1. Introduction

Remote sensing reflectance (R_rs(λ), in sr⁻¹), defined as the ratio of water-leaving radiance (L_w(λ), in µW/cm²/nm/sr) to the downwelling irradiance just above the surface (E_d(0⁺, λ), in µW/cm²/nm), is a fundamental property in ocean color remote sensing [1]. The spectrum of R_rs(λ) is governed by water constituents in the upper water column and can be used to interpret in-water properties from satellite observations, such as the inherent optical properties (IOPs) and chlorophyll concentration [2–5]. Accurate determination of R_rs(λ) is, therefore, crucial for the vicarious calibration of satellite sensors [6], as well as validation of ocean color algorithms, including the atmospheric correction and the retrieval of various optical and biogeochemical parameters [7–11].

The experimental determination of R_rs(λ) requires measurements of both L_w and E_d. Accurate measurement of spectral E_d(0⁺, λ) in the field depends on the rigorous calibration of an irradiance sensor as well as its deployment [12], while the determination of L_w(λ) is more complicated [13,14]. There are three conventional approaches to measure L_w(λ) in the field, i.e., the above-water approach (AWA), the in-water approach (IWA), and the recently advocated on-water approach (OWA) [14–16]. The AWA measures total upwelling radiance (L_t(λ)) and sky radiance (L_i(λ)) from reciprocal zenith angles and derives L_w(λ) after removal of the surface-reflected skylight from L_t(λ) [17]. However, AWA has strict requirements on observing geometry, and a complete correction for the surface-reflected sun and sky radiance remains challenging, especially when the ambient light field is not stable, or the sea surface is roughened [18–20]. The IWA derives L_w(λ) from the profile measurements of the upwelling radiance (L_u(z, λ)) within the water column. The IWA assumes an exponential decreasing of L_u(z, λ) with depth and requires sophisticated postprocessing to obtain L_w(λ), including the extrapolation of L_u(z, λ) to the upwelling radiance just beneath the surface (L_u(0⁻, λ)) and the conversion of L_u(0⁻, λ) to L_w(λ). More importantly, the assumption of exponential variation of L_u(z, λ) may cause uncertainties in conditions such as near-surface optical stratification, inelastic scattering (Raman, fluorescence), shallow bottom, or variability in the angular distribution of the upwelling radiance [14,21]. Note that both AWA and IWA do not directly measure L_w(λ), but rather measure various related components and subsequently derive L_w(λ) from these components. The OWA, particularly the skylight-blocked approach (SBA), can directly measure L_w(λ) as the surface-reflected skylight is blocked off by a custom apparatus [21,22]. However, this “extra” component also introduces self-shading to the measured L_w(λ) that needs to be corrected.

To meet this requirement related to the SBA measurements, a shade correction scheme, termed as Shang17 hereafter, has been proposed. The shade error ɛ(λ) of obtained R_rs(λ) by SBA is defined as (the wavelength argument λ is omitted in the following for brevity when not necessary) [23],

While ${R}_{{rs}}^{{shade}}$ is available from SBA measurements, the shade error (ɛ) is required to be determined to obtain ${R}_{{rs}}^{{true}}$.

The impact of self-shading was first discussed and evaluated for the in-water instruments where the presence of a sensor, particularly its housing, will affect the in-water light field and result in errors in the measured upwelling radiance [24–27]. Gordon and Ding [25] approximated the shade related error as a function of the radius of sensor housing (R, in m), the in-water solar zenith angle (θ_w), and the total absorption coefficient (a, in m⁻¹). Note that θ_w can be converted from solar zenith angle (θ_s) by ${\theta _w}\textrm{ = }\arcsin ({\theta _s}/{n_w})$ with n_w as the refractive index of water and taken as 1.34. Although the SBA system includes a partial immersed cone, its impact on the self-shading is negligible based on Monte Carlo simulations (unpublished results). Therefore, Shang et al. [23] expanded the approximation of Gordon and Ding [25] and obtained a model for ɛ for the SBA system as,

