Optical closure is essential for the determination of biogeochemical properties from ocean color remote sensing information. Mie scattering theory, a radiative transfer model, and a semi-analytical inversion algorithm were used to investigate the influence of particles and their properties on optical closure. Closure results were generally poor. Absorption coefficient (at) inversions were more accurate for moderate particle size distribution slopes (3.50 ≤ ξ ≤ 3.75). The degree of success in the derivation of the backscattering coefficient (bbp) was highest at moderate indices of refraction (1.15 ≤ np ≤ 1.20) and high values of ξ (> 3.75). Marked improvements in the estimates of bbp were enabled by a priori knowledge of bbp at one wavelength. At moderate values of np, derivations of at and bbp were within 25% of Mie-modeled values when Gershun’s relationship was used in combination with the semi-analytical algorithm.
©2009 Optical Society of America
Optical closure involves solutions to the forward and inverse problems in ocean optics. Both problems are twofold: Forward - the determination of the inherent optical properties (IOPs) from characteristics of the particulate and dissolved material and the prediction of the apparent optical properties (AOPs) from the IOPs; Inverse - the derivation of the IOPs from the AOPs, and the determination of biogeochemical properties from the IOPs. The latter component of the inverse problem is especially important for interpretation of ocean color remote sensing data to synoptically observe and monitor interdisciplinary processes such as biogeochemical cycling, particle transport, and ecosystem dynamics (e.g., see ). Other than the computation of the AOPs from the IOPs, i.e. component #2 of the forward problem , optical closure has proven problematic - particularly for global scale applications.
Optical theory suggests that the optical properties of particles depend on characteristics such as particle size, shape, composition, and index of refraction [3, 4]. Mie scattering theory involves the estimation of the IOPs from characteristics for a single particle or a population of mono-dispersed or poly-dispersed particles. Briefly, Mie theory determines the scattering and extinction efficiency factors (and absorption by difference) for a homogeneous sphere at a given wavelength, index of refraction, and particle diameter. Ensembles of efficiency factors for spheres of different sizes can then be applied, using a specified particle size distribution (PSD) and number concentration to derive the respective IOPs, i.e. the absorption, scattering, and attenuation coefficients for a particle population.
The IOPs are in turn directly related to remote sensing reflectance through radiative transfer theory, which can be numerically modeled (e.g., ). A simpler approach to radiative transfer has been proposed, with remote sensing reflectance correlated to the absorption and backscattering coefficients and represented by:
where bbt(λ) is total spectral backscattering, at(λ) is total spectral absorption, and the f/Q ratio (wavelength notation hereafter suppressed) is a parameter that depends on the shape of the upwelling light field and the volume scattering function (VSF) . The parameter Q(λ) is a measurable quantity: the ratio of upwelling irradiance to upwelling radiance. However the value, f(λ), is often approximated for a single VSF  or using assumed scattering properties and sky conditions [7, 8], which oftentimes leads to erroneous computations of rrs(λ) from the IOPs and vice versa when inverting this relationship (i.e. the inverse problem).
Empirical and semi-analytical algorithms have been developed to invert remote sensing reflectance signals to obtain the IOPs and biogeochemical constituents (e.g., chlorophyll concentration; ). Empirical algorithms have proven challenging for Case II waters, whose optical properties are not dominated by chlorophyll-containing particles but rather by inorganic particles, colored dissolved organic matter (CDOM), or both quantities. Semi-analytical algorithms based on Eq. (1) (e.g., see ) have been more successful when applied to widely varying optical water types, however, assumptions about the spectral and angular scattering properties and the upwelling light field can negatively impact inversion results.
The effects of particles and particle characteristics on the variability of remote sensing reflectance need to be understood in order to solve the forward and inverse problems in ocean optics and importantly, to better utilize ocean color data to obtain information about biogeochemistry and ecosystem dynamics. We present results from forward and backward approaches to optical closure with emphasis on the effects of particles and their characteristics on backscattering, the backscattering ratio, and ocean color. We build upon decades of theoretical, laboratory, and field results (e.g., see [3, 4, 11–20]).
Mie scattering theory and the radiative transfer model, Hydrolight, were used to compute the bulk IOPs and AOPs for a set of hypothetical water masses containing particle populations with variable bulk indices of refraction and PSDs. The influence of particles and their properties (e.g., bulk real index of refraction and PSD slope, np and ξ, respectively) on the IOPs and AOPs and on inversion algorithms to derive the IOPs from AOPs was investigated using a semi-analytical remote sensing inversion algorithm.
