evidence for wavelength dependence of the scattering phasefunction and its implication for modeling...

Evidence for wavelength dependence of thescattering phase function and its implication formodeling radiance transfer in shelf seas

David McKee and Alex Cunningham

More than 90% of stations from the Irish and Celtic Seas are found to have significantly higher back-scattering ratios in the blue (470 nm) than in the red (676 nm) wave band. Attempts to obtain opticalclosure by use of radiance transfer modeling were least successful for stations at which backscatteringratios are most strongly wavelength dependent. Significantly improved radiance transfer simulationresults were obtained with a modified scattering correction algorithm for AC-9 absorption measurementsthat took wavelength dependency in the scattering phase function into account. © 2005 Optical Societyof America

OCIS codes: 000.0000, 010.4450, 290.5820.

1. Introduction

Using commercially available software such as Hy-drolight1 to combine radiance transfer computationswith in situ measurements from modern optical in-strumentation, it should be possible to construct re-alistic mathematical models of underwater lightfields and to predict remote-sensing reflectance accu-rately. Recent results indicate unexpected variabilityin the degree to which this aim can be achieved.2–5

The variability has been attributed to poor qualitycontrol of either the inherent optical property mea-surements used to constrain the models or the radio-metric measurements used for model validation.4,5

However, since the degree of closure obtained variesfrom one station to another in a given cruise and evenbetween wavelengths for a given station, it is likelythat the source of variation lies in the optical char-acteristics of the water body in which the measure-ments are being made rather than in themeasurement procedures. In this paper, we considerwhether the difficulties encountered in obtaining con-sistent closure between models and measurements inshelf seas can be attributed to wavelength depen-

dence of the scattering phase function. The method-ology employed was developed for modelingcoccolithophore blooms,3 but it can be applied to otherwaters where standard inherent optical property(IOP) measurements are difficult to reconcile with insitu radiometric profiles.

2. Theory

The IOPs of seawater are commonly measured withan AC-9 dual-tube spectrophotometer (WET Labs,Philomath, Oregon) and a Hydroscat backscatteringmeter (HOBI Labs, Inc., Tucson, Arizona). The AC-9provides coefficients of attenuation (cn) and absorp-tion (an) for materials other than water, and the Hy-droscat provides total backscattering coefficients (bb).Data from both instruments are subject to correctionprocedures that make assumptions about the opticalproperties of the medium in which the measurementsare made. These assumptions may be valid for oce-anic waters, but they are largely untested for shelfseas. In this paper, we consider possible modificationof the correction procedure for the AC-9 because themanufacturer has published a critical analysis of thegeometric and optical principles involved. It is likelythat Hydroscat correction procedures should also beexamined critically in coastal waters, but we onlyconsider whether the errors involved invalidate themain thrust of the argument.

A. AC-9 Correction

It has been reported3 that removal of the assumptionof a wavelength-independent scattering phase func-

The authors are with Department of Physics, University ofStrathclyde, 107 Rottenrow, Glasgow, G4 0NG, Scotland. D Mc-Kee’s e-mail address is [email protected].

Received 23 December 2003; revised manuscript received 7 Sep-tember 2004; accepted 21 September 2004.

0003-6935/05/010126-10$15.00/0© 2005 Optical Society of America

126 APPLIED OPTICS � Vol. 44, No. 1 � 1 January 2005

tion from the analysis by Zaneveld et al.6 leads to amodified version of the scattering correction algo-rithm for reflecting tube absorption measurements ofthe form

an�� ai�� ai��r�ci�� ai��ci��r� � ai��r�

F��, �r�, (1)

where an is the corrected estimate of the absorptioncoefficient for materials other than water, ai and ci

are the uncorrected instrument measurements of ab-sorption and attenuation, � denotes the wavelengthat which the corrected value is calculated, and �r isthe reference wavelength (715 nm) at which an isassumed to be zero. Equation 1 differs from the orig-inal expression by the presence of an extra factorF��, �r� that can be written in full as

