1. Introduction

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 [1]). Other than the computation of the AOPs from the IOPs, i.e. component #2 of the forward problem [2], 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., [2]). A simpler approach to radiative transfer has been proposed, with remote sensing reflectance correlated to the absorption and backscattering coefficients and represented by:

where b_bt(λ) is total spectral backscattering, a_t(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) [5]. 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 [6] or using assumed scattering properties and sky conditions [7, 8], which oftentimes leads to erroneous computations of r_rs(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; [9]). 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 [10]) 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]).

2. Methods

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, n_p 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 [4] 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 n_p 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 n_p are indicative of biological particles (n_p ^{phytoplankton} = 1.02 - 1.07; [23]) because of their high water content, and minerogenic particles are represented by higher values of n_p (n_p ^minerogenic = 1.14 to 1.26; [24]). 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).

 

Fig. 1. Measured IOPs (Santa Barbara Channel). Thick black lines indicate the spectral mean and dashed black lines denote one standard deviation from the mean. The lower right-hand panel shows a data series of bulk n_p (red) and ξ (blue) computed using measured IOPs and methods presented by Boss et al. [25] and Twardowski et al. [12]. [The IOP property subscript ‘p’ denotes particulate material and ‘g’ represents dissolved matter.]

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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 [10] in order to examine the influence of particles and their properties (n_p and ξ) on derivations of the IOPs from AOPs. This algorithm uses the relationship presented by Gordon et al. [26]:

where

and the g-constants are dependent on the particle phase function, oftentimes represented as g₀ = 0.084 and g₁ = 0.17 [10], which are the values we used in our computations. Due to the relatively large values of a_t(440) (> 0.3 m^-1), a_t(650) was first parameterized as a function of Hydrolight-derived r_rs(λ), then b_bt(650) was derived from Eq. (2), as opposed to using the 555 nm wavelength method [10]. Next, b_bt(λ) was computed; it was assumed to monotonically decrease with increasing wavelength:

where λ ₀ = 650 nm and η is a function of r_rs(λ). Spectral a_t(λ) was calculated following Eq. (2). Details regarding empirical parameterizations of the IOPs and r_rs(λ) can be found in Lee et al. [10].

3. Results and discussion

To verify data and model integrity, we provide an independent parameterization of a_t(λ) derived from Gershun’s equation. The exact relationship (wavelength notation suppressed):

(where K_d is the diffuse attenuation coefficient for downwelling irradiance, μ_d and μ_u 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 K_d, μ_d, μ_u, and R at Z = 4 m. Comparisons between Mie-modeled absorption coefficients, a_{t Mie}(λ), and absorption coefficients inverted using Gershun’s equation, a_t ^Ger(λ), at 4 m are shown in Fig. 2. The exact 1:1 correlation between a_t ^Mie(λ) and a_t ^Ger(λ) for all n_p, ξ, and λ indicates that no errors were made in Mie computed inputs to Hydrolight or Hydrolight modeling.

 

Fig. 2. Comparison between a_t ^Mie(λ) and a_t ^Ger(λ) [Eq. (4)] at nine wavelengths, five different values of n_p, and six different values of ξ.

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3.1 Forward approach - IOPs

The effects of variable n_p and ξ on spectral IOPs are shown in Fig. 3. The absorption, attenuation, scattering, and backscattering coefficients (minus water) all increased with increasing n_p at all wavelengths. Harder particles (i.e. material with higher n_p-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 n_p and decreased with increasing ξ; spectral variability was more or less constant across all n_p and ξ. PSD slope effects were similar between c_pg(λ) and b_p(λ), i.e. attenuation was controlled more by scattering processes. In general, c_pg(λ) and b_p(λ) decreased with increasing ξ but at n_p > 1.10, c_pg(λ) and b_p(λ) increased in the blue wavelengths as ξ exceeded 4.0 (Fig. 3). Wozniak and Stramski [17] also showed higher blue-peaked spectra for mass-specific scattering coefficients at high values of ξ. Effects of n_p variability were more pronounced at higher ξ for c_pg(λ) and b_p(λ). This implies that variability in c_pg(λ) and b_p(λ) was more strongly affected by the presence of smaller, harder (i.e. minerogenic) particles.

 

Fig. 3. Spectral IOPs and IOP ratios as a function of n_p (legend in top left panel; colors correspond to plot lines and not fill colors) and ξ (x-axes).

