SOLWEIG 1.0 – Modelling spatial variations of 3D radiant fluxes and ...

Int J Biometeorol (2008) 52:697–713 

DOI 10.1007/s00484-008-0162-7 

ORIGINAL PAPER 

SOLWEIG 1.0 – Modelling spatial variations 

of 3D radiant fluxes and mean radiant temperature 

in complex urban settings 

Fredrik Lindberg & Björn Holmer & Sofia Thorsson 

Received: 6 August 2007 /Revised: 18 March 2008 /Accepted: 28 April 2008 / Published online: 4 June 2008 

# ISB 2008 

Abstract The mean radiant temperature, T mrt, which sums 

up all shortwave and longwave radiation fluxes (both direct 

and reflected) to which the human body is exposed is one 

of the key meteorological parameters governing human 

energy balance and the thermal comfort of man. In this 

paper, a new radiation model (SOLWEIG 1.0), which 

simulates spatial variations of 3D radiation fluxes and T mrt 

in complex urban settings, is presented. The Tmrt is derived 

by modelling shortwave and longwave radiation fluxes in 

six directions (upward, downward and from the four 

cardinal points) and angular factors. The model requires a 

limited number of inputs, such as direct, diffuse and global 

shortwave radiation, air temperature, relative humidity, 

urban geometry and geographical information (latitude, 

longitude and elevation). The model was evaluated using 

7 days of integral radiation measurements at two sites with 

different building geometries – a large square and a small 

courtyard in Göteborg, Sweden (57°N) – across different 

seasons and in various weather conditions. The evaluation 

reveals good agreement between modelled and measured 

values of Tmrt, with an overall good correspondence of R 2 = 

0.94, (p

698 Int J Biometeorol (2008) 52:697–713 

sky the upper hemisphere, both with an angle factor of 0.5 

(e.g. Jendritzky et al. 1990; PickupanddeDear1999). 

Although easier to use, this method is only reliable for 

unobstructed open spaces, and obstruction effects should 

be added using sky view factors. 

Over the years, several different software have been 

developed to simulate Tmrt in outdoor urban settings. 

TOWNSCOPE (Teller and Azar 2001) is a CAD-based 

software which, among other things, simulates spatial 

variations of solar access and Tmrt in complex urban 

environments. Thermal comfort can only be calculated on 

a daily, monthly or annual basis and, since vector data are 

utilised, the spatial extension is limited. The RayMan 

software (Matzarakis 2000; Matzarakis et al. 2000, 2007) 

is a tool which, along with Tmrt, simulates different thermal 

indices. RayMan is a stationary model and can be run on 

very few input meteorological parameters. RayMan is very 

user-friendly and thus a popular tool for researchers, urban 

planners and practitioners. Although commonly used, 

RayMan has some principal shortcomings regarding the 

calculation of the three-dimensional radiation flux densities 

and surface temperatures and, consequently, the resulting 

Tmrt (Thorsson et al. 2007). The three-dimensional ENVImet 

model (Bruse 1999, 2006) simulates radiation fluxes, 

temperature, wind flow, turbulence and humidity, as well as 

Tmrt with high spatial and temporal resolution. The model is 

grid-based. Because of its complexity, the model’s domain 

size is restricted to 250×250×25 grids. It can be run on a 

regular PC. Like the other models mentioned above, ENVImet 

calculates Tmrt according to Fanger (1972); the 

surrounding environment is divided into building surfaces, 

the free atmosphere (sky) and the ground surface for which 

the direct, diffuse and diffusely reflected shortwave and the 

total longwave radiation components are taken into account. 

At street level, the total longwave radiation fluxes 

are assumed to originate from the upper hemisphere (sky 

and buildings) and from the ground (lower hemisphere). 

ENVI-met is still under development. To date, the model 

has only been evaluated with respect to Tmrt for an east–west 

oriented urban canyon (H/W=1) in Freiburg, Germany, 

during a hot summer’s day (Ali Toudert 2005). The 

evaluation showed good agreement between simulated 

shortwave radiation fluxes and field data. However, the 

simulated longwave radiation fluxes revealed discrepancies 

with the field data by up to 50 Wm −2 , which played the main 

role in the differences observed in T mrt, (–8°C in daytime) 

(Ali Toudert 2005). 

In this paper, the development of a new radiation model, 

SOLWEIG 1.0 (solar and longwave environmental irradiance 

geometry-model), which simulates three-dimensional 

daytime radiation fluxes and Tmrt in complex urban settings 

is presented. The first part of the paper presents the features 

of the SOLWEIG 1.0 model and is followed by an 

evaluation of the model using three-dimensional integral 

radiation measurements. 

