1. Introduction

Ocean color products contain a certain degree of uncertainty resulting from imperfect calibration, sensor noise, uncertainty in ancillary data, and retrieval algorithms [1–5]. Providing retrieval-level uncertainty estimates within ocean color products has been recommended by Group on Earth Observations (GEO) and International Ocean-Colour Coordinating Group (IOCCG) within the quality assurance framework for earth observation and should be a general requirement of any satellite missions [3,6]. Traditionally, uncertainty in remote sensing reflectance (R_rs) retrievals is derived through statistical comparison of satellite retrievals with collocated in situ measurements. Due to the limited availability of such matchups and their sparse distribution in space and time, statistical measures over all matching pairs or large groupings are typically used to gauge the uncertainty in the retrieved R_rs. Such derived uncertainties have at least three issues. First, the statistical measures represent an averaged value for the measurement conditions when and where the in situ data are collected. In practice, however, every pixel within the satellite image represents a different set of observing conditions, including radiant path geometry between the sensor, surface, and Sun, aerosol type and concentration, and surface conditions (e.g., Sun glint), and these different observational conditions result in different retrieval uncertainties. The averaged uncertainty doesn’t represent the value at a specific pixel. Second, in situ data have uncertainties [7,8] that contribute to the perceived mismatch and are typically included in the statistical measures. Although it is possible to account for the in situ data uncertainties in evaluating the mismatch [9], accurate estimates of in situ data uncertainty are not always available. Last, the spatial and temporal differences between satellite retrievals and in situ data could result in additional uncertainty in the statistical measures. While in situ data are measured at one point, satellite values used in the matchup represent the measurement average over the satellite pixel footprint (e.g., ∼1 km²), and are typically derived by averaging over even larger areas (e.g., 5×5 satellite pixels centered on the location of the in situ measurement, as recommended by [10]). Furthermore, the satellite and in situ measurements are rarely collected at exactly the same time, and within the matchup time window (e.g., 3 hours recommended by [10]) the optical properties could change, especially in coastal waters [11]. Together, these factors mean it is not an apples-to-apples comparison, and especially the effect from spatial and temporal differences is hard to quantify. The European Space Agency’s Ocean Color Climate Change Initiative program (OC_CCI) does provide pixel-level uncertainty in R_rs merged from various missions [12], as computed using the weighted average of the uncertainty within optical water classes associated with that pixel. As the uncertainty of each class is derived from the validation against in situ data, issues with the validation described above still exist. Furthermore, optical water classes do not capture spatiotemporal variations in some key drivers of variability in atmospheric correction (AC) uncertainty, such as geometry and atmospheric turbidity.

Given the issues with the validation against in situ data as a measure of uncertainty, several image-based approaches were developed to estimate uncertainty in satellite retrieved R_rs. For example, [13] used a bias-resistant algorithm for concentration of chlorophyll-a (chl-a) to determine R_rs with the highest quality, which was then used as a surrogate for “ground truth” to estimate R_rs uncertainty. Using coincident daily R_rs from two sensors or matching satellite retrieved and in situ R_rs, [14] established an approach based on collocation analysis to generate R_rs uncertainty associated with random effects. Using geostationary measurement from Geostationary Ocean Color Imager (GOCI) collected over the course of a day, and with the assumption that no detectable changes occur in the optical properties over waters with low productivity during the daytime period, uncertainty in R_rs is calculated as twice the standard deviation of multiple observations in one day [15]. Although some issues with validation using in situ data could be resolved by the image-based approaches, the uncertainty derived is either valid for a specific dataset or only includes the random uncertainty.

Although there have been some studies on the pixel-level uncertainty in inherent optical properties (IOPs) [16,17], few studies focus on R_rs, which is critical for calculating physical, biological and biogeochemical products, including IOPs. Uncertainty in R_rs is either neglected [16] or assumed constant [17], which is unrealistic [13]. These studies need a relatively realistic pixel-level uncertainty in R_rs. A derivative approach was developed to propagate sensor noise into uncertainty in R_rs [18], as retrieved from Ocean and Land Colour Instrument (OLCI) onboard Sentinel-3 using an AC algorithm for clear water [19]. A Monte Carlo (MC) approach was used to propagate sensor noise into uncertainty in R_rs [20], as retrieved from the standard NASA AC algorithm [21], which is based on the algorithm of Gordon &Wang (1994) [22] (GW94) but includes an iterative method to improve performance in highly productive or turbid waters. With the assumption that R_rs can be expressed as first-order approximation of top-of-atmosphere (TOA) radiance (L_t), Gillis et al. propagated sensor noise into R_rs [23], as retrieved using the Tafkaa AC algorithm [24]. Only sensor noise is included in those studies, while in reality there are other significant uncertainty sources that should be considered including instrument systematic uncertainties, ancillary data uncertainties, model uncertainties and assumptions in the AC algorithms [3], some of which have been indicated to play a more significant role than sensor noise in R_rs uncertainty [25]. The AC algorithms used by [18,23] are different from the algorithm used operationally by the Ocean Biology Processing Group (OBPG) at NASA for processing ocean color data. Since uncertainty propagation depends on the AC algorithm, the propagation method developed by [18,23] cannot be directly applied to the NASA algorithm. MC approaches such as [20] provide a generalized mechanism for calculating pixel-level R_rs uncertainties for any algorithms, but they are computationally intensive and therefore impracticable for routine production.

OBPG has been distributing global ocean color products for more than two decades. While these products have been used widely, pixel-level uncertainties in R_rs have not yet been provided. OBPG is planning the next ocean color reprocessing using a Multiple-Scattering EPSilon AC algorithm (MSEPS) [25,26], which has been shown to perform better than GW94 [27]. In this study, we will establish a derivative method to propagate instrument random noise, instrument systematic uncertainty, and forward model uncertainty through MSEPS, with the goal of generating and verifying pixel-level uncertainty in R_rs retrieved from MODIS and establishing a framework for computationally efficient generation of pixel-level R_rs uncertainties that can be applied for all ocean color missions processed and distributed by NASA.

