## Abstract

Absorption (*a*) and backscattering (*b _{b}*) 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

*b*with hyperspectral remote-sensing reflectance (

_{b}*R*) and diffuse attenuation coefficient (

_{rs}*K*) 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

_{d}*a*and

*b*from hyperspectral

_{b}*R*and

_{rs}*K*, as an independent data source for cross-evaluation of

_{d}*in situ*measurements of

*a*and

*b*and for the development and/or evaluation of remote sensing algorithms for such optical properties.

_{b}© 2018 Optical Society of America

## 1. Introduction

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 (*R _{rs}*, sr

^{−1}), and the diffuse attenuation coefficient of downwelling irradiance (

*K*, m

_{d}^{−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 [7]. 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 (*b _{b}*, 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 [10]. And, there is still no instrument available yet that can measure hyperspectral

*b*, 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

_{b}*a*and

*b*.

_{b}in situCurrently, 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*, *R _{rs}*, and

*K*. Previous studies have also shown that these AOPs are well related to

_{d}*a*and

*b*[13–16]. Based on this knowledge, Gordon and Boynton [17] used radiance-irradiance profiles to estimate

_{b}*a*and

*b*in stratified waters. To simplify the calculations, Loisel and Stramski [18] developed a model to retrieve hyperspectral

_{b}*a*,

*b*and

*b*from the combination of

_{b}*R*just beneath the sea surface and

*K*of the first optical depth. The limitation of this scheme is that most instruments provide measurements of

_{d}*R*, not

_{rs}*R*. It is thus necessary to convert

*R*to

_{rs}*R*via some empirical models [19, 20] if one is to apply this model for global measurements. The conversion factor between

*R*and

_{rs}*R*is not a constant though, thus extra uncertainties could be introduced into the estimates of

*a*and

*b*if this scheme is applied.

_{b}In this paper, with an updated model for *R _{rs}* and the

*K*model presented in Lee et al. [21], an updated scheme to derive hyperspectral

_{d}*a*and

*b*with hyperspectral (350-700 nm, ~2 nm step)

_{b}*R*and

_{rs}*K*as inputs is presented, where values of hyperspectral

_{d}*a*and

*b*spectra were obtained in oceanic and turbid inland waters. These estimates were further compared with available

_{b}*in situ*measurements from commercial instruments to check constancy. Furthermore, limitations and sources of uncertainties associated with both strategies for

*in situ a*and

*b*are discussed.

_{b}## 2. Data

#### 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. [15], and (2) to evaluate the updated scheme for derivation of *a* and *b _{b}* from purely radiative-transfer simulated

*K*and

_{d}*R*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 [25].

_{rs}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/m^{3} (0.01, 0.05, 0.1, 1, 2, 5, 10, 20, 50 mg/m^{3}); 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 [26]) 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 [27]. 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 (*r _{rs}*, sr

^{−1}), and the one with Raman was used for the application/evaluation of the

*R*-

_{rs}*K*scheme.

_{d}#### 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 *b _{b}*),

*R*and

_{rs}*K*, was obtained from the NASA bio-Optical Marine Algorithm Data set (NOMAD), which covers a

_{d}*K*(490) range from ~0.016 to ~4.6 m

_{d}^{−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

*b*were obtained using HOBI Labs Inc. HS6, or WET Labs ECO-BB or ECO-VSF. The concurrent measurements of

_{b}*R*and

_{rs}*K*were acquired from profile measurements of upwelling radiance and downwelling irradiance, respectively. A scheme of quality control for

_{d}*R*was also conducted following the method of Wei et al. [30], and here we only used

_{rs}*R*values with a quality assurance score higher than 0.7.

