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Research article 12 Jun 2018
Research article  12 Jun 2018
Novel automated inversion algorithm for temperature reconstruction using gas isotopes from ice cores
 ^{1}Climate and Environmental Physics, University of Bern, Bern, Switzerland
 ^{2}Oeschger Centre for Climate Change Research (OCCR), Bern, Switzerland
 ^{1}Climate and Environmental Physics, University of Bern, Bern, Switzerland
 ^{2}Oeschger Centre for Climate Change Research (OCCR), Bern, Switzerland
Correspondence: Michael Döring (doering@climate.unibe.ch)
Hide author detailsCorrespondence: Michael Döring (doering@climate.unibe.ch)
Greenland past temperature history can be reconstructed by forcing the output of a firndensification and heatdiffusion model to fit multiple gasisotope data (δ^{15}N or δ^{40}Ar or δ^{15}N_{excess}) extracted from ancient air in Greenland ice cores using published accumulationrate (Acc) datasets. We present here a novel methodology to solve this inverse problem, by designing a fully automated algorithm. To demonstrate the performance of this novel approach, we begin by intentionally constructing synthetic temperature histories and associated δ^{15}N datasets, mimicking real Holocene data that we use as “true values” (targets) to be compared to the output of the algorithm. This allows us to quantify uncertainties originating from the algorithm itself. The presented approach is completely automated and therefore minimizes the “subjective” impact of manual parameter tuning, leading to reproducible temperature estimates. In contrast to many other icecorebased temperature reconstruction methods, the presented approach is completely independent from icecore stablewater isotopes, providing the opportunity to validate waterisotopebased reconstructions or reconstructions where water isotopes are used together with δ^{15}N or δ^{40}Ar. We solve the inverse problem T(δ^{15}N, Acc) by using a combination of a Monte Carlo based iterative approach and the analysis of remaining mismatches between modelled and target data, based on cubicspline filtering of random numbers and the laboratorydetermined temperature sensitivity for nitrogen isotopes. Additionally, the presented reconstruction approach was tested by fitting measured δ^{40}Ar and δ^{15}N_{excess} data, which led as well to a robust agreement between modelled and measured data. The obtained final mismatches follow a symmetric standarddistribution function. For the study on synthetic data, 95 % of the mismatches compared to the synthetic target data are in an envelope between 3.0 to 6.3 permeg for δ^{15}N and 0.23 to 0.51 K for temperature (2σ, respectively). In addition to Holocene temperature reconstructions, the fitting approach can also be used for glacial temperature reconstructions. This is shown by fitting of the North Greenland Ice Core Project (NGRIP) δ^{15}N data for two Dansgaard–Oeschger events using the presented approach, leading to results comparable to other studies.
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Holocene climate variability is of key interest to our society, since it represents a time of moderate natural variations prior to anthropogenic disturbance, often referred to as a baseline for today's increasing greenhouse effect driven by mankind. Yet, highresolution studies are still very sparse and therefore limit the investigation of decadal and even centennial climate variations over the course of the Holocene. One of the first studies about changes in the Holocene climate was conducted in the early 1970s by Denton and Karlén (1973). The authors investigated rapid changes in glacier extents around the globe potentially resulting from variations of Holocene climatic conditions. Mayewski et al. (2004) used these data as the base of a multiproxy study identifying rapid climate changes (socalled RCCs) globally distributed over the whole Holocene time period. Although not all proxy data showed an equal behaviour in timing and extent during the quasiperiodic RCC patterns, the authors found evidence for a highly variable Holocene climate controlled by multiple mechanisms, which significantly affects ecosystems (de Beaulieu et al., 2017; Crausbay et al., 2017; Pál et al., 2016) and human societies (Holmgren et al., 2016; Lespez et al., 2016). Precise highresolution temperature estimates can contribute significantly to the understanding of these mechanisms. Icecore proxy data offer multiple paths for reconstructing past climate and temperature variability. The studies of Cuffey et al. (1995), Cuffey and Clow (1997) and DahlJensen et al. (1998) demonstrate the usefulness of inverting the measured boreholetemperature profile for surfacetemperaturehistory estimates for the investigated drilling site using a coupled heat and iceflow model. Because of smoothing effects due to heat diffusion within an ice sheet, this method is unable to resolve fast temperature oscillations and leads to a rapid reduction of the time resolution towards the past. Another approach to reconstruct past temperature is based on the calibration of waterstable isotopes of oxygen and hydrogen (δ^{18}O_{ice}, δD_{ice}) from icecore watersamples assuming a constant (and mostly linear) relationship between temperature and isotopic composition due to fractionation effects during ocean evaporation, cloud formation and snow and ice precipitation (Johnsen et al., 2001; Stuiver et al., 1995). This method provides a rather robust tool for reconstructing past temperature for times where large temperature excursions occur when an adequate relationship is used (Dansgaard–Oeschger events, glacial–interglacial transitions; Dansgaard et al., 1982; Johnsen et al., 1992). Also, in the Holocene where Greenland temperature variations are comparatively small, seasonal changes of precipitation as well as of evaporation conditions at the source region may contribute to waterisotopedata variations (Huber et al., 2006; Kindler et al., 2014; Werner et al., 2001). A relatively new method for icecorebased temperature reconstructions uses the thermal fractionation of stable isotopes of air compounds (nitrogen and argon) within a firn layer of an ice sheet (Huber et al., 2006; Kindler et al., 2014; Kobashi et al., 2011; Orsi et al., 2014; Severinghaus et al., 1998, 2001). The measured nitrogen and argonisotope records of air enclosed in bubbles in an ice core can be used as a paleothermometer due to (i) the stability of isotopic compositions of nitrogen and argon in the atmosphere at orbital timescales and (ii) the fact that changes are only driven by firn processes (Leuenberger et al., 1999; Mariotti, 1983; Severinghaus et al., 1998). To robustly reconstruct the surface temperature for a given drilling site, the use of firn models describing gas and heat diffusion throughout the ice sheet is necessary to decompose the gravitational from the thermaldiffusion influence on the isotope signals.
This work addresses two issues relevant for temperature reconstructions based on nitrogen and argon isotopes. First, we introduce a novel, entirely automated approach for inverting gasisotope data to surfacetemperature estimates. For that, we force the output of a firndensification and heatdiffusion model to fit gasisotope data. This methodology can be used for many different optimization tasks not restricted to icecore data. As we will show, the approach works in addition to δ^{15}N for all relevant gasisotope quantities (δ^{15}N, δ^{40}Ar, δ^{15}N_{excess}) and for Holocene and glacial data as well. Furthermore, the possibility of fitting all relevant gasisotope quantities, individually or combined, makes it possible, for the first time, to validate the temperature solution gained from a single isotope species by comparison to the solution calculated from other isotope quantities. This approach is a completely new method which enables the automated fitting of gasisotope data without any manual tuning of parameters, minimizing any potential “subjective” impacts on temperature estimates as well as working hours. Also, except for the model spinup, the presented temperature reconstruction approach is completely independent from waterstable isotopes (δ^{18}O_{ice}, δD_{ice}), which provides the opportunity to validate waterisotopebased reconstructions (e.g. MassonDelmotte, 2005) or reconstructions where water isotopes are used together with δ^{15}N or δ^{40}Ar (e.g. Capron et al., 2010; Huber et al., 2006; Landais et al., 2004). To our knowledge, there are only two other reconstruction methods independent from waterstable isotopes that have been applied to Holocene gasisotope data, without a priori assumption on the shape of a temperature change. The studies from Kobashi et al. (2008a, 2017) use the secondorder parameter δ^{15}N_{excess} to calculate firntemperature gradients, which are later temporally integrated from past to future over the time series of interest using the firndensification and heatdiffusion model from Goujon et al. (2003). Additionally Orsi et al. (2014) use a linearized firnmodel approach together with δ^{15}N and δ^{40}Ar data to extract surfacetemperature histories. The method presented here can be used when no δ^{40}Ar data are available, which is often the case because δ^{40}Ar is a more analytically challenging measurement and is not as commonly measured as δ^{15}N and further allows a comparison among solutions obtained from any of the available isotope quantities.
