December 1988 K. Nakagawa 903. Estimation of the Sky View-factor from a Fish-eye Lens Image,

Transcription

1 December 1988 K. Nakagawa 903 Estimation of the Sky View-factor from a Fish-eye Lens Image, Considering the Anisotropy of the Downward Longwave Radiation By Kiyotaka Nakagawa Division of Science, Joetsu University of Education, Niigata 943, Japan (Manuscript received 23 December 1987, in revised form 5 October 1988) Abstract An analytic expression is derived to estimate the zenith angle distribution of the downward longwave radiation from sky elements in terms of the surface air temperature, surface water vapor pressure and zenith angle, taking into account the anisotropy of the downward longwave radiation. In addition, a formula for estimating the sky view-factor is also derived for the special case in which the zenith angle of the skyline is constant. It is shown, on the basis of this formula, that the additional downward longwave radiation is a function of the surface air temperature, surface water vapor pressure, surface temperature of obstructions, and the zenith angle of the skyline. The values estimated by use of this function fit well with Kondo's (1982) screening factor F2, assuming that the surface air temperature is equal to the surface temperature of the obstructions. The relative deviation of the screening factor for isotropic radiation from that for anisotropic radiation is larger with respect to increasing surface water vapor pressure and decreasing skyline elevation. Applying this formula, a new method is proposed to calculate the view-factors of visible and obstructed skies from the fish-eye lens image using a digitizer in conjunction with a computer. 1. Introduction If Ls* is the ideal downward infrared flux density from an unobscured sky and L0* is that from a sky perfectly obstructed by surface construction, then the actual downward infrared flux density L* can be expressed as follows; where *s and *0 are ratios of the actual radiation from visible and obstructed skies to Ls and L0* respectively. In the present paper, * and *0 are termed the sky and wall *s viewfactor, respectively. If the sky is obstructed by surface construction, then the total downward infrared flux density reaching the ground surface increases remarkably, since the radiative temperature of the sky is relatively lower than that of the 1988, Meteorological Society of Japan obstructions. Because this effect is expected to exist in urban canyons, it has been pointed out that the decreased net longwave radiation loss from urban canyons due to the reduction in the sky view-factor by construction, is one of causes of the urban heat island (e.g. Oke,1978). Although many workers (Fleagle, 1950; Unsworth, 1975; Kobayashi, 1979, 1984; Steyn, 1980; Oke, 1981; Kondo, 1982; Johnson and Watson, 1984, 1986; Steyn et al., 1986; Park, 1987a, b) have tried to represent the effects of surface obstructions on the additional flux of downward longwave radiation in terms of various paremeters, all, with the exception of Unsworth (1975) and Kondo (1982), have assumed that both the radiation from the visible sky and that from the obstructions are isotropic. Under the assumption of isotropy of the longwave radiation, both parameters *s and *0 are complementary, i.e. *s+*o=1.

2 904 Journal of the Meteorological Society of Japan Vol. 66, No. 6 f I the downward longwave radiation from a visible sky is isotropic and the zenith angle of the skyline * is constant, the sky view-factor *s(*) can be expressed by However, the downward longwave radiation from a visible sky is anisotropic, and increases as the zenith angle increases. Kondo (1982) introduced another parameter F2, for the portion of the obstructed sky that takes into account the dependence of the downward longwave radiation on the elevation angle. Using this, he numerically calculated some characteristic values of this parameter assuming a typical zenith angle distribution of the downward longwave radiation. Although his concept is both physical and reasonable, he did not present a formula for F2. In general, the zenith angle of a skyline varies widely with respect to the azimuth angle. Therefore, his concept can not be applied to an arbitrary site with a complex skyline. The present study has three purposes. The first is to derive an analytical formula for estimating the zenith angle distribution of the downward longwave radiation from a cloudless sky. The second is to derive formulae for estimating the sky and wall view-factors, *s and o, and the additional longwave radiation due * to obstructions for the special case in which the zenith angle of the skyline is constant. The third is, on the basis of a fish-eye lens image, to develop a method of calculating the actual sky and wall view-factors of an arbitrary site with a complex skyline. II. The zenith angle distribution of the downward longwave radiation Consider a visible sky element d*d* with a azimuth angle * and zenith angle *, whose solid angle is sin *d*d*. If the atmosphere is assumed to be a grey-body emitter, the infrared flux density from this sky element received on a horizontal surface element d Ls (*, *) is given as follows; where * is Stefan-Boltzmann constant, T the absolute temperature, z the altitude, *I the emissivity of the column, and u the effective water vapor content. Elsasser (1942) pointed out that the following relationship holds between emissivities both of a column and of a slab, i.e. *I(u) and *(u)such as; Here 5/3 is a diffusivity factor. According to Nakagawa and Kayane (1979), the following equations hold; As a result, from Eqs. (4) and (5), the beam emissivity is a function of the effective water vapor content as follows; The effective water vapor content u was defined by Yamamoto (1952) as; where * is the water vapor density, and p the atmospheric pressure. Subscript 0 denotes the earth's surface. In the lower troposphere, vertical profiles of temperature, pressure and water vapor density are rather well approximated by the following equations (Brutsaert,1975); and

