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 A simple parameterization to derive the broadband effective albedo over highly reflecting surfaces under overcast conditions is presented. High spatial variability in the surface albedo affects the downwelling solar irradiance in neighboring regions via the multiple reflections of light between the surface and the cloud base. The effective albedo is defined as the albedo of a homogeneous surface that would result in the same downwelling irradiance as observed at the observation point in the presence of a heterogeneous surface. The proposed method parameterizes the effective albedo using the cloud base height and a surface albedo map as inputs. The parameterization is based on the spatial distribution of surface reflections contributing to the downwelling irradiance at the observation site, which is approximated with a gamma distribution. The parameterization was validated against reference values of effective albedo derived from three-dimensional backward Monte Carlo and one-dimensional DISORT radiative transfer calculations for four idealized surface albedo maps and various specifications of cloud properties. It gave values of effective albedo very close to the reference calculations, performing substantially better than any other approach tested, also when applied to the retrieval of cloud optical depth. The method can be implemented into one-dimensional radiative transfer models or used to interpret broadband irradiance measurements in Polar coastal regions, in the marginal sea ice zones, or in patchy terrain with forests and snow-covered fields.
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 Over highly reflecting surfaces, such as snow and ice covered areas, the effect of clouds on the broadband shortwave radiation budget at the surface is complicated by the occurrence of multiple reflections between the cloud base and the surface. Because of multiple reflections the downward solar irradiance at the surface during overcast condition is mostly controlled by the cloud optical depth and the surface albedo [Shine, 1984; Leontyeva and Stamnes, 1994].
 For the interpretation and analysis of ground-based or satellite irradiance measurements and in single-column radiative transfer calculations, the surface albedo measured at a single site or in a single surface pixel is generally used. However, over snow covered areas the downwelling irradiance at one site is affected by the albedo of the surrounding areas, as the multiple reflections between the surface and the cloud base efficiently enhance the horizontal transport of photons. Thus the presence of high albedo contrasts can have a significant impact on the local irradiance field. This effect has been investigated through two-dimensional (2-D) and three-dimensional (3-D) radiative transfer models [Ricchiazzi and Gautier, 1998; Podgorny and Lubin, 1998; Degünter and Meerkötter, 2000; Benner et al., 2001; Barker et al., 2002; Ricchiazzi et al., 2002; Chiu et al., 2004; McComiskey et al., 2006], and also through experiments [Smolskaia et al., 1999]. A conclusion that can be drawn from these studies is that, in case of small-scale (from few meters to hundreds of meters) and rather uniformly distributed albedo heterogeneity, such as over sea ice with melt ponds in summer, the effect of the local albedo contrasts can be accounted for in single-column calculations simply by performing an area-weighted average of the albedos of the various snow/ice/water surfaces [Benner et al., 2001; Barker et al., 2002; Chiu et al., 2004]. When the scale of the horizontal albedo heterogeneity is ≥100 times the cloud base height, as in uniform areas located hundreds of kilometers away from sharp albedo contrasts, cloud base and surface can be treated as infinite surfaces and radiative transfer is correctly calculated with the Independent Column Approximation (ICA) [Chiu et al., 2004]. In case of intermediate-scale albedo heterogeneity (∼1–100 km), as for instance in the proximity of Arctic and Antarctic coastal stations with offshore open water, neither the average surface albedo approach nor the ICA are appropriate [Chiu et al., 2004]. The sharp albedo contrast affects the downwelling irradiance up to distances of several or even tens of kilometers from the coastline, depending on the cloud base height, cloud optical thickness, and on the considered wavelength bands [Barker and Davies, 1989; Ricchiazzi and Gautier, 1998; Podgorny and Lubin, 1998; Degünter and Meerkötter, 2000]. To account for this effect in 1-D models, Li et al.  proposed a method to obtain the “effective albedo”, namely the albedo that must be applied in the 1-D model to reproduce the observed incident irradiance measured at one site and affected by the albedo heterogeneity of the surrounding area. Using a 1-D radiative transfer model, Li et al. derived the effective albedo by attributing the discrepancy between modeled and observed downwelling irradiance to the effect of albedo heterogeneity. The weakness of this method is that it requires very accurate and complete specification of atmospheric and cloud parameters, which in reality are often poorly known.
 The need of a practical and accurate method to estimate the local effective surface albedo is also motivated by its role in the retrieval of cloud optical depth, a parameter that is of primary interest for cloud and radiative modeling in general and has recently also become an issue for data assimilation for numerical weather prediction [Benedetti and Janiskova, 2008]. Ricchiazzi and Gautier  showed that, in the vicinity of a high albedo contrast, the use of locally measured surface albedo in a 1-D model can cause the retrieved cloud optical depth to differ from the true value by as much as a factor of two, for clouds with base at 1 km and optical depth of 10. Moreover, high contrasts in surface albedo cause horizontal variations of the cloud base brightness, and the effects of surface albedo heterogeneity can be hard to distinguish from the effect of cloud optical depth heterogeneity [Ricchiazzi et al., 2002]. For these reasons, cloud optical depth has been difficult to retrieve over areas with discontinuous and highly variable albedo, as for instance in the Canadian Arctic coast [Barker et al., 1998].
