1. Introduction

Estimation of phytoplankton chlorophyll a concentration (Chl) plays a significant role in determining the rates and magnitudes of ocean primary productivity [1]. Observations from satellite-based instruments allow the measurements of varieties of ocean color (OC) information at spatial and temporal scales that cannot be substituted by other techniques. In the traditional Chl retrieval algorithms, the atmosphere and ocean system are decoupled using atmospheric correction scheme to mainly aim at estimating the influence of aerosol scattering. The procedure is conducted by comparing measurements with calculated radiance in red and near infrared bands, where the ocean can be assumed black. Then the best fitting aerosol modes are selected and extrapolated to estimate the radiance in shorter wavelengths [2–9]. Finally the Chl is determined from empirical or semi-empirical formulations related to the retrieved water leaving radiance [10–12]. Moreover, there are other algorithms using the direct inversion method, which is simultaneous retrieval of atmospheric and oceanic optical parameters using coupled radiative transfer (RT) model and statistically optimized schemes from multiple spectral or geometry information in one step [13–22]. It has been proved that all of those approaches have good accuracy and availability in the estimation of Chl.

Due to the fairly complicated process of those two types of algorithms using atmospheric correction procedure or nonlinear iteration optimization approach, a more simple method employing linear combination index (LCI) scheme was proposed to estimate the surface chlorophyll a concentration [23]. This approach aims at stabilization of accuracy by reducing processing time significantly for the satellite data, particularly for the high-resolution image data. In this method, aerosol scattering component is eliminated by a technique of mathematic transformation with a set of linear combination coefficients to make this method simpler and easier than the conventional atmospheric correction scheme. Then the Chl is derived from the relationship between LCI and Chl, which is determined by the bio-optical model or statistical regression from measurements. It has been proved that this algorithm has a good performance in the retrieval of Chl based on the simulated imagery [23], however, rare research has been done in its extension to real satellite data and validation of the retrieval from the perspective of radiative transfer model, which promoting a comprehensive investigation on its availability using actual data.

In this study, the LCI method is implemented to retrieve the surface chlorophyll a concentration in waters near Hokkaido using satellite observation data from MODerate resolution Imaging Spectroradiometer (MODIS)/Aqua instrument. To have a better estimation of current algorithm and assess the sensitivity of LCI to Chl and other influence factors, an oceanic numerical RT model coupled with a comprehensive bio-optical module is developed, where the roughness of air-water interface, influence of oceanic temperature and salinity are considered. Moreover, the error analysis and investigation of validity of this algorithm within a variety of atmospheric condition are also conducted. To estimate the retrieval accuracy from LCI method, the derived Chl are compared with the MODIS operational data products determined by the standard OC processing procedure. Moreover, validation of the retrieval from in situ measurements is performed using different bio-optical ocean models and linear combination coefficients. Finally, conclusion and perspectives are provided in the last section.

2. Data and methods

2.1 Satellite and in situ data

Due to the highly accurate instrument calibration and wide observation bands used, MODIS/Aqua level 1b calibrated radiance and geometry information obtained from the collection 6 data set derived from the MODIS land and atmosphere team are used in this study. In term of in situ measurement, the AERONET OC level 2 data [24] from a quality-assured scheme of the NASA Goddard space flight center are adopted for validation.

2.2 Linear Combination Index method and data preprocessing

The linear combination index method is a simplified scheme in the determination of Chl, which is based on the assumption that the measured reflectance combined in the visible and near infrared spectral bands can be related to Chl by a low-order polynomial, without explicit correction for aerosols.

For the atmosphere-ocean system, the top-of-atmosphere reflectance, $R^{*}$, after the correction of gas absorption and ignoring the influence of direct sun glitter and whitecaps, can be expressed as [25]:

In this study, diffuse transmittance of aerosols, $t_a(\lambda )$, is defined as being 1 [26], and$t_m(\lambda )=\exp [-{\tau }_m(\lambda )(\mu +{\mu }_0)/(2\mu {\mu }_0)]$, where $\mu$ and ${\mu }_0$ are the cosine of satellite zenith angle and solar zenith angle, respectively. The gas absorption coefficients are calculated by a correlated k-distribution approach [27] where several main absorptive gases of water vapor, carbon dioxide, ozone, nitrous oxide, carbon monoxide, methane, and oxygen are considered. To calculate $R_m$, a Rayleigh scattering look-up table (LUT), which considers the effects of polarization, surface pressure, and wind speed, is generated from a vector RT model [28] with the atmosphere divided into 30 layers using the 1976 U.S. standard model profile. The model has proven to be highly accurate in computing radiance and polarized radiance in the atmosphere [29]. The Lagrange polynomial interpolation method is then used to obtain $R_m$ from LUT. Other supplementary data, such as pressure and wind speed, are taken from the National Centers for Environmental Prediction reanalysis data. It is noted that the pixels with wind speed over 12 m s⁻¹ are excluded so that the influence of whitecap can be ignored. In addition, LCI is determined as the linear combination of two or more spectral bands of $R_c$ given by:

where the aerosol scattering component, $R^{\prime }$, which is the sum of pure aerosol scattering and molecule-aerosol interaction, is approximated by a polynomial of wavelength. To remove the influence of aerosol, the coefficients, $a_i$, are then selected to enable the accumulation of $a_i{\lambda }_i^{n_j}$ is 0. Due to systematic variations in aerosols and chlorophyll a concentrations, use of best fit from a combination of visible and near infrared measurement is suggested to determine the selected wavelengths [23]. It is studied that the aerosol types characterized by “$n_j$” are typically −1 and 0.3 near the Japanese area, and the combination of wavelength using “${\lambda }_i=487,547,866nm$” for MODIS/Aqua shows a good performance in the retrieval of Chl [30], since such a spectral combination can be not only useful to determine Chl from blue and green bands, but also beneficial in the correction of aerosols from infrared spectrum. As a result, the coefficients, $[a_i;i=1,3]$, are determined as $[a_{2,3}=-1.3150,0.3042]$ by equations of $[{\displaystyle \sum _{i=1}^3{a}_i{\lambda }_i^{n_j}}=0;j=1,2]$ under the assumption of “$a_1=1.0$”. After substantially reducing the effect of aerosols and other perturbing effects, the values of the LCI index are calculated by the sum of a polynomial of $R_w$, which is related to Chl and can be derived by bio-optical RT model.

2.3 Radiative transfer model

2.3.1 A Chl-controlled bio-optical module

In this study, a RT model coupled with a comprehensive bio-optical module is developed. Three optical property-altering components (pure seawater, phytoplankton, and CDOM) are considered for CASE 1 waters. The inherent optical properties (IOPs) of these substances are calculated from a set of empirical or semi-empirical formulas that are based on extensive observations or statistic theories proposed by previous studies. The imaginary part of the refractive index of pure seawater is taken from several data set that have been consistently merged from [31–33], with the absorption coefficient computed using$a_w=4\pi n_i(\lambda )/\lambda$, where $\lambda$ is the wavelength and $n_i(\lambda )$ is the imaginary part of the refractive index. The real part of the seawater refractive index is calculated using formulation [34] derived from [35] expression for the 300–800 nm range. For longer wavelengths, data from [31] are used. Moreover, it is studied that seawater absorption and scattering coefficient are influenced by ocean temperature ($T$) and salinity ($S$) with large changes at specific wavelengths [36]. Hence, the spectral dependent temperature and salinity are modeled based on the research of [37] and [38] as follows:

Figure 1 shows the variation of reflectance ($\rho =\pi I/[\cos ({\theta }_0)E_d]$) in nadir just above the ocean surface for various temperature and salinity in different Chl conditions, where $E_d$ is the incident solar irradiance in the direction of zenith angle of ${\theta }_0$ just above the ocean surface. It is demonstrated that $\rho$ increases almost linearly with an increase of oceanic salinity; the relative difference of $\rho$ in 412 nm can reach 17.5% and 10.1% (as shown by the solid-dotted line) when the salinity changes from 0 practical salinity units (PSU) to 40 PSU in chlorophyll a concentrations of 0.5 and 3.5 mg m⁻³, respectively. However, inconspicuous changes are shown for the dependence of $\rho$ on temperature; the relative difference is only about 2% when $T$ changes from 0°C to 27°C (as shown by the dashed line with dots). As there is a similar rangeability among the different viewing zenith angles [39], it is noted that salinity imposes a more significant effects on the radiation process in the ocean body than the variation in temperature. In addition, similar change patterns are also demonstrated in the relatively longer wavelength (674 nm) (shown by the line with triangles), although the extent of variation is a little less significant; this may be related to the more obvious absorption effects of seawater in those bands region.

