## Abstract

The study proposes the simplification of the multilayer urban canopy model. Four types of multilayer urban canopy models—level 4, level 3, level 2, and level 1—are developed to reduce the computational load of the heat budget calculations at the wall surface. The level 4 model, which accounts for the wall directions and the vertical layer, is simplified in three ways: the level 3 model only accounts for the vertical layers, the level 2 model accounts for the wall directions, and the level 1 model accounts for neither the wall directions nor the vertical layer. From the simplification, compared to the level 4 model, the memory is reduced by 57%, 65%, and 72% for the level 3–level 1 models, respectively, when the vertical canopy layer is seven. At the same time, the CPU time is reduced by 67%, 70%, and 78% for the level 3–level 1 models. Then, each canopy model is compared with observations in Tokyo. The results show that the simulations from the four models are close to the observed ones, and the differences among the four models are very small. An additional model intercomparison study based on idealized simulations indicates that the level 3 model can be used instead of the level 4 model in any condition, whereas the level 2 and level 1 models are proposed to be used under conditions with a large sky view factor.

## 1. Introduction

Numerical studies on the urban heat island phenomenon have been actively performed to investigate its formation mechanisms. There are three main ways to describe the urban thermal environment: 1) slab model, 2) single-layer urban canopy model (e.g., Masson 2000; Kusaka et al. 2001; Harman et al. 2004; Best 2005; Kanda et al. 2005; Lee and Park 2008), and 3) multilayer urban canopy model (MUCM; e.g., Kondo and Liu 1998; Brown 2000; Hagishima et al. 2001; Vu et al. 2002; Martilli et al. 2002; Dupont et al. 2004; Otte et al. 2004; Chin et al. 2005; Kondo et al. 2005). The slab model treats the urban geometry as a flat surface with a large roughness length, small albedo, and small moisture availability. Even now this approach is widely used, as the treatment is very simple. However, the slab model underestimates the nocturnal cooling (Kusaka et al. 2001; Zehnder 2002; Kusaka and Kimura 2004a; Kusaka and Hayami 2006).

Thus, several urban canopy models have been developed. Some of the single-layer urban canopy models have been incorporated into mesoscale models (e.g., Lemonsu and Masson 2002; Rozoff et al. 2003; Kusaka and Kimura 2004a,b; Chen et al. 2004; Kusaka et al. 2005; Chen et al. 2006; Zhang et al. 2008; Miao et al. 2009; Kusaka et al. 2009). Kusaka and Kimura (2004a) coupled their urban canopy model with a simple mesoscale model, and Kusaka and Kimura (2004b) examined the mechanism of the urban heat island formation. Lemonsu and Masson (2002) simulated the heat island circulation over Paris, France, using the mesoscale model with their urban canopy model [Town Energy Balance (TEB)]. Rozoff et al. (2003) incorporated the TEB with the Regional Atmospheric Modeling System (RAMS) and simulated thunderstorms over St. Louis, Missouri. Zhang et al. (2008) coupled the urban canopy model of Kusaka et al. (2001) into RAMS and improved the reproducibility of the heat island in Chongqing, China. Chen et al. (2004, 2006), Kusaka et al. (2005, 2009), Lin et al. (2008), and Miao et al. (2009) officially coupled the urban canopy model of Kusaka et al. (2001) with the Weather Research and Forecasting model (WRF), and they confirmed that the reproducibility of the heat island in Houston, Texas; Tokyo, Japan; Taipei, Taiwan; and Beijing, China was improved.

The multilayer urban canopy model (MUCM) has many advantages for simulating the features of the urban climate. Previous studies of the MUCM involved one-way nesting (e.g., Roulet et al. 2005; Ohashi et al. 2007), idealized simulation (Martilli 2002, 2003), and coupling between MUCM and mesoscale meteorological model (Martilli et al. 2003; Dupont et al. 2004; Otte et al. 2004). Dupont et al. (2004) and Otte et al. (2004) incorporated the MUCM in the fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5) and evaluated the model for a real case above Philadelphia, Pennsylvania. The simulated results including the wind velocity, friction velocity, turbulent kinetic energy, and potential temperature vertical profiles were more consistent with the observations than with using the roughness approach. Recently, the MUCM of Martilli et al. (2002) has officially been adopted into the WRF model, version 3.1. However, the computational efficiency (demand) of the current MUCM is high in general (particularly when the MUCM has a three-dimensional urban geometry). Thus, the computational load of the MUCM is preferred to be smaller when it is incorporated into a mesoscale meteorological model.

