Role of measurement uncertainties in observedvariability in the spectral backscattering ratio:a case study in mineral-rich coastal waters
David McKee,1,* Malik Chami,2 Ian Brown,1 Violeta Sanjuan Calzado,3
David Doxaran,2 and Alex Cunningham1
1Physics Department, University of Strathclyde, 107 Rottenrow, Glasgow, G4 ONG, Scotland2Laboratoire d’Océanographie de Villefranche, Université Pierre et Marie Curie–Paris 6,
Unité Mixte de Recherche CNRS 7093, BP 28, F-06234Villefranche-sur-Mer Cedex, France
3National Oceanography Centre Southampton, University of Southampton, Waterfront Campus,European Way, Southampton SO14 3ZH, United Kingdom
The radiative transfer equation, which describes thepropagation of photons through natural waters, re-quires knowledge of both the absorption and theangular scattering characteristics of the medium.
Despite recent advances, in situ measurements ofthe volume scattering function (VSF) over the fullrange of scattering angles are technically demandingand remain relatively scarce [1–4]. It has beenshown, however, that the scattering phase functioncan be reasonably approximated from the back-scattering ratio [5]. A new generation of commercialin situ sensors has been developed that providesmeasurements of the required optical parameters:
the coefficients of absorption, attenuation, and back-scattering (with scattering obtained by subtraction ofabsorption from attenuation). These sensors havebeen deployed in a wide variety of natural watersacross the globe and large data sets now appear inthe literature [6–8].An understanding of systematic and random mea-
surement uncertainties is required prior to analysisof the measured optical properties. For example, anymeasurement of an inherent optical property (IOP),such as absorption, is likely to be subject to systema-tic error due to incomplete accounting for scatteringcollection error [9–12]. Other IOPs are derived frommeasurements that require assumptions about theshape of the VSF that may not be universally applic-able [13]. Furthermore, each instrument will have acharacteristic noise level that introduces a degree ofrandom uncertainty. For parameters such as thebackscattering ratio, where combinations of mea-surements are involved, the random uncertainty ispotentially further enhanced by inhomogeneity ofmaterial distributions on small physical scales andthe fact that each contributing measurement is notmade on exactly the same sample of water [14].Furthermore, pumped flow-through instruments(such as the WETLabs AC9) could be affected by dis-ruption of aggregates of particles, while backscatter-ing sensors are much less likely to disturb theparticle population in this manner.The backscattering ratio is a particularly impor-
tant parameter as it is widely used as a proxy todetermine scattering phase functions [5]. The parti-culate backscattering ratio (bbp=bp) may also be use-ful as a proxy for particle composition as it can berelated to particle size distribution (PSD) and refrac-tive index [15]. Theoretical studies have suggestedthat the particulate backscattering ratio should bewavelength independent if the particles are notstrongly pigmented, and the particle size distribu-tion follows the commonly assumed power-law distri-bution [16]. However, it has been shown thatpigmented particles following a power-law size dis-tribution would have a spectrally variable backscat-tering ratio through the influence of the imaginarycomponent of the refractive index [17,18]. The litera-ture contains conflicting views on whether fieldmeasurements support a wavelength-dependentbackscattering ratio. Studies in coastal waters havefound evidence for wavelength-dependent scatteringphase functions [4,7,10,19], while other studies withmore open ocean stations suggest that bbp=bp is wa-velength independent [8,18]. There are a number ofreasons why it is important that these differingviews are resolved. For example, the widely usedscattering correction procedure for AC9 absorptionmeasurements proposed by Zaneveld et al. [20] isbased on an assumption of a wavelength-independent scattering phase function. Models ofunderwater light fields for remote sensing andprimary productivity studies are also dependent on
assumptions about the nature of the scattering phasefunction.
This study examines the role of measurement un-certainty in the determination of bbp=bp. Althoughattempts have already been made to quantify instru-ment uncertainties, a new approach was adoptedusing the analysis of in situ signals. The aim is to de-velop an understanding of observed variability inparticulate backscattering ratios derived from in situIOP measurements and to establish a robust methodfor determining the wavelength variability of bbp=bp.Data are considered both as individual sets ofmeasurements for each location and depth, and asan assemblage of observations representative ofthe optical properties of the region.