2.1 Forward approach
Mie scattering theory was used to generate bulk IOPs for a series of hypothetical water masses. Mie model input values were determined based on four years of time series measurements of optical properties collected on a mooring at 4 m water depth in shallow coastal waters of the Santa Barbara Channel (Fig. 1). Optical water types in this region of the Santa Barbara Channel are highly biogeochemically complex and can vary rapidly from relatively clear waters dominated by biogenic particles to very turbid and comprised of mostly inorganic particles [21, 22].
We used the Mie code provided by Bohren and Huffman  for scattering by homogeneous spheres (routines BHMIE and CALLBH), translated from Fortran language into Matlab by E. Boss (U Maine). This code computes the elements and efficiencies of the amplitude scattering matrix. The modeled PSDs were restricted to a size range of diameters, D, ranging from 0.2 - 100 μm in 35 logarithmically spaced size bins. The PSDs followed a differential size distribution function where f(D) = N × D-η, where N is the particle concentration (held constant at 5 × 105 mL-1), i.e. we assumed Junge-type PSDs. We used np values of 1.01, 1.05, 1.10, 1.15 and 1.20 and ξ, values of 3.0, 3.25, 3.5, 3.75, 4.0 and 4.25 at nine wavelengths (λ = 412, 440, 488, 510, 532, 555, 650, 676, and 715 nm). The imaginary index of refraction of particles was held constant at n’ = 0.01 for all Mie theory calculations. Flatter PSD slopes, i.e. lower values of ξ, generally imply more large particles are present in a population, whereas steeper PSD slopes specify smaller particles. Lower values of np are indicative of biological particles (np phytoplankton = 1.02 - 1.07; ) because of their high water content, and minerogenic particles are represented by higher values of np (np minerogenic = 1.14 to 1.26; ). Two hundred and seventy sets of IOPs were determined from Mie modeling efforts. These Mie-computed IOPs were within the range of IOPs measured in the Santa Barbara Channel (Fig. 1).
Phase functions, together with Mie-computed scattering and attenuation (and absorption by difference) coefficients, were then inputted into the numerical radiative transfer model, Hydrolight, to compute the AOPs at the surface, just below the surface, and at 4 m geometric water depth in optically deep waters. Particle phase functions were generated from Mie-calculated VSFs using relevant phase function discretization operations in Hydrolight. Inelastic scattering processes were not included in computations and the following environmental conditions were assumed: wind speed = 5 m s-1, solar angle = 30°, and 0% cloud cover. Hydrolight calculations resulted in 270 arrays of AOPs representing five different bulk indices of refraction, six different PSD slopes, and nine different wavelengths.
2.2 Inverse approach
We employed a semi-analytical remote sensing inversion algorithm  in order to examine the influence of particles and their properties (np and ξ) on derivations of the IOPs from AOPs. This algorithm uses the relationship presented by Gordon et al. :
and the g-constants are dependent on the particle phase function, oftentimes represented as g0 = 0.084 and g1 = 0.17 , which are the values we used in our computations. Due to the relatively large values of at(440) (> 0.3 m-1), at(650) was first parameterized as a function of Hydrolight-derived rrs(λ), then bbt(650) was derived from Eq. (2), as opposed to using the 555 nm wavelength method . Next, bbt(λ) was computed; it was assumed to monotonically decrease with increasing wavelength:
where λ 0 = 650 nm and η is a function of rrs(λ). Spectral at(λ) was calculated following Eq. (2). Details regarding empirical parameterizations of the IOPs and rrs(λ) can be found in Lee et al. .
3. Results and discussion
To verify data and model integrity, we provide an independent parameterization of at(λ) derived from Gershun’s equation. The exact relationship (wavelength notation suppressed):
(where Kd is the diffuse attenuation coefficient for downwelling irradiance, μd and μu are the average cosines for downwelling and upwelling light, respectively, R is irradiance reflectance, and Z is depth) was applied to our Hydrolight-derived values for Kd, μd, μu, and R at Z = 4 m. Comparisons between Mie-modeled absorption coefficients, at Mie(λ), and absorption coefficients inverted using Gershun’s equation, at Ger(λ), at 4 m are shown in Fig. 2. The exact 1:1 correlation between at Mie(λ) and at Ger(λ) for all np, ξ, and λ indicates that no errors were made in Mie computed inputs to Hydrolight or Hydrolight modeling.
