- Email: [email protected]
Comment on “radiative properties of snow for clear sky solar radiation”
177
CoM Regions Science and Technology, 5 (1981) 177-180 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
Short Communication COMMENT ON " R A D I A T I V E PROPERTIES OF SNOW FOR CLEAR SKY SOLAR RADIATION
"
Stephen G. Warren Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO., 80309 (U.S.A.)
and Warren J. Wiscombe Department of Applied Science, New York University, New York City, N. Y. 10003 (U.S.A.)
(Received August 3, 1981; accepted August 7, 1981)
ABSTRACT Choudhury 's (1981) addition of a special "surface reflection" term to a theoretical model for snow albedo is both unnecessary and incorrectly formulated.
Choudhury has published a series of papers, many of them with his colleague Chang, on the interaction of solar radiation with snow. They at first used two-stream methods for the multiplescattering calculations (Choudhury and Chang, 1979) but have now switched to the delta-Eddington method recommended by Wiscombe and Warren (1980; hereafter WW). Choudhury (1981; hereafter Ch), as well as Choudhury and Chang (198lab) used approximations to the single-scattering quantities for ice spheres instead of the exact Mie calculations of these quantities used by WW. However, these approximations probably lead to little error in the wavelength range (0.4 ~< ;~ ~< 2.4/am) which Ch considers, at least for the larger snow grain sizes. Thus the spectral albedo calculations of Ch which omit "surface reflection" differ hardly at all from those of WW. Choudhury's method, however, diverges from that of WW, and gives different albedos, when the additional "surface reflection" term is put into the model. The purpose of this note is to state that (1) a separate accounting of surface reflection is unnecessary for the snow situation Ch con-
siders and (2) even if one does want to employ a surface reflection treatment, it is formulated incorrectly by Ch (and by Choudhury and Chang). The snowpack was modeled by WW as a scattering medium with no special dielectric discontinuity at the surface that is any different from the dielectric discontinuities between air and ice within the snowpack. Of course the topmost grains do reflect radiation, but this is already accounted for within the multiple-scattering approximation. The "glitter" that one sees from individual snow grains is not computed explicitly (since the deltaEddington method gives only fluxes, not intensities), but it is included implicitly. The glitter off a snow surface, which Ch mentions, is qualitatively no different than the glitter off "diamond dust" or off the top of a cirrus cloud. And in modeling the reflection of sunlight from these media, no special surface reflection term is ever invoked. The nature of the interaction of electromagnetic radiation with a snowpack depends on the magnitude of the wavelength of light relative to the interparticle (center-to-center) separation. The snow is effectively, blurred to a resolution comparable to the wavelength. This means that whereas solar radiation experiences dielectric discontinuties between air and ice throughout the snowpack, radiowaves by contrast experience a dielectric discontinuity only at the top of the snowpack. In the latter case, the snowpack as a whole is then regarded as having some effective complex dielectric constant which is intermediate between those
178 of ice and air (e.g. Evans, 1965). The snowpack thus exhibits a "surface" to radiowaves but not to sunlight. [There is an intermediate wavelength regime, around the microwave, where neither limiting case is valid, so that both internal scattering and surface reflection must be considered (cf. England, 1974)] Although the surface need not be treated separately for reflection of sunlight by ordinary snow, there are some less common snow conditions for which it may indeed have to be considered. A case in point is glazed snow, for example the "firnspiegel" shown in Fig. 59 of LaChapelle (1969). The question then becomes: Is Choudhury's formulation of surface reflection applicable to such a situation? The answer is no, because the coupling of the surface layer to the underlying scattering layer is done improperly. First, Ch ignores the bending of rays at the surface. In his delta-Eddington calculation he simply uses the incident solar zenith angle instead of using Shell's law to transform that angle at the surface. Second, the "surface" layer is sometimes present,. sometimes absent in Ch's formulation. For example, it is assumed to be present for direct radiation but absent for diffuse radiation (which is actually nothing but the sum of direct beams from all possible angles). And for the case of direct radiation, Ch assumes the "surface" layer to be present for downward radiation but absent for upward radiation emerging from the multiple-scattering medium, viz: F f = asrF,~ + (1 - otsr)F~, ams
(1)
where F f and F+ are respectively the upflux and downflux at the surface, t~st is Ch's albedo due to "surface reflection", and Otms is his albedo due to multiple scattering. Ch thus allows the "surface" layer to reflect radiation back up from its top surface but not back down into the snow from its lower surface. This may or may not be important quantitatively, depending on the wavelength*; the point is that the formulation is inconsistent. Proper treatments of the coupling of a dielectric interface to a multiple-scattering medium are available in the literature. This has been a subject of active research in optical oceanography, a recent careful formulation being that of Tanaka and Nakajima (1977).
