JOURNAL
VOL. 79, NO. 27
The Determination
OF
GEOPHYSICAL
of the Index of Refraction
RESEARCH
Distribution
SEPTEMBER
20, 1974
of Oceanic Particulates
J. RONALD V. ZANEVELD, DAVID M. ROACH, AND HASONG PAK
Schoolof Oceanography,OregonState University, Corvallis, Oregon 97331
A method is describedfor determiningthe index of refraction distribution and the particle size distribution of suspendedparticles.The distributionsare obtained by breakingdown an observedvolume
scatteringfunction into its contributingcomponents.The componentscatteringfunctionsare calculated
usingMie theory. The componentfunctionsincludeall sizedistributionsand indicesof refractionthat can
be expectedto be present.The method was applied to a volume scatteringfunction observedin the
SargassoSea. Forty componentswere usedwith five differentindicesof refractionand eightdilYerentparticle size distributions.The resultant index of refraction distribution was bimodal. Components with indicesof 1.05 and 1.15 dominate the calculatedvolume scatteringfunction. The calculatedparticle sizedistribution falls within experimentally determined limits for the size distribution.
The parametersthat determinethe light scatteringand attenuationpropertiesof a particle suspendedin the oceanare
Calculation time increasesdramatically for multilayered
spheres,however.
shape,size,and internaldistributionof the indexof refracFor a collectionof spheres
onecandefinea particlesizedistion. Lightscattering
propertiesof theindividualparticlesare tributionf(D) dD giving the numberof particlesper unit
difficultto measure.Techniquesfor studyinglargenumbersof volume with diametersbetweenD and D +dD. For nonmoredifficult.
particles simultaneouslyhave thus been developed as a sphericalparticlesthe definitionof sizebecomes
necessity.Optical measurementsare particularly well suited Sinceparticlesizedistributionsare usuallyobtainedwith an
for the study of particles in the ocean, sincethere is no destructiveinteractionbetweenthe particlesand the light. Light
scatteringby oceanicsuspensions
is usuallydescribedin terms
of the volume scatteringfunction/•(0) defined by
-
dI(O)
counter. In our theoreticalconsiderationsof light scatteringit
E dV
wheredl(O)is the radiant intensityscatteredat an angle0 with
respectto the incidentbeamfrom the volumeelementdV when
it is illuminatedby a monochromaticirradianceE. Exceptfor
the extremelynear-forwardangles(0 < 1o) the volumescattering functioncan readily be determined.Determinationof the
particle characteristics
from the volume scatteringfunctionis
extremelydifficult owing to the great variety of shapes,sizes,
and indicesof refraction. It is necessaryto make severalsimplifying assumptionsif any particle propertiesare to be calculated from observedoptical parameters.
The theoretical
link between observed and calculated
instrument that measuresthe volume of a particle, the diam-
eterof a nonspherical
particlewill be takento be the diameter
of a sphericalparticlewith the samevolumeas the observed
particle. The instrumentmost commonly used for measurementsof the particle size distribution is the Coulter
scat-
is not necessary
to know the sizedistributionbeforehand.
The index of refraction of the suspendedparticlesis still
largelyunknown,althoughgreatprogress
in thisareahasbeen
made in recentyears.The indexof refractionof particlesmay
become a valuable indicator of the nature of the suspended
particles.Alreadya qualitativecorrelationhasbeenfoundbetweendynamicfeaturesin the oceanandthe distributionof the
index of refraction [Pak and Zaneveld, 1973].
Even if a givensuspended
particleis homogeneous
and can
be characterized
by one indexof refraction,a sizedistribution
of particlescan only be characterizedcompletelyby a twodimensional index,0f refraction distribution, since for each size
tering functionsis providedby Mie's [1908] theory. By using
Mie's theory it is possibleto calculatethe light scatteringcharacteristicsof a singlesphericalparticlewith a uniform index of
refraction.The light scatteringcharacteristics
of a collectionof
spherescan then be obtainedby summingthe contributionsof
the individual spheresprovided they are separatedby a distance of about three radii or more [Van de Hulst, 1957]. The
scatteringcalcutationshave been extended to include other
simpleshapesand layeredspheres.It hasbeenshown,however
[Holland and Gagne, 1970; Plass and Kattawar, 1971], that
when one considersany sizedistributionof randomly oriented
particles,the samesize distributionof sphericalparticleswill
give comparablevolume scatteringfunctions,particularly in
the forward hemisphere,when the other particle parameters
of particlethereexistsa distributionof indicesof refraction.It
are similar.
results of the various authors and shows that the average in-
is this two-dimensional distribution that one must ultimately
obtain in order to fully understandlight scatteringby particulate matter in the ocean.The presentpaper is a steptoward
obtainingmethodsfor the determinationof sucha distribution.