2. Methods
A. Optical Measurements
Absorption (an) and attenuation (cn) coefficients fornon-water materials were measured at nine wave-lengths (412, 440, 488, 510, 532, 555, 650, 676,and 715nm) in the visible–NIR with a WETLabsAC9 calibrated with Milli-Q ultrapure water.Absorption was corrected for scattering effects usingthe method of Zaneveld et al. [20]. This method wasused for consistency with other data sets, but it isbased on assumptions of wavelength independencefor the scattering phase function and zero absorptionat 715nm, which may not hold for the waterssampled [10,21]. Data were corrected for tempera-ture and salinity effects using the coefficients ofSullivan et al. [22] and data from a SeaBird SBE19-Plus CTD (conductivity, temperature, and depth) in-strument. Particulate scattering (bp) was obtainedfrom bp ¼ cn − an. Visible–NIR particulate backscat-tering was determined from VSF measurementsmade with a WETLabs ECO-BB9 (412, 440, 488,510, 532, 595, 660, 676, and 715nm). The instrumentwas calibrated by the manufacturer prior to shippingshortly before the cruise, and data were corrected forpath-length absorption effects using the correctionfactor provided by the manufacturer. The conversionfrom VSF to bbp was performed using the χp factorfrom Boss and Pegau [23], though we note that thisis subject to some variability [13]. The AC9 and BB9did not have completely matched wavelengths, anddata were interpolated to give bbp at 555 and650nm. All data were averaged into 1m depth bins.The number of samples varies between 572 and 742data points, according to wave band, as the BB9 sen-sor saturates at different backscattering levels foreach channel. A HOBI Labs Hydroscat-2 was usedat a subsample of stations (334 data points per chan-nel) to provide backscattering data at 470 and676nm. Hydroscat-2 data were corrected for path-length absorption effects using coefficients providedby the manufacturer. AC9 data were linearly inter-polated to provide bp to match Hydroscat-2 bbp at470nm. The Hydroscat-2 676nm channel has a20nm FWHM filter to permit dual use as a
chlorophyll fluorometer. There is, therefore, poten-tial for fluorescence contamination of this channel.However, it should be noted that the signal was domi-nated by mineral backscattering in this region andthat the effect of any fluorescence contaminationwould be to increase the apparent backscatteringat this wavelength. Correction for fluorescence ef-fects on the 676nm Hydroscat-2 backscatteringchannel would, therefore, enhance any observedwavelength dependency in the backscattering ratio.The WETLabs BB9 and HOBI Labs Hydroscat-2
sensors have different optical geometries, thoughboth make wide angle measurements of volume scat-tering that require extrapolation to give bbp. Theyare calibrated using quite different methodologies.The BB9 is calibrated using polystyrene micro-spheres in a series of dilutions, with Mie theory usedto calculate volume scattering function values foreach set of dilutions. This calibration method re-quires knowledge of the scattering collection geome-try and wavelength bandwidth for each opticalchannel, the refractive index and size distributionof the beads, and a method to normalize measure-ments to the total scattering coefficient (currentlyrealized by making simultaneous measurements ofattenuation with a WETLabs AC9). Each of these as-pects of the manufacturer’s calibration procedurecarries an associated uncertainty that is currentlydifficult to quantify. The Hydroscat-2 is calibratedby measuring the scattered signal from a submergedreflectance target that traverses the scattering vo-lume of the sensor [24]. Here the reflectance of thetarget is critical in determining the uncertainty inthe calibration. In both cases, the purpose of the ca-libration exercise is to determine the calibrationslope that relates the measured signal (minus a darksignal) to the VSF corresponding to the sensor opticalgeometry. Whitmire et al. [8] provided a detailed ana-lysis of the likely errors affecting Hydroscat and AC9calibration and reached the conclusion that the likelymaximum error in bbp=bp would be around 20%.Their estimate erred on the side of caution in the pro-pagation of errors and was greater than the ∼10%differences in bbp=bp found in comparisons madeamong different instruments and methods [14].The present study focuses primarily on the effectof other measurement uncertainties, such as limita-tions in the performance of the AC9 scattering cor-rection and random uncertainties due to differentsensors not measuring the same sample volume.However, uncertainty in the calibration slope isclearly a serious issue, particularly when discussingobservations of spectral variability in bbp=bp.