 Here, we present a simple parameterization of the effective surface albedo under overcast conditions based on reference calculations done utilizing a 3-D Monte Carlo radiative transfer model and the 1-D radiative transfer model DISORT, both described in section 2. The surface, cloud, and atmospheric properties applied in the simulations are described in section 3: four idealized surfaces were chosen, which included a sharp albedo contrast between 0.1 and 0.8 arranged in different patterns. A homogeneous, plane-parallel cloud layer is assumed, and an Arctic summer atmospheric profile is applied. In section 4 we report the effective albedo calculated with the Monte Carlo code over the four idealized surfaces, for a wide range of cloud optical depths and cloud base heights. These constitute the “true” values of effective albedo used to validate our parameterization. In section 5 we describe the method followed to derive the parameterization. First, we calculate with the Monte Carlo model the normalized spatial probability density function of the surface reflections experienced by the photons that reach the observation site, assuming a homogeneous surface albedo. We then parameterize the resulting distribution with a Gamma probability density function. Finally, the effective albedo is calculated by integrating the probability density distribution over the albedo maps. The results are then compared with two alternative approaches and validated against the Monte Carlo results of section 4. Since one of the most important foreseen applications of the effective albedo parameterization is in the remote sensing retrieval of cloud optical depth τ, in section 6 we investigate the implications of our findings for the retrieval of τ, in the special cases of our four surface patterns. A final section of summary and conclusions follows.
2. Radiative Transfer Models
 Radiative transfer calculations for cases with horizontally heterogeneous surface albedo were performed using a Monte Carlo model. This model was introduced as a monochromatic forward Monte Carlo model by Barker [1992, 1996]. It was updated by Räisänen et al.  for broadband shortwave calculations, and here, it was further modified into a backward Monte Carlo model. As noted by Gordon  and Ricchiazzi et al. , backward Monte Carlo models provide much greater efficiency than forward models in the computation of radiances or irradiances at a specific point at the surface.
 The backward Monte Carlo model exploits the principle of reciprocity in radiative transfer theory [e.g., Gordon, 1985]. Instead of following the trajectories of solar photons entering the top-of-the-atmosphere and evaluating their contribution to the downwelling irradiance at the observation point, the problem is reversed. Photons are ejected from the observation point, with an angular distribution proportional to the cosine of photon zenith angle [Gordon, 1985]. They are then followed until they escape to space or are absorbed at the surface or in the atmosphere. At each scattering event (either at the surface or in the atmosphere), the contribution to the radiance toward the sun is computed using the local estimation method [Marchuk et al., 1980; see below].
 The downwelling irradiance at the observation site is obtained as a sum of two terms:
 Here, the first term is the contribution by direct (i.e., non-scattered) solar radiation:
where μ0 is the cosine of solar zenith angle, S0 is the solar constant, and wk are the spectral weights for the total of K = 72 pseudo-monochromatic intervals used in the scheme by Freidenreich and Ramaswamy . τk is the total column optical depth corresponding to interval k, and is assumed horizontally homogeneous in this study.
 The contribution by “diffuse” radiation (i.e., radiation scattered at least once) is obtained from backward Monte Carlo calculations:
where Np = 107 is the total number of photons, Nk = wkNp is the number of photons used for interval k, Sk,p is the number of scattering events experienced by photon p, and ΔLk,p,ssun is the contribution to the radiance scattered toward the sun by the sth scattering event. This contribution is estimated using the local estimation method as
where θs is the scattering angle toward the sun, P is the scattering phase function relevant for this scattering event, and is the optical depth of the atmospheric column between the top of the atmosphere and the scattering point at an altitude zs.
 Finally, for each case considered, the effective surface albedo αeff was determined by repeating the calculations for horizontally homogeneous surfaces using the 1-D DISORT model [Stamnes et al., 1988] with 32 streams. The homogeneous surface albedo was varied iteratively until the downwelling irradiance matched that obtained with the Monte Carlo model for the horizontally heterogeneous case. As a consistency check, many cases with horizontally homogeneous surface albedo were also performed using the Monte Carlo code. The differences between Monte Carlo and DISORT results were always within the expected random errors of the Monte Carlo calculations. For Np = 107 photons, the typical one-sigma uncertainty of the derived effective albedo is Δαeff ∼ 0.002.
3. Surface, Cloud, and Atmospheric Properties
3.1. Surface Properties
 Four heterogeneous albedo patterns were considered (Figure 1): in each of them the light gray area corresponds to a surface albedo α of 0.8, while the dark area to a surface albedo of 0.1. These albedo values have been chosen to represent the typical albedo contrast between snow covered areas and open sea, for which the methodology is intended to be applied. The four cases are studied as if the observation site were placed at the center of the surface area. Surfaces 1 and 2 are mostly drawn to maximize the effect of the surrounding albedo on the observation point, since these cases are radially symmetric. The inner highly reflecting area has a radius of 1.5 km, and the dark annulus has a width of 1 km and 3.5 km in case 1 and 2, respectively. In nature, case 1 could roughly correspond to a snow covered valley surrounded by bare rocky mountains, which are again surrounded by snow covered areas, while case 2 could represent a snow covered field or frozen lake surrounded by a thick spruce forest where snow has melted from the trees. Surfaces 3 and 4 represent the most common cases of a sea ice area with an open lead (case 3) and of the edge between open sea and snow-covered sea ice or land (case 4). In case 3, the dark area is at a 1 km distance from the observation site and has a width of 2 km, while in case 4, it extends from 1 km to the edge of the computational domain.