3 December 1988 K. Nakagawa 905 Here, * is the temperature lapse rate, g the acceleration of gravity, R the gas constant for dry air, * the ratio of the densities of water vapor and dry air, e the water vapor pressure and, if zisinkm, areas such as a campus (Hayashi and Taguchi, 1983). Therefore, use can be made of these profiles for the problem of determining the sky view-factor in an urban canyon. Substitution of Eqs. (9) and (10) into Eq. (7) leads to with the exception of a surface inversion, calculated profiles given the surface values, are very good approximations to the actual profiles. Lapse rates can be obtained even during nighttime in built-up areas, not only of a large city (e.g. Aida and Yaji, 1979) but also of smaller where In addition, substitution of Eqs. (12) and (6) into Eq. (3) leads to where B(a,b) is known as the beta function, and Since, according to Nakagawa and Kayane (1979), the downward longwave radiation from an entirely cloudless sky Ls* is given by Eq. (14) can be rewritten as follows; In other words, Eq. (17) expresses the dependence not only of the flux of the radiation beam from the visible sky element but also of the view-factor of the sky element on the zenith angle *. The constant in Eq. (16) is derived for the lapse rate of the standard atmosphere, i. e. *=6.5K/km. Since this constant depends on k1 and k2, which are function of *, the constant becomes larger as * decreases. However, it should be noted that the view-factor of the sky element is independent *. Upon changing the upper limit of the integral in Eq. (3) to z, the longwave radiation originating below the height z is given by where I* (a,b) is the incomplete beta function ratio. Compared with Eq. (3), it should be noted that the incomplete beta function ratio in Eq. (18) denotes the contribution of the emission originating below z to the total flux density. Fig. 1 shows this contribution as a function of z. It is clear that most of the downward longwave radiation comes from the lowest 100m of the atmosphere. Therefore, even if a nocturnal upper inversion occurs over a large city (Kawamura, 1979), most of the additional radiation due to the upper inversion is absorbed by the atmosphere along the path, and the contribution from this inversion may be neglected.

4 906 Journal of the Meteorological Society of Japan Vol. 66, No. 6 Fig. 1. Contribution of the emission originating below a given height to the total flux density. Solid line stands for the case of A=0.723 and m=0.090, even in the lowest atmospheric layer, and the dashed lines for the case of A=0.837 and m=0.154 in the lowest atmospheric layer on the basis of Eq. (5). Numbers by the dashed lines indicate the surface water vapor pressure in hpa. On the other hand, the radiation beam from obstructions may be treated as being isotropic (Aids, 1982). Therefore, the longwave radiative flux density from the obstruction element received by a horizontal surface element dl0(*,,d) is given as follows; * where Ts is the radiative temperature of the surface of the obstructions, and d is the horizontal distance from the obstruction element. The first term in Eq. (19) represents the absorption attenuation of the radiant energy from the obstruction surface by the atmosphere along its path. The second term represents the emission contribution from the atmosphere. Fig. 2 shows the intensities of both terms in a unit of the intensity of a black body radiating at the surface air temperature and the surface temperature of the obstruction, respectively, as a function of distance from the obstruction d, zenith angle * and surface water vapor pressure e0. Here, since in an urban canyon the horizontal distance from the obstruction is not very long and the effective water vapor content is usually less than 0.1cm, Fig. 2 is shown for A=0.837 and m=0.154 on the basis of Eq. (5). It is clear from Fig. 2 that the emission contribution from the atmosphere along the path increases with respect to distance and surface water vapor pressure, and decreases with respect to the zenith angle. However, if the surface air temperature can be assumed to be equal to the surface temperature of the obstruction, both terms in Eq. (19) are almost complementary. In addition, for the case that the distance from obstruction is not very long, most longwave radiation comes directly from the obstruction. Since the distance and temperature difference are not very large in an urban canyon, the radiative beam from an obstruction can be regarded as the black body radiation at its surface temperature, and it is given by

