Stochastic simulations of the Madden–Julian oscillation activity
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DOI: 10.1007/s00382-009-0660-2
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- Jones, C. & Carvalho, L.M.V. Clim Dyn (2011) 36: 229. doi:10.1007/s00382-009-0660-2
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Abstract
The Madden–Julian oscillation (MJO) is the most prominent form of tropical intraseasonal variability. This study investigated the following questions. Do interannual-to-decadal variations in tropical sea surface temperature (SST) lead to substantial changes in MJO activity? Was there a change in the MJO in the 1970s? Can this change be associated to SST anomalies? What was the level of MJO activity in the pre-reanalysis era? These questions were investigated with a stochastic model of the MJO. Reanalysis data (1948–2008) were used to develop a nine-state first order Markov model capable to simulate the non-stationarity of the MJO. The model is driven by observed SST anomalies and a large ensemble of simulations was performed to infer the activity of the MJO in the instrumental period (1880–2008). The model is capable to reproduce the activity of the MJO during the reanalysis period. The simulations indicate that the MJO exhibited a regime of near normal activity in 1948–1972 (3.4 events year^{−1}) and two regimes of high activity in 1973–1989 (3.9 events) and 1990–2008 (4.6 events). Stochastic simulations indicate decadal shifts with near normal levels in 1880–1895 (3.4 events), low activity in 1896–1917 (2.6 events) and a return to near normal levels during 1918–1947 (3.3 events). The results also point out to significant decadal changes in probabilities of very active years (5 or more MJO events): 0.214 (1880–1895), 0.076 (1896–1917), 0.197 (1918–1947) and 0.193 (1948–1972). After a change in behavior in the 1970s, this probability has increased to 0.329 (1973–1989) and 0.510 (1990–2008). The observational and stochastic simulations presented here call attention to the need to further understand the variability of the MJO on a wide range of time scales.
Keywords
Madden–Julian oscillationTropical intraseasonal variationsStochastic simulationClimate changeGlobal warming1 Introduction
The Madden–Julian oscillation (MJO) is the most prominent form of tropical intraseasonal variability (Madden and Julian 1994; Lau and Waliser 2005; Zhang 2005). The oscillation propagates eastward in the tropical region with phase speeds on the order of 5–10 m s^{−1} and influences the monsoons in Asia, Australia and Americas (Mo 2000; Nogues-Paegle et al. 2000; Goswami and Mohan 2001; Higgins and Shi 2001; Jones and Carvalho 2002). Moreover, several studies have shown that the MJO can affect the distribution of rainfall and extreme events in many locations around the world (Mo and Higgins 1998; Higgins et al. 2000; Jones 2000; Carvalho et al. 2004; Jones et al. 2004a). Since the MJO involves intense tropical convective heating anomalies (Kiladis et al. 2005), tropical–extratropical interactions are significant during its life cycle. Therefore, previous investigations have shown modulations on weather forecasts skills (Ferranti et al. 1990; Lau and Chang 1992; Hendon et al. 2000; Jones and Schemm 2000; Seo et al. 2005) and potential predictability (Waliser et al. 2003; Jones et al. 2004a, b).
An interesting aspect about the MJO is that it was discovered with rigorous statistical data analysis of a few sparse tropical weather stations available at the time (Madden and Julian 1971, 1972). In that context, spectral and cross-spectral analyses of relatively short time series (~5–9 years) revealed the presence of a significant intraseasonal oscillation in the tropical atmosphere. In contrast, characterization of its spatial structure, identification of individual events and variations on seasonal-to-interannual time scales are more easily attained with gridded atmospheric data (e.g., Jones and Carvalho 2006). Presently, knowledge about the history of the MJO is limited to the period 1948-present when reanalyzed data became available (Kalnay et al. 1996; Bengtsson et al. 2004). It is important to note that uncertainties about the variability of the MJO in the reanalysis period are still present, since significant changes in the observational system have occurred. For example, the number of tropical weather stations has varied substantially over time and satellite data became available in the reanalysis after 1978 (Kistler et al. 2001).