Thus, it is necessary to determine both a and b_b to estimate ɛ, where a spectral optimization scheme for ${R}_{{rs}}^{{shade}}$ is used for this goal in Shang17 [23]. In Shang17, modeling ${R}_{{rs}}^{{shade}}$ requires parameterizations of the absorption and backscattering coefficients of primary water optical constituents, i.e., water molecule, phytoplankton, colored dissolved organic matter (CDOM), and non-algal particles (NAP, including mainly the detritus and mineral). While contributions from water molecules can be considered constant [28–31], parameterizations of the other components IOPs could be environment-dependent, and there are no guarantees of perfect matching with the target water [32,33]. As indicated in many studies, a mismatch in the spectral shapes of the component IOPs will result in errors in the derived a and b_b spectra from spectral optimization [34–36]. These errors will further be propagated into the estimated shade error, and subsequently, the desired ${R}_{{rs}}^{{true}}$. This is particularly true for the modeling of the a_ph spectrum, which varies widely in natural waters in both magnitude and spectral shapes due to different pigment compositions and cell sizes [37–39]. It is, therefore, impossible to use a single mathematic expression to parameterize spectral a_ph for all types of waters.

Given that the shade error is dependent on the bulk absorption and that the a_ph spectral shape of the target water is not always known a priori, a correction scheme avoiding the parameterizations of component IOPs is strongly desired. In this study, a novel scheme is proposed to correct the shade error for R_rs obtained by SBA without any assumptions of the spectral shapes of the component absorption coefficients. Evaluations with both simulated and in-situ datasets demonstrate that the proposed scheme is robust for waters from clear to very productive. The potential factors that could affect the validity of this novel scheme are also evaluated and discussed.

2. Novel shade correction scheme

One of the key steps in Shang17 is to model spectral ${R}_{{rs}}^{{shade}}$ from parameterized spectral a, b_b, and ɛ according to Eqs. (1)–(4). Specifically, ${R}_{{rs}}^{{true}}$ can also be modeled from a and b_b based on theoretical analyses and numerical simulations of the radiative transfer equation [2,40],

Here b_bw and b_bp are the backscattering coefficients of water molecule and particles (b_b= b_bw + b_bp), g_w is the model parameter for molecular scattering, and g_p is a model parameter for particle scattering phase function. The values of spectral b_bw are known [29], while g_w, g₀, g₁, and g₂ are constants for a given light geometry and particle phase function and are taken as 0.113, 0.197, 0.636, and 2.552 for nadir-viewed r_rs in this study [41].

As shown in Eqs. (3)–(4), one can estimate the shade error when a and b_b are known. To avoid the uncertainties associated with the modeling of the component absorption coefficients, we propose to derive spectral a and b_b from ${R}_{{rs}}^{{shade}}$ following the merits of the quasi-analytical algorithm (QAA, Lee et al. [2]). This innovative scheme is termed ShadeCorr_QAA, with its detailed steps described in the following.

Step 1: Assuming the absorption contributions from non-water constituents at 750 nm are negligible, so that a(750) = a_w(750), with a_w the pure water absorption coefficient and is adopted from Smith and Baker [42] for the 700 - 750 nm domain. Since SBA blocks off the surface-reflected light, at least under ideal conditions, SBA-obtained R_rs is free from the surface-reflected sun and sky radiance and can be considered of high precision in the near-infrared band (e.g., 750 nm), while the self-shading will bring a bias on the desired R_rs value. As a(750) is already known, the only unknown for ɛ(750) is b_bp(750). With only one variable (i.e., b_bp(750)) for ${R}_{{rs}}^{{shade}}\textrm{(750)}$, one can solve b_bp(750) numerically by minimizing the difference between measured ${R}_{{rs}}^{{shade}}\textrm{(750)}$ and modeled ${R}_{{rs}}^{{shade}}\textrm{(750)}$. The cost function (err) for this minimization is defined as,

Step 2: With derived b_bp(750) from Step 1, b_bp at the shorter wavelengths can be calculated following the power-law expression,

Step 3: For any wavelength where ${R}_{{rs}}^{{shade}}\textrm{(}\mathrm{\lambda }\textrm{)}$ is available, when b_bp(λ) is known after Step 2, the only unknown for ɛ(λ) and ${R}_{{rs}}^{{true}}\textrm{(}\mathrm{\lambda }\textrm{)}$ becomes a(λ). Similar to Step 1, a(λ) can be numerically solved using minimization. The same cost function of Eq. (8) is used in this step. Therefore, spectral a can also be derived.

Step 4: The shade error is subsequently computed using Eqs. (3)–(4) from the derived spectral a and b_b. Consequently, the shade-corrected R_rs for SBA measurements can be obtained following Eq. (2).

3. Data used for evaluation

To evaluate the robustness of ShadeCorr_QAA, two datasets from HydroLight simulations and controlled experiments in a custom tank and a lake were acquired in this study.

