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

Effect of suspended particulate-size distributionon the backscattering ratio in the remote sensingof seawater

Dubravko Risovic

Mie theory is used to study the influence of the particle-size distribution �PSD� on the backscattering ratiofor 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 backscat-tering 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 fittedwith hyperbolic �Junge� and the two-component model of the PSD. It is shown that the result ofextrapolation, hence the backscattering ratio, critically depends on the chosen PSD model. For aparticular PSD model the role of submicrometer particles and the applied integration limits on thebackscattering ratio is discussed. The use of the hyperbolic PSD model largely overestimates thenumber of small �submicrometer� particles that significantly contribute to backscattering and conse-quently leads to an erroneously high backscattering ratio. The two-component model proves to be anadequate PSD model for use in backscattering�scattering calculations providing satisfactory resultscomplying with experimental data. The results are relevant for the inversion of remotely sensed dataand the prediction of optical properties and the concentration of phytoplankton pigments, suspendedsediment, and yellow substance. © 2002 Optical Society of America

OCIS codes: 010.4450, 290.1350, 290.5850, 290.4020, 280.0280.

1. Introduction

It is well established that the remote sensing of seacolor is a valuable tool in optical oceanography thatyields information on the concentration of phyto-plankton pigments, suspended sediment, and yellowsubstance in a euphotic layer.1 Depending onwhether these concentrations are correlated amongthemselves, water is classified as case 1 and case 2.Oceans are typical case 1 waters whereas coastalwaters are often referred to as case 2 water. �Hereoften a high degree of correlation may be found al-though subject to large spatial and temporal varia-tions due to local phenomena.2�

A number of different algorithms are used for theinversion of remotely sensed radiance to obtain the

The author is with Molecular Physics Laboratory, Ru�er Bosk-ovic Institute, POB 180, HR-10002 Zagreb, Croatia �e-mail,[email protected]�.

Received 11 July 2002; revised manuscript received 11 July2002.

0003-6935�02�337092-10$15.00�0© 2002 Optical Society of America

7092 APPLIED OPTICS � Vol. 41, No. 33 � 20 November 2002

oceanic pigment concentrations or the total absorp-tion coefficients.3–13

It is common practice to assume that the remote-sensing reflectance R�� is related to the ratio of thetotal volume backscattering coefficient bb to the totalvolume absorption coefficient a. Hence R�� C��bb��a��, where the proportionality functionC�� is often presumed constant.6–8,14 Scattering byhigh concentrations of inorganic particles, especiallyin coastal waters, strongly influences the submarinelight field in some cases to an even greater extentthan absorption processes; thus bb is a major compo-nent of the reflectance equation.7,9,10 One of theproblems is that until recently bb�� was difficult tomeasure and generally unknown. Hence the behav-ior of R�� and C�� was investigated in terms ofeasily measured parameters such as b��, a��, diffuseattenuation coefficient Kd��, and the particle-sizedistribution.15,16 It seems that the shape of particle-size distributions has a profound influence on bb��,hence on R, although some questions remain. Sim-ulation of light scattering from oceanic waters17 in-dicate a major influence and contribution ofsubmicrometer particles to bb, while for turbid case 3waters it has been suggested18 that R should depend

only on the total cross-sectional area of the particu-late per unit volume and the diffuse attenuation co-efficient. To our knowledge neither hypothesis hasbeen verified experimentally. On the other hand, itwas found that for case 2 waters R was insensitive tothe natural fluctuations in particle-size distribu-tions,15 at least for � � 650 nm. Modeling the back-scattering ratio showed that the influence of verysmall �submicrometer� particles is dominant.17 Inall these studies and simulations the Junge�hyperbolic�-type distribution19,20 was used to modelthe particle-size distributions �PSDs�. This oftenused, analytically simple, and easy to handle func-tion, although appropriate for a description of PSDsin narrow size ranges, is inadequate for the descrip-tion of PSDs in a very broad size range such as en-countered in seawater.21,22 Furthermore, becausethe measured particle-size range is usually narrowerthan required in light-scattering calculations, extrap-olations �to smaller and larger sizes� are in order.Since the number of particles is an inverse powerfunction of particle size r, the model tends to overes-timate the number of small particles �r3 0, N3 ��.Hence at least to a degree the calculated influence ofvery small �submicrometer� particles on backscatter-ing could be a somewhat synthetic result. To com-pensate for this effect, the use of a hyperbolic PSDmodel imposes the necessity of establishing an ap-propriate lower cutoff in integration limits; otherwisethe integral diverges. The question of cutoff and itsinfluences on the calculation of the backscatteringratio for the Junge model was considered by Ulloa etal.17 who found that the reasonable lower limit ofintegration is r � 0.005 �m.