Materials and methods 

Model structure 

The framework theory for Tmrt calculations used in this 

study is based on the measuring procedure proposed by 

Höppe (1992), in which each of the six longwave and 

shortwave radiation fluxes (upward, downward and from 

the four cardinal points) is considered. The meteorological 

input parameters are direct, diffuse and global shortwave 

radiation, air temperature and relative humidity. Spatial 

variations of urban geometry are represented by a highresolution 

urban digital elevation model (DEM) covering 

the central parts of Göteborg (Fig. 1). SOLWEIG 1.0 is a 

2.5-dimensional model in the sense that it applies a 2.5dimensional 

DEM (i.e. x and y coordinates with height 

attributes) in the calculation of Tmrt. However, the output 

from the current version of SOLWEIG is represented in two 

dimensions (x and y). The model was evaluated by using 

three-dimensional radiation data from two different locations 

within the centre of Göteborg, Sweden (57°N), during 

different weather conditions and at different times of the 

year. Radiation data obtained from the Swedish Meteorological 

and Hydrological Institute (SMHI) are used as 

parameterisation data in the cloudiness calculations. Sur- 

Fig. 1 A DEM covering the central parts of Göteborg. The two 

locations where integral radiation measurements were conducted are 

marked (SITE 1 and SITE 2)

Int J Biometeorol (2008) 52:697–713 699 

face and air temperature measurements were made at Site 1 

(Fig. 1) on 11 clear days throughout the year in order to 

parameterise daytime variations of surface temperatures. 

Mean radiant temperature 

In order to determine Tmrt first, it is necessary to consider 

the mean radiant flux density (S str), which is defined as sum 

of all fields of long and shortwave radiation in three 

dimensions, together with the angular and absorption 

factors of an individual (VDI 1994): 

Sstr ¼ z k 

X 6 

i¼1 

X 

KiFiþ"p 

6 

LiFi 

i¼1 

ð1Þ 

where Ki and Li are the short and longwave radiation fluxes 

respectively (i=1–6) and Fi are the angular factors between 

a person and the surrounding surfaces. For a (rotationally 

symmetric) standing or walking person, Fi is set to 0.22 for 

radiation fluxes from the four cardinal points (east, west, 

north and south) and 0.06 for radiation fluxes from above 

and below. ζk the absorption coefficient for shortwave 

radiation (standard value 0.7) and ɛp is the emissivity of the 

human body (standard value 0.97). When Sstr is known, 

Tmrt is calculated using the Stefan-Boltzmann law: 

Tmrt ¼ 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

4 

þ 273:15 ð2Þ 

Sstr= "ps 

where σ is the Stefan-Boltzmann constant (5.67× 

10 –8 Wm −2 K −4 ). 

Shortwave radiation fluxes 

The calculation of shortwave radiation is fairly straightforward, 

and data on direct, diffuse and global radiation are 

used as inputs in the model. The equation of incoming 

shortwave radiation at a specific location (K↓ij) in an urban 

setting is a modification from Lindberg (2007): 

K#ij ¼ Kdir Shij sin h þ Kdiff


All temperatures are given in Kelvin. The first term on the 

right-hand side is the direct sky longwave radiation, the 

second is the wall radiation and the third is the reflected sky 

radiation. The Prata (1996) formula is used to estimate ɛsky 

during clear-sky conditions: 

"sky ¼ 1 1 þ 46:5 ea 

Ta 

exp 1:2 þ 3:0 46:5 ea 

Ta 

! 

0:5 

ð8Þ 

where ea is the vapour pressure (hPa), derived from Ta and 

RH. The Prata equation has been thoroughly tested at 

different geographical sites and over a large range of 

temperatures. Jonsson et al. (2006) found that the Prata 

formula overestimated the sky emissivity during daytime by 

0.04. This is taken into account in the model presented in 

this paper. The Prata formula also gives a general overestimation 

of incoming longwave radiation of 25 Wm −2 (e.g. 

Jonsson et al. 2006; Duarteetal.2006). Calculated values of 

L↓ are therefore reduced by 25 Wm −2 . Since the presence of 

clouds significantly increases the total effective emissivity 

of the sky, modifications must be made to the existing clearsky 

formulations (Crawford and Duchon 1999): 

L# ¼ L#cð1cÞþcsT 4 a ð9Þ 

where c is the fractional cloud cover (0 ≤ c ≤ 1). In general, 

data on cloud cover is sparse. Thus, an objective method for 

deriving c is used in this paper. Fractional cloud cover is 

calculated as follows: 

c ¼ 1 

S# 

S#c 

ð10Þ 

where S↓/S↓c is the ratio of observed solar radiation to a 

modelled clear-sky solar radiation, i.e. Clearness Index (CI). 

When modelled values of clear-sky solar radiation based on 

the work of Crawford and Duchon (1999) are used, a clear 

diurnal variation is observed. Corrections for this will be 

considered below (Section Adjustment of modelled clearness 

index at high sun zenith angles). 

For calculation of outgoing longwave radiation (L↑), the 

Stefan-Boltzmann law is applied. Shadow patterns are 

considered, so that sun-exposed surfaces emit a larger 

amount of longwave radiation due to the expected higher 

surface temperatures (Ts): 

L"ij ¼ "gs Ta þ ShijðTsTaÞ 4 

ð11Þ 

where ɛ g is the emissivity of the ground surface. The 

temperatures on shadowed surfaces are set to equal Ta. The 

temperatures on sun-exposed surfaces are estimated based 

on a linear relationship between maximum solar elevation 

and maximum difference between Ta and Ts during clear 

day conditions (Tdiffmax). This is based on the work of 

Bogren et al. (2000), who found a linear relationship 

between the difference of sunlit and shadowed ground 

surface temperatures and maximum solar elevation for clear 

day conditions. The data used for parameterisation are 

taken from Ts and Ta measurements which were made at 

Site 1 on 11 clear days throughout the year (Fig. 2). Thus, 

the parameterisation of T diffmax leads to: 