3.1. Simulated dataset

The simulated dataset included ${R}_{{rs}}^{{shade}}\textrm{(}\mathrm{\lambda }\textrm{)}$ and ${R}_{{rs}}^{{true}}\textrm{(}\mathrm{\lambda }\textrm{)}$, with ${R}_{{rs}}^{{true}}\textrm{(}\mathrm{\lambda }\textrm{)}$ simulated using HydroLight 5.3 (Sequoia Scientific, Inc.) with input IOPs from user-supplied total absorption (a) and attenuation (c) coefficients. The spectral a and c used in HydroLight were adopted from the published dataset of Craig et al. [43]. It is worthy to point out that a_ph spectra in the Craig dataset are mined from extensive in-situ measurements across the ocean to coastal waters that are archived on NASA’s SeaBASS repository (SeaWiFS Bio-optical Archive and Storage System, https://seabass.gsfc.nasa.gov/). The phase function for detritus-sediment particles adopts the ‘Petzold average’ with a backscattering ratio of 0.018 [44], while a 1% Fournier-Forand function is used for phytoplankton particles [45]. Note that the same values of a_w and b_bw used in the shade correction scheme were used in the HydroLight simulations. The wind speed is set as 5 m/s, and the solar zenith angle (θ_s) is set as 30°. The water column is assumed infinitely deep and homogeneous. For the inelastic scatterings, chlorophyll fluorescence is included, while the Roman scattering and CDOM fluorescence are not considered. In total, 720 ${R}_{{rs}}^{{true}}\textrm{(}\mathrm{\lambda }\textrm{)}$ covering 350 to 800 nm with an interval of 10 nm were simulated.

${R}_{{rs}}^{{shade}}\textrm{(}\mathrm{\lambda }\textrm{)}$ spectra were further synthesized based on Eq. (2), with the shade error computed from the known a and b_b following Eqs. (3)–(4) with θ_s as 30° and cone radius as 45 mm. Note that the dataset in Craig et al. [43] includes many points having extremely high absorption coefficients, such as a(440) value is over 20.0 m⁻¹ (effectively ‘black’ water), which results in the computed shade error close to 100% for a radius of 45 mm and unrealistic shade-corrected R_rs by both ShadeCorr_QAA and Shang17. Therefore, simulations with a(440) larger than 20.0 m⁻¹ (N = 46) are discarded. As a result, a total number of 674 simulations are used in the subsequent analysis, with the simulated ${R}_{{rs}}^{{true}}\textrm{(}\mathrm{\lambda }\textrm{)}$ showing in Fig. 1(a).

 

Fig. 1. Spectral ‘true’ R_rs (a) of the HydroLight simulated dataset and (b) measured by the above-water approach from controlled lake and tank experiments.

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3.2. In-situ dataset

In a previous effort [46], several controlled experiments were carried out in a custom tank and a lake to evaluate the factors that could contribute to the self-shading effects of SBA-obtained R_rs, such as θ_s (∼18° - 64°), cone radius (22 mm and 45 mm), and IOPs. The set-up of the tank and lake experiments is described in detail in Lin et al. [46]. Briefly, four experiments were conducted in a lake to evaluate mainly the impacts of θ_s and cone radius. The ‘true’ R_rs was first determined by AWA under well-controlled conditions, i.e., clear sky and calm water, followed by SBA-obtained R_rs using two cones with different sizes, respectively. Besides, four extra sets of radiometric measurements were taken from a large black water tank (2.2 m in diameter and 2 m in height), filled by well-mixed lake waters and tap waters. The proportion of tap waters was set from 0% to 30% to form an IOPs gradient of the tank waters. The same measure sequence of ‘true’ R_rs and shade-bearing R_rs as that in the lake experiments was used. Collectively, 16 pairs of AWA-determined ‘true’ R_rs and SBA-obtained shade error-included R_rs are employed in this study to evaluate the performance of ShadeCorr_QAA. The spectra of AWA-determined ‘true’ R_rs are presented in Fig. 1(b).

3.3. Evaluation metrics

Statistical measures are introduced in this study to quantitatively evaluate the performance of shade correction algorithms, which include the normalized root mean square difference(nRMSD), the median percentage difference (MPD), and the absolute percentage difference (APD). These metrics are defined as:

4. Results

4.1. Evaluation with the simulated dataset

The performance of ShadeCorr_QAA is first evaluated with the HydroLight simulated dataset. ShadeCorr_QAA is applied to the synthesized ${R}_{{rs}}^{{shade}}$, and the shade-corrected R_rs is then compared with the HydroLight simulated R_rs, which is considered as true. The Shang17 is also applied to the same dataset for comparison. Figure 2 shows the evaluation results of ShadeCorr_QAA and Shang17 at six selected wavelengths at 400, 440, 490, 550, 690, and 750 nm, respectively.

