Calculation of melt pond albedos on arctic sea ice ALEXANDER P. MAKSHTAS and IGOR A . PODGORNY Makshtas, A . P. & Podgorny, I. A . 1996: Calculation of melt pond albedos on arctic sea ice. Polar Research 15(1), 43-52. An analytical approximation of spectral albedo is derived for a melt pond with a Lambertian bottom assuming that Rayleigh scattering in the water is small compared to absorption. A Monte Carlo method is used to verify that scattering can he ignored in the water. This enables us to calculate pond albedos in the 4OC-700 nm wavelength hand using the analytical approximation. Model calculations and observations indicate that a step-decrease in albedo is likcly t o occur when a melt pond initially forms, and melt pond albedos i n the visible depend more on the structural and optical properties of the bottom than on the depth of the pond. A . P. Makshtus, Arctic & Antarctic Reseurch Institute, 38 Bering s t . , 199397, St. Petershurg, Russia; I . A . Podgorny, Universiry of Wushingron, JlSAO, P . O . Box 354235, Seattle, W A 98195, USA Introduction Melt ponds represent an important component of the arctic climate system. They greatly affect both the surface albedo and redistribution of solar energy in the sea ice cover during the melt season (e.g. Maykut 1986; Maykut & Perovich 1987; Barry et al. 1993; Ebert & Curry 1993; Ebert et al. 1995). Over the past few decades a large quantity of albedo measurements have been reported for melt ponds (e.g. Nazincev 1964; Langleben 1971; Grenfell & Maykut 1977; Grenfell & Perovich 1984; Ivanov & Alexandrov 1994; Perovich 1994; Morassutti & LeDrew 1995). The measurements span the entire season and include observations for a variety of ice types and illumination conditions. For example, according to Nazincev’s (1964) observations in the central part of the Arctic basin, melt ponds having an albedo as low as 0.3 covered up to 45% of the surface. As a result, area-averages albedos of the surface were as low as 0.45 in July. In addition t o empirical studies, the effect of melt ponds on the short- wave radiative transfer has been studied using a coupled atmosphere-ocean-sea ice model (Jin et al. 1994). Despite this considerable effort, there is a need for better knowledge of how solar energy is redis- tributed in the ponded ice (Moritz et al. 1993). While there do exist some empirically obtained parameterisations for melt pond albedos and radi- ative fluxes into the melt water layer (Ebert & Curry 1993; Ebert et al. 1995; Morassutti & LeD- rew 1995), a theoretical basis for these par- ameterisations is still absent. As outlined by Moritz et al. (1993), spectral albedo models for melt ponds must be formulated taking into account ice thickness, melt pond area and thick- ness, lead fraction, ice age, salinity and snow characteristics. Investigation of how melt pond albedo is affected by each of these parameters will yield an improved understanding of the relationship between summer albedos and melt processes. This paper presents new data on pond albedos in the Arctic Ocean in the 40G700nm band and describes a method for calculating spec- tral albedos of melt ponds. Only the visible spec- trum is considered because pond albedos in the solar infrared are determined primarily by Fresnel reflection and are nearly independent of wave- length (e.g. Grenfell & Maykut 1977). Both measured and calculated albedos are analyscd as a function of pond depth. We do not attempt to present a definitive parameterisation of pond albedo that could he incorporated into the sea-ice models. Instead, the objectives of this paper are twofold: (1) to increase the understanding of the short-wave radi- ative transfer in melt ponds and (2) to develop a forward model that can be applied in estimating 44 the vertical partitioning of solar energy in a melt pond and underlying ice from measurements of pond albedo and depth. A . P . Makshtas & I . A . Podgorny Spectral albedo of melt ponds Statement of the p r o b l e m Because many details of the inherent optical properties of ponded ice are still uncertain, a priori prescribed spectral albedos of the melt pond bottom are used in numerical simulations of radi- ative transfer