Many studies have consistently found that the particle backscattering coefficient (bbp) in oligotrophic oceans estimated from remote-sensing reflectance (Rrs) using semi-analytical algorithms is higher than that from in situ measurements. This overestimation can be as high as ~300% for some oligotrophic ocean regions. Various sources potentially responsible for this discrepancy are examined. Further, after applying an empirical algorithm to correct the impact from Raman scattering, it is found that bbp from analytical inversion of Rrs is in good agreement with that from in situ measurements, and that a closure is achieved.
© 2014 Optical Society of America
The spectrum and amplitude of the light leaving the ocean depends on the constituents therein and therefore can be used to obtain information about these constituents. For remote sensing application, the water-leaving radiance (Lw, W m−2 sr−1) is used to describe this light, and because Lw is proportional to the incoming light on the ocean, described by the downwelling irradiance (Ed, W m−2), Ed is used to normalize Lw and remove the dependence of variable incident irradiance. The ratio Lw/Ed, termed the remote sensing reflectance (Rrs, sr−1) , depends on the concentration of optically active constituents in the water and the geometry of observation (angular distribution of the light field and observation direction of the sensor).
The impact of the optically active constituents on Rrs can be described by inherent optical properties (IOPs) . The IOPs depend only on the type and concentration of constituents and not on the ambient light field . These IOPs are the absorption coefficient (a, m−1) which describes the loss of photon by absorption, and the volume scattering function (VSF, m−1 sr−1) which represents the angular distribution of scattered photons. Because photons leaving the ocean are incoming from the atmosphere and returning to the atmosphere (i.e., returning backward), the total backscattering coefficient (bb, m−1), which is the integration of the VSF in the backward direction, is commonly used to parameterize the impact of the VSF on Rrs [4,5]. Such that a function of the form Rrs = f(a, bb) is generally used to describe the relationship between a, bb and Rrs in the ocean [6,7]. This function will also have to describe the observation geometry, which can be approximated at first order by the satellite viewing angle and solar angle. Most of these variables have a spectral dependence. In addition to a and bb, other processes contribute to the Rrs, and these process are trans-spectral and will transfer energy from shorter to longer wavelengths, which lead to a larger amount of light in the longer wavelengths than would be expected solely from the absorption and backscattering processes [6,7]. One such trans-spectral process is Raman scattering [8,9], in which a fraction of the photons incident on water at a shorter wavelength lead to emission at a longer wavelength (with a constant wavenumber difference from the incident photon). Another trans-spectral process is fluorescence where photons are absorbed by a molecule and reemitted at longer wavelengths. The main constituents in ocean waters that fluoresce are chlorophyll (emission light is centered near 685 nm)  and CDOM with a broad emission band throughout the visible .
The particulate backscattering coefficient (bbp, m−1), the fraction of the total backscattering coefficient that is not due to water molecules, is an important IOP. Not only does it describes how light is returned to the ocean surface and thus forms the signal collected by satellite sensors, but it’s value, when interpreted in terms of particle population, has also been proposed to infer concentrations of organic and inorganic constituents in the oceans [12–15]. Because of this importance, both remote sensing [13,16] and in situ methods  have been developed to obtain measurements of bbp in surface waters in the global oceans.
Remote sensing methods to retrieve bbp are based on its relationship to Rrs. For empirical algorithms , functional relationships are described based on concurrent measurements of bbp and Rrs. For such empirical algorithms, there is no involvement of the radiative transfer equation (RTE), where all the information required for the algorithms is provided by the measurements and consequently any measurement bias are directly reflected in the output. Empirical algorithms will also include all relevant trans-spectral effects as they covary with the input Rrs and do not require knowledge of the optical properties of waters. Semi-analytical methods [16,18], on the other hand, obtain bbp based on approximations of relationships found through the radiative transfer theory between Rrs and the IOPs and employ a few optical and bio-optical properties and/or relationships. Thus, bbp derived semi-analytically from Rrs is independent of measured bbp in situ but depends on using an accurate approximation of the relationship between Rrs, IOPs and trans-spectral processes.