 

Fig. 1 Variation of nadir reflectance in different salinity (symbolized by “S”) and temperature (symbolized by “T”) in chlorophyll a concentrations (symbolized by “C”) of 0.5 and 3.5 mg m⁻³. Solar zenith angle was set to 45°.

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In relation to the optical properties of phytoplankton, the absorption coefficient is computed using [40] empirical formula,

The scattering coefficient of phytoplankton, $b_{ph}(\lambda )$, and the backscattering fraction, $B_{ph}$, were calculated using [44] method:

To characterize the scattering angular distribution of phytoplankton, the Fournier–Forand (FF) phase function [45] is used. It is a Mie-theory-based analytical form, and has been proved to be more realistic than other analytical models developed thus far [46]. FF is written as follows,

where

where $n$ is the real part of the particles’ refractive index, $\mu$ is the slope parameter of the Junge distribution varied mostly between 3.0 and 5.0 [47], $\Theta$ is the scattering angle, and ${\delta }_{180}$ is the value of $\delta$ at 180°. With the FF scattering function, the backscattering fraction can also be obtained analytically [46],

Where ${\delta }_{90}^{}$ is $\delta$ evaluated at 90°. Furthermore, a simple linear relationship between $\mu$ and $n$ is proposed [46] to reproduce the FF fit to observation from Petzold [48] data, as expressed below,

Based on the formulas of Eqs. (7)-(11), the refractive index and slope parameter can be determined using $B_{bp}=B_{bp-FF}$ for a given Chl, which is similar to that of method [21,49].

The wavelength-dependent absorption coefficient of yellow substance $a_y(\lambda )$ related to the chlorophyll a concentration is parameterized by an exponential empirical relation based on the absorption coefficient of phytoplankton at 440 nm [50],

In order to decrease the influence of oceanic depth uncertainty in the upwell radiance, several correction procedures are conducted in this study. The depth of surface layer, $Z_{pd}$, which is defined as having a thickness corresponding to one attenuation length, is approximated from [51] as:

where $Z_e$ is the euphotic depth, which is defined as the depth where the downwelling irradiance is reduced to 1% of its value at the surface based on the formulation from [52].

The vertical distribution of $[\text{Chl}]$ is assumed to be Gaussian, where each layer of chlorophyll a concentration ${[\text{Chl}]}_{layer}(z)$ is defined as follows:

where $C_b$, $\xi$, ${\xi }_{\max }$, and $\Delta \xi$ are calculated according to the method of [51].

Given these assumptions, the bio-optical oceanic module for CASE 1 water is developed, in which all IOPs of hydrosol are determined by the value of $[Chl]$. Wavelength-dependent parameters, such as the whole absorption coefficient $a(\lambda )$, scattering coefficient $b(\lambda )$, backscattering coefficient $b_b(\lambda )$, single scattering albedo $\omega (\lambda )$, phase function $P(\Theta )$ and optical thickness $\tau (\lambda )$ at the geometric depth $Z$, can be expressed as follows:

2.3.2 Radiative transfer scheme

The RT in the ocean system is very similar to that in the atmosphere. However, the optical properties are different, particularly those of the refractive index owing to the refractive effects of the ocean surface. In this study, the RT scheme is developed based on the formulation of [28, 53, 54] using discrete ordinate and matrix operator hybrid method, which has been used in several studies of satellite-based and ground-based remote sensing in the atmosphere [55, 56]. The discrete ordinate method is used because it can transform the RT equation into a set of coupled ordinary differential equations; these formulations are related into matrix form and reduced to eigenvalue problem of symmetric matrices appearing in the basic matrix equation. The reflection and transmission matrices for each oceanic layer are then obtained. Then, adding theory is applied to determine the unknown integral constants and solve the inhomogeneous oceanic layer including the rough ocean surface. Finally, an analytical interpolation scheme is used to compute the radiance in any viewing angle and layer. The detailed RT scheme can be referred to [53, 54] and a validation of RT scheme is performed for several standard underwater optical problems summarized in Appendix A. It is noted that we use a full physical technique to calculate the water leaving radiance ($L_w$) as:

where $\mu$ and $\varphi$ are the cosine of the viewing zenith angle and relative azimuth angle, respectively. $L_w^m(\mu )$ is the mth order Fourier component of $L_w$ as calculated by:

where $L_u^m({\tau }_0^{-};{\mu }_i)$ means the mth order Fourier component of upwelling radiance just below the ocean surface; ${\mu }_i$ are the points for a discrete quadrature order of $N_s$. $T$ is the diffuse transmissivity function of the rough ocean surface, which is calculated using the formulation of [53].

