# AOPs Are Not Additive: On the Biogeo-Optical Modeling of the Diffuse Attenuation Coefficient

^{1}School for the Environment, University of Massachusetts Boston, Boston, MA, United States^{2}State Key Laboratory of Marine Environmental Science, College of Ocean and Earth Sciences, Xiamen University, Xiamen, China^{3}Department of Biology, The University of North Carolina at Greensboro, Greensboro, NC, United States

Commonly we see the diffuse attenuation coefficient of downwelling irradiance (*K*_{d}) expressed as a sum of the contributions of various constituents. We show here that, both theoretically and numerically, because *K*_{d} is an apparent optical property (AOP), this approach is not consistent with radiative transfer. We further advocate the application of models of *K*_{d} developed in past decades that are not only consistent with radiative transfer but also provide more accurate estimates, in particular for coastal turbid waters.

## Background

Solar radiation is the energy source for the entire earth system. In aquatic environments, unlike terrestrial environments, solar radiation can penetrate to great depths to fuel photosynthesis and to heat up the upper layer (Zaneveld et al., 1981; Platt, 1986; Lewis et al., 1990). The propagation of solar radiation from surface to greater depths can be expressed as (Mobley, 1994)

Here *E*_{d} (W/m^{2}/nm) is the downwelling irradiance, *z* (m) is the depth from the surface (0^{−} for subsurface), *K*_{d} (m^{−1}) is the attenuation coefficient of downwelling irradiance between surface and depth *z*, and λ (nm) for wavelength. Since the variation of *E*_{d}(0^{−}) is independent of water properties (except extremely turbid waters where the enhanced upwelling flux will result in significant contributions to *E*_{d}(0^{−}) due to internal reflection), it is then imperative to describe the variation of *K*_{d} for various aquatic environments when quantifying the impact of water constituents on the heat budget (Morel and Antoine, 1994; Ohlmann et al., 2000), the feedback of oceanic systems on climate changes (Rochford et al., 2001; Gnanadesikan and Anderson, 2009), as well as the vertical variation of primary production (Sathyendranath and Platt, 1995).

Historically, with an objective of easy modeling and efficient calculation for large scale applications, *K*_{d} is commonly expressed as (Smith and Baker, 1978; Morel, 1988; Morel and Maritorena, 2001),

with *K*_{w} the contribution of pure (sea)water, and *K*_{bio} the contributions of phytoplankton. In this expression, i.e., the so-called “Case-1” scheme (Morel and Prieur, 1977), the attenuation of pure (sea)water is considered as a background, while other constituents that are actively changing, such as phytoplankton and suspended mineral solids, are considered as added contributions. In addition, the contributions of colored dissolved organic matter (CDOM) and organic detritus are considered as co-varying with phytoplankton, and lumped into the *K*_{bio} term. So their contributions are not ignored or omitted as might be implied by the equation, although its application is limited to “Case-1” waters.

In order to explicitly evaluate and understand the impact of constituents such as CDOM and/or suspended mineral particles or particulate inorganic matter (PIM) on the propagation of solar radiation, *K*_{d} in many studies is expanded as a sum of more components, although there are subtle variations among these models (Smith and Baker, 1978; Baker and Smith, 1982; Gallegos et al., 1990; Devlin et al., 2009; Kim et al., 2015),

Here *K*_{CDOM} and *K*_{PIM} are the diffuse attenuation coefficients resulted from CDOM and PIM, respectively. In essence, these biogeo-optical models of *K*_{d} effectively treat *K*_{d}, an apparent optical property (AOP) (Preisendorfer, 1976), as an inherent optical property (IOP) (Preisendorfer, 1976), which is not consistent with the definitions and the nature of variations of *K*_{d} (Stavn, 1988). The attitude of treating *K*_{d} as an IOP might stem from that *K*_{d} of “Case 1” water, after correcting for the sun angle effect, can be considered as a “quasi” IOP (Gordon, 1989). However, it was never claimed that this would work in any other water types than “Case 1” water. Many subsequent studies have, for the most part, somehow ignored these limitations in applications.

Fundamentally *K*_{d} is sun-angle dependent (Stavn, 1988; Mobley, 1994) (also weakly dependent on atmospheric properties). So, considering the model of Morel and Maritorena (2001), it is specifically stated that the model and the empirical coefficients (Equation 3 in Morel and Maritorena, 2001) are *valid just for low zenith sun angles*. But this restriction has in fact largely been ignored by the research community, which leads to inconsistent applications and errors. For instance, if we use this model for early morning or late afternoon situations, because of the likely large sun angle, this can easily result in 30% or greater errors in estimating *K*_{d} (Morel et al., 2002; Lee et al., 2005b). In the following, we demonstrate the non-additive nature of *K*_{d} theoretically and numerically.

