Spectra of particulate backscattering in natural waters

Hyperspectral profiles of downwelling irradiance and upwelling radiance in natural waters (oligotrophic and mesotrophic) are combined with inverse radiative transfer to obtain high resolution spectra of the absorption coefficient (a) and the backscattering coefficient (bb) of the water and its constituents. The absorption coefficient at the mesotrophic station clearly shows spectral absorption features attributable to several phytoplankton pigments (Chlorophyll a, b, c, and Carotenoids). The backscattering shows only weak spectral features and can be well represented by a power-law variation with wavelength (λ): bb ~λ −n , where n is a constant between 0.4 and 1.0. However, the weak spectral features in bb suggest that it is depressed in spectral regions of strong particle absorption. The applicability of the present inverse radiative transfer algorithm, which omits the influence of Raman scattering, is limited to λ < 490 nm in oligotrophic waters and λ < 575 nm in mesotrophic waters. ©2009 Optical Society of America OCIS codes: (010.4450) Oceanic optics; (010.4588) Oceanic scattering; (010,5620) Radiative transfer; (280.4991) Passive remote sensing. References and links 1. G. J. Kirkpatrick, D. F. Millie, M. A. Moline, and O. Schofield, “Optical discrimination of a phytoplankton species in natural mixed populations,” Limnol. Oceanogr. 45, 467–471 (2000). 2. N. Hoepffner, and S. Sathyendranath, “Determination of major groups of phytoplankton pigments from absorption spectra of total particulate matter,” J. Geophys. Res. 98(C12 C12), 22789–22803 (1993). 3. C. O. Davis, J. Bowles, R. A. Leathers, D. Korwan, T. V. Downes, W. A. Snyder, W. J. Rhea, W. Chen, J. Fisher, P. Bissett, and R. A. 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Gordon, “The sensitivity of radiative transfer to small-angle scattering in the ocean: A quantitative assessment,” Appl. Opt. 32(36), 7505–7511 (1993). 10. S. Hooker, S. McLean, J. Sherman, M. Small, G. Lazin, G. Zibordi, and J. Brown, “The seventh SeaWiFS intercalibration round robin experiment (SIRREX-7),” NASA Technical Memorandum 2002–206892, Vol 17, SeaWiFS Postlaunch Technical Report Series (1999). 11. J. Mueller, C. Pietras, S. Hooker, R. Austin, M. Miller, K. Knobelspiesse, R. Frouin, B. Holben, and K. Voss, “Ocean optics protocols for satellite ocean color sensor validation, Revision 4, Volume 2: Instrument specifications, characterization, and calibration,” NASA-TM-2003–21621/Rev4-Vol II, (2003). #113277 $15.00 USD Received 24 Jun 2009; revised 14 Aug 2009; accepted 21 Aug 2009; published 27 Aug 2009 (C) 2009 OSA 31 August 2009 / Vol. 17, No. 18 / OPTICS EXPRESS 16192 12. M. S. Twardowski, J. M. Sullivan, P. L. Donaghay, and J. R. V. 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Twardowski, S. Freeman, X. Zhang, S. Vagle, and R. Zaneveld, “Resolving surf zone particle dynamics with high sampling rate volume scattering function measurements,” Proc. Ocean Optics XIX, October 6–9, Barga, Italy (2008). 18. M. S. Twardowski, H. Claustre, S. A. Freeman, D. Stramski, and Y. Huot, ““Optical backscattering properties of the “clearest” natural waters,” Biogeosciences 4, 1041–1058 (2007). 19. C. Moore, M. S. Twardowski, and J. R. V. Zaneveld, “The ECO VSF A multi-angle scattering sensor for determination of the volume scattering function in the backward direction,” Proc. Ocean Optics XV, 16–20 October, Monaco, Office of Naval Research, USA (2000). 20. X. Zhang, and L. Hu, “Estimating scattering of pure water from density fluctuation of the refractive index,” Opt. Express 17(3), 1671–1678 (2009). 21. X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: effect of salinity,” Opt. Express 17(7), 5698–5710 (2009). 22. T. Oishi, “Significant relationship between the backward scattering coefficient of sea water and the scatterance at 120°,” Appl. Opt. 29(31), 4658–4665 (1990). 23. E. Boss, and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40(30), 5503–5507 (2001). 24. J. M. Sullivan, M. S. Twardowski, P. L. Donaghay, and S. A. Freeman, “Use of scattering to discriminate particle types in Coastal Waters,” Appl. Opt. 44, 1667–1680 (2005). 25. J.-F. Berthon, E. Shybanov, M. E.-G. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46(22), 5189–5203 (2007). 26. T. J. Petzold, “Volume scattering functions for selected natural waters,” Scripps Institution of Oceanography, Visibility Laboratory, San Diego, CA. 92152, SIO Ref. 72−78 (1972). 27. F. M. Sogandares, and E. S. Fry, “Absorption spectrum (340-640 nm) of pure water. I. Photothermal measurements,” Appl. Opt. 36(33), 8699–8709 (1997). 28. R. M. Pope, and E. S. Fry, “Absorption spectrum (380-700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36(33), 8710–8723 (1997). 29. A. Morel, B. Gentili, H. Claustre, M. Babin, A. Bricaud, J. Ras, and F. Tieche, “Optical properties of the ‘clearest’ natural waters,” Limnol. Oceanogr. 52, 217–239 (2007). 30. S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for globalscale applications,” Appl. Opt. 41(15), 2705–2714 (2002). 31. R. H. Stavn, and A. D. Weidemann, “Optical Modeling of Clear Ocean Light Fields: Raman Scattering Effects,” Appl. Opt. 27(19), 4002–4011 (1988). 32. B. R. Marshall, and R. C. Smith, “Raman scattering and in-water ocean optical properties,” Appl. Opt. 29(1), 71– 84 (1990). 33. Y. Ge, K. J. Voss, and H. R. Gordon, “In situ measurements of inelastic light scattering in Monterey Bay using solar Fraunhofer lines,” J. Geophys. 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Introduction
Space-borne sensors such as the Coastal Zone Color Scanner (CZCS), the Sea-viewing Wide Field-of-view Sensor (SeaWiFS), and the MODerate resolution Imaging Spectroradiometer (MODIS) have provided ocean color imagery from which the concentration of the phytoplankton photosynthetic pigment chlorophyll a (Chl a) has been estimated on a global scale since 1978.Phytoplankton are the first link in the marine food chain and play a significant role in the global carbon cycle and upper ocean heating.CZCS was a four-band radiometer operating from 443 to 670 nm, while SeaWiFS and MODIS are eight-band radiometers operating from 412 to 865 nm.It is reasonable to expect that an instrument with a larger number of spectral bands might provide more information regarding phytoplankton.For example, with a more detailed rendition of the ocean color spectrum it should be possible to apply advanced spectroscopic tools such as derivative analysis to determine the relative concentrations of the various accessory pigments; this would provide information regarding the size and species composition of the phytoplankton community [1].Laboratory measurements clearly show spectral features that enable modeling of the plankton absorption with individual pigment components [2], suggesting that remote measurements could do the same.Indeed, "hyperspectral" reflectance measurements might enable a wide range of spectroscopic analyses of absorption and backscattering spectra to derive the phytoplankton species composition or community structure, an important determinant of the upper ocean carbon cycle and something that is not possible with the limited number of spectral channels available on most current satellite sensors.Such hyperspectral aircraft instruments have been developed [3], and space borne missions optimized for ocean color have been proposed.
The spectral diffuse reflectance of the natural waters (the "ocean color") is proportional to the ratio of the backscattering (b b ) coefficient to the absorption coefficient (a), i.e., b b /a [4].High spectral resolution absorption data are available for particles suspended in natural waters, but there is no instrumentation available to provide high spectral resolution backscattering data.Such data are needed to understand the hyperspectral diffuse reflectance of natural waters, for both interpretation and for modeling of hyperspectral data.In this paper we provide examples of the derivation of high resolution spectra of a and b b from hyperspectral depth profiles of downwelling irradiance and upwelling radiance obtained in oligotrophic and mesotrophic waters, via methods of inverse radiative transfer [5,6].
We start by reviewing radiative transfer, defining the inherent and apparent optical properties of natural waters, and describing the inversion algorithm used to retrieve the inherent optical properties from the apparent optical properties.Next we describe the hyperspectral radiance/irradiance data and present the resulting inherent optical properties.Then we show how Raman scattering contamination limits the spectral range of applicability of the inversions.Finally, we provide a discussion of the results and some concluding remarks.