One advantage of current model is that it only needs one input data (surface chlorophyll a concentration) in general cases, since other optical setting, such as vertical distribution, size parameters, refractive index and phase function of Chl are all automatically processed in this model to make it more flexible to be used in the field of retrieval. Another merit of this model is that it can also process more complex oceanic conditions, such as extreme salinity water.

3. Results and discussion

3.1 Sensitivity analysis of LCI index to chlorophyll concentration

To investigate the sensitivity of LCI index to the variation of Chl, we simulate RT in the model. The solar zenith angle is taken as an averagely 45° near the Hokkaido ocean region on 18 September 2007. The averaged oceanic salinity of 35.5 PSU and wind speed of 5 m s⁻¹ are also used. In addition, the ocean is divided into four layers with infinite depth. The vertical distribution of pigment is assumed to be Gaussian, and the Chl in each layer is determined by surface Chl based on Eq. (14). The solid line of Fig. 2 shows the variation of surface Chl as a function of the LCI index related to the water leaving reflectance, $R_w$, in a nadir direction. It is demonstrated that the dependence of the logarithm of chlorophyll a concentration is in significant agreement with the values of the LCI index, with a determination coefficient of up to 0.9943. The chlorophyll a concentration increases with a decrease of the LCI index, where the relationship between LCI and Chl can be summarized as:

 

Fig. 2 Sensitivity of LCI index to chlorophyll a concentration (Chl) simulated by the oceanic radiative transfer model for different wind speed, oceanic salinity and scattering angle (symbolized by “W”, “PSU” and “SCAG”, respectively)

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To further assess the stability of LCI index to other influence factors, we simulated the variation of LCI with Chl for different wind speed, oceanic salinity and scattering angle, as also shown in Fig. 2. The results indicate that LCI index is hardly susceptible to wind speed, with the relative difference of retrieved Chl from LCI less than 3% when the wind speed changes from 2 to 10 m s⁻¹. In terms of oceanic salinity, more impact on the value of LCI index than wind speed is demonstrated, particularly in much low oceanic salinity conditions, where the averaged relative difference of determined Chl is over 12% when the salinity varied from 5.5 to 35.5 PSU. The scattering angle, characterizing the geometry information of solar and satellite, influences the LCI index significantly in both low LCI and scattering angle conditions, with the relative difference of retrieved Chl over 20% when scattering angle changes from 155° to 100° in the condition of LCI less than 0. Nevertheless, the rangeability of LCI is not obvious in the common salinity and scattering angle conditions. With regard to the regional satellite imagery, the LCI index is significantly linear to the variation of Chl and not sensitive to other influence factors generally, which can be adopted as an independently representative index to retrieve Chl. However, correction for the LCI index should be conducted in much low salinity or scattering angles conditions.

3.2 Application to MODIS Imagery

The algorithm is then applied to a real satellite imagery obtained around the Hokkaido ocean region using MODIS/Aqua data. A comparison between the spatial distribution of Chl from MODIS level 2 products and those derived by the LCI method are shown in Fig. 3. The results demonstrate that relatively high Chl values (2 mg m⁻³ ~) are mostly located off the east coast of Hokkaido and Sakhalin, while low values (~0.2 mg m⁻³) are observed in the northeast and northwest ocean region of Hokkaido, as shown in Fig. 3(b). The general spatial distribution of retrieved Chl is significantly consistent with that generated from MODIS Chl products, which are determined by the standard atmosphere and Chl processing procedure, as shown in Fig. 3(a), even though the LCI method uses a simpler scheme to retrieve the Chl and does not have an extra atmosphere correction process.