Choudhury's apparent reason for introducing "surface reflection" into his model was in order to match Liljequist's (1956) observations of spectrally-integrated snow albedo made at large zenith angles. Figure 8 of Ch shows that without "surface reflection" his model's prediction of that albedo exhibits virtually no zenith angle dependence (to within 1%). By contrast, Liljequist's measurements showed a substantial zenith angle dependence. But we think that this is a weak justification, at best, for introducing "surface reflection," which in our view is no more than an extra tuneable parameter with no real physical basis. The model.measurement discrepancy may be due to any of a number of errors, both experimental and theoretical. But even assuming no experimental error, the discrepancy cannot be considered a telling indictment of the snow albedo model. In order to obtain the zenith-angle dependence of spectrallyaveraged albedo from the spectral snow albedo model, one needs to know the incident radiation spectrum. This is because the incident spectrum changes with zenith angle, and also because the zenith-angle dependence of albedo is different at each wavelength, becoming stronger the higher the ice absorptivity (Fig. 1 la of WW). But Liljequist did not measure the incident radiation spectrum, so Ch had to calculate an incident spectrum using his atmospheric radiation model. It is not possible to be 100% confident that the atmospheric model reproduces the spectral downflux at Maudheim for the time that Liljequist was there. (For example, the transmission functions for the atmospheric gases may be inaccurate at low temperatures.) The cause of a discrepancy between observation and calculation of spectrally integrated albedo may thus be due to a deficiency either in the snow albedo model or in the atmospheric radiation model; *The error caused by using eqn. (1) will be insignificant at wavelengths where ice is relatively transparent [because as ares -~ 1, eqn. (1) reduces to F t = ares F~ ] but not in highly absorbing regions. At X = 1.5 ~m and 2.0 ~m, for example, where snow albedo is only a few percent, the snow behaves like an inf'mitely thin highly-absorbing layer exhibiting reflection only from the topmost grains. The multiple-scattering model behaves properly in this limit. By artificially adding a second infinitely-thin reflective layer which is not allowed to absorb radiation coming from the snowpack below, Ch obtains albedos which axe too high by about a factor of 2 at these wavelengths.
