Abstract

A simple yet accurate parameterization of spectral and broadband ocean surface albedo has been developed. To facilitate the parameterization and its applications, the albedo is parameterized for the direct and diffuse incident radiation separately, and then each of them is further divided into two components: the contributions from surface and water, respectively. The four albedo components are independent of each other, hence, altering one will not affect the others. Such a designed parameterization scheme is flexible for any future update. Users can simply replace any of the adopted empirical formulations (e.g., the relationship between foam reflectance and wind speed) as desired without a need to change the parameterization scheme. The parameterization is validated by in situ measurements and can be easily implemented into a climate or radiative transfer model.

© 2011 OSA

1. Introduction

Over 70% of the Earth surface is covered by water. The Ocean surface albedo (OSA), defined as the ratio of reflected radiation from the ocean surface to incident radiation upon it, is a key parameter to calculate the atmospheric radiation budget and the solar heating in the upper ocean. Different from most land surfaces, the volume scattering of waters below the ocean surface also contributes to the OSA. In situ measurements have clearly shown that the OSA varies significantly with solar elevation, spectral band, wind speed, atmospheric conditions, and ocean optical properties [1]. However, most OSA parameterizations generally ignore the spectral dependence or ocean optics [24]. On the other hand, climate models have been evolved to include more biological processes and more accurate radiative energy computation is required to understand the atmosphere-ocean interactions. All these studies require an accurate parameterization of spectral and broadband OSA.

Based on years of observation data and the coupled ocean atmosphere radiative transfer (COART) model [1], Jin et al. [5] developed an OSA parameterization through the look up table approach. Using the albedo look up table, users can obtain the OSA at given spectral band, solar zenith angle, wind speed, and ocean chlorophyll concentration [5]. However, this table format is not very convenient and may not be computationally efficient enough for some applications. For example, climate modeling desires a simpler and faster parameterization in its intensive computations and some radiative transfer calculations require the albedo input separated into direct and diffuse components. To satisfy these new requirements, this study presents a simplified spectral and broadband OSA parameterization.

2. Parameterization

The OSA depends on a number of parameters, which include several atmospheric and oceanic properties, solar zenith angle (SZA), ocean surface roughness (wind speed) and wavelength. Parameterizing OSA as a single function of all these dependents directly would be formidable. However, the total OSA can be partitioned into several independent components that depend on different parameters. Each albedo component can be parameterized separately with its own dependents to simplify the parameterization process. Therefore, we separate the OSA into the direct (αdir) and diffuse (αdif) contributions first; each of them is then further divided into two components: the surface reflection and the ocean volume scattering (as illustrated in Fig. 1 ). These four different albedo components (αdirs,αdirw,αdifs,αdifw) are formulated in the following sub-sections, respectively.

 

Fig. 1 The four components of ocean surface albedo (OSA).

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2.1 Surface direct

The OSA component corresponding to the surface Fresnel reflection of the direct solar incidence (αdirs) is resulted from the refractive index change at the air-water interface. This albedo component depends on incident angle (θ), wind speed (w) and the relative refractive index of water and air (n). The spectral wavelength (λ) dependence of αdirs is taken into account implicitly through the relationship of n and λ. Given the surface slope distribution or the wind speed, αdirscan be calculated explicitly [1]. Figure 2 shows an example of αdirs as a function of incident angle for two refractive indices (1.34 and 1.20) and three wind speeds (0, 3 and 12 m/s).

 

Fig. 2 The calculated direct surface albedo, αdirs, due to the Fresnel reflection for two refractive indices (n) and three wind speeds.

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As shown in Fig. 2, this direct Fresnel surface albedo has a slight increase as wind increases at small to moderate incident angles but it decreases as wind at large incident angles. This is due to the two different and competitive impacts of the water surface roughness: the multiple scattering, which enhances surface reflection, and the shading effect, which damps reflection, among the surface wave facets [1]. Through the multiple regression over the albedo data set from explicit COART computations, αdirs can be expressed as

αdirs(λ,θ,w)=αdirs(n(λ),μ(θ),σ(w))=rf(n,μ)rf(n,μ)rf(n0,μ)f(μ,σ),
where μ = cos(θ) and θ is the incident angle, equivalent to the solar zenith angle here. σ is the mean slope distribution width of the Gaussian function defining the surface roughness and is related to the wind speed (w). If the Cox-Munk model [6] is used, this relationship is
σ2=0.003+0.00512w.
rf(n,μ) represents the Fresnel reflectance. It can be computed analytically and the formulation can be found in any textbook of optics [7]. n0 = 1.34 is approximately the refractive index of water in the visible spectrum. When surface is flat (i.e., σ = 0), is reduced to rf(n,μ), that is

αdirs(n,μ,O)=rf(n,μ).

