45‰ (mean -48.5‰, SD 1.9‰). The layering appears strongly perturbed in the depth of ∼60–100cm for profile positions < 30m. Here, a broad and diffuse region of rather constant δ18O values around -40‰ is observed, together with a prominent, 20cm thick feature of high delta values between 18 and 28m.

The four δ18O profiles obtained from trench T2 (Fig. ) show similar features as trench T1. We can identify roughly five cycles in each profile. However, the profiles diverge considerably at depths of 50–90cm, which coincides with the region of strong perturbations identified in T1.

To further analyse the isotopic layering, we determine the pronounced local maxima and minima of each T1 δ18O profile and visually assign summer and winter to the depths of these extrema. This results in consecutive horizontal curves tracing the vertical positions of seasonal extrema along the trench (seasonal layer profiles, Fig. b). Assuming that respective isotopic extrema occur at the same point in time (summer/winter), the seasonal layer profiles reflect the surface height profile for a given season. However, considering the highly variable isotopic composition observed at the current trench surface (Fig. a), this is a rough approximation and the seasonal layer profiles will likely overestimate the past surface height profiles. Nevertheless, the vertical undulations of the layer profiles show peak-to-peak amplitudes of 6–24cm (average SD 3.7cm), comparable to the present surface undulations, and the layers are vertically separated by approximately 20cm, in accord with the local mean annual accumulation rate of snow (64mm w.e. yr-1) and the mean firn density measured in trench T1 (ρfirn=340kg m-3). To study the similarity between the seasonal layer profiles and the present surface height profile, we calculate the standard deviation of their height differences (SDsurface, hence, the SD of each layer profile on surface coordinates). This is compared to the standard deviation of the layer profiles on absolute coordinates (SDhoriz). We find that the first layer profile closely follows the present surface (SDhoriz-SDsurface=1.8cm). For the second layer profile, the link with the surface is weaker (SDhoriz-SDsurface=1.5cm), and the layer profiles below 20cm are on average horizontally aligned (SDhoriz-SDsurface=-0.6cm). This can be explained by an annual reorganisation of the stratigraphy so that aligning the isotopic variations on absolute coordinates is on average more appropriate than the alignment according to a specific surface height profile. The positive autocorrelation with a decorrelation length of ∼ 6cm that is found from the vertical T1 δ18O variations after subtraction of the mean trench profile is consistent with this hypothesis.

Due to the on average horizontal stratigraphy of the isotopic composition in the larger part of the trench record all further plots and calculations will be evaluated on absolute coordinates.

The isotope record of trench T1 (Fig. a) allows the quantification of the horizontal isotopic variability of the snow and firn column in our study region. We observe considerable horizontal variability with a mean variance of σh,T12≃5.9(‰)2, directly affecting the representativity of single trench profiles. To mimic the potential result obtained from correlating two snow pits taken at a distance of 500m, similarly done in many firn-core studies e.g., we calculate the pairwise Pearson correlation coefficient between single profiles of T1 and single profiles of T2. We account for potential surface undulations between the trenches by allowing bin-wise vertical shifts of ±12 cm between the T1 and T2 profiles to maximise their correlation. The estimated correlations (Fig. ) are substantially scattered around a mean correlation of ∼ 0.50 (SD=0.13). The relative majority (∼43%) of all possible profile pairs (N=152) shows a maximum correlation at a shift of +3cm, which is well below the estimated upper vertical height difference of the trenches.

Spatial averaging is expected to improve the correlation between the trenches compared to the single profiles. We therefore correlate the mean trench profiles of T1 and T2, allowing again for bin-wise vertical shifts of the T2 profile to maximise the correlation.

The mean trench profiles (Fig. ) are highly correlated (rT1,T2=0.81 for an optimal shift of +3cm; p=0.01, accounting for the full autocorrelation structure and allowing for vertical shifting), indicating a common isotopic signal reproducible over a spatial scale of at least 500m. It is interesting to note that this value is above most of the single inter-trench correlations (Fig. ).

