Estimation of the Inherent Optical Properties of Natural Waters from the Irradiance Attenuation...

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Estimation of the inherent optical properties of naturalwaters from the irradiance attenuation coefficient andreflectance in the presence of Raman scattering

Hubert Loisel and Dariusz Stramski

By means of radiative transfer simulations we developed a model for estimating the absorption a, thescattering b, and the backscattering bb coefficients in the upper ocean from irradiance reflectance justbeneath the sea surface, R~02!, and the average attenuation coefficient for downwelling irradiance, ^Kd&1,between the surface and the first attenuation depth. The model accounts for Raman scattering by water,and it does not require any assumption about the spectral shapes of a, b, and bb. The best estimationsare obtained for a and bb in the blue and green spectral regions, where errors of a few percent to ,10%are expected over a broad range of chlorophyll concentration in water. The model is useful for satelliteocean color applications because the model input, R~02! and ^Kd&1, can be retrieved from remote sensingand the model output, a and bb, is the major determinant of remote-sensing reflectance. © 2000 OpticalSociety of America

OCIS codes: 010.4450, 030.5620, 280.0280, 290.5860.

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1. Introduction

The propagation of light within the ocean is describedby a radiative transfer equation ~RTE!. The mostcommon form of RTE describes the change of radi-ance with depth, which is caused by scattering andabsorption of light by seawater ~see, e.g., Ref. 1!.The various constituents of seawater that are opti-cally significant include water molecules, particles,dissolved substances, and air bubbles. The pro-cesses of light absorption and scattering by theseconstituents are described by the inherent opticalproperties ~IOP’s!. The most fundamental IOP’s arethe spectral absorption coefficient a~l, z! and thepectral volume scattering function b~l, c, z!, where

l is the wavelength of light, c is the scattering angle,and z is the water depth. Three other importantOP’s, i.e., the backscattering coefficient bb~l, z!, theotal scattering coefficient b~l, z!, and the beam at-enuation coefficient c~l, z! 5 a~l, z! 1 b~l, z!, can be

The authors are with the Marine Physical Laboratory, ScrippsInstitution of Oceanography, University of California, San Diego,La Jolla, California 92093-0238. The e-mail address for H. Loiselis [email protected].

Received 12 August 1999; revised manuscript received 3 Febru-ary 2000.

0003-6935y00y183001-11$15.00y0© 2000 Optical Society of America

erived from a~l, z! and b~l, c, z!. Because all op-tical quantities are functions of wavelength l anddepth z, for simplicity these arguments are omittedfrom our notation unless they are specifically re-quired ~see Table 1 for definitions and units!.

Until recently, beam attenuation was the only IOProutinely measured in situ, and these measurementswere usually limited to a single wavelength ~see, e.g.,Ref. 2!. Therefore significant efforts have been de-voted to developing models for estimating a, b, and bbfrom radiometric measurements of the light-fieldcharacteristics. The models that are of direct inter-est to this study are those that do not involve anyassumptions about the IOP’s and are based on ap-parent optical properties ~AOP’s! such as the diffuseattenuation coefficient for downwelling irradiance,Kd, and irradiance reflectance, R.3–6 Kd @52d~lnEd!ydz!# and R ~5EuyEd! can be obtained from in situ

easurements of downwelling plane irradiance Edand upwelling plane irradiance Eu. Recent develop-ment of new commercial instruments, such as theac-9 and the Hydroscat-6, has provided capabilitiesfor routine in situ measurements of a, c, and bb ateveral wavelengths.7–9 However, because of poten-

tial errors in these measurements10 and because theOP’s are generally easier to measure with good ac-uracy than the IOP’s, there is still a need for modelshat relate IOP’s to AOP’s. Such models can be usedot only for estimating the IOP’s that are missing or

20 June 2000 y Vol. 39, No. 18 y APPLIED OPTICS 3001

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Table 1. Notation Used in This Paper

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not measured directly but also to verify the quality ofthe IOP measurements and consistency between themeasured IOP’s and the measured light-field charac-teristics. In addition, and importantly, the modelscan be used to retrieve the IOP’s from AOP’s that areestimated from remote sensing of ocean color. Inthis application, both AOP’s of interest to us, Kd andR, are determined from algorithms based on water-leaving radiances.

The AOP’s depend on the IOP’s and also, in con-trast to the IOP’s, on the ambient light field.11 Twoapproaches to estimating the IOP’s from AOP’s orfrom measurements of the light-field characteristicsare currently in use. In the first approach, the re-lationships between the IOP’s and AOP’s are devel-oped through extensive radiative transfersimulations, which ideally represent a broad range ofIOP’s and boundary conditions necessary for solutionof the RTE. One of the most common results ofthese simulations were the relationships that permit

Symbol Definition Unit

a Absorption coefficient m21

b~c! Volume scattering function m21 sr21

b~c! Scattering phase function@5b~c!yb#

sr21

bw~c! Molecular-scattering phasefunction @5bw~c!ybw#

sr21

bp~c! Particle-scattering phase func-tion @5bp~c!ybp#

sr21

b Scattering coefficient m21

bw Molecular-scattering coefficient m21

bp Particle-scattering coefficient m21

c Beam attenuation coefficient~5a 1 b!

m21

bb Backscattering coefficient m21

h Ratio of molecular scattering tototal scattering ~5bwyb!