The f(μ,σ) in Eq. (1) is the regression function and it reduces to 0 when w=0.

f(μ,σ)=(p0+p1μ+p2μ2+p3μ3+p4σ+p5μσ)×exp(p6+p7μ+p8μ2+p9σ+p10μσ),
where the fitting coefficients pi(i = 0,1,2, ..., 10) = (0.0152, -1.7873, 6.8972, -8.5778, 4.071, -7.6446, 0.1643, −7.8409, −3.5639, −2.3588, 10.0538). Detailed description of the multiple regression technique can be found in a number of books, for example, those by Freedman [8] and by Robert [9].

Note, we used the surface roughness parameter (σ) instead of the wind speed (w) in the parameterization of Eq. (1). This is to facilitate any possible update or replacement of the function σ(w) as needed by users. The surface slope distribution describing the ocean surface roughness is commonly considered as a Gaussian function. However, the formulation to relate the distribution width (σ) to wind speed (w) varies and Eq. (2) is one of those. The parameterization formulated as Eq. (1) allows the users to choose this relationship without a need to change the parameterization.

A comparison between the parameterized direct surface albedo using Eq. (1) and the explicit model calculations using COART are presented in Fig. 3 . The upper panel shows the exact albedo for wind range from 0 to 24 m/s for all incident angles. The middle panel shows the parameterization. The x and y axes represent μ and w, respectively. The lower panel shows the relative error of the parameterization in percentage projected on the μ-w plane. This error is generally less than 3% of the exact value.

 

Fig. 3 Comparison of the parameterization by Eq. (1) and the exact calculations for the surface direct albedo, αdirs, The lower panel shows the relative error (in percentage) of the parameterization.

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2.2 Surface diffuse

In principle, given the angular distribution of diffuse incidence, the diffuse albedo component from surface reflection (αdifs) can be calculated as an integral of the direct surface albedo (αdirs) expressed by Eq. (1). In reality, however, the downwelling radiance distribution is usually unknown, though it can be obtained through an atmospheric radiative transfer model or from measurements. Because this distribution varies with atmospheric components, it is not practical to make a parameterization for all atmospheric conditions. Here we provide the parameterizations for two popular cases, the isotropic incidence and overcast condition, for situations when the distribution is not known. Under clear skies, the nearly isotropic Rayleigh scattering makes the downwelling diffuse radiance distribution nearly isotropic. When the scattering is weak, the diffuse incident energy is small and the direct component becomes dominant. These negate the effect of the isotropic assumption. The αdifs for isotropic incidence can be parameterized as

αdifs(λ,w)=αdifs(n(λ),σ(w))=0.14820.012σ+0.1608n0.0244nσ.

Figure 4 compares the parameterized diffuse surface albedo by Eq. (5a) with the explicit model calculations. The relative error of this parameterization (lower panel) is less than 2% for refractive index from 1.20 to 1.45 (approximately the variation range of water refractive index in the solar spectrum) and for wind speed from 0 to 24 (m/s).

 

Fig. 4 Comparison of the parameterization by Eq. (5a) and the explicit calculations for the surface diffuse albedo, αdifs, under clear skies.

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Under cloudy skies, the downwelling radiance distribution varies as cloud properties. However, as cloud optical depth increases, this radiance distribution will reach an asymptotic shape, regardless of solar zenith angle and azimuth angle. This asymptotic distribution depends only on the single scattering albedo and the azimuthally averaged phase function of cloud [10]. Figure 5 shows the downwelling radiance distribution under a water cloud (normalized to nadir view). The solar zenith angle of 30 degree and wavelength of 0.55 micrometer are used in this example. The upper panel in Fig. 5 is for the solar principal plane and the lower panel is for the perpendicular plane. The radiance distribution in the principal plane reaches the asymptotic shape slower than other azimuth planes due to the effect of direct solar incidence in this plane. Since clouds have high single scattering albedo and similar scattering characteristics (strong forward scattering) over most of the solar spectrum, the diffuse radiance distribution in Fig. 5 can be used for overcast conditions. Using this asymptotic radiance distribution, we can parameterize the diffuse surface albedo under cloudy skies as

 

Fig. 5 Downwelling diffuse radiance distribution under clouds with different optical depths (different lines). SZA = 30.

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αdifs(λ,w)=αdifs(n(λ),σ(w))=0.1479+0.1502n0.016nσ. (5b)

Figure 6 shows the comparison of the parameterized diffuse surface albedo by Eq. (5b) with the explicit model calculations. The relative parameterization error (lower panel) is less than 0.6% of the exact value for this albedo component.

 

Fig. 6 Comparison of the parameterization by Eq. (5b) and the explicit calculations for the surface diffuse albedo, αdifs, under cloudy skies.

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For the same reason as for the direct, the surface roughness parameter (σ) instead of the wind speed (w) is used in the parameterizations of (5a) and (5b).

