Atmospheric methane (CH4) recorded in Antarctic ice cores represents the closest ice proxy available for Greenland temperature changes beyond times when Greenland climate records are available. The record over four climatic cycles from the Vostok ice core offers the opportunity to study the phase relationship between Greenland and Antarctic climate changes through detailed CH4 profiles. Combining American and French analytical efforts, we have improved the time resolution of the existing CH4 record from Petit et al. . Spectral analyses reveal high- and low-frequency variability (including a strong precessional component). The phase relationship between CH4 and the Antarctic temperature proxy (δD) shows a systematic lag of CH4 versus temperature by 1100 ± 200 years, on long timescales (50–400 kyr) and a more complex behavior over shorter timescales (i.e., ≤25 kyr), suggesting that Dansgaard/Oeschger-type of climatic variability and associated interhemispheric linkage are robust features of late Quaternary climate.
 In the context of global climate change, the study of greenhouse gases such as carbon dioxide (CO2), methane (CH4), and other biogenic gases is of major interest [Intergovernmental Panel on Climate Change, 2001]. Records of atmospheric trace gas mixing ratios over tens of millennia from ice cores allow to better document the natural cycles of these gases and to identify mechanisms and interactions responsible for climate change and/or variability. They are unique complements to direct atmospheric measurements (covering only the last 50 years or so) and numerous studies over the past 2 decades have demonstrated that ice cores (in particular in Antarctica) faithfully record the atmospheric history of CH4 and other trace gases [Etheridge et al., 1996, 1998]. These studies have contributed significantly to our understanding of climate and of the biogeochemical cycles that control the level of various trace gas species in the atmosphere [Berner et al., 1978; Brook et al., 1996; Raynaud et al., 1988].
 Here we present a refined high-resolution record of atmospheric CH4 from the extended Vostok ice core (78°S, 106°E, 3488 m elevation). Initial work conducted at Vostok provided long-term records of CO2 and CH4 [Barnola et al., 1987, 1991; Chappellaz et al., 1990; Jouzel et al., 1993; Raynaud et al., 1988] covering the last 220 kyr. These studies demonstrated a strong relationship between temperature and greenhouse gas (CO2 and CH4) variations on glacial-interglacial timescales, with low mixing ratio during cold periods and high mixing ratio during warm ones. The extended 5Γ ice core, drilled down to a depth of 3623 m, provides a record of the past four glacial-interglacial cycles (∼420 kyr of atmospheric history [Petit et al., 1999]). We have improved the resolution of the CH4 record, which allows us to discuss CH4 variations on a variety of timescales (typically, 2–100 kyr). We first examine CH4 variability on the glacial-interglacial timescale and then focus in more detail on the shorter-term variability. These observations are used, with the paleotemperature record derived from the D/H ratios in Vostok ice (δD), to further constrain the CH4-climate phase relationship during short-term and long-term climate events.
2. Experimental Procedure and Age Scale
 The high-resolution CH4 record from Vostok presented in this paper is based on measurements at more than 700 depth levels (more than 1000 individual samples) conducted both at Laboratoire de Glaciologie et Géophysique de l'Environnement (LGGE, Grenoble, France) and at Washington State University (WSU, Vancouver, Washington, USA). CH4 mixing ratios were measured on three different Vostok cores over the last 15 years (core 3Γ and 4Γ for the upper 2500 m and 5Γ below 2500 m). Depth differences between ice cores were corrected as described by Petit et al. . The GT4 timescale (Glaciological Timescale 4) represents the reference timescale used in this study (see [Petit et al., 1999] for details). Other timescales proposed more recently and derived from marine chronology tuning [Shackleton, 2000] or inverse modeling [Parrenin et al., 2001] are also used for sensitivity studies. The gas-ice age difference (hereafter Δage), due to the fact that air bubbles are trapped around 100 m below the surface (about 2 to 6 kyr after the surrounding ice was deposited, depending on the time period considered), has been estimated using a firn densification model, including different assumptions summarized in Table 2 [Arnaud et al., 2000; Barnola et al., 1991]. We obtain the gas age by subtracting the Δage with values at each depth from the assigned ice ages.
