In everyday life, it is common knowledge that rain starts at a certain moment and stops some time later. However, due to its discreet nature, rain (which generally consists of a very large number of raindrops) is not an easy concept to define. Indeed, the exact definition of a rain event will depend on the sensor's characteristics (specific surface capture, detection threshold, instrumental noise) as well as the spatial or temporal resolution chosen for the study. This definition may also depend on the purpose of the study and thus on the scientific community behind it. There is thus a wide range of criteria used to break down precipitation records into rain events. For this reason, it is important to define and apply an unambiguous definition of a “rain event”.

In this study, the pattern produced by the 1-minute rain rate time series RRt(t) can be simplified by grouping non-null rain rates into a set of separate “primitive events” (Brown et al., 1985). On the basis of an assigned minimum inter-event time (MIT; Coutinho et al., 2014) each rain rate value, corresponding to a specific 1-minute period of observation, is assigned to a given rainfall event, i.e., either the rainfall event in progress or a subsequent event that is considered to be independent and “new”. The MIT could also be defined as the duration of a dry period Ddry following which the next occurrence of non-null rainfall marks the beginning of a new event (Driscoll et al., 1989). For dry periods shorter than the MIT, rain rates from either side of this period are considered to belong to the same “composite event”. Various authors have proposed different values of MIT that ensure event independence. Llasat (2001) noted that “The definition of an episode is quite subjective. In this case it was felt possible to distinguish between two different episodes, when the time which elapses between them without rainfall exceeds 1 h, which ensures that the two episodes come from different “clouds”. Moussa and Bocquillon (1991) wrote, “the constant rain observations on less than 30 min represent only 5 % of all the rainy periods. The representative threshold of the discretization of the data is 30 min to an hour.”

Dunkerley (2008a, b) carried out an analysis of the inter-event time (IET) in order to check the influence of this variable on the definition of rainfall events and its influence on the average rainfall rate. As emphasized in this study, when determining a value for the MIT, it is crucial to find an appropriate compromise between the independence of rain events and the intra-event variability of rain rates. The choice of MIT thus has a direct impact on the macrophysical characteristics that are ultimately determined by the analysis. Other researchers have proposed to use MIT values of 20 min, 1 h or even 1 day (see Dunkerley, 2008a, for a detailed list). In the present study it was decided to set the MIT to 30 min. This is in agreement with the value used by Coutinho et al. (2014), Haile et al. (2011), Dunkerley (2008a, b), Balme et al. (2006) and Cosgrove and Garstang (1995).

When applied to our data set, this choice leads to the identification of 545 rain events, which can be divided up into two subsets, i.e., one for learning and the other for testing (Table 1). The learning data set is composed of observations collected over a 2-year period between 2013 and 2014, with an availability of 96.4 %, whereas the test data set collected during the 2008–2012 period contains periods with missing data due to a malfunction of the recording device.

Rain events contain a wealth of information, which generally needs to be condensed into a limited set of well-chosen features. However, there is no conventional or commonly accepted list or specific set of macrophysical features that can be used to accurately describe and summarize an event. In the present study it was thus decided to consider a large number of features, allowing the macrophysical rain event information described in the literature to be correctly represented. Seventeen characteristics were selected and identified (Llasat, 2001; Moussa and Bocquillon, 1991) and are listed in Table 2. Some of these are parameter dependent, such as Pc, which uses three values of the parameter c. These three values lead to three Pc indices, namely Pc1, Pc2 and Pc3. Finally, a total of 23 descriptors were defined and numbered from 1 to 23 (column 1 in Table 2).

