Conversion between upwelling irradiance and radiance at the sea surface, essential for some aspects of remote sensing of ocean colour, requires knowledge of the Q factor. The Q factor depends on solar zenith angle, satellite viewing angle and the optical properties of the water. For simulated data for two fixed solar zenith angles, we express the Q factor as a function of total absorption and backscattering coefficients. The new parameterisation is used in an algorithm to retrieve total water backscattering from in situ ocean-colour data. The algorithm performs better than those in which the Q factor is expressed as a function of chlorophyll-a concentration alone.
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Morel and Gentili [1991, 1993, 1996] developed a parameterisation of the Q factor as a function of chlorophyll-a concentration, solar zenith angle and viewing angle, based on Monte Carlo simulations. Limitations of this parameterisation, as acknowledged by the authors, were: (1) the model was designed for case-1 waters (optical properties covary with chlorophyll-a concentration); (2) use of a single volume scattering function regardless of the particle size distribution; and (3) the narrow range of chlorophyll-a concentration (0.03 to 3 mg m−3). The last two limitations were addressed in a subsequent work [Morel et al., 2002] using a phase function that was related to chlorophyll-a concentration and a range of chlorophyll concentrations extending to 10 mg m−3. This work also incorporated Raman scattering, which improved the parameterisation at low chlorophyll-a concentration. Loisel and Morel  studied the Q factor as a function of the total scattering (b) and absorption (a), Q = f(1 + b/a), based on radiative transfer simulations that included case-2 waters (waters in which phytoplankton, detrital and other suspended particles and dissolved organic matter vary independently of each other).
 In this paper, a bivariate polynomial regression is performed on simulated Q data for open-ocean and coastal waters, at various wavelengths and two solar zenith angles. The performance of the new parameterisations is tested against in situ data. Total backscattering at 555 nm is retrieved using the reflectance model of Sathyendranath and Platt  and the new Q-factor parameterisation. Our results are also compared with derivations from the method of Morel et al. , which is designed for case-1 waters.
2. Data and Methods
 The synthesised dataset used in this study was generated by a Working Group of the International Ocean-Colour Coordinating Group (IOCCG). It contains five hundred cases, comprising reflectances (Eu/Ed, where Ed is downwelling irradiance) and remote-sensing reflectances (Lu/Ed) (outputs) from 400 to 800 nm (10 nm resolution) with the associated backscattering and absorption coefficients (inputs) for two solar zenith angles, 30 and 60°, and chlorophyll concentrations ranging from 0.03 to 30 mg−3. The simulated data compared well with in situ data [IOCCG, 2006, Figure 2.3] and have been used in a previous study [Lee et al., 2005]. The data set, along with a detailed description of the various inputs, is available at http://www.ioccg.org/groups/OCAG_data.html. The radiative transfer model Hydrolightsupscr>®/supscr> was used to simulate reflectances. Random variations were introduced to the inputs to simulate the natural variability of oceanic waters. Total backscattering and absorption were not correlated with phytoplankton concentration in the data, such that for a given chlorophyll-a concentration many sets of inherent optical properties are possible. Using the IOCCG dataset, the Q factor was computed as the ratio of irradiance reflectance to remote sensing reflectance. Propagation of light through the air-sea interface has a negligible impact on the Q factor for a nadir viewing angle [Lee et al., 2002], so that reflectances below the sea surface could be used to infer the nadir-viewing Q factor. For viewing angles different from nadir, additional geometrical considerations have to be taken into account when deriving the Q factor. This case is not treated in the present study.
 Because in situ Q-factor values were not available, performance of our parameterisation could not be tested by comparison of computed Q with measured Q. Therefore, our parameterisation was evaluated by inverting remote-sensing reflectances to retrieve total backscattering coefficient at 555 nm, which requires computation of the Q factor. Because the Q factor is highly correlated with the backscattering coefficient, one anticipates that improved retrieval of the Q factor would lead to improved retrieval of the backscattering coefficient. In situ data were downloaded from the NASA bio-Optical Marine Algorithm Data set (NOMAD) at NASA's Ocean Color website (http://seabass.gsfc.nasa.gov/restrict/nomad.cgi). Only data collected at sun zenith angles ranging from 24 to 36° and 54 to 66° were used, to be consistent with the parameterisation developed using the synthesised dataset for θs = 30 and 60°. A total of 54 water-leaving radiances at five wavelengths (i.e., 412, 443, 490, 510 and 555 nm) and associated total backscattering coefficients at 555 nm were used (27 data for θs ≈ 30° and 27 data for θs ≈ 60°).