Radiative transfer, inherent and apparent optical properties
Consider a plane parallel, vertically stratified water-body with the z-coordinate measured from zero at the surface downward.Let the direction of propagation of light be specified by the unit vector ξ , defined as follows.With the x-y plane parallel to the water surface, let ˆx e , ˆy e and ˆz e be unit vectors in the x, y, and z directions, respectively.Then ξ is given by ˆˆˆŝin cos sin sin cos , It is independent of the magnitude of the scattering.The IOPs are linearly summable over those of the water and of the constituents suspended or dissolved in the water, e.g., ), where the subscripts "w" and "p" refer, respectively, to water and suspended particles.For convenience later, we define component IOPs, some of which are directly amenable to measurement with existing in situ instruments.The absorption coefficient a is separated into that due to water, particles, and dissolved material, denoted respectively by the subscripts "w", "p", and "g", respectively: In a homogeneous water body, R L (z), K x (z), and ( ) z µ are all only weakly dependent on depth; hence they are termed "apparent" optical properties.Of course they are not properties of the medium alone because they depend on the external illumination, e.g., the solar zenith angle, as well.
If the RT equation is multiplied by d ( ) and integrated over all directions ξ , the result can be put in the form This exact relationship is referred to as Gershun's equation.
Given the IOPs and the distribution of radiance incident on the medium, it is possible to show [8] that the solution to the RT equation is unique as long as a(z) ≠ 0, which is always the case if the medium (or part of the medium) is water.In the case of interest here, the medium is the ocean-atmosphere system illuminated by the solar beam; hence the actual IOPs will yield a unique solution to the radiance.