Second, we investigate the accuracy of our novel fitting approach by examining the method on different synthetic nitrogenisotope and temperature scenarios. The aim of this work is to study the uncertainties emerging from the algorithm itself. Furthermore, the focal questions in this study are what the minimal mismatch in δ^{15}N for Holocenelike data we can reach is and what the implication for the final temperature mismatches is. Studying and moreover answering these questions makes it mandatory to create welldefined δ^{15}N targets and related temperature histories. It is impossible to answer these questions without using synthetic data in a methodology study. The aim is to evaluate the accuracy and associated uncertainty of the inverse method itself to then later apply this method to real δ^{15}N, δ^{40}Ar or δ^{15}N_{excess} datasets, for which of course the original driving temperature histories are unknown.
2.1 Reconstruction approach
The problem that we deal with is an inverse problem, since the effect, observed as δ^{15}N variations, is dependent on its drivers, i.e. temperature and accumulationrate changes. Hence, the temperature that we would like to reconstruct depends on δ^{15}N and accumulationrate changes. To solve this inverse problem, the firndensification and heatdiffusion model (from now on referred to as firn model), which is a nonlinear transfer function of temperature and accumulation rate to firn states and relates to δ^{15}N values, is run iteratively to match the modelled and measured δ^{15}N values (or other gas species). The automated procedure is significantly more efficient and less time consuming than a manual approach. The Holocene temperature reconstruction is implemented by the following four steps (Fig. 1):

Step 1: a prior temperature input (first guess) is constructed, which serves as the starting point for the optimization.

Step 2: a longterm solution which passes through the δ^{15}N data (here synthetic target data) is generated following a Monte Carlo approach. It is assumed that the smooth solution contains all longterm temperature trends (centuries to millennia) as well as firncolumn height changes (temperature and accumulationrate dependent) that drive the gravitational background signal in δ^{15}N.

Step 3: the longterm temperature solution is complemented by superimposing shortterm information directly extracted from the δ^{15}N data (here synthetic target data). This step adds shortterm temperature changes (decadal) in the same time resolution as the data.

Step 4: the gained temperature solution is further corrected using information extracted from the mismatch between the synthetic target and modelled δ^{15}N time series.
The functionality of the presented inversion algorithm is schematically displayed in Fig. 1. It guides the reader through chapters and documents where variables, listed in Table 1, are in use. In the following, a detailed description of each step is given.
2.1.1 Step 1: prior input
The starting point of the optimization procedure is the first guess. To construct the firstguess temperature input T_{g,0}(t), a constant temperature of −29.6 ^{∘}C is used for the complete Holocene section, which corresponds to the last value of the temperature spinup (Fig. 2b).
2.1.2 Step 2: Monte Carlo type input generator – generating longterm solutions
During the second step of the optimization, the prior temperature input T_{g,0}(t) from step 1 is iteratively (j) changed following a Monte Carlo approach. The basic idea of the Monte Carlo approach is to generate smooth temperature inputs T_{mc,j}(t) by superimposing lowpassfiltered values P_{j} of uniformly distributed random values P_{r,j} on the prior input ${T}_{\mathrm{mc},j\mathrm{1}}$. Then, the new input is fed to the firn model and the mismatch ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},j}}$ (with X≡δ^{15}N_{mc,j}) between the modelled δ^{15}N_{mc,j} (here X_{mod}), calculated from the model output, and the synthetic δ^{15}N_{syn} (here X_{target}) is computed for every time step (i) of the target data δ^{15}N_{syn} according to
(Note: if not otherwise stated, all mismatches in this study labelled with “D” are calculated similar to Eq. 1.)
${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc}}}$ serves as the criterion which is minimized during the optimization in step 2. If the mismatch ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},j}}$ decreases compared to the prior input (${T}_{\mathrm{mc},j\mathrm{1}}$, ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},j\mathrm{1}}}$), the new input is saved and used as the new guess (${T}_{\mathrm{g},j}={T}_{\mathrm{mc},j}$). This procedure is repeated until convergence is achieved, leading to the final longterm temperature T_{mc,fin}(t). Table 2 lists the number of improvements and iterations performed for the different synthetic datasets.
The perturbation of the current guess T_{g,j} is conducted in the following way: let ${\mathit{T}}_{\mathrm{g},\mathrm{0}}={T}_{\mathrm{g},\mathrm{0}}\left(t\right)$ be the vector containing the prior temperature input. A second vector (P_{r,1}) with the same number of elements n_{mc} as T_{g,0} is generated, containing n_{mc} uniformly distributed random numbers within the limits of an also randomly (equally distributed) chosen standard deviation (SD). SD is chosen from a range of 0.05–0.50, which means that the maximum allowed perturbation of a single temperature value T(t_{0}) is in a range of ±5 to ±50 %. Creating the synthetic frequencies, P_{r,1} is lowpass filtered using cubicspline filtering (Enting, 1987) with an equally distributed random cutoff period (COP) in the range of 500 to 2000 years generating the vector P_{1}. Hereby the lowpass filtering of P_{r,1} reduces the amplitudes of the perturbation of T_{g,0}. The new surface temperature input T_{mc,1} is calculated from P_{1} according to
The superscript “T” stands for transposed and $\widehat{\mathrm{1}}$ is the n by 1 matrix of ones.
This approach provides a high potential for parallel computing. In this study, an eightcore computer was used, generating and running eight different inputs of T_{mc} simultaneously, minimizing the time to find an improved solution. For example, during the 706 iterations for scenario S2, about 5600 different inputs were created and tested, leading to 351 improvements (see Table 2). Since it is possible to find more than one improvement per iteration step due to the parallelization on eight CPUs, the solution giving the minimal misfit ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},j}}$ is chosen as new firstguess for the next iteration step. This leads to a decrease of the used improvements for the optimization (e.g. for S2, 172 of the 351 improvements were used). Additionally, a first gasage scale (Δage_{mc,fin}(t)) is extracted from the model using the last improved conditions, which will then be used in step 3.
