## 1. Introduction

The long-term predictability of the variability generated by the interaction between the atmosphere and the ocean has been one of the central issues of our field. The topic acquired a renewed interest when it was clear that phenomena such as El Niño–Southern Oscillation (ENSO) were capable of affecting weather variability and seasonal anomalies on global scales causing significant impacts to societies and economic activities.

The basic nature of the predictability problem exhibited by chaotic systems like the atmosphere and ocean has been stated by Lorenz (1963, 1984, 1987) and Ferrari and Cessi (2003). Because of small, but inevitable, errors committed in defining the initial state, the solution of the simulation quickly moves away from the real solution. He also estimated that the small errors generally tend to amplify exponentially in time; he verified that on synoptic scales they tend to double in 2.5 days (Lorenz 1969).

Some years later, Shukla (1981) extended the concept of predictability to the predictability of time averages. He showed that whereas the predictability limit for the synoptic-scale wave is only about two weeks, the predictability limit of the low-frequency planetary waves could be longer. These results laid the physical foundations to investigate the feasibility of dynamical prediction of monthly means.

Two years later, Miyakoda et al. (1983) highlighted that some large-scale quasi-stationary anomalous circulation features could still be successfully predicted using an initial value problem, but for longer time scales the role of the ocean, and therefore of the lower boundary conditions, becomes prominent (Shukla 1985). The feasibility of seasonal forecasting using multiple general circulation model (GCM) simulations by prescribing the global sea surface temperature (SST) was then shown by Stern and Miyakoda (1995), following a large number of results indicating the large control exercised by SST on atmospheric seasonal anomalies (Shukla 1998; Trenberth et al. 1998; Kang et al. 2002; Wang and Zhang 2002; Wang et al. 2004). Recently, the investigations have started to consider even longer time scales (Meehl et al. 2009).

Even though there have been significant advances in understanding predictability, several issues remain obscure, in particular the seasonal variation of predictability in the case of the ENSO [also known as the spring predictability barrier (SPB)]. Several works have been done on the predictability of ENSO (Bellenger et al. 2014; Chen et al. 2015; Samelson and Tziperman 2001; Newman et al. 2009) and in particular on the relation between ENSO and the SPB (Lopez and Kirtman 2014; Mu et al. 2007; Yu et al. 2012; Levine and McPhaden 2016). This phenomenon is one of the largest causes of prediction uncertainties in ENSO forecasting. It is characterized by an apparent drop in prediction accuracy during April and May (Webster and Yang 1992; Webster 1995). After the spring (or the autumn in the Southern Hemisphere), the ability of the models to predict the evolution of ENSO becomes increasingly better. SPB appears to be a moment where the system loses part of the information about its state and prediction becomes more difficult.

We are presenting here an analysis that aims to investigate the seasonal variation of ENSO from a different viewpoint. Navarra et al. (2013) prompted the idea that ENSO could be a system described by a sequence of states, rather than a simple oscillation. It is then possible to define in a consistent manner transition probabilities associated with ENSO and to characterize them. Each individual transition probability from one state to another (transition amplitude) can be organized in transition matrices that describe the overall range and possibility of such transitions. This method highlights the persistence, or the ability to transition, of the different states of the system. This can be a useful tool for simple diagnostics of a GCM. Leveraging these matrices, in fact, one can investigate the period and for which particular states of the system the model has more difficulties in representing nature. An interesting feature that has emerged is the trace of the SPB in these matrices. This feature has been exploited to define a predictability index of ENSO.

The transition amplitude, representing the probability that one phenomenon can transit from different states, has been considered before in meteorological problems. Brooks and Carruthers (1953) suggested these matrices in order to study meteorological phenomena, and Gabriel and Neumann (1962) started to use this method for a systematic investigation of data of daily rainfall occurrence.