B. Sample Analyses
Chlorophyll samples were filtered through 25mmGF/F filters and immediately frozen. Once in the la-boratory, the filter papers were soaked for 24h inneutralized 90% acetone, and the absorbance ofthe extract measured in a Shimadzu UV-2501PCspectrophotometer using 1 cm path-length cuvettes
before and after acidification with dilute hydrochlo-ric acid. The trichromatic equations of Jeffrey andHumphrey [25] were used to convert absorbancespectra to concentrations of chlorophyll a (Chl). Allsamples were measured in triplicate. Total sus-pended solids (TSS) samples were obtained by filter-ing 5 liters of seawater through preweighed 90mmGF/F filters and rinsing with 50ml of distilled water.Samples were stored frozen until returned to the la-boratory, where they were dried in an oven at 100 °Cfor 3h and reweighed. The concentration of mineralsuspended solids (MSS) was obtained by reweighingsamples after they had been placed in a furnace at500 °C for 3h, at which point it was assumed thatall organic materials had been combusted. Coloreddissolved organic materials (CDOM) samples werefiltered through 0:2 μm membrane filters, with thefiltrate being collected in acid-rinsed glass bottleswith Nalgene caps and stored under refrigeration.Absorption by CDOM was measured in theShimadzu UV-2501PC spectrophotometer using10 cm cuvettes and UV treated ultrapure water asa reference. Given the unknown and probably com-plex chemical composition of CDOM, the absorptioncoefficient of the filtrate material at 440nmwas usedas a proxy for the concentration of CDOM.
3. Results and Discussion
A. Location of Experiment
Previous studies have presented data from a widerange of water types. Indeed, one study claims tobe representative of the global ocean [8]. There aretwo issues with this approach. The first is the under-lying assumption that a global trend exists that canbe exposed by sampling enough points. There is noreason to suppose that there is a globally relevantbbp=bp anymore than there is an equivalently globalspecific absorption coefficient. The second issue is thedifficulty in identifying local trends and relation-ships in the midst of very large data sets. In thisstudy we examine data from a single cruise in theBristol Channel (Fig. 1). This shallow, macrotidal es-tuary experiences very strong resuspension of sedi-ment, giving a broad range of scattering signalswithin a limited geographic range. The mineral com-ponent of total suspended particulate material(MSS) reached concentrations as high as 15 gm−3
for the optical data presented here (at which pointall of the BB9 optical channels were saturated),corresponding to a median value of 82% of the totalsuspended particulate material. The maximum Chlconcentration was less than 3:5mgm−3 with a med-ian value of 1:25mgm−3. Figure 2 shows all the ab-sorption, attenuation, scattering, and backscatteringspectra obtained for this study. Figure 2(d) illus-trates the manner in which each backscatteringchannel reached saturation at different levels of tur-bidity, with blue-green channels able to operate overconsiderably greater dynamic ranges than red-NIRchannels. Figures 2(b) and 2(c) show attenuation
and particulate scattering spectra with consistentspectral shapes across the full range of observations.Absorption spectra [Fig. 2(a)] show strong increasestoward blue wavelengths, consistent with strong con-tributions from mineral particles and/or CDOM.There are also significant absorption peaks at676nm for some of the samples, suggesting an influ-ence of phytoplankton on red absorption values. Inthis study we are primarily interested in particulatescattering and backscattering, and it is necessary toassess to what extent these parameters are influ-enced by either MSS or Chl. Coefficients of determi-nation obtained by linear regression of bbp, bp, and anagainst MSS and Chl are given in Table 1. MSS ac-counts for between 65% and 87% of observed varia-bility in bbp, with lower values in the red-NIR due toreduced dynamic range at these wavelengths as a re-sult of sensor saturation. MSS accounts for ∼85% ofobserved variability in bp for all wavelengths. The in-fluence of Chl on either bbp or bp is minimal acrossthe data set, with maximum values of 5% for red-NIR bbp and <1% for all other bbp and bp. MSS alsoaccounts for 85% of observed variability in an, withthe exception of 676nm, where it only accounts for65% of the variability. Chlorophyll accounts for31% of variability in an at 676nm, 5% at 650nm,and is elsewhere insignificant (<1% observed varia-bility). The median absorption by CDOM at 440nmwas 0:16m−1 with a maximum value of 0:50m−1.