v0 Single-scattering albedo ~5byc!u0 Solar zenith angle in airmw Cosine of solar zenith angle in

waterl Wavelength of light in vacuum nmta~550! Aerosol optical thickness at 550

nmChl Chlorophyll concentration mg m23

z Geometrical depth mz1 First attenuation depth mEu Upwelling irradiance on a hori-

zontal surfaceW m22

Ed Downwelling irradiance on ahorizontal surface

W m22

Kd Vertical attenuation coefficientfor Ed ~52d ln Edydz!

m21

^Kd&1 Average Kd within the layerbetween the surface and z1

m21

R Irradiance reflectance ~5EuyEd!DR Contribution to irradiance re-

flectance associated with Ra-man scattering

Re Irradiance reflectance in theabsence of Raman scattering

002 APPLIED OPTICS y Vol. 39, No. 18 y 20 June 2000

the estimation of IOP’s from measurements of Eu andEd.3,4,6,12–14 An important advantage of this ap-proach is that, once the relationships are established,their subsequent use to calculate the IOP’s is a simpletask.

The second approach involves an iterative processof solving the RTE.15,16 For given boundary condi-tions, the IOP’s are varied at each iteration until thecalculated downwelling and upwelling irradiances ~ordownwelling irradiance and upwelling radiance!match their measured counterparts. The search forthe IOP’s by this method requires measurements ofthe depth profiles of the radiometric quantities, and itis computationally much more demanding than theuse of modeled relationships obtained with the firstapproach.

In this paper we apply the first approach with thegoal of developing a simple model to estimate a~l!,~l!, and bb~l! within the upper ocean from in situeasurements or remote-sensing estimates of two

asic AOP’s, the irradiance reflectance just beneathhe sea surface, R~l, 02! 5 Eu~l, 02!yEd~l, 02!, and

the average irradiance attenuation coefficient withinthe upper layer, ^Kd~l!&1 5 1yz1, where 02 indicatesthe depth just beneath the sea surface and z1 is thefirst attenuation depth at which the downwelling ir-radiance is reduced to 37% of its surface value. Thedevelopment of the model is based on R~l, 02! and^Kd~l!&1 taken directly from the output of our radia-tive transfer simulations. An important feature ofthe model is that both R~l, 02! and ^Kd~l!&1 can bestimated from satellite measurements of oceanolor.

The relationship between R~l, 02! and theatellite-derived water-leaving radiance, Lw~l, 01!,

where z 5 01 is just above the sea surface, is wellunderstood ~see, e.g., Ref. 1!. The estimation of R~l,

2! from Lw~l, 01! involves the propagation of Lw~l,1! through the air–water interface to obtain in-

water upwelling radiance, Lu~l, 02!, and then thealculation of Eu~l, 02! by use of Q factor, which is

Eu~l, 02!yLu~l, 02!. The radiances Lw~l, 01! andLu~l, 02! are directional quantities determined forthe satellite viewing angle, but the angle argumentsare here omitted for simplicity. The Q factor can beestimated from satellite measurements with a sensorsuch as the POLDER ~Polarization and Directionalityf the Earth’s Reflectances!17 or from the procedure

described by Morel and Gentili.18,19 The feasibilityof estimating the attenuation coefficient ^Kd&1 at l 5490 nm from an ocean-color algorithm based on theempirical relationship between ^Kd~490!&1 and the ra-tio Lu~443, 02!yLu~550, 02! has been demonstratedby Austin and Petzold.20 High correlation between^Kd~490!&1 and ^Kd~l!&1 at other wavelengths21–23 sug-ests the possibility of estimating the spectralKd~l!&1 from radiances detected by ocean-color sat-

ellite sensors. We note that the upper oceanic layerbetween the surface and z1 generates 90% of thewater-leaving upwelling photons, which can be de-tected by satellite sensors.24

In radiative transfer simulations leading to our

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model, we consider two types of effect on the relation-ships between IOP’s and AOP’s, which were not in-cluded in the development of similar models in thepast.3,4,6 First, variations in the shape of the totalscattering phase function caused by variable contri-butions of molecular scattering14,25,26 are consideredin an explicit manner in the development of ourmodel. Second, the effects of Raman scattering,which can make a significant contribution to the lightfield in the red and green spectral regions,27–30 arealso included in our model. In addition, in contrastto the models of Gordon et al.3 and Gordon,4 ourmodel does not require data on the beam attenuationcoefficient. The general value of the model of Kirk6

is limited, inasmuch as the simulations were madeonly for a black sky and the relationships representone depth within the water column, where the irra-diance level is 10% of its surface value.

2. Radiative Transfer Simulations

The development of our model is based on two sets ofradiative transfer simulation: first, the simulationswithout Raman scattering that were described byMorel and Loisel26 and, second, a new set of simula-tions with Raman scattering. The simulations ofMorel and Loisel were carried out for a homogeneousand infinitely deep ocean in the absence of inelasticprocesses, that is, no Raman scattering and no fluo-rescence. These simulations were run for a nearlyflat sea surface under no wind ~sea surface slopescharacterized by the residual term in the Cox–Munk31 distribution! and six solar zenith angles thatvaried from 0° to 75° at 15° intervals. The RTE wassolved by either the invariant embedding method~Hydrolight 3.0 code! or the Monte Carlo method.32

The Hydrolight simulations were made for a sun in ablack sky, that is, with no atmosphere. The MonteCarlo simulations were made for the coupledatmosphere–ocean system with clear skies. The at-mospheric model used in this case is described byMorel and Loisel.26