A much more realistic and flexible description ofPSD in a broad-size range is achieved with the two-component model introduced by Risovic.22 Thismodel, although more complex than the Junge model,provides a better fit over a wider size range thanother models21 and also greater flexibility in opticalmodeling through the possible use of different indicesof refraction for each component. Furthermore itdoes not impose limitations on integration.

In this study we used the Mie -theory to investigatethe backscattering properties of the marine-particleensemble by applying the Junge and the two-component �2C� PSD model. In particular the influ-ence of the PSD model �chosen to fit and extrapolatein situ measured PSD� on the calculated backscatter-ing ratio in case 1 and 2 waters was investigated.Also predictions of B, b, and bb based on the 2C andthe Junge PSD models are compared with actual insitu measured values for case 1 and 2 waters.

2. Theory

A. Backscattering Ratio for Polydispersions

The backscattering coefficient bb is an inherent opti-cal property that can be partitioned as

bb � bbw � bbp , (1)

where bbw is the backscattering coefficient of pureseawater and bbp is the backscattering coefficient ofthe suspended particles. In the visible part of thespectrum bbw is small23; hence the dominant contri-bution to bb comes from bp.

For a spherical particle of size �radius� r the effi-ciency factor for scattering is given by

Qs � Cs��r2 , (2)

where Cs is the scattering cross section.The backscattering efficiency factor Qbs is given by

Qbs � Cbs��r2 , (3)

where Cbs is the backscattering cross sectionExact �Mie� efficiency factors are given by24

Qs �22

n�1

�

�2n � 1��ans�2 � �bn

s�2� , (4)

Qbs �1

�2��

n�1

�

�2n � 1�� 1�n �an � bn�� 2 , (5)

where an and bn are the scattering coefficients and is the size parameter given by

�2�mw r

�0,

where mw 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 bbp is commonly mod-eled as

bbp � Bbp ,

where B is backscattering ratio B � bbp�bp, where bpis the �total� scattering coefficient of the particles.

The ensemble average of backscattering ratio B foran ensemble of particles, described by the size distri-bution N��, is given by

B �

�0

�

Qbs�m, �2N��d

�0

�

Qs�m, �2N��d

, (6)

where m � n � i is the complex refractive index ofthe particles.

The size distribution is given by

N�� Ntotf �� , (7)

where Ntot is the total number of particles per unitvolume and f�� is the probability density functionsuch that

�0

�

f ��d � 1.

20 November 2002 � Vol. 41, No. 33 � APPLIED OPTICS 7093

Note that the backscattering ratio B �Eq. �6�� doesnot depend on the absolute number of particlespresent in the scattering volume �or for that matter ineach size class� but only on the shape of the sizedistribution �or the relative distribution of abundancebetween the size classes�.

The cumulative percent contribution to the back-scattering coefficient from the size fraction of rmin tor �assuming a single value for the index of refractionof the particles collection� is given by

Cbb�r� � 100 �

�rmin

r

QbsN�r�r2dr

�rmin

rmax

QbsN�r�r2dr

. (8)

B. Particle-Size Distributions

Particulate matter in the sea consists of biogenic andterrigenous material that spans a broad size rangewith abundance and relative concentrations thatvary considerably. However, measurements of thePSD indicate that the number of particles increasesrapidly with decreasing size. PSDs were modeled bydifferent distributions including hyperbolic,19,20,25

segmented hyperbolic with two or three seg-ments,26,27 lognormal28,29 or its combinations,30 ageneralized gamma distribution,22 and a combinationof segmented hyperbolic with Gaussian.31,32 Herewe 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 pro-vides better fit and more flexibility in modeling. Adetailed discussion and a comparison of the perfor-mance of various PSD models performance in the fitof numerous measured PSDs �including hyperbolic,segmented hyperbolic, lognormal, and 2C models�can be found elsewhere.21,22

