- Email: [email protected]
Long memory effect of past climate change in Vostok ice core records
Thermochimica Acta 532 (2012) 41–44
Contents lists available at SciVerse ScienceDirect
Thermochimica Acta journal homepage: www.elsevier.com/locate/tca
Long memory effect of past climate change in Vostok ice core records Yuuki Yamamoto a,∗ , Naoki Kitahara b , Makoto Kano a a b
Department of Mechanical Engineering, Tokyo University of Science, Yamaguchi, Japan Department of Electronics and Computer Science, Tokyo University of Science, Yamaguchi, Japan
a r t i c l e
i n f o
Article history: Available online 6 December 2011 MSC: 60G05 60G18 60J65 Keywords: Palaeoclimate Proxy data Time series analysis Spectrum estimate Detrended fluctuation analysis
a b s t r a c t Time series analysis of Vostok ice core data has been done for understanding of palaeoclimate change from a stochastic perspective. The Vostok ice core is one of the proxy data for palaeoclimate in which local temperature and precipitation rate, moisture source conditions, wind strength and aerosol fluxes of marine, volcanic, terrestrial, cosmogenic and anthropogenic origin are indirectly stored. Palaeoclimate data has a periodic feature and a stochastic feature. For the proxy data, spectrum analysis and detrended fluctuation analysis (DFA) were conducted to characterize periodicity and scaling property (long memory effect) in the climate change. The result of spectrum analysis indicates there exist periodicities corresponding to the Milankovitch cycle in past climate change occurred. DFA clarified time variability of scaling exponents (Hurst exponent) is associated with abrupt warming in past climate. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction To understand the mechanism of climate change, long term data observations, analyses and an appropriate interpretation are indispensable. However, it has just past about 100 yrs since climate data observations by using instruments began. 100-Yr-long data is not sufficient to find out hidden mathematical rules in climate. To obtain long term climate records, some proxies in which old climate is recorded indirectly (i.e. ice sheet cores [1–4] and tree-rings [5,6]) are required. In this paper, Vostok ice core data with a period of 420,000 yrs has been used to reconstruct past temperature.1 In the following sections, time series analysis for the Vostok temperature records will be conducted. 2. Spectrum estimate and a result Fig. 1 shows a time series of the local temperature deviation from the present temperature at the atmospheric level reconstructed from the deuterium content of the ice (ıDice , a proxy of local temperature change) [1]. It is very obvious that the time series has some periodicities as an asymmetric sawtooth, not as symmetric sinusoidal. From a long term view, the past temperature tends to rise steeply until it reaches a peak, then drops gradually.
For determining periodicities of the local temperature change in Vostok, we performed a spectrum estimate using the Yule–Walker method [7]. Since intervals of time series in Fig. 1 are irregular, these were regulated to 1-kyr-intervals using linear interpolation, before the estimate. The result of the spectrum estimate is shown in Fig. 2a. In the profile, there are three large peaks indicating periodicities of 105 kyrs, 38 kyrs and 20 kyrs. These estimated periodicities are presumably correlated with the Milankovich cycle (changes in orbital eccentricity (19, 22, 24 kyrs), changes in inclination (41 kyrs) and precession of the equinoxes (95, 125, 400 kyrs)) and approximately consistent with Ref. [1,8]. Incidentally, in Ref. [1] the spectrum analysis was done using the Blackman–Tukey method. However, dramatic changes between glacial and interglacial occurred several times in past are not completely explained by only the periodicities of changes in orbital eccentricity, changes in inclination and precession of the equinoxes, since the variability of insolation of 10-kyr-period is very small compared to the temperature deviation between glacial and interglacial [9,10]. Fig. 2b shows the comparison between the real data of the temperature variability after the interpolation (solid line) and the reproduced one from the inverse Fourier transform imaging major periodicities and some minor periodicities (broken line).
3. Detrended fluctuation analysis (DFA) and results ∗ Corresponding author. E-mail address: [email protected] (Y. Yamamoto). 1 Vostok ice core data for 420,000 yrs. Available at ftp://ftp.ncdc.noaa.gov/pub/ data/paleo/icecore/antarctica/vostok/deutnat.txt.
