Drought lacunarity around the world and its classiﬁcation

. Drought duration strongly depends on the deﬁnition thereof. In meteorology, dryness is habitually measured by means of ﬁxed thresholds (e.g. 0.1 or 1 mm usually deﬁne dry spells) or climatic mean values (as is the case of the Standardised Precipitation Index), but this also depends on the aggregation time interval considered. However, robust measurements of drought duration are required for analysing the statistical signiﬁcance of possible changes. Herein we have climatically classiﬁed the drought duration around the world according to their similarity to the voids of the Cantor set. Dryness time structure 5 can be concisely measured by the n-index (from the regular/irregular alternation of dry/wet spells), which is closely related to the Gini index and to a Cantor-based exponent. This enables the world’s climates to be classiﬁed into six large types based upon a new measure of drought duration. We performed the dry-spell analysis using the full global gridded daily Multi-Source Weighted-Ensemble Precipitation (MSWEP) dataset. The MSWEP combines gauge-, satellite-, and reanalysis-based data to provide reliable precipitation estimates. The study period comprises the years 1979-2016 (total of 45165 days), and a spatial 10 resolution of 0.5°, with a total of 259,197 grid points. Data set is publicly available at https://doi.org/10.5281/zenodo.3247041

ruler, the more infinite the British coastline appears to be, since the immeasurable curves of the coast situate it betweena line (one dimension) and a surface (two dimensions), i.e. with a fractal or fractional dimension (Mandelbrot, 1967).
A commonly used method for measuring the dimension of fractal objects involves box counting, which is similar to using a ruler for measuring a coastline. Given an object embedded in an N-volume (N=1, length; N=2, area; N=3, volume; etc.), the 25 method consists of covering the object several times, using unitary (N-1)-volume boxes of different sizes r for each completed covering, and counting how many covering boxes are required in each case (Olsson et al., 1992;Sakhr and Nieminen, 2018).
As the box size becomes smaller, the total (N-1)-volume of the fractal object tends towards the infinite rather than converging towards a finer value, and the N-volume is zero. For instance, the Cantor set is embedded in the [0, 1] segment with infinite (0-dimensional) points, but its total (1-dimensional) length is zero. Formally, an object (embedded in an N-volume) possesses 30 a fractal (non-integer) dimension B between N-1 and N if there exists a well-defined B-volume V = M r r B , where M r is the number of boxes with size r (Imre and Bogaert, 2006). A well-defined B-volume means that V and B remain constant for small values of r.
Another related measure involves the Lyapunov exponent, which indicates the rate of separation of infinitesimally close trajectories, or involves the inverse, sometimes referred to as Lyapunov time, since it indicates the timeexpected to become 35 a chaotic trajectory (Boeing, 2016;Kuznetsov, 2016;Gaspard, 2005;Bezruchko and Smirnov, 2010). The Hurst exponent is also related to the fractal dimension of chaotic time series, providing possible long-term memory throughout autocorrelation (Mandelbrot, 1985;Feder, 1988;Yu et al., 2015).
The fractal behaviour of dry spells can be observed in a Richardson's log-log plot of cumulative dry durations with regard to different unit durations (Sen, 2008;Meseguer et al., 2017). Similarities with the Cantor set (positive Lyapunov exponents) 40 were found for dry-spell sequences in Europe (Lana et al., 2010).
According to a multifractal analysis of the standardised precipitation, power-law decay distribution describes well the probability density function of return intervals of drought events (Hou et al., 2016). The Hurst exponent was also used to analyse the fractal persistence of the Palmer Drought Severe Index, providing values close to 1 (i.e. long-term positive autocorrelation) throughout Turkey (Tatli, 2015). The concept of persistence of dryness is used by some authors as an early indicator of drought, 45 according to the upper-order Markov Chain model (? Lana et al., 2018;López de la Franca Arema et al., 2015).
In addition, the fractal density of wet (or dry) spells can be estimated according to the n-index (Monjo, 2016). This index measures the persistence of records (or lengths) of a sequence of wet (or dry) spells similar to how regularity is measured in a Lorenz curve, whilst preserving the time structure of the events. A value of n <0.5 implies that a time series is persistent (consecutive similar values), while for n >0.5 the time series is anti-persistent. This regularity measure is closely related to 50 the Shannon entropy, the Gini index (G) and the box-counting dimension of rainfall (Monjo, 2016;Monjo and Martin-Vide, 2016). For this reason, the n-index constitutes the main measure chosen in our work for analysing drought lacunarity around the world, also compared with a Cantor-based exponent (C e ). The main fractal measure was estimated for dry-spell density by means of the n-index. For this propose, each spell duration D i was taken as a precipitation value, considering the minimum value D 0 = 1 as the dry value by definition. For instance, let D = (3, 4, 7, 1, 3, 10, 12) be a time series of consecutive dry spells. Subsequently, two independent events ("spells of spells") are built around the dry value (D 0 = 1) as (3, 4, 7), (3, 10, 12) ( Supplementary Fig. A2). In the present study, only dry spells were considered for building events; thus, each separated event is referred to as a "dry-spell spell" (DSS).

