# Multi-temporal drought rarity curves-A yearly classification of meteorological drought severity in France

Juliette Blanchet, Baptiste Ainési, Sandra Rome, Jean-dominique Creutin

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## To cite this version:

Juliette Blanchet, Baptiste Ainési, Sandra Rome, Jean-dominique Creutin. Multi-temporal drought rarity curves-A yearly classification of meteorological drought severity in France. Advances in Water Resources, 2024, 193, pp.104829. 10.1016/j.advwatres.2024.104829. hal-04785095

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Submitted on 14 Feb 2025

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1 Graphical Abstract

2 Multi-temporal drought rarity curves - a yearly classification of meteorological drought severity in France

3

4 Juliette Blanchet, Baptiste Ainési, Sandra Rome, Jean-Dominique Creutin

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5 Highlights

6 Multi-temporal drought rarity curves - a yearly classification of meteorological drought severity in France

8 Juliette Blanchet, Baptiste Ainési, Sandra Rome, Jean-Dominique Creutin

9 • Rarity curves to analyze how drought events build and persist across time.

10 • Applied to meteorological drought in France over 1950-2022.

11 • The shape of rarity curves discriminate short- and long-term historical droughts.

12 • The methodology is generic and could be applied to other drought variables.
Multi-temporal drought rarity curves - a yearly classification
of meteorological drought severity in France

15 Juliette Blanchet<sup>a,*</sup>, Baptiste Ainési<sup>a,b</sup>, Sandra Rome<sup>a</sup>, Jean-Dominique Creutin<sup>a</sup>

<sup>a</sup>Univ. Grenoble Alpes, CNRS, G-INP, IRD, IGE, Grenoble, F-38000, France

<sup>b</sup>Geosciences Department, Ecole Normale Superieure - PSL University, Paris, France

16 Abstract

17 Droughts are recurrent phenomena that present a large variety of space and time
18 patterns making rather difficult the assessment of their rarity and the comparison
19 between events.

20 Our study focuses on the “memory effect” of meteorological drought over France
21 using the daily raingauge network maintained by Météo-France over 1950-2022. The
22 proposed easy tool of rarity curves analyzes how drought events build and persist
23 across time. The approach is purely statistic, assuming that drought consequences
24 depend on the probability of non exceedance (“rarity”) of antecedent rainfall accu-
25 mulations. In order to cover a large spectrum of “memory effects”, we consider a
26 continuum of accumulation periods ranging from a few weeks to several years. The
27 rarity curve of a given year displays the most severe rarity values encountered during
28 the year as a function of the various accumulation periods (256 values ranging from
29 4 to 260 weeks in our case).

30 Over the study period of 1950-2022 we show how the shape of rarity curves dis-
31 criminate short- and long-term historical droughts. A k-means algorithm proves to
32 be useful to classify the different years into typical drought situations from contin-

<footnote>*Corresponding author</footnote>

<footnote>Preprint submitted to *Advances in Water Resources*</footnote>

<footnote>July 19, 2024</footnote>

<footnote>Email address: juliette.blanchet@univ-grenoble-alpes.fr (Juliette Blanchet)</footnote>
33

34

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uous drying to continuous wetting over the antecedent five years. Looking at the
chronology of the found clusters also allows studying how droughts build and persist
across time.

Keywords: Meteorological drought, rarity, classification, France

37

# 1. Introduction

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Several severe drought periods impacted Europe since the beginning of the 20th Century (Stahl et al., 2016). Since the end of the 1980s, and more markedly since the turn of the century, a variety of successive drought events triggered an almost continuous series of impacts (Spinoni et al., 2015) that is quite distinct from the more individualized impacts of the droughts of 1947 over Central Europe (Brázdil et al., 2016) or 1976 over Western Europe for instance (Zaidman et al., 2002).

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Even if some years are specially studied - e.g. the droughts of 2003 (Fink et al., 2004) or 2015 (Fink et al., 2004) -, various studies attracted attention on “multi-year droughts” such as for instance in 2014-2018 (Moravec et al., 2021) and 2018-2020 (Rakovec et al., 2022). The multi-year approach recognizes that even if eco-and socio-systems can recover from elementary single-year droughts, they are badly impacted if the relief time between droughts is too short compared to their “recovery time” (Schwalm et al., 2017).

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Droughts are basically described in reference to the main compartments of the water cycle (see definitions in Box 3.3 of Chapter 3 of the SREX IPPC Report IPCC, 2012): precipitation (“meteorological drought”), soils and vegetation (“agricultural drought”), water tables and rivers (“hydrological drought”). Each compartment combines processes with “characteristic times and sizes” ranging over several orders
of magnitude as classically presented in Stommel diagrams (Stommel, 1963).

