1. Introduction
When a climate system is forced to exceed its critical transition point, abrupt climate change usually occurs (Alley et al. 2003). In recent years, research on early warning signals for abrupt climate change has attracted increasing attention, as this change can have catastrophic effects on human societies and natural systems (Alley et al. 2003; Bathiany et al. 2018b; Duarte et al. 2012; Kortsch et al. 2012; Lenton et al. 2012; Lenton 2011, 2013; Livina and Lenton 2007; Overpeck and Cole 2006; Sinha et al. 2019; Taylor et al. 2004; Williams et al. 2011). Numerical models are effective tools for simulating past behaviors in a climate system and can also be used to project future climate change (Bathiany et al. 2018a; Drijfhout et al. 2015; Hansen et al. 2016; He et al. 2019). In recent years, with the development of science and technology and the deepening of human understanding about the mechanisms of climate change, the simulation performance and prediction skills of various numerical models have been continuously improved. However, due to the nonlinear and complex characteristics of climate systems, it is still difficult to accurately predict abrupt climate change through the numerical models at present (Thompson and Sieber 2012).
Research shows that when a system approaches its critical transition point, a small disturbance may cause a sudden change in the state of the system (Rial et al. 2004). Therefore, if a general characteristic can be revealed when a dynamic system approaches its critical transition point, it may provide an early warning signal to identify whether the system is approaching its critical point. Wissel (1984) found that there was a universal law—the phenomenon of “critical slowing down” appears when approaching the critical point of a dynamic system—which provides a new way to investigate early warnings of abrupt changes. Subsequently, numerous studies on critical transitions were carried out (Lindegren et al. 2012; Scheffer et al. 2009; Sun et al. 2014).
A series of statistical indicators based on the phenomenon of “critical slowing down” have been proposed. For instance, Kleinen et al. (2003) found that the power spectrum would become “red” when approaching the bifurcation point of a simple two-box model of hemispheric thermohaline circulation (THC). Held and Kleinen (2004) proposed that the minimum decay rate of a climate system could be used as an indicator to determine the distance from a critical point of the climate system. Livina and Lenton (2007) modified the degenerate fingerprinting method by employing detrended fluctuation analysis (DFA) for detecting the transition from glacial to interglacial conditions. Then, the DFA exponent was calibrated by the autocorrelation function at lag 1 [AR(1)], and a new early warning signal “DFA-propagator” was obtained. Based on eight paleoclimate reconstruction time series, Dakos et al. (2008) found that there was an increase in autocorrelation before the onset of abrupt climate change events. Lenton et al. (2012) systematically examined the robustness of two kinds of early warning signals, namely, the autocorrelation function (ACF) indicator and the DFA indicator. Paleoclimate data from three different sites that recorded the end of the last ice age, and three groups of numerical experiments were used as experimental platforms. They concluded that ACF and DFA methods have their own advantages and disadvantages, and they can be combined as a cross-check. Rypdal (2016) analyzed the δ18O records from the North Greenland Ice Core Project (NGRIP) and found that the high-frequency bands of δ18O records showed an increasing level of fluctuation with the approach of the interstadial period (warm period). Then, Rypdal (2016) proposed new early warning signals based on wavelet analysis—
For a stationary time series, the term “long-term memory” (LTM) is also commonly called as “long-range correlation” or “long-range persistence,” which implies that there is nonnegligible dependence between the present and all points in the past (Graves et al. 2017). LTM exists universally in many complex systems, such as DNA sequences in genomics (Peng et al. 1993a, 1994, 1992), electrocardiogram time series in cardiology departments (Mäkikallio et al. 1999; Peng et al. 1993b), asset return data in finance (Cajueiro and Tabak 2004; Weron 2002), price fluctuation sequences in stock markets (Serinaldi 2010), traffic flows (Musha and Higuchi 1976), hydrological streamflow records (Hurst 1951, 1954, 1956; Mandelbrot and Wallis 1969a,b; Mandelbrot and Wheeler 1983), temperature and precipitation records, and ozone concentration series in the climate system (Bunde et al. 2005; Fraedrich and Blender 2003; Fraedrich et al. 2004; Varotsos and Kirk-Davidoff 2006). A large number of previous studies have proved that there is short-term memory enhancement [AR(1) approaches 1] in systems slowly approaching the critical transition (Held and Kleinen 2004; Lenton et al. 2012; Lenton 2011; Lenton et al. 2009; Livina and Lenton 2007; Scheffer et al. 2009), so will LTM be the same? Therefore, an open question is raised: are there general characteristic changes in the LTM of a dynamic system as it approaches its critical transition point? There are many methods to estimate LTM (Hurst 1951, 1954; Kantelhardt et al. 2001; Mandelbrot and Wallis 1969a; Taqqu et al. 1995). In this paper, we choose rescaled range (R/S) analysis with a simple calculation process to study the feasibility of changing LTM as an early warning indicator of abrupt climate change resulting from a critical transition. Moreover, the performance of R/S is compared with existing indicators, including AR(1), variance, and DFA-propagator.