2. Data and methodology

2.1. MODIS data

Uncalibrated (Level-1A) data from MODIS aboard the Aqua satellite were downloaded from NASA’s Ocean Biology Distributed Active Archive Center (OB.DAAC), and processed into calibrated and geolocated (Level-1B) data using the SeaDAS software package (seadas.gsfc.nasa.gov) and latest instrument calibration coefficients, as also distributed by the OB.DAAC.

2.2. In situ data

Using the OB.DAAC’s in situ data archive and validation search utility tool (SeaBASS, https://seabass.gsfc.nasa.gov/search#val), coincident matchups spanning the years 2002-2019 were collected between MODIS-Aqua R_rs retrievals and in situ data from three sources: Marine Optical Buoy (MOBY) [28], Acqua Alta Oceanographic Tower (AAOT) [29], and BOUée pour l'acquiSition d'une Série Optique à Long termE (BOUSSOLE) [30].

2.3. MSEPS atmospheric correction

The purpose of AC is to retrieve spectral water-leaving radiance, L_w, from observed radiance, L_t, at the top of the atmosphere (for a complete list of symbols describing the AC process, see Table 4 in Appendix A). The NASA standard atmospheric correction algorithm is detailed in [21]. Briefly, L_t can be expressed as:

MSEPS is based on the relationship between aerosol reflectance, ρ_a, and aerosol optical thickness, τ_a:

2.4. Uncertainty propagation through AC

The major categories contributing to R_rs uncertainties as shown by Eq. (4) are:

In general, a variable y that is a function of variables x_i can be expressed as:

To calculate u_c(R_rs), partial derivatives of R_rs with respect to each term with known uncertainty on the right side of Eq. (4) (L_t, t_g, L_r, TL_g, t_vr, L_f, L_a, t_v, t_s, and f_b) are needed. For those terms, uncertainty in L_r results from the uncertainty in wind speed (ws) and surface pressure (pr). Uncertainty in t_vr results from uncertainty in pr. Uncertainty in L_f results from the uncertainty in ws [40]. Uncertainty in t_g results from uncertainty in gas concentration (including ozone (oz), water vapor (wv), nitrogen dioxide (no₂)). Uncertainty in f_b results from chl-a uncertainty. The uncertainty in the model of L_r, L_f, t_g, L_g, f_b, and t_vr are included in the forward model uncertainty calculated through the system vicarious calibration (SVC) that is described in Appendix B, which is applied to L_t. Uncertainty in L_t, L_r, t_vr, L_f, and t_g can be calculated relatively straightforwardly without aerosol information. Grouping these terms together, Eq. (4) at band ${\mathrm{\lambda }_i}$ can be rewritten as:

While Fig. 1 shows a general flow chart, the calculation of u_c(R_rs) is detailed step by step as follows:

 

Fig. 1. The flow chart for the calculation of u_c(R_rs). For detailed convergence criteria of the iteration to account for non-zero L_w(NIR), readers are referred to [37].

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2.5. Estimation of uncertainty sources for MODIS

Uncertainty in L_t comes from sensor noise and systematic error in instrument calibration. For MODIS, the sensor noise ($\mathrm{\chi }$) is modeled as:

Table 1. Coefficients in Eq. (11) for calculating sensor noise of MODIS.

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The instrument systematic uncertainty is more difficult to quantify, but the largest source is the absolute instrument calibration that relates measured counts to radiance in geophysical units. For MODIS ocean color processing, NASA updates the prelaunch counts to radiance conversion using SVC approach [42]. In the SVC process, in situ measurements of L_w from MOBY are matched-up in time and space with satellite observations, and the coincident MOBY measurements are propagated to the TOA using forward model of the AC algorithm to derive an expected TOA radiance at each visible band. The ratio of expected TOA radiance to MODIS-observed L_t is a measure of the absolute calibration gain, and many such samples are collected and averaged over the mission lifetime to derive the mean SVC gain that effectively replaces the pre-launch calibration. We thus adopt here the uncertainty in SVC gain as a first-order estimate of systematic uncertainty on L_t. It should be noted from Table 2 the low systematic uncertainty at bands 412-748 nm with respect to 869-nm band, which results from the low percentage of aerosol radiance in the TOA radiance. As the systematic uncertainty is mainly due to the uncertainty in aerosol radiance that is extrapolated from 869-nm band, the low percentage of aerosol radiance will result in a low systematic uncertainty.

Table 2. Instrument systematic uncertainty (Sys) and forward model uncertainty (Mod).

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While sensor noise is spectrally independent, uncertainty from SVC, as with systematic instrument calibration errors in general, will exhibit some level of spectral covariance that should be considered in the uncertainty propagation. Specifically, in the SVC process, MODIS 748-nm band is calibrated relative to the 869-nm band, and the VIS bands are calibrated relative to those two NIR bands. Due to the spectral dependence of the u(L_t), the covariance of error in L_t at VIS and at NIR bands should be included, i.e., u(L_t(λ_i), L_t(748)), u(L_t(λ_i), L_t(869)), and u(L_t(748), L_t(869)) should be included in calculating u_c(R_rs(λ_i)). Using the correlation coefficient (r) between ρ_t at bands λ_i and 748 nm derived in Appendix C, u(L_t(λ_i), L_t(748)) can be calculated from:

Forward model uncertainty is also difficult to estimate. It derives from algorithm assumptions and modeling errors in determining L_a, L_r, L_f, f_b, and L_g, as well as ancillary data uncertainties and other unknown sources. Here we again take advantage of the SVC process and assume that the variance in the individual SVC gain samples provides an estimate of the total uncertainty on L_t, combined with the uncertainty in the in situ measurements after propagation to TOA. Thus, forward model uncertainties are derived by removing other terms from the standard deviation of the SVC gains, including (1) standard error of the mean SVC gain, (2) sensor noise, (3) uncertainty in ancillary data, and (4) uncertainty in MOBY L_w. For the uncertainty in ancillary data, we adopt the local temporal variability (i.e., the difference between the two temporal samples that bound the time of satellite observation) as a first-order estimate. The estimation of the forward model uncertainty is further detailed in Appendix B. Table 2 lists the instrument systematic uncertainty and the forward model uncertainty, expressed as a percentage of L_t. The high forward model uncertainty at band 748 nm is due to the fixed aerosol model used in the SVC [42], as the true aerosol type in the SVC region of the SPG (see Appendix B) may vary with time.