Tdiff max ¼ 0:37 h max 3:41 ð12Þ 

T diffmax is considered to occur 2 h after the solar 

elevation maximum is reached. The Ts wave for a clear 

day is then specified as a sinusoidal function, where the 

amplitude is taken from the linear relationship in Eq. 12 and 

the period for a certain day of the year is established based 

on the time between sunrise of the day of interest and 

1400 hours. The initial morning value of T s is set at – 

3.41 K lower than Ta, based on the theoretical difference 

between Ts and Ta at a sun elevation of zero degrees at an 

open square (Fig. 2). For non-clear model runs, Tdiffmax is 

multiplied by CI. A condition is also set such that if a 

certain ground surface pixel should change from sunexposed 

to shadowed area, the new surface temperature 

value would not be equal to Ta the following hour. Instead, 

the new assumed value would be 75% of the calculated 

sun-exposed surface temperature value for that specific 

hour. This is done to consider the thermal properties of the 

ground and the fact that a gradual temperature drop is 

Fig. 2 Observed relation between sun elevation and maximum 

difference between ground surface (T s) and air temperature (T a)on 

clear days at a large open square in Göteborg, Sweden (SITE 1)


evident when a surface location is changing into a 

shadowed position. The opposite condition is also included 

in SOLWEIG 1.0, where a gradual temperature rise is 

evident when a certain pixel is changing from shadowed 

into a sun-exposed area. 

To estimate the longwave radiation fluxes from the four 

cardinal points, the same formula is used in all directions. 

Thus the current model does not take variations in building 

geometry in different directions into account, but instead 

uses y as a non-directional factor of building geometry. 

The total flux from one cardinal point is the sum of the 

fluxes from five different sections: 

wsky 

L!SKY ¼ L# 

wtotal 

0:5 ð13aÞ 

L!REFLECTED ¼ L# þ L" ð1 "wÞ 

wwall 

wtotal 

L!WALLshadow ¼ "wsT 4 w 

L!WALLsun ¼ "wsT 4 w 

wwall 

wtotal 

wwall 

wtotal 

0:5 ð13bÞ 

fsh 0:5 ð13cÞ 

1 fsh cos h sun 

0:5 ð13dÞ 

L!GROUND ¼ L" 0:5 ð13eÞ 

All terms are multiplied by 0.5 due to the fact that only 

half of the hemisphere is taken into account for each 

cardinal point, while w are angular weighting factors of the 

amount of radiation originating from either building walls 

or sky. In order to establish these weighting factors, the 

different areas of the elements (buildings and sky) are 

calculated using simple spherical geometry. These areas are 

then sinus weighted so that radiation on a fictitious, sidefacing 

instrument receives more radiation from perpendicular 

elements. The average angles based on the heights of 

the buildings (β y) are calculated according to the relation- 

Fig. 3 Schematic image for the 

calculation of the fraction of 

shadowed building walls (dark 

gray). The left figure shows the 

cylindrical wedge calculated 

from sun elevation (η) and 

average building angle heights 

(β) in a circular yard and the 

right figure shows the recalculated 

non-directional fraction of 

shadowed building walls 

ship y =cos 2 βy, representing the case of a basin (Oke 

1987). Holmer (1992) argued that the y could be used as a 

good approximation of the fluxes of longwave radiation as 

compared to an open horizontal surface, assuming the same 

temperature and emissivity. A statistical relationship between 

y and summed sinus-weighted sub-areas on a 

hemisphere is then established by a fifth-order polynomial 

with a determination coefficient of 0.999: 

wsky ¼ 26:16< 5 

53:29< 4 

39:33< 3 

9:71< 2 

þ 1:98< 0:05 ð14Þ 

Consequently, only y is needed to derive the different 

values of w. By inserting y =1, wtotal is set to 4.417 and 

wwall is simply the difference between wtotal and wsky. 

The fraction of shadowed wall surfaces (f sh) (Eqs. 13c 

and 13d), seen from a specific location (pixel) was 

estimated by using information of sun zenith angle (90-η) 

and the average angle based on the heights of the buildings, 

derived from βy. The calculation of fsh was based on a 

fictitious circular yard which was surrounded by buildings 

with constant roof heights. The geometrical properties of a 

cylindrical wedge could therefore be applied by using 

information about η and βy (Fig. 3, left). If wall surfaces 

facing the sunbeams are exposed to a higher proportion of 

sunlight then fsh is recalculated so that the parameter is the 

same in all four cardinal directions (Fig. 3, right). Possible 

values of f sh range between 1 (all building walls in the 

shade) and 0.5 (half of the building walls in the shade). The 

cosine factor used in Eq. 13d is included in order to obtain 

an average angle from the point of origin of the radiation. 

ηsun is derived using simple trigonometry based on 

information about βy and fsh. 

