10Be in late deglacial climate simulated by ECHAM5-HAM - Part 2: isolating the solar signal from 10Be deposition

This study investigates the effect of deglacial climate on the deposition of the solar proxy 10Be globally, and at two specific locations, the GRIP site at Summit, Central Greenland, and the Law Dome site in coastal Antarctica. The deglacial climate is represented by three 30 year time slice simulations of 10 000 BP (years before present = 1950 CE), 11 000 and 12 000 BP, compared with a preindustrial control simulation. The model used is the ECHAM5-HAM atmospheric aerosol-climate model, driven with sea-surface temperatures and sea ice cover simulated using the CSIRO Mk3L coupled climate system model. The focus is on isolating the 10Be production signal, driven by solar variability, from the weather- or climate-driven noise in the 10Be deposition flux during different stages of climate. The production signal varies at lower frequencies, dominated by the 11 year solar cycle within the 30 year timescale of these experiments. The climatic noise is of higher frequencies than 11 years during the 30 year period studied. We first apply empirical orthogonal function (EOF) analysis to global 10Be deposition on the annual scale and find that the first principal component, consisting of the spatial pattern of mean 10Be deposition and the temporally varying solar signal, explains 64% of the variability. The following principal components are closely related to those of precipitation. Then, we apply ensemble empirical decomposition (EEMD) analysis to the time series of 10Be deposition at GRIP and at Law Dome, which is an effective method for adaptively decomposing the time series into different frequency components. The low-frequency components and the long-term trend represent production and have reduced noise compared to the entire frequency spectrum of the deposition. The high-frequency components represent climate-driven noise related to the seasonal cycle of e.g. precipitation and are closely connected to high frequencies of precipitation. These results firstly show that the 10Be atmospheric production signal is preserved in the deposition flux to surface even during climates very different from today's both in global data and at two specific locations. Secondly, noise can be effectively reduced from 10Be deposition data by simply applying the EOF analysis in the case of a reasonably large number of available data sets, or by decomposing Abstract. This study investigates the effect of deglacial climate on the deposition of the solar proxy 10 Be globally, and at two speciﬁc locations, the GRIP site at Summit, Central Greenland, and the Law Dome site in coastal Antarctica. The deglacial climate is represented by three 30 year time slice simulations of 10 000 BP (years before present = 1950 CE), 11 000 and 12 000 BP, compared with a preindustrial control simulation. The model used is the ECHAM5-HAM atmospheric aerosol–climate model, driven with sea-surface temperatures and sea ice cover simulated using the CSIRO Mk3L coupled climate system model. The focus is on isolating the 10 Be production signal, driven by solar variability, from the weather- or climate-driven noise in the 10 Be deposition ﬂux during different stages of climate. The production signal varies at lower frequencies, dominated by the 11 year solar cycle within the 30 year timescale of these experiments. The climatic noise is of higher frequencies than 11 years during the 30 year period studied. We ﬁrst apply empirical orthogonal function (EOF) analysis to global 10 Be deposition on the annual scale and ﬁnd that the ﬁrst principal component, consisting of the spatial pattern of mean 10 Be deposition and the temporally varying solar signal, explains 64 % of the variability. The following principal components are closely related to those of precipitation. Then, we apply ensemble empirical decomposition (EEMD) analysis to the time series of 10 Be deposition at GRIP and at Law Dome, which is an effective method for adaptively decomposing the time series into different frequency components. The low-frequency components and the long-term trend represent production and have reduced noise compared to the entire frequency spectrum of the deposition. The high-frequency components represent climate-driven noise related to the seasonal cycle of e.g. precipitation and are closely connected to high frequencies of precipitation. These results ﬁrstly show that the 10 Be atmospheric production signal is preserved in the deposition ﬂux to surface even during climates very different from today’s both in global data and at two speciﬁc locations. Secondly, noise can be effectively reduced from 10 Be deposition data by simply applying the EOF analysis in the case of a reasonably large number of available data sets, or by decomposing the individual data sets to ﬁlter out high-frequency ﬂuctuations.