 CH4 measurements were conducted in both laboratories using wet extraction procedures previously described [Brook et al., 2000; Chappellaz et al., 1997]. At LGGE, seven samples can be measured during a single run, and each extracted gas sample is measured three times during the run. The final CH4 value arises from a weighted mean of the three measurements [Chappellaz et al., 1997] taking into account their decreasing precision with decreasing gas pressure injected into the gas chromatograph. At WSU a single measurement is made on each extracted gas sample. At each depth, two separate ice sample were systematically measured. In both laboratories, the CH4 sample mixing ratios are calculated using a calibration procedure conducted with a standard gas reference for each set of measurements and corrected for the blank contamination [Sowers et al., 1997]. For consistency with previously published Vostok CH4 data sets [Chappellaz et al., 1990; Petit et al., 1999; Raynaud et al., 1988], the CH4 mixing ratio presented here are set on the LGGE scale, which is 2% (∼10 ppbv) higher than the NOAA scale used at WSU [Sowers et al., 1997]. New tests and calibration conducted at LGGE during this study generally confirm these results. However, mixing ratios measured at WSU are on average higher than those determined at LGGE. The average shift observed between both data sets over the 250–430 kyr period (same depth resolution between the two laboratories) is 17 ppbv after correcting for offset in the calibration scale. A slight standard drift and/or under or over-estimation of the blank in either laboratory may cause the 17 ppbv shift. The consistency between old and new LGGE CH4 measurements leads us to correct the WSU values by subtracting 17 ppbv from the measured mixing ratios. The mean reproducibility on duplicates at WSU is better than ±20 ppbv (2σ). At LGGE more than 25% of the depth intervals were measured at least twice to check the reproducibility of the method, leading to a comparable reproducibility of ±25 ppbv (2σ). In the following, we refer to the mean values when replicate measurements were performed.
3. High-Resolution Methane Record
 Near the bottom of the Vostok core (about 430 kyr, 3320–3330 m, Figure 1) there is an abrupt and large change in the CH4 mixing ratio and δD of the ice. This feature described by Petit et al.  and Raynaud et al.  is thought to be a result of flow disturbance of the stratigraphic record and therefore not a true climate signature [Jouzel et al., 1999]. We will thus limit the following discussion to the CH4 record over the upper 3310 m (426 kyr) of the core.
 The mean temporal resolution of the Vostok global CH4 profile has been improved by 30% compared to the initial published record [Petit et al., 1999], especially over the interglacial period MIS 9 (550 years) and during the Climate Cycle 2 (CC2) and full glacial section of climate cycle 3 (see Table 1 for definitions and resolutions of the different climate cycles). The improved resolution of the record confirms that CH4 levels in interglacial periods MIS 7.5, 9 and 11 do not exceed ∼780 ppbv. This observation confirms the inference, from lower-resolution data, that the current atmospheric CH4 levels (>1700 ppbv) are highly anomalous with respect to the last 420 kyr of atmospheric history.
Table 1. Spectral Analysis Results of the Vostok CH4 Record: Full Record (Vostok Four Cycles), Individual Climate Cycle (CC1, CC2, CC3, CC4), and Full Glacial Period of Individual Climate Cycle (CC1 FG, CC2 FG etc.)a
For comparison, the results for Greenland over the last climatic cycle are also presented. The reference age scales considered for the analysis are presented in column 2; the time period considered and the resolution of the record over that time period are presented in columns 3 and 4. We define a climatic cycle as the period between two absolute isotopic maxima (see also Figure 1). Abbreviations are as follows: GT4, Glaciological Timescale 4; NJS, Vostok timescale of N. J. Shackleton; and FG, full glacial conditions.
Main periodicities are obtained with the Multitaper Method. Boldface indicates significant periodicities common or close to at least the several different time periods considered.