Although seasonal variations in the MJO are relatively well understood (Wang 2005), physical mechanisms involved in interannual and longer variations are much less clear (Hendon et al. 1999; Slingo et al. 1999; Lau 2005; Slingo et al. 2005). Jones and Carvalho (2006) and Pohl and Matthews (2007), for instance, found that the activity of the MJO has increased in recent years. Moreover, Jones and Carvalho (2006) showed that the MJO appears to exhibit regime changes on low-frequency time scales (e.g., 5–15 years). Consequently, some fundamental aspects about the temporal behavior of the MJO remain unexplored, especially how decadal climate changes and global warming might affect the MJO.
To investigate mechanisms associated with the MJO on different time scales, one would ideally like to use a comprehensive global coupled climate model and perform a large ensemble of numerical experiments. Although substantial progress has been made over the years, only a very limited number of climate models exhibit intraseasonal variability that resembles the observed MJO (Zhang 2005). Currently, global climate models are unable to accurately represent all characteristics of the MJO (Lin et al. 2006).
This paper investigates the long-term activity of the MJO employing a probabilistic approach. We refer here to activity as the number of events occurring in a period of time. Reanalysis data were used to develop a stochastic model capable to simulate the non-stationary behavior of the MJO during 1948–2008. The model is driven by observed sea surface temperature (SST) anomalies in the tropical Indian and Pacific Oceans warm pool region (hereafter warm pool) and performed a large ensemble of simulations to infer the activity of the MJO in the instrumental period (1880–2008). Specifically, this study investigated the following questions. Do interannual-to-decadal variations in SST in the warm pool lead to substantial changes in MJO activity? Was there a change in the MJO in the 1970s? Can this change be associated to SST anomalies in the warm pool region? What was the level of MJO activity in the pre-reanalysis era?
The paper is organized as follows. Section 2 describes data sets and Sect. 3 discusses identification of MJO events. Section 4 summarizes the basic core of a stochastic model previously published. Section 5 discusses extensions performed in the model to account for non-stationarity of the MJO. Results are presented in Sect. 6. Discussion and conclusions are provided in Sect. 7.
2 Data
The primary data set used in this study is the National Centers for Environmental Prediction/National Center for Atmospheric Research reanalysis (NNR) (Kalnay et al. 1996; Kistler et al. 2001). Daily averages of zonal wind components at 850-hPa (U850) and 200-hPa (U200) were used for the period 1 January–31 December 1948–2008. In general, interannual changes in the MJO based on NNR are consistent with the 40-year European Centre for Medium Range Forecasts (ECMWF) Re-Analysis (ERA-40; Slingo et al. 2005; Jones and Carvalho 2006). To complement the analysis, daily averages of outgoing longwave radiation (OLR; Liebmann and Smith 1996) during 1 January–31 December 1979–2008 were used to characterize the convective signal associated with the MJO.
Daily climatologies of OLR, U850 and U200 were computed by averaging each calendar day and smoothing the resulting time series with 300 passes of a 1-2-1 moving average. Daily climatologies were subtracted from the original time series to remove the annual cycle. To isolate the MJO signal, the time series were detrended and filtered in frequency domain to retain variations between 20 and 200 days. This procedure follows Matthews (2000), who determined that the wide 20–200 day band more accurately represents isolated MJO events.
Interannual-to-decadal changes and long-term trends in the warm pool region (15°S–15°N; 50°E–150°W) were analyzed with monthly SST from four different data sets during January 1880–December 2008 (Kaplan et al. 1998; Smith and Reynolds 2003; Ishii et al. 2005; HadISST 2006). A time series of SST anomalies in the warm pool was obtained by averaging the four data sets and removing the mean seasonal cycle.
3 Identification of the MJO during 1948–2008
Identification of MJO events follows the procedure discussed in Jones (2009) and is summarized here for completeness. A combined empirical orthogonal function (EOF) analysis (Wilks 1995) was performed on U200 and U850 anomalies averaged in latitude (15°S–15°N). The first two EOFs, which account for 23.8 and 19.5% of the total variance respectively, were used to represent the MJO. The spatial structures of the first two EOFs can be seen in Jones (2009) and Wheeler and Hendon (2004).
