Texture and Radiation4

8/12/2019 Texture and Radiation4

1/46

1

Urban texture and radiation exchange

Carlo Rattia

, Nick Bakerb

, Koen Steemersb

aSENSEable City Laboratory, Massachusetts Institute of Technology, 77 Massachusetts Avenue,

Cambridge, 02139 MA USA

bThe Martin Centre for Architectural and Urban Studies, Department of Architecture, University

of Cambridge, 6 Chaucer Road, Cambridge CB2 2EB, UK

Address for correspondence: Carlo Ratti, SENSEable City Laboratory, Massachusetts Institute of

Technology, 77 Massachusetts Avenue, Cambridge, 02139 MA USA; e-mail: [email protected]

ABSTRACT: This paper explores how raster-based models of urban form (DEMs) and software algorithms derived from

image processing can successfully be used to compute radiative parameters. It starts by reviewing an innovative and

fast algorithm to cast shadows over extensive urban areas. This algorithm is then elaborated into a more complexfunction to calculate the urban sky view factors, a well-established parameter which is central to the comprehension

of the urban microclimate. From the sky view factor the analysis extends to some estimates of illumination falling on

the urban surface under different sky configurations, such as the uniform diffuse sky and the standard CIE overcast

sky. The distribution and variation of illuminance values both outside and inside the buildings is calculated, allowing

the comparison of the environmental characteristics of different urban textures. Results suggest that the analysis of

DEMs could open the way to a new paradigm for urban analysis.


2/46

2

1. Introduction

Sunlighting as form-giver in architecture (Lam, 1986). Or, more poetically Sun rhythm form

(Knowles, 1981). These are the titles of two books, amongst many publications on solar energy

in architecture and planning, which reflect a similar concept: the importance of sunlight and

daylight in shaping the built environment.

This very idea traversed the whole history of architecture and survived even the otherwise

iconoclastic Modern Movement. For instance, the third Congrs International dArchitecture

Moderne (CIAM), held in Brussels in 1930, addressed, among others, the problem of

accessibility of daylight in modern buildings. An influential speech was delivered there by

Walter Gropius, who tried to demonstrate analytically the relationships between building

height, open space, sunlighting and orientation1.

Le Corbusiers attitude towards sunlight and daylight was similarly central to his architecture,

though possibly more poetic. In the city of Chandigarh, the capital of the Indian state of

Punjab which he designed from 1948, the middle of the governmental Capitol complex is

occupied by a Tower of Shadows. Often described as a cosmic sign alongside the monumental

symbols of secular power, the tower is an airy structure entirely clad with brise-soleils, which

cut the sun in each direction from the internal space. In Corbus words, this was to demonstrate

that "that one can control the sun on the four cardinal points of an edifice and that one can play

with it even in a torrid country and obtain lower temperatures(Chandigarh Planning &

Architecture, 2001). This play was to be repeated on most of Chandigarhs faades through the

sculptural use of concrete brise-soleils, supposedly contributing to the overall environmental

well-being of the town (this fact was recently confirmed by monitoring carried out by Faruqui

Ali, 1998).

The pressure on the use and control of sunlight and daylight in architecture will possibly be

greater in the future than it is today, due to the increasing recognition of the resulting energy-

1His findings were later reworked by Martin and March (1972), p. 71 ff.


3/46

3

saving implications. In the words of Littlefair (2000) solar energy in its various forms will be

more and more important in the buildings of tomorrow. In fact, increasing importance is placed

in public policy, on the utilisation of renewable energy through solar water panels or

photovoltaics (PVs). The European White Paper on renewable sources of energy (EU, 1997), forinstance, envisages 500,000 1KW building-integrated PV systems installed in Europe by 2010.

Subsidised action is currently being undertaken by countries as diverse as the USA and India

and is expected to have a considerable impact on the overall energy budget of cities.

In addition to energy considerations, sunlight and daylight have implications for the quality of

urban life. Baker (2000) argues that rich and varied urban environments, with a close

relationship with nature and natural light, generate a higher degree of satisfaction and comfort

than artificial environments. Quantitative measurement to support this thesis is provided by

Nikolopoulou (1998) and Nikolopoulou and Steemers (2000), who carried out a general study on

urban comfort in outdoor spaces. A number of medical studies also exist, suggesting that the

reduction of daylight in urban environments might have serious health consequences on the

population, such as loss of visual acuity2.

Despite the well-established importance of sunlight and daylight conditions in the urban

environment, tools for measuring and simulating them over extensive portions of cities are

inadequate. This was noted by Compagnon (1999), who referred in particular to three limits:

the simplified approach in modelling the sky-vault, the neglect of inter-reflections amongst

buildings and the difficulties of performing simulations on extensive urban areas, which can

occupy several hectares.

The latter task namely the difficulty of modelling geometry on large portions of cities is

partly related to computing difficulties. Traditional models work at the scale of the building but

fail on larger areas because of too much vectorial complexity. This paper aims to test the

possibility of using a very simple raster model: the so-called Digital Elevation Model (DEM),

which is shown in Figure 1. The DEM is a compact way of storing urban 3D information using a

2The extensive use of artificial lighting might have lessened the importance of the considerations below. However,in the early days of studies on daylight, research showed that reduced illumination causes a lack of visual acuity. Inthe preparation of The hygiene of the eye in schools (1886), Hermann Cohn (1838-1906) studied the visual acuity of10,600 children in relation to lighting conditions. He noted the number, size and position of windows and thepresence of urban obstructions. By standardising other factors to which children were exposed, so that findings


4/46

4

2D matrix of elevation values; each pixel represents building height and can be displayed in

shades of grey as a digital image. The analysis of DEMs with image processing techniques has

already proven to be an affective way of storing and handling urban 3-D information, and being

very conducive to a number of urban analyses (Ratti and Richens, 2004). Could it contribute tothe study of daylight and sunlight conditions in cities? This paper will address that question by

describing a nmber of algorithms that have been written using the Matlab software (2004), a

well known package for doing numerical computations with matrices and vectors. Matlab's

extensive matrix capabilities are supplemented by different toolboxes, among which is the

image processing toolbox, with elaborate graphics outputs.

A number of image processing techniques are developed below and applied to three case study

sites in London, Toulouse and Berlin. Standard measures used to quantify the luminous flux will

be examined in urban areas, as well as more general parameters that characterise the radiative

exchange between surfaces (in this case the urban surface). Among these parameters, the sky

view factor has a prominent role; it is a well-known urban climatology variable, whose easy

calculation might prove extremely beneficial to urban planning.

Aside from its computational interest, this paper also presents some comparative results on

different urban textures as a preliminary insight into the interrelationship between urban

texture and radiation.

The study of the impact of urban geometry on the penetration and distribution of sunlight,

daylight, etc. might go some way to clearing up the confusion that occurs in present scientific

literature. In a recent publication aimed at informing solar design of cities, for instance, it was

found: Compared to open country, built urban sites have a larger area of exposed surfaces per

unit area of ground covered. Because of this larger area, potentially more solar radiation can be

collected on a built urban site than on a flat open terrain, especially in winter3. This assertion is

clearly not true: the total radiation falling on a unit area of ground built or un-built is

clearly the same. Furthermore, solar energy in built areas can only be collected with more

difficulty than on flat open terrain, because of the effects of overshadowing and the changing

could be attributed generally to school illumination conditions only, he found that the closer and higher the urbanobstruction, the higher the percentage of myopics among the pupils.3the authors name could be omitted, since this statement is used here in a symptomatic way.


5/46

5

distribution of radiation in space and time. This is caused by the complex patterns produced by

urban geometry. We will try to quantify them below.