Introduction
Reconstruction of solar activity has so far only been possible for the Holocene (e.g. Steinhilber et al., 2012;Vonmoos et al., 2006). Evidence of the existence of solar cycles during the last ice age was found by Wagner et al. (2001) in the 10 Be record from the GRIP ice core between 25 and 50 kyr BP, but a continuous record extending from the Holocene into the preceding ice age is still missing. During the last deglaciation the solar proxies 10 Be and 14 C exhibited significant, climatedriven, differences, which complicates the extraction of the solar signal (e.g. Muscheler et al., 2004). In order to study the climate impact on 10 Be during the last deglaciation we perform time slice model simulations during three stages: 10 000 ("10k"), 11 000 ("11k") and 12 000 ("12k") BP (years before 1950 CE), compared with a control ("ctrl") simulation during the preindustrial climate. The mean climate change as well as the mean difference in 10 Be deposition and atmospheric distribution has been analysed in an accompanying manuscript . The main findings are that the lower greenhouse gas concentrations in the deglaciation simulations influence the climate, leading to a tropospheric cooling and drying and changes in sea ice cover which affect atmospheric circulation patterns. However, these changes were found to cause 10 Be deposition to fluctuate by no more than 50 % locally, although changes in air concentrations and dry deposition were significantly larger than that. The results indicate that 10 Be deposition is mostly driven by mass balance. The amount of 10 Be produced in the atmosphere is deposited onto the surface within a few years and therefore, averaged over a few years, the deposition equals production.
While the accompanying study concentrates on spatial differences as influenced by the mean state of the climate, this study focuses on temporal changes and investigates how the 11 year solar signal in 10 Be production is preserved in the global and local deposition flux. The aim is to assess how much the production signal is distorted by climatic noise in these simulations. We first focus on global deposition and quantify the different components of the variability, production and climate-related "noise", with the aid of empirical orthogonal function (EOF) analysis. Then, because observations do not cover the entire globe but are limited to a few locations we analyse time series of the modelled 10 Be deposition at two locations: the GRIP drilling site in Greenland and the Law Dome site in Antarctica.
The traditional approach to detect solar cycles in 10 Be records has been to create a frequency, for example a Fourier, spectrum which reveals the known solar cycles, e.g. ∼ 11 (Schwabe), ∼ 22 (Hale), ∼ 88 (Gleissberg), ∼ 205 (de Vries) and ∼ 2300 (Hallstatt) years (e.g. McCracken et al., 2012). Bandpass filtering has been used to distinguish between solar and geomagnetic modulation of 10 Be production by assuming that fluctuations with frequencies below a given threshold, typically 1000 years, are due to geomagnetic variations whereas high-frequency fluctuations are due to solar variability (Beer et al., 1994(Beer et al., , 2002Wagner et al., 2001). The drawback of using the Fourier spectrum to detect frequency peaks is that the length of each solar cycle is assumed to be constant in time. However, the length of the cycles has been found to be non constant and is currently under much investigation (e.g. Richards et al., 2009). Already during the 30 year period investigated within this study each of the three ca. 11 year cycles varies by ±1 year in length. Bandpass filtering, on the other hand, requires a priori knowledge of the frequencies of the cycles to set the frequency limits. To overcome these potentially limiting assumptions we propose the ensemble empirical mode decomposition (EEMD) method (Huang and Wu, 2008;Huang et al., 1998;Wu and Huang, 2009) in this study. EEMD decomposes the 10 Be signal into a set of frequency components, termed intrinsic mode functions (IMFs). As this decomposition is based on the local characteristics of the data, it offers a potentially viable method for nonlinear and nonstationary data analysis, especially for timefrequency representation. The IMFs, therefore, have no set frequency but are allowed to vary with time. Moreover, the IMFs represent the entire frequency spectrum of the data and not only a preset frequency range. The IMFs resulting from EEMD analysis can then be combined to be associated with solar or geomagnetic forcing on 10 Be data.
EEMD has widely been used in time series analysis, such as surface temperature (Franzke, 2012), tree ring data (Shi et al., 2012) and changes in onset of seasons (Qian et al., 2009), but, to our knowledge, never in combination with 10 Be. This study focuses on the level of distortion of the solar signal in 10 Be deposition due to deglacial climate changes. Model data are useful to test the suitability of EEMD for this study because the solar signal used is known. While the length of the model data (30 years) restricts the type of solar cycles to be studied to only the 11 year one, EEMD can be applied to realworld data including a larger number of cycles in the future. Only the 10 Be observations available from the last deglaciation are from the GISP2 ice core (Finkel and Nishiizumi, 1997), but their temporal resolution of 20-50 years does not compare with the monthly resolution of this study. Holocene observations covering several solar cycles typically have an annual or longer temporal resolution. Sub-annually resolved observations are limited in length and typically include up to one 11 year cycle only. This prohibits a direct comparison with the current model data.

