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Evidence for wavelength dependence of the scattering phase function and its implication for modeling radiance transfer in shelf seas David McKee and Alex Cunningham More than 90% of stations from the Irish and Celtic Seas are found to have significantly higher backscattering ratios in the blue (470 nm) than in the red (676 nm) wave band. Attempts to obtain optical closure by use of radiance transfer modeling were least successful for stations at which backscattering ratios are most strongly wavelength dependent. Significantly improved radiance transfer simulation results were obtained with a modified scattering correction algorithm for AC-9 absorption measurements that took wavelength dependency in the scattering phase function into account. © 2005 Optical Society of America OCIS codes: 000.0000, 010.4450, 290.5820. 1. Introduction Using commercially available software such as Hydrolight1 to combine radiance transfer computations with in situ measurements from modern optical instrumentation, it should be possible to construct realistic mathematical models of underwater light fields and to predict remote-sensing reflectance accurately. Recent results indicate unexpected variability in the degree to which this aim can be achieved.2–5 The variability has been attributed to poor quality control of either the inherent optical property measurements used to constrain the models or the radiometric measurements used for model validation.4,5 However, since the degree of closure obtained varies from one station to another in a given cruise and even between wavelengths for a given station, it is likely that the source of variation lies in the optical characteristics of the water body in which the measurements are being made rather than in the measurement procedures. In this paper, we consider whether the difficulties encountered in obtaining consistent closure between models and measurements in shelf seas can be attributed to wavelength depen- The authors are with Department of Physics, University of Strathclyde, 107 Rottenrow, Glasgow, G4 0NG, Scotland. D McKee’s e-mail address is [email protected]. Received 23 December 2003; revised manuscript received 7 September 2004; accepted 21 September 2004. 0003-6935/05/010126-10$15.00/0 © 2005 Optical Society of America 126 APPLIED OPTICS 兾 Vol. 44, No. 1 兾 1 January 2005 dence of the scattering phase function. The methodology employed was developed for modeling coccolithophore blooms,3 but it can be applied to other waters where standard inherent optical property (IOP) measurements are difficult to reconcile with in situ radiometric profiles. 2. Theory The IOPs of seawater are commonly measured with an AC-9 dual-tube spectrophotometer (WET Labs, Philomath, Oregon) and a Hydroscat backscattering meter (HOBI Labs, Inc., Tucson, Arizona). The AC-9 provides coefficients of attenuation (cn) and absorption (an) for materials other than water, and the Hydroscat provides total backscattering coefficients (bb). Data from both instruments are subject to correction procedures that make assumptions about the optical properties of the medium in which the measurements are made. These assumptions may be valid for oceanic waters, but they are largely untested for shelf seas. In this paper, we consider possible modification of the correction procedure for the AC-9 because the manufacturer has published a critical analysis of the geometric and optical principles involved. It is likely that Hydroscat correction procedures should also be examined critically in coastal waters, but we only consider whether the errors involved invalidate the main thrust of the argument. A. AC-9 Correction It has been reported3 that removal of the assumption of a wavelength-independent scattering phase func- tion from the analysis by Zaneveld et al.6 leads to a modified version of the scattering correction algorithm for reflecting tube absorption measurements of the form an共兲 ⫽ ai共兲 ⫺ ai共r兲 ci共兲 ⫺ ai共兲 ci共r兲 ⫺ ai共r兲 F共, r兲, (1) where an is the corrected estimate of the absorption coefficient for materials other than water, ai and ci are the uncorrected instrument measurements of absorption and attenuation, denotes the wavelength at which the corrected value is calculated, and r is the reference wavelength (715 nm) at which an is assumed to be zero. Equation 1 differs from the original expression by the presence of an extra factor F共, r兲 that can be written in full as F共, r兲 ⫽ 冋 ka共兲 1 ⫺ ka共兲 ⫺ kc共兲 冒 ka共r兲 1 ⫺ ka共r兲 ⫺ kc共r兲 册 Fig. 1. Sigma-corrected total backscattering values show increasing divergence from uncorrected total backscattering as the turbidity of the water increases. , (2) where ka共兲 is the fraction of scattered light that is not collected by the absorption sensor and kc共兲 is the