5 December 1988 K. Nakagawa 907 Fig. 2. Intensities of radiation from the surface of the obstruction and the atmosphere along the path in a unit of the intensity of a black body radiating at the surface air temperature and the surface temperature of the obstruction, respectively, as a function of zenith angle, surface water vapor pressure and distance from the obstruction. Distances from the obstruction are 1m (a), 5m (b), 10m (c), and 100m (d), respectively. Solid lines stand for the absorption attenuation by the atmosphere along the path, and dashed lines for the emission contribution from the atmosphere. Numbers by the lines indicate the surface water vapor pressure in hpa. In other words, since the view-factor of the obstructed sky element is given by It should be noted that the difference between Eqs. (14) and (20) is the additional radiative flux due to the existence of the obstruction element. Therefore, the additional radiative flux d*l (*,*) can be expressed as follows;

6 908 Journal of the Meteorological Society of Japan Vol. 66, No. 6 III. The sky and wall view-factors for the case of a simple skyline with a constant zenith angle Consider the special case in which the zenith angle * of the skyline is constant throughout the azimuth *. In this case, the sky view-factor *s (*) can be obtained by integrating Eq. (17) with respect to inside of the skyline; For the entire hemispherical sky, since the zenith angle * of the skyline is equal to */2, it follows that parameters reaches a maximum of 1.7% at a zenith angle of 53 degrees. In the same way, the wall view-factor *0(*) for the above special case is given by It should be noted that the sky view-factor *s (*) by Eq. (26) and the wall view-factor *0 (*) by Eq. (27) are not complementary. The additional radiative flux *L (*) due to the existence of the obstruction is expressed as follows; In a physical sense, *s (*/2) should be unity. Indeed, with m equal to on the basis of Eq. (5), the value of Eq. (25) becomes This fact means that the approximation of the emissivity function of a column by Eq. (6) holds to a very good degree of accuracy. Since the value of Eq. (25) is unity, Eq. (24) is rewritten as follows; Assuming that the surface air temperature is equal to the surface temperature of the obstructions, Kondo (1982) defined another parameter for the portion of the obstructed sky, i. e. F2 (*) defined as follows; Therefore, from Eq. (28), Eq. (29) is rewritten as Although this expression for the sky view-factor is very similar to Eq. (2) based on an assumption of isotropy, Eq. (26) yields values slightly less than Eq. (2,). The difference between both It is clear from Eq. (30) that Kondo's screening factor F2 has to be a function of not only the zenith angle but also of the surface water vapor pressure. For the purpose of comparison, the Table 1 Comparison of the screening factors between Kondo's(1982) original values and the present study