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In a similar way as for precipitation, the maximum accumulated dry-spell duration (P i ) of a DSS event is defined as: where i is the number of accumulated events, and N is total considered events. For each DSS event, maximum average duration Y i at i-step is: Therefore, the maximum average duration satisfies a scaling relationship with respect to this event number: where Y 1 is the maximum expected dry length per year and n is the n-index, which is bounded as d ≤ n ≤ 1, i.e. between the fractal dimension (d) of the spells considered and the dimension of the time series (Monjo, 2016). The parameters Y 1 and n were fitted for each DSS and averaged for each timeseries of grid points. Taking Eq. 2 and ??, maximum accumulated dry-spell 70 duration (P i ) is: Due to the low probability of the longest spells, a high maximum duration Y 1 implies a big difference in relation to the previous and subsequent durations, i.e. it implies high values for n. Therefore, a statistical link is expected between the probability distributions of Y 1 and n. In order to test this hypothesis, it suffices to set a distribution for one and a fit for the other. For 75 example, if a distribution over Y 1 is set as 1 − 1/Y 1 , we can find a distribution F(n) of n such as: In particular, two-parametric versions of three theoretical distributions were fitted: Exponential (F 1), Classical Gumbel (F 2 ) and Opposite Gumbel (F 3 ) distributions (Monjo and Martin-Vide, 2016):

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F 2 (n) = exp(− exp(−α 2 n + β 2 )) (7) The Akaike Information Criterion was applied to each fitted model using the log-likelihood function according to the equa- whereL is the maximum value of the likelihood function for the model fitted, p m is the number of parameters in the model, and  Mandelbrot, 1985;Feder, 1988).

Cantor-based exponent
Finally, the lacunarity of the Cantor set was compared with the frequency distribution of dry-spell durations for a given timeseries of L days. To this end, a Cantor-based time series was built using 'segments of zeros' or gaps {G kj } found between consecutive Cantor points (represented by 'segments of ones') obtained by the k-th iteration given by k = log(T )/ log (3), where T 100 is the length of the binary time series considered. For example, for the first iteration, k = 1, only the gap G 1i = T /3 is obtained; for k = 2, three gaps are found, G 2i = {T /9, T /3, T /9}; for k = 3, seven gaps {G 3i } = {T /27, T /9, T /27, T /3, T /27, T /9, T /27}; and so on. The set of gaps greater than one, was compared with that obtained from the set of dry spells, The value of the iteration k was chosen as the minimum iteration, when the total number of elements of ∆ is less than, or equal to, the total number of elements (cardinal) of Γ k , i.e. |∆| ≤ |Γ k |. Finally, we defined a Cantor-based exponent C e by the quantile-quantile map

Data availability
The data used in this study was obtained from the full global gridded daily Multi-Source Weighted-Ensemble Precipitation (MSWEP) dataset (Beck et al., 2017). The MSWEP combines gauge-, satellite-, and reanalysis-based data to provide reliable   Fig. A1). Therefore, six large drought types can be defined based upon the combination of both criteria (Fig. 3): -Type L . Occurrence of very short dry spells alternating with longer wet spells.
-Tropical examples: The main rainforest cores of the world (within the Amazon, the Congo and Southeast Asia among others).

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-Subpolar examples: The Southern Ocean and some regions of the North Atlantic and North Pacific oceans.
-Type Ls. Occurrence of very short dry spells alternating with short wet spells. Examples: northeast America and northeast Asia, especially Japan.
-Type M . Median dry spells alternating with longer wet spells. Examples: The Artic Ocean and North Asia.