Over the entire data set, CDOM absorption onlyaccounted for ∼7% of the observed variability innon-water absorption at 440nm. From this we canconclude that IOPs in the Bristol Channel arestrongly influenced by mineral particles, althoughthere are instances where absorption at 676nm is ob-viously affected by algal pigments. The fact that par-ticulate scattering and backscattering are dominatedby a population of nonbiogenic mineral particles isimportant in this context as it reduces the complexityof the data set for subsequent analysis, and we cananticipate that there ought to be less variability inbbp=bp than would be observed in a data set coveringwaters with a wide variety of particle composition. Ofcourse, there may remain instances in the data set,particularly specific locations and/or depths whereMSS is low, where phytoplankton may have a signif-icant influence on bp or bbp, but we have demon-strated that, when we treat these parameters asan assemblage representative of the region, the influ-ence of MSS is dominant.
B. Statistical Approaches
In this study we analyze variability in the particulatebackscattering ratio using two different approaches.The first is a point-by-point method where individualmeasurements of particle backscattering, bbp, are di-vided by corresponding individual measurements ofparticle scattering, bp, and variability is assessedusing descriptive statistics of the resulting distribu-tions of particulate backscattering ratio, bbp=bp.The second approach uses linear regression to findbest-fit values of bbp=bp for the data set as a whole.
C. bbp=bp from a Point-by-Point Approach
Figure 3 shows the distribution of particulatebackscattering ratios calculated from individualmeasurements of bp and bbp for this data set. The532nm channel has been selected as generally repre-sentative of the distributions found for all the mea-sured wavelengths. bbp=bp appears to vary over anorder of magnitude with values ranging from0.0065 to 0.0675, a mean value of 0.031, and a med-ian value of 0.029. This is a very broad range, evengreater than that presented byWhitmire et al. for theglobal ocean [8], and is most surprising given the re-stricted geographic range of the sample locations. Ifthese data were taken at face value, they wouldsuggest marked variability in material composition,particle size distribution, or both. Of course, it isalso necessary to consider other potential sourcesof apparent variability, including measurementuncertainties.
D. bbp=bp from a Regression Approach
The backscattering and scattering values used togenerate the distribution of bbp=bp shown in Fig. 3are plotted against each other in Fig. 4. As there ismeasurement uncertainty in both x and y variables,geometric mean regression was used to determinethe best-fit line through the data. The best-fit line
Fig. 1. Map of the British Isles with the location of the samplingarea in the Bristol Channel marked with a rectangle. The lowerpanel shows station positions in the Bristol Channel marked bycrosses.
accounts for 96% of the observed variability and has aslope of 0.0479 with a 95% confidence interval of�0:0007. The best-fit offset (−0:0109) is both largeand statistically significant. This is unexpected (zero
values of bbp and bp ought to coincide) and indicates asystematic offset in the measurement data, either anunderestimate of bbp or an overestimate of bp. Giventhat the calibration of the backscattering sensor
Fig. 2. IOP spectra for the full range of data collected in the Bristol Channel up to the point where all BB9 backscattering channelsreached saturation. (a) Non-water absorption spectra increase strongly toward blue wavelengths, typical of strong mineral and/or CDOMabsorption. Some stations show the influence of phytoplankton pigment absorption at 676nm. (b), (c) Non-water attenuation and parti-culate scattering spectra show strong spectral consistency across the range of signals encountered. (d) Particulate backscattering spectradecrease toward both the blue- and red-NIR ends of the spectrum. Note that red-NIR channels reached saturation at much lower levels ofturbidity than blue-green channels.
