Wavelet based fractal analysis of El Niño/La Niña episodes

The fractal dimension H, which is frequency-dependent within Quasi-Biennial (QB) period, was explored to measure the noise characteristics of Nino 3.4 Sea Surface Temperature (SST), where “noise” specifies the cycles within QB such as the Tropospheric Biennial Oscillation (TBO). The results show that the oscillation pattern of H corresponds mostly to development of El Nino, particularly during two strong Tropical Pacific Decadal Oscillation (TDO) periods of 1894 to 1923 and 1978 to 2000. This represents a stochastic resonance mechanism when a positive-phase noise overlaps a stronger positive-phase of TDO. In this case, SST would exceed a critical value to trigger an El Nino. The mechanism provides a favorable condition by which the onset of El Nino becomes more sensitive to noise. Self-organized criticality (SOC) explains that a small disturbance on an uncertain system will result in an avalanche, including scale-invariance (scaling) and criticality (threshold) features. The results show that strong and medium El Nino events regularly show scaling within QB period especially after the 1970s. Therefore, scaling is a critical state for onset of a strong El Nino and noise modulation by SOC within QB period plays a significant role in the El Nino developments.


INTRODUCTION
El Ni no-Southern Oscillation (ENSO) has been widely studied for sixty years and has been regularly observed for more than a couple of decades (TAO). The ENSO variability affects global precipitation and hydrological cycle affecting the water resources distribution. The frequency and strength of El Ni no have been reported to have increased since the late 1970s together with global warming (Wang, 1995). These changes usually result in variation in fractal characteristics, and changes in fractal characteristics may also indicate changes in the background of El Ni no (Tsonis et al., 1998).
The fractal dimension, also known as the Hurst coefficient H (first proposed by Hurst, 1951), is important for climate study (Koutsoyiannis, 2006). It has been applied to fields including the study of precipitation and river discharge (Hurst, 1951), ice and clouds (see Ausloos, 2004 for a review). H is dependent on time-space frequency. Both data-based and model-based studies (for example, Fraedrich et al., 2004) have documented the temporal and spatial fractal dimensions of sea surface temperature (SST). The relationships between the timefrequency dependent H for SST and the development of an El Ni no event have not been documented so far. The importance and necessity of it is based on the following experiences.
Noise in the ENSO signal refers to high-frequency forcing and small spatial scale processes such as a westerly wind burst or Madden-Julian Oscillation (MJO, 30 60 days cycle). The importance of noise for forecasting El Ni no is controversial with some researchers supporting its use (Dijkstra et al., 2002, amongst others) and others not (e.g. Chen et al., 2004).
Additionally, the Tropospheric Biennial Oscillation (TBO; Meehl et al., 1997) greatly affects ENSO. The Indian Ocean Dipole (IOD, Saji et al., 1999) is mainly due to Quasi-Biennial (QB) cycle and as an aspect of TBO (Rao et al., 2002), interacting with ENSO (Behera et al., 2006) and influencing ENSO through the Darwin pressure (Behera et al., 2003). The biennial variability of ENSO is a part of TBO (Li et al., 2006). TBO and IOD are important factors for ENSO variability.
An essential element of any TBO mechanism is the memory of the ocean, and certain short-term oscillations such as Kelvin waves also contribute to the TBO mechanism (Meehl et al., 2003). This TBO mechanism provides us with possibilities to explore an index H of a short-term memory, which only depends on frequencies within QB period, to investigate the noise characteristics of SST accompanying the development of El Ni no.
ENSO has a cycle of two to seven years. In this study, cycles under two years are considered "noise" and cycles longer than seven years are the "background state". Therefore, H, which has restricted frequencies within QB period, is used to index the noise characteristics (the mean power of noise) of SST. This study aims to find the relationship of H with the development of an El Ni no/La Ni na event and the way noise and background interact leading up to the onset of an El Ni no event.
Another profile of fractal, Self-organized criticality (SOC) (Bak et al., 1987) is one possible explanation for fractal behavior (Fraedrich et al., 2004). SOC explains that a little disturbance on an uncertain system will result in an avalanche. SOC has also been applied to study soil moisture, ice surges, and so on (see the review of Turcotte, 1999). Scale-invariance (scaling) and critical state (threshold) are the two main independent components of SOC; SOC should include both. Andrade et al. (1995) suggested that an El Ni no event was a phenomenon of SOC. Physically, El Ni no can be viewed as a series of events when the ocean releases energy into the atmosphere. This study also aims to determine whether an avalanche of energy spectrum in time-frequency space on a specific timescale, for example QB period, will result in an El Ni no event, and also define the role of noise based on SOC.

DATA SOURCES
Sea surface temperature data was sourced from the Ni no 3.4 region covering 5°N 5°S, 120°W 170°W, comprising monthly data from the Hadley Centre Sea Ice and SST Data Set (HadISST), over the period from 1870 to 2005. The data was provided by the Hadley Centre, Met Office, UK (Rayner et al., 2003).
The ENSO event-index is a value representing the intensity of an El Ni no/La Ni na event for a given year (Severov et al., 2004), which is one of the following (shown in Table I): 1.5 (strong La Ni na), 1 (medium strength La Ni na), 0.5 (weak La Ni na), 0 (neutral year), 0.5 (weak El Ni no), 1 (medium strength El Ni no), and 1.5 (strong El Ni no), respectively.