F��, �r� �

� ka��1 � ka�� kc�� ka��r�

1 � ka��r� � kc��r��, (2)

where ka�� is the fraction of scattered light that is notcollected by the absorption sensor and kc�� is thefraction of scattered light that is collected by the at-tenuation sensor. Zaneveld et al.6 assumed ka�� ka��r� and kc�� kc��r�, and consequently F��, �r�� 1, but this assumption does not hold if the scatter-ing phase function varies significantly with wave-length. Unfortunately, none of the terms on the right-hand side of Eq. 2 are readily obtained from in situoptical measurements, and F��, �r� must be deter-mined by a computationally intensive iterative pro-cedure.3 The magnitude of the error that results fromignoring wavelength dependence depends on theproduct of F��, �r� and ai��r�, and it is greatest inturbid waters, where ai��r� values tend to be highest.In waters in which wavelength dependency was ob-served, the ratio of the total backscattering coefficientto the total scattering coefficient (bb�b) generally de-creased with increasing wavelength. From Eqs. 1 and2, the effect is to produce an overestimate of an andconsequently an underestimate of bn.

B. Hydroscat Correction

The sigma correction procedure for Hydroscat datacan be written as

bb � � bbu, (3a)

� � k0 � k1Kbb � k2Kbb2 , (3b)

where bbu is the uncorrected signal and k0,1,2 are cal-ibration coefficients supplied by the manufacturer.Kbb is a measure of the attenuation of the signal bywater within the sensor’s measurement geometryand is calculated for each wavelength with7

Kbb � a � 0.75b. (3c)

Figure 1 shows the effect of sigma correction onmeasurements of backscattering for surface watersat all stations occupied in the Irish Sea. Significantdivergence between bb and bbu occurs as the level ofturbidity increases, and so backscattering measure-ments become increasingly sensitive to correction er-rors in strongly attenuating shelf seas. TheHydroscat-2 data presented in this paper have beencorrected with values of total absorption and scatter-ing derived from AC-9 measurements in which thestandard scattering correction algorithm was used.Since positive errors in a and negative errors in bpartially cancel in Eq. 3c, and k0 � k1 � k2 in Eq. 3b,the sigma correction procedure is rather insensitiveto the artifacts in the AC-9 data discussed in theprevious section. In the extreme case of a being over-estimated by 50% and b being underestimated by 5%,bb would be calculated to be 2.5% above its true valuefor this data set.

C. Wavelength Dependence of the Scattering PhaseFunction

The wavelength dependence of the scattering phasefunction can be characterized in terms of availableIOP measurements by

B ��bb�b�470

�bb�b�676

. (4)

B is subject to errors in the correction proceduresapplied to measurements of both total backscattering(bb) and total scattering (b). The values of a, b andsigma-corrected bb at 676 nm are assumed to be rel-atively unaffected by correction errors because thiswave band is close to the AC-9 reference channel.Consequently our worst-case example of a 5% under-estimate in b and a 2.5% overestimate in bb at 470 nmproduces bb�b and B values that are overestimated byapproximately 8%.

Fig. 1. Sigma-corrected total backscattering values show increas-ing divergence from uncorrected total backscattering as the tur-bidity of the water increases.

1 January 2005 � Vol. 44, No. 1 � APPLIED OPTICS 127

D. Possible Influence of Fluorescence on bb676

Much of our discussion here rests on the observationthat measured values of bb�b470 tended to be higherthan those for bb�b676 and that their ratio [B in Eq.4] was usually greater than unity. However, the Hy-droscat 676-nm channel covers a relatively broadwave band (20 nm FWHM) that partly overlaps withthe fluorescence emission spectrum of chlorophyll-ain vivo. It must therefore be presumed that Hy-droscat bb676 measurements contain a fluorescencecomponent, though no quantitative study has beenpublished. The effect of fluorescence would be to raisethe bb676 signal and to generate erroneously highapparent values of bb�b676. As a result, actual Bvalues would be even greater than those measuredhere. The observed wavelength dependence in thebackscattering ratio is therefore qualitatively robustin the presence of fluorescence artifacts in bb676.