The input IOP’s in this first set of simulations with-out Raman scattering are represented by two dimen-sionless quantities, the single-scattering albedo v0and the scattering phase function b~c! 5 b~c!yb.We changed the single-scattering albedo by varyingthe scattering-to-absorption ratio bya according to

v0 5b

a 1 b5

bya1 1 bya

. (1)

The total absorption coefficient was fixed at 1 m21,and the bya ratio varied from 0 to 10. Phase func-tion b~c! characterizes the angular distribution ofsingle-scattering events and is derived from aweighted sum of the particle-scattering phase func-tion bp~c! and the molecular-scattering phase func-ion bw~c!:

b~c! 5 hbw~c! 1 ~1 2 h!bp~c! , (2)

here

h 5bw

bw 1 bp5

bw

b(3)

and bw and bp are the scattering coefficients that aredue to water molecules and particles, respectively.The molecular phase function was calculated fromthe appropriate formula.1 For particles, the averageparticle phase function proposed by Mobley et al.,32

which is based on measurements by Petzold,33 wasused. Note that, for a given value of h, only one totalphase function, b~c!, was generated. The variationin b~c! was driven solely by the variation in h, whichranged from 0 to 0.2.

The bya and h values used in the simulations coverthe range that is representative of the visible part ofthe spectrum for most oceanic waters, especially case1 waters ~see Fig. 2 of Ref. 26!. Note, however, thatthere is no explicit dependence of the input IOP’s onthe wavelength of light in these simulations. Acloser analysis for different wavelengths and a rangeof natural waters indicates that the combination ofrelatively high bya and high h appears to be unreal-istic. Although a small portion of our simulations isbased on such combinations ~for example, bya . 6 forh 5 0.2!, these calculations have no significant effecton our model and its performance, as was verifiedwith the sensitivity analysis.

To take into account Raman scattering in ourmodel development, we made a new set of radiativetransfer simulations, using the Hydrolight code. Inthis second set of simulations the boundary condi-tions and the IOP’s of seawater were modeled withthe library of subroutines provided with the Hydro-light code.34 Specifically, the sky model predictedthe sky radiance distribution for clear maritime at-mosphere with a visibility of 15 km, which providesgood conditions for remote sensing of ocean color.Several values of the solar zenith angle from 0 to 60°were used. The roughness of the sea surface wasmodeled for a wind speed of 5 m s21. The ocean wasassumed to be optically homogeneous and infinitelydeep. The IOP’s were defined as a function of wave-length by use of the bio-optical models parameterizedin terms of chlorophyll concentration, Chl. The val-ues of Chl were 0.02, 0.05, 0.1, 0.2, 0.3, 1, and 3 mgm23. We did not include higher values for Chl be-cause the prediction of IOP’s from chlorophyll-basedrelationships at high Chl is subject to relatively largeuncertainty. The values for the absorption coeffi-cient of pure water were taken from Sogandares andFry35 ~350–370 nm! and Pope and Fry36 ~380–700

m!.The simulations were carried out from 355 to 705

m with 10-nm bands, and the results were producedor the central wavelength of each band, that is 360,70, 380, . . . , 700 nm. For each set of boundary

conditions and IOP’s, simulations with and withoutRaman scattering were made. The Raman wave-length redistribution function fR~l, l9!, which speci-

es the wavelength l where radiance of wavelength


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l9 ends up after being Raman scattered, was modeledfollowing the description by Mobley1 ~Sec. 5.14, withtypographical errors corrected!. The Raman absorp-tion coefficient, aR~l9!, was assumed to change asl924, and its value at 488 nm was 2.6 3 1024

m21.1,28,37 The spectral variation of the Raman scat-ering coefficient, bR~l, l9!, was calculated as theroduct of aR~l9! and fR~l, l9!.1

3. Formulation of the Model

A. Without Raman Scattering

Irradiance reflectance R increases with bya, and therate of this increase diminishes as bya becomes high-er.6,26 Kirk6 determined the relationship betweenbya and R at a depth zm, where 10% of the surfaceirradiance is observed:

bya 5103Re~zm!

1 2 Re~zm!. (4)

This relation is based on Monte Carlo simulations fora sun in a black sky ~no atmosphere! and for a scat-ering phase function represented solely by particleomponent ~no molecular scattering!. The super-

script e indicates that the reflectance was calculatedin the absence of Raman scattering.

Our numerical simulations show that a similar re-lationship holds just beneath the sea surface ~Fig. 1!.

owever, these results also indicate that the rela-ionship between bya and Re~021!y@1 2 Re~02!# is

dependent on the shape of the phase function throughthe parameter h. Specifically, the smaller the valueof h, the steeper the slope of the relationship is. Fig-ure 1 also shows that for a fixed value of h the slopeof the relationship is dependent on solar zenith angleu0. When u0 increases, the slope decreases. As aesult, instead of a constant value of 103 as in Eq. ~4!,he relationship involves a variable slope represented

Fig. 1. Relationship between bya on Re~02!y@1 2 Re~02!# at sev-eral solar zenith angles u0 and values of h. The circles representesults of the radiative transfer simulations, and the solid lines arehe least-squares linear regressions. For each regression the de-ermination coefficient r2 is greater than 0.99.


y a certain function g~h, mw!, which depends on bothh and mw:

ba

5 g~h, mw!Re~02!