Using the estimation of uncertainty sources described above, u_c(X) can be calculated:

2.6. Verification of uncertainty propagation using Monte Carlo analysis

Monte Carlo analysis is used to verify u_c(R_rs) derived from the derivative method when only instrument random noise is included. A Gaussian random noise is generated as:

2.7. Evaluation of u_c(R_rs) using validation results

Following the approach presented by [44,45], u_c(R_rs) is evaluated using the matchups between MODIS retrieved and in situ R_rs at MOBY, AAOT, and BOUSSOLE. By adding in quadrature u_c(R_rs) from the derivative method, which represents uncertainty in MODIS retrieved R_rs, uncertainty in in situ R_rs, and the spatial and temporal difference between these two measurements, we calculate an expected discrepancy (Δ_D) between MODIS-retrieved and in situ R_rs. The uncertainty-normalized difference, Δ_N, is defined as the ratio of actual retrieval difference to Δ_D, i.e.,

3. Results

3.1. Evaluation of the derivative method against MC analysis for instrument random noise

Figure 2 shows one example of u_c(R_rs(443)) calculated from the derivative method for MODIS data over the South Pacific ocean with very clear waters. Only instrument random noise is included to compare with uncertainty derived from MC. We can see from Fig. 2 that u_c(R_rs(443)) from these two methods shows a similar spatial pattern, higher at the edge than at the center of the swath. The higher u_c(R_rs) at edges is due to the longer path length and higher relative contribution of path radiance to L_t, increasing the uncertainty in L_a, which is then propagated to R_rs. Higher L_t also means larger random noise as A₁ in Eq. (11) is positive. Figure 3 shows the quantitative comparison of spectral u_c(R_rs) derived from these two methods over 4 pixels, ranging from high to low values. The spectral u_c(R_rs) agree very well. The higher jump in uncertainty at 469 nm and 555 nm is due to the lower signal-to-noise ratio (SNR) at ocean signal levels for those bands, which were designed with a much higher dynamic range to support land applications. Figure 4 shows the mean ratio between u_c(R_rs) from the derivative method against that from MC over all valid pixels in Fig. 2. This is generally between 0.9 and 1.1 across the VIS bands, with the derivative method tending to underestimate u_c(R_rs) at blue bands compared with MC. Note that 2000 random samples were used for the MC calculations, as this seems sufficient to get stable results for this type of scene (Fig. 5). Figs.2-4 show that u_c(R_rs) derived from the derivative method compares reasonably well with that from MC method, indicating the reliability of the derivative method when only instrument random noise is included in the uncertainty budget.

 

Fig. 2. u_c(R_rs(443)) derived by applying the derivative method and MC to MODIS data over South Pacific ocean on Apr. 19, 2017. Only instrument random noise is included. The ratio is calculated from dividing u_c(R_rs(443)) from derivative method by that from MC.

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Fig. 3. Spectral u_c(R_rs) from the derivative method compared with that from MC over pixels denoted by (a) ‘A’, (b) ‘B’, (c) ‘C’, and (d) ‘D’ in Fig. 2. Only instrument random noise is included in the calculation.

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Fig. 4. Mean ratio of derivative to MC u_c(R_rs) over all the valid pixels in Fig. 2. Error bars indicate the standard deviations.

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Fig. 5. Variation of u_c(R_rs(443)) and u_c(R_rs(547)) from instrument random noise, as a function of the number of random samples used in the MC calculation. The calculation is based on pixel ‘D’ in Fig. 2.

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3.2. Evaluation of u_c(R_rs) including all modeled uncertainty sources

3.2.1. Evaluation of spatial patterns

We first assess u_c(R_rs) estimates, including all modeled uncertainty sources, through the expected spatial patterns. Figure 6 shows one example of u_c(R_rs) in absolute terms and expressed as relative uncertainty, δ (u_c(R_rs)×100/ R_rs). u_c(R_rs) is high at the edge of sun glint and over regions near thin clouds. The higher u_c(R_rs(443)) than u_c(R_rs(547)) results from the large sensor noise as well as the large systematic and forward model uncertainty (see Table 2). Note from Fig. 6(b) that some pixels in the circled area lack valid values due to the absence of an upper bounding aerosol model. As described in Section 2.3, a lower bounding aerosol model x (with the corresponding epsilon ɛ_x) and upper bounding aerosol model y (with the corresponding epsilon ɛ_y) are selected to interpolate ρ_a using a ratio of $\frac{{\varepsilon - {\varepsilon _x}}}{{{\varepsilon _y} - {\varepsilon _x}}}$, where $\varepsilon $ is calculated from L_rfc(748) and L_rfc(869), and ɛ_x and ɛ_y are calculated from L_rfc(869) using the coefficients in the LUTs. Model x (or y) is considered as the aerosol for this pixel when there is no upper bounding aerosol (or no lower bounding aerosol). Then, the ratio is assumed to be 1 and L_a in other wavelengths is calculated from L_rfc(869) without using L_rfc(748), which means that the uncertainty in L_rfc(748) cannot be propagated to L_a, resulting in the underestimation of u_c(R_rs). Pixels without lower or upper bounding aerosol models are currently masked out until an improved implementation can be realized.

 

Fig. 6. Spatial analysis of a MODIS scene over the Gulf of Mexico on May 6, 2017 showing (a) true-color image, (b) u_c(R_rs(443)), (c) δ(R_rs(443)), (d) u_c(R_rs(547)), and (e) δ(R_rs(547)). Uncertainty sources include instrument random noise, instrument systematic uncertainty, and forward model uncertainty.

