# Extracting subseasonal scenarios: an alternative method to analyze seasonal predictability of regional-scale tropical rainfall.

Vincent Moron, Pierre Camberlin, Andrew W. Robertson

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Vincent Moron, Pierre Camberlin, Andrew W. Robertson. Extracting subseasonal scenarios: an alternative method to analyze seasonal predictability of regional-scale tropical rainfall. Journal of Climate, 2013, 26 (8), pp.2580-2600. 10.1175/JCLI-D-12-00357.1. hal-00828733

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<figure><img src="image_1.png" /></figure>
# Extracting Subseasonal Scenarios: An Alternative Method to Analyze Seasonal Predictability of Regional-Scale Tropical Rainfall

VINCENT MORON

Aix-Marseille Université, CEREGE UM 34 CNRS, Aix en Provence, France, and International Research Institute for Climate and Society, Columbia University, Palisades, New York

PIERRE CAMBERLIN

Biogéosciences, Centre de Recherches de Climatologie, Université de Bourgogne, Dijon, France

ANDREW W. ROBERTSON

International Research Institute for Climate and Society, Columbia University, Palisades, New York

(Manuscript received 20 June 2012, in final form 3 October 2012)

## ABSTRACT

Current seasonal prediction of rainfall typically focuses on 3-month rainfall totals at regional scale. This temporal summation reduces the noise related to smaller-scale weather variability but also implicitly emphasizes the peak of the climatological seasonal cycle of rainfall. This approach may hide potentially predictable signals when rainfall is lower: for example, near the onset or cessation of the rainy season. The authors illustrate such a case for the East African long rains (March-May) on a network of 36 stations in Kenya and north Tanzania from 1961 to 2001. Spatial coherence and potential predictability of seasonal rainfall anomalies associated with tropical sea surface temperature (SST) anomalies clearly peak during the early stage of the rainy season (in March), while the largest rainfall (in April and May) is far less spatially coherent; the latter is shown to contain a large noise component at the station scale that characterizes interannual variability of the March-May seasonal total amounts. Combining the empirical orthogonal function of both interannual and subseasonal variations with a fuzzy k-means clustering is shown to capture the most spatially coherent subseasonal "scenarios" that tend to filter out the noisier variations of the rainfall field and emphasize the most consistent signals in both time and space. This approach is shown to provide insight into the seasonal predictability of long dry spells and heavy daily rainfall events at local scale and their subseasonal modulation.

## 1. Introduction

Seasonal forecasts of tropical rainfall are now issued routinely by global centers using general circulation models (GCMs; e.g., Kumar et al. 1996; Livezey et al. 1996; Goddard et al. 2001, 2003; Barnston et al. 2003, 2010; Friederichs and Paeth 2005; Saha et al. 2006; Batté and Déqué 2011). Forecasts are also issued on a regional basis through regional climate outlook forums (RCOFs) such as the Programme de Renforcement et de Recherche sur la Sécurité Alimentaire en Afrique de l'Ouest

(PRESAO) for West African countries or the Greater Horn of Africa Climate Outlook Forum (GHACOF) for East African countries, often using statistical methods or hybrid statistical-dynamical methods (Berri et al. 2005; Ogallo et al. 2008). The forecasts are typically made for running 3-month seasonal rainfall totals expressed as tercile-category probabilities relative to a 30-yr historical climatology. Seasonal predictability of tropical rainfall is largely a product of sea surface temperature (SST) forcing associated with the El Niño-Southern Oscillation (ENSO) phenomenon (Ropelewski and Halpert 1987, 1996) but other SST anomaly patterns such as north-south gradients in the tropical Atlantic (e.g., Folland et al. 1986; Enfield 1996) or east-west gradients in the Indian Ocean (e.g., Saji et al. 1999; Behera et al. 2005) also contribute. Considering seasonal

<footnote>*Corresponding author address:* Pr. Vincent Moron, Aix-Marseille Université, CEREGE UM 34 CNRS, Europôle méditerranéen de l'Arbois, BP 80 Aix en Provence, F-13545, France.
E-mail: moron@cerege.fr</footnote>
total amounts of rainfall acts as a spatiotemporal filter that increases the signal-to-noise ratio, by enhancing the slow SST-forced larger-scale component of rainfall variability, while averaging out the smaller-scale unpredictable weather variability.

It is now well established that the frequency of wet days within the season is potentially more predictable than the mean daily intensity of rainfall, due to the higher spatial coherence of the former compared to the latter, while both components contribute rather equally to the interannual variance of seasonal rainfall anomalies at a local scale (Moron et al. 2006, 2007). While seasonal averaging generally acts as an effective weather-noise filter, several studies have shown clear within-season modulation of the ENSO impact on tropical rainfall. Lyon et al. (2006) found a seasonal reversal in the sign of the ENSO-related Philippines rainfall anomalies, with El Niño events associated with negative rainfall anomalies around the onset of the southwest monsoon in May (Moron et al. 2009a) and positive anomalies during the core of the rainy season in July–September. Similarly, negative rainfall anomalies are widespread during the onset stage of the monsoon over Indonesia in September–December of El Niño years (Haylock and McBride 2001; Moron et al. 2009b, 2010) but less coherent during the core of the rainy season in December–March (Chang et al. 2005; Moron et al. 2009b, 2010). These rainfall anomalies tend to become positive over mountainous parts of Java (Qian et al. 2010). In such cases, seasonal averaging is likely to obscure predictability that is specific to particular stages of the seasonal cycle. Reducing the length of the time average such as using fixed calendar months would adversely enhance the noise and may also artificially cut consistent periods during a given season.

There are several physical reasons to consider sub-seasonal modulation of the predictable component across a wet season lasting 3–8 months. First, the strength of the SST anomaly forcing (e.g., ENSO) and the associated atmospheric anomalies exhibit a strong seasonal modulation, with ENSO events tending to emerge/decay in April–May and to peak near the end of the calendar year (Balmaseda et al. 1995). Second, atmospheric teleconnections are to first order well described by linear theory and are thus sensitive to the mean seasonal cycle “basic state” (Gill 1980; Robertson and Frankignoul 1990). For example, Hendon (2003) showed that low-level easterly wind anomalies across Indonesia associated with El Niño events strengthen the trade winds prior to the rainy season but subsequently decrease the strength of the northwest monsoon flow, with consequences for both local-scale air–sea coupling and local-scale rainfall anomalies (Hendon 2003; Moron et al.

2010; Qian et al. 2010). Third, the impact of soil moisture anomalies could be strongly modulated during the wet season. There is some evidence that soil moisture anomalies as well as their gradients play a role in the formation and/or intensity of mesoscale systems across the Sahel through a positive feedback loop of soil moisture on precipitation toward the end of the rainy season. Such positive feedback is able to lead to larger spatial coherence of rainfall anomalies, hence increasing potential predictability during the second part of the rainy season (e.g., Douville et al. 2001; Clark et al. 2004; Dirmeyer et al. 2009).

