the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Forecasting green roof detention performance by temporal downscaling of precipitation timeseries projections
Rasmus Benestad
Edvard Sivertsen
Tone Merete Muthanna
JeanLuc BertrandKrajewski
A strategy to evaluate the suitability of different multiplicative random cascades to produce rainfall time series, taking into account climate change, inputs for green infrastructures models. The multiplicative random cascades reproduce a (multi)fractal distribution of precipitation through an iterative and multiplicative random process. In the current study, the initial model, a flexible cascade that deviates from multifractal scale invariance, was improved with (i) a temperature dependency and (ii) an additional function to reproduce the temporal structure of rainfall. The structure of the models with depth and temperature dependency was found to be applicable in eight locations studied across Norway and France. The resulting time series from both reference period and projection based on RCP 8.5 were applied to two green roofs with different properties. The different models led to a slight change in the performance of green roofs, but this was not significant compared to the range of outcomes due to ensemble uncertainty in climate modelling and the stochastic uncertainty due to the nature of the process. The hydrological dampening effect of the green infrastructure was found to decrease in most of the Norwegian cities due to an increase in precipitation, especially Bergen (Norway), while slightly increasing in Marseille (France) due to decrease in rainfall event frequency.
Hydrologic performance of stormwater green infrastructure (GI) is usually divided between retention and detention. Retention refers to water stored, infiltrated, or evapotranspirated. Actual evapotranspiration can be estimated from a water balance including potential evapotranspiration, accumulated precipitation, a soil moisture evaluation function, and a crop factor (Johannessen et al., 2017; Oudin et al., 2005). The temporal resolution for modelled evapotranspiration process for green infrastructure is typically daily (Stovin et al., 2013) or hourly (Kristvik et al., 2019). Detention refers to water temporarily stored in the GI before being discharged into a downstream stormwater network. The process temporal resolution is typically minutes. Consequently, modelling GI detention performance requires higherresolution data to estimate its outflow (Schilling, 1991). Therefore, both highresolution climate data and projections at subdaily and subhourly scales are needed in order to model GI and to estimate their potential as a climate change adaptation measure.
In Norway and most of the European countries, precipitation has been measured with tipping buckets in numerous cities from years to decades. Moreover, climate projections at daily resolution for future precipitation and temperature from the EUROCORDEX project are available at 1×1 km spatial resolution in Norway (Dyrrdal et al., 2018) and 12×12 km resolution in France (Jacob et al., 2014). Consequently, the use of such data by urban hydrologists to assess the resilience of GI solutions to face climate change is conditioned by the possibility to downscale them to a subhourly resolution.
Downscaling includes two families of methods: dynamical downscaling and statistical downscaling (Benestad, 2016). Dynamical downscaling methods use physically based equations and are usually computationally expensive specially to generate highresolution data. Statistical downscaling consists in improving the resolution of data based on statistical properties observed from a lowerresolution dataset. The computational cost is lower; therefore, statistical methods might still be used to fill the gap in the next decades until the computational power is sufficient for routine dynamical downscaling. In addition, the use of a stochastic approach is necessary due to the current limitation in parametrization of smallscale processes (below the truncation scale) and the lack of coupling between resolved and parameterized scales (Sanchez et al., 2016).
Statistical downscaling has already been extensively used to temporally downscale data for various temporal resolutions, usually hourly or daily data. Three popular methods can be mentioned: (i) the method of fragment, (ii) the method based on point process theory, and (iii) the method of multiplicative random cascades. The method of fragment (Li et al., 2018; Lu et al., 2015) is a resampling method based on knearest neighbours (Kalra and Ahmad, 2011), which has been applied to derive hourly data from daily data. It can be accurate and effective due to its resampling nature, but it requires a large dataset, and by its design it cannot ensure extrapolation from observed data. Therefore, it might not be suitable to downscale climate projections. Methods based on point process theory have been used (Glasbey et al., 1995; Onof et al., 2000). The main principle is to generate storm occurrences and then describe them based on rain cells and statistical distribution based on Poisson point process. Multiplicative random cascades (MRCs) consist of using successively random cascades to split data in N data of finer resolution (N=2 in most of the cases). It is a very popular method that deserves further investigation (Gaur and Lacasse, 2018; Rupp et al., 2012; Thober et al., 2014). They were originally based on the hypothesis of multifractal scale invariance (Schertzer and Lovejoy, 1987) and were further developed by Gupta and Waymire (1993) and Olsson (1998). While multifractal scale invariance continues to be studied (Gires et al., 2020), several studies noted a deviation from that behaviour which led to the use of more flexible models (Koutsoyiannis and Langousis, 2011; Veneziano et al., 2006). Multiplicative random cascades can be divided between canonical and microcanonical types. The canonical one ensures conservation on average, while the microcanonical one ensures exact conservation. The parameters of the canonical MRC are often calibrated by fitting between observed and simulated noncentred moments of depths or intensity through the timescale (Paschalis et al., 2012). The principle of microcanonical MRC is usually based on reverse cascades: studying how the data are split and then reproducing the properties of the weights distribution depending on different quantities. The influences of timescale, rainfall intensity (Paschalis et al., 2012; Rupp et al., 2009) or season (McIntyre et al., 2016) have been extensively studied. Lombardo et al. (2012) suggested that the commonly used MRC suffers from conceptual weaknesses due to the nonstationary process of autocorrelation and proposed a method to improve the model. More recently, Bürger et al. (2014, 2019) suggested to include a temperature dependency in MRC models to make them more robust. This also enables them to be used with projections.
Green infrastructures, due to their retention and detention capacities, are seen as a promising solution to manage storm water and cope with climate change, especially in cities where urbanization increases. Among green infrastructures, green roofs are especially suitable for dense urban centres. They are designed to retain daytoday rain by evapotranspiration and attenuate major rainfall events (Stovin, 2010). Depending on their characteristics they can also help to detain extreme rainfall (Hamouz et al., 2020). Due to the timescale of their detention process, and their sensitivity to initial water content at the beginning of a rainfall event, they are suitable for evaluating downscaled time series. Moreover, it is especially relevant to evaluate their detention performance by the end of the century under a scenario such as RCP 8.5 (Thorndahl and Andersen, 2021). The results could be used to evaluate, at strategic level, their potential in mitigating storm water in order to make robust decision (Walker et al., 2013).
While downscaling models have been used to model the performance of green infrastructure under current climate (Stovin et al., 2017) or applied to intensity–duration–frequency (IDF) curves to do an eventbased simulation of local stormwater measures (Kristvik et al., 2019), none has been developed to produce future highresolution time series as input for green infrastructure models. The aim of this research is to evaluate different MRC downscaling models and their potential to produce input time series to predict the performance of stormwater green infrastructure, for the case of green roofs. In order to achieve this aim, different parts are detailed in the paper: (i) the development of a general structure of MRC, (ii) the improvement of the MRC structure by adding a temperature dependency, (iii) the addition of an ordering function to improve the temporal structure of the produced rainfall time series, (iv) the evaluation of the capability to reproduce the performance of GI based on observed data, and finally (v) the analysis of a possible shift in performance of GI at the end of the century.