2.3 Ocean volume direct

For the albedo component contributed by the volume scattering of water below the surface, we consider the so-called case 1 waters which consist 99% of the ocean and the variation of optical properties in case 1 water is associated with the chlorophyll concentration (i.e., Chl) [11]. Coastal waters are usually case 2 water and their constituent and optical properties are more complex. However, the contribution from water volume scattering is limited in the visible spectrum and is usually smaller than the contribution by the Fresnel surface reflection described above, especially for large incident angles. The water volume albedo for direct incidence, αdirw, can be expressed as

αdirw(λ,μ,w,chl)=R0(1rw)(1αdirw)1rwR0,
where rw represents the water-to-water reflectance at the air-water interface for upwelling diffuse incidence from water below, which is usually considered as a constant of 0.48 in visible spectrum, but it actually varies slightly with surface roughness [1]. Using COART model, we can fit this term as

rw=0.48170.0149σ0.207σ2.

R0 in Eq. (6) is the irradiance reflectance just below the surface. This is a classic apparent optical property (AOP) of ocean optics and has been studied extensively [11,12]. It is proportional to the backscattering coefficient, bb, inversely proportional to the absorption, a, and can be expressed as

R0(λ,μ,chl)=β(μ)bb(λ,chl)a(λ,chl),
where β is the proportionality constant and is expressed as [1]
β=0.62790.2227ηb0.0513ηb2+(0.2465ηb0.3119)μ,
where ηb is the ratio of backscattering by water molecules to total backscattering. bb and a depend on Chl and their formulations are available from Morel and Maritorena [11] (see their Eqs. (13) and (16)).

2.4 Ocean volume diffuse

For diffuse incidence, the ocean volume albedo can be simply represented by the direct ocean volume albedo at an effective incident direction, μe. Based on Morel and Gentili [12], μe = 0.676. Therefore, the ocean volume diffuse albedo, αdifw, is

αdifw(λ,w,chl)=αdifw(λ,μc,w,chl).

Figure 7 shows some examples of the water volume albedo calculated by the Eqs. (6) and (10). Each line in a panel is for a different chlorophyll concentration (Chl), which is in mg/m3 and is represented by the number on the left of each line. It is noted that this albedo component decreases as Chl increases in the blue, but increases as Chl increases in the green. Consequently, the combined effect of Chl on the broadband albedo is small.

 

Fig. 7 Examples of the parameterized albedo due to the water volume scattering. The upper two panels are based on Eq. (6) for the direct component and the lower panel is from Eq. (10) for the diffuse.

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2.5. Total spectral albedo

Having the four components of the surface albedo given above, we can now obtain the parameterized direct, diffuse, and total spectral surface albedo as

αdir=αdirs+αdirw
αdif=αdifs+αdifw
α=fdirαdir+fdifαdif
fdir+fdif=1.0
Where fdir and fdif represent the direct and diffuse fractions of the incident flux at surface in the specified spectrum, respectively. Note, all the variables in Eqs. (11)-(14) are wavelength dependent. The fdir and fdif can be obtained from a separate parameterization, a radiative transfer simulation, or directly from measurements as in the examples shown in the following section 3. Because fdir and fdif are complementary (see Eq. (14), only one is required.

2.6 Broadband albedo

Major portion of the solar incidence at surface is within the visible spectrum with maximum at around 500 nm. For broadband solar radiation, the direct and diffuse albedo components from surface reflection can be represented by the spectral parameterizations of Eqs. (1) and (5), respectively, with n = n0 = 1.34 (i.e., the refractive index in visible spectrum). The ocean volume component for broadband albedo is small and is approximately 0.006 for Case 1 waters. Therefore, the broadband albedo can be written as

αb=fdirαdirs(n0,σ)+fdifαdifs(n0,σ)+0.006,
where fdir and fdif are the direct and diffuse fractions of the incident broadband solar radiation at surface.

2.7 Foam adjustment

In addition to roughening the surface, wind can also lead to the formation of foams or whitecaps. The parameterizations above have not considered the effect of whitecaps, which could be significant at high wind speeds. However, measured foam reflectances differ greatly with each other and have large uncertainties [1416]. Here we only provide a flexible scheme of foam correction so that users can replace it easily when desired measurement data are available. In the example here, we use the simple foam correction proposed by Koepke [14], which assumes a constant foam reflectance, αwc = 0.55, and relates the fractional surface coverage of white-caps, fwc, to the wind speed (w) as

fwc=2.95×106w3.52.

The albedo after the foam correction is simply the area averaged foam albedo and the parameterized albedo above as

αc=fwcαwc+(1fwc)α.