In this paper, two ecological models and a zero-dimensional climate model are adopted to simulate a large number of stationary time series. Each value of the control parameter corresponds to a time series. When the control parameter approaches the critical transition point of each folding model, the change of LTM in the time series can be obtained. Finally, the different early warning signals are reused to analyze previously studied paleoclimate data, namely, the early warning of the termination of the Younger Dryas (YD) period and the onset of the Holocene. The rest of this paper is organized as follows: section 2 introduces the methods, models, and paleoclimate records used in this study. Section 3 provides the results of the artificial models and paleoclimate records. The conclusions and a brief discussion are given in section 4.
2. Methods, models, and data
a. Rescaled range (R/S) analysis
R/S analysis is a classical statistical method, which can quantitatively estimate the LTM intensity of a stationary time series. It was first proposed by the hydrologist Harold Edwin Hurst (Hurst 1951, 1954, 1956). Since then, R/S has been constantly modified (Lo 1991; Mandelbrot and Wallis 1969a; Mandelbrot 1975; Mandelbrot and Wallis 1969b,c). The R/S algorithm is briefly introduced as follows (Hurst 1956; Mandelbrot and Wallis 1969a).
First, for a time series with length N, {xi, i = 1, 2, …, N}, divide it into n adjacent nonoverlapping subsequences of equal length s, {yj, j = 1, 2, …, s}. Here, n is equal to [N/s], and the square brackets indicate rounding.
Third, calculate the statistical average values of the R/S statistics of all subsequences.
Finally, we change the length s of the subsequences and repeat the three steps mentioned above to obtain the relationship between the R/S statistic and s. For a time series with fractal characteristics, Hurst (1956) indicated that the R/S statistic and s should satisfy the following relationship:
For stationary time series, H is limited to the range 0 ≤ H ≤ 1, with the DFA exponent α ranging from 0 to 1.5 (Kantelhardt et al. 2001). The evolution of a time series has the characteristics of antipersistence when 0 < H < 0.5 (0 < α < 0.5). In contrast, the time series has long-term memory (or persistence) for 0.5 < H < 1 (0.5 < α < 1.5). For uncorrelated records, H = 0.5 (Samorodnitsky 2006). The closer the H value is to 1, the stronger the long-term memory of a dynamic system is (Bhattacharya et al. 1983; Davies and Harte 1987; Graves et al. 2017; He et al. 2016; Mandelbrot and Wallis 1969b; Millán et al. 2021). The autocorrelation function can be characterized by a power law C(s) ∼ s−γ for large scales s, where γ = 2 − 2H if the series is stationary (0 < γ < 1) (Kantelhardt et al. 2002; Movahed et al. 2006). And its power spectra can be characterized by S(ω) ∼ ω−β with frequency ω, β = 2H − 1, and β = 2α − 1. In the nonstationary case, the relationship of the exponent is γ = −2H and β = 2H + 1, respectively (Movahed et al. 2006). It should be emphasized that all analysis objects in this paper are developed based on the time series of stationary assumption. Although it has been pointed out that the R/S method is somewhat similar to the DFA0 (no trends elimination) analysis (Eichner et al. 2003; Kantelhardt et al. 2001, 2002), we believe that the two are not exactly the same. Since the R/S method is suitable for stationary records, it is necessary to filter out the trend of the original time series properly before analyzing the nonstationary time series.