 Note that in the backward Monte Carlo model, the computational domain can be made arbitrarily large, which avoids the problem related to the lateral boundary conditions [Ricchiazzi and Gautier, 1998]. A domain size of 2000·2000 km2 was used in our computations. Hence the albedo patterns continue as depicted in Figure 1 up to a distance of 1000 km from the observation point both in x and y directions.
 For simplicity, Lambertian surface was assumed, as the influence of anisotropic snow reflectance on radiative transfer is minimal compared to the 2-D effect of sharp albedo contrasts [Degünter and Meerkötter, 2000].
3.2. Cloud Properties
 As our baseline case, we consider a water cloud with a vertical profile of liquid water content (LWC) that increases from 0 at the cloud bottom to a maximum at the cloud top with a vertical gradient χ = dLWC/dz = 1 g m−3 km−1. This corresponds to the adiabatic cloud water profile for a cloud base pressure of 950 hPa and a temperature of about 263 K. A gamma distribution of droplet sizes with a droplet number concentration of Nd = 100 cm−3 and an effective variance νeff = 0.1 is assumed. Five different cloud optical depths (τ = 2.5, 5, 10, 20, and 40) are considered, where τ is the spectral-mean optical depth weighted by the top-of-the-atmosphere insolation. Cloud geometric thickness Δz, liquid water path LWP and cloud-mean droplet effective radius re (ratio of third to second moment of the droplet size distribution integrated throughout the cloud) increase with increasing τ as indicated in Figure 2.
 Below, results for five cloud base heights (0.2, 0.5, 1.0, 2.0 and 3.0 km) are considered. The sensitivity to assumptions about liquid water vertical profile and droplet number concentration is also discussed in section 4.
 Cloud optical properties (extinction coefficient, single-scattering albedo, and asymmetry parameter) are computed using Mie theory. For simplicity, the phase function of Henyey and Greenstein  is employed instead of the full Mie phase function. This has generally little impact on radiative fluxes, but it accelerates considerably the convergence of the backward Monte Carlo results, because the random errors in Fdif↓ in equation (3) are reduced substantially.
3.3. Atmospheric Properties
 Atmospheric properties representative of Arctic summer conditions were assumed. A composite of radiosoundings made during the Surface Heat Budget of the Arctic Ocean (SHEBA) campaign [Uttal et al., 2002] was used to define the vertical profiles of temperature and specific humidity up to the 250 hPa level. The subarctic summer atmosphere of McClatchey et al.  was used to define the ozone profile as well as temperature and humidity profiles above 250 hPa. The surface air temperature is 269 K, and the vertically integrated water vapor amount is ≈10 kg m−2.
 Gaseous absorption and molecular Rayleigh scatttering are computed using the k distribution scheme of Freidenreich and Ramaswamy , while aerosols are neglected for simplicity. The atmosphere is divided into 19 layers in the vertical, 10 of which are located in the cloud to resolve the cloud structure properly. Each layer is treated as vertically and horizontally homogeneous in the Monte Carlo calculations.
 In the calculations reported in this paper, a solar zenith angle of 60° is assumed, and the sun is located to the left of the albedo patterns in Figure 1. Alternative values for solar zenith and azimuth angle were experimented with, but the impact on the effective albedo was very minor, generally ≤0.01. Weak sensitivity to solar position is expected: for plane-parallel clouds with optical depths of 2.5–40 as considered here, very little direct radiation penetrates to the surface, so virtually all the illumination at the surface is from a diffuse field.
4. Effective Surface Albedo: Monte Carlo Results
 In this section, the dependence of the effective albedo αeff on various parameters is reported briefly. The physical interpretation of these results is discussed in section 5.2.
 The impact of cloud base height zb and optical depth τ is considered in Figures 3 and 4. Baseline values are assumed for the vertical gradient of LWC (χ = 1 g m−3 km−1) and droplet number concentration (Nd = 100 cm−3). There is a substantial dependence of αeff on zb especially in cases 2 and 4 (Figure 3). For example, for a typical cloud optical depth of τ = 10, the values of αeff for case 2 range from 0.452 to 0.675 (Δαeff ≈ 0.22), while those for case 4 range from 0.541 to 0.729 (Δαeff ≈ 0.19). Moreover, in the first three cases, the dependence on zb is non-monotonic: αeff attains its minimum value for zb = 0.5 km (case 1) or zb = 1.0 km (cases 2 and 3). Only in case 4 is there a monotonic decrease of αeff with increasing zb.
 The influence of cloud optical depth is most distinct in case 4, in which αeff increases monotonically with τ for all values of zb, by 0.04–0.07 when going from τ = 2.5 to τ = 40. In the other cases the dependence on τ is smaller and often non-monotonic.
 To explore the sensitivity of αeff to the specification of cloud structure, five alternative assumptions were tested (Table 1). These include halved (experiment χ0.5) and doubled (experiment χ2.0) vertical gradient of LWC, 70% lower (experiment ND30) and threefold (experiment ND300) droplet number concentration, and the use of a vertically homogeneous cloud (experiment VHOMO) instead of one where the LWC increases linearly with height (Table 1). A common feature of the χ2.0, ND300, and VHOMO experiments is that the center of cloud mass is located slightly lower than in the baseline case, while the opposite holds true for χ0.5 and ND300. The impact of these assumptions depends substantially on the cloud base height but is typically rather small, with RMS differences of 0.005−0.021 to the baseline case. Thus, to conclude this section, cloud base height zb is clearly the most important parameter determining the effective albedo.