C. Hyperbolic �Junge� Particle-Size Distribution

This is a one-parameter distribution given by

dN�r� � Cr�kdr , (9)

where dN�r� is the total number of particles per unitvolume with radii between r and r �dr, C is a con-stant depending on the particle concentration, r is theparticle radius, and k is a parameter that depends onparticle type, size range, and measurement site.For the PSD encountered in seawater, k has a valuebetween 2.5 and 6. This type of distribution provedto be useful in modeling the PSD in a micrometer-sizerange of 0.5 � r � 25. Usually this formula is usedto fit the results from the Coulter counter; in whichcase the measurement size intervals � �bins� increaselogarithmically. The number of particles within agiven size interval, � � rmax � rmin �bin� is given by

�rmin

rmax

dN�r� .

The integral diverges when rmin3 0; hence a suit-able, physically justifiable lower-limit cutoff is re-quired for actual integration.

D. Two-Component Model of the Particle SizeDistribution

The PSD is given by

dN�r� � dNA�r� � dNB�r�

� CA FA�r�dr � CB FB�r�dr,

FA�r� � r2 exp��52r�A�,

FB�r� � r2 exp��17r�B� , (10)

where FA�r� and FB�r� are the size-distribution com-ponents and particle size �radius� r is expressed inmicrometers. The parameter � varies around meanvalues ��A � 0.137 and ��B � 0.226 �values obtainedfrom the fit to the numerous measured PSD distri-butions21,22�, Ci �i � A, B� are the constants of thedistribution determined from experimental data anddirectly related to particle concentration �orders ofmagnitude: CA, 1025–1027 cm�3�m�3; CB, 109–1011

cm�3 �m�3�. The A component �the small compo-nent� is dominant in the small size range ��1 �m�while the B component �the large component� is dom-inant in medium and large size ranges ��3 �m�.

The number of particles within a given size inter-val, rmax � rmin, is again given by

�rmin

rmax

dN�r� ,

but, contrary to the situation with a hyperbolic dis-tribution, for rmin 3 0 the integral

�rmin

�

dN�r�

converges.

Fig. 1. Schematic representation of Junge and the 2C PSD mod-els.


3. Calculation of the Backscattering Ratio for Case 1and 2 Waters

A. Modeling of the Particles-Size Distribution

Usually the measured PSD data are limited to theCoulter-counter size range spanning roughly r �0.5–50 �m. But for light-scattering applicationssuch as calculation of the backscattering ratio, amuch broader size range, particularly to the side ofsmall particles, should be included. This is a resultfrom most of the backscattering contribution comingfrom abundant submicrometer particles.33 Hence,to calculate the backscattering ratio we must extrap-olate the measured PSD by using some PSD modelproviding a reasonable fit.

To study the effect of a suspended PSD on thebackscattering ratio, we used several in situ mea-sured PSDs covering a very broad size range thatcould be considered typical and representing case 1and 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 incoastal Mediterranean water �southern France, in-shore Villefranche� that were used in the calculationspresented, are shown in Fig. 2 along with the fit ofmeasured data with the Junge-type distribution �Eq.�9�� and the 2C model of the PSD �Eqs. �10��. Thesedistributions were measured in a broad size range�0.2 �m � r � 40 �m� and consequently consideredsuitable for the study of small-particle influence onbackscattering.

The other PSD data that were used are from theNorth Atlantic and North Pacific, representing case 1waters. These PSDs are measured in a somewhatnarrower, but still broad, size range spanning from0.5 �m � r � 40 �m. The PSDs with correspondinghyperbolic and 2C model fit are depicted in Fig. 3.

