# Ensemble Radar-Based Rainfall Forecasts for Urban Hydrological Applications
## 2 Materials and Methods
The nowcasting module isolates small characteristics (estimation of advection field, temporal evolution of rainfall and spectral decomposition) into multiplicative cascades. This ensures that features which cannot be accurately predicted by the model are substituted by stochastic noise. The model assumes that the rate of temporal evolution of rainfall and temporal development of velocity fields remain stationary during the forecast. Even with the uncertainties inherent to this model, radar nowcasting produces more skilful forecasts than NWP forecasts up to 2 h lead-time, and therefore STEPS uses a multi-cascade approach to blend the two components [7; 36]. For this study, the nowcasting component of the STEPS model was used to producethe forecasts up to 3 h ahead. [PERSON] and [PERSON] [43] produced probabilistic nowcasts using the STEPS model and concluded that with a number of ensembles larger than 10-20 members it does not effectively increase the accuracy of the forecast. In different research papers, a number of ensembles between 20 and 30 are commonly used [44; 45; 46; 47; 48; 49; 50]. In this research, each probabilistic forecast is formed by 25 ensemble members. The radar rainfall errors were modelled using the model proposed by [PERSON] et al. [14]. This model uses the difference between the radar measurements and a reference (e.g., rain gauge observations) obtained from a large historical data set. The perturbation fields are calculated through the covariance matrix of the residual errors, by using radar and rain gauge measurements. Values with no rainfall are excluded from the calculations. Temporal correlation of the errors is imposed using an autoregressive model AR(2) [37]. The perturbation fields (\(\delta\)) are added to the radar rainfall (\(R\)) in the log domain, resulting in radar rainfall ensemble members (\(\phi\)). Radar rainfall ensembles (RE) attempt to assess in a realistic way the residual uncertainties that remain even after the correction algorithms are applied to the radar data. These radar rainfall ensembles are used instead of the original radar images to generate forecasts using the nowcasting model from STEPS. A deterministic forecast is produced using each of the 25 radar ensemble members. As a result, each of these 25 forecasts become the ensemble forecasts. Ensemble forecasts based on this method will be referred to as radar ensembles in this paper, while ensemble forecasts generated using the stochastic ensemble generator by STEPS will be referred as STEPS ensembles. Note that the STEPS ensembles are generated by adding spatially correlated noise to the deterministic radar forecasts. The rainfall forecast output was used as an input in a hydrodynamic sewer network model built in the Infoworks CS software package to simulate the flows in the sewer network. Rainfall-runoff processes and the flow through the sewer network conduits were modelled utilising the Infoworks model provided by Yorkshire Water for research purposes. The sewer system is mainly combined, being used to carry both wastewater and rainfall runoff. The urban area, Ilkley, is located in the Pennine hills and has an area of 11.06 km\({}^{2}\) (Figure 1). This study utilises the geographic grid reference of the Ordnance National Grid for the radar rainfall data [51]. Infoworks CS uses both rainfall-runoff volume and runoff routing models to simulate flows in the catchment and in this study the New UK percentage Runoff model, the Wallingford model and the Double Linear Reservoir model, were used. The full St Venant equations are used in the model to calculate the flows in the sewer conduits [9]. [PERSON] et al. [52] and [PERSON] et al. [9] provide further information about the Infoworks CS model used in this study. The calibration of the hydrological urban model was performed using current industrial standards [53] and data from three storm days and one dry day from events that happened between March and April 2000 were used.