In this paper, we propose the simplification of the multilayer urban canopy model. Four types of multilayer urban canopy models—level 4, level 3, level 2 and level 1—are developed to reduce the computational load of the heat budget calculations at the wall surface. Additionally, the validity of the simplifications to perform the model intercomparison is investigated.

## 2. Model description

### a. Atmospheric model

The canopy model used in this study consists of one-dimensional diffusion equations for momentum, potential temperature, and specific humidity based on Kondo et al. (2005):

where *u* and *υ* are the wind velocity components, *θ* is the potential temperature, *q* is the specific humidity, *f* is the Coriolis parameter, *u _{g}* and

*υ*are the geostrophic wind components,

_{g}*m*is the volume porosity (e.g., Kondo et al. 2005),

*c*is the drag coefficient,

_{d}*ρ*is the air density,

*l*is the evaporative latent heat, and

*A*=

*b*/[(

*b*+

*w*)

^{2}−

*b*

^{2}], where

*b*is the average building width and

*w*is the average road width. Here

*Q*

_{AS}and

*Q*

_{AL}are the sensible heat and the latent heat exchanges, respectively, between the building’s walls and the atmosphere. The symbols

*K*,

_{m}*K*, and

_{h}*K*are the momentum, heat, and water vapor diffusivities, respectively, which are calculated from the Mellor–Yamada model at level 2 (Mellor and Yamada 1974). The turbulent length scale

_{q}*L*is given by Watanabe and Kondo (1990) and was derived from the consideration of the forest canopy,

where *κ* is the von Kármán constant. Here *L* above the canopy layer is limited by Blackadar’s formula (Blackadar 1962). The value of *K _{q}* is equivalent to

*K*.

_{h}### b. Urban model (base model)

Figure 1 shows a schematic diagram of our parameterization models, consisting of a three-dimensional, infinitely extended regular array of buildings with a square horizontal cross section. The height of the buildings is uniform. In these models, three types of ground conditions can be considered, and it is possible to change the orientation of the buildings (Fig. 1a). On this account, we explain the level 4 model (Fig. 1b), which is the base model for the simplification. The level 4 model solves the heat budget equation for each vertical layer of the wall and for each wall orientation. This model is similar to the models developed by Kondo et al. (2005) and Hagishima et al. (2001). Kondo et al. (2005) consider the transmission through the buildings, whereas our models do not consider this factor because the building heights are uniform.

The level 4 model estimates the surface temperatures of the road, roof, and each level of the building’s wall in four separate directions from the surface energy balance, as well as the individual fluxes from the various surfaces. The surface energy budgets for the various surfaces are

where Rnet, *H*, *lE*, and *G* are the net radiation, sensible heat flux, latent heat flux, and conductive heat flux, respectively. The subscript *i* indicates the type of surface, which can represent the roof, wall, or road. The sensible heat flux from each surface uses Jürges’s formula when the surface skin temperature is higher than the air temperature,

When the surface skin temperature *T _{s}* is lower than the air temperature

*T*, the surface-layer formulas (Louis 1979) are used to compute the sensible heat flux. When a part of the ground is covered by vegetation (Fig. 1a), the latent heat flux is calculated by

_{a}where *β* is evaporation efficiency, *C _{H}* is bulk transfer coefficient,

*U*is wind speed, and

*q*

_{sat}is saturated specific humidity. Total heat fluxes from the ground are obtained by the average of the fluxes on each type of ground surface (Kimura 1989). The conductive heat flux

*G*and interior temperature

_{z}*T*at depth

_{z}*z*to each surface are calculated by the one-dimensional thermal conduction equations

where *λ* is the interior thermal conductivity and *ρc* is volumetric heat capacity.

For the calculation of the net shortwave radiation Snet on each surface, we consider direct, diffuse, and reflected solar radiation. The reflection in the canyon is considered only once within the model,

where the subscripts *g*, *w*, *r*, and *s* denote ground, wall, roof, and sky, respectively. The terms with ∑_{k}∑_{j} are the reflection from other surfaces. Here *I _{d}*,

*I*, and

_{p}*S*are the direct solar radiation on each surface, direct solar radiation on the horizontal plane, and global solar radiation on the horizontal plane, respectively,

_{p}*α*is albedo,

*F*is view factor, and

*R*is the ratio of the area under sunlight (refer to the appendix).