Methods
Here we only give general information on the model simulations and refer to the accompanying paper  for details. The model used is the ECHAM5-HAM atmospheric aerosol-climate model which incorporates radionuclide production, transport and deposition processes. To produce deglacial climate the model is driven with seasurface temperatures and sea ice cover obtained from simulations using the CSIRO Mk3L climate system model version 1.2 (Phipps et al., 2011(Phipps et al., , 2012. Each of the ECHAM5-HAM model simulations (ctrl, 10k, 11k and 12k) represents a 30 year time slice of an equilibriated state of climate during these periods. We refer to  for details on the 10 Be production rate used in this study. Analysis of the mean changes in climate and in atmospheric 10 Be transport and deposition are provided by .
We first apply the empirical orthogonal functions (EOF) analysis, also known as principal component analysis, to global 10 Be data in this study. This method was introduced by Lorenz (1956) and has since been widely used in climate data analysis to detect patterns such as the North Atlantic oscillation or the Southern Annular Mode in sea level pressure data, among various others. It creates a linear combination of a number of orthogonal spatial patterns (referred to as EOFs in this manuscript), multiplied by a time series component (referred to as PCs). Because the global 10 Be deposition fields comprise a temporally varying component, the production signal, but also vary spatially due to differences in the precipitation patterns and location of the stratospheretroposphere exchange, this method seems suitable for removing noise and reducing the dimensionality of 10 Be data.
The words "signal" and "noise" will be used throughout the manuscript to refer to the solar variability-driven atmospheric production (signal) and climate-driven fluctuations (noise) in 10 Be deposition data. Both components typically have very distinctive timescales. The production varies on multi-year timescales, such as the 11 year cycle. Shorterterm fluctuations in the solar activity parameter cause highfrequency fluctuations in the production rate, but these are efficiently filtered out by the atmospheric transport from the stratosphere to the troposphere. Climate-related changes, the largest of which is the seasonal cycle of e.g. the precipitation rate, act on sub-annual timescales. Long-term trends in climatic variables are also possible, but were not found during the relatively short simulations of 30 years each. In order to decompose the 10 Be deposition into various frequencies we apply the EEMD method to 10 Be deposition and the precipitation rate at GRIP and Law Dome, for each of the four simulations. We aim to analyse the raw data without applying any averaging or filtering. However, seasonal fluctuations of 10 Be data are of much larger amplitude than solar modulation and have to be removed. We apply a simple 25 month running mean to smooth out the seasonal cycle but only minimally reduce the length of the data set, consistent with . Different lengths of the running mean window were tested, but no strong influence on the results was found. Use of a running mean introduces some high-frequency noise (less than annual) into the data set, but only variations longer than annual will be analysed.
The EEMD method decomposes time series into intrinsic mode functions (IMF), each of which represents a specific frequency range, and a long-term trend. The first IMF has the highest frequency and so on. The sum of these IMFs and the long-term trend reproduces the original time series. The length of the time series determines the number of IMFs. Our time series consist of monthly 30 year data, smoothed with a 25 month running mean, adding up to 336 data points. This creates seven IMFs and a long-term trend. Each of the model simulations is analysed separately, because the data have to be continuous for EEMD, and then combined.
The EEMD analysis can be briefly summarised as follows: 1. Add white noise with a predefined noise amplitude to the data to be analysed.