0

50

100

150

200

250

Figure 1. Case study site in central London in the Digital Elevation Model DEM format (left) and its axonometric view

(right)

2. Context

Before embarking on the derivation of radiative parameters in cities, it is useful to review some

basic terminology:

- Radiant fluxis the power emitted, transmitted or received in the form of electromagnetic

radiation [Unit: W];

- Radiance(in a given direction, at a given point of a real or imaginary surface) is the

radiant flux in a given direction per unit solid angle, per unit area perpendicular to that

direction (Figure 2). Expressed as a formula:

ddx

dL

cos= [Unit: Wm

-2sr

-1]

where d is the radiant flux transmitted by an elementary beam passing through the given

point and propagating in the solid angle d , containing the given direction; dx is the

area of a section of that beam containing the given point; is the angle between the

normal to that section and the direction of the beam;

- Irradiance(at a point of a surface) is the quotient of the radiant flux d incident on an

element of the surface containing the point, by the area dA of that element [Unit: Wm-2].

- Irradiationis the product of irradiance and time, i.e. surface density of the radiant energy

received [Unit: Wsm-2].


6/46

6

The same amount of electromagnetic radiation, however, has substantially different effects on

the human eye according to its wavelength. No effect at all is recorded outside the so-called

visible band, while maximum efficacy is noted around a wavelength of 555 nanometres. Thisfundamental mechanism of human vision must be taken into account when studying lighting in

architecture and is dealt with in building science with the introduction ofphotometric

parameters. These correspond exactly to the radiative parameters defined above, except that

the radiant flux is weighted by its spectral response to the human eye. In other terms,

radiation is evaluated according to its action upon the so-called CIE (Commission

Internationale de lEclairage, 2004) standard observer. The photometric quantities that

correspond to the radiant flux, radiance, irradiance and irradiation are named respectively:

- Luminous flux

- Luminance

- Illuminance

- Illumination(not very common, but used in this paper)

When radiant or luminous flux falls on a surface it is partly reflected, partly absorbed and partly

transmitted (the latter if the medium is semitransparent). The surface is therefore characterised

by three adimensional ratios, namely the reflectivity, absorptivityand transmittivity ,defined as fractions of the total incident radiation. These ratios are usually a function of the

wavelength and of the angle of incidence of the incoming radiation. However, the following

equation holds:

1=++

An alternative name for reflectivity, ratio of the reflected to incident radiation, is albedo,

which is commonly used by the climatology and earth sciences community.

It is now possible to review some methods which are found in the literature for assessing the

distribution of solar energy in urban areas. The most used quantities are irradiance and

illuminance either of these being chosen according to the focus of the analysis: solar energy

or lighting. Each of these two terms is then separated into two contributions: the direct


7/46

7

component, coming directly from the suns disc, and the diffuse component, received from the

whole sky hemisphere.

The direct component requires knowledge of shadowing in the city. It is zero on all surfaces inshadow, while it has a value which is a function of the angle of incidence on all lit surfaces. In

fact, all lit surfaces at a certain time of the day receive from the sun the same amount of

incident solar radiation (per unit area perpendicular to the radiation beam).

This simple problem of detecting areas in shadow might prove difficult in urban areas with

complex geometry. A review of methods to quantify solar access can be found in Littlefair

(1998). From a single faade, traditional graphic methods are generally used, based on sunpath

diagrams. A wide variety of them exist in the literature; in addition to the indication of sun

and shadows, in some cases they allow the estimation of the energy that can be collected at

any point.

Despite their convenience, graphic diagrams can be used only on one faade at a time.

Furthermore, they require the plotting of the urban geometry against the sun path, and this

process can be time-consuming. Consequently, their use in architectural design is usually

limited to the building scale, or to repetitive and simplified urban layouts. It cannot be easily

extended to urban areas of complex geometry.

In the latter case, the use of computer software becomes imperative. To date, this is not a

problem in terms of computational power: purpose-built programs for architects and urban

designers are readily available. An example is Shadowpack(Peckham, 1985), developed at the

CEC research centre in Ispra, Italy, which utilises its own CAD-type program to generate a

layout and then allows the evaluation of the amount of solar radiation received by each

surfaces. Other software developed specifically for shadow calculations is Townscope(Laboratory

of Architectural Methodology, 2001), Sombrero(Niewienda and Heidt, 1996) and Shading

(Yezioro and Shaviv, 1994). Furthermore, it should be noted that today most CAD packages

incorporate their own tools for shadow casting.

The most serious limitation of all these computer methods is that they require full 3-D urban

models in vectorial form, something that can be extremely costly; in this sense, DEMs have a


8/46


9/46

9

A similar approach will be followed below for urban areas, where techniques developed in the

geosciences have not yet been used. However, due to the discontinuous nature of urban DEMs,

it should be noted that a simple transposition of algorithms developed in the geosciences to

urban areas is not possible.

Figure 2.Symbols used in the definition of radiance. Image from Sillion and Puech (1994).

3. How to calculate shadows on a DEM

Let us start with the simplest problem: shadow casting on a DEM. Even such a simple problem

can become challenging in urban areas. Traditional vectorial model fails, developed to work at

the scale of the building, fail because of excessive geometric complexity.

Shadow casting is the first macro that was written, at the beginning of this investigation into

potential uses of image processing DEMs for urban analysis. The approach is to compute

shadow volumes, that is, the upper surface of the volume of air that is in shadow. As

explained in Ratti and Richens (2004), this can be done by repeatedly shifting the DEM and

reducing its height.

This algorithm is very simple and impressively fast (it allows processing acres of city at a time,

something unthinkable with traditional geometric models; Figure 3). It is used below as a basis

for a number of more complex analyses.


10/46

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50 100 150 200 250

50

100

150

200

250

Figure 3.Shadow casting on the London DEM, sun position: azimuth=30, altitude=30.

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Figure 4.Shadow casting in central London on 31 December, hourly intervals.

4. Animating the sun: dynamic calculations and solar envelopes

The algorithm described above computes shadows for an arbitrary lighting angle; the next stage

is to add a procedure to calculate shadows from the sun for any given latitude, time of year,


11/46

11

and time of day, using the usual astronomical formulae. This allows the possibility of

performing simulations for a whole day or number of days. Results are shown for instance in

Figure 4, which represents the patterns of sun and shadows in central London on 21 December.

The task of converting latitude, time of year, and time of day into altitude and azimuth of the

sun has been carried out in this case by a subroutine. This subroutine however elaborate it

might be is not novel, as it codes well-established astronomical data. Therefore it need not

be described in detail here; references on the formulas used can be found in Muneer (1997).

The first operation made possible by this subroutine is the dynamic representation of

shadowing. Different images can be produced and subsequently animated in Matlab, creating

short movies which give the architect or planner an idea of the shadowing conditions of an

urban site during an entire day.

From these images, simple quantitative parameters have been defined by Steemers and Ratti

(1999a), by the analysis of all the hourly frames of a single day. Summing all black and white

shadowing images, each made of 0s and 1s, it is possible to obtain grayscale images which

have values in the range 0..n, where n is the number of sun positions considered for the day

(equivalent to the number of hours of sun, if the sun positions are taken at hourly intervals).

The resulting image portrays in an elementary way the number of hours of shadow for each

pixel (this value is simply given by the resulting value of that pixel). Contours can be added to

improve visualisation: a standard function in Matlab allows the detection of value changes on

images such as between a pixel with x and x+1 hours of sun.

A mean shadow density can also be defined as the average number of hours of shadow on a

certain region: its value is simply the mean of all pixel values in that region.