The inversion algorithm
The inversion algorithm is a procedure for the retrieval of a(z) and b b (z) from the downwelling E d (z) and upwelling E u (z) irradiances.As in the present application, when the upwelling radiances L u (z) are measured rather than E u (z), the code retrieves Q(z) = E u (z)/L u (z) as well.The inversion algorithm is described in detail in Gordon and Boynton [5].
The basic idea of the inversion algorithm is to use the measured AOPs to find IOPs that reproduce the AOPs when inserted into the RT equation.This is effected through an iterative procedure in which trial IOPs are introduced into the RT equation, which is then solved for the AOPs.Based on the differences between the measured and calculated AOPs, a new trial set of IOPs is determined and the RT equation solved again, etc. Gordon and Boynton [5] found that their method of varying the IOPs usually "converges" to a solution.However, one must not be too ambitious, as the AOPs depend only weakly on b(z) but strongly on a(z) and b b (z) [9].Thus, neither b(z) nor c(z) can be obtained except in rare situations: when the scattering phase function of the particles is known a priori.Although E d (z) and L u (z) are input into the code, the AOPs that actually drive the solution are K d (z) and R L (z).E d (z) and L u (z) are usually measured by independent sensors.Their absolute calibration is unimportant; however, because of the importance of R L (z), the relative calibration of E d (z) and L u (z) is of paramount importance.
The solution to the RT equation (at each iteration) requires a guess for the scattering phase function; Gordon and Boynton [5] however showed that the accuracy of the guess was not crucial, and that excellent retrievals could still be obtained with an incorrect phase function.Indeed, if the phase function actually used in the retrievals is close to being correct, then excellent retrievals of both b(z) and c(z) are obtained as well.Usually, the algorithm is operated with a depth-independent phase function.However, in clear water there is a considerable contribution to β from β w , particularly into the backward hemisphere.This can cause significant variations in the phase function with depth in a stratified medium, and this degrades the performance of the algorithm.Thus, the following modification to the algorithm was developed [6].A fixed phase function was first chosen for the particles, and then the algorithm was operated to completion resulting in a(z) and b b (z).From b b (z) and the phase function, b(z) can be computed, along with the fraction η(z) = b w /b(z).Then the phase function is modified to is the phase function for pure water, and ˆ( ) is the depthindependent phase function for particles.This new depth-dependent phase function was then used in the RT code and after operating the full algorithm with the modified phase function, new values of a(z) and b b (z) were obtained, which yield new estimates of η(z), etc.We call these iterations on η(z) "external" iterations to differentiate between the iterations in the Gordon and Boynton [5] algorithm which we refer to as "internal" iterations.Boynton and Gordon [6] showed that this procedure yielded retrievals in clear water at the accuracy seen earlier in the constant-P p algorithm.In the results presented in Section 5, up to a maximum of 50 internal iterations were employed and 6 external iterations were used.
This algorithm has the attribute that it provides IOPs that reproduce the AOPs as accurately as possible; however, if the physics used to relate the AOPs and IOPs is incomplete, the retrieved IOPs will be incorrect, although the reconstructed irradiances will be excellent.As Raman scattering and fluorescence are important, particularly at longer wavelengths, they should be included; however, here we ignore both transspectral terms, but assess the influence of Raman scattering on the final results in Section 6.