However, it is not clear if ENSO can be considered as a damped oscillator sustained by noise or a fully nonlinear system driven by noise. The works written about the representation of ENSO, in fact, can be loosely grouped into two big frameworks: a deterministic one (e.g., Cane and Zebiak 1985; Cane et al. 1986; Zebiak and Cane 1987) and a stochastic one (Penland 1996; Penland and Sardeshmukh 1995; Levine et al. 2016). Both frameworks involve a positive ocean–atmosphere feedback (Bjerknes 1969). In the former case, ENSO is seen as a self-sustained, naturally oscillatory mode of the coupled ocean–atmosphere system (Wang et al. 2012). Several mechanisms, able to reproduce the basic characteristics of ENSO, have been proposed for this deterministic oscillation, explaining the dynamics in terms of a delayed oscillator (Schopf and Suarez 1988; Suarez and Schopf 1988; Battisti and Hirst 1989; Philander 1990; Jin and Neelin 1993; Neelin et al. 1994; Tziperman et al. 1994; Jin et al. 1994; Neelin et al. 1998), in terms of a recharge mechanism (Jin 1996, 1997), in terms of an advective–reflective oscillator (Picaut et al. 1997), and with other conceptual models. These models basically differ in the negative feedback used to limit the growth of the oscillation, using reflected Kelvin waves, discharging process due to Sverdrup transport, or anomalous zonal advection, respectively. Wang (2001) showed the existence of the unified oscillator model that includes the physics of all oscillator models discussed above. On the other hand ENSO could be also represented as a linear system with noise and recently this model has been used to investigate the SPB in ENSO (Levine and McPhaden 2015). A nonlinear system with noise has been developed by Navarra et al. (2013) and highlights the possibility to describe ENSO as a sequence of discrete states.

The last observation inspired this work, in which we are proposing to define four states for sea surface temperature anomaly (SSTA) and then to compute transition probability matrices for these states for different transitions periods. These matrices are general and they do not necessarily describe a Markovian process. Confidence intervals can also be obtained using appropriate statistics. Although the initial idea was inspired by a nonlinear stochastic theory, the transition matrices may also be constructed from nonstochastic time series. As shown in the following section, these matrices can be useful, but without further investigation it is not really possible to quantify the importance of noise.

Previous works have tried to link Markovian processes to the climate one. Nicolis (1990) developed a method for mapping deterministic chaos in general and atmospheric and climate dynamics in particular, into a Markovian process. She described a multivariate system continuously evolving in time with a discrete iterative form, monitoring successive extrema of some of the variables, as was also done by Fraedrich (1988). From the map obtained, using an appropriate partition of the possible states, Fraedrich (1988) mapped the deterministic chaotic system into a stochastic one, described as a Markov process. However, our approach does not assume a Markovian nature for the underlying stochastic process, and so it is somewhat more general.

This statistical approach is applied both to the Niño-3.4 index computed from the observations and to the same index computed from a long simulation run of a GCM, yielding matrices that can be used to compare models and observations. From the comparison, it is possible to see in detail the statistic related to the dynamics of ENSO, the persistence of a state, or its capabilities to transit in another one. The information obtained is not limited to the stationary distribution for every season, but it can also give information on time-dependent transitions. Comparing the matrices obtained from the model and observations, it is possible to have a more detailed picture of the transitions.

The paper is organized as follows. The method is described in section 2, and the results from observations and models are presented in sections 3 and 4. The concluding remarks in section 5 will close the paper.

## 2. Methods

We have assumed that the SSTA can fall just in four states defined in Table 1. The choice of this number of states is dictated by limitations on the amount of available data. A higher number of states would lead to more detailed information of the transitions. However, the number of counts in the different entries of the matrices, for the calculation of the probability, must be such as to allow a statistical analysis to compute the confidence intervals for the probabilities. For this reason we have not chosen a larger number of states. On the other hand, a number of states that is too small (e.g., two) does not allow one to appreciate the redistribution of the transition probability in the different seasonal transitions. For this reason four states have been considered a fair compromise for the computation.

Division of the continuous interval of SSTA variation in four different discrete states.

The definitions for the states are based on the conventional SST ranges. The *A*^{−} state represents a strong negative anomaly that could be considered a La Niña event, while *A*^{+} is a strong warm event that could be considered an occurrence of El Niño. The intermediate states *N*^{−} and *N*^{+}, respectively negative and positive, are the so-called neutral states, characterized by anomalies too small to consider the Pacific Ocean in a state of La Niña or El Niño. This allows an easy and direct understanding about the transitions between different states and makes it easy to understand how the states of El Niño or La Niña are transformed over time. Clearly, other state definitions are possible because the thresholds for the states can be arbitrarily chosen. For example, an alternative could be to consider the quartiles of the ENSO PDF distribution integrated over time. If we use PDFs, which also vary seasonally, the definition of the quartiles would also vary over time, as would the definitions for the states. This would result in a difficult and confusing interpretation of the behavior of the system. The intervals to define the states should be fixed in time. However, the matrices would be modified; in particular, if for example we extended one of the states that we call *N* in a region of the current states *A*, you would get a greater persistence in time for these new states *N*. In our opinion a division of this kind would result in a difficult understanding of the system, and furthermore new definitions of El Niño and La Niña would be necessary. Also, in the case of using multiple states, we believe that their division should always be carried out in compliance with the conventional ranges of SST.

