The region of interest extends over 36 700 km2, dominated by the Baetic mountainous system (Fig. 1). This highly rugged relief is very close to the coast, directly sinking into the Mediterranean Sea. As a result, many small-to-medium ephemeral catchments – called Ramblas in eastern Spain – spread over the selected area. Annual rainfall amounts are characterized by a decrease from north to south, passing from nearly 700 mm to less than 300 mm (Fig. 2). Conversely, annual mean temperature increases from 16 ∘C in the north to 18 ∘C in the south. According to the Köppen–Geiger classification, the dominant climate is either hot or cold semi-arid (BSh/BSk), although the northernmost part is categorized as hot- and dry-summer Mediterranean (CSa) while some limited areas of the Almería coast are classified as hot desert (BWh; Chazarra-Bernabé et al., 2018).

The northernmost Cànyoles watershed settles over extensive karstified limestone, carbonate and dolomitic fractured bedrock, favouring the recharge of deep calcareous aquifers (Sese-Minguez et al., 2017). The Rambla Salada basin lies on bedrock weakness primarily made up of marls in a badland area. Being located in a semi-arid climate, the long and recurrent drought periods produce the generation of a dense network of desiccation cracks on the surface. As a consequence, the hydrological response of the Rambla Salada is strongly modulated by (i) exacerbating infiltration rates and (ii) stemming overland flow (Cerdà, 1995). The Rambla del Albujón features deep and silty soils with low perviousness (Fig. 2). However, current intensive agricultural practices have notably increased soil drainage and water storage (García-Pintado et al., 2009).

Soil profiles are profound and have medium to high permeability over the Rambla de Benipila. Nowadays, its eastern part is also devoted to intensive agricultural production. This watershed lies over marble, mica schist, quartzite, gneiss, phyllite and plaster bedrock. The Rambla de Canalejas is mainly located on dendritic and quaternary formations with medium-to-high perviousness. Other portions of the drainage area lie on mica schist, quartzite, phyllite and plaster bedrock (IGME, 1993). The mountainous headwaters of the Almanzora catchment are also settled over carbonate, limestone and dolomitic fractured bedrock, leading to large infiltrabilities that feed deep aquifers (Vallejos et al., 1994). Being hydraulically disconnected of the underlying aquifers, all these river basins feature very irregular regimes. Most of their tributaries are dry and hydrologically active during flash floods. Note that for discussion purposes the Ramblas Salada, del Albujón and de Benipila are grouped as the central catchments. Accordingly, the Rambla de Canalejas and Almanzora watersheds are labelled as the southern basins (Fig. 2).

QPEs are derived from the reflectivity volume scans of the Almería, Murcia and València Doppler C-band radars from 11 to 14 September at 00:00 UTC (Fig. 1). Spatial resolution is 1 km in range and 0.8∘ in azimuth. A complete volume scanning is performed every 10 min, with a maximum range of 240 km. Volume scanning is affected by the complex orography of the region. Therefore, partial beam occlusion is amended by numerically simulating the beam power percentage blocked by the rugged topography. This amelioration is carried out by numerically modelling beam propagation over a high-resolution digital terrain model (Pellarin et al., 2002). Signal attenuation by heavy rain is also corrected by means of the mountain reference technique (Bouilloud et al., 2009). Next, quantitative rainfall estimations are obtained by applying the standard WSR-88D convective rainfall-rate–reflectivity relationship (i.e., Z=300R1/4; Hunter, 1996; Fulton et al., 1998).

As radar quantitative precipitation estimation entails large uncertainties (e.g., Gochis et al., 2015), additional inaccuracies in the hourly cumulative rains and patterns are amended by using a dynamical fitting to the 369 automatic pluviometers available over the region of interest (Fig. 1). This dynamic method consists of a pluviometer-adjusted radar rainfall estimator by employing a gridded multi-quadric surface fitting which varies in time and space. The gauge-adjustment factors are computed as the modified ratio of the observed and radar-estimated rainfall rates at a certain rain gauge by using two incidental parameters (Cole and Moore, 2008). Finally, an independent safety check is performed by comparing the 48 h radar-derived precipitation against observations by 227 independent daily pluviometers belonging to a different rain-gauge network than those measuring the automatic pluviometric data (Figs. 1 and 3). Statistical comparison between both databases exhibits a strong positive correlation (R2=0.88; Fig. 3), even if QPEs feature a slight mean underestimation of 11.1 %.