179 we think it premature to lay the blame on the snow albedo model. In fact, not only Liljequist's albedo measurements, but almost all of the published measurements of zenith-angle dependence are insufficiently monochromatic to provide a crucial test of a theoretical snow albedo model. What is badly needed are measurements of the zenith-angle dependence at a few discrete wavelengths. Then any discrepancy could be unambiguously assigned to errors in the snow albedo model. However, if future monochromatic albedo measurements do indeed show a stronger zenith-angle dependence than the delta-Eddington model predicts, the discrepancy can very likely be explained on more rational grounds than "surface reflection": (1) The delta-Eddington approximation increasingly underestimates the albedo compared to an exact multiple-scattering solution as the zenith angle increases (Fig. 3 of Joseph et al., 1976); hence Ch's "discrepancy" may at least partly be numerical error rather than an omitted physical mechanism. (2) As zenith angle increases, the nonsphericity of snow grains becomes increasingly important, especially if they are not randomly oriented; this is because there are fewer scattering events before a photon emerges and thus less opportunity for the effects of nonsphericity to be washed out. If one is to make an adjustment for nonsphericity, it should be done throughout the scattering medium, as Liou (1973) and Pollack and Cuzzi (1980) have done. Basically, Ch's "surface reflection" is just a deus ex rnachina invoked to explain something which should first be examined within the context of multiple-scattering theory (albeit not necessarily at the crude level of approximation embodied in delta-Eddington). There is one other related point in Ch on which we might briefly comment. Ch has misinterpreted Bryazgin and Koptev's (1969) measurements. Bryazgin and Koptev's Figure la, both in the Russian original and in the English translation, shows that the all-wave albedo was independent of sun angle, but that the 0 . 6 - 1 . 2 /am albedo dropped steeply with increasing solar elevation (about 0.01 albedo change per degree of sun angle). This is an illustration of the fact that albedo is more sensitive to zenith angle at 0 . 6 - 1 . 2 /am than at ~ < 0.6 /am,
which is well-explained by the delta-Eddington model (Fig. l l a of WW). Choudhury's statement (last paragraph of p. 113) that weaker surface reflection explains weaker zenith-angle dependence at 0 . 6 - 1 . 2 /am shows just how artifical the "surface reflection" concept is; here he has used it to explain the exact opposite of what Bryazgln and Koptev observed.
ACKNOWLEDGMENT
Our research on the optical properties o f snow and ice is supported by NSF grant ATM-80-24641.
REFERENCES
Bryazgin, N., and Koptev, A. (1969), O Spektral'nom arbedo snezhno-ledyanogo pokrova, ProbL Arktiki i Antark., 3 1 : 7 9 - 8 3 [English Translation (1970), Spectral albedo of snow-ice cover, in Problems of the Arctic and the Antarctic, 29-32. Israel Program for Scientific Translations, pp. 355-360]. Choudhury, B. (1981), Radiative properties of snow for clear sky solar radiation, Cold Regions Sci. Technol., 4: 103-120. Choudhury, B.J. and Chang, A.T.C. (1979), The solar reflectance of a snow field, Cold Regions Sci. Technol., 1: 12t-128. Choudhury, B.J. and Chang, A.T.C. (1981a), On the angular variation of solar reflectance of snow, J. Geophys. Res., 86: 465-472. Choudhury, B.J. and Chang, A.T.C. (1981b), The albedo of snow for partially cloudy skies, Boundary-Layer Meteorology, 20: 371-389. England, A.W. (1974), Thermal microwave emission from a halfspace containing scatterers, Radio Science, 9: 447 -454. Evans, S. (1965), Dielectric properties of ice and snow a review, J. Glaciol., 5: 773-792. Joseph, J.H., Wiscombe, W.J. and Weinman, J.A. (1976), The delta-Eddington approximation for radiative flux transfer, J. Atmos. Sci., 33: 2452-2459. LaChapelle, E.R. (1969), Field Guide to Snow Crystals, Univ. of Washington Press, 101 pp. Liljequist, G.H. (1956), Energy exchange of an Antarctic snow-field. Short-wave radiation, in Norwegian-BritishSwedish Antarctic Expedition, 1949-52, Scientific Results, Vol. 2, Pt. 1A, Oslo, Norsk Polarinstitutt, 107 PP.
Liou, K.-N. (1973), Transfer of solar irradlance through cirrus cloud layers, J. Geophys. Res., 78: 1409-1418.
180
Pollack, J.B. and Cuzzi, J.N. (1980), Scattering by nonspherical particles of size comparable to a wavelength: A new semi-empirical theory and its appfication to tropospheric aerosols, J. Atmos. Sci., 37: 868-881. Tanaka, M. and Nakajima, T. (1977), Effects of oceanic turbidity and index of refraction of hydrosols on the
flux of solar radiation in the atmosphere-ocean system, J. Quant. Spect. Radiat. Trans., 18: 93-111. Wiscombe, W.J. and Warren, S.G. (1980), A model for the spectral albedo of snow. I. Pure snow, J. Atmos. Sci., 37: 2712-2733.