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3.2.2. Closure analysis with results from validation against in situ data

u_c(R_rs) calculated using all uncertainty sources is further evaluated using the validation results derived from matchup comparison between MODIS retrieved R_rs and in situ measurements at MOBY, AAOT, and BOUSSOLE. Following the approach presented by [10], a spatial window of 5×5 pixels centered on the location of in situ data and time window of ± 3h were used to search matching pairs. A valid matching pair also requires spatial homogeneity with a coefficient of variation (i.e., ratio of standard deviation to mean over a 5×5 pixels region) smaller than 15%. For each matching pair, we have MODIS retrieved R_rs, in situ R_rs, and u_c(R_rs). As described in Section 2.7, our approach requires knowledge of uncertainty in in situ R_rs and the spatial and temporal difference between MODIS retrieved and in situ R_rs to evaluate u_c(R_rs).

MOBY includes three L_u sensors deployed at depths of 1 m (top), 5 m (middle), and 9 m (bottom), with the uncertainty in L_u measured by the top sensor increasing from 2.1% in blue wavelengths to 3.3% in red wavelengths, for good scans on good days [46]. Combining this L_u uncertainty with the environmental uncertainty (personal communication with Kenneth J Voss) as well as the uncertainty in downward irradiance just above the sea surface (E_d), we used a constant 5% uncertainty in MOBY R_rs at VIS bands in this study. For BOUSSOLE, Białek et al. show R_rs uncertainty of less than 4% in blue and green wavelengths and less than 5% in red wavelengths [47], and we adopt those values here. The SeaPRISM system used to collect data at AAOT has an uncertainty of 5.3%, 4.8%, 4.6%, 4.9%, and 7.3% in wavelengths of 412 nm, 443 nm, 488 nm, 551 nm, and 667 nm respectively [48], and we adopt those values here for AAOT data. Note 4.9% is used for bands at 531 nm, 547 nm and 555 nm. 7.3% is used for bands at 667 nm and 678 nm. A temporal variation of 2%, 3%, 4% is indicated for normalized water-leaving radiance at 551nm, L_wn(551), at time difference of 0.5, 1.0, 1.5 h at AAOT [49]. As a result, we add a 3% per hour spectrally independent uncertainty to account for the time difference between satellite and in situ data at this site. Temporal variation is neglected at MOBY and BOUSSOLE due to the stability of the optical properties [50]. The standard deviation over the box of 5×5 pixels centered on the location of in situ data is used to represent the spatial variation between retrieved and in situ R_rs. Then, the Δ_D for a given matchup is calculated by adding in quadrature u_c(R_rs), uncertainty in in situ R_rs, and the spatial and temporal variability estimates. Because uncertainty is a measure of the statistical dispersion of retrievals relative to truth, the evaluation needs to be done on a statistical rather than pairwise basis. Figure 7 shows two methods for this with the number of matchups listed in Table 3. The left column shows the PDF of normalized difference (Eq. (16)) and the theoretical Gaussian distribution. These two distributions should ideally match if the uncertainty estimates are reliable [44]. Results are reasonable at band 443 nm at BOUSSOLE and at bands 412 nm-531 nm at MOBY. R_rs at bands 547 nm and 555 nm at MOBY tend to be biased (PDFs not centered around zero) and Δ_D estimates are overconfident (PDFs wider than expected). Δ_D estimates are underconfident in red wavelengths at MOBY and BOUSSOLE (PDFs narrower than expected). R_rs in all wavelengths tends to be biased at AAOT, probably due to the different conditions (including water and aerosol) from that at the SVC site (i.e., MOBY). The different conditions complicate the calculation of L_a in the atmospheric correction, either because L_w(NIR) is not well represented or because the aerosol models are not able to properly model the actual aerosol condition. The right column shows binned Δ_D vs. 1σ of the absolute difference between retrieved and in situ R_rs within each bin. At least 100 matchups are needed for each bin, for better statistical robustness. Please note the exception of 412 nm and 667 nm at BOUSSOLE, with the former having one bin with 88 matchups and the latter having two bins with 81 matchups for each. There is only one bin with 65 matchups for 678-nm band at AAOT. Overall, Δ_D agrees reasonably well with 1σ points of absolute difference, especially at MOBY. This shows that the derivative method has skill in distinguishing relatively low-uncertainty cases from high-uncertainty cases and capturing the spectral dependence of uncertainty. The underestimate of Δ_D at AAOT could be partly due to the approximation of temporal and spatial variation, which are challenging to quantify considering the complicated water environment in a transitional zone from coastal to open ocean. The underestimate of Δ_D could also result from the bias shown in Fig. 7(c). This hypothesis is supported by Fig. 8, which is the same as Fig. 7(d), except for subtracting the mean R_rs bias (i.e., bias-correction) before calculating the absolute difference. In this case the points are much closer to the 1:1 line, suggesting some systematic error in the retrieval at this site but a reasonable estimate of dispersion.

 

Fig. 7. Evaluation of u_c(R_rs) using matchup comparison between MODIS retrieved and in situ R_rs at MOBY (1^st row), AAOT(2^nd row), and BOUSSOLE (3^rd row). The left column shows the PDF of uncertainty-normalized difference, with the black line representing theoretical Gaussian distribution with mean 0 and variance 1. The right column shows the Δ_D versus 1σ absolute difference between retrieved and in situ R_rs; the 1:1 line is dashed.

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Fig. 8. As Fig. 7(d) but subtracting mean R_rs bias at AAOT before calculating the absolute difference.

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Table 3. Number of matchups between MODIS retrieved and in situ R_rs used in Fig. 7.

