There is a need in atmospheric sciences to obtain a measure of the degree of sky visibility. This is especially true for studies conducted over complex topography (mountains) or complex land use (towns) (Chapman et al., 2001; Blankenstein and Kuttler, 2004). The sky view factor (SVF) is, for example, widely used in urban climatology in order to explain the urban heat island (Oke, 1981; Svensson, 2004; Unger, 2004). In these cases, the limitation of the sky visibility by the surroundings can significantly reduce the amount of received atmospheric infrared radiation and scattered solar radiation. The mask effect also has an impact on the amount of radiation emitted towards the sky by an object and increases significantly the radiative exchanges between the terrestrial surfaces, a key mechanism in urban climatology. Actually, the SVF expresses the proportion (ratio) of radiation leaving the sky, assumed isotropic, that is able to reach a ground surface tilted at an arbitrary angle. Its value must vary between the minimum of 0, when the sky is not visible at all and to the maximum of 1, when the ground surface is horizontal and the sky entirely visible. Several methods found in the literature can be applied to determine the SVF (Johnson and Watson, 1984; Unger, 2009). Initially, they were essentially manual procedures (Watson and Johnson, 1987) until they become automatic during the last decade (Grimmond et al., 2001). They employ fish eye photographs (Chapman and Thornes, 2004; Chapman et al., 2007) or digital elevation models (DEMs) (Teller and Azar, 2001; Souza et al., 2003; Lindberg, 2004; Gal et al., 2007). However, these techniques may suffer from limitations: either they give only sparse values (fish eye), or can only be applied on horizontal surfaces, or can not take obstacles like bridges or trees into account. The purpose of the present paper is to describe a numeric method capable of estimating the SVF from a DEM, for any arbitrarily tilted ground surface, for any geometric environment and with a dense spatial coverage.

2. View factors and radiation transfers

The procedure takes benefit of progresses in several scientific domains like physics of radiative heat transfer (Siegel and Howell, 2002), illumination engineering (Choi, 2002) and computer graphics (Schröder and Hanrahan, 1993). The common problem in all these fields is to predict the amount of radiation that can be exchanged between an emitting plate (hereafter called X) and a receiving plate (hereafter called Y), assuming the presence of a non-participating medium between them (inert gas). The solution consists in obtaining the irradiance of the receiving plate Y according to

(1)

Where I_{X→Y} is the irradiance of plate Y (W m^{−2}), E_{X} is the emittance of plate X (W m^{−2}), A_{X} and A_{Y} are respectively the areas (m^{2}) of the emitting and receiving plates and F_{X→Y} (0–1) the view factor between the two plates (sometimes called ‘form factor’ or ‘shape factor’ in the literature). This fundamental equation is widely used in all scientific fields that deal with light, illumination and radiation (e.g. Gebhart, 1960; Clark and Korybalski, 1974; Michael and Wallace, 1993). The view factor (F_{X→Y}) is terms of special importance because it parametrises the fraction of radiation leaving the ideal diffuse emitter that actually reaches the receiver. It is developed on the base of purely geometric considerations. Equation (2) gives an example of how it is possible to compute the differential view factor between two infinitely small (differential) elements located on two plates:

(2)

In this equation, dF_{dX→dY} represents the differential view factor between a differential element (dX) of plate X and a differential element (dY) of plate Y (Figure 1). θ is the angle (rad) between the normal of a differential element and the segment connecting them. S is the distance (m) and dA_{Y} is the area of the differential receiver (m^{2}). By analogy, this equation can also be used to estimate the view factor between a differential emitter (dX) and a finite receiver (Y):

(3)

To apply Equation (3), the finite receiver (Y) is divided into a finite number n of differential elements (dY) (Figure 2) for which it is possible to compute the differential view factor associated with dX. Thus, Equation (3) can be rewritten in discrete form:

(4)