The role of the spatial and temporal variability of rainfall in controlling flood response at the catchment scale is assessed by means of the spatial moments of catchment rainfall (Zoccatelli et al., 2011). The spatial moments detail the rainfall organization over the basin in terms of concentration and dispersion statistics, as a function of the distance measured along the flow path. Next, the mathematical formulation of the spatial moments of catchment rainfall is briefly revised.

The flow distance (d(x,y)) is defined as the path length from a given position (x,y) to the catchment outlet. The nth spatial moment of the rainfall field (r(x,y,t)) is introduced as 1pn(t)=|A|-1∫Ar(x,y,t)⋅d(x,y)ndA, where A stands for the drainage area and t for time. The zeroth-order spatial moment (p0(t)) renders the average catchment rainfall intensity at time t. Likewise, the nth-order moment of the flow distance (gn) is introduced as 2gn=|A|-1∫Ad(x,y)ndA. The first two orders of the non-dimensional spatial moments of catchment rainfall are given by 3δ1(t)=1g1p1(t)p0(t),4δ2(t)=1g2-g12p2(t)p0(t)-p1(t)p0(t)2. The first-order scaled moment (δ1(t)) measures the distance between the catchment rainfall and basin centroids. Values of δ1(t) close to 1 indicate either a rainfall distribution located near the position of the watershed centroid or spatially uniform. δ1(t) smaller (greater) than unity reflects that rainfall is distributed near the basin outlet (headwaters). The second-order scaled moment (δ2(t)) accounts for the dispersion of the precipitation-weighted flow distances about their mean value with respect to the scattering of the flow distances. Values of δ2(t) close to the unity point out to a uniform-like precipitation pattern, with values less (greater) than 1 indicating that the rainfall pattern features a unimodal (multimodal) distribution along the flow distance (Zoccatelli et al., 2011).

Hydrologic response is analysed by using the event-based and fully distributed kinematic local excess model (KLEM; Da Ros and Borga, 1997). This hydrological model accounts for properties in topography, soil and vegetation. Runoff rate (q(x,y,t)) at a given location and time (x,y,t) is computed from precipitation rate (P(x,y,t)) by employing the Soil Conservation Service curve number method (CN; USDA, 1986). The identification of the drainage network requires the characterization of hillslope and channelled paths by means of a threshold area procedure for catchment channelization. Next, the description of drainage system response is used to represent runoff routing (Giannoni et al., 2003). Discharge (Q(t)) at any location along the stream is calculated as 5Q(t)=∫∫Aq[x,y,t-τ(x,y)]dxdy, where A represents the drainage area to the specific outlet location and τ(x,y) is the routing time from point (x,y) to the outlet of the catchment. τ(x,y) is defined as 6τ(x,y)=Lh(x,y)vh+Lc(x,y)vc, where Lh(x,y) stands for the distance from the generic basin location (x,y) to the channel network following the steepest descent path and Lc(x,y) is the length of the subsequent channel path to basin outlet. That is, surface and channel flow routing depend on two constant velocities along the hillslopes (vh) and channels (vc) of the drainage network. KLEM models baseflow by using a linear conceptual reservoir based on the Horton–Izzard equation (Moore and Bell, 2002).

Landscape morphologies and soil properties are described by a 25 m grid size cell. Curve numbers are derived from lithology and land use maps, set to dry antecedent moisture conditions (i.e., AMC I) and kept constant during the calibration process (Table 2). The hydrological model is driven by the 10 min radar-derived QPEs from 18:00 UTC on 11 September to 00:00 UTC on 15 September. The computational model time step is the same that the 10 min radar-observing frequency.