 

Fig. 3 Comparison between Chl from (a) MODIS standard products and those derived from (b) LCI method on 18th September, 2007 in waters near Hokkaido

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The correlation coefficient (R), root mean square deviation (RMSD) and average percentage difference (APD) for those two products are up to 0.9702, 0.3756 mg m⁻³ and 13.89%, respectively. It is demonstrated that current algorithm has a good accuracy in the retrieval of Chl in comparison to that from the standard operational scheme. However, the Chl values in this study are a little bit underestimated in high Chl conditions compared with those retrieved from the MODIS standard algorithm, as shown in Fig. 4. A larger dispersion of retrieved Chl from these two algorithms in high Chl values is also demonstrated. To investigate possible reasons causing such a shape of scatter, a simple simulation experiment is conducted using the MODIS observed geometry information and Chl values to generate water-leaving radiances from the bio-optical ocean model. Then the retrieved Chl are achieved using Eq. (18). Figure 5 shows the distribution of retrieved relative error as a function of Chl. It is demonstrated that larger errors (over 40%) can be seen in higher Chl values, while the relative error is generally less than 20% when Chl is less than 2 mg m⁻³. In addition, more contaminated substance, such as non-algal and suspended particles, may be existed in the higher Chl water body to introduce more errors for the calculation of water leaving radiance using CASE 1 module.

 

Fig. 4 Scatter plot comparisons of Chl between LCI predictions and MODIS standard products (from Fig. 3)

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Fig. 5 Distributions of retrieved relative error from LCI method in each Chl value

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It is noted that the influence of aerosol scattering is eliminated using the linear combination coefficients in this algorithm. However, the availability and reasonability of such an approach in the correction of aerosols should be further assessed. In order to investigate the accuracy of this method in a variety of atmospheric conditions, particularly in a varying aerosol loading circumstance, the “real” aerosol scattering contributions, ${\displaystyle \sum _i[a_i{R}^{\prime }({\lambda }_i)]}$, to the LCI index are calculated by Eq. (3), where values of $R_w({\lambda }_i)$ are computed from MODIS standard OC data. It is demonstrated that the influence of aerosol scattering components in LCI index are mostly small in magnitude shown in Fig. 6(a), generally within ± 0.001, allowing the estimated retrieval error of Chl less than 25% in most cases. However, some relatively larger errors are still existed, where several values over ± 0.003, which may be caused by the existence of complex aerosol or incompatible geometry conditions. Figure 6(b) shows the spatial distribution of aerosol optical thickness (AOT) at 869 nm, results indicate that values of ${\displaystyle \sum _i[a_i{R}^{\prime }({\lambda }_i)]}$ in Fig. 6(a) are not significant even in the relative high AOT (~0.2) conditions. Compared with the individual $R^{\prime }({\lambda }_i)$, the linear combination is typically one to two orders of magnitude of smaller. Generally, such a mathematic transformation to remove the aerosol influence is reasonable and available in a variety of atmospheric conditions.

 

Fig. 6 Spatial distributions of aerosols scattering contribution to the LCI index (a) and aerosol optical thickness (AOT) at 869 nm (b).

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3.3 Validation from in situ measurements

To enable a more sufficient estimation of algorithm accuracy, the in situ data set from AERONET OC products [24] are collected from the USC_SEAPRISM site as validation. It is of note that the same values of parameters of “$n_j$” in Eq. (3) are adopted, as the site has relatively clear atmospheric conditions with an averaged aerosol optical thickness in 550 nm of 0.090, based on the AERONET observation; therefore, the influence of the aerosol type characterized by $n_j$ in the calculation of LCI index is small. Since the above analysis are performed based on the linear combination coefficients of “${\lambda }_i=487,547,866nm$” and formulation of Eq. (18) derived from the developed bio-optical RT model, we compare the retrieval results determined by other RT model and different linear combination coefficients of “${\lambda }_i=465,554,857nm;a_i=1.0,-1.5025,0.4582$” (hereafter C2) as a further investigation of current algorithm in this section. The model named Pstar [26] is adopted as comparison, which is a coupled atmosphere-ocean model where effects of polarization are considered. Different from current RT model, Pstar use different IOPs data sets for Chl without considering the absorption of detritus as well as CDOM, and calculating methods for absorption and scattering coefficients of seawater that influence of temperature and salinity are neglected. The Henyey-Greenstein phase function is used instead of FF function in Pstar model. To have a better comparison, the vertical distribution of Chl from Eq. (14) is also adopted. Figure 7(d) shows the sensitivity of LCI index to Chl using Pstar model (blue: polarization is considered; green: polarization is neglected) and C2. It is demonstrated that effects of polarization on the value of LCI index are very small and can be generally neglected. Pstar model estimates a lower Chl in the region of LCI larger than 0.004 and higher Chl in other regions than those derived by current model, which is caused by the different setting of IOPs of oceanic substances. The inversion results derived from those two models are shown in Figs. 7(a) and 7(b), which indicates that current bio-optical module have a better retrieval of Chl, with the RMSD and APD of 0.3075 mg m⁻³ and 22.39% in comparison to the in situ measurements. The retrieval results using C2 are also shown in Fig. 7(c), where the estimated Chl are generally overestimated among Chl lower than 1.0 mg m⁻³. Based on the above analysis, it is indicated that the retrieval accuracy of Chl from LCI method depends on the bio-optical model selected, particularly the setting of IOPs of oceanic substance, and linear combination coefficients used. In general, current bio-optical module combined with the coefficients of “${\lambda }_i=487,547,866nm;n_i=1.0,-1.3150,0.3042$” generate a more accurate retrieval of Chl, while more uncertainties might be introduced for the inversion of Chl in high concentrations circumstance.