 

Fig. 2. Evaluation of the shade-corrected R_rs by ShadeCorr_QAA and Shang17 using the simulated dataset. The black dash line represents the 1:1 line. Blue crosses represent the results from Shang17, while red dots for ShadeCorr_QAA.

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As shown in Fig. 2, shade-corrected R_rs by ShadeCorr_QAA matches the true R_rs very well at all selected wavelengths, with all data points closely distributed along the 1:1 line. In contrast, Shang17 has larger errors compared with ShadeCorr_QAA and has more scattered data points, especially at the two blue bands of 400 and 440 nm. More importantly, the outliers in Fig. 2(a) demonstrates that Shang17 could be subject to large errors if the a_ph model used in spectral optimization is not appropriate.

The performances of ShadeCorr_QAA and Shang17 are also quantitatively evaluated using MPD, with calculated MPD for six selected wavelengths tabulated in Table 1. Specifically, MPD is calculated for all the simulations and two subgroups, representing the relatively clear (defined as a(440) < = 0.3 m⁻¹) and productive waters (defined as a(440) > 0.3 m⁻¹), to highlight the robustness of the two schemes in different types of waters. Consistent with the results shown in Fig. 2, shade-corrected R_rs by ShadeCorr_QAA for all the simulations has overall smaller MPD at each wavelength, except for 690 nm (see details later about this band in Section 5.2.4). For the entire simulated dataset, MPD of corrected R_rs by ShadeCorr_QAA at the two blue bands (i.e., 400 and 440 nm) are less than 0.3%, and MPD at longer wavelengths are generally around 0.1%, except for 690 nm where MPD = -3.0%. In contrast, corrected R_rs by Shang17 is systematically underestimated, and the absolute values of MPD suggest that ShadeCorr_QAA outperforms Shang17, except for 690 nm.

Table 1. MPD between shade-corrected R_rs and true R_rs at six selected wavelengths for ShadeCorr_QAA and Shang17. Statistics are shown for all the simulations, and two subgroups representing the clear waters (a(440) < = 0.3 m⁻¹) and productive waters (a(440) > 0.3 m⁻¹), respectively.

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Excluding the 690 nm, ShadeCorr_QAA and Shang17 are quite comparable for clear waters with very small MPD for all the wavelengths. This is mainly because the shade error in clear waters is negligible due to weak absorption and scattering. For the productive waters, both schemes have shown increased MPD of the corrected R_rs. However, ShadeCorr_QAA still outperforms Shang17 for shade-corrected R_rs at all wavelengths, excluding 690 nm. For example, Shang17-corrected R_rs, on average, are systematically underestimated compared to true R_rs, with much greater MPD in absolute value compared to that by ShadeCorr_QAA. Results in Table 1 suggest that Shang17 may have limited applicability in productive waters, which could be attributed to the mismatch in the modeled component IOPs, particularly the a_ph.

At 690 nm, a band chlorophyll fluorescence contributes to elevated reflectance, MPD of corrected R_rs by ShadeCorr_QAA is much larger than that by Shang17 in all water groups. Therefore, a refinement of ShadeCorr_QAA is necessary for the R_rs bands containing chlorophyll fluorescence. Detailed discussions regarding the impacts of chlorophyll fluorescence on ShadeCorr_QAA and its remediation are provided in Section 5.2.4.

4.2. Evaluation with the in-situ dataset

ShadeCorr_QAA is also evaluated using the in-situ dataset, and the spectral differences between shade-corrected R_rs and the ‘true’ R_rs measured by AWA are presented in Fig. 3. Shang17 is also employed for inter-comparison. Four experimental cases are selected and presented in Fig. 3, with two from the lake experiments and the other two from the tank experiments. The selection of these four cases is trying to demonstrate the performance of shade correction algorithms for SBA measurements under various scenarios, such as using cones with different sizes (Cone22 and Cone45, representing the cone radius of 22 and 45 mm, respectively), under various θ_s, or measuring water of different IOPs. As shown in Fig. 3, ShadeCorr_QAA has consistently good performance, with the corrected R_rs matching very well with the ‘true’ R_rs in all the four experimental cases. Shang17, on the other hand, has shown relatively large uncertainties in the corrected R_rs, especially for the tank experiment using Cone45 and under an θ_s of 18°, where the corrected R_rs is systematically underestimated compared to the ‘true’ R_rs.