fraction of scattered light that is collected by the attenuation sensor. Zaneveld et al.6 assumed ka共兲 ⫽ ka共r兲 and kc共兲 ⫽ kc共r兲, and consequently F共, r兲 ⫽ 1, but this assumption does not hold if the scattering phase function varies significantly with wavelength. Unfortunately, none of the terms on the righthand side of Eq. 2 are readily obtained from in situ optical measurements, and F共, r兲 must be determined by a computationally intensive iterative procedure.3 The magnitude of the error that results from ignoring wavelength dependence depends on the product of F共, r兲 and ai共r兲, and it is greatest in turbid waters, where ai共r兲 values tend to be highest. In waters in which wavelength dependency was observed, the ratio of the total backscattering coefficient to the total scattering coefficient (bb兾b) generally decreased with increasing wavelength. From Eqs. 1 and 2, the effect is to produce an overestimate of an and consequently an underestimate of bn. B. Hydroscat Correction The sigma correction procedure for Hydroscat data can be written as bb ⫽ bbu, 2 ⫽ k0 ⫹ k1Kbb ⫹ k2Kbb , (3a) (3b) where bbu is the uncorrected signal and k0,1,2 are calibration coefficients supplied by the manufacturer. Kbb is a measure of the attenuation of the signal by water within the sensor’s measurement geometry and is calculated for each wavelength with7 Kbb ⫽ a ⫹ 0.75b. (3c) Figure 1 shows the effect of sigma correction on measurements of backscattering for surface waters at all stations occupied in the Irish Sea. Significant divergence between bb and bbu occurs as the level of turbidity increases, and so backscattering measurements become increasingly sensitive to correction errors in strongly attenuating shelf seas. The Hydroscat-2 data presented in this paper have been corrected with values of total absorption and scattering derived from AC-9 measurements in which the standard scattering correction algorithm was used. Since positive errors in a and negative errors in b partially cancel in Eq. 3c, and k0 ⬎ k1 ⬎ k2 in Eq. 3b, the sigma correction procedure is rather insensitive to the artifacts in the AC-9 data discussed in the previous section. In the extreme case of a being overestimated by 50% and b being underestimated by 5%, bb would be calculated to be 2.5% above its true value for this data set. C. Wavelength Dependence of the Scattering Phase Function The wavelength dependence of the scattering phase function can be characterized in terms of available IOP measurements by B⫽ 共bb 兾b兲470 . 共bb 兾b兲676 (4) B is subject to errors in the correction procedures applied to measurements of both total backscattering (bb) and total scattering (b). The values of a, b and sigma-corrected bb at 676 nm are assumed to be relatively unaffected by correction errors because this wave band is close to the AC-9 reference channel. Consequently our worst-case example of a 5% underestimate in b and a 2.5% overestimate in bb at 470 nm produces bb兾b and B values that are overestimated by approximately 8%. 1 January 2005 兾 Vol. 44, No. 1 兾 APPLIED OPTICS 127 D. Possible Influence of Fluorescence on bb676 Much of our discussion here rests on the observation that measured values of bb兾b470 tended to be higher than those for bb兾b676 and that their ratio [B in Eq. 4] was usually greater than unity. However, the Hydroscat 676-nm channel covers a relatively broad wave band (20 nm FWHM) that partly overlaps with the fluorescence emission spectrum of chlorophyll-a in vivo. It must therefore be presumed that Hydroscat bb676 measurements contain a fluorescence component, though no quantitative study has been published. The effect of fluorescence would be to raise the bb676 signal and to generate erroneously high apparent values of bb兾b676. As a result, actual B values would be even greater than those measured here. The observed wavelength dependence in the backscattering ratio is therefore qualitatively robust in the presence of fluorescence artifacts in bb676. The magnitude of fluorescence augmentation of the backscattering signal in the present data set is difficult to measure directly. However, its likely significance can be judged from an analysis of seawater composition. Chlorophyll-a concentrations measured by acetone extraction were generally low, with 95% of the stations having values of less than 2 mg m⫺3. The bb676 signal was poorly correlated with chlorophyll concentration (r2 ⫽ 0.29) for stations with significant chlorophyll (1–7 mg m⫺3) and low mineral (⬍5 g m⫺3) concentrations but was highly correlated with suspended mineral concentration (r2 ⫽ 0.92) for all stations. It therefore appears that backscattering in this data set is dominated by the suspended mineral component and that chlorophyll fluorescence is likely to be of minor significance. E. Fig. 2. Backscattering ratios are generally greater in the blue (470 nm) than in the red (676 nm) for most stations in the Irish and Celtic Seas. the upward radiance (Lu) is also underestimated. For coccolithophore blooms, we were able to show that the use of Eq. 1 with computationally derived values of F共, r兲 for correcting AC-9 data produced improved fits between radiance transfer models and in situ measurements in turbid waters. We extend these observations to a larger geographical area (the Irish and Celtic Seas) whose optics are not dominated by coccolithophores. 