2.1.3 Step 3: adding shortterm (highfrequency) information
In step 3, the missing shortterm temperature history providing a suitable fit between modelled and synthetic δ^{15}N data is directly extracted from the pointwise mismatch ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},\mathrm{fin}}}\left(t\right)$, between the modelled δ^{15}N_{mc,fin}(t) obtained in step 2 and the synthetic δ^{15}N_{syn} target. Note that for a real reconstruction, this mismatch is calculated using the measured δ^{15}N_{meas} dataset instead of the synthetic one. ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},\mathrm{fin}}}\left(t\right)$ can be interpreted in first order as the detrended highfrequency signal of the synthetic δ^{15}N_{syn} target. ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},\mathrm{fin}}}\left(t\right)$ is transferred to the gasage scale using Δage_{mc,fin}(t) provided by the firnmodel output for the smooth temperature input T_{mc,fin}(t). This is needed to insure synchroneity between the highfrequency temperature variations ΔT(t) extracted from the mismatch ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},\mathrm{fin}}}\left(t\right)$ on the iceage scale and the smooth temperature solution T_{mc,fin}(t). Additionally, the signal is shifted by about 10 years towards modern values to account for gas diffusion from the surface to the lockin depth (Schwander et al., 1993), which is not yet implemented in the firn model. This is necessary for adding the calculated shortterm temperature changes ΔT(t) to the smooth signal T_{mc,fin}(t). The ΔT values are calculated according to Eq. (3):
using the thermaldiffusion sensitivity ${\mathrm{\Omega}}_{{\mathrm{N}}_{\mathrm{2}},i}$ for nitrogenisotope fractionation from Grachev and Severinghaus (2003):
${\stackrel{\mathrm{\u203e}}{T}}_{i}$ is the mean firn temperature in Kelvin which is calculated by the firn model for each time point i. To reconstruct the final (highfrequency) temperature input (T_{hf}(t)), the extracted shortterm temperature signal ΔT(t) is simply added to the longterm temperature input T_{mc,fin}(t):
2.1.4 Step 4: final correction of the surface temperature solution
For a further improvement of the remaining δ^{15}N and resulting surfacetemperature misfits (${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{hf}}}\left(t\right)$, ${D}_{{T}_{\mathrm{hf}}}\left(t\right)$), it is important to find a correction method that contains information that is also available when using measured data. The benefit of the synthetic data study is that several laterunknown quantities can be calculated and used for improving the reconstruction approach (see Sects. 3 and 4). For instance, it is possible to split the synthetic δ^{15}N_{syn} data in the gravitational and thermodiffusion parts or to use the temperature misfit, which is unknown in reality. The idea underlying the correction algorithm explained hereafter is that the remaining misfits of δ^{15}N (${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{hf}}}\left(t\right)$) and temperature (${D}_{{T}_{\mathrm{hf}}}\left(t\right)$) are connected to the Monte Carlo (step 2) and highfrequency part (step 3) of the reconstruction algorithm. In the present inversion framework, it is not possible to find a longterm solution δ^{15}N_{mc,fin} (or T_{mc,fin}) which exactly passes through the δ^{15}N_{syn} (or T_{syn}) target in the middle of the variance in all parts of the time series. This leads to a slight over or underestimation of δ^{15}N_{mc,fin}(t) and their corresponding temperature values T_{mc,fin}(t). For example, a slightly too low (or too high) smooth temperature estimate T_{mc,fin} leads to a small increase (or decrease) of the firncolumn height, creating a wrong gravitational background signal in δ^{15}N_{mc,fin} on a later point in time (because the firn column needs some time to react). An additional error in the thermaldiffusion signal is also created due to the highfrequency part of the reconstruction (step 3), because the highfrequency information is directly extracted from the deviation of the synthetic target δ^{15}N_{syn}(t) and the modelled δ^{15}N_{mc,fin}(t) from the final longterm solution T_{mc,fin}(t) of the Monte Carlo part. Therefore, this error is transferred into the next step of the reconstruction and partly creates the remaining deviations.
To investigate this problem, the deviations ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},\mathrm{fin}}}\left(t\right)$ of the synthetic target data δ^{15}N_{syn} to δ^{15}N_{mc,fin} of the Monte Carlo part are numerically integrated over a time window of 200 years (Sect. 4, Supplement Sect. S3), and thereafter the window is shifted from past to future in 1year steps resulting in a time series called IF(t). IF(t) equals a 200year running mean of ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},\mathrm{fin}}}\left(t\right)$. For t, the middle position of the window is allocated. The time evolution of IF(t) is a measure for the deviation of the longterm solution δ^{15}N_{mc,fin}(t) (or T_{mc,fin}(t)) from the perfect middle passage through the target data δ^{15}N_{syn}(t) (or T_{syn}(t)) and for the slight over and underestimation of the resulting temperature.
Next, the sample crosscorrelation function (xcf) (Box et al., 1994) is applied to IF(t) and the remaining misfits ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{hf}}}\left(t\right)$ of δ^{15}N after the highfrequency part. The xcf shows two extrema (Fig. 3a), a maximum (xcf_{max}) and a minimum (xcf_{min}) at two certain lags (lag${}_{max,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$ at xcf${}_{max,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$ and lag${}_{min,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$ at xcf${}_{min,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$). Now, the same analysis is conducted for IF(t) versus the temperature mismatch ${D}_{{T}_{\mathrm{hf}}}\left(t\right)$ (Fig. 3b), which shows an equal behaviour (two extrema, lag${}_{max,T}$ at xcf${}_{max,T}$ and lag${}_{min,T}$ at xcf${}_{min,T}$). Comparing the two cross correlations shows that lag${}_{max,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$ equals the negative lag${}_{min,T}$ and lag${}_{min,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$ corresponds to the negative lag${}_{max,T}$ (Fig. 3d, e). The idea for the correction is that the extrema in the crosscorrelation IF(t) versus ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{hf}}}\left(t\right)$ with the positive lag (positive means here that ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{hf}}}\left(t\right)$ has to be shifted to past values relative to IF(t)) creates the misfit of temperature ${D}_{{T}_{\mathrm{hf}}}\left(t\right)$ on the negative lag (modern direction) of IF(t) versus ${D}_{{T}_{\mathrm{hf}}}\left(t\right)$, and vice versa. So, IF(t) yields information about the cause and allows us to correct this effect between the remaining mismatches, ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{hf}}}\left(t\right)$ and ${D}_{{T}_{\mathrm{hf}}}\left(t\right)$, over the whole time series. The lags are not sharp signals, due to the fact that (i) the cross correlations are conducted over the whole analysed record, leading to an averaging of this causeandeffect relationship and that (ii) IF(t) is a smoothed quantity itself. The correction of the reconstructed temperature after the highfrequency part is conducted in the following way: from the two linear relationships between IF(t) and ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{hf}}}\left(t\right)$ at the two lags (lag${}_{max,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$ at xcf${}_{max,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$, lag${}_{min,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$ at xcf${}_{min,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$), two sets of δ^{15}N correction values (Δδ^{15}N_{max}(t) from xcf${}_{max,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$ and Δδ^{15}N_{min}(t) from xcf${}_{min,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$) are calculated. Then, the lags are inverted (Fig. 3c, e), shifting the two sets of the δ^{15}N correction values to the attributed lags of the cross correlation between IF(t) and ${D}_{{T}_{\mathrm{hf}}}\left(t\right)$ (e.g. Δδ^{15}N_{min}(t) to lag from xcf${}_{max,T}$ from the cross correlation between IF(t) and ${D}_{{T}_{\mathrm{hf}}}\left(t\right)$), therefore changing the time assignments of Δδ^{15}N_{min}(t) and Δδ^{15}N_{max}(t) to Δδ^{15}N${}_{min}(t+{\mathrm{lag}}_{max,T}$) and Δδ^{15}N${}_{max}(t+{\mathrm{lag}}_{min,T}$). Now, the Δδ^{15}N_{max}(t) and Δδ^{15}N_{min}(t) are summed up componentwise, leading to the time series Δδ^{15}N_{cv}(t). From Eq. (3) with Δδ^{15}N_{cv,i} instead of ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},\mathrm{fin},i}}$, the corresponding temperature correction values are calculated and added to the highfrequency temperature solution T_{hf}(t), giving the corrected temperature T_{corr}(t). Finally, T_{corr}(t) is used to run the firn model to calculate the corrected δ^{15}N_{corr}(t) time series. This causeandeffect relationship found in the cross correlations between IF(t) and ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{hf}}}\left(t\right)$, and IF(t) and ${D}_{{T}_{\mathrm{hf}}}\left(t\right)$, is exemplarily shown in Fig. 3 for scenario S1 and was found for all eight synthetic scenarios. The derived correction algorithm leads to a further reduction of the mismatches of about 40 % in δ^{15}N and temperature (see Sect. 3.2).