 

Fig. 3. Comparisons of the spectral R_rs measured by AWA and SBA, along with shade-corrected R_rs using Shang17 and ShadeCorr_QAA for four experimental cases.

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To more explicitly demonstrate the performance of the two shade-correction schemes, APD between shade-corrected R_rs and ‘true’ R_rs are calculated and presented in Fig. 4. The APD between SBA-obtained R_rs and ‘true’ R_rs is also presented in Fig. 4 for reference. It can be found that ShadeCorr_QAA can significantly reduce the shade error of SBA-obtained R_rs, with APD generally less than 5% at all wavelengths for the four experiments. Most importantly, the shade error of SBA-obtained R_rs under unfavorable scenarios (e.g., small θ_s, large cone size, or large IOPs) can also be adequately corrected to less than 5%, as shown in Fig. 4(d). The Shang17 scheme is overall comparable with ShadeCorr_QAA for the experimental cases using the smaller cone (e.g., Fig. 4(a) and Fig. 4(c)), while it shows degraded performances when the larger cone is used (e.g., Fig. 4(b) and Fig. 4(d)).

 

Fig. 4. Spectral APD of SBA-obtained R_rs (R_rs-shade) and shade-corrected R_rs by Shang17 and ShadeCorr_QAA, referring to ‘true’ R_rs, for the four experimental cases in Fig. 3.

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The robustness of ShadeCorr_QAA and Shang17 are quantitatively evaluated using nRMSD, which implies the spectral closeness between shade-corrected R_rs and ‘true’ R_rs. Table 2 tabulates the calculated nRMSD of the shade-corrected R_rs for the four experimental measurements in Fig. 3, as well as the averaged nRMSD for all the 16 measurements of the in-situ dataset. nRMSD of the SBA-obtained R_rs, referring to the ‘true’ R_rs, is also included in Table 2 for comparison. Water component IOPs of these four cases derived from Shang17, including a, b_b, and a_ph at 440 nm, are tabulated in Table 2 and used in the subsequent analysis.

Table 2. The spectral averaged difference (nRMSD) between spectral R_rs obtained by SBA and that corrected by Shang17 and ShadeCorr_QAA for the four cases presented in Fig. 3. IOPs of the corresponding waters are derived from Shang17 for qualitative analysis, including a(440), b_b(440), and a_ph(440).

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The statistics in Table 2 show that ShadeCorr_QAA outperforms Shang17 in all four experimental cases with smaller nRMSD. The shade error can be reduced from 13.6% to 3.4% in terms of the averaged nRMSD when ShadeCorr_QAA is employed. In contrast, the Shang17 scheme has a larger averaged nRMSD of 4.9% for all experimental measurements.

Although we deem R_rs measured by AWA is the ‘true’ R_rs, these measurements could still include uncertainties. For example, the fraction of surface-reflected light was taken as a constant 0.022 to compute AWA measured R_rs for all in-situ measurements in Lin et al. [46], but it could vary with solar zenith angle [17]. The errors in measured ‘true’ R_rs will be propagated to the calculated metrics, which could also partly explain why ShadeCorr_QAA and Shang17 have much smaller errors in the ‘error-free’ simulated dataset (see Table 1).

5. Discussion

5.1. Impact of a_ph spectral shape on Shang17

Evaluations with both simulated and in-situ datasets show that relatively larger uncertainties of shade-corrected R_rs are obtained when Shang17 is used. The relatively poor performance could be attributed to a mismatch between modeled component IOPs and the ‘true’ component IOPs of target waters. Note that spectral optimization weighs more on the spectral shape of component IOPs [36]. While spectral b_b and absorption coefficients of CDOM and NAP are generally inverse of the wavelength, spectral a_ph cannot be modeled using a simple mathematical expression as it has unique spectral curvatures that vary largely in natural waters. Therefore, there is likely a poor closure between modeled and measured ${R}_{{rs}}^{{shade}}$ if there is a significant mismatch in spectral a_ph.