 The overall correlation between CH4 and δD long-term records remains but weakens (R2 = 0.61 instead of 0.73 with the previous CH4 data set). The rapid jump of CH4 characterizing the second part of the deglaciation appears sharper than previously shown by the lower-resolution record (less than 400 years, or 800 years for termination IV, MIS 9). Last, a high-frequency variability shows up. CH4 fluctuates within the range 350–600 ppbv during glacial periods, with a typical duration of 1.5 to 3 kyr for each rapid event. Most of this CH4 structure is also depicted by the deuterium record.
 We applied a Multitaper Method (MTM) to the CH4 series (after a linear interpolation of the signal at a 100-year resolution) using the MTM-SSA Toolkit (available at www.atmos.ucla.edu/tcd/ssa). The MTM method was retained over other techniques ((Blackmann-Tukey; Maximum Entropy Method) because it has a better spectral resolution and permits two types of statistical tests to differentiate a time series from background noise [Ghil et al., 2002; Mann and Lees, 1996]. Nevertheless, other spectral methods have been tested and give similar results albeit with a lesser resolution (not shown). Sensitivity tests using different averaging procedures and time steps for resampling did not significantly change the results. On the other hand, using other Vostok timescales (e.g., NJS in Table 1) can significantly affect spectral properties. This is particularly true for Climatic Cycle 2 (CC2) and Climatic Cycle 3 (CC3) where most dating discrepancies appear (Figures 2 and 3). However, the recurrence of the same peaks at high-frequency bands, being statistically significant whatever the dating used, preclude them from being artifacts.
 Apart from the orbital periodicities of 100, 41, 23 and 19 kyr remaining unchanged compared to Petit et al.  (Table 1 and Figure 2), significant peaks around 12 and 10 kyr and even shorter periods (2–7 kyr, see Table 1) appear now. The 10- to 12-kyr period has already been found in climatic records and is thought to be a nonlinear combination from harmonics of the precession band (19–23 kyr) [Hagelberg et al., 1994; Yiou et al., 1995]. In other words, the Sun passes overhead in the tropics twice a year, so that the Sun is overhead at perihelion every 11 kyr. As the tropical regions are particularly sensitive to seasonal insolation changes in the precessional band (affecting the monsoon strength) and are thought to be the major contributing source regions for the CH4 budget [Chappellaz et al., 1993b], it is not surprising to recover this kind of variability within the CH4 record. Similar conclusions were drawn for the last climatic cycle on the Greenland Ice Sheet Project 2 (GISP2) core [Brook et al., 1996].
 At the scale of each single climatic cycle, typical periods of high-frequency variability are 9–7 kyr and between 2 and 5 kyr (Table 1), i.e., much similar to those found in earlier studies conducted on the CH4 records from Greenland Ice Core Project (GRIP) and GISP2 cores [Brook et al., 1996; Mazaud et al., 2000; Yiou et al., 2001] for the last glacial period and which were associated with Dansgaard/Oeschger (D/O) events. They reflect millennial-scale CH4 changes that apparently characterize each glacial period investigated here. Indeed the association of the CH4-Greenland isotope variations over the last glacial is not linear; we note that the maximum and/or average amplitude of CH4 changes could vary by more than a factor of 2 or 3 between D/O events of similar isotopic amplitude [Brook et al., 1999; Flückiger et al., 2004]. Although the smoothing of atmospheric CH4 fluctuations during gas trapping at Vostok (probably of the order of 500–700 years during glacial times [Barnola et al., 1991]) partly dampened the true atmospheric CH4 fluctuations, the D/O structure in the signal still shows up as a strong component in the CH4 spectrum. Assuming that such relationship held for earlier times, our extended Vostok record indicates that high-frequency northern hemisphere climate variability characterized the climate system during all climatic cycles. In this sense, we are using CH4 as a proxy for northern hemisphere temperature fluctuations.