It is opportune to mention a few issues regarding the identification of MJO events. Since this study focuses on the long term variability of the MJO and there is no OLR data prior to 1977, events are identified here based only on circulation characteristics. Small differences in the number of events (less than 10%) can arise when one uses OLR anomalies as another metric for the MJO. In addition, several previous studies have shown that the first two EOFs represent the bulk of the MJO variability (e.g., Jones and Carvalho 2006; Pohl and Matthews 2007). While the third EOF can sometimes be separated from the fourth and might be important in interannual shifts in the eastward propagation of the MJO (Kessler 2001), the actual initiation of MJO events is well represented by the first two EOFs. This issue was further confirmed by analyzing MJO occurrences during significant ENSO events (not shown). Additional discussions are provided in Jones and Carvalho (2006), Pohl and Matthews (2007) and Jones (2009).
4 Homogeneous stochastic model of the MJO
The variability of the MJO in the instrumental period was investigated in the context of stochastic simulations. Jones (2009) introduced a homogeneous stochastic model capable of simulating the temporal and spatial variability of the MJO. The model consists of three components which are briefly summarized here.
Transition probabilities of homogeneous stochastic model
Xt | Xt + 1 | ||||||||
---|---|---|---|---|---|---|---|---|---|
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
0 | 0.984351 | 0.011913 | 0.001772 | 0.001062 | 0.000902 | 0 | 0 | 0 | 0 |
1 | 0 | 0.847352 | 0.152648 | 0 | 0 | 0 | 0 | 0 | 0 |
2 | 0 | 0.011289 | 0.829030 | 0.159681 | 0 | 0 | 0 | 0 | 0 |
3 | 0 | 0 | 0.010669 | 0.831843 | 0.157488 | 0 | 0 | 0 | 0 |
4 | 0.006752 | 0 | 0 | 0.010353 | 0.816064 | 0.166831 | 0 | 0 | 0 |
5 | 0.007918 | 0 | 0 | 0 | 0.007475 | 0.847031 | 0.137577 | 0 | 0 |
6 | 0.014670 | 0 | 0 | 0 | 0 | 0.021244 | 0.823718 | 0.140368 | 0 |
7 | 0.007758 | 0 | 0 | 0 | 0 | 0 | 0.011430 | 0.836571 | 0.144241 |
8 | 0.094074 | 0.084706 | 0 | 0 | 0 | 0 | 0 | 0.009110 | 0.812111 |
Recently, Matthews (2008) investigated the characteristics of primary and successive MJOs and found that 40% of the MJOs are primary events (1974–2005). Despite differences in methodology, this study is in good agreement, since the percentage of observed primary events found was 63% in 1948–2008 and 36% in 1974–2008.
Once the temporal evolution of the MJO is simulated, the second component of the model determines the spatial structure. Given the time series of simulated phase transitions (X_{t−simul}), the spatial structures of OLR, U200 and U850 anomalies (or any other field) from observed phase composites are assigned to each phase S of X_{t−simul}, where S ∈ [1, 8]. The third component of the model assigns amplitudes to the events with an amplitude factor that follows a Gaussian distribution. This implies that the amplitudes of the simulated events can be weaker (stronger) than the “canonical” MJO (Fig. 1). Examples of MJO simulations are shown in Jones (2009).
In summary, events generated by the stochastic model occur irregularly in time and can appear as isolated events or successive MJOs. The oscillations in the model can have different zonal propagations, that is, the MJO can start in any of the phases 1–4 (i.e., Indian Ocean to Indonesia), propagate eastward and end in any of the phases 4–8. The zonal scale of simulated events is consistent with observations, since the spatial structure is assigned from observed composites. Moreover, MJOs in the model have different durations lasting between 30 and 90 days and each event can be stronger or weaker than the mean composite according to a normal distribution. Since the present study focuses on the variability of the MJO in the instrumental period, only the temporal component of the model is relevant here.