In East Africa, the boreal spring rains of March–May (MAM), also called the “long rains,” are an example of generally low seasonal prediction skill of seasonal rainfall totals (Moron et al. 1998; Friederichs and Paeth 2005; Batté and Déqué 2011). It has been hypothesized that this lack of skill is partly because the season is not homogeneous. The relationship with ENSO, besides being weak, tends to switch sign between March and May (Indeje et al. 2000). Camberlin and Philippon (2002) and Mutai and Ward (2000) also found that the relationships between MAM rainfall and large-scale circulation patterns and SST were different between the early and late stages of the season.

We present here a new method that considers both interannual rainfall variations together with their sub-seasonal modulation as a basis for estimating the potential predictability at a seasonal scale, in contrast to the usual practice of considering only interannual variations of seasonal total rainfall amounts. The approach consists of empirical orthogonal function (EOF) analysis of low-pass-filtered daily rainfall station data, followed by fuzzy clustering to identify the leading modes of interannual variability and its subseasonal evolution (referred to as typical subseasonal “scenarios”). The aim is to focus on interannual rainfall anomalies that covary in space but also systematically in time within each season while minimizing the anomalies which are purely local in space or transient in time, such as those associated with the Madden–Julian oscillation (MJO), which are not phase locked to a particular stage of the rainy season.

Our approach considers both interannual variability and its subseasonal evolution in order to cluster years into similar seasonally evolving scenarios and associate them with SST forcings. The data and statistical approach are detailed in sections 2 and 3, respectively. The results are presented in section 4, first in terms of the identified seasonal scenarios and second in terms of the impact of SST anomalies on them. The summary and discussion follow in section 5.
<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Location of the 36 rain gauges with their mean February–June rainfall (in mm day$^{-1}$). The altitude (from 1-km elevation terrain model) is displayed as light (500 m), medium (1000 m), and dark (2500 m) gray shadings. (b) Mean daily rainfall for each rain gauge (dotted lines) and for the average of the 36 rain gauges (bold full line).</figcaption></figure>

## 2. Rainfall data and sea surface temperatures

The daily rainfall dataset, covering Kenya and northern Tanzania, was assembled from previous studies on East Africa variability (Camberlin and Okoola 2003; Camberlin et al. 2009), with the raw material obtained from the Kenya Meteorological Department and the Tanzania Meteorological Agency. We retained the 36 rain gauges having less than 5% of missing entries (Fig. 1a) from February 1961 to December 2001. Leap days have been simply removed from the database. The missing entries are filled using an analog method (see appendix A). The mean seasonal cycle (Fig. 1b) shows that the peak of the rainy season is not always found in April and that the onset and withdrawal stages are rather smooth since some rainfall is still observed in February as well as in June. The detection of onset and withdrawal dates would be rather challenging in that

context (Boyard-Micheau et al. 2012, manuscript submitted to *J. Climate*).

Observed monthly sea surface temperatures covering the same period as the rainfall data, at a resolution of 5° in latitude and longitude, are taken from the extended reconstructed SST (ERSST) v3 dataset (Smith et al. 2008).

## 3. Methodology

The procedure for identifying subseasonal scenarios of interannual rainfall variability consists of three steps: (i) smoothing out the fast unpredictable variations of daily rainfall; (ii) computing the leading seasonally evolving EOF modes of the smoothed rainfall variations by considering both interannual and subseasonal time scales to emphasize the regional-scale variations and minimize the purely local-scale ones; and (iii) identifying the subseasonal scenarios to differentiate seasons with quasi-constant (i.e., dry or wet) anomalies from seasons when the anomaly is temporally modulated. Each step is outlined below, with technical details given in appendix B.

The first step is to minimize the fast variations of the rainfall field. There is no unique definition of what should be considered as fast but the limit of deterministic predictability of the atmosphere is ~15 days (Lorenz 1963). A 31-day running mean filter is used here, as a balance between fast unpredictable variations and the seasonal time scale. However, we will show that the subseasonal scenarios extracted based on this monthly filtering can be easily translated into purely local-scale sequences at daily time scales (see section 4c) and the sensitivity of the results to the temporal smoothing is briefly discussed in appendix B. The running 31-day mean rainfall at the 36 rain gauges is computed over the whole series of 41 × 365 days (1961–2001), to avoid possible edge effects on each February–June (FMAMJ) season. The 150-day period from 1 February to 30 June (with each day as the center of running 31-day sequence) is selected to include the MAM wet season, with one month on either side. The resulting time series are then square rooted to decrease the positive skewness of the rainfall data. The long-term mean for each day of the smoothed rainfall data is removed to consider the anomalies relative to the mean seasonal cycle. We then standardize these local-scale anomalies by the overall standard deviation (across days and years) to give all stations an equal weight.

The second step is to apply EOF prefiltering to extract the leading seasonally evolving modes of the interannual anomalies defined above, as they evolve on a daily basis from 1 February to June 30 over the period 1961–2001 in the smoothed rainfall data. We thus formed an extended
<figure><img src="image_3.png" /><figcaption>FIG. 2. (a) Mean monthly rainfall amounts (averaged across the 36 rain gauges). (b) Spatial DoF of interannual variations of standardized anomalies of monthly rainfall (perfect covariant anomalies result in a unitary DoF while independent variations between the 36 rain gauges result in DoF ~ 19). (c) Skill (= correlations between observed and hindcast anomalies) of monthly rainfall specified by synchronous tropical sea surface temperatures (30°N–30°S). The hindcast is done using a cross-validated canonical correlation analysis (with EOF prefiltering) with one year left out at each turn. The bars are for the regional-scale index and the bold line with open circles is the average of local-scale correlations.</figcaption></figure>

data matrix with $N = 41$ rows describing years from 1961 to 2001 and $M (=36$ stations) $\times L (=150$ days) columns describing each daily FMAMJ sequence for all stations [see Eq. (B1)]. Thus, each station on each calendar day is treated as a separate variable (column), with the rows containing the data points for these variables (e.g., station 10 on 16 May) for each consecutive year. By doing so, we consider the spatiotemporal variability between stations across years and also across the seasons. The leading principal components (PCs) will then extract the covariant modes of variations in space and in time while the purely local-scale variations in space but also in time will be accounted by the higher PCs not considered in the following analyses. Note also that any intraseasonal transients still present in the 31-day running averages, such as the MJO, will be filtered out by this approach if they are not phase locked to the climatological seasonal cycle.