The agricultural drought is governed by vertical water transfers at the local scale of soil patches and vegetation and has characteristic response times to rain ranging from hours to days (soil moisture) and from weeks to years (grass, crops, trees, see Figure 2 of Hunt et al., 2021, for instance). The hydrological drought depends on the slow response time of watersheds to rain ranging from days to months (horizontal channel flow from upstream basins to main rivers) and from month to decades (accompanying horizontal water transfers in aquifers, see Figure 2 of Blöschl and Sivapalan, 1995).

The meteorological drought is to a large extent the primary cause of the above two types of droughts in the sense that it concerns the upstream atmospheric compartment. Its effect is only modulated by potential evapotranspiration that depends on radiation, wind and air humidity. There is a kind of “resonance” between precipitation deficits over a range of accumulation periods (say from days to decades) and the response of downstream surface processes. Depending on the duration of precipitation shortage different types of drought will occur because soils dry up faster than aquifers, just as streams dry up faster than large rivers, or grasslands suffer before forests. Over this range of possible shortage durations, rainfall variability is governed by the complete cascade of atmospheric waves (Orlanski, 1975).

Compared to other climatic hazards such as flooding or wildfires, droughts respond to persistant anomalies of the atmospheric circulation (see Figure 3 of Leonard et al., 2014). In North-western Europe for instance a favorable weather type must persist over several weeks to trigger a hydrological drought and this duration varies
substantially (6 to 30 weeks) depending on the basin storage characteristics (Fleig 79 et al., 2011). At mid-latitudes such as in Europe, this persistence results from stationary Rossby waves and associated anti-cyclonic anomalies over several months 80 that are governed by global scale atmospheric and oceanic conditions (Fink et al., 81 2004; Bakke et al., 2020). A central point in drought studies is thus long “memory” 82 effects in all the compartments, starting with lasting rainfall shortage and heat waves 83 and then impacting soils, vegetation, aquifers and rivers. 84

The various indices assessing drought severity and space-time characteristics that 86 are presented in a vast body of literature take this ”memory” effect into account 87 (Mishra and Singh, 2010). Some studies consider cumulative shortage over a fixed 88 period - for example, a year or a season. The choice of a fixed period may be limiting 89 since by construction it does not take into account the variety of characteristic times 90 seen above. It may also miss local minima when a shortage is moderate over a year 91 and yet has been very intense over several months. 92

Other studies consider the cumulative shortage over a variety of moving an-
tecedent period lengths. For example, Tijdeman et al. (2022) consider periods of
1, 3 and 12 months at a monthly time step - moving month after month they deter-
mine a new cumulative shortage based on the previous k months, k = 1, 3, 12. They
visually identify three distinct types of drought in southwestern Germany: intense
multi-seasonal drought episodes such as in 2015, long-term and less intense drought
episodes such as in 1990-1993, short-term episodes such as in 2011. These years
are somehow the "champions" of the corresponding drought types. Earlier, McKee
et al. (1993) used a comparable approach considering a larger number of moving
antecedent period lengths (3, 6, 12, 24 and 48) aiming at covering five types of what they call usable water sources: namely the soil moisture, ground water, snowpack, streamflow and reservoir storage. In both studies, the moving accumulation avoids the local minima issues but the choice of the lengths is arbitrary in front of the great and continuous variability of characteristic times inside and among the different water sources or compartments - e.g. the huge variability of permeability in soils and aquifers. Vidal et al. (2010a) consider a larger number of 24 consecutive period lengths from 1 to 24 months aggregated at the monthly time step, giving each year a total of 288 series of drought indices. They show in particular that the 1976 drought was in 2010 the prevalent benchmark for meteorological drought in France for period lengths from 3 to 15 months, while the 1989-1990 meteorological drought took over for longer periods.

The aim of this paper is to analyze how drought builds and persists across time over a given space and to classify year after year the state of drought over a deep memory spectrum. We follow a comparable approach used to qualify storm severity in flood studies Ramos et al. (2005). We applied our precipitation shortage analysis to metropolitan France during the years 1950-2022. This area of roughly 55,0000 km² receives the annual volume of precipitation of about 450 km³ of which more than 60% returns to the atmosphere by evapotranspiration, leaving about 165 km³ for runoff and aquifers. During severe droughts this volume can be divided by a factor 2 (Duband, 2010).

The proposed easy tool of rarity curves is purely statistic, assuming that drought consequences depend on its probability of non exceedance at all scales. We work at
a weekly time-step on antecedent rainfall accumulations over time periods ranging
from 1 month to 5 years. Each year, the rarity curve analyses drought conditions as
a continuum without assumptions about drought event definitions and about “typ-
ical” temporal scales. A classification of the curves allows to compare the drought
severity of the 70 years under study. In this first application we chose to deal with
precipitation shortage over a fixed area, but the methodology can be extended to a
range of area sizes and applied to other dryness variables such as soil moisture for
instance.