within the ice layer. The ice is assumed to be optically thick, while the reflec- tance distribution of the pond bottom is assumed to be Lambertian and independent of the angular distribution of the incident radiance. Optical properties of the melt water are postulated to be similar to the optical properties of the clearest water observed in the central part of the Arctic Basin (Smith 1973). The configuration of the model used in this study is shown in Fig. 1. The model consists of two plane-parallel layers, hereafter referred to as the water and ice layers. Both layers are assumed to be optically homogeneous in the vertical and horizontal directions. Spectral albedo (cu) is denoted by the subscript w for the water in the melt pond itself and by the subscript i for the ice at the pond bottom. Melt pond depth is denoted The directions defining the incoming and out- coming light beams in the atmosphere are expressed by ( p , @ ) , where p is the cosine of the zenith angle and @ is the azimuthal angle. Fur- thermore, p and - p denote upward and downward directions. The direction of the incoming solar beam is denoted by (-k,, &). Similarly, the direc- by hw. / .c=O z=Tw h W ICE Fig. 1. Configuration of the one-dimensional melt-pond model. tions defining the light beams in the water layer are expressed by (q,@), with q and -7 denoting upward and downward directions respectively. The direct solar radiation reaching the air-water interface from above (F,) is refracted into the direction (-qo, 4) in accordance with Snell's law where n, is the refractive index of the water. The n, value is taken t o be 1.33 in this study, irrespective of wavelength. We start by considering the monochromatic radiative transfer equation in one spatial dimen- sion for the water layer. Specifically. ( 2 ) - 4n where y and y' are determined by ( 3 ) Here, I ( T , ~ , @ ) is the unpolarised diffuse spectral radiance at wavelength A (omitted for brevity) at optical depth T. A, is the single-scattering albedo, x(y) is the phase function, y is the scattering angle (0 y ISOO), R ( h ) denotes the Fresnel reflection coefficient for the air-water boundary, and F,, is the incident spectral irradiance. As seen from Fig. 1, optical depth increases with the depth of the water layer. In addition to Eq. ( 2 ) , which governs the trans- fer of diffuse radiation, we introduce the transfer equation for direct radiation in the water layer: The direct component of the radiation field is associated with that part of the unscattered solar flux which penetrates through the air-water interface and is attenuated exponentially. The boundary condition at the top of the water layer is given by Calciilution of melt pond albedos on arctic sea ice 45 while downwelling radiation at the top of the ice layer is Here, t, = (K, + a,,,)lhwl is the optical thickness of the water layer, K~ and ow are the beam absorp- tion and scattering coefficients in the water, and R(q) is the Fresnel reflection coefficient for the water-air interface. Eq. (6) describes the Fresnel reflection of the upwelling diffuse radiation from the top surface of the water layer. Eq. (7) des- cribes the diffuse reflection of the downwelling direct and diffuse radiation from the top surface of the ice layer. IT and I J. denote the upwelling and downwelling radiances respectively. Because the pond bottom is assumed to be a Lambertian reflector, r/ and 9 are omitted in the left side of The problem is formulated as follows: Under the assumption that scattering is small compared to absorption, one needs to find an approximate (analytical) solution to Eq. (2) with the boundary conditions given by Eqs. (6) and (7). The accuracy of the analytical solution must then be estimated for actual rW and A, values. Eq. (7). Anutyfic solution to the radiutiue transfer eqitation When scattering in the water layer is small com- pared t o absorption, the diffuse radiation field is independent of the azimuthal angle for any 71 and 5 because the only source of diffuse radiation in this case is the reflection from the melt pond bottom in accordance with Eq. (7). Inserting a delta-function for the phase function into the right side of