Calibration tasks addresses peak discharge, time to peak and runoff volume, which are mainly modulated by the infiltration and the hillslope and channel flow velocities. The CN model is considered as a suitable conceptual scheme to describe the runoff-generating processes associated with the 12–13 September 2019 episode. Recall that direct runoff volume (Vr(t)) only depends on cumulative precipitation (P(t)) at a specific time from the beginning of the storm (t; Maidment, 1993): 7Vr(t)=P(t)-Ia2P(t)-Ia+SifP(t)>Ia0ifP(t)≤Ia, where Ia denotes the runoff threshold and S represents the soil retention capacity. S is a site storage parameter described by the curve number, while Ia is defined as a fraction of S (Ponce and Hawkins, 1996): 8S=S0⋅100CN-1,9Ia=λ⋅S, where S0 stands for the infiltration storativity and λ is the initial abstraction ratio. S0 and λ are considered as adjustment parameters so as to better deal with the initial very dry soil conditions and large storage capacities. Calibration of S0 and λ enables the correct simulation of the observed water balance (Borga et al., 2007). Heterogeneities in hydraulics have been encompassed by calibrating the hillslope and channel flow velocities (Table 2).

A fundamental research topic in hydrometeorology is to delve into the physical basis of observed self-similarity linking peak discharge to other physical variables. As runoff and flow routing processes are scale-dependent on cumulative precipitation in extreme flash flooding, the connection between flood magnitude and total rainfall amount is investigated through scale invariance. Schumer et al. (2014) observed that a power-law relationship links peak discharge (Qp [m3 s-1]) to total flow volume (Vr [m3]) for a given episode duration in arid and semi-arid basins dominated by infiltration-excess runoff generation: 10Vr=a⋅Qpb, where the regression coefficient (a, [sb m-3(b-1)]) is the reduced discharge volume and b is the flood-scaling exponent. Direct runoff volume and cumulative precipitation for the same episode and duration are also related as 11Vr=13.6A⋅Cr⋅P, where A [km2] is the drainage basin, Cr [–] is the event runoff coefficient, P [mm] is the total catchment-average rainfall amount and 1/3.6 is for unit conversion. By combining Eqs. (10) and (11), specific peak discharge (qp [m3 s-1 km-2]) is related to cumulative precipitation by the following scaling law: 12qp=A1-bb⋅Cr3.6⋅a1/b⋅P1/b≡α⋅Pβ, where the regression coefficient (α [m3 s-1 km-2 mm-β]) is the reduced specific peak discharge and β is the flood-scaling exponent. Equations (10) and (12) exhibit an inverse relationship between the algebraic exponents of both power laws. Schumer et al. (2014) also indicated that parameters a and b are related to catchment and rainfall properties. This assumption can be readily checked by considering the limiting case of a linear dependence between flood magnitude and total precipitation (i.e., β=1). In this particular limit, Eqs. (10) and (12) become 13Vr=a⋅Qp,14qp=13.6⋅a⋅Cr⋅P, where a in Eq. (13) matches the definition of flood timescale (TQ [s]) introduced by Gaál et al. (2012): 15TQ=VrQp. Therefore, Eq,~̇(14) can be expressed as follows: 16qp=13.6⋅TQ⋅Cr⋅I⋅Tm=13.6⋅VartQ⋅R⋅Tm, where I [mm s-1] is the storm- and basin-average rainfall rate and Tm [s] denotes the storm duration. R [mm s-1] stands for the storm- and catchment-average excess rainfall, tQ [s] corresponds to the timing of runoff (i.e., the characteristic time over which runoff is distributed), and Var(tQ) is the temporal dispersion of the runoff hydrograph. The inverse of the temporal dispersion of the runoff hydrograph measures the peakedness of the hydrograph. Viglione et al. (2010) found that an almost mathematical equality stands between the flood timescale and standard deviation of timing in runoff. Equation (16) is the response number, a measure of flood peak magnitude also defined by Viglione et al. (2010). In this limiting case, the scaling relationship becomes the response number, which integrates the joint effect of runoff coefficient and hydrograph peakedness in determining flood peak.

In the general case, Eq. (12) assesses the nonlinear interaction between storm properties and hydrologic processes for particular arid or semi-arid catchment specificities by only considering two integrated variables: specific peak discharge and cumulative precipitation. Regression coefficients and flood-scaling exponents differing from unity quantify the degree at which basins filter spatial and temporal variability in rainfall. On the one hand, as an increased precipitation amount results in an enhanced efficiency of the rainfall–runoff processes, basins are expected to have flood-scaling exponents greater than 1. On the other hand, heterogeneities in soil and underlying substrate anticipate regression coefficients much smaller than unity. The smaller the reduced specific peak discharge, the larger the impact of these features on the nonlinear hydrological response. Finally, the power-law relationship also allows examining how precipitation amount modulates flood magnitude for different return periods.