Among the 23 indicators (hereafter referred to as variables) corresponding to the previously defined features, some are very well known. These include the event duration (De), the quartile (Qi), the mean event rain rate (Rm) and the standard rain rate deviation (σR), as well as other, less traditional parameters such as the parameter βL (indicator for the convective nature of the rain; see Llasat, 2001), the absolute rain rate variation of order c (Pc) or the absolute rain rate variation (Ps,c). Some variables that are usually used to describe time series, such as the fractal dimension, multi-fractal parameters, trend, seasonality and autocorrelation, require a long series of data and are not well suited to an event-by-event analysis. This set of 17 features is not exhaustive, and some other features could also be included, depending on the application. One example is the case of hydrology, for which the positions of the intensity peaks inside the event could be a relevant feature. Although, for events comprising a very small number of samples (very low value of variable De), the computation of some indicators (σR, Qi) is questionable, in the present study the 23 variables were computed for each of the 545 rain events.

It is important to note that very few of these 23 variables are compatible with the probabilistic assumptions generally associated with exploratory statistical methods. They are often highly variable, with highly skewed distributions, and therefore do not have normal distributions. It is thus more difficult to make direct use of standard statistical methods with these data, as it may lead to misleading interpretations (Daumas, 1982). It is thus necessary to introduce an additional step in order to transform the original distributions into tquasi normally distributed distributions. The most suitable type of normalizing transformation for each of these variables was selected empirically by testing seven different possible transformations (Table 3). For each variable, the retained transformation is that leading to a distribution with the strongest similarity to a normal distribution, i.e., with a kurtosis close to 3 and a skewness close to 0. For each indicator, the selected transformation is provided in the last column of Table 2.

Following the normalization step, PCA was carried out on the learning data set (see end of Sect. 2.1 and Table 1). It follows that the two principal axes contain 73 % of the total information, whereas the first five principal axes are needed to represent 90 % of the total information. The IETp variable (no. 7) is very well correlated with axis 5, whereas the other variables are not. This means that there is no linear relationship between IETp and the other variables. For this reason, this variable was not considered as a possible candidate in the variable selection process during the remainder of the study. The results obtained in Sect. 4.2 confirm that there is no relationship between this variable and the other 22 variables. The correlation circle on axes 1 and 2 (Fig. 1a) shows that among the 23 variables, 16 are well correlated with the axis (close to a unit circle) and are distributed in approximately five groups (hereafter referred to as PCA groups). The first PCA group (G1) can be identified by the variables, which are grouped close to the first axis and are well correlated with it. As an example, this is the case for the variables σR (no. 9), PCN (nos. 17–18) and β (nos. 21 to 23). A second PCA group (G2) comprises the variables Rmax (no. 11) and Pc3 (no. 16), just above axis 1. The third PCA group (G3) is formed by the variable Pc2 (no. 15) only. The fourth PCA group (G4) comprises the variables Pc1(no. 14) and Ps,c (no. 20) and is well correlated with axis 2. The last PCA group (G5) is formed by the variable De (no. 1). The correlation circle on axis 1 and 3 (Fig. 1b) shows that the variables Q1 (no. 4), Q2 (no. 5) and M0 (no. 10) are quite well represented by these two axes. A similar remark can be made for variables Dd (no. 3) and βL1 (no. 21) on axes 1 and 4 (not shown).

Finally, PCA analysis clearly shows that, within each PCA group, many variables are highly intercorrelated, i.e., linearly dependant on each other. This means that several variables could be removed with no substantial loss of information. This leads to the following question: which variables can be removed in order to retain the most parsimonious subset of variables representative of the full data set? The PCA extracts summary variables, which are a linear combination of original variables, but does not allow for the selection of variables. To answer to this question, we propose a method for the global selection of variables that seeks to identify the relevant variables in a data set. As it appears to be intuitively more advantageous to select variables with a physical sense, rather than using dimension reduction methods (e.g., PCA, which is more suitable for the detection of linear relationships), the proposed method is based on the use of a GA. The following section provides a brief introduction into the concept of GAs and shows how they can be advantageously used for the selection of variables in the context of the present study.