_{rs}Set two: Data collected from NOAA VIIRS ocean color validation cruise in 2014 (21 stations during 10 days). Profiles of downwelling irradiance (*E _{d}*, w/m

^{2}/nm) and upwelling radiance (

*L*, w/m

_{u}^{2}/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

*r*was calculated as the ratio of

_{rs}*L*(0

_{u}^{-}) to

*E*(0

_{d}^{-}) just below the water surface, which were obtained via extrapolation of measured

*L*(z) and

_{u}*E*(z) vertical profiles, respectively. The spectral

_{d}*K*was calculated by a nonlinear fitting of

_{d}*E*(z) between 0

_{d}^{-}and z

_{10%}, which is the depth where

*E*(z) is 10% of

_{d}*E*(0

_{d}^{-}). To reduce the impact of wave-focusing effect [31] on derived

*R*and

_{rs}*K*, 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. [9]. 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 [32] and Lee et al. [33]. 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 [34].

_{d}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*, *R _{rs}* and

*K*) were compiled, where the

_{d}*K*(490) value ranged from ~2.0 to ~20.7 m

_{d}^{−1}. The particulate absorption was measured using the quantitative filter-pad technique. The above-water measurement of

*R*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

_{rs}*E*at different depths between 0 to 1 m with a depth resolution of ~5 cm. The spectral

_{d}*K*was then calculated by a nonlinear fitting of

_{d}*E*(z) between 0

_{d}^{-}and z

_{10%}.

## 3. Methods

#### 3.1 Two-term r_{rs} model

In this study, we adopted the two-term *r _{rs}* model in Lee et al. [15] to link

*r*with IOPs, which is expressed as

_{rs}*a*and

*b*the total absorption and backscattering coefficients of the bulk water; while

_{b}*b*=

_{b}*b*+

_{bw}*b*, and the subscript

_{bp}*w*and

*p*represent water and particles, respectively. Here Ω represents the sun-sensor angular geometry for

*r*. ${G}_{0}^{w},{G}_{1}^{w},{G}_{0}^{p},{G}_{1}^{p}$ are model parameters that are functions of sun angle (

_{rs}*θ*) and viewing angles (

_{s}*θ*and

_{v}*φ*) [15]. A LUT for discrete sun-sensor angular geometry has been developed in Lee et al. [35]. 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 (${G}_{0}^{w},{G}_{1}^{w},{G}_{0}^{p},{G}_{1}^{p}$) of each view geometry (

_{v}*θ*and

_{v}*φ*), respectively, as a function of sun angle.

_{v}*θ*is a reference sun angle and fixed here at 30°. The values of model parameters (

_{s0}*x*and

_{0-3}*y*, Table 2) were obtained by a nonlinear fitting between parameters (${G}_{0}^{w},{G}_{1}^{w},{G}_{0}^{p},{G}_{1}^{p}$) and

_{0-5}*θ*via Eqs. (2.1) – (2.4). Note that the values of both

_{s}*θ*and

_{v}*φ*are 0° when applied to profiler measurements (

_{v}*L*(z) and

_{u}*E*(z)).

_{d}#### 3.2 Model for K_{d}

We employed the latest *K _{d}* model as presented in Lee et al. [21], where

*K*is expressed as

_{d}_{0-3}and γ) are presented in Lee et al. [21]. Note that this model can be used to estimate

*K*from ultraviolet to the visible bands.

_{d}#### 3.3 Steps to estimate a and b_{b}

As described by Eqs. (1) and (3), both *r _{rs}* and

*K*are functions of

_{d}*a*and

*b*; therefore, for any wavelength having measurements of both

_{b}*r*and

_{rs}*K*, there are just two unknowns, thus it is straightforward to calculate

_{d}*a*and

*b*from the two equations. Note that

_{b}*θ*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

_{s}*R*following the approach of Lee et al. [21], and the conversion between

_{rs}*r*and

_{rs}*R*is given by the following equation

_{rs}*r*(λ) and

_{rs}*K*(λ) are provided,

_{d}*a*(λ) and

*b*(λ) can be calculated straightforwardly. Note that it is not straightforward to correct the contributions from chlorophyll fluorescence, therefore retrievals of IOPs in the longer wavelengths (around 683 nm) in field-measured data could contain larger uncertainties.