The data sets
The hyperspectral AOP data we examine here consist of depth profiles of the downwelling spectral irradiance, E d , and the upwelling spectral radiance toward the zenith, L u , measured at 3.3 nm increments from 350 to 700 nm with a wavelength accuracy of ± 0.1 nm.Each spectral band was approximately 9 nm wide.The data were obtained at two locations.The first station was approximately 12 miles west of the north coast of Lanai, Hawaii in oligotrophic waters (Chl a ~0.1 mg/m 3 ) occupied during February/March, 2007, and the second station was approximately 11 miles west of San Diego, California in mesotrophic waters (Chl a ~1 mg/m 3 ) and occupied in January, 2008.
The measured values of E d and L u as a function of depth and wavelength at the Hawaii station are provided in Figs. 1 and 2, respectively.These were obtained using a free-fall profiling radiometer system (HyperPro II, Satlantic Inc.).Sensors were characterized and calibrated using National Institute for Standards and Technology (NIST) standard lamps preand post-cruise following standard protocols developed in partnership with NASA and NIST (e.g., [10,11]).For the Hawaii experiment, sensors were intercompared with NIST-provided standards of spectral irradiance and radiance at the field site as well.Based on these documented results, absolute radiometric uncertainty is assessed at < 2.8% for radiance, and < 2.1% for irradiance.
Dark corrections are made using an internal optical shutter operated throughout each profile.An above-water downwelling irradiance sensor mounted in the ship's superstructure is used to ensure stable surface values as the profile is taken, and tilt and temperature measurements are made using sensors onboard the instrument.Measured radiometric quantities are corrected for out-of-band stray light and thermal responsivity.
During data processing, all raw sensor data are first converted into engineering units using the calibration coefficients, and corrected for dark values, stray light and thermal responsivity.Quality control procedures include elimination of data with instrument tilts > 5°, and any profiles with significant fluctuations in surface irradiance due to clouds.Data from the radiance and irradiance sensors are then interpolated to common depth and wavelength bins, and referenced to absolute depth by way of surface pressure tare values.
To avoid clutter on the graphs in Figs. 1 and 2, only the data at approximately 6 nm intervals from 353 to 552 nm are plotted (every other wavelength is omitted).Noting that the algorithm requires vertical derivatives of E d and L u , some data smoothing is required.The smoothing for the Hawaii data was effected by computing a running average of these quantities taken over 11 m (the value at z is taken to be the average of the values obtained from z − 5 m to z + 5 m).For the San Diego station, the data were smoothed over a depth of 5 m.
Examination of the E d profiles shows that near the surface in some parts of the profiles the irradiance actually increases with depth.Such artifacts are likely due to surface waves.The inversion algorithm cannot handle such occurrences, so these regions must be omitted from the inversion.For the Hawaii station, the depth range actually used was 20 to 90 m, while for the San Diego station the range was 5 to 19 m.The maximum depth at each station is restricted to ensure that retrievals at all wavelengths cover the same depth range.Fig. 2. Vertical profiles of Lu off Lanai, Hawaii (same location as Fig. 1).
Contemporaneous in situ data were obtained providing a pg (z) and c pg (z) at the Hawaii station using an ac-9 (WET Labs, Inc.).A purified water blank (Barnstead Nanopure 4cartridge water purification system) was obtained with the ac-9 following the protocol of Twardowski et al. [12] 2 days before the presented in situ data were collected to quantify any drift from the manufacturer's calibration.Corrections for time lags, the temperature and salinity dependence of pure water absorption and attenuation [13,14], and drift were applied in post-processing.Scattering errors associated with the reflective tube absorption measurement [15] were obtained independently by convolving directly measured volume scattering functions with the scattering error weighting function presented in McKee et al. [16] for the ac-9 reflective tube.Volume scattering functions were measured at 10° increments with a prototype device called the MASCOT [17].
Measurements with an ECO-BB3 (WET Labs, Inc.) provided b b (z) at 462, 527, and 657 nm.For each wavelength, a single measurement of broadly weighted scattering in the backward direction, centered at 124°, is made.The sensor was calibrated with NISTtraceable, 0.2 µm microspherical beads (Duke Scientific) according to the method of Twardowski et al. [18].A correction for absorption of incident and scattering beams was applied [18,19], but had a negligible effect (<1%).Backscattering coefficients were obtained from these measurements of scattering by first subtracting β w (124°) using the pure water backscattering values of Zhang and Hu [20] with salt effect added according to Zhang et al. [21].The resulting particulate fraction β p (124°) was used to compute b bp according to the relationship b bp = 2π χ p (124°) β p (124°) [22,23], where χ p (124°) was set to 0.89 following the empirical analysis by Sullivan et al. [24] for the ECO-BB sensors.After Berthon et al. [25], the χ p was assumed to be spectrally invariant.Pure seawater backscattering, b bw [20,21] At the San Diego station in situ data were obtained providing a pg (z), a g (z), c pg (z), a p (z), and c p (z) along with b b (z) at 527 and 657 nm.Consecutive casts were made with the same ac-9 with and without a 0.2 µm prefilter to obtain the "g" component and "pg" components, respectively.The "p" components were obtained by difference.An ECO-BB2C (WET Labs, Inc.) was used for the b b (z) measurements, following the protocol used for the ECO-BB3 deployed for the Hawaii station.At 527 nm, b bw was approximately 25% of b b near the surface.The data acquired with these state-of-the-art in situ instruments is considered to constitute a validation data set for the inversion retrievals.