We define by xk the chromosome number k: xk=x1,x2,…,x23. xk is a binary vector in {0,1}23 space such that each component has the following meaning: for alliin1,…,23,xi=1The    variable    numberi in  Table 2  column  1is  selected,xi=0The    variable  numberiin  Table 2  column  1is  not  selected. The word “selected” in Eq. (1) means that the corresponding variable will be used, both in the learning step described in “Step 2” below and for performance evaluation. Otherwise, if the corresponding variable is not selected it will be used only for performance evaluation.

As previously stated, the fitness function allows a measure to be provided of how well a minimal subset of variables can represent the entire data space (in dimension 23). The fitness function f is thus defined as follows: f(xk)=1nxkte(xk), where xk is chromosome k, n(xk) is the number of selected variables in chromosome xk and te(xk) is the topological error associated with chromosome xk.

As the aim of this approach is to minimize the number of selected variables n and the topological error te, we seek to maximize the fitness function. The estimation of the topological error made from an SOM is somewhat complicated and requires some explanation. The notion of an SOM, introduced by Kohonen (1982, 2001), makes use of a popular clustering and visualization algorithm. SOM is a neural network algorithm based on unsupervised learning, derived from the technique of competitive learning (Kohonen, 1982, 2001; Vesanto and Alhoniemi, 2000). It may be considered as a nonlinear generalization, which has many advantages over the conventional feature extraction techniques such as empirical orthogonal functions (EOF) or PCA (e.g., Liu et al., 2006). SOM applications are becoming increasingly useful in geosciences (e.g., Liu and Weisberg, 2011). As stated by Uriarte and Martín (2008): “The SOM provides a nonlinear, ordered, smooth mapping of high-dimensional input data manifolds onto the elements of a regular, low-dimensional array. The main characteristic of the projection provided by this algorithm is the preservation of neighborhood relationships; as far as possible, nearby data vectors in the input space are mapped onto neighboring locations in the output space.” This property makes it straightforward to compute a topological error (see Uriarte and Martín, 2008, Eq. 2). For each of the xk chromosomes, an SOM M(xk) is learned on the learning data set. Only the selected variables are used during the learning process. Finally, for each Map M(xk), the topological error te(xk) can be computed in accordance with Eq. (2) in Uriarte and Martín (2008). Section 4 provides additional information concerning SOMs.

The genetic algorithm is based on the following five steps (Fig. 2.):

In the first step, initialization, a (initial) population {xk,k=1,…,60} of 60 chromosomes of dimension 23 is randomly generated.

The GA is applied to our data sets in order to obtain an optimal subset of variables forming a subspace, which can (in a certain sense) provide relatively accurate information concerning the global space, whilst having the particularity of containing non-redundant information. At the 187th generation the algorithm produces a subspace comprising five variables, namely, event duration De (no. 1), standard deviation σR (no. 9), maximum rain rate during event Rmax (no. 11), rain event depth Rd (no. 13) and absolute rain rate variation Pc1 (no. 14).

The three variables (De, Rmax, Rd) selected using this data-driven approach are commonly used in the study of hydrological processes (Haile et al., 2011). Moreover, it should be noted that the commonly used variable Rm, which is computed simply by dividing the rain event depth (Rd) by the duration (De), was not selected by the algorithm. This result could be expected, since it is correlated with the latter variables, and the algorithm provides a parsimonious description. Concerning the absolute rain rate variation (Pc1), this variable was proposed by Moussa and Bocquillon (1991). It tends to provide information on the structure of the events, more specifically related to smooth events with a small number of sharp peaks. In fact, this variable promotes low variations of RRt because Pc1 is in a certain sense a structure function of order c1 of the variable RRt (see no. 14, column 4 in Table 2), with a low value for the exponent (c1= 0.5). Finally, the standard deviation variable (σR), which is a second-order moment, is the most commonly used indicator to describe the variability of the precipitation rate within the rain event.