2.1 Meteorological data
Time series of precipitation and temperature from six locations in Norway and two in France, representing four different climates (Table 1) according to the Köppen–Geiger classification (Peel et al., 2007), were used to apply the downscaling method. In Norway, the precipitation was measured by 0.2 mm plumatic tipping rain gauge manufactured by Kongsberg Våpenfabrikk. The rain gauges were not heated and thus did not operate in cold temperature. They were successively replaced by Lambrecht 1518H3 (measuring tip of 0.1 mm) in the 1990s and 2000s. The stations were operated by the Norwegian Water Resources and Energy Directorate (NVE) and the Norwegian Meteorological Institute (MET). The data were quality checked by the Norwegian Meteorological institute (MET) (Lutz et al., 2020). In Lyon and Marseille, precipitation was measured by 0.2 mm PrécisMécanique tipping bucket rain gauges. Ten climate projections (temperature and precipitation) on daily resolution with the RCP 8.5 for the period from 2071 to 2099 for Norwegian cities were available online https://nedlasting.nve.no/klimadata/kss (last access: 26 May 2022; Dyrrdal et al., 2018). For Lyon and Marseille (France), 12 climate projections at daily resolution were available for the same period and RCP (2071 to 2099, RCP 8.5) from http://www.driasclimat.fr/ (last access: 25 May 2022). The RCP 8.5, a scenario with a high gasemission baseline leading to a radiative forcing of 8.5 W m^{−2} in 2100, and the end of the century were chosen to test the methods since it was the scenario and period that deviate the most from the current climate among the available data in both countries. In practice, it is relevant to evaluate GI performance at the end of the century, but their design could be based on a different period depending on their lifetime.
2.2 Downscaling models and workflow
2.2.1 Data aggregation and processing
The historical data were aggregated two by two from 1 min resolution (and accordingly 6 min) to more than 1 d resolution in order to capture a part of the uncertainty linked to the estimation of the parameter of the models. The aggregation was done for each possible time steps: all multiples of 2 smaller than 1500 min (as there are 1440 min in a day). During the process of aggregation, both the weights, Eq. (1), and the rainfall continuity indicator, Eq. (2), measuring the proportion of high weight on the side of the highest neighbouring depth were computed. Given j∈[1…750] a timescale in minutes, i∈[0…N_{2j}] a time step with N_{2j} the number of time steps at scale 2j, and d_{i,2j} a rainfall depth, the weight w_{i,2j}, and the indicator S_{i,2j} of the side of the neighbour were calculated according to
2.2.2 Downscaling process
The MRC downscaling process consists of transforming daily rainfall depths to rainfall depths at lower timescale, e.g. 1 min, by means of successive distribution of the depth of a parent time steps between its two children time steps. The process is repeated by iteration until the desired timescale is reached. Figure 1 describes the downscaling process. In practice, the downscaling started at 1440 min (1 d) time step with eight iterations to reach a time step of 5.625 min. The results were interpolated and scaled to a 6 min time step for comparison with observed data. The final time step of 6 min was chosen based on the resolution of original datasets in Lyon and Marseille. Three steps are necessary to downscale a parent time step to two children time steps. The occurrence of a zero weight, i.e. the probability to assign all the water from the parent time step to only one of the children time steps (Fig. 1, centre left), is tested. This property is especially important and acknowledged by other studies. If a zero weight does not occur, a nonzero weight ${w}_{i,\mathrm{2}j}\in ]\mathrm{0},\mathrm{0.5}]$ is generated from a probability distribution (Eq. 3b). It distributes the depth from the parent time step between the two children time steps, as illustrated in Fig. 1, centre right. Finally, the weights w_{i,2j} and $\mathrm{1}{w}_{i,\mathrm{2}j}$ have to be assigned to the children time steps. The occurrence of S_{W} (Eq. 4), i.e. allocating the highest weight to the children with the neighbour with the highest depth, is tested (Fig. 1, bottom).
2.2.3 Downscaling models conceptualization and calibration
Based on the observed data, six different MRC models were developed. Different mathematical expressions and probability distributions, detailed in Appendix A, were defined to represent Eqs. (3a), (3b) and (4), depending on the hypothesis inherent to the later described models (Table 2). The models consist in three generators: a zeroweight generator, a nonzeroweight generator and a stochastic element permutation generator (SEP generator). Each of the zeroweight and nonzeroweight generators (Eqs. 3a and 3b) were considered to vary with timescale (indicated with S in the model naming). The letter I in the nomenclature indicates a depth/Intensity for the zeroweight generator (Eqs. A2a–A2d). The temperature dependency for the zeroweight generator (Eqs. A3a–A3e) was indicated by the letter T in the nomenclature. The temperature dependency was added in an attempt to improve the robustness of the model toward climate change under the hypothesis that the change in rainfall pattern would be correlated to the shift in temperature and that the existing observed datasets already carry the necessary information for calibration. In the models MRC_{SSEP}, MRC_{SISEP} and MRC_{SITSEP} (Table 2) the weights generated were permuted stochastically depending on the neighbour (indicated with SEP, Eqs. 4 and A5), while the MRC_{S}, MRC_{SI}, and MRC_{SIT} model considered equal probability (0.5) to permute the two children weights.
Similarly to Rupp et al. (2009), the generators of the MRC models include timescale dependency through analytical formulas. In practice it means that there is a single set of parameters per generator and not a different set at each cascade step. It ensured a relatively parsimonious number of parameters compared to other recent works on microcanonical MRC where a set per cascade step is often used (e.g. 12 to 36 total parameters by Bürger et al., 2019 or from 6 to 224 parameters per cascade step by MüllerThomy, 2020). It should be noted that based on dataset analysis and as advised by Serinaldi (2010), despite the fact that universal and canonical MRC represent the most parsimonious approach parameterwise, their microcanonical counterpart was preferred. This choice of model that deviates from the hypothesis of multifractal scale invariance was supported by several other studies (Koutsoyiannis and Langousis, 2011; Veneziano et al., 2006). The number of parameters can be lower with the use of universal MRCs which were excluded from this study due to lack of flexibility (Serinaldi, 2010). It also allows the model to be used with any desired initial resolution lower than 1500 min. Homogeneity of the resolution in the input datasets was not required for calibration and data processing (i.e. the model can be calibrated using multiple datasets with different resolutions between 1 min and 1 d). The parameters of each generator of MRC models and each location required calibration. A singlestep calibration, based on the processed data, was sufficient for generators with only timescale dependency. A multiplestep calibration with data manipulation was necessary for generators with depth/intensity, temperature dependency, and for the nonzeroweight generator. This choice was motivated by the development of the model through data analysis and conceptualization of the model. Especially, the steps and calibration of the three generators were chosen to avoid compensation between processes using a bottomup approach (i.e. starting from local properties and then add dependencies to progressively upscale the model). Additionally, the choice of regression over timescale was used to avoid parameter sets that lead to the correct distribution of precipitation intensities without temporal consistency. Later studies can further improve the procedure to make it more easily calibrated. The optimizations were based on nonlinear least squares in the standard library scipy.optimize with default parameters in Python (Virtanen et al., 2020).