The total basin-average cumulative precipitation fluctuated roughly from 135 to 226 mm, while runoff ratios were remarkably small, varying from 0.04 to 0.16 (except for the Rambla Salada; Table 1). The runoff coefficients do not exhibit any clear relationship with maximum hourly rainfall rate or total accumulation across the examined basins. These highly nonlinear hydrological responses can be ascribed to the combination of very dry initial soil conditions and large soil water capacities. Indeed, runoff deficits were especially severe on these basins with underlying fractured and karstified bedrock (i.e., the Cànyoles and Almanzora catchments) or with soils strongly altered by anthropogenic pressures (i.e., the Rambla del Albujón). As a result, specific peak discharges were uneven and rather small at the basin scale, being only notable on the Ramblas de Benipila and de Canalejas. Similar hydrologic behaviour has been reported in previous flash floods over the western Mediterranean region. For instance, the Verdouble basin featured a very dampened hydrologic response during the 12 and 13 November 1999 flash flood over the Aude river, France. The runoff ratio was probably much lower than 10 % throughout the entire episode (Gaume et al., 2004). When analysing the 29 September 2003 flash flood in the eastern Italian Alps, a region characterized by karstified limestone, Borga et al. (2007) reported runoff ratios ranging from less than 0.1 to 0.25, strongly depending on the initial soil moisture conditions.

The catchment response time is linked to basin size, runoff generation, and hillslope and channel network routing. In turn, runoff triggering is intimately connected to the spatial and temporal distribution of the rainfall patterns. Herein, lag time is used as a proxy to characterize basin dynamics (Creutin et al., 2009). It is defined as the temporal difference between the centres of mass of the rainfall hyetograph and hydrograph, measuring the catchment response time from the beginning of precipitation. Besides acute heterogeneities in the rainfall–runoff transformation, pronounced nonlinearities also arose in the hydraulics of the selected basins. Lag times were a little sensitive to drainage area, being more dependent on features in rainfall (Fig. 9a–d). This behaviour would reflect the composite effects of several factors: an increased amount of sheet flows on the hillslopes with cumulative precipitation, an expansion of the stream networks to previously unchannelled topographic elements and an increase in flow velocity with discharge (Borga et al., 2007).

According to the lag time as function of basin size, three different modes of hydrological response were present during the unfolding of the 12–13 September 2019 widespread flash flooding (Fig. 9a). Despite the initial high abstractions, the Rambla de Canalejas and Cànyoles watersheds exhibited fast hydrological responses, with lag times of 2.7 and 3.5 h, respectively. For the latter catchment, features in precipitation did not stand out in comparison to the Almanzora basin, with similar size and physiography (Fig. 9b–d). However, heavy rainfall impacted the Cànyoles watershed during 7 consecutive hours, being mostly focused on its lower part, according to the temporal average of δ1 over the most intense period (i.e., 00:00–12:00 UTC on 12 September; Figs. 6c, d and 9e, f). In the end, the rapid response of this basin can be mainly attributed to the particular spatial and temporal distributions of the rainfall patterns. The Rambla de Canalejas offered an exemplary case of fast infiltration-excess runoff generation to extreme precipitation rate: rainfall was concentrated in just 3 h, leading to a total catchment-average amount of 147.6 mm, with relative basin coverages above 70 % and a strong unimodal distribution along the flow path (Figs. 6i. j and 9b–f).

After the end of phase 2, the catchment-average total rains were of 44.4 mm in the Rambla de Benipila, 61.6 mm over the Rambla de Albujón, and 37.5 mm in the Almanzora watershed, but no appreciable runoff had yet been produced. Subsequent bursts of heavy precipitation during phase 3 led to fast infiltration-excess production, resulting in widespread flooding on these watersheds. Their hydrological responses were relatively slower than these observed in the Cànyoles and Rambla de Canalejas catchments, with lag times ranging from almost 5 to 8 h (Table 1; Fig. 9a). The delay in runoff activation points to cumulative precipitation as another essential factor for flood control during this episode: as long as the runoff thresholds were not surpassed, sudden hydrological response did not start. It is likely that total rainfall amount was also the main ingredient triggering sudden infiltration-excess runoff generation on the Cànyoles basin. Camarasa-Belmonte and Beltrán Segura (2001), Gaume et al. (2004), Borga et al. (2007) and Zanon et al. (2010) have also reported rainfall accumulation as a paramount ingredient to trigger other flash floods in the western Mediterranean region.