The magnitude of fluorescence augmentation of thebackscattering signal in the present data set is difficultto measure directly. However, its likely significancecan be judged from an analysis of seawater composi-tion. Chlorophyll-a concentrations measured by ace-tone extraction were generally low, with 95% of thestations having values of less than 2 mg m�3. Thebb676 signal was poorly correlated with chlorophyllconcentration (r2 � 0.29) for stations with significantchlorophyll (1–7 mg m�3) and low mineral (�5 g m�3)concentrations but was highly correlated with sus-pended mineral concentration (r2 � 0.92) for all sta-tions. It therefore appears that backscattering in thisdata set is dominated by the suspended mineral com-ponent and that chlorophyll fluorescence is likely tobe of minor significance.

E. Predicted Effects on Radiance Transfer Calculations

The qualitative effect of wavelength-dependent scat-tering phase functions on attempts to achieve opticalclosure can be predicted from established relationsbetween IOPs and apparent optical properties. Toreasonable approximations, the radiance reflectance(RL) is given by8

RL �f

Q

bb

a, (5)

and the diffuse attenuation coefficient for downwardirradiance (Kd) by9

Kd �a � bb

�d. (6)

Overestimation of a therefore leads to values of RL

that are too low and to values of Kd that are too high.As a result, the computed downward irradiance (Ed)decreases more rapidly with depth than actual mea-surements, and since

RL �Lu

Ed, (7)

the upward radiance (Lu) is also underestimated.For coccolithophore blooms, we were able to show

that the use of Eq. 1 with computationally derivedvalues of F��, �r� for correcting AC-9 data producedimproved fits between radiance transfer models andin situ measurements in turbid waters. We extendthese observations to a larger geographical area (theIrish and Celtic Seas) whose optics are not dominatedby coccolithophores.

3. Materials and Methods

Data were obtained during a series of cruises in theIrish and Celtic Seas between May 2001 and July

Fig. 2. Backscattering ratios are generally greater in the blue(470 nm) than in the red (676 nm) for most stations in the Irish andCeltic Seas.

Fig. 3. Ratio of backscattering ratios at 470 and 676 nm, B, showssignificant variability for surface waters of the Irish and CelticSeas. Less than 5% of stations have a value of B between 0.9 and1.1, where the scattering phase function might be considered wave-length independent. Most stations have stronger backscatteringratios in the blue than in the red. It can be concluded that thescattering phase function is generally wavelength dependent forthese waters.


2002. A total of 120 stations were occupied covering awide range of water types.

A. Inherent Optical Property Measurements

A 25-cm path-length WET Labs AC-9 was used tomeasure the absorption coefficient (an) and beamattenuation coefficient (cn) of materials other thanwater at nine wavelengths (10 nm FWHM) acrossthe visible spectrum. Optical blanks for the AC-9were regularly measured with ultrapure Milliporewater treated with ultraviolet light, and calibrationof the two optical channels remained within themanufacturer=s specifications of �0.005 m�1. Ab-sorption and attenuation signals at 715 nm werecorrected for temperature-dependent water absorp-tion,10 and the data were averaged over 1-m depthintervals. We assumed that the standard correctionalgorithm always suffices for AC-9 measurements at676 nm, since this is close to the 715-nm referencewavelength. Total absorption (a) and attenuation (c)coefficients were obtained by addition of partial coef-ficients for water obtained from the literature.11,12

Scattering coefficients were obtained from b � c

Fig. 4. Significant degree of variability is exhibited in the relationbetween backscattering ratios at 470 and 676 nm for the top 10 mof four sample stations. Lines (gradients between 1.0 and 1.8)indicate the strength of the wavelength dependency in the back-scattering ratio, B.