We consider that a clue to solve the question about the drastic climate change is correlated to a long memory effect of the fluctuation (random force) in the climate. Suppose that the climate change
0040-6031/$ – see front matter. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.tca.2011.11.033
42
Y. Yamamoto et al. / Thermochimica Acta 532 (2012) 41–44
behavior, negative correlation); H > 1.0 implies a non-stationary state with inflow of external energy. To determine the Hurst exponent of random force in Vostok records, we employed detrended fluctuation analysis (DFA) technique [13,14]. For applying DFA to Vostok records, the increment of one step in n is defined as follows:
4
Temperature Deviation (°C)
2 0 −2
n = n+1 − n .
−4
n and their histogram are shown in Fig. 3. Standard deviation of the increments is 0.85, skewness, measure of symmetry, is almost 0, and kurtosis, measure of sharpness, is 0.97. The kurtosis greater than 0 implies the distribution concentrates around the mean more than that of the Gaussian distribution. Incidentally, skewness and kurtosis are defined as follows:
−6 −8 −10 450
400
350
300
250
200
150
100
50
3 N Xi − X (skewness) = ,
0
Age (kyr BP) Fig. 1. Time series of the local temperature change. The deuterium content of the ice (ıDice ) is employed as a proxy of local temperature change.
is regarded as a stochastic process, time series of temperature n may be given by the following: n =
p
ai n−i + n ,
(1)
i=1
where ai is autoregressive coefficient, p is an autoregressive order and n is random force. If there is not memory effect in the climate time series, time evolution of variance of n is proportional to time n as follows: n2 ∼n.
(0 ≤ H ≤ 1, H = / 0.5),
4 N Xi − X (kurtosis) = − 3,
where Xi are stochastic variables, X is a mean of Xi , X is a standard deviation of Xi and N is a number of stochastic variables. DFA is applied to n according to the following steps: 1. Define the average of n over a specific period ( is less than N: length of n ), m+
(m = 1, 2, 3, . . . , N − ).
n=m
2. Take sum of deviations: m − m ,
Zk =
k
[m − m ].
m=1
Zk can be considered as the position of a random walker on a linear chain after k steps.
(b) 4
105,000 years
2
Temperature Deviation (°C)
Spectrum (arbitrary unit)
1 n
m =
(3)
100
80
60
40 38,000 years
20
0 0.00
(6)
X2
i=1
where H is called the Hurst exponent, which characterizes a scaling property of random variables [11,12]. If H is close to 0.5, random variables in the time series have no memory; as H is between 0.5 and 1.0, random variables in the time series have persistence (long memory or correlation); as H is between 0.0 and 0.5, random variables in the time series have anti-persistency (rebounding
(a) 120
(5)
X2
i=1
(2)
Conversely, if there exists memory effect, time evolution of the variance is given by: n2 ∼n2H
(4)
20,000 years
0 −2 −4 −6 −8
0.05
0.10
kyrs-1)
0.15
−10 450
400
350
300
250
200
150
100
50
0
Age (kyr BP)
Fig. 2. (a) Spectrum estimate of Vostok records by the Yule–Walker method. There exist salient periodicities of 105 kyrs, 38 kyrs and 20 kyrs correlated to the Milankovich cycle. (b) Comparison between the real data of the temperature variability and the reproduced one from the Fourier image. Solid line is for the real data of the temperature variability and broken line is for the reproduced one by inverse Fourier transform.
Y. Yamamoto et al. / Thermochimica Acta 532 (2012) 41–44
4
(a)
(b)
140 120
2
100
Frequency
Temperature Difference (°C)
3
43
1 0
80 60
−1 40
−2
20
−3 −4 450
400
350
300
250
200
150
100
50
0 −4
0
−3
−2
−1
0
1
2
3
4
Temperature Difference ( °C)
Age (kyr BP)
Fig. 3. Time series of (a) temperature increments of the interpolated Vostok records and (b) its histogram.
3. Divide Zk into non-overlapping s segments denoted by Zk (j) (j: segment index, j = 1 to s), and subtract each local linear trend Zk,lin (j, s) (least square fitting line) from each segment.
Temperature Deviation (°C)
4. Calculate the root mean square of Fk (j, s),
N 1 f (j, s) = Fk 2 (j, s). k=1
5. Take average of f(j, s) with respect to j. 1 f (j, s). s
4
s
f (s) =
Shift
2
Fk (j, s) = Zk (j, s) − Zk,lin (j, s).
