Absorption (a) and backscattering (bb) coefficients play a key role in determining the light field; they also serve as the link between remote sensing and concentrations of optically active water constituents. Here we present an updated scheme to derive hyperspectral a and bb with hyperspectral remote-sensing reflectance (Rrs) and diffuse attenuation coefficient (Kd) as the inputs. Results show that the system works very well from clear open oceans to highly turbid inland waters, with an overall difference less than 25% between these retrievals and those from instrument measurements. This updated scheme advocates the measurement and generation of hyperspectral a and bb from hyperspectral Rrs and Kd, as an independent data source for cross-evaluation of in situ measurements of a and bb and for the development and/or evaluation of remote sensing algorithms for such optical properties.
© 2018 Optical Society of America
The inherent optical properties (IOPs) include absorption (a, m−1), scattering (b, m−1), and beam attenuation (c, m−1) coefficients of water, which determine the magnitude and spectral signature of the light field in water. The apparent optical properties (AOPs), such as the irradiance reflectance (R, dimensionless), the remote sensing reflectance (Rrs, sr−1), and the diffuse attenuation coefficient of downwelling irradiance (Kd, m−1), are connected to the IOPs via the radiative transfer equation [1–3]. To simplify the calculations, various models have been developed to provide quick, and more straightforward cause-effect, relationships between IOPs and AOPs. Subsequently, IOPs can be retrieved from these AOPs through various algorithms [4–6].
During the past decades, many commercial hydrologic optical instruments (e.g., ACS or AC9 from WET Labs Inc., HS6 from HOBI Labs Inc.) have been widely employed in the field to obtain in situ values of IOPs. These instruments not only facilitate the study of IOPs of water but also provide ′′ground truths′′ for evaluation of the IOPs derived from inversion algorithms . However, many limitations still remain in the operation of these instruments. First, it is found that uncertainties of these measurements sometimes can be very large (e.g., uncertainties of ACS measurements resulted from pure-water calibration and scattering corrections could reach more than 30% [8, 9]). Further, current backscattering (bb, m−1) meters (e.g., HS6) have difficulty in obtaining reliable measurements in some complex waters (e.g., Lake Taihu, China) because of an inappropriate sigma-correction . And, there is still no instrument available yet that can measure hyperspectral bb, which could provide valuable information on particle size and its composition [11, 12]. Therefore, it is useful and important to find alternative ways to obtain reliable hyperspectral a and bb in situ.
Currently, commercial spectroradiometers can provide simultaneous hyperspectral measurements of radiance and irradiance in the upper water column, which can be used to obtain AOPs such as R, Rrs, and Kd. Previous studies have also shown that these AOPs are well related to a and bb [13–16]. Based on this knowledge, Gordon and Boynton  used radiance-irradiance profiles to estimate a and bb in stratified waters. To simplify the calculations, Loisel and Stramski  developed a model to retrieve hyperspectral a, b and bb from the combination of R just beneath the sea surface and Kd of the first optical depth. The limitation of this scheme is that most instruments provide measurements of Rrs, not R. It is thus necessary to convert Rrs to R via some empirical models [19, 20] if one is to apply this model for global measurements. The conversion factor between Rrs and R is not a constant though, thus extra uncertainties could be introduced into the estimates of a and bb if this scheme is applied.
In this paper, with an updated model for Rrs and the Kd model presented in Lee et al. , an updated scheme to derive hyperspectral a and bb with hyperspectral (350-700 nm, ~2 nm step) Rrs and Kd as inputs is presented, where values of hyperspectral a and bb spectra were obtained in oceanic and turbid inland waters. These estimates were further compared with available in situ measurements from commercial instruments to check constancy. Furthermore, limitations and sources of uncertainties associated with both strategies for in situ a and bb are discussed.
2.1 Hydrolight simulations
As in many studies [13, 22–24], a series of radiative transfer simulations (Table 1) were carried out with Hydrolight (http://www.sequoiasci.com). The simulation data are used to (1) obtain model parameters of remote sensing reflectance for the geometry angles not included in the look-up table (LUT) presented in Lee et al. , and (2) to evaluate the updated scheme for derivation of a and bb from purely radiative-transfer simulated Kd and Rrs to check model consistency. Note that the current Hydrolight implements a scalar (unpolarized) version of invariant imbedding calculations, thus some uncertainties could be introduced. In addition, the sea surface modeling for wave elevation and slope could also introduce some potential uncertainties .