The five variables De, σR, Rmax, Rd and Pc1 used for learning are referred to as “selected” variables, whereas the remaining 18 variables are referred to as “unlearned” variables. In order to study the relationship between these variables, Fig. 4 shows the projections for each of the variables in the M(xBest) map obtained with the aforementioned GA selection algorithm. The variables are discussed individually, by considering their structure, as well as the relationships between them. We note that the map is well structured for the majority of variables. This advantageous structuration of most of the variables confirms the ability of the selected variables to summarize all of the significant characteristics of rain events. Only a small number of characteristics are not adequately represented. It should be noted that almost all variables are structured according to the first or second diagonal. Among these, one may consider an initial subset comprising variables that are more or less structured according to the first diagonal. This is the case for the unlearned variable Dd, as well as for the selected variables Pc1 and De. A second subset comprising variables that are structured in approximate accordance with the second diagonal can be identified. This is the case for the unlearned variables Rm,r, Rm and PCNi, which are very similar to the selected variable σR. The unlearned variables Q3,Pc3 and PS,C also belong to the second subset and have a structure close to that of the selected variable Rmax.

The map can be related to the previously implemented PCA (Fig. 1). As can be seen in Fig. 4, the variables Pc3 (no. 16) and Rmax (no. 11), which have a similar structure, also belong to the same PCA group, namely group G1 (see Sect. 2.3, Fig. 1a). It is interesting to note that the variables PS,C (no. 20) and Rmax(no. 11), which also have a similar structure, do not belong to the same PCA group (groups G4 and G2 respectively) and are uncorrelated (they are orthogonal in Fig. 1a). This remark means that the topological map reveals a relationship that cannot be detected using PCA. As the rain event depth (Rd) depends on both the duration and the intensity of the events, the corresponding map has a top-down structure. Two distinct situations thus occur:

Those events which contribute the greatest quantities of water (Fig. 4., brown neuron at the bottom right of Rd) are among the longest (see corresponding neuron of De), but they do not have an extremely high peak rain rate (see corresponding neuron of Rmax) and are quite smooth (see corresponding neuron of Pc1 and σR).

Concerning the variable IETp (previous IET), the map is not structured, reflecting the independence of the characteristics of a rain event with respect to the drought period preceding the event. This corroborates the results of several previous studies (Lavergnat and Gole, 1998, 2006; Akrour et al., 2015; de Montera et al., 2009) dealing with rain support simulations. When studying temperate midlatitudes for relatively short periods, these authors noticed that successive rain and no-rain periods are uncorrelated, such that a rain time series could be considered as an independently drawn, alternating series of rain events and periods without rain. This is equivalent to an IET that does not characterize the rain events. The same effects are not necessarily observed at other locations and under different climatological conditions. Brown et al. (1983) also investigated a possible correlation between IETs and the intra-event characteristics and concluded that their data provided no evidence of this.

The variable βL (Llasat, 2001) is considered to represent a measure of the convective nature of the rain, it makes sense that the three variables βL1, βL2 and βL3 are structured similarly, with the peak rain rate variable Rmax. This relationship is clearly visible on the maps.

Several other relationships, which are not described in detail here, can be observed. These include the correlation between the normalized absolute rain rate variation (PCNi) and the standard deviation of the intensity (σR). We conclude that the combination of the five selected variables provides a relatively accurate summary of the information needed to describe the rain events. The poor structuring of some variables is justified by the independence of these variables with respect to the properties of the rain events; this is the case for the variable dry percentage in event Dd or the variable IETP.