Adjustment of modelled clearness index at high sun zenith 

angles 

The objective method for deriving fractional cloud cover (c) 

includes modelled values of CI (Crawford and Duchon 

1999). An analysis of the hourly data of incoming 

shortwave radiation at the SMHI station from days with 

only clear weather conditions revealed that CI was under-


estimated. This was especially evident in the mornings and 

evenings throughout the annual cycle and was particularly 

pronounced during the winter season. The underestimation 

was shown to be as much as 50% during the very first and 

very last hour of daylight. Thus a relationship was 

established between CI during clear days and the altitude 

of the sun. Clear days were objectively selected by fitting a 

dome-shaped polynomial of the second degree with a 

determination coefficient (R 2 ) greater than 0.98. The entire 

dataset (1986–2005) obtained from the SMHI station was 

used. A total of 96 days was derived from the dataset. The 

logarithmic relationship in Fig. 4 was then used as a 

correction factor for the clear-sky solar irradiance: 

CIcorr ¼ CI þ ð1ð0:15 ln h þ 0:35ÞÞ 

ð15Þ 

Model domain and data 

Göteborg (57°42′N, 11°58′E) is situated on the west coast 

of Sweden in an aligned joint valley landscape. The city is 

the second largest city in the country, with nearly 500,000 

inhabitants. The model domain covers an area measuring 

1400×1400 m in the central part of the city as shown in 

Fig. 1. This central area has a classical European design, 

consisting of three to five-storey buildings, narrow street 

canyons and canals. The mean building height is 16.5 m, 

with a standard deviation of 6.0 m. Two low hills are found 

within the model’s domain. The DEM consists of both 

ground topography within the study area that ranges from 0 

to 35 m a.s.l. and building structures that range from 1 to 

100 m a.s.l. The DEM is derived from local governmental 

digital data according to a method presented by Lindberg 

Fig. 4 Clearness index versus sun elevation on 96 clear days between 

1986 and 2005 in Göteborg, Sweden 

(2005) and the spatial resolution is 1 m. Trees and bushes 

are not included in the current DEM (Fig. 1). 

The input data used in the SOLWEIG model were 

collected hourly and consisted of direct, diffuse and global 

shortwave radiation obtained from a nearby weather station 

(1 km west of the study area). The station is run by the 

SMHI. The air temperature and relative humidity were 

taken from the same weather station. The available data 

were collected between 1986 and 2005. The same dataset 

was used as parameterisation data for cloudiness calculations. 

The T s data used to parameterise daytime variations 

of surface temperatures during clear daytime weather 

conditions were acquired at Site 1 on 13 occasions using 

a handheld IR instrument (AMiR 7811–20). The surface 

material at Site 1 is very homogenous consisting of large 

cobblestones. 

For model validation, data from integral radiation 

measurements within the model domain was used. In total, 

measurements taken over seven days from sunrise to sunset 

were compared with the results obtained from the model. 

Three net radiometers (Kipp & Zonen, CNR 1) were 

mounted on a steel stand in order to measure the threedimensional 

radiation fields. Shortwave and longwave 

radiation fluxes from the four cardinal points, as well as 

those from the upper and lower hemisphere were measured. 

The instruments had an offset of approximately 20° from 

the north and were instead positioned perpendicular to the 

surrounding building walls (Thorsson et al 2006, 2007). 

Two locations were used, one large open square (Site 1) 

with a y of 0.95 and a smaller courtyard (Site 2), y =0.65, 

as shown in Fig. 5. Both sites have cobblestone surfaces 

and almost no vegetation present. 

Results 

In this section, 4 days of measurements covering different 

weather conditions, locations and the time of year are 

compared with the results from model results in detail. One 

clear autumn day, one clear summer day and one semicloudy 

day at Site 1 are presented, together with one clear 

autumn day at Site 2. Corrections for clear-sky calculations 

at high sun zenith angles are also presented. Finally, the 

overall model performance and spatial variations of 

modelled Tmrt are shown. 

Radiation fluxes on clear days 

Clear summer day at a large square (SITE 1) 

Figure 6a–c show modelled and measured radiation fluxes 

and T mrt at Site 1 on 26 July 2006 between sunrise and 

sunset. This was a clear summer day with large variations


Fig. 5 The two locations where 

integral radiation measurements 

were conducted are marked. 

A Square (Site 1) and B courtyard 

(Site 2) 

of both short and longwave radiation fluxes at different 

times of the day. In general, the modelled values of both 

shortwave and longwave fluxes were at the same levels as 

measured values, except for a few special occasions 

(Fig. 6a,b). Kwest is underestimated at 1900 hours LST by 

250 Wm −2 and all longwave fluxes are underestimated 

during the morning and evening hours by up to 50 Wm −2 . 

Ldown during the morning hours (0500–0700 hours LST) is 

underestimated due to fog. The underestimation of shortwave 

radiation at 1900 hours LST is due to the hourly time 

resolution and is further discussed in “Discussion: The 

temporal resolution”. On average, the difference between 

modelled and measured values of Tmrt on a clear summer 

day at Site 1 is 2.3 K. The Tmrt values are underestimated 

during the first 2 h after sunrise (approximately 5 K) as well 

as during the last 2 h before sunset (approximately 7 K) 

(Fig. 6c). 

Clear autumn day at a large square (Site 1) 

Figure 7a–c shows modelled and measured radiation fluxes 

and Tmrt at Site 1 on 11 October 2005 between sunrise and 

sunset. This was also a clear day, but during the autumn 

season, with higher sun zenith angles and a shorter daytime 

period. In general, the model simulates all six shortwave 

radiation fluxes well (Fig. 7a). Modelled values of K south 

show a slight underestimation by up to 30 Wm −2 , especially 

during midday. The largest overestimation of any of the 

shortwave fluxes is found for the K west component 

(142 Wm −2 ) at 1600 hours LST (Fig. 7a), due to the time 

resolution. The differences in the six longwave fluxes are 

relatively small (Fig. 7b). The modelled longwave fluxes 

from the four cardinal points give underestimated values 

during morning and evening hours by up to 10 Wm −2 , and 

the upward component is overestimated during lunch hours 

by 10 Wm −2 . The model performance of Tmrt on a clear 

autumn day at Site 1 shows that the difference between 

modelled and measured values is 1.7 K (Fig. 7c), on 

average. The largest differences between modelled and 

measured values (3.5 K) are found during the first 2 h of 

the day. 

