Effect of suspended particulate-size distribution on the backscattering ratio in the remote sensing of seawater

Transcription

1 Effect of suspended particulate-size distribution on the backscattering ratio in the remote sensing of seawater Dubravko Risović Mie theory is used to study the influence of the particle-size distribution PSD on the backscattering ratio for case 1 and 2 waters. Several in situ measured PSDs from coastal water and the open ocean, representing typical case 2 and 1 waters, were used in this investigation. Calculation of the backscattering ratio requires integration of the PSD over a much broader size range than is usually measured. Consequently extrapolation from fitted data is necessary. To that purpose the measured data are fitted with hyperbolic Junge and the two-component model of the PSD. It is shown that the result of extrapolation, hence the backscattering ratio, critically depends on the chosen PSD model. For a particular PSD model the role of submicrometer particles and the applied integration limits on the backscattering ratio is discussed. The use of the hyperbolic PSD model largely overestimates the number of small submicrometer particles that significantly contribute to backscattering and consequently leads to an erroneously high backscattering ratio. The two-component model proves to be an adequate PSD model for use in backscattering scattering calculations providing satisfactory results complying with experimental data. The results are relevant for the inversion of remotely sensed data and the prediction of optical properties and the concentration of phytoplankton pigments, suspended sediment, and yellow substance Optical Society of America OCIS codes: , , , , Introduction It is well established that the remote sensing of sea color is a valuable tool in optical oceanography that yields information on the concentration of phytoplankton pigments, suspended sediment, and yellow substance in a euphotic layer. 1 Depending on whether these concentrations are correlated among themselves, water is classified as case 1 and case 2. Oceans are typical case 1 waters whereas coastal waters are often referred to as case 2 water. Here often a high degree of correlation may be found although subject to large spatial and temporal variations due to local phenomena. 2 A number of different algorithms are used for the inversion of remotely sensed radiance to obtain the The author is with Molecular Physics Laboratory, Ru er Bošković Institute, POB 180, HR Zagreb, Croatia , drisovic@rudjer.irb.hr. Received 11 July 2002; revised manuscript received 11 July $ Optical Society of America oceanic pigment concentrations or the total absorption coefficients It is common practice to assume that the remotesensing reflectance R is related to the ratio of the total volume backscattering coefficient b b to the total volume absorption coefficient a. Hence R C b b a, where the proportionality function C is often presumed constant. 6 8,14 Scattering by high concentrations of inorganic particles, especially in coastal waters, strongly influences the submarine light field in some cases to an even greater extent than absorption processes; thus b b is a major component of the reflectance equation. 7,9,10 One of the problems is that until recently b b was difficult to measure and generally unknown. Hence the behavior of R and C was investigated in terms of easily measured parameters such as b, a, diffuse attenuation coefficient K d, and the particle-size distribution. 15,16 It seems that the shape of particlesize distributions has a profound influence on b b, hence on R, although some questions remain. Simulation of light scattering from oceanic waters 17 indicate a major influence and contribution of submicrometer particles to b b, while for turbid case 3 waters it has been suggested 18 that R should depend 7092 APPLIED OPTICS Vol. 41, No November 2002

2 only on the total cross-sectional area of the particulate per unit volume and the diffuse attenuation coefficient. To our knowledge neither hypothesis has been verified experimentally. On the other hand, it was found that for case 2 waters R was insensitive to the natural fluctuations in particle-size distributions, 15 at least for 650 nm. Modeling the backscattering ratio showed that the influence of very small submicrometer particles is dominant. 17 In all these studies and simulations the Junge hyperbolic -type distribution 19,20 was used to model the particle-size distributions PSDs. This often used, analytically simple, and easy to handle function, although appropriate for a description of PSDs in narrow size ranges, is inadequate for the description of PSDs in a very broad size range such as encountered in seawater. 