For these simulations, class-1 IOP models (classic case-1 option in Hydrolight) were adopted; chlorophyll-a concentration was set to a wide range from 0.01 to 50 mg/m3 (0.01, 0.05, 0.1, 1, 2, 5, 10, 20, 50 mg/m3); wavelengths were set in a range of 360 ‒ 700 nm (20 nm step); the sun was positioned at a range of 0° – 80° from zenith (10° step); and an averaged phase function for particle scattering (Petzold ) was used. A wind speed of 5 m/s was assumed, and the sky was set cloud free. For all simulations, geophysical depths ranged from 0 ‒ 200 m (with 1 m depth resolution for 0 ‒ 20 m range and 5 m depth resolution for the 20 ‒ 200 m range, respectively). The skylight was simulated by RADTRAN developed by Gregg and Carder . The contributions of fluorescence from chlorophyll-a or colored-dissolved organic matter were not included in this study. For each IOP case, two simulations were carried out separately (with and without Raman). The data set without Raman scattering was used for the modeling of sub-surface remote sensing reflectance (rrs, sr−1), and the one with Raman was used for the application/evaluation of the Rrs-Kd scheme.
2.2 Field measurements
To evaluate the new scheme, measurements from a wide range of aquatic environments were used, which include measurements collected in oceanic, coastal, and highly turbid inland waters (Fig. 1).
Set one: A data set, including concurrent IOPs (a and bb), Rrs and Kd, was obtained from the NASA bio-Optical Marine Algorithm Data set (NOMAD), which covers a Kd(490) range from ~0.016 to ~4.6 m−1. The total absorption coefficients were derived from measurements of the two major components: absorption coefficient of suspended particles and colored dissolved organic matter (CDOM), where the particle absorption was measured with quantitative filter-pad technique [28, 29]. The measurements of the spectral bb were obtained using HOBI Labs Inc. HS6, or WET Labs ECO-BB or ECO-VSF. The concurrent measurements of Rrs and Kd were acquired from profile measurements of upwelling radiance and downwelling irradiance, respectively. A scheme of quality control for Rrs was also conducted following the method of Wei et al. , and here we only used Rrs values with a quality assurance score higher than 0.7.
Set two: Data collected from NOAA VIIRS ocean color validation cruise in 2014 (21 stations during 10 days). Profiles of downwelling irradiance (Ed, w/m2/nm) and upwelling radiance (Lu, w/m2/nm/sr) were collected with a Satlantic hyperspectral profiler, which is equipped with up to 135 bands covering a wavelength range from ~352 nm to 800 nm. The radiometry sensors were calibrated a few days before the cruise. The rrs was calculated as the ratio of Lu(0-) to Ed(0-) just below the water surface, which were obtained via extrapolation of measured Lu(z) and Ed(z) vertical profiles, respectively. The spectral Kd was calculated by a nonlinear fitting of Ed(z) between 0- and z10%, which is the depth where Ed(z) is 10% of Ed(0-). To reduce the impact of wave-focusing effect  on derived Rrs and Kd, multiple casts of profile measurements were collected for each station (normally 4-8 casts for each station). In addition, the non-water absorption coefficients were measured by two ACS (WETLabs Inc.) meters, where the two ACS instruments were mounted inside the same profiling cage. The pure-water calibration was carried out on the first day and the middle of this cruise. The scattering correction was conducted with the method (Eq. (4) of Röttgers et al. . Finally, the total absorption coefficients were calculated by the sum of non-water absorption coefficients and pure seawater absorption coefficients with the values of later taken from Pope and Fry  and Lee et al. . The backscattering coefficients of seawater were measured by an ECO-BB7FL2 (WETLabs Inc.). The dark offsets were obtained by covering the detectors with electrical tape on the last day of the cruise. The values of pure seawater backscattering coefficients were taken from Morel .