Table 1. Coefficients of Determination for Linear Regressions of IOPs against MSS and Chl
λ (nm) bbp Versus MSS bbp Versus Chl bp Versus MSS bp Versus Chl an Versus MSS an Versus Chl
determines a slope factor, it is difficult to see how thisprocess could generate a systematic offset error.Furthermore, the AC9 calibration was performed towithin the manufacturer’s specified limits in ourlaboratory. One possible explanation is that the bp va-lues are overestimated as a result of the AC9 scatter-ing correction, which assumes zero absorption at715nm. If this assumption is not valid in thesewaters[21], the absorption would be underestimated and,since bp ¼ cn − an, scattering values would be overes-timated. One advantage of the regression approach isthat the slope of the best-fit line is unaffected by thepresence of an offset error in either parameter, andrepresents an estimate of the best-fit bbp=bp for theentire population. Of course, the regression slopecould still potentially be affected by errors in the cali-bration slope of the backscattering sensor, but a valueof bbp=bp ¼ 0:0479 is consistent with the presence ofnonbiogenic mineral particles [15] and with previousvalues found in similar water types elsewhere in theIrish Sea [19].Another potential benefit of the regression ap-
proach is the predictive power of the resultingbest-fit parameters. Figure 4(b) shows that the meanpoint-by-point estimate of bbp=bp has poor predictivequalities, and that the apparent range of possible va-lues suggested by the point-by-point range of bbp=bpstrongly overstates the true variability observed inthe overall population. This should be compared withthe 95% prediction interval shown in Fig. 4(a), whichgives a significantly better representation of wherefuture measurements are expected to fall. It shouldbe noted that these point-by-point values of bbp=bpare probably subject to underestimation due to the
potential offset error in bp. Offset correction of bpand recalculation of bbp=bp results in a significantlymodified distribution that is less skewed toward lowvalues but retains a very broad range of values. Themedian of the distribution changes from 0.0294 to0.0473, much closer to the best-fit regression valueof 0.0479. These results emphasize the potential im-pact of offset errors on point-by-point estimates ofbbp=bp, even though it has to be noted that errorsin the calibration slope for bbp measurements couldcause significant differences for both point-by-pointand regression estimates of bbp=bp.
Fig. 3. Distribution of particulate backscattering ratios at532nm calculated on a point-by-point basis for Bristol Channeldata shows an apparent order of magnitude variability.
Fig. 4. Particulate backscattering plotted against particulatescattering with (a) best-fit regression (solid line), 95% confidenceinterval for the slope (dashed line) and 95% prediction interval(dashed-dotted line), and (b) point-by-point mean (solid line)and minimum/maximum range values (dashed lines).