Earlier investigations[Sasakiet al., 1960;Kullenberg,1972;
Gordon and Brown, 1972; Zaneveld and Pak, 1973; Carder et
al., 1972;Morel, 1973a]have dealt with the determinationof
onevalueof the indexof refracti'onthat wouldbe representative of the entirecollectionof particles.The value of the index
of refraction chosenis the value that best reproducesan observedsetof light scatteringcharacteristics,
givenan observed
or postulatedparticlesizedistribution.Table 1 summarizes
the
Nonuniformity of the index of refractionwithin a particle dicesof refractioncalculatedby variousauthorsfall either in
will causethe light scatteringcharacteristics
to differ markedly the range1.15-1.25or 1.01-1.05.This resultindicatesthat parfrom thosefor a particlewith a uniform index [Mueller, 1973]. ticle distributions are either dominated by inorganic and
organic
skeletal
material
witha highindexof refraction
or by
Copyright ¸
1974 by the American GeophysicalUnion.
the watery protoplasmof organic matter with a low index.
4091
4092
lANEVELD ET AL.: OCEANIC PARTICULATE
This view is supportedby the fact that low index dominated and organic material. We then have
particulateswere mostly observedin regionsof high productivity.
•(0) =
,4,]1(D)•(0, D, m,) dD
It has usually been assumedthat a low index of refraction is
associatedwith particulatesof organicorigin and a high index
of refraction with an inorganicorigin. This assumptionhasnot
qA2]2(D)•( O, D, m2) dD
been thoroughly tested. Organic particles such as
phytoplanktonfrequentlycontainskeletalmaterialwith a high and
index of refraction, so that assigningextremely low indices
(1.01-1.02) to organic matter is probably incorrect.
J(D) dD = f•(D) dD + A(D) dD
11
(3)
where subscripts1 refer to organicmatter and subscripts2
referto inorganicmatter.One may vary the indicesm• and m•.
If a volume elementcontainsN particles,each with a scat- as well as the limits D•u, D•, D•,, and D•.• until again the extering function fiE(O),then the scattering function for the perimentalcurvemost nearly matchesthe observedone. In the
pastthis has beendone by trial and error; a more deterministic
volume element is given by
approach will be presentedin this paper.
It is natural to try to expand the model to contain a large
N
number
of components.We can group particlesaccordingto
•(0) = •i=1 •,(0)
(1)
size distribution and index of refraction given
THEORETICAL CONSIDERATIONS
The problemto whichwe directourselvesis, given/?(0)(but
not necessarily
the sizedistributionof the N particles),to obtain maximum information concerningthe indicesof refraction of the N particlesand the sizedistributionof the particles
if it is not known. If we supposethat the indexof refractionof
all the particles is the same, then
(4)
and
N
](D) dD = •
i=1
M
• A•if,(D)dD
i=1
(5)
In many casesthe total sizedistributionfiD)dD is not known,
in which case(5) is not a condition that needsto be satisfied.
•(0) =
A/(D)•(O, D, m) dD
(2)
As onlyfiD) dD and/?(0)or/?(0) alone are known, it is clearly
impossibleto determinetheft(D) dD without a number of apwherefiD) dD is the normalizedsizedistribution (one particle proximations, and even then one cannot be assuredof ohper unit volume) of the particles,A is the total number of taininguniquesolutions,sincethe shapeof the volumescatterparticlesper unitvolume,m is the index of refraction of the par- ing function is-only a weak function of the size distribution
ticles relative to water, and fl(0, D, m) is the scatteringfunc- and index of refraction.