Eq. (2), integrating over q' and 4' and dividing by (1 - Aw), we reduce the transfer equation for the diffuse radiation t o the dif- ferential form of Beer's law: where t = (1 - Aw)r. (9) In terms of t, boundary condition (7) is expressed as XI? (t,)= a; 2n I " ~ ~ , r l ' ) r l ' d r l ' + F ~ , ~ ( $ ) ) , (10) ( i : where the optical thickness of the water layer i; is equal to K,lhwl. According to Eq. (8) and boundary condition ( 6 ) , the diffuse radiance 1 7 ( t,) is decreased by a factor exp(-:) along the direction ( q , $ ~ ) between the pond bottom and top of the water layer. It is then reduced by a factor R(q) at the air-water interface and, finally, by another factor exp( -:) along the direction (-q,$) before reaching the pond bottom again. 2tw Inserting I (t,)R($)exp( -7) into the right rl side of Eq. (10) and integrating over q', the solution to Eq. (8) is written as follows a;(l - R(p,,))Foexp(-- f - -7 tw - vo rl I t ( 2 , v ) = n( 1 - 2c~iR3(22,)) and where is a generalisation for the third exponential inte- gral (Abramowitz & Stegun 1964) Integrating (1 - R(q)IT(O,q)q over the upper hemisphere yields F (0), the upwelling irradiance just above the air-water interface. Dividing FT (0) derived from Eq. (11) by Fo and adding the Fresnel reflection coefficient for the air-water interface, we obtain an expression for a, the spectral albedo for the melt pond of optical thick- ness 't, under direct illumination for the case of a 46 A . P. Makshtas & I . A . Podgorny Lambertian bottom: As earlier, ,LQ a n d qo a r e related by E q . ( 1 ) . Parameterisation of the function R3(t) T h e goal of this section is t o parameterise the function R3(t) so that % can be computed efficiently. Because melt ponds a r e unlikely t o b e deeper than 1 m in the Arctic, actual values of ‘t, a r e unlikely t o exceed 1 in t h e visible spectrum according t o data on K~ compilated by Smith & Baker (1981). Consequently, the following series expansion can b e used for calculating E3(t): (-1)”t” m i 2 where C is t h e Euler constant. A convenient where denotes t h e cosine of the critical angle for total internal reflection. Expanding exp (-:) in a power series, we get 5 E3(t) - R,(t) = (-l)mrmtn’, (19) I11 = 0 where rm is given by When Eqs. (16) and (19) a r e employed to cal- culate E3(t) o r E3(t) - R3(t), the summation over index m is performed from 0 to a finite numbe1 M. Finally, let us evaluate R3(0) which is a pre- requisite for calculating radiometric quantities for an optically thin melt pond. I t is seen from E q . (14) that E,(O) = 0.5. According t o E q . (13), 2 R 4 0 ) is equal to the internal reflectance of radi- ation at the air-water interface for radiation inci- dent from below r + , for the case of a uniform radiance distribution at a plane water surface (Preisendorfer 1976). T h e r + values are in the range from 0.47 to 0.49 for the visible spectrum (Preisendorfer 1976). Consequently, the value of R3(0) is about half t h e value of E3(0). Monte Carlo simulations Monte Carlo method was employed t o find an “exact” solution to Eq. (2) with the boundary conditions given by Eqs. (6) and (7). Comparison of t h e Monte Carlo solution t o E q . (2) and the analytical solution to E q . (8) enables us t o esti- mate errors of the analytical solution. These errors arise d u e to scattering of radiation in the water layer. T h e estimated errors give a measure of the precision of E q . (15) in calculating the pond albedo over the wavelength region of interest. T h e method t o b e applied in Monte Carlo siinu- lations is similar t o t h e methods reported earlier in t h e context of computing underwater light fields (see Mobley 1994 for references). In particular, a weight Wo is assigned to each photon as follows Wo = exp(-K,L) R(%). (21) H e r e gw is the path length of the photon, K is the number of Fresnel reflections from the water- air interface, and R ( q k ) is t h e Fresnel reflection coefficient for radiation incident from below the interface. Two detectors are used: the first to determine the upwelling irradiance F T (0) just above the water surface, and the second to deter- mine t h e downwelling irradiance F .1 ( tw) just above the