 On the other hand, the high-latitude dynamics associated with the strengthening/weakening of the global thermohaline circulation, driving freezing/thawing extent and amplitude of discontinuous permafrost, could have also contributed to the millennial variability in the CH4 signal. Recent studies on the interhemispheric CH4 gradient [Brook et al., 2000; Dällenbach et al., 2000] suggest that even if the tropics remained the major contributor to the global CH4 budget during glacial periods, the higher CH4 concentration during the warm D/O events could be caused by higher source strength at mid/high northern latitudes. Other possible contribution to the CH4 budget could also be found through clathrate degassing [Kennett et al., 2000]. As pointed out by Flückiger et al. , it is still not possible to draw definitive conclusions as to where the CH4 variability mainly originated from and which climatic mechanisms or feedbacks could drive these fluctuations.
5. Leads and Lags Between CH4 and Vostok Temperature
5.1. Age Scale and Uncertainty
 The accuracy of comparing climate signals recorded respectively in the gas bubbles and in the ice essentially depends on the uncertainty in the age and Δage calculations. The ice core chronology appears crucial on this regard because it is linked with the accumulation rate, which is among the parameters needed to calculate Δage. To investigate this issue we conducted sensitivity tests using three different ice chronologies and five different estimates of Δage based on various parameterizations in the densification model used here (Figure 3 and Table 2).
Table 2. Summary of the Different Ice and Gas Chronologies Used in This Study for the Sensitivity Experimenta
Reference Ice Chronology
Parameterization of the Densification Model
Surface Temperature, °C
Physics of the Firn
Source of Densification Model Adaptation
The Δage calculations (i.e., the gas age chronologies) are based on the densification model originally developed by [Barnola et al., 1991], forced by the surface temperature deduced from the δD signal using the δD-T slope of 6.04/°C (taking into account a correction between the surface and the inversion layer and the variation of the isotopic composition of the water source for past temperatures [Jouzel et al., 2003]) and by the accumulation rate based on δD [Barnola et al., 1991], except for the PN and NJS gas chronologies which have their own prescribed temperature and accumulation (see Parrenin et al.  and Shackleton , respectively).
glaciological inversion plus chronological constraints
 The differences between Δages calculated with the various scenarios are significant (Figure 3b). The largest difference lies in full glacial conditions (up to 2.5 kyr) whereas warm periods show a reduced discrepancy (less than 800 years); the deglaciations are associated with comparable Δages whatever the chronology considered.
 Indeed, Δages can be evaluated at Vostok through permanent gas isotope anomalies (δ15N, δ40Ar) recording surface temperature or accumulation changes which then affect the firn thickness and/or generate a thermal anomaly (see Caillon et al.  for further discussions). Such constraints are only available at the MIS 5d/5c transition (110 kyr BP), at the start of termination III (240 kyr BP [Caillon et al., 2003b]) and at the MIS 7.5 optimum. These independent Δage estimates agree well with all the Δage estimates given by the models (Figure 3b). The maximum difference obtained between both approaches happens at 1570 m of depth; a Δage of 5350 ± 250 years is given by δ15N measurements whereas the modeled Δages range from 4700 to 5360 years. The underestimated ages come from PN, GT4, G57 and A55 (Table 2 for the definition of scenarios). For the two other Δage estimates (at 2775.75 m, Δage = 2600 ± 250 years and at 2802 m, Δage = 4100 ± 250 years) most modeled Δage are within errors of the δ40Ar-based estimate constraint, except for the NJS chronology and the PN chronology at 2802 m of depth. Overall, this comparison indicates that the modeled Δages are correct to within a few centuries where permanent gas estimates are available. Unfortunately, the lack of independent estimates from gas isotopes during MIS 2, 3 and 6 (where differences between models are the most significant) does not yet allow picking up the most probable scenario of Δages.
 Recently, ages at Vostok were evaluated against the Byrd CH4 and δ18O records, pointing out that the surface temperature changes deduced from water isotopes were more probable than some borehole temperature interpretation [Blunier et al., 2004]. The Vostok “experimental” Δages thus inferred from Byrd range between ∼4.5 and ∼6 kyr for Vostok depths between 500 and 750 m. In our case, this evaluation would tend to favor the scenarios GT4, G57, A55 and NJS.