3. Validation

Ultimately, parameterization accuracy has to be evaluated by comparison against observations. In this section, we compare the parameterized spectral and broadband albedo with in situ measurements at the CERES Ocean Validation Experiment (COVE) site, which is 25km from the Virginia Beach in the Atlantic ocean [17]. The downwelling broadband shortwave measurements include the direct and diffuse fluxes by a pyrheliometer and a pyranometer, respectively. The narrowband global and diffuse measurements are made using a multifilter rotating shadowband radiometer (MFRSR) in six spectral bands. Except for the spectral and broadband flux measurements, co-incident measurements for aerosol, precipitable water, and wind speed are also available at COVE. These observation data have been used extensively for the validation of COART model used here [1,18]. Figure 8 is an example to compare the measured and parameterized spectral albedo in four clear days. The wind speed, measured by NOAA meteorology station at COVE, varied greatly in the selected four days (shown in the middle panels). In this example, the measurements in two MFRSR channels (614 nm and 865 nm) are used. The chlorophyll concentration from SeaWiFS satellite retrieval is used for the parameterization input, however, it has little impact on the parameterized albedo here because of the small water volume reflectance in the selected two channels. The direct flux fraction (lower panels) for the parameterization is derived from the diffuse and global MSFSR measurements of the downwelling irradiance. As expected, this energy fraction in the 614 nm band is always lower than that in the 865 nm band because of the stronger scattering at shorter wavelength. Figure 8 shows that the albedo at low sun and its maximum decrease significantly as wind speed increases from day 1 to day 4. The parameterization correctly captures the albedo variation as the wind, the solar elevation, and the direct/diffuse ratio of incident flux.

 

Fig. 8 Comparison of measured and parameterized albedo in two spectral bands (614 nm and 865 nm) in four clear days. The wind speed (middle panels) and the fraction of direct incidence at surface (lower panels) for parameterization input are also from measurements at the same site as for albedo.

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Similar to Fig. 8, Fig. 9 shows the comparison for four cloudy days with different winds and with varying direct fraction of incidence. It shows that the albedo variation with wind under cloudy skies is not as significant as that under clear skies shown in Fig. 8, but its correlation with the direct fraction of incident flux (lower panel) is more obvious, especially under low sun condition where the direct albedo is much different (higher) from the diffuse albedo. Same as in the clear days, the fraction of direct incidence in the 614 nm channel is constantly lower than that in the 865 nm channel, but different from the clear skies, this fraction varies wildly as time because of the constant variation of clouds. For visual clarity, only the energy fraction for 865 nm band is shown in Fig. 9. Though the clouds and the associated direct energy fraction vary greatly, the parameterization still agrees with the measurements well. The somewhat large difference between parameterization and measurements in the late afternoon (low sun) is due to the higher measurement uncertainty in the low energy condition near sunset, which affects the accuracy of both measured and parameterized albedo.

 

Fig. 9 Same as Fig. 8, but for comparison in four cloudy days. The direct flux fraction (lower panels) is from measurements at 865 nm.

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Figure 10 compares the measured and parameterized broadband shortwave albedo for two years (2000-2001) of data (15 minute averaged) under all skies. As for the Figs. 8-9, the wind speed is also from the NOAA meteorology station at COVE. The direct fraction of incident flux is derived from the direct and diffuse downwelling fluxes measured by the pyrheliometer and pyranometer, respectively. Larger discrepancy in low sun conditions near sunset occurred for the broadband albedo too.

 

Fig. 10 Comparison of measured and parameterized broadband albedo under all skies.

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4. Conclusion

A parameterization for the spectral and broadband, direct and diffuse ocean surface albedo has been developed. To simplify the parameterization, the albedo (spectral and broadband) is divided into four different components: surface direct, surface diffuse, water volume direct, and water volume diffuse. Each component is parameterized separately as a function of its own dependent parameters. Since the total albedo is assembled from the four independent components, a change or update of any component will not affect the others.

The parameterization scheme is designed to be flexible in order to facilitate any desired update on the empirical formulations of dependent parameters, specifically, the surface roughness dependence on wind speed, the ocean optical property dependence on chlorophyll concentration, and the foam reflectance dependence on wind speed. There is no need to modify the parameterization scheme when new measurement data become available for the improvement of any of those dependence relationships in the future.

Comparisons with in situ measurements show that the parameterization is accurate and that it correctly captures the albedo variations with wind, solar zenith, direct/diffuse fraction of incident flux, and wavelength. This parameterization provides a simple and fast way to obtain the spectral and broadband ocean surface albedo at various atmospheric, surface and oceanic conditions. It can be readily implemented into a climate or radiative transfer model.

The basic data set required for the parameterization (e.g., the spectral refractive indices and the absorption and scattering coefficients of sea water) and the module for the parameterization are available from the authors and online (Media 1, Media 2, Media 3, and Media 4).

Acknowledgments

We thank the CERES COVE team at NASA Langley Center for the observation data. Particularly, Ken Rutledge kindly provided us two years of carefully calibrated data of spectral and broadband irradiances. Mr. Jun Wu helped to prepare some of the figures. We also thank the three anonymous reviewers for their thoughtful comments. The initiation of this study was supported by NASA’s CERES project. It was continued and finished at the Hefei Institutes of Physical Science at the Chinese Academy of Sciences (CASHIPS) and supported by the director foundation to the Key Laboratory of Optical Calibration and Characterization at CASHIPS.