E. Measurement Uncertainty and Apparent Point-by-PointVariability
Figure 4 demonstrated that the point-by-point ap-proach significantly overstated the variability inbbp=bp for regional predictive purposes. One potentialmechanism for the apparent variability given by thepoint-by-point approach is random measurementuncertainty in both bbp and bp. After all relevant cor-rections have been applied, the outputs from in situinstruments are measured values of scattering(bpm) andbackscattering (bbpm). Thesediffer fromtruevalues of bp and bbp throughmeasurement uncertain-ties that we can call εbp and εbbp. As a result, the trueparticulate backscattering ratio is given by
bbpbp
¼ bbpm � εbbpbpm � εbp
: ð1Þ
If εbp and εbbp are small, bbpm=bpm will tend toward thetrue value. However, if measurement errors are sig-nificant, bbpm=bpm may deviate significantly fromthe true value. Manufacturers provide estimates ofinstrument noise that are related to electrical signalnoise, optical detector noise, etc. However, when in-struments are deployed in natural waters additionalfactors come into play (e.g., measurements are notmade on identical sample volumes, theremay be localheterogeneity of the sample, and pumping samplesinto the AC9 may cause aggregate disruption) sothat themanufacturer’s noise specifications for an in-dividual channel are not necessarily relevant in thiscontext. Instead, a measure of the overall randommeasurement uncertainty due to instrument noisecombined with sample volume heterogeneity is re-quired. Oneway to estimate this is to examine signalsfrom two channels of the same instrument that areclosely spaced spectrally. Assuming that wavelengthdependence of bp and bbp for the population is smalland reasonably uniform for a small wavelength differ-ence, we can use residual analysis to estimate thecombined measurement uncertainty of the bp andbbp signals.Figure 5(a) shows particulate scattering at two
wavelengths (510 and 532nm) plotted against oneanother for the entire data set. The geometric meanregression line has a very small offset and a coeffi-cient of determination almost equal to unity, suggest-ing that spectral variability between the twowavelengths for this population is well accountedfor by the slope value of 1.0094. The correspondingplot of bbp532 versus bbp510 [Fig. 5(b)] demonstratesvery similar characteristics, with the spectral varia-bility between these two sets of measurementsalmost completely accounted for (r2 ¼ 0:99) by thegeometric mean regression. Residual variability be-tween each set of measurements can then be attrib-uted to random measurement uncertainties.Figures 5(c) and 5(d) show residual bp and bbp values,obtained by subtracting values predicted from the re-gressions shown in Figs. 5(a) and 5(b), plottedagainst measured bp and bbp, respectively. These
plots show that the measurement uncertainty rangefor each parameter does not increase significantlywith increasing scattering signal, indicating thatthe range has a constant value, rather than a percen-tage or fractional figure. The magnitude of the rangecan be approximately estimated by taking the 95thpercentile points of the absolute distribution, giving�0:009m−1 for bbp and �0:026m−1 for bp.
The fact that the measurement uncertainty rangefor each signal is constant, rather than proportionalto the signal, is highly significant when we assess thesource of apparent variability in the point-by-pointbackscattering ratio. In this analysis we take thebest-fit relation between bbp and bp from Fig. 4(a),use it to correct measured bp values by subtractingthe observed offset (0:0109=0:0479 ¼ 0:2276), andthen calculate point-by-point bbp=bp values usingthe corrected bp data. Figure 6 shows point-by-pointbbp=bp plotted against bp for the entire data set at532nm. Also shown is the best-fit estimate ofbbp=bp (¼ 0:0479) from the regression in Fig. 4(a).It can be seen that the greatest apparent variabilityin point-by-point bbp=bp occurs for small bp and thatvalues tend toward the best-fit regression valuewhen the scattering signal increases. We can now de-monstrate that this behavior is directly attributableto random measurement uncertainties. Using Eq. (1)with values of bpm from 0 to 5m−1, corresponding pre-dicted values of bbpm obtained using the best-fit slopeof 0.0479, and the uncertainty ranges for εbp and εbbp(�0:026 and �0:009m−1) found previously, it is pos-sible to determine the bounds of parameter spacethat can be attributed to measurement uncertain-ties. These are shown as dashed curves in Fig. 6.A large number (>60%) of observations fall withinthese bounds, and the bounds increase in widthrapidly as the scattering signals decrease. This isa direct consequence of constant values of εbp andεbbp. In effect, much of the apparent variability ob-served in point-by-point particulate backscatteringratios can be attributed to signal-to-noise issues be-coming significant at low signal values.