21,22 Furthermore, because the measured particle-size range is usually narrower than required in light-scattering calculations, extrapolations to smaller and larger sizes are in order. Since the number of particles is an inverse power function of particle size r, the model tends to overestimate the number of small particles r 3 0, N 3. Hence at least to a degree the calculated influence of very small submicrometer particles on backscattering could be a somewhat synthetic result. To compensate for this effect, the use of a hyperbolic PSD model imposes the necessity of establishing an appropriate lower cutoff in integration limits; otherwise the integral diverges. The question of cutoff and its influences on the calculation of the backscattering ratio for the Junge model was considered by Ulloa et al. 17 who found that the reasonable lower limit of integration is r m. A much more realistic and flexible description of PSD in a broad-size range is achieved with the twocomponent model introduced by Risović. 22 This model, although more complex than the Junge model, provides a better fit over a wider size range than other models 21 and also greater flexibility in optical modeling through the possible use of different indices of refraction for each component. Furthermore it does not impose limitations on integration. In this study we used the Mie -theory to investigate the backscattering properties of the marine-particle ensemble by applying the Junge and the twocomponent 2C PSD model. In particular the influence of the PSD model chosen to fit and extrapolate in situ measured PSD on the calculated backscattering ratio in case 1 and 2 waters was investigated. Also predictions of B, b, and b b based on the 2C and the Junge PSD models are compared with actual in situ measured values for case 1 and 2 waters. 2. Theory A. Backscattering Ratio for Polydispersions The backscattering coefficient b b is an inherent optical property that can be partitioned as b b b bw b bp, (1) where b bw is the backscattering coefficient of pure seawater and b bp is the backscattering coefficient of the suspended particles. In the visible part of the spectrum b bw is small 23 ; hence the dominant contribution to b b comes from b p. For a spherical particle of size radius r the efficiency factor for scattering is given by Q s C s r 2, (2) where C s is the scattering cross section. The backscattering efficiency factor Q bs is given by Q bs C bs r 2, (3) where C bs is the backscattering cross section Exact Mie efficiency factors are given by 24 Q s 2 2n 1 a s 2 n 2 b s n 2, (4) n 1 Q bs 1 2n 1 1 n a 2 n b n n 1 2, (5) where a n and b n are the scattering coefficients and is the size parameter given by 2 m wr, 0 where m w is the index of refraction of medium seawater, 0 is the wavelength of light in vacuum, and r is the particle radius. In current bio-optical models b bp is commonly modeled as b bp Bb p, where B is backscattering ratio B b bp b p, where b p is the total scattering coefficient of the particles. The ensemble average of backscattering ratio B for an ensemble of particles, described by the size distribution N, is given by B 0 0 Qbs m, 2 N d Qs m, 2 N d, (6) where m n i is the complex refractive index of the particles. The size distribution is given by N N tot f, (7) where N tot is the total number of particles per unit volume and f is the probability density function such that 0 f d November 2002 Vol. 41, No. 33 APPLIED OPTICS 7093

3 Schematic representation of Junge and the 2C PSD mod- Fig. 1. els. C. Hyperbolic Junge Particle-Size Distribution This is a one-parameter distribution given by dn r Cr k dr, (9) where dn r is the total number of particles per unit volume with radii between r and r dr, C is a constant depending on the particle concentration, r is the particle radius, and k is a parameter that depends on particle type, size range, and measurement site. For the PSD encountered in seawater, k has a value between 2.5 and 6. This type of distribution proved to be useful in modeling the PSD in a micrometer-size range of 0.5 r 25. Usually this formula is used to fit the results from the Coulter counter; in which case the measurement size intervals bins increase logarithmically. The number of particles within a given size interval, r max r min bin is given by r maxdn r. Note that the backscattering ratio B Eq. 6 does not depend on the absolute number of particles present in the scattering volume or for that matter in each size class but only on the shape of the size distribution or the relative distribution of abundance between the size classes. The cumulative percent contribution to the backscattering coefficient from the size fraction of r min to r assuming a single value for the index of refraction of the particles collection is given by Cbb r 100 r r min r min Q bs N r r 2 dr r max Qbs N r r 2 dr. (8) B. Particle-Size Distributions Particulate matter in the sea consists of biogenic and terrigenous material that spans a broad size range with abundance and relative concentrations that vary considerably. However, measurements of the PSD indicate that the number of particles increases rapidly with decreasing size. PSDs were modeled by different distributions including hyperbolic, 19,20,25 segmented hyperbolic with two or three segments, 26,27 lognormal 28,29 or its combinations, 30 a generalized gamma distribution, 22 and a combination of segmented hyperbolic with Gaussian. 