N
Window with 80-kyr-duration
0 −2 −4 −6 −8
(7) −10 450
j=1
400
350
300
6. Repeat 3–5, varying s ranged from 2 to [N/5]. If a stochastic process has a scaling property, f (s) satisfies the following:
200
150
100
50
0
Fig. 4. Data window with the duration of 80 kyrs shifting in time series of temperature increments.
(8)
Hurst exponent Past temperature 1.0
0.5
4.0
0.0
0.0
−4.0
−8.0
−12.0
450
400
350
300
250
200
150
100
50
Temperature Deviation (°C)
The Hurst exponent can be estimated from the slope of linear fitting line of f (s) versus s plot in log–log scale. To obtain time variability of Hurst exponents, the data window with 80-kyr-duration was shifted sequentially from past to present in n as shown in Fig. 4. Each Hurst exponent for the interpolated data in the window was estimated. In all the estimations of the Hurst exponent, the scaling property of the fluctuation was confirmed in the time scale ranging from 1 to 50 kyrs. Fig. 5 shows the comparison between time variability of Hurst exponents and past temperatures in Vostok. In the figure, the variability of Hurst exponents shows rectangular (or trapezoidal) increase-and-decrease behavior ranging from 0.23 to 0.97. In the variability profile, during the interval in which the temperature goes down from a peak of warming, Hurst exponent remains almost constant, which is greater than 0.5, implying the stochastic process is of a long memory. On the other hand, end of decrease (or beginning of drastic increase) in the Hurst exponent (V-shaped points, vertical dotted lines in the figure) corresponds closely to beginning of drastic increase in temperature. This means that fluctuation of temperature once loses memory or becomes anti-persistent before temperature gets into warming state. Additionally, a histogram was constructed to seize distribution of Hurst exponents in Fig. 6. The peak of Hurst exponent distribution is
H
f (s)∼s−H .
250
Age (kyr BP)
0
Age (kyr BP) Fig. 5. Time variability of Hurst exponents. Vertical dotted lines are for ending time of decrease (or beginning time of drastic increase) in Hurst exponent.
44
Y. Yamamoto et al. / Thermochimica Acta 532 (2012) 41–44
3. The end of decrease (or beginning of drastic increase) in the Hurst exponent (V-shaped points, vertical dotted lines in the figure) corresponds well to the beginning of drastic warming, that is, temperature fluctuation once loses memory or becomes antipersistent before drastic warming begins.
700 600
Frequency
500
These conclusions imply the clue of global climate change mechanism is in fluctuation, thus further analysis of fluctuation likely enables us to understand even the mechanism of global warming.
400 300
References
200 100 0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Hurst Exponent Fig. 6. Distribution of Hurst exponents. Peak of distribution is around 0.75.
around 0.75, implying the time series have a long memory through whole period. 4. Conclusions In this study, we have examined periodicity and long memory property in Vostok records using spectrum estimate and DFA. Conclusions are summarized as follows: 1. There exist three salient peaks with the periodicities of 105 kyrs, 38 kyrs and 20 kyrs. These periodicities are probably related to the Milankovich cycle. However, the variability of insolation in the Milankovich cycle cannot completely explain the variability of past climate. 2. While the temperature goes down from a peak of warming, Hurst exponent remains almost constant, which is greater than 0.5, implying the stochastic process is of a long memory.