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3.2.3. Comparison with uncertainty estimates from other studies

Hu et al. calculated uncertainty in R_rs as the standard deviation of the difference between MODIS retrieved and reference R_rs [13]. The reference R_rs for a given chl-a level (with ±2% range to find enough pixels for statistical analysis) is derived by averaging all the R_rs that produce chl-a from two algorithms matching within 5%, where one of the algorithms has been shown to be highly resistant to spectrally correlated bias in R_rs. Figure 9 shows the comparison between u_c(R_rs) derived from the derivative method and the uncertainty presented by Hu et al. MODIS data over the North Atlantic and South Pacific subtropical gyres during Dec. 3-10, 2019 are used to calculate u_c(R_rs). We can see from Fig. 9(b) that uncertainty from these two approaches both show a higher value for chl-a level of 0.05 mg/m³ than that for chl-a level of 0.03 mg/m³. While uncertainty from these two approaches show a similar spectral pattern that is decreasing with wavelength, uncertainty derived from the derivative method is higher than that from Hu et al. at bands 412 nm and 443 nm. These two compare reasonably well at 488 nm, 531 nm, and 547 nm. The lower uncertainty in Hu et al. is likely primarily due to the uncertainty in the reference R_rs and spatial/temporal variations between specific R_rs and reference R_rs, which are not accounted for. The uncertainty in Hu et al. only captures the model uncertainty and noise, not the instrument systematic uncertainties. Differences between the AC algorithms (MSEPS for this study vs. GW94 for Hu et al.) and the assumptions in the derivative method could also contribute to the difference in the uncertainty from these two approaches.

 

Fig. 9. Comparison between u_c(R_rs) and uncertainty estimates from [13] over (a) North Atlantic subtropical gyre, and (b) South Pacific subtropical gyre. The numbers in the legend refer to chl-a. Derivative_0.05 is calculated by averaging u_c(R_rs) over all the pixels in the region with chl-a in the range of 0.05×(1 ± 2%). The other values are from Table 3 in Hu et al. MODIS data from Dec. 3-10, 2019 are used to calculate mean u_c(R_rs) for derivative method.

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Another approach has been presented by [14], which uses coincident R_rs data between different satellite missions and between satellite missions and in situ measurements. Figure 10 shows the comparison of the uncertainty derived from averaging u_c(R_rs) over all the matchups described in Section 3.2.2 with that presented by Mélin et al. at MOBY and AAOT. While Fig. 10 shows that these two compare reasonably well at MOBY, uncertainty derived from the derivative method tends to be lower than that from Mélin et al. especially at AAOT. The higher value from Mélin et al. may be partly due to the contribution from the spatiotemporal variation between MODIS retrieved and in situ R_rs. The difference may also result from different AC algorithms used for generating R_rs (again MSEPS vs. GW94) and the assumptions in the uncertainty estimate techniques between those studies. Please note from Fig. 10(b) the lower u_c(R_rs) from the derivative method than uncertainty in in situ R_rs [48], which means that u_c(R_rs) is probably underestimated. The underestimation may result from the forward model uncertainty that is estimated at MOBY which is representative of open ocean. However, the forward model uncertainty is likely larger in coastal waters as AAOT than that in open ocean due to the complexity in atmosphere (e.g., presence of absorbing aerosol) and water optical properties (e.g., bidirectional reflectance correction).

 

Fig. 10. Comparison between u_c(R_rs) and the uncertainty estimates from Mélin et al. at (a) MOBY, and (b) AAOT. u_c(R_rs) is derived by averaging over all the matchups described in Section 3.2.2. Uncertainty values for Mélin et al. are estimated from Fig. 9 in [14]. Uncertainty in in situ R_rs at AAOT is from [48].

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3.3. Global u_c(R_rs) maps

Figure 11 shows 8-day global u_c(R_rs) and δ at 412, 443, 488, 531, and 547 nm, as well as chl-a calculated with the OCI algorithm [51]. δ at 412, 443, and 488 nm show a similar spatial pattern to chl-a. This spatial pattern results from chl-a absorption. The low R_rs due to chl-a absorption results in a high δ over waters with high chl-a and vice versa for waters with low chl-a. The increased atmospheric turbidity could also increase the u_c(R_rs) in coastal regions. This spatial pattern is not obvious at bands 531 nm and 547 nm as these bands are only weakly dependent on chl-a. u_c(R_rs) doesn’t show as much spatial variability as δ, which is also presented by [14]. It should be noted that the 8-day u_c(R_rs) is simply the average uncertainty in each bin over that period and does not represent the uncertainty in an 8-day (Level-3) R_rs mean. Figure 12 shows the cumulative distribution function (CDF) of δ over clear water pixels (chl-a ≤ 0.1 mg/m³) with valid data in Fig. 11. Overall, around 7.3%, 17.0%, and 35.2% of all the valid pixels with clear waters have δ ≤ 5% at bands 412 nm, 443 nm, and 488 nm respectively, which is a common goal of ocean color retrievals for clear waters [52]. Those percentage numbers are different from the conclusion reached by [13,14] that the goal of 5% is fulfilled at blue bands over clear waters. The difference is primarily due to the methods used to generate u_c(R_rs). Those methods have different assumptions. The difference could also be due to the different satellite data used to generate the R_rs uncertainty. While MODIS global data during Dec. 3-10, 2019 are used in this study, Hu et al. (2013) use data over the North Atlantic and South Pacific subtropical gyres in 2006 and Melin et al. (2016) use global data during 2003-2007.

 

Fig. 11. 8-day R_rs uncertainty in both absolute (left column) and relative term (right column), calculated from the derivative method, and chl-a calculated using R_rs retrieved using MSEPS from MODIS data during Dec. 3-10, 2019. Gray means land and black means no valid data.

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Fig. 12. Cumulative distribution function (CDF) of δ at bands 412 nm, 443 nm, 488 nm, 531 nm, and 547 nm for all the pixels with valid data and with chl-a ≤ 0.1 mg/m³ in Fig. 11.

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4. Discussions and conclusions

We present and perform an initial evaluation of a derivative-based method to calculate the uncertainty in R_rs retrieved from MSEPS atmospheric correction algorithm. Distinct from the (diagnostic) uncertainty products derived from statistics of validation against in situ data, which represent an overall summary uncertainty for an entire dataset, this (prognostic) method estimates a pixel-level uncertainty. It accounts for uncertainty sources including instrument random noise, instrument systematic uncertainty, and forward model uncertainty.