2.2 Firndensification and heatdiffusion model
Surfacetemperature reconstruction relies on firn densification combined with gas and heat diffusion (Severinghaus et al., 1998). In this study, the firndensification and heatdiffusion model, developed by Schwander et al. (1997), is used to reconstruct firn parameters for calculating synthetic δ^{15}N values depending on the input time series. It is a semiempirical model based on the work of Herron and Langway (1980) and Barnola et al. (1991), and implemented using the Crank and Nicholson algorithm (Crank, 1975), and was also used for the temperature reconstructions by Huber et al. (2006) and Kindler et al. (2014). Besides surfacetemperature time series, accurate accumulationrate data are needed to run the model. The model then calculates the densification and heatdiffusion history of the firn layer and provides parameters for calculating the fractionation of the nitrogen isotopes for each time step, according to the following equations:
δ^{15}N_{grav}(t) is the component of the isotopic fractionation due to the gravitational settling (Craig et al., 1988; Schwander, 1989) and depends on the lockin depth (LID) z_{LID}(t) and the mean firn temperature $\stackrel{\mathrm{\u203e}}{T}\left(t\right)$ (Leuenberger et al., 1999). g is the gravitational acceleration, Δm the molar mass difference between the heavy and light isotopes (equal to 10^{−3} kg mol^{−1} for nitrogen) and R the ideal gas constant. z_{LID} is defined as a density threshold ρ_{LID}, which is slightly sensitive to surface temperature, following the formula from Martinerie et al. (1994), with a small offset correction of 14 kg m^{−3} to account for the presence of a nondiffusive zone (Schwander et al., 1997):
where
The thermalfractionation component of the δ^{15}N signal (Severinghaus et al., 1998) is calculated using Eq. (8), where T_{surf}(t) and T_{bottom}(t) stand for the temperatures at the top and the bottom of the diffusive firn layer. In contrast to T_{surf}(t), which is an input parameter for the model, T_{bottom}(t) is calculated by the model for each time step. The thermaldiffusion constant α_{T} was measured by Grachev and Severinghaus (2003) for nitrogen (Eq. 12):
The firn model used here behaves purely as a forward model, which means that for the given input time series the output parameters (here, finally δ^{15}N_{mod}(t)) can be calculated, but it is not easily possible to construct from measured isotope data the related surfacetemperature or accumulationrate histories. The goal of the presented study is an automatization of this inversemodelling procedure for the reconstruction of the rather small Holocene temperature variations.
2.3 Measurement, input data and timescale
2.3.1 Timescale
For the entire study, the GICC05 chronology is used (Rasmussen et al., 2014; Seierstad et al., 2014). During the whole reconstruction procedure, the two input time series (surface temperature and accumulation rate) are split into two parts. The first part ranges from 20 to 10 520 years b2k (called the “Holocene section”) and the second one from 10 520 to 35 000 years b2k (“spinup section”). The entire accumulationrate input, as well as the spinup section of the surfacetemperature input, remains unchanged during the reconstruction procedure.
2.3.2 Accumulationrate data
Besides surface temperatures, accumulationrate data are needed to drive the firn model. In this study, we use the original accumulation rates, reconstructed in Cuffey and Clow (1997), produced using an iceflow model adapted to the Greenland Ice Sheet Project Two (GISP2) location but adapted to the GICC05 chronology (Rasmussen et al., 2008; Seierstad et al., 2014). A detailed description of the adaption procedure can be found in Sect. S1 of the Supplement. The raw accumulationrate data for the main part of the spinup section (12 000 to 35 000 years b2k) are linearly interpolated to a 20year grid and lowpass filtered with a 200year COP using cubicspline filtering (Enting, 1987). For the Holocene section (20–10 520 years b2k) and the transition part between Holocene and spinup section (10 520 to 12 000 years b2k), the raw accumulationrate data are linearly interpolated to a 1year grid to obtain equidistant integer pointtopoint distances which are necessary for the reconstruction, and to preserve as much information as possible for this time period (Fig. 2a). Except for these technical adjustments, the accumulationrate input remains unmodified, assuming high reliability of these data during the Holocene. The accumulation data were reconstructed using annual layer counting, and a thinning model which should lead to maximum relative uncertainty of 10 % for the first 1500 of the 3000 m ice core (Cuffey and Clow, 1997). From the three accumulationrate scenarios reconstructed in Cuffey and Clow (1997) and adapted here to the GICC05 chronology, the intermediate one is chosen (red curves in Fig. S1 in the Supplement). Since the differences between the scenarios are not important for the evaluation of the reconstruction approach, they are not taken into account for this study.
Additionally, two sensitivity experiments were conducted (see Sect. S2 in the Supplement) in order to investigate (i) the influence of lowpass filtering of the highresolution accumulation rates on the model outputs and (ii) the possible contribution of the accumulationrate variability to the δ^{15}N data during the Holocene. The first experiment shows that filtering the accumulation rates with cutoff periods in the range of 20 to 500 years has nearly no influence on the modelled δ^{15}N or lockin depth as long as the major trends are being conserved. The second experiment leads to the finding that the accumulationrate variability explains about 12 to 30 % of δ^{15}N variability. A total of 30 % corresponds to the 8.2 kyr event and 12 % to the mean of the whole Holocene period including the 8.2 kyr event. Hence, the influence of accumulation changes, excluding the extreme 8.2 kyr event, is generally below 10 % during most parts of the Holocene.
2.3.3 δ^{18}O_{ice} data
Oxygenisotope data from the GISP2 icecorewater samples measured at the University of Washington's Quaternary Isotope Laboratory are used to construct the surfacetemperature input of the model spinup (12 to 35 kyr b2k, Grootes et al., 1993; Grootes and Stuiver, 1997; Meese et al., 1994; Steig et al., 1994; Stuiver et al., 1995; data availability: Grootes and Stuiver, 1999). The raw δ^{18}O_{ice} data are filtered and interpolated in the same way as the accumulationrate data for the spinup part.
2.3.4 Surfacetemperature spinup
The surfacetemperature history of the spinup section (Fig. 2b) is obtained by calibrating the filtered and interpolated δ^{18}O_{ice} data (Eq. 13) using the values for the temperature sensitivity α_{18O} and offset β found by Kindler et al. (2014) for the North Greenland Ice Core Project (NGRIP) ice core assuming a linear relationship of δ^{18}O_{ice} with temperature.
The values 35.2 ‰ and −31.4 ^{∘}C are moderntime parameters for the GISP2 site (Grootes and Stuiver, 1997; Schwander et al., 1997). The spinup is needed to bring the firn model to a welldefined starting condition that takes possible memory effects (influence of earlier conditions) of firn states into account.