If every step necessary for both determinations is well controlled and understood, bbp from both methods should agree with each other, i.e. there should be a closure [19–21], as demonstrated in Gordon et al. . But the study 22 covered just two cases, and the inversion scheme uses vertical profiles of upwelling radiance and downwelling irradiance as inputs, which is very different from remote sensing inversion. Empirical approaches also indicated good matches [13,15], but such results do not indicate a closure between two independent determinations. Various studies, in particular those using Rrs as the input to analytically retrieve bbp [23–25], however, have found that bbp values derived from semi-analytical algorithms are generally higher than those estimated from in situ measurements, i.e. there is no closure on bbp. For instance, it was found that bbp derived from MODIS-Aqua Rrs is systematically higher than bbp inferred from chlorophyll-a concentration ([Chl], mg/m3) by a factor of 2-3 (see Fig. A1 in Huot et al.  and Fig. 11 in Brown et al. ), where the relationship between bbp and [Chl] was developed based on concurrent measurements of [Chl] obtained by HPLC and bbp obtained by a BB3 sensor (Wetlabs, Inc.) . Beyond the study by Huot et al. , the Rrs-derived bbp in oligotrophic waters via spectral optimization algorithms is also found much higher (up to a factor of 2) than that measured in situ in recent evaluations of algorithm performances [25,27].
In principle, the uncertainty in the bbp derived analytically from Rrs are small at least for oligotrophic waters  where the absorption coefficients in the longer wavelengths (550 nm and longer) are nearly constant (see more detailed discussions below). This is well demonstrated when the semi-analytical algorithms [25,29] were applied to a data set  obtained from radiative transfer simulations with the Hydrolight software 30 when trans-spectral effects were not included. Comparison showed that the retrieved bbp generally contains errors ~20% or less for oligotrophic conditions [16,25]. Errors in measured Rrs will certainly contribute to analytically derived bbp, but the errors in measured Rrs are not systematic [31,32], therefore we should not expect systematically and significantly higher bbp from errors in Rrs. It is necessary to address such large systematic overestimations of bbp. And, more importantly, it is necessary to demonstrate if a closure can be achieved after these sources of error are taken into consideration. Here, we focus on a semi-analytical algorithm, the quasi-analytical algorithm (QAA) developed by Lee et al. , to examine these discrepancies. After rapidly reviewing the algorithm, we further address the sources that could contribute to the non-closure in bbp, and then followed with evaluations of ocean color bbp products after the sources of uncertainties/errors are considered.
2. Brief description of the QAA
For the retrieval of inherent optical properties (IOPs) from spectral Rrs, a quasi-analytical algorithm (QAA) was developed by Lee et al. . QAA uses the relationship between IOPs and Rrs derived from the RTE, and makes very few assumptions in the process of algebraically solving for bbp from Rrs. The uncertainty in QAA-derived bbp, at least for oligotrophic waters, is small and well understood.
From the RTE, a simplified relationship between IOPs and reflectance derived from radiative transfer computations, where the trans-spectral effects are omitted, is [2,6]6,33]. The coefficients g0 and g1 are wavelength-independent model parameters, which depend on sun-sensor angular geometry as well as the shape of particle phase functions [7,34]. bb can be expressed as35,36], although bbw varies slightly with temperature and salinity . Therefore, when bb is known, it is straightforward to derive bbp using Eq. (2).
As Eq. (1) indicates, once g0 and g1 are known, that there are just two unknowns for any given rrs: a and bb, so bb can be easily derived when a is known. In the QAA scheme, a at a reference wavelength (λ0) is estimated with
The first step of the algorithm consists in estimating a(λ0) using an empirical algorithm based on Rrs band ratios to estimate Δa(λ0). As described in Lee et al. , for oligotrophic waters a(λ0) is dominated (>95%) by aw(λ0), thus a(λ0) can be accurately estimated; errors in estimating Δa(λ0) have negligible impact on a(λ0) for such waters. This is particularly true for the oceanic gyres, where the amount of optically active constituents other than water (such as phytoplankton and gelbstoff) are scarce [37,38]. Therefore, as long as Rrs(550) (or Rrs(555)) is measured accurately and Eq. (1) is valid, bb(550) (or bb(555)) can be estimated accurately. If bbw is accurately known, or at the very least is the same value as that used when processing the in situ validation measurements, bbp(550) (or bbp(555)) estimated from Rrs should match that from in situ measurements.