pond bottom. T h e one-dimensional ( t h e coordinate t) path of each photon is followed until t h e photon strikes t h e bottom twice. T h e coordinate q is used explicitly t o determine t h e direction of t h e photon travel. In turn, the coor- dinate q5 is employed implicitly to compute a new value of q in accordance with Eqs. (3) and (4) when a scattering interaction takes place. G o r d o n & Brown (1974) introduced a modi- fication of the Monte Carlo method to assess the influence of bottom albedo o n the diffuse K k = l Calculation of melt pond albedos on arctic sea ice 47 reflectance of a plane-parallel ocean as a function of bottom depth and albedo. As this problem is very close to that addressed here, the method of Gordon & Brown (1974) is applicable to our study. In particular, an expression for the upwell- ing irradiance measured by the first detector (above the water surface) is written as where F O t (0) is the contribution to FT (0) from photons that do not strike the bottom, F1 (0) is the contribution to F (0) from photons that strike the bottom once for a; = 1, and p is the ratio of the contribution to FL(z,) from photons that strike the bottom twice to the contribution from photons striking the bottom once for nl = 1. So, the value a, = 1 is applied to both contributions which form the ratio p, but the actual value of a; is then used to determine FT (0) from Eq. (22). The important thing to mention is that the factor (1 - a,p)-' in Eq. ( 2 2 ) accounts for all the mul- tiple reflections. The computations were carried out for the case of Rayleigh scattering. Accordingly, the phase function was specified by 3 4 x( y) = -( 1 + cosq y ) ) . (231 Although the A, values for the clearest water do not exceed 0.31 in the 40G700 nm wavelength region (Smith & Baker 1981), our calculations were made for an overestimated value of A, = 0.5. Moreover, the maximum possible value for the spectral albedo of melting white ice, ai = 0.8 (Grenfell & Maykut 1977), was used. Thus, the effect of scattering on the pond albedo may be slightly overestimated. The maximum relative statistical error in estimating a,,, by the Monte Carlo method was about 1.5%. Shown in Fig. 2 are the a; values versus tw calculated for a solar zenith angle of 60". Curve I (dashed line) represents the "exact" Monte Carlo solution. Curves I1 and 111 represent albedos cal- culated for the non-scattering water layer, i.e. either directly from Eq. (15) (solid line) or using parameterisations (16) and (19) with M = 4 (dot- ted line). z, is determined from Z, using Eq. (9). As seen from Fig. 2, scattering has no effect on a; whenever t, < 1. Beyond the visible spectrum tw can exceed 1 for actual ponds, but A, is neg- ligible for wavelengths exceeding 700 nm. Eq. 0.01 0.1 1 10 OPTICAL THICKNESS Fig. 2 . Melt-pond albedo versus optical thickness of the water layer calculated using ( I ) the Monte Carlo method, (11) ana- lytical expression (15), and (111) calculations using analytical expression (15) where functions E, (t) and R, (t) are computcd using expressions (16) and (19). (15) and, thus, parameterisations (16) and (19) (M = 4) are used below to cohpute a;. 0 bservations Measurements in the Barents Sea The observational data presented in this study were collected during the R/V LANCE cruise in the northern part of the Barents Sea in August 1993 under the Russian-Norwegian Oceanographic Programme (RUSNOP). Measurements of pond albedo were carried out on 12 August 1993 over a small multiyear floe with a diameter of about 150 m located near 78.8"N, 34.8"E. Measure- ments were also made of the albedo of the melting bare ice. The ice thickness was approximately 2 m. Glitter patterns on the pond surfaces were not observed. Wind speed was about 2 m/s. A quantum sensor LI-190 SB with a spectral window of 40G700 nm was employed to measure both the incident and reflected irradiances over a sequence of melt ponds of varying depths. The sensor was located about 0.2 m above the water surface and between 1.5 and 2.0 m away from the bare ice to avoid reflected radiance from outside the melt pond surface. 