5.2. Leads and Lags
 The Greenland/Byrd-record comparison shows that during major climatic events of the last glacial, Antarctic temperature rises when Greenland experiences a stadial event and decreases at the time of abrupt Greenland warming [Blunier and Brook, 2001; Blunier et al., 1998]. This expression of the so-called “bipolar seesaw” shows up in a phase comparison between the two records as an apparent lead of Antarctica on Greenland climate, by 1.5 to 3 kyr. As initiated by Bender et al.  a bipolar comparison can also be produced using Vostok data over the last glacial, but it is affected by the Δage uncertainty.
 In order to determine leads and lags between CH4 and Vostok Antarctic temperature at different timescales and taking into account the different Δage scenarios (Table 2), we apply a multiscale lag regression method (detailed in Appendix A) where maximum correlation between the two signals is determined within a given time window by varying the time axis of one variable with respect to the other. The method is applied to each ice chronology and Δage scenario, providing an optimal time lag and correlation coefficient in each case (Figures 4 and 5). The results presented here were tested using two different approaches that produced similar results (see Appendix A). For clarity, Figures 4, 5, and 6 only present results obtained with the square correlation (R2) maximization method. Several time windows (400, 200, 100, 50, 25 and 12 kyr) were used in order to investigate leads and lags over different timescales. Where no estimate of the time lag or correlation coefficient is provided (Figures 4, 5, and 6), the maximum correlation was reached on the edge of the time window, and therefore discarded (see Appendix A for more details).
 Several interesting observations can be made from this analysis. First, time windows between 50 and 400 kyr all indicate a coherent lag of CH4 versus Antarctic temperature ranging from 100 to 2100 years (Figure 4, mean value of 900 to 1300 years depending on the time window considered). The 12- and 25-kyr windows show much more variability in the phase relationship at different time periods and depending on the chronology used. Similar conclusions can be drawn for the correlation coefficient (Figure 5).
Figure 4 provides three major features on the CH4/Antarctic temperature phase relationship:
 1. A CH4 time lag between 700 and 1600 years on the scale of two or four climatic cycles (200- and 400-kyr window); this time lag is mostly forced by the variance of both signals linked with terminations. A similar feature was observed during termination I where Antarctic temperature starts to raise about 1000 years before the CH4 early increase and where the Younger Dryas/Preboreal CH4 increase takes place when Antarctica nearly ends its deglacial warming [Blunier et al., 1998]. This also confirms the general feature of deglacial events considered by Petit et al. .
 2. With time windows of 100 and 50 kyr, a significant decrease of the CH4 lag with time is observed; in addition within the 50-kyr windows, the first halves of glaciations reveal a smaller lag than the second halves. There are three possible explanations to such feature: a climatic mechanism varying bipolar energy exchanges in relation with ice sheet extents, an artifact due to smaller amount of captured variability for both CH4 and Antarctic isotope with increasing time, especially during MIS 10, or a variable bias in the ice chronology possibly due to a badly estimated accumulation rate upstream from Vostok or due to ice flow. This last explanation can a priori be discarded because the decreased time lag trend appears for all chronologies and especially for the NJS chronology, which is orbitally tuned (Figure 4).
 3. The 25- and 12-kyr time windows show on average a reduced CH4 lag than longer time windows. This arises from frequent jumps between CH4 lags and CH4 leads over Vostok temperature; the lengths of the correlation windows being close to the duration of rapid events, best signal coherence can be found either by increasing or decreasing the relative age of the CH4 record with respect to the temperature record, with the possibility of highest R2 through anticorrelation of the two signals (Figure 5). This phenomenon is even more obvious for the 12-kyr window and reveals the limit of the correlation method used here, as detailed in Appendix A.