Set three: A data set obtained from Lake Taihu, China, which is a typical turbid shallow lake with significant spatial and seasonal variation of optical properties. A total of 113 concurrent measurements (a, Rrs and Kd) were compiled, where the Kd(490) value ranged from ~2.0 to ~20.7 m−1. The particulate absorption was measured using the quantitative filter-pad technique. The above-water measurement of Rrs was collected with an ASD (Analytical Spectral Devices) field spectrometer (Analytical Devices, Inc, Boulder, CO). Separately, an SAM 8180 sensor (TriOS) was used to measure the Ed at different depths between 0 to 1 m with a depth resolution of ~5 cm. The spectral Kd was then calculated by a nonlinear fitting of Ed(z) between 0- and z10%.
3.1 Two-term rrs model
In this study, we adopted the two-term rrs model in Lee et al.  to link rrs with IOPs, which is expressed as15]. A LUT for discrete sun-sensor angular geometry has been developed in Lee et al. . To effectively determine model parameters for sun angles not included in the LUT and to obtain a smooth transition among the sun angles, we here express the model parameters () of each view geometry (θv and φv), respectively, as a function of sun angle.Table 2) were obtained by a nonlinear fitting between parameters () and θs via Eqs. (2.1) – (2.4). Note that the values of both θv and φv are 0° when applied to profiler measurements (Lu(z) and Ed(z)).
3.2 Model for Kd
We employed the latest Kd model as presented in Lee et al. , where Kd is expressed as21]. Note that this model can be used to estimate Kd from ultraviolet to the visible bands.
3.3 Steps to estimate a and bb
As described by Eqs. (1) and (3), both rrs and Kd are functions of a and bb; therefore, for any wavelength having measurements of both rrs and Kd, there are just two unknowns, thus it is straightforward to calculate a and bb from the two equations. Note that θs is obtained according to the time and location (longitude and latitude) of the field measurements. In addition, a correction of contribution from Raman scattering was also applied to Rrs following the approach of Lee et al. , and the conversion between rrs and Rrs is given by the following equation
4. Evaluation of the derived IOPs
4.1 Self-consistency test
With data obtained from Hydrolight simulations (Raman scattering included), a self-consistency test was first conducted and the results are shown in Fig. 2. The impact of Raman scattering on Rrs was first corrected with the approach of Lee et al. . Note that a and bb (from 400 nm to 700 nm) are also the input parameters for the Hydrolight simulations, which are further compared with a and bb estimated from Rrs and Kd that are calculated from the simulated light field. It is found that the mean absolute percentage difference (MAPD) is very small (< 1% on average), which indicates that errors in the derived a and bb resulted from the model formulation and model parameters employed for rrs and Kd are extremely small. The MAPD here is defined asTable 3. It is found that generally the error in derived a is primarily resulted from Kd, while the error in derived bb is contributed by both Rrs and Kd, with the compound error generally less than ~11% for a and less than ~20% for bb for ± 10% errors in both Rrs and Kd.
The spectrum of particle backscattering coefficient (bbp, m−1) is generally considered following a power-law equation 34]. Further, η was calculated by a nonlinear fitting of bbp(λ) between 400 nm and 700 nm. For the simulation data set, the range of η of input bbp(λ) varies from ‒ 0.5 to 2.5. Fig. 3 shows a histogram of the ratios of η calculated from retrieved bbp(λ) to those from known bbp(λ). It is found that the two η data sets agree with each other very well with the ratios ranging from 0.75 to 1.18 (mean value as 1.0). This result indicates that as long as rrs and Kd are of high quality, it is adequate to obtain reliable hyperspectral bbp from the hyperspectral measurements of rrs and Kd.
4.2 Evaluation with field measurements
Further, hyperspectral data (Table 1) obtained from field measurements were used to evaluate the performance of the updated scheme. These include global measurements (NOMAD), measurements from VIIRS ocean color validation cruise in 2014 (VIIRS2014) and measurements collected in highly turbid inland water (Lake Taihu, China).
First, the NOMAD data set was used to evaluate the performance of the updated scheme. The data were collected in various waters covering a wide range of optical properties with a(490) from ~0.01 to 2.0 m−1 . An example of estimated and measured IOPs (a(λ) and bb(λ)) is shown in Fig. 4. It is found that good agreements were obtained between the estimates and measurements in the entire wavelength range of ~400 to ~670 nm, but bb at ~683 nm is overestimated because of the contribution of chlorophyll-a fluorescence to rrs at this wavelength, where no attempt (e.g., Huot et al. ) was taken here to remove this contribution in the process of deriving bb.