CoM Regions Science and Technology, 5 (1981) 177-180 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
Short Communication COMMENT ON " R A D I A T I V E PROPERTIES OF SNOW FOR CLEAR SKY SOLAR RADIATION
"
Stephen G. Warren Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO., 80309 (U.S.A.)
and Warren J. Wiscombe Department of Applied Science, New York University, New York City, N. Y. 10003 (U.S.A.)
(Received August 3, 1981; accepted August 7, 1981)
ABSTRACT Choudhury 's (1981) addition of a special "surface reflection" term to a theoretical model for snow albedo is both unnecessary and incorrectly formulated.
Choudhury has published a series of papers, many of them with his colleague Chang, on the interaction of solar radiation with snow. They at first used two-stream methods for the multiplescattering calculations (Choudhury and Chang, 1979) but have now switched to the delta-Eddington method recommended by Wiscombe and Warren (1980; hereafter WW). Choudhury (1981; hereafter Ch), as well as Choudhury and Chang (198lab) used approximations to the single-scattering quantities for ice spheres instead of the exact Mie calculations of these quantities used by WW. However, these approximations probably lead to little error in the wavelength range (0.4 ~< ;~ ~< 2.4/am) which Ch considers, at least for the larger snow grain sizes. Thus the spectral albedo calculations of Ch which omit "surface reflection" differ hardly at all from those of WW. Choudhury's method, however, diverges from that of WW, and gives different albedos, when the additional "surface reflection" term is put into the model. The purpose of this note is to state that (1) a separate accounting of surface reflection is unnecessary for the snow situation Ch con-
siders and (2) even if one does want to employ a surface reflection treatment, it is formulated incorrectly by Ch (and by Choudhury and Chang). The snowpack was modeled by WW as a scattering medium with no special dielectric discontinuity at the surface that is any different from the dielectric discontinuities between air and ice within the snowpack. Of course the topmost grains do reflect radiation, but this is already accounted for within the multiple-scattering approximation. The "glitter" that one sees from individual snow grains is not computed explicitly (since the deltaEddington method gives only fluxes, not intensities), but it is included implicitly. The glitter off a snow surface, which Ch mentions, is qualitatively no different than the glitter off "diamond dust" or off the top of a cirrus cloud. And in modeling the reflection of sunlight from these media, no special surface reflection term is ever invoked. The nature of the interaction of electromagnetic radiation with a snowpack depends on the magnitude of the wavelength of light relative to the interparticle (center-to-center) separation. The snow is effectively, blurred to a resolution comparable to the wavelength. This means that whereas solar radiation experiences dielectric discontinuties between air and ice throughout the snowpack, radiowaves by contrast experience a dielectric discontinuity only at the top of the snowpack. In the latter case, the snowpack as a whole is then regarded as having some effective complex dielectric constant which is intermediate between those
178 of ice and air (e.g. Evans, 1965). The snowpack thus exhibits a "surface" to radiowaves but not to sunlight. [There is an intermediate wavelength regime, around the microwave, where neither limiting case is valid, so that both internal scattering and surface reflection must be considered (cf. England, 1974)] Although the surface need not be treated separately for reflection of sunlight by ordinary snow, there are some less common snow conditions for which it may indeed have to be considered. A case in point is glazed snow, for example the "firnspiegel" shown in Fig. 59 of LaChapelle (1969). The question then becomes: Is Choudhury's formulation of surface reflection applicable to such a situation? The answer is no, because the coupling of the surface layer to the underlying scattering layer is done improperly. First, Ch ignores the bending of rays at the surface. In his delta-Eddington calculation he simply uses the incident solar zenith angle instead of using Shell's law to transform that angle at the surface. Second, the "surface" layer is sometimes present,. sometimes absent in Ch's formulation. For example, it is assumed to be present for direct radiation but absent for diffuse radiation (which is actually nothing but the sum of direct beams from all possible angles). And for the case of direct radiation, Ch assumes the "surface" layer to be present for downward radiation but absent for upward radiation emerging from the multiple-scattering medium, viz: F f = asrF,~ + (1 - otsr)F~, ams
(1)
where F f and F+ are respectively the upflux and downflux at the surface, t~st is Ch's albedo due to "surface reflection", and Otms is his albedo due to multiple scattering. Ch thus allows the "surface" layer to reflect radiation back up from its top surface but not back down into the snow from its lower surface. This may or may not be important quantitatively, depending on the wavelength*; the point is that the formulation is inconsistent. Proper treatments of the coupling of a dielectric interface to a multiple-scattering medium are available in the literature. This has been a subject of active research in optical oceanography, a recent careful formulation being that of Tanaka and Nakajima (1977).