[1] J. Petit, J. Jouzel, D. Raynaud, N. Barkov, J. Barnola, I. Basile, M. Bender, J. Chappellaz, M. Davis, G. Delaygue, et al., Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica, Nature 399 (6735) (1999) 429–436. [2] J. Jouzel, C. Waelbroeck, B. Malaize, M. Bender, J. Petit, M. Stievenard, N. Barkov, J. Barnola, T. King, V. Kotlyakov, et al., Climatic interpretation of the recently extended Vostok ice records, Climate Dynamics 12 (8) (1996) 513–521. [3] J. Jouzel, N. Barkov, J. Barnola, M. Bender, J. Chappellaz, C. Genthon, V. Kotlyakov, V. Lipenkov, C. Lorius, J. Petit, et al., Extending the Vostok ice-core record of palaeoclimate to the penultimate glacial period, Nature 364 (6436) (1993) 407–412. [4] J. Jouzel, C. Lorius, J. Petit, C. Genthon, N. Barkov, V. Kotlyakov, V. Petrov, Vostok ice core: a continuous isotope temperature record over the last climatic cycle (160,000 years), Nature 329 (1987) 402–408. [5] J. Kurths, C. Spiering, W. Müller-Stoll, U. Striegler, Search for solar periodicities in Miocene tree ring widths, Terra Nova 5 (4) (1993) 359–363. [6] G. Gordon, M. Hughes, P. Kelly, J. Pilcher, V. LaMarche, Climate from Tree Rings, Cambridge University Press, Cambridge, 1982. [7] G.E.P. Box, G.M. Jenkins, G.C. Reinsel, Time Series Analysis, fourth ed., Wiley, Hoboken, NJ, 2008. [8] R. Benzi, A. Sutera, A. Vulpiani, The mechanism of stochastic resonance, Journal of Physics A: Mathematical and General 14 (1981) L453. [9] J. Imbrie, K. Imbrie, Ice Ages: Solving the Mystery, Harvard University Press, 1986. [10] M.E. Raymo, K. Nisancioglu, The 41 kyr world: Milankovitch’s other unsolved mystery, Paleoceanography 18 (1) (2003) 1011. [11] H.E. Hurst, Long-term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers 116 (1951) 770–808. [12] B.B. Mandelbrot, J.W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review 10 (1968) 422–437. [13] C.K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Mosaic organization of DNA nucleotides, Physical Review E 49 (2) (1994) 1685–1689. [14] A. Bashan, R. Bartsch, J. Kantelhardt, S. Havlin, Comparison of detrending methods for fluctuation analysis, Physica A 387 (21) (2008) 5080–5090.
Contents lists available at SciVerse ScienceDirect
Thermochimica Acta journal homepage: www.elsevier.com/locate/tca
Long memory effect of past climate change in Vostok ice core records Yuuki Yamamoto a,∗ , Naoki Kitahara b , Makoto Kano a a b
Department of Mechanical Engineering, Tokyo University of Science, Yamaguchi, Japan Department of Electronics and Computer Science, Tokyo University of Science, Yamaguchi, Japan
a r t i c l e
i n f o
Article history: Available online 6 December 2011 MSC: 60G05 60G18 60J65 Keywords: Palaeoclimate Proxy data Time series analysis Spectrum estimate Detrended fluctuation analysis
a b s t r a c t Time series analysis of Vostok ice core data has been done for understanding of palaeoclimate change from a stochastic perspective. The Vostok ice core is one of the proxy data for palaeoclimate in which local temperature and precipitation rate, moisture source conditions, wind strength and aerosol fluxes of marine, volcanic, terrestrial, cosmogenic and anthropogenic origin are indirectly stored. Palaeoclimate data has a periodic feature and a stochastic feature. For the proxy data, spectrum analysis and detrended fluctuation analysis (DFA) were conducted to characterize periodicity and scaling property (long memory effect) in the climate change. The result of spectrum analysis indicates there exist periodicities corresponding to the Milankovitch cycle in past climate change occurred. DFA clarified time variability of scaling exponents (Hurst exponent) is associated with abrupt warming in past climate. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction To understand the mechanism of climate change, long term data observations, analyses and an appropriate interpretation are indispensable. However, it has just past about 100 yrs since climate data observations by using instruments began. 100-Yr-long data is not sufficient to find out hidden mathematical rules in climate. To obtain long term climate records, some proxies in which old climate is recorded indirectly (i.e. ice sheet cores [1–4] and tree-rings [5,6]) are required. In this paper, Vostok ice core data with a period of 420,000 yrs has been used to reconstruct past temperature.1 In the following sections, time series analysis for the Vostok temperature records will be conducted. 2. Spectrum estimate and a result Fig. 1 shows a time series of the local temperature deviation from the present temperature at the atmospheric level reconstructed from the deuterium content of the ice (ıDice , a proxy of local temperature change) [1]. It is very obvious that the time series has some periodicities as an asymmetric sawtooth, not as symmetric sinusoidal. From a long term view, the past temperature tends to rise steeply until it reaches a peak, then drops gradually.