Figure 5 shows a comparison of measured and estimated IOPs (a and bb) at selected wavelengths (411, 490, 555 and 665 nm). The retrievals in general agree very well with those from in situ measurements. Taking 490 nm as an example, in linear regression analysis, a slope of 1.12 with an intercept of ‒ 0.02 m−1 was obtained (R2 = 0.93, N = 739) for a(490), with a mean absolute deviation (MAD) of 0.04 m−1 and a MAPD of 24.4% (Fig. 5(b)). Separately, a slope of 0.95 and an intercept of 0.001 m−1 (R2 = 0.82, MAD = 0.0007 m−1, MAPD = 18.3%,N = 181) were obtained for bb(490) (Fig. 6(b)). In view both in situ measurements and derivation from AOPs have errors/uncertainties, these results suggest that the a and bb values derived from the combination of Rrs and Kd match well with instrument measurements.
The power-law formula (Eq. (6)) is still generally adopted by the community to describe the spectral variation of bbp , although other schemes were also proposed . The parameter η in Eq. (6) is important in ocean color remote sensing in many aspects, as it is required in inversion algorithms [5, 40, 41] and it is an indicator of particle size distributions [11, 12], but there has been limited reporting in the literature regarding its range and its variation for different waters or particulates. As a consequence, crude estimates have to be employed in ocean color algorithms in order to extend the analytically-derived bbp at one wavelength (λ0) to the rest of the visible bands . Here, to evaluate the robustness of Eq. (6) for bbp(λ), nonlinear regression was carried out between bbp(λ) and λ (resulting slope is η) for in situ Rrs&Kd derived hyperspectral bbp(λ). The wavelength range is ~400 - 670 nm, thus avoiding the impact of chlorophyll-a fluorescence around 683 nm. It is found that more than ~83% of the waters studied have R2 > 0.85 from this nonlinear regression analysis. Such a result echoes that Eq. (6) is in general quite reliable in describing the spectral dependence of bbp , although there are exceptions (~5% of this data set with R2 less than 0.5). Further, it is found that the derived η (R2 > 0.85) of this data set is in a range of ~0 – 2.4 and centered ~0.8 (Fig. 7(a)), with smaller (~0) values for coastal turbid waters and higher (~1.5 – 2.4) values for clear oceanic waters. Such a range of η is generally consistent with historical perceptions regarding the range of this property , but narrower than the range presented in Kostadinov et al. , especially for clear oceanic waters. Further, a histogram of the ratios of all retrieved η from Rrs&Kd to those calculated from field-measured bbp(λ) is shown in Fig. 7(b), and it is found that there are more than 80% of the points have the ratio within the range of ~0.5 - 1.5 (with a mean value as 0.93). Such results further support the use of hyperspectral Rrs&Kd for the derivation of hyperspectral bbp.
Further, a and bb values were also estimated by the updated scheme from the measured Kd and rrs of the VIIRS2014 data set. Because multiple-cast measurements were collected at each station, here we only used averaged values of these measurements and estimates for scheme evaluation and comparisons. Similarly, Fig. 8 shows an example of estimated and measured a and bb spectra. It is found that the spectral features of a(λ) and bb(λ) can be retrieved well by the new scheme. In particular, with this new scheme, the a and bb values in UV wavelengths can also be estimated, while it is not possible to measure these yet with current commercial instruments (e.g., ACS, BB9).
Figure 9 shows a comparison of ACS-measured a with those derived from Kd and rrs at selective wavelengths. Excellent agreements between the two independent determinations can be found where there are low MAD and low MAPD values (e.g., a(490) has MAD = 0.007 m−1 and MAPD = 8.5%). Larger uncertainties were observed at 665 nm, which is likely due to contributions of chlorophyll fluoresces at this wavelength, but the MAPD is still smaller than 6%. Likely in part due to more careful data processing and quality control, the agreement between the derived a and ACS-measured a is much better than that showing in the NOMAD data set. Figure 10 shows a comparison of BB9-measured bb(λ) and the retrievals from rrs and Kd. For all wavelengths from 411 nm to 665 nm, a very good agreement was obtained where MAD is commonly less than 0.001 m−1 and MAPD < 30%. These results further demonstrate a robust performance of this updated scheme.