31,32 Here we consider only two models of the PSD Fig. 1 : hyperbolic, which is by far the most frequently used, and the 2C model, which although more complex provides better fit and more flexibility in modeling. A detailed discussion and a comparison of the performance of various PSD models performance in the fit of numerous measured PSDs including hyperbolic, segmented hyperbolic, lognormal, and 2C models can be found elsewhere. 21,22 r min The integral diverges when r min 3 0; hence a suitable, physically justifiable lower-limit cutoff is required for actual integration. D. Two-Component Model of the Particle Size Distribution The PSD is given by dn r dn A r dn B r C A F A r dr C B F B r dr, F A r r 2 exp 52r A, F B r r 2 exp 17r B, (10) where F A r and F B r are the size-distribution components and particle size radius r is expressed in micrometers. The parameter varies around mean values A and B values obtained from the fit to the numerous measured PSD distributions 21,22, C i i A, B are the constants of the distribution determined from experimental data and directly related to particle concentration orders of magnitude: C A, cm 3 m 3 ; C B, cm 3 m 3. The A component the small component is dominant in the small size range 1 m while the B component the large component is dominant in medium and large size ranges 3 m. The number of particles within a given size interval, r max r min, is again given by r max dn r, r min but, contrary to the situation with a hyperbolic distribution, for r min 3 0 the integral converges. r min dn r 7094 APPLIED OPTICS Vol. 41, No November 2002

4 3. Calculation of the Backscattering Ratio for Case 1 and 2 Waters A. Modeling of the Particles-Size Distribution Usually the measured PSD data are limited to the Coulter-counter size range spanning roughly r m. But for light-scattering applications such as calculation of the backscattering ratio, a much broader size range, particularly to the side of small particles, should be included. This is a result from most of the backscattering contribution coming from abundant submicrometer particles. 33 Hence, to calculate the backscattering ratio we must extrapolate the measured PSD by using some PSD model providing a reasonable fit. To study the effect of a suspended PSD on the backscattering ratio, we used several in situ measured PSDs covering a very broad size range that could be considered typical and representing case 1 and 2 water. 34,35 The measured PSDs were modeled fitted by a hyperbolic distribution and the 2C model. The PSDs representing case 2 water, measured in coastal Mediterranean water southern France, inshore Villefranche that were used in the calculations presented, are shown in Fig. 2 along with the fit of measured data with the Junge-type distribution Eq. 9 and the 2C model of the PSD Eqs. 10. These distributions were measured in a broad size range 0.2 m r 40 m and consequently considered suitable for the study of small-particle influence on backscattering. The other PSD data that were used are from the North Atlantic and North Pacific, representing case 1 waters. These PSDs are measured in a somewhat narrower, but still broad, size range spanning from 0.5 m r 40 m. The PSDs with corresponding hyperbolic and 2C model fit are depicted in Fig. 3. In all represented cases the 2C model provides a better fit than the Junge model. For the case 2 water samples considered here the average R 2 values are and for the 2C and the Junge model, respectively. The average R 2 values for the representative case 1 water samples are and for the 2C and the Junge model, respectively. However, these differences in the quality of the fit are more pronounced for very small and very large size ranges. The Junge model of the PSD provides a relatively good fit to the measured data in the central size range of r 1 15 m hence its widespread use. Outside this size range the hyperbolic Junge-type distribution Eq. 9 overestimates the number of particles, while the 2C model provides a good fit in the whole measured size range of 0.2 m r 40 m. Fig. 2. Measured particle-size distributions in symbols, case 2 water and, curves, corresponding fits and extrapolations with hyperbolic and 2C PSD models. Surface particle-size distribution from the bay at Villefranche SE France: a, Sample #15. Parameters of the corresponding fitted integral distributions are as defined: exponent k 3.09 for the hyperbolic model R , A 0.108, B 0.196, C A :C B for the 2C model R b, Sample #13. Parameters of the corresponding distributions are as follows: exponent k 3.0 for the hyperbolic model R , A 0.093, B 0.202, and C A :C B for the 2C model R B. Calculation of the Backscattering Ratio Question of Integration Limits The significant contribution to the backscattering comes from submicrometer particles. However, the experimental data on the PSD in the submicrometer size range are most often sparse, and one must extrapolate from the fit to a measured PSD. The question is how far this extrapolation is to be executed and how good the approximation is for the actual PSD it represents. In this context the limits of integration in Eq. 6 play a significant role in the determination of B. It was shown 17 that in case of the Junge-type distribution Eq. 9 