## 2. Data and method

### 2.1. *Matrix of antecedent precipitation*

We consider the daily raingauge network over the Metropolitan France maintained
by Météo-France over 1950-2022, keeping for consistency the 330 stations with less
than 1% of missing data over the period (Figure 1). The daily values at day *d*
correspond to the precipitation accumulated from day *d* at 06:00 UTC to day *d* + 1
at 06:00 UTC. We compute the daily averages of the station values from the beginning
of 1950 to the end of 2022. It gives a unique 73-year long series of daily data roughly
representing daily rainfall accumulated over the whole of France. The years 2021
and 2022 of the obtained daily times series is shown in Figure 2 for illustration.

In order to study the rareness of precipitation shortage across different accumu-
lation periods, we sum, day by day, the antecedent rainfall over periods of 4 to 260
weeks (about 5 years). Considering this set of durations allows a multi-temporal
approach, i.e. accounting for the different characteristic times of droughts as a con-
<figure><img src="image_4.png" /><figcaption>Figure 1: Location of the 330 daily raingauges in Metropolitan France used to compute daily rainfall accumulated over the whole of France over 1950-2022.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 2: Years 2021 and 2022 of the daily time series of precipitation accumulation in France. The three red horizontal lines correspond to periods of 4, 25 and 52 weeks showing the lowest NEPs at these scales in 2022. By construction, the corresponding NEP values are reported in the rarity curve of 2022.</figcaption></figure>
tinuum. For example, the antecedent rainfall accumulation on the 1st of July 2022 at the 52-week accumulation period is the precipitation accumulated from 3rd of July 2021 to 1st of July 2022. A given day is then associated to 257 values; each value represents the rainfall accumulated over the preceding $j$ weeks, $j = 1, \dots, 260$. Said otherwise, we obtain a matrix of antecedent precipitation accumulation of 260 rows (one for each accumulation period) and 26547 columns (one for each day from the beginning of 1950 to the end of 2022). The $(i, j)$-element of the matrix is given by

$$M_{ij} = \sum_{k=0}^{7j-1} R_{i-k},$$

where $R_i$ is the daily average rainfall (in mm) at day $i$, for $i = 1, \dots, 26547$ and $j = 1, \dots, 260$ is the accumulation period (in weeks). Note that the antecedent precipitation accumulation $M_{ij}$ can only be computed when $i > 7j - 1$; otherwise it is set to NA (missing value) - this case only happens during the first five years of the record. Note also that considering a weekly accumulation period with daily steps instead of monthly accumulation periods with monthly steps as usually done allows for a better estimation of the extremes of precipitation deficit, particularly for the short periods (say, subannual).

Figure 3 shows matrix of antecedent precipitation accumulation, zooming over the 52 first accumulation periods (rows) to ease visualization. It shows some variability among years, e.g. the dark red and orange peaks in year 1976 correspond to weak antecedent precipitation accumulation up to about 40 weeks. However the rarity of antecedent precipitation for different accumulation periods is difficult to assess due
<figure><img src="image_6.png" /><figcaption>Figure 3: Matrix of antecedent precipitation (mm) for accumulation periods of 4 to 52 weeks over 1950-2022. The gray lines show every 1st of January (in bold gray every 5 years). The white triangle spanning the first five years correspond to missing values when antecedent precipitation cannot be computed.</figcaption></figure>

## 2.2. Rareness of precipitation shortage: non-exceedance probability

One way of assessing the rareness of the precipitation shortage is to transform the matrix of antecedent precipitation accumulation into non-exceedance probabilities (NEPs) accounting for seasonality (calendar month). For example, a NEP of 1% on the 1st of July 2022 for the 52-week accumulation period means that the precipitation accumulated from 3rd of June 2021 to 1st of July 2022 is among the 1% smallest values over all 52-week antecedent precipitation accumulation ending in July.

A critical step in the computation of NEP is to choose a well-adapted distribu-
tion for each accumulation period and each calendar month in order to accurately
estimate extremely low NEPs. Given the central limit theorem, antecedent precipi-
tation over long accumulation periods (say, over one year) is expected to be normally
distributed. However this does not apply to antecedent precipitation over short pe-
riods that are likely asymmetric. Guttman (1999) claims that Pearson Type III
distribution is the “best” universal model. Vicente-Serrano (2006) confirms it is well
adapted for accumulation periods of 1 to 36 months in the Iberian Peninsula. How-
ever Lloyd-Hughes and Saunders (2002) find the Gamma distribution suitable for
monthly periods for the larger part of Europe. Soľáková et al. (2014) select, for each
month of the year and each accumulation period (1, 3, 12 months) the best cumu-
lative distribution function (CDF) among nine candidates: Burr, Erlang, Gamma,
generalized Gamma, inverse Gaussian, log-normal, Pearson, Weibull, and general-
ized extreme value distributions. The results show that different distributions are
selected depending on the month and the accumulation period. In order to avoid
distribution choice, Vidal et al. (2010b) apply a flexible nonparametric approach by
fitting the empirical density of each accumulation period and month with a kernel
density estimate. We follow a similar approach in this article but in a more parsimo-
nious way: the CDFs of the antecedent precipitation are considered nonparametric,
one for each accumulation period, with monthly-dependent means. For this, we first
deseasonalize antecedent precipitation by subtracting the mean value of the current
month: for a given day i in month m (January-December) and a given accumulation
period j, we subtract from M<sub>ij</sub> the mean of all M<sub>kj</sub> for days k in month m. We
then estimate the CDF associated to each accumulation period (regardless of the
188 month) using a Gaussian kernel based on the 26547 deseasonalized values. The NEP
189 associated to $M_{ij}$ is then this CDF applied to the deseasonalized $M_{ij}$.