In the Shang17 scheme, spectral a_ph is modeled as [47],

For the in-situ dataset, the improvement by ShadeCorr_QAA is particularly significant for the tank measurement with Cone45 and θ_s of 18°, where nRMSD decreased from 25.4% to 2.4%. In contrast, nRMSD of Shang17-corrected R_rs is decreased from 25.4% to 11.3% (see Table 2). The composition of water constituents may provide insight and explanation on the performance of Shang17 for this particular case. As shown in Table 2, a_ph(440) is 1.14 m⁻¹ and accounts for ∼ 67.5% of a(440), and b_b(440) is relatively small (∼ 0.12 m⁻¹) in this case, suggesting that the water is highly absorptive and dominated by phytoplankton. A reasonable explanation for the poor performance of Shang17 would be that there is most likely a significant mismatch between a_ph in the water and that modeled in Shang17.

Results from both the simulated and in-situ datasets highlight the necessity of using an optimal a_ph model for a better performance of Shang17. However, spectral a_ph vary broadly in natural waters in both the spectral shape and magnitude [37–39]. It is a challenge to obtain a perfect a_ph model for the implementation of Shang17 for each R_rs measurement. The mismatch of spectral shape between modeled a_ph and that in the target waters will inevitably result in uncertainties for the shade-corrected R_rs by Shang17, which is why ShadeCorr_QAA is proposed to skip the modeling of a_ph.

5.2. Important components of the ShadeCorr_QAA scheme

It is demonstrated in both the simulated and in-situ datasets that ShadeCorr_QAA outperforms Shang17, especially in productive waters. The validity of ShadeCorr_QAA, however, depends on the two assumptions made in the scheme, i.e., negligible absorption by non-water components at 750 nm and the empirical estimation of the power-law angstrom Y. The assumption of negligible absorption by non-water components (a_nw) at the near-infrared band (i.e., a(NIR) = a_w(NIR)) in general valid in most natural waters, which has been widely adopted in inverse algorithms [48–50]. However, the initiation at 750 nm requires a radiometer with highly precise and sensitive radiance measurements at 750 nm. In cases of a less sensitive radiometer at 750 nm or measured radiance at 750 nm is close to 0, a shift of the initial wavelength to a shorter region, such as 640 nm or 550 nm, is also valid if a_nw can be neglected. For waters where a_nw(750) can not be neglected, ShadeCorr_QAA can initiate at longer wavelengths, such as 860 nm, as long as the radiometer is well calibrated in this spectral domain. The impact of the empirically calculated Y will be detailed discussed in Section 5.2.2. Moreover, the r_rs-IOPs relationship and chlorophyll fluorescence may also impact the shade-corrected R_rs by ShadeCorr_QAA, which need to be evaluated and rectified for the broader applicability of ShadeCorr_QAA.

5.2.1. Impacts of r_rs-IOPs relationships

The relationship between r_rs and IOPs is essential to model r_rs and then ${R}_{{rs}}^{{shade}}$ from a and b_b. It is of interest to evaluate the impacts of r_rs-IOPs relationships on the shade corrected R_rs by ShadeCorr_QAA. In addition to the formula used in this study (i.e., Eq. (6) from Lee et al. [41], termed as Lee04 hereafter), a quadratic expression is also widely used by the community with r_rs-IOPs relationship formulated as [40],

Table 3. MPD of shade-corrected R_rs by ShadeCorr_QAA using two different r_rs-IOPs relationships.

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It can be found from Table 3 that ShadeCorr_QAA using Lee04 has much better performance than that using Gordon88. The key difference between Lee04 and Gordon88 is that their model parameters (i.e., g₀, g₁, and g₂ in Eq. (7); g₃ and g₄ in Eq. (15)) were optimized from datasets that are characteristic of different particle phase functions. Note that the r_rs-IOPs relationship is strongly dependent on the particle phase function [32,41]. The better performance of Lee04 is expected because Lee04 explicitly separated the phase function effects between molecular and particle scatterings, which likely play different roles for different wavelengths. In contrast, Gordon88 does not separate the phase function effects of molecular and particle scatterings. Nevertheless, given that the particle phase function is not known a prior in natural waters, the shade correction scheme may always be subject to some degree of uncertainties, including ShadeCorr_QAA. The relatively larger difference between ShadeCorr_QAA-corrected R_rs and ‘true’ R_rs for the in-situ dataset could also be partly attributed to the mismatch in particle phase function.

5.2.2. Impact of Y on Shade-Corrected R_rs

According to Eq. (9), as b_bp is first estimated at 750 nm and then extended to shorter wavelengths, it can be expected that the change of Y would result in different spectral b_bp values in the shorter wavelengths, particularly the blue bands. The formula to estimate Y in ShadeCorr_QAA is adopted from QAA and is purely empirical (i.e., Eq. (10)). The sensitivity of Y on the shade-corrected R_rs, therefore, requires further evaluation. A simple sensitivity analysis approach was used here by setting Y to half and double of its values estimated from Eq. (10) and then re-run ShadeCorr_QAA. Figure 5 shows the resultant shade-corrected R_rs in response to changing Y values.