In the third step, the leading six unstandardized PCs (their standard deviations equal their corresponding eigenvalues) accounting for 50% of the total variance are clustered using a fuzzy k-means cluster analysis (Bezdek 1974, 1981). The fuzzy partition uses Euclidean distance as the metric between the observations: that is, the unstandardized PC time series describing the interannual and subseasonal rainfall variability in Kenya and north Tanzania. As we use unstandardized PCs, the distances are weighted by the eigenvalues conveyed by each PC. Fuzzy k-means defines the centroid (i.e., the center of each cluster) as the weighted average of all the 41 years, according to the membership probability that

each year belongs to a particular cluster [see Eqs. (B4) and (B5)]. The number of clusters $c$ and the fuzziness of the clustering $m$ are parameters that need to be defined a priori [see Eqs. (B4) and (B5)]. We used an ad hoc approach of the choice of $c$ and $m$ following, for example, McBratney and Moore (1985) and De Bruin and Stein (1998), among others, in which the values of $c$ and $m$ are based on an explanatory hypothesis that underlies the clustering. In other words, a partition is useful only if the results can be understood within the context of the research question at issue. Our main hypothesis here is that typical subseasonal scenarios of interannual rainfall variability arise as a response to SST anomalies. In our case, a reasonable test is thus to find the combination of $c$ and $m$ that maximizes the area of significant SST anomalies across the tropical zone (30°N–30°S), from which it is assumed that most of the boundary forcing arises. Our partition of the interannual/subseasonal variations across the network of 36 stations is most useful if the yearly membership grades show a strong relationship with tropical SST anomalies.

## 4. Results

### a. Evidence of subseasonal modulation of rainfall spatial coherence and skill from SST

Figure 2 compares the monthly mean rainfall spatially averaged across the 36 stations (Fig. 2a) with the spatial coherence of the interannual variability of the monthly
rainfall (Fig. 2b) and regional-scale interannual varia-
tions “predicted” linearly from the contemporaneous
tropical SST anomalies (Fig. 2c). The number of spatial
degrees of freedom (DoF) indicates how heterogeneous
are interannual variations between the stations and is
a relative measure of the strength of spatially coherent
variations across a network (Fraedrich et al. 1995;
Bretherton et al. 1999; Moron et al. 2007). It is computed
from monthly rainfall anomalies standardized to zero
mean and unit variance. Its minimal value equals 1 if all
stations show exactly the same interannual variations.
The maximal value depends on the rank of the matrix
(here = 36) but also on its size [here max(DoF) ~ 19
estimated from 10 000 Monte Carlo simulations of
a white noise matrix with 41 rows and 36 columns] when
all stations are linearly independent from each other.
The amount of interannual variance of monthly rainfall
anomalies predicted (or accounted for) linearly from
contemporaneous tropical SST is computed with a ca-
nonical correlation analysis (CCA) using monthly tropi-
cal SST as the predictors at zero lag. It should be viewed
as the optimal or potential skill coming linearly from
tropical SST. The CCA is cross validated with one year
left out at each turn and the predictors and predictands
are prefiltered with EOF. The number of EOF modes
(from 1 to 10) and CCA modes (from 1 to 10) retained
in the model was determined by maximizing a station-
averaged Spearman's correlation score, under cross vali-
dation, using the Climate Predictability Tool (CPT) tool-
box (Linux version 12.04) from the International Research
Institute for Climate and Society (IRI; http://portal.iri.
columbia.edu/portal/server.pt?open=512&objID=697&
PageID=7264&mode=2).

April is confirmed to be the wettest month and
May ranks second (Fig. 2a). The DoFs are smallest in
March, denoting higher spatial coherence, and then
strongly increase toward June, as already noted by
Camberlin et al. (2009). In that case, the months with
the largest rainfall amounts (April and then May) are
not the most spatially coherent ones and thus could be
viewed as a source of noise in the seasonal amounts. In
other words, the seasonal amounts are not necessarily the
most predictable variables since they are impacted by
large amounts of rainfall, which are not spatially coherent
at a regional scale (Moron et al. 2007). The skill associ-
ated with concurrent SST anomalies is fully consistent
with this behavior (Fig. 2c); the skill is highest when DoFs
are lowest (high spatial coherence) and decreases as DoF
increases. These analyses suggest that some seasonal
predictable signals are at least partly hidden in the sea-
sonal amounts of rainfall and that we need to emphasize
the beginning stage of the rainy season in order to get
a proper view of the seasonal predictability.

b. *Identification of subseasonal scenarios and modulation of spatial coherence*

The EOF prefiltering described in section 3 and in
appendix B extracts the leading modes of spatially and
subseasonally covarying interannual variability among
the rain gauges. The temporal modulation of the spatial
covariance has already been provided by Fig. 2b. Figure 3
shows the seasonal evolution of the interannual vari-
ability captured by subsets of PCs 1–6 [see Eq. (B3)]. The
leading PC (explained variance = 17.9%) emphasizes the
variance of the period from mid-February to mid-April.
The second PC (explained variance = 10.3%) adds some
variance around this period (Fig. 3), while the recon-
structed variance from May is still heavily reduced. The
following PCs take into account this noisier time period
around May and the sum of PCs 1–6 captures almost
evenly the observed variance (Fig. 3).

The cluster analysis first requires determining the
optimal values of the parameters *c* and *m* of the fuzzy
*k*-means algorithm (see appendix B). We computed the
seasonal (FMAMJ) SST anomalies weighted by the
membership probabilities *u*<sub>*ki*</sub> defined in Eqs. (B4) and
(B5). Figure 4 shows the proportion of area where
the weighted tropical seasonal sea surface temperature
anomalies (SSTA) of the clusters (from *c* = 2 to *c* = 6
and from *m* = 1.1 to *m* = 2.2 with a step of 0.1) are
significant at the two-sided 90% significance level (see
appendix B). A clear maximum emerges for *c* = 4
clusters around *m* = 1.5, and this solution is retained in
the subsequent analysis. Figure 5 shows the membership
probabilities of each of the four clusters for each year; 18
out of 41 FMAMJ seasons belong to a cluster with
a probability exceeding 0.75 and thus provide clear cases
of seasons where a specific space-time scenario prevails.
Cluster 4 is highly specific to 1966, 1968, 1978, and 1990
(membership > 0.86), while 30 other seasons have
a membership <0.1 for this cluster. Several other sea-
sons are more ambiguously defined, as in 1961–64, 1979,
1986, 1991, 1998, and 1999, where no single cluster is
really dominant (Fig. 5). The attribution of these seasons
to a single cluster, as done in “hard” *k*-means, would be
unsatisfactory.