48 A . P . Makshtus & I . A . Podgorny All albedo measurements were made under clear skies at about 1200 solar time. The solar zenith angle was about 60" and the full solar disc was visible. Weather and illumination conditions were stable during the observations. Measure- ments of upwelling and downwelling irradiances for each individual pond were made between three and five times in order to estimate mean values and standard deviations of albedo. More- over, before measurements were made, every melt pond was tested visually for the presence of bottom contamination, and only uncontaminated, clean ponds were selected. No clean ponds were found with a depth exceeding 0.35m; one ref- erence measurement was made for a com- paratively deep (0.5 m) contaminated pond. As seen from Table 1, the measured albedo at h, = 0.5 m is as low as 0.35 compared to 0.59 observed at h, = 0.35 m. This is clearly connec- ted, in our opinion, with cryoconite holes (Eicken et al. 1994) that were observed on the bottom of the deepest pond. According to Eicken et al. (1994), the occurrence of cryoconite holes is a consequence of contaminants deposited o n the pond bottom and is linked, therefore, to a decreased value of pond albedo. At the same time, the albedos reported in Table 1 for the shallow (h, < 0.35 m) ponds are nearly inde- pendent of pond depth. Comparison with other duta sets These albedos are in reasonable agreement with observations made by other investigators. In par- ticular, Ivanov & Alexandrov (1994) carried out measurements north of Franz Josef Land during an expedition aboard R/V POLARSTERN in August 1993 and found spectrally-averaged albedos of clean ponds to be 0.30 ? 0.05 for depths less than 0.3 m, and 0.26 ? 0.03 for depths in the range of 0.30 t o 0.50 m. Although these albedo values are lower than those shown in Table 1 for the shallow ponds, the difference can be partly attributed to employing the model PP-1 pyranometer which has a spectral window of 390-780nm, for the measurements performed by Ivanov & Alex- androv (1994). Morassutti & LeDrew (1995) calculated regression curves for pond albedo in the 400- 700 nm wavelength band versus pond depth based on an analysis of several hundred field measure- ments. The regression curve, reported for the case of clear sky, demonstrates a sharp decrease in a depth range from 0 to 0.1 m but is almost constant at larger depths. Spectrally-averaged albedo of melt ponds We used a simplified approach to compute the incident irradiance reaching the surface. Because the atmosphere was assumed to be cloudless and aerosol-free, only the absorption of solar radi- ation by O3 was taken into account. Absorption by water vapor and other gases was ignored because their effect on solar radiation is insig- nificant over the spectral region of interest (e.g. Liou 1992). Furthermore, Rayleigh scattering in the atmosphere was also ignored. First, the direct incident irradiance reaching the surface exceeds the downwelling diffuse irradiance by an order of magnitude for the solar zenith angle of 60". Second, the Fresnel external reflectance for the case of uniform radiance at a plane water surface is about 0.066 (Preisendorfer 1976) and is close to the observed value of R(KJ of about 0.060 for e,, = 600. Table 1 . Albedo of melt ponds in the 4 W 7 0 0 nm wavelength band versus pond depth Pond depth (m) Melt pond bottom (top surface of the bare ice) Albedo Standard deviation bare ice 0.02 0.10 0.19 0.35 0.50 clean clean clean clean clean contaminated 0.84 0.54 0.53 0.56 0.59 0.35 0.05 0.02 0.01 0.02 0.02 0.02 Calculation of melt pond albedos on arctic sea ice 49 The measurements of spectral albedo reported by Grenfell & Maykut (1977) for melting mul- tiyear white ice were used to specify a;. Absorp- tion of solar radiation by O3 in the Chappuis band for the total ozone content equivalent to 0.5 cm was calculated according to Nicolet (1981). The K, values were taken from Smith & Baker (1981). Spectrally averaged broadband albedo, ( G), for a spectral region (A1,&) is introduced as fol- lows where Fo(A) is the distribution of the incident spectral irradiance. The integration was performed from dl = 400 nm to A, = 700 nm with a step of 10 nm. 00 = 60” was used. The zero value of