2.3.5 Generating synthetic target data
In order to develop and evaluate the presented algorithm, eight temperature scenarios were constructed and used to model synthetic δ^{15}N data, which serve later as targets for the reconstruction. From these eight synthetic surfacetemperature and related δ^{15}N scenarios (S1–S5 and H1–H3), three datasets (later called Holocenelike scenarios H1–H3) were constructed in such a way that the resulting δ^{15}N time series are very close to the δ^{15}N values measured by Kobashi et al. (2008b) in terms of variability (amplitudes) and frequency (data resolution) of the GISP2 nitrogenisotope data (Figs. 4, 5).
The synthetic surfacetemperature scenarios S1–S5 are created by generating a longterm temperature time series (T_{syn,smooth}) analogous to the Monte Carlo part of the reconstruction procedure for only one iteration step (see Sect. 2.1). The values for the cutoff period used for the filtering of the random values, and the SD values (standard deviation of the random values; see Sect. 2.1) for the first five scenarios can be found in Table 3. The longterm temperatures (Fig. 4I) are calculated on a 20year grid, which is nearly similar to the time resolution of the GISP2 δ^{15}N measurement values of about 17 years (Kobashi et al., 2008b). For the Holocenelike scenarios, the smooth temperature time series were generated from the temperature reconstruction for the GISP2 δ^{15}N data (not shown here). The final Holocene surfacetemperature solution was filtered with a 100year cutoff to obtain the longterm temperature scenario.
Following this, highfrequency information is added to the longterm temperature histories. A set of normally distributed random numbers with a zero mean and a standard deviation (1σ) of 1 K for scenarios S1–S5 and 0.3 K for Holocenelike scenarios H1–H3 is generated on the same 20year grid and added up to the longterm temperature time series. Finally, the resulting synthetic targettemperature scenarios (Figs. 4II, 5I) are linearly interpolated to a 1year grid.
These synthetic temperatures are combined with the spinup temperature and are used together with the accumulationrate input to feed the firn model. From the model output, the synthetic δ^{15}N targets are calculated according to Sect. 2.2. The firnmodel output provides iceage as well as gasage information. The final synthetic δ^{15}N target time series (δ^{15}N_{syn}) are set intentionally on the iceage scale to mirror measured data, because no prior information is available for the gasage – iceage difference (Δage) for icecore data.
3.1 Monte Carlo type input generator
Figure 6 shows the evolution of the misfit ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},j}}$ between the synthetic target data (δ^{15}N_{syn}) and the modelled output δ^{15}N_{mc,j} of the Monte Carlo part (step 2) as a function of the applied iterations (j) for all synthetic scenarios. One can easily see that all scenarios show a steep decline of the mismatch during the first 50 to 200 iterations followed by a rather moderate decrease, which finally leads to a constant value. During the Monte Carlo part, it was possible to reduce the misfit ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc}}}$ compared to the firstguess solution ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{g},\mathrm{0}}}$ by about 15 to 75 % depending on the scenario and the mismatch of the firstguess solution (see Table 2). This leads to a reduction of the temperature mismatches ${D}_{{T}_{\mathrm{mc}}}$ compared to the firstguess temperature ${D}_{{T}_{\mathrm{g},\mathrm{0}}}$ mismatch of about 51 to 87 %.
Figure 7 provides the comparison between the firstguess (g,0; step 1) and Monte Carlo (mc,fin; step 2) solution versus the synthetic target data (syn) for the modelled δ^{15}N (Fig. 7a–c) and surfacetemperature values (Fig. 7d–f) for scenario S5. Subplots (a) and (d) show the time series of the synthetic target (black dotted line), the firstguess solution (blue line) and the Monte Carlo solution (red line) for δ^{15}N and temperature. In subplots (b) and (e), the distribution of the pointwise mismatch D_{i} of the firstguess (blue) and the Monte Carlo solutions (red) versus the synthetic target data for δ^{15}N (${D}_{{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$) and temperature (D_{T}) can be found. Subplots (c) and (f) contain the time series for ${D}_{{\mathit{\delta}}^{\mathrm{15}}\mathrm{N},i}$ and ${D}_{{T}_{i}}$. The ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},\mathrm{fin}}}\left(t\right)$ data (red) are used to calculate the highfrequency signal that is superimposed to the longterm temperature solution T_{mc,fin} according to Eqs. (3) and (5) (see Sect. 2.1, step 3). From Fig. 7, it can be concluded that the Monte Carlo part of the reconstruction algorithm (step 2) leads to two major improvements of the firstguess solution. First, it is obvious that the Monte Carlo approach corrects the offsets of the firstguess input (g,0), which shifts the midpoint of the distributions of ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{mc},i}}$ and ${D}_{{T}_{\mathrm{mc},i}}$ to zero (see blue against red in Fig. 7b, e). The second improvement is that the distributions become more symmetric and the misfit is overall reduced (the distributions become narrower) compared to the first guess due to the middle passage through the δ^{15}N_{syn} targets. These improvements can be observed for all eight synthetic scenarios, showing the robustness of the Monte Carlo part (see Table 2, Fig. 7).
3.2 Highfrequency step and final correction
Figure 8 provides the comparison between the Monte Carlo (mc,fin; step 2), the highfrequency (hf; step 3) and the correction (corr; step 4) parts of the reconstruction procedure for the scenario S5. Additional data for all other scenarios can be found in Table 4. The upper four plots (Fig. 8a–d) illustrate each reconstruction step and their effect on the modelled δ^{15}N; the bottom four plots (Fig. 8e–h) show the corresponding results on the temperature. Plots (a) and (e) contain the time series of the synthetic δ^{15}N_{syn} or T_{syn} target (syn; black dotted line), the highfrequency solution (hf; blue line) and the final solution after the correction part (corr; red line). For visibility reasons, subplots (b) and (f) display a zoomin for a randomly chosen time window of about 500 years for the same quantities, which shows the excellent agreement in timing and amplitudes of the modelled δ^{15}N and temperature compared to the synthetic target data. Histograms (c) and (g) and subplots (d) and (h) show the distribution and the time series of the pointwise mismatches (${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{i}}$ for δ^{15}N; ${D}_{{T}_{i}}$ for temperature) between the modelled and the synthetic target data in δ^{15}N and temperature for each reconstruction step.
Compared to the Monte Carlo solution, the highfrequency part leads to a large refinement of the reconstructions. For the mean δ^{15}N misfits ${D}_{{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$, the improvement between the Monte Carlo and the highfrequency parts is in the range of 64 to 76 % (see Table 4). This leads to a reduction of the temperature mismatches D_{T} of 43 to 67 %. The standard deviations (1σ) of the pointwise mismatches (Fig. 8c, d, g, h) in δ^{15}N and temperature after the highfrequency parts are in the range of about 2.7 to 5.4 permeg (one permeg equals 10^{−6}) for δ^{15}N and 0.22 to 0.40 K for the reconstructed temperatures depending on the scenario, which is clearly visible in the decreasing width of the histograms (Fig. 8c and g, blue against grey).