## Theoretical Model of *K*_{d}

_{d}

Based on radiative transfer, *K*_{d} is a function of IOPs (especially the absorption, *a*, and backscattering, *b*_{b}, coefficients) as (Lee et al., 2005b),

Here μ_{d} (μ_{u}) is the average cosine and *r*_{d} (*r*_{u}) is the shape factor for the downwelling (upwelling) light field (Stavn and Weidemann, 1989), respectively. *R* is the irradiance reflectance (Gordon et al., 1975). Through numerical simulations via Hydrolight, it was found that the above expression could be simplified as (Lee et al., 2005b)

with m_{0−3} constants that are independent of wavelength and water properties. Note that these model parameters vary weakly with depth (Lee et al., 2005b) due to changes of light field structure, consistent with the change of μ_{d} with depth (Stavn, 1988; Berwald et al., 1995; McCormick, 1995). Also note that for large zenith angles, the forward scattering coefficient will also contribute to the diffuse attenuation coefficient through its contribution to μ_{d}, μ_{u}, *r*_{d} and *r*_{u} (Stavn and Weidemann, 1989). Mathematically, Equation (5) can be rewritten as,

Consequently, although *a*(λ) and *b*_{b}(λ) are additive, a nature of IOPs, the interaction term between *a*(λ) and *b*_{b}(λ) (the third term on the right side of Equation 6) is *not* additive, thus *K*_{d} cannot be additive—a general nature of AOPs. This characteristic is further highlighted in details below.

For simplicity, let's consider a medium has just two constituents: pure seawater and suspended inorganic mineral particles (PIM). For pure seawater alone, following Equation (6), there is

Here *a*_{w}(λ) and *b*_{bw}(λ) are the spectral absorption and backscattering coefficients of pure seawater.

For suspended inorganic mineral particles alone,

with *a*_{PIM} and *b*_{bPIM} being the absorption and backscattering coefficients of suspended mineral particles.

Therefore, a sum (${K}_{d}^{sum}(\lambda )$) of the two contributions to *K*_{d} following Equations (2) and (3) resulted in,

However, when the medium is composed of both pure seawater and suspended mineral particles, its *K*_{d} following radiative transfer (Equation 6) is

Clearly, as shown above, when there are more constituents, because the light field is determined by the bulk properties (Stavn, 1988; Stavn and Weidemann, 1989; Lee et al., 2005b), *a*_{w} and *a*_{PIM} will affect the contribution of both *b*_{bw} and *b*_{bPIM} to *K*_{d}. However, when *K*_{d} is treated as an additive property of *K*_{w} and *K*_{PIM}, the effect of *a*_{w} on the contribution of *b*_{bPIM} and the effect of *a*_{PIM} on the contribution of *b*_{bw} are excluded.

## Numerical Demonstration

To demonstrate the above point numerically, Figure 1 compares *K*_{d} spectra from Hydrolight (Mobley and Sundman, 2013) simulations with ${K}_{d}^{sum}$, where the two component spectra (*K*_{w} and *K*_{PIM}) were also obtained from Hydrolight simulations using the same constituents as for *K*_{d}. Specifically, spectral (400–800 nm, 10 nm interval) *E*_{d}(*z*) were simulated with Hydrolight, and *K*_{d} between surface and *z* is calculated following

**Figure 1**. Comparison of *K*_{d} spectra between Hydrolight simulation (blue), sum of individual components (red), and that from semi-analytical model based on bulk IOPs (green). The range for *K*_{d} is between surface and 5 m.

For the derivation of *K*_{w} from Hydrolight, all other constituents were held to 0 except for the properties of pure seawater. Values of *a*_{w} are a combination of Lee et al. (2015) and Pope and Fry (1997) while values of *b*_{bw} are those of Morel (1974). For the derivation of *K*_{PIM} from Hydrolight, PIM was set as 10 g/m^{3} and the default optical model parameters for suspended minerals included in Hydrolight were used to get the absorption and scattering coefficients of PIM. Note that this PIM concentration is just a low-medium value for turbid coastal waters (Babin et al., 2003; Doxaran et al., 2009). For this simulation, an idealized “transparent pure seawater” was used where very low values of *a*_{w} (0.1 × 10^{−4} m^{−1}) and *b*_{bw} (0.5 × 10^{−5} m^{−1}) were employed. With such a setup the contribution of this “transparent pure seawater” to the calculated *K*_{d} (Equation 11) is then negligible, and the resultant *K*_{d} from Hydrolight simulations can be considered as *K*_{PIM}. The sun angle for all simulations for both *K*_{w} and *K*_{PIM} was set as 30° from zenith along with a clear sky.