h, corresponds to both bare ice and ponds of infinitesimally small depth, hereafter referred to as “newly formed” melt ponds. The calculated (a,,,) are plotted in Fig. 3 together with the measurements reported in Table 1. Discussion Although the differences between calculated and observed albedos are significant, two conclusions can still be drawn from Fig. 3. The first conclusion is that a step-decrease in albedo is likely t o occur when a melt pond forms. In order to address this issue in more detail, we write Eq. (15) for t, = 0. Clearly, this is the case of the newly formed melt pond. Taking into account that 2R3(0) = r+ for the case of a Lambertian bottom, we obtain from Eq. (15) Eq. (25) allows us to determine Aml, the dif- ference between ~ ( 0 ) and q, 0.0 0.2 0.4 0.6 0.8 1.0 POND DEPTH (m) Fig. 3. Measured and calculated albedos for melt ponds and bare ice in the. -700 nm wavelength band. Vertical bars represent standard deviations. As seen from Eq. (26), Aa; is negative if a;r+ > R ( h ) . This is satisfied for actual values of a; and 0,. The maximum absolute value of A % is about 0.12 for q = 0.6 and 0, = 60”. According to Mullen & Warren (1988), who calculated the spectral albedo of lake ice, the external surface reflectance tends to increase albedo over what it would be without the surface layer (air-ice interface, in our case), whereas the internal reflection directs radiation back down tending t o decrease albedo. With respect t o the problem addressed in this paper, one may con- clude that the spectral albedo of the ice layer beneath the pondis larger than the spectral albedo of the bare ice with the same optical properties under conditions where the solar zenith angle is not too large. On the other hand, an analysis of Table 1 of this study and fig. B8a from Morassutti & LeDrew (1995) suggests that there exists a difference between the albedos of newly formed pond and bare ice; the magnitude of this difference is about 0.1-0.3 in the 400-700 nm wavelength band. Moreover, a similar sharp decFease in spectral albedo was reported. by Perovich (1994) for a shallow (0.03m) melt pond. Finally, this dif- ference is also observed in our calculations of melt pond albedo reported in Fig. 3. Thus, the only way to explain both observed and modelled 50 A . P . Makshtas & I . A . Podgorny step-decreases in albedo is to postulate that the changes in structural and optical properties of the bottom, which occur when the melt pond forms, cause a greater decrease than predicted by Eq. To assess the influence of the assumption that the melt pond bottom is a Lambertian reflector, the case where this assumption is relaxed is exam- ined. A convenient way to parameterise the upwelling radiance for the newly formed pond is to use the cardioidal radiance distribution (26). where E is a parameter. Lambertian reflectance is the special case of E = 0. Under the assumption that the law of diffuse reflectance for the pond bottom is independent of the direction of the incident radiation, Eq. (25) is still valid for the radiance distribution given by Eq. (27). This can be shown by applying Eq. (22) to the newly formed pond. When q = 1 and absorption of the radiation in the water layer is negligible, the average contribution t o the upwelling radiance just above the air-water interface from photons that do not strike the bottom is exactly R ( h ) , while the contribution from photons striking the bottomonceis(1 - R ( h ) ) ( l - r+).Furthermore, the ratio of the contribution to the downwelling irradiance at the pond bottom from photons that strike the bottom twice to the contribution from photons striking the bottom once is r + for the case of a; = 1. Incorporating p = rr and the above considerations into Eq. (22) yields Eq. (25). In accordance with Eq. (271, r + calculated for negative (positive) E is larger (smaller) than r + calculated for the case of a Lambertian bottom. When E > 0, radiation is redirected into a more nearly normal direction and it is easier, therefore, for it to escape from the melt pond due to the reduced Fresnel reflection at the small zenith angles. Alternatively, for E < 0, radiation is more likely to be reflected back to the