2. Run EMD to decompose the data with added white noise into IMFs.
3. Repeat the above steps several times to create the ensembles.

EOF analysis of global 10 Be deposition
In the following we analyse the temporal variability of the simulated global 10 Be deposition flux. In order to detect the solar cycle in the 10 Be flux it is necessary to remove the seasonal cycle which has been found to dominate temporal variability on short timescales . Therefore, we analyse 25 month running mean values. The mean distribution of 10 Be deposition over each 30 year simulation is shown in Fig. 1. We apply EOF analysis for the three-dimensional deposition field, with all four simulations combined to a continuous time series to produce the common EOFs for each simulation. Because of this, the amount of variability explained by these patterns can be assessed during different states of climate. Analysing each run separately was also tested, but all simulations resulted in very similar spatial patterns. The first EOF obtained is shown in Fig. 2 together with the three first principal components. The first EOF (top panel) explains 64 % of the variability and is very similar to the mean 10 Be deposition pattern (Fig. 1). The first principal component, shown in blue, correlates strongly (r = 0.92) with the 11 year solar cycle (green). The delay of ca. 1 year between the production (solar) and the deposition signal reflects the atmospheric residence time of 10 Be (e.g. Beer et al., 1990). The following EOFs are fairly patternless and exhibit significant variability only in the tropics (not shown). The tropics are generally not best suited for recording 10 Be as a solar proxy due to the low production variability and the uplifting of air due to the Brewer-Dobson circulation. Therefore the tropical tropospheric air is less enriched by stratospheric 10 Be, which exhibits the largest production variability, and the 10 Be signal in tropical tropospheric air  includes more noise. The corresponding two following PCs (in blue) explain 23 and 5 % of the variability and the rest of them less than 5 % each. It seems that the variability of the internal climate modes, described by these PCs, was not amplified in the deglaciation simulations. The standard deviations ("SD" shown in the figure) are slightly reduced relative to ctrl in the deglaciation simulations, especially at 12k. However, the mean value of the second PC at 12k is higher. The first PC thus represents the production signal and the following PCs the climate-related noise. In order to investigate if the noise components are related to climatic variability we perform EOF analysis for the 25 month running mean precipitation fields. The PCs are very similar to the noise components of the 10 Be deposition, suggesting that the climatic noise is closely related to precipitation variability. The first three precipitation PCs are shown in Fig. 3 (green) together with the second to fourth PCs of 10 Be deposition (blue). The PCs of precipitation explain 59 % (1st), 13 % (2nd) and 11 % (3rd) of the variability. The correlation coefficients (0.98 to 0.81) suggest that these are closely related to the second to fourth PCs of 10 Be deposition.
Given these results the 10 Be deposition can be decomposed into a spatial deposition pattern which is similar to the climatological mean deposition pattern multiplied by the temporally varying production signal plus noise, which can thus be discarded. Even during deglacial climate the production signal seems large enough to make full use of the method. These results suggest that this method can therefore be applied to observations as well. However, observational  records might not be easily combined due to their different temporal resolution and coverage, variable quality and very limited spatial coverage. In reality, insufficient observations are available to fully distinguish signal from noise.

EEMD analysis of 10 Be deposition at GRIP and at Law Dome
Typically these complications restrict the number of time series which can be analysed collectively. Hence, principal component analysis might not be able to reveal the solar signal during periods when observations disagree. Therefore we apply an alternative method, the EEMD, to analyse time series separately at two particular locations: the GRIP site in central Greenland ( ). Both are characterised by relatively high snow accumulation, and therefore a number of high-resolution time series exist (e.g. Muscheler et al., 2005;Pedro et al., 2011;Yiou et al., 1997). We first present the modelled time series of 10 Be deposition at both sites (Fig. 4) for all four simulations. Both monthly mean and 25 month running mean values are shown. The monthly fluctuations are considerable in all simulations at