Similarly, the range of the retrieved η based on rrs-Kd derived bbp(λ) is found to vary from ‒ 0.2 to 1.1 (Fig. 11(a)) with a mean value of ~0.55, which is consistent with the general nature that most of the stations are nearshore. Further, a histogram of the ratio of estimated η to that calculated from BB9-measured bbp(λ) is shown in Fig. 11(b), where ~75% of the ratios are within the range of 0.5 – 1.5 (average is 0.91).
With multicast measurements of Ed(z) and Lu(z), the standard deviation (STD) of a(λ) estimated from rrs-Kd is assessed, which are further compared with the STD of a(λ) obtained from multicasts of ACS measurements. Note that because Kd is an average value in the upper water column, we here used the mean ACS-a(λ) between 0- and z10% for comparisons. Figure 12 shows the histograms of STD for these determinations. For the wavelengths ranging from 411 to 555 nm, the STD of Rrs-Kd estimates is equivalent or smaller than that of the ACS measurements (at least for the data set evaluated). Taking 411 nm as an example, the STD of Rrs-Kd estimated a(411) varies from ~0.001 to ~0.022 m−1with a mean value as 0.003 m−1, but the ACS-measured a(411) had a range of ~0.001 to ~0.034 m−1 with a mean value as 0.008 m−1, indicating a more reliable a was estimated from the combination of Rrs and Kd for this data set. Similarly, Fig. 13 shows the histograms of the coefficient of variation (CV) for these determinations, where CV is defined as the ratio of STD to mean value. We obtained similar results that the CV of Rrs-Kd estimates is equivalent or smaller than that of the ACS measurements. The higher STD (or CV) of ACS measurements could result from the processes involved with the pure-water calibration and/or the scattering correction. These issues have been widely reported and investigated in the last decades [9, 43, 44].
4.2.3 Lake Taihu
Finally, the new scheme was evaluated with the measurements collected in Lake Taihu, China, where the optical properties are more complicated due to the highly dynamic environment (e.g., sediment resuspension due to wind and waves , complex network of rivers or channels, and rich terrestrial input ). Figure 14 shows a comparison of measured and estimated a at selected wavelengths (411, 443, 490, 530, 555, and 665 nm). Although extremely high absorption coefficients were observed in this turbid water (a(443) can be as high as ~20 m−1, for instance, see Fig. 14(b)), the two sets of a values (measured and retrieved a) agree with each other well (e.g., MAD = 1.3 m−1, MAPD = 18.1% for a(443), see Fig. 14(b)). These results provide us further confidence in applying this updated scheme to obtain important IOPs even for extremely turbid waters.
5. Comparison with other approaches
The concept of deriving IOPs from a combination of AOPs is not new, where many previous studies [17, 18, 47–49] combined irradiance reflectance (R) and Kd for the derivation of absorption and backscattering coefficients, but R is seldom measured in the field anymore. We here briefly compare the results of the Rrs-Kd scheme with those of the two earlier R-Kd approaches [18, 47], as these approaches can be more easily implemented than other similar approaches [17, 48, 49]. The approach of Loisel and Stramski  was developed based on radiative transfer simulations, with much simpler relationships between IOPs and R (or Kd) compared to the approach of Morel . We first evaluated the R-Kd approaches with the same data of rrs and Kd calculated from the Hydrolight simulations, where a simple relationship (R = 3.5 rrs) was used to convert rrs to R. Figure 15 shows a comparison of estimated a and bb with the Hydrolight-input a and bb. As indicated in Loisel and Stramski , the a can be well retrieved with a small MAPD (4.76%), while the bb was generally overestimated for bb lower than ~0.002 m−1 and underestimated for bb higher than ~0.004 m−1. The reasons for such (small) discrepancies may include the imprecise empirical conversion from rrs to R and/or the approximations in the models between bb and R (or Kd). For the approach of Morel , as simple relationships between IOPs and R (or Kd) were adopted, both a and bb were estimated with relatively larger MAPDs (> 13%). In particular, most of the absorption coefficients were overestimated (Fig. 15(c)).