the determination of b b is particularly sensitive to the lower limit of integration r min because lim r30 N r, hence a sensible lower limit of integration, r min 0 m, must be established. For one to obtain a backscattering ratio that is independent of the lower limit of integration, r min must be at least 0.05 m. 17 Unfortunately, as demonstrated with the examples above, the hyperbolic PSD significantly overestimates the number of particles especially in such a small size range. Therefore the extrapolation of PSD to such small sizes based on a hyperbolic PSD model may result in a significant orders of magnitude! overestimation of the number of particles, hence an unrealistic B. For that matter this model also overestimates the number of particles in the large size range, but that is not so important in con- 20 November 2002 Vol. 41, No. 33 APPLIED OPTICS 7095

5 Fig. 3. Measured particle-size distributions in symbols, case 1 water and, curves, corresponding fits and extrapolations with hyperbolic and 2C PSD models. Surface particle-size distributions from North Atlantic N:62 45 W and North Pacific 4 43 N: W : a, North Atlantic sample #2. Parameters of the corresponding fitted integral distributions are as follows: exponent k 2.92 for the hyperbolic model R , A 0.094, B 0.198, and C A :C B for the 2C model R b, North Pacific sample #4. Parameters of the corresponding distributions are as follows: exponent k 2.97 for the hyperbolic model R , A 0.087, B 0.187, and C A :C B for the 2C model R sideration of backscattering, because the contribution of very large particles compared with submicrometer particles can be neglected owing to their inherently smaller contribution and lower numbers of several orders of magnitude abundance. These problems are not present with the 2C model of PSD for which lim r30 N r 0 Fig. 1. The upper limit of integration is usually not taken to be infinite but fixed at some finite value. This is imposed by practical reasons connected with timeconsuming calculations and problems in recursion algorithms for amplitude functions in Mie theory that often occur at very high. However, as stated above, this upper limit cutoff is justified with a rapid decrease in particle number with an increase in size and an inherently smaller contribution of larger particles to backscattering. 4. Results and Discussion Before investigation of the influence of small particles and the lower limit of integration on B, we must establish a reasonable upper limit of integration. For that purpose we fixed the lower limit of integration to r min m and varied the upper limit from r max 1 m upward until the corresponding increase in B fell below The integration was Fig. 4. Dependence of backscattering ratio B on the upper limit of integration for the hyperbolic and 2C model of PSDs corresponding to case 2 and 1 water samples depicted in Figs. 2a and 3a, respectively: 532 nm, n Symbols, calculated values; curves, corresponding fit with the B-spline function. carried out for the relative index of refraction, m i the average value for a large number of marine particles and 532 nm the approximate mean value of the relevant spectral range. The results are summarized for both PSD models considered in Fig. 4 for case 2 and case 1 water. Although the values of the backscattering ratio differ as expected, both curves for B versus r max show a similar shape. For the 2C model the approximately constant value of B is obtained for r max 40 m, while the hyperbolic model requires a somewhat larger upper limit of r max 45 m consistent with overestimation of the large particle number. Hence for the considered PSDs the contribution to the backscattering ratio of particles with r 50 m can be neglected. This result is consistent with earlier findings for the Junge-type distribution. 17 However, if the PSD is broad but not so steep, e.g., a corresponding Junge exponent of k 3.2, the upper integration limit should be greater to include a contribution from large particles that in this case is not negligible. A. Case 2 Water The dependence of backscattering ratio B on the lower limit of integration for both PSD models that that were considered, fitted to the measured data from Fig. 2a coastal, case 2 water, is shown in Fig. 5. The upper limit of integration was fixed at r max 50.0 m, and the lower limit was varied in range, m r min 1.0 m. The integration was 7096 APPLIED OPTICS Vol. 41, No November 2002