b. Folding bifurcation models
It is important to emphasize that none of the time series analyzed here has mutated. In the process of numerical calculation, we checked the time series but did not consider the time series with critical transition. At the same time, for all the numerical simulations, our control parameters did not change with time; that is, under different control parameters, the model iterates for the same time step to test the LTM of the model under specific control parameters. Therefore, the relationship between the control parameters and the H cannot be directly deduced. For the convenience of comparison, in addition to the boxplot of multiple tests, we calculate the relative growth rate in the supplemental material based on the first value, that is, [x(μ) − x(1)]/x(1), where x refers to four early warning signals and μ refers to specific control parameters. MATLAB-R2015b is used to solve the stochastic folding models numerically, and Intel FORTRAN is used to calculate the early warning signals.
c. Paleoclimate data
The YD event is a typical abrupt climate change, which can be confirmed in paleoclimate data of different sites referring to a sudden cooling event in the continuous warming process of the last deglaciation period in particular (Alley 2000; Alley and Ágústsdóttir 2005; Alley et al. 1993; Cheng et al. 2020; Firestone et al. 2007). In the North Atlantic region, the event occurred approximately 12.8 kyr ago (Cheng et al. 2020). After the end of the YD period, the temperature began to rise rapidly around 11.7 kyr BP (Andersen et al. 2004), Greenland warmed by about 8°–12°C (Wolff et al. 2010), the sea surface temperature in the northeast Pacific Ocean and North Atlantic Ocean rose by 4°–5°C, and the monsoon effect was similar to Bølling–Allerød warming (Brovkin et al. 2021), which marked the beginning of the Holocene.
The grayscale data used in this study are updated from the early published data of core PL07-56PC in the Cariaco Basin, and a new age model is adopted (Hughen et al. 2000). It is a high-resolution proxy for local productivity in the tropical Atlantic, and its records are considered to be related to climate change in Greenland (Lenton et al. 2012). The grayscale is affected by productivity in response to wind-driven upwelling, with lower numbers corresponding to enhanced windiness, and the time scale for this record was derived independently of those for the ice core (Alley 2000). These records cover a period of 14 952 to 426 years BP, including 10 424 time points. Following the footsteps of predecessors, we consider the interval of 12.5–11.6 kyr BP (n = 2111 points), which spans the Younger Dryas but excludes the sudden decline in productivity at its end. To achieve stationarity, we refer to the practice of Dakos et al. (2008), remove the long trend in the sequence, and analyze the residual sequence. To test the warning performance of R/S analysis for real abrupt climate change events, this paper analyzes whether H can present a precursor signal for this abrupt climate change from the termination of the YD to the onset of the Holocene, that is, before the global climate enters the Holocene, whether the H of the grayscale time series has some statistically significant change characteristics.
d. Statistical significance test
3. Results
a. The performance of the new early warning signal in stochastic folding models
Figure 1 is the bifurcation diagram of the first folding model [(Eq. (4)], in which solid lines show two different stable equilibrium points of the model and the dotted line represents unstable equilibrium points. There are two critical transition points,
Bifurcation diagram of the univariate vegetation model [Eq. (4)]. The solid lines indicate that the system is in a stable equilibrium state. The upper branch is a high-density state, and the lower branch represents a low-density state. The dotted line between the two solid lines represents the unstable equilibrium point of the vegetation model.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
Figure 2 shows the R/S analysis results of the vegetation biomass density under relatively small random external forcing, namely, σV = 0.05. The results show that all the fitting degrees (R2) of the linear fitting lines of the log–log curves are greater than 99%, indicating that the vegetation biomass density has typical fractal characteristics. By comparing the results of R/S analysis under four different parameter c values, it can be found that all of the Hurst exponents are significantly greater than 0.5 (Fig. 2), which indicates that the vegetation biomass density has typical LTM characteristics. At the same time, the closer the parameter c is to the critical point (
The R/S results of the univariate vegetation model under relatively small external forcing (σV = 0.05). The Hurst exponent is obtained by least squares linear fitting, and R2 is the fitting degree in the graph. The black line and squares are the R/S results for the maximum grazing rate parameter c = 1.0; the red line and circles are the results for c = 2.0; the green line and positive triangles are the results for c = 2.5; the blue line and inverted triangles are the results for c = 2.6.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
In addition, we also consider the case of a relatively large random external forcing intensity, that is, σV increases from 0.05 to 0.25. In this case, the larger noise (σV = 0.25) makes the attractive region of the model critical point easier to approach. This means that the integral of Eq. (4) near the critical point is easy to collapse over time. Therefore, the R/S result for c = 2.6 is not applicable to σV = 0.25. Figure 3 shows the R/S results of vegetation biomass density for four different values of parameter c, including c = 1.0, 1.5, 2.0, and 2.4. It is not difficult to find that the univariate vegetation model still presents typical LTM characteristics. Similar to the results of σV = 0.05, under the relatively large random external forcing σV = 0.25, the Hurst exponent also shows an obvious increasing trend with the increase of parameter c.