A seasonal stratification can be defined (Table 2) so that the states have a seasonal dependence. We have chosen the usual boreal season definitions to compute the transition matrices. The choice of the seasons is arbitrary, but as is shown in the next section, it does not modify the conclusions of the paper. Seasonal means for the Niño-3.4 index have been obtained from a monthly mean time series of 159 yr, from January 1856 to December 2014. The observations from 1950 to 2014 (NOAA/NCEP 2015) are patched with a reconstruction (Kaplan et al. 1998) for the period 1856–1949. We arbitrarily will take this time series as meter of measure for ENSO.

Conventions for the variables used to indicate different seasons.

^{nk}the matrix that contains the transition probabilities between states considering the starting season

*s*

_{n}and the ending season

*s*

_{k}. If

*n*<

*k*, both seasons belong to the same year, if

*n*≥

*k*the convention used here is that

*s*

_{k}belongs to the next year. The elements of this matrix are

*j*during the season

*s*

_{k}starting from the state

*i*in the season

*s*

_{n}, with the convenience that the time of the transition must be less than one year. Note that notation adopted here for the conditional probability is reversed from the standard notation.

*i*during one of the months belonging to the season

*s*

_{n}to the state

*j*in one of the three months of the season

*s*

_{k}and the total number of transitions made out of state

*i*during this period

*c*

_{1}=

*c*

_{2}= (1 − CL)/2. This system can be solved with a simple recursive algorithm of dichotomous search. Solving the problem for each value of the transition matrix, it is possible to assume that

These transition matrices may show some important features about ENSO. For example, the spring barrier could be highlighted using just four states by the values of the probabilities and by their movement through the matrix considering different seasonal transitions.

_{E}is the matrix representing the uniform distribution, the one with all entries equal to 0.25. This index measures the distance of the transition matrix from a matrix representing the transitions between states that are uniformly probable and therefore have no information content. A value of zero would mean that all the states have the same probability in the transition, whereas the values closer to one will indicate there is a certain transition. We would like to see the degradation of the information contained in these matrices. Since entropy is a measure of the unpredictability information content, it is natural to consider the uniform transition matrix as the one that contains less information. So it is interesting seeing how much our matrices are different from this one and not from a matrix derived from the climatology, which would contain much more information. Using the uniform matrix one has an idea of the spread of the information due to a particular transition; in particular, one can see how the spring shakes the system completely. It is important to stress the fact that this helps us to see not how far the transitions are from the climatological matrix but how the information is degraded and spread to avoid the possibility of doing some predictions.

*d*

_{2}between matrices:

If the entropy allows us to describe the intrinsic loss of information (the unpredictability) of the system, using a well-known quantity in statistical physics, the distance *d*_{2} allows a direct comparison between the matrices, and in particular between the same matrix computed from the observation and from the model.

Other indices, as well Niño-3.4, are used to characterize ENSO, for example Niño-1+2 and Niño-4. The domains for the latter two indices are defined as 10°N–10°S, 80°–90°W and 5°N–5°S, 160°E–150°W, respectively. In Fig. 1 are shown the discrete PDF obtained by the three indices. Using just four states, defined considering the conventional SST ranges, only the discrete PDF of the Niño-3.4 index seems to be able to capture the variability of the system during the different seasons. In fact, the change in concavity for the discrete PDF in Figs. 1b,c is very reduced. This is the reason that motivated us to use only the first index in the paper and not to further consider the other two indices. It is probable that, with other choices of states based on PDF data (as the quartiles for example) or a large number of states, this limit may be exceeded.