3.1 Forward approach - IOPs
The effects of variable np and ξ on spectral IOPs are shown in Fig. 3. The absorption, attenuation, scattering, and backscattering coefficients (minus water) all increased with increasing np at all wavelengths. Harder particles (i.e. material with higher np-values) resulted in higher scattering and less transmission of light, which has been predicted from Mie theory (e.g., see [11, 12]). The absorption coefficient was minimally affected by variations in np and decreased with increasing ξ; spectral variability was more or less constant across all np and ξ. PSD slope effects were similar between cpg(λ) and bp(λ), i.e. attenuation was controlled more by scattering processes. In general, cpg(λ) and bp(λ) decreased with increasing ξ but at np > 1.10, cpg(λ) and bp(λ) increased in the blue wavelengths as ξ exceeded 4.0 (Fig. 3). Wozniak and Stramski  also showed higher blue-peaked spectra for mass-specific scattering coefficients at high values of ξ. Effects of np variability were more pronounced at higher ξ for cpg(λ) and bp(λ). This implies that variability in cpg(λ) and bp(λ) was more strongly affected by the presence of smaller, harder (i.e. minerogenic) particles.
Magnitudinal variability in bbp(λ) behaved as expected from Mie theory; bbp(λ) values increased with increasing np. Increasing ξ from 3.0 to 3.5 resulted in decreased bbp(λ) and then this trend reversed between ξ = 3.5 and 4.25. Mie-derived backscattering spectral shapes were not always represented by Eq. (3) and at np > 1.05, bbp(λ) exhibited spectral inflections, particularly at ξ = 3.0 (Fig. 3). The spectral shape of bbp(λ) became less steep with increasing np, with the mean power-law exponent changing from η = -2.601 to -0.1056, averaged over six different ξ. The mean of η over the 270 Mie-computed bbp(λ) spectra was calculated to be -0.8991 with a standard deviation of 1.0844. In comparison, Snyder et al.  found that the mean and standard deviation of η for over 6000 field measurements of bbp(λ) in coastal waters was -0.942 ± 0.210. They stated that, “… the spread in this value is not random and represents small, but real, spectral changes in the backscattering spectral properties of the water” . Their results show that significant departures from the average power-law function occurred where biological particulates dominated. We also find that the spectral shape of bbp(λ) deviated from a power-law function at lower np (at np < 1.10, bbp(λ) exponentially decayed with increasing wavelength; not visible in Fig. 3), which is representative of particles with higher water content, i.e. biological particles. Spectral inflections in the particulate backscattering coefficient have been observed in modeling results that used input parameters based on measurements of phytoplankton cultures [27–30]. Direct measurements of the spectral backscattering properties of marine phytoplankton cultures have also shown spectral inflections . Reduced backscattering in spectral regions of strong absorption (e.g. peaks in absorption by Chlorophyll-a around 440 and 670 nm) is caused by the change in the imaginary index of refraction at these wavelengths, an effect known as anomalous dispersion [3, 32]. However, spectral inflections present in our results are somewhat surprising, given that we kept the real and imaginary portions of the index of refraction constant across wavelengths in our model runs.
Based on theoretical studies [11, 12], the backscattering ratio, bbp(λ)/bp(λ), has traditionally been assumed to be spectrally flat. However, Ulloa et al.  assert that in the case of monodispersions or in waters dominated by minerogenic particles, bbp(λ)/bp(λ) can vary strongly with wavelength depending on the size of the particles and their refractive index. We found that Mie-modeled bbp(λ)/bp(λ) was spectrally flat only at ξ ≤ 3.25 for all np, i.e. larger particles. During all other conditions, bbp(λ)/bp(λ) varied spectrally with increasing np and ξ. The spectral shape of bbp(λ)/bp(λ) exhibited increasing values with increasing wavelength, with the steepest spectral slopes observed at ξ = 4.25. The particulate backscattering ratios observed by Snyder et al.  also exhibited a wide range in wavelength dependence and were highly variable in space and time. However, others have found little spectral variation in bbp(λ)/bp(λ). Boss et al.  and Mobley et al.  reported that bbp(λ)/bp(λ) spectra varied by less than 10% and 24% for all values of backscattering measured in coastal New Jersey and modeled using Hydrolight, respectively. Whitmire et al.  found no significant spectral variability in a dataset of over 9000 observations that included data from diverse aquatic environments and particle populations, with rare exceptions in instances of monodispersions of large particles. Risović  found that Mie-modeled bbp(λ)/bp(λ) was only weakly dependent on wavelength but suggested additional research on the spectral behavior of bbp(λ)/bp(λ).