both stations but smoothing the seasonal cycle out (25 month running mean) reveals the solar cycle. The three ca. 11 year solar cycles are seen in all simulations at both stations, however some distortion is visible, especially at 12k. The mean value of 10 Be deposition only varies by ca. 5 % between these stations. While the global mean deposition has to be constant in all simulations, local changes of up to 50 % could have been expected based on the analysis of the mean climate . The precipitation rate (Fig. 5) does vary more at GRIP, exhibiting reduced monthly  variability and a lower mean at 12k than in other simulations. At Law Dome, the mean precipitation rate and the standard deviation are gradually reduced at 12k relative to ctrl. It seems that the reduced precipitation rate at 12k does not therefore affect the mean 10 Be deposition at 12k; however, it might contribute to the high amplitude of variability in the reconstructed production signal at 12k. Figure 6 shows the data as input for the EEMD analysis, with the 25 month running mean 10 Be production, deposition and precipitation rate. The production rate shown is the global mean. The data are normalised through division by the mean, showing that the amplitude of the deposition variability is comparable with the global mean production rate  variability. The 10 Be deposition follows the three solar cycles shown by the production rate in all simulations at both stations. The 10 Be deposition is delayed in exhibiting the second production minimum in the 11k simulation at GRIP, which could be due to the large simultaneous peak in the precipitation rate. Given the length of the time series, the EEMD analysis results in seven intrinsic mode functions (IMFs). These are shown in Figs. 7 and 8. In addition, a long-term trend is obtained (IMF 8). The sum of these IMFs and the trend reproduces the original data. The first three IMFs are interpreted as climate-related noise as their frequency is less than annual. The following five IMFs (4-8) are considered to   represent the reconstructed solar signal, or 10 Be production rate. Which IMFs are attributed to signal and noise is ambiguous and depends on the time resolution of the data. In our case, sub-annual variations can only be of climatic origin and can be discarded as noise. It might be advantageous to vary the number of the IMFs used to reconstruct the production as closely to the original solar signal as possible in each simulation, but in the case of observations the actual signal is not known. We therefore aim to create a standard methodology based on physically justified thresholds which can be applied to any data without prior knowledge of the reconstructed signal. The high-frequency components IMF 1-2 do not vary significantly between the simulations at GRIP. Only IMF 3 fluctuates more strongly at 12k (standard deviation 10-60 % higher). IMF 5 is closest to the 10 Be production signal, exhibiting the three ca. 11 year solar cycles. However, the first cycle of IMF 5 is shorter than the solar one, for which IMF 6 contributes by creating the broader shoulder seen during the first third of each 30 year period. This suggests a stronger climatic impact on 10 Be deposition during this period, seen as anomalously low precipitation rate at ctrl, 10k and 11k (Fig. 6). IMF 7 has a similar form but is flat in 10k, however its amplitude is negligible compared with other IMFs. At Law Dome the noise components (IMF 1-3) are fairly similar in amplitude in all simulations. IMF 4 has a lower frequency in ctrl than in the other simulations, and it contributes more to the two last solar cycles than IMF 5, which nearly misses them. Such a shift towards higher-frequencies suggests stronger climate impact during the last solar cycle. This is consistent with the higher amplitude of the highfrequency IMFs of precipitation (not shown) which seems to distort the reconstructed solar signal in 10 Be deposition. Also ctrl 10k 11k 12k ctrl 10k 11k 12k Fig. 8. The seven intrinsic mode functions (IMF1-7) and the long-term trend (IMF8) of 10 Be deposition (atoms/m 2 /s) at the Law station.  9. The normalised reconstructed solar, or 10 Be production, 'signal' (IMFs 4-8) from the 10 Be deposition with the high-frequen removed for the four simulations, compared with the normalised original production signal (black). The percentages show the va explained by the 'signal' components in each simulation. Fig. 9. The normalised reconstructed solar, or 10 Be production, "signal" (IMFs 4-8) from the 10 Be deposition with the highfrequency noise removed for the four simulations, compared with the normalised original production signal (black). The percentages show the variability explained by the "signal" components in each simulation.