The parameters of zeroweight generator with only timescale dependency (Eq. A1) followed a singlestep calibration against observed zeroweight proportions by nonlinear least squares.

The parameters of the zeroweight generator with timescale and depth dependency (Eqs. A2a–A2d) followed a twostep calibration: (i) for each timescale, the proportion of zeroweight depending on depth was evaluated using a weighted running window to compensate for rare occurrence of extreme depths. The proportion of zero weight depending on depth was then fitted to a function (Eq. A2a). (ii) The functions modelling the parameters depending on timescale were then calibrated by least squares (parameters of Eqs. A2b, A2c and A2d).

The parameters of zeroweight generator with timescale, depth and temperature dependency (Eqs. A3a–A3e) followed a similar calibration procedure. (i) Using running windows of temperature, the proportion of zeroweight depending on depth was fitted by least squares for different temperatures (Eq. A3a). (ii) Given a timescale the parameters depending on temperature were fitted to a Gaussian function (Eq. A3b). (iii) The parameters of the Gaussian function depending on timescale were then fitted to set of functions by least squares (Eqs. A3c, A3d and A3e).

The nonzeroweight generator consisted of a truncated normal distribution on [0, 0.5] with μ=0.5 (Eq. 3b) and a function σ depending on timescale (Eq. A4). It was chosen against more commonly used beta distributions (McIntyre et al., 2016) after a goodnessoffit test was applied to the historical data. The calibration was done in two steps. (i) σ was evaluated by nonlinear least squares for each timescale. (ii) The parameters of Eq. (A4) were calibrated against the evaluated σ depending on timescale by least squares.

The parameters of the SEP generator (Eq. A5) followed a singlestep calibration by least squares with processed proportion of high weight on the side of highest neighbour depending on timescale.
2.3 Green Infrastructure modelling
In order to quantify the influence of rainfall input in green roof performance estimation, two green roofs located in Trondheim were modelled. They were selected due to data availability and the contrast of their behaviours: (i) a typical extensive green roof (EGreen roof) with sedum vegetation, 30 mm of substrate, and 10 mm “eggbox” drainage layer (Hamouz and Muthanna, 2019), and (ii) a detentionbased extensive green roof (DGreen roof) with sedum vegetation, 30 mm of substrate, and 100 mm of lightweight clay aggregates (Hamouz et al., 2020). The model (Eqs. 5a–5c) was a simple reservoir model with smoothed linear function (Eq. 5c) for the outflow, Oudin's model for potential evapotranspiration (PET, Eq. 5b) and a soil moisture evaluation function to estimate actual evapotranspiration (AET) (Johannessen et al., 2017).
WC_{i} is the water content (mm) at time t_{i}. P_{i} is the precipitation (mm min^{−1}). The discharge Q_{i} (mm min^{−1}) is based on the empirical (Eq. 5c). The temperature T_{mean} is in degrees Celsius, and the extraterrestrial radiation R_{a} is derived from the latitude and the Julian day. The constant $\frac{\mathrm{1}}{\mathit{\lambda}\mathit{\rho}}\approx \mathrm{0.408}$ depends on latent heat and volumetric mass of water. The factor C is a calibrated factor depending on the maximum storage and the crop factor. The smoothed linear function (Eq. 5c) has three parameters: K the conductivity slope, S_{K} the smoothing factor and WC_{K} the starting delay. The model was developed based on data from extreme tests with artificial precipitation (Hamouz et al., 2020) by establishing a relationship between water content and runoff. The outflow depending on water content was used as input for calibration of the parameters of the discharge function using a Bayesian calibration with DREAM setup (Laloy and Vrugt, 2012). It should be noted that the model remains limited as it lumps processes and neglects dynamical effect, i.e. the wetting of the aggregates and substrate and the spatial distribution of water content within the roof (Hamouz et al., 2020). The DGreen roof's model was tested against measured discharge with, as input, a rainfall series of 2.5 months from 10 July to 25 September 2018 and a 1month series from 5 September to 5 October 2019. The EGreen roof's model was tested against measured discharge with a rainfall series from April 2017 to September 2017 as input. Snow periods were mostly excluded for the evaluation.
2.4 Evaluation of the downscaling models
For each location, the observed precipitations were aggregated to daily resolution and downscaled to obtain 200 time series of 6 min time step. They were used to model the extensive and detentionbased extensive green roofs in parallel. It should be noted that irrigation needs and snow periods were neglected since the primary objective of the study was to evaluate the produced time series. There were 10 projections available in Norway for the RCP 8.5 and 12 in France with the EUROCORDEX project. Each projected time series was downscaled 20 times (200 simulations for Norwegian locations, and 240 simulations for French locations) to capture the following: (i) the variability between the projections and (ii) the variability due to the nature of the downscaling model. The number of simulations per location and per period was chosen to ensure reasonably low simulation time and represent the stochastic uncertainty inherent to the downscaling process. The stability of the percentile estimator with 200 simulations was verified against 1000 simulations in one model and one location to validate the choice.
To evaluate the performance of the downscaling model and the projected performance of green roofs, different indicators were used:

The lag1 autocorrelation depending on timescale was evaluated. It was chosen to assess the temporal structure of the produced time series. The autocorrelation depending on lag time for timescale 6, 48 and 180 min were used for an indepth analysis.

The survival distribution of precipitation and discharge from both roofs were assessed at 6 min time steps. This approach is similar to the use of flow duration curves recently applied to green roofs by Johannessen et al. (2018). The exceedance probabilities were presented with a log axis to account for extreme probabilities. The median, 5th and 95th percentiles of the downscaled time series were represented. The survival distribution of discharge from the roofs with downscaled time series compared to the distribution based on observed data indicates the applicability of the downscaled time series as an input for green infrastructure modelling.

Along with the survival distribution, a performance indicator derived from the Kolmogorov–Smirnov (KS) distance was used. The KS distance was indeed not relevant for the survival distributions where the extreme probabilities are of prime importance. The authors did not find a standard indicator for such cases in the literature; therefore the following indicator, which penalizes more errors for extreme probabilities, was developed:
$$\begin{array}{}\text{(6)}& {\mathrm{KS}}_{\mathrm{rel}}=max\left({\displaystyle \frac{{\mathrm{Distrib}}_{\mathrm{Sim},\mathrm{median}}{\mathrm{Distrib}}_{\mathrm{Obs}}}{{\mathrm{Distrib}}_{\mathrm{Obs}}}}\right).\end{array}$$ 
Three different discharge thresholds were used to report exceedance frequency on different operating modes: 1 L s^{−1} ha^{−1} for small events, 10 L s^{−1} ha^{−1} for major events and 100 L s^{−1} ha^{−1} for extreme events. Those thresholds were chosen in common for all roofs to facilitate comparison. They represent a compromise to have the same operating modes for each location even if the occurrence of those modes differs due to different climate conditions. Small event durations were counted in days per year, major events in hours per year and extreme events in minutes per year.

The distribution of dry periods and the retention fraction were computed. They are not expected to be affected by the downscaling process since the dry periods affecting the roofs can be observed on daily resolution, and the retention fraction can be estimated with conceptual models using daily timestep data. However, they provide additional information to analyse the behaviour of the roofs.