The above considerations lead us to a more articulated concept: that of the solar envelope. This

was first introduced by Knowles (1981, 2000, 2004) as the geometric envelope that describes

the volumetric limits of building that will not shadow surroundings at specified times.In other

words, given a certain site in an urban context, the solar envelope defines the maximum built

height that can be reached on that site without compromising the neighbouring buildings solar


12/46

12

accessibility. This is defined as the minimum number of hours of sun, during critical periods of

the day and the year.

The imaginary boundaries specified by the solar envelope can act, according to Knowles, as anoperational tool to inform urban design and planning. They open new aesthetic possibilities in

architecture and urban design, by bringing rhythm and a closer sense of nature into our

buildings and urban spaces, and could be used by public administration as an instrument of

zoning to regulate urban development. In some countries bylaws already acknowledge this fact:

for instance, in the Netherlands, every living room should receive a minimum of three hours of

direct sunlight on its faade every day, from 21 March to 21 September (MVRDV, 1999).

The emphasis on solar rights considerations in the design of urban fabric has recently led to

the extension of the solar envelope concept by Capeluto and Shaviv (2000). They make the

distinction betweensolar rights envelope andsolar collection envelope:

1) thesolar rights envelopedefines the maximum height of a building in order not to violate

the solar rights of any of the neighbouring buildings during a given period of the year; it is

basically equivalent to Knowles definition;

2) thesolar collection envelopedefines the lowest possible surface to locate windows and solar

collectors so that they are not obstructed by neighbouring buildings, during a given period

of the year. This is, in a certain sense, a symmetrical parameter to the solar right

envelope: it describes the overshadowing of a neighbourhood on a given site.

These two envelopes can be interpreted as the upper and lower boundary of asolar volume4,

which represents the portion of space where new developments could be allowed without

reducing the solar access of the neighbouring buildings and guaranteeing sufficient solar

access.

The subroutine that governs shadowing on a DEM, together with the dynamic calculation of sun

positions, allows the detection of solar envelopes. For instance, it will be shown here how this

can be done in the case of the solar collection envelope. Let us consider a fictitious open site


13/46

13

in central London depicted in Figure 5 and let us ask the following question: which is the

boundary surface where there are at least n hours of sun on 21 March?

The algorithm is particularly simple and elegant. It is based on running the shadowing volume

program (see section 3 above) on a DEM, for all n hourly sun positions on 21 March. Each of

these n shadowing volume images represents the height of the separation surface between

light and shadow at a certain time. If on each pixel of the DEM, these n heights are sorted out

by increasing values, n 3-D surfaces one on the top of the other are obtained. Each of these

images can be read as a 3-D map (or DEM) which define the portion of space where there are 1,

2, 3,, n hours of shadows. In other words, each of these images represents in the DEM form

the solar collection envelope which corresponds to the required condition of n hours of sun.

Figure 6 represents some of these surfaces on 31 December on the urban site in central London.

As expected, the height of the solar collection envelope increases with the increase in the

number of hours of sun.

In its standard form the solar envelope criterion has a limit: it does not take into account the

angle of incidence of sunlight. In other words it is just binary, 0 or 1, sun or shade. An hour of

sun at midday and one at dawn count the same, although they are very different from an

energy point of view. This limitation, however, could simply be overcome by introducing a

weighting function based on the sun altitude.

The information embedded in images such as the one shown above have direct applications in

architectural design. This aspect was investigated, amongst others, by Birks (2000), who has

been using the above shadow casting algorithms to translate images in proposals of building

form. He concluded that software of this kind could play an active role in generating design.

His analysis drew on a wider approach to the synthesis of architectural and urban form, based

on the use of the datascapes technique. This has been theorised by the Dutch practice MVRDV

(1999), as an observation technique which should let the limits and gravities inherent a certain

design situation emerge. Constraints (such as shadowing conditions, noise level regulations,

etc.) are laid onto space and used to inform the design process.

4It should be noted, however, that when the solar collection condition is particularly restrictive, the solarcollection envelope might be higher than the solar rights envelope. In this case the solar volume does not exist.


14/46

14

Although MVRDVs design proposals often look like far too naive responses to context

constraints as when they take plans of noise contours produced from a motorway as faade

generators for architectural shape their analysis proves that solar envelopes and similartechniques have potential for exploring primary design solutions.

20 40 60 80 100 120 140

20

40

60

80

100

120

140

Figure 5.Experimenting with solar envelopes: the image represents a fictitious open site in central London.

20 40 60 80 100 120 140

20

40

60

80

100

120

140

20 40 60 80 100 120 140

20

40

60

80

100

120

140

20 40 60 80 100 120 140

20

40

60

80

100

120

140

Figure 6.Experimenting with solar envelopes: the image represents the height of the surface where there are

1, 3, 6 hours of sun respectively (values for 31 December); on the far right the surface where there are 6

hours of sun is plotted in axonometric.


15/46

15

5. The sky view factor

All the algorithms presented above shadow casting and its derivatives are based on the

following principle: fix an arbitrary sun position and from this position trace back rays to the

city. The DEM pixels that are intercepted by these rays are illuminated by the sun (and

therefore white). All other pixels are in shadow (black).

At the end of the previous section, however, another approach emerged, based on the following

principle: fix a point on the DEM and count the number of previously defined sun positions

that can be seen.

This method can have very interesting applications, such as determining the amount of sky that

is visible from a given pixel. The mechanism is the following: spread a number of sun positions

on the sky-vault, cast shadows each time on the DEM and count the number of whites and

blacks obtained on each pixel. If the sun positions are distributed uniformly on the sky-vault,

the count of whites on a given pixel divided by the total number of sun positions taken into

consideration is directly the solid angle of view of the sky (with a proportionality factor of

2 ).

By a slight refining of this algorithm, a very interesting parameter, often used in urban

climatology, can be derived: thesky view factor(also called configuration, shape or form

factor). This parameter is similar to the solid angle of view of the sky, although two weighting

coefficients are applied in order to weight different parts of the sky-vault in different ways.

Expressed as a formula, the sky view factorskydAi

F

reads:

=sky

j

ji

skydA dA

R

Fi 2

coscos

The integration is performed onto all elemental surfacesjdA which compose the sky-vault; idA

is an elemental surface of the city,i and j are the angles between the vectors normal to

idA and jdA and the line connecting them, whose length is R (Figure 8).


16/46

16

In fact, the sky view factor is simply the sum of cosine weighted elemental solid angles on a

whole hemisphere. The reason for this weighting comes from heat transfer theory, where the

view factor was first introduced to model radiative exchange between surfaces. Given two

diffuse surfaces5

i and j , the view factor simply defines the fraction of the total radiation

leaving surface i which is intercepted by surface j . A more complete discussion of this aspect

can be found in textbooks of heat transfer theory such as Incropera and DeWitt, 1990.

In practice, the sky view factor is quite difficult to derive. Analytical solutions and numerical

tables are available only in a number of simple geometrical configurations. In the urban

context, Glenn and Watson (1984) derived an analytical expression to determine it for standard

canyon arrangements. In more general cases, approximate graphic methods can be used, based

on the manual or computer-based analysis of zenithal fish-eye lens photographs (see for

instance Steyn, 1980; Rich, 1989; Chalfoun, 1998, who used it in the assessment of outdoor

thermal comfort; Brown et al., 2001) as well as numerical techniques. The latter, however, can

be quite time-consuming.

Let us look at how the sky view factor can be calculated on a DEM. As anticipated, the

algorithm is based on the repeated application of the shadow algorithm. We simply compute

the shadows for a large number of light sources, distributed over the sky, and for each pixel

count the number of times they are in light. So if we use 1000 samples, any pixel whose count

is 1000 can see all the sky and has a sky view factor 1, while a count of 0 means that it cannot

see the sky at all (Figure 7).

