Phytoplankton are an integral part of the ocean ecosystem and play a fundamental role in global biogeochemical cycles such as carbon export from the well-lit surface ocean to deeper waters [1]. Traditional methods of analyzing temporal and spatial variations in the marine food web structure and the efficiency of the ocean as a sink for atmospheric carbon dioxide are costly, time consuming, and sparse [2,3]. Satellite observations of phytoplankton optical properties are a necessary next step in improving this type of analysis. A combination of the Lorenz–Mie theory and optical observing systems has been previously used to retrieve the variability in the abundance and size of different classes of oceanic particulate constituents [4]. However, most bio-optical models used in such retrievals still assume that oceanic constituents consist only of homogeneous spheres. This simplification leads to biases in retrieved inherent optical properties (IOPs) such as the hemispherical backscatter coefficient [5,6]. The NASA Plankton, Aerosol, Cloud, ocean Ecosystem (PACE) mission will deliver UV–VIS hyperspectral, multiangle radiance and polarization measurements that will provide more information for the characterization of oceanic constituents [7]. To take full advantage of the information contained by these measurements in real (operational) time requires the use of more advanced particle models and inversion techniques to derive phytoplankton size distributions (PSDs) and photophysiological characteristics. In particular, using coated spheres to represent phytoplankton particles leads to less biased retrievals in the hemispherical backscatter coefficient [5,6]. Here we present a coated sphere lookup table (LUT) enabling fast retrievals of UV–VIS phytoplankton IOPs using measurements from PACE’s and other existing and future remote sensing instruments.

Our LUT is designed to provide a fast way to compute the phytoplankton IOPs based on their physical properties. We model phytoplankton as coated spheres with a pigment-like shell surrounding a cytoplasm-like core. We assume that the ratio of the core radius to the whole particle radius equals 85%, which is a static value but consistent with the dynamic range [6]. Covered phytoplankton radii range from 0.15, the minimum size of Prochlorococcus under high light [8,9], to 100 $\mathrm {\mu }$m. The list of IOPs covered by this phytoplankton LUT is similar to our aerosol and cloud LUTs [10,11]. These phytoplankton-specific IOPs include the normalized scattering matrix; the absorption, extinction, and scattering coefficients; and the hemispherical backscatter coefficient. The single-scattering IOPs are then used as inputs for multiple-scattering radiative transfer simulations [12,13]. Our LUTs can be used with advanced single- and multi-instrument remote sensing retrievals as a part of the forward model (e.g., radiative transfer model or lidar equation) for an arbitrary lidar, polarimeter, or radiometer wavelength in the range from 0.355 to 1.065 $\mathrm {\mu }$m [14,15].

For radiative transfer calculations, one needs the normalized scattering matrix that relates the incident and scattered Stokes parameters. In the Lorenz–Mie theory of light scattering by coated spheres, this matrix can be represented as [12,13]

The elements of the matrix $\mathbf{P}\left (\psi,m,\lambda,q\right )$ can be computed as

The phytoplankton scattering coefficient $b_{\rm ph}\left (m,\lambda,q\right )$ that appears in Eq. (2) as well as the extinction (attenuation) coefficient $c_{\rm ph}\left (m,\lambda,q\right )$ can be computed as

We compute the phytoplankton absorption coefficient as

The best way to organize the Lorenz–Mie scattering LUT is to use the scale invariance rule [10,19,20]. The scale invariance rule (SIR) exploits the fact that Lorenz–Mie computations are actually done using the size parameter $x=2\pi r/\lambda$ that relates radius and wavelength [12,13]. For a given CRI, we can establish a direct connection between the efficiencies [see Eqs. (2) and (3)] at wavelengths $\lambda$ and $\lambda _{\rm r}$ using a scaling in the radius domain as

The scattering angle quadrature consists of 123 angles $\psi$ in the range from 0 to 180° (see Table 1). As in our aerosol-focused LUT [10], the quadrature spacing gradually reduces down to 0.2° near angles of 0 and 180° because the elements of the scattering matrix $\mathbf{P}$ [see Eq. (1)] can rapidly change there [21]. The rate of change in $\mathbf{P}$ is smaller between the angles of 10 and 170° which allows for a coarser 2° spacing. The IOP values for the other angles $\psi$ can be estimated using interpolation [10].