Figure 6 shows the centroids of each cluster in terms of
the subseasonal rainfall evolution for each station
(dotted line), together with the spatial average (solid
line). For each station and day of the season, the cluster
centroids are given by their projection from the EOF to
the physical space of Y [see Eqs. (B1)–(B6)], while Fig. 7
displays station maps of the seasonal FMAMJ anomalies
for each cluster, weighted by the cluster membership
probabilities of each year. Cluster 1 is rather symmet-
rical to cluster 4, and cluster 2 is symmetrical to cluster 3
<figure><img src="image_4.png" /><figcaption>FIG. 3. Spatial averages of the local-scale interannual standard deviation of rainfall for sliding 31-day windows (all) and the same from EOF reconstruction from one EOF-1 to 1–6 EOFs (symbols) [see Eq. (B6)].</figcaption></figure>

in terms of both subseasonal evolution (Fig. 6) as well as spatial pattern (Fig. 7). This is partly expected since the data are standardized to zero mean and anomalies thus tend to be symmetrical around zero. What is not necessarily expected is the fact that the amplitudes of the centroids tend to zero in May–June for all clusters (Fig. 6). This is in accordance with the lower spatial coherence at the end of the rainy season. The spatial deviations versus the spatial average (range of dots above and below the bold line) are also larger in April–May than in March, especially for clusters 1 and 3 (Figs. 6a,c). Cluster 1 is dominated by negative rainfall anomalies in February–March, while the positive rainfall anomalies peak in early March in cluster 4. These clusters may be associated with an abnormally late (cluster 1) and early and more abundant (cluster 4) onset of the MAM rains. Cluster 2 (cluster 3) are associated with negative (positive) rainfall anomalies from mid-March to early May and reversed anomalies before and after this core period. In other words, cluster 2 reflects an attenuated seasonal cycle, with below-normal rainfall near the peak month (April) and slightly above-normal rainfall before and after, resulting in a protracted rainy season. Cluster 3 shows the opposite, with an enhanced (positive anomaly peaking in early April) but shortened rainy season.

Figure 7 shows that clusters 2 and 3 reflect near-normal seasons overall, while cluster 4 is strongly wet and cluster 1 is anomalously dry. Note also from Fig. 7 that the local-scale anomalies are rather homogeneous across the network, with the exception of the coastal

area where near-zero anomalies are observed in cluster 1 (Fig. 7a) and weak negative but significant anomalies are observed in cluster 3 while weak positive anomalies are observed over the remaining stations (Fig. 7c). As stated before, the centroids provide an efficient way to summarize the complex intraseasonal–interannual and spatial variability and do not necessarily reveal “true” clusters from a topological point of view (e.g., Ghil and Robertson 2002).

<figure><img src="image_5.png" /><figcaption>FIG. 4. Area of composite tropical (30°N–30°S) SSTA (in %) in FMAMJ significant at the two-sided 90% level according to the fuzifier (from 1.1 to 2.2 with a step of 0.1) and the number of clusters (from 2 to 6). The composite is computed as the weighted average [weight = membership of each year defined in Eqs. (B4) and (B5)] of SSTA and its significance is estimated with a Monte Carlo permutation of year 1000 times (see text).</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>FIG. 5. Cumulative membership [defined in Eqs. (B4) and (B5)] of each FMAMJ season.</figcaption></figure>

Figure 8 illustrates the data-adaptive weighting of the rainfall anomalies provided by the cluster analysis, in terms of the correlation between the observed low-pass-filtered daily rainfall anomalies and those reconstructed from the cluster centroids, averaged over the 36 stations. The cluster centroids are firstly projected from the PC-space to the physical space using Eq. (B6). We thus obtained a 4 by 150 days × 36 yr matrix giving the local-scale time behavior of each cluster across each FMAMJ season. The cluster membership probabilities of each year (which sum to one) are then multiplied with these centroids to get interannual local-scale time series accounted by each subseasonal scenario. The correlations are rather high during the first half of the rainy season but decrease rapidly after mid-April. The correlation computed using the standardized anomaly index (SAI) is higher relative to the simple spatial average of station correlations during the first half of the rainy season, while there are no differences between them during its second half (Fig. 8).

The difference between the first and second halves of the season could be related to several factors. We can hypothesize that the spatial scale of rainfall events differs between the two periods. Figure 9 shows the spatial correlations between stations versus distance between stations, for the raw daily rainfall records and the low-pass-filtered rainfall anomalies, for 1 February–14 April (Fig. 9a) and 15 April–30 June (Fig. 9b) (cf. Moron et al. 2007, among others). To construct Fig. 9, we first computed the correlations between the daily amounts of

each pair of stations provided that both stations are wet (i.e., receiving at least 1 mm). Then we did the same using running 5-, 11-, 21-, and 31-day amounts of rainfall expressed as anomalies from the local-scale climatology. These correlations are then averaged by nonoverlapped segments of 100-km distances (the first one is 1–100 km, then 101–200 km, etc.), and these mean correlations for a fixed set of distances between two stations are plotted on Fig. 9. There are no marked differences between both periods when raw daily rainfall are considered (black dots), suggesting that the spatial scale of daily rainfall events is similar throughout the season. However, the differences between the two periods become large when time averages are considered (open symbols in Fig. 9). Correlations are higher and decline much more slowly with increasing distance between stations in the 1 February–14 April period than in the 15 April–30 June period. Thus, time-averaged rainfall anomalies are seen to be much more spatially coherent during the first half than during the second half of the rainy season. This explains why the subseasonal scenarios extract more variance during the first half of the season, even if it does not necessarily match with the wettest part of the season.

### c. Subseasonal scenarios and local-scale properties

The seasonal scenarios are estimated from local-scale time series through the construction of extended matrix **Y** [see Eq. (B1)], but they intrinsically emphasize regional-scale signals through the EOF. It is important to
<figure><img src="image_7.png" /><figcaption>FIG. 6. Running 31-day local-scale (dotted lines) rainfall anomalies and its spatial average (full line) associated with the centroids of the four-cluster solution. The units are standard deviation. The years at the top of each panel are those having a membership >0.75 (first row) and between 0.5 and 0.75 (second row). The SAI is the weighted mean of the standardized anomaly index [= spatial average of standardized local-scale seasonal (FMAMJ) anomalies] for each cluster. The weights are the membership defined in Eqs. (B4) and (B5).</figcaption></figure>

assess how much local-scale information can be re-
trieved from these scenarios and whether they have any
specific spatial signature. Figure 10 shows the temporal
(150 days) correlation between the local-scale centroids
and their spatial average (Fig. 6). It is clear that most
(77%) of the local-scale correlations are >0.8, especially
in the well-documented area of southern and western
Kenya but also over the less-documented area of north-
ern and northeastern Kenya, which is far drier and where
rainfall is more scattered than in southwestern Kenya
(Fig. 1a). An area that is less related to the spatial average
is the Indian Ocean coast (Fig. 10), as already seen in Figs.
7a,c. It is not a real surprise since there are specific
meteorological features over this region: the seasonal
rains peak later (May), and MAM rainfall variability is
distinct from that of the inland areas, at both inter-
annual (Ogallo 1989; Camberlin and Philippon 2002)
and intraseasonal time scales (Camberlin and Planchon
1997). The largest differences are observed for cluster 3

(mean correlation of 0.42 between the coastal stations
and the spatial average) with drier conditions overall
than the spatial average (Fig. 7c), especially from mid-
April to late May (not shown). For comparison, the
mean correlations between the coastal stations and the
spatial average reach 0.91 (cluster 1: Fig. 7a), 0.71
.cluster 2: Fig. 7b), and 0.74 (cluster 4: Fig. 7d). How-
ever, in mean, the clusters are fairly representative of
rainfall variations over most of Kenya and northern
Tanzania.