Table 1. Mean and Root Mean Square Differences in αeff From the Baseline Case (χ = 1 g m−3 km−1, Nd = 100 cm−3) due to Changed Assumptions About Cloud Structurea
In the experiments χ0.5 and χ2.0, the vertical gradient of LWC is modified (χ = 0.5 and 2.0 g m−3 km−1, respectively). In the experiments ND30 and ND300, the droplet number concentration is changed (Nd = 30 and 300 cm−3, respectively). In the last experiment VHOMO, a vertically homogeneous cloud is assumed, with the same cloud-mean LWC, Nd, and cloud geometric depth as in the baseline case. The statistics include calculations for five cloud base heights (zb = 0.2/0.5/1.0/2.0/3.0 km) and five cloud optical depths (τ = 2.5/5/10/20/40).
5. Parameterization of the Surface Effective Albedo
5.1. Theoretical Background
 We are searching a parameterization that calculates the effective albedo αeff by integrating the contribution of each surface point surrounding the observation site. To derive such a parameterization, two issues need to be considered. First, we have to determine where the surface reflections relevant for F↓ at the observation site occur. To quantify this, we use the normalized probability density function of surface reflections p(r), defined by
where r is the distance from the observation site, and f(r) is defined in section 5.2.
 The second issue that needs consideration is how to use p(r) to derive αeff. In a study of the effects of surface albedo heterogeneity on cloud shortwave absorption, Chiu et al.  considered two limiting cases. First, in the limit of small surface heterogeneity elements, the albedo of each point can be weighted linearly. This results in
where α(r) is the mean albedo at a distance r from the observation site.
 Second, when the surface heterogeneity elements are very large, the Independent Column Approximation can properly reproduce the cloud-surface radiative interactions. In this case, αeff is obtained by taking an average of the multiple reflection factor describing the series of reflections between the cloud base and the surface:
 Here, αc is the cloud base albedo, and atmospheric absorption and scattering between the cloud base and the surface is neglected for simplicity (or incorporated implicitly into αc). Using Jensen's inequality, it is easy to show that αeff(ICA) ≥ αeff(lin).
 The results by Chiu et al.  indicate that the linear weighting scheme is appropriate when the scale ratio
where zb is the cloud base height and d is the horizontal scale of surface heterogeneity elements. Conversely, the ICA is valid when sr ≤ 0.01. In the intermediate range (0.01 < sr < 1) neither method is strictly valid. For a cloud base height of zb = 1 km, this corresponds to surface heterogeneities in the scales of about 1 to 100 km. Thus, in general we expect that αeff(lin) will underestimate the true αeff, while αeff(ICA) will overestimate it.
5.2. Derivation of the Probability Density Function f(r)
 The function f(r) appearing in equation (5) is defined such that f(r)dr is the probability that a photon reaching the observation site has experienced a surface reflection at a distance between r and r + dr from the observation site. A homogeneous surface is assumed, on one hand to avoid double-counting the effect of surface heterogeneities in equations (6) and (7), and on the other hand, because we do not wish the parameterized p(r) to depend on the nature of surface heterogeneity.
 The backward Monte Carlo model was utilized to derive f(r). Along the path of each photon, track was kept of the locations of surface reflections. As explained above, at each scattering event s, the contribution to the radiance toward the sun ΔLk,p,ssun is evaluated, which factually gives us a contribution to the downwelling diffuse irradiance Fdif↓ at the observation site (see equation (3)). For any discrete distance bin Δr, the corresponding contribution to f(Δr) is proportional to nk,p,s(Δr)ΔLk,p,ssun, where nk,p,s(Δr) is the number of surface reflections in the distance bin Δr that have occurred along the photon path between the observation site and this scattering event s. Thus, taking into account all spectral intervals k, photons p and scattering events s, and after normalization by the total downwelling irradiance at the observation site, we obtain
 Distance bins of Δr = 100 m were used to compute the distribution of surface reflections up to a distance of 200 km. These calculations were repeated for various homogeneous surface albedos, cloud optical depths and cloud base heights, assuming baseline values for the vertical gradient of LWC (χ = 1 g m−3 km−1) and the droplet number concentration (Nd = 100 cm−3). Some of the results are illustrated in Figure 5.
Figure 5a indicates that for a fixed τ and zb (here τ = 10 and zb = 0.5 km), f(r) increases rather uniformly with increasing surface albedo α because higher α enhances the multiple reflections between the cloud and the surface. The corresponding differences in the normalized probability distribution p(r) are quite small for the range of α = 0.5−0.8 considered here (Figure 5b). It can, however, be discerned that, with increasing α, the peak of the distribution becomes slightly lower, and the tail slightly longer (i.e., the role of far-away surface reflections increases with increasing surface albedo, as noted by Ricchiazzi and Gautier ). The small but systematic change in the shape of the distribution with surface albedo is also visible in the cumulative normalized probability density function
 Similar to increased surface albedo, increased cloud optical depth τ enhances multiple reflections, thereby increasing f(r) (Figure 5d). The corresponding normalized probability distribution p(r) (Figure 5e) and the cumulative distribution P(r) (Figure 5f) converge quite tightly for τ ≥ 10. However, for smaller τ, the peak of p(r) is slightly lower and the tail of the distribution is longer. This tail is related to Rayleigh scattering above the cloud. When the cloud is partly transparent, there is a contribution to F↓ by photons that have been reflected at the surface far away from the observation site and have then been backscattered by air molecules above the cloud. This effect was also suggested by Ricchiazzi and Gautier .