We first assessed the derivative method by comparing estimates considering only instrument random noise with Monte Carlo analysis, which showed reasonable (within 10% on average) spatial and spectral agreement. We then performed a deeper closure analysis, comparing MODIS u_c(R_rs) against statistical analysis of matchups between MODIS R_rs retrievals and coincident in situ measurements at MOBY, AAOT, and BOUSSOLE, while also accounting for uncertainties in in situ measurements and effects of spatial and temporal sampling differences. The closure analysis demonstrates the capability of the derivative method at characterizing the relative magnitude and spectral dependence of R_rs uncertainty. However, the uncertainty is systematically overestimated or underestimated at some wavelengths and sites, showing the need for a better understanding of the uncertainty model and contributions from in situ data and spatial/temporal variation.

u_c(R_rs) presented above includes multiple uncertainty sources, which may raise questions about the contribution from each source. MODIS scene from Fig. 6 is used to examine the contribution from instrument random noise, instrument systematic uncertainty and forward model uncertainty to u_c(R_rs). Results indicate that instrument random noise is generally a much smaller contribution than either instrument systematic or forward model uncertainty sources. It is not trivial to further disentangle the instrument systematic and forward model uncertainty due to considerable spectral covariance between the terms.

While sensor noise is reasonably well understood, the other sources involve simplifications and assumptions including uncertainty in L_t, calibration for band 869 nm, and MOBY L_w. For the uncertainty in L_t, only sensor noise and systematic error are accounted for in this study, but there could be other sources, e.g., structured errors [53] that haven’t been quantified. The calibration uncertainty on MODIS 869-nm band is assumed to be 2% in this study. The effect from this assumption is assessed by calculating the mean ratio and standard deviation of u_c(R_rs) assuming a 2.5% vs. 2% calibration uncertainty for that band over all the valid pixels in Fig. 6. The resulting changes in uncertainty are 1.20 ± 0.06 at band 443 nm and 1.40 ± 0.07 at band 547 nm. The uncertainty in MOBY L_w is estimated at between 2.3%−4.4% in the blue-red wavelengths using L_u uncertainty presented by [46] and the environmental uncertainty (personal communication with Kenneth J Voss). The effect of uncertainty in MOBY L_w is evaluated by comparing u_c(R_rs) derived using those values with that derived using a 5% constant uncertainty. The mean ratio of the former to the latter (and standard deviation) of u_c(R_rs) over all the valid pixels in Fig. 6 is 1.28 ± 0.051 at band 443 nm and 1.11 ± 0.20 at band 547 nm. The effect is more significant at 443 nm than at 547 nm, due to the small L_w(547) at MOBY.

While the evaluations using MC and validation results indicate the derivative method established in this study can provide reasonable u_c(R_rs), some issues need further investigation, including the need for more specific quantitative knowledge of the uncertainty in ancillary data, calibration at the 869-nm band, uncertainty in in situ measurements at MOBY. Forward model uncertainty is affected by the uncertainty in in situ L_w at MOBY, which is significant at blue bands. Despite this, the method shows significant progress towards providing useful pixel-level R_rs uncertainty estimates and can be updated as our knowledge of the contributing terms improves.

Appendix A. Calculation of partial derivative of R_rs

As described in Section 2.3, aerosol calculation starts with L_a(748) and L_a(869) (Eq. (9)), from which L_a at all bands are derived using MSEPS. During the AC process, the partial derivative of L_a(λ), t_v(λ), t_s(λ), and τ_a(λ) with respect to L_rfc(NIR), $\tau _a^{\prime}$(869), chl-a, and rh can be derived, denoted by $\frac{{\partial {L_a}(\lambda )}}{{\partial {L_{rfc}}({NIR} )}}$, $\frac{{\partial {L_a}(\lambda )}}{{\partial \tau _a^{\prime}({869} )}}$, $\frac{{\partial {L_a}(\lambda )}}{{\partial ch{l_a}}}$, and $\frac{{\partial {L_a}(\lambda )}}{{\partial rh}}$ (the same notation is adopted for the derivative of t_v(λ), t_s(λ), and τ_a(λ) by replacing L_a(λ)). Then, the partial derivatives of R_rs with respect to L_rfc(NIR), $\tau _a^{\prime}$(869), chl-a, and rh can be derived:

(A1a)$$\begin{aligned} &\frac{{\partial {R_{rs}}(\lambda )}}{{\partial {L_{rfc}}({NIR} )}} = \frac{{ - \partial {L_a}(\lambda )}}{{\partial {L_{rfc}}({NIR} )}}{f_b}(\lambda )/[{{t_v}(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}} ] - \frac{{\partial {t_s}(\lambda )}} {{\partial {L_{rfc}}({NIR} )}} \\ &[{{L_{rfc}}(\lambda )- T{L_g}(\lambda )- {L_a}(\lambda )} ]{f_b}(\lambda )/[{t_s^2(\lambda ){t_v}(\lambda ){F_0}(\lambda )cos{\theta_s}} ] - \frac{{\partial {t_v}(\lambda )}} {{\partial {L_{rfc}}({NIR} )}} \\ &[{{L_{rfc}}(\lambda )- T{L_g}(\lambda )- {L_a}(\lambda )} ]{f_b}(\lambda )/[{t_v^2(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}}] \end{aligned}$$

(A1b)$$\begin{aligned} &\frac{{\partial {R_{rs}}(\lambda )}}{{\partial \tau _a^{\prime}({869} )}} = \frac{{ - \partial {L_a}(\lambda )}}{{\partial \tau _a^{\prime}({869} )}}{f_b}(\lambda )/[{{t_v}(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}} ]- \frac{{\partial {t_s}(\lambda )}} {{\partial \tau _a^{\prime}({869} )}} \\ &[{{L_{rfc}}(\lambda )- T{L_g}(\lambda )- {L_a}(\lambda )} ]{f_b}(\lambda )/[{t_s^2(\lambda ){t_v}(\lambda ){F_0}(\lambda )cos{\theta_s}} ]- \frac{{\partial {t_v}(\lambda )}} {{\partial \tau _a^{\prime}({869} )}} \\ &[{{L_{rfc}}(\lambda )- T{L_g}(\lambda )- {L_a}(\lambda )} ]{f_b}(\lambda )/[{t_v^2(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}} ]- \\& \frac{{\partial T{L_g}(\lambda )}}{{\partial \tau _a^{\prime}({869} )}}{f_b}(\lambda )/[{{t_v}(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}} ]\end{aligned}$$