To get meaningful results, according to the above definition of view factors, it is necessary to

distribute the sample points over the sky in the correct manner. If a uniform distribution is

used, then what is measured is not the view factor but the solid-angle of sky visible from each

point. In order to take into account the cosine correction, the density of samples must be

higher at the zenith than towards the horizon. It is easy to see that the correct distribution to

compute the view factor from the city to the sky can be obtained by spreading points evenly

over a unit circle in the horizontal plane, and then projecting up to a unit hemisphere.

5A diffuse (or lambertian) surface transmits or reflects light adhering to Lambert's cosine law. This law states thatthe reflected or transmitted luminous intensity in any direction from an element of a perfectly diffusing surfacevaries as the cosine of the angle between that direction and the normal vector of the surface. As a consequence, theluminance of that surface is the same regardless of the viewing angle.


17/46

17

The number of points being very large, it is possible to chose a uniform and random

distribution of points on the circle. This will also avoid a well-known problem in image

processing, namely the emergence of patterns of interference. The following question, however,should be answered: how can a uniform distribution of points on a hemisphere be achieved?

The Matlab function RAND produces random entries chosen from a uniform distribution on the

interval [0,1]. An elemental area in polar co-ordinates is written:

rdrddA =

If a uniform distribution is taken on rand , the resulting distribution of points on the circle

will not be uniform. The density of points will be higher towards the centre and lower at the

periphery, following an inverse proportionality to r(Figure 9).

In order to have a uniform distribution of points on the circle a function zr= must be used,

where zfollows a uniform distribution. A proof of this fact is given in the footnote6, results

are shown in Figure 9.

From this uniform distribution of points on a disc of radius 1, the positions of the sun are

derived by projecting on a hemisphere. The iterative application of the shadow-casting routine

allows the calculation of the sky view factor. The higher the number of calculations, the more

accurate the results. 100 iterations are usually enough to produce satisfactory sky view factors,

such as those for London, Toulouse and Berlin shown in Figure 10.

6The proof can be given as follows:

Make a change of variable so that dA becomes independent from r. If a new function )(rzz= is chosen, thisimplies:

kdzrdr=

Where kis a constant of proportionality. The above equation can be integrated:

= kdzrdr and gives:

cr

kz +=2

2

Any function will satisfy our condition, so we choose 2r= . This gives:

zr=


18/46

18

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50 100 150 200 250

50

100

150

200

2500

0.5

1

1.5

2

2.5

3

50 100 150 200 250

50

100

150

200

250

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

50 100 150 200 250

50

100

150

200

2500

1

2

3

4

5

6

7

8

9

10

50 100 150 200 250

50

100

150

200

250

Figure 7.Viewing the sky from the city: the process is based on the iterative sum of shadow images. A similar

procedure, based on a multi-sun heliodon, had been used by Lionel March in the 1970s at the Martin Centre

to add shadows on physical models. The figures above show results for the 1st, 3

rd, 5

thand 10

thiteration; from

the 100thiteration (sum of 100 shadow images) results are very accurate; the figures in Figure 10were

stopped at the 1000thiteration.

Figure 8.Symbols used in the sky view factor formulas, taken from Incropera and De Witt (1990).


19/46

19

Figure 9.Distribution of points on a disc. The image on the left shows a uniform distribution of points on

polar co-ordinates rand ; the image on the right shows a uniform distribution of points per unit area.


20/46

20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50 100 150 200 250

50

100

150

200

250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50 100 150 200 250

50

100

150

200

250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50 100 150 200 250

50

100

150

200

250

Figure 10. Sky view factors in central London, Toulouse and Berlin obtained after 100 shadowing

calculations.


21/46

21

6. The meaning of sky view factor: the urban heat island

The view factor from the city to the sky can be intuitively understood as a measure of the

openness of the urban texture to the sky. Its interest, however, lies in its capacity to explain a

number of climatological phenomena in cities, such as the urban heat island.

The urban heat island is a well-documented example of modification of the atmosphere by

urbanisation, which is accompanied by an increase in air temperature. While the trend toward

higher temperatures in cities was first noticed by Howard (1833) during the first half of the 19th

century, the term urban heat island appeared in the scientific literature towards the end of the

1950s. It refers to the characteristic shape that isotherms assume in urban areas, an island of

higher urban temperatures can be recognised in the sea of lower rural temperatures.

The features of the urban heat island vary both in space and time, as a result of meteorological,

locational and urban characteristics. The maximum temperature differences are found with clear

sky conditions and wind calm, while they diminish with cloudy and windy weather. The urban

heat island peaks are generally in the core of the city.

A well known formula was introduced by Oke (1981) to relate the maximum heat island

intensity between urban and rural sites and the sky view factor:

skyruralurbanT 88.1327.15-max =

wheresky is the view factor from the middle of the canyon floor to the sky and ruralurbanT -max

the maximum air temperature difference measured between that canyon and a rural site7. The

formula was obtained by results from physical scale models representing rural and urban

surfaces, to mimic the passive radiative cooling of these environments following sunset on a

calm and cloudless night. It was also verified using experimental data from a number of cities.

It might be useful here to review the physics that is behind the dependency between urban

heat island and sky view factor. Two processes should be mentioned:


22/46

22

1) The reduction of long-wave radiation from the street canyons. As has been seen (see section

5 above), the concept of sky view factor was originally introduced in heat transfer literature

to model the radiative exchange between surfaces. The higher the sky view factor in astreet, the higher the radiative exchange between that street and the sky. Because of its

low temperature, the clear sky is a very important energy sink in the infra-red region of the

electromagnetic spectrum. Therefore, a high view of the sky means elevated heat losses,

especially during the night when there is no solar radiation.

In densely built areas the ground screening by flanking buildings reduces the sky view

factor. An increasingly larger proportion of the cold sky is replaced by the relatively warm

sides of buildings, reducing radiative losses. This phenomenon is in accordance with the

occurrence of the urban heat island at night-time, when the long-wave heat loss is very

important in the energy balance of cities.

2) The increase of short-wave energy absorption due to multiple reflections. The city surface is a

mixture of vertical and horizontal elements that create urban canyons; this has

consequences on the amount of solar radiation that is absorbed in urban areas. The

physical mechanism can be described as follows: when solar radiation falls on a flat surface,

it is partly absorbed and partly reflected to the whole hemisphere. When solar radiation

falls in a urban canyon part of the reflected radiation will hit the canyon surface again.

This results in a sequence of multiple reflections which increases the total amount of solar

radiation absorbed (Figure 11).

The phenomenon of trapping solar radiation occurs on any crenellated or textured surface

and has been discussed at length in the scientific literature. All approaches show that in

general the albedo of a crenellated surface is lower than that of a flat plane composed of

the same material. Numerical simulation was carried out by Aida and Gotoh (1982), who

found that the urban reflectivity decreases as the urban irregularity increases. Their overall

results were confirmed by physical model experiments carried out by Steemers et al.(1997).

Reflectances on the London, Berlin and Toulouse case studies, whose DEM models are

examined here, were measured. Findings can be summarised as follows: In broad terms the

7The definition of an urban temperature is quite difficult. Okes formula (1981) is based on air temperature measures


23/46

23

study has shown that, for all paint reflectances, urban forms absorb more sunlight energy

than flat planes Furthermore, it is apparent that the complexity, or occlusivity [it might be

said view factor] of the urban texture affects the amount of light that is absorbed. Thus the

reduction of light reflected from the modelled surface (compared with plane surface) is muchgreater for Toulouse than for London, which in turn is greater than for Berlin. (Steemers et

al., 1997).