## 3. ENSO transition probability matrices

Figure 2 shows the transition probability matrices for Niño-3.4 for a transition period less than one year. We have preferred to keep different color-bar ranges to better estimate the values of the probabilities into the matrices by visual inspection. From the color bar one can estimate the differences between the matrices. These matrices are found using Eq. (2) and the confidence interval with CL = 95% using Eq. (4). The numerical values of these matrices with the errors are reported in the appendix. Figures 2a,d,g,j show the shortest transitions, which are the transitions that involve two seasons: the starting season and the ending season. The highest values of the probabilities are denser around the diagonal of these matrices. Thus, this first set of matrices shows the persistence of the system. After one season, the system does not change much. This is particularly true for the autumn–winter transition matrix *A*^{+} (*A*^{−}) during the autumn *s*_{4} it is likely to be in *A*^{+} (*A*^{−}) also in winter *s*_{1}. This is not true for the neutral states that tend to be less persistent. A similar stability for the extreme states is also present between summer *s*_{3} and autumn. This is reflected in the matrix elements

Transition probability matrices computed with Eq. (2) for the 159-yr Niño-3.4 time series.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Transition probability matrices computed with Eq. (2) for the 159-yr Niño-3.4 time series.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Transition probability matrices computed with Eq. (2) for the 159-yr Niño-3.4 time series.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Figures 2b,e,h,k show the three-season transition. The effect of the loss of information, because of the SPB, appears more evident here, in particular in the matrix *A*^{−} is the most persistent. Matrix *A*^{+} is more stable than *A*^{−}, but the diagonal value

For the four-season transition, Figs. 2c,f,i,l highlight the features discussed above. The matrix

Applying Eq. (7), we have been able to compute a clear index for the predictability of the ENSO evolution, as shown in Fig. 3. For every starting season, ordered in the abscissa, we have computed the ENSO_{idx} for each transition, represented with a different color line. The longer the period of the transition is, the less information remains in the system, which means that the phenomenon is less predictable. We need to point out that this work is based just on the transitions of the SST. This is enough to show the role that transition matrices can play as a diagnostic or descriptive tool. However, considering other interesting quantities, for example the equatorial heat content, and then building enlarged matrices, the predictability of this phenomenon could be improved.

ENSO index computed using Eq. (7). On the abscissa the starting seasons are indicated, while different lines represent the number of seasons involved in the transition. The entropy used for every matrix is the mean of the entropy ensembles obtained varying the matrices according to their 95% confidence level interval. The errors on the entropies are one standard deviation of these ensembles. The effect of the SPB is particularly evident in the two-season transition starting from the season *s*_{2} (MAM).

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

ENSO index computed using Eq. (7). On the abscissa the starting seasons are indicated, while different lines represent the number of seasons involved in the transition. The entropy used for every matrix is the mean of the entropy ensembles obtained varying the matrices according to their 95% confidence level interval. The errors on the entropies are one standard deviation of these ensembles. The effect of the SPB is particularly evident in the two-season transition starting from the season *s*_{2} (MAM).

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

ENSO index computed using Eq. (7). On the abscissa the starting seasons are indicated, while different lines represent the number of seasons involved in the transition. The entropy used for every matrix is the mean of the entropy ensembles obtained varying the matrices according to their 95% confidence level interval. The errors on the entropies are one standard deviation of these ensembles. The effect of the SPB is particularly evident in the two-season transition starting from the season *s*_{2} (MAM).

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

The effect of the SPB is clearly evident in the two-season transition (blue line) starting from spring *s*_{2}. In fact, for this matrix the curve has a well-pronounced minimum. The stability of the autumn–winter transition can be appreciated looking at the maximum of the blue line.

The effect of the SPB is also evident in the magenta line, representing the three-season transition. Now it is the first point of the line that passes through spring and loses more information in respect to the other starting point, while the transition involving summer, as the starting point, goes through the autumn–winter transition and limits the loss of information more than the others.

This index gives an idea of the predictability for different transition periods, starting in different periods of the year. This is useful to compare the different seasons of the year. Clearly, this plot contains the global information of the different matrices. The index gives relative information between different seasonal transitions, but for different matrices the real discrimination for a good prediction is the initial state. Let us consider the matrix *A*^{−} or *A*^{+}, the peaks in the probability for the final favorite state in winter can reach 90% of the total probability. If you consider as initial states the neutral ones, the peaks in the final probability represent just 40%, more or less, of the total probability. This is the reason the global index shown in the plot for this matrix is the highest one, but the value is small. The matrix

We have performed also another analysis to test the sensitivity of the results in respect the current choice of the seasons. We have considered just two long seasons, let us call them *s*^{c} [November–March (NDJFM)] and *s*^{w} [April–October (AMJJASO)], and we have computed the correspondent transitions matrices Fig. 4. Also in this case we have preferred to keep different color-bar ranges to better estimate the values of the probabilities into the matrices by visual inspection. The matrix *s*^{c} → *s*^{w}, while the matrix *s*^{w} → *s*^{c}. Starting from the season *s*^{c}, which contains the first month of the boreal spring, the correspondent matrix experiences something that spreads the transition probability around the matrix. Conversely, the other transition, which starts from a season that now contains two months of the usual boreal spring (the whole summer and almost all of autumn) shows a really good persistence, at least for the states *A*^{+} and *A*^{−}. In between March and April something happens, and the persistence of the different states changes. In particular, the persistence for the extreme states in the matrix