References and links

1. Z. Jin, T. P. Charlock, K. Rutledge, K. Stamnes, and Y. Wang, “Analytical solution of radiative transfer in the coupled atmosphere-ocean system with a rough surface,” Appl. Opt. 45(28), 7443–7455 (2006). [CrossRef]   [PubMed]  

2. B. P. Briegleb, P. Minnis, V. Ramanathan, and E. Harrison, “Comparison of regional clear-sky albedos inferred from satellite observations and model computations,” J. Clim. Appl. Meteorol. 25(2), 214–226 (1986). [CrossRef]  

3. J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983). [CrossRef]  

4. J. P. Taylor, J. M. Edwards, M. D. Glew, P. Hignett, and A. Slingo, “Studies with a flexible new radiation code. II: Comparisons with aircraft shortwave observations,” Q. J. R. Metro, Soc. 122, 839–861 (1996).

5. Z. Jin, T. Charlock, W. Smith Jr, and K. Rutledge, “A parameterization of ocean surface albedo.” Geophys. Res. Let. 31, L22301, doi:. [CrossRef]  

6. C. Cox and W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44(11), 838–850 (1954). [CrossRef]  

7. E. Hecht, Optics, Second Edition (Addison Wesley, 1990).

8. D. A. Freedman, Statistical Models: Theory and Practice (Cambridge University Press, 2005).

9. M. G. Robert, Second-Semester Applied Statistics (Kendall/Hunt Publishing Company, 2004).

10. H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic Press, 1980).

11. A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: a reappraisal,” J. Geophys. Res. 106(C4), 7163–7180 (2001). [CrossRef]  

12. J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29(2), 350–356 (1984). [CrossRef]  

13. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30(30), 4427–4438 (1991). [CrossRef]   [PubMed]  

14. P. Koepke, “Effective reflectance of oceanic whitecaps,” Appl. Opt. 23(11), 1816–1824 (1984). [CrossRef]   [PubMed]  

15. R. Frouin, M. Schwindling, and P.-Y. Deschamps, “Spectral reflectance of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications,” J. Geophys. Res. 101(C6), 14,361–14,371 (1996), doi:. [CrossRef]  

16. K. D. Moore, K. J. Voss, and H. R. Gordon, “Spectral reflectance of whitecaps: Their contribution to water-leaving radiance,” J. Geophys. Res. 105(C3), 6493–6499 (2000), doi:. [CrossRef]  

17. C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith Jr, B. E. Fabbri, J. J. Madigan Jr, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006). [CrossRef]  

18. Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

References

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  1. Z. Jin, T. P. Charlock, K. Rutledge, K. Stamnes, and Y. Wang, “Analytical solution of radiative transfer in the coupled atmosphere-ocean system with a rough surface,” Appl. Opt. 45(28), 7443–7455 (2006).
    [Crossref] [PubMed]
  2. B. P. Briegleb, P. Minnis, V. Ramanathan, and E. Harrison, “Comparison of regional clear-sky albedos inferred from satellite observations and model computations,” J. Clim. Appl. Meteorol. 25(2), 214–226 (1986).
    [Crossref]
  3. J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
    [Crossref]
  4. J. P. Taylor, J. M. Edwards, M. D. Glew, P. Hignett, and A. Slingo, “Studies with a flexible new radiation code. II: Comparisons with aircraft shortwave observations,” Q. J. R. Metro, Soc. 122, 839–861 (1996).
  5. Z. Jin, T. Charlock, W. Smith, and K. Rutledge, “A parameterization of ocean surface albedo.” Geophys. Res. Let. 31, L22301, doi:.
    [Crossref]
  6. C. Cox and W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44(11), 838–850 (1954).
    [Crossref]
  7. E. Hecht, Optics, Second Edition (Addison Wesley, 1990).
  8. D. A. Freedman, Statistical Models: Theory and Practice (Cambridge University Press, 2005).
  9. M. G. Robert, Second-Semester Applied Statistics (Kendall/Hunt Publishing Company, 2004).
  10. H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic Press, 1980).
  11. A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: a reappraisal,” J. Geophys. Res. 106(C4), 7163–7180 (2001).
    [Crossref]
  12. J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29(2), 350–356 (1984).
    [Crossref]
  13. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30(30), 4427–4438 (1991).
    [Crossref] [PubMed]
  14. P. Koepke, “Effective reflectance of oceanic whitecaps,” Appl. Opt. 23(11), 1816–1824 (1984).
    [Crossref] [PubMed]
  15. R. Frouin, M. Schwindling, and P.-Y. Deschamps, “Spectral reflectance of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications,” J. Geophys. Res. 101(C6), 14,361–14,371 (1996), doi:.
    [Crossref]
  16. K. D. Moore, K. J. Voss, and H. R. Gordon, “Spectral reflectance of whitecaps: Their contribution to water-leaving radiance,” J. Geophys. Res. 105(C3), 6493–6499 (2000), doi:.
    [Crossref]
  17. C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
    [Crossref]
  18. Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

2006 (2)

Z. Jin, T. P. Charlock, K. Rutledge, K. Stamnes, and Y. Wang, “Analytical solution of radiative transfer in the coupled atmosphere-ocean system with a rough surface,” Appl. Opt. 45(28), 7443–7455 (2006).
[Crossref] [PubMed]