5 Non-homogeneous stochastic model of the MJO
5.1 MJO activity and variability in the warm pool
The annual number of events shown in Fig. 2a clearly indicates that the occurrence of the MJO is a non-stationary process. In this section, we discuss modifications done to the temporal component of the stochastic model to incorporate this non-stationarity. We begin by examining how transition probabilities of MJO initiation vary over time. We first define a constant parameter given by P_{HMG} = P_{01} + P_{02} + P_{03} + P_{04}, that is, the sum of conditional probabilities of primary events in the homogeneous model (Table 1, first row, P_{HMG} = 0.015649). Next, we define P_{NHM} = P_{01} + P_{02} + P_{03} + P_{04}, which is the sum of conditional probabilities of primary events estimated in 3-year moving windows during 1948–2008. The selected size of 3-year moving windows was tested in many different ways, for example, increasing from 1-year, 2-year,…15-year, 20-year etc. As the size of the window increases, P_{NHM} converges to P_{HMG}. On the other hand, if the size of the window is too small (e.g., 1-year) the results can be sensitive due to the non-stationary nature of the MJO. The 3-year size was deemed optimal and, as it is shown next, this window size reproduces some important multi-year variations in the activity of the MJO.
As discussed in the introduction, this paper explores relationships between interannual-to-decadal variations in SST and activity of the MJO. For this purpose, Fig. 4 also shows 3-year running means (middle) and 3-year running variances (bottom) of SST anomalies in the warm pool region. In addition to the obvious long-term trend, the running means show high and low periods of SST anomalies lasting roughly 3–7 years. Note additionally, that the warm pool domain does not encompass the Niño3.4 region and therefore El Niño/Southern Oscillation (ENSO) effects in the running means and variances are not too evident (i.e., opposite SST anomalies in the Indian Ocean and western Pacific associated with ENSO may cancel each other out). Another important feature is that, although 3-year running variance does not reveal significant trend, it is highly non-stationary and shows several periods of high variance lasting between ~5 and 10 years. The reasons for selecting this domain are threefold. First, this is the region where the MJO initiates and its convective signal is most strong (Madden and Julian 1994). Second, some studies indicated weak to no contemporaneous correlation between ENSO and MJO (Hendon et al. 1999; Slingo et al. 1999). Third, this study specifically investigates the hypothesis that low-frequency variations and trends in the warm pool interact with the activity of the MJO.
The next step consisted in exploring relationships between 3-year running means (M_{SSTA}) and variances (V_{SSTA}) of SST anomalies and P_{NHM}/P_{HMG}. The time series of P_{NHM}/P_{HMG}, M_{SSTA} and V_{SSTA} were first detrended by removing the respective 30-year running means (brackets are used to differentiate detrended series from raw series). The reason for removing 30-year running means instead of linear trends is explained later in this section.
5.2 Sensitivity analysis
The empirical result shown above brings an important new insight relating multi-annual changes in the behavior of the MJO and low-frequency SST variability in the warm pool. Since there was no a priori hypothesis for this finding, we discuss now several a posteriori sensitivity tests to quantify the robustness of the relationships between 〈P_{NHM}/P_{HMG}〉, M_{SST} and V_{SST}.
5.3 Error analysis
5.4 A model for the non-stationary behavior of the MJO
We finally discuss the implementation of a non-stationary component in the stochastic model of the MJO. Two aspects were considered: a long-term trend in the probability of initiation of primary events (Fig. 4 top) and low-frequency changes in the probability ratio 〈P_{NHM}/P_{HMG}〉 (Fig. 6). They are further discussed as follows.
As a predictor for the long-term changes in P_{NHM}/P_{HMG} (Fig. 4 top), we considered the trend in SST anomalies in the warm pool (Fig. 4 middle). The correlation between 30-year running means of P_{NHM}/P_{HMG} and 30-year running means of SST anomalies is 0.962. To estimate the statistical significance, we generated 1,000 pairs of synthetic time series of P_{NHM}/P_{HMG} and SST anomalies using first order auto-regressive processes and computed the correlations between the 30-year running means of these time series (see Carvalho et al. 2007 for details on similar analysis). The observed correlation was compared with the frequency distribution of the correlations from synthetic time series and determined to be statistically significant at 2% level (statistics were stable after 100 realizations of synthetic time series).
Likewise, the 0.4 constant was obtained to minimize the overall bias of the stochastic model. Upper/lower bounds in P81_{NHM}/P81_{HMG} were constructed based on the corresponding values for P_{NHM}/P_{HMG}.