 Consistent with the results for αeff in section 4, the cloud base height zb has a strong impact on the distribution of surface reflections (Figures 5g, 5h, and 5i). With increasing zb, the peaks of f(r) and p(r) become smoother and move further away from the observation site. In other words, the higher is the cloud base, the larger is the region whose surface conditions influence the downwelling irradiance at the observation site [cf. Barker and Davies, 1989; Ricchiazzi and Gautier, 1998; Podgorny and Lubin, 1998]. In Figure 5i, the size of this region is characterized in terms of the distance corresponding to the 90th percentile of the cumulative distribution P(r) of surface reflections. This distance increases roughly linearly with the cloud base height, ranging from 2.5 km for zb = 0.2 km to 23 km for zb = 3.0 km, for the parameter values considered here (α = 0.6 and τ = 10).
 The strong dependence of p(r) on zb provides a simple explanation for the corresponding dependence of αeff on zb in Figure 3. Thus, considering for example case 2, the minimum in αeff occurs for zb ≈ 1 km because this maximizes the relative contribution by the dark annulus extending from 1.5 to 5 km from the observation site. When the cloud base is very low, the dominant contribution comes from the bright region in the immediate vicinity of the observation site. Conversely, for zb much higher than 1 km, the bright region outside the dark annulus plays the dominant role. Roughly similar arguments apply to cases 1 and 3. In case 4, however, the weight of the dark region increases monotonically with increasing zb.
 The monotonic increase of αeff with increasing τ in case 4 (Figure 4) is partly explained by the change of p(r) with τ: for optically thicker clouds, the tail of p(r) is shorter, which reduces the contribution from the dark regions south of the observation site. However, another part of the explanation is related directly to the cloud optical depth. For large τ, the cloud albedo αc in equation (7) is larger, which acts to increase αeff(ICA). Thus, even if p(r) were independent of τ, αeff(ICA) would increase with τ, while αeff(lin) would stay constant. Since the true αeff lies between these two extremes, this contributes to an increase in αeff with increasing τ. As noted above, in cases 1−3 the dependence of αeff on τ is generally weaker and less regular. For brevity, we omit a detailed discussion of this feature.
5.3. New Parameterization
 To derive the new parameterization, we started with the normalized distribution of surface reflections p(r) for a cloud optical depth of τ = 10, averaged over constant surface albedos of 0.5, 0.6, 0.7, and 0.8 (hereafter, this distribution is referred to as pMC(r)). A gamma distribution was fit to pMC(r), separately for the five cloud base heights (from 0.2 to 3 km):
 Here, r is the distance from the observation site, α and β are the shape and scale factors of the Gamma distribution, respectively, and Γ is the Gamma function. A least-square fit for the coefficients α and β as a function of cloud base height zb was then derived:
where p1 = 2.0248, p2 = −0.12426, p3 = 0.80449, and p4 = 16.412. For numerical integration, pΓ(r) is calculated at a 100 m interval up to a distance of 200 km from the observation site, for various cloud base heights.
Figure 6 presents a comparison between pΓ(r) and pMC(r). The parameterized pΓ(r) curves (thin lines in Figure 6) capture the location of the pMC(r) maxima and the general shape of the pMC(r) distribution. However, for zb between 0.5 and 2.0 km, pΓ(r) tends to underestimate the maximum of pMC(r), while it overestimates it for zb = 0.2 km. Moreover, the pΓ(r) distribution has generally a wider bell-shaped part and a shorter tail compared to the pMC(r) distribution.
Figure 7 shows the contributions p(r)α(r) to the linearly averaged effective albedo (equation (6)) when using pMC(r) and pΓ(r), for three cloud base heights. The overall shapes of these curves follow Figure 6, except that the contributions from the dark regions are decreased, and the differences between α(r)pMC(r) and α(r)pΓ(r) are likewise damped down.
 A final choice regarding the parameterization concerns the use of αeff(lin) vs. αeff(ICA), as discussed above. In general, the true effective albedo should lie between these two extremes, that is
where 0 ≤ w ≤ 1. In principle, one could think of parameterizing the weight factor w as a function of the scale ratio sr (equation (8)). However, a practical difficulty is that, unlike for the checkerboard patterns of Chiu et al. , for the cases considered here, it is not obvious how to define the effective scale of surface heterogeneity d. Additional tests (not reported here) suggested that d in fact tends to increase with increasing zb. Therefore, as a pragmatic compromise, we recommend to simply use w = 0.5, that is, take an average of αeff(lin) and αeff(ICA):
 This choice is supported by the Monte Carlo results in Figure 3. The true value of αeff indeed lies consistently between αeff(lin) and αeff(ICA), although closer to the former in the first three cases. This is not an entirely self-evident result because use of p(r) derived for horizontally homogeneous cases involves the implicit assumption that the downwelling radiation field is horizontally uniform. This is, of course, not strictly true for cases with surface heterogeneity.