 The CH4/temperature relationship deduced from the Byrd Antarctic ice core, with its better precision on Δage estimates (∼1 order of magnitude better than Vostok) provides an independent check of these findings over the last 90 kyr, through our multiscale lag regression method. The Byrd phasing also reveals a consistent lag of CH4 versus Byrd Antarctic temperature and thus gives support to our analyses; but the magnitude of the lag is slightly different compared to Vostok (Figures 4 and 5). Considering two different Byrd chronologies [Blunier and Brook, 2001] does not affect significantly this observation. The worst agreement between Vostok and Byrd lag estimates arises for the 50-kyr window covering MIS 4 and 5, with a difference between both estimates of about 600 years. This is the time period where the available Vostok CH4 record provides a poor time resolution (mean of 1800 years with a range between 500 and 2800 years) and does not resolve all the variability observed in higher resolution records such as GISP2, Byrd and Taylor Dome.
 To investigate more specifically the leads and lags on rapid events, we applied the multiscale regression method on the residuals of the CH4 and δD series, after removal of the variability linked with orbital frequencies. Two methods were used: (1) using a low-pass filter and (2) reconstructing the Vostok series after deconvolution and removal of the first four principal frequencies for each signal using the Singular Spectrum Analysis method ((SSA) [Ghil et al., 2002]). Using the first method ensures the removal of a similar frequency on each time series, but possibly not all the orbital variability, whereas the second method ensures the cutting of all the orbital frequencies but possibly with different spectral signature for each time series. We used the GT4 age scale, but a similar conclusion would be drawn from other chronologies. The leads and lags resulting from both approaches are coherent and agree well with those for the unfiltered original data, as illustrated in Figure 6. This confirms that much of the lead/lag relationship pointed out with the full records is linked with rapid Dansgaard/Oeschger-type events.
 Overall, although the magnitude of the lag is still a matter for discussions and improvements, our analyses of the temporal relationship between atmospheric CH4 and Antarctic temperature supports a systematic apparent lead of the latter in the course of the last four climatic cycles, both at low and high frequencies. As CH4 has been shown to follow Greenland temperature changes at the scale of a few decades over the last glaciation and deglaciation [Severinghaus and Brook, 1999; Severinghaus et al., 1998], our observation implies that a sustained climatic mechanism (or a suite of them) drove the sequence of rapid events consistently throughout the last 420 kyr. Arguing for instance about the relative contribution of the thermohaline circulation and tropical water cycle feedbacks is beyond the scope of our findings. As already pointed out by Caillon et al. [2003a], the phase relationship between the two poles cannot be considered diagnostic of the location of origin of millennial-scale climate changes, nor the physical mechanism involved. However, it provides constraints to test the validity of global climate models that simulate the sequence of events.
 The improved temporal resolution of our Vostok CH4 record corroborates prior findings that the high present-day CH4 mixing ratio is unprecedented during the last 420 kyr and confirms the strong imprint of orbital frequencies in this biogeochemical signal. Most importantly, it reveals a CH4 high-frequency variability over the past four climatic cycles, which had been previously depicted by Greenland records only over the last climatic cycle. This suggests that the climate system not only behave as an oscillator over long time periods (i.e., orbital frequencies) but also shows up with a coherent and cyclic high-frequency variability in the course of each climate cycle.
 Phase correlation analyses between CH4 and Vostok temperature indicate a systematic apparent lag of CH4 by ∼1100 ± 200 years, similar to the one observed during the last glacial period and expressing the so-called “bipolar seesaw.” This result has been tested against different ice and trapped gas chronologies, and appears robust for long time windows (200 kyr to 50 kyr). Over shorter time windows (25 kyr and 12 kyr), the CH4-δD phase correlation shows much more variability as the window length gets closer to the duration of events. Overall, the improved Vostok CH4 profile and its correlation analysis with Vostok temperature, either with the full record or with residuals deconvolved from orbital frequencies point toward the existence of internal climatic mechanisms that acted as high-frequency pacemakers of glacial climate variability over at least the last four glaciations. Although the exact mechanisms remain unclear and unconstrained through our approach, it represents a challenging observation for future climate modeling efforts.
