We computed spatial-scale characteristics based on a climatological variogram, following the approach outlined by . presented the theoretical spatial rainfall resolution required for an hydrological model in urban area, deriving it starting from a climatological (semi-) variogram. The (semi-) variogram γ was calculated at each time step as follows: 1γ=12n∑tnR(x)-R(x+h)2, where n is the number of radar pixel pairs located at a distance h, R is the rainfall rate and x is the centre of the given pixel, normalized by the sample variance and averaged over the time period. The obtained variogram, characteristic of the averaged rainfall spatial structure during the peak period, was then fitted with an exponential variogram and the area A under the correlogram was calculated for the exponential variogram as Ar=2πr29. Ar can be considered as the average area of spatial rainfall structure estimated with radar measurements over the study area . Characteristic length scale rc [L] of a rainfall event was defined as rc=2π3r, where r [L] is the variogram range. Minimum required spatial resolution Δsr was defined in this work as half of the storm characteristic length scale: 2Δsr=rc2≅0.418r.

This parameter describes the spatial variability of the rainfall event core.

Another parameter to quantify and compare the spatial variability of rainfall is the spatial rainfall variability index Iσ. This parameter was at first proposed by , called index of rainfall variability, and then recently redefined by . This index was estimated as follows: 3Iσ=∑tσtRt∑tRt, where σt is the standard deviation of spatially distributed hourly rainfall across all pixels in the basin, per time step t, and Rt represents the spatially averaged rainfall intensity per time step. As can be seen, Iσ corresponds to a weighted average, based on instantaneous intensity, of the standard deviation of the rainfall field during a given storm event. Small values of Iσ indicate a low rainfall variability, typical of stratiform rainfall events. Large values of Iσ generally represent convective storms, characterized by high spatial variability. In the study presented by , Iσ was applied to rainfall data measured in a French region with a resolution of 1000 m–5 min and it varied between 0 and 5.

presented a characterization of storm motion and a definition of the minimum required temporal resolution. Storm motion was defined applying the TREC method (TRacking Radar Echoes by Correlation) proposed by This method allows a vector representing storm motion velocity magnitude and direction of the rainfall event to be obtained at each time step. The minimum required temporal resolution, Δtr, was obtained considering time that a storm needs to pass over the storm event characteristic length scale rc. The term Δtr can be written as follows: 4Δtr=rc|v¯|, where |v¯| [L T-1] corresponds to the mean storm motion velocity magnitude, and |v¯| is obtained from the average of the storm motion velocity vectors, estimated at each time step during the peak period.

In this work, a different approach to classify rainfall events is presented, considering storm spatial and temporal variability in combination with rainfall intensity thresholds. To select the thresholds Z for the nine rainfall events over the radar grid (6 km × 6 km), percentiles at 25, 50, 75 and 95 % of the entire 100 m–1 min resolution rainfall dataset were calculated. In this way it was possible to calculate the different thresholds Z25,Z50,Z75 and Z95, corresponding to the 25th, 50th, 75th and 95th percentiles.

Fractional coverage was largely studied in the literature and it was shown that it has a strong influence on flood response . The percentage of coverage %cov used in this study, was defined as the sum of the number of pixels Nt above a threshold at each time step t divided over the total number of pixels of the catchment Ntot and over the total number of time steps d of the event: 5%cov=∑tNtNtot⋅d.

The percentage of coverage was calculated for each event, in order to give a first classification of the spatial rainfall variability.

Since variograms provide a strongly smoothed measure of rainfall field, we used alternative metrics to characterize the space scale and timescale of storm events based on cluster identification. To analyse the spatial variability of the storm core, we identified, for each rainfall event, the main rainfall cluster dimension SZ above the selected thresholds Z, as defined in Sect. .

For each time step, the area covered by rainfall above a certain threshold was considered. Main clusters were defined as the union of rainfall pixels above a given threshold. To identify the clusters, an algorithm based on has been used. The algorithm executes the following rules:

All pixels above a certain threshold are considered.

To obtain a characteristic number for each storm, cluster sizes per time step were averaged over the entire duration of rainfall event. Figure  presents an example of rainfall coverage at a time step t. Rainfall was divided considering different thresholds and the red line highlights the cluster for Z75 in Fig. a and for Z95 in Fig. b. The clusters identified with yellow circles are ignored because they are too small to give a considerable contribution. In a case in which there is more than one cluster, as for Fig. b, the average of the main clusters is considered.