2.5 Hybrid eventbased downscaling
In order to assess the applicability of downscaled time series to predict the future performance of green infrastructure, the methods were compared to the current recommended practice in the locations: the use of an eventbased design method based on IDF curves with a climate factor (CF) (Kristvik et al., 2019). In particular, the variational method (Alfieri et al., 2008) is applied. It consists, given a return period, in considering the constantintensity rainfall leading to the highest discharge. It should be noted that the comparison intended to follow the recommended design method and not to follow the guidelines of a specific city since they can differ in terms of regulation. For instance, in Trondheim a threshold for maximum discharge has to be fulfilled (Trondheim Kommune, 2015), while in Lyon the first 15 mm of a 20year return period has to be retained, and beyond those 15 mm a threshold is set for maximum discharge from the parcel (Greater Lyon council, 2020). The longest available time series, which originated from Trondheim, was the most adequate for this example. For 2, 5 and 10year return period rainfall and runoff events, three approaches were compared: (i) peak runoff of runoff events based on an observed precipitation time series (reference), (ii) peak runoff of rainfall events based on variational method, the IDF curves and with and without climate factor (typical design approach), and (iii) a hybrid approach based on downscaling 10^{5} rainfall events with a daily depth based on the return period curves with and without climate factors. This last approach used the MRC_{SITSEP} model, the initial water content was set to the most probable value based on analysis of a long time series. According to the current recommendation in Norway for Trondheim municipality, a climate factor of 1.4 was applied (Dyrrdal and Førland, 2019).
3.1 Green infrastructure model
The parameterized empirical reservoir model was applied to the extensive green roof and the detentionbased extensive green roof. The performance was evaluated both on the time series and individual events extracted from the time series. The criteria were (i) Nash–Sutcliffe efficiency (NSE) indicator on time series for both discharge and water content, (ii) NSE for rainfall events defined with a minimum inter events time of 6 h to analyse further the behaviour of the model, and (iii) the volumetric error on the time series to account for model retention evaluation. The water content was estimated directly from discharge measurement using the empirical curve. The performance was as follows:

NSE>0.8 for both discharge and water content for the extensive green roof. For the three most intense events the NSE ranged from 0.9 to 0.75. The water balance error was found to be 2.1 %.

NSE>0.94 for both discharge and water content for the detentionbased extensive roof. For the three most intense events the NSE ranged from 0.96 to 0.85. The water balance error was found to be 5 %.
The conceptual limitation of the model can be seen in Fig. 2 at the beginning of the events of the testing period. It suggests that short events with low intensity are not reproduced well by the model as it cannot represent the delay induced by the wetting of the different layers of the roofs. Since the objectives of this study involve the use of a simple model to reproduce the behaviour of two roofs, the model was not further improved.
3.2 Analysis of climates properties
Figure 3 presents the zeroweight proportion depending on timescale, depth and temperature for two different datasets (Bodø and Hamar). In Fig. 3a the proportion of zeroweight decreases with increasing timescale for Bodø. In Hamar the proportion decreases until 45 min and increases for higher timescales. Based on this observation, two types of datasets were identified in terms of zeroweight occurrence. For data from Bodø, Bergen, Kristiansund and Trondheim, the proportion of weights that equalled zero decreased with increasing timescale. For the data from Hamar, Lyon, Marseille and Kristiansand, the proportion decreased until 45 min timescale and increased afterwards (Fig. B1a). Given a timescale, the proportion of weight equal to zero was not uniform depending on the weights (e.g. Bodø and Hamar Fig. 3b with a timescale of 48 min). Therefore, the monotony or nonmonotony of the proportion of weights equalling to zero depending on timescale can be explained by different distributions of depth in the observed data. The proportion depended on depth, which is consistent with previous work (Rupp et al., 2009). It should be noted that a high proportion of zeroweight is linked to shorter and more intense rainfall events. It could explain why the proportion is higher in Lyon than in Bergen (cf. Appendix B, Fig. B1a).
In Fig. 3b, the zeroweights proportion decreases with increasing depth for the case of Bodø. In the case of Hamar, it increases for depth higher than 2 mm. Figure 3c and d show that a temperature dependency may explain this behaviour. In Bodø, the proportion depending on depth gives similar results for different ranges of temperature at 48 min resolution (Fig. 3c). On the contrary, in Hamar, the subsets with lower temperature lead to a lower proportion of weights being equal to zero, compared to subsets with higher temperature (Fig. 3d). Moreover, the higher depths were observed in subsets with higher temperature. The increase observed in Hamar can be explained by the distribution of observed values. It is consistent with the observation of different temporal distributions of rainfall for different temperature ranges such as convective rains (Berg et al., 2013; Zhang et al., 2013). If, given a depth of 10 mm at resolution of 48 min, the probability to have a weight equal to zero is higher, then there is a higher probability to have an intense rainfall. The nonhomogeneity of observed datasets and the shift in temperature with climate change might lead to inconsistency in time series produced by the downscaling methods that exclude depth and/or temperature dependency. The 48 min timescale was chosen to exemplify these properties. The same properties can be observed for different timescales, but the magnitude differs and tends to lower with higher timescale (Fig. B1b, c and d). Developing a model without temperature dependency might prevent comparability of parameters between locations and does not necessarily lead to parameter parsimonious models. Moreover, a model such as MRC_{SI} can result in overfitting when used with datasets like Hamar. The functions necessary to represent the behaviour without considering the temperature dependency are more complex and less explanatory. Based on this analysis it was possible to add the temperature dependency and conceptualize a more explanatory model (MRC_{SIT}, with Eqs. A3a–A3e) with more robust results for the influence of climate change. This underlies the hypothesis that the information about the correlation between rainfall and precipitation will be expressed in the same way through those variables in the future.
3.3 Evaluation of the downscaling methods
An overview of the performance of the downscaling and green roof models in Bergen is presented on Fig. 4. All the downscaling models performed similarly in terms of dryperiod distribution and slightly underestimated the dry periods in observed data (Fig. 4b). The dry periods were directly linked to the zeroweight probability. In green infrastructure modelling, the length of the dry periods influences the retention performance as it can lead to water stress hindering evapotranspiration. However, dry periods leading to water stress can be also evaluated with daily timestep series (there is no need for minute timestep series). Therefore, dry periods longer than the initial daily resolution are not significantly affected by downscaling.