 Note that in Figure 3, αeff(lin) and αeff(ICA) have been computed as exactly as possible: the normalized distribution of surface reflections p(r) was adjusted for actual values of αeff and τ in each case, and the dependence of cloud albedo αc on τ was fully taken into account. This differs from the Γ-function parameterization, which is based on p(r) averaged over α = 0.5, …, 0.8, for a cloud optical depth of τ = 10. Furthermore, assuming that τ is generally not known a priori, a constant value αc = 0.5343 is used in equation (7), which is also based on calculations for τ = 10. These simplifications and the numerical inaccuracy of the Gamma fit make the Gamma-function parameterization to deviate from the Monte Carlo results for αeff(mix). The related RMS differences in αeff amount to 0.013 (case 1), 0.034 (case 2), 0.011 (case 3) and 0.024 (case 4), when including calculations for five cloud base heights (0.2, 0.5, 1.0, 2.0 and 3.0 km) and five cloud optical depths (τ = 2.5, 5, 10, 20 and 40).
5.4. Alternative Parameterizations for Effective Albedo
 To put the results produced by the new parameterization into a perspective, two other formulations are considered.
 First, Li et al.  employed the following equation to calculate the effective albedo αeff from a surface albedo map, in an effort to validate a more general method that derives αeff without resorting to an albedo map:
 Here, αi,j is the albedo of pixel (i,j), ri,j is the horizontal distance of pixel (i,j) from the observation site, and dΩ is the differential solid angle. In discrete terms, ΔΩ becomes ΔSi,j cosθi,j/(zb2 + ri,j2), where ΔSi,j is the surface area of pixel (i,j), and θi,j = tan−1(ri,j/zb) is the corresponding viewing zenith angle, as seen by a hypothetical sensor located at the cloud base directly above the observation site. In our calculations, a pixel size of 50 m was utilized. The probability density distribution pLi(r) corresponding to equation (15) clearly overestimates the contribution of the area close to the observation site (Figure 8). Li et al.  used their formula to calculate the effective areal-mean surface albedo in a vegetated area, where the scale of the albedo heterogeneity was of the order of few hundred meters. We speculate that in their case the shape of p(r) did not really matter and that a simple non-weighted areal-mean albedo would have given a very similar result. This does not, however, compromise the validity of Li et al.'s main result, which is a general method to derive the surface effective albedo by inverting a 1-D radiative transfer model.
 Second, we also test the simple method of area-averaged surface albedo, which matches the pixel averaging approach utilized in satellite remote sensing of surface and cloud radiative properties. We chose to perform the areal average of albedo over a radius equal to 10 times the cloud base height zb. For the lowest zb considered here (0.2, 0.5, and 1.0 km), this roughly corresponds to the surface area where 90% of surface reflections contributing to F↓ at the observation site occur (cf. Figure 5i).
5.5. Comparison of Parameterized Effective Albedo With Monte Carlo Results
 The performance of several approaches in estimating the effective surface albedo is summarized in Table 2. For each surface albedo pattern, mean and RMS differences to the true αeff derived from Monte Carlo calculations are given. These statistics include calculations for five cloud base heights (zb = 0.2/0.5/1.0/2.0/3.0 km) and five cloud optical depths (τ = 2.5/5/10/20/40), assuming baseline values for liquid water vertical gradient and droplet concentration (χ = 1 g m−3 km−1, Nd = 100 cm−3). Furthermore, the skill of many of the parameterizations in representing the dependence of αeff on zb is illustrated in Figure 9.
Table 2. Mean and Root Mean Square Differences Between the Effective Albedo Calculated With Various Parameterizations and Integration Techniques and αeff Obtained With Monte Carlo Calculationsa
The statistics include calculations for five cloud base heights (zb = 0.2/0.5/1.0/2.0/3.0 km) and five cloud optical depths (τ = 2.5/5/10/20/40). Baseline values are assumed for the vertical gradient of LWC (χ = 1.0 g m−3 km−1) and the droplet number concentration (Nd = 100 cm−3). See text for the definitions.
 The following approaches are considered: (1) αeff(mix) is the result of the “mixture integration” (equation (14)) using directly the Monte Carlo results for p(r); (2) αΓ(mix) is the new parameterization introduced in section 5.3, using the mixture integration of pΓ(r); (3) αΓ(lin) and (4) αΓ(ICA) are the corresponding results obtained using the linear and ICA integration of pΓ(r); (5) αLi is computed using equation (15); and finally (6) αave is the result of the spatial averaging method.
 Most importantly, Table 2 and Figure 9 demonstrate that the new parameterization αΓ(mix) performs very well. The RMS differences to the true αeff are only 0.008–0.026, depending on the surface albedo pattern, while the mean differences are marginally positive (0.002–0.007). In fact, these mean differences are slightly smaller than those for αeff(mix), and the same is true for the RMS differences in cases 1 and 3. Thus, fortuitously, it appears that the errors related to the gamma function fit (equations (11) and (12)) do not degrade the overall accuracy of the parameterization. Figure 9 also indicates that generally, αΓ(mix) captures the dependence of αeff on zb very nicely.