Table 1. Scattering Angles Included in the SIR-Ph LUT

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The quadrature for the complex refractive index consists of 12,800 values and is designed to cover a range of CRIs modeled for phytoplankton for the core [6,18] and for the shell [6,16–18]. The quadrature for the shell’s real part $\left (m_{\rm R}^{\rm s}\right )$ is 20 values ranging between 1.05 and 1.24 with a step of 0.01, and for the shell’s imaginary part $\left (m_{\rm I}^{\rm s}\right )$, it is 64 values including 10^-7 along with 63 log-equidistant values between 10^-5 and 0.3. For the core, the real part $\left (m_{\rm R}^{\rm c}\right )$ is fixed at 1.02, and there are 10 values for the imaginary part $\left (m_{\rm I}^{\rm c}\right )$ including 0 along with 9 log-equidistant values between 10^-5 and 10^-3.

The radius quadrature consists of 650 log-equidistant grid bins to cover the phytoplankton radii from $r_{\rm min}=0.05$ to $r_{\rm max}=100\,\mathrm {\mu }$m at the reference wavelength $\lambda _{\rm r}=0.355\,\mathrm {\mu }$m. The SIR-Ph LUT coefficients are computed over the integration intervals formed by the neighboring radius grid bins [10]. We use Simpson’s rule with 10⁵ points over each interval and the Peña et al. program [22–24]. Let us mention that at the wavelength $\lambda =1.065\,\mathrm {\mu }$m, the covered phytoplankton radii after SIR adjustment will range from $r_{\rm min}^{\lambda =1.065\,\mathrm {\mu }{\rm m}}=\frac {1.065}{0.355}\times 0.05=0.15$ to $r_{\rm max}^{\lambda =1.065\,\mathrm {\mu }{\rm m}}=\frac {1.065}{0.355}\times 100=300\,\mathrm {\mu }$m, which is the feature of our SIR LUTs [10]. The wavelength $\lambda =1.065\,\mathrm {\mu }$m is the maximal wavelength of our interest for PACE ocean color applications. We choose the maximum wavelength for the LUT based on the concept of the “skin depth,” the nominal depth of the water column where particles can contribute significantly to the water leaving radiance signal. At wavelength $1.065\,\mathrm {\mu }$m, the skin depth of the water column decreases to about 10 cm, and the particle load would have to increase to well outside our valid parameter space for retrievals. By selecting the minimal SIR-Ph LUT radius as $r_{\rm min}=0.05\,\mathrm {\mu }$m, we guaranteed that the targeted radii range from 0.15 to 100 $\mathrm {\mu }$m is covered for all desired wavelengths.

To examine the capabilities of SIR-Ph LUT, we compute the IOPs [see Eqs. (2)–(5)] using the LUT and compare them to the simulated truth values. We compute the simulated truth using the Peña et al. program [22–24] and Simpson’s rule for integration with 10⁸ radius grid bins from 0.15 to 100 $\mathrm {\mu }$m [10,11]. Earlier we set a target of $\pm 1{\% }$ precision at all wavelengths between 0.355 and 1.065 $\mathrm {\mu }$m, which cover the channels provided by most existing and planned lidars, spectrometers, and polarimeters including for the NASA PACE and Atmosphere Observing System (AOS) missions [7,25].

For the comparisons, we conducted 10⁵ random $\lambda$–CRI–PSD unit tests. We use monomodal lognormal PSDs $n\left (r\right )$ defined by effective radius $r_{\rm eff}$ and effective variance $\nu _{\rm eff}$ [10]. A random number generator provides uniformly distributed values of wavelength from 0.355 to 1.065 $\mathrm {\mu }$m, effective radius from 0.1 to 5 $\mathrm {\mu }$m, effective variance from 0.05 to 0.6, shell’s real part of the CRI from 1.05 to 1.24, shell’s imaginary part of the CRI from 10^-7 to 0.3, and core’s imaginary part of the CRI from 0 to 0.001. The core’s real part of the CRI is fixed to 1.02. Using these inputs, we compute the simulated truth values of the IOPs [see Eqs. (2)–(5)] and compare them to the corresponding SIR-Ph LUT values.