190 It is worth stressing that considering NEPs is equivalent – up to a bijective trans-
191 formation – to considering standardized precipitation indices (SPIs) that are stan-
192 dardly used in drought studies. Thus rather than NEPs, another possibility would
193 have been to compute SPIs by transforming the fitted distributions into standard
194 Gaussian distributions as $SPI = \Phi^{-1}(NEP)$, where $\Phi^{-1}$ is the quantile function
195 of the standard Gaussian distribution (Soľáková et al., 2014). All the methodology
196 introduced in this article could be used likewise to SPI values; however we find the
197 NEPs more directly interpretable in terms of rareness since they explicitly give prob-
198 abilities. For example, a NEP of 1% for a given accumulation period means that only
199 1% of the precipitation accumulation where dryer that the considered precipitation
200 shortage; saying that SPI is -2.33 is equivalent but less directly interpretable. For
201 this reason, all this article will be based on NEP values but the corresponding SPI
202 values will be indicated in the plots for the readers familiar with this index.

## 2.3. Multiscale rarity curve

203 A given column of the NEP matrix may be seen as a curve representing, for
204 the corresponding day, the NEP of antecedent precipitation accumulation over the
205 range of considered accumulation periods. The multiscale rarity curve of a given
206 year is then defined as the lower envelope of the 365 daily NEPs values of the year.
207 It represents the maximum rarity of antecedent precipitation during that year over
208 the continuum of accumulation periods. For example, a value of 0.05 for a given
209 accumulation period means that on a given day of the year (that the rarity curve
210
does not show), the antecedent precipitation was among the 5% lowest value among 211 212 1950-2022. Figure 2 shows for illustration periods of 4, 25 and 52 weeks showing 213 the lowest NEPs at these scales in 2022. By construction, the corresponding NEP 214 values are reported in the rarity curve of 2022. In the next section, we analyze the 215 rarity curves of the different years and use them to produce a multiscale typology of 216 droughts in France. 217

## 2.4. Multiscale typology of drought rarity: classification of rarity curves

In order to summarize the main multiscale features shown by the different years, 218 an objective classification of drought rarity curves into a reasonable number of groups 219 is performed. At this step, the five first years that have missing values are excluded, 220 thus starting in 1955. We use a k-means algorithm with the Euclidean distance. The 221 algorithm requires to set the number of clusters beforehand, that we note *k*. Numer- 222 ous indices have been developed to measure the quality of a classification and help 223 in the choice of *k*. Three frequently used and well performing indices are the Davies- 224 Bouldin, Calinski-Harabasz and the silhouette indices (Arbelaitz et al., 2013). They 225 respectively select *k* =7, 4 and 4 clusters. However, inspection of the corresponding 226 clusters revealed large within-class variability, impeding discriminating between the 227 most severe curves and the mild ones (see Figures A.11 and A.12). For this reason, 228 we developed our own criteria: given that our main interest is in the most severe 229 drought curves, we consider for a given *k* the three most severe clusters (i.e. the three 230 centroids with lowest average curves over accumulation periods). We then determine 231 the lowest *k* such that increasing *k* does not modify these three centroids. The idea 232 behind this is that if these clusters were too mixed, increasing *k* would lead to divid- 233
ing them – or respectively, if they are stable when increasing the number of clusters, we can reasonably assume they are coherent. We apply this simple procedure for $k$ ranging from 4 to 16, which leads us to select $k = 12$ (see Section 3.2). This classification defines a multiscale typology of drought rarity in France. This differs from the typology defined in Tijdeman et al. (2022) on essentially three main points. First, we apply an automatic classification whereas the identification in Tijdeman et al. (2022) seems to be manual. Second and more importantly, our typology is based on the overall rarity curve over a continuum of accumulation periods whereas Tijdeman et al. (2022) uses three periods (1, 3, 12 months) with monthly (versus daily) steps. Finally, we classify here years (from January 1st to December 31st) whereas Tijdeman et al. (2022) is able to identify flexible periods of droughts - we will be back to this point in the discussion section.