2.2.1. Model for Q Factor Retrieval
 Here, we assume that the Q factor, an apparent optical property, can be expressed as a function of backscattering and absorption. The justification for this parameterisation is shown in Figure 1. Q is correlated with both log-transformed a (r2 = 0.81) and log-transformed bb (r2 = 0.91). The spread of data points in the plot is uniform for all values of a, whereas the spread is small for low values of bb, and increases with bb. At low backscattering coefficients, backscattering by pure sea water represents the main contribution to the upwelling irradiance, and the upward radiative field tends toward isotropy. As backscattering by particles becomes dominant, bidirectional effects in the light field lead to the high scatter observed in Figure 1b. Changes in the shape of the volume scattering function for particles amplify this effect.
 A bivariate polynomial function was fitted to the synthesised dataset. The Q factor was computed as the ratio between Eu(λ, θs, θv) and Lu(λ, θs, θv), where the wavelength, λ, is equal to 440, 550 or 670 nm, the sun zenith angle, θs, is equal to 30 or 60° and the viewing angle, θv, is equal to 0° (i.e., nadir). Higher correlation coefficients and smaller root-mean-square errors were found when a third-degree polynomial function was chosen with both a and bb as independent variables (Figure 2). The Q factor is expressed as:
The values of the parameters mi and ni and the regression statistics are summarized in Table 1. When p-values of the estimated coefficients were not significant (p-value < 0.05), the regression was recomputed with those coefficients set to zero (shown as “-”in Table 1).
Table 1. Coefficients and Standard Errors of the Bivariate Polynomial Regression to Estimate the Q Factora
Bivariate polynomial regression is equation 1. The units of the independent variables (a and bb) in equation 1 are in m−1. The p-value for all coefficients is <0.001, meaning that the test of significance would succeed by chance only once in a thousand trials. The standard error refers to the standard error on the estimates of the coefficients (square root of the estimated variance).
Sun Zenith Angle = 30°
Sun Zenith Angle = 60°
2.2.2. Inversion of Reflectance
 The effect of the parameterisation of the Q factor on the retrieval of total backscattering coefficient from reflectance was studied using the model of Sathyendranath and Platt [1997, 1998] coupled with the optimisation method described by Devred et al. . The model of Sathyendranath and Platt  relates reflectance, R, just beneath the sea surface to inherent optical properties. For some semi-analytical reflectance models, such as the one used in this study, knowledge of the Q factor is required to convert irradiance reflectances to remote-sensing reflectances in the optimization procedure used to derive IOPs. Two parameterisations of the Q factor were used in the procedure, the one presented here and that of Morel et al. .
3. Discussion and Conclusion
 Although parameterisations were performed from 400 to 700 nm at 10 nm resolution, only results for 440, 550 and 670 nm are shown here (Figure 2). The values of Q derived from the synthesised dataset are representative of measurements collected in natural waters. The model of Åas and Højerslev [1999, equation 26], yields values of Q ranging from 3.17 to 4.08 for θs = 30°, in agreement with the synthesised dataset in which Q varies from 3.41 to 4.64. One observes a degradation of the polynomial regression in the near-infrared (λ = 670 nm), although the r2 are still high (0.947 and 0.883 respectively for θs = 30 and 60°), revealing a possible limitation of the polynomial fit. The solar-zenith angle dependence of the Q factor established by Åas and Højerslev  does not apply to our parameterisations (results not shown), perhaps because their data were collected in case-1 waters and no spectral resolution was accounted for. This implies that the results presented here for sun zenith angles of 30 and 60° cannot be generalised to all sun angles.
 Retrieval of backscattering coefficient at 555 nm using the new Q parameterisation was improved compared with the case-1 approach (Figure 3). The root-mean-square-errors (RMSE, see IOCCG [2006, equation 2.1]) for retrieval of the backscattering coefficient are 0.096 (θs = 30°) and 0.123 (θs = 60°) using the new parameterisation, whereas the RMSE are 0.125 and 0.214 using the case-1 approach. Differences between the two approaches at low backscattering coefficients are small and increase as the backscattering coefficient increases. This is not surprising since a higher backscattering coefficient is often an indication of coastal, or case-2 waters where the case-1 approach is known to be inadequate. This demonstrates the potential of the two-variable Q-factor model to succeed not only in case-1 waters but also in case-2 waters.
 The goal of the study presented here was to demonstrate the utility of modeling the Q factor as a function of total backscattering and absorption. The success of the result for the two solar zenith angles for which simulated data are available suggests that it would be worthwhile to extend the model to other zenith and viewing angles. The results presented here emphasize the importance of the Q-factor to convert upwelling irradiance to upwelling radiance. This step is required in some reflectance models when retrieving inherent optical properties from ocean colour radiometry. Since the Q-factor is sensitive to the shape of the volume scattering function of particles, this study also highlights the requirement for more measurements of particle volume scattering coefficients at sea to achieve further improvements in our understanding of the underlying causes for variability in the Q-factor.
 This work was supported by the Canadian Space Agency through the GRIP program. The authors thank IOCCG and Z.P. Lee for making the dataset available for this study.