The mismatches after the correction part of the reconstruction approach show clearly a further decrease of the misfits. This means that the width of the distributions of the pointwise mismatches ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{i}}$ as well as ${D}_{{T}_{i}}$ is further reduced, and the distributions become more symmetric (long tales disappear; see histograms c and g; red against blue of Fig. 8). The time series of the mismatches (Fig. 8d and h) clearly illustrate that the correction approach mainly tackles the extreme deviations (sharp reduction of extreme values' occurrence in the red distribution compared to the blue distribution) leading to a further improvement of about 40 % in δ^{15}N and temperature. Finally, the 95 % quantiles (2${\mathit{\sigma}}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{corr},\mathrm{95}}}$, 2${\mathit{\sigma}}_{{T}_{\mathrm{corr},\mathrm{95}}}$) of the remaining pointwise mismatches of δ^{15}N and temperature (${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{i}}$ or ${D}_{{T}_{i}}$) were calculated for the final solutions for all scenarios and are used as an estimate for the 2σ uncertainty of the reconstruction algorithm (see Fig. 8c, g and Table 4). The final uncertainties (2σ) are of the order of 3.0 to 6.3 permeg for δ^{15}N and 0.23 to 0.51 K for the surface temperature misfits. It is noteworthy that the measurement uncertainties (per point) of stateoftheart δ^{15}N measurements are of the same order of magnitude, i.e. 3 to 5 permeg (Kobashi et al., 2008b), highlighting the effectiveness of the presented fitting approach. Table 5 contains the final mismatches (2σ) in Δage between the synthetic target and the final modelled data after the correction step for all scenarios, and shows that with a known accumulation rate and assumed perfect firn physics, it is possible to fit the Δage history in the Holocene with mean uncertainties better than 2 years. In other words, the uncertainty in Δage reconstruction due to the inversion algorithm alone is of the order of 2 years.
4.1 Monte Carlo type input generator
Figure 9 shows the distribution of the COPs (I) and SD values (II) used to create the improvements (Sect. 2.1, step 2) for all scenarios. The cutoff periods are more or less evenly distributed, which shows that nearly all of the allowed frequency range (500 to 2000 years) was used to create the improvements during the iterations. In contrast, the distributions of the SD values show clearly that mostly small SD values are used to create the improvements, which implies that iterations with small perturbations will more likely lead to an improvement than larger ones.
Figure 6 reveals a weak point of the Monte Carlo part, namely the absence of a suitable termination criterion for the optimization. The implementation until now is conducted such that the maximum number of iterations is given by the user or the iterations are terminated after a certain time (e.g. 15 h). Figure 6 shows that for nearly all scenarios it would be possible to stop the optimization after about 400 iterations, due to rather small additional improvements later on. This would decrease the time needed for the Monte Carlo part to about 10 h (a single iteration needs about 90 s). Since the goal of the Monte Carlo part is to find a temperature realization that leads to an optimal middle passage through the δ^{15}N target data, it would be possible to use the mean difference between the δ^{15}N target and splinefiltered δ^{15}N data using a certain cutoff period as a termination criterion. This issue is under investigation at the moment. Another possibility to decrease the time needed for the Monte Carlo part could be an increase in the number of CPUs used for the parallelization of the model runs. For this study, an eightcore parallelization was used. A further increase in numbers of workers would improve the speed of the optimization.
4.2 Highfrequency step and final correction
To investigate the timing and contributions of the remaining mismatches in δ^{15}N and temperature for scenario S1 after the highfrequency (step 3) and correction parts (step 4), different crosscorrelation experiments were conducted (see Sect. S3 in the Supplement). The experiments led to equal results. The major fraction of the final mismatches of δ^{15}N emerges from mismatches in the thermaldiffusion component ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{therm}}}$. Also a cancellation effect between the gravitational component ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{grav}}}$ and ${D}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{therm}}}$ of the total mismatch in δ^{15}N became obvious, affecting the calculation of lag${}_{max,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$ and lag${}_{min,{\mathit{\delta}}^{\mathrm{15}}\mathrm{N}}$ and most likely leading to a fundamental residual uncertainty in the lowpermeg level for the corrected δ^{15}N data. The same analyses were conducted for all synthetic scenarios, leading to similar results.
Additionally, the influence of the window length, used for the calculation of IF(t), on the correction was analysed, showing that for all investigated window lengths the correction reduces the mismatches of δ^{15}N and temperature, whatever correction mode was used (calculated with xcf_{max}, xcf_{min} or both quantities). Moreover, the correction is most efficient for window lengths in the range of 100 years to 300 years with an optimum at 200 years for all cases.
4.3 Key points to be considered for the application to real data
4.3.1 Benefits of the novel gasisotope fitting approach
In addition to the fitting of δ^{15}N data, the algorithm is able to fit δ^{40}Ar and δ^{15}N_{excess} data as well using the same basic concepts (Fig. 10). Here, the δ^{40}Ar and δ^{15}N_{excess} data from Kobashi et al. (2008b) were used as the fitting targets. We reach final mismatches (2σ) of 4.0 permeg for δ^{40}Ar/4 and 3.7 permeg for δ^{15}N_{excess}, which are for both quantities below the analytical measurement uncertainty of 4.0 to 9.0 permeg for δ^{40}Ar/4 and 5.0 to 9.8 permeg for δ^{15}N_{excess} measured data (Kobashi et al., 2008b).
The automated inversion of different gasisotope quantities (δ^{15}N, δ^{40}Ar, δ^{15}N_{excess}) provides a unique opportunity to study the differences in the gained solutions using different targets and to improve our knowledge about the uncertainties of gasisotopebased temperature reconstructions using a single firn model. Next, the presented algorithm is not dependent on the firn model, which leads to the implication that the algorithm can be coupled to different firn models describing firn physics in different ways. Furthermore, an automated reconstruction algorithm avoiding manual manipulation and leading to reproducible solutions makes it possible for the first time, to study and learn from the differences between solutions matching different targets. Finally, differences obtained by applying different firn physics (densification equations, convective zone, etc.) but the very same inversion algorithm may help to assess firnmodel shortcomings, resulting in more robust uncertainty estimates than it was ever possible before.
In this publication, we show the functionality and the basic concepts of the automated inversion algorithm using wellknown synthetic δ^{15}N fitting targets. In this “perfectworld scenario”, the forward problem, converting surface temperature to δ^{15}N, as well as the inverse problem, converting δ^{15}N to surface temperature, are completely described by the used firn model. Consequently, all sources of signal noise are ignored. For the later use of the algorithm on δ^{15}N, δ^{40}Ar or δ^{15}N_{excess} measured data, this will not be the case anymore due to different sources of signal noise in the used measured data. As a result, differences between temperature solutions obtained from individual targets (δ^{15}N, δ^{40}Ar, δ^{15}N_{excess}) will become obvious. These differences will allow us to quantify the uncertainties associated with different unconstrained processes. Next, we will list and discuss potential sources of uncertainties and try to provide suggestions for their handling and quantification in our approach.