## 3. Results

### 3.1. Rarity curve

Figure 4 shows the daily NEP curves of year 2022 (in gray) and its lower envelope (in red) for illustration. This envelope that we call rarity curve shows small values up to long accumulation periods - NEPs are below 0.1 for 4 to 100-week accumulation periods. We see here the benefit of the rarity curve showing "visually" over which antecedent period lengths we had precipitation accumulation shortage. More precisely, let consider the "mild", "moderate", "severe" and "extreme" drought categories defined by McKee et al. (1993) and associate them the corresponding NEP values (see Table 1). We read on the rarity curve of Figure 4 that year 2022 shows at least one
<figure><img src="image_7.png" /><figcaption>Figure 4: Red: rarity curve of year 2022, defined as the lower envelope of the 365 daily NEP curves of each year (one curve for each day, in gray). The four dotted lines show the four upper levels associated to the drought category of McKee et al. (1993): mild (blue), moderate (green), severe (red) and extreme (black) drought categories.</figcaption></figure>

<table><caption>Table 1: Drought categories according to McKee et al. (1993) and corresponding SPI and NEP values.</caption><thead><tr><th>SPI values</th><th>Drought categories</th><th>NEP values</th></tr></thead><tbody><tr><td>&gt; 0</td><td>Wet</td><td>0.5 to 1</td></tr><tr><td>0 to -1</td><td>Mild drought</td><td>0.16 to 0.5</td></tr><tr><td>-1 to -1.5</td><td>Moderate drought</td><td>0.07 to 0.16</td></tr><tr><td>-1.5 to -2</td><td>Severe drought</td><td>0.02 to 0.07</td></tr><tr><td>&lt; -2</td><td>Extreme drought</td><td>&lt; 0.02</td></tr></tbody></table>

256 day with moderate drought up to the antecedent 160 weeks, severe drought up to the antecedent 96 weeks and extreme drought up to the antecedent 73 weeks. More generally, the rarity curve being always below the mild drought upper level, year 257 258 259 2022 saw for all accumulation periods at least one day with precipitation shortage.

260 Figure 5 generalizes this by showing the rarity curve of each year of the study period. There is obviously a lot of variability between the years but a common feature 261 262 is that all the rarity curves concentrate around 0 for short accumulation periods.
This is also visible in the summarized rarity curves in the left panel of Figure 6.
Mathematically, this is because each value of the curve corresponds to the minimum
out of 365 daily NEPs. The less dependent the daily NEPs for a given accumulation
period -and thus the less dependent the daily precipitation accumulations, the more
the minimum concentrates around 0 (in case of independence and given that daily
NEPs for a given accumulation period are by construction uniformly distributed
over (0,1), the CDF of the minimum would be $1 - (1-x)^{365}$ for $0 \le x \le 1$ which
has a mean of $1/366 \approx 0.003$). As a result, for our data, the rarity curves for
monthly (4-5 weeks) accumulation periods are almost all below 0.15, i.e. there is
almost always at least one month per year showing at least moderate drought; see
the red, green and blue curves in the right panel of Figure 6. The extreme drought
category occurs in 70% of the years for monthly accumulation periods and in 10%
of years for 5-year accumulation periods (260 weeks). The small values shown by
the rarity curve of year 2022 with severe or extreme drought up to 100 weeks is
not common but not extraordinary either - this applies to about 20% of the years.
The fact that "too many" droughts are shown by the rarity curve should not be
seen as an artefact of the methodology. On the contrary, it is a signature of the
strength of dependence of precipitation accumulation and hence of the climatology
of precipitation. For example for our data, 4 years out of 73 show mild droughts at
10-week scale, whereas shuffling the data in time gives on average 10 years out of 73
in this category: although daily accumulations are weakly dependent, precipitation
shortage shows some persistence.
<figure><img src="image_8.png" /><figcaption>Figure 5: Rarity curves of all years between 1950 and 2022.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6: a) Mean and 90% envelope of the rarity curves over 1950-2022. The colored ranges show the drought categories of McKee et al. (1993). b) Percentage of rarity values belonging to each drought category for each accumulation period. The sum of the five curves gives 100% for each accumulation period.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 7: Three most severe drought rarity centroids obtained by K-mean classification with 4 to 16 clusters. The four dotted lines show the four levels associated to the drought category of McKee et al. (1993). The classifications leading to the same centroids are indicated in the legend as $k_1 - k_2$ (e.g. Cluster 1 is the same whether we consider 4 to 6 clusters).</figcaption></figure>

## 3.2. Multiscale typology of drought rarity

The procedure to select the number of clusters in the typology is illustrated in Figure 7. It shows that in the most severe cluster (Cluster 1), the centroid curve decreases from moderate drought for 4 clusters to severe drought for 12 clusters and it stabilizes then. In the second most severe cluster (Cluster 2) the rarity curve also decreases from 4 to 12 clusters when it stabilizes. In Cluster 3 the same applies but 16 cluster gives slightly more severe centroids. However the classification into 16 clusters gives several very small clusters (3 clusters comprise 1 to 3 years) so it is probably more sensitive to sampling. As a tradeoff between having balanced and stable clusters and reasonable within-cluster variability, we select the typology with 12 clusters.