To identify the characteristic timescale of rainfall events, maximum wetness periods were defined as the number of time steps estimated for which rainfall at a pixel is constantly above a given threshold. With this aim, every pixel in the catchment was analysed and maximum number of consecutive time steps above the chosen threshold was retrieved. Figure c illustrates the process followed to select the maximum duration Twmax above the threshold Z. For each pixel, the value of the maximum duration above the threshold is identified. These values are averaged over the whole catchment to obtain a temporal length scale that characterizes rainfall event TwZ.

For each pixel n, the maximum wetness period TwZ above a selected threshold Z is defined as ∑nNtotTwmax∑Ntot, where Ntot is the total number of pixels.

In order to characterize the intermittency of rainfall events, the maximum dry period Tdmax, defined as the maximum number of time steps during which the threshold Z was not exceeded, was also identified. Figure c shows how these lengths, TwZ and TdZ, were selected. The combination of these two parameters gives an indication of how constant or intermittent is the rainfall event.

Several studies have shown that drainage area is one of the dominating factors affecting the variation in urban hydrological responses resulting from using rainfall at different spatial and temporal resolutions as input . Considering a larger drainage area implies aggregating and averaging rainfall and consequently smoothing rainfall peaks, with the result of having large areas that are less sensitive to high-resolution measurements.

In order to compare spatial scale of models and rainfall spatial variability, the average dimension of sub-catchments was analysed to characterize the model spatial scales. To investigate the effects of the drainage area Ad on hydrological response sensitivity, 13 locations, with connected surface that varies from less than 1 ha to more than 600 ha, were considered. Given that the coarser resolution model (SD1) does not contain small drainage areas (< 35 ha), only 8 of the 13 selected locations were available for SD1. To compare FD with SD models, we assumed that FD sub-catchments have the same dimension of SD2 sub-catchments. Table b presents the drainage area Ad connected to each location, while in Fig.  the location of the selected pipes is highlighted on the catchment with a thick red line.

Dimensionless parameters as proposed by and were determined to investigate the interaction and relation between rainfall resolution and different model properties and characteristics. The catchment sampling number ΔsLC was introduced as the ratio of the rainfall spatial resolution Δs to the characteristic length of the catchment LC (square root of the total area). This parameter describes the interaction between rainfall resolution and study area. If the catchment sampling number is higher than 1, rainfall variability is insufficiently captured and for small rainfall events the position might not be properly represented. The runoff sampling number was defined as ΔsLRA, where LRA indicates the spatial resolution of the runoff model, defined as the square root of the averaged sub-catchment size . Lower values of this ratio indicate that the model is unable to capture rainfall variability, while higher values indicate possible incorrect transformation of rainfall into runoff. The sewer sampling number ΔsLS describes the interaction between rainfall resolution and sewer length LS, indicating higher sensitivity to rainfall variability with increasing values of this ratio.

In the literature, there is no unique parameter to characterize the temporal variability of the model. Several authors have proposed different timescale characteristics (see for a review), but no unique formulation has been chosen yet, especially for urban areas. Time of concentration and lag time are the most commonly used temporal model scales, but other time lengths have been proposed in the literature . In this study, temporal variability of the three models was classified using lag time tlag, which describes the runoff delay compared to rainfall input. The variable tlag can be defined in different ways: as the difference between the centroid of the hyetograph and the centroid of the hydrograph , or as the distance between rainfall and flow peaks . The hyetograph in a specific location was estimated as the average of rainfall intensity in the considered sub-catchment, while the hydrograph was represented using the flow in selected pipes. The lag time can be considered as a characteristic basin element. It depends on drainage area size, slope and imperviousness , but it is also influenced by rainfall characteristics. For this reason, tlag was calculated for the nine rainfall events and the average of these values was taken as the representative number.

Lag time increases with drainage area, following a power law as proposed by . For urban areas, an empirical relation between catchment area A (ha) and lag time tlag (min) was presented: 6tlag=3A0.3.

This relation was confirmed, incorporating results obtained by and . tlag was calculated for each selected sub-catchment, and then compared with the rainfall temporal scale, to investigate the interaction between model and rainfall scale. The relation between averaged lag time and connected drainage area was studied at each location.