The R/S results of the univariate vegetation model under relatively large external forcing (σV = 0.25). The black line and squares are the R/S results for the maximum grazing rate parameter c = 1.0; the red line and circles are the results for c = 1.5; the green line and regular triangles are the results for c = 2.0; the blue line and inverted triangles are the results for c = 2.4.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
When the univariate vegetation model approaches its critical point, is it a general rule that the H of the model state variable tends to increase? To address this question and compare different early warning signals, Fig. 4 shows the boxplots of the four kinds of early warning signals [AR(1); variance; H; DFA-propagator] under different maximum grazing rate c values. In the process of the univariate vegetation system approaching the critical point
Early warning signals of abrupt changes occur in the univariate vegetation model as the maximum grazing rate parameter c slowly approaches the critical transition point
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
When the intensity σV of the random external forcing increases from 0.05 to 0.25, compared with σV = 0.05, the AR(1), H, and DFA-propagator have almost no change in value (Fig. 4). However, the value of variance has increased by 10 times in general. This means that the variance cannot be used as an early warning signal alone; otherwise, there is a high probability of a false positive. Kendall’s τ for AR(1) (Fig. 5a) and variance (Fig. 5b) are still 1 (p < 10−4) for both, and for H (Fig. 5c) and DFA-propagator (Fig. 5d) are 0.970 44 (p < 10−4). Thus, for σV = 0.25 (Fig. 5c), the increasing trend of the H is still statistically significant at a significance level of α = 0.05. Under the external forcing of different intensities, the relative growth rate of all early warning signals is almost unchanged (Fig. S1).
As in Fig. 4, but for the stronger intensity of the random external forcing σV = 0.25.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
The bifurcation diagram of the bivariate vegetation model [Eqs. (5) and (6)] is shown in Fig. 6. The upper and lower solid lines of the bifurcation diagram represent two stable states of the vegetation model, i.e., the state of dense vegetation and the state of desolation, respectively. The dotted line represents the unstable equilibrium points of the vegetation model in Fig. 6. The two critical transition points of the bivariate vegetation model are related to the rainfall rate, which are
Bifurcation diagram of the bivariate vegetation model. The upper and lower solid lines represent the vegetation system in a stable equilibrium state, in which the upper branch represents a dense vegetation state and the lower branch represents the barren state without vegetation. The dotted line between the two solid lines represents the equilibrium point of the unstable state of the model.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
Figure 7 shows the boxplot of the four kinds of early warning signals, which change with the rainfall rate R of the bivariate vegetation model. When the fluctuation intensity of the random external forcing σB is relatively small (σB = 0.05), the early warning signals of the vegetation biomass density increase with the decrease of rainfall rate R, and Kendall’s τ of the AR(1), variance, and H are equal to 1 (p < 10−4) (Figs. 7a–c). However, Kendall’s τ of the DFA-propagator (Fig. 7d) is equal to 0.977 21 (p < 10−4). When the fluctuation intensity of the random external forcing σB is relatively large (σB = 0.25), the early warning signals of the vegetation biomass density also increase with the decrease in the rainfall rate R, and Kendall’s τ of the AR(1) (Fig. 8a) and variance (Fig. 8b) are equal to 1 (p < 10−4). Kendall’s τ of H (Fig. 8c) and the DFA-propagator (Fig. 8d) are equal to 0.982 91 (p < 10−4) and 0.954 42 (p < 10−4), respectively. The results of the relative growth rate are shown in Fig. S2. It can be found that under different external forcing intensities, the difference in the relative growth rate of H is the smallest (Fig. S2).