Results
We now describe the results of the inversion for the two data sets.

The Hawaii data set
The inversion algorithm was operated with the smoothed E d (z) and L u (z) data along with the wavelength and solar zenith angle.The Petzold phase function [7,26] was used for the particles.It was assumed that the atmosphere was aerosol free.(If aerosols are included, their influence is to decrease E d (z) and L u (z) by the same fraction; hence they have little or no influence on the resulting a and b b .Simulations we carried out suggest that increasing the aerosol optical depth at 500 nm from zero, as assumed in these retrievals, to 0.25, which is actually high for a marine environment, will cause the retrieved values of a and b b to increase by about 1%.) Figs. 3 and 4  respectively.These retrievals were typical of those over the entire spectrum.The depth variations of a(z) follow almost exactly the depth variations of K d .Also provided on Figs. 3  and 4 are the in situ measurements of a(z) and b b (z) made with the ac-9 and ECO-BB3, respectively.The IOP data were averaged over 1 m depths to reduce noise.Overall, Fig. 3 suggests that the retrieved and in situ a(z) are in excellent agreement for most of the profile, with no obvious trend with depth, except that the retrieved is usually smaller than the in situ.
The differences are remarkably small, and usually less than ± 0.004 m −1 , within the (worst case) stated accuracy of the ac-9 instrument ( ± 0.005 m −1 ).
In the case of b b (z) at 462 nm (Fig. 4), there are still obvious small-scale depth variations that correlate with the inversion-retrieved a.The figure shows that there is again a bias in the b b (z) results with the in situ values being ~0.0002 m −1 or less below the inversion-retrieved results.Clearly, the inversion-retrieved b b increases at a slightly faster rate with depth than its in situ counterpart.This is likely due to the increasing importance of Raman scattering with depth at this wavelength (see Section 6).
To present the spectral distribution of the retrieved IOPs, we averaged the inversion retrievals over a depth range 20 to 90 m at each wavelength and compared them with similarly-averaged values of the in situ measurements.These are provided in Fig. 5 along with a w and b bw .Two representations of the absorption of pure water a w are provided.The first ("Water Absorption -F") is taken from Sogandares and Fry [27] for the spectral region 340-380 nm and Pope and Fry [28] for 380-700 nm.However, there is evidence that, in both Refs.27 and 28, a w is overestimated for λ ≤ 420 nm [29].For this reason, we include a second estimate of a w suggested by Morel et al. [29] in this spectral region ("Water Absorption -M").The backscattering of pure seawater is taken from Zhang and Hu [20] and Zhang et al. [21].The corresponding values of a pg and b bp are provided in Fig. 6.The retrieved quantities are remarkably smooth functions of wavelength and clearly display some of the spectral features of pure water.In particular, the absorption due to the seventh harmonic of the O-H stretching mode near 449 nm [28] is clearly revealed in the inversions (Fig. 5).In addition, the overall spectral variation in b b is clearly dominated by the water itself.The inversion-retrieved absorption coefficient falls below that of pure water at wavelengths greater than 510 nm.Similarly b bp near 490 nm (Fig. 6) departs from its spectral behavior at shorter wavelengths and begins to increase rapidly into the green region of the spectrum.This is due to the influence of Raman scattering (Section 6).The retrieved a pg shows absorption similar to that characteristic of phytoplankton; however, when "Water Absorption -F" is used for a w , strange inexplicable features are seen between 350 and 400 nm.In contrast, when "Water Absorption -M" is used for a w , the spectrum is smooth and displays an exponentially increasing absorption with decreasing wavelength in the UV characteristic of detrital particles and/or dissolved organic material.This suggests that a w given in Ref. 29 more faithfully describes the absorption of water in the UV.
The backscattering coefficient appears to be more-or-less featureless in Fig. 6.Many investigators suggest that b bp ~λ−n , e.g., Maritorena et al. [30].Fitting b bp to such a power-law over the range λ = 350 to 450 nm (to minimize the Raman contribution, longer wavelengths were omitted in the fit) we find n = 0.97 with an R 2 of 0.87.