Spatial variability index values for each of the nine rainfall events are presented in Table  for the observed rainfall at 100 m–1 min (Iσ) and at 1000 m–5 min (Iσ1000m). The last two columns on the right were added to have a direct comparison with the values presented by , who used the same resolution. Iσ values are generally high when compared to values found by for all the investigated regions. This indicates that most events are characterized by high spatial variability. Aggregation has a strong impact on this parameter, which becomes smaller with a coarser resolution, highlighting the fact that information about rainfall variability is lost during the coarsening process. Iσ1000m values are generally higher than values presented for the northern region, where values are below 1, but are comparable to the Mediterranean area, where Iσ reaches values around 4.

Values obtained based on variogram analysis (spatial range) and storm tracking (temporal development) following are also presented in Table .

Results show that the spatial variability index tends to increase as well as the required spatial resolution for storms larger than 2500 m spatial range, while events with small spatial range (E5, E7 and E9, spatial range below 2500 m) are characterized by relatively high spatial variability indexes. Required temporal resolution Δtr, obtained from the combination of storm motion velocity and required spatial resolution (see Sect. ) varies between 1.7 and 5.9 min; the lowest values of Δtr are associated with fast storm events (e.g. E8 and E5) and small-scale events (e.g. E9 and E7).

The first step in obtaining cluster dimensions is to identify rainfall thresholds (Z) characterizing the rainfall values' distribution (see Sect. ). Table  shows rainfall threshold values corresponding to the 25th, 50th, 75th and 90th percentiles for the nine rainfall events.

The 25th percentile of the rainfall values distribution is zero, indicative of strong intermittency and small areal coverage of some of the events (especially events E7 and E9). The 95th percentile is 22 mm h-1 (over a 1 min time window), corresponding to a recurrence interval of less than 6 months , indicating that the selected events are representative of frequently occurring events. For this region, rainfall intensities above 25 mm h-1, over a 15 min time window, correspond to a return period of once per year, indicating an intense rainfall event. For only few rainfall events, E1, E2, E3 and E7, the 25 mm h-1 threshold is exceeded over a 15 min time window, for few time steps and, in particular, for E7 this happens only at the peak. This implies that rainfall events considered in this study are not classifiable as extreme.

The percentage of areal coverage, estimated for the catchment, is presented in Fig. a, d, g, j. Areal coverage associated with 25th percentile values provides an indication of event-scale intermittency. Events with 25th percentiles close to 1 cover the entire catchment most of the time, while smaller and more intermittent events, especially E7 and E9, are characterized by lower 25th percentile values. Areal coverage for 95th percentile thresholds indicates the size of storm cell cores: E1 and E2 have storm cores covering up to 65–70 % of the catchment; E4 and E6 have median coverage values close to zero, indicating that these are mild events without an intense storm core.

Box plots in Fig. b, e, h and k show the number of time steps above selected thresholds as a percentage of total event duration, to enable comparison between events. Results confirm patterns identified based on areal coverage: events E7 and E9 are identified as high-intermittency events (based on 25th percentile threshold). Maximum percentage of time steps above the highest threshold is 30 % for events E1 and E2. Each box plot represents the spatial variability of rainfall between pixels. Thresholds Z50 and Z75 present a high intra-event variability, highlighting the differences between rainfall events. For the other two thresholds, the intra-event variability is not high, suggesting that the rainfall event characteristics might not be well represented. For Z95, all events present a coverage variability lower than 30 %, and differences between events are not properly defined. Thresholds Z50 and Z75 present also a high inter-event variability, indicating that in these cases the spatial variability of the rainfall event above the catchment area is high.

Dimensions of the main cluster were determined for each of the four thresholds and for all time steps of the nine events. Results are presented in Fig. c, f, i and l, where the red line indicates the median and the blue dot the average.

The plots show that for Z25 only intermittent events, like E7 and E9, present a median below 861 ha (entire catchment area). The intra-event variability is generally quite high for most of the events, especially for the 50th and 75th percentiles, indicating that clusters change their dimension and shape during the event. Only a couple of events, E4 and E2, do not show high variability above Z25 and Z50 threshold. For Z95, the cluster dimension variability is relatively small, suggesting that the average or the median can be a good approximation of the storm core dimension. Values above Z50 present high inter-event variability. There is a clear distinction between constant events, such as E2 and E4, and intermittent events, E7 and E9, which show low median and average values.

Intense and constant rainfall events are also characterized by median values being generally higher than the mean. However, intermittent events, such as E9, have an average higher than the median, especially for the 50th and 75th percentiles. These results suggest that Z50 and Z75 are able to describe rainfall spatial and temporal scale well.