 

Fig. 7 Comparison of LCI estimated Chl (mg m⁻³) from (a: current model; b: Pstar model; c: C2 coefficients and current model) with those of in situ products and sensitivity of LCI index to Chl for different models and linear combination coefficients (d)

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4. Discussion and Conclusion

In this paper, an inversion algorithm is implemented to retrieve the surface chlorophyll a concentration using linear combination method without explicit correction of aerosol scattering. To extend the LCI method to real satellite data and validation of retrieval, the MODIS/Aqua and AERONET in situ data set are selected to investigate the availability of current algorithm. Moreover, an oceanic radiative transfer model coupled with a comprehensive bio-optical module is developed to estimate the sensitivity of LCI to Chl and other influence factors. It is studied that the LCI index is significantly linear to the variation of Chl and not sensitive to other influence factors in common circumstances. However, there is over 12% uncertainty for the inversion of Chl from LCI index in much low oceanic salinity conditions, besides, the geometry also influences the LCI index significantly in both low LCI and scattering angle conditions, with the relative difference of estimated Chl over 20%. Retrieval results indicated that the derived Chl by Eq. (18) are highly consistent with MODIS operational OC products in waters near Hokkaido, with the correlation coefficient, root mean square deviation and average percentage difference of 0.9702, 0.3756 mg m⁻³ and 13.89%, respectively, even though there is not additional aerosol scattering correction in this study. Nevertheless, current algorithm seems to underestimate the Chl in high value conditions compared with the standard products. Computation of the “real” aerosol scattering radiance also proves that the influence of aerosols on LCI index is generally within ± 0.001, allowing the estimated retrieval error of Chl less than 25% in most cases. Finally, validation from in situ measurements is conducted using different RT models and coefficients, where a better retrieval case is indicated using current bio-optical module and coefficients of “${\lambda }_i=487,547,866nm$”.

In this study, we force the aerosol diffuse transmittance $t_a(\lambda )$ to be 1 and neglect the multiple scattering between atmosphere and ocean in Eq. (1), which may impact the algorithm accuracy to some extent, since the LCI index is rigorously expressed as $\text{LCI}={\displaystyle \sum _i[a_i{t}_a({\lambda }_i)R_w({\lambda }_i)/(1-s({\lambda }_i)R_w({\lambda }_i))]}$ derived from Eqs. (1)-(3). Simulation results demonstrate that the relative difference of retrieved Chl is generally less than the relative variation of $t_a(\lambda )$ in percentage, except for extreme high or small LCI values conditions with higher retrieval errors. Mostly, the uncertainty of $t_a(\lambda )$ exerts little influence on the determination of Chl with less than 10% variation among Chl ranges from 0.5 to 2.5 mg m⁻³ when $t_a(\lambda )$ change from 1.0 to 0.8, since the LCI values are near zero in those conditions. Moreover, the effect of atmospheric reflectance for the water leaving radiance at its base, $s(\lambda )$_, on LCI index is not obvious, with the relative difference of derived Chl less than 3% in most cases when values of $s(\lambda )$ increase from 0 to 0.3.