In both mean 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 depth range of 65–90cm. Interestingly, this obscured part without clearly depleted δ18O “winter” values is found in both trenches, indicating that this feature persists over at least 500m and is thus likely of climatic origin, e.g. a winter with unusually low precipitation. Despite the statistically significant correlation, pronounced differences between the mean profiles are present, such as a significantly more depleted, and partially more enriched, isotopic composition of the T2 mean between 50 and 80 cm.

To analyse annual-mean δ18O time series we use different binning methods to average the seasonal trench data with bins defined by (1) the six local maxima determined from the average of the two mean trench profiles, (2) the five local minima, (3) the midpoints of the ascending slopes flanking the maxima and (4) the midpoints of the descending slopes. To display the data on an absolute time axis we assign the year 2012 to the first annual bin. The annual-mean time series derived from the four possible binning sets are averaged to obtain a single time series for each trench (Fig. ). The correlation of the average annual-mean δ18O time series of rT1‾,T2‾=0.87-0.20+0.07 (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.

We have shown that spatial averaging significantly increases the correlation between the trenches. To learn more about the spatial correlation structure of the trench isotope record, we investigate (1) the inter-profile correlation as a function of profile spacing for T1 and (2) the inter-trench correlation between different sets of mean profiles from T1 and the mean T2 profile.

The inter-profile correlation is estimated as the mean of the correlations obtained from all possible T1 profile pairs separated by a given spacing, allowing a tolerance in the horizontal position of 5%. For the inter-trench correlation, we define a T1 profile stack as the spatial average across a certain number of T1 profiles separated by a given distance, and determine all possible equivalent stacks. The inter-trench correlation with the mean T2 profile is then recorded as the mean across the correlations between the mean T2 profile and all possible T1 stacks.

The inter-profile correlation approaches 1 for nearest neighbours and rapidly drops with increasing inter-profile spacing before it stabilises around a value of ∼ 0.5 for spacings ≳10m (Fig. ). For the inter-trench correlation, we find a steady increase in the correlation with the T2 reference with increasing number of profiles used in the T1 stacks (Fig. ). Additionally, the correlation increases with a wider spacing between the individual profiles of a stack.

The observed decrease of the inter-profile correlation with distance suggests a horizontal autocorrelation of the isotopic composition. Indeed, a positive autocorrelation of the horizontal δ18O variations of T1 with a decorrelation length of λ≃1.5m is found by applying a Gaussian kernel correlation which accounts for the irregular horizontal sampling. As we do not expect any climate-related part of the isotopic record to vary on such small spatial scales, we attribute the observed autocorrelation to noise features.

The inter-profile correlation rXY provides an estimate of the signal-to-noise variance ratio F of single profiles , F=rXY/(1-rXY). Neglecting the small-scale correlation, we estimate F from the data using the mean inter-profile correlation for the profile spacings between 10 and 35 m and find F=0.9±0.1.

Based on our findings, we develop a simple statistical model: we assume that each trench profile consists of the sum of a common climate signal S and a noise component w independent of the signal. The noise component is modelled as a first-order autoregressive process (AR(1)) in the horizontal direction. Then, the inter-profile correlation coefficient between profiles X and Y becomes a function of their spacing d (see Appendix ), rXY=11+F-11+F-1exp⁡-dλ. Here, F-1=var(w)/var(S) is the inverse of the signal-to-noise variance ratio. Using our estimate for F and the value for λ obtained in the previous section, the model reproduces the observed inter-profile correlations (Fig. ). Applying the same parameter values, the theoretical inter-trench correlation (Eq. ) is also in good agreement with the empirical results (Fig. ). This validates the model and the parameter values (F,λ) from the intra-trench (∼10m) to the inter-trench spatial scale (∼500m).