Figure 11 shows the probability of observing a dry
spell lasting at least 14 days at a local scale. Such a dry
spell could be detrimental for crops especially near the
onset and grain filling stages (Sivakumar 1993; Barron
et al. 2003; Marteau et al. 2011). It is computed as the
weighted average [weights are as before the member-
ships defined in Eqs. (B4) and (B5)] of the local-scale
probability. The climatology shows two maxima near
the starting and ending dates of the rainy season as
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

FIG. 7. Weighted standardized seasonal (FMAMJ) anomalies for each cluster. The weights are the membership defined in Eqs. (B4) and (B5). The upper (lower) triangles denote positive (negative) anomalies, and the gray symbols indicate significant anomalies at the 90% two-sided level according to 1000 random simulations of reshuffled yearly memberships. The scale displayed in the bottom-right corner of (a) is the same for the four panels.

would be expected. The minimum probability is also logically observed near the peak seasonal rainfall from late March to mid-May (Fig. 11). We estimate the significance using a Monte Carlo resampling approach in which the data are randomly resampled in year-long chunks to create 1000 simulated datasets. The membership matrix $u_{ki}$ is randomly reshuffled by year. We then computed the probability of observing a local-scale dry spell lasting at least 14 days using the reshuffled memberships as weights. This is repeated 1000 times. By

doing so, we expect that the PDF of the reshuffled set will tend to the climatological probability distribution, including stochastic interannual variations. Finally, the significance of the observed probability is given by its rank in the reshuffled set. The probability of observing a dry spell lasting at least 14 days is, as expected, higher (lower) for the main anomalous negative (positive) rainfall anomalies revealed by Fig. 6. However, what is interesting here is that the local-scale risk attached to different clusters featuring very dissimilar seasons
<figure><img src="image_12.png" /><figcaption>FIG. 8. Daily correlation between the *k*-means reconstructed (see text) and observed rainfall anomalies. The full light line is the spatial average of the local-scale correlations while the full thick line is the correlations computed with the standardized anomaly index (=spatial average of local-scale standardized anomalies).</figcaption></figure>

(e.g., dry and average seasons, for clusters 1 and 2, respectively) may be identical during certain periods (e.g., enhanced risk of a dry spell in late March in both clusters 1 and 2; Figs. 11a,b). This risk is very different for cluster 3 (reduced probability of a dry spell over much of the peak

rainfall period; Fig. 11c), although this cluster depicts, as cluster 2, a seasonal rainfall amount close to normal (Fig. 7c). Note that considering other dry spell lengths as 7 or 21 days leads to a very similar behavior in terms of subseasonal modulation as well as time phasing of significant anomalies associated with each cluster (not shown).

The same technique is applied on the risk of heavy rainfall, expressed as the probability to get a daily rainfall amount above the 95% percentile (computed from raw sequences of daily rainfall over the whole period February–June for 1961–2001) in running 31-day sequences (Fig. 12). As before, the relationship between seasonal mean anomalies and the subseasonal behavior is far from being trivial: for example, the probability of observing an extreme rainfall in the late season, which could be detrimental to harvesting, is enhanced for cluster 2 (Fig. 12b) and not for cluster 4 (Fig. 12d), which includes wetter seasons. These examples show that these subseasonal scenarios of rainfall variability are relevant for agricultural applications and that it is rather easy to extract local-scale properties useful for such applications. The approach could easily be extended to a wide range of purposes.

<figure><img src="image_13.png" /><figcaption>FIG. 9. Scatterplot of the rainfall amount correlation vs distance between couples of stations in (a) 1 February–14 April and (b) 15 April–30 June. The values are averaged for nonoverlapped sets of 100 km (the first one is 1–100 km, then 101–200, etc.). The black dots are for daily rainfall using only days receiving at least 1 mm of rainfall at both stations. The curves with symbols are for the standardized anomalies (zero mean and unit variance) from running 5–31-day averages.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>FIG. 10. Correlation between the local-scale centroids and the spatial average for each cluster. The median of the local-scale correlations is indicated in each panel.</figcaption></figure>

### d. Links with tropical SST

Figure 13 shows the SST anomalies associated with each cluster, derived by weighting each SST field for each FMAMJ season 1961–2001 by the cluster membership probabilities in Fig. 5. These SST composites are tested using the same Monte Carlo resampling procedure detailed in section 4c, except that it is applied to interannual FMAMJ SST fields. It is clear that warm ENSO events tend to coincide with cluster 2 (Fig. 13b) and that cold ENSO tend to coincide events with cluster 3 (Fig. 13c). Both of these clusters are associated with near-normal seasonal rainfall totals (Fig. 7), but with substantial changes in seasonality (Fig. 6). Thus, warm ENSO events (cluster 2) are associated with a protracted but less intense rainy season (Fig. 6b), while cold ENSO (cluster 3) events tend to be associated with shorter but more intense rainy seasons (Fig. 6c). Note that such signal would be hard to extract using only seasonal total

amounts since the seasons included in clusters 2 and 3 are close to the long-term mean (Figs. 7b,c). We checked this apparent dependence between ENSO events and rainfall seasonality with simple composites. Figure 14 shows the composite daily evolution of spatially averaged low-pass-filtered rainfall anomalies for warm versus cold ENSO years, based on the upper and lower tercile categories of the FMAMJ Niño-3.4 index (5°N–5°S, 170°–120°W). It is clear that the spatial average of rainfall anomalies observed during warm (cold) ENSO events matches well with cluster 2 (3) (Fig. 14).