The combined effect of the two factors above the reduction of long-wave radiation from the

street canyons and the increase of short-wave energy absorption due to multiple reflections is

at the basis of the urban heat island. Both factors are well taken into account by the value of

the sky view factor8.

Taking the results for our three urban case studies (average view factor in the urban canyons,

Figure 12), it can be seen that they rank in the order London, Toulouse and Berlin, with

average values of the sky view factor at the street level of 0.529, 0.646 and 0.720 respectively.

By applying the above formula9by Oke (1981) it is possible to predict values of the maximum

urban heat island, which are respectively 7.9, 6.3 and 5.3C a considerable difference, which

is simply produced by urban morphology.

These results, however, pose a series of questions: How good is it to have a urban heat island

in cities? Is there an optimum value for it, which maximises, say, energy savings or thermal

comfort? And, in this case, would designers be able to implement urban design strategies in

order to modify the sky view factor and therefore the intensity of the urban heat island ?

The answers to all these questions are likely to be climate dependent. Little doubt exists that in

cold climates the urban heat island has a beneficial effect in energy saving. This is clearly

stated by Oke (1981): Further work on the effects of geometry could also be of practical value.

For example in mid- and high-latitude cities fostering the winter heat island effect can reduce

taken near the ground in the middle of urban canyons.8It is noted here that the sky view factor is a rather coarse parameter at the small scale, as it is orientation-insensitive and unable to predict precisely where solar absorption occurs. However, it has a fundamental role to playat the urban level, where the focus is on spatially averaged parameters.9By using average values of the sky view factor at street level we are slightly underestimating the mid-street skyview factor used in Okes formula. The approximation, however, is negligible. Furthermore the sky view factor valuesin Okes formula were derived from measured height-to-width ratios with a little geometrical simplification, whichtends to compensate the approximation.


24/46

24

space heating requirements by 5-15 per cent (Summers, 1974) and in cities subject to excessive

heat in summer minimising this effect could save on energy used for air conditioning and perhaps

even reduce fatalities due to heat stress (Clarke, 1972). If the role of geometry is as important as

suggested in this study it should be regarded as a fundamental input in urban design.

In another paper, the same author suggests that for mid-latitude cities with no overheating

problems the minimum acceptable H/W ratio of the urban canyon would be approximately 0.4

(which for an infinite canyon corresponds to a sky view factor of 0.78). This value would

maintain about one third of the heat island potential for a given city (Oke, 1988).

The interpretation of the effects of the sky view factor and the subsequent urban heat island in

hot climates is more controversial. In fact, minimising the urban heat island would suggest

sparse and scattered urban developments. However, this is the opposite of what is commonly

found in vernacular architecture in hot arid regions (such as Arabic countries). There, a

courtyard-based and compact urban matrix, which tends to minimise the sky view factor, is

mostly adopted (Figure 13). Would this be a case of anticlimatic solution, i.e. of irrational

response to climate10(Rapoport, 1969)? Or could it be justified in urban climatology terms?

Let us consider in more detail the urban heat island phenomenon. This is often described using

average temperatures: in the definition of Givoni (1998) on average the diurnal temperature, in

a densely-built urban area, is warmer than the surrounding open (rural) country. The average

however, does not take into account peaks. The urban heat island usually presents two of them:

a maximum during the night and a minimum during the day, often described as an urban cool

island.

In hot arid climates the unmodified night-time temperatures are usually low, and an increase in

them would probably be accepted11if it can concomitantly alleviate extreme temperature stress

during the day. Furthermore, urban pedestrian comfort is not only based on air temperature,

10In fact, rarely is an architectural response to climate completely rational or completely irrational. More often it is a

question of balance, where some design variables are privileged in comparison to others. For instance, an urban formcould be optimised in terms of thermal mass and not in terms of sky view factor. It will be shown below that thelatter is not the case for the dense vernacular developments which are found in hot-arid regions.11It should also be noted that in some hot-arid countries, such as Morocco, people have developed the habit of

sleeping on building roofs, to maximise radiative losses.


25/46

25

but also on radiative exchange: this benefits from low sky view factors, which mean an increase

in direct shading and a reduction in reflected radiation.

Quantitative descriptions of these phenomena, which justify the adoption of compact urbanstructures in hot arid climates, are given by Pearlmutter et al.(1999), who simulated the

energy exchange between a cylindrical body, representing a pedestrian, and the canyon

environment. Pearlmutter (1998) analyses new Israeli towns, planned according to the garden

city model imported from Europe, and concludes that they are climatically inappropriate, as

they lack the thermal moderation effect of traditional compact developments.

A recent study of the city of Fez, Morocco, based on field measurements, confirms the same

results. Rosenlund et al.(2000) monitored temperatures in two districts of the city, associated

with different housing types: a traditional one, based on the compact assembly of buildings

and a courtyard structure, and a more recent one, based on modern two- to three-storey houses

arranged alongside wide streets. Temperature results are markedly different in the two sites. In

the traditional district temperatures are higher during the night, but during the day a

favourable cool island appears. Overall conditions are more stable than in the modernist

development, with the tendency to smooth down maximum and minimum temperatures.

Interesting data have also been obtained by Perez-de-Lama and Cabeza (1998). They examined

the environmental performance ofpatiosin Seville using the sky view factor (which they call

configuration factor). They focussed on maximum temperatures inside the patios during the day

and concluded that in all cases but one maximum temperatures measured in the set of patios

have been below the reference temperature considered. They also suggest that patio cooling

performance in summer is proportional to the envelope to plan surface ratio. The smaller the

opening to the sky in relation to the general envelope surface (i.e. the average sky view factor)

the better the performance is.

These conclusions reassuringly show that vernacular urban structures found in hot arid regions

are well adapted to the climatic context at least as far as air temperature and the urban heat

island are concerned.


26/46

26

Figure 11.Schematic distribution of solar radiation in open flat country (top), built-up area with height-to-

width ratio of about 1 (middle) and high-density urban area with height-to-width ratio of about 4

(bottom); from Givoni (1998).

London Toulouse Berlin

Average sky view factor at ground level 0.529 0.646 0.720

Predicted increase in temperature [C] using the

formula:

skyruralurbanT 88.1327.15-max = 7.9 6.3 5.3

Figure 12.Data for London, Toulouse and Berlin.

Figure 13.Courtyards in central Marrakesh, from Rudofsky (1964).


27/46

27

7. The meaning of sky view factor: urban illumination

Aside from its ability to explain the urban heat island, the sky view factor has a further

meaning: it is proportional to the amount of light that would fall on a city under a uniform sky

i.e. a sky that has the same radiance in all directions12. The sky view factor can therefore be

considered as a preliminary indicator of urban daylight conditions: when it assumes the values

0 and 1, it represents, respectively, nil and maximum illumination (where maximum

illumination means an unobstructed view of the sky hemisphere).

Using this key, the grayscale views of London, Toulouse and Berlin shown in Figure 10 are

easily interpreted. Small courtyards receive little light from the sky and are therefore dark.

Roofs have an almost unobstructed view of the whole sky hemisphere, and are therefore very

white. Street junctions are particularly interesting, because they make evident the linear

additive nature of the radiation processes: their luminous intensity is approximately double of

that of streets as two strips of the sky are visible at the same time.

In architectural and urban design, however, the greatest interest is not in illumination values

in the streets, but on those on faades. These can be calculated on a DEM by cutting it at

different heights (for instance each 3 metres, a value that approximates a standard floor

height) and processing it slice by slice. Results represent sky view factors from an elemental

horizontal area placed at a certain height on the faades.