Fig. 5. IOPs of stations B4 and ST19 are significantly different in terms of (a) absorption at 488 nm, (b) scattering at 488 nm, (c)backscattering at 470 nm, and (d) backscattering ratio at 470 nm. B4 is a relatively clear station, whereas ST19 is quite turbid.


� a. Values of absorption and scattering at 470 nm,which are used at various points in the paper, wereobtained by interpolation between readings at 440and 488 nm. Sets of modified AC-9 data were gener-ated with Eq. 1 with values of F��, �r� ranging be-tween 1 and 1.5. Total backscattering (bb) wasmeasured at 470 and 676 nm by use of a Hydroscat-2(HOBI Labs) and was corrected with calibration con-stants supplied by the manufacturer.

B. Radiometric Measurements

Downward irradiance (Ed) and upward radiance (Lu)were measured in seven wavebands (10 nm FWHM)across the visible spectrum by use of a Satlantic Sea-WiFS Profiling Multi-Channel Radiometer (SPMR).The SPMR was deployed at a distance of at least 20 mfrom the ship to avoid shadowing. A reference radi-ometer measuring surface irradiance (Es) in the samewave bands was mounted on the superstructure ofthe ship. The stability of the irradiance sensors was

monitored at the start and the end of each cruise byuse of a 100-W standard lamp, and the radiance sen-sors were checked by use of the same lamp to illumi-nate a Spectralon reflectance target. All sensorsremained within factory specifications. Signals fromthe SPMR were processed with ProSoft, a MATLAB

module supplied by the manufacturers. Data process-ing steps included application of calibration con-stants and averaging over 1-m depth intervals.

C. Radiance Transfer Modeling

Radiance transfer calculations were carried out withHydrolight (Sequoia Scientific, Bellevue, Washing-ton).1 Surface irradiance data for each station wereobtained from the SPMR deck reference. The choice ofwavelength for this study was limited by the fact thatwe had access only to a two-channel Hydroscat back-scattering meter operating at 470 and 676 nm. Theanalysis was limited to the blue wave band to avoid thecomplicated effects of inelastic processes on the radio-

Fig. 6. Standard IOPs provide satisfactory matches between measured (SPMR) and modeled (Hydrolight) (a) downward irradiance, Ed,and (b) upward radiance, Lu, for station B4. The match between measured and modeled Ed (c) is also satisfactory for ST19. The smalloverestimate of modeled upward radiance for ST19 (d) may be attributable to limitations in the performance of sigma correction ofbackscattering data for very turbid water such as this.


metric measurements. A direct comparison was madebetween Hydrolight computations at 488 nm andSPMR profiles at the same wavelength. The IOP in-puts were bb values at 470 nm measured with theHydroscat 2 and a and b values at 488 nm measuredwith the AC-9. It is assumed that the 18-nm wave-length discrepancy between the backscattering dataand the other IOPs has a negligible effect on theresults. We performed separate runs with AC-9 inputfiles generated by use of different values of F��, �r� forthe modified scattering correction algorithm [Eq. 1].Fournier–Forand13,14 scattering phase functions foreach station and depth were chosen using sigma-corrected Hydroscat-2 values for the backscatteringcoefficient. The match between the Hydrolight outputand the in situ measurements from the SPMR wasevaluated by use of the average standard percentageerror ε, defined as

� ��n

�xin � xmod� � xin 100

n, (8)

where n is the number of observations, xin refers to insitu measurements, xmod refers to modeled values,and x is the parameter being investigated (either Ed

or Lu).

4. Results

A. Backscattering Ratios in the Irish Sea

Figure 2 shows bb�b at 470 nm, plotted against bb�bat 676 nm for surface waters at 120 stations in theIrish Sea, derived from Hydroscat and AC-9 datacorrected with the standard procedures. For most sta-tions the points lie above the 1:1 line, indicating thatbackscattering ratios are higher at shorter wave-lengths. Figure 3 shows the distribution of B for thedata in Fig. 2. For 95% of the stations, B � 1.1, andfor 50% B � 1.3. These percentages are not changedsignificantly by the possible existence of an 8% over-estimate in B values for the most wavelength-dependent stations. The implications of this result forthe correction of AC-9 measurements and the abilityto obtain optical closure were therefore investigated.