METHODOLOGY
We selected (Torrence and Compo, 1998) the second order of the Daubechies orthogonal wavelet function (shown in Figure 1) for estimating H (see Huang et al. (2006) and references therein for details of the definition of H).
The plot of the logarithm of wavelet power spectrum against the logarithms of timescale is called a "log-log plot", and is presented in the results below (e.g. Figure  4).
The following procedure was used to analyze the Ni no 3.4 SST in this study: 1. A time window of the SST time series was selected with a period of 64 months, equivalent to the average ENSO cycle, starting from January. The years used for definition of H are supposed to be the start years. 2. The Daubechies wavelet based spectral analysis was applied to the selected SST time window. 3. The Hurst coefficient H was calculated using H = (K 1)/2, where K is the slope rate of the linear fitted line of a log-log plot, shown in Figure 4. 4. The time window was then shifted forward one year and steps 1 to 3 were repeated.
In the log-log plot ( Figure 4) the slope shows the average energy of wavelet spectra subjected to noise in that frequency inside QB periodicity and H relates to the slope of the log-log plot. Therefore, H is an index of the mean power of noises.
The Morlet wavelet was also applied to estimate H, but the results were a little different. We also applied the Morlet wavelet to investigate the power spectra of a Ni no3.4 SST.

Fractal characteristics of SST
Sensitivity of the noise characteristic index H to El Ni no/La Ni na Figure 2 shows the relationship between the oscillations of H and that of the ENSO event-index. The shaded areas indicate the in-phase oscillations. The oscillation patterns of H and the ENSO event-index were well-matched in phase oscillation, meaning that noise corresponds well to the development of an El Ni no event. Two periods longer than twenty years with a cooscillation rate of at least 85 percent between the two time series were selected for further analysis: 1894 to 1923 and 1978 to 2000.
An AR(1) process model (the univariate lag-1 autoregressive process) (Torrence and Compo, 1998) was used to simulate both the ENSO-event index and H time series, and the Monte Carlo method employed to perform statistical significance tests. The tests showed that these two periods occurred significantly at the five percent level. A Chi-square test was then used to find the goodness of phase-fit between the two time series during these two periods. The test also showed a 95 percent confidence level. However, the Monte Carlo test for investigating the amplitude-fitness of the two time series shows an 85 percent confidence level. A little lower confidence here reflects that the ENSO eventindex is influenced by TDO and inter-annual cycles but H is not.
For example, there are seven strong and medium El Ni no years after 1968 such as 1972, 1982, 1986, 1987, 1991, 1994 and 1997. We compared H and ENSO eventindex of these years with the intensity of IOD respectively. It is worth to note that all of them are positivephase IOD years except 1986. We found the same in phase oscillation patterns happened in these years between them except 1986.
The difference in phase oscillation between the H and ENSO event-index in Figure 2, such as 1978, 1984 and 1988 is due to negative-phase of IOD accompanying a La Ni na. This shows that the noise modulated by IOD interacts with ENSO (Rao et al., 2002;Behera et al., 2006). Similarly, Torrence and Compo (1998) also found that during 1880 to 1920 and 1960 to 1990 respectively, significantly higher powers existed in the Southern Oscillation (SOI) and the Ni no3 SST. This implies that noise contributes more to the onset of an El Ni no event in the presence of a stronger positive-phase TDO background. This is a stochastic resonance mechanism (Nicolis, 1993) when a positive-phase noise overlaps with a stronger positive-phase TDO, the Ni no 3.4 SST would exceed easily a critical state to launch an El Ni no. This mechanism gives a condition when the onset of El Ni no is more sensitive to noise. A strong TDO was evident after 1978 though the TDO signal was not so clear from 1894 to 1923. In fact, the significance test of wavelet spectrum was sensitive to the type of red-noise background selected (Torrence and Compo, 1998). A 95 percent confidence level was achieved when a propriety rate of noise to signal was selected.

Scaling features of SOC for El Ni no events
Scaling or scale-invariance means that an invariant property exists in the distribution of energy to frequency. Log-log plots of energy versus frequency will show an almost-straight line in such a case. Figure 4 shows the log-log plot for 1997, and the almost-straight    During the periods 1870 to 1899 and 1960 to 2000, the number of scaling to the number of strong El Ni no events had ratios of 5:5 and 4:4 respectively. Medium El Ni no events had ratios of 2:3 and 4:4, respectively. Thus, strong and medium El Ni no events mainly showed scaling during these two periods, for example after 1968 all seven events witness scaling. Scaling represents a strong signal of the short-term memory of the Pacific Ocean that is an essential element of the IOD/TBO mechanism and interacts with IOD (Meehl et al., 2003). Incidentally, except 1986, all of these years are strong or medium positive IOD years.
The ratios of the total number of scaling to the number of events decreased from strong (12:15) to medium (11:18) to weak (3:12) El Ni no events. This shows that stronger El Ni no events had a higher occurrence of scaling. Scaling is a critical state for strong El Ni no.