Cluster 1 (dry, especially in the first half of the rainy season) tends to be associated with negative SST anomalies over much of the trade wind regions of the Northern Hemisphere (especially the northern tropical Pacific and Atlantic Oceans but also the Arabian Sea; Fig. 13a). Cluster 4 (wet, especially in the first half of the rainy season) is associated with a weak warm ENSO signal, with mostly positive SSTA in the northern
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FIG. 11. Weighted probability to observe a dry spell lasting at least 14 days counted from local-scale raw rainy sequences for each cluster. Weights are membership defined in Eqs. (B4) and (B5). Upper and black (lower and open) triangles indicate significant (at the two-sided 90% level) positive (negative) anomalies relatively to a random permutation of the membership matrix.

tropical Atlantic and weak negative SSTA in much of the Southern Hemisphere (Fig. 13d).

The subseasonal modulation revealed by the clustering (Fig. 6) could arise from many mechanisms. It could be partly related to a subseasonal modulation of the SST forcing itself. Figure 14 shows the evolution of monthly cluster-weighted SST anomalies for seven regional-scale indices chosen in important areas for teleconnection (central and eastern equatorial Pacific: 5°N–5°S, 180°–120°W; tropical western North Pacific: 5°–20°N, 140°E–180°; tropical eastern North Pacific: 10°–25°N, 160°–100°W; tropical North Atlantic: 10°–25°N, 70°–30°W; tropical South Atlantic: 20°–10°S, 15°W–15°E; eastern Indian Ocean: 10°S–10°N, 40°–70°E; and Arabian Sea: 12°–25°N, 60°–80°E), based on the above results (Fig. 13) and previous studies on the East African MAM rains (Mutai and Ward 2000; Camberlin and Philippon 2002). The month of January is included to give some insights about the potential predictability of the probability of each scenario from SST. Figure 15a shows that the ENSO forcing is rather constant throughout the

season for the contrasted clusters 2 (i.e., warm ENSO events) and 3 (i.e., cold ENSO events). By contrast, cluster 1 is mostly related to decaying cold ENSO events while cluster 4 shows a reversed behavior but SSTA in central tropical Pacific remains insignificant at a monthly time scale (Fig. 15a). For the other SST indices, it seems that each region is usually able to discriminate between one or two scenarios. While some SST indices exhibit a subseasonal modulation [e.g., tropical western North Pacific for cluster 2 (Fig. 15b) or northern tropical Atlantic for clusters 3 and 4 (Fig. 15d)], others do not.

# 5. Summary and discussion

## a. Summary

In this paper, we have presented a conceptual analysis of the seasonal predictability from a rather new perspective. Instead of considering a priori the seasonal total amounts as the desired variable to predict, we use
<figure><img src="image_19.png" /><figcaption>FIG. 12. Weighted probability to observe locally a daily rainfall >95th percentile in running 31 days counted from local-scale raw rainy sequences for each cluster. Upper and black (lower and open) triangles indicate significant (at the two-sided 90% level) positive (negative) anomalies relatively to a random permutation of the membership matrix.</figcaption></figure>

a statistical method to objectively minimize the noisier and/or transient variations in rainfall, which are not phase locked to a specific stage of the season each year, and to emphasize the most spatially coherent subseasonal signals. The underlying hypothesis is that high spatial coherence is a necessary (though not sufficient) condition for potential seasonal predictability at the station scale. The use of seasonal amounts can be detrimental when the largest rainfall amounts near the seasonal peak are not spatially coherent and may hide potentially predictable signals, notably at the beginning and end of the rainy seasons, that can be important for crops (Sivakumar 1993; Barron et al. 2003; Sultan et al. 2005; Marteau et al. 2011).

Our results demonstrate that the interannual variability of the long (FMAMJ) rains in Kenya and north Tanzania is consistent with such behavior, with the most spatially coherent signals confined to the early stage or at least the first half of the rainy season (Figs. 2, 6). The largest rainfalls in April and especially in May (Fig. 1) are not spatially coherent (Figs. 2, 3, 6) and, while they contribute the most to the seasonal amounts, they tend

to hide the interannual variability at the beginning of the season. The beginning of the season is far more spatially coherent (Figs. 2, 6) and is likely to contain the most predictable signals at seasonal time scale. The four subseasonal scenarios identified by the cluster analysis represent typical evolutions of rainfall anomalies across a network of 36 stations. The most spatially consistent rainfall anomalies are mainly concentrated in the first half of the long rains for all clusters while the second part is characterized mostly by localized anomalies and near-zero regional-scale anomalies (Figs. 2, 3, 8). There are two clusters that are anomalously dry (cluster 1) and anomalously wet (cluster 4) during approximately the first half of the rainy season, while clusters 2 and 3 characterize protracted and shorter rainy seasons, respectively. As said before, these clusters are mostly defined by what happens during the first half of the rainy season since the second part is characterized mostly by localized anomalies and when regional-scale anomalies tends to zero (Fig. 6).

The seasonal modulation of the spatial coherence and consequently the different amplitude revealed by the
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FIG. 13. Weighted standardized SST anomalies (FMAMJ) for each cluster [the weights are memberships defined in Eqs. (B4) and (B5)]. Positive anomalies are shown by the solid lines (interval = 0.1 std), negative anomalies are shown by the dashed lines (interval = 0.1 std), and the zero line is shown by the bold line. The small plus signs (dots) indicate significant positive (negative) anomalies at two-sided 90% level from a random permutation of the year's membership 1000 times.

seasonal scenarios do not seem to be related to a differ-
ing spatial scale of the daily rainy events (Fig. 9). It
could not be firmly proved with daily records only since
it is well known that the duration of individual rainy
events is far shorter than a day but the spatial scale of
daily rainfall is at least not significantly smaller after
mid-April than before (Fig. 9). However, the rainy
anomalies summed up over running 5–31-day windows
seem far less synchronized in time during the second
half of the rainy season, since the correlations drop
sharply below 0.3 beyond 200 km and using a longer
time window does not radically increase the spatial
scale of interannual anomalies (Fig. 9). On the con-
trary, the spatial scale of rainfall anomalies progress-
ively increases with averaging time during the first half
of the rainy season (Fig. 9). From the point of view of
the seasonal predictability, this increase is relevant to
the downscaling of regional-scale anomalies toward sub-
regional or even local scales.