C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
[Crossref]

2005 (1)

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

2001 (1)

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: a reappraisal,” J. Geophys. Res. 106(C4), 7163–7180 (2001).
[Crossref]

2000 (1)

K. D. Moore, K. J. Voss, and H. R. Gordon, “Spectral reflectance of whitecaps: Their contribution to water-leaving radiance,” J. Geophys. Res. 105(C3), 6493–6499 (2000), doi:.
[Crossref]

1996 (2)

R. Frouin, M. Schwindling, and P.-Y. Deschamps, “Spectral reflectance of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications,” J. Geophys. Res. 101(C6), 14,361–14,371 (1996), doi:.
[Crossref]

J. P. Taylor, J. M. Edwards, M. D. Glew, P. Hignett, and A. Slingo, “Studies with a flexible new radiation code. II: Comparisons with aircraft shortwave observations,” Q. J. R. Metro, Soc. 122, 839–861 (1996).

1991 (1)

1986 (1)

B. P. Briegleb, P. Minnis, V. Ramanathan, and E. Harrison, “Comparison of regional clear-sky albedos inferred from satellite observations and model computations,” J. Clim. Appl. Meteorol. 25(2), 214–226 (1986).
[Crossref]

1984 (2)

P. Koepke, “Effective reflectance of oceanic whitecaps,” Appl. Opt. 23(11), 1816–1824 (1984).
[Crossref] [PubMed]

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29(2), 350–356 (1984).
[Crossref]

1983 (1)

J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
[Crossref]

1954 (1)

Briegleb, B. P.

B. P. Briegleb, P. Minnis, V. Ramanathan, and E. Harrison, “Comparison of regional clear-sky albedos inferred from satellite observations and model computations,” J. Clim. Appl. Meteorol. 25(2), 214–226 (1986).
[Crossref]

Charlock, T.

Z. Jin, T. Charlock, W. Smith, and K. Rutledge, “A parameterization of ocean surface albedo.” Geophys. Res. Let. 31, L22301, doi:.
[Crossref]

Charlock, T. P.

Z. Jin, T. P. Charlock, K. Rutledge, K. Stamnes, and Y. Wang, “Analytical solution of radiative transfer in the coupled atmosphere-ocean system with a rough surface,” Appl. Opt. 45(28), 7443–7455 (2006).
[Crossref] [PubMed]

C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
[Crossref]

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

Cota, G.

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

Cox, C.

Denn, F. M.

C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
[Crossref]

Deschamps, P.-Y.

R. Frouin, M. Schwindling, and P.-Y. Deschamps, “Spectral reflectance of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications,” J. Geophys. Res. 101(C6), 14,361–14,371 (1996), doi:.
[Crossref]

Edwards, J. M.

J. P. Taylor, J. M. Edwards, M. D. Glew, P. Hignett, and A. Slingo, “Studies with a flexible new radiation code. II: Comparisons with aircraft shortwave observations,” Q. J. R. Metro, Soc. 122, 839–861 (1996).

Fabbri, B. E.

C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
[Crossref]

Frouin, R.

R. Frouin, M. Schwindling, and P.-Y. Deschamps, “Spectral reflectance of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications,” J. Geophys. Res. 101(C6), 14,361–14,371 (1996), doi:.
[Crossref]

Gentili, B.

Glew, M. D.

J. P. Taylor, J. M. Edwards, M. D. Glew, P. Hignett, and A. Slingo, “Studies with a flexible new radiation code. II: Comparisons with aircraft shortwave observations,” Q. J. R. Metro, Soc. 122, 839–861 (1996).

Gordon, H. R.

K. D. Moore, K. J. Voss, and H. R. Gordon, “Spectral reflectance of whitecaps: Their contribution to water-leaving radiance,” J. Geophys. Res. 105(C3), 6493–6499 (2000), doi:.
[Crossref]

Hansen, J. G.

J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
[Crossref]

Harrison, E.

B. P. Briegleb, P. Minnis, V. Ramanathan, and E. Harrison, “Comparison of regional clear-sky albedos inferred from satellite observations and model computations,” J. Clim. Appl. Meteorol. 25(2), 214–226 (1986).
[Crossref]

Hignett, P.

J. P. Taylor, J. M. Edwards, M. D. Glew, P. Hignett, and A. Slingo, “Studies with a flexible new radiation code. II: Comparisons with aircraft shortwave observations,” Q. J. R. Metro, Soc. 122, 839–861 (1996).

Jin, Z.

Z. Jin, T. P. Charlock, K. Rutledge, K. Stamnes, and Y. Wang, “Analytical solution of radiative transfer in the coupled atmosphere-ocean system with a rough surface,” Appl. Opt. 45(28), 7443–7455 (2006).
[Crossref] [PubMed]

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

Z. Jin, T. Charlock, W. Smith, and K. Rutledge, “A parameterization of ocean surface albedo.” Geophys. Res. Let. 31, L22301, doi:.
[Crossref]

Kahn, R.