 As expected, when the gamma function fit pΓ(r) was combined with the linear integration (αΓ(lin)), a slight overall negative bias in effective albedo resulted (Table 2). Conversely, use of the Independent Column Approximation (αΓ(ICA)) led to a slightly larger positive bias. For both of these approaches, the RMS errors are somewhat larger than for the mixture integration.
 For αLi, the overall biases are rather small (0.002–0.020 depending on the surface albedo pattern), but this is somewhat misleading. For all four surface patterns, αLi has a positive bias in case of zb < 1 km, and a negative bias of comparable magnitude for zb > 1 km (Figure 9). Consequently, the RMS errors in Table 2 (0.032–0.098) are three to five times larger than those for αΓ(mix), save for case 4 for which the difference is smaller. Clearly, the αLi biases evidenced in Figure 9 are not just due to the linear integration method, which would generate a slight negative bias at all zb (as demonstrated in Figure 3). Rather, they are mostly caused by the inaccuracy in the probability distribution pLi(r) shown in Figure 8, which gives too much weight to the immediate vicinity of the observation site.
 Finally, for the spatial averaging method (αave), the errors in effective albedo are generally large, reaching −0.36 in the worst case (case 2, zb = 0.5 km in Figure 9). The RMS errors range from 0.05 (in case 3) to 0.23 (in case 2; see Table 2).
6. Implications for the Retrieval of Cloud Optical Depth
 An important application of the effective albedo is in the remote sensing of cloud optical depth. Over bright surfaces, such as snow in the visible region, the increase in the top-of-the-atmosphere reflectance caused by the cloud presence is small compared to the case of a dark surface. This makes the optical depth retrieval more sensitive to errors in reflectance measurements and to surface albedo uncertainty, the latter being potentially large because of the high temporal and spatial variability of the snow albedo. In satellite retrieval of τ, the surface albedo is measured during clear sky and is accounted for on a pixel by pixel basis. Similarly, surface-based retrieval of τ traditionally involves the use of the locally measured surface albedo, which has been proved to be by far the most important variable affecting the inferred τ over snow covered areas [Pinto et al., 1997; Benner et al., 2001].
 Both the satellite and the ground-based retrieval of τ include the implicit assumption that the surface albedo is homogeneous over the whole surface that contributes, via multiple reflections, to the irradiance received by the measurement device. If there are albedo heterogeneities in scales comparable to those in our idealized albedo maps, we expect the surface pattern to have a significant impact on the retrieved τ. To demonstrate this, we consider in Figure 10 the remote sensing of τ based on the broadband downwelling irradiance at the surface. Three parameterizations for effective albedo introduced in section 5 are tested (αΓ(mix), αLi, αave), along with the use of the local albedo at the observation site (αloc = 0.8). For each method x, a retrieved value of cloud optical depth τx was derived using
where the “real” downwelling irradiance on the right-hand side is obtained from the Monte Carlo model for a cloud optical depth of τ, for a heterogeneous surface albedo pattern. The differences between τx an τ thus represent the retrieval errors related to the treatment of surface albedo. Other error sources are ignored.
Figure 10 displays, for the four surface albedo maps, the relative (i.e., fractional) errors in retrieved τ as a function of cloud base height zb and the parameterization method for effective albedo. Almost always, the smallest retrieval errors (generally <10%) are obtained when using αΓ(mix). The relative RMS errors (including calculations for five values of zb and five values of τ) range from 2.2% (in case 3) to 5.8% (in case 4) depending on the surface albedo pattern. The errors in τ progressively increase when using αLi, αave, and αloc. Generally, the largest errors occur in case 2 (with relative RMS errors of 22%, 50% and 98% for αLi, αave, and αloc). However, substantial errors also occur in the cases that more closely approach the typical surface heterogeneity encountered at coastal high-latitude stations (cases 3 and 4). Thus, for example, in case 4, the area-averaged albedo αave utilized in satellite retrievals results in a relative RMS error of 19%, while for the locally measured albedo generally utilized in the ground-based retrievals, the relative RMS error amounts to 74%. Moreover, the magnitude and often even the sign of the errors depends strongly on the cloud base height, although, using αloc always results in significantly overestimated τ. These results agree qualitatively with the error estimates made by Ricchiazzi and Gautier , who demonstrated that, close to the sharp albedo contrast of the Antarctic coast, the error done in retrieving τ from satellite observations when applying the single pixel mean surface albedo can vary from 50% to 200% depending on the cloud base height.
7. Summary and Conclusions
 The motivation of this study was the need of a simple 1-D parameterization that accounts for the 2-D effect of surface albedo heterogeneity over highly reflecting surfaces under overcast conditions. Such a method is needed in single-column radiative transfer calculations and in the interpretation of surface or aircraft-based measurements of downwelling irradiances.
 First, to derive the “true” effective albedo αeff, a backward Monte Carlo model was employed to calculate the downwelling irradiance received at an observation site surrounded by heterogeneous surface, followed by 1-D DISORT calculations to determine the homogeneous albedo that results in the same downwelling irradiance. Such calculations were performed for four idealized surface albedo maps, for various specifications of cloud properties. These tests showed that the cloud base height zb is clearly the most important parameter determining αeff, while changes in cloud optical depth and assumed microphysical properties only play a secondary role.