(A1c)$$\begin{aligned} &\frac{{\partial {R_{rs}}(\lambda )}}{{\partial ch{l_a}}} = \frac{{ - \partial {L_a}(\lambda )}}{{\partial ch{l_a}}}{f_b}(\lambda )/[{{t_v}(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}} ]- \frac{{\partial {t_s}(\lambda )}}{{\partial ch{l_a}}} \\ &[{{L_{rfc}}(\lambda )- T{L_g}(\lambda )- {L_a}(\lambda )} ]{f_b}(\lambda )/[{t_s^2(\lambda ){t_v}(\lambda ){F_0}(\lambda )cos{\theta_s}} ] - \frac{{\partial {t_v}(\lambda )}}{{\partial ch{l_a}}} \\ &[{{L_{rfc}}(\lambda )- T{L_g}(\lambda )- {L_a}(\lambda )} ]{f_b}(\lambda )/[{t_v^2(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}} ] + \frac{{\partial {f_b}(\lambda )}}{{\partial ch{l_a}}} \\ &[{{L_{rfc}}(\lambda )- T{L_g}(\lambda )- {L_a}(\lambda )} ]/[{{t_v}(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}} ]\end{aligned}$$

(A1d)$$\begin{aligned} &\frac{{\partial {R_{rs}}(\lambda )}}{{\partial rh}} = \frac{{ - \partial {L_a}(\lambda )}}{{\partial rh}}{f_b}(\lambda )/[{{t_v}(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}} ]- \frac{{\partial {t_s}(\lambda )}}{{\partial rh}} \\ &[{{L_{rfc}}(\lambda )- T{L_g}(\lambda )- {L_a}(\lambda )} ]{f_b}(\lambda )/[{t_s^2(\lambda ){t_v}(\lambda ){F_0}(\lambda )cos{\theta_s}} ] - \frac{{\partial {t_v}(\lambda )}}{{\partial rh}} \\ &[{{L_{rfc}}(\lambda )- T{L_g}(\lambda )- {L_a}(\lambda )} ]{f_b}(\lambda )/[{t_v^2(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}} ]\end{aligned}$$

Based on Eq. (7), the partial derivative of R_rs(λ) with respect to L_rfc(λ) can be calculated as

(A2)$$\frac{{\partial {R_{rs}}(\lambda )}}{{\partial {L_{rfc}}(\lambda )}} = {f_b}(\lambda )/[{{t_v}(\lambda ){t_s}(\lambda ){F_0}(\lambda )cos{\theta_s}} ]$$

Table 4. Glossary of symbols

View Table | View all tables in this article

Appendix B. Estimation of instrument systematic uncertainty and forward model uncertainty

In order to quantify instrument systematic uncertainty and forward model uncertainty, we need to go through the SVC process following the approach presented by [42]. Due to a stable aerosol loading and negligible L_w(NIR), South Pacific Gyre (SPG) region was selected to calibrate the 748-nm band. As the pixels used for calibration are required to be free of sun glint, the predicted $L_t^t$(748) can be expressed as:

(B1)$$L_t^t({748} )= [{{L_r}({748} )+ {L_a}({748} )+ {t_v}({748} ){L_f}({748} )} ]{t_g}({748} ){f_p}({748} )$$

where f_p is polarization correction factor [54]. L_a(748) is extrapolated from L_rfc(869) using the aerosol model determined by the time series of aerosol measurements over SPG. The partial derivatives of L_a(748) and t_v(748) with respect to L_rfc(869) can be derived during the extrapolation of L_rfc(869) to L_a(748), denoted by $\frac{{\partial {L_a}({748} )}}{{\partial {L_{rfc}}({869} )}}$ and $\frac{{\partial {t_v}({748} )}}{{\partial {L_{rfc}}({869} )}}$, from which the partial derivative of $L_t^t$(748) can be expressed as:

(B2a)$$\frac{{\partial L_t^t({748} )}}{{\partial {L_{rfc}}({869} )}} = {t_g}({748} ){f_p}({748} )[\frac{{\partial {L_a}({748} )}}{{\partial {L_{rfc}}({869} )}} + {L_f}({748} )\frac{{\partial {t_v}({748} )}}{{\partial {L_{rfc}}({869} )}}$$

The partial derivative of $L_t^t$(748) with respect to L_r(748), L_f(748), and t_g(748) can be written as:

(B2b)$$\frac{{\partial L_t^t({748} )}}{{\partial {L_r}({748} )}} = {t_g}({748} ){f_p}({748} )$$

(B2c)$$\frac{{\partial L_t^t({748} )}}{{\partial {L_f}({748} )}} = {t_v}({748} ){t_g}({748} ){f_p}({748} )$$

(B2d)$$\frac{{\partial L_t^t({748} )}}{{\partial {t_g}({748} )}} = [{{L_r}({748} )+ {L_a}({748} )+ {t_v}({748} ){L_f}({748} )} ]{f_p}({748} )$$

Combining Eq. (B2) with u_c(L_rfc(869)), u_c(L_r(748)), u_c(L_f(748)), and u_c(t_g(748)), the uncertainty in $L_t^t$(748) can be derived, denoted by ${u_c}(L_t^t({748} ))$. It should be noted that the uncertainty in L_t(869) used to calculate u_c(L_rfc(869)) only include sensor noise, which is a limitation of this approach considering the possibility of a systematic error component in L_t(869). A vicarious calibration gain sample (g_i) can be derived from:

(B3)$$g_i(748) = \frac{{L_t^t({748} )}}{{{L_t}({748} )}}$$

where ${L_t}({748} )$ is the measured value with the uncertainty coming from sensor noise. Combining the partial derivative of g_i with respect to $L_t^t({748} )$ and ${L_t}({748} )$ derived from Eq. (B3) with ${u_c}(L_t^t({748} ))$ and u(L_t(748)), the uncertainty in g_i can be derived, denoted by u_c(g_i(748)).