3. Impact of errors in the QAA components on the bbp retrieval
As mentioned above, the QAA algorithm when applied to data simulated by the radiative transfer equation retrieves bbp with very low errors , as is expected in waters where the optical properties are consistent with those of oligotrophic waters. Because QAA is essentially based on such simulations, this highlights that the algorithm structure and approximations allow accurate retrieval of bbp for such conditions. The discrepancy found when compared with in situ measurements must thus originate from one of three sources. The first is that the radiative transfer simulations are not representing accurately the relationships between Rrs and IOPs in situ as expressed in Eq. (1) (and those that allow going from rrs to Rrs). The second is that the optical properties of pure saltwater, used in Eqs. (2) and (3), are incorrect. The third is that there are errors or biases present in measured bbp. We now examine these different potential sources of variability.
3.1 Rrs-IOP relationship
The relationship between Rrs and bb/(a + bb) as described by Eq. (1) is not exact and other mathematical models were also developed in the past decades [39–42]. Instead, it is an approximate solution of the RTE with no trans-spectral effects present. For an hypothetical ocean without trans-spectral effects, this simplification results in errors up to ~12% compared with the simulations . Depending on inputs used in data simulations (e.g., phase function of particle scattering, range of IOPs, etc.), slightly different g0 and g1 values are obtained. By changing the values of g0 and g1 using two sets of published values (i.e. from 0.0949 and 0.0794  to 0.089 and 0.125 ), it is found that the retrieved bbp values with the latter coefficients are higher than that with the former, and this difference is higher for smaller bbp (see Fig. 1), with the maximum difference around 12% (for bbp(555) ~0.0005 m−1). This analysis shows that the selection of values to represent g0 and g1 cannot be the reason leading to ~200-300% difference between Rrs-retrieved bbp and in situ measured bbp. This is further supported by the fact that in more eutrophic waters the retrieved and measured bbp are much closer; therefore, for errors in g0 and g1 to be the source of the significant differences in bbp observed it would require substantial changes in g0 and g1 with trophic status. The values of g0 and g1 vary mostly due to the geometry of observation (including angular distribution of the incident light field) and angular shape of the particle phase function in the backward domain [5,34]. While the former does not changes with trophic status, observations [23,43,44] suggest that the latter also does not change strongly across trophic states.
A major caveat of Eq. (1) in modeling Rrs is the omission of contributions from in-elastic processes, which include those from chlorophyll a and CDOM fluorescence and Raman scattering . Chlorophyll a fluorescence is centered around 685 nm (with a roughly Gaussian shape at a half-height width around 25 nm), so its influence in the QAA-retrieved bbp is nil as QAA uses measurements around 550 nm to retrieve bbp. For oceanic waters, the contribution from the CDOM fluorescence is small, so the impact of this effect is also negligible [11,45]. However, the contribution from the Raman scattering to Rrs can be as high as 20% in the longer wavelengths for oceanic waters [8,9,46], thus the impact of Raman scattering on the retrieval of bbp could be significant for clear oceanic waters [47,46,48]. Indeed for simulated waters with Raman contribution so extreme that bbp = 0 and an absorption consistent with a chlorophyll concentration of 0.01 mg/m3, the QAA scheme returns a bbp(550) value of ~0.00025 m−1 as it interprets the light from Raman scattering to particulate backscattering 22; this value is approximately half of the lowest values of bbp(550) measured in the NOMAD data set .
To effectively process ocean color satellite imageries, an empirical algorithm has been developed to correct this contribution in Rrs inversion , and Fig. 2 presents bbp retrievals before and after the correction of Raman contribution for measurements made in the South Pacific gyre. For these “clearest” water of the global oceans [37,51], the ratio of Rrs -derived bbp to in situ bbp decreased from ~300% to ~200% (for lowest bbp(555)), removing on average 0.00022 m−1 of retrieved bbp(555) from Rrs, clearly echoing the importance of removing Raman effect in the analytical retrieval of bbp from Rrs [22,46–48]. As presented, however, bbp from Rrs is still generally higher than that from in situ measurement, and this difference is ~0.00025 m−1 for all samples.