4.3.2 Measurement uncertainty and firn heterogeneity (centimetrescale variability)
Many studies have investigated the influence of firn heterogeneity (or density fluctuations) on measurements of air compounds and quantities (e.g. δ^{15}N, δ^{40}Ar, CH_{4}, CO_{2}, O_{2} ∕ N_{2} ratio, air content) extracted from ice cores resulting in centimetrescale variability and leading to additional noise on the measured data (e.g. Capron et al., 2010; Etheridge et al., 1992; Fourteau et al., 2017; Fujita et al., 2009; Hörhold et al., 2011; Huber and Leuenberger, 2004; Rhodes et al., 2013, 2016). Using discrete measurement techniques instead of continuous sampling methods makes it difficult to quantify these effects. However, during discrete analyses of icecore air data, it is common to measure replicates for given depths, from which the measurement uncertainties of the gasisotope data are calculated using pooled standard deviation (Hedges and Olkin, 1985). Often, it is not possible to take real replicates (same depth), and instead the replicates are taken from nearby depths. Hence, any potential centimetrescale variability is to some degree already included in the measurement uncertainty, because each measurement point represents the average over a few centimetres of ice. This is especially the case for lowaccumulation sites or glacial ice samples for which the vertical length of a sample (e.g. 10–25 cm long for the glacial part of the NGRIP ice core; Kindler et al., 2014) covers the equivalent of 20 to 50 years of ice at approximately 35 kyr b2k. Increasing the depth resolution of the samples would increase our knowledge of centimetrescale variability, for, e.g. identifying anomalous entrapped gas layers that could have been rapidly isolated from the surface due to an overlying highdensity layer (e.g. Rosen et al., 2014). As this variability is likely due to heterogeneity in the density profile, modelling such heterogeneities (if possible at all) may not help to better reconstruct a meaningful temperature history but rather to reproduce the source of noise. This means that the potential centimetrescale variability, in many cases, is already incorporated in the analytical noise obtained from gasisotope measurements, due to analytical techniques themselves. Assuming the measurement uncertainty as Gaussian distributed, it is easy to incorporate this source of uncertainty in the inversemodelling approach presented here. This will increase the uncertainty of the temperature according to Eq. (3).The same equation can also be used for the calculation of the uncertainty in temperature related to measurement uncertainty in general.
To answer the pertinent question of how to better extract a meaningful temperature history from a noisy icecore record, an excellent – but costly – solution is of course to use multiple ice cores. For example, a δ^{15}Nbased temperature reconstruction from the combination of data from the GISP2 ice core with the “sister ice core” GRIP drilled 30 km apart is likely one of the best ways to overcome potential centimetrescale variability. A comparison of ice cores that were drilled even closer might be even more advantageous.
4.3.3 Smoothing effects due to gas diffusion and trapping
It is known that gasdiffusion and trapping processes in the firn can smooth out fast signals and result in a damping of the amplitudes of gasisotope signals (e.g. Grachev and Severinghaus, 2005; Spahni, 2003). The duration of gas diffusion from the top of the diffusive column to the bottom where the air is closed off in bubbles is for Holocene conditions in Greenland approximately of the order of 10 years (Schwander et al., 1997), whereas the data resolution of the synthetic targets was set to 20 years to mimic the measurement data from Kobashi et al. (2008b) with a mean data resolution of about 17 years (see Sect. 2.3: “Generating synthetic target data”). In the study of Kindler et al. (2014), it was shown that a glacial Greenland lockin depth leads to a damping of the δ^{15}N signal of about 30 % for a 10 K temperature rise in 20 years. We further assume that the smoothing according to the lockin process is negligible for Greenland Holocene conditions according to the much smaller amplitude signals and shallower LID. Yet, for glacial conditions, it requires attention.
4.3.4 Accumulationrate uncertainties
For the synthetic data study presented in this paper, it is assumed that the used accumulationrate data are well known with zero uncertainty. This simplification is used to show the functionality and basic concepts of the presented fitting algorithm in every detail on wellknown δ^{15}N and temperature targets and to focus on the final uncertainties originating from the presented fitting algorithm itself. For the later reconstruction using measured gasisotope data together with the published accumulationrate scenarios shown in Sect. S1 of the Supplement, this will not be the case anymore. Uncertainties in layer counting and corrections for ice thinning will lead to a fundamental uncertainty. Especially in the early Holocene, this can easily exceed 10 %. As the accumulationrate data are used to run the firn model, all potential accumulation uncertainties are in part incorporated into the temperature reconstruction. On the other hand, as we discussed in Sect. S2 of the Supplement, the accumulationrate variability has a minor impact compared to the input temperature on the variability of δ^{15}N data in the Holocene. The influence of these quantities, accumulation rate or temperature, on the temperature reconstruction is not equal; during the Holocene, accumulationrate variability explains about 12 to 30 % of δ^{15}N variability. A total of 30 % corresponds to the 8.2 kyr event and 12 % to the mean of the whole Holocene period including the 8.2 kyr event. Hence, the influence of accumulation changes, excluding the extreme 8.2 kyr event, is generally below 10 % during the Holocene. If the accumulation is assumed to be completely correct, then the missing part will be assigned to temperature variations. Nevertheless, for the fitting of the Holocene measurement data, we will use all three accumulationrate scenarios as shown in Sect. S1 of the Supplement. The difference in the reconstructed temperatures arising from the differences of these three scenarios will be used for the uncertainty calculation as well and is most likely higher than the uncertainty arising from uncertainties due to the process of producing the accumulationrate data and from the conversion of the accumulationrate data to the GICC05 timescale.
4.3.5 Convective zone variability
Many studies have shown the existence of a wellmixed zone at the top of the diffusive firn column, called the convective zone (CZ). The CZ is formed by strong katabatic winds and pressure gradients between the surface and the firn (e.g. Kawamura et al., 2006, 2013; Severinghaus et al., 2010). The existence of a CZ changes the gravitational background signal in δ^{15}N and δ^{40}Ar as it reduces the diffusivecolumn height. The presented fitting algorithm was used together with the two most frequently used firn models for temperature reconstructions based on stable isotopes of air, the Schwander et al. (1997) model which has no CZ built in (or better – a constant CZ of 0 m) and the Goujon firn model (Goujon et al., 2003) (which assumes a constant convective zone over time, that can easily be set in the code). This difference between the two firn models only changes significantly the absolute temperature rather than the temperature anomalies as it was shown by other studies (e.g. Guillevic et al., 2013, Fig. 3). In the presented work, we show the results using the model from Schwander et al. (1997), because the differences between the obtained solutions using the two models are negligible, except a constant temperature offset. Also noteworthy is that there is no firn model at the moment which uses a dynamically changing CZ. Indeed, this should be investigated but requires additional intense work. Additionally, the knowledge of the time evolution of CZ changes for time periods of millennia to several hundreds of millennia (in frequency and magnitude) is too poor to estimate the influence of this quantity on the reconstruction. In principle, it is possible to cancel out the influence of a potentially changing CZ by using δ^{15}N_{excess} data for temperature reconstruction, as due to the subtraction of δ^{40}Ar/4 from δ^{15}N the gravitational term of the signals is eliminated. From that point of view, it will be interesting to compare temperature solutions gained from δ^{15}N_{excess} fitting with the solutions based on δ^{15}N or δ^{40}Ar alone. This can offer a useful tool for quantifying the magnitude and frequency of CZ changes in the time interval of interest.
It is known that for some very low accumulationrate sites in areas with strong katabatic winds (e.g. “megadunes”, Antarctica) extremely deep CZs can occur, which are potentially able to smooth out even decadalscale temperature variations (Severinghaus et al., 2010). For this, its deepness would need to be of several dozens of metres, which is highly unrealistic even for glacial Summit conditions (Guillevic et al., 2013; see discussion in Annex A4, p. 1042) as well as for the rather stable Holocene period in Greenland for which no low accumulation and strong katabatic wind situations are to be expected.