A horizontally stratified trench without horizontal isotopic variations would yield perfectly correlated single profiles. Opposed to that, our records (Table ) show a significant variability in horizontal direction with mean variances σh,T12≃5.9(‰)2, σh,T22≃5.3(‰)2 that are smaller but of the same order of magnitude as the mean down-core variances σv,T12≃9.5(‰)2, σv,T22≃7.3(‰)2. In consequence, coherent isotopic features between single profiles separated by the trench distance are only found by chance (Fig. : the median correlation is 0.49, only for two pairs (∼1.3%) the correlation is > 0.8). Thus, single firn profiles from our study region are no representative recorders of climatic isotope signals on the vertical scales analysed here.

On the horizontal scale of the trenches (∼10–500m), we expect that stratigraphic noise dominates the isotopic variations . The observed length scale of the horizontal decorrelation of the noise (λ∼1.5m) is similar in magnitude as that on which the local small-scale surface height variations occur, indicating that stratigraphic noise is in fact the prominent noise component in our data.

Despite the low single-profile representativity, the trench record contains a climate signal becoming apparent through the inter-profile correlation of ∼ 0.5 remaining on scales on which the stratigraphic noise is decorrelated (≳ 10m). It appears to be regionally (≲ 1km) coherent as suggested firstly by the comparable values of the inter-profile correlation for spacings ≳ 10m and the mean correlation between single T1–T2 records (Fig. ), and secondly by the common seasonal signal observed in the mean trench profiles (Fig. ).

Noise is always reduced by averaging profiles; here, the autocorrelation causes nearby profiles to share more common noise variance than profiles at a larger spacing. Therefore, albeit the same number of profiles is averaged, stacks using a larger profile spacing will exhibit less common noise variance and hence a larger proportion of the underlying signal (Fig. ). Our results show a minimum profile spacing of ∼10m to be optimal.

For quantitative climate reconstructions from proxy data, a robust estimate of the climate signal is necessary. Based on our statistical noise model, we can estimate the isotopic climate signal content of a profile stack for our study region depending on the number of averaged profiles and their spacing.

To this end, we define the climate representativity of a trench profile stack as the correlation between the stack and a common climate signal (Eq. ). This signal is identified with the coherent isotope signal of the trench records. A physical interpretation of the climate representativity is then the upper bound of the correlation with a local temperature record, for example from a weather station. However, bearing in mind other influences such as meteorology (variable storm tracks, changing moisture source regions, precipitation-weighting), the true correlation will be lower. In the limit of independent noise our definition of climate representativity is equivalent to the expression derived by .

In general, climate signals are timescale-dependent. For example, the seasonal amplitude of the isotopic signal is much larger than any variations between the years. On the other hand, one expects larger changes of the climate signal on longer timescales, such as glacial–interglacial cycles. Moreover, not only the climate signal but also the noise can be a function of the timescale. One extreme example for this is the non-climate oscillations of the isotopic composition on up to centennial timescales which have been indicated by snow-pit studies around Vostok station and linked to the movement of accumulation waves on various scales . Since the climate representativity (Eq. ) depends on the ratio F of signal and noise variance, it is in consequence also a function of the timescale.

Here, we assess the climate representativity of firn isotope profiles from our study region for two specific timescales: (1) the original resolution of the trench data and (2) an annual resolution based on binning the trench data.

Analysing the original data, which is dominated by variations on seasonal timescales, the climate representativity can be readily calculated with the model parameters obtained in Sect. . For the analysis at annual resolution, estimates of both annual signal and noise variance are necessary to assess the variance ratio F. However, the shortness of our trench data only allows heuristic estimates (see Appendix  for details). Specifically, for the annual noise variance we discuss two limiting cases: for case I we assume that the vertical noise is white (best-case scenario), for case II that the vertical noise shows complete inter-dependence on the sub-annual timescale (worst case). The inverse of the annual signal-to-noise variance ratio, Fannual-1=var(w)annual/var(S)annual, used in the model is then ∼ 1.2 for case I and ∼ 8.7 for case II. A summary of the noise levels is given in Table .