The scheme to empirically correct the Raman effect has a maximum error of about 15% in estimating the Raman contribution to Rrs . The impact of a 15% adjustment in the Raman correction, however, results in just ~3% uncertainty in the Rrs derived bbp. This indicates that the empirical scheme is sufficient for the correction of Raman scattering contribution in oceanic waters and that the origin of the remaining differences cannot originate from incorrect Raman correction.
3.2 Pure-water absorption coefficient
Another source of error in the QAA-derived bbp is the uncertainty in the absorption coefficient of pure water at the reference wavelength (550 or 555 nm), since this value is used in the estimation of a(λ0) before the analytical derivation of bbp(λ0). Based on Eq. (1), higher a(555) will result in higher bb(555) and consequently higher bbp(555). Modern laboratory and field measurements (see a list of aw values in Mobley 1) of aw(555) span approximately 0.0596 m-1  to 0.0673 m−1 (spectrally interpolated from Smith and Baker ). QAA uses the lowest published aw values , leading to the lowest values of bbp(555). Pope and Fry  also provide the standard deviation of the error on their measurement at 555 nm as 0.0012 m−1, this translates in QAA in an error of ~4% on the determination of bbp(555) for the data shown in Fig. 2. Obtaining optically pure water is very challenging and the value published may still suffer from minor contamination by absorbing components (as was the case with previous determinations). However, above ~450 nm , the aw(555) values of Pope and Fry  are consistent with values obtained from the diffuse attenuation coefficient of the clearest water which imposes a strong upper limit on the aw value and strongly support the assertion that these are nearly devoid contribution from colored substance at 555 nm.
4. Uncertainties in bbp measured in situ
The non-closure of bbp does not have to originate only from errors or uncertainties in Rrs inversion; it also includes errors from in situ measurements . While significant efforts were deployed to obtain accurate values in the clearest Pacific waters, these measurements are extremely challenging and push the sensitivity and resolution of the instruments to their limit.
Since the advent of commercial instruments, measurements of bbp in situ are generally made with active instruments (e.g., Hydroscat, HOBI Lab, Inc.; BB3, WET Labs, Inc.). In these sensors, light from a pulsed LED is emitted into the water and the backscattered light (usually with a scattering angle of ~120° to 140°) from this source is recorded, and this energy is converted to bbp after subtracting the contribution of pure saltwater. Calibration of this system is usually carried out in the lab with well-characterized materials (reflecting surface or beads) . Apart from possible calibration errors, there are two main sources of errors that can arise from these measurements. The first is caused by an estimation of bbp from a measurement of the volume scattering function at 120þ or 140þ, while the other may arise from the undersampling of the backscattering from larger and less abundant particles due to the limited observation volumes.
The error caused by the estimation of bbp from a measurement of the VSF between 120þ to 140þ (a constant factor χ is used for this conversion) has been studied by different authors [57–59]. All authors report that errors caused by this approximation should be less than ~10%. This conclusion arises in all cases from measured VSF in coastal waters as well as simulated VSF from Mie scattering. Although a thorough study of the conversion factor hasn’t been carried out in oligotrophic ocean waters it appears unlikely that the error could increase by a factor of 10 and could explain the discrepancies observed. Therefore, the measurement strategy at a single angle does not seem to be a plausible source of error to explain the differences observed.