F. Wavelength Dependence of the ParticulateBackscattering Ratio
Aprevious study using a point-by-point approach andwavelengths between 442 and 620nm found thatspectral variability in mean values of bbp=bp wasdwarfed by variability in the magnitude of thebbp=bp estimate at each wavelength [8]. This resultcanbe reproducedwithdata fromtheBristolChannel.Figure 7(a) shows mean values of bbp=bp, plotted as afunction ofwavelength, togetherwith standard devia-tions and maximum/minimum range values. Usingthe point-by-point approach, there is appreciablespectral variability between blue- and red-NIR va-lues, but it is largelywithin theboundsof the standarddeviations of each measurement. Interestingly, Huotet al. [18] also show significantly lower bbp=bp at650nm compared to blue and green wavelengthsusing a point-by-point approach for data from very
clear South Pacific Ocean waters. Extension of thewavelength range out to the NIR does give a greatersense of wavelength dependency than both the Huotet al. [18] and Whitmire et al. [8] observations, butit would be difficult to justify a definitive conclusionof wavelength dependency taking this approach.However, performing geometric mean regressions inthe same manner as Fig. 4(a) for each wavelength,and taking the best-fit slope as the best estimate ofbbp=bp for the population, we obtain an alternativeview of the spectral dependency of the particulatebackscattering ratio. Figure 7(b) shows best-fit esti-mates of bbp=bp as a function of wavelength and asso-ciated 95% confidence intervals for the regressionslopes. Coefficients of determination vary between0.98 and 0.88 across the spectral range (decreasingwith wavelength as scattering signals decrease) withoffsets of the same magnitude and sign (−0:006 to−0:015) as the 532nm value. These regression-basedestimates of bbp=bp for the Bristol Channel data aregenerally higher than point-by-point mean values
(which are potentially underestimated due to the off-set error in bp discussed earlier), but show a similarspectral dependence, with bbp=bp generally decreas-ing toward the red-NIR. The regression value ofbbp=bp at 715nm is ∼57% of the value at 440nm, re-presenting a significant reduction in magnitude withwavelength. These values of bbp=bp are statisticallysignificant, have good predictive power [Fig. 4(a)],and their wavelength dependence is consistent withprevious results from similar waters elsewhere inthe Irish Sea area [10]. Figure 7(b) also showsbbp=bp at 470 and 676nm calculated using indepen-dent measurements of bbp from a Hydroscat-2 instru-ment deployed at the same time as the BB9. TheHydroscat-2 observations confirm the magnitude ofthe BB9 signals (to within ∼10%) and also the wave-length dependence. Correction for the effect of possi-ble fluorescence contamination of the Hydroscat-2676nm channel would further reduce bbp=bp relativeto the 470nm channel. The fact that such similarresults are obtained independently from two back-
Fig. 5. Plots of (a) bp532 versus bp510 and (b) bbp532 versus bbp510 show that virtually all the spectral variability between these data setscan be accounted for with geometric mean regression best-fit lines. These regressions are used to derive residual values of (c) bp532 and(d) bbp532, which show that measurement uncertainty ranges are constant for each parameter.
scattering meters calibrated by separate manufac-turers using significantly different procedures pro-vides a degree of confidence in our results. It isinteresting to note that the discrepancy in our esti-mates of bbp=bp from the two different sets of instru-ments is rather close to the∼10% found by Boss et al.[14].Thus the regression-derived values ofbbp=bp sug-gest that the particle backscattering ratio is wave-length dependent for these mineral-rich waters.There remains the possibility, however, that our datacould be affected by systematic biases in the calibra-tion slopes of both of the backscattering sensors. Herewe examine themagnitude of backscattering slope ca-libration error that would be required to maintain aspectrally flat model for bbp=bp. Figure 7(b) showsdotted curves corresponding to�30% errors in the ca-libration slope for the BB9 and Hydroscat-2 sensors.This level of calibration slope error would be just suf-ficient to eliminate spectral dependency in bbp=bp.However, even with this magnitude of error incalibration slope (which is significantly greater thanWhitmire et al. [8] predicted for the Hydroscat-2),we would require a systematic overestimate forblue-green wavelengths and an underestimate forred wavelengths that cannot readily be explained.Alternatively, the 715nm channel of the BB9 sensorwould have to underestimate bbp by a factor of 0.6in order to maintain a model of spectrally flatbbp=bp. We conclude that the field data indicate thatthe particulate backscattering ratio is wavelength de-pendent in theBristolChannelunless therearesignif-icant undocumented uncertainties in the calibration
of two widely used oceanographic backscatteringsensors.