bottom. Thus, deviations from Lambertian behaviour can cause either an increase or reduction of A q . A negative E has a stronger effect on the absolute value of A&. For example, when E = -0.9 (which is representative for the upwelling radiance dis- tribution observed in the upper ocean), and a; = 0.8, ~ ( 0 ) = 0.57 because r+ would be about 0.7 (Preisendorfer & Mobley 1985). The reference value of k ( 0 ) for E = 0 is as large as 0.7 As mentioned above, wind speeds during our observations were too small ( 2 m/s) to produce capillary wave slopes sufficient to cause a signifi- cant effect on the penetration of radiation into the melt pond. The effect of capillary waves can be stronger, however, when a high wind speed is combined with a comparatively large (>60") solar zenith angle. The major contribution to the albedo change in this case would be due to the reduced Fresnel reflection of the incident radiation, whereas the effect of capillary waves on the internal reflection would be smaller accord- ing to Preisendorfer & Mobley (1985), Therefore, with respect to the melt pond albedo, possible deviations from Lambertan behavior of the pond bottom seem to be more important than the effect of capillary waves. The step-decrease in albedo is associated with a variation in F (tw), the downwelling irradiance entering the ice layer. For the case of direct illumination, contribution t o F 4 (t,,,) from the diffuse radiation field can be found by integrating I L ( t W , q ) q (see Eq. (12)) over the lower hemi- sphere. The contribution from the direct radiation fieldis F&(tw) (see Eq. ( 5 ) ) . As a result, we have where denotes spectral transmissivity of the water layer, It can be shown using Eqs. (15) and (29) that spectral albedo of the newly formed pond and its spectral transmissivity are related by the expression Thus, Tw(0) > 1 if A q < 0. The second conclusion from Fig. 3 is that the melt pond albedo depends more on the optical properties of the bottom than on pond depth. To evaluate the effect of absorption of the short- wave radiation in the water on pond albedo, let us determine the spectral radiative energy flux into the water layer (H,,,). The upwelling irradiance at the ice-water interface can be found Calculation of melt p o n d albedos on arctic sea ice 51 Table 2. Spectral albedo of a melt pond, spectral transmissivity of the water layer and spectral radiative energy fluxes into the water and ice layers at A = SO0 nm as functions of pond depth (a, = 0.8) Pond dcpth (m) n, T, H,/Fo H,/Fo 0.0 0.70 1.51 0 0.30 0.1 0.69 1.49 0.01 0.30 1 .o 0.62 1.36 0.11 0.27 by multiplying Eq. (28) by a;: FT(t,) = a;T,(t,)Fo. (31) The difference between F ( t,) and F T ( t,) is the special radiative energy flux into the ice layer: H, = (1 - dTw(%)Fo. ( 3 2 ) Finally, H, can be obtained by subtracting Hi from (1 - q ) F o : H, = (1 - a;, - (1 - a ; ) ~ ~ ( t ~ ) ) ~ ~ . ( 3 3 ) Table 2 shows the spectral melt pond albedos, spectral transmissivities of the water layer and spectral radiative energy fluxes into the ice and water layers computed for three different values of h, using Eqs. (15), (29), ( 3 2 ) and ( 3 3 ) . These values represent a newly formed pond, a shallow pond and a deep pond respectively. Calculations are performed for A = 500 nm and a; = 0.8. For convenience, the radiative fluxes are presented relative to Fo so that knowledge of the absolute value of Fo is unnecessary. The results reported in Table 2 enable us to evaluate the effect of absorption of short-wave radiation in the water layer on the melt pond albedo. The effect is negligible for shallow ponds, and should be taken into account only for deep ponds. Nevertheless, the usually observed dif- ference between spectral albedos of melting bare ice (as high as 0.8 at A = 500 nm for the white ice) and deep ponds (as low as 0.2-0.3 at A = 500 nm for the old ponds) (e.g. Grenfell & Maykut 1977) is unlikely to be explained only by absorption of the radiation in the water layer which is rather transparent to the radiation in the visible spectrum. The major contribution to this dif- ference should be attributed to the optical and structural properties of t h e underlying