_{b}## 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 *R _{rs}* was first corrected with the approach of Lee et al. [21]. Note that

*a*and

*b*(from 400 nm to 700 nm) are also the input parameters for the Hydrolight simulations, which are further compared with

_{b}*a*and

*b*estimated from

_{b}*R*and

_{rs}*K*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

_{d}*a*and

*b*resulted from the model formulation and model parameters employed for

_{b}*r*and

_{rs}*K*are extremely small. The MAPD here is defined as

_{d}*x*is a known value and

_{k}*x*is a modeled value. To understand the error propagation in the scheme, a test was further conducted with ± 5% or ± 10% errors included in

_{m}*R*or

_{rs}*K*. The results are shown in Table 3. It is found that generally the error in derived

_{d}*a*is primarily resulted from

*K*, while the error in derived

_{d}*b*is contributed by both

_{b}*R*and

_{rs}*K*, with the compound error generally less than ~11% for

_{d}*a*and less than ~20% for

*b*for ± 10% errors in both

_{b}*R*and

_{rs}*K*.

_{d}The spectrum of particle backscattering coefficient (*b _{bp}*

_{,}m

^{−1}) is generally considered following a power-law equation [36]

*b*(λ) was first calculated as

_{bp}*b*(λ) ‒

_{b}*b*(λ), where

_{bw}*b*(λ) is the backscattering coefficients of pure seawater with values taken from Morel [34]. Further, η was calculated by a nonlinear fitting of

_{bw}*b*(λ) between 400 nm and 700 nm. For the simulation data set, the range of η of input

_{bp}*b*(λ) varies from ‒ 0.5 to 2.5. Fig. 3 shows a histogram of the ratios of η calculated from retrieved

_{bp}*b*(λ) to those from known

_{bp}*b*(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

_{bp}*r*and

_{rs}*K*are of high quality, it is adequate to obtain reliable hyperspectral

_{d}*b*from the hyperspectral measurements of

_{bp}*r*and

_{rs}*K*.

_{d}#### 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).

### 4.2.1 NOMAD

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} [37]. An example of estimated and measured IOPs (*a*(λ) and *b _{b}*(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

*b*at ~683 nm is overestimated because of the contribution of chlorophyll-a fluorescence to

_{b}*r*at this wavelength, where no attempt (e.g., Huot et al. [38]) was taken here to remove this contribution in the process of deriving

_{rs}*b*.

_{b}Figure 5 shows a comparison of measured and estimated IOPs (*a* and *b _{b}*) 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 (

*R*

^{2}= 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}(

*R*

^{2}= 0.82, MAD = 0.0007 m

^{−1}, MAPD = 18.3%,N = 181) were obtained for

*b*(490) (Fig. 6(b)). In view both

_{b}*in situ*measurements and derivation from AOPs have errors/uncertainties, these results suggest that the

*a*and

*b*values derived from the combination of

_{b}*R*and

_{rs}*K*match well with instrument measurements.

_{d}The power-law formula (Eq. (6)) is still generally adopted by the community to describe the spectral variation of *b _{bp}* [7], although other schemes were also proposed [39]. 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

*b*at one wavelength (λ

_{bp}_{0}) to the rest of the visible bands [5]. Here, to evaluate the robustness of Eq. (6) for

*b*(λ), nonlinear regression was carried out between

_{bp}*b*(λ) and λ (resulting slope is η) for

_{bp}*in situ R*&

_{rs}*K*derived hyperspectral

_{d}*b*(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

_{bp}*R*> 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

^{2}*b*[36], although there are exceptions (~5% of this data set with

_{bp}*R*less than 0.5). Further, it is found that the derived η (

^{2}*R*> 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 [42], but narrower than the range presented in Kostadinov et al. [12], especially for clear oceanic waters. Further, a histogram of the ratios of all retrieved η from

^{2}*R*&

_{rs}*K*to those calculated from field-measured

_{d}*b*(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

_{bp}*R*&

_{rs}*K*for the derivation of hyperspectral

_{d}*b*.