3. The SVF equation

The analogy with the SVF becomes evident if the differential element dX is located somewhere on the ground and if the finite element Y represents the whole sky (as in Figure 2). For this special case, all the radiation emitted by dX must strike Y and, as a consequence, F_{dX→Y} must be equal to 1 (to preserve the principle of energy conservation of Equation (1)). This is exactly the value of the SVF of a point located on a horizontal terrain with no obstacles around it. Now, if for some reasons (because of the presence of obstacles between them) parts of the receiving area (Y) are not visible from dX, the irradiance of Y must decrease and the value of F_{dX→Y} must be less than 1. At the extreme case, when Y is not visible at all from dX, the value of F_{dX→Y} must be equal to 0. To take this phenomena into account, a visibility parameter (v) is introduced to cancel out the differential view factors of the elements of Y that are not visible from dX:

(5)

The visibility parameter (v) is set to 1 when there are no obstacles between dX and dY and to 0 otherwise. Equation (5) is exactly the expression needed to estimate the SVF. Hence, the SVF equation can be written as:

(6)

However, validity of Equation (6) implies that several preconditions are fulfilled: (1) the receiving surface Y properly replaces the sky; (2) Y is centred over each calculation point for which the SVF is estimated; (3) Y can be discretized into n small elements for which the differential form factors can be evaluated and (4) the visibility parameter can be properly estimated.

4. The visibility parameter

To account for the necessary conditions prior to applying Equation (6), a hemispheric virtual sky dome is created (Figure 2). This sky dome is discretized into (n) small planar polygons by a tessellation algorithm (e.g. Okabe et al., 2000) and is centred over each calculation point for which SVF needs to be estimated according to the DEM. Then, the dF_{dX→dYi} terms of Equation (6) are computed between the calculation point (dX) and all the n sky dome polygons (dY_{i}). The θ, S and dA_{Y} terms necessary for this operation are obtained thanks to some trivial trigonometric or geometric functions.

The most important point consists in estimating the visibility parameter between the calculation point and the sky dome polygons. This is essential to get the correct SVF value in case of rough topography or if obstacles are introduced into the DEM (e.g. buildings, trees, etc.). A technique applied in 3D computer graphics and derived from ray tracing can be used to obtain these values (Cook et al., 1984; Glassner, 1989). It consists in shooting virtual rays from the calculation point trough the midpoint of each sky dome polygon. Each ray is tested for potential intersection with the surfaces of the DEM (Badouel, 1990). A sky dome polygon is assumed to be entirely visible if the ray that passes through its midpoint is not intercepted by the DEM (v = 1). If the ray hits the DEM, the sky can be still visible according to the nature of the object that has been hit. There are two possibilities: (1) the object is opaque (like a wall or a terrain slope) and v takes the value 0 and (2) the object is transparent (like glass) and v is set to the transmittance of the object (multiple transmittance values can be taken into account along the path of the ray). Only the polygons with a visibility value greater than 0 will contribute to the SVF in Equation (6).

5. Discussion

It is possible to slightly simplify the computation of the differential form factors since, sky dome being centred over the position where the SVF is computed, the normal vector of each sky dome polygon points strictly towards the calculation point and all the θY angles are always 0. Then:

(7)

The accuracy of Equation (7) increases when the size of the sky dome polygons diminishes. This is because the polygons become closer to a differential area and the shape of the sky dome closer to a hemisphere. Table I shows that the SVF for a horizontal slope without obstacles around it is indeed closest to one when the number (n) of the sky dome polygons is large. Another remark can be done: the value of S (the distance between the calculation point and the midpoint of a polygon) is the same for each polygon if the sky is approximated with a hemisphere (this is not the case if the sky is replaced by an hemicube like in some Computer Graphics methods). However, several values of S can be used. A sensitivity test was performed to obtain the SVF for a horizontal surface without obstacles around it. When the actual radius of the sky dome (r) is assigned to S (method 1, Table I), the SVF is systematically slightly underestimated. This is because the area of a polygon is not exactly at a distance r of the calculation point, but a little more closer (as it can be seen in Figure 3). If S is replaced with the real distance of the midpoint of the polygons (S2), the SVF is systematically overestimated (method 2). In fact, the area of the polygon is located at a distance between r and S2. The best approximation for triangular polygons is found to be obtained with a weighted average of the two distances (method 3):

(8)

It is not really a problem to compute the SVF for a point located on an tilted surface (e.g. slope, wall, roof; Figure 3). The plane that contains dX, separates the space into two half spaces (front or back) according to the normal of the tilted surface. When a sky dome polygon lies on the backspace of the plane, its visibility (v) is simply set to 0. This technique is tested against different slope angles and the results compared to the theoretical values (Table II). The error with respect to the theoretical values is less than − 0.0255% if the sky dome is discretized in 4096 polygons.