The distribution of precipitation (Fig. 4a) was properly reproduced by MRC_{SI}, MRC_{SISEP}, MRC_{SIT}, and MRC_{SITSEP} (KS_{rel}=1 in this case, indicating that the maximum distance has the same order of magnitude in data and model results) models, while MRC_{S} and MRC_{SSEP} underestimated low precipitation and overestimated high precipitation depths (KS_{rel}=10^{2} meaning that the maximum distance reached 2 orders of magnitude). This was expected as the time steps with high depth have higher probability to not be split in the observed data. This is not the case for MRC_{S} and MRC_{SSEP} models, for which probability is uniformly distributed. In Bergen, the observed precipitations were within the range of 90 % coverage interval for MRC_{SI}, MRC_{SISEP}, MRC_{SIT}, and MRC_{SITSEP}. For the four later mentioned models, the discharge of the DGreen roof was underestimated by 1 order of magnitude with a KS_{rel} of 1.7×10^{1} (Fig. 4c), due to the behaviour of the roof with rare high discharge. The hyetographs produced by downscaling probably tend to generate less favourable hyetographs for this roof. Although the discharge of the EGreen roof did not fall in the 90 % coverage interval, it can be considered as slightly underestimated since the magnitude is similar with a KS_{rel} of 2.0 (Fig. 4d). However, this was not the case for all locations, as in Hamar the most extreme precipitation tended to be underestimated, while the discharge from both roofs had the same order of magnitude as the observed data but tended to be overestimated. These findings could suggest inconsistency in the temporal structure of rainfall. This hypothesis can be confirmed by the autocorrelation (Fig. 5) being overestimated at 6 min time step.
The autocorrelation was underestimated by MRC_{S} and MRC_{SSEP} models. The use of the rainfall continuity indicator increased the lag1 autocorrelation for all models but did not improve the overall performances. The models MRC_{SI}, MRC_{SISEP}, MRC_{SIT}, and MRC_{SITSEP} underestimated the lag1 autocorrelation between 48 and 300 min timescales, but an indepth analysis with different lags at 48 and 180 min timescale shows that despite that underestimation for lag1 the general behaviour of the observed time series is reproduced. Similar observations were done for other locations.
To evaluate the produced time series it is necessary to compare the discharge with observed time series to the discharge with downscaled time series. For most of the locations, the predicted range of precipitation or discharge deviated for lowest probabilities from the values obtained with observed time series: (i) when the precipitation range matched with the observed distribution, the discharge tended to be overestimated; (ii) when the precipitation was underestimated, the discharge with observed data tends to lie in the range obtained from downscaled time series. While the downscaled time series suffer from some limitation when compared to results obtained from the observed time series, the raw discharge time series might as well not be suitable for robust decision making in green infrastructure implementation as it does not represent the natural variation of performance of green infrastructure.
In order to evaluate the potential of discharge from downscaled time series to approach the range of performance linked to natural variability, a 3year moving window was used on precipitation and discharge time series resulting from observed precipitation. The resulting 5th and 95th percentiles of the annual duration exceeding 1, 10 and 100 L s^{−1} ha^{−1} are presented in Fig. 6 to evaluate the time series in different operating modes of the roofs. It is compared to the stochastic variability (5th and 95th percentiles) from the six models. Each horizontal line in Fig. 6 represents the range between 5th and 95th percentiles for the threshold and model considered. The different thresholds represent the discharge for respectively small events, for major events and extreme events. In Fig. 4, the thresholds correspond to 0.006, 0.06 and 0.6 mm min^{−1}. A good estimate is defined by a complete or partial overlap between the observed natural variability and the stochastic variability range; the order of magnitude of the estimates should be similar. For instance, in Bergen (first column), the observed range of the EGreen roof higher than 100 L s^{−1} ha^{−1} (third row) is predicted, based on observed input, from 4 to 10 min; the MRC_{S} model provided values around 200 min; it is not a good estimate as there is no overlap and the order of magnitude varies; the MRC_{SI} model resulted in a range from 10 to 20 min. It is a good estimate as the ranges are overlapping, and the orders of magnitude are similar. The MRC_{S} and MRC_{SSEP} models tend to underestimate the order of magnitude of the range of exceedance frequencies of the small events (1 L s^{−1} ha^{−1}) (in Bergen, Hamar and Marseille) but tend to overestimate major (10 L s^{−1} ha^{−1}) (Hamar) and extreme events (100 L s^{−1} ha^{−1}) (Bergen, Bodø, Hamar and Marseille). The other models gave mostly good estimates for each of the thresholds (Figs. 6, C1). In Marseille, the models MRC_{SI}, MRC_{SISEP}, MRC_{SIT}, and MRC_{SITSEP} tended to underestimate the higher bound of the extreme event precipitation with values lower than 50 min yr^{−1}, whereas the observed time series led to a maximum of 90 min yr^{−1}. However, those models kept the order of magnitude, while MRC_{S} and MRC_{SSEP} models estimated it higher than 10^{2} min. The same behaviour was observed with Hamar (Fig. 6) and Lyon datasets (Appendix C, Fig. C1). This suggests that the models performed worse for dryer locations, possibly due to the calibration procedure since fewer wet days are available for calibration. MRC_{SI} and MRC_{SIT} performed similarly, but due to its structure, MRC_{SI} may overfit to the calibration data. It could result in an inaccurate prediction in the case of significant temperature shift between the calibration and prediction datasets. To conclude, MRC_{S} and MRC_{SSEP} lead to overestimation of the natural variability, while MRC_{SI}, MRC_{SISEP}, MRC_{SIT}, and MRC_{SITSEP} give more accurate estimates.
3.4 Assessment of green roof future performance
All six models were used to assess the performance of green roofs for future climate as illustrated for Bergen in Fig. 7. It was nevertheless acknowledged that MRC_{S} and MRC_{SSEP} models gave less accurate estimates. The four model MRC_{SI}, MRC_{SISEP}, MRC_{SIT}, and MRC_{SITSEP} lead to similar results in Bergen (Fig. 7). The difference in estimates between the models with coherence indicator (MRC_{SISEP}, MRC_{SITSEP}) and without (MRC_{SISEP}, MRC_{SITSEP}) was negligible in comparison to the stochastic uncertainty inherent to the models and the variability linked to the different projections available under RCP 8.5 (Fig. 7). In Bergen, according to the projections, the performance of the two solutions is likely to lead to worse performance: under the current climate, the 100 L s^{−1} ha^{−1} exceedance was lower than 1 min for the DGreen roof; according to the MRC_{SITSEP} model it might reach between 5 and 19 min in future climate. It suggested a shift in the order or magnitude from 10^{0} to more than 10^{1} min. Similarly, the EGreen roof might have a 100 L s^{−1} ha^{−1} exceedance shift from 10^{1} to 10^{2} min. It means that the threshold would regularly be reached. As illustrated by Figs. 8 and C2, the performance shift depends highly on the location. While the 100 L s^{−1} ha^{−1} exceedance of the green roofs was likely to get worse in Bergen, it was found to stay stable despite a small increase in Bodø and to improve in Hamar and Marseille. The increase of exceedance frequency in the Norwegian cities was due to an increase in precipitation (Table 1). However, the increase in temperature led to an increase in potential evapotranspiration and therefore might have attenuated or even counterbalanced the effect of rainfall increase by lowering the initial water content in the roofs at the beginning of a rainfall event. The Table 3 shows that the retention fraction was likely to decrease in Bergen, Bodø, Hamar, Kristiansand and Kristiansund. It was found to increase more significantly in Lyon, Marseille and slightly in Trondheim. The models with temperature dependency performed similarly to the model with only depth dependency in most of the locations. However, in Lyon and Marseille, the 100 L s^{−1} ha^{−1} exceedance or precipitation predicted differed from 16–27 to 21–50 min (14–30 to 14–43 in Marseille). This suggests that some locations are more sensitive than others to temperaturedependent patterns. The models MRC_{SI}, MRC_{SISEP}, MRC_{SIT}, and MRC_{SITSEP} allow the evaluation of shift in performance for the different roofs using exceedance range.