Choudhury's apparent reason for introducing "surface reflection" into his model was in order to match Liljequist's (1956) observations of spectrally-integrated snow albedo made at large zenith angles. Figure 8 of Ch shows that without "surface reflection" his model's prediction of that albedo exhibits virtually no zenith angle dependence (to within 1%). By contrast, Liljequist's measurements showed a substantial zenith angle dependence. But we think that this is a weak justification, at best, for introducing "surface reflection," which in our view is no more than an extra tuneable parameter with no real physical basis. The model.measurement discrepancy may be due to any of a number of errors, both experimental and theoretical. But even assuming no experimental error, the discrepancy cannot be considered a telling indictment of the snow albedo model. In order to obtain the zenith-angle dependence of spectrallyaveraged albedo from the spectral snow albedo model, one needs to know the incident radiation spectrum. This is because the incident spectrum changes with zenith angle, and also because the zenith-angle dependence of albedo is different at each wavelength, becoming stronger the higher the ice absorptivity (Fig. 1 la of WW). But Liljequist did not measure the incident radiation spectrum, so Ch had to calculate an incident spectrum using his atmospheric radiation model. It is not possible to be 100% confident that the atmospheric model reproduces the spectral downflux at Maudheim for the time that Liljequist was there. (For example, the transmission functions for the atmospheric gases may be inaccurate at low temperatures.) The cause of a discrepancy between observation and calculation of spectrally integrated albedo may thus be due to a deficiency either in the snow albedo model or in the atmospheric radiation model; *The error caused by using eqn. (1) will be insignificant at wavelengths where ice is relatively transparent [because as ares -~ 1, eqn. (1) reduces to F t = ares F~ ] but not in highly absorbing regions. At X = 1.5 ~m and 2.0 ~m, for example, where snow albedo is only a few percent, the snow behaves like an inf'mitely thin highly-absorbing layer exhibiting reflection only from the topmost grains. The multiple-scattering model behaves properly in this limit. By artificially adding a second infinitely-thin reflective layer which is not allowed to absorb radiation coming from the snowpack below, Ch obtains albedos which axe too high by about a factor of 2 at these wavelengths.