For field measurements, Fig. 16 shows histograms of the ratio of estimated a(490) and bb(490) to in situ measurements. For the NOMAD data set, the approach of Loisel and Stramski  obtained similar a(490) estimates when compared with the Rrs-Kd scheme (Fig. 16(a)), but more bb(490) are underestimated (Fig. 16(b)). For the VIIRS2014 data set, where most of the measurements were collected in coastal waters, both approaches obtained similar estimates (only limited data points available, N = 21). For the Lake Taihu data set, since there were no measurements of backscattering coefficients in this turbid water, comparisons are made regarding the absorption coefficients only. In such a turbid region, the Rrs-Kd scheme appears to derive more reliable a(411) and a(490) than the approach of Loisel and Stramski  (Figs. 16(e-f)). Similarly, Fig. 17 compares the Rrs-Kd scheme with the approach of Morel . Generally, both a(490) and bb(490) were overestimated when estimated with the R-Kd approach of Morel .
Oceanography is a science where field measurements are critical to describe not only the spatial-temporal variations of various physical and biogeochemical properties, but also to understand the relationships and processes associated with such changes. This is also true for the inherent optical properties of the bulk waters, and it is always desirable to obtain these properties directly from proven instruments. For such, a few optical instruments (e.g., ACS, ECO-BB) are now widely used in the field to obtain information of water’s IOPs (e.g., a and bb). During ship-based surveys, these instruments can be employed in either profile or underway mode to provide data of high vertical or horizontal resolutions, therefore there have been lots of measurements of world oceans (especially those by AC9 or ACS) in the past decades.
However, it is necessary to keep in mind that these data are associated with various level of uncertainties inherent in the measurement strategy. In addition to the requirement of absolute calibration for the ACS or AC9 system that is not easy to perform accurately for clear oceanic waters (low optical properties approaching present instrument precision), a key aspect is that the direct measurements from these instruments are never IOPs, rather attenuated or scattered light intensities, where IOPs are calculated from these light signals. For instance, for the AC9 or ACS systems, the absorption (a) and the beam attenuation (c) coefficients are calculated based on light intensities measured at the two sides of the tubes holding the water samples. In particular, the calculation of a depends on a correction of the scattered signals in the tube for the measurement of a. Inevitably, for such calculations, assumptions and models (empirical or semi-empirical) have to be developed [8, 50]. These assumptions and models will introduce various levels of uncertainties to the final desired properties [9, 51]. Especially, as discussed in detail in a series of articles [52–54], a major challenge in the AC9 or ACS system for the ‘measurement’ of a is the correction of the scattering contaminations. Various schemes have been developed for such corrections [8, 9, 43] with varying performances found for different waters. Uncertainties in the obtained a can be up to 80% .
On the other hand, the present bb sensors (e.g., HS6, ECO-BB) never provide direct measurements of bb, and are at a few spectral bands only. These bb sensors not only just measure light scattered by water samples, but also at a fixed angle (varying from 120° to 140° for different instruments) that provides information for the determination of the volume scattering function (VSF, β, m−1 sr−1) at this angle. It is then necessary to use models to correct the contaminations due to light absorption and scattering [55, 56] and then to convert this β at an angle to bb [57, 58]. Because of these models and assumptions, uncertainties of the ‘measured’ bb can easily be ~20% . These practices and observations advocate the development of more robust instruments to accurately ‘measure’ IOPs in situ .
Similarly as the systems for in situ IOPs, the AOPs to IOPs schemes (including the updated Rrs-Kd scheme described here) do not provide a direct measurement of IOPs, rather they calculate IOPs based on models and assumptions using AOPs as inputs. Consequently, the accuracy of the derived IOPs depends on 1) accuracy of the AOPs, and 2) robustness of the assumptions and models used in the IOPs-AOPs relationships (e.g., no polarization included in current numerical simulations, etc.). Presently with the advancement of optical-electronic systems, as long as proper procedures and protocols are followed for instrument calibrations, hyperspectral upwelling radiance and downwelling irradiance in the water column can be measured well in the field. However, due to wave-introduced motions, these measured signals could be contaminated by sensor tilting and wave focusing/de-focusing [31, 60]. Although significant progress has been achieved in deriving both rrs and Kd from such data [61, 62], it is still an ongoing research subject to improve the data quality of rrs and Kd obtained from such profiling measurements [63, 64].