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Magnitudinal variability in b_bp(λ) behaved as expected from Mie theory; b_bp(λ) values increased with increasing n_p. Increasing ξ from 3.0 to 3.5 resulted in decreased b_bp(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 n_p > 1.05, b_bp(λ) exhibited spectral inflections, particularly at ξ = 3.0 (Fig. 3). The spectral shape of b_bp(λ) became less steep with increasing n_p, 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 b_bp(λ) spectra was calculated to be -0.8991 with a standard deviation of 1.0844. In comparison, Snyder et al. [20] found that the mean and standard deviation of η for over 6000 field measurements of b_bp(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” [20]. 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 b_bp(λ) deviated from a power-law function at lower n_p (at n_p < 1.10, b_bp(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 [31]. 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, b_bp(λ)/b_p(λ), has traditionally been assumed to be spectrally flat. However, Ulloa et al. [11] assert that in the case of monodispersions or in waters dominated by minerogenic particles, b_bp(λ)/b_p(λ) can vary strongly with wavelength depending on the size of the particles and their refractive index. We found that Mie-modeled b_bp(λ)/b_p(λ) was spectrally flat only at ξ ≤ 3.25 for all n_p, i.e. larger particles. During all other conditions, b_bp(λ)/b_p(λ) varied spectrally with increasing n_p and ξ. The spectral shape of b_bp(λ)/b_p(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. [20] also exhibited a wide range in wavelength dependence and were highly variable in space and time. However, others have found little spectral variation in b_bp(λ)/b_p(λ). Boss et al. [15] and Mobley et al. [33] reported that b_bp(λ)/b_p(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. [19] 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 [13] found that Mie-modeled b_bp(λ)/b_p(λ) was only weakly dependent on wavelength but suggested additional research on the spectral behavior of b_bp(λ)/b_p(λ).

Risović [13] also suggested that b_bp(λ)/b_p(λ) can be overestimated with Mie theory when assuming a Junge-type particle distribution. However, our resultant b_bp(λ)/b_p(λ) values are quite similar to those measured in the Santa Barbara Channel (Figs. 1 and 3). The magnitude of b_bp(λ)/b_p(λ) generally increased with increasing n_p and ξ, which has been shown by Mie theory (e.g., Fig. 5 in [11] and Fig. 1 in [12]). Theoretical studies have also indicated that b_bp(λ)/b_p(λ) is strongly affected by the presence of sub-micrometer particles and that its magnitude varies with the slope of the PSD [11]. Observed magnitudinal variability of b_bp(λ)/b_p(λ) has been related to particle composition. Loisel et al. [34] indicated that, based on data collected in European waters, the amount of organic material has a strong influence on b_bp(λ)/b_p(λ) and detrital particles resulted in higher values of b_bp(λ)/b_p(λ). Boss et al. [15] also report that b_bp(λ)/b_p(λ) is strongly related to the index of refraction particles. Snyder et al. [20], however, do not find any quantifiable evidence of particle type (organic vs. inorganic) with the magnitude of b_bp(λ)/b_p(λ) and suggest that other factors need to be considered in quantifying the variability in b_bp(λ)/b_p(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, b_bp(λ)/b_p(λ) was not always spectrally flat and spectral b_bp(λ) did not always follow the widely used expression shown in Eq. (3). This suggests that spectral variability, in addition to magnitudinal variability in b_bp(λ)/b_p(λ) and b_bp(λ) could contain valuable information about the characteristics of particles. Importantly, variable spectral shapes (e.g., exponential versus power-law) at low n_p 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. [35] reported field-measured and simulated values of Q(λ) between 0 and 8 sr and Loisel and Morel [36] 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 [36] also reported f/Q values between 0.06 and 0.20 sr^-1 for the same simulations. K_d(λ) values and spectral shapes in Fig. 4 are typical of turbid coastal waters (e.g., Fig. 2 in [36]).

 

Fig. 4. 3-D plots of spectral AOPs as a function of n_p (legend in top left panel; colors correspond to plot lines and not fill colors) and ξ (x-axes).