Risović  also suggested that bbp(λ)/bp(λ) can be overestimated with Mie theory when assuming a Junge-type particle distribution. However, our resultant bbp(λ)/bp(λ) values are quite similar to those measured in the Santa Barbara Channel (Figs. 1 and 3). The magnitude of bbp(λ)/bp(λ) generally increased with increasing np and ξ, which has been shown by Mie theory (e.g., Fig. 5 in  and Fig. 1 in ). Theoretical studies have also indicated that bbp(λ)/bp(λ) is strongly affected by the presence of sub-micrometer particles and that its magnitude varies with the slope of the PSD . Observed magnitudinal variability of bbp(λ)/bp(λ) has been related to particle composition. Loisel et al.  indicated that, based on data collected in European waters, the amount of organic material has a strong influence on bbp(λ)/bp(λ) and detrital particles resulted in higher values of bbp(λ)/bp(λ). Boss et al.  also report that bbp(λ)/bp(λ) is strongly related to the index of refraction particles. Snyder et al. , however, do not find any quantifiable evidence of particle type (organic vs. inorganic) with the magnitude of bbp(λ)/bp(λ) and suggest that other factors need to be considered in quantifying the variability in bbp(λ)/bp(λ).
The most striking results for the effects of particle characteristics on our set of hypothetical optical water types was the highly variable spectral shapes of the backscattering coefficient and the backscattering ratio. Contrary to assumptions made in the past, bbp(λ)/bp(λ) was not always spectrally flat and spectral bbp(λ) did not always follow the widely used expression shown in Eq. (3). This suggests that spectral variability, in addition to magnitudinal variability in bbp(λ)/bp(λ) and bbp(λ) could contain valuable information about the characteristics of particles. Importantly, variable spectral shapes (e.g., exponential versus power-law) at low np and spectral inflections at low ξ-values could have profound impacts on remote sensing and inversion algorithms for the derivation of the IOPs and biogeochemical properties.
3.2 Forward approach - AOPs
Hydrolight-computed AOP values were within the ranges of those reported in the literature (Fig. 4). For example, Morel et al.  reported field-measured and simulated values of Q(λ) between 0 and 8 sr and Loisel and Morel  found Q(λ) to vary between about 3 and 7 sr from numerical simulations of case II coastal waters with variable solar angles and scattering events. Loisel and Morel  also reported f/Q values between 0.06 and 0.20 sr-1 for the same simulations. Kd(λ) values and spectral shapes in Fig. 4 are typical of turbid coastal waters (e.g., Fig. 2 in ).
Spectral AOPs for variable np and ξ-values are shown in Fig. 4. Kd(λ) behaved similar to apg(λ), increasing with increasing np and decreasing with increasing ξ at all wavelengths. Remote sensing reflectance and R(λ) (irradiance reflectance) spectral effects were as expected with increasing np and ξ; spectral shapes shifted from blue-peaked (case I-like) at low np and ξ to green-peaked (case II-like) at high n and ξ. The variability in the f/Q ratio was highly dependent on wavelength, np, and ξ. The f/Q ratio increased with increasing np at all wavelengths, however effects of variable ξ were spectrally and np dependent. This suggests that f/Q is strongly related to scattering processes and should not be treated as a constant (e.g., [37, 38]) [note that the computation of f assumes that the relationship in Eq. (1) is true]. This has important implications for inversion algorithms to determine the IOPs and biogeochemical constituents from remotely sensed ocean color data as many radiative transfer-based inversion models employ empirically or numerically determined constants in the place of f/Q, e.g., Eqs. (1) and (2).
3.3 Inverse approach
Absorption and backscattering coefficients derived using a semi-analytical algorithm  were within 150% of “true” values, i.e. Mie-computed (Fig. 5). Lower values of at(λ) were estimated more accurately than higher values, likely because the algorithm was formulated based on at(λ) values of less than 0.3 m-1 whereas our Mie-computed at(λ) values were much higher (mean of 0.5 m-1 at 412 nm; Fig. 1). The degree of success of derivation of at(λ) was more influenced by ξ as compared to np or wavelength. For derivations of at(λ), the inversion algorithm performed best at moderate ξ-values (3.50 ≤ ξ ≤ 3.75) for all values of np and at all wavelengths. Absorption was slightly overestimated for high values of ξ (i.e. smaller particles) and underestimated for low ξ (i.e. larger particles). On the other hand, successful derivations of bbp(λ) were dependent on np and ξ but not necessarily on wavelength or the magnitude of bbp(λ). We show that more accurate bbp(λ) estimations were achievable at 1.15 ≤ np ≤ 1.20 and at higher values of ξ (> 4.0; i.e. smaller particles). Derivations were underestimated for lower ξ.