The San Diego data set
The retrievals for the San Diego data are presented in Fig. respectively.We note that a pg (λ) becomes negative at about 595 nm, i.e., the retrieved a(λ) becomes less than that of pure water.In addition, b bp (λ) near 590 nm begins to depart significantly from its weak decrease with increasing wavelength.As in the case of the Hawaii station, these effects are due to the increasing importance of Raman scattering as a w (λ) increases in this region.Clearly, the retrievals become poor at wavelengths somewhat less than 595 nm; however, Fig. 8 suggests that they should still be valid near 570 nm.For this data we find that b bp ~λ−n with n ≅ 0.40 over the spectral range 400 to 570 nm.We assess the influence of Raman scattering in the next Section with the aid of synthetic data, and show that reliable values of a(λ) and b b (λ) can be obtained at wavelengths less than 575 nm at this station.

Effect of Raman scattering
The fact that the retrieved a becomes < a w at wavelengths ≥ 510 nm for the Hawaii station and ≥ 595 nm for the San Diego station, is manifest in the ever-increasing contribution of Raman scattering with increasing wavelength.In Raman scattering, photons at a wavelength λ e < λ are inelastically scattered to a wavelength, λ [31][32][33][34].Assuming that the RT equation refers to radiance at a wavelength λ (the dependence of the radiance on λ was suppressed in Section 2), Raman scattering adds a term ( , , )

∫∫
where the Raman scattering function β r is given by ( ) where ρ is the Raman depolarization (~0.17 where the wavelength is in nm.Thus, in the presence of Raman scattering, Gershun's equation for the absorption coefficient ( , ) and if the last term is ignored, the retrieved absorption coefficient is too small.Although ( , ) r b z λ in the visible is small compared to ( , ) a z λ , at depths where the scalar irradiance ratio is large, i.e., 0 a z λ can be significant.Note that at wavelengths where the contribution to Raman scattering is large, e.g., in the red, ( , ) z µ λ and ( , ) V K z λ will differ significantly from the values that would be obtained in its absence, as in the inversion algorithm.
Although some corrections can be made for ( , ) a z λ at Raman-compromised wavelengths using the above equation, there is no simple method for assessing the Raman-induced error in the retrieved absorption and backscattering coefficient.To provide an estimate of the error, we performed a series of RT simulations, using an RT code [34] that includes Raman scattering, in which the water's IOPs were similar to those measured or retrieved here.