 

Fig. 5. Evaluation of the shade-corrected R_rs by ShadeCorr_QAA with spectral b_bp parameterized by different values of Y. Blue crosses represent correction results when Y is halved, while red points indicate correction results when Y is doubled.

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Statistical metrics (MPD) for the shade-corrected R_rs using different Y values are tabulated in Table 4. The impact of Y on shade-corrected R_rs is more significant at shorter wavelengths, as it is further away from the reference wavelength at 750 nm. However, MPD for the shade-corrected R_rs at blue bands is still within a small range. For instance, MPD for the shade-corrected R_rs changed from 0.3% to 2.5% at 400 nm when Y is doubled, while much smaller MPD are observed for longer wavelengths. The variation of MPD for the shade-corrected R_rs is even smaller when Y is halved (see also Fig. 5).

Table 4. Calculated MPD of shade-corrected R_rs by ShadeCorr_QAA at six wavelengths using different inputs of Y.

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The Y values are, in general, relatively larger (around 2.0) in clear waters and are much smaller in productive and turbid waters (close to 0). This is because Y values are associated with the particle sizes, which are smaller in the oligotrophic ocean but larger in coastal waters. Therefore, doubling or halving the Y values will impact more of the derived b_b in the shorter wavelength in clear waters. For instance, for clear waters in the simulated dataset (i.e., a(440) < = 0.3 m⁻¹), the derived b_bp and a at 400 nm would be increased by 88.8% and 90.6%, respectively, when Y is doubled. Consequently, the estimated shade error is increased by 84.5%. However, since the value of shade error in clear waters is very small due to low absorption (e.g., median ε(400) ∼ 0.03), an increase of 84.5% in the estimated ɛ still has minimal impact on the shade-corrected R_rs as the change in 1-ɛ is negligible (see Eq. (2)). Note that it is very rare for natural waters with Y as large as 3.0 or 4.0, evaluation results shown here represent the extreme situations that may not happen in natural waters.

For productive waters, Y values are generally small. For instance, estimated Y from Eq. (10) varies between −0.24 and 0.55 for simulations with a(440) > 0.3 m⁻¹, with a median value of only 0.10 and a standard deviation of 0.17. As a result, doubling Y would have limited impacts on the derived b_b and a, thus the estimated ɛ. Statistically, derived b_b and a are increased only by 2.4% and 3.4% when Y is doubled, respectively. Therefore, the errors in the estimation of Y would have very limited impacts on the ShadeCorr_QAA-corrected R_rs for both clear and productive waters. Therefore, the empirical calculation of Y in QAA can be considered adequate for the shade correction using ShadeCorr_QAA.

5.2.3. Impact of chlorophyll fluorescence

The inelastic scattering induced by chlorophyll fluorescence will enhance the reflectance at red bands and forms a reflectance peak around 685 nm [51–53], which can be observed from the spectral R_rs in Fig. 1. In ShadeCorr_QAA, since there is no correction of this extra source for R_rs, the fluorescence peak around 685 nm will result in an underestimation of derived a, thus an underestimated shade error. Such a mechanism explains the large MPD for the shade-corrected R_rs(690), as shown in Table 1. For Shang17, the impact of chlorophyll fluorescence on the shade-corrected R_rs around 690 nm is small, which is because that a(690) and b_b(690) are determined by the spectral eigenvectors for the component IOPs, rather than derived from R_rs(690).

Since the estimation of a is affected by chlorophyll fluorescence in ShadeCorr_QAA, one way to minimize the impact is to find an alternative approach to calculate a for the bands around chlorophyll fluorescence. Lee et al. [54] proposed an empirical approach to construct hyperspectral absorption coefficients from multispectral absorption coefficients using a look-up-table (LUT) of spectral transfer coefficient (STC). Specifically, for the total absorption coefficient at any given wavelength (λ_j) between 400 and 700 nm, it can be constructed from the absorption coefficient at blue-green bands following,

The above-mentioned approach is valid for all wavelengths affected by chlorophyll fluorescence. Here we use 690 nm as a demonstration, as chlorophyll fluorescence has the overall most dominant impact on R_rs(690) in the Craig dataset. As shown in Fig. 6(a), constructed a(690) agrees very well with known a(690), with an MPD of only −2%. Note that the Craig dataset is independent of the dataset used by Lee et al. [54] to develop the LUT. The small MPD of constructed a(690) for the Craig dataset suggests that the LUT developed by Lee et al. [54] could be, in general, valid and appropriate for various water types. However, further improvement of the LUT with more in-situ measurements could still be required as it tends to underestimate a(690) for most of the productive waters (see Fig. 6(a)).