We found that the $P_{12}\left (\psi \right )$ element of the normalized scattering matrix is the most difficult IOP to estimate, as in our previous LUT studies [10,11]. The $\pm 1{\% }$ precision was not achieved at multiple angles in 0.1% (or 84) of the test cases. Figure 1 shows the distribution of problematic cases with all of them having the shell’s real part below 1.08 and the imaginary part below 0.013 [see Fig. 1(a)]. The relative difference in Fig. 1(e) is computed as

 

Fig. 1. Distribution of problematic cases with errors larger than 1% for (a) and (b) microphysical properties, (c) wavelength, (d) scattering angles, and (e) relative difference for scattering matrix element $P_{12}\left (\psi \right )$.

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All other IOPs were computed using the SIR-Ph LUT within the targeted $\pm 1{\% }$ precision for all test cases.

Let us provide an example of a practical application for the SIR-Ph LUT using the hyperspectral synthetic dataset for PACE algorithm development available from the PANGAEA database [26]. This dataset was constructed using a selection of 720 phytoplankton absorption spectra from over 4,000 spectra available in the NOMAD SeaBASS database [27]. These selected spectra were retained as baseline variables and were associated with a concurrently measured chlorophyll a (Chl a) concentration. Synthetic spectra were computed using parameterizations generated in accordance with the constraints outlined in the International Ocean Colour Coordinating Group (IOCCG) Report 5 [28]. Specifically, phytoplankton beam attenuation spectra were derived as a power law distribution with respect to wavelength, with the parameters generated randomly from the IOCCG distributions. This beam attenuation was then used to calculate backscattering spectra for phytoplankton, which were set as 1% of the value of the beam attenuation minus the absorption spectra. For the purposes of this Letter, we conducted inversions on three test cases associated with low, medium, and high Chl a concentrations of about 0.1, 1.5, and 3 mg m^-3, respectively. The synthetic $a_{\rm ph}$ and $b_{\rm bph}$ data chosen for these Chl a concentrations are shown by the solid curves in Fig. 2.

 

Fig. 2. PANGAEA database’s (solid lines) and Monte Carlo retrieved (dashed lines) phytoplankton absorption and hemispherical backscatter spectra for the three test cases representing low, medium, and high Chl a concentrations.

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For each test case, we attempted to reproduce the synthetic IOCCG data for the phytoplankton absorption and hemispherical backscatter coefficients at 61 wavelengths ranging from 0.4 to 0.7 $\mathrm {\mu }$m at 0.005 $\mathrm {\mu }$m resolution. We used the Monte Carlo method under the assumption that at least some of the PANGAEA optical datasets can be reproduced if we model phytoplankton as coated spheres. To make our model more realistic, we considered the shell’s CRI and the core’s imaginary part of the CRI to be size- and wavelength-dependent [29]. We used the power law distribution to model the phytoplankton PSD with randomly generated slope parameters ranging from 2.5 to 6 [4,30]. The integration in terms of radii [see Eqs. (2)–(5)] was done from 0.15 to 100 $\mathrm {\mu }$m.

Table 2 shows that the retrieved Chl a concentrations have the same order of magnitude as the PANGAEA database’s and the difference between them stays under 48%. The retrieved phytoplankton absorption and hemispherical backscatter spectra stay within 25% for each of the 61 measurement wavelengths (see Fig. 2). We consider the 25% error bar to be reasonable for PANGAEA’s optical datasets. The Monte Carlo retrieved slope parameters are inversely related to Chl a concentration (see Table 2), suggesting that as Chl a concentrations increase, the slope parameters decrease, and the PSD, as expected, becomes dominated by larger size phytoplankton classes [31].

Table 2. Results of Phytoplankton Property Retrievals

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In conclusion, the SIR-Ph LUT was specially designed for the NASA PACE mission, but the single-scattering phytoplankton IOPs have a number of potential applications. The LUT can be applied to interpret radiometric, polarimetric, lidar, and in situ ocean remote sensing measurements at wavelengths in the UV–NIR. The precision of LUT-provided IOPs is nearly equivalent to the direct integration, but the calculations are done up to 10⁴ times faster, within a fraction of a second instead of hours.