## 3 Methodology  
The runoff coefficient is a dimensionless parameter that represents the relationship between the total runoff volume from the roof and the total amount of precipitation in a certain time period [33]. In impervious roofs, where there is no loss of water by absorption and where water trapped or evaporated is not significant, it has a value near one. For a given reference period (usually day, month, or year), the multiplication of the runoff coefficient by the amount of precipitation in that period corresponds to the volume of water that can be used by the rainwater harvesting system. In the case of green roofs, average annual values between 0.4 and 0.6 for extensive green roofs (green roofs with a maximum soil depth of about 150 mm), or between 0.1 and 0.4 for intensive green roofs (green roofs with a soil depth of 150 mm or more), are generally adopted in central Europe and the UK, according to the FLL (_Forschungsgesellschaft Landschaftsentwicklung Landschaftsbau_) guidelines [33]. In fact, these values depend on the characteristics of the roof, such as the type of plants used and the characteristics of the substrate, and are very dependent on the climatic conditions in the region, especially temperature and precipitation diagrams. In Mediterranean climates, monthly runoff coefficients are particularly important for sizing the storage tanks of rainwater harvesting systems [34] considering the existence of long dry periods, extending, in general, throughout the summer period. The monthly runoff coefficient represents the relationship between the total runoff volume from the green roof during a given month and the total amount of precipitation on the roof in that month added to the volume of possible watering done in that period. From the perspective of integrating green roofs with rainwater harvesting systems, previous research has been conducted by the authors and other researchers on a conventional extensive green roof system in Oporto city (Portugal) [35]. This study revealed low runoff coefficient values, but allowed the development of an expression to predict the monthly runoff coefficients for this type of conventional green roof. The extensive pilot system adopted (Figures 1 and 2) followed the typical extensive green roof structure: geotextile membranes, a water holding capacity layer using expanded clay, and a 10 cm growing substrate composed of a mixture of expanded clay and organic matter. The pilot green roof was established with three different common aromatic plant species: _Satureja montana_, _Thymus caespititius_, and _Thymus pseudolanguinosus_. Figure 1: Green roof schematic representation [35]. Figure 2: Green roof experimental set up [35]. Although the pilot system was relatively small (\(0.70\times 0.70\) m\({}^{2}\)) and could lead to scale distortions in the results, it was thought that these errors would not be significant and that the results met the objective of obtaining a practical mathematical expression that would allow, with acceptable approximation, the determination of average values of the monthly runoff coefficient for a typical extensive green roof [35]. Measurements of the pilot green roof allowed the development of the following expression for monthly runoff coefficient prediction [35]:  
\[C_{M}=\text{K}\frac{\left(P_{M}+R_{M}\right)}{\left(2T_{M}-T_{M-1}\right)^{1.2}} \tag{1}\]  
where K = 0.016 (\({}^{\circ}\)C\({}^{1.2}\) mm\({}^{-1}\)), \(C_{M}\) is the runoff coefficient of month \(M\), \(P_{M}\) is the precipitation of month \(M\) (mm), \(R_{M}\) is the watering in month \(M\) (mm), \(T_{M}\) is the mean air temperature during month \(M\) (\({}^{\circ}\)C), and \(T_{M-1}\) is the mean air temperature during month \(M\)-1 (\({}^{\circ}\)C). The expression obtained, which significantly depends on the temperature in previous periods and precipitation, has similarities with the well-known Turc formula [36] that has been widely used in hydrological studies to determine flow deficit, which can be considered an indicator of its consistency [35].