In an effort to provide additional information for validation of the map, we compared each of the 23 variables with their corresponding value given by the SOM, for the learning data set and the test data set. For each of the 311 events of the test data set, the best matching unit of the SOM, i.e., the neuron that is the closest to the event, is determined with respect to the five selected variables. As an example, for each event Fig. 5 shows the current value of the unlearned variable βL3 as a function of the corresponding value given by the best matching unit of the event. A spread can be seen, in particular in the central zone, whereas the spread is relatively small for values located near to the edges (which are more numerous). A linear regression leads to a relatively good determination coefficient (R2; 0.96 and 0.89, respectively, for the learning and test data sets). Table 4 lists the value of R2 for the 23 variables obtained with the learning and test data sets. As expected, the coefficient of determination of the variable IETp is very poor (0.31/0.26), since this variable is not related to the five selected variables and as a consequence cannot be well represented by the SOM (Fig. 4). The selected variables have good determination coefficients, with both the learning and the test data sets; this confirms the quality of the learning and the generalization ability of the SOM. The quality of the learning step is confirmed by the fact that the R2 values of the selected variables obtained on the test set are close to those obtained on the learning set. The R2 corresponding to the unlearned variables obtained on the learning data set emphasize the ability of the selected variables to provide the information contained in the unlearned variables; in the case of the test data set it denotes the ability of the SOM to derive all event characteristics from the selected variables only.

We have shown that the distance matrix (Fig. 3) confirms the successful deployment of the map. Based on the distance between neurons, it appears that neurons can be grouped to obtain a limited number of classes, each with its own characteristics. In order to group the 64 neurons into a small number of classes, a hierarchical cluster analysis was carried out (Everitt, 1974). Only the five selected variables were used for the classification, and a Euclidian distance was selected for the hierarchical algorithm. Figure 6 shows the resulting dendrogram, applied to the 64 neurons.

Depending on the physical processes involved, experts tend to separate rain events into two different classes: stratiform and convective events. Although this classification is relatively crude, since stratiform and convective events can sometimes exist inside the same rain event, it is very commonly used. Concerning the time series, most authors use a very simple scheme to distinguish between stratiform and convective rain types. For reasons of simplicity, rain classification is sometimes defined using the instantaneous rain rate and the standard deviation estimated over consecutive samples. As an example, Bringi et al. (2003) defined stratiform rain samples when the standard deviation of the rain rate, taken over five consecutive 2 min samples, is less than 1.5 mm h-1, the convective rain samples are defined for a rain rate greater than or equal to 5 mm h-1, and the standard deviation of the rain rate over five consecutive 2 min samples is greater than 1.5 mm h-1.

Firstly, we separate the dendrogram into two classes. The first class contains 51 neurons and 79 % of the observations, whereas the second class contains 13 neurons and 21 % of the observations. The solid black line in Fig. 3 corresponds to the dividing line between these two classes. The first class, containing the greatest number of neurons, is in most cases characterized by relatively low rain rates. This can be seen by examining the structure of the map, according to the mean rain rate variable (Rm). Moreover, through analysis of the standard deviation (small values of σR), absolute rain rates Pc (high values of Pc1 and low values of Pc3) show that this class is more or less characterized by quiet, homogeneous events. Our analysis of event durations (De) shows that this class contains both short and long durations but is dominated by the latter. These characterizations are relatively well matched to a description involving stratiform and stable precipitations, which are often the consequence of the slow, large-scale uprising of a large mass of moist air which then condenses uniformly.

The second group is characterized by a smaller number of neurons. This corresponds to the higher values of the mean rain rates (Rm) and peak rain rates (Rmax). The variables σR and Pc have the opposite values with respect to those of the previous group. Most of the event durations (De) in this group are short, with the exception of neuron no. 64 (bottom right on the maps). This group fits well with the definition of convective events resulting from the rapid rise of air masses loaded with moisture for buoyancy. This convective moist air can lead to the development of cumulus clouds up to an altitude in excess of 10 km and to heavy rain.

Our analysis of the structure of the variables βL1, βL2 and βL3 in Fig. 4 confirms the previous interpretation of the two groups. These three variables, which are representative of convective rain, have high values for the neurons belonging to this group.

Figure 7a and b show the neurons in the Rm, βL3 and Pc2 subspace. These three variables were not used in the learning step. Nevertheless, the two classes are well separated, although an overlap does occur in Fig. 7a due to neuron no. 64 (bottom right on the map, Fig. 4). Although it belongs to the convective class, this neuron nevertheless has some characteristics of the stratiform class.