the percentage of total variability explained by noise is larger at 7 % than in other simulations (see the following subsection). IMF 7 is very flat in ctrl and 10k but has a distinctive pattern in 11k and 12k but again the amplitude is too small to be detected in the total signal. The reconstructed production signal from the 10 Be deposition (IMF 4-8) is illustrated in Fig. 9 for both stations, together with the original production rate. Removing the highfrequency noise flattens the signal and increases the agreement with production (compare with Fig. 6). The data sets have been normalised for comparison. The amplitude of the reconstructed production agrees reasonably well with the actual production, but is slightly underestimated at GRIP and overestimated at Law Dome in 11k. The second production maximum is underestimated in 12k at both stations. This was already seen in the original data (Fig. 6) and cannot be improved by removing the high-frequency noise. Figure 10 shows the reconstructed high-frequency part of the spectrum (IMF 1-3) of both the 10 Be deposition and precipitation. They have been standardised to allow for comparison. Generally the variability seems similar in all simulations and both stations. Both noise components seem correlated, especially in the case of 10k, 11k and 12k at GRIP and 10k and 11k at Law Dome. In ctrl the precipitation noise fluctuates more strongly than 10 Be deposition noise at both stations. The variability explained by the signal and the noise components is shown in the figures as well. The signal components dominate the variability, explaining 93-97 % of total variability at both stations. The variability contribution of all IMFs is shown in Fig. 11. The first three IMFs are negligible at both stations and all simulations. At GRIP, IMF 5, which is very closely related to the solar signal (see Fig. 7 IMF1  IMF2  IMF3  IMF4  IMF5  IMF6  IMF7  IMF8   0   20   40   60   80   GRIP   IMF1  IMF2  IMF3  IMF4  IMF5  IMF6  IMF7  IMF8   0 Fig. 10. The standardised reconstructed high-frequency "noise" (IMFs 1-3) from the 10 Be deposition for the four simulations, shown with the same standardised high-frequency components of precipitation (black). The percentages show the variability explained by the "noise" components in each simulation.
70 % of total variability in ctrl. However, in the deglacial simulations this contribution is reduced and IMF 4 and the longterm trend get more weight. At Law Dome there is no single dominant IMF, but IMFs 4-5 (and the long-term trend in the case of deglacial simulations) are most dominant. Apparently IMF 7, albeit exhibiting distinct differences between the simulations, is not of importance for the total variability in any of the simulations. Combining results of both stations suggests that in the 12k simulation there is a significant long-term trend, which is absent in ctrl. Furthermore, the Law Dome station seems more strongly affected by the climatic noise than GRIP in these simulations. Figures 12 and 13 show scatter plots of normalised 10 Be deposition and precipitation, and 10 Be production and deposition at GRIP and Law Dome, respectively. In order to distinguish the effect of noise reduction from production signal, we analyse correlations between variables. Figure 12 shows scatter plots of normalised 10 Be deposition and precipitation, and 10 Be production and deposition at GRIP. 10 Be deposition and production (signal) are shown without (IMF 1-8; blue) and with (IMF 4-8; red) filtering of high-frequency noise, 10 Be deposition and precipitation (noise) only without filtering because of the different scales of the variables (IMF 1-3 have zero mean due to the subtraction of the longterm mean and thus different scale). However, correlation coefficients are shown for both the unfiltered (IMF 1-8; first) and the filtered (IMF 1-3; second) data. Comparison of the unfiltered and filtered correlations indicates that filtering the high-frequency noise improves the agreement between 10 Be deposition and production signals; however, the difference is not large. This is due to the strong 11 year cycle which causes the data to be autocorrelated, dominating the correlation. Fig. 10. The standardised reconstructed high-frequency 'noise' (IMFs 1-3) from the 10 Be deposition for the four simulations the same standardised high-frequency components of precipitation (black). The percentages show the variability explained b components in each simulation.
Therefore the correlation coefficients should be interpreted as indicative only. However, we do not attempt to remove the autocorrelation because the 11 year cycle is the very part of the 10 Be production signal which we are trying to detect. In case of the noise components ( 10 Be deposition and precipitation) there seems to be no connection between them in the unfiltered data, indicated by the low correlation coefficients.  case of 10 Be, the correlation increases significantly. At Law Dome the results are similar to GRIP. The correlation of the filtered signal with the production signal is improved from the unfiltered data, albeit only slightly. The noise components of 10 Be deposition and precipitation correlate fairly strongly when unfiltered, but the correlation is reduced when the data are filtered. This is due to a similar long-term trend, which, when filtered out, reduces the correlation. Also the fact that white noise is added into the data by the EEMD might reduce the correlation between filtered data sets. Looking at Fig. 6 the precipitation rate at Law Dome seems to exhibit cycles similar in length to the 11 year cycle, especially in ctrl and 11k. This, however, is coincidental, as the model employs a standard radiation scheme with a constant value for solar irradiation. In an atmospheric-only model the climate is constrained mostly by the sea-surface temperatures and sea ice and the solar irradiation play a minor role.