In all represented cases the 2C model provides abetter fit than the Junge model. For the case 2 wa-ter samples considered here the average R2 valuesare 0.971 and 0.733 for the 2C and the Junge model,respectively. The average R2 values for the repre-sentative case 1 water samples are 0.947 and 0.756for the 2C and the Junge model, respectively. How-ever, these differences in the quality of the fit aremore pronounced for very small and very large sizeranges. The Junge model of the PSD provides a rel-atively good fit to the measured data in the centralsize range of r � 1–15 �m �hence its widespread use�.Outside this size range the hyperbolic �Junge-type�distribution �Eq. �9�� overestimates the number ofparticles, while the 2C model provides a good fit inthe whole measured size range of 0.2 �m � r � 40�m.

B. Calculation of the Backscattering Ratio—Question ofIntegration Limits

The significant contribution to the backscatteringcomes from submicrometer particles. However, theexperimental data on the PSD in the submicrometersize range are most often sparse, and one must ex-trapolate from the fit to a measured PSD. The ques-

tion is how far this extrapolation is to be executed andhow good the approximation is for the actual PSD itrepresents. In this context the limits of integrationin Eq. �6� play a significant role in the determinationof B. It was shown17 that in case of the Junge-typedistribution �Eq. �9�� the determination of bb is par-ticularly sensitive to the lower limit of integrationrmin because

limr30

N�r� � � ,

hence a sensible lower limit of integration, rmin � 0�m, must be established. For one to obtain a back-scattering ratio that is independent of the lower limitof integration, rmin must be at least � 0.05 �m.17

Unfortunately, as demonstrated with the examplesabove, the hyperbolic PSD significantly overesti-mates the number of particles especially in such asmall size range. Therefore the extrapolation ofPSD to such small sizes based on a hyperbolic PSDmodel may result in a significant �orders of magni-tude!� overestimation of the number of particles,hence an unrealistic B. For that matter this modelalso overestimates the number of particles in thelarge size range, but that is not so important in con-

Fig. 2. Measured particle-size distributions in symbols, case 2water and, curves, corresponding fits and extrapolations with hy-perbolic and 2C PSD models. Surface particle-size distributionfrom the bay at Villefranche SE France: a, Sample #15. Param-eters of the corresponding fitted �integral� distributions are asdefined: exponent k � 3.09 for the hyperbolic model �R2 � 0.766�,�A � 0.108, �B � 0.196, CA:CB � 5.045 � 1015 for the 2C model�R2 � 0.960�. b, Sample #13. Parameters of the correspondingdistributions are as follows: exponent k � 3.0 for the hyperbolicmodel �R2 � 0.746�, �A � 0.093, �B � 0.202, and CA:CB � 5.834 �1015 for the 2C model �R2 � 0.931�.


sideration of backscattering, because the contribu-tion of very large particles compared withsubmicrometer particles can be neglected owing totheir inherently smaller contribution and lower num-bers of several orders of magnitude �abundance�.These problems are not present with the 2C model ofPSD for which limr30 N�r� � 0 �Fig. 1�.

The upper limit of integration is usually not takento be infinite but fixed at some finite value. This isimposed by practical reasons connected with time-consuming calculations and problems in recursionalgorithms for amplitude functions in Mie theory thatoften occur at very high . However, as statedabove, this upper limit cutoff is justified with a rapiddecrease in particle number with an increase in sizeand an inherently smaller contribution of larger par-ticles to backscattering.

4. Results and Discussion

Before investigation of the influence of small particlesand the lower limit of integration on B, we mustestablish a reasonable upper limit of integration.For that purpose we fixed the lower limit of integra-tion to rmin � 1 � 10�3 �m and varied the upper limitfrom rmax � 1 �m upward until the correspondingincrease in B fell below 10�5. The integration was

carried out for the relative index of refraction, m �1.05 � 0i �the average value for a large number ofmarine particles36–38 and � � 532 nm �the approxi-mate mean value of the relevant spectral range�.The results are summarized for both PSD modelsconsidered in Fig. 4 for case 2 and case 1 water.Although the values of the backscattering ratio differas expected, both curves for B versus rmax show asimilar shape. For the 2C model the approximatelyconstant value of B is obtained for rmax � 40 �m,while the hyperbolic model requires a somewhatlarger upper limit of rmax � 45 �m �consistent withoverestimation of the large particle number�. Hencefor the considered PSDs the contribution to the back-scattering ratio of particles with r � 50 �m can beneglected. This result is consistent with earlierfindings for the Junge-type distribution.17 How-ever, if the PSD is broad �but not so steep, e.g., acorresponding Junge exponent of k � 3.2�, the upperintegration limit should be greater to include a con-tribution from large particles that in this case is notnegligible.