The hypothesis that the two categories of precipitation events corresponding to different dynamic regimes can be identified solely on the basis of hydrometeorological variables is in agreement with the findings of Molini et al. (2011). These authors have shown that there is a strong agreement between the hydro-meteorological classification (based on the duration and extent of events from rain gauge network data) and dynamic classifications (the convective adjustment timescale identified to distinguish between equilibrium and non-equilibrium convection derived from ECMWF analysis). We conclude that this unsupervised automatic clustering, based on the five selected variables, makes it possible to correctly implement a classification with these two well-known classes (stratiform and convective). It should be noted that, unlike other classifications described in the literature, this was established without making use of a priori information, since it is produced by an unsupervised process.

From the stratiform and convective classification described above, it is interesting to refine the two classes into a set of subclasses. The synoptic rainfall associated with midlatitude depressions provides an example of stratiform precipitation, which forms in depressions in the vicinity of warm and cold fronts. The very light type of rainfall (drizzle) associated with stratus or stratocumulus is included in the class of stratiform precipitation. This can occur under anticyclonic conditions,or in the warm region of a depression. The associated rain depths (Rd) are minimal and usually have no hydrological impact other than superficial wetting. In order to identify relevant subclasses, our classification was broken down into a number of unknown subclasses, such that n > 2.

An important step in hierarchical clustering is the selection of an optimal number of partitions (nopt) in the data set (Grazioli et al., 2015). Many indices can be used to evaluate each partition, from the point of view of data similarity only. Most of these evaluate the scattering inside each cluster, with respect to the distance between clusters, and assign relatively favorable scores to partitions with compact and well-separated clusters. Although different indices were tested, these did not provide the same number of subclasses (between 2 and 32 with the indices tested in this study). It should be noted that these did not take the physical meaning of each class into account. Finally, we chose nopt=5, since higher values led to classes with the same physical sense. The new classification based on the use of five subclasses is shown in Fig. 8.

From these five subclasses, two belong to the stratiform class and the other three belong to the convective class. In the learning data set, the first subclass represents 12 % of all events and 68, 1.2, 6.8 and 12 % for subclasses 2, 3, 4 and 5 respectively. The characteristics of these five subclasses are summarized below and in Table 5. The five selected variables are remarkably heterogeneous between classes, meaning the accuracy of these variables for clustering:

Subclass 1 (drizzle and very light rain): the main feature of this class is the very low mean value (Rm) and standard deviation σR of rain rate events, in addition to the features of the superclass. The mean rain rate events lie in the range [0, 0.5] mm h-1, with a mean value of 0.36 mm h-1 and σR in the range [0, 3] mm h-1, with a mean value equal to 0.1 mm h-1. Although this event has a significant duration, the corresponding subclass, which corresponds to drizzle, involves only small quantities of water. It can also be noted that a low value of βL3 is a good indicator (< 0.01) for drizzle.

The three remaining subclasses correspond to convective classes of events, which are characterized by a strong temporal heterogeneity and significant intensities. Depending on the depth of rain events, this convective class is subdivided into three subclasses.

Subclass 3 contains relatively long events (De) with high values for the rain event depth (Rd) variable and Pc1. This class represents events with a very small likelihood of occurrence (1.2 %).

To conclude this section, this new classification allows the conventional definition for stratiform events to be refined. The convective classification can be subdivided into five different subclasses, each of which is homogeneous. This classification is obtained for midlatitude climates. As the data set used in this study is representative of only one specific region and topography (i.e., the temperate climate encountered in the Île-de-France region, France), its analysis cannot reveal information related to different processes, i.e., those which are not sampled in the data set. Such processes could lead to the identification of additional specific clusters of events. In particular, there are no orographic rainfall events or oceanic observations. The final step in this study involves assessing whether the homogeneous character of each class is preserved at the microphysics scale and attempting to identify any relationships between the information present at the scale of both the microphysics and the macrophysics of these events (hydrological information).