As in Fig. 4, but for the bivariate vegetation model of the random external forcing σB = 0.05 and σw = 0.01.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
As in Fig. 4, but for the bivariate vegetation model of the random external forcing σB = 0.25 and σw = 0.01.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
When the magnitude of the random external forcing σw is 0.05 and 0.25, similar results are obtained. That is, as the bivariate vegetation model approaches its critical transition point
As in Fig. 4, but for the bivariate vegetation model of the random external forcing σw = 0.05 and σB = 0.01.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
As in Fig. 4, but for the bivariate vegetation model of the random external forcing σw = 0.25 and σB = 0.01.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
The intensity of solar radiation is critical to the distribution of the global climate, and the reduction of solar radiation intensity will change Earth’s climate. Based on a zero-dimensional climate model, we linearly reduce the relative intensity of solar radiation, so that the model slowly approaches the critical transition point
Bifurcation diagram of the zero-dimensional climate model. The upper and lower solid lines represent the stable equilibrium temperature, in which the upper branch indicates stable high temperature and the lower branch is the “deep freeze” climate under the limiting planetary albedo. The dotted line between the two solid lines represents the unstable equilibrium temperature (Fraedrich 1978).
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
Figures 12 and 13 show the boxplots of four kinds of early warning signals as the solar radiation relative intensity μ decreases in the zero-dimensional climate model under different intensities of external forcing. Under the two fluctuation intensities (σT = 0.05 and 0.25), the AR(1), variance, H, and DFA-propagator of the zero-dimensional climate model increase significantly with the decrease in the relative intensity μ of solar radiation, and their Kendall’s τ values are all equal to 1 (p < 10−4) (Figs. 12a–d). Figure S4 shows the relative growth rate of all the early warning signals. All the relative growth rates are greater than zero. This means that when the zero-dimensional climate model approaches its critical transition point, all the compared early warning signals are valid.
As in Fig. 4, but for the zero-dimensional climate model of the random external forcing σT = 0.05.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
As in Fig. 4, but for the zero-dimensional climate model of the random external forcing σT = 0.25.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
As mentioned by Ditlevsen and Johnsen (2010) and Lenton (2011) due to the limitations of early warning signals, at least two kinds of signals should be considered in cross-checking to avoid false positives. Based on the above numerical model tests, the early warning performance of H is similar to that of DFA-propagator in most cases, and the calculation of R/S is relatively simple. Therefore, R/S is a completely effective alternative signal for both DFA-propagator and AR(1). When the control parameters of the system are close to the critical point, the LTM of the relatively stable system increases.
b. The influence of extra noise on warning signals
In addition to the external forcing that promotes the evolution of the system, there may be annoying noise. By analogy, for instrument meteorological data, this extra noise may be caused by equipment failure, which has nothing to do with the meteorological data itself. To test whether this annoying noise affects the value of early warning signals, Gaussian white noise with the same length to the original time series and different signal-to-noise ratios (SNR = 20, 10, and 5 dB) was added to the original numerical calculation results of the models for comparison. For each model, we only select a set of numerical experiments and consider the fluctuation intensity of the random external forcing σ = 0.25. For the convenience of comparison, we only show the change in the average value of each set of analysis results, instead of the boxplot. Since the influence of extra Gaussian noise on the value of the warning signal is mainly discussed here, we only pay attention to the change in the value of the warning signal under the same value of c and do not pay attention to the increased amplitude of different c values under the same intensity of extra noise.
Figure 14 shows the results of model 1 (σV = 0.25). All of the early warning signals changed when extra false noise with different SNR was added. For AR(1), the value of AR(1) decreases with the increase of extra noise. With the enhancement of the extra noise, the reduction range of AR(1) is reduced near the critical point. For variance, no matter what the value of c is, the increase of extra noise will increase the variance by the same amount. For H, the effect of smaller extra noise on H does not seem to be obvious. Only when SNR = 5 dB, there is a significant decrease in the region far from the critical point. For DFA-propagator, the value of DFA-propagator increases with the enhancement of extra noise. As Livina and Lenton (2007) pointed out, calibration is not suitable for highly nonlinear cases. Due to the increase of extra noise, it is possible that the original signal is completely covered, so that the value of the warning signal is affected by the addition of extra noise. By comparing the four subgraphs in Fig. 14, we find that only the value of R/S has the smallest change when c = 2.4 after adding extra noise, which means that R/S can eliminate the influence of extra noise to some extent.