Based on g(748) calculated by averaging all g_i(748), VIS are vicariously calibrated using in situ L_w at MOBY. The uncertainty in L_w is derived from the uncertainty in L_u presented by [46] and the environmental uncertainty during the propagation of L_u to L_w. Pixels used for calibration are again required to be free from sun glint, so $L_t^t({VIS} )$ can be expressed as:

(B4)$$L_t^t({\textrm{VIS}} )= [{{L_r}({\textrm{VIS}} )+ {L_a}({\textrm{VIS}} )+ {t_v}({\textrm{VIS}} ){L_f}({\textrm{VIS}} )+ {t_v}({\textrm{VIS}} ){L_w}({\textrm{VIS}} )} ]{t_g}({\textrm{VIS}} ){f_p}({\textrm{VIS}} )$$

With the assumption of negligible L_w(NIR), L_a(NIR) are equal to L_rfc(NIR) and then applied to calculate L_a(λ) using MSEPS, from which the partial derivative of L_a(λ) and t_v(λ) with respect to L_rfc(NIR) can be calculated, denoted by $\frac{{\partial {L_a}({\textrm{VIS}} )}}{{\partial {L_{rfc}}({NIR} )}}$ and $\frac{{\partial {t_v}({\textrm{VIS}} )}}{{\partial {L_{rfc}}({NIR} )}}$. The derivative of $L_t^t({\textrm{VIS}} )$ with respect to L_rfc(NIR) can be derived as:

(B5a)$$\frac{{\partial L_t^t({\textrm{VIS}} )}}{{\partial {L_{rfc}}({NIR} )}} = \left[ {\frac{{\partial {L_a}({\textrm{VIS}} )}}{{\partial {L_{rfc}}({NIR} )}} + ({{L_w}({\textrm{VIS}} )+ {L_f}({\textrm{VIS}} )} )\frac{{\partial {t_v}({\textrm{VIS}} )}}{{\partial {L_{rfc}}({NIR} )}}} \right]{t_g}({\textrm{VIS}} ){f_p}({\textrm{VIS}} )$$

The derivative of $L_t^t(\mathrm{\lambda } )$ with respect to L_r($\mathrm{\lambda }$), L_f(VIS), L_w(VIS), t_g(VIS) can be expressed as:

(B5b)$$\frac{{\partial L_t^t({VIS} )}}{{\partial {L_r}({VIS} )}} = {t_g}({VIS} ){f_p}({\textrm{VIS}} )$$

(B5c)$$\frac{{\partial L_t^t({VIS} )}}{{\partial {L_f}({VIS} )}} = \frac{{\partial L_t^t({VIS} )}}{{\partial {L_w}({VIS} )}} = {t_v}({VIS} ){t_g}({VIS} ){f_p}({\textrm{VIS}} )$$

(B5d)$$\frac{{\partial L_t^t({VIS} )}}{{\partial {t_g}({VIS} )}} = [{L_r}({VIS} )+ {L_a}({VIS} )+ {t_v}({VIS} ){L_f}({VIS} )+ {t_v}({VIS} ){L_w}({VIS} )]{f_p}({\textrm{VIS}} )$$

Combining Eq. (B5) with u_c(L_rfc(NIR)), u_c(L_r(VIS)), u_c(L_f(VIS)), and u(L_w(VIS)), the uncertainty in $L_t^t({VIS} )$ can be calculated, denoted by ${u_c}(L_t^t({VIS} ))$. Using the same approach as that for 748 nm, g_i(VIS) and u_c(g_i(VIS)) can be derived. Standard deviation, σ, is derived from all g_i at each band. The standard error (SE) is derived from:

(B6)$$SE = \frac{\mathrm{\sigma }}{{\sqrt N }}$$

where N is the number of vicarious calibration gain samples, which are 221 and 63 for 748 nm and VIS bands respectively. The forward model uncertainty is derived from subtracting in quadrature u_c(g) and SE from σ. u_c(g) is derived by averaging those N u_c(g_i).

Appendix C. Correlation between ρ_t

As the 748-nm band is calibrated against the 869-nm band, and VIS bands are calibrated by NIR bands, covariance exist between ρ_t(VIS) and ρ_t(NIR) and between ρ_t(748) and ρ_t(869). Following the approach presented by [18], the correlation coefficients between ρ_t(VIS/748) and ρ_t(869) and between ρ_t(VIS) and ρ_t(748) are calculated using vicariously calibrated ρ_t over a box of 5×5 pixels in the SPG (26.5°−27.5°S, 124.5-123.5°W) with very clear waters. A valid box requires that all pixels are free of level 2 ocean color flags indicating processing problems (https://oceancolor.gsfc.nasa.gov/atbd/ocl2flags/) and the coefficient of variation of R_rs at 412 nm, 443 nm, and 488 nm is smaller than 1% (i.e. spatial stability). Using those 25 ρ_t, correlation coefficients between different bands can be derived. Table 5 lists the mean correlation coefficients over around 500 MODIS granules during 2002-2019. The correlation coefficients are used to calculate the covariance u(L_t(λ_i), L_t(748)), u(L_t(λ_i), L_t(869)), and u(L_t(748), L_t(869)) based on Eq. (12), which should be included when calculating u_c(R_rs(λ_i)). It should be noted here that the correlation between ρ_t is assumed to be a good approximation of inter-band error correlation.

Table 5. Correlation coefficients between ρ_t at VIS and NIR bands.

View Table | View all tables in this article

Funding

National Aeronautics and Space Administration Terra and Aqua Senior Review for MODIS algorithm maintenance and the NASA PACE Project.

Acknowledgements

We acknowledge NASA’s Ocean Biology Distributed Active Archive Center for making available all the in situ data and MODIS data used in our analysis. We also thank the in situ data collection teams (and Principal Investigators) from MOBY (K. Voss), AAOT (G. Zibordi), and BOUSSOLE (D. Antoine) for the collection, processing, and quality control of those datasets. We further wish to acknowledge K. Voss for helpful discussion and insight into the uncertainties on the MOBY measurements.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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