1 2 Re~02!, (5)

where mw is the cosine of the refracted solar beam justbeneath the surface. This equation is the first com-ponent of the present model.

The next step is to develop a relationship betweenIOP’s and the average vertical attenuation coefficientfor downwelling irradiance, ^Kd&1, within the upperoceanic layer from the surface, z 5 02, to the firstattenuation depth, z1. Morel and Loisel26 showedthat variations in ^Kd&1, as affected by the absorptioncoefficient, the scattering coefficient, the sun angle,and the contribution of the molecular scattering tototal scattering, can be described as

^Kd&1 5a

mwF1 1

ba

G~h, mw!G0.5

, (6)

where G~h, mw! is a function fitted to the results ofsimulations obtained with various IOP’s, sun angles,and atmospheric aerosol optical thicknesses. At anygiven solar angle, the function G~h, mw! accounts forthe increase in ^Kd&1 with the increase in h and can bewritten as

G~h, mw! 5 ~g1 1 g2h! 1 ~g3 1 g4h!mw. (7)

he coefficients g1, g2, g3, and g4 depend on the at-mospheric properties parameterized in terms of theaerosol optical thickness, and their values are givenin Table 4 of Morel and Loisel.26

After bya is substituted from Eq. ~5! into Eq. ~6!,he total absorption coefficient a is

a 5mw^Kd&1

F1 1 h~h, mw!Re~02!

1 2 Re~02!G0.5 , (8)

where h~h, mw! 5 g~h, mw! G~h, mw!. For a givenvalue of the solar zenith angle ~mw fixed at someconstant value!, the function g~h, mw! decreases withh, whereas G~h, mw! increases with h. Therefore onecan expect that the product of these two functions,that is, h~h, mw!, will be weakly dependent on orindependent of h. For any values of h, mw, and bya,one can calculate h~h, mw! by rearranging Eq. ~8!:

h~h, mw! 5@~^Kd&1ya!mw#2 2 1

Re~02!

1 2 Re~02!

. (9)

The values of h calculated from Eq. ~9! are plotted inig. 2 as a function of h for various values of u0. Asan be seen, h is approximately independent of h for. 0.03. For a given solar zenith angle, h can thus

e approximated by an average value h# over therange of h from 0.03 to 0.20. In the visible part ofthe spectrum this situation corresponds to case 1 wa-ters with Chl less than approximately 1.0 mg m23

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~see Fig. 3 of Ref. 14!. Such waters represent morethan 97% of oceanic surface waters from 50 °N to50 °S.38

Next we determine a relation between h# and theSun position in the clear sky, using radiative transfersimulations with realistic sky conditions. The pre-vious study26 demonstrated that the values of Re~02!and ^Kd&1 depend weakly on the atmospheric aerosol’sptical thickness ta at 550 nm, which varies from 0.1

to 0.8, corresponding to the atmospheric visibilityfrom 39 to 5 km. Therefore in this determination weconsider only the radiative transfer simulations for arelatively clear atmosphere with ta~550! 5 0.2 and aisibility of 20 km. Such atmospheric conditions areavorable for ocean-color remote sensing. The datarom these simulations indicate that h# can be esti-ated as a function of mw by use of the second-order

polynomial:

h# ~mw! 5 2.54 2 6.54mw 1 19.89mw2, (10)

where the determination coefficient r2 is 0.99. Thisesult is plotted in Fig. 3.

Substituting h# ~mw! for h~h, mw! in Eq. ~8! gives thenal formula for estimating the total absorption co-fficient from Re~02! and ^Kd&1 for a given sun posi-

tion parameterized in terms of mw:

a 5mw^Kd&1

F1 1 h# ~mw!Re~02!

1 2 Re~02!G0.5 . (11)

Having determined the absorption coefficient, wean estimate the total scattering coefficient b. The

analysis of results from radiative transfer simula-tions leads to the parameterization of the functiong~h, mw!:

g~h, mw! 5

1~0.0215 2 0.0149mw! 1 ~0.1652 2 0.0358mw!h

. (12)

Fig. 2. Changes in the h function with h for several values of u0

ranging from 0° to 75° at 15° intervals.

This function g~h, mw! is substituted into Eq. ~5!, andhen, after substituting bwyb for h, we obtain

b 5

aRe~02!

1 2 Re~02!2 bw~0.165 2 0.0358mw!

0.0215 2 0.0149mw. (13)

Note that, with this approach, once the absorptioncoefficient is known ~and for a given sun position!, theotal scattering coefficient is obtained from the irra-iance reflectance only, Re~02!. We can also esti-

mate the scattering coefficient from ^Kd&1 bycombining Eqs. ~3!, ~6!, and ~7!:

b 5

aFS^Kd&1

amwD2

2 1G 2 bw~2.0303mw 2 0.3005!

0.00054 1 0.1183mw,

(14)

where the coefficients g1, g2, g3, and g4 were taken forthe aerosol optical thickness ta~550! 5 0.2. Al-though it is possible that the numerators in Eqs. ~13!and ~14! become negative numbers that give unreal-stic estimates of b, sensitivity analysis showed thathis problem will not occur in case 1 waters, whoseptical properties are consistent with the chlorophyll-ased bio-optical relationships that are used to pa-ameterize the IOP’s in our simulations.