_{bp}### 4.2.2 VIIRS2014

Further, *a* and *b _{b}* values were also estimated by the updated scheme from the measured

*K*and

_{d}*r*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

_{rs}*a*and

*b*spectra. It is found that the spectral features of

_{b}*a*(λ) and

*b*(λ) can be retrieved well by the new scheme. In particular, with this new scheme, the

_{b}*a*and

*b*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).

_{b}Figure 9 shows a comparison of ACS-measured *a* with those derived from *K _{d}* and

*r*at selective wavelengths. Excellent agreements between the two independent determinations can be found where there are low MAD and low MAPD values (e.g.,

_{rs}*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

*b*(λ) and the retrievals from

_{b}*r*and

_{rs}*K*. For all wavelengths from 411 nm to 665 nm, a very good agreement was obtained where MAD is commonly less than 0.001 m

_{d}^{−1}and MAPD < 30%. These results further demonstrate a robust performance of this updated scheme.

Similarly, the range of the retrieved η based on *r _{rs}*-

*K*derived

_{d}*b*(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

_{bp}*b*(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).

_{bp}With multicast measurements of *E _{d}*(z) and

*L*(z), the standard deviation (STD) of

_{u}*a*(λ) estimated from

*r*-

_{rs}*K*is assessed, which are further compared with the STD of

_{d}*a*(λ) obtained from multicasts of ACS measurements. Note that because

*K*is an average value in the upper water column, we here used the mean ACS-

_{d}*a*(λ) between 0

^{-}and z

_{10%}for comparisons. Figure 12 shows the histograms of STD for these determinations. For the wavelengths ranging from 411 to 555 nm, the STD of

*R*-

_{rs}*K*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

_{d}*R*-

_{rs}*K*estimated

_{d}*a*(411) varies from ~0.001 to ~0.022 m

^{−1}with 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

*R*and

_{rs}*K*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

_{d}*R*-

_{rs}*K*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].

_{d}### 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 [45], complex network of rivers or channels, and rich terrestrial input [46]). 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 *K _{d}* 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

*R*-

_{rs}*K*scheme with those of the two earlier

_{d}*R*-

*K*approaches [18, 47], as these approaches can be more easily implemented than other similar approaches [17, 48, 49]. The approach of Loisel and Stramski [18] was developed based on radiative transfer simulations, with much simpler relationships between IOPs and

_{d}*R*(or

*K*) compared to the approach of Morel [47]. We first evaluated the

_{d}*R*-

*K*approaches with the same data of

_{d}*r*and

_{rs}*K*calculated from the Hydrolight simulations, where a simple relationship (

_{d}*R*= 3.5

*r*) was used to convert

_{rs}*r*to

_{rs}*R*. Figure 15 shows a comparison of estimated

*a*and

*b*with the Hydrolight-input

_{b}*a*and

*b*. As indicated in Loisel and Stramski [18], the

_{b}*a*can be well retrieved with a small MAPD (4.76%), while the

*b*was generally overestimated for

_{b}*b*lower than ~0.002 m

_{b}^{−1}and underestimated for

*b*higher than ~0.004 m

_{b}^{−1}. The reasons for such (small) discrepancies may include the imprecise empirical conversion from

*r*to

_{rs}*R*and/or the approximations in the models between

*b*and

_{b}*R*(or

*K*). For the approach of Morel [47], as simple relationships between IOPs and

_{d}*R*(or

*K*) were adopted, both

_{d}*a*and

*b*were estimated with relatively larger MAPDs (> 13%). In particular, most of the absorption coefficients were overestimated (Fig. 15(c)).

_{b}For field measurements, Fig. 16 shows histograms of the ratio of estimated *a*(490) and *b _{b}*(490) to

*in situ*measurements. For the NOMAD data set, the approach of Loisel and Stramski [18] obtained similar

*a*(490) estimates when compared with the

*R*-

_{rs}*K*scheme (Fig. 16(a)), but more

_{d}*b*(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

_{b}*R*-

_{rs}*K*scheme appears to derive more reliable

_{d}*a*(411) and

*a*(490) than the approach of Loisel and Stramski [18] (Figs. 16(e-f)). Similarly, Fig. 17 compares the

*R*-

_{rs}*K*scheme with the approach of Morel [47]. Generally, both

_{d}*a*(490) and

*b*(490) were overestimated when estimated with the

_{b}*R*-

*K*approach of Morel [47].