The passive measurement systems have a sampling volume in the range of 10s to 1,000s m3, while the instruments measuring bbp in situ have a sampling volume around 10−6 m3, i.e. there is a many orders of magnitude difference in the sampling volume between the active backscatter sensors and the passive remote systems. In oceanic waters, larger particles are generally scarcer, thus the sampling volume of an active sensor will likely miss some of the larger particles during individual observations. This is demonstrated in Briggs et al.  where the observed spikes in bb time series are indicators of large particles, although larger particles generally have a smaller backscattering efficiency . The difference in sampling volume may help to explain the slightly (~0.0003 m−1) higher bbp by Hydroscat than that by BB3 in the clearest waters, although both sensors were carefully calibrated for the field measurements (see Stramski et al. [54,51] for details). This is because Hydroscat has larger sampling volume than BB3 (or BB9) due to measurement geometry, thus more particles might be “counted” for by a Hydroscat sensor, although Hydroscat will not be able to cover all particles in the bulk water during a measurement scan. Indeed, if Hydroscat underestimated bbp(555) by 0.0002 m−1 (a very small number) , much better closure of bbp is achieved between Rrs inversion and in situ measurements for waters in the South Pacific gyre (see Fig. 3), with the ratio of Rrs-derived bbp(555) to the adjusted (adding 0.0002 m−1 to the Hydroscat measured value) in situ bbp(555) in a range of ~0.8-1.2 (which was ~1.1 - 2.2, see Fig. 2). The ± 20% difference (transfers to ~0.0001 m−1 absolute difference) is beyond the precision of either remote sensing measurements or in situ determinations. Indeed, even the value of 0.0002 m−1 that we added is within the instrument offset uncertainty (specified as between 5x10−5 to 5 x 10−4 m−1, http://www.hobilabs.com/). These results suggest that it is necessary to take appropriate average over a long time or wide spatial range of in situ bb measurements to have a “ground truth” to match the large volume observed by remote sensing systems. In particular, if the objective is to compare products of backscattering sensors with that from passive sensors, whether remotely or in situ, the removal of “spikes” must be carried out only if the spikes are caused by instrument malfunction and not by rare large particles (see user manual of the backscattering sensors).
In an attempts to examine the impact of Raman scattering on the remote sensing retrieval globally, we carried out an analysis similar as Huot et al. , where we compared the MODIS derived bbp using QAA with in situ statistical relationships that have been derived (Fig. 4). As opposed to Huot et al.  the comparison is done at 555 nm instead of 443 nm to avoid spectral errors when extrapolating the value from 555 nm to 443 nm; furthermore, care was taken to use the same value of the backscattering coefficient of pure saltwater (though at constant salinity) for the remote sensing estimates as was used for the processing of the Hydroscat data underlying the Huot et al.  relationships. Before the Raman corrections there is clearly an overestimation of bbp(555) (Fig. 4(a), blue line vs pink line); this overestimation disappears after the correction, and the running median of the remote sensing data follows very closely the Hydroscat parameterization of the backscattering and is just below the global statistical relationship derived by Brewin et al. . These results confirm that the Raman effect is a big contributor to the non-closure of bbp in previous studies and that the empirical scheme to correct this effect is plausible .
The results of this study highlighted the two major sources that contributed to the non-closure of bbp in oligotrophic oceans shown in previous studies. One is the Raman scattering, which resulted in an increased bbp when it is analytically derived from Rrs when the Raman effect was not considered; another is associated with the uncertainty in in situ measured bbp (see Table 1 for summary). In particular, it is extremely difficult to get the “ground truth” of bbp of such clear waters, as the values of bbp are generally smaller than 0.001 m−1. A closure in bbp for oligotrophic water is achieved after correcting the Raman contribution in the inversion process and using the adjusted bbp values (adding 0.0002 m−1 to that measured by Hydroscat, a value well within the measurement uncertainty). This closure in bbp demonstrated the effectiveness of the empirical scheme in correcting the Raman scattering contribution, and provided a confidence in satellite ocean-color bbp products, as long as all relevant processes are considered in its analytical inversion.
Financial support from the NASA Ocean Biology and Biogeochemistry and Water and Energy Cycle Programs and the JPSS VIIRS Ocean Color Cal/Val Project is greatly appreciated. We are grateful for the comments, and discussions, about bb measurements from Drs. Jim Sullivan, Robert Brewin, and Giorgio Dall'Olmo. The acquisition of BIOSOPE data where funded through CNRS(1)-INSU(2) grants; we thank Marlon Lewis for sharing the Rrs data and Dariusz Stramski for sharing the Hydroscat data. Comments and suggestions from two anonymous reviewers significantly improved this article and are greatly appreciated.
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