The obtained typology with 12 clusters is shown in Figure 8 - for ease of interpretation and without loss of generality, the clusters are ordered as in Figure 7 with respect to the area under the curve of the centroid of the class (i.e. the mean curve
<figure><img src="image_11.png" /><figcaption>Figure 8: K-mean classification of drought rarity curves into 12 clusters. The four dotted lines show the four levels associated to the drought category of McKee et al. (1993). The bold black line shows the centroid of the class. The gray lines show the rarity curves of the years in the class. In parenthesis is indicated the % of years classified in each cluster.</figcaption></figure>

of Cluster 1 is the lowest one on average). This elementary ordering looks accept-
able. It ranks the years depending on the length of the antecedent shortage. To
help understanding the physical meaning of the clusters, Figure 9 shows the rainfall
anomaly corresponding to each of them.

Cluster 1 is associated to severe to extreme drought over the antecedent 4 to
260 weeks with negative anomalies at all temporal scales. We find in this cluster
<figure><img src="image_12.png" /><figcaption>Figure 9: Rainfall anomaly associated to the 12 clusters. The bold black line shows the centroid of the class. The gray lines show the anomalies of the years in the class.</figcaption></figure>
<table><caption>Table 2: Cluster names associated to the 12 clusters.</caption><thead><tr><td>Cluster number</td><td>Cluster name</td></tr></thead><tbody><tr><td>1</td><td>Dry cluster</td></tr><tr><td>2,4,6,9,11</td><td>Drying cluster</td></tr><tr><td>3</td><td>Recovering cluster</td></tr><tr><td>5</td><td>Balanced cluster</td></tr><tr><td>7</td><td>Wet-to-balanced cluster</td></tr><tr><td>10,12</td><td>Wet cluster</td></tr><tr><td>8</td><td>Wetting cluster</td></tr></tbody></table>

most of the severe years mentioned above (namely 1955 to 1958, 1965, 1973-1974, 1976, 1990-1992 and 2006, see Figure 10) - we can call it the "dry" cluster (Table 2). Clusters 4/2/6,9,11 are respectively associated to moderate to extreme drought with decreasing anomalies up to the antecedent 100-150/50-100/20-50 weeks - we can call them the "drying" clusters. Cluster 3 is associated to moderate to extreme droughts over long accumulation periods of 150-260 weeks and mild drought with small anomalies for shorter accumulation periods - we can call it the "recovering" cluster. The other clusters correspond to drought over short accumulation periods only, which is actually very frequent for yearly periods (see Figure 6). Cluster 5 is a "balanced" cluster with small anomalies over all scales - its centroid is very close to the mean rarity curve. Cluster 7 is a "wet-to-balanced" cluster - close to the average curve up to 200 weeks of accumulation and wet after that. Clusters 10 and 12 are "wet" clusters - their centroids are constantly above the mean rarity curve and their anomalies are almost always positive. Finally Cluster 8 can be qualified as "wetting" - going from moderately dry (slightly negative anomalies) to moderately wet (slightly positive anomalies) over the last year (0-50 weeks).
<figure><img src="image_13.png" /><figcaption>Figure 10: Cluster chronology over the study period. The colors code the cluster names (Table 2): black for the dry cluster (1), red for the drying clusters (2,4,6,9,11), green for the recovering cluster (3), blue for the balanced cluster (5), cyan for the wet-to-balanced cluster (7), orange for the wet cluster (10,12), gray for the wetting cluster (8).</figcaption></figure>
321 **4. Discussion**

322 A couple of years stand out from Figure 5 because they show severe drought for
323 short to very long accumulation periods (up to 260 weeks) : 1956, 1957, 1973, 1976,
324 1991, 1992, 2006. We find here the most famous European droughts. The 1955-1957
325 drought is referenced as the longest drought in Northern Europe in Spinoni et al.
326 (2015); Figure 5 shows that it also strongly affected France. The 1975-1976 drought is
327 referenced in Spinoni et al. (2015) as one of the major droughts over Central Europe
328 and it has been much documented (Ratcliffe, 1978; Rodda and Marsh, 2011; Parry
329 et al., 2012). The 1989-1991 drought impacted mainly Central and Eastern Europe;
330 it is referenced as the most severe drought in France-Benelux since 1950 (Spinoni
331 et al., 2015). The 2006 drought does not seem to have been much documented from
332 a meteorological perspective. However it impacted crop yield comparably to the 1976
333 drought (Debaeke and Bertrand, 2008). As expected, all the aforementioned years
334 are classified in the dry cluster (Cluster 1) of long-lasting severe drought, validating
335 the ability of our approach to automatically detect the most severe long-lasting
336 droughts. But maybe more interestingly, looking at the chronology of the clustering
337 in Figure 10 allows understanding how these droughts built through years, which a
338 usual drought analysis based on individual indexes (such as SPI) cannot allow.