The operation described above can be done with much the same computational time as for one

single image: instead of first cutting the DEM to obtain multiple images and then running the

sky view factor algorithm on each one, the slicing process is introduced at the end of the

shadowing subroutine. This does not result in an increase in the total number of iterations.

View factors with height for the London, Toulouse and Berlin case studies are presented in

Figure 14, Figure 15, Figure 16. It is interesting to see how available daylight increases with

height. This is quantified in Figure 17, which show the variation of the average sky view factor

in the streets with height. As expected, values approximate 1 at the top of the urban canopy.


28/46

28

A better evaluation of available daylight, however, is obtained by considering only faade

values. These have been averaged and plotted with height (Figure 18 and Figure 19). As

expected, they approximate the value 0.5 at the top of the urban canopy (an unobstructed

faade will always see just half of the sky hemisphere). In Figure 18 a sudden reductionsometimes appears in the upper part of the city. This is due to the progressive cut-off of

buildings with increasing slicing height; as a result of this the faades on which the average is

calculated varies and so does the mean value.

Also, numerical results showing the average variation of the sky view factor on building faades

can be linearly interpolated (Figure 20). The slope of the line is proportional to the variation of

illumination with height: the steeper the slope, the smaller the difference in lighting levels

between the top and the bottom of urban canyons. Results show that London, which has a

more articulated texture (i.e. a complex mix of buildings heights) has the maximum slope (and

therefore the most even vertical distribution of light). Toulouse, which is a more uniform city

but still shows some vertical differences, follows and then Berlin, where most of the buildings

are the same height. These results will be discussed in the next section.

Another parameter that would be of great use to architects is the distribution of illumination

inside the buildings. This can be done using the sky view factor parameter, after some

assumptions about the structure of the urban matrix: floor height and faade transmittance. In

this case they have been assumed respectively 3 m and 100% (in reality faades will be

partially windowed with glass of transmittance less than 100% and partially obstructed by

opaque materials such as masonry; however, as far as they can be considered homogeneous and

isotropic, a simple proportionality constant applies). A further assumption is that there are no

partitions inside the buildings. With these hypotheses the city can be imagined as a series of

slabs floating in the air at a distance of 3 metres.

It is worth noting here that under the condition of uniform sky, the view factor inside the

buildings is the same as what architects call the sky component of daylight factor. This is

defined as the ratio of that part of the illuminance at a given point on a given plane which is

received directly from the sky, to the illuminance on a horizontal plane due to an unobstructed

hemisphere of this sky (Baker et al., 1993).

12This meaning of the sky view factor directly follows from its definition in terms of radiative exchange with the sky.


29/46

29

Results are presented in Figure 21, Figure 22, Figure 23. It is interesting to note not only the

increase of illumination with height, but also its different penetration inside the buildings. The

latter clearly is at a maximum on the top floors, which can see the largest area of sky.Important information can also be extracted from these data by applying a threshold (cf.

Compagnon and Raydan (2000). If it is assumed that an area inside a building relies solely on

daylight if it has a sky view factor (in the absence of a faade) greater than a given value h ,

all urban portions which are potentially naturally lit can be detected on the DEM.

Results shown by Figure 21, Figure 22, Figure 23 are of course very high, because of the

hypothesis of dealing with buildings with completely trasmittive skin. In reality, a major

reduction of illumination is caused by the glazing ratio of faades, the obstructions of framing

and the transmittivity of glass: lighting levels would therefore be much lower. Also, a more

accurate algorithm might take into account the directional properties of glass transmittivity:

light falling from patches of sky high on the horizon has a low angle of incidence on vertical

windows and is therefore mostly likely to be reflected than light falling with high angles of

incidence (Figure 24 shows the variation of glass transmittivity with angle of incidence).

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Figure 14.Sky view factors on the London DEM computed at 0,3,6,9,12,,39 m; results were obtained by

spreading 1000 fictitious sun-positions on the sky-vault.


30/46

30

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Figure 15.Sky view factors on the Toulouse DEM computed at 0,3,6,9,12,,30 m; results were obtained by


50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Figure 16.Sky view factors on the Toulouse DEM computed at 0,3,6,9,12,,18 m; results were obtained by


0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

Figure 17.Average sky view factor in the street canyons (from left to right: London, Toulouse and Berlin). As

expected, it increases with heights and approximates 1 at the top.


31/46

31

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

35

40

Figure 18.Average sky view factor on the building faades. As expected it increases with height and

approximates 0.5 at the top (where faades view half of the sky-vault); the slight decrease observed at the

top and the various changes in slope are due to the fact that the average is taken on different urban

portions, due to the progressive cut-off of buildings.

London Toulouse Berlin London Toulouse Berlin

0 m 0.20 0.18 0.24 21 m 0.40 0.38 -

3 m 0.23 0.21 0.26 24 m 0.41 0.34 -

6 m 0.27 0.25 0.27 27 m 0.42 0.36 -

9 m 0.31 0.28 0.29 30 m 0.44 0.38 -

12 m 0.34 0.33 0.31 33 m 0.44 - -

15 m 0.35 0.36 0.34 36 m 0.44 - -

18 m 0.40 0.37 0.38 39 m 0.43 - -

Figure 19.Value of the average sky view factor on the building faades for London, Toulouse and Berlin.

0

2

4

6

8

10

12

14

16

18

20

0.2000 0.2200 0.2400 0.2600 0.2800 0.3000 0.3200 0.3400

London

Touluse

Berlin

Figure 20.Linearly interpolated variation of sky view factors with height in London, Toulouse and Berlin

(showing just the first 20 metres near the ground).


32/46

32

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Figure 21.Sky view factors inside the buildings on the London DEM, computed at 0,3,6,9,12,,39 m; results

were obtained by spreading 1000 fictitious sun-positions on the sky-vault.

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Figure 22.Sky view factors inside the buildings on the Toulouse DEM computed at 0,3,6,9,12,,30 m;

results were obtained by spreading 1000 fictitious sun-positions on the sky-vault.


33/46

33

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250

Figure 23.Sky view factors on the Berlin DEM computed at 0,3,6,9,12,,18 m; results were obtained by


8. A brief mathematical diversion: optimisation of urban daylight

This section contains some elementary mathematical developments, inspired by previous

results. It was shown in section 7 above that London behaves better than Toulouse, which in

turns behaves better than Berlin in terms of daylight and sky view factor distribution. Although

the difference is not striking, it prompts the following question:

Is there an urban texture that optimises the distribution of daylight?

This question could be addressed in a numerical way by taking different DEMs of both real and

imaginary urban cities, by analysing them and by comparing the results. However, under a

number of simplifying assumptions, the problem can be treated mathematically.

The concept of optimum daylight distribution should be discussed first. Clearly, urban texture

does not affect the total amount of radiation falling on a city, but only the way this radiation

is distributed amongst faades, streets or roofs (the image of Marrakesh, Figure 13, is anexample where radiation is mostly intercepted at roof level). Furthermore, the notion of

optimum is likely to be climate dependent: introverted urban texture with reduced light

admittance might prove effective in hot and sunny climates, while open textures with large

faade areas might be preferred in cold climates.


34/46

34

Here a simple and rather general definition of optimum urban texture is proposed: that where

available daylight is most evenly distributed, as in a sort of democratic access to light

principle. We will test against this law a simplified urban component, widely used in urban

climatology: the infinite and symmetrical urban canyon. Under these conditions the analysisbecomes simply 2-D, and the previous question can be more accurately rephrased:

What section of an urban canyon results in the most uniform distribution of sky view factors on

faades?