The impact of extra false noise at different signal-to-noise ratios on the value of early warning signals for model 1 (σV = 0.25). (a) The results of AR(1). (b) The results of variance. (c) The results of H. (d) The results of DFA-propagator. The black circle represents the result of original data, the green five-pointed star represents the results of Gaussian white noise with signal-to-noise ratios of 20 dB, the red triangle represents the results of Gaussian white noise with signal-to-noise ratios of 10 dB, and the blue square represents the results of Gaussian white noise with signal-to-noise ratios of 5 dB.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
Figure 15 shows the results of model 2 (σB = 0.25, σw = 0.01). All of the early warning signals changed when extra noise with different SNR was added as in Fig. 14. For AR(1), the value of AR(1) decreases with the increase of noise. And the decline of AR(1) decreases near the critical point. For variance, the range of variance change remains constant, while the overall value decreases, which is similar to the case of model 1. For H, the influence of smaller noise on H is not obvious. When SNR = 5 dB and the system is far away from the critical point, there is a significant reduction. Like model 1, the value of DFA-propagator increases with the enhancement of extra noise.
As in Fig. 14, but for model 2 (σB = 0.25, σw = 0.01).
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
Figure 16 shows the results of model 3 (σT = 0.25). Here, the addition of extra white noise reduces the value of AR(1) as well. For variance, as in the previous two models, only the size of the value is affected, but the range of change has no effect. For H when SNR = 20 and 10 dB, the value of H is hardly affected by extra noise. And when SNR = 20 dB, the value of DFA-propagator is almost unaffected by extra noise. But for SNR = 10 and 5 dB, the value of DFA-propagator increases due to the introduction of extra noise.
As in Fig. 14, but for model 3 (σT = 0.25).
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
c. Warning of Holocene onset based on the increasing H
Early warning results based on the three folding models show that when the key control parameters of the dynamic system are close to the critical transition point, H presents a significant increasing trend which is basically consistent with AR(1), variance, and DFA-propagators. Will H also be effective in paleoclimate data, and does H have different early warning performance compared with other early warning signals used by predecessors? To assess this, the R/S analysis is used for an early warning of YD termination/Holocene onset based on the core grayscale data in the Cariaco Basin, following the example of Lenton et al. (2012). The reconstructed data during 11.6–12.5 kyr BP (n = 2111) are used to calculate different signals, and the fixed sliding window size is 1055, which is half of the total time series. According to the practice of Dakos et al. (2008), we filter the original data with a bandwidth of 100 by Gaussian kernel smoothing function to obtain a stationary sequence, and analyze the early warning signal from the residual sequence. The subsequence length s for R/S and DFA analysis increases gradually at intervals of 1 from 10 to 100. Figure 17a shows the evolution curve of the core grayscale data over time in the Cariaco Basin. By analyzing the residual time series, we found that almost all of the early warning signals had a certain degree of warning performance before the onset of the Holocene. Meanwhile, H and DFA-propagator have a more obvious changing trend before the onset of the Holocene. The H and DFA-propagator began to increase continuously at about 12 kyr BP, lasted for about 250 years, and decreased at about 11.7 kyr BP. Kendall’s τ of the AR(1), variance, H, and DFA-propagator are equal to 0.4853 (Fig. 17b), −0.439 01 (Fig. 17c), 0.542 34 (Fig. 17d), and 0.634 22 (Fig. 17e), respectively. Figure S5 shows the result of original data (no detrend). The comparison between Fig. S12b and Fig. 17b shows that AR(1) for the residual sequence can provide some warning performance, that is, an overall increased AR(1), while it does not work for the original sequence. According to the analysis results of AR(1) (Fig. 17b), the signal changes in three stages, and there are two minima around 1.18 and 1.17 kyr BP. The performance of other indicators is similar. The possible reason is that simple statistical indicators are not enough to analyze the real system. The reason for these inconsistent changes may be that the real system has multiple time scales that need to be separated; otherwise, the warning signals may not have consistent changes as expected (Williamson and Lenton 2015; Williamson et al. 2016). However, this does not affect our conclusion that H is a completely valid alternative signal for AR(1) and DFA-propagator because R/S is a lot easier to calculate than DFA-propagator. And from the simple Kendall’s τ trend of changing, the early warning of DFA-propagator and H has some effect.