Transition probability matrices computed for the two long seasons *s*^{c} and *s*^{w} to test the sensitivity of the season definitions used in the paper.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Transition probability matrices computed for the two long seasons *s*^{c} and *s*^{w} to test the sensitivity of the season definitions used in the paper.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Transition probability matrices computed for the two long seasons *s*^{c} and *s*^{w} to test the sensitivity of the season definitions used in the paper.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Finally, we show a small example of forecast using these matrices (Fig. 5). We have tried to analyze the year 1997, in which a strong ENSO event caught the scientific community by surprise (McPhaden 1999). Several dynamical and statistical forecast models predicted, one to three seasons in advance, some anomalies (Anderson and Davey 1998; Barnston et al. 1999), but all these anomalies developed too slow and were too weak until the onset of El Niño. To perform these forecasts, we simply multiply the row vector that represents the starting probability distribution for the states (during *s*_{1}, *s*_{2}, and *s*_{3} in this year), readable from the plot of the anomalies, by the right transition matrix (

(a) Forecast of the preferred state of *s*_{4} (SON) season starting from three antecedent seasons for the year 1997. (b) Only the forecast *s*_{3} → *s*_{4} is able to show a peak probability on the correct state.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

(a) Forecast of the preferred state of *s*_{4} (SON) season starting from three antecedent seasons for the year 1997. (b) Only the forecast *s*_{3} → *s*_{4} is able to show a peak probability on the correct state.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

(a) Forecast of the preferred state of *s*_{4} (SON) season starting from three antecedent seasons for the year 1997. (b) Only the forecast *s*_{3} → *s*_{4} is able to show a peak probability on the correct state.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Only the forecast *s*_{3} → *s*_{4} is able to show a peak probability on the correct state. Not only is this true for the year 1997, but it will be true for all the transitions that start with similar conditions. Clearly with the advance of the computation power, and with models that are able to take into account more physics, better results than those obtained in 1997 could be reached; in any case this simple computation shows a sort of intrinsic lack of predictability of this phenomenon. This is also in accord with the work of Vecchi et al. (2006) and Levine and McPhaden (2016).

## 4. Model comparison

The procedure applied to the observations above is repeated also for a 500-yr run of the coupled CMCC-CMS model, used in Davini et al. (2014). The model had a horizontal spectral resolution of T63 and a vertical resolution of 95 levels (L95). The top of the atmosphere was at 0.01 hPa. The ocean model, OPA version 8.2, had a 2° resolution, as did the sea ice model simulated with LIM. This long run was made with preindustrial conditions. The model used allows us to make a comparison with the observations and easily evaluate if the model is able to reproduce the same transitions as in nature. From the model the Niño-3.4 index is obtained by averaging the temperature (within 5°N–5°S, 170°W–120°W) and subtracting the climatological seasonal cycle.

For the benefit of the reader, in Fig. 6 we show a comparison between the Niño-3.4 time series computed from the observations and from the model with a standard metric, such as the PDF and the Fourier spectrum. This model exhibits more variability in respect to the observations, as highlighted by the PDF, and the Fourier spectrum shows that only one oscillation frequency is really captured. This can lead to some problems in the ENSO forecasting; however, some particular transitions exhibit the same transition probability both for the model and for the observations as underlined by the transition matrices below.

(top) Comparison between the PDF of the Niño-3.4 index obtained from observations and CMCC-CMS. (bottom) Comparison of the Fourier spectrum for the two time series.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

(top) Comparison between the PDF of the Niño-3.4 index obtained from observations and CMCC-CMS. (bottom) Comparison of the Fourier spectrum for the two time series.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

(top) Comparison between the PDF of the Niño-3.4 index obtained from observations and CMCC-CMS. (bottom) Comparison of the Fourier spectrum for the two time series.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Figure 7 shows the transition probability matrices for transitions that occur in a period shorter than one year. Figures 7a,d,g,j show the two-season transitions. As for the observation, the model shows a persistent behavior also if the matrix

Transition probability matrices for the SSTA obtained with CMCC-CMS.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Transition probability matrices for the SSTA obtained with CMCC-CMS.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Transition probability matrices for the SSTA obtained with CMCC-CMS.