Our adaptive approach is also able to extract a rather
clean ENSO signal (Figs. 13, 14) in the two intermediate
clusters 2 and 3, where seasonal rainfall amounts are
close to the long-term mean (Figs. 7b,c) but distinct
anomalies are found at subseasonal scales (Figs. 6b,c).
Warm ENSO events seem to be associated with a di-
luted rainy season, with above-normal rainfall near the
beginning and the ending stages of the rains and below-
normal rainfall during the peak (Figs. 6b, 14) while cold
ENSO events seem to be associated with a shortened but
more intense rainy season (Figs. 6c, 14). The SST signal
itself is rather constant throughout the season in the
equatorial central and eastern Pacific (Fig. 15) and the
seasonal modulation is perhaps not directly linked to
the forcing of the tropical Pacific. The two other clusters
reveal different SST patterns and it seems that dry (wet)
clusters (Figs. 6a,d) are mostly related to negative SST
anomalies in the northern (southern) subtropics (Figs.
13a,d). It remains to be investigated whether this could
<figure><img src="image_21.png" /><figcaption>FIG. 14. Spatial average of the running 31-day rainfall anomalies (running 31-day sums are square rooted and standardized according to the mean seasonal cycle for each rain gauge) during the 14 coldest (lower open triangle) and the 14 warmest (upper black triangle) ENSO events (defined from the SST averaged over the Niño-3.4 box in FMAMJ). The thin black lines are the 80%, 90%, and 95% confidence intervals defined from 14-yr samples randomly extracted from the set of 41 yr from 1961 to 2001.</figcaption></figure>

be associated with an anomalous shift of the ITCZ, especially during the early and middle stages of the rainy season over East Africa.

b. Discussion

Our results may be useful to teams engaged in operational seasonal rainfall forecasting over the greater Horn of Africa region. The significant SST anomalies found ahead (i.e., in January) of the rainy season for some clusters of subseasonal rainfall variability suggest a predictability potential of the membership probabilities, which needs to be further explored before any operational use. Another general message of this paper is that we need to analyze carefully the subseasonal time scales to estimate the potential predictability at seasonal-to-interannual time scale. Considering subseasonal patterns is useful in that it enables us to detect nonlinear relationships and to better extract the predictable component of seasonal rainfall variations. It is also in line with the seasonal prediction of integrative variables such as crop yields, which aggregate different components of the rainfall variations across the season, not necessarily in a linear way. In fact, several previous studies demonstrated that yield could be more predictable than seasonal rainfall amounts (e.g., Cane et al. 1994). There are many previous studies that indirectly point to the temporal modulation of predictability across the season. For example, several studies have discussed the potential predictability of the onset of the rainy season, over the Sudan–Sahel (e.g., Fontaine and

Louvet 2006; Laux et al. 2008; Marteau et al. 2009), as well as in other tropical regions (e.g., Nicholls 1984; Wu and Wang 2000; Marengo et al. 2001; Moron et al. 2009a,b). Others documented a clear intraseasonal modulation of the predictable signals related to SST forcing (e.g., Haylock and McBride 2001; Lyon et al. 2006; Moron et al. 2010). We find that such approaches focusing on a particular property of the rainy season (viz., the phase of the rainy season) are complementary to the approach developed in this paper, which does not consider a priori a specific stage of the rainy season but rather identifies the signal versus noise components in a data-adaptive way. We can imagine other statistical solutions to minimize the local-scale noise and next analyses may focus on, for example, Multichannel Singular Spectrum Analysis (M-SSA) to emphasize the signals (Moron et al. 2012).

It is important to note that we did not try to estimate the “true” number of clusters from a topological or dynamical point of view (i.e., Ghil and Robertson 2002). We do not know if, for example, the trajectories of subseasonal sequences of local-scale rainfall are concentrated in certain area of the phase space. In fact, it is clearly unfeasible to analyze the phase space of such small matrix. Our approach is more pragmatic and uses EOF and fuzzy clustering scheme to summarize complex variations, our keystone being to minimize the noisier components of the space-time variations. The fuzzy approach is especially relevant in that context: the membership probabilities (varying from 0 to 1) emphasize well-defined seasons (i.e., those with a membership close to 1) from other seasons that are less well defined (i.e., when two or more memberships tend to 1/c). These poorly defined seasons may be viewed either as being close to the overall mean behavior or exotic relatively to other seasons and thus do not reveal any systematic boundary forcing. These seasons would be mixed with well-defined seasons in “hard” k-means.

A last important issue that should be dealt with in subsequent studies is the physical links between the subseasonal scenarios and the SST forcings. It seems that the contrasted clusters 1 (dry) and 4 (wet) are mostly associated with negative SST anomalies in the northern and southern tropics, respectively. We can hypothesize that there is some link between this SST anomaly pattern and the phase of the northward ITCZ migration (abnormally late in cluster 1 and early in cluster 4), but the SST anomalies are not so strong across Indian Ocean (especially in cluster 4). It is possible that continental surfaces play a role in that behavior by enhancing/reducing sea continent thermal contrasts. The fact that spatial coherence abruptly decreases from mid-April onward is also intriguing. It is possible that the
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FIG. 15. Weighted standardized monthly SST anomalies (from January to June) for each cluster [the weights are memberships defined in Eqs. (B4) and (B5)] in (a) central and eastern equatorial Pacific: 5°N–5°S, 180°–120°W; (b) tropical western North Pacific: 5°–20°N, 140°E–180°; (c) tropical eastern North Pacific: 10°–25°N, 160°–100°W; (d) tropical North Atlantic: 10°–25°N, 70°–30°W; (e) tropical South Atlantic: 20°–10°S, 15°W–15°E; (f) eastern Indian Ocean: 10°S–10°N, 40°–70°E; and (g) Arabian Sea: 12°–25°N, 60°–80°E. The filled markers indicate significant anomalies at two-sided 90% level from a random permutation of the year's membership 1000 times.
$Z_{1-6}$ and $V_{1-6}$, respectively, in the following. This truncated set could be used to reconstruct the most co-variant information of $\mathbf{Y}$,

$$ \mathbf{Y}_{\text{rec}} = Z_{1-6} \times V'_{1-6}, \qquad (B3) $$

and it is then easy to reshape this reconstruction to get a truncated reconstruction of $X$ from Eq. (B1).

The third step is to cluster the rows of $Z_{1-6}$ according to the squared Euclidean distances in the EOF space using a fuzzy $k$-means dynamical clustering. The $k$-means method (Diday and Simon 1976) has been used for a long time in climatology (e.g., Michelangeli et al. 1995: Moron and Plaut 2003; Fauchereau et al. 2009). Given a number $k$ of clusters, the algorithm will find a partition of the data points into $k$ clusters so that the squared Euclidean distances within each cluster are minimized. Note that $k$ indicates now the index of the clusters rather than the space component of $X$. The algorithm is iterative and starts from random seeds to define the clusters centroids since it is usually impossible to examine the full distribution of partition of $N$ observations into $k$ clusters. The algorithm stops when the sum of the within variance converged toward a local minimum: that is, the next iteration does not decrease the total within variance summed over $k$ clusters. The maximum number of iterations is fixed to 1000 in our case.