Clear autumn day at a small courtyard (Site 2) 


and Tmrt at Site 2 on 7 October 2005 between sunrise and 

sunset. The measurement point (instrumentation) was in 

shade all day, except for 2 h in the afternoon (1500 and 

1600 hours LST). When the measurement point was 

shaded, the shortwave radiation fluxes were in agreement 

with modelled values. However, during the 2 h with direct 

shortwave radiation evident, the model overestimates all the 

components which are exposed to direct shortwave radiation 

(Kwest, Ksouth and K↓) by as much as 200 Wm −2 

(Fig. 8a). The longwave fluxes are more complex in an 

environment with higher y. Since the SOLWEIG-model 

utilises non-directional calculations of the longwave fluxes 

from the four cardinal points, it is not able to capture the 

high measured values of L north (Fig. 8b). The modelled 

values of Lup are overestimated by 10–15 Wm −2 when the 

validation pixel is in shadow and by 25 Wm −2 when the 

validation point is exposed to direct sunlight. T mrt at Site 2 

is overestimated by an average of 6.5 K throughout the day. 

During the two sun-exposed hours at 1500 and 1600 hours 

LST, T mrt is overestimated by as much as 17 K (Fig. 8c). 

Radiation fluxes on a semi-cloudy day 


and Tmrt at Site 1 on 1 August 2006 between sunrise and 

sunset. This was the measurement day with the highest 

variability due to cloudiness of all the days included in the 

validation data. It was characterised by occasional cloudiness 

up to 1600 hours LST, when a fully cloud-covered sky 

became evident. A light drizzle of rain occurred between 

1700 and 1800 hours LST, which explains the missing data 

at 1700 hours LST in Fig. 9c. The SOLWEIG model 

appears to estimate both shortwave and longwave radiation 

fluxes reasonably well, except for a few hours (0700, 0800, 

1200–1400 hours LST) at which the shortwave fluxes were 

incorrectly calculated (Fig. 9a) due to variations of 

cloudiness between Site 1 and the weather station located 

1 km west of Site 1. However, the overall model


Fig. 6 A–C Comparison between 

measured and modelled 

(m) three dimensional radiation 

fluxes and T mrt at the large open 

square in Göteborg (Site 1), 

Sweden, on a clear day (26 July 

2006). A Shortwave radiation 

fluxes, B longwave radiation 

fluxes, C T mrt for a standing 

man 

performance of Tmrt on a semi-cloudy day is satisfactory, 

with an average difference of 3.1 K between the modelled 

and measured values of Tmrt. The largest discrepancies 

between modelled and measured values of Tmrt are found 

during 1300–1500 hours LST (3.0 K), when the shortwave 

fluxes are insufficiently modelled. 

Overall model performance 

Figure 10 shows modelled versus measured upward, 

downward, sideward and total shortwave and longwave 

hourly radiation fluxes from both sites for all seven days of 

measurements. As shown, SOLWEIG works very well,





fluxes, T mrt at the large open 


Sweden, on a clear day (11 

October 2005). A Shortwave 

radiation fluxes, B longwave 

radiation fluxes, C) T mrt for a 

standing man 

with a good overall correspondence between the modelled 

and measured fluxes, explaining 96% of the variance in total 

shortwave radiation fluxes (RMSE=152.2 Wm −2 ) and 93% 

of the variance in total longwave radiation fluxes (RMSE= 

70.6 Wm −2 ). The model is found to simulate the downward 

and upward shortwave fluxes well (K ↓:R 2 =0.97, RMSE= 

42.1 Wm −2 ; K↓ R 2 =0.97, RMSE=7.0 Wm −2 ). However, 

simulated sideward shortwave fluxes are slightly underestimated 

(R 2 =0.93, RMSE=126.0 Wm −2 ). The large 

scattering, which is particularly evident for the downward 

and sideward shortwave fluxes, is mainly due to the hourly 

time resolution. SOLWEIG is found to simulate longwave 

fluxes fairly well (Ldown: R 2 =0.73, RMSE=17.5 Wm −2 ;Lup: 

R 2 =0.94, RMSE=15.6 Wm −2 ; Lside: R 2 =0.92, RMSE= 

48.9 Wm −2 ), however it slightly overestimates the downward 

and sideward fluxes. 

A comparison of modelled versus measured downward, 

sideward and total shortwave and longwave radiation fluxes





fluxes and T mrt at the courtyard 

in Göteborg (Site 2), Sweden, 

on a clear day (7 October 2005). 

A Shortwave radiation fluxes, 

B) longwave radiation fluxes, 

C) T mrt for a standing man 

absorbed by a standing man was conducted, in other words, 

the angular factors and absorption coefficients of shortwave 

and longwave radiation were included (see “Materials and 

methods: Mean radiant temperature”). The plot of the 

modelled versus measured total absorbed shortwave and 

longwave fluxes (not shown) shows a similar pattern to that 

of Fig. 10, but with RMSEs of 20.3 Wm −2 and 11.8 Wm −2 

for shortwave and longwave radiation fluxes, respectively. 