We now consider the estimation of the backscatter-ng coefficient bb. Gordon et al.3 showed that bb can

be estimated from the irradiance reflectance and theattenuation coefficient for downwelling irradiance.This estimation is based on the facts that R is pro-portional to bbya and that Kd is, to a first approxima-tion, controlled by a. Based on Monte Carlosimulations, the formula for bb was proposed previ-ously4:

bb 51

D0^Kd& (

i51

3

ri9Re~1!i, (15)

where D0 is the ratio of the downwelling scalar irra-diance to the downwelling plane irradiance just be-

Fig. 3. Variation of h# with mw. The circles represent the valuesbtained from radiative transfer simulations, and the curve wasalculated with Eq. ~10!.


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neath the sea surface in the absence of scattering bywater ~for a sun in a black sky, D0 equals mw!, ri9 arethe coefficients obtained from a least-squares analy-sis of simulation results, and Re~1! is the reflectancejust beneath the sea surface for a sun at zenith in theabsence of the atmosphere. The value of Re~1! muste estimated by extrapolation of the actual measure-ents of R made at several different sun angles.lthough Eq. ~15! gives a good approximation of bb

with an average error of 10.6%,4 bb cannot be esti-mated from data obtained at a single sun position.Our goal is to develop a model for estimating bb frommeasurements of R~02! and ^Kd&1 made at any sunangle, that is, without the necessity to make mea-surements over the range of sun angles. As in thederivation of the absorption and scattering coeffi-cients, the effect of molecular scattering is explicitlyincluded in our model for retrieving bb.

The ratio bby^Kd&1 is plotted in Fig. 4 as a functionof Re~02! for three solar zenith angles u0 and at eachu0 for two extreme values of h. The relationshipbetween bby^Kd&1 and Re~02! depends on h, especiallyfor large values of u0. A closer examination of thedata for a given u0 and a given h reveals that log~bby^Kd&1! increases approximately linearly withog@Re~02!#:

log@bby^Kd&1# 5 a 1 d log@Re~02!#. (16)

The determination coefficient between these vari-ables is always greater than 0.99. Accordingly, thefinal formula for estimating the backscattering coef-ficient is

bb 5 ^Kd&1 10a@Re~02!#d, (17)

where a is a function of mw and h, and d is only aunction of h ~meaning that d is independent of theistribution of the incident radiance at the sea sur-ace!:

a 5 @20.83 1 5.34h 2 12.26h2#

1 [email protected] 2 4.124h 1 8.088h2#, (18)

d 5 0.871 1 0.40h 2 1.83h2. (19)

The values of h are determined as bwyb, where b isthe estimate of the scattering coefficient obtainedfrom either Eq. ~13! or Eq. ~14!. Below, we comparehe estimates of b and bb based on these two equa-

tions.

B. With Raman Scattering

Up to this point we have considered the results of theradiative transfer simulations without Raman scat-tering. These considerations led to the equations forestimating a, b, and bb, which can be applied when

aman scattering has a negligible or small effect onhe light field. However, the above model cannot bepplied without any correction if Raman scatteringontributes significantly to the light field. Gener-lly, the upwelling light field at wavelengths longerhan ;520 nm can be significantly affected by Raman


scattering.28,39,40 In the case of pure water, the Ra-man contribution to water-leaving radiance at 550nm can reach 25%.30 Although Raman scatteringcan have a significant effect on the upwelling light,and hence on R~02!, the downwelling irradiance and^Kd&1 in the upper oceanic layers are only weaklyaffected by this inelastic process. Therefore our ap-proach to including Raman scattering in our model isto correct the actual reflectance for the Raman con-tribution such that the corrected values represent thereflectance that would be measured if there were noRaman scattering by water. Then the corrected re-flectance is used in the final equations of the model@Eqs. ~11!, ~13!, and ~17!#, which were derived without

aman scattering.The procedure for correcting the reflectance for Ra-an scattering is as follows: We first define the

ctual reflectance R~02! in the presence of both elas-tic scattering and Raman scattering as a sum of twocomponents:

R~02! 5 Re~02! 1 DR~02!, (20)

where Re~02! is the reflectance that would be mea-sured if there were no Raman scattering and DR~02!is the contribution associated with Raman scattering.

Fig. 4. Relationship between bby^Kd&1 and Re~02! for two extremealues of h, as indicated. Each figure corresponds to a differentolar zenith angle u0, as indicated.

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We emphasize that DR~0 ! merely represents a dif-ference between the reflectance in the presence ofboth elastic and Raman scattering, R~02!, and thehypothetical reflectance in the absence of Ramanscattering, Re~02!, and that DR~02! does not repre-sent the reflectance that is due to Raman scattering.In other words, DR~02! Þ Eu

RyEdR, where Eu

R andd

R are the upwelling and downwelling irradiances,respectively, that are due to Raman scattered pho-tons. In the applications of our model, R~02! repre-sents the reflectance that is actually obtained eitherfrom in situ measurements or from remote sensingbecause Raman scattering is always present in nat-ural waters. Our goal is thus to develop a procedurefor determining Re~02! from R~02!. This is why wemade a new set of simulations ~as described in detailabove! with Raman scattering that provided R~02!nd without Raman scattering that provided Re~02!.

From these simulations we generated a lookup tablewith the values of k:

k 5R~02! 2 DR~02!