A. Case 2 Water

The dependence of backscattering ratio B on thelower limit of integration for both PSD models thatthat were considered, fitted to the measured datafrom Fig. 2a �coastal, case 2 water�, is shown in Fig.5. The upper limit of integration was fixed at rmax �50.0 �m, and the lower limit was varied in range, 1 �10�3 �m � rmin � 1.0 �m. The integration was

Fig. 3. Measured particle-size distributions in symbols, case 1water and, curves, corresponding fits and extrapolations with hy-perbolic and 2C PSD models. Surface particle-size distributionsfrom North Atlantic �25°54�N:62°45�W� and North Pacific �4°43�N:149°58�W�: a, North Atlantic sample #2. Parameters of the cor-responding fitted �integral� distributions are as follows: exponentk � 2.92 for the hyperbolic model �R2 � 0.495�, �A � 0.094, �B �0.198, and CA:CB � 4.58 � 1015 for the 2C model �R2 � 0.995�. b,North Pacific sample #4. Parameters of the corresponding distri-butions are as follows: exponent k � 2.97 for the hyperbolic model�R2 � 0.927�, �A � 0.087, �B � 0.187, and CA:CB � 1.39 � 1016 forthe 2C model �R2 � 0.947�.

Fig. 4. Dependence of backscattering ratio B on the upper limit ofintegration for the hyperbolic and 2C model of PSDs correspondingto case 2 and 1 water samples depicted in Figs. 2a and 3a, respec-tively: � � 532 nm, n � 1.05. Symbols, calculated values;curves, corresponding fit with the B-spline function.


conducted for � � 532 nm with a relative index ofrefraction, m � 1.05 � 0i.

For the hyperbolic model the backscattering ratiostrongly depends on the lower limit of integration,and inclusion of particles with sizes �radii� of � 0.01�m is necessary to obtain constant �integration limitindependent� B. For the 2C model representation ofthe same PSD the dependence of the calculated back-scattering ratio on the lower limit of integrationshows behavior similar to that calculated for the hy-perbolic �Junge� model. However, the variation isless pronounced, and the final �integration limit in-dependent� backscattering ratio is obtained at ahigher lower limit than in the case of the hyperbolicdistribution: rmin � 0.03 �m.

Additional insight is obtained by considering thecumulative percent contribution to the backscatter-ing coefficient from the particular size fraction in therange from rmin to r�.

The contribution of particles smaller than a givenr� to the backscattering coefficient calculated fromEq. �8� is depicted in Fig. 6 for the Junge and the 2CPSD models corresponding to Villefranche #15 dataand calculated for � � 532 nm and n � 1.05. Mostof the contribution to backscattering comes fromsmall particles. For the case considered the contri-bution of submicrometer particles to backscatteringin the 2C model is �71%, while particles with r � 0.1�m contribute �32%. Particles larger than 35 �mcontribute less than 1%. Compared with that, a con-tribution to bb from submicrometer particles in a hy-perbolic model constitutes more than 92%, and thesmall particles with r � 0.1 contribute more than67%. Even particles smaller than 0.03 �m still con-tribute �6% �compared with � 0.5% in the 2C model�.Similar results for the cumulative contribution to bbin the case of a Junge-type distribution with an ex-ponent of k � �4 and for � � 550 nm were obtainedby Stramski and Kiefer.33