The longwave absorbed radiation flux becomes increasingly 

important since ɛ p is 0.97, whereas ζ k is 0.7. 

Figure 11 shows modelled and measured hourly values 

of Tmrt using all data obtained from both sites for all 7 days 

of measurements (n=95). The determination coefficient of 

the linear regression is 0.94 (p





fluxes and T mrt at the large open 


Sweden, on a semi-cloudy day 

(8 August 2006). A Shortwave 

radiation fluxes, B longwave 

radiation fluxes, C) T mrt for a 

standing man 

overestimated by approximately 1.5 K at the higher 

temperature span (T mrt>50°C). On average, modelled 

values in a dense urban environment (e.g. Site 2) are 7 K 

higher than measured values of Tmrt and modelled values at 

Site 1 are about the same as measured values across the 

temperature range. A cluster of higher scattering is found in 

Tmrt at around 25–40°C. Furthermore, one significantly 

underestimated value is found at the lower part of the 

temperature range of Tmrt (Fig. 11). These features are dealt 

with in the “Discussion”. 

Spatial variations of Tmrt 

The SOLWEIG model is a non-stationary model that is able 

to calculate the spatial variation of Tmrt on different 

temporal scales. Figure 12 shows an example of this spatial

708 Int J Biometeorol (2008) 52:697–713


Fig. 10 Modelled versus measured hourly data of all the shortwave 

(n=92) and longwave (n=96) fluxes at two sites, one large square and 

one small courtyard, in Göteborg Sweden 

variation of T mrt at ground level for a part of the model 

domain. The example is taken from an afternoon occasion 

during a clear autumn day (1500 hours LST, 11 October 

2005). The air temperature for the modelled hour was 19.5°C 

and the global radiation was 274 Wm −2 at the weather 

station located 1 km west of Site 1. The first obvious features 

are the shadow patterns, which are essential for the spatial 

estimation of Tmrt: sunlit areas, show considerably higher 

values of Tmrt. The red areas are exposed to direct sunlight, 

while the blue areas are in shade. Another clear feature is 

that Tmrt is relatively high close to a building wall (e.g. 

Fig. 12a) and decreases as the distance from the buildings 

increases (e.g. Fig. 12b). This is evident in both shadowed 

and sunlit areas. 

One can also observe that the areas that shift from being 

exposed to sunlight to becoming shadowed or vice versa 

experience a slower rise/drop in temperature, Tmrt, than 

areas that have been exposed to sunlight or in shade for 

more than 1 h. This is illustrated by the letters c, d, e and f 

in Fig. 12. The area around c, which just became exposed 

to the sun, reflects lower values of Tmrt in comparison to 

that around d, which has been in direct sunlight for more 

than 1 h. The area around e, which consequently just 

became shadowed, shows higher values of T mrt in comparison 

to f, which has been in the shade for more than 1 h. 

Due to the fact that SOLWEIG model simulates all 

longwave fluxes from the four cardinal points similarly, 

façades facing north have the same surface temperature as 

those facing south, which is not the case in a practical 

situation. This may be illustrated at letter g in Fig. 11, 

which signifies a small, shadowed yard surrounded by 

buildings. The Tmrt value at g is relative constant within the 

yard. In reality, T mrt is probably higher near the walls facing 

north and east. Direct sunlight reaching parts of these two 

walls will increase the surface temperature and consequently, 

the longwave irradiance originating from these two walls. 

Discussion 

In this section, the SOLWEIG 1.0 model is compared with 

other models that could be used to estimate radiation fluxes 

and Tmrt. Model performance and sensitivity of SOLWEIG 

1.0 is also discussed. 

Model intercomparison 

Compared to other available models, the technique of 

estimating the three-dimensional radiation fluxes used in 

the SOLWEIG model appears to improve the accuracy of 

T mrt. This is especially evident for high sun zenith angles, 

since the radiation fluxes are not only considered for 

horizontal surfaces, but also from the four cardinal points. 

Thorsson et al. (2007) validated the RayMan 1.2 model and 

found that it underestimates Tmrt significantly at high sun 

zenith angles. The average differences between measured 

and modelled T mrt at Site 1 using RayMan 1.2 for 11 

October 2005 and 26 July 2006 were 10.1 K and 6.8 K, 

respectively. The corresponding values for the SOLWEIG 

1.0 model at the same location and dates were 1.7 K and 

2.3 K. So far, the ENVI-met model has not been tested in 

the city of Göteborg. However, Ali-Toudert (2005) compared 

the T mrt modelled by ENVI-met to measured values 

of Tmrt throughout the diurnal cycle for a street canyon (H/ 

W=1) in Freiburg (Germany). She found that ENIV-met 

was well able to represent the trends of Tmrt with its two 

contrasting periods (day and night). However, the values 

that were returned by the ENVI-met model were significantly 

overestimated during the morning hours by up to 

15 K and underestimated from noon and throughout the 

night by up to 7 K (Ali-Toudert 2005, p. 154). 

Model performance 

The current SOLWEIG 1.0 model estimates spatial variations 

of Tmrt relatively well for most of the different urban 

geometries and weather conditions used in the current 

validation dataset. Below, the performance and some 

limitations of the SOLWEIG 1.0 model are discussed. 