The Hawaii station
For the Hawaii station, the simulations were carried out for λ = 412, 490, 510, and 556 nm, for which 363, e λ ≈ 420, 435, and 471 nm.At wavelengths where the Raman contribution was expected to be small, λ < 510 nm, the depth-averaged retrieved values of the IOPs were used (along with the Petzold phase function for the particles).At wavelengths where the Raman contribution was expected to be large, i.e., 510 and 556 nm, the measured values of the IOPs were used (along with the Petzold phase function for the particles).For the purposes of computation of the Raman source function, J at λ, the Raman contribution to the light field at e λ was assumed to be negligible (i.e., processes in which photons are Raman-scattered from λ a → λ b and again from λ b → λ c , are ignored).These computations produced simulated values of L u (z) and E d (z) at each wavelength that were similar to those measured in the field.These simulated L u (z) and E d (z) values were then treated in a manner identical to the field data, i.e., entered into the inversion algorithm after smoothing, and the IOPs were retrieved.Contrary to the situation with field data, in the simulations we know the true values of all of the IOPs.The result of this exercise is presented in Table 1.As with the inversions presented in Fig. 5, the entries in the table for the inversions are averages over the depth range 20 to 90 m.

Discussion
A shortcoming of our inversions is our inability to separate the absorption coefficient into its constituent parts.However, the in situ measurements of a g allow the separation of dissolved absorption from particulate absorption.The a g data at the San Diego station were found to fit an exponential function of wavelength: .Over the range 412 to 555 nm the maximum difference between the measured and the exponential fit was 0.002 m −1 .To try to estimate a p , we interpolated and extrapolated the exponential fit for a g to the rest of the spectrum and subtracted it from a pg (Fig. 8).The resulting a p is provided Fig. 9.This procedure has rendered the absorption bands Chlorophyll a (382, 416 and 436 nm), Chlorophyll b and Chlorophyll c (466 and 462 nm, respectively), and Carotenoids (489 nm) clearly visible, and demonstrates that even in mesotrophic waters the in-water diffuse light field contains considerable information regarding the details of the pigmentation of the particles.
An important result of this study is the observation that the backscattering coefficient is almost void of any spectral features, i.e., the spectra confirm that to a good approximation b bp ~λ−n .However, closer examination of Fig. 8 reveals that b bp (λ) appears to be slightly depressed over the spectral region attributed to phytoplankton (Fig. 9).It is well known that phytoplankton spectral absorption features are muted in spectra of the beam attenuation coefficient c(λ).This implies that the spectral scattering coefficient must be depressed near absorption features (for several examples of spectra obtained for phytoplankton cultures, see Bricaud et al. [36], and for backscattering by particles in eutrophic waters, see Morel et al. [37]).This effect has been called anomalous dispersion [38,39], and results from the real part of the refractive index of a particle changing in the vicinity of an absorption band.The influence of anomalous dispersion on phytoplankton b bp (λ) was examined theoretically by Zaneveld and Kitchen [39], who predicted localized spectral depressions, particularly pronounced at wavelengths shorter than an absorption band peak.

(
θ is the angle between the propagation direction and the + z axis, and φ is the azimuth angle in the x-y plane.The radiative transfer (RT) equation for radiance ( , )L z ξ propagating in the direction ξ in the medium is[7] #113277 -$15.00USD Received 24 Jun 2009; revised 14 Aug 2009; accepted 21 Aug 2009; published 27 Aug 2009 z) is the beam attenuation (extinction) coefficient of the medium and ˆfunction (differential scattering cross section per unit volume) for scattering from the direction ξ ′ to the direction ξ .Although not specified in the argument lists, all quantities also depend on the wavelength λ of the radiation.The quantity d angle surrounding ξ ′ , and the integration is carried out over the entire range of solid angles.The total scattering coefficient b(z) is given by An important quantity in ocean color remote sensing is the backscattering coefficient defined inherent optical properties (IOPs) of the medium.It is convenient to define a normalized function, the scattering phase function IOPs, radiometric quantities derived from the radiance in the medium, e.g.

Fig. 3 .Fig. 4 .
Fig. 3. Comparison of retrieved and in situ vertical profiles of absorption at 443 nm at the station off Lanai, Hawaii.

Fig. 7 .Fig. 8 .
Fig. 7. Retrieved absorption (left scale) and backscattering (right scale) coefficients for waters off San Diego.The retrieved quantities are averaged from 5 to 19 m.The measured backscattering plotted at 650 nm is really that at 657 nm.

J z ξ λ
to the right-hand-side of the equation: As in the derivation of Gershun's equation, if we integrated J over solid angle we find inside the integral has been replaced by 0 ( , ) e E z λ outside the integral, with e λ being the mean wavelength over the excitation band.This is acceptable because 0 ( , ) of this term on ( , )

Fig. 9 .
Fig. 9.Estimated absorption coefficient (averaged from 5 to 19 m) of particles for the San Diego station.