The physical meaning of these findings is that the temporal variability of 10 Be deposition into ice is mostly dominated by the production signal on an annual scale. The correlation between 10 Be deposition and production is high, but deteriorates because of fluctuations caused by short-term changes in precipitation rate. If this short-term "climatic" noise is filtered out, the 10 Be production signal, reconstructed from the 10 Be deposition flux, agrees better with the actual production signal. However, this method only corrects for highfrequency noise, but cannot distinguish longer-term climatic noise from the production signal. This is shown by the fact that the excessively low or high amplitudes of the solar cycles, or the delays in the response to production minima or maxima in 10 Be deposition, cannot be corrected. Still, the EEMD-filtered signal explains more than > 95 % of total variability, a result which cannot be achieved by simple bandpass filtering.

Summary and conclusions
This study analyses four time slice (30 years each) simulations of the solar proxy 10 Be at different stages of climate: 10 000 BP ("10k"), 11 000 BP ("11k") and 12 000 BP ("12k") during the last deglaciation, compared with a control simulation during the preindustrial period ("ctrl"). We investigate to what extent the different climatic conditions distort the solar signal in the 10 Be deposition flux to the surface and how the distortion can be corrected by analysing the frequency spectrum of the 10 Be deposition. During the relatively short period studied (30 years) the climatic distortion, called noise, is assumed to be represented by the highest frequencies, whereas the solar signal is known to vary on a longer timescale. In order to remove the seasonal cycle from the data we first smooth it using 25 month running means.
First, the global field of 10 Be deposition is analysed to study the temporal and spatial variability by means of EOF (empirical orthogonal function) analysis, also known as PC (principal component) analysis. We find that the first spatial pattern closely resembles the global deposition field, and the first temporal pattern correlates with the solar signal with r = 0.92. 64 % of the total variability of 10 Be deposition can be attributed to solar, or production, variability, and 36 % to noise. Analysing the noise components we find close connections between the second and higher temporal patterns and all temporal patterns of precipitation, suggesting that precipitation variability drives the noise part of the 10 Be deposition variability after the production signal has been removed. This method allows for noise reduction, as the noise components can be removed. It can be applied to observational data as well, if sufficient spatial coverage is provided and the temporal coverage matches.
As in reality the number of 10 Be observations is limited, EOF analysis can produce unreliable results during periods when observations disagree. We propose the use of the ensemble empirical decomposition (EEMD) method, which analyses one-dimensional data. EEMD decomposes the data into intrinsic frequency components without requiring any prior knowledge of these frequencies. Furthermore, it has the advantage of allowing the amplitude and the length of the cycles in the data to vary over time. We decompose the modelled 10 Be deposition and precipitation at two particular locations from which a number of ice core records exist: the GRIP site at the summit of Greenland and the Law Dome site in coastal Antarctica. The results are composed of seven intrinsic mode functions with decreasing frequencies and a long-term trend. The high-frequency components are interpreted as climate-driven noise and the low-frequency ones as the solar signal plus any long-term climatic variability. The results for GRIP and Law Dome are fairly similar, and removing the high-frequency noise improves the agreement between the 10 Be deposition flux and the production signal. 10 Be deposition at Law Dome includes slightly more climatic noise in these simulations. The amplitude of the reconstructed production signal from the deposition is very similar to the original production. Comparison of the highfrequency noise components of 10 Be deposition with those of precipitation suggests they are interconnected, in agreement with the results of the global EOF analysis.
These findings support the assumption that, regardless of the state of climate, the variability of 10 Be deposition is dominated by the production variability on annual and longer timescales, simply due to mass conservation. Locally significant fluctuations from the global mean could have been expected but were not found, although the precipitation rate was reduced in the deglacial climate. The EEMD method proved useful in analysing single data series. It was successful in noise reduction and resulted in a deposition signal closer to production, explaining > 95 % of total variability in each simulation than can be obtained by a simple lowpass filtering or smoothing. However, it was only able to remove high-frequency noise and could not correct for all spurious forms at lower frequencies. EEMD thus seems well suited for noise reduction in single 10 Be time series. We propose it for analysing multi-annually resolved 10 Be records including several solar cycles of various frequencies. Seasonal noise with its amplitude of several factors larger than production variability complicates the analysis of high-resolved records. The strength of EEMD will be the decomposition of the entire frequency spectrum allowing for a distinction of solar cycles of various lengths, as well as the slowly varying strength of the geomagnetic field.