For single profiles, the climate representativity estimated at seasonal resolution is 0.69 (Fig. ). At annual resolution, single profiles show a representativity of 0.67 in the best-case scenario (Fig. a) and of 0.32 in the worst-case scenario (Fig. b).

Similar to the correlation between the trenches (Fig. ), the representativity increases with the number of profiles averaged with a stronger increase for larger inter-profile spacings. However, spacings above 10m do not yield a further increase as the stratigraphic noise is largely decorrelated. To obtain a climate representativity of 0.8 at annual resolution with profiles separated by 10m, a minimum of 3–16 cores are needed (from best to worst case). Demanding a representativity of 0.9, the number of cores required increases to 6–37.

The modelled single-profile climate representativity at annual resolution appears consistent with previous findings from Dronning Maud Land. estimated a low signal-to-noise variance ratio of F=0.14 obtained from the cross-correlations of 16 annually resolved δ18O records from an area of 500km× 200km. Due to the large inter-core spacings, the stratigraphic noise in the records is decorrelated and the variance ratio F can be translated into a single-profile representativity of rSX=1/1+F-1≃ 0.35, consistent with our findings for the worst-case scenario. However, the records analysed in are also subject to dating uncertainties, additional variability caused by spatially varying precipitation-weighting and other effects. Therefore, the similar representativities are not necessarily caused by the high stratigraphic noise level assumed in the worst-case scenario. In addition, our trench data indicate vertical autocorrelation of the noise (Fig. b and Sect. ). Thus, the true climate representativity for our study region will likely be in between our limiting estimates.

Stratigraphic noise affects not only isotopes but also other parameters measured in ice cores, such as aerosol-derived chemical constituents. investigated the seasonal to inter-annual representativity of ion records from five Greenland firn cores taken at varying distances from 7 to 10 m in the vicinity of the NEEM drilling site. Using the definition of representativity based on , they found inter-annual representativities of ∼0.55–0.95, depending on the number of averaged cores and the ion species considered. These numbers are slightly higher than our best-case-scenario results for δ18O, which is expected since the accumulation rate at the NEEM site is about 3 times higher than at Kohnen Station .

Our estimates for the climate representativity of firn cores hold as long as the signal-to-noise variance ratio F does not change. Variance-affecting processes such as diffusion and densification have equal influence on signal and noise and thus do not alter the ratio F. On the other hand, only one component might change over time; for example, the noise variance might vary due to changing environmental conditions, or the variability of the climate could have been different in the past for certain time periods. Nevertheless, given the stability of the Holocene climate, we do not expect first-order changes of the signal and noise properties over time. However, we do expect a timescale dependency of the climate signal with more variance associated with longer timescales e.g.. The signal-to-noise variance ratio and the climate representativity of firn cores will improve considerably on these scales.

Our noise level and implied climate representativity estimates underline the challenge of firn-core-based climate reconstructions at seasonal to annual resolution in low-accumulation regions. For our study site, we now discuss implications of our noise model concerning (1) the required measurement precision of water isotopes in the case of classical isotope thermometry, (2) the potential noise fraction in isotope signals of the EDML ice core and (3) the detectability of a temperature trend.

Our estimates of the stratigraphic noise level are based on the upper 1 m of firn. Due to the shortness of the data our results are limited by insufficient knowledge of the vertical noise covariance structure for inter-annual and longer timescales for which we now assume white-noise behaviour. The noise of isotopic data obtained from deeper parts of the firn column is affected by diffusion and densification. Densification is only of importance when studying the isotopic time series in the depth domain since in that case constantly sampled data will contain noise levels on varying timescales. We estimate the effect of diffusion and find that at decadal resolution below the firn–ice transition the noise level at Kohnen Station is only 5% smaller as compared to the undiffused case (Appendix  and Table ).