 Second, to give a theoretical basis for the parameterization of αeff, we calculated with the Monte Carlo model the normalized probability density p(r) of the surface reflections experienced by the photons that reach the observation site, assuming a homogeneous surface albedo. The maximum of this distribution was located within a few kilometers of the observation site, being highest and closest to observation site for the lowest cloud base heights. Also, the role of far-away reflections increased noticeably with decreasing cloud optical depth and increasing surface albedo. These results, theoretically well explainable and qualitatively partly illustrated also by some previous authors [Barker and Davies, 1989; Ricchiazzi and Gautier, 1998; Podgorny and Lubin, 1998; Degünter and Meerkötter, 2000], are here for the first time thoroughly quantified, allowing the search for a fitting parameterization.
 A gamma distribution pΓ(r) was fitted to the Monte Carlo simulated distribution p(r), with coefficients parameterized as a function of cloud base height. To complete the parameterization of effective albedo, pΓ(r) is integrated over the area surrounding the measurement site. In principle, the optimal integration method depends on the scale of surface heterogeneities [Chiu et al., 2004]: simple linear integration is appropriate for very small-scale heterogeneities and the Independent Column Approximation for very large-scale heterogeneities; the latter approach always leading to a higher estimate of αeff than the former. As a pragmatic compromise, we recommend the use of the average of these two. The resulting parameterization αΓ(mix) performed very well, with RMS errors in αeff of less than 0.03 for all four surface patterns considered (including statistics for five cloud base heights and five optical depths). These results are substantially better than those for any of the other approaches we tested. For example, a non-weighted spatial average of the surface albedo (αav) resulted in RMS errors of up to 0.23.
 A straightforward and relevant application of our parameterization lies in the retrieval of cloud optical depth. For a set of idealized remote sensing simulations based on the surface downwelling irradiance, the use of the αΓ(mix) parameterization instead of the true αeff derived from Monte Carlo simulations resulted in relative RMS errors of only 2–6% in the retrieved cloud optical depth, depending on the assumed surface albedo pattern. Use of other treatments for surface albedo degraded the accuracy of the retrieval, with worst-case errors exceeding 100% when applying a single point locally measured albedo (αloc).
 The proposed parameterization of the effective albedo is very simple and requires the knowledge of the surface albedo heterogeneity (albedo map) and cloud base height. This information is usually available for manned ground-based measurement stations. The surface albedo map can also be obtained from clear-sky satellite observations, but in general, cloud base height cannot be estimated from satellites with reasonable accuracy. However, even in the case of satellite remote sensing our method could be used to give the range of the possible surface effective albedo, which in turn could be utilized to estimate the uncertainty range of the retrieved cloud optical properties. Because of its simplicity, we also think that this parameterization can be directly implemented into high-resolution numerical weather prediction models that use broadband irradiances.
 While the method described in the present study appears promising, a couple of areas for further improvement can be pointed out. First, the method only considers the broadband effective albedo and as such does not account for differences in the horizontal distance traveled by photons of different wavelengths. Yet it has been shown that at the near-infrared wavelengths the horizontal path length of photons and, consequently, the extension of the surface area relevant for calculating the effective albedo is much shorter than at the visible wavelengths [Ricchiazzi and Gautier, 1998; Podgorny and Lubin, 1998]. We plan to address the spectral dependence of the effective albedo in a continuation study, in order to extend the applicability of the proposed method also to those 1-D radiative transfer models that include a spectral treatment of the irradiance. In that case the method could also be used to create look-up tables of surface effective albedo under overcast conditions for various cloud base heights, in order to improve the satellite retrieval of cloud optical properties.
 Second, all our calculations are based on the assumption of a plane-parallel horizontally homogeneous cloud layer. Deviations from this idealization could cause substantial errors in the estimated downwelling irradiance and in the inferred cloud optical depth [Boers et al., 2000; Varnai and Marshak, 2001], even when the surface is homogeneous. In Polar Regions, however, clouds tend to conform to the plane-parallel assumption better than at lower latitudes, as the prevalent cloud types are stratiform and of limited vertical extension [Lubin and Frederick, 1991; Dong and Mace, 2003]. Moreover, overcast conditions are by far more common than partly covered skies [Intrieri et al., 2002; Fitzpatrick and Warren, 2005]. Thus the impact of 3D cloud heterogeneity on radiative transfer in Polar Regions is typically less important than the effect of sharp contrasts in the 2-D distribution of surface albedo. Nevertheless, the combined effects of surface and cloud heterogeneity could be worth studying (e.g., how does cloud heterogeneity influence the spatial distribution of surface reflections).
 As most of the Arctic and Antarctic measurement stations are located in the vicinity of the coast or on ice shelves, where the contrast between snow and open water is prevalent during spring and summer, our proposed method should have a potentially wide applicability in Polar Regions. Moreover, large albedo contrasts at scales of a few kilometers, as in the cases illustrated here, can be observed at middle latitudes as well, for example during winter and spring in areas with snow covered fields, swamps or lakes adjacent to conifer forests with no snow on the trees.
 This work was supported by the Academy of Finland (project 210794) and by the DAMOCLES project, which is financed by the European Commission in the 6th Framework Programme for Research and Development. Timo Vihma and Hannu Savijärvi are acknowledged for their support and comments on the manuscript. The comments of the anonymous reviewers led to a great improvement of the manuscript.