Drought lacunarity
Since the total (1-dimensional) length of the Cantor set is zero, the total length of the complementary gaps is equal to one. That is, following the analogy between the drought duration and the Cantor set lacunarity, the total duration of a dry spell series approaches one when the size of the measurement box is accurate. For instance, one can find dry days in a wet month, and on rainy days, there can be several hours with no rain. If a ground point is used for measurement, the duration of a rain drop hitting 160 the ground (from leading surface to trailing surface) tends towards zero and thus the dry pauses are distributed paradoxically throughout an entire rainy day.
According to this idea, the dimension of the drought duration is practically one (the length of the time series), and the box-counting dimension therefore makes more sense for measuring rainfall duration than for estimating drought duration.
However, the lacunarity of the drought can be analysed by means of other measures, such as the Gini index (G) and the Cantor-165 based exponent (C e ), both of these related to the frequency of dry spell durations. In particular, the Cantor-based exponent indicates how likely it is to find longer dry spells over time. For instance, if C e ∼ 0, all dry spells will present similar lengths and therefore the standard deviation will be constant (i.e. extreme values are normally distributed). However, if C e ∼ 1, the distribution of lengths is similar to the Cantor lacunarity, and the standard deviation will therefore increase over time. In this case, extreme dry spells show a linear increase for longer time series (in the same way that the maximum Cantor gap is set at 170 1/3 of the total length), and the Gini index also tends to approach 1. Indeed, the correlation between the G and C e is R 2 = 0.86 (p-value <0.0001) and the approximation C e = G only provides an error of 10%, thus reinforcing the lacunar interpretation of the dryness. Accordingly, irregular values imply that long dry spells are followed by much shorter dry sequences. It should be kept in mind a high degree of irregularity is correlated with the longest dry spells (Supplementary Fig. A3).
In short, the time patterns of the dryness of a climate can be characterised in a simple manner by means of the n-index. This  Fig. A4).
The DSS n-index therefore provides information on the relative distribution of dry spells, on the longest duration and on the 185 total accumulated duration. Indeed, the start/end of a synthetic DSS event can be established by the number 'i' of dry-spell events, so that the averaged maximum duration Yi is a particular threshold (e.g. one day, which is defined as the dry threshold of a DSS event). This idea enables the concept of a DSS event to be employed as an alternative definition of meteorological drought, with diffuse borders that, paradoxically, are well defined by the n-index. For instance, if Y i = 0 is considered as drought borders, the total duration of the dryness coincides with the total length of a time series (as the length of the supplementary 190 gaps of the Cantor set). In this case, the difference between one drought or another is given by the decay rate of the maximum dry spell (well measured by the n-index).
As in (fractal) wet spells, the behaviour of dryness is self-similar on all timescales, that is to say, dry spells can be used at daily, monthly, yearly resolution, etc., considering specific dryness thresholds. This is guaranteed by the goodness-of-fit of the n-index model (p-value <0.0001). The proposed mathematical definition is complementary to the previous ideas on the 195 persistence of dryness, for example, according to upper-order Markov chains26. Indeed, the order of chains depends on the alternation and frequency of short and long dry spells, as with the rest of the measures (Gini index, Hurst exponent, n-index, etc.).

Conclusions
As a principal conclusion, the study demonstrates that drought lacunarity can be analysed with the use of self-similarity features 200 obtained from the DSS n-index. This measure is useful for characterising the temporal and spatial patterns of drought, which are consistent with the climatic features established worldwide. For instance, localities presenting very frequent wet spells alternating with short dry spells provide low values of the DSS n-index (this is the case, for example, of the Amazon, the Congo and other rainforests). This results from an accumulation of isolated short dry spells, all presenting the same duration.
Additionally, places with longer dry spells (such as deserts) scored higher values on the DSS n-index. This result is logical 205 because the n-index is strongly correlated with the maximum and average lengths of dry spells, as well as with the Gini index. The third and most important outcome refers to the fact that consideration of dryness lacunarity provides a better understanding of drought duration and helps to predict when droughts start and finish. In particular, the value of the DSS n-index is related to the effective duration (according to borders determined by a threshold of the "minimum expected dry spell"). Indeed, 215 the best analogy with regard to understanding the features of the DSS is Cantor lacunarity, i.e. total dry spells are almost equal to the length of the entire series (in the same manner that the total length of Cantor voids is equal to one).

Code and data availability
The datasets from Figures (1-3