339 The classification of Figure 10 of years 1973 and 1974 in the dry cluster (Cluster
340 1), year 1975 in the recovering cluster (Cluster 3) and year 1976 back in the dry
341 cluster describes the temporal dynamics of the multi-year drought in the mid-1970s
342 in France with dry years from 1972 to 1976 only interspersed by the year 1975 that
343 was normally wet, in accordance with Brochet (1977). These years show the records
<table><caption>Table 3: Rarity records across scales and associated clusters.</caption><thead><tr><th>Accum. period</th><th>Year</th><th>Cluster number</th><th>Cluster name</th></tr></thead><tbody><tr><td>4 to 25 weeks</td><td>1978</td><td>6</td><td>Drying cluster</td></tr><tr><td>26 to 51 weeks</td><td>1976</td><td>1</td><td>Dry cluster</td></tr><tr><td>52 to 56 weeks</td><td>1989</td><td>2</td><td>Drying cluster</td></tr><tr><td>57 to 58 weeks</td><td>1990</td><td>1</td><td>Dry cluster</td></tr><tr><td>59 to 62 weeks</td><td>1989</td><td>2</td><td>Drying cluster</td></tr><tr><td>63 to 121 weeks</td><td>1990</td><td>1</td><td>Dry cluster</td></tr><tr><td>122 to 144 weeks</td><td>1991</td><td>1</td><td>Dry cluster</td></tr><tr><td>145 to 161 weeks</td><td>1992</td><td>1</td><td>Dry cluster</td></tr><tr><td>162 to 167 weeks</td><td>1991</td><td>1</td><td>Dry cluster</td></tr><tr><td>168 to 224 weeks</td><td>1992</td><td>1</td><td>Dry cluster</td></tr><tr><td>225 to 257 weeks</td><td>1993</td><td>3</td><td>Recovering cluster</td></tr><tr><td>258 to 260 weeks</td><td>1976</td><td>1</td><td>Dry cluster</td></tr></tbody></table>

of low rarity values for both the shorter accumulation periods (from 4 to 51 weeks for years 1976 and 1978, see Table 3) and the longest ones (from 258 to 260 weeks for year 1976).

The classification of year 1989 in the drying cluster (Cluster 2), years 1990 to 1992 in the dry cluster (Cluster 1) followed by year 1993 in the recovering cluster (Cluster 3) shows that the 1992 drought impacting Central Europe (Spinoni et al., 2015) resulted in France in severe drought conditions over the five years prior to 1992 that ended in 1993 with a normally wet year. The years 1990 to 1993 show actually the records of low rarity values from 52 to 257 weeks (see Table 3).

The aforementioned examples correspond to multi-year droughts but how shorter droughts build can also be commented based in the chronology of the clusters. Year 2003 is classified in the shortly drying cluster (Cluster 9) due to a persistent heat wave during the summer that impacted the whole of Europe (Spinoni et al., 2015) and aggravated the precipitation shortage impacts experienced during the summer
and the antecedent spring (Fink et al., 2004). However, thanks to precipitation excess in the next autumn, it resulted in a relatively short drought of about one year (52 weeks), classifying year 2004 in the wet-to-balanced cluster (Cluster 7). It can be noted that year 2002 shows a similar rarity curve as 2003 over subannual scale (Cluster 11, shortly drying cluster), yet its impact seems to have been much less due to less extreme temperature. Shortly after, the 2005-2006 drought impacted strongly from the Iberian Peninsula to the Balkan Republics (Spinoni et al., 2015). In France, the precipitation deficit during the previous years of 2002 and 2003 produced a severe shortage over the antecedent three-four years, making 2006 be classified in the dry cluster (Cluster 1).

Another example of short drought is the summer drought of 2015 that affected a large portion of continental Europe and was characterized by very high temperatures and a lack of precipitation (Ionita et al., 2017). However this lack does not seem to have been very severe in France according to the 2015 rarity curve, given also that the previous 3-4 years were particularly wet (see the large values of the 2015 curve around 150-200 weeks). As a consequence, the classification of the rarity curves puts year 2013 in the wetting cluster (Cluster 8), year 2014 in the wet cluster (Cluster 12) and year 2015 in the shorty drying cluster (Cluster 11).