For the hyperbolic �Junge� fit of the consideredPSD the backscattering ratios calculated at � � 532

nm and for n � 1.05 � 0i are BJ�#13� � 1.4 � 10�2

and BJ�#15� � 1.8 � 10�2. The values of the back-scattering ratios calculated from the 2C model, as-suming the same and equal index of refraction forboth components, are significantly lower: B2C�#13�� 3.76 � 10�3 and B2C�#15� � 6.7 � 10�3. Theinfluence of the refractive index and the exponent onbackscattering in the Junge model was discussed ear-lier.17,38 In contrast to the Junge PSD model thatpermits only one �average� index of refraction for thewhole particle population the 2C model permits dif-ferent indices of refraction for each component. Thisprovides an additional degree of freedom in modelingby providing the means for inclusion of, e.g., the min-eral �the high refractive index� fraction into the smallcomponent in addition to the predominantly plank-tonic �a low to medium index of refraction� large com-ponent. The influence of various combinations ofthe components’ refractive indices on B at 532 nm forthe 2C model of the Vilefranche #15 PSD sample isdepicted in Fig. 7. It is obvious that the indices ofrefraction associated with PSD components �orrather their combinations� have significant influenceon the backscattering ratio. This in particular isvalid if a fraction with high refractive index �mineral�is present in the small A component. That, owing tothe high relative index of refraction of mineral mate-rial �1.15–1.20�, results in an increase in the averagebulk index of refraction of the A component and con-sequently a substantial increase in B. The otherinfluential factors are the shape �skewness� of thecomponents �governed by a corresponding � and therelative ratio of the component abundances �CA:CB�.The influence of the latter quantity on the backscat-tering ratio is rather straightforward as discussed byTwardowski et al.39 The influence of the �-parametervalues of the components on the backscattering ratio inthe 2C model is depicted in Fig. 8. We see that thebackscattering ratio is more sensitive to changes in �A

Fig. 6. Contribution of particles smaller than the given size �r��to the backscattering coefficient bb calculated from the hyperbolic�Junge� and the 2C model corresponding to the case 2 water PSD�sample #15 from Fig 2a�: � � 532 nm, m � 1.05 � 0i. Symbols,calculated values; curves, corresponding fit with a B-spline func-tion.

Fig. 5. Dependence of backscattering ratio B on the lower limit ofintegration for the hyperbolic and 2C model of the PSD correspond-ing to the case 2 water sample depicted in Fig. 2a �#15�: � � 532nm, n � 1.05. Symbols, calculated values; curves, correspondingfit with a B-spline function.


than �B, hence to the shape of the size distribution ofthe small particles.

The calculated backscattering ratio depicted in Fig.9 exhibits a weak spectral dependence in compliancewith earlier calculations and recent observa-tions.9,17,40 The values of the backscattering ratiocalculated from the 2C model for the representativecases are similar to the backscattering ratios of case2 waters measured at the surface in the Gulf of Cal-ifornia: B � 5 � 10�3 and B � 9.5 � 10�3 at 532nm.39

Somewhat higher backscattering ratios, such asmeasured offshore in Southern California14 �B �0.014, b � 0.275 m�1, � � 400–600 nm� and nearshore at St. Luis bay and the Mississippi Sound wa-ters11,15 �B � 0.012 at 660 nm� and the famous Pet-zold data41 �B � 9.5 � 10�3–1.25 � 10�2, b � 0.22m�1 at 530 nm offshore �in Southern California� are

easily accounted for in the 2C model by allowing aslightly higher index of refraction �1.08–1.09� of thesmall A component �Fig. 7�. This increase in theaverage small component’s index of refraction corre-sponds to the presence of a mineral fraction �n �1.1–1.2� in addition to organic material �bacteria,virions, etc.� in the small component. Also, a higherabundance of small particles �higher �A and�or CA�would have a similar effect on B �Fig. 8�.

Although the Junge model predictions of B are con-siderably higher than those of the 2C model, they arenot completely beyond the range of realistic possibil-ities. Also, although there is evidence that the 2Cmodel provides a better fit of the PSD and conse-quently a better extrapolation that should lead to abetter estimation of the backscattering ratio, the factremains that measurements of the PSD in the sub-micrometer range are relatively sparse and that ex-trapolation into the unknown �very small� size rangeremains somewhat ambiguous. Moreover the back-scattering ratios that comply with measured data canalso be obtained from the totally unrealistic values ofthe scattering and backscattering coefficients in ad-equate proportion. Hence it is worthwhile to com-pare predictions of the models with actual measuredscattering and backscattering coefficients associatedwith a measured PSD in order to check the compli-ance not only with measured B but also with b and bb.To that purpose we used the data measured in theWashington East Sound.42 The PSD measured nearthe surface �2 m� in the East Sound is depicted in Fig.10 along with corresponding Junge and 2C fits. The2C model provides a good fit in the whole measuredparticle-size range �R2 � 0.987�. When the simplehyperbolic �Junge� model is used, the best �whole-sizerange� fit is obtained for k � 2.113 �R2 � 0.685�. Abetter fit to the measured PSD could be obtained witha segmented hyperbolic distribution with two seg-ments, one with k � 6.06 for the smallest measuredsizes �r � 1.5 �m, R2 � 0.982� and the other withexponent k � 1.94 �R2 � 0.985� for particles with r �

Fig. 7. Influence of the index of refraction �of the components� onthe backscattering ratio in the 2C model �abscissa, the real part ofthe small component’s index of refraction; parameter, the large Bcomponent’s index of refraction� and comparison with the Jungemodel. Calculated values at � � 532 nm for the parameters cor-responding symbols, to #15 PSD and curves, to the B-spline fit.

Fig. 8. Influence of the � parameters on the backscattering ratioin the 2C model. B versus �A �the parameter is �B�. Symbols,calculated values and, curves, the corresponding fit with theB-spline for CA:CB � 2 � 1016, � � 532 nm, and real indices ofrefraction, nA � nB � 1.05.

Fig. 9. Spectral dependence of the backscattering ratio B for therepresentative case 2 water samples calculated from the Junge andthe 2C model. Symbols, calculated values and, curves, corre-sponding fit with the B-spline function.


1.5 �m. This situation is depicted in the inset inFig. 10 where the extrapolations to submicrometersizes for both cases are presented. We can see that,if the two-segment hyperbolic model is used, the ex-trapolation from the �small-size� segment with expo-nent k � 6.06 results in a tremendous overestimationof small particles and consequently to absurd valuesof the calculated scattering and backscattering coef-ficients and ratio, even if the lower limit of integra-tion is set to 0.1 �m. Hence we have accomplishedcalculations with the original �single-segment� Jungefit with k � 2.11.

The results of 2C and hyperbolic model calculationsalong with the corresponding measured values ofB��, b��, and bb�� are presented in Fig. 11. Thecalculations were done at matching Hydroscat-6 andAC-9 wavelengths used in the measurements. Theresults with the 2C model are in far better agreementwith the measured values than results from theJunge model. Although providing acceptable re-sults for the backscattering ratio, the Junge modelpredicts values of scattering and backscattering coef-ficients that are significantly different from the mea-sured values in the whole spectral range �bb, almostan order of magnitude!�.

B. Case 1 Water

The dependence of the backscattering ratio B on thelower limit of integration for both considered PSDmodels fitted to the measured data from Fig. 2b �oce-anic, case 1 water� are shown in Fig. 12. The upperlimit of integration was fixed at rmax � 50.0 �m, andthe lower limit was varied in the range of 5 � 10�3

�m � rmin � 1 �m. The calculation was accom-plished at � � 532 nm and the relative index of re-fraction, m � 1.05 � 0i.

The behavior of the considered models is similar tothat obtained for case 2 water: The hyperbolic distri-

bution is more sensitive to the lower limit of integra-tion, and obtaining a limit-independent B inclusion ofvery small particles is necessary. Some other as-pects of more general nature discussed in subsection4, A apply also for case 1 waters.

Fig. 10. Symbols, Measured particle-size distribution from theWashington East Sound at a depth of 2 m; curves, corresponding fitwith the Junge and the 2C model. Junge distribution: k � 2.11,R2 � 0.685; 2C model: �A � 0.130, �B � 0.159, CA:CB � 2.132 �1016, R2 � 0.987. Inset: extrapolations to the submicrometerparticle-size range from the hyperbolic fit �k � 2.11�, segmentedhyperbolic fit �small segment, k � 6.06�, and the 2C model.

Fig. 11. Spectral dependence of the measured �Hydroscat-6 andAC-9, East Sound, 2 m� and b, calculated scattering coefficient; bb,backscattering coefficient; B, backscattering ratio. a, Comparisonof measured values with results from the 2C model for nA � 1.085and nB � 1.03. b, Comparison of measured values with resultsfrom the Junge model for n � 1.05. Measured values are repre-sented by larger, symbols while the calculated values are repre-sented by small solid symbols and lines corresponding to theB-spline fit. The measured and calculated scattering parameterscorrespond to the PSD shown in Fig. 10.