R~02!. (21)

This table provides k at various discrete wavelengthsl, and for each l the k values are listed as functionsf bbya and solar zenith angle u0. We note that the

parameter h is not explicitly included in the lookuptable because, at any fixed wavelength, only onevalue of h corresponds to a given value of bbya, whichis related to the use of chlorophyll-based relation-ships to generate IOP’s.

Knowing the k value, we can calculate Re~02! fromR~02!:

Re~02! 5 kR~02!. (22)

These calculations are based on a simple iterativescheme that must be carried out at each wavelengthof interest, l. At the first iteration we assume thatthe reflectance R~l, 02! represents the case with elas-ically scattered light only, which means that no Ra-an scattering occurs and that k~l! 5 1. Under this

assumption we obtain the first estimates of a~l!, b~l!,nd bb~l! from Eqs. ~11!, ~13!, and ~17!. The next

iteration begins with selecting the appropriate valueof k~l! from the precomputed lookup table, using theestimated value of bb~l!ya~l! for a given solar zenithangle u0. With this value of k~l! we then calculateRe~l, 02! from Eq. ~22!. We finally use this newestimate of Re~l, 02! to obtain new estimates of a~l!,~l!, and bb~l! from Eqs. ~11!, ~13!, and ~17!. This

iterative process is repeated until a change in theestimated ratio bb~l!ya~l! becomes arbitrarily small.Typically, this change was less than 0.1% after two orthree iterations. We made tests showing that, if theiterative process were based on evaluating the con-vergence for a, b, and bb separately rather than forthe bbya ratio, the final results would be virtuallydentical. In this iterative procedure the ratio bya

could have been used instead of bbya. But, as will beshown below, bb is estimated with better accuracy

than b, so the bbya ratio is preferable. The improve-ment of the IOP estimates obtained with this itera-tive procedure compared with the straightforwardapplication of our model without the Raman-scattering correction is discussed below.

4. Error Analysis and Sensitivity to Scattering PhaseFunction

The radiative transfer calculations without Ramanscattering26 that were used to develop the model ofEqs. ~11!, ~13!, and ~17! included 528 simulations forhe range of bya from 0 to 10, h from 0 to 0.2, and u0

from 0 to 75°. The data set of R~02! and ^Kd&1 gen-erated by these simulations can be used to test theperformance of our model in terms of the retrieval ofthe three IOP’s, a, b, and bb. The average error ~in

ercent! for the retrieved IOP based on all 528 sim-lations can be calculated as

errIOP~%! 51

528 (j51

528 uIOPmodel~ j! 2 IOPtrue~ j!uIOPtrue~ j!

3 100,

(23)

here IOPmodel~ j! is the value of a, b, or bb estimatedfrom the model for the jth radiative transfer simula-tion and IOPtrue~ j! is the value of the correspondingIOP that was used as input to the jth simulation.Table 2 provides the values for the average error andfor the maximum error that were found among these528 simulations. As shown, the errors increase sig-nificantly if large solar zenith angles, u0 . 60°, areincluded in the analysis. For the absorption coeffi-cient, however, the average errors, erra, are alwaysvery small, only a few percent. The maximum errorfor a is 6.5% if the sun altitude is greater than 30°,but if we include lower sun positions the maximumerror can reach 15%. The errors for the total scat-tering coefficient are larger, and Eq. ~13! results inbetter estimates than Eq. ~14!. If Eq. ~13! is used,he average error, errb, is 8.5% and the maximum

error is 23%. The difference in the performance ofEqs. ~13! and ~14! is likely associated with the pa-rameterization of g~h, mw! used in Eq. ~13! as opposedto the parameterization of G~h, mw! used in Eq. ~14!.

inally, Table 2 shows that the backscattering coef-cient is estimated with relatively good accuracy.he average error, errbb, is ;6% if u0 , 60°. The

choice between Eq. ~13! and ~14! has a small effect onhe accuracy of backscattering estimates, but theath involving Eq. ~13! is preferred in the applicationf the model.Our model could be expected to perform with the

bove errors if there were no Raman scattering byater and if measurements of R~02! and ^Kd&1 wereerfectly accurate. Using the Hydrolight simula-ions with Raman scattering ~referred to earlier as aew set of simulations!, we can examine how theerformance of our model improves when the correc-ion procedure for Raman scattering is applied. Be-ause these simulations were run as a function ofavelength and the input IOP’s were obtained from

hlorophyll-based relationships, we present the error


tftuttslRefiloa^

mbcbtccsats

s

Table 2. Average and Maximum Errors ~in percent! for the IOP Estimation from the Model

3

analysis for the three selected wavelengths, 440, 550,and 660 nm, as a function of chlorophyll concentra-tion Chl ~Fig. 5!. Specifically, this figure compareshe errors, ~IOPmodel 2 IOPtrue!yIOPtrue, obtainedrom the model without the Raman-scattering correc-ion ~left! with the errors obtained when the modelses the correction ~right!. Regardless of whetherhe Raman-scattering correction is applied, absorp-ion coefficient a is retrieved from the model with aimilar good accuracy. This result is seen even atow Chl and long visible wavelengths ~660 nm!, whenaman scattering is expected to have the greatestffect on the underwater light field. This small ef-ect of Raman scattering on the absorption retrievals caused primarily by the fact that ^Kd&1, whichargely controls the absorption, is weakly dependentn Raman scattering. In addition, although R~02! isffected by Raman scattering to a greater extent thanKd&1 is, the retrieval of a is less sensitive to R~02!

than to ^Kd&1. We also observe that, at each wave-length, the error of absorption estimate increaseswith an increase in Chl ~that is, with decreasing h!but always remains below 6%.