The integral
tion for a particle with diameter D and relative index of refraction m. The D, and Du are the smallestand largestdiameters,
respectively,of particlesthat are presentor from which a con(6)
I, i =
],(Dffi(O, D, mi) dD
tribution to the scatteringfunction is expected.The fl(0, D, m)
functionsare obtainedfrom Mie theory. By performingthe integration for severalvalues of the index of refraction relative has been shown to be [Morel, 1973b]a smooth function of C
to water m, one may choosethat value of m for which the ex- for the range of valuesexpectedin the oceanwhenJ•(D) dD is
perimental volume scatteringfunction fl(0) most closely ap- Junge-typedistributionJ•(D) = ND -cdD. Similarly, I o is a
proximatesthe theoreticalone. This approach is the classical smooth although possiblydouble-valuedfunction of mi. It is
one for which the resultsare shown in Table 1. An improve- thus reasonableto assumethat a limiti•d number of theseIo
ment over assumingthat all particleshave the same index of componentscan be used to reconstru,t observedscattering
refraction is to postulatethat the particlesconsistof inorganic functions that are also reasonably smooth. We may then
fD
•u
1
fD
D•u
choosethe appropriatem• andJ•(D) dD in (4). The valuesof m•
TABLE 1.
Calculated
Relative
Indices
Particulate
or
Material
Japan Trench
600 m
1500 m
2650 m
3000 m
Sargasso Sea
Mediterranean
Baltic Sea
Phytoplankton
Sargasso Sea
Eastern equatorial
Index
Relative
of
Refraction
1.25
1.25
1.25
1.20
1.20
1.1S to 1.20
1.04 to 1.0S
Pacific
of Suspended
Matter
Calculated
Location
of Refraction
1.02 to 1.04
1.0S - 0.01i
1.02 to 1.05
Reference
Sasaki et al. [1960]
Sasaki et al. [1960]
Sasaki et al. [1960]
Sasaki et al. [1960]
Kullenberg [1972]
Kullenberg [1972]
Kullenberg [1972]
Carder et al. [1972]
Gordon and Brown [1972]
laneveld and Pak [1973]
Sargasso Sea
Diameter of 52.S v
Diameter
of
>2.S
V
Numerous areas
Tongue of the ocean
1.01 - 0.01i
Brown and Gordon [1973a]
1.1S
1.03 to 1.0S
1.1S to 1.20
Morel [1973b]
Brown and Gordon [1975b]
mustspanthe rangeof indicesthat oneexpects
to find;this
range is 1.00-1.20 relative to water. Ideally, one would constructtheft(D) to be very narrow distributions,sinceeachsize
range has its own index of refraction distribution; N would
then be the number of size steps.Anothe• lessgeneral but
more tractable approach is to let theJ•(D) be Junge-typedistributions.The limits of integrationalsoinfluencethe shapeof
the scatteringfunction; the summationin (4) thus may include
the same size distribution but with various upper and lower
limits of integration.
When the particle size distributions appropriate to the
problem under considerationhave been chosenas well as the
indicesm•, one can then determinethe index of refraction distribution by minimizing the difference between the experimentally observedvolume scatteringfunction and the sum
ZANEVELD
ETAL.'OCEANIC
PARTICULATE
4093
of the chosencomponents.That is, if the observedvolume fore not imposed.The programwasfreeto choosetheAo that
scatteringfunctionis givenby/•e(O)and L is the numberof minimizethe squareof the differencebetweenthe observed
data points, the quantity
L
k--1
ß
A,ii,(D)/•(Ok, D, mi)dD -- •Se(Ok)
(7)
is minimizedby adjustingthe coe•cients Ao. The technique
employedis a rapidlyconvergingdescentmethoddescribedby
Fletcher and Powell [1963]. The determination of the
coe•cients A o then constitutesa determinationof the indexof
refractiondistributionas well as the particlesizedistribution
by means of (5).