7 December 1988 K. Nakagawa 909 values calculated by Eq. (30) and Kondo's original values are shown in Table 1. Kondo also tabulated values of his parameters for several zenith angles, but did not present data on the surface water vapor pressure. The present study shows four calculated values, i.e. for surface water vapor pressures of 5, 10, 20, 30 hpa. It is clear that values produced by Eq. (30) agree well with Kondo's original values. In addition, the screening factor for the case of isotropic radiation is also shown in Table 1. The relative deviation of the screening factor for isotropic radiation from Kondo's screening factor is larger with respect to decreasing elevation angle. It should be noted that the screening factor for isotropic radiation is equal to Kondo's screening factor for e0=0 (not shown). Therefore, the difference in screening factors between isotropic and anisotropic radiation is larger with respect to increasing surface water vapor pressure. IV. Method of calculating the sky view-factor using a fish-eye lens image In the special case mentioned in the previous section, with a constant zenith angle of the skyline with the air and obstruction temperatures and water vapor pressure given at the ground, the utilization of above formulae (Eqs. (26) through (28)) enables one to calculate the sky and wall view-factors and the additional radiative flux density as well as the total downward infrared flux density. However, with a complex skyline such as in an urban canyon, the zenith angle of the skyline *(*) is a function of the azimuth angle *. Therefore, it is necessary to collect zenith angle data for numerous azimuths. This difficulty is alleviated by the utilization of a fish-eye lens image of the city skyline. For example, some techniques have been proposed by several workers, such as Ito (1976), Steyn (1980) and Johnson and Watson (1984, 1986) to name a few. Photo. 1 is an example of a fish-eye lens image of a city skyline, taken with a 35-mm camera equipped with a standard fish-eye lens (Canon 7.5mm, f/5.6). The photograph was taken in the central business district of Joetsu City, Niigata Prefecture, Central Japan, on 6 June The Photo. 1. Fish-eye lens photograph of an urban canyon, taken at the central business district of Joetsu City, Niigata Prefecture, Central Japan, on 6 June 1987 by the author. field of view of this lens is 180*, and the azimuth angle can be determined based upon the angle around the center of the picture. Because the projection of this lens is equiangular, the zenith angle is proportional to the distance from the center. From this, the zenith angle and azimuth of an arbitrary point on the skyline can be determined by measuring the angle around the center and the distance from the center. Most existing studies that calculate the sky view-factor under the assumption of isotropy utilize an annulus model (Kobayashi, 1979; Steyn, 1980;.Johnson and Watson, 1984). This method first divides the fish-eye lens image into several annuli, determines the angular width of the visible sky part of each annulus, calculates the view-factor of each visible sky part, and finally accumulates these view-factors to obtain the sky view-factor. The method mentioned above is suitable for manual labor with polar graph paper. However,

8 910 Journal of the Meteorological Society of Japan Vol. 66, No. 6 Fig. 3. Model for calculating the view-factor of a fan-shaped sky element within a circular image. when using a digitizer, this annulus model is not necessarily best, since it is not easy to determine the angular width of the visible sky part of each annulus with a computer. The present study proposes a quasi-fan model for calculating the sky and wall view-factors, instead of using the existing annulus model. When a sky element, consisting of a zenith angle and a skyline element is assumed as an quasi-fan, the shape of the sky element can be approximated by a perfect fan. If *0 is the radius of the plane circle of the fish-eye lens image and (*1,*1) and (*2,*2) are the polar coordinates of both ends of the skyline element as shown in Fig. 3, then the average zenith angle of the skyline element * is obtained by and the view-factor of this visible sky element d*s is obtained from Eq. (26) as follows; Therefore, dividing the visible sky into numerous quasi-fans and summing up view-factors of all the visible sky elements leads to the value of the sky view-factor. In order to actually obtain the value of the sky view-factor from the fish-eye lens image, it is very important to determine the values of parameters *1, *2, *1, *2. The present study proposes a method of determining these parameter values using a digitizer controlled by a computer. After setting the fish-eye lens photograph on the digitizer, the first step is to read the coordinates of three points on the plane circle. Next, from the components of these three points, two chords have to be determined. The coordinates of the intersection of their perpendicular bisectors (*0, y0) is the center of the plane circle of the fish-eye lens image. The radius of the circle *0 can be easily obtained. Therefore, if (*i, yi) are the coordinates of any point on the skyline Pi read by the digitzer, then the polar coordinate components of this point can be obtained as follows; and Using the method mentioned above, in the case of Photo. 1, the sky and wall view-factors are measured as 77.2% and 21.7%, respectively. Furthermore, under the assumption that the radiative temperature of the obstructions is equal to the surface air temperature, the additional downward longwave radiation from the obstructions is calculated by Eq. (28) to be 21Wm-2 for air having a temperature of 5* with water vapor pressure of 5hPa and 17Wm-2 for 25* and 25 hpa. V. Concluding remarks The present study showed that the sky and wall view-factors and the additional downward longwave radiation due to the existence of obstructions can be expressed as a function of the zenith angle of the skyline, surface air temperature, surface water vapor pressure and the radiative temperature of the surface of the obstructions, taking the anisotropy of the longwave radiation in the atmosphere into account. A new method was proposed to calculate these