6 Fig. 5. Dependence of backscattering ratio B on the lower limit of integration for the hyperbolic and 2C model of the PSD corresponding to the case 2 water sample depicted in Fig. 2a #15 : 532 nm, n Symbols, calculated values; curves, corresponding fit with a B-spline function. Fig. 6. Contribution of particles smaller than the given size r to the backscattering coefficient b b calculated from the hyperbolic Junge and the 2C model corresponding to the case 2 water PSD sample #15 from Fig 2a : 532 nm, m i. Symbols, calculated values; curves, corresponding fit with a B-spline function. conducted for 532 nm with a relative index of refraction, m i. For the hyperbolic model the backscattering ratio strongly depends on the lower limit of integration, and inclusion of particles with sizes radii of 0.01 m is necessary to obtain constant integration limit independent B. For the 2C model representation of the same PSD the dependence of the calculated backscattering ratio on the lower limit of integration shows behavior similar to that calculated for the hyperbolic Junge model. However, the variation is less pronounced, and the final integration limit independent backscattering ratio is obtained at a higher lower limit than in the case of the hyperbolic distribution: r min 0.03 m. Additional insight is obtained by considering the cumulative percent contribution to the backscattering coefficient from the particular size fraction in the range from r min to r. The contribution of particles smaller than a given r to the backscattering coefficient calculated from Eq. 8 is depicted in Fig. 6 for the Junge and the 2C PSD models corresponding to Villefranche #15 data and calculated for 532 nm and n Most of the contribution to backscattering comes from small particles. For the case considered the contribution of submicrometer particles to backscattering in the 2C model is 71%, while particles with r 0.1 m contribute 32%. Particles larger than 35 m contribute less than 1%. Compared with that, a contribution to b b from submicrometer particles in a hyperbolic model constitutes more than 92%, and the small particles with r 0.1 contribute more than 67%. Even particles smaller than 0.03 m still contribute 6% compared with 0.5% in the 2C model. Similar results for the cumulative contribution to b b in the case of a Junge-type distribution with an exponent of k 4 and for 550 nm were obtained by Stramski and Kiefer. 33 For the hyperbolic Junge fit of the considered PSD the backscattering ratios calculated at 532 nm and for n i are B J # and B J # The values of the backscattering ratios calculated from the 2C model, assuming the same and equal index of refraction for both components, are significantly lower: B 2C # and B 2C # The influence of the refractive index and the exponent on backscattering in the Junge model was discussed earlier. 17,38 In contrast to the Junge PSD model that permits only one average index of refraction for the whole particle population the 2C model permits different indices of refraction for each component. This provides an additional degree of freedom in modeling by providing the means for inclusion of, e.g., the mineral the high refractive index fraction into the small component in addition to the predominantly planktonic a low to medium index of refraction large component. The influence of various combinations of the components refractive indices on B at 532 nm for the 2C model of the Vilefranche #15 PSD sample is depicted in Fig. 7. It is obvious that the indices of refraction associated with PSD components or rather their combinations have significant influence on the backscattering ratio. This in particular is valid if a fraction with high refractive index mineral is present in the small A component. That, owing to the high relative index of refraction of mineral material , results in an increase in the average bulk index of refraction of the A component and consequently a substantial increase in B. The other influential factors are the shape skewness of the components governed by a corresponding and the relative ratio of the component abundances C A :C B. The influence of the latter quantity on the backscattering ratio is rather straightforward as discussed by Twardowski et al. 39 The influence of the -parameter values of the components on the backscattering ratio in the 2C model is depicted in Fig. 8. We see that the backscattering ratio is more sensitive to changes in A 20 November 2002 Vol. 41, No. 33 APPLIED OPTICS 7097

7 Fig. 7. Influence of the index of refraction of the components on the backscattering ratio in the 2C model abscissa, the real part of the small component s index of refraction; parameter, the large B component s index of refraction and comparison with the Junge model. Calculated values at 532 nm for the parameters corresponding symbols, to #15 PSD and curves, to the B-spline fit. than B, hence to the shape of the size distribution of the small particles. The calculated backscattering ratio depicted in Fig. 9 exhibits a weak spectral dependence in compliance with earlier calculations and recent observations. 