The maximum wet period TwZ and maximum dry period TdZ were calculated for four rainfall intensity thresholds in order to represent temporal variability of a rainfall event. Table  presents maximum wetness period TwZ and maximum dry period TdZ, normalized by total duration of the rainfall event, to enable comparison between events and to investigate how long the main core is in relation to the total duration of the event.

For some events TwZ decreases depending on the threshold, passing from values close to 1 for Z25 to values close to 0 for Z95. The change between different thresholds can be gradual, as for example for E2, E8 or E5, or sharp, as is the case of E3 or E4. For intermittent events, however, the maximum wet period does not vary too much, and it is relatively short, like E7 or E9. This implies that there are probably multiple short periods above the threshold. When comparing TwZ and TdZ, we can observe that some events show a symmetrical behaviour, when a decrease in wet period coincides with an increase in dry period, with the increase in the threshold (E4, E3). E7 and E9 present a moderate decrease in TwZ while they have a steep increase in TdZ, indicative of strong intermittency. For the other events, the behaviour is generally the opposite, indicative of a concentrated storm core.

Dimensionless sampling numbers, presented at first by , and then re-proposed by , are presented in Table  for the three models (for underlying equations see Sect. ). SD2 and FD model have the same contributing area and network length, hence they show that values for the catchment sampling number and sewer sampling number are the same.

Catchment sampling numbers higher than 1 indicate that models can not properly represent rainfall variability . In this study, for 3000 m spatial rainfall resolution values are bigger than 1, so poor model performance at this resolution is expected. The runoff sampling number suggests that SD1 will not be able to capture rainfall variability, because it presents low values for all spatial resolutions, while FD has high values of this parameter, which highlights some uncertainty in rainfall–runoff transformation. SD2, instead, presents runoff sampling numbers similar to the values found by , where this parameter varied between 2.6 for high resolution and 93 for lower resolution. The sewer sampling number applied to SD2 and FD presents similar results to , where the values were varying between 2 for high resolution and 77 for low resolution. However, the sewer sampling number is pretty low for SD1, which indicates a low sensitivity of this model to rainfall variability. This parameter increases with coarsening of spatial resolution, suggesting a high sensitivity to coarser rainfall resolutions.

The catchment sampling number can be applied also to the selected sub-catchments, comparing spatial resolution with the sub-catchments dimension reported in Table b. Also in this case, when the ratio is bigger than 1 the rainfall might not be well represented. This happens for sub-catchment L1, which is smaller than 100 m, and for all locations when they have to deal with 3000 m rainfall resolution. Locations from L2 to L5, presenting a drainage area between 100 and 500 m, should show the effects of aggregation for spatial resolution of 500 and 1000 m, when the catchment sampling coefficient is higher than 1, and the variability is not well captured. When the catchment sampling number is lower than 0.2, the catchment is too large to be compared to the rainfall input, and the effects of averaging over the area should be visible, as for example for L13 when considering a 100 m input resolution.

Lag time tlag was computed for 9 storms for each model at 12 sub-catchments and at the catchment outlet, as explained in Sect. . Results, presented in Fig. a, show that tlag increases with drainage area and varies from just above 1 min for FD at L1 (upstream location with the smallest Ad) to over 100 min for the coarsest model and largest catchment scale.

For only a few locations, tlag is lower than 10 min and for this reason a low sensitivity to temporal variability of rainfall events is expected. However, lag times vary over a wide range between events, and this highlights a strong influence of event characteristics. Model scale clearly influences computed lag times, which are generally larger for coarser models, where sub-catchments are bigger. However, for locations with smaller drainage area (<245 ha), SD1 presents tlag values comparable with the other models, but with a much lower variability compared to the finer-scale models.

As discussed in Sect. , tlag strongly depends on drainage area. Figure b shows how lag time varies, as a function of drainage area, for SD2, based on average, median, minimum and maximum values across rainfall events. Results confirm that tlag increases with the drainage area, fitting a power law, similar to the one suggested by (Eq. ). In this case the power law that fits at best the average of empirical data is tlag=8.9⋅Ad0.27 (R2=0.841), an equation that presents the same exponent of the one proposed by and a slightly higher coefficient. The power law proposed by represents a wider range of surface areas wider than what is presented in this work; hence, only a small part of it is considered.