The errors or uncertainties in rrs and Kd directly affect the accuracy of a&bb derived from the Rrs-Kd scheme (see Table 3), but they have different impacts on the derived a and bb values. Specifically, because Kd is primarily a sum of a and bb [13, 65] and generally a >> bb for most natural waters, the accuracy of Kd has more impact on the accuracy of a; on the other hand, since rrs is proportional to bb, the accuracy of rrs has stronger impact to the derived bb. Also, as indicated in the in situ measurement of bb where assumptions have to be made in order to convert single angle β to bb, it is also required to assume a particle scattering phase function for the relationship between rrs and IOPs [15, 16]. The uncertainty due to this assumption is found to be < 20% for most waters , and some closure studies have shown that the rrs can be well reconstructed from in situ IOPs (a and bb) if an appropriate phase function is used [66–68]. It is thus clear that inherently there will be some uncertainties between the in situ measured bb and the Rrs&Kd derived bb, because both systems actually obtain β at an angle (and not necessarily the same angle) and both have to assume some kind of phase function (again, not necessarily the same) in order to convert β at an angle to bb. In view of the above, and keep in mind that many times during field measurements the waters targeted by Rrs&Kd were not concurrent with those targeted by the IOPs instruments or those used to get component IOPs (e.g., the particulate absorption coefficients included in NOMAD), the less than ~20% difference in a values (except Lake Taihu) and less than ~30% in bb values between those derived from Rrs&Kd and those from IOPs instruments indicate excellent agreements from the two independent determinations. More importantly, even with the various sources of uncertainties associated with the Rrs&Kd system, the precisions of IOPs derived from the Rrs&Kd system are equivalent with the precisions of IOPs obtained from the present IOPs instruments (see Figs. 12-13). These results further support the generation of a&bb with the Rrs&Kd system, where the latter could be further improved in the future with more realistic numerical models for aquatic environments [2, 25] and better understanding and determination of particle scattering phase functions .
Note that here the Rrs&Kd system aims at a well-mixed upper water column, and it does not produce depth-resolved solutions. If the upper water column is stratified, the derived a&bb should be compared with averages in the upper water column (normally between 0- and z10%, as presented in this study) and larger uncertainties would be expected as the models of Rrs and Kd are based on vertically mixed waters. For stratified waters, a solution of the depth-dependent radiance-irradiance field is likely required in order to get depth-resolved IOPs as demonstrated in Gordon and Boynton  and Gordon et al. .
An updated scheme is presented in this study to derive a and bb of bulk water from a combination of field-measured Rrs and Kd. Applications to both Hydrolight simulations and field measurements show that the retrieved a and bb values match known or measured values very well for both clear oceanic waters and turbid coastal/inland waters. In particular, an analysis about product precisions indicates that the a and bb values obtained from the Rrs&Kd system have equivalent precisions compared to those from present IOPs instruments, at least for data sets in this study. Therefore, with the existence of large AOP data sets, the updated Rrs&Kd system here not only provides an alternative and robust way to obtain a&bb where there might be no measurements of such data from IOPs instruments, and the derived IOPs will be important and useful for studies of remote sensing and biogeochemical processes in the aquatic environments. More importantly, with hyperspectral Rrs and Kd, this scheme will fill the gap that the current bb sensors cannot provide a measurement of hyperspectral bb. On the other hand, it is always desired to have more robust instruments to measure hyperspectral IOPs in situ.
National Oceanic and Atmospheric Administration (NOAA) JPSS VIIRS Ocean Color Cal/Val Project (NA11OAR4320199); National Aeronautics and Space Administration (NASA) Ocean Biology and Biogeochemistry and Water and Energy Cycle Programs (NNX14AK08G, NNX14AQ47A, NNX15AC84G).
We are in debt to colleagues provided valuable field data to NASA SeaBASS (the base for NOMAD), and we thank Emmanuel Boss and an anonymous reviewer for constructive comments and suggestions that greatly improved this manuscript.
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