Fig. 11 Modelled versus measured hourly data of T mrt (n=95). The 

regression line is valid for data from Site 1 and Site 2 combined


Fig. 12 Spatial variations of 

T mrt (°C) in the city centre of 

Göteborg, Sweden at 1500 hours 

LST on 11 October 2005. The 

spatial resolution is 1 m. The 

letters a through f are points of 

interest referred to in the text 

Shortwave radiation fluxes 

The shortwave fluxes are relatively easy to estimate, since 

actual values of shortwave radiation are used as input in 

SOLWEIG. However, some features of the shortwave fluxes 

should be discussed. Firstly, the reflection term in Eq. 3 is a 

simplification and some differences between modelled and 

measured shortwave fluxes could therefore arise. This is most 

obvious for the southerly component of shortwave radiation 

(asseeninFigs.7a and10). Since the instruments were 

situated at a measuring height of 1.1 m – the centre of gravity 

for a standing person – and therefore reflection from ground 

surfaces was present, the shortwave fluxes for all the sidefacing 

instruments were influenced. In particular, modelled 

values of K south turned out to be lower than measured values. 

Secondly, all three components of shortwave radiation 

(diffuse, direct and global) are required as input in the model. 

It could sometimes be difficult to obtain such data from a 

nearby weather station. However, there are many existing 

methods for the calculation of shortwave radiation fluxes 

which could be used instead of actual measured values (e.g. 

Olseth and Skartveit 1993; Roderick 1999; Gueymard 2000). 

Longwave radiation fluxes 

The process of estimating longwave fluxes is much more 

complicated than is the case for shortwave fluxes. None of 

the longwave fluxes is measured directly and used as input 

data in the model. Instead they are estimated based on other 

meteorological parameters and empirical constants. The 

model developed by Jonsson et al. (2006) estimates the 

downward fluxes of longwave radiation reasonably well, 

after the modelled values have been reduced by 25 Wm −2 . 

The reason for the general overestimation of L↓ when using 

the empirical relationship set up by Prata (1996) is 

unknown. However, this aspect is shown by both Jonsson 

et al. (2006) and Duarte et al. (2006). L↑ is calculated based 

on the value of Ta and the relationship is derived from the 

linear relationship between the solar elevation and the 

maximum difference between the ground surface and air 

temperatures on clear days (Fig. 2). The daily temperature 

wave follows a sinusoidal path, where the amplitude is 

taken from Fig. 2 and the period is twice the length of time 

between sunrise and 1400 hours LST, which is the peak of 

the surface temperature wave. There are a few uncertain 

variables in the calculation of Ts. The initial value of Ts at 

sunrise is unknown and set at −3.4 K lower than Ta. This is 

based on the difference between T s and T a at a sun elevation 

of zero degrees (Eq. 12). This initial value appears to be 

underestimated, especially during clear weather conditions 

with intensive radiative cooling during the night (e.g. 

Figs. 6b and 7b). The relationship in Fig. 2 is established 

from data measured at Site 1. In reality, however, this figure 

should vary depending on the material, slope and aspect of


the surface of interest. The same relationship is also giving 

a small underestimation of L ↑ during a clear day in July at 

Site 1 (Fig. 6b) and a small overestimation of L↑ during a 

clear day in October at Site 1 (Fig. 7b). More parameterised 

data should be included in Fig. 2 in order to obtain a better 

estimation of L↑. Further, in the current version Ts is set 

equal to Ta in shaded areas. This will underestimate Ts. 

However, the effect will be reduced since shaded areas are 

often in dense urban environments where nocturnal cooling 

is reduced due to low values of y . 

According to Ali-Toudert (2005), a standing body 

absorbs more than 70% of the energy in the form of 

longwave irradiance in the daytime, which demonstrates the 

importance of accurate longwave radiation simulations/ 

measurements for the estimations of Tmrt. This is especially 

important in complex urban settings where emitted longwave 

radiation from the surrounding walls represents a 

large part of the human energy balance. When no direct 

shortwave radiation is evident, the longwave fluxes become 

even more important for a correct estimation of Tmrt. The 

general overestimation of Tmrt at SITE 2 could be explained 

by an overestimation of the longwave radiation fluxes from 

south, east, west and up at the more dense urban location 

(Fig. 8b). The main reason why SOLWEIG 1.0 overestimates 

longwave radiation at SITE 2 is because of the 

properties depicted in Eq. 13e. Although the distance from 

a specific pixel and the average distance to the building 

walls diminishes, L →GROUND remains the same. In reality, 

as a building wall is approached, the ground surface area 

decreases in the direction of the approached wall. At 

present, this is the main setback in SOLWEIG 1.0 and 

should be improved in subsequent versions of the model 

(see “Conclusions and future prospects”). Another reason 

why large discrepancies are found at Site 2 could be that the 

modelled value of y is too high. High trees rise above the 

roofs of buildings to the west and block parts of the visible 

sky. If vegetation were included as an additional set of 

information then y would probably be lower at Site 2. The 

absence of vegetation is therefore another drawback of the 

current model, which will affect T mrt when present. Apart 

from the trees mentioned above, almost no vegetation is 

found at the two validation sites used in this study. In 

addition, the air mass between the pixel of interest and the 

building’s walls are not considered, which could influence 

the longwave fluxes from the four cardinal points. 

Nevertheless, this effect is considered to be minimal. 