9,17,40 The values of the backscattering ratio calculated from the 2C model for the representative cases are similar to the backscattering ratios of case 2 waters measured at the surface in the Gulf of California: B and B at 532 nm. 39 Somewhat higher backscattering ratios, such as measured offshore in Southern California 14 B 0.014, b m 1, nm and near shore at St. Luis bay and the Mississippi Sound waters 11,15 B at 660 nm and the famous Petzold data 41 B , b 0.22 m 1 at 530 nm offshore in Southern California are Fig. 8. Influence of the parameters on the backscattering ratio in the 2C model. B versus A the parameter is B. Symbols, calculated values and, curves, the corresponding fit with the B-spline for C A :C B , 532 nm, and real indices of refraction, n A n B Fig. 9. Spectral dependence of the backscattering ratio B for the representative case 2 water samples calculated from the Junge and the 2C model. Symbols, calculated values and, curves, corresponding fit with the B-spline function. easily accounted for in the 2C model by allowing a slightly higher index of refraction of the small A component Fig. 7. This increase in the average small component s index of refraction corresponds to the presence of a mineral fraction n in addition to organic material bacteria, virions, etc. in the small component. Also, a higher abundance of small particles higher A and or C A would have a similar effect on B Fig. 8. Although the Junge model predictions of B are considerably higher than those of the 2C model, they are not completely beyond the range of realistic possibilities. Also, although there is evidence that the 2C model provides a better fit of the PSD and consequently a better extrapolation that should lead to a better estimation of the backscattering ratio, the fact remains that measurements of the PSD in the submicrometer range are relatively sparse and that extrapolation into the unknown very small size range remains somewhat ambiguous. Moreover the backscattering ratios that comply with measured data can also be obtained from the totally unrealistic values of the scattering and backscattering coefficients in adequate proportion. Hence it is worthwhile to compare predictions of the models with actual measured scattering and backscattering coefficients associated with a measured PSD in order to check the compliance not only with measured B but also with b and b b. To that purpose we used the data measured in the Washington East Sound. 42 The PSD measured near the surface 2m in the East Sound is depicted in Fig. 10 along with corresponding Junge and 2C fits. The 2C model provides a good fit in the whole measured particle-size range R When the simple hyperbolic Junge model is used, the best whole-size range fit is obtained for k R A better fit to the measured PSD could be obtained with a segmented hyperbolic distribution with two segments, one with k 6.06 for the smallest measured sizes r 1.5 m, R and the other with exponent k 1.94 R for particles with r 7098 APPLIED OPTICS Vol. 41, No November 2002

8 Fig. 10. Symbols, Measured particle-size distribution from the Washington East Sound at a depth of 2 m; curves, corresponding fit with the Junge and the 2C model. Junge distribution: k 2.11, R ; 2C model: A 0.130, B 0.159, C A :C B , R Inset: extrapolations to the submicrometer particle-size range from the hyperbolic fit k 2.11, segmented hyperbolic fit small segment, k 6.06, and the 2C model. 1.5 m. This situation is depicted in the inset in Fig. 10 where the extrapolations to submicrometer sizes for both cases are presented. We can see that, if the two-segment hyperbolic model is used, the extrapolation from the small-size segment with exponent k 6.06 results in a tremendous overestimation of small particles and consequently to absurd values of the calculated scattering and backscattering coefficients and ratio, even if the lower limit of integration is set to 0.1 m. Hence we have accomplished calculations with the original single-segment Junge fit with k The results of 2C and hyperbolic model calculations along with the corresponding measured values of B, b, and b b are presented in Fig. 11. The calculations were done at matching Hydroscat-6 and AC-9 wavelengths used in the measurements. The results with the 2C model are in far better agreement with the measured values than results from the Junge model. Although providing acceptable results for the backscattering ratio, the Junge model predicts values of scattering and backscattering coefficients that are significantly different from the measured values in the whole spectral range b b, almost an order of magnitude!. Fig. 11. Spectral dependence of the measured Hydroscat-6 and AC-9, East Sound, 2 m and b, calculated scattering coefficient; b b, backscattering coefficient; B, backscattering ratio. a, Comparison of measured values with results from the 2C model for n A and n B b, Comparison of measured values with results from the Junge model for n Measured values are represented by larger, symbols while the calculated values are represented by small solid symbols and lines corresponding to the B-spline fit. The measured and calculated scattering parameters correspond to the PSD shown in Fig. 10. B. Case 1 Water The dependence of the backscattering ratio B on the lower limit of integration for both considered PSD models fitted to the measured data from Fig. 2b oceanic, case 1 water are shown in Fig. 12. The upper limit of integration was fixed at r max 50.0 m, and the lower limit was varied in the range of m r min 1 m. The calculation was accomplished at 532 nm and the relative index of refraction, m i. The behavior of the considered models is similar to that obtained for case 2 water: The hyperbolic distribution is more sensitive to the lower limit of integration, and obtaining a limit-independent B inclusion of very small particles is necessary. Some other aspects of more general nature discussed in subsection 4, A apply also for case 1 waters. Fig. 12. Dependence of the backscattering ratio B on the lower limit of integration for the hyperbolic and the 2C model of the PSD corresponding to the case 1 water sample depicted in Fig. 3a: 532 nm, m 1.05 Oi. Symbols, calculated values; curves, corresponding fit with the B-spline function). 20 November 2002 Vol. 41, No. 33 APPLIED OPTICS 7099