Early warning of the YD termination/Holocene beginning based on different signals. (a) The solid black line is the original reconstructed grayscale data from the Cariaco Basin core PL07-56P, 11.6–12.5 kyr BP (n = 2111). The smooth red line passing through the time series is the Gaussian kernel function used to filter out the slow trends. The horizontal dashed line indicates the length of sliding window (1055 points) for analysis, and the red vertical dotted line marks the cutoff time for analysis. (b) The changing AR(1) propagator of the grayscale time series. (c) The results of variance. (d) The results of Hurst exponent H. (e) The results of DFA-propagator. For R/S and DFA analysis, the subsequence length s increases gradually at intervals of 1 from 10 to 100.
Citation: Journal of Climate 37, 2; 10.1175/JCLI-D-22-0263.1
4. Conclusions and discussion
Many complex systems in nature and society have the characteristics of LTM, such as air temperature, humidity, river runoff, and human heart rate (Hurst 1951; Pelletier and Turcotte 1997; Peng et al. 1992; Yang et al. 2020). Therefore, as these systems approach their critical transition point, the LTM characteristics will change accordingly. For the estimation of LTM, many different calculation methods have been proposed by predecessors, among which the most well-known one is DFA. Based on the DFA method, Livina and Lenton (2007) proposed a modified early warning signal, named DFA-propagator. It is obtained by using AR(1) to empirically calibrate the results of DFA analysis. The LTM can also be calculated by wavelet analysis. For example, Rypdal (2016) proposed new early warning signals based on wavelet analysis—
Although numerical experiments have confirmed that LTM characteristic H can be used as an early warning signal, the warning performance in actual data remains to be studied. To compare the methods, we analyzed the residual sequence of grayscale time series reconstructed from the Cariaco Basin core (Hughen et al. 1996, 2000) based on the R/S method. We selected time series with the same time interval as Lenton et al. (2012) and filtered the original data with a bandwidth of 100 to obtain a stationary sequence like Dakos et al. (2008) did. By comparing the analysis results of the residual series and the original series, we find that simple statistical indicators are far from enough for the real time series, and other methods are needed to separate the time scale. This will be discussed in our subsequent research. By changing the size of the integral time step and the length of the integration time of the zero-dimensional climate model, it can be found that the early warning of AR(1) and variance appears to be ineffective when the integral time step dt is equal to 0.001 (Fig. S8). However, from the simple Kendall’s τ trend of changing, the early warning of H still has some effect.
Before the critical transition, the LTM characteristics of the state variables of the three folding models and core grayscale proxy time series all show a statistically significant increasing trend, and H is a completely valid alternative signal for AR(1) and DFA-propagator. Although the results in this study are only obtained based on limited sample of three folding models and paleoclimate reconstruction data, many abrupt changes in nature and society show a typical folding bifurcation, including climate systems, ecosystems, Arctic sea ice, and financial crises (Fraedrich 1978; Steele and Henderson 1994; Guttal and Jayaprakash 2008; Scheffer et al. 2001, 2009; Dakos et al. 2008; Diks et al. 2019). Therefore, in simple folding models, the theoretical basis of the early warning signals is quite strong (Scheffer et al. 2009). Of course, for real complex systems with multiple time scales, some other analysis methods are needed to further separate the time scales, such as Fourier analysis, which will be studied in our subsequent work. In addition, the precise relationship between H and the stability of the underlying state approaching the transition will be further studied as well.
Acknowledgments.
This work was jointly supported by the Guangdong Basic and Applied Basic Research Foundation (2021A1515011428), the National Natural Science Foundation of China (Grants 41975086, 42175067, 42075051, and 41775092), and the Fundamental Research Funds for the Central Universities (Grant 20lgzd06). The authors declare that they have no conflicts of interest.
Data availability statement.
The artificial critical transitions data used in this study can be obtained by numerically solving each stochastic differential model(s) based on a simple Euler algorithm and the Ito formula. The grayscale data in the Cariaco Basin core records are included in the paper of Hughen et al. (2000) and can be downloaded from the website https://www.ncei.noaa.gov/access/paleo-search/study/5853. All codes can be obtained directly by contacting the corresponding author.
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