_{d}## 6. Discussions

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 *b _{b}*). 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% [9].

On the other hand, the present *b _{b}* sensors (e.g., HS6, ECO-BB) never provide direct measurements of

*b*, and are at a few spectral bands only. These

_{b}*b*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

_{b}^{−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

*b*[57, 58]. Because of these models and assumptions, uncertainties of the ‘measured’

_{b}*b*can easily be ~20% [55]. These practices and observations advocate the development of more robust instruments to accurately ‘measure’ IOPs

_{b}*in situ*[59].

Similarly as the systems for *in situ* IOPs, the AOPs to IOPs schemes (including the updated *R _{rs}*-

*K*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

_{d}*r*and

_{rs}*K*from such data [61, 62], it is still an ongoing research subject to improve the data quality of

_{d}*r*and

_{rs}*K*obtained from such profiling measurements [63, 64].

_{d}The errors or uncertainties in *r _{rs}* and

*K*directly affect the accuracy of

_{d}*a*&

*b*derived from the

_{b}*R*-

_{rs}*K*scheme (see Table 3), but they have different impacts on the derived

_{d}*a*and

*b*values. Specifically, because

_{b}*K*is primarily a sum of

_{d}*a*and

*b*[13, 65] and generally

_{b}*a*>>

*b*for most natural waters, the accuracy of

_{b}*K*has more impact on the accuracy of

_{d}*a*; on the other hand, since

*r*is proportional to

_{rs}*b*, the accuracy of

_{b}*r*has stronger impact to the derived

_{rs}*b*. Also, as indicated in the

_{b}*in situ*measurement of

*b*where assumptions have to be made in order to convert single angle β to

_{b}*b*, it is also required to assume a particle scattering phase function for the relationship between

_{b}*r*and IOPs [15, 16]. The uncertainty due to this assumption is found to be < 20% for most waters [16], and some closure studies have shown that the

_{rs}*r*can be well reconstructed from

_{rs}*in situ*IOPs (

*a*and

*b*) if an appropriate phase function is used [66–68]. It is thus clear that inherently there will be some uncertainties between the

_{b}*in situ*measured

*b*and the

_{b}*R*&

_{rs}*K*derived

_{d}*b*, 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

_{b}*b*. In view of the above, and keep in mind that many times during field measurements the waters targeted by

_{b}*R*&

_{rs}*K*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

_{d}*a*values (except Lake Taihu) and less than ~30% in

*b*values between those derived from

_{b}*R*&

_{rs}*K*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

_{d}*R*&

_{rs}*K*system, the precisions of IOPs derived from the

_{d}*R*&

_{rs}*K*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

_{d}*a*&

*b*with the

_{b}*R*&

_{rs}*K*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 [69].

_{d}Note that here the *R _{rs}*&

*K*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

_{d}*a*&

*b*should be compared with averages in the upper water column (normally between 0- and z

_{b}_{10%}, as presented in this study) and larger uncertainties would be expected as the models of

*R*and

_{rs}*K*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 [17] and Gordon et al. [70].

_{d}## 7. Summary

An updated scheme is presented in this study to derive *a* and *b _{b}* of bulk water from a combination of field-measured

*R*and

_{rs}*K*. Applications to both Hydrolight simulations and field measurements show that the retrieved

_{d}*a*and

*b*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

_{b}*a*and

*b*values obtained from the

_{b}*R*&

_{rs}*K*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

_{d}*R*&

_{rs}*K*system here not only provides an alternative and robust way to obtain

_{d}*a*&

*b*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

_{b}*R*and

_{rs}*K*, this scheme will fill the gap that the current

_{d}*b*sensors cannot provide a measurement of hyperspectral

_{b}*b*. On the other hand, it is always desired to have more robust instruments to measure hyperspectral IOPs

_{b}*in situ*.

## Funding

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).

## Acknowledgment

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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