The longwave radiation flux Lnet on each surface is solved by

where *L _{a}* is the longwave radiation from the atmosphere, ɛ is emissivity, and

*T*is the surface skin temperature.

### c. Simplification method

This section introduces three types of simplified models: the level 3, level 2, and level 1 models. The level 3 model (Fig. 1c) considers each vertical layer but does not take into account the wall orientation. The level 2 and level 1 models (Figs. 1d and 1e, respectively) do not divide the wall into many vertical layers when the heat budget at the wall surface is solved. The level 2 model considers the wall orientation but the level 1 model does not consider this factor.

The shortwave radiation of level 1–level 3 models is calculated in the same way as the level 4 model. Then, the averaged value is given to each facet.

In the level 3 model, which does not take into account the wall orientation, the shortwave radiation flux on the wall is calculated as the averaged value of the four wall orientations:

The longwave radiation fluxes on each surface are calculated in the same way as Eq. (19). However, the wall surface temperature *T*_{wj,k} is not dependent on any orientation:

In the level 2 model, which does not take into account several vertical layers for the wall, the shortwave radiation flux on the wall surface is calculated by

where *w _{k}* is calculated following Eq. (21). Here

*dz*is the grid space of the

_{k}*k*th vertical layer:

Equations (20) and (21) indicate that the averaged values of the vertical layers are given to each wall.

The longwave radiation fluxes on each surface are calculated in the same way as Eq. (20), despite *T*_{wj,k} not being dependent on the vertical layer.

In the level 1 model, which takes into account neither the wall orientation nor the wall’s vertical layer, the shortwave radiation flux on the wall is calculated by

where *w _{k}* is calculated by Eq. (21). Equation (22) indicates that the averaged values of all wall facets are given to the wall. The longwave radiation fluxes on each surface are calculated in the same way as Eq. (22). However,

*T*

_{wj,k}is not dependent on the vertical layer and wall orientation.

For the level 2 and level 1 models, the input radiations mentioned earlier are given to the heat budget equation, and then the surface skin temperature is predicted. Next, the surface sensible heat flux is calculated for each layer using each layer’s wind speed and air temperature (Figs. 1d and 1e).

Table 1 shows the number of facets of the different models for which the heat budget equation is solved. The level 4 model needs to solve 4*N* heat budget equations on the walls (*N* represents the number of layers), but the level 1 model only requires one heat budget equation to be solved. Compared to the level 4 model, the memory is reduced by 57%, 65%, and 72% for the level 3–level 1 models, respectively, when *N* is seven. At the same time, the CPU time is reduced by 67%, 70%, and 78% for the three models (Table 1).

## 3. Simulated results

### a. Verification of the proposed four canopy models against observations in Kanda, Tokyo

Each model is compared with the observations in Kanda (office area), Tokyo, for the verification. These observation data were already used by Kondo et al. (2005) and Ohashi et al. (2007) for testing their multilayer urban canopy model. The observations were carried out during the periods 29–30 July 2002 (case 1) and 10–11 August 2002 (case 2). The Japan islands were covered by a high pressure system for both periods. Table 2 lists the initial and boundary conditions used in the model calculations. The values for the input data and boundary conditions are shown in Fig. 2. The downward radiation is measured on the building roof in the Kanda area. The upper boundary conditions for the wind and temperature were obtained from routine measurements at a height of 250 m on Tokyo Tower, situated about 3 km south from the Kanda area. Figure 3 shows the diurnal variation of the anthropogenic heat given in the models. This anthropogenic heat flux was originally estimated by Mizuno et al. (1997) but was updated for 2006. The surface and urban structure parameters, tabulated in Tables 3 and 4, respectively, were defined following Ohashi et al. (2007).

Figure 4a (case 1) and Fig. 5a (case 2) show the simulated surface air temperature 1 m above ground level (from the level 4 model), which is compared with the observations (symbols). The simulated results are in good agreement with the observations. The root-mean-square-error (rmse) values for the level 1–level 3 models against the level 4 model are listed in Table 5. The rmse values are very small in both cases, lower than 0.1°C in all of the models. Figures 4b and 5b show the temperature difference between the simplified models and the level 4 model [(level 3–level 1 models) − (level 4 model)]. From the simulated results of case 1, the maximum difference of the surface air temperature between the level 1 and level 4 models is 0.039°C, and the maximum difference between the level 3 and level 4 models is 0.004 9°C (Fig. 4b). In case 2, the maximum difference of the surface air temperature is 0.088°C between the level 1 and level 4 models, and 0.018°C between the level 3 and level 4 models (Fig. 5b). The performance of the level 3 model is higher than the level 2 and level 1 models.