The noise of an isotopic signal includes the stratigraphic noise as well as noise caused by the measurement process. Since the stratigraphic noise is a function of the number of analysed cores, and measurement precision is often related to measurement time, obtaining the best signal is a trade-off between measurement precision and the amount of analysed samples.

At seasonal as well as annual resolution, the measurement uncertainty of the trench data of σCRDS=0.09‰ is much lower (∼4–10%) than the standard deviation of the stratigraphic noise (Table ). This ratio is independent of the temporal resolution if a lower resolution is obtained by averaging annually resolved data as both contributions 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-precision measurements on single cores. As an example, with the cavity ring-down spectrometers used for this work faster measurements are possible by reducing the number of repeated measurements per sample and applying a memory correction . We explicitly note that this possibility is limited to classical single-isotope (δ18O) reconstructions as it can affect the data usability for diffusion- or deuterium-excess-based inferences.

If a lower temporal resolution is obtained by a coarser sampling of the cores, the measurement error to stratigraphic noise ratio will depend on the analysed resolution (Table ). For a resolution corresponding to 10 years, our measurement uncertainty might amount to up to 32% of the stratigraphic noise level, accounting for full diffusion. The noise level of single cores would become comparable to the measurement uncertainty for averages over ∼ 104 or ∼ 735 years (best- or worst-case scenario of annual noise level).

The deep EPICA DML ice core obtained in the vicinity of Kohnen Station reflects the climate evolution in Antarctica over the last 150 000 years . studied a core section covering the last 6000 years at decadal resolution. We find a δ18O variance for this section of ∼ 0.57(‰)2. Using our diffusion-corrected stratigraphic noise variance estimates would imply that ∼ 15–100% (from best to worst case) of the observed decadal variance in the core might be noise (Table ), masking the underlying climate variability. We note that this is only a rough estimate as the shortness of the trench data does not allow one to fully assess the decadal noise covariance. In any case, averaging across multiple cores seems necessary in low-accumulation regions to reconstruct the climate variability of the last millennium. Alternatively, if only the magnitude of variability is of interest, the proxy variability has to be corrected for the noise contribution e.g..

As a final example of applying our noise model, we test the influence of stratigraphic noise on the detectability of a linear trend at Kohnen Station. This is motivated by the finding of that in the last 50 years the surface temperature over East Antarctica has warmed by about half a degree. While both the climate signal as well as the relationship between local temperature and isotopic signal are complex, we assess the detectability with a simplified model experiment. For this, we assume the climate signal to be a purely linear trend (0.5∘C (50 yr)-1) and a linear isotope-to-temperature relationship (1‰ K-1), further influenced only by post-depositional noise. In a Monte Carlo approach repeated 105 times, we create stacks from 50yr long δ18O profiles with post-depositional noise variances based on our two limiting cases (Table ), accounting for an average effect of diffusion on the annual noise level over the 50 years (Appendix ) and assuming independent noise between the profiles (inter-profile spacings ≳ 10m), and vary the number of averaged profiles. A trend in the stacked profile is successfully detected for an estimated trend that is significantly larger than zero (p=0.05); the estimated slope is defined to be correct when it lies in a range of 25% around the true slope. The probability of trend detection/slope determination is then the ratio of successful reconstructions to total number of realisations.

Drilling a single core, the probability of detecting the trend or reconstructing its slope is around 25% in the best-case and below 10% in the worst-case scenario (Fig. ). To reliably (>80% of the cases) detect the warming over the East Antarctic Plateau, our results suggest that averaging across at least ∼5–35 firn cores taken at spacings of 10m (Fig. ) is needed, depending on the scenario for the annual noise variance. Inferring the right slope would need 3 times that number of cores. We note that more realistic assumptions about the isotopic signal (natural climate and atmospheric variability, varying isotope–temperature relationship, etc.) further complicate the trend detectability.