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Spectral AOPs for variable n_p and ξ-values are shown in Fig. 4. K_d(λ) behaved similar to a_pg(λ), increasing with increasing n_p and decreasing with increasing ξ at all wavelengths. Remote sensing reflectance and R(λ) (irradiance reflectance) spectral effects were as expected with increasing n_p and ξ; spectral shapes shifted from blue-peaked (case I-like) at low n_p 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, n_p, and ξ. The f/Q ratio increased with increasing n_p at all wavelengths, however effects of variable ξ were spectrally and n_p 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 [10] were within 150% of “true” values, i.e. Mie-computed (Fig. 5). Lower values of a_t(λ) were estimated more accurately than higher values, likely because the algorithm was formulated based on a_t(λ) values of less than 0.3 m^-1 whereas our Mie-computed a_t(λ) values were much higher (mean of 0.5 m^-1 at 412 nm; Fig. 1). The degree of success of derivation of a_t(λ) was more influenced by ξ as compared to n_p or wavelength. For derivations of a_t(λ), the inversion algorithm performed best at moderate ξ-values (3.50 ≤ ξ ≤ 3.75) for all values of n_p 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 b_bp(λ) were dependent on n_p and ξ but not necessarily on wavelength or the magnitude of b_bp(λ). We show that more accurate b_bp(λ) estimations were achievable at 1.15 ≤ n_p ≤ 1.20 and at higher values of ξ (> 4.0; i.e. smaller particles). Derivations were underestimated for lower ξ.

 

Fig. 5. Derived a_t ^QAA(λ) and b_bp ^QAA(λ) compared to a_t ^Mie(λ) and b_bp ^Mie(λ). Wavelengths are represented by different colors from blue (400 nm range) to red (600 nm range). Open symbols in the first column denote the different ξ-values (triangles = 3.0, circles = 3.25, squares = 3.50, pluses = 3.75, stars = 4.0, and asterisks = 4.25) and closed symbols in the second column represent the different n_p-values (triangles = 1.01, circles = 1.05, squares = 1.10, crosses = 1.15, and stars = 1.20).

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To investigate potential sources of error in the semi-analytical algorithm for these particular IOPs and AOPs, we assumed a priori knowledge of b_bp(650) and then estimated b_bp(λ) and a_t(λ) using Eqs. (3) and (2). Results of this exercise are shown in Fig. 6a–d. Not surprisingly, derivations of spectral b_bp(λ) are much more accurate with knowledge of b_bp(λ) at one wavelength (Fig. 6c, d). Other than for n_p = 1.01, b_bp(λ) was generally overestimated, more so at the blue wavelengths when b_bp(650) is known. Thus, a_t(λ) was also overestimated. The absorption coefficient was generally overestimated by up to 130%, even at a_t(650) when b_bp(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.

The right two columns of Fig. 6 (e, f) show derivations of a_t(λ) using a simplified version of Eq. (4), where μ_u = 0.40, μ_d = 0.90, and dR/dZ is neglected [39]:

and by assuming R(λ) = 3.5 × r_rs(λ). Eq. (3) was then used to estimate b_bp(λ) (Fig. 6g, h). The use of a simplified Gershun’s equation yielded more accurate derivations of a_t(λ), mostly within 25% of Mie-computed values (Fig. 6e, f). The largest deviations in a_t(λ) and b_bp(λ) were found at higher n_p values. Unsuccessful derivations of b_bp(λ) at n_p = 1.01 in both cases are due to the fact that the spectral shape of b_bp(λ) at the lowest n_p-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 [17] 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 b_bp at one wavelength or by applying a simplified Gershun’s equation to first derive a_t(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.

 

Fig. 6. (a–d) Same as Fig. 5 but assuming b_bp(650) is known; (e–h) Same as Fig. 6 but for a_t(λ) derived using Eq. (5); b_bp(λ) was derived using Eq. (3).

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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 n_p, 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, b_bp(λ)/b_p(λ) varied spectrally with increasing n_p and ξ. In evaluating the effect of particle properties on AOPs, we found that the f/Q ratio exhibited escalating spectral variability with increasing n_p and ξ. In tests of the performance of a semi-analytical algorithm in predicting IOPs, the degree of success in the derivation of a_t(λ) was more influenced by the magnitude of a_t(λ) and ξ as compared to n_p or wavelength. Accurate derivations of b_bp(λ) were dependent on n_p and ξ but not necessarily on wavelength or the magnitude of b_bp(λ). Marked improvements in estimates of b_bp(λ) were enabled by a priori knowledge of b_bp at one wavelength. At 1.05 < n_p < 1.15, derivations of at(λ) and b_bp(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 n_p 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.