### b. Verification of the proposed four canopy models against observations in Kugahara, Tokyo

The heat budget and the vertical profile of the temperature, wind speed, and specific humidity are compared with observations on 1 September 2005 in Kugahara (residential area), Tokyo, for the model verification. The observations were obtained in the Kugahara project (e.g., Moriwaki and Kanda 2004; Moriwaki et al. 2006). The residential area mainly consists of densely built-up houses, paved roads, and small playgrounds. The instruments were attached to a 29-m tower installed in a backyard of one of the homes. Table 6 lists the observation height of temperature, wind speed, and specific humidity used for the model verification. Table 7 lists the initial and boundary conditions used in the model calculations. The values for the input data and boundary conditions are shown in Fig. 6. The downward shortwave and longwave radiations were measured at 25 m above the ground. The upper boundary conditions for the wind, temperature, and specific humidity were measured at 29, 28, and 28 m above the ground, respectively. Figure 7 shows the diurnal variation of the anthropogenic heat flux given in the models. This anthropogenic heat flux was originally estimated by Mizuno et al. (1997), but it was updated for 2006. The surface parameters, the urban structure parameters, and land cover rate are shown in Tables 8 –10, respectively.

Next, the verification results for the level 4 model are described. The upward shortwave and longwave radiations are in good agreement with the observations (Fig. 8a). Figure 8b shows the diurnal variation of heat fluxes: net radiation, sensible heat flux, latent heat flux, and conductive heat flux. The conductive heat flux is determined as the energy balance residual term from direct observation of net radiation, sensible heat, and latent heat fluxes. The simulation results of net radiation and sensible heat flux are in good agreement with the observations. The latent heat flux agrees with the observation during nighttime. However, during the daytime, the simulated result is lower than the observation. In particular, the calculated value for 0700, 1100, and 1500 LT are almost half of the observed ones. The simulated conductive heat flux is higher than the observations from 0900 to 1100 LT. Figures 9 –11 show the vertical profiles of temperature, specific humidity, and wind speed, respectively. The solid line indicates the simulated profile, and the open circles indicate the observations. The vertical profiles of temperature (Fig. 9) are in good agreement with the observations during the daytime. However, during the nighttime, the calculated profiles near the ground (within the canopy layer) are slightly higher than the observations. The vertical profiles of specific humidity are in good agreement with the observations. Figure 11 shows the vertical profiles of wind speed. The simulated wind speed of 11 m above the ground is stronger than the observations during the course of the day. In particular, the differences with the observations at 11 and 15 m above the ground at 1200 LT are large. We performed the same verification test against the level 3–level 1 models. The results are nearly equal with the results of the level 4 model (figure not shown).

### c. Intercomparison of four canopy models for idealized summer and winter cases

Our multilayer urban canopy models (level 1–level 3) are each compared with the level 4 model. We set the same physical constants, surface parameters, urban geometry, and atmospheric conditions for a typical clear summer day and a winter day. The initial conditions, surface parameters, and urban geometry parameters are shown in Tables 11 –13, respectively. The calculation was executed for 6 days, and the results of the last day were used for the intercomparison. We investigated the difference of each model for the different sky view factors (SVFs).