An interesting geometrical property is noted first: all points on a semicircle, such as that shown

in Figure 25, see the segment AB with a constant angle (if AB is the diameter, then

2/= ). If this law is extended to a semicircular urban canyon of infinite length, i.e. half an

infinite cylinder, it is possible to conclude that all points of the canyon would see the sky

under the same solid angle or, in other words, would see the same amount of sky.

Going back to the sky view factor, which can be thought of as a kind of sky-view solid angle

with a cosine weighting imposed by radiative transfer principles (see section 5), by analogy, it

is possible to infer that the canyon profile that results in a uniform sky view factor on the

faades should have a shape similar to that of Figure 25: a smooth and concave symmetrical

curve.

Analysing the problem in more detail, mathematical tables exist that give analytical expressions

for calculating sky view factor between surfaces. Using the expression reproduced in Figure 26,

it is possible to impose the condition of uniform view factor from an elemental horizontal

surface to the sky. This leads to the following equation13(in a Cartesian co-ordinate system):

13The equation can be obtained in the following way.

Figure 26 gives the view factor between an elemental surface1dA and a rectangular surface 2A parallel to it at a

distance h :

++

+

++

=

22

1

2222

1

22tantan

2

121

hb

a

hb

b

ha

b

ha

aF AdA

Due to the additive property of view factors (Figure 26) it is possible to write:

514131211 AdAAdAAdAAdAAdA FFFFF

tot +++=

In an infinite canyon it can be assumed that b . Taking the limit for each term gives:


35/46

35

constyx

x

yx

x=

++

++

+

2222 )1(2

1

)1(2

1

Plotting this equation gives the curve shown in Figure 27 (for 5.0=const ), somewhat similar

to a parabola. Any elemental horizontal surface (which could represent, for instance, a working

desk) placed on this curve will have a constant view factor of the sky14.

The profile of Figure 27 is certainly difficult to realise in architectural practice, as it would

require tilted and glazed building faades. Despite Frank O. Gehrys efforts, most of todays

buildings still consist of vertical and horizontal surfaces. It can be pleasantly noted, however,

that two environmentally conscious projects dating back to the late 60s namely the

Brunswick Centre in London and the archetypal courtyard forms proposed by Leslie Martin and

others, Figure 28 and Figure 29 tried to take advantage of sloping faades. In the Brunswick

centre, in particular, the upper part of the glazed verandas is tilted and the resulting canyon

profile somewhat approximates to the curve of Figure 29.

The above analysis, however, would not be valid if a zig-zag faade is considered, made of

horizontal and vertical surfaces, however close it might be to the profile of Figure 27. The sky

view factor would then not be constant anymore, but higher in the upper part of the canyon

than in the lower one. The beneficial increase in view of the sky provided by the tilt of the

faade would be lost by the view cutting effect of the horizontal and vertical surfaces.

The interest of the above analysis, however, might lie elsewhere. If extended, it might confirm

something that was already suggested by the comparison of sky view factor distributions in

+=

22

1

lim 2221

ha

a

F AdAb

Adding together all 4 terms and adopting a Cartesian co-ordinate system, where the y axis is vertical and the top of

the canyon has co-ordinates )0,1(A and )0,1(B , gives:

constyx

x

yx

x=

++

++

+

2222)1(2

1

)1(2

1

14If instead of the sky view factor the CIE overcast sky model were used (see section 9), the canyon profile would be

changed further (it would be deeper, as the CIE model weights patches of the sky high on the horizon more than thesky view factor).


36/46

36

London, Toulouse and Berlin: that a mixed distribution of building heights, as in a parabola, is

beneficial to a more uniform accessibility to daylight on faades. This would mean that, for a

given built density, cities with irregular skylines might be better than homogeneous ones.

Formal proof of this fact, however, could only be given by a complex statistical analysis, whichlies beyond the scope of this work.

Figure 24.The variation of the trasmittance of ordinary clear window glass with angle of incidence, according

to different authors. Almost 100% of light is reflected when the angle of incidence is very low, compared

with just 8% when it is 90. Image from Hopkinson et al.(1966).

A B

P1

P2

Figure 25.It is easy to prove with elementary geometry that is constant in all triangles inscribed within a

circle; in particular, when AB is the diameter, 2/ = .


37/46

37

Figure 26.Calculating the view factor between two surfaces. Left: the view factor between a rectangular

surface2A and an elemental surface 1dA parallel to it at the distance h can be written as

++

+

++

=

22

1

2222

1

22tantan

2

121

hb

a

hb

b

ha

b

ha

aF AdA

.

Right: the view factor from1dA to totA is additive and can be written as the sum of the partial view factors

514131211 AdAAdAAdAAdAAdA FFFFF

tot +++= . From Boffa and Gregorio (1976).

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

y

1 0.5 0.5 1x

Figure 27.Plot which represents the section of an infinite long urban canyon where all points have the same

view of the sky (i.e. the sky view factor, and therefore illumination under a uniform sky, is constant). The

function in its implicit mathematical formulation is given in the footnotes and is plotted here for a value of

5.0=const . The plot was produced using the mathematical computer program Maple (see, for example,

Nicolaides and Walkington, 1996).


38/46

38

Figure 28.The court form, or pavilion antiform, according to Martin (1972).

Figure 29.The Brunswick Centre, London, an icon of British architecture of the 1970s. Designed by Patrick

Hodgkinson, it is characterised by its slanted glass faade. Images from Progressive Architecture(1973).

9. A more precise estimate of illuminance in urban areas

It has been seen above that the sky view factor is a precise estimate of illuminance in urban

areas if the sky has a uniform luminance. This condition, however, is not very realistic. Even

under optimal overcast conditions, the sky is brighter at the zenith than near the horizon. More

generally, the sky luminance changes from point to point and is a complex function of the

azimuth and elevation of the sky patch considered. Expressed as a formula:

),( LLsky =


39/46

39

The functionskyL , representing sky luminance at a certain time, can be given either with an

analytical expression or with a table of measured values, from which intermediate values are

calculated by interpolation. In general, one of the following is used:

1) The CIE overcast skyis the simplest sky luminance model, which is bound to represent ideal

overcast conditions. The zenith is 3 times brighter than the horizon. The luminance

distribution has cylindrical symmetry (i.e. it is solely dependent on the elevation ) and is

given by the relation below:

3

)sin21()(

+=

ZenithLL

where ZenithL is the luminance of the zenith.

2) The CIE clear sky, a model like the one above but representing ideal clear sky conditions; its

full formulation can be found in Commission Internationale de lEclairage (1973).

Both models above represent ideal overcast or clear sky conditions. More generally, however,

the sky luminance will take intermediate values and present more complex patterns. This can be

accounted for with the formulations below:

3) Tables with measured sky luminance distributions at given locations. Although these tables

are the most accurate and realistic sky representations, they are available only for a

handful of geographical locations. This has prompted researchers to produce empirical

models that can produce sky luminance from easily-measured quantities, such as direct and

global irradiance at a given location, as reviewed below.

4) A number of sky luminance distribution models existing in the scientific literature. For acomparison of their respective performances see Ineichen et al.(1994). The model most

referred to in the scientific literature is the one by Perez et al.(1993). It is based on an

experimental data set of more than 16000 full-sky scans from Berkeley, California, and

provides an all-weather luminance distribution based on measured diffuse and global

radiation at certain locations (these quantities are commonly available for a large number

of locations across the globe). This model has been incorporated into a number of software


40/46

40

packages such as Radiance, by the Fraunhofer Institute for Solar Energy Systems in Freiburg

(Germany).

Assuming that the function ),( LLsky = is available from one of the formulations above, how

can it be used to calculate illuminance values on a DEM ?