3. Materials and Methods Data were obtained during a series of cruises in the Irish and Celtic Seas between May 2001 and July Predicted Effects on Radiance Transfer Calculations The qualitative effect of wavelength-dependent scattering phase functions on attempts to achieve optical closure can be predicted from established relations between IOPs and apparent optical properties. To reasonable approximations, the radiance reflectance (RL) is given by8 RL ⫽ f bb Q a , (5) and the diffuse attenuation coefficient for downward irradiance (Kd) by9 Kd ⫽ a ⫹ bb d . (6) Overestimation of a therefore leads to values of RL that are too low and to values of Kd that are too high. As a result, the computed downward irradiance (Ed) decreases more rapidly with depth than actual measurements, and since RL ⫽ 128 Lu Ed , APPLIED OPTICS 兾 Vol. 44, No. 1 兾 1 January 2005 (7) Fig. 3. Ratio of backscattering ratios at 470 and 676 nm, B, shows significant variability for surface waters of the Irish and Celtic Seas. Less than 5% of stations have a value of B between 0.9 and 1.1, where the scattering phase function might be considered wavelength independent. Most stations have stronger backscattering ratios in the blue than in the red. It can be concluded that the scattering phase function is generally wavelength dependent for these waters. 2002. A total of 120 stations were occupied covering a wide range of water types. A. Fig. 4. Significant degree of variability is exhibited in the relation between backscattering ratios at 470 and 676 nm for the top 10 m of four sample stations. Lines (gradients between 1.0 and 1.8) indicate the strength of the wavelength dependency in the backscattering ratio, B. Inherent Optical Property Measurements A 25-cm path-length WET Labs AC-9 was used to measure the absorption coefficient (an) and beam attenuation coefficient (cn) of materials other than water at nine wavelengths (10 nm FWHM) across the visible spectrum. Optical blanks for the AC-9 were regularly measured with ultrapure Millipore water treated with ultraviolet light, and calibration of the two optical channels remained within the manufacturer=s specifications of ⫾0.005 m⫺1. Absorption and attenuation signals at 715 nm were corrected for temperature-dependent water absorption,10 and the data were averaged over 1-m depth intervals. We assumed that the standard correction algorithm always suffices for AC-9 measurements at 676 nm, since this is close to the 715-nm reference wavelength. Total absorption (a) and attenuation (c) coefficients were obtained by addition of partial coefficients for water obtained from the literature.11,12 Scattering coefficients were obtained from b ⫽ c Fig. 5. IOPs of stations B4 and ST19 are significantly different in terms of (a) absorption at 488 nm, (b) scattering at 488 nm, (c) backscattering at 470 nm, and (d) backscattering ratio at 470 nm. B4 is a relatively clear station, whereas ST19 is quite turbid. 1 January 2005 兾 Vol. 44, No. 1 兾 APPLIED OPTICS 129 Fig. 6. Standard IOPs provide satisfactory matches between measured (SPMR) and modeled (Hydrolight) (a) downward irradiance, Ed, and (b) upward radiance, Lu, for station B4. The match between measured and modeled Ed (c) is also satisfactory for ST19. The small overestimate of modeled upward radiance for ST19 (d) may be attributable to limitations in the performance of sigma correction of backscattering data for very turbid water such as this. ⫺ a. Values of absorption and scattering at 470 nm, which are used at various points in the paper, were obtained by interpolation between readings at 440 and 488 nm. Sets of modified AC-9 data were generated with Eq. 1 with values of F共, r兲 ranging between 1 and 1.5. Total backscattering (bb) was measured at 470 and 676 nm by use of a Hydroscat-2 (HOBI Labs) and was corrected with calibration constants supplied by the manufacturer. monitored at the start and the end of each cruise by use of a 100-W standard lamp, and the radiance sensors were checked by use of the same lamp to illuminate a Spectralon reflectance target. All sensors remained within factory specifications. Signals from the SPMR were processed with ProSoft, a MATLAB module supplied by the manufacturers. Data processing steps included application of calibration