Fig. 7. IOPs of stations ST12 and ST16 indicate (a) similar levels of absorption at 488 nm, (b) significantly higher levels of scattering at 488nm, (c) backscattering at 470 nm, and (d) backscattering ratio at 470 nm for ST12. All IOPs were generated with standard correctionalgorithms.


B. Case Studies

Figure 4 shows backscattering ratios at 470 nm plot-ted against backscattering ratios at 676 nm for thetop 10 m of four stations that cover a representativerange of B values. These ratios were derived fromIOP data corrected with standard procedures. Sta-tions B4 and ST19 are examples with low wavelengthdependency in the backscattering ratio (B � 1.2),whereas stations ST12 and ST16 are examples withmarked wavelength dependency (1.2 � B � 1.8).Equations 5–7 predict that radiance transfer modelsin which standard IOP inputs are used will producesatisfactory outputs for stations B4 and ST19 andwill underestimate both Ed and Lu for ST12 andST16.

C. Closure for Stations with B � 1.2

Figure 5 shows IOP profiles generated with AC-9data corrected by use of the standard algorithm forstations B4 and ST19. B4 was a relatively clear sta-tion in the middle of the Irish Sea, occupied in April

2002, with low values of absorption, scattering, andbackscattering throughout the top 25 m of the watercolumn. ST19 was a turbid station close to the Welshcoast, occupied in November 2001, with significantlyhigher values of all three IOPs. The backscatteringratio for ST19 was almost double that at B4. TheIOPs in Fig. 5 were used as inputs for Hydrolightsimulations, and the modeled profiles of Ed and Lu forthe top 10 m of the water column compared with thein situ measurements obtained from the SPMR radi-ometer system (Fig. 6). For both these stations theaverage standard percentage error was less than 6%for measurements of Ed. There was a tendency tooverestimate Lu, with values of � � 11% for B4 and�32% for ST19. However, the apparently poor per-formance of the simulation for Lu at ST19 corre-sponded to an average absolute error of less than 5 �10�5 W m�2 nm�1 sr�1 and represented the smallestabsolute error of any of the simulations presented.Light levels were very low for this station, whichprovided a severe test of modeling and measurement

Fig. 8. Use of standard IOPs in radiance transfer calculations results in systematic underestimates of both Ed [(a) and (c)] and Lu [(b) and(d)] for both ST12 and ST16. This underestimate is consistent with the effects of a wavelength-dependent scattering phase function on thescattering correction of in situ measurements of absorption.


Fig. 9. Average percentage error over the top 10 m of the water column between measured and modeled values of both Ed and Lu can beminimized for stations ST12 and ST16 by varying only the value of F��, �r� used in the scattering correction for the AC-9 measurements.The optimal value of F��, �r� varies with station, and the magnitude of the initial error varies with the size of the scattering signal.

Fig. 10. Improved matches between modeled and measured values of Ed and Lu are obtained with values of F��, �r� � 1.2 for ST12 [(a)and (b)], and F��, �r� � 1.3 for ST16 [(c) and (d)].


precision. In both cases the simulations provided areasonable degree of optical closure by use of stan-dard AC-9 data.

D. Closure for Stations with B � 1.2

Figure 7 shows profiles of IOPs for the top 25 m ofstations ST12 and ST16. Both stations were fairlyhomogeneous over the top 15 m, with similar absorp-tion coefficients. The scattering and backscatteringcoefficients and backscattering ratios were all signif-icantly higher for ST12 than for ST16, but in bothcases backscattering ratios were higher in the bluethan in the red (Fig. 4). Radiance transfer modelingwith AC-9 data corrected by use of the standard al-gorithm [F��, �r� � 1.0] produced systematic under-estimates of both Ed and Lu (Fig. 8) for both thesestations. The discrepancy was most obvious for ST12for which the average standard percentage errorswere 38% and 54% for Ed and Lu, respectively. Theerrors for ST16 were lower at 13% for Ed and 17% forLu. As predicted, the standard [F��, �r� � 1.0] AC-9scattering correction algorithm for these stations re-sults in a systematic overestimate of absorption atblue wavelengths.