 On the ice core side, the next step will consist of (1) decreasing the noise of the CH4 and Antarctic temperature records, especially on the basis of the new EPICA/Dome C ice core, (2) improving the quantitative estimate of the CH4 lag with respect to temperature, using chronological constraints between Vostok and Dome C, and (3) extending back in time our observations into epochs when the 40-kyr period dominated the Earth's climate variability, again with the new Dome C ice core.
Appendix A.: Multiscale Lag Regression Between Two Time Series
 The starting assumption of the method is that a time series Y(t) is obtained from a reference X(t) by a constant linear transform and random phase shifts. Hence a global coherence and phase analysis between the series X(t) and Y(t) might be quite irrelevant because the phase lags are certainly not constant through time (and can change signs). The general idea is to cut each time series into nonoverlapping intervals, and perform a phase analysis on each interval. Borrowing the concept of multiscale analysis from wavelet transforms, we vary the length of the intervals with a geometrical sequence. Thus slow covariance will be captured by large scales, and fast covariance by small scales. For instance, it is not a priori clear that temperature and CH4 share the same relationship on glacial to interglacial cycles, and rapid events such as the Dansgaard-Oeschger oscillations. Thus, using a multiscale analysis is useful to investigate whether the CH4-temperature relationship comes from one or several biophysical processes. The stability of the results will be investigated by using different chronologies for the time series.
 On each time interval T, we determine an optimal phase lag between X(t) and Y(t + τ) by maximizing the square correlation R2 between the two sub time series:
Here τ varies between −30 and 30 time steps, corresponding to a lag range of −3 kyr to 3 kyr. A more rigorous way of optimizing the lag is to make a linear regression of X(t) and Y(t) on each interval T, with varying phase lags, compute a likelihood function of these regressions Λ:
where N is the number of data points in the interval T, D(t) is the residual of the linear regression between Y(t) and X(t + τ) in T:
and σD2 is the variance of D(t). Then we compute the maximum (log) likelihood over the phase lag range τ and regression parameters a and b. Both correlation and likelihood approaches give similar results (albeit not strictly identical) and thus we will mention the local correlation optimization because it is more straightforward to interpret.
 It is important to note that the optimum correlation (or likelihood) should be reached within the lag range, and not on the edge of a given lag interval, to ensure that the maximized function (R2 or Λ) is concave. Otherwise, the maximum might depend on the length of the scale, which is undesirable. Thus, whenever a maximum was reached on the edge of a lag interval, we discarded the corresponding scale and time interval, and rejected the hypothesis that there is a local linear relation between the two time series.
 Sensitivity tests made using lag intervals of different length shows that the choice of the lag interval can also affect the optimum correlation (or likelihood, not shown). Hence we advocate the use of lag intervals that are no larger than one quarter of the scale, especially in the presence of large noise levels and sampling heterogeneity between the two time series. All the lag computations where run with Scilab (available at www.scilab.org), and the routines can be obtained from the authors.
 The Vostok ice coring program is a multiyear joint effort between Russia, France and the USA. We thank ice drillers and all participants in the fieldwork and ice sampling, the Russian Antarctic Expeditions (RAE), the Institut Français Paul Emile Victor (IPEV) and the Office of Polar Programs (NSF) for logistic support. This work is a contribution to the EC project Pole-Ocean-Pole (EVK2-2000-22067). Additional financial support in France was provided by the Programme National d'Etudes de la Dynamique du Climat (INSU-CNRS). The US National Science Foundation, Office of Polar Programs, provided financial support for the WSU portion of the project. We are indebted to Susan Harder who contributed significantly to the CH4 measurements at WSU. We thank Jean Jouzel, Valérie Masson-Delmotte and Amaëlle Landais for fruitful discussion and comments. We are deeply grateful to Todd Sowers, Bernhard Stauffer and an anonymous reviewer for constructive and helpful reviews.