constants and averaging over 1-m depth intervals. B. C. Radiometric Measurements Downward irradiance (Ed) and upward radiance (Lu) were measured in seven wavebands (10 nm FWHM) across the visible spectrum by use of a Satlantic SeaWiFS Profiling Multi-Channel Radiometer (SPMR). The SPMR was deployed at a distance of at least 20 m from the ship to avoid shadowing. A reference radiometer measuring surface irradiance (Es) in the same wave bands was mounted on the superstructure of the ship. The stability of the irradiance sensors was 130 APPLIED OPTICS 兾 Vol. 44, No. 1 兾 1 January 2005 Radiance Transfer Modeling Radiance transfer calculations were carried out with Hydrolight (Sequoia Scientific, Bellevue, Washington).1 Surface irradiance data for each station were obtained from the SPMR deck reference. The choice of wavelength for this study was limited by the fact that we had access only to a two-channel Hydroscat backscattering meter operating at 470 and 676 nm. The analysis was limited to the blue wave band to avoid the complicated effects of inelastic processes on the radio- Fig. 7. IOPs of stations ST12 and ST16 indicate (a) similar levels of absorption at 488 nm, (b) significantly higher levels of scattering at 488 nm, (c) backscattering at 470 nm, and (d) backscattering ratio at 470 nm for ST12. All IOPs were generated with standard correction algorithms. metric measurements. A direct comparison was made between Hydrolight computations at 488 nm and SPMR profiles at the same wavelength. The IOP inputs were bb values at 470 nm measured with the Hydroscat 2 and a and b values at 488 nm measured with the AC-9. It is assumed that the 18-nm wavelength discrepancy between the backscattering data and the other IOPs has a negligible effect on the results. We performed separate runs with AC-9 input files generated by use of different values of F共, r兲 for the modified scattering correction algorithm [Eq. 1]. Fournier–Forand13,14 scattering phase functions for each station and depth were chosen using sigmacorrected Hydroscat-2 values for the backscattering coefficient. The match between the Hydrolight output and the in situ measurements from the SPMR was evaluated by use of the average standard percentage error ε, defined as ⫽ 兺n ⱍ共xin ⫺ xmod兲 兾 xinⱍ ⫻ 100 n , (8) where n is the number of observations, xin refers to in situ measurements, xmod refers to modeled values, and x is the parameter being investigated (either Ed or Lu). 4. Results A. Backscattering Ratios in the Irish Sea Figure 2 shows bb兾b at 470 nm, plotted against bb兾b at 676 nm for surface waters at 120 stations in the Irish Sea, derived from Hydroscat and AC-9 data corrected with the standard procedures. For most stations the points lie above the 1:1 line, indicating that backscattering ratios are higher at shorter wavelengths. Figure 3 shows the distribution of B for the data in Fig. 2. For 95% of the stations, B ⬎ 1.1, and for 50% B ⬎ 1.3. These percentages are not changed significantly by the possible existence of an 8% overestimate in B values for the most wavelengthdependent stations. The implications of this result for the correction of AC-9 measurements and the ability to obtain optical closure were therefore investigated. 1 January 2005 兾 Vol. 44, No. 1 兾 APPLIED OPTICS 131 Fig. 8. Use of standard IOPs in radiance transfer calculations results in systematic underestimates of both Ed [(a) and (c)] and Lu [(b) and (d)] for both ST12 and ST16. This underestimate is consistent with the effects of a wavelength-dependent scattering phase function on the scattering correction of in situ measurements of absorption. B. Case Studies Figure 4 shows backscattering ratios at 470 nm plotted against backscattering ratios at 676 nm for the top 10 m of four stations that cover a representative range of B values. These ratios were derived from IOP data corrected with standard procedures. Stations B4 and ST19 are examples with low wavelength dependency in the backscattering ratio (B ⬍ 1.2), whereas stations ST12 and ST16 are examples with marked wavelength dependency (1.2 ⬍ B ⬍ 1.8). Equations 5–7 predict that radiance transfer models in which standard IOP inputs are used will produce satisfactory outputs for stations B4 and ST19 and will underestimate both Ed and Lu for ST12 and ST16. C. Closure for Stations with B ⬍ 1.2 Figure 5 shows IOP profiles generated with AC-9 data corrected by use of the standard algorithm for stations B4 and ST19. B4 was a relatively clear station in the middle of the Irish Sea, occupied in April 132 APPLIED OPTICS 兾 Vol. 44, No. 1 兾 1 January 2005 2002, with low values of absorption, scattering, and backscattering throughout the top 25 m of the water column. ST19 was a turbid station close to the Welsh coast, occupied in November 2001, with significantly higher