There are distinct differences in *K*_{d} (at least for this case) in the longer wavelengths (~10–15% for the 600–800 nm range), where *a*_{w} makes significant contributions to the total *a*; and this contribution, when there are sediments, to *K*_{d} is not represented in the additive descriptions of *K*_{d} (the red curve). For the shorter wavelengths (<~500 nm), because most (>~98%) of the contributions to *K*_{d} comes from PIM, the sum of the two terms match the bulk results well. Certainly the impact of the non-additive nature of *K*_{d} depends on the values of both *a* and *b*_{b}. For “Case-1” waters or waters where the scattering coefficients are relatively small, it might be applicable, without great errors, to treat *K*_{d}(λ) as an additive property. However, this will depend on the validity of the above-mentioned assumptions. While not based on any assumptions of “Case-1” conditions or dependencies, the modeled *K*_{d} spectrum following Equation 5 is in an excellent agreement with the Hydrolight *K*_{d} spectrum (~1% differences, see Figure 1), which highlights the much wider applicability of models based on radiative transfer. And, the robust performance of this model was also demonstrated in Zimmerman et al. (2015) for the quite turbid Chesapeake Bay waters.

Historically (Lorenzen, 1972; Smith and Baker, 1978; Woodruff et al., 1999; Gallegos, 2001; Devlin et al., 2008), there are also studies that treat the attenuation coefficient (*K*(PAR)) of the photosynthetic available radiation (PAR) as being additive of the contributions of individual constituents,

with *K*_{x}(PAR) for contributions except phytoplankton and pure (sea)water. Following the above logic and discussion regarding spectral *K*_{d}, we easily observe that this model is not consistent with radiative transfer either (Morel, 1988). In particular, it is ambiguous of the light spectra that should be used for the calculation of *K*_{w}(PAR) or *K*_{bio}(PAR). Further, because *K*(PAR) is the attenuation coefficient of solar radiation of a wide spectral range (400–700 nm, i.e., the PAR spectral range), while the spectral quality of *E*_{d}(*z*) changes significantly from surface to depths, which then causes *K*(PAR) to change greatly (as much as a factor of 4) from surface to depth (Lee et al., 2005a; Lee, 2009). Consequently, the applicability of such biogeo-optical models for *K*(PAR) is ambiguous at the very least.

## Conclusions

Because the interaction term (the third term on the right side of Equation 6) of *K*_{d}(λ) (or *K*(PAR)) depends on the values of both *a* and *b*_{b}, the contribution of this term to *K*_{d} is not always small or negligible. Also, this interaction term is not a linear function of *a* and *b*_{b}. Therefore, for consistency with radiative transfer and for more accurate estimation, and also to incorporate advancements in ocean optics of recent decades, it is better to get bulk IOPs first from biogeochemical properties, and then to calculate *K*_{d} based on IOPs. In short, IOPs are additive, but AOPs are not.

## Author Contributions

All authors contributed to the hypothesis and overall discussions regarding diffuse attenuation of solar radiation. ZL drafted the manuscript and both SS and RS commented and edited the manuscript before submission.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Acknowledgments

Funding support provided by the National Key Research and Development Program of China (2016YFC1400905, 2016YFC1400904, #2016YFA0601201, SS), the National Oceanic and Atmospheric Administration (NOAA) JPSS VIIRS Ocean Color Cal/Val Project (NA11OAR4320199, ZL), the National Aeronautic and Space Administration (NASA) Water and Energy Cycle, Ocean Biology and Biogeochemistry Programs (NNX14AK08G, NNX14AQ47A, ZL), and the University of Massachusetts Boston are greatly appreciated.

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Keywords: solar radiation, apparent optical properties, inherent optical properties, diffuse attenuation coefficient, optical additivity

Citation: Lee Z, Shang S and Stavn R (2018) AOPs Are Not Additive: On the Biogeo-Optical Modeling of the Diffuse Attenuation Coefficient. *Front. Mar. Sci*. 5:8. doi: 10.3389/fmars.2018.00008

Received: 14 July 2017; Accepted: 11 January 2018;

Published: 30 January 2018.

Edited by:

Kevin Ross Turpie, University of Maryland, Baltimore County, United StatesReviewed by:

Emmanuel Devred, Fisheries and Oceans Canada, CanadaKnut Barthel, University of Bergen, Norway

Copyright © 2018 Lee, Shang and Stavn. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Zhongping Lee, zhongping.lee@umb.edu