For determining periodicities of the local temperature change in Vostok, we performed a spectrum estimate using the Yule–Walker method [7]. Since intervals of time series in Fig. 1 are irregular, these were regulated to 1-kyr-intervals using linear interpolation, before the estimate. The result of the spectrum estimate is shown in Fig. 2a. In the profile, there are three large peaks indicating periodicities of 105 kyrs, 38 kyrs and 20 kyrs. These estimated periodicities are presumably correlated with the Milankovich cycle (changes in orbital eccentricity (19, 22, 24 kyrs), changes in inclination (41 kyrs) and precession of the equinoxes (95, 125, 400 kyrs)) and approximately consistent with Ref. [1,8]. Incidentally, in Ref. [1] the spectrum analysis was done using the Blackman–Tukey method. However, dramatic changes between glacial and interglacial occurred several times in past are not completely explained by only the periodicities of changes in orbital eccentricity, changes in inclination and precession of the equinoxes, since the variability of insolation of 10-kyr-period is very small compared to the temperature deviation between glacial and interglacial [9,10]. Fig. 2b shows the comparison between the real data of the temperature variability after the interpolation (solid line) and the reproduced one from the inverse Fourier transform imaging major periodicities and some minor periodicities (broken line).
3. Detrended fluctuation analysis (DFA) and results ∗ Corresponding author. E-mail address: [email protected] (Y. Yamamoto). 1 Vostok ice core data for 420,000 yrs. Available at ftp://ftp.ncdc.noaa.gov/pub/ data/paleo/icecore/antarctica/vostok/deutnat.txt.
We consider that a clue to solve the question about the drastic climate change is correlated to a long memory effect of the fluctuation (random force) in the climate. Suppose that the climate change
0040-6031/$ – see front matter. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.tca.2011.11.033
42
Y. Yamamoto et al. / Thermochimica Acta 532 (2012) 41–44
behavior, negative correlation); H > 1.0 implies a non-stationary state with inflow of external energy. To determine the Hurst exponent of random force in Vostok records, we employed detrended fluctuation analysis (DFA) technique [13,14]. For applying DFA to Vostok records, the increment of one step in n is defined as follows:
4
Temperature Deviation (°C)
2 0 −2
n = n+1 − n .
−4
n and their histogram are shown in Fig. 3. Standard deviation of the increments is 0.85, skewness, measure of symmetry, is almost 0, and kurtosis, measure of sharpness, is 0.97. The kurtosis greater than 0 implies the distribution concentrates around the mean more than that of the Gaussian distribution. Incidentally, skewness and kurtosis are defined as follows:
−6 −8 −10 450
400
350
300
250
200
150
100
50
3 N Xi − X (skewness) = ,
0
Age (kyr BP) Fig. 1. Time series of the local temperature change. The deuterium content of the ice (ıDice ) is employed as a proxy of local temperature change.
is regarded as a stochastic process, time series of temperature n may be given by the following: n =
p
ai n−i + n ,
(1)
i=1
where ai is autoregressive coefficient, p is an autoregressive order and n is random force. If there is not memory effect in the climate time series, time evolution of variance of n is proportional to time n as follows: n2 ∼n.
(0 ≤ H ≤ 1, H = / 0.5),
4 N Xi − X (kurtosis) = − 3,
where Xi are stochastic variables, X is a mean of Xi , X is a standard deviation of Xi and N is a number of stochastic variables. DFA is applied to n according to the following steps: 1. Define the average of n over a specific period ( is less than N: length of n ), m+
(m = 1, 2, 3, . . . , N − ).
n=m
2. Take sum of deviations: m − m ,
Zk =
k
[m − m ].
m=1
Zk can be considered as the position of a random walker on a linear chain after k steps.
(b) 4
105,000 years
2
Temperature Deviation (°C)
Spectrum (arbitrary unit)
1 n
m =
(3)
100
80
60
40 38,000 years
20
0 0.00
(6)
X2
i=1
where H is called the Hurst exponent, which characterizes a scaling property of random variables [11,12]. If H is close to 0.5, random variables in the time series have no memory; as H is between 0.5 and 1.0, random variables in the time series have persistence (long memory or correlation); as H is between 0.0 and 0.5, random variables in the time series have anti-persistency (rebounding
(a) 120
(5)
X2
i=1
(2)
Conversely, if there exists memory effect, time evolution of the variance is given by: n2 ∼n2H
(4)
20,000 years
0 −2 −4 −6 −8
0.05
0.10
kyrs-1)
0.15
−10 450
400
350
300
250
200
150
100
50
0
Age (kyr BP)
Fig. 2. (a) Spectrum estimate of Vostok records by the Yule–Walker method. There exist salient periodicities of 105 kyrs, 38 kyrs and 20 kyrs correlated to the Milankovich cycle. (b) Comparison between the real data of the temperature variability and the reproduced one from the Fourier image. Solid line is for the real data of the temperature variability and broken line is for the reproduced one by inverse Fourier transform.