3.5 Design perspectives
The potential of downscaling models to improve the current practices was investigated. Figure 9 presents results based on continuous simulation, on the variational method and on the hybrid approach with downscaled events. It shows that the variational method underestimated the peak runoff with observed data, and the distribution from the hybrid approach covered them. It suggests that the variational method might not be enough conservative when compared to peak runoff from runoff events instead of rainfall events. Even if the results from the hybrid eventbased downscaling lead to realistic distribution based on probable rainfall events, the downscaling models might need a different calibration or conceptualization to be optimized specifically for extreme events. Moreover the initial water content for the events remains a limitation of this method. The observed peaks show a range of possible outcomes, which highlights the limitations of the variational method with a single estimate, whereas the hybrid downscalingevent based method, leading to a range of probable outcomes, gave promising results that can lead to more robust design and decision making. Due to its characteristics, the shift in performance between current climate and future climate is higher for the EGreen roof than for the DGreen roof. This is due to the detention layer in the DGreen roof which is not saturated by a 10year return period event (Hamouz et al., 2020).
In this study, multiplicative random cascades models with different variable dependency were developed. They were based on a study of timescale, depth, and temperature characteristics of the datasets to ensure a consistent structure in the view to apply them to dailyresolution climate projections. The applicability of the synthetic time series to be used as input for performance modelling of green infrastructure was evaluated. They were used to predict the shift in runoff exceedance under a future climate.
Six downscaling models were developed: two models with only timescale dependency (MRC_{S} and MRC_{SSEP}), two models with timescale and depth dependency (MRC_{SI} and MRC_{SISEP}), and two models with timescale, depth and temperature dependency (MRC_{SIT} and MRC_{SITSEP}). The models MRC_{SSEP}, MRC_{SISEP} and MRC_{SITSEP} include a rainfall continuity property with the intention of improving the temporal structure of the rainfall. The parametrization of the models ensures the continuity of the different properties modelled and a low number of parameters.
The MRC_{S} and MRC_{SSEP} were not sufficient to predict the future performance of green infrastructure as they lead to an overestimation of runoff; the MRC_{SI}, MRC_{SISEP}, MRC_{SIT}, and MRC_{SITSEP} lead to better performance: it was possible to predict runoff exceedance frequency with similar order of magnitude as the estimate of the natural variability of performance based on observed time series. The structure of the MRC_{SI} and MRC_{SISEP} models makes them more vulnerable to overfitting than MRC_{SIT} and MRC_{SITSEP}, which makes them less reliable for future performance estimates. However, the differences between them were negligible compared to the variability linked to the different outcome of climate models, the variability inherent to the model and its accuracy. The MRC_{SSEP}, MRC_{SISEP} and MRC_{SITSEP} add an equation to improve the temporal structure of downscaled rainfall. The models predicted higher runoff from the detentionbased extensive green roof, which is consistent with their properties; however the change in performance was not significant compared to stochastic uncertainty.
Using the RCP 8.5, the different downscaling and the green roof models suggest that the performance shift due to climate change highly depends on the location. The runoff exceedance is likely to increase in Bergen while slightly decreasing in Lyon and Marseille and keeping the same order of magnitude in the other locations. The results were compared to one of the current practices: the use of the variational method with a climate factor. It highlighted the limitation of this practice that provides a singular estimate and underestimates the observed peaks. A hybrid method using downscaling on extreme events led to promising results by estimating a distribution of performance of peak runoff.
The models performed well in the eight locations and four different climates. The use of a more advanced calibration procedure with Bayesian methods should improve the results. Similarly, a sensitivity analysis could improve the parametrization, especially for the models with depth and temperature dependency in order to fix nonbehavioural parameters. The current study does not include irrigation, and snow modelling a study centred on green infrastructure modelling is therefore needed to extend the results. In order to be applied in practice on eventbased simulation for design perspectives, the downscaling models need to be improved with a calibration procedure developed for extreme events and not on the complete spectrum of observation as in the current study.
A1 Zeroweight generator with only timescale dependency
A2 Zeroweight generator with both timescale and depth dependency
A3 Zeroweight generator with timescale, depth and temperature dependency
A4 Nonzeroweight generator
It consists in a truncated normal distribution described by Eq. (3b). The function σ depends on timescale:
A5 SEP generator
The stochastic element permutation follows a function generating the threshold to be compared to a uniformly generated random number depending on timescale:
Abbreviation  Meaning 
GI  Green infrastructures 
MRC  Multiplicative random cascades 
IDF curves  Intensity–duration–frequency curves 
NVE  Norwegian Water Resources and Energy Directorate 
MET  Norwegian Meteorological Institute 
PET  Potential evapotranspiration 
AET  Actual evapotranspiration 
EGreen roof  Extensive green roof 
DGreen roof  Detentionbased extensive green roof 
RCP 8.5  Representative Concentration Pathway scenario with a 8.5 W m^{−2} radiative forcing in 2100 
NSE  Nash–Sutcliffe efficiency 
VM  Variational method 
S_{i,2j}  Rainfall continuity indicator at time step i and timescale 2j 
d_{i,2j}  Depth [mm] at time step i and timescale 2j 
w_{i,2j}  Minimum weight at time step i and timescale 2j from aggregation of depths D_{2i,j} and ${D}_{\mathrm{2}i+\mathrm{1},j}$ 
S_{W}  Random discrete variable of neighbour of the highest weight 
CF  Climate factor 
The code is still under development and not sufficiently commented at the moment for dissemination. It is the intent of the authors to make it publicly accessible in the future. In the meantime code is available by communication to the author under reasonable request.
The data from the Norwegian locations can be provided by the authors upon request. The Lyon and Marseille rainfall data, owned by the municipalities, have been made available to the authors. The authors do not own the data and cannot make them publicly accessible.
VP was responsible for developing and programming the downscaling and green roofs models. RB provided his expertise in downscaling. JLBK came up with the idea of comparison with the variational method. The Norwegian Meteorological Institute, represented by RB, provided the Norwegian data, JLBK provided the French data. TMM, ES and JLBK supervised each step of the study. VP wrote the first manuscript. The manuscript was revised by all coauthors.
The contact author has declared that neither they nor their coauthors have any competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The study was supported by the Klima2050 Centre for Researchbased Innovation (SFI) and financed by the Research Council of Norway and its consortium partners (grant no. 237859/030). This research was partly performed within the framework of the OTHU (Field Observatory for Urban Water Management – http://othu.org, last access: 26 May 2022) and realized within the Graduate School H2O'Lyon (ANR17EURE0018) and Université de Lyon (UdL) as part of the program “Investissement d'Avenir” and by Agence Nationale de la Recherche (ANR). The authors acknowledge the Water Department of Lyon Metropole and the Water and Sewerage Department of Marseille Provence Metropole for sharing their rainfall data. The authors would like to thank the Norwegian Meteorological Institute for providing the Norwegian data and especially Cristian Lussana and Lars Grinde for their advice.