#### 1) Summer case

First, we report the results under the summer conditions. Figure 12a shows the diurnal variations of the surface air temperature (1 m above ground level) simulated in the level 4 model. It shows that for the case when the SVF is 0.22, there is a delay at the start of the increase in surface air temperature, generally a lower amplitude during the course of the day and a high nighttime temperature compared with the case of when the SVF is 0.52. This is the result of less incoming radiation at daytime and less losses by infrared emissions at nighttime. Figure 12b shows the rmse values for the surface air temperature simulated from the level 1–level 3 models against the level 4 model. The result of the level 3 model is almost the same as that of the level 4 model. The maximum rmse is 0.0030°C when the SVF is 0.15. On the other hand, the rmse for the level 2 and level 1 models against the level 4 model becomes larger as the sky view factor becomes smaller. When the value of the SVF is greater than 0.19, the rmse for the surface air temperature of the level 1 model is less than 0.05°C. On the other hand, when the sky view factor is less than 0.19, the rmse increases rapidly (Fig. 12b). When the SVF is 0.15, the rmse of the level 1 model is 0.087. Figures 13a and 13b show the diurnal variation of energy balances calculated from the level 4 model when the SVF is 0.22 and 0.52, respectively. When the SVF is 0.22, the nocturnal sensible heat flux is larger than that with a SVF of 0.52, and the conductive heat flux into the buildings and roads (ground heat flux) before noon has a larger value than that with a SVF of 0.52. Figures 13c–e show the rmse values for the net radiation, sensible heat flux, and conductive heat flux against the level 4 model, respectively. The rmse values for the net radiation, sensible heat flux, and conductive heat flux of the level 3 model are very small regardless of the SVF with rmse values lower than 0.10 W m^{−2}. On the other hand, the rmse values of the level 2 and level 1 models become larger as the SVF becomes smaller. When the SVF is greater than 0.3, the rmse values for net radiation, sensible heat flux, and conductive heat flux are less than 1.00 W m^{−2}. When the SVF is 0.15 (the smallest SVF in this case), the rmse values of the level 1 model is 1.29 W m^{−2} for net radiation, 2.01 W m^{−2} for sensible heat flux, and 3.05 W m^{−2} for conductive heat flux. For this SVF value, the maximum difference of net radiation, sensible heat flux, and conductive heat flux for the level 1 model compared to the level 4 model are 1.90, 3.47, and 4.16 W m^{−2}, respectively. Figure 14 shows the diurnal variation of upward shortwave and longwave radiations calculated from the level 4 model under the SVF of 0.52. The upward shortwave radiation has no model difference, as the shortwave radiation is calculated in the same way as the level 4 model on each facet. The upward longwave radiation has model differences, because it is dependent on the surface skin temperature for each facet (Fig. 14b). Here, the rmse values for the upward longwave radiation are equivalent to the rmse values of net radiation (Fig. 13c). With regard to the radiation budget, the net shortwave radiation has no model difference and the downward longwave radiation gives a constant value (Fig. 14a). Therefore, the fluctuation of upward longwave radiation and net radiation takes the same value.

#### 2) Winter case

Then we report the results under the winter conditions. Figure 15a shows the diurnal variations of the surface air temperature (1 m above ground level) simulated in the level 4 model. The characteristics are similar to the summer case, but the maximum temperature simulated from the SVF of 0.22 is lower than the case for the SVF of 0.52. This indicates a smaller amount of total insolation reaching the ground and lower building canyons when the SVF is 0.22 than for 0.52. This tendency becomes remarkable in winter when the solar elevation is low. Figure 15b shows the rmse values for surface air temperature for the level 1–level 3 models against the level 4 model. The results indicate that the differences in surface air temperature between the level 4 model and the level 2 and level 1 models are almost 2–3 times larger than that of the summer case. When the sky view factor is lower than approximately 0.4, the difference rapidly increases (Fig. 15b). In the level 3 model, the rmse values are less than 0.1°C for any SVF. The maximum surface air temperature difference compared with the level 4 model is also lower than 0.1°C for any SVF. Figures 16a and 16b show the diurnal variation of energy balances calculated from the level 4 model under the SVF of 0.22 and 0.52, respectively. It is different from the summer case in that the value of the sensible heat flux is negative during the night and the negative value is larger for a large SVF. Figures 16c–e show the rmse values for the net radiation, sensible heat, and conductive heat fluxes against the level 4 model, respectively. The rmse value for the net radiation is nearly equal to that of the summer case, but the values for the sensible and the conductive heat fluxes are 2–3 times larger than that of the summer case. The rmse values of the level 3 model remain small regardless of the SVF, with the rmse value for sensible heat and conductive heat fluxes being smaller than 1.00 W m^{−2}. For the level 2 and level 1 models, the rmse values for the sensible heat and conductive heat fluxes are smaller than 1.00 W m^{−2} when the SVF is larger than 0.4. However, the rmse values rapidly increase when the SVF is smaller than 0.4. When the SVF is 0.15 (which is the smallest SVF in this case), the rmse values for the sensible heat and conductive heat fluxes of the level 1 model are 6.05 and 7.21 W m^{−2}, respectively.

From these results, the performance of the level 3 model is nearly equal to that of the level 4 model in both seasons. However, the performance of the level 2 and level 1 models decreases as the SVF becomes smaller. This indicates that the differences in the simulated surface air temperature and heat fluxes are mainly dependent on whether the wall’s heat budget equation is solved for each layer or not. In other words, models that do not consider orientation—such as the level 3 model and the model proposed by Martilli et al. (2002)—would practically suffice when the multilayer urban canopy model is incorporated into the mesoscale model.