Unlike for absorption retrieval, the application ofthe Raman-scattering correction makes a significantdifference in the accuracy of the estimation of scat-tering and backscattering coefficients. Significantimprovement associated with the use of this correc-tion is seen in the estimates of b and bb in the greenspectral band, l 5 550 nm, where the errors in thesecoefficients are generally only a few percent. Al-though the Raman-scattering correction also im-proves the estimates of b and bb in the blue and thered, the errors at 440 and 660 nm may still be signif-icant. For example, at 440 nm the error in b canexceed 20% at low Chl. At 660 nm, our model withRaman-scattering correction underestimates b atmoderate to high Chl by ;15% and overestimates bbby 10% to more than 20% at Chl , 0.1 mg m23. The

ost important conclusion from this analysis is thatoth the absorption coefficient and the backscatteringoefficient are retrieved with very small errors in thelue and green spectral regions. This attribute ofhe model is essential from the standpoint of ocean-olor remote-sensing applications because these twooefficients are major determinants of remote-ensing reflectance and blue–green spectral bandsre commonly used in ocean-color algorithms for re-rieving ocean products. The result that the totalcattering coefficient appears to be more difficult to

IOP Eqs.

AverageError,

0° , u0 , 75°

a ~11! 3.3b ~13! 11b ~14! 16bb

d ~17! and ~13! 7.6bb ~17! and ~14! 8.5


retrieve with high accuracy than a and bb from mod-els based on the measurements of Ed and Eu ~andhence of R and Kd! is consistent with previous find-ings by Gordon.41

Because we developed our model by using a single-particle phase function bp~c! based on Petzold’s mea-surements, which is commonly assumed to representan average function for marine particles, we testedthe sensitivity of the model retrieval to possible vari-ations in bp~c!. These tests were made with twophase functions that differ significantly from the Pet-zold phase function. These two functions were de-rived from Mie scattering calculations assuming thesize distribution and refractive index of particles.42

Fig. 5. Comparison of errors ~in percent! in the values of a, b, andbb estimated from our model with ~left! no correction for Ramancattering and with ~right! the correction for Raman scattering ~see

text for details about the calculations of errors!. The errors areplotted as a function of chlorophyll concentration Chl for the threeselected wavelengths, 440, 550, and 660 nm, as indicated. Thecircles indicate the errors for the absorption coefficient; thesquares, those for the scattering coefficient; the triangles, those forthe backscattering coefficient.

aximumError,

u0 , 75°

AverageError,

0° , u0 , 60°

MaximumError,

0° , u0 , 60°

15 2.4 6.530 8.5 2335 12 2626 5.9 2027 6.1 20

M

0° ,

tert

Cu

T

s

Specifically, one function was derived to representscattering by a polydispersed assemblage of organicparticles with a broad size distribution ~Junge-typedistribution with the exponent 24! and low refractiveindex ~n 5 1.04 relative to water!. Such particles mayinclude organic detritus as well as planktonic micro-organisms. The other function was derived to repre-sent the same Junge-type size distribution, but therefractive index of the particles was high ~n 5 1.18!,which is typical of minerals. These functions are re-ferred to here as detrital and mineral scattering phasefunctions. A comparison of these functions with thePetzold phase function reveals significant differences~Fig. 6!. It is remarkable that at scattering anglesc . 10° the phase function for minerals is highest andthat for detritus is lowest. The backscattering ratiosare 0.034, 0.0054, and 0.019 for the mineral, detrital,and Petzold phase functions, respectively. At verysmall angles, c , 1°, the Petzold function is morestrongly peaked than the two other functions.

We made special radiative transfer simulationswith the detrital and mineral phase functions, usingthe Hydrolight code. Other IOP’s were generated bythe Hydrolight subroutines from chlorophyll-basedrelationships. The simulations were run at severalwavelengths covering the blue, green, and red spec-tral regions and with the assumption that there is noRaman scattering by water. This assumption is jus-tifiable, as we focus here just on the effect of theparticle phase function. The realistic sky modelwith the sun at 30° from zenith and the wind speed of5 m s21 were used as input to these simulations. Wehen used the calculated values of Re~02! and ^Kd&1 tostimate a, b and bb from Eqs. ~11!, ~13!, and ~17!,espectively. A comparison of these estimates withhe true values of a, b and bb that were used as input

to the Hydrolight simulations is shown in Fig. 7 forone selected wavelength, l 5 550 nm, as a function of

hl. When the detrital or mineral phase function issed @Figs. 7~a! and 7~b!# the errors in the absorption

and backscattering estimates remain relativelysmall. The error in a is less than 10% for the exam-ined range of Chl. For bb the error exceeds 10% onlyat high Chl. These errors are similar to or slightlylarger than those that were obtained with similarsimulations for the Petzold phase function @Fig. 7~c!#.

his result indicates that the estimation of a and bbfrom our model shows relatively little sensitivity tochanges in the scattering phase function, at least forthe range of variability characterized by our detritaland mineral functions. This conclusion holds notonly at l 5 550 nm but also at other wavelengths.