The temporal resolution 

In general, the performance of the model at SITE 1 (SVF= 

0.95) is very good, with some exceptions (Figs. 6, 7 and 9). 

The largest discrepancies of T mrt are found at late 

afternoons and early evenings and are mainly due to large 

differences between the modelled and measured values of 

shortwave radiation fluxes. This is mainly because of the 

temporal resolution of the meteorological input data. Since 

just one image of shadow patterns represents a 60-min 

period in SOLWEIG 1.0, a pixel is either in the shade or 

exposed to the sun for a full hour, which might not be 

the case in reality. The shadow pattern is generated in the 

middle of each hour. In reality, a location could be in the 

shade during the first 40 min of an hour and then exposed 

to sunlight during the remaining 20 min. The SOLWEIG 

model calculates this location as if it were in the shade for 

the entire hour, even if that is not the case. This may result 

in underestimations or overestimations of the shortwave 

radiation fluxes and consequently of T mrt if a location is 

moving either out of, or into, the shade (e.g. Fig. 6a at 

1900 hours LST and Fig. 8a at 1500 and 1600 hours LST). 

For very complex urban structures, this issue could result in 

large discrepancies throughout the daily cycle if a location 

of interest is constantly moving in and out of the shade. 

There is a quite simple solution to this problem, which is 

not included in the current version of the SOLWEIG model. 

Generating several images of shadow patterns for each hour 

(e.g. each 15 min) would include information about the 

amount of time a pixel is in the shade during a specific 

hour; this information could then be used to arrive at better 

estimations of the radiation fluxes for each specific hour. 

This, however, will complicate the modelling and increase 

the computation time for each model run, since more 

images of shadow patterns must be generated. 

Model sensitivity 

A number of parameterisation parameters in the SOLWEIG 

1.0 model are only established from published material and 

it is necessary to understand the model’s sensitivity if these 

parameters are altered. Examples of these input parameters 

are the surface albedo, emissivity of walls and ground, 

accuracy and precision of the DEM used and the calculation 

of the position of the sun. The albedo was originally set 

to 0.15 and ground and wall emissivity to 0.95 and 0.90, 

respectively, according to Oke (1987). A small sensitivity 

test was carried out on Site 1 on 11 October 2005, during 

which some of these parameters were altered. When the 

albedo was increased from 0.15 to 0.20, Tmrt increased by 

0.2°C and the total shortwave radiation flux increased by 

20.4 Wm −2 during midday. When ground emissivity was 

increased from 0.95 to 0.98, Tmrt increased by 0.9°C and 

the total shortwave radiation flux increased by 51.4 Wm −2 . 

Consequently, these parameters have a limited effect on the 

estimation of Tmrt in the SOLWEIG 1.0 model. The overall 

performance of the fluxes shown in Fig. 10 is relatively 

good. The higher RMSE values found in the lateral 

shortwave fluxes due to the effect of the temporal


resolution discussed above, will not affect the final outcome 

of T mrt to any significant extent. Since the longwave fluxes 

account for the bulk of the absorbed radiation, the 

SOLWEIG model’s overall performance of Tmrt estimations 

is good (Fig. 11). 

Adjustment of clearness index 

SOLWEIG 1.0 could also be used during non-clear weather 

conditions. The model provides an objective method of 

measuring cloudiness, where the ratio of observed solar 

radiation to modelled clear-sky solar radiation is applied. 

The clear-sky formulations presented by Crawford and 

Duchon (1999) have some setbacks. Clear-sky solar 

radiation is underestimated at very low sun elevations 

during morning and evening, as well as during the winter 

season, especially at high latitudes. This is accounted for in 

SOLWEIG 1.0 by the established relationship illustrated in 

Fig. 4 (Eq. 15). The data scattering in Fig. 4 could be 

explained by the transmission coefficients for gases and 

aerosols in the atmosphere that are used to assess clear-sky 

solar irradiance. The actual amounts of gases and aerosols 

are not taken into account in the formulations. However, the 

formulas are advantageous, since they constitute an 

objective method of determining cloudiness, as opposed 

to using subjective cloudiness data which are usually 

sparse. The more complex and variable radiation conditions 

in semi-cloudy weather conditions result in higher scattering 

in the data and the distance between the model domain 

and the weather station used becomes more significant. In 

this study, the distance between the two is only 1 km, but 

large variations in shortwave radiation are still evident (e.g. 

Fig. 9a at 1300hours LST). 

Conclusions and future prospects 

In this paper, the new computer model SOLWEIG 1.0 is 

described. The model simulates spatial variations of 3D 

radiation fluxes and mean radiant temperature based on 

simple meteorological parameters and urban geometry 

represented by building structures and ground topography. 

In general, the correspondence between modelled and 

measured values of T mrt are high (R 2 =0.94, p


5. The model will be evaluated and certified in different 

urban settings and under different weather conditions 

through measurements and model comparisons, in 

order to arrive at a greater understanding of the spatial 

variations of T mrt within the urban environment. 

6. The SOLWEIG model will be transformed into a userfriendly 

interface. 

Acknowledgements Financial support for this project was provided 

by FORMAS, the Swedish Research Council for Environment, 

Agricultural Sciences and Spatial Planning in their key action area 

Urban Public Places. Thanks to Ingegärd Eliasson and Sven Lindqvist 

who supervised the project, and to Alexander Walther for programming 

support. 

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