APPLICATION
A volumescatteringfunctionmeasuredby Kullenberg[1968]
in the SargassoSea at a wavelengthof 632.8 nm has been
analyzed extensivelyby Gordonand Brown [1972] and Brown
and Gordon[1973a]. In the earlier paper theseauthors concluded that the best fit (averageerror of about 20%) to the
scatteringfunction was obtained by a distribution of
•D)
dD = 3.3 X 10• D -• dD
and an index of refraction of m = 1.05 - 0.01•. The size limits
consideredwere0.08 and 10 u. In the later paperthe authors
concludedthat a bestfit (averageerror of about 19%)wasobtained
and calculatedvolume scatteringfunctions.The program not
only determinesthe index of refraction distribution but also
the resultant particle size distribution. The resultant particle
size distribution and its componentsare shown in Figure 1.
The componentsas well as the result of the 40-componentfit
using the Fletcher-Powellminimization technique are shown
in Table 2. Although all 40 componentswere found to be present, only 7 contribute significantly to the calculated volume
scatteringfunction. These are componentswith a relative index of refraction of 1.05 and exponentsof 3.5, 3.7, and 3.9,
componentswith an index of 1.075 with exponentsof 3.7 and
3.9, and componentswith an index of 1.15 with exponentsof
3.5 and 3.7. Componentswith indicesof 1.02 and 1.10 were essentially not present. The calculated and observedvolume
scattering functions and the componentsof the calculated
scatteringfunction are shown in Figure 2. The averageerror
was only 13.4%, and the maximum error was 28.3%. The experimental error was estimatedto be 30% by Kullenberg.
The major contribution to the calculatedvolume scattering
function is due to the particleswith an index of refraction of
1.15, especiallyat the larger angles.At the smaller anglesthe
contribution of the 1.05 component rapidly increases.The
1.075componentcontributesbut little at any angle,and ignoring this component would not greatly affect the calculated
scattering function. The most striking result is that so few
componentsare chosen by the minimization program, es-
with
•D)
,o8I
dD = 4.83 X 10• D -• dD
The indices of refraction
used were 1.01 - 0.01i from 0.1 to
2.5 • and 1.15from 2.5 to 10.0•. Kullenberg'sscatteringfunction thusappearedidealto testour method,sinceit wouldpermit comparisonwith an earlier determination of an index of
refraction
distribution.
The Mie scatteringcurveswere obtained from an excellent
set of tablescomputedby Morel [1973a].The tablescontain
volume scattering functions for Junge distributions with
parameters C of 3.1, 3.3, 3.5, 3.7, 3.9, 4.2, 4.5, and 5.0 and for
indicesof refraction relative to water of 1.02, 1.05, 1.075, 1.10,
and 1.15. The integrationsin the table are carried out between
limits of a = 0.2 and a = 200, where a = 2=D/X. These values
representdiametersof D• = 0.04 um and Du = 40 um at the
f(D)
wavelengthunderconsideration.Although thesetablesdo not
permit us to vary the limits of integration (which do have a
slight influenceon the shapeof the scatteringfunction) and
also do not contain imaginary componentsof the index of
refraction,the choiceof scatteringfunctionsstill permitsan
excellentfit to the experimental distribution.
The calculatedscatteringfunction was thus given by
//50
/050
•c(O)
= • '• Aii13•.i(O
)
i=l
1075
i=l
10-4
i
i
1
i
001
Ol
io
io
where /•o(0) representsa tabulatedvolume scatteringfuncD
tion. The/•o were normalizedto 104particles/mi.
In our minimizationapproachfor this particularproblem,
Fig. 1. The calculatedtotal particle size distribution(uppermost
no conditionswere imposedon the sizedistributions.We did
curve) for the scatteringfunction shownin Figure 2 and the particle
not assumethat a size distributionfor the particulatematter size distributions for the componentswith indices of refraction of
was known. A conditionsuchas that shownin (5) wasthere- 1.050, 1.075, and 1.150.
4094
ZANEVELD
ET AL.:OCEANICPARTICULATE
TABLE 2.