It is of noted that the LCI method is able to estimate Chl very effectively and efficiently, however, more attention is required to use this method in more complex circumstances, such as in turbid waters or absorption aerosol loading, where more advanced iterative algorithms, such as GRASP [18] or standard atmosphere correction schemes are suggested. Besides, future work is required to further decrease the influence of aerosol scattering on LCI index using optimization procedure for linear combination coefficients or more spectral information, particularly in complicated geometry conditions.

Appendix A Validation of radiative transfer scheme

Validation of RT scheme is performed using the underwater light fields of several standard RT problems. Details of this model comparison project can be found in [57]. Here five standard problems covering specific aspects of RT in the ocean, including the effects of high absorbing and high scattering waters, scattering by molecules and particulates, surface roughness and finite-depth bottom are used to estimate the accuracy of model. In brief, the five problems used are as follows:

• Problem 1 (P1): an unrealistically simple problem
• Problem 2 (P2): a base problem using actual IOPs for the seawater
• Problem 3 (P3): a base problem but with stratified water
• Problem 4 (P4): a base problem but with a finite depth bottom
• Problem 5 (P5): a base problem but with a wind-blown sea surface

For each of these problems, the water body is taken to be horizontally homogeneous with the real part of the refractive index of water as 1.340. The depth below the ocean surface can be specified using either the non-dimensional optical depth, $\tau$, or the geometric depth, $z$, in meters. The first four problems assume that: (a): the ocean surface is flat; (b): there is no atmosphere; (c): the solar zenith angle is 60°; (d): the incident solar irradiance just above the ocean surface is 1 w m⁻² nm⁻¹ on a surface perpendicular to the sun’s ray; (e): there is no inelastic scattering or other source of light in the ocean body; and that (f): the water is either highly scattering ($\omega =0.9$) or highly absorbing ($\omega =0.2$). In addition, each of the five standard problems is defined using exceptions to the above assumptions. These specific problem definitions are as follows: (P1) the water is vertically homogeneous and infinitely deep where the Rayleigh phase function is used; (P2) is similar to P1, but the realistic Petzold phase function [48] is used; (P3) is similar to P2, but the water body is assumed as a highly stratified medium with non-homogeneous layers and the absorption and scattering coefficients of Chl are calculated as [58, 59]:

(19)$$a_{ph}(z)=0.04{[Chl(z)]}^{0.602};b_{ph}(z)=0.33{[Chl(z)]}^{0.620}.$$

In addition, a pigment profile with depth is taken from [60], and the absorption and scattering coefficients for pure sea water at 500 nm are 0.0257 and 0.0029 m⁻¹, respectively. Furthermore, (P4) is similar to P2 except that a finite-depth bottom is imposed (the bottom is assumed to be an opaque, Lambertian reflector surface with reflectance albedo $A_{lambert}=0.5$ for irradiance at a depth of ($\tau =5$); (P5) is similar to P2, but the ocean surface is assumed to be rough with a wave slope standard deviation of 0.2, corresponding to a wind velocity of 7.23 m s⁻¹ based on [61] capillary-wave spectrum at a solar zenith angle of 80°.

Figure 8 shows a comparison of the results of downward irradiance ($F_d$), upward scalar irradiance ($F_{0u}$), and upward nadir radiance ($L_{nadir}$) at different ocean depths calculated by the RT model, with standard values denoted by the symbol of “AVE.” It is demonstrated that the results calculated by the RT model have a good accuracy for all the standard underwater radiation problems shown in this study, with the average difference of radiance and irradiance are both less than 1.0%, indicating that the present model simulated the radiation process well in ocean body.

 

Fig. 8 Inter-comparison between the predictions in this study (shown as continuous line) with standard values [57] (shown as dot) for several problems (a: P1; b: P2; c: P3; d: P4; e: P5)

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Funding

MOEJ/GOSAT/GOSAT2; JAXA/EarthCARE/GCOM-C; MOEJ/ERTDF/S-12; JST/CREST/JPMJCR15K4; National Natural Science Foundation of China (NSFC) (41590875, 41475031, 41571130024); Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology (KLME1509).

Acknowledgments

The authors express their sincere thanks to the AERONET and MODIS science teams. The authors are grateful to Dr. Robert Frouin, Dr. Hiroshi Murakami, Dr. Nan Li for useful discussions. We thank the anonymous reviewers for providing constructive comments, which strengthen the manuscript largely.

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