The probability ratios shown in Fig. 13 were used in the stochastic simulations of the MJO. The standard deviation of the rms error (σ = 0.13) shown in Figs. 10 and 11 was used to represent random errors in the regression model (3) that captures low-frequency changes in the MJO. This procedure was implemented as follows. For any given day during 1880–2008, the probability ratio of primary MJO initiation is represented as P_{NHM}/P_{HMG} ± ε, where ε has a Gaussian distribution with zero mean and σ standard deviation. Note that P_{NHM}/P_{HMG} ± ε is bounded by the upper/lower limits shown in Fig. 13 top. This is necessary because ε randomly drawn could be very large just by chance, which would generate unrealistic values of P_{NHM}/P_{HMG}. Once P_{NHM}/P_{HMG} is determined, model (5) specifies P81_{NHM}/P81_{HMG} and the temporal component of the stochastic model is fully implemented.
Lastly, it is important to mention that the model described above is intended to simulate the non-stationarity of the MJO. In this context, we used the entire record of available MJO observations (1948–2008) to construct an empirical model relating probabilities of MJO initiation to low-frequency and long-term changes in SST in the warm pool. The construction of probabilities of MJO activity before 1948 is based on the statistical relationships derived during 1948–2008. Although the model presented here takes into account uncertainties in SST observations and errors in the multi-linear regression model (3), the stochastic simulations of MJO activity in the pre-reanalysis era should be interpreted with caution.
6 Variability of the MJO during 1880–2008
The non-homogeneous stochastic model of the MJO was used to perform simulations during 1880–2008. Here, monthly SST anomalies in the warm pool region drove changes in conditional probabilities of primary and successive MJOs as described in the previous section. To illustrate the process, the model was initialized on 1 January 1880 with observed SST anomaly in the warm pool and model (4) determined P_{NHM}/P_{HMG}. A random error was generated and added to P_{NHM}/P_{HMG}. We recall that the random error is bounded by the upper/lower limits in P_{NHM}/P_{HMG} (Fig. 13 top) and therefore represents the uncertainty in the SST anomalies and the error in the regression model (3). Next, since P_{HMG} is constant, P_{NHM} is known. The increase (decrease) in P_{NHM} was then used to change P_{01}, P_{02}, P_{03} and P_{04} values in Table 1. For instance, if P_{NHM}/P_{HMG} is 1.2, the sum of P_{01}, P_{02}, P_{03} and P_{04} increase by 20% (the increase in each individual term is done maintaining the same relative ratios) and P_{00} decreases by 20%. Likewise, the corresponding change in P_{81} in this case is 1.072 (model 5). Thus, P_{81} increases by 7.2% and P_{80} decreases by the same amount. All the remaining probabilities in Table 1 remain unaltered so that we can specifically investigate changes in MJO initiation associated with SST variability in the warm pool. Note that when P_{NHM}/P_{HMG} is equal to 1, the MJO in the non-homogeneous model behaves as a stationary process. Other details about the homogeneous model are discussed in Jones (2009).
An ensemble of 1,000 members was constructed such that each member runs for 129 years at daily resolution. Each simulation resulted in a time series of phases and represents situations of active MJO (phases 1–8) and quiescent periods (phases equal to 0) (see Jones 2009 for examples). Next, we computed the number of MJO events per calendar year during 1880–2008; thus, each year has a frequency distribution of 1,000 data points.
7 Discussion and conclusions
This study investigated several questions associated with long-term variability of the MJO. Do interannual-to-decadal variations in SST in the warm pool lead to substantial changes in MJO activity? Was there a change in the MJO in the 1970s? Can this change be associated to SST anomalies in the warm pool region? What was the level of MJO activity in the pre-reanalysis era? These questions were investigated from a probabilistic point of view using stochastic simulations.
In order to represent the non-stationarity of the MJO, modifications were done to the stochastic model described in Jones (2009). A significant finding shown here is the empirical result that relates low-frequency variations in SST anomalies to conditional probabilities of MJO initiation. Cold (warm) SST anomalies in warm pool, expressed as 3-year running means, lead to high (low) MJO activity by about 1.7 years. Likewise, 3-year running variances indicate that high (low) SST variability in the warm pool lead to high (low) MJO activity ~3.3 years later. Furthermore, the 30-year running mean trend in SST anomalies in the warm pool is significantly correlated with the trend in the MJO in the reanalysis period.