Fig. 12. Dependence of the backscattering ratio B on the lowerlimit of integration for the hyperbolic and the 2C model of the PSDcorresponding 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).


The backscattering ratios calculated from the hy-perbolic model at � � 532 nm and for m � 1.05 � 0ifor the considered PSDs are BJ�NA#2� � 1.2 � 10�2

and BJ�NP#4� � 1.3 � 10�2. If the same index ofrefraction is used for both components of the 2Cmodel, substantially lower values for B are obtained,B2C�NA#2� � 3.6 � 10�3 and B2C�NP#4� � 6.9 �10�3. The calculated spectral dependence B�� isshown in Fig. 13. The selected wavelengths corre-spond to those usually used in AC-9 and Hydroscatinstruments.

The values for B, b, and bb calculated from the 2Cmodel for North Pacific surface sample #4 �the corre-sponding differential Junge PSD exponent, k � 3.97�at 532 nm and for mA � mB � 1.05 � 0i are B � 6.9 �10�3, b � 0.37 m�1, and bb � 2.6 � 10�3 m�1. Thesevalues calculated from the 2C model are similar tothe values of B, b, and bb measured at 530 nm for case1 water in the Gulf of California39 for a similar PSD�an estimated corresponding differential Junge PSDexponent, k � 3.9, and estimated n � 1.05�: B � 7 �10�3, b � 0.26 m�1, and bb � 1.9 � 10�3 m�1. Thecorresponding values obtained from the Junge modelare B � 1.3 � 10�2, b � 9.6 � 10�2 m�1, and bb �1.3 � 10�3 m�1.

The representative value of B for an oceanic envi-ronment between oligotrophic and eutrophic, comply-ing with Morel’s model of case 1 waters,43 can betaken as �0.01.11 However, this model also esti-mates bb�b as a function of chlorophyll concentrationand predicts values higher than 0.01 with chlorophyllconcentrations typical of oligotrophic case 1 water.Similar �higher� B values are found in Petzold’sTongue of the Ocean data �B � 1.9–2.3 � 10�2 at 530nm�.41 Such results are obtained from the 2C modelif one assumes a somewhat higher index of refractionfor the small component �Fig. 7� or higher abundanceof small particles �larger �A and�or CA, Fig. 8�.

Hence the results of calculations and predictionsobtained from the 2C model could be considered as

adequate and compliant with measurements of case 1water. The source of differences in predictions fromthese two PSD models comes obviously from the over-estimated number of submicrometer particles thatoccurs in extrapolation from the hyperbolic PSDmodel.

5. Conclusions

1. Small �submicrometer� particles significantlycontribute to the backscattering ratio.

2. For calculation of the backscattering ratio amuch broader PSD is needed than is usually mea-sured. Hence extrapolation from the fit of the mea-sured data is necessary.

3. Results of extrapolation critically depends onthe chosen PSD model.

4. A hyperbolic-type PSD largely overestimatesthe number of small �and also large� particles.

5. Calculation of the backscattering ratio with ahyperbolic-type PSD model requires inclusion of verysmall particles ��0.1 �m�. Consequently the cal-culated backscattering ratio could be much higherthan the real value corresponding to the measuredPSD.

6. The 2C model provides an adequate model ofthe PSD that assures a good fit even in the broadestmeasurement size ranges and facilitates reasonableextrapolations.

7. The backscattering ratio and the values of thescattering coefficients calculated with the 2C modelare in agreement with the measured and the ex-pected values.

8. For case 2 water it seems that B is only weaklysensitive to a change in wavelength �at least in theconsidered spectral range�. This result is consistentwith earlier observations.9 However, a more de-tailed analysis of the spectral behavior of the back-scattering ratio and the total scattering coefficient aswell as a sensitivity to change in the parameter val-ues of the 2C model, which are beyond the scope ofthis study, are subject of further investigation.

Support from the Croatian Ministry of Science andTechnology, grant 00980303, is acknowledged.

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