The Rambla Salada basin had a very different hydrological response owing to its particular physiography. The total basin-average rainfall estimates were of 31.4, 77.5 and 116.7 mm during phases 1, 2 and 3, respectively. In spite of featuring the largest amount of precipitated water, the second largest hourly rainfall rate and basin coverage, and the highest runoff coefficient, lag time was 14.5 h (Table 1; Fig. 9). The rising and recession limbs of the observed hydrograph were more gradual (Fig. 8). The catchment response departed from the paroxysmal runoff processes and overland and channel flows typical of flash floods, as subsurface processes modulated flux dynamics (Table 1; Fig. 9). Owing to the important contribution of subsurface flow, it took around 8 h for the flow to recede to 30 % of the second peak discharge. The exponential decay of the remaining recession limbs was much faster. As an illustrative example, it only took 1.2 h for the observed hydrographs to recede in the same proportion over the Cànyoles and Rambla de Canalejas watersheds, exemplifying the predominant contribution of fast overland flow during the 12–13 September 2019 episode.

The performance of QPE-driven runoff simulations is evaluated by means of the Nash–Sutcliffe efficiency criterion (NSE; Nash and Sutcliffe, 1970). The skill of the hydrological experiments is also analysed in terms of the relative errors in peak discharge and total direct runoff volume, expressed as a percentage. The calibrated KLEM simulations succeed when performing the different hydrological responses (Tables 2 and 3, and Fig. 8). The remarkably high NSE scores indicate a good general reproduction of the observed hydrographs in terms of peak discharge and timing. However, the observed water balances are considerably overestimated in the Cànyoles, Rambla de Benipila and Rambla de Canalejas catchments. The overall successful performance of the numerical experiments can be partially ascribed to the strong role of the heavy rainfall when modulating flood response for the case under study, as QPEs have been estimated with a relatively good accuracy over the region of interest (Figs. 3 and 4). It seems unlikely that imprecisions in the simulated water balances emerge from large errors in rainfall estimates over the concerned watersheds. Automatic rain-gauge density is relatively high on and around the catchments, minimizing possible biases. As pointed out by Zanon et al. (2010), these inaccuracies would be more attributable to errors in the precise description of soil and geological properties, which are always more difficult to assess.

Imprecisions in the simulated water balances mainly arise due to inaccuracies in the reproduction of the observed hydrograph limbs (Table 3 and Fig. 8). Errors in the simulated recession branches for the Cànyoles and Rambla de Canalejas basins would denote that soil did not reach saturation. Abstractions remained despite the large volume of water previously precipitated on both watersheds. For the former watershed, these losses may be ascribed to the underlying karst geology over large portions of the basin. Imprecisions also emerge when simulating the observed rising limb in the Rambla de Benipila, resulting in an excessive initial runoff volume and a double-peak discharge (Fig. 8d). This watershed combined high initial abstractions with a sudden subsequent overflow production, resulting in a remarkable peak discharge once runoff thresholds were exceeded. These outcomes are in line with the information reported for the 12–13 November 1999 and 8–9 September 2002 floods in France. For the former, the runoff ratio was lower than 10 % on the Verdouble basin during the height of the flood (Gaume et al., 2004). For the latter, the Vidourle catchment featured a notable rainfall water retention capacity, with a runoff coefficient not exceeding 50 % during the peak of the flood (Gaume and Borga, 2008). In addition, it is arduous to accurately reproduce the fast processes involved in increasing the efficiency of the rainfall–runoff transformation and flow routing, while notable abstractions still remain. According to the observed hydrograph in the Rambla de Canalejas, a sharp decrease in catchment response to increased rainfall amount occurred: the main peak discharge was just 1 h apart from the previous relative maximum, being quite sudden and narrow (Fig. 8e).