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

Kirk, J. T. O.

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29(2), 350–356 (1984).
[Crossref]

Knapp, R. J.

C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
[Crossref]

Koepke, P.

Lacis, A.

J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
[Crossref]

Lebedeff, S.

J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
[Crossref]

Madigan, J. J.

C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
[Crossref]

Maritorena, S.

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: a reappraisal,” J. Geophys. Res. 106(C4), 7163–7180 (2001).
[Crossref]

Minnis, P.

B. P. Briegleb, P. Minnis, V. Ramanathan, and E. Harrison, “Comparison of regional clear-sky albedos inferred from satellite observations and model computations,” J. Clim. Appl. Meteorol. 25(2), 214–226 (1986).
[Crossref]

Moore, K. D.

K. D. Moore, K. J. Voss, and H. R. Gordon, “Spectral reflectance of whitecaps: Their contribution to water-leaving radiance,” J. Geophys. Res. 105(C3), 6493–6499 (2000), doi:.
[Crossref]

Morel, A.

Munk, W.

Ramanathan, V.

B. P. Briegleb, P. Minnis, V. Ramanathan, and E. Harrison, “Comparison of regional clear-sky albedos inferred from satellite observations and model computations,” J. Clim. Appl. Meteorol. 25(2), 214–226 (1986).
[Crossref]

Redemann, J.

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

Rind, D.

J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
[Crossref]

Rose, F.

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

Ruedy, R.

J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
[Crossref]

Russell, D.

J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
[Crossref]

Rutan, D. A.

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

Rutledge, C. K.

C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
[Crossref]

Rutledge, K.

Z. Jin, T. P. Charlock, K. Rutledge, K. Stamnes, and Y. Wang, “Analytical solution of radiative transfer in the coupled atmosphere-ocean system with a rough surface,” Appl. Opt. 45(28), 7443–7455 (2006).
[Crossref] [PubMed]

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

Z. Jin, T. Charlock, W. Smith, and K. Rutledge, “A parameterization of ocean surface albedo.” Geophys. Res. Let. 31, L22301, doi:.
[Crossref]

Schuster, G. L.

C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
[Crossref]

Schwindling, M.

R. Frouin, M. Schwindling, and P.-Y. Deschamps, “Spectral reflectance of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications,” J. Geophys. Res. 101(C6), 14,361–14,371 (1996), doi:.
[Crossref]

Slingo, A.

J. P. Taylor, J. M. Edwards, M. D. Glew, P. Hignett, and A. Slingo, “Studies with a flexible new radiation code. II: Comparisons with aircraft shortwave observations,” Q. J. R. Metro, Soc. 122, 839–861 (1996).

Smith, W.

Z. Jin, T. Charlock, W. Smith, and K. Rutledge, “A parameterization of ocean surface albedo.” Geophys. Res. Let. 31, L22301, doi:.
[Crossref]

Smith, W. L.

C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
[Crossref]

Stamnes, K.

Stone, P.

J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
[Crossref]

Taylor, J. P.

J. P. Taylor, J. M. Edwards, M. D. Glew, P. Hignett, and A. Slingo, “Studies with a flexible new radiation code. II: Comparisons with aircraft shortwave observations,” Q. J. R. Metro, Soc. 122, 839–861 (1996).

Travis, L.

J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
[Crossref]

Voss, K. J.

K. D. Moore, K. J. Voss, and H. R. Gordon, “Spectral reflectance of whitecaps: Their contribution to water-leaving radiance,” J. Geophys. Res. 105(C3), 6493–6499 (2000), doi:.
[Crossref]

Wang, Y.

Zhang, T.

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

Appl. Opt. (3)

Bull. Am. Meteorol. Soc. (1)

C. K. Rutledge, G. L. Schuster, T. P. Charlock, F. M. Denn, W. L. Smith, B. E. Fabbri, J. J. Madigan, and R. J. Knapp, “Offshore radiation observations for climate research at the CERES Ocean Validation Experiment,” Bull. Am. Meteorol. Soc. 87(9), 1211–1222 (2006).
[Crossref]

J. Atmos. Sci. (1)

Z. Jin, T. P. Charlock, K. Rutledge, G. Cota, R. Kahn, J. Redemann, T. Zhang, D. A. Rutan, and F. Rose, “Radiative Transfer Modeling for the CLAMS Experiment,” J. Atmos. Sci. 62, 1052–1070 (2005).