Figure 9 shows the effect of varying F��, �r� on theperformance of radiance transfer simulations. ForST12 the minimum average error for both Ed and Lu

occurred when F��, �r� � 1.2. For ST16 the optimalvalue of F��, �r� was closer to 1.3. It is interesting tonote that the overall magnitude of the error withF��, �r� � 1.0 was greater for ST12 than for ST16even though the optimal value of F��, �r� was less forST12 than for ST16. This occurrence was probablydue to a higher residual absorption [ai��r�] for ST12 asa result of significantly higher scattering at this sta-tion. When the optimal values of F��, �r� derivedabove were incorporated in the AC-9 correction algo-rithm (Fig. 10), the average standard percentage er-rors for the radiance transfer simulations were less

than 10% for ST12 and less than 5% for ST16. Thisresult represents a significant improvement in mod-eling accuracy over the results from use of standardAC-9 data.

Figure 11 illustrates the effect of using optimizedF��, �r� values on absorption coefficients for thesetwo stations. In both cases the modified AC-9 absorp-tion signals were lower than the standard values,with the average percentage difference varying from27% for ST12 to 11% for ST16. The resulting scatter-ing coefficients (not shown) were approximately 4%greater than the standard values for both stations.

5. Discussion

The observations presented here show that it can bedifficult to obtain optical closure in waters wherescattering coefficients are high relative to oceanic val-ues. The nature of the errors observed in radiancetransfer calculations and the changes in AC-9 correc-tion procedures necessary to obtain closure were bothconsistent with the hypothesis of wavelength depen-dence in the scattering phase function. In the IrishSea, where the backscattering ratio was generallyfound to decrease with increasing wavelength, thestandard AC-9 scattering correction produces overes-timates in the absorption coefficient and underesti-mates in the scattering coefficient at blue and greenwavelengths. These errors can be reduced by intro-duction of a modified scattering correction algorithm[Eq. 1] with an additional term F��, �r� whose mag-nitude can be determined computationally by mini-mization of the discrepancies between radiancetransfer models and IOP measurements. However,the significance of the IOP measurement errors willbe determined not only by F��, �r� but also by ai��r�,the magnitude of the residual scattering signal at 715nm. Since the latter term generally varies with sus-pended particle concentration, the greatest errors areexpected in coastal and shelf seas.

Fig. 11. Absorption values obtained with the modified scattering correction algorithm and F��, �r� � 1.0 are generally lower thanstandard (F��, �r� � 1.0) values of absorption. Higher levels of scattering at ST12 than at ST16 are responsible for the greater differencebetween standard and modified absorption values at ST12 despite the optimal value of F��, �r� being lower for this station.


One approach to obtaining appropriately correctedmeasurements of the absorption and scattering coef-ficients in these waters would be to extend themethod described here to all wave bands by deployingspectrally matched instrument packages. It may benecessary to carry out further Monte Carlo simula-tion of the reflecting tube design15,16 with appropri-ate wavelength dependency being prescribed for allIOPs, including the scattering phase function.17 Al-ternatively, a number of relations between IOPs andapparent optical properties have been published thatwould allow IOPs to be derived from radiometricmeasurements with reduced precision but greaterfreedom from systematic errors.18–20 Whichevermethod is adopted, improvement in the accuracy ofIOP measurements is urgently required for validat-ing remote-sensing products and for interpretingdata from moored arrays of optical sensors in coastalseas.

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evidence for wavelength dependence of the scattering phasefunction and its implication for modeling...

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