179 we think it premature to lay the blame on the snow albedo model. In fact, not only Liljequist's albedo measurements, but almost all of the published measurements of zenith-angle dependence are insufficiently monochromatic to provide a crucial test of a theoretical snow albedo model. What is badly needed are measurements of the zenith-angle dependence at a few discrete wavelengths. Then any discrepancy could be unambiguously assigned to errors in the snow albedo model. However, if future monochromatic albedo measurements do indeed show a stronger zenith-angle dependence than the delta-Eddington model predicts, the discrepancy can very likely be explained on more rational grounds than "surface reflection": (1) The delta-Eddington approximation increasingly underestimates the albedo compared to an exact multiple-scattering solution as the zenith angle increases (Fig. 3 of Joseph et al., 1976); hence Ch's "discrepancy" may at least partly be numerical error rather than an omitted physical mechanism. (2) As zenith angle increases, the nonsphericity of snow grains becomes increasingly important, especially if they are not randomly oriented; this is because there are fewer scattering events before a photon emerges and thus less opportunity for the effects of nonsphericity to be washed out. If one is to make an adjustment for nonsphericity, it should be done throughout the scattering medium, as Liou (1973) and Pollack and Cuzzi (1980) have done. Basically, Ch's "surface reflection" is just a deus ex rnachina invoked to explain something which should first be examined within the context of multiple-scattering theory (albeit not necessarily at the crude level of approximation embodied in delta-Eddington). There is one other related point in Ch on which we might briefly comment. Ch has misinterpreted Bryazgin and Koptev's (1969) measurements. Bryazgin and Koptev's Figure la, both in the Russian original and in the English translation, shows that the all-wave albedo was independent of sun angle, but that the 0 . 6 - 1 . 2 /am albedo dropped steeply with increasing solar elevation (about 0.01 albedo change per degree of sun angle). This is an illustration of the fact that albedo is more sensitive to zenith angle at 0 . 6 - 1 . 2 /am than at ~ < 0.6 /am,
which is well-explained by the delta-Eddington model (Fig. l l a of WW). Choudhury's statement (last paragraph of p. 113) that weaker surface reflection explains weaker zenith-angle dependence at 0 . 6 - 1 . 2 /am shows just how artifical the "surface reflection" concept is; here he has used it to explain the exact opposite of what Bryazgln and Koptev observed.
ACKNOWLEDGMENT
Our research on the optical properties o f snow and ice is supported by NSF grant ATM-80-24641.
REFERENCES
Bryazgin, N., and Koptev, A. (1969), O Spektral'nom arbedo snezhno-ledyanogo pokrova, ProbL Arktiki i Antark., 3 1 : 7 9 - 8 3 [English Translation (1970), Spectral albedo of snow-ice cover, in Problems of the Arctic and the Antarctic, 29-32. Israel Program for Scientific Translations, pp. 355-360]. Choudhury, B. (1981), Radiative properties of snow for clear sky solar radiation, Cold Regions Sci. Technol., 4: 103-120. Choudhury, B.J. and Chang, A.T.C. (1979), The solar reflectance of a snow field, Cold Regions Sci. Technol., 1: 12t-128. Choudhury, B.J. and Chang, A.T.C. (1981a), On the angular variation of solar reflectance of snow, J. Geophys. Res., 86: 465-472. Choudhury, B.J. and Chang, A.T.C. (1981b), The albedo of snow for partially cloudy skies, Boundary-Layer Meteorology, 20: 371-389. England, A.W. (1974), Thermal microwave emission from a halfspace containing scatterers, Radio Science, 9: 447 -454. Evans, S. (1965), Dielectric properties of ice and snow a review, J. Glaciol., 5: 773-792. Joseph, J.H., Wiscombe, W.J. and Weinman, J.A. (1976), The delta-Eddington approximation for radiative flux transfer, J. Atmos. Sci., 33: 2452-2459. LaChapelle, E.R. (1969), Field Guide to Snow Crystals, Univ. of Washington Press, 101 pp. Liljequist, G.H. (1956), Energy exchange of an Antarctic snow-field. Short-wave radiation, in Norwegian-BritishSwedish Antarctic Expedition, 1949-52, Scientific Results, Vol. 2, Pt. 1A, Oslo, Norsk Polarinstitutt, 107 PP.
Liou, K.-N. (1973), Transfer of solar irradlance through cirrus cloud layers, J. Geophys. Res., 78: 1409-1418.
180
Pollack, J.B. and Cuzzi, J.N. (1980), Scattering by nonspherical particles of size comparable to a wavelength: A new semi-empirical theory and its appfication to tropospheric aerosols, J. Atmos. Sci., 37: 868-881. Tanaka, M. and Nakajima, T. (1977), Effects of oceanic turbidity and index of refraction of hydrosols on the
flux of solar radiation in the atmosphere-ocean system, J. Quant. Spect. Radiat. Trans., 18: 93-111. Wiscombe, W.J. and Warren, S.G. (1980), A model for the spectral albedo of snow. I. Pure snow, J. Atmos. Sci., 37: 2712-2733.