J. Clim. Appl. Meteorol. (1)

B. P. Briegleb, P. Minnis, V. Ramanathan, and E. Harrison, “Comparison of regional clear-sky albedos inferred from satellite observations and model computations,” J. Clim. Appl. Meteorol. 25(2), 214–226 (1986).
[Crossref]

J. Geophys. Res. (3)

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: a reappraisal,” J. Geophys. Res. 106(C4), 7163–7180 (2001).
[Crossref]

R. Frouin, M. Schwindling, and P.-Y. Deschamps, “Spectral reflectance of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications,” J. Geophys. Res. 101(C6), 14,361–14,371 (1996), doi:.
[Crossref]

K. D. Moore, K. J. Voss, and H. R. Gordon, “Spectral reflectance of whitecaps: Their contribution to water-leaving radiance,” J. Geophys. Res. 105(C3), 6493–6499 (2000), doi:.
[Crossref]

J. Opt. Soc. Am. (1)

Limnol. Oceanogr. (1)

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29(2), 350–356 (1984).
[Crossref]

Mon. Weather Rev. (1)

J. G. Hansen, D. Russell, D. Rind, P. Stone, A. Lacis, S. Lebedeff, R. Ruedy, and L. Travis, “Efficient three-dimensional global models for climate studies: Models I and II,” Mon. Weather Rev. 111(4), 609–662 (1983).
[Crossref]

Q. J. R. Metro, Soc. (1)

J. P. Taylor, J. M. Edwards, M. D. Glew, P. Hignett, and A. Slingo, “Studies with a flexible new radiation code. II: Comparisons with aircraft shortwave observations,” Q. J. R. Metro, Soc. 122, 839–861 (1996).

Other (5)

Z. Jin, T. Charlock, W. Smith, and K. Rutledge, “A parameterization of ocean surface albedo.” Geophys. Res. Let. 31, L22301, doi:.
[Crossref]

E. Hecht, Optics, Second Edition (Addison Wesley, 1990).

D. A. Freedman, Statistical Models: Theory and Practice (Cambridge University Press, 2005).

M. G. Robert, Second-Semester Applied Statistics (Kendall/Hunt Publishing Company, 2004).

H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic Press, 1980).

Supplementary Material (4)

» Media 1: TXT (2 KB)     
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» Media 3: TXT (0 KB)     
» Media 4: TXT (9 KB)     

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Figures (10)

Fig. 1
Fig. 1

The four components of ocean surface albedo (OSA).

Fig. 2
Fig. 2

The calculated direct surface albedo, α dir s , due to the Fresnel reflection for two refractive indices (n) and three wind speeds.

Fig. 3
Fig. 3

Comparison of the parameterization by Eq. (1) and the exact calculations for the surface direct albedo, α dir s , The lower panel shows the relative error (in percentage) of the parameterization.

Fig. 4
Fig. 4

Comparison of the parameterization by Eq. (5a) and the explicit calculations for the surface diffuse albedo, α dif s , under clear skies.

Fig. 5
Fig. 5

Downwelling diffuse radiance distribution under clouds with different optical depths (different lines). SZA = 30.

Fig. 6
Fig. 6

Comparison of the parameterization by Eq. (5b) and the explicit calculations for the surface diffuse albedo, α dif s , under cloudy skies.

Fig. 7
Fig. 7

Examples of the parameterized albedo due to the water volume scattering. The upper two panels are based on Eq. (6) for the direct component and the lower panel is from Eq. (10) for the diffuse.

Fig. 8
Fig. 8

Comparison of measured and parameterized albedo in two spectral bands (614 nm and 865 nm) in four clear days. The wind speed (middle panels) and the fraction of direct incidence at surface (lower panels) for parameterization input are also from measurements at the same site as for albedo.

Fig. 9
Fig. 9

Same as Fig. 8, but for comparison in four cloudy days. The direct flux fraction (lower panels) is from measurements at 865 nm.

Fig. 10
Fig. 10

Comparison of measured and parameterized broadband albedo under all skies.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

α dir s (λ,θ,w)= α dir s (n(λ),μ(θ),σ(w))= r f (n,μ) r f (n,μ) r f ( n 0 ,μ) f(μ,σ),
σ 2 =0.003+0.00512w.
α dir s (n,μ,O)= r f (n,μ).
f(μ,σ)=( p 0 + p 1 μ+ p 2 μ 2 + p 3 μ 3 + p 4 σ+ p 5 μσ) ×exp( p 6 + p 7 μ+ p 8 μ 2 + p 9 σ+ p 10 μσ),
α dif s (λ,w)= α dif s (n(λ),σ(w))=0.14820.012σ+0.1608n0.0244nσ.
α dir w (λ,μ,w,chl)= R 0 (1 r w )(1 α dir w ) 1 r w R 0 ,
r w =0.48170.0149σ0.207 σ 2 .
R 0 (λ,μ,chl)=β(μ) b b (λ,chl) a(λ,chl) ,
β=0.62790.2227 η b 0.0513 η b 2 +(0.2465 η b 0.3119)μ,
α dif w (λ,w,chl)= α dif w (λ, μ c ,w,chl).
α dir = α dir s + α dir w
α dif = α dif s + α dif w
α= f dir α dir + f dif α dif
f dir + f dif =1.0
α b = f dir α dir s ( n 0 ,σ)+ f dif α dif s ( n 0 ,σ)+0.006,
f wc =2.95× 10 6 w 3.52 .
α c = f wc α wc +(1 f wc )α.

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