With the exception of the Rambla Salada, the simulated overland flow speeds vary from 0.18 to 0.35 m s-1, whereas channel flow velocities range from 3.5 to 4.0 m s-1 (Table 2). The overland flow velocities are high as a result of the sparse vegetation and large amounts of sheet flow produced on the hillslopes during flash flooding. Since KLEM assumes the overland flow to be constant in space and time, high overland flow speeds allow accounting for sheet flow as well as concentrated overland flow in not previously channelized areas (Borga et al., 2007). Channel flow velocities are also high, as a consequence of the lack of vegetation in ephemeral stream beds and on banks, relatively steep slopes, and increase in flow velocity with discharge.

The 12–13 September 2019 episode offers a suitable benchmark to preliminarily explore the general features in the scale dependency between flood peak and rainfall amount. The unusual spatial and temporal extension of the responsible HPE, together with the contrasting physical features of the examined watersheds, permits cross-validations and inter-comparisons. Amengual et al. (2021) devised a set of distinct ensemble prediction systems (EPSs) based on a convection-permitting numerical weather prediction model in order to examine predictability of the 12 and 13 September 2019 HPE at the regional and catchment scale. These ensemble strategies also provide valuable information about plausible and equally likely rainfall scenarios for this particular episode. Therefore, it is possible to examine how each basin integrates the diverse collection of rainfall intensities and amounts. Consequently, all the quantitative precipitation forecast (QPF)-driven runoff simulations are used to further examine the scaling properties linking peak discharge and total rainfall amount.

In the end, five different 50-member EPSs were generated for two consecutive 24 h periods aiming to cope with all meteorological uncertainties. Next, the QPFs were used to force the KLEM model. Note that this procedure considered up to 500 different rainfall scenarios (i.e., 5 EPSs × 50 ensemble members each × 2 initialization days) impacting on each individual watershed. Again, regression coefficients and flood-scaling exponents are calculated by using OLS regression; uncertainty is quantified by applying a 1000-sample bootstrap with replacement, while statistical significance is checked by means of the p value. Certainly, although this numerical exploration is exclusively limited to the fixed set of hydrological model parameters describing each basin response for the 12–13 September 2019 episode (Table 2), it can be considered as a valuable experiment so as to provide some guidance in relation with the different basin responses.

The Rambla de Canalejas and Almanzora watersheds feature the most efficient hydrological responses to large cumulative precipitation (Table 4 and Fig. 10). As expected, the most effective catchments are the steepest (Fig. 2), as they promote a fast generation of runoff, sheet flow, and concentrated overland flow, as well as a quick flow channel routing. Despite being the most effective, these catchments evidence the largest nonlinear rainfall–runoff transformations: besides having the highest flood-scaling exponents, they also exhibit very low regression coefficients. The acute nonlinearity of the Almanzora response may be ascribed to the dominant presence of highly karstified and fractured bedrock. The efficient rainfall–runoff conversion for the Rambla de Canalejas is probably related to the fast processes involved in increasing runoff generation and propagation with enhanced cumulative precipitation. Nevertheless, it also features the weakest squared correlation coefficient, as a high variability is found for the smallest specific peak discharges in terms of rainfall amount (Table 4, Fig. 10e).

The Ramblas del Albujón and de Benipila feature a smaller enhancement in the effectiveness of the runoff and routing processes at increased precipitation, partially because both watersheds are flatter (Table 4; Fig. 2). In addition, the Rambla del Albujón has a comparatively small scaling coefficient, probably because it is a highly human modified basin. The hydrological response over the Cànyoles basin is halfway to the behaviour of the previous watersheds. Besides having comparable mean slopes, this outcome may also be attributed to the fact that rainfall amounts are not so extreme over this catchment. Indeed, only one rainfall scenario renders cumulative precipitation above the 10-year return period (Fig. 10a).

The Rambla de Benipila yields more substantial peak discharges for a particular rainfall amount than the remaining basins. Specific peak discharges are always above 0.02 m3 -1 km-2, independently of the cumulative precipitation (Fig. 10d). This output reflects the aforementioned difficulties when simulating high initial abstractions and sudden subsequent runoff productions in this watershed. The Rambla Salada departs from the paroxysmal dynamics of arid and semi-arid basins under flash-flood conditions owing to the dominance of subsurface processes (Tables 1 and 2). Accordingly, this watershed does not follow a scaling flood law with cumulative precipitation, as its internal hydrological processes result in less efficient rainfall–runoff conversion and flow propagation at enlarged rainfall amount (Fig. 10b). When subsurface flow processes are dominant, the scaling law no longer remains.