The Values of the Coefficients
of the 40 Components Providing the Best Fit
Scattering
Function Observed in the Sargasso Sea
Relative
Exponent of
Junge Distribution
1.02
3.1
3.3
3.5
3.7
3.9
4.2
4.5
5.0
1.64
5.83
1.39
7.74
2.60
1.80
x
x
x
x
x
x
1.79 x
4.58 x
Index
1.05
10-7
10-7
10-16
10 -21
10-14
10-12
10-11
10-24
1.45
1.30
4.41
6.30
4.98
1.42
1.68
7.02
x
x
x
x
x
x
x
x
of
Refraction
1.075
10-7
10 -7
10 -2
10-2
10-3
10-9
10-14
10-13
4.92
1.65
2.88
6.02
3.06
9.75
7.34
3.30
x
x
x
x
x
x
x
x
to a Volume
1.10
10-7
10 -12
10 -10
10 -3
10-2
10-13
10-8
10-12
2.92
1.69
4.06
3.49
2.31
1.28
4.17
3.12
x
x
x
x
x
x
x
x
1.15
10-13
10-10
10 -9
10 -12
10-20
10-14
10-8
10-11
9.33
1.43
1.10
6.27
2.13
6.54
3.81
8.70
x
x
x
x
x
x
x
x
10-9
10-8
10 -1
10 -2
10-8
10-9
10-11
10-10
The coefficients are normalized to 104 particles/ml greater than 1 • in diameter.
peciaily since the shapesof the componentscatteringfunc
tions do not changevery much when one of the parametersis
varied.
Since the slope of the Junge distribution is different for the
componentswith indicesof refraction of 1.05, 1.075, and 1.15,
the percentagesof the total representedby a given index of
The presentresultsclearly indicate a bimodal distribution,
with a 1.15 index of refractioncomponentand a 1.05-1.075 index of refraction component; all otherøindices are unimportant. This result is not unexpected,since previous calculations
summarized
in Table
I showed a similar distribution.
One interpretation of this observation is that the organic
refractionare a functionof size.Thesepercentages
are shown matter present had an index of around 1.050 and the inin Figure 3, which indicatesthat the high-indexparticlescon- organic and organic skeletal material had an index of 1.15.
stitutea larger percentageof the total with increasingparticle
Our average error was only about 13% as compared with
size. The total particle size distribution had an exponent of 19% for the model of Brown and Gordon [1973a]. Their conabout 3.6, which falls in the range(3.4-4.1) observedby Gor- tention that the large particles must consist mostly of indon and Brown [1972] for the particulate matter in the surface organic material is supported by our results. Attaching
water of the SargassoSea. The amount of particles per physicalsignificanceto a bestfit must be done with great care,
milliliter larger than I/• was found to be 3.2 X 103.This quan- however, sincethe many independentparametersinvolved in
tity is a factor of 4 lessthan the minimum observedby Gordon the determinationsof volume scatteringfunctionscan be comand Brown [1972]. As the scatteringfunction and the particle bined in severalways to obtain calculatedscatteringfunctions
size distribution
were measured neither at the same time nor at
the samelocation,this resultliesentirely within expectedfluctuations in the particle concentration.
that are similar
to observed
ones.
As our calculationsindicatethat almostno particleswith an
index of 1.02 were present,the choiceof an indexof refraction
of around 1.01 for organic material is probably too low. Furthermore, organic matter containsskeletalmaterial that has an
index of refraction similar to that of inorganic matter. Our
results show that the index of refraction
for skeletal and in-
organic material must be about 1.15 or higher. It is thus likely
that a pure plankton culture would show a bimodal index of
70--
io 2
•3(e)
•%
,o
-3
so
t t
//50
50
4O
Ii
• //50
%
/050
30
20
/ 050
/075
io
10-6/
0
,1
I
20
I
I
40
I
I
60
I
I
80
J
1075
o o•
Fig. 2. The experimental
volumescatteringfunctionobservedby
Kullenberg(dots), the calculated40-componentbestfit (uppermost
curve) to the experimentalcurve, and the contributionsto the total
scatteringfunction by the 1.050, 1.075, and 1.150 index of refraction
components.
o•
•o
•o
•oo
D
Fig. 3. The percentageof the total numberof particlesas a function of particle diameter in microns with the componentindex of
refraction as a parameter.
ZANEVELD
ETAL.:OCEANIC
PARTICULATE
4095
refraction distribution, most material (watery protoplasm) the sizedistribution of the suspendedparticles. If the sizedishaving an index of refraction between 1.02 and 1.05 and a tribution has also been determined experimentally, the
smaller fraction of the material having an index of refraction difference between the calculated and the observed size disaround
1.15.
tribution may also be minimized.