Citation: Journal of Climate 30, 13; 10.1175/JCLI-D-16-0490.1

Summary table of comparison between the transition probability matrices computed from the Niño-3.4 series and the one computed from the SSTA series of the CMCC-CMS. The numbers in the table are the mean distances of the distances ensembles generated varying the two matrices according to the their 95% confidence level interval. The error is one standard deviation computed from these ensembles. The boldface numbers highlight the highest distance once the starting seasons have been chosen.

Table 3 shows the distance between the probability transition matrix computed from the observations and the one computed from CMCC-CMS. It is remarkable that the transitions that bring the system into, or past, the spring present the biggest distance.

The advantage of the transition probability matrices is that the whole process is seen in a more detailed way. The matrix

## 5. Conclusions

We have shown that is possible to analyze ENSO by means of probability transition matrices. For each season we have computed the transition probability matrices for transitions between two, three, and four seasons. These matrices are not necessarily Markovian. The choice of the number of states has been dictated by limitations on the amount of available data.

Considering the length of the available Niño-3.4 series (159 yr) four states can be reliably defined, in such a way that a set of 4 × 4 matrices are defined. In this case the statistics for small populations is necessary to identify the confidence interval for the entries of the matrices that represent the probability to move from one state to another over the selected interval. It is difficult to use more states because for some transitions we have too few samples. Important features of the transitions, such as the persistence and the loss of information, are found, and a general way to write an index for the predictability of a phenomenon is presented. The SPB appears in these matrices and more clearly in the index found using the entropy of these matrices. This is in accord with previous work (Lopez and Kirtman 2014; Mu et al. 2007; Yu et al. 2012; Levine and McPhaden 2016). Also, these simple matrices are able to capture this peculiarity that makes the ENSO’s predictions in that period a real challenge.

The ENSO index for the predictability presented here contains the information for the whole matrix considered without distinguishing among different transition states. It is a measure of the predictability that considers all the possible initial and final states.

The intrinsic predictability limits of ENSO are highlighted with the case of the ENSO event of 1997/98. In this particular case, the two-season transition, from summer to autumn, is the only forecast that is able to catch the correct final state with a clear peak in the probability distribution for that state. Clearly these results could be improved with the advance of computational power and with models that have more physics; better results than those obtained in 1997 could be reached. However, this simple computation shows a sort of intrinsic lack of predictability of this phenomenon. This is also in accord with the work of Vecchi et al. (2006) and Levine and McPhaden (2016). We need to point out that this work is based on just the transitions of the SST. This is enough to show the role that transition matrices can play as a diagnostic or descriptive tool. However, considering other interesting quantities (e.g., the equatorial heat content) and then building enlarged matrices, the predictability of this phenomenon could be improved.

Finally, the transition matrices have been used to compare the Niño-3.4 index computed from a GCM and the index obtained with the observations. Once the distance from the respective matrices is computed, one can have an idea of how well ENSO is represented by the model. It is remarkable that the transitions that bring the system into the spring, or past the spring, present the largest differences. Using these matrices it is also possible to observe, in a more detailed way, in which state and in which period the model has a different behavior with respect to the observations. The GCM turns out to underestimate the one-season persistence of the four states. In general the model has shown less persistence behavior, allowing also improbable transitions as in the case of the matrix

## Acknowledgments

Thanks to the reviewer for important suggestions that made the manuscript more solid. The support of the GEMINA project funded by the Ministry for University and Research and by the Ministry of Environment, Land and Sea of Italy is gratefully acknowledged.

## APPENDIX

### Observed Transition Probability Matrices

#### a. Two-season transition (persistence)

These are the two-season transition probability matrices computed for the observed Niño-3.4 series of 159 yr. The confidence level used to find the error is CL = 95%. As explained in section 2, the error limits shown for each entry in the matrices appear to be sometimes asymmetric because, for the small population statistic, the binomial distribution is asymmetric and there is a strong dependence on the number of counts in each matrix bin and on the probability; see Eq. (4) and Rotondi et al. (2001).

#### b. Three-season transition

These are the three-season transition probability matrices computed for the observed Niño-3.4 series of 159 yr. The confidence level used to find the error is CL = 95%.

#### c. Four-season transition

These are the four-season transition probability matrices computed for the observed Niño-3.4 series of 159 yr. The confidence level used to find the error is CL = 95%.

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