values of all three IOPs. The backscattering ratio for ST19 was almost double that at B4. The IOPs in Fig. 5 were used as inputs for Hydrolight simulations, and the modeled profiles of Ed and Lu for the top 10 m of the water column compared with the in situ measurements obtained from the SPMR radiometer system (Fig. 6). For both these stations the average standard percentage error was less than 6% for measurements of Ed. There was a tendency to overestimate Lu, with values of ⬃ 11% for B4 and ⬃32% for ST19. However, the apparently poor performance of the simulation for Lu at ST19 corresponded to an average absolute error of less than 5 ⫻ 10⫺5 W m⫺2 nm⫺1 sr⫺1 and represented the smallest absolute error of any of the simulations presented. Light levels were very low for this station, which provided a severe test of modeling and measurement Fig. 9. Average percentage error over the top 10 m of the water column between measured and modeled values of both Ed and Lu can be minimized for stations ST12 and ST16 by varying only the value of F共, r兲 used in the scattering correction for the AC-9 measurements. The optimal value of F共, r兲 varies with station, and the magnitude of the initial error varies with the size of the scattering signal. Fig. 10. Improved matches between modeled and measured values of Ed and Lu are obtained with values of F共, r兲 ⫽ 1.2 for ST12 [(a) and (b)], and F共, r兲 ⫽ 1.3 for ST16 [(c) and (d)]. 1 January 2005 兾 Vol. 44, No. 1 兾 APPLIED OPTICS 133 Fig. 11. Absorption values obtained with the modified scattering correction algorithm and F共, r兲 ⬎ 1.0 are generally lower than standard (F共, r兲 ⫽ 1.0) values of absorption. Higher levels of scattering at ST12 than at ST16 are responsible for the greater difference between standard and modified absorption values at ST12 despite the optimal value of F共, r兲 being lower for this station. precision. In both cases the simulations provided a reasonable degree of optical closure by use of standard AC-9 data. D. Closure for Stations with B ⬎ 1.2 Figure 7 shows profiles of IOPs for the top 25 m of stations ST12 and ST16. Both stations were fairly homogeneous over the top 15 m, with similar absorption coefficients. The scattering and backscattering coefficients and backscattering ratios were all significantly higher for ST12 than for ST16, but in both cases backscattering ratios were higher in the blue than in the red (Fig. 4). Radiance transfer modeling with AC-9 data corrected by use of the standard algorithm [F共, r兲 ⫽ 1.0] produced systematic underestimates of both Ed and Lu (Fig. 8) for both these stations. The discrepancy was most obvious for ST12 for which the average standard percentage errors were 38% and 54% for Ed and Lu, respectively. The errors for ST16 were lower at 13% for Ed and 17% for Lu. As predicted, the standard [F共, r兲 ⫽ 1.0] AC-9 scattering correction algorithm for these stations results in a systematic overestimate of absorption at blue wavelengths. Figure 9 shows the effect of varying F共, r兲 on the performance of radiance transfer simulations. For ST12 the minimum average error for both Ed and Lu occurred when F共, r兲 ⫽ 1.2. For ST16 the optimal value of F共, r兲 was closer to 1.3. It is interesting to note that the overall magnitude of the error with F共, r兲 ⫽ 1.0 was greater for ST12 than for ST16 even though the optimal value of F共, r兲 was less for ST12 than for ST16. This occurrence was probably due to a higher residual absorption [ai共r兲] for ST12 as a result of significantly higher scattering at this station. When the optimal values of F共, r兲 derived above were incorporated in the AC-9 correction algorithm (Fig. 10), the average standard percentage errors for the radiance transfer simulations were less 134 APPLIED OPTICS 兾 Vol. 44, No. 1 兾 1 January 2005 than 10% for ST12 and less than 5% for ST16. This result represents a significant improvement in modeling accuracy over the results from use of standard AC-9 data. Figure 11 illustrates the effect of using optimized F共, r兲 values on absorption coefficients for these two stations. In both cases the modified AC-9 absorption signals were lower than the standard values, with the average percentage difference varying from 27% for ST12 to 11% for ST16. The resulting scattering coefficients (not shown) were approximately 4% greater than the standard values for both stations. 