The estimation of the total scattering coefficient isgreatly affected by variations in the scattering phasefunction, however. For the detrital phase functionthe retrieved b is underestimated by several tens ofpercent. The use of the mineral phase function re-sulted in large overestimation of b, by more than100% at high Chl. Therefore the application of ourmodel for the estimation of b is limited by such highensitivity to variations in the scattering phase func-

Fig. 7. Effects of the use of mineral- and detrital-scattering phasefunctions on the estimation of a, b, and bb from our model. Thepercent errors in the estimated values of the absorption coefficient~circles!, the scattering coefficient ~squares!, and the backscatter-ing coefficient ~triangles! are plotted as a function of chlorophyllconcentration Chl for a wavelength l 5 550 nm. For comparison,the errors obtained with the use of the Petzold phase function arealso shown.

Fig. 6. Comparison of the average particle-scattering phase func-tion obtained from Petzold measurements32 with the calculatedphase functions representing assemblages of low-refractive-indexparticles ~detritus! and high-refractive-index particles ~minerals!.c is the scattering angle.


s

oja^

oftycR

re

o

s

g

orstt~V

R

3

tion. This result can be attributed to the fact thatthe magnitude of b is governed mainly by forwardcattering and that neither R~02! nor ^Kd&1 can ac-

count adequately for variations in scattering at smallangles. The variations of the phase function in oce-anic waters are rather poorly characterized, so thereis a need for further studies in this area. As dis-cussed above, we also emphasize that, even when theactual scattering phase function agrees with the Pet-zold function, the errors in the estimates of b canexceed 20%. We therefore caution against indis-criminate use of the model for estimating the totalscattering coefficient.

5. Conclusions

Using radiative transfer simulations, we developed amodel for estimating the absorption a, the scatteringb, and the backscattering bb coefficients in the uppercean from measurements of irradiance reflectanceust beneath the sea surface, R~02!, and the averagettenuation coefficient for downwelling irradiance,Kd&1, between the surface and the first attenuation

depth. In addition to these two measured quanti-ties, our model requires knowledge of the solar zenithangle. Because the simulations were made for clearskies and both R~02! and ^Kd&1 can be derived fromremote sensing of ocean color, the model is suited tosatellite applications. The model requires the hypo-thetical reflectance Re~02! that would be measured ifthere were no Raman scattering by water rather thanthe actual reflectance R~02!. Therefore an iterativeprocedure is built into the model that permits thecalculation of Re~02! from R~02!.

For a given solar zenith angle the coefficients a, b,and bb are estimated from Re~02! and ^Kd&1 accordingto the scheme depicted in Fig. 8. The absorption co-efficient is calculated first @Eq. ~11!#, and this determi-nation is weakly affected by whether the correction for

Fig. 8. Schematic diagram summarizing the estimation of ab-sorption coefficient a, scattering coefficient b, and backscatteringcoefficient bb from our model based on the attenuation coefficientfor downwelling irradiance, ^Kd&1, and the irradiance reflectance,

e~02!. This schematic applies to a given solar zenith angle.


Raman scattering is applied. The total scattering co-efficient is then estimated from a and Re~02! @Eq. ~13!#r from a and ^Kd&1 @Eq. ~14!#. The determination of brom Eq. ~13! is more sensitive to Raman scatteringhan from Eq. ~14!, but Eq. ~13! is recommended as itields better estimates of b, provided that the Ramanorrection is used. Only if the errors in the measured~02! are expected to be significantly larger than in

the measured ^Kd&1 should b be estimated from Eq. ~14!ather than Eq. ~13!. Finally, the backscattering co-fficient is estimated from Re~02! and ^Kd&1. This es-

timation also requires knowledge of h 5 bwyb, which isbtained from the estimate of b. Because b can be

estimated in two different ways, we can also obtain twoestimates of bb, but the recommended approach is touse the path involving Eq. ~13!.

The error analysis showed that the best retrieval isobtained for the absorption and backscattering coeffi-cients. Our radiative transfer simulations suggestthat errors in the estimates of a and bb in the blue andgreen spectral regions are generally of the order of afew percent over a broad range of chlorophyll concen-tration in water. This is an important attribute of themodel because these two IOP’s are major determinantsof remote-sensing reflectance and the blue–green spec-tral region plays a critical role in the ocean color algo-rithms. In addition, we found that the retrieval of aand bb is weakly dependent on possible variations ofthe particle-scattering phase function in the ocean.This finding is reassuring because such variationswere not taken into account in the development of ourmodel. In contrast to those for a and bb, errors in theestimates of the total scattering coefficient can be sig-nificant. These errors can exceed 50% or even 100%when the actual particle phase function differs consid-erably from the Petzold phase function that was as-sumed in the development of our model. Wetherefore caution against indiscriminate use of themodel for estimating b.

Owing to recent developments in optical instru-mentation, it is now possible to make simultaneous initu measurements of a, b, bb, Ed, and Eu routinely at

several wavelengths. As the appropriate databasegrows to include various sky conditions and a varietyof water bodies with different optical properties, itwill be possible to test the performance of our modelagainst in situ observations. Efforts toward thisoal are under way.

We thank A. Morel for discussions and commentsn the manuscript and C. D. Mobley for his help withadiative transfer computations. This research wasupported by the Agence Spatiale Europeenne ~con-ract 11.878y96yNLyGS!, by the Environmental Op-ics Program of the U.S. Office of Naval Researchgrant NOOO14-98-1-0003!, and by the NASA EOSalidation Program ~grant NAG5-6466!.

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