Threshold features of SOC for El Ni no events subject to QB
The tail of a log-log plot with a timescale of 32 months is the top-right point (for example, Figure 4). The height of the tail represents the average energy of the QB signal over 64 months.
The threshold refers to the existence of a critical state in the process of energy spectrum oscillations at the QB frequency, where the energy spectrum is raised from a low level before an El Ni no event, and then reaches the straight line of a log-log plot during El Ni no evolution, and finally drops after the event. The straight line of the log-log plot (scaling) is the critical state of SOC.
For example, in Figure 4, the "tail" goes from 1996 (normal year) up to that of 1997 (strong El Ni no year) and drops in 1998 (La Ni na year). Therefore, 1997 is the threshold state. Plots for other years are not shown here. Table III summarizes the ratios of the number of thresholds to the number of El Ni no events in each period. The ratio of the total number of thresholds to the total number of strong and medium events is 16:33, showing that the threshold features of SOC exist in an El Ni no episode. This is particularly the case after 1990, with a ratio of 3:3. A 95 percent confidence level is observed for values in both Tables II and III through the Monte Carlo method based on the simulation of Ni no3.4 SST using the AR(1) process model (Torrence and Compo, 1998).
Threshold feature represents that the system tends to flip-flop back and forth from year to year similar to the TBO mechanism (Meehl at al., 2003). As an example, we have discussed the interactions between H and the ENSO event-index and the intensity of IOD by the phase-fitness in section 4.1.1.

SOC for El Ni no events subjected to QB
Tables II and III imply that some strong and medium El Ni no events are the result of SOC subjected to QB. In particular, most strong and medium El Ni no events indicate SOC after the 1970s. Strong and medium El Ni no events have higher ratios of scaling (23:33) than weak El Ni no events (3:12). Strong and medium El Ni no events have higher ratios of the threshold (16:33) subjected to QB than a weak El Ni no (3:12) (Table III). Therefore, a strong QB does not necessarily imply that an El Ni no event will occur, but a strong El Ni no event regularly accompanies a strong QB. This result supports previous studies, for example Meehl et al. (2003). Whether QB will cause a strong El Ni no depends on whether the system reaches a critical state and is ready for an avalanche.

DISCUSSION
The roles of annual and intra-annual cycles for El Ni no based on SOC were discussed as follows. From the view of SOC, a little disturbance will result in a largescale avalanche for a sandpile only situation when the system is near the critical point (Bak et al., 1987). El Ni no can be viewed as a result of a chain reaction of energy releases on a large scale, without being sensitive to the "avalanche" at the high frequencies. High frequencies may play a role in the build-up of energy, starting the first chain reaction of energy releases on a large scale, leading to the onset of an El Ni no event.
This view supports previous findings. Intra-seasonal cycles, i.e. high frequency events such as winds related to MJO, tropical cyclones and Yanai waves, alter the SST anomaly, which may help trigger the development and demise of an El Ni no event (Bergman et al., 2001). However, they do not directly cause El Ni no as they do not produce sufficient conditions for El Ni no to occur. El Ni no occurs when energy in time-frequency space reaches a critical state, and when background oceanic and atmospheric conditions are conducive to the rapid growth of random disturbances (Moore and Kleeman, 1999).

CONCLUSION
This study has discussed the ENSO variability, which has a significant role in surface water budget through changes in mean precipitations. Roles of noise with frequency within QB oscillations and background state including the TDO are investigated for El Ni no evolutions based on fractal analysis and SOC. The main results are summarized below: The oscillation of the noise characteristics, represented by H, which was frequency-dependent within QB, mostly corresponds with the development of El Ni no, particularly for periods 1894 to 1923 and 1978 to 2000. This was due to the contribution of strong signals of the TDO background. A stronger TDO background provides the conditions for high frequencies to be more sensitive to the onset of an El Ni no. This represented the stochastic resonance mechanism and gave a condition when the onset of El Ni no was more sensitive to noise.
Strong and medium El Ni no events regularly show scaling within QB period and on QB threshold, especially after the 1970s. Scaling is a critical state for the onset of a strong El Ni no. From the view of SOC, TDO provides the background for the onset of an El Ni no. Interannual cycles directly affect El Ni no and noise may be the first in the chain-reaction leading to the onset of an El Ni no event. SOC gives a condition in timefrequency space for noise to trigger an El Ni no, confirming QB is significant for ENSO.
Many areas remain to be researched. Scaling and the critical state of SST oscillation exist during El Ni no episodes, whereas the exact critical points of spectrum variation for El Ni no/La Ni na are not yet clear. Moreover, the way that El Ni no and La Ni na affect the shortterm memory of the ocean interacting with TBO or IOD requires further investigation.