Meteorological drought lacunarity around the world and its classiﬁcation

. The measure of drought duration strongly depends on the deﬁnition considered. 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 5 time structure 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. To conclude, outcomes provide ability to determine when droughts start and ﬁnish. 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 10 reliable precipitation estimates. The study period comprises the years 1979-2016 (total of 45165 days), and a spatial resolution of 0.5°,

with gaps (Martínez et al., 2007;Feng et al., 2015;Dayeen and Hassan, 2016;Lucena et al., 2018). According to Mandelbrot, fractality can be found by measuring. He noted that, the more accurate the measurement ruler, the more infinite the British coastline appears to be, since the immeasurable curves of the coast situate it between a 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 25 ruler for measuring a coastline. Given an object embedded in an N-volume (N=1, length; N=2, area; N=3, volume; etc.), the 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 30 (0-dimensional) points, but its total (1-dimensional) length is zero. Formally, an object (embedded in an N-volume) possesses 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 35 trajectories, or involves the inverse, sometimes referred to as Lyapunov time, since it indicates the time expected to become 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 40 different unit durations (Sen, 2008;Meseguer-Ruiz et al., 2017). Similarities with the Cantor set (positive Lyapunov exponents) were found for dry-spell sequences in Europe (Lana et al., 2010). In this sense, dry pauses of the rainfall can be compared with the gap distribution of fractal objects, which is also known as lacunarity (Martínez et al., 2007;Lucena et al., 2018). The lacunarity analysis is used to characterise 'spatial' patterns (such as invariance, density and heterogeneity) of fractal objects, which represent attractor solutions of nonlinear dynamical systems (Plotnick et al., 1996;Wilkinson et al., 2019). If a time 45 series of precipitation is solution of the climatic system in a given point, the dryness distribution informs about (climate) average features of the system (e.g. surface convergences/divergences of moisture flows and latent energy fluxes or speed of the hydrological cycle).
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 50 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, according to the upper-order Markov Chain model (Martín-Vide and Gomez, 1999;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 . This index 55 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 the Shannon entropy, the Gini index (G) and the box-counting dimension of rainfall Monjo and Martin-Vide, 2016). For this reason, the n-index constitutes the main measure chosen in our work for analysing drought lacunarity around 60 the world, also compared with a Cantor-based exponent (C e ).

Dry-spell n-index
The main fractal measure was estimated for dry-spell density by means of the n-index.In a similar way as n-index describes the decrease rate (power law) of the maximum average intensities of rainfall over time (within a particular meteorological event), 65 it also can be applied to analyse how dry-spell lengths decrease around a maximum value. 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 . 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 80 duration (P i ) is:

Statistical analysis
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 85 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 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): 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 k = 2 is used for the usual AIC, or k = log(N ) (with N equal to the number of observations) for the so-called BIC (Bayesian Information Criterion) ( Supplementary Fig. A3).

Cantor-based exponent 100
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 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}; 105 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 110 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 The results of the dry-spell n-index were compared with other dryness fractal measures in each grid point: the Cantor-based exponent (C e ), the Hurst exponent (H) and the Gini index (G), all the cases estimated considering the entire time-series of 115 dry spells {D j }. The Gini Index is defined as the area under the Lorenz curve, which describes the relative accumulation of the variable D j and its cumulative frequency (Monjo and Martin-Vide, 2016). Alternatively, the Hurst exponent measures the possible long memory in time series. Given a set {A i } formed by T anomalies A i of the variable (D j ) and the corresponding k H is a constant, S is the standard deviation of the set {A i } and R is the range (i.e. maximum minus minimum value) of the set 120 {C i } (Mandelbrot, 1985;Feder, 1988).

Data considered
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 precipitation estimates. The study period comprises the years 1979-2016 (total of 45165 days), and a spatial resolution of 0.5°, 125 with a total of 259,197 grid points ( Supplementary Fig. A1).