To investigate potential sources of error in the semi-analytical algorithm for these particular IOPs and AOPs, we assumed a priori knowledge of bbp(650) and then estimated bbp(λ) and at(λ) using Eqs. (3) and (2). Results of this exercise are shown in Fig. 6a–d. Not surprisingly, derivations of spectral bbp(λ) are much more accurate with knowledge of bbp(λ) at one wavelength (Fig. 6c, d). Other than for np = 1.01, bbp(λ) was generally overestimated, more so at the blue wavelengths when bbp(650) is known. Thus, at(λ) was also overestimated. The absorption coefficient was generally overestimated by up to 130%, even at at(650) when bbp(650) is known, which suggests that g0 and g1 (Eq. 2) are not entirely accurate and should not be treated as constants, at least for this set of IOPs. This is evidenced by the highly variable f/Q ratio shown in Fig. 4 and discussed in the previous section.
and by assuming R(λ) = 3.5 × rrs(λ). Eq. (3) was then used to estimate bbp(λ) (Fig. 6g, h). The use of a simplified Gershun’s equation yielded more accurate derivations of at(λ), mostly within 25% of Mie-computed values (Fig. 6e, f). The largest deviations in at(λ) and bbp(λ) were found at higher np values. Unsuccessful derivations of bbp(λ) at np = 1.01 in both cases are due to the fact that the spectral shape of bbp(λ) at the lowest np-value (Fig. 6d, h) is represented by an exponential, rather than a power-law function. This implies that for complex coastal waters, improvements need to be made to the empirical parameterization of the power-law exponent, η.
Wozniak and Stramski  also show that mineral particles in seawater can result in significant errors in the derivation of, in their case chlorophyll concentration from ocean color data. Marked improvements in the inversion algorithm are found by assuming a priori knowledge of bbp at one wavelength or by applying a simplified Gershun’s equation to first derive at(λ). This shows that minimal in situ measurements can result in accurate assessments of the IOPs and hence biogeochemical properties from ocean color remote sensing data.
4. Summary and conclusions
Mie scattering theory and the radiative transfer model, Hydrolight, were used to compute the IOPs and AOPs for a set of hypothetical optical water types with variable bulk real indices of refraction and particle size distribution slopes. The influence of particles and their properties on inversion algorithms to derive the IOPs from modeled AOPs was investigated using a semi-analytical remote sensing inversion algorithm. Notable conclusions are highlighted here.
The spectral shape and magnitude of the absorption coefficient was minimally affected by variations in np, but its magnitude decreased significantly with increasing ξ. Variability in the attenuation and scattering coefficients was strongly influenced by the presence of smaller, harder particles. The spectral shape of the particulate backscattering coefficient was not always represented by a power-law function, and often exhibited spectral inflections. The particulate backscattering ratio was spectrally flat only in the presence of larger particles. During all other conditions, bbp(λ)/bp(λ) varied spectrally with increasing np and ξ. In evaluating the effect of particle properties on AOPs, we found that the f/Q ratio exhibited escalating spectral variability with increasing np and ξ. In tests of the performance of a semi-analytical algorithm in predicting IOPs, the degree of success in the derivation of at(λ) was more influenced by the magnitude of at(λ) and ξ as compared to np or wavelength. Accurate derivations of bbp(λ) were dependent on np and ξ but not necessarily on wavelength or the magnitude of bbp(λ). Marked improvements in estimates of bbp(λ) were enabled by a priori knowledge of bbp at one wavelength. At 1.05 < np < 1.15, derivations of at(λ) and bbp(λ) were well within 25% of true values when Gershun’s equation was used in combination with the semi-analytical algorithm.
Our efforts here are focused on improving and quantifying understanding of the relationships between the bulk optical properties and the particle characteristics. The IOPs form the link between biogeochemical constituents and the vertical structure of the radiance field; therefore successful inversions of satellite-measured ocean color properties to derive biogeochemical products are dependent on efforts like these. We identify the need to better comprehend the effects of individual and bulk particle characteristics that are more detailed than np and ξ (e.g., individual particle composition and size) on the spectral shapes of the backscattering coefficient and backscattering ratio and on the variability of the f/Q ratio (or the g-constants) in order to properly parameterize inversion models. We also show that significant improvements to inversions can be achieved with minimal in situ measurements, i.e. ocean color remote sensing ground-truthing.
GC was supported by the National Oceanographic Partnership Program as part of the MOSEAN program. AW was supported by a NOAA Sea Grant Industry Fellowship. The authors would like to acknowledge MOSEAN PIs Tommy Dickey, Casey Moore, Al Hanson, and Dave Karl. The authors also thank Tim Cowles, Andrew Barnard, Frank Spada, and an anonymous reviewer who provided an excellent, insightful and thorough review of an earlier version of this paper.
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