5. Discussion The observations presented here show that it can be difficult to obtain optical closure in waters where scattering coefficients are high relative to oceanic values. The nature of the errors observed in radiance transfer calculations and the changes in AC-9 correction procedures necessary to obtain closure were both consistent with the hypothesis of wavelength dependence in the scattering phase function. In the Irish Sea, where the backscattering ratio was generally found to decrease with increasing wavelength, the standard AC-9 scattering correction produces overestimates in the absorption coefficient and underestimates in the scattering coefficient at blue and green wavelengths. These errors can be reduced by introduction of a modified scattering correction algorithm [Eq. 1] with an additional term F共, r兲 whose magnitude can be determined computationally by minimization of the discrepancies between radiance transfer models and IOP measurements. However, the significance of the IOP measurement errors will be determined not only by F共, r兲 but also by ai共r兲, the magnitude of the residual scattering signal at 715 nm. Since the latter term generally varies with suspended particle concentration, the greatest errors are expected in coastal and shelf seas. One approach to obtaining appropriately corrected measurements of the absorption and scattering coefficients in these waters would be to extend the method described here to all wave bands by deploying spectrally matched instrument packages. It may be necessary to carry out further Monte Carlo simulation of the reflecting tube design15,16 with appropriate wavelength dependency being prescribed for all IOPs, including the scattering phase function.17 Alternatively, a number of relations between IOPs and apparent optical properties have been published that would allow IOPs to be derived from radiometric measurements with reduced precision but greater freedom from systematic errors.18 –20 Whichever method is adopted, improvement in the accuracy of IOP measurements is urgently required for validating remote-sensing products and for interpreting data from moored arrays of optical sensors in coastal seas. 8. 9. 10. 11. 12. 13. 14. References 1. C. D. Mobley, “Light and Water: Radiative Transfer in Natural Waters,” (Academic, San Diego, 1994). 2. A. Cunningham, J. C. Boyle, and P. Wood, “Radiative transfer modelling of the relationship between seawater composition and remote sensing reflectance in sea lochs and fjords,” Int. J. Remote Sensing 23, 3713–3724 (2002). 3. D. McKee, A. Cunningham, and S. Craig, “Semi-empirical correction algorithm for AC-9 measurements in a coccolithophore bloom,” Appl. Opt. 42, 4369 – 4374 (2003). 4. T. J. Smyth, G. F. Moore, S. B. Groom, P. E. Land, and T. Tyrrell, “Optical modeling and measurements of a coccolithophore bloom,” Appl. Opt. 41, 7679 –7688 (2002). 5. G. C. Chang, T. D. Dickey, C. D. Mobley, E. Boss, and W. S. Pegau, “Toward closure of upwelling radiance in coastal waters,” Appl. Opt. 42, 1574 –1582 (2003). 6. J. R. V. Zaneveld, J. C. Kitchen and C. M. Moore, “The scattering error correction of reflecting-tube absorption meters,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE 2258, 44 –55 (1994). 7. M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. 15. 16. 17. 18. 19. 20. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, C7, 14129 –14142 (2001). A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. II. Bidirectional aspects,” Appl. Opt. 32, 6864 – 6879 (1993). H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389 –1409 (1989). W. S. Pegau, D. Gray, and J. R. V. Zaneveld, “Absorption and attenuation of visible and near-infrared light in water: dependence on temperature and salinity,” Appl. Opt. 36, 6035– 6046 (1997). R. M. Pope and E. S. Fry, “Absorption spectrum (380 –700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36, 8710 – 8723 (1997). R. C. Smith and K. Baker, “Optical properties of the clearest natural waters (200-800 nm),” Appl. Opt. 20, 177–184 (1981). G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE 2258, 194 –201 (1994). C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002). J. T. O. Kirk, “Monte Carlo modeling of the performance of a reflective tube absorption meter,” Appl. Opt. 31, 6463– 6468 (1992). J. Piskozub, P. J. Flatau, and J. R. V. Zaneveld, “Monte Carlo study of the scattering error of a quartz reflective absorption tube,” J. Atmos. Oceanic Technol. 18, 438 – 445 (2001). D. Stramski and J. Piskozub, “Estimation of scattering error in spectrophotometric measurements of light absorption by aquatic particles from three-dimensional radiative transfer simulations,” Appl. Opt. 42, 3634 –3646 (2003). R. A. Leathers, C. S. Roesler, and N. J. McCormick, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38, 5096 –5103 (1999). M. Stramska, D. Stramski, B. G. Mitchell, and C. D. Mobley, “Estimation of the absorption and backscattering coefficients from in-water radiometric measurements,” Limnol. Oceanogr. 45, 628 – 641 (2000). D. McKee, A. Cunningham, and S. Craig, “Retrieval of inherent optical properties from in situ radiometric measurements in case II waters,” Appl. Opt. 42, 2804 –2810 (2003). 1 January 2005 兾 Vol. 44, No. 1 兾 APPLIED OPTICS 135