Drought patterns detected
By analysing Dry Spell Spells (DSS), the first overview of the spatial distribution of the DSS n-index is given by seven quasilatitudinal bands ( Fig. 1): Three low-value bands in the equatorial zone and at medium latitudes in both hemispheres, and 130 four high-value bands in the two tropical areas and in the two polar areas. These bands generally correspond to the large climatic areas of the world. Indeed, there is a statistically significant relationship between annual dry days and the n-index ( Supplementary Fig. A1).
More specifically, rainforests like those in the Amazon, the Congo or Southeast Asia present values of n <0.3. This is consistent with the low values also found for other rainforest zones such as those in Madagascar, Central America and South

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America. This is due to the high degree of persistence of very short dry spells, alternating with very frequent wet days. Low followed by shorter dry events. The Polar Regions score a secondary maximum of n due to the usually long but irregular dry events. Similar spatial results are found if G and C e are used (Fig. 2), while the Hurst exponent displays noisier patterns around to distinguish between the alternation with longer ( ) or shorter (s) wet events; for example, considering the threshold of three consecutive wet days ( Supplementary 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. -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. -Type Hs. Occurrence of very long dry spells alternating with short wet events. Examples: All the desert regions around 165 the world, including the eastern fringe of the tropical oceanic areas.

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 170 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 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.

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However, the lacunarity of the drought can be analysed by means of other measures, such as the Gini index (G) and the Cantorbased 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 180 case, extreme dry spells show a linear increase for longer time series (in the same way that the maximum Cantor gap is set at 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.

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The DSS n-index provides information on the structure of drought lacunarity, in particular measuring the probability of regularity (if n~0) or irregularity (if n~1). Regular values of dry spells indicate that similar dry-spell lengths are usually consecutive.
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).

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The DSS n-index therefore provides information on the relative distribution of dry spells, on the longest duration and on the 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 200 borders, the total duration of the dryness coincides with the total length of a time series (as the length of the supplementary 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 205 the n-index model (p-value <0.0001). The proposed mathematical definition is complementary to the previous ideas on the persistence of dryness, for example, according to upper-order Markov chains (Lana et al., 2018). 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.).
Finally, the main limitation in this study is the possible errors derived from the used precipitation dataset. However, the 210 source errors are generally smaller for the breaks between dry and wet spells because they are not influenced by the absolute precipitation amount.

Conclusions
As a principal conclusion, the study demonstrates that drought lacunarity can be analysed with the use of self-similarity features obtained from the DSS n-index. This measure is useful for characterising the temporal and spatial patterns of drought, which 215 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 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. Further works may use other dataset at a global or regional scale (with more spatial resolution) to analyse past and future climate change (e.g. by using CMIP6 climate models). Moreover, future studies may focus on designing specific indicators to monitor hydrological and agricultural 230 droughts, especially involving data sets on river discharges, soil moisture, water stress or related damages. 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, 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). The methodology presented here provides a set of potentialities: From analyzing the change of drought regimes 235 to feeding indicators for monitoring impacts on agricultural or hydrological droughts, with the possibility of using data with a higher spatial resolution.

Code and data availability
The datasets from Figures (1-3  in the Asia-Pacific region (monzonic region), reflecting the division of the year into two halves, the rainy summer, with long rains, and the dry winter, with an absence of precipitation for many days in a row. This pattern can be detected in other areas presenting a humid-dry tropical climate, such as the southernmost area of the Sahel, north-eastern Brazil, an area in Africa (south of the equator) and northern Australia. The SW-NE diagonals of mid-latitudes with decreasing values of the MDS in the Atlantic and the Pacific are clearly appreciated, according to the westerlies and the track of the low pressure systems.   (3), β 1 = -9.543 (7), AIC ≈ BIC =˘1.4 · 10 6 , R 2 = 0.91, p-value <0.0001), Gumbel (α 2 = 1.225 (3), β 2 = -10.01(1), AIC ≈ BIC =˘1.2 · 10 6 , R 2 = 0.88, p-value <0.0001) and Opposite Gumbel (α 3 = 3.829(2) and β 3 = -0.5347(9), AIC =˘1.1 · 10 6 , R 2 = 0.87, p-value <0.0001), according to the Eqs. 6-8. Note that the minimum DSS n-index corresponds to the Amazon, where almost every DSS event lasts for only one day, and the accumulation of dry spells (Y i ) is therefore very regular. Furthermore, the maximum n-index (close to the Zambezi River) 265 represents an average DSS event with a maximum dry spell of 33% of the total accumulated dry-spell length (most of the entire time series), and successive dry spells with shorter durations (as in the Cantor gaps).