9 adequate and compliant with measurements of case 1 water. The source of differences in predictions from these two PSD models comes obviously from the overestimated number of submicrometer particles that occurs in extrapolation from the hyperbolic PSD model. 5. Conclusions Fig. 13. Spectral dependence of the backscattering ratio B for representative case 1 water samples calculated from the Junge and the 2C model. Symbols, calculated values; curves, corresponding fit with the B-spline function. The backscattering ratios calculated from the hyperbolic model at 532 nm and for m i for the considered PSDs are B J NA# and B J NP# If the same index of refraction is used for both components of the 2C model, substantially lower values for B are obtained, B 2C NA# and B 2C NP# The calculated spectral dependence B is shown in Fig. 13. The selected wavelengths correspond to those usually used in AC-9 and Hydroscat instruments. The values for B, b, and b b calculated from the 2C model for North Pacific surface sample #4 the corresponding differential Junge PSD exponent, k 3.97 at 532 nm and for m A m B i are B , b 0.37 m 1, and b b m 1. These values calculated from the 2C model are similar to the values of B, b, and b b measured at 530 nm for case 1 water in the Gulf of California 39 for a similar PSD an estimated corresponding differential Junge PSD exponent, k 3.9, and estimated n 1.05 : B , b 0.26 m 1, and b b m 1. The corresponding values obtained from the Junge model are B , b m 1, and b b m 1. The representative value of B for an oceanic environment between oligotrophic and eutrophic, complying with Morel s model of case 1 waters, 43 can be taken as However, this model also estimates b b b as a function of chlorophyll concentration and predicts values higher than 0.01 with chlorophyll concentrations typical of oligotrophic case 1 water. Similar higher B values are found in Petzold s Tongue of the Ocean data B at 530 nm. 41 Such results are obtained from the 2C model if one assumes a somewhat higher index of refraction for the small component Fig. 7 or higher abundance of small particles larger A and or C A, Fig. 8. Hence the results of calculations and predictions obtained from the 2C model could be considered as 1. Small submicrometer particles significantly contribute to the backscattering ratio. 2. For calculation of the backscattering ratio a much broader PSD is needed than is usually measured. Hence extrapolation from the fit of the measured data is necessary. 3. Results of extrapolation critically depends on the chosen PSD model. 4. A hyperbolic-type PSD largely overestimates the number of small and also large particles. 5. Calculation of the backscattering ratio with a hyperbolic-type PSD model requires inclusion of very small particles 0.1 m. Consequently the calculated backscattering ratio could be much higher than the real value corresponding to the measured PSD. 6. The 2C model provides an adequate model of the PSD that assures a good fit even in the broadest measurement size ranges and facilitates reasonable extrapolations. 7. The backscattering ratio and the values of the scattering coefficients calculated with the 2C model are in agreement with the measured and the expected values. 8. For case 2 water it seems that B is only weakly sensitive to a change in wavelength at least in the considered spectral range. This result is consistent with earlier observations. 9 However, a more detailed analysis of the spectral behavior of the backscattering ratio and the total scattering coefficient as well as a sensitivity to change in the parameter values of the 2C model, which are beyond the scope of this study, are subject of further investigation. Support from the Croatian Ministry of Science and Technology, grant , is acknowledged. References 1. L. Prieur and S. Sathyendranath, An optical classification of coastal and oceanic waters based on the specific spectral absorption of phytoplankton pigments, dissolved organic matter, and other particulate materials, Limnol. Oceanogr. 26, H. R. Gordon and A. Morel, Remote assessment of ocean color for interpretation of satellite visible imagery: a review, in Lecture Notes on Coastal and Estuarine Studies, M. Bowman, ed. Springer-Verlag, Berlin, J. R. V. Zaneveld, Remotely sensed reflectance and its dependence on vertical structure: theoretical derivation, Appl. Opt. 21, H. R. Gordon, O. B. Brown, and M. M. Jacobs, Computed relationship between the inherent and apparent optical properties of flat homogenous ocean, Appl. Opt. 14, H. R. Gordon, Ocean color remote sensing: influence of the 7100 APPLIED OPTICS Vol. 41, No November 2002

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