9 December 1988 K. Nakagawa 911 parameters on the basis of the fish-eye lens image, by use of a digitizer in conjunction with a computer. This new model, termed a quasi-fan model, is superior in determining the sky and wall view-factors with a digitizer to the existing annulus model. However, this method still contains manual operations, such as printing the fish-eye lens image and tracing the skyline by use of a digitizer. The manual processing involved makes the technique time-consuming, and leads to errors due to individual differences. This disadvantage has to be overcome in order to calculate the sky and wall view-factors automatically. Recently, a new type of video camera module, which enables a fish-eye lens to be attached, and a video digitizer have become available. In addition, an algorithm for calculating the viewfactor of pixels in a video image has been proposed under the assumption of isotropy (Johnson and Watson, 1986), and was applied in three urban locations (Steyn et al., 1986). However, it was found to be severely difficult to distinguish pixels representing sky from those representing non-sky. Therefore, one of the most important recommendations is to develop an automatic method for distinguishing sky pixels from non-sky pixels for use in calculating the parameters discussed in the present study. References Aida, M., 1982: The Atmosphere and Radiative Processes (in Japanese). Tokyo-Do Press, Tokyo, 280pp. Aida, M. and Yaji, M., 1979: Observations of atmospheric downward radiation in the Tokyo area. Boundary-Layer Meteorol., 16, Brutsaert, W., 1975: On a derivable formula for longwave radiation from clear skies. Water Resour. Res., 11, Elsasser, W.M., 1942: Heat transfer by infrared radiation in the atmosphere. Harvard Meteorol. Studies, 6, Fleagle, R.G., 1950: Radiation theory of local temperature differences. J. Meteorol., 7, Hayashi, Y. and Taguchi, A., 1983: On roughness length in the campus of the University of Tsukuba (in Japanese). Study Environ. Tsukuba, 7C, Ito, K., 1976: How to Look at and Use Figures and Tables Concerned with Sunshine (in Japanese). Ohm-sha, Tokyo, l41pp. Johnson, G.T. and Watson, ID., 1984: The determination of view-factors in urban canyons. J. Climate AppL Meteor., 23, Johnson, G.T. and Watson, I.D., 1986: Estimating view-factors using video imagery. in Zannetti, P. (ed.): ENVIROSOFT 86. Proceedings of International Conference on Development and Application o f Computer Techniques to Environmental Studies. Comput. Mech. Publications, Southampton, Kawamura, T. (ed.), 1979: The Atmospheric Environment of Cities (in Japanese). Univ. Tokyo Press, Tokyo, 185pp. Kobayashi, M., 1979: Comparative observation of longwave radiation balance on ground-surface and on roof level in the urban area (in Japanese with English abstract). Geogr. Rev. Japan, 52, Kobayashi, M., 1984: Temperature properties of urban canyon elements at nighttime (in Japanese). Study Environ. Tsukuba, 8C, Kondo, J., 1982: Preliminary theoretical study on nocturnal cooling over complex terrain (in Japanese). Tenki, 29, Nakagawa, K. and Kayane, I., 1979: Theoretical derivation of a formula for estimating the downward longwave radiation in the lower troposphere under the overcast conditions (in Japanese with English abstract). Geogr. Rev. Japan, 52, Oke, T.R., 1978: Boundary Layer Climates. Methuen,. London, 372pp. Oke, T.R., 1981: Canyon geometry and the nocturnal urban heat island: comparison of scale model and field observations. J. Climatol., 1, Park, H., 1987a: City size and urban heat island intensity for Japanese and Korean cities (in Japanese with English abstract). Geogr. Rev. Japan, 60A, Park, H., 1987b: Sky view factor of urban canyon and long-wave radiation balance caused by the nocturnal heat island (in Japanese). Tenki, 34, Steyn, D.G., 1980: The calculation of view factors from fisheye-lens photographs. Atmos.-Ocean, 18, Steyn, D.G., Hay, J.E., Watson, I.D. and Johnson, G.T., 1986: The determination of sky view-factors in urban environments using video imagery. J. Atmos. Ocean. Tech., 3, Unsworth, M.H., 1975: Long-wave radiation at the ground II. Geometry of interception by slopes, solids, and obstructed planes. Quart. J. Roy. Met. Soc., 101, Yamamoto, G., 1952: On a radiation chart. ScL Rep. Tohoku Univ., Ser. 5, Geophys., 4, 9-23.

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