4.4 Proof of concept for glacial data
For glacial conditions, the task of reconstructing temperature (with correct frequency and magnitude) without using δ^{18}O_{ice} information is much more challenging due to the highly variable gasage – iceage differences (Δage) between stadial and interstadial conditions. Here, contrary to the rather stable Holocene period, the Δage can vary by several hundreds of years. Also, the accumulationrate data are more uncertain than for the Holocene. To prove that the presented fitting algorithm also works for glacial conditions, we inverted the δ^{15}N data measured for the NGRIP ice core by Kindler et al. (2014) for two Dansgaard–Oeschger events, namely DO6 and DO7. Since the magnitudes of those events are higher and the signals are smoother than in the Holocene, we only had to use the Monte Carlo type input generator (see Sect. 2.1.2) for changing the temperature inputs. To compare our results to the δ^{18}O_{ice}based and manually calibrated values from Kindler et al. (2014), we use the ss09sea06bm timescale (NGRIP members, 2004; Johnsen et al., 2001) as it was done in the Kindler et al. (2014) publication. For the model spinup, we use the accumulationrate and temperature data from Kindler et al. (2014) for the time span 36.2 to 60 kyr. The reconstruction window (containing DO6 and DO7) is set to 32 to 36.2 kyr. As the firstguess (starting point) of the reconstruction, we use the accumulationrate data (Acc_{g,0}) for NGRIP from the ss09sea06bm timescale together with a constant temperature of about −49 ^{∘}C for this time window. As minimization criterion D_{g} for the reconstruction, we simply use the sum of the root mean squared errors of the δ^{15}N and Δage mismatches weighted with their uncertainties (wRMSE) according to the following equation, instead of the mean δ^{15}N misfit alone as used for the Holocene (Eq. 1).
Here, ${\mathit{\epsilon}}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{i}}$ and ε_{Δage,k} are the uncertainties in δ^{15}N and Δage for the measured values i or k (Δage match points: Guillevic, 2013, p. 65, Table 3.2) and N, M the number of measurement values. We set ${\mathit{\epsilon}}_{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{i}}=\mathrm{20}$ permeg for all i (Kindler et al., 2014) and ${\mathit{\epsilon}}_{\mathrm{\Delta}\mathrm{age},k}=\mathrm{50}$ years for all k. The relative uncertainties in Δage can easily reach up to 50 % and more in the glacial period using the ss09sea06bm timescale which results in a preeminence of the δ^{15}N misfits over the Δage misfits (10 to 20 % when using GICC05 timescale; Guillevic, 2013, p. 65 Table 3.2). Due to this issue, we have to set Δage uncertainties to 50 years to make both terms equally important for the fitting algorithm. In Fig. 11, we show preliminary results. The δ^{15}N and Δage fitting (Fig. 11a, b) and the resulting gained temperature and accumulationrate solutions (Fig. 11c, d) using the presented algorithm are completely independent from δ^{18}O_{ice}, which provides the opportunity to evaluate the δ^{18}O_{ice}based reconstructions. In this study, the algorithm was used in three steps. First, starting with the first guess (constant temperature), the temperature was changed as explained before. The accumulation rate was changed in parallel to the temperature, allowing a random offset shift (up and down) together with a stretching or compressing (in y direction) of the accumulationrate signal over the whole time window (32 to 36.2 kyr). This first step leads to the “Monte Carlo solution 0” (MCS0) which provides a first approximation and is the base for the next step. For the next step, we fixed the accumulation rate and let the algorithm only change the temperature to improve the δ^{15}N fit (MCS1). Finally, we allow the algorithm to change the temperature together with the accumulation rate using the Monte Carlo type input generator for both quantities. This allows to change the shape of the accumulationrate data. This final step can be seen as a fine tuning of the gained solutions from the steps before. The obtained mismatches in δ^{15}N and Δage of all steps are at least of the same quality or better than the δ^{18}O_{ice}based manual method from Kindler et al. (2014) (see Table 6). The gained temperature solutions show a very good agreement in timing and magnitude compared to the reconstruction of Kindler et al. (2014). Also, the accumulationrate solutions show that the accumulation has to be reduced significantly compared to the ss09sea06bm data to allow a suitable fit of the δ^{15}N and Δage target data, a result highly similar to Guillevic et al. (2013) and Kindler et al. (2014). The mismatches in δ^{15}N and Δage of the final (MCS FIN) solution show a 15 % smaller misfit of δ^{15}N (2σ) and an about 31 % smaller misfit of Δage (2σ) compared to the Kindler et al. (2014) solution. Keeping in mind that the used approach is completely independent from δ^{18}O_{ice} strengthens the functionality and quality of the presented gasisotope fitting approach also for glacial reconstructions. As this section contains a proof of concept of the presented automated gasisotope fitting algorithm on glacial data, preliminary results and ongoing work were shown here. Furthermore, as the presented fitting algorithm was developed and tested in first order for Holocenelike data, it is highly probable that the functionality of the algorithm using glacial data will be further extended and adjusted in future studies.
A novel approach is introduced and described for inverting a firndensification and heatdiffusion model to fit small gasisotopedata variations as observed throughout the Holocene. From this new fitting method, it is possible to extract the surfacetemperature history that drives the firn status which in turn leads to the gasisotope time series. The approach is a combination of a Monte Carlo based iterative method and the analysis of remaining mismatches between modelled and target data. The procedure is fully automated and provides a high potential for parallel computing for time consumption optimization. Additional sensitivity experiments have shown that accumulationrate changes have only a minor influence on shortterm variations of δ^{15}N, which themselves are mainly driven by highfrequency temperature variations. To evaluate the performances of the presented approach, eight different synthetic δ^{15}N time series were created from eight known temperature histories. The fitting approach leads to an excellent agreement in timing and amplitudes between the modelled and synthetic δ^{15}N and temperature data. The obtained final mismatches follow a symmetric standarddistribution function. A total of 95 % of the mismatches compared to the synthetic data are in an envelope between 3.0 to 6.3 permeg for δ^{15}N and 0.23 to 0.51 K for temperature, depending on the synthetic temperature scenarios. These values can therefore be used as a 2σ estimate for the reconstruction uncertainty arising from the presented fitting algorithm itself. For δ^{15}N, the obtained final uncertainties are of the same order of magnitude as stateoftheart measurement uncertainty. The presented reconstruction approach was also successfully applied to δ^{40}Ar and δ^{15}N_{excess} measured data. Moreover, we have shown that the presented fitting approach can also be applied to glacial temperature reconstructions with minor algorithm modifications. Based on the demonstrated flexibility of our inversion methodology, it is reasonable to adapt this approach for reconstructions of other nonlinear physical processes.
The synthetic δ^{15}N and temperature targets, the reconstructed δ^{15}N and temperature data (using the synthetic δ^{15}N as fitting targets), and the used accumulation rates can be found in the data supplement of this paper available at https://doi.pangaea.de/10.1594/PANGAEA.888997 (Döring and Leuenberger, 2018). The GISP2 δ^{18}O_{ice} data used in this study for calculating the temperature spinup can be found in Grootes and Stuiver (1999). The source code for the inversion algorithm and additional auxiliary data are available upon request.
The supplement related to this article is available online at: https://doi.org/10.5194/cp147632018supplement.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Global Challenges for our Common Future: a paleoscience perspective” – PAGES Young Scientists Meeting 2017. It is a result of the 3rd Young Scientists Meeting (YSM), Morillo de Tou, Spain, 7–9 May 2017.
This work was supported by SNF grants (SNF159563 and SNF172550). We
would like to thank Takuro Kobashi, Philippe Kindler and Myriam Guillevic
for helpful discussions about the icecore data and model
peculiarities.
Edited by: Emily Dearing Crampton Flood
Reviewed by: three anonymous referees
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