The method was applied to a volume scatteringfunction obDISCUSSION AND CONCLUSIONS
servedin the SargassoSea by using40 componentscovering
Although the principalpurposeof this paperis to presenta the most likely indices of refraction and particle size dismethod for determiningindex of refraction distributions,the tributions. The method was successful in predicting a
bimodal index of refraction distribution deserves some compreviously observedparticle size distribution and index of re.
The resultant index of refraction
disment. Phytoplankton consist of hard calcareousor siliceous fraction distribution.
shellsand soft watery protoplasm.The hard shellsprobably tribution was bimodal, with a component in the range
have a relative index of refractionin the neighborhoodof 1.15, 1.05-1.075 and a componentof 1.15.The total particle sizediswhereas the protoplasm has a relative index of refraction of tributions had an exponent of 3.6 in the Junge distribution.
1.01-1.05. If phytoplanktonare present,one might expectto
Acknowledgment. This investigation has been supported by the
find particleswith an index of refraction that is somevalue be-
tween the indices for hard shells and protoplasm. Optical
propertiesdo not, however,averagein a linear fashion.A par-
Office of Naval Research through contract N00014-67-A-0369-0007
under project NR 083-102 with Oregon State University.
REFERENCES
ticle that consists for 50% of material with an index of 1.04 and
for 50% of material
with an index of 1.10 does not have a
scatteringfunction like that of a particlewith an index of 1.07.
According to our present resultssuch a particle appearsto
scatter more like two particles, one with an index of 1.04 and
one with an indexof 1.10.This hypothesiscan be testedby examination of the scatteringfunctionsof phytoplanktoncultures and the determination
of their index of refraction
dis-
tributions.
The observationthat mostof the larger particlesappearto
be inorganicor consistof phytoplanktonshellsis contraryto
expectation.Yet the sameresult was obtainedby Brownand
Gordon[1973a], usinga different method. These authorshave
also soughtto explainthe observation.Of all the scattering
functionsanalyzedby using our method, the SargassoSea
sample is the only one in which the inorganicsdominate the
large size ranges. No general conclusionsconcerningparticulate matter in the ocean should thus be drawn from the
analysisof the SargassoSeasample.The SargassoSeasample
provides, however, a useful test of our method, since an unusual observation
was confirmed.
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1972.
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en vue de l'applicationaux particulesmarines,Rapp.10, Lab. d'Ocfianogr.Phys.,Univ. de Paris,
The choice of component particle size distributions is
governedby many factors.Ideally, one would choosea large
Centre Nat. de la Rech. Sci., Paris, 1973a.
number of very narrow monodispersesizedistributions.OwMorel, A., Diffusion de la lumi•re par les eaux de met, Resultatsexing to the computation time involved one is forced to search
perimentauxet approchethfiorique,Optics of the Sea, Lect. Ser. 61,
for alternativebut reasonablesize distributions.Very dense
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Brussels, 1973b.
populations of organic material, such as can be found in a
J. L., Theinfluence
ofphytoplankton
onocean
colorspectra,,
plankton bloom, displaya distributionthat is normal or skew Mueller,
Ph.D. thesis,Oregon State Univ., Corvallis, i973.
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population will lead to erroneous results.
It has not been possiblewith present instrumentation to
separate the organic from the inorganic fraction before a
reliable particle count is made. If as is the case with the
SargassoSea example the total particle size distribution is a
Junge type, it is thus reasonable to assume that the components also have this type of distribution.
By meansof the Fletcher-Powellminimizationtechniqueit
is possible to determine the best fit to an observed volume
scatteringfunction by using a large number of component
scatteringfunctions.By judiciouslychoosingthe component
scatteringfunctionsone can interpret the fit in termsof the index of refractiondistributionof the suspendedparticlesand
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Wiley, New York, 1957.
Zaneveld, J. R. V., and H. Pak, Method for the determination of the
indexof refractionof particlessuspended
in the ocean,J. Opt. Soc.
Amer., 63, 321-324, 1973.
(Received March 14, 1974;
acceptedApril 14, 1974.)