 

Fig. 6. Evaluation of (a) the error of constructed a(690) using the spectral transfer coefficients in Lee et al. [54] for the Craig dataset and (b) the performance of shade-corrected R_rs(690) by ShadeCorr_QAA using constructed a(690).

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The consistency between constructed and known a(690) would ensure an improved accuracy of the estimated shade error at the chlorophyll fluorescence bands. As shown in Fig. 6(b), shade-corrected R_rs(690) by the original ShadeCorr_QAA are systematically underestimated, while the use of constructed a(690) significantly improves the accuracy of shade-corrected R_rs(690) with MPD improved from −3.0% to 0.04%. The encouraging result demonstrates that, with a proper approach, the impact of chlorophyll fluorescence on the shade correction by ShadeCorr_QAA can be adequately addressed. It is worthy to point out that it is assumed in this study that the shade effect at the fluorescence band is similar to that described in Shang et al. [23]. However, the impact of fluorescence also involves light in the blue-green domain, and the shade impacts on fluorescence may not be as simple as assumed in this study. Further research, including analysis using both Monte Carlo simulations and in-situ measurements, is required to explore the mechanism of shade impacts of chlorophyll fluorescence.

5.2.4 Limitations in the shade correction schemes

As mentioned in Section 3.1, both ShadeCorr_QAA and Shang17 failed to obtain reasonable shade correction results for waters with extremely high absorption coefficients (e.g., a(440) > 20.0 m⁻¹), which could be most likely because of the ɛ-IOPs relationships in Eqs. (3)–(4) are not applicable to calculate the shade error for such ‘black’ waters. One explanation could be that the development of Eqs. (3)–(4) did not employ samples from extremely absorptive waters [23]. Moreover, the relationship between r_rs and IOPs are dependent on the particle phase function [32,41], which may also have large uncertainty for such ‘black’ waters without tuning the model parameters in Eq. (6) and Eq. (7).

Separately, both simulated and the in-situ datasets do not include samples of high backscattering waters (extremely turbid waters). The validity of the r_rs-IOPs relationship of Eq. (6) and the ɛ-IOPs relationship of Eq. (4) in extremely turbid waters are yet to be confirmed. However, it is reasonable to speculate that the shade correction schemes, including both ShadeCorr_QAA and Shang17, could also be subject to large uncertainty in extremely turbid waters. Given that extremely absorptive or turbid waters only account for a fraction of a percent of global waters, it is safe to say that ShadeCorr_QAA is applicable and robust for SBA measurements in most natural waters.

6. Summary

ShadeCorr_QAA is proposed in this study to correct the shade error of SBA-obtained R_rs following the merits of QAA. Total absorption and backscattering are directly derived from SBA-obtained shaded R_rs using minimization, in which way parameterizations of component absorption coefficients can be avoided. Evaluations with both simulated and in-situ datasets show that ShadeCorr_QAA outperforms the presently adopted Shang17 scheme. Moreover, the validity of ShadeCorr_QAA is evaluated for impacts by the relationship between r_rs and IOPs, as well as the two assumptions made in ShadeCorr_QAA, including the negligible absorption by non-water components at 750 nm and empirical estimation of the power-law angstrom Y. Evaluation results show that the r_rs-IOPs may have significant impacts on shade-corrected R_rs, while the two assumptions could have minimal impacts in most natural waters. Also, a scheme using constructed absorption coefficients is proposed to reduce the shade error at the chlorophyll fluorescence bands, which show excellent performance for shade-corrected R_rs at 690 nm. However, further studies are still required to more accurately characterize the impact of an SBA system on the measurement of fluorescence. From the results of this study, we advocate the proposed ShadeCorr_QAA, with broad applicability in natural waters and fewer limitations than Shang17, to be used to correct the shade error of SBA-obtained R_rs. At last, it is worthy to point out that the overall concept of ShadeCorr_QAA is equally applicable to shade correction of R_rs data by in-water approach, as long as there is a robust model to describe the shaded R_rs (or r_rs) as a function of IOPs and known parameters related to the instrument apparatus.