Figure  presents rainfall peak attenuation ratios ReR for the range of spatial and temporal aggregation levels investigated. The plot shows the median over the nine events (marker) and the variability of the data (from 25 to 75 %: solid lines; total range: dotted lines).

Rainfall peaks are reduced up to 80 % when aggregating in space or time and up to 88 % when combining the spatial and temporal aggregation at the coarsest resolution. For high resolution, aggregation over time seems to play a larger role than over space. Approximately half of the rainfall peak is lost when aggregating from 1 to 3 min, while from 100 to 500 m peak attenuation is relatively smaller (40 %). For lower resolutions, spatial aggregation has a slightly stronger attenuating effect than temporal aggregation. At 3000 m spatial resolution, rainfall peaks are strongly underestimated, independent of the temporal resolution.

In this sub-section, we compare effects of spatial and temporal aggregation on rainfall variability and peak intensity across sub-catchment scales. Figure  shows examples of rainfall aggregation effects, as a function of the drainage area. Results for two rainfall events are shown: E4 is a constant, low-intensity event, which has a low variability in time and space, while E9 is an intermittent event, with multiple peaks. The plots clearly show that rainfall variability for the constant event is less sensitive to aggregation than that for the intermittent event. Rainfall sensitivity to aggregation decreases for larger sizes. ReR and RR2 results for all the nine studied events are available in the Supplement.

Figure  shows results for statistical indicators ReQ and RQ2 for 16 combinations of rainfall resolution and in relation to catchment area. Results are shown for a stratiform low-intensity rainfall event (E4) and a convective intermittent storm (E9) for increasing catchment size. For both events, the sensitivity to rainfall input resolution generally decreases with increasing catchment size. The variability of ReQ and RQ2 is much stronger for E9 than E4, pointing out the important role of rain event characteristics.

Comparing Fig.  with Fig. , similar patterns are observed for rainfall and flow. In both cases, sensitivity to rainfall aggregation in space and time decreases with increases in the drainage area. Moreover, in both cases, the small and constant event (E4) is less sensitive to aggregation than the intermittent one (E9). Rainfall patterns are more sensitive to aggregation than flow, due to smoothing induced by rainfall–runoff processes.

To investigate the influence that model complexity has on hydrological response sensitivity, results obtained with the three models are analysed. Figure  compares the influence of model complexity to the impact of spatial rainfall variability on the sensitivity of hydrological response. For each model, outputs at all locations are plotted for the 16 different rainfall input resolutions. There is not a clear behaviour that characterizes differences between sensitivity of the three models. All models appear sensitive to 3000 m spatial resolution and 10 min temporal resolution: in these cases the performance is lower. For upstream locations, SD1 seems to be slightly more sensitive than the other models to spatial coarsening for the upstream location, while FD performs worse for L13. The plot shows that there are some minor differences between the outputs of the three models, but the strongest sensitivity is connected to the rainfall scale as characterized by the cluster dimension. All models show higher sensitivity to small clusters, especially for cluster sizes below 100 ha. For small clusters, SD1 presents a higher sensitivity for both statistical indicators, while it is less sensitive than SD2 and FD for large clusters.

Model complexity does not have a large influence on sensitivity to rainfall resolution coarsening, while other characteristics, such as rainfall parameters or catchment details, seem to have a higher impact.

Several approaches to classifying rainfall variability have been presented and discussed in Sects.  and . In these sections, their influence on the hydrological response will be analysed.

Figure  compares the influence of spatial and temporal required resolutions (Δsr and Δtr), spatial variability index Iσ, cluster above Z75 and Z95, and the maximum wet period TwZ75 to model performance at different resolutions. Sensitivity to rainfall input resolution generally increases for smaller required spatial and temporal resolution, for higher spatial variability index, and for smaller cluster size. The clearest relationships are observed for required temporal resolution and cluster size above Z75. This parameter seems to represent spatial scale of the rainfall events quite well, and therefore it is chosen in this work to characterize the spatial scale of rainfall events.

Figure  compares the influence of rainfall spatial scale, based on cluster size above Z75, with drainage area size. The variability of RQ2 is higher for lower values of both rainfall scale and drainage area and decreases in a similar way with increases in both rainfall and catchment dimensions.

For this case study, we can conclude that sensitivity to rainfall resolution depends mainly on the scale of rainfall events and study catchment, and much less on the complexity of the models used. Choosing a complex model is useful only when studying small-scale events and catchments and only if high-resolution rainfall data are available.