## 2 Calculation Model  
In this iterative model combining the radiosity method and image source method (IMCRI), street canyons were modeled as uniform rectangles with limited lengths. Sound energy transferred to the upper side and the end of the streets was assumed to be absorbed. The surfaces were divided into small patches (\(2\times 2\) m). The calculation model was implemented as an iterative model, where each iteration models all of the sound energy transferred after a diffuse reflection occurs. For each iteration, the received energy of each receiver position was calculated and stored in a matrix with a time factor. With the increase in the reflection order, the received energy converged toward the full impulse response that was used to calculate the acoustic parameters. Figure 1 shows the schematic diagram of the sound transfer path and the structures of IMCRI and existing hybrid models (EHM). The sound energy was first emitted from the source to each patch and then separated into two independent components: the specular reflected energy and the diffuse reflected energy. The specular reflected energy could be modeled easily by the ISM, while the diffuse reflected energy was saved as patch sources for the next sound energy transfer modeled by ARM. The process in the 1 st reflection was the same in IMCRI and EHM. The difference between IMCRI and EHM began in the 2 nd reflection. As shown in Figure 1b, the entire sound propagation was modeled in EHM by a specular reflection process (modeled by the ISM) and a diffuse reflection process (modeled by the ARM). Figure 1c shows the full sound transfer path, where the red line and blue line represent the components calculated by ISM and ARM in EHM, respectively. As the order increased, there were more neglected components in EHM. Only 75% and 50% of the components were calculated in the 2 nd reflection and in the 3 rd reflection, respectively. It could lead to considerable errors with the increase in the reflection order, especially for acoustic parameters determined by the high-order reflections [26]. This problem was solved by applying 'Image source' in IMCRI, as shown in Figure 2. By creating the image sources of patches, the subsequent specular reflected energy after diffuse reflections, the 'D-S' and 'D-S-S' paths in Figure 1c, were calculated. The 'D-S-D' path was then calculated by ARM in the next iteration. Therefore, all sound transfer paths were considered in the IMCRI. Figure 1: Schematic diagram of (**a**) iterative model combining the radiosity method and image source method (IMCRI); (**b**) existing hybrid models (EHM) and (**c**) sound energy transfer paths. D and S represent the specular reflected energy and the diffuse reflected energy. The red line and blue line represent the components calculated by image source model (ISM) and acoustic radiosity model (ARM) in EHM, respectively. The energy emitted from an image source is calculated as follows:  
\[E_{image\ source_{n}}=E_{source}\prod_{i=1}^{n}(1-\alpha_{i})\cdot(1-s_{i}) \tag{1}\]  
where \(\alpha_{i}\) and \(s_{i}\) are the absorption and scattering coefficients of the surface where the reflection occurs, respectively. Moreover, \(E_{source}\) and \(E_{image\ source_{n}}\) represent the energy of the source and that of the \(n\)th order image source, respectively. The sound intensity created by a point source (sound source and image sources) \(I\) can be calculated as follows:  
\[I=\frac{E_{Source}}{4\pi d^{2}} \tag{2}\]  
Therefore, the received energy of a patch \(E_{received}\) can be calculated as follows:  
\[E_{received}=\int\frac{cos\theta}{4\pi d^{2}}dS_{j}\cdot E_{source} \tag{3}\]  
The specular reflected part of the received energy can be modeled by the image source method.