This research has been supported by the Norges Forskningsråd (grant no. 237859/030).
This paper was edited by MarieClaire ten Veldhuis and reviewed by two anonymous referees.
Alfieri, L., Laio, F., and Claps, P.: A simulation experiment for optimal design hyetograph selection, Hydrol. Process., 22, 813–820, https://doi.org/10.1002/hyp.6646, 2008. a
Benestad, R.: Downscaling Climate Information, in: Oxford Research Encyclopedia of Climate Science, Oxford University Press, https://doi.org/10.1093/acrefore/9780190228620.013.27, 2016. a
Berg, P., Moseley, C., and Haerter, J. O.: Strong increase in convective precipitation in response to higher temperatures, Nat. Geosci., 6, 181–185, https://doi.org/10.1038/ngeo1731, 2013. a
Bürger, G., Heistermann, M., and Bronstert, A.: Towards subdaily rainfall disaggregation via clausiusclapeyron, J. Hydrometeorol., 15, 1303–1311, https://doi.org/10.1175/JHMD130161.1, 2014. a
Bürger, G., Pfister, A., and Bronstert, A.: Temperaturedriven rise in extreme subhourly rainfall, J. Climate, 32, 7597–7609, https://doi.org/10.1175/JCLID190136.1, 2019. a, b
Dyrrdal, A. V. and Førland, E. J.: Klimapåslag for korttidsnedbørAnbefalte verdier for Norge, NCCS report 5, https://klimaservicesenter.no/kss/rapporter/rapporterogpublikasjoner_2 (last access: 26 May 2022), 2019. a
Dyrrdal, A. V., Stordal, F., and Lussana, C.: Evaluation of summer precipitation from EUROCORDEX finescale RCM simulations over Norway, Int. J. Climatol., 38, 1661–1677, https://doi.org/10.1002/joc.5287, 2018. a, b
Gaur, A. and Lacasse, M.: Multisite multivariate disaggregation of climate parameters using multiplicative random cascades, Urban Climate, https://doi.org/10.1016/j.uclim.2018.08.010, 2018. a
Gires, A., Tchiguirinskaia, I., and Schertzer, D.: Blunt extension of discrete universal multifractal cascades: development and application to downscaling, Hydrolog. Sci. J., 65, 1204–1220, https://doi.org/10.1080/02626667.2020.1736297, 2020. a
Glasbey, C. A., Cooper, G., and McGechan, M. B.: Disaggregation of daily rainfall by conditional simulation from a pointprocess model, J. Hydrol., 165, 1–9, https://doi.org/10.1016/00221694(94)025986, 1995. a
Greater Lyon council: Règlement du service public de l'assainissement collectif [Stormwater and wastewater service regulation], City guideline, Greater Lyon, https://www.grandlyon.com/services/gestiondeseauxdepluie.html (last access: 10 November /2021), 2020. a
Gupta, V. K. and Waymire, E. C.: A Statistical Analysis of Mesoscale Rainfall as a Random Cascade, J. Appl. Meteorol. Clim., 32, 251–267, https://doi.org/10.1175/15200450(1993)032<0251:ASAOMR>2.0.CO;2, 1993. a
Hamouz, V. and Muthanna, T. M.: Hydrological modelling of green and grey roofs in cold climate with the SWMM model, J. Environ. Manage., 249, 109350, https://doi.org/10.1016/j.jenvman.2019.109350, 2019. a
Hamouz, V., Pons, V., Sivertsen, E., Raspati, G. S., BertrandKrajewski, J.L., and Muthanna, T. M.: Detentionbased green roofs for stormwater management under extreme precipitation due to climate change, BlueGreen Systems, 2, 250–266, https://doi.org/10.2166/bgs.2020.101, 2020. a, b, c, d, e
Jacob, D., Petersen, J., Eggert, B., Alias, A., Christensen, O. B., Bouwer, L. M., Braun, A., Colette, A., Déqué, M., Georgievski, G., Georgopoulou, E., Gobiet, A., Menut, L., Nikulin, G., Haensler, A., Hempelmann, N., Jones, C., Keuler, K., Kovats, S., Kröner, N., Kotlarski, S., Kriegsmann, A., Martin, E., van Meijgaard, E., Moseley, C., Pfeifer, S., Preuschmann, S., Radermacher, C., Radtke, K., Rechid, D., Rounsevell, M., Samuelsson, P., Somot, S., Soussana, J.F., Teichmann, C., Valentini, R., Vautard, R., Weber, B., and Yiou, P.: Erratum to: EUROCORDEX: new highresolution climate change projections for European impact research, Reg. Environ. Change, 14, 579–581, https://doi.org/10.1007/s101130140587y, 2014. a
Johannessen, B. G., Hanslin, H. M., and Muthanna, T. M.: Green roof performance potential in cold and wet regions, Ecol. Eng., 106, 436–447, https://doi.org/10.1016/j.ecoleng.2017.06.011, 2017. a, b
Johannessen, B. G., Muthanna, T. M., and Braskerud, B. C.: Detention and Retention Behavior of Four Extensive Green Roofs in Three Nordic Climate Zones, Water, 10, 671, https://doi.org/10.3390/w10060671, 2018. a
Kalra, A. and Ahmad, S.: Evaluating changes and estimating seasonal precipitation for the Colorado River Basin using a stochastic nonparametric disaggregation technique, Water Resour. Res., 47, W05555, https://doi.org/10.1029/2010WR009118, 2011. a
Koutsoyiannis, D. and Langousis, A.: 2.02 – Precipitation, in: Treatise on Water Science, edited by: Wilderer, P., 27–77, Elsevier, Oxford, https://doi.org/10.1016/B9780444531995.000270, 2011. a, b
Kristvik, E., Johannessen, B. G., and Muthanna, T. M.: Temporal Downscaling of IDF Curves Applied to Future Performance of Local Stormwater Measures, Sustainability, 11, 1231, https://doi.org/10.3390/su11051231, 2019. a, b, c
Laloy, E. and Vrugt, J. A.: Highdimensional posterior exploration of hydrologic models using multipletry DREAM (ZS) and highperformance computing, Water Resour. Res., 48, W01526, https://doi.org/10.1029/2011WR010608, 2012. a
Li, X., Meshgi, A., Wang, X., Zhang, J., Tay, S. H. X., Pijcke, G., Manocha, N., Ong, M., Nguyen, M. T., and Babovic, V.: Three resampling approaches based on method of fragments for dailytosubdaily precipitation disaggregation, Int. J. Climatol., 38, e1119–e1138, https://doi.org/10.1002/joc.5438, 2018. a
Lombardo, F., Volpi, E., and Koutsoyiannis, D.: Rainfall downscaling in time: theoretical and empirical comparison between multifractal and HurstKolmogorov discrete random cascades, Hydrolog. Sci. J., 57, 1052–1066, https://doi.org/10.1080/02626667.2012.695872, 2012. a
Lu, Y., Qin, X. S., and Mandapaka, P. V.: A combined weather generator and Knearestneighbour approach for assessing climate change impact on regional rainfall extremes, Int. J. Climatol., 35, 4493–4508, https://doi.org/10.1002/joc.4301, 2015. a
Lutz, J., Grinde, L., and Dyrrdal, A. V.: Estimating rainfall design values for the City of Oslo, Norwaycomparison of methods and quantification of uncertainty, Water (Switzerland), 12, 1735, https://doi.org/10.3390/W12061735, 2020. a