Classical $k$-means is an example of a “hard” or “crisp” partition: that is, each observation belongs to only one cluster. In other words, the membership of each observation equals 1 for one cluster and 0 for the other ones. This approach could lead to misleading results when the clusters are not so well defined. This is especially critical for small matrices as here. Moreover, we are here looking at typical sequences of rainfall variations and it is hard to think that few well-defined sequences would be properly defined using short records. We thus used a more flexible approach (Zadeh 1965): that is, the fuzzy $k$-means, also called fuzzy $c$-means (Bezdek 1974, 1981). Whereas the $k$-means algorithm computes the average of the vectors in a cluster as the center, fuzzy $k$-means find the center as a weighted average of all observations, using the membership probabilities for each point as weights. For a given number of clusters $c$ and the fuzzifier $m$, the fuzzy $k$-means minimize the objective cost function $J(c, m)$ as

$$ J(c, m) = \sum_{k=1}^{c} \sum_{i=1}^{N} (\mathbf{u}_{ki})^{m} (Z_{i} - C_{k})^{2}, \qquad (B4) $$

where we used Euclidean metrics for the distances between centroids $C_k$ and years $Z_i$, which are in our case

$Z_{1-6}$. Here, $\mathbf{u}_{ki}$ denotes the membership value of FMAMJ season $i$ to the cluster $k$, satisfying the following criteria:

$$ \sum_{k=1}^{c} \mathbf{u}_{ki} = 1, \quad \text{where} \quad 0 \le \mathbf{u}_{ki} \le 1. \qquad (B5) $$

The centroid (i.e., the center of a cluster) corresponds to the center of all objects of the dataset, with each contributing with its own membership value. The centroids could be projected from the EOF space onto the “physical” space of $\mathbf{Y}$ by

$$ C_{Y} = C_{k} \times V'_{1-6}. \qquad (B6) $$

There are myriad internal indices, based on the centroids and/or the membership's matrix, to determine an optimal value of either the fuzzifier $m$ or the number of clusters $c$. Usually, the largest number of clusters should be less than the square root of $N$ (i.e., 6 here). Here, $m=1$ leads to a hard partition while increasing $m$ toward infinity increases the fuzziness. For a given value depending on the size and the structure of the matrix, all membership tends to $1/c$ and the partition is then useless. Here, this values is variable and depends on the size and intrinsic covariance of the matrix (Schwämmle and Jensen 2010) and occurs here near $m \sim 2.3$. In consequence, we need to test clusterings based on $m$ between ~1.1 and ~2.2 (using, e.g., a step of 0.1) and $c$ between 2 and 6: that is, $12 \times 5 = 60$ solutions. The published validity indices (Pakhira et al. 2004; Yang and Wang 2008) are usually tested on real and artificial datasets, which contain usually well-defined clusters. A lot of these indices have been recently reviewed by Yang and Wang (2008), who concluded that these validity indices do not usually converge to a single choice.

Another approach of the choice of $c$ and $m$ has been proposed by McBratney and Moore (1985) and De Bruin and Stein (1998), among others. Here, the goal is to estimate an ad hoc value based on explanatory hypotheses that underline the clustering. In our case, we choose to estimate the extent of significant sea surface temperature anomalies (SSTA) associated with the clusters, as explained in the main text. The significance is estimated through a random permutation of the sequence of the yearly memberships 10 000 times. The weighted SSTA in the reshuffled set are compared with those in the observed one and their rank in the sorted sequence of random set gives the one-sided significance level associated with the SSTA obtained from any observed membership being different from noise.
## b. Sensitivity analyses

Our approach involves three user-defined parameters: the length of time window to smooth out the daily rainfall, the value of the fuzzifier *m*, and the number of clusters *c*. We used an ad hoc approach to define the latter two parameters (Fig. 4). A posteriori, it is rather easy to check out the sensitivity of the results to the value of the fuzzifier. We consider the four-clusters solution with *m* varying from 1.1 to 2.2 (as in Fig. 4) to simply compute the cross-correlation between the membership **u**<sub>ki</sub> of the solution considered in our analyses (i.e., *m* = 1.5) and the other 11 solutions. The averaged correlations across the four clusters is always > 0.81 (>0.93 for *m* between 1.2 and 1.8), except with *m* = 1.1 (*r* = 0.76) and *m* = 2.2 (*r* = 0.56), where the clustering tends to be useless (no membership > 0.5). So a reasonable choice of *m* (i.e., below the limit when **u**<sub>ki</sub> tends to 1/c) has a weak impact on the final results. A related topic is the difference between fuzzy and hard k-means. An hard k-means with *c* = 4 leads to a similar clusters as those displayed on Fig. 6 (not shown), at least for the seasons having one membership >0.5 (Fig. 5), but the seasons that are less clearly associated with a single cluster (e.g., 1961–64) are, by definition, grouped into a single cluster in hard k-means: for example, 1961–63 falls within the equivalent of cluster 2 and 1964 falls within the equivalent of cluster 3. The issue is not the allocation to one or another cluster but the fact that these not well-defined subseasonal scenarios (otherwise they will get a higher membership in a given cluster) are as important as well-defined ones in a given cluster. One consequence is that the warm ENSO signal associated with cluster 2 is strongly weakened in the hard k-means (not shown). The fuzzy approach is clearly more flexible than a hard approach and is especially relevant for a relatively short record.

The issue of the sensitivity to the number of clusters is not as simple to solve since the final “quality” of the clusters depends on what we want to diagnose, beyond the fact that a minimum of two clusters and a maximum of $\sqrt{N}$ (=6 in our case) could adequately limit the choice. It is quite clear that the mean temporal behavior revealed by each cluster (Fig. 6) is somehow constrained by the method. Considering, for example, only the leading PC will naturally lead to two symmetrical clusters whose amplitude will peak where the spatial covariance is at its maximum. For example, we can consider that the four-cluster solution is valuable since it is able to extract a significant ENSO forcing. It is probably possible to define more objectively the number of clusters based on a skill score: for example, the retrospective hindcast of the membership **u**<sub>ki</sub> from the anterior SST. This issue will

be considered in a forthcoming work, and it is perhaps safer to examine all solutions between 2 and $\sqrt{N}$.

The issue of the degree of smoothing depends ultimately on which time scales should be predictable at interannual time scales and/or which time scales could be phase locked with the seasonal cycle. To check the sensitivity of the results due to the temporal smoothing, we perform a fuzzy k-means considering running mean of 11, 21, and 41 days with *m* = 1.5 and *c* = 4 clusters. The mean of the correlations (across the four clusters) between **u**<sub>ki</sub> of the same clusters and the 31-day solution considered in the paper equal 0.77 (11 days), 0.92 (21 days), and 0.96 (41 days). So, as for *m*, it seems that the degree of smoothing does not critically modify the final results as soon as a sufficient degree of smoothing is reached.

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