ice. Conclusions In this study, we have both empirically and theor- etically examined melt pond albedos in the 4 0 G 700 nm wavelength band as a function of the pond depth. An analytical approximation of spectral albedo was derived for a pond with a Lambertian bottom, assuming that Rayleigh scattering in the water is small compared to absorption. Using a Monte Carlo method, it was shown that scattering in the water can be ignored. Measurements of albedo in clean ponds were made in the Barents Sea in August 1993 under clear sky conditions. Comparison of observations and theoretical calculations indicates that (1) a step-decrease in albedo is likely to occur when a melt pond initially forms, and (2) melt pond albedos depend more on the structural and optical properties of the underlying ice than on pond depth. The effect of absorption of short-wave radiation in the water on melt pond albedos needs to be taken into account for the deep ponds only. The analytical approximation to the melt pond spectral albedo can be inverted and applied to estimating the spectral albedo of melt pond bot- toms as described by Podgorny (1995). This enables us to determine the redistribution of solar energy between the melt pond and underlying ice from measurements of pond albedo and depth using the forward model developed in this paper. Acknowkdgernenu. ~ The authors wish to acknowledge T. Vinje for the organisation of the expedition aboard RIV LANCE and J . H ~ k e d a l for his assistance in performing measurements of melt pond albedo. We are also grateful to T. Grenfell, B . Ivanov, D. Perovich and G. Shved for many suggestions and discussions and t o two anonymous reviewers for valuable com- ments on this paper. Assistance of M. Berge and T. Grenfell in preparing the manuscript for publication is highly appreci- ated. This research was partly supported by the Russian Fund of Fundamental Research under grant 95-0.5-15315, by the Inter- national Science Foundation under grant NSFOOO and by the Joint Institute for the Study of the Atmosphere and Ocean contribution 339. References Abramowitz, M. & Stegun, I . A . (Eds.) 1964: Handbook of mathematical functions with formulas, graphs and math- ematical tables. National Bureau of Standards, Applied Mathematics Scries - SS. Barry, R. G . , Serreze, M. C . , Maslanik, J . 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Notation the third exponential integral incident spectral irradiance (W m-* nm-I) downwelling spectral irradiance (W rn-' nm-I) upwelling spectral irradiance (W m-* nm-') thickness of the water layer or pond depth (m) spectral radiative energy flux into the ice layer (W m-2 nm-I) spectral radiative energy flux into the water layer (W nm-') diffuse spectral radiance (W m-* nm-I sterad-') diffuse downwelling radiance (W m-* nm-l sterad-') diffuse upwelling radiance (W m-z nm-' sterad-I) refractive index of the water internal reflectance of the air-water for uniformillumi- nation Fresnel reflection coefficient for irradiance incident on the air-water from above Fresnel reflection coefficient for irradiance incident on the air-water from below Fresnel reflection integral spectral transmissivity of the water layer for the downwelling irradiance weight assigned t o a photon phase function spectral albedo of the ice layer, ( F f ( z w ) / F i ( ~ ) spectral albedo of the melt pond spectral albedo of the newly formed pond spectrally averaged albedo of the melt pond scattering angle parameter of the cardioidal radiance distribution difference between a , ( O ) and LU, path length of photon in the water layer cosine of zenith angle in the water layer cosine of zenith angle for direct solar radiation in the water layer cosine of the critical angle for total internal reflection solar zenith angle (degrees) spectral absorption coefficient of the water (m-I) wavelength (nm) single-scattering albedo of the water cosine of zenith angle in the atmosphere cosine of the solar zenith angle scattering coefficient of the water (m-') optical depth optical thickness of the water layer (exact solution) optical thickness of the water layer (analytical solu- tion) azimuthal angle (degrees) solar azimuthal angle (degrees)