Although not classified as the most severe droughts, remarkable is the fact that the five last rarity curves (2019-2022) are always below the 0.5 NEP level of mild drought category, classifying these years in either the shortly drying (Clusters 2 and 4) or balanced (Cluster 5) clusters. This shows in particular that year 2022 was not extrem in terms of precipitation shortage across the antecedent months to years,
381

however the summer heatwave of 2022 surely exacerbated its impacts on soil and vegetation. Although climate variability cannot be excluded, this is notably the only case in the chronology with five consecutive years not belonging to neither the recovery (Cluster 3), nor the wet (Clusters 10 and 12) nor the wetting (Cluster 8) clusters.

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The above examples examined the chronology of clusters around peculiar drought years but other comments can be made from a more statistical point of view. Although the rarity curves are clustered independently of the chronology, it can be noted in Figure 10 some temporal regularity in the clustering, which was expected since each rarity curve goes back five years in time. For instance, the dry cluster (Cluster 1) is the most persistent as it corresponds to at least severe drought over the previous five years. When it is not preceded by a year in Cluster 1, it is almost always preceded by Cluster 2 (drying cluster) - corresponding to a continuously drying process over the five anteceding years. Also Clusters 12, 11, 9 are always consecutive, corresponding to wet periods getting more and more dry (wet and drying clusters). These few examples show an advantage of our methodology that allows, not only to find typical drought situations, but also to study how they build and persist through time.

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# 5. Conclusion

400

We have shown in this article that an easy manipulation of daily antecedent precipitation accumulated over a continuum of time periods allows assessing what we called the multiscale rarity of drought - representing how droughts build and persist

401

402
across years and time periods. The rarity curves provide a summarized but more
detailed representation of drought rarity than fixing a small number of accumulation
durations (e.g. 3, 6, 12 months) as usually done. Applied to France, we showed that
a classification of the rarity curves between 1955 and 2022 allows discriminating the
short to long-term drought lasting from about one year to five years. The presented
methodology is generic and it could likewise be applied to other regions in the world.
Although is was used here at a single space scale - France - it could also be applied
to areas of different sizes to compare drought dynamics across space scales. It could
also be applied to other drought variables such as discharge and soil moisture con-
tent to study more specifically hydrological and agricultural droughts. Nevertheless
we believe that considering a deep enough “memory effect” brings relevant informa-
tion about soil and hydrological droughts by merely considering varied characteristic
times.

The greatest merit of the proposed methodology is its simplicity while allowing
for a multiscale characterization of droughts. The continuum nature of this charac-
terization has also to merit to show that extreme droughts occur counter-intuitively
quite often - e.g. in 70% of the years for monthly accumulation periods. However the
methodology shows also some limitations. First, the proposed methodology some-
how double counts the events. For example, by construction of the rarity curves over
the antecedent five years, the drought event of 2003 contributes to the low values
of rarity curves up to 52 weeks in 2003, but it also contributes to the low values
at higher accumulation periods for the rarity curves of 2005 to 2009. The choice
to work on five-year curves impacts the obtained typology of droughts that would
likely change if we would consider a different range of accumulation periods, e.g. up to two or ten years. There is no universal best choice and it has to be guided by the application - e.g. soil moisture respond faster than water table storage so the antecedent period of the former could be chosen shorter. Second, we have chosen to calculate the rarity curves by years, i.e. scanning all the days from January 1st to December 31st. Doing so, we loose the seasonality and we possibly mix-up drought of similar rarity occurring at different seasons. An easy way to solve this would be to compute the rarity curves by season - with the cost of having to analyze four curves instead of one. Doing so, we would abandon the idea of studying drought of flexible durations.

*Acknowledgment.* This work was initiated by the Master 1 thesis of Aymeric Badel in University Grenoble Alpes, France. Part of this study has received funding from Agence Nationale de la Recherche - France 2030 as part of the PEPR TRACCS programme under grant number ANR-22-EXTR-0005.

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<figure><img src="image_14.png" /><figcaption>Figure A.11: K-mean classification of drought rarity curves into 4 clusters, as selected by Calinski-Harabasz and the silhouette indices. The three dotted lines show the four levels associated to the drought category of McKee et al. (1993). The bold black line shows the centroid of the class. The gray lines show the rarity curves of the years in the class.</figcaption></figure>

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## Appendix A. K-means classification with different number of classes

Figures A.11 and A.12 show the clusters obtained with respectively 4 and 7 classes, as selected by Calinski-Harabasz and the silhouette indices (4 classes), or Davies-Bouldin index (7 classes) (Arbelaitz et al., 2013). Compared to Figure 8, the classifications are unable to discriminate the most severe rarity curves that are mixed up mainly in different clusters with much milder curves. In particular there is in Figures A.11 and A.12 no such very dry cluster as Cluster 1 in Figure 8.
<figure><img src="image_15.png" /><figcaption>Figure A.12: K-mean classification of drought rarity curves into 7 clusters, as selected by Davies-Bouldin index. The three dotted lines show the four levels associated to the drought category of McKee et al. (1993). The bold black line shows the centroid of the class. The gray lines show the rarity curves of the years in the class.</figcaption></figure>