Table I. Sky view factors of a horizontal surface. The values are given according to the number (n) of sky dome meshes and for three methods to estimate S. Method 1: S is the radius of the sky dome; method 2: S is the distance between the centre of the sky dome and the midpoint of each polygon; method 3: S is a weighted average (see the text)

N

Method 1

Method 2

Method 3

256

0.989790

1.011034

1.000111

1024

0.997429

1.002748

1.000030

4096

0.999356

1.000687

1.000009

16384

0.999839

1.000170

1.000004

Table II. Sky view factors for several tilted slopes (the angles are given in degrees). Comparison between the method described in the text (Calc) and the theoretical values (Ref). The sky dome is discretized into 4096 polygons and method 3 is applied to obtain S

Angles

15°

30°

45°

60°

75°

90°

Calc

0.9829

0.9329

0.8533

0.7499

0.6293

0.5000

Ref

0.9830

0.9330

0.8536

0.7500

0.6294

0.5000

Diff (%)

− 0.0082

− 0.0084

− 0.0255

− 0.0105

− 0.0133

0.0008

To demonstrate the full capacity of the method to approximate the SVF in the presence of obstacles, the computation is performed for an infinite urban canyon with variable height/width ratios (Table III). The mask effect is correctly taken into account and the results are very close to the theoretical values given by Oke's formula (Oke, 1981). To complete this test, the method is evaluated against finite urban canyons of a district in Strasbourg city (France) for which a simplified DEM including buildings was acquired. All the elements of the DEM are discretized into small patches of a few square meters and the SVFs are computed for the centre of each patch (Figure 4). As expected, the resulting values are the highest (equal to one) for the flat roofs at the top of the highest buildings (located at the top right corner of the figure), and the lowest (≈ 0.045) for the façades of the buildings located into narrow streets (black values). Of course, the results in such complex environments are strongly influenced by the quality of the DEM, its spatial resolution and the degree of details it contains. To demonstrate the ability of the method to obtain the SVF in the presence of suspended obstacles or even negative slopes (surfaces looking downward) the SVF is computed for a 3D model of a bridge (Figure 5). The method is robust enough to obtain the SVF everywhere in the DEM (for the bridge deck, the bricks of the bridge that are looking downward and also for the water surfaces under the arches).

Table III. Comparison of sky view factors obtained for an urban canyon of infinite length and different height/width ratios. The method (Calc) is compared to Oke's formula (Oke, 1981). The infinite canyon is simulated with the typical dimensions of 6-km long, 10-m height and the width is fitted to obtain the correct height/width ratio. The sky dome is discretized into 16 384 polygons and method 3 is applied to obtain S. The computation is done for a point located on the street at the middle of the canyon

H/W

0.25

0.5

1

2

3

4

Calc

0.895

0.699

0.447

0.243

0.164

0.124

Oke (1981)

0.89

0.71

0.45

0.24

0.16

0.12

6. Conclusion

The method described in this article is specifically developed to compute the SVF with DEMs. It can be used for any type of tilted surface and in the presence of obstacles. The tests and the comparisons with theoretical values demonstrate the capacity of the method to predict the correct values for complex environments. This can be useful for a lot of applications in the field of atmospheric sciences, as in urban climatology (Gal et al., 2009), in meteorological modelling (Müller and Scherer, 2005), biometeorology (Holmer et al., 2001) or in radiation transfer models (Kastendeuch and Najjar, 2009; Gros et al., 2011; Matzarakis and Matuschek, 2011). With such a method, these studies can take advantage of the fact that DEMs become more and more available, complete and accurate.