The algorithm is similar to the one developed to calculate sky view factors, and is based on

casting shadows from a number of points spread over the sky-vault. However, to calculate the

sky view factor (see section 5 above), these points were distributed uniformly on a circle and

then projected onto a hemisphere, in order to obtain the appropriate cosine weighted function.

Now, in order to calculate the luminous energy falling on the DEM from a sky of given

luminance, it is necessary to distribute points uniformly on the sky-vault and then weight the

shadow images using the function ),( LLsky = .

A random distribution of points on a hemisphere with a uniform density can be obtained as

explained below.

An elemental area on a sphere of unit radius is (in spherical co-ordinates):

dddA )cos(=

where is the azimuth and the elevation.

If a uniform distribution is taken on and , the resulting distribution of points on the circle

will not be uniform. The density of points will be higher towards the poles and lower at the

equator, following an inverse proportionality to )cos( (Figure 30). In order to have a uniform

distribution of points on the circle a function )arcsin(z= must be used. A proof of this fact

is given the footnotes15; results are shown in Figure 30.

15The proof can be given as follows:

Make a change of variable so that dA becomes independent from . If a new function )(zz= is chosen, this

implies:kdzd =cos

Where kis a proportionality constant. The above equation can be integrated:


41/46


42/46

42

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50 100 150 200 250

50

100

150

200

2500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

50 100 150 200 250

50

100

150

200

250

Figure 31.Illumination distribution on the London case study under the CIE overcast sky (left). The image

has been normalised, so that pixels with an unobstructed view of the sky have the value 1. Although at a

first sight this image resembles to the sky view factor image (illumination under a uniform sky, see Figure

10), a more accurate examination shows diversity: the image on the right represents the difference between

uniform and CIE sky. This is approximately nil on roofs, which have an almost unobstructed view of the sky-

vault, and maximum for streets and highly obstructed courtyards, due to the fact that the CIE sky is brighter

near the zenith than low on the horizon.

10. Further developments: integrated energy values and methods to

estimate light inter-reflections

An algorithm to calculate illuminance values on a DEM for a given sky luminance function

),( LLsky = was described above. Exactly the same algorithm, where just the weighting

function has been altered, can be used to calculate a wider set of parameters. The use of

radiance ),( RRsky = , instead of a luminance sky distribution function, allows the calculation

of received irradiance on the urban surface a variable which is of great interest in estimating

solar gains in buildings.

Furthermore, instead of an instantaneous function ),( LLsky = (or ),( RRsky = ), a cumulative

one can be selected to represent sky luminance (or radiance) distributions integrated over a

given period of time: for instance, a week, a month or a year. This leads to the construction of

what Compagnon and Raydan (2000) call statistical photometric (or radiometric, if dealing with

radiance) sky models (Figure 32). These sky models simply show average luminance (or


43/46

43

radiance) values computed over a number of hours. Their use allows the calculation of total

illumination (or irradiation) received on the urban surface during a certain period of time. As

explained above, the resulting images can be thresholded by detecting all pixels higher than a

given value. This produces the immediate identification, for instance, of all roof areas whereannual irradiation is higher than 1000 KWhm

2and therefore PV applications are economically

viable (Compagnon and Raydan, 2000).

Another extension of the analysis techniques developed so far allows the modelling of light

inter-reflections on the DEM. In fact, all the algorithms above deal with illuminance (or

irradiance) falling directly from the sky. This is the main contribution to the total, especially in

cities at northern latitudes, where the reflected component is limited due to a relatively low

reflectivity (dark buildings and streets). More accurate estimates, however, should take into

account inter-reflections of radiation.

This can be done on a DEM. However, the procedure is complicated, time-consuming and poses

problems of accuracy, as the DEM is not well-suited to the representation of vertical surfaces.

This makes DEM analysis less competitive in accounting for light inter-reflections than other

kinds of modelling tools, such as Radiance and its derivatives (Compagnon and Raydan, 2000;

Mardaljevic and Rylatt, 2000) and will thus not be examined here.

Simplified alternatives could also be envisaged, analogous to methods that are of common use

in the calculation of daylight factors inside buildings. These methods are based on the exact

determination of the so-called direct component (radiation falling directly from the sky) and a

rough estimate of the reflected component (amount of radiation coming from internal

reflections) via the introduction of an average surface albedo. Transposing a similar approach

to urban areas would not allow a precise knowledge of illumination at a given point, but might

instead suggest an average correction to be applied to a whole portion of a city in order to take

into account the effects of inter-reflections.


44/46

44

Figure 32.Statistical sky model for Fribourg, Switzerland, for June. The image represents mean radiance

values computed over 445 daylit hours; the image was kindly provided by Raphael Compagnon, Ecole

dIngnieurs et darchitectes de Fribourg (Switzerland); a similar diagram can be found in Compagnon and

Raydan (2000).

11. Conclusions

This paper opened with a question: could the analysis of DEMs with image processing

techniques contribute to the study of daylight and sunlight conditions in cities? The answer is

in the affirmative: the DEM has proven an extremely versatile support for the calculation of a

number of parameters from simple shadowing to more elaborate view factors and illuminance

distributions under different skies.

Various algorithms have been tested on different case study images, showing a significant

variation of the parameters considered with urban morphology. This is particularly evident in

the case of the sky view factor, which accounts for differences in the magnitude of the urban

heat island in different cities; it varies greatly between London, Toulouse and Berlin and

therefore suggests a significant difference in urban temperatures.

Nevertheless, the link between morphological parameters and environmental variables is

complex. Many processes are involved, such as the absorption of radiant and luminous energy

from the sun and the emission of long wave radiation. The different behaviours of urban texture

cannot be described with a single morphological parameter. Furthermore, varying and

conflicting requirements are faced: the time of the day and the year, the patterns of occupancy

of the building, the overall external environmental conditions, etc. redefine the optimum each


45/46

45

time. For instance, in a cold climate maximising the accessibility to daylight (which is

regulated to a certain extent by a high view factor) conflicts with urban warmth from the urban

heat island (which requires a low view factor).

For this reason it is impossible to establish a general equation valid in all cases of urban

planning and design. Ad-hoc simulations are required for decision-making as in the case of

the solar envelope. The importance of the DEM is that it constitutes an accessible medium to

support this process. Its increasing availability at low cost due to the rapid development of

surveying techniques such as Synthetic Aperture Radar or Laser Altimetry is bound to make it

a key tool for urban analysis in the coming years.

12. References

Baker N., 2000, We are all outdoor animals, Proceedings of the International Conference on Passive and Low Energy ArchitecturePLEA 2000(Cambridge, UK, July 2000).Baker N., Fanchiotti A., Steemers K. (editors), 1993, Daylighting in Architecture - A European Reference Book, (James & James,London).Brown M., Grimmond S., Ratti C., 2001, Comparison of Methodologies for Computing Sky View Factor in Urban Environments,Proceedings of the 2001 International Symposium on Environmental Hydraulics (Tempe, AZ, December 2001).Burrough P A, McDonnell R A, 1998, Principles of Geographical Information Systems (Oxford University Press, Oxford).Chalfoun, N.V., 1998, Outdoor thermal comfort assessment using MRT and fish-eye lens photography, Proceedings of the

International Conference on Passive and Low Energy Architecture PLEA 1998(Lisbon, Portugal, June 1998).Clarke J. F., 1972, Some effects of the urban structure on heat mortality, Environmental research 5, pp 93-104.

Cohn H., 1886, The hygiene of the eye in schools, (Simpkin,

Date post:	03-Jun-2018
Category:	Documents
Upload:	ioana-sararu
View:	220 times
Download:	0 times

Texture and Radiation4

Documents