A multi-linear regression model that uses 3-year running means (M_{SSTA}) and variances (V_{SSTA}) of SST anomalies in the warm pool as predictors for detrended variations in the probability of primary MJO initiation, 〈P_{NHM}/P_{HMG}〉, was discussed in detail. When the model is trained in one time period, it is relatively successful in reproducing changes in 〈P_{NHM}/P_{HMG}〉 in independent data. The relationships between 〈P_{NHM}/P_{HMG}〉, M_{SSTA} and V_{SSTA}, however, do not appear to hold in the early part of the reanalysis record (before 1960s), when 〈P_{NHM}/P_{HMG}〉 exhibited small variations. Whether or not the NNR data realistically represent MJO variability before the introduction of satellite data is a topic that needs to be further evaluated. Coupled ocean-atmosphere reanalysis using only conventional data could possibly shed additional insights into this problem.
A large ensemble of stochastic simulations shows that the non-homogeneous model is capable to reproduce the activity of the MJO during the reanalysis period. The MJO exhibited a regime of near normal activity in 1948–1972 (3.4 events year^{−1}) and two regimes of high activity in 1973–1989 (3.9 events) and 1990–2008 (4.6 events). Additionally, this is the first study to derive statistical inferences about the activity of the MJO before the availability of reanalysis data. Stochastic simulations indicate decadal shifts with near normal levels in 1880–1895 (3.4 events), low activity in 1896–1917 (2.6 events) and a return to near normal levels for an extended period from 1918 until the beginning of reanalysis data.
Variations in the activity of the MJO were also found in decadal changes of cumulative probabilities of number of events per year. Simulations indicate that probabilities of very active years (5 or more events) varied as: 0.214 (1880–1895), 0.076 (1896–1917), 0.197 (1918–1947) and 0.193 (1948–1972). After the changepoint in the 1970s, it has increased to 0.329 (1973–1989) and 0.510 (1990–2008). Evidently, the stochastic simulations of MJO activity before the reanalysis period need to be further evaluated. It would be interesting to compare the results shown here with coupled ocean-atmosphere simulations from a model that realistically represents the MJO.
In a separate study, (Jones and Carvalho, manuscript in preparation) performed stochastic simulations similar to the ones described here and derived projections of changes in MJO 840 activity in the A1B global warming scenario (IPCC 2007). In that case, conditional probabilities of MJO initiation were driven by projections of SST anomalies from five different global coupled climate models participating in the 4th Assessment report of the Intergovernmental Panel on Climate Change. Stochastic simulations suggest substantial increases in the probabilities of very active years (6 or more events): 0.39 (2009–2048) and 0.56 (2049–2099). Thus, if the current warming rate in the warm pool continues throughout the 21st century, the impact in the MJO will likely be strong as well. It is reasonable to expect that very active years of MJO activity will be associated with other changes in the climate system too.
In the context above, several studies have shown that the MJO may exert an import role as stochastic forcing of ENSO variability (Moore and Kleeman 1999; Batstone and Hendon 2005; Zavala-Garay et al. 2005; Marshall et al. 2009). A unique aspect of our model is that it generates MJO irregularity represented as primary and successive events with different durations. This model can be used to investigate the stochastic forcing of the MJO by coupling it to an ocean model. Since conditional probabilities of MJO initiation in the stochastic model vary with SST anomalies, possible feedback processes could be examined in detail. For instance, as the MJO forces the ocean model, low-frequency changes in tropical SSTs would modify the probabilities of MJO initiation and therefore change the nature of the stochastic forcing.
Lastly, as pointed out in the introduction, the sparseness and irregularity of tropical weather station data make it difficult to characterize the activity of the MJO before the reanalysis era. The stochastic simulations presented in this study differ substantially from traditional techniques used in previous observational MJO studies. The authors are presently developing additional statistical methods to improve the reconstruction of the activity of the MJO in the pre-reanalysis era.
Acknowledgments
The authors would like to thank two anonymous reviewers for valuable comments and suggestions for this study. This research was funded by NOAA Office of Global Programs (NOAA/NA05OAR4311129, NA07OAR4310211 and NA08OAR4310698). NCEP/NCAR Reanalysis and OLR data provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their Web site at http://www.cdc.noaa.gov.
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