Y. Yamamoto et al. / Thermochimica Acta 532 (2012) 41–44
4
(a)
(b)
140 120
2
100
Frequency
Temperature Difference (°C)
3
43
1 0
80 60
−1 40
−2
20
−3 −4 450
400
350
300
250
200
150
100
50
0 −4
0
−3
−2
−1
0
1
2
3
4
Temperature Difference ( °C)
Age (kyr BP)
Fig. 3. Time series of (a) temperature increments of the interpolated Vostok records and (b) its histogram.
3. Divide Zk into non-overlapping s segments denoted by Zk (j) (j: segment index, j = 1 to s), and subtract each local linear trend Zk,lin (j, s) (least square fitting line) from each segment.
Temperature Deviation (°C)
4. Calculate the root mean square of Fk (j, s),
N 1 f (j, s) = Fk 2 (j, s). k=1
5. Take average of f(j, s) with respect to j. 1 f (j, s). s
4
s
f (s) =
Shift
2
Fk (j, s) = Zk (j, s) − Zk,lin (j, s).
N
Window with 80-kyr-duration
0 −2 −4 −6 −8
(7) −10 450
j=1
400
350
300
6. Repeat 3–5, varying s ranged from 2 to [N/5]. If a stochastic process has a scaling property, f (s) satisfies the following:
200
150
100
50
0
Fig. 4. Data window with the duration of 80 kyrs shifting in time series of temperature increments.
(8)
Hurst exponent Past temperature 1.0
0.5
4.0
0.0
0.0
−4.0
−8.0
−12.0
450
400
350
300
250
200
150
100
50
Temperature Deviation (°C)
The Hurst exponent can be estimated from the slope of linear fitting line of f (s) versus s plot in log–log scale. To obtain time variability of Hurst exponents, the data window with 80-kyr-duration was shifted sequentially from past to present in n as shown in Fig. 4. Each Hurst exponent for the interpolated data in the window was estimated. In all the estimations of the Hurst exponent, the scaling property of the fluctuation was confirmed in the time scale ranging from 1 to 50 kyrs. Fig. 5 shows the comparison between time variability of Hurst exponents and past temperatures in Vostok. In the figure, the variability of Hurst exponents shows rectangular (or trapezoidal) increase-and-decrease behavior ranging from 0.23 to 0.97. In the variability profile, during the interval in which the temperature goes down from a peak of warming, Hurst exponent remains almost constant, which is greater than 0.5, implying the stochastic process is of a long memory. On the other hand, end of decrease (or beginning of drastic increase) in the Hurst exponent (V-shaped points, vertical dotted lines in the figure) corresponds closely to beginning of drastic increase in temperature. This means that fluctuation of temperature once loses memory or becomes anti-persistent before temperature gets into warming state. Additionally, a histogram was constructed to seize distribution of Hurst exponents in Fig. 6. The peak of Hurst exponent distribution is
H
f (s)∼s−H .
250
Age (kyr BP)
0
Age (kyr BP) Fig. 5. Time variability of Hurst exponents. Vertical dotted lines are for ending time of decrease (or beginning time of drastic increase) in Hurst exponent.
44
Y. Yamamoto et al. / Thermochimica Acta 532 (2012) 41–44
3. The end of decrease (or beginning of drastic increase) in the Hurst exponent (V-shaped points, vertical dotted lines in the figure) corresponds well to the beginning of drastic warming, that is, temperature fluctuation once loses memory or becomes antipersistent before drastic warming begins.
700 600
Frequency
500
These conclusions imply the clue of global climate change mechanism is in fluctuation, thus further analysis of fluctuation likely enables us to understand even the mechanism of global warming.
400 300
References
200 100 0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Hurst Exponent Fig. 6. Distribution of Hurst exponents. Peak of distribution is around 0.75.
around 0.75, implying the time series have a long memory through whole period. 4. Conclusions In this study, we have examined periodicity and long memory property in Vostok records using spectrum estimate and DFA. Conclusions are summarized as follows: 1. There exist three salient peaks with the periodicities of 105 kyrs, 38 kyrs and 20 kyrs. These periodicities are probably related to the Milankovich cycle. However, the variability of insolation in the Milankovich cycle cannot completely explain the variability of past climate. 2. While the temperature goes down from a peak of warming, Hurst exponent remains almost constant, which is greater than 0.5, implying the stochastic process is of a long memory.
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