## 4. Conclusions

We propose the simplification of the multilayer urban canopy model. Four types of multilayer urban canopy models—level 4, level 3, level 2, and level 1—are developed to reduce the computational load of the heat budget calculations at the wall surface. Compared to the level 4 model, the amount of memory is reduced by 57%, 65%, and 72% for level 3, level 2, and level 1 models, respectively, when the walls are discretized with 7 vertical layers. At the same time, the CPU time is reduced by 67%, 70%, and 78% for the three models. Then the model results were compared to the observations in Kanda (office area) and Kugahara (residential area), Tokyo. The simulated surface air temperature agreed closely to the observed ones. Moreover, we performed a model intercomparison. The results are summarized as follows:

The performance of the level 3 model is nearly equal to the level 4 model, whereas the level 1 and level 2 models produce lower performance for small SVFs. This indicates that the difference in the simulated surface air temperature mainly depends on whether or not the wall’s heat budget equation is solved for each vertical layer.

In the summer case, the difference in the surface air temperature between the level 3 model and the level 4 model is very small. This tendency only slightly depends on the sky view factor. The maximum rmse for surface air temperature is 0.003 0°C when the SVF is 0.15. The rmse values for all terms of the surface heat budget are lower than 0.10 W m

^{−2}for any SVFs. On the other hand, the difference between the level 2 and level 1 models and the level 4 model becomes large when the sky view factor is smaller than 0.19. When the SVF is 0.15, the rmse values of the level 1 model are 1.29 for the net radiation, 2.01 for sensible heat flux, and 2.01 W m^{−2}for conductive heat flux into the buildings and roads. Even if the level 1 and level 2 models give higher rmse, these rmse values remain very small.The results from the winter case show that the rmse values for the surface air temperature, sensible heat flux, and conductive heat fluxes into the buildings and roads are almost 2–3 times larger than that of the summer case. However, the rmse value of the level 3 model is still lower than 0.10°C for any SVF.

From these results, the level 3 model can be used instead of the level 4 model. The level 2 and the level 1 models can also be used if the sky view factor is larger than 0.19 in summer or approximately 0.4 or more in winter. Even for the winter case, the air temperature difference between the level 4 and level 1 model simulations is smaller than 0.3°C (Fig. 15b). Such a small difference might be important for some urban modeling applications; however, for weather and climate models, they can be easily compensated by errors in other physics parameterizations within the models, such as radiation.

## Acknowledgments

The present study was supported by the Global Environment Research Fund (S-5-3) of the Ministry of the Environment, Japan. This work was partially supported by a Grant-in-Aid for Scientific Research from JSP [Grant-in-Aid for Young Scientists (B) 20700667]. We thank Prof. Yukitaka Ohashi of the Okayama University of Science for providing the observational data of Kanda area. We also thank Prof. Yukihiro Kikegawa of Meisei University for providing the original anthropogenic heat dataset. Free software Generic Mapping Tools (GMT) was used in drawing the figures.

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### APPENDIX

#### Calculation of the Shadow Area and Shape Factor of the Multilayer Urban Canopy Models for Levels 1–4

##### Calculation of the shadow area

The shadow area on the ground is computed from a simple calculation method (Kawamoto and Ooka 2006). The shadow area on the ground *A*_{0} is described as follows:

where *h* is the average building height, *b* is the average building width, *w* is the average road width, *θ* is the solar elevation, and *ϕ* is the solar angle. When *x* > *w* or *y* > *w*, the shadow reaches the other buildings. Therefore, there are four shadow patterns considered for the calculation.

- When the shadow reaches another building (Fig. A1b), the overlapping part is subtracted: when
*x*>*w*; when*y*>*w*.

##### Calculation of the shape factor

The view factor is calculated by the analytical solution between an area element and a rectangle (Figs. A1d and A1f):

and

First, the values of to (Fig. A1f) are calculated by using Eq. (A10). Then, *F*_{g→w} is calculated by using Eqs. (A12)–(A14) (Kawamoto and Ooka 2006):

Here, is calculated by using similar aspects of the three-dimensional diagram as follows (Fig. A1g):

The wall view factor of each wall orientation for the ground is a quarter of the value of Eq. (14). The sky view factor for the ground is

## Footnotes

*Corresponding author address:* Ryosaku Ikeda, Graduate School of Life and Environmental Sciences, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8572, Japan. Email: s0820923@u.tsukuba.ac.jp