McIntyre, N., Shi, M., and Onof, C.: Incorporating parameter dependencies into temporal downscaling of extreme rainfall using a random cascade approach, J. Hydrol., 542, 896–912, https://doi.org/10.1016/j.jhydrol.2016.09.057, 2016. a, b
MüllerThomy, H.: Temporal rainfall disaggregation using a microcanonical cascade model: possibilities to improve the autocorrelation, Hydrol. Earth Syst. Sci., 24, 169–188, https://doi.org/10.5194/hess241692020, 2020. a
Olsson, J.: Evaluation of a scaling cascade model for temporal rain fall disaggregation, Hydrol. Earth Syst. Sci., 2, 19–30, https://doi.org/10.5194/hess2191998, 1998. a
Onof, C., Chandler, R. E., Kakou, A., Northrop, P., Wheater, H. S., and Isham, V.: Rainfall modelling using poissoncluster processes: A review of developments, Stoch. Env. Res. Risk A., 14, 384–411, https://doi.org/10.1007/s004770000043, 2000. a
Oudin, L., Hervieu, F., Michel, C., Perrin, C., Andréassian, V., Anctil, F., and Loumagne, C.: Which potential evapotranspiration input for a lumped rainfallrunoff model? Part 2 – Towards a simple and efficient potential evapotranspiration model for rainfallrunoff modelling, J. Hydrol., 303, 290–306, https://doi.org/10.1016/j.jhydrol.2004.08.026, 2005. a
Paschalis, A., Molnar, P., and Burlando, P.: Temporal dependence structure in weights in a multiplicative cascade model for precipitation, Water Resour. Res., 48, W01501, https://doi.org/10.1029/2011WR010679, 2012. a, b
Peel, M. C., Finlayson, B. L., and McMahon, T. A.: Updated world map of the KöppenGeiger climate classification, Hydrol. Earth Syst. Sci., 11, 1633–1644, https://doi.org/10.5194/hess1116332007, 2007. a
Rupp, D. E., Keim, R. F., Ossiander, M., Brugnach, M., and Selker, J. S.: Time scale and intensity dependency in multiplicative cascades for temporal rainfall disaggregation, Water Resour. Res., 45, W07409, https://doi.org/10.1029/2008WR007321, 2009. a, b, c
Rupp, D. E., Licznar, P., Adamowski, W., and Leśniewski, M.: Multiplicative cascade models for fine spatial downscaling of rainfall: parameterization with rain gauge data, Hydrol. Earth Syst. Sci., 16, 671–684, https://doi.org/10.5194/hess166712012, 2012. a
Sanchez, C., Williams, K. D., and Collins, M.: Improved stochastic physics schemes for global weather and climate models, Q. J. Roy. Meteor. Soc., 142, 147–159, https://doi.org/10.1002/qj.2640, 2016. a
Schertzer, D. and Lovejoy, S.: Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes, J. Geophys. Res.Atmos., 92, 9693–9714, https://doi.org/10.1029/JD092iD08p09693, 1987. a
Schilling, W.: Rainfall data for urban hydrology: what do we need?, Atmos. Res., 27, 5–21, https://doi.org/10.1016/01698095(91)90003F, 1991. a
Serinaldi, F.: Multifractality, imperfect scaling and hydrological properties of rainfall time series simulated by continuous universal multifractal and discrete random cascade models, Nonlin. Processes Geophys., 17, 697–714, https://doi.org/10.5194/npg176972010, 2010. a, b
Stovin, V.: The potential of green roofs to manage Urban Stormwater, Water Environ. J., 24, 192–199, https://doi.org/10.1111/j.17476593.2009.00174.x, 2010. a
Stovin, V., Poë, S., and Berretta, C.: A modelling study of long term green roof retention performance, J. Environ. Manage., 131, 206–215, https://doi.org/10.1016/j.jenvman.2013.09.026, 2013. a
Stovin, V., Vesuviano, G., and DeVille, S.: Defining green roof detention performance, Urban Water J., 14, 574–588, https://doi.org/10.1080/1573062X.2015.1049279, 2017. a
Thober, S., Mai, J., Zink, M., and Samaniego, L.: Stochastic temporal disaggregation of monthly precipitation for regional gridded data sets, Water Resour. Res., 50, 8714–8735, https://doi.org/10.1002/2014WR015930, 2014. a
Thorndahl, S. and Andersen, C. B.: CLIMACS: A method for stochastic generation of continuous climate projected point rainfall for urban drainage design, J. Hydrol., 602, 126776, https://doi.org/10.1016/j.jhydrol.2021.126776, 2021. a
Trondheim Kommune: VAnorm. Vedlegg 5. Beregning av overvannsmengde Dimensjonering av ledning og fordrøyningsvolum [Water and Wastewater Norm. Attachment 5. Calculation of stormwater flows. Design of pipes and detention basins], City guideline, The City of Trondheim, https://www.vanorm.no/trondheim/ (last access: 10 November 2021), 2015. a
Veneziano, D., Furcolo, P., and Iacobellis, V.: Imperfect scaling of time and space–time rainfall, J. Hydrol., 322, 105–119, https://doi.org/10.1016/j.jhydrol.2005.02.044, 2006. a, b
Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S. J., Brett, M., Wilson, J., Millman, K. J., Mayorov, N., Nelson, A. R. J., Jones, E., Kern, R., Larson, E., Carey, C. J., Polat, İ., Feng, Y., Moore, E. W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E. A., Harris, C. R., Archibald, A. M., Ribeiro, A. H., Pedregosa, F., van Mulbregt, P., and SciPy 1.0 Contributors: SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nat. Methods, 17, 261–272, https://doi.org/10.1038/s4159201906862, 2020. a
Walker, W. E., Haasnoot, M., and Kwakkel, J. H.: Adapt or Perish: A Review of Planning Approaches for Adaptation under Deep Uncertainty, Sustainability, 5, 955–979, https://doi.org/10.3390/su5030955, 2013. a
Zhang, Q., Li, J., Singh, V. P., and Xiao, M.: Spatiotemporal relations between temperature and precipitation regimes: Implications for temperatureinduced changes in the hydrological cycle, Global Planet. Change, 111, 57–76, https://doi.org/10.1016/j.gloplacha.2013.08.012, 2013. a
 Abstract
 Introduction
 Methods
 Result and discussion
 Conclusions
 Appendix A: Generators description
 Appendix B: Data analysis
 Appendix C: Other locations performance
 Appendix D: Table of abbreviations
 Code availability
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Abstract
 Introduction
 Methods
 Result and discussion
 Conclusions
 Appendix A: Generators description
 Appendix B: Data analysis
 Appendix C: Other locations performance
 Appendix D: Table of abbreviations
 Code availability
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References