G. Relationship between Spectral Shape of bbp=bp andParticle Size Distribution
Morel and Bricaud [17] demonstrated that the parti-culate backscattering ratio depends on the real and
Fig. 6. Point-by-point values of bbp=bp calculated with offset-corrected bp values show strong apparent variability at lowscattering signals but tend toward the best-fit regression valueof bbp=bp as scattering increases. More than 60% of observedpoint-by-point bbp=bp values fall within the region (dashed curves)that is accounted for by measurement uncertainties.
Fig. 7. (a) Wavelength dependence of point-by-point bbp=bp is in-significant compared to the apparent variability in magnitude ateach wavelength. Standard deviations are shown with thicker er-ror bars, maximum and minimum range values with thinner bars.(b) bbp=bp from regression slopes have tight confidence intervalsand show potentially statistically significant wavelength depen-dency. bbp=bp calculated using Hydroscat-2 data (black circles)shows similar magnitude (�10%) and wavelength dependency.Dotted lines indicate intervals representing �30% uncertaintyin the slope calibration of the BB9. This is the minimum calibra-tion slope error that could result in a spectrally flat particulatebackscattering ratio.
imaginary components of the refractive index andthe particle size distribution. Ulloa et al. [16] showedthat assumptions of a power-law size distributionand negligible particle absorption generated spec-trally flat particulate backscattering ratios, whichis an attractive simplification for radiative transfermodeling. Given that our field data apparently di-verge strongly from that scenario, it would be usefulto consider what possible characteristics of naturalparticle populations could produce wavelength de-pendence in the particulate backscattering ratio.Consequently, Mie theory was used to calculateparticulate backscattering ratio spectra for selectedhypothetical particle size distributions and particlerefractive indices to determine if our observationsof wavelength-dependent bbp=bp are physicallyrealistic. Calculations were performed for particlediameters from 0.01 to 100 μm and particulate back-scattering ratios were calculated from
bbpbp
¼R ππ=2 βðθÞ sin θdθR π0 βðθÞ sin θdθ ; ð2Þ
where βðθÞ is the VSF at scattering angle θ. Moreland Bricaud [17] showed that bbp=bp decreases withincreasing absorption (and, hence, imaginary refrac-tive index, n0) for both monodispersed and polydis-persed particle populations. The results of Ulloa et al.[16] suggested that the opposite is true for power-lawparticle size distributions (Ref. [15], Fig. 3), but thisis not supported by results presented here. Babinet al. [26] suggested that the imaginary refractive in-dex for mineral particles could be modeled so that n0increases exponentially with decreasing wavelengthin a manner consistent with observed spectral ab-sorption signals [Fig. 8(a)]. Figure 8(b) shows that in-creasing n0 in the manner of Babin et al. [26] formineral particles with an assumed real refractive in-dex, n ¼ 1:15 (relative to seawater), and a power-lawsize distribution with a slope factor, ξ ¼ 3, results ina reduction in bbp=bp in the blue relative to the red.In contrast, our field data shows bbp=bp in the blue tobe generally greater than at red wavelengths. Thus,although we cannot afford to ignore absorption ef-fects, they are insufficient on their own to explainour observed spectral variability in bbp=bp.Chami et al. [4] previously found that major angu-
lar features observed in spectral ratios of the VSFcould only be accounted for by including both absorp-tion effects and non-power-law size distributions intheir Mie calculations. In order to observe the effectof varying particle size distribution in isolation, weset the imaginary refractive index to a constant va-lue, n0 ¼ 0:001. Figure 9 shows hypothetical particlesize distributions and corresponding bbp=bp spectra.Increasing the slope of a power-law particle size dis-tribution increases the magnitude of bbp=bp but doesnot introduce any significant wavelength depen-dence, which is consistent with the results of Ulloaet al. [16]. However, introduction of an additionalmode of particles, in this case a lognormal component
centered in the submicrometer diameter range[Fig. 9(b)], generates considerable wavelength varia-bility in bbp=bp, which is more consistent with ourfield observations. These calculations are not pre-sented as a fit to our data. We have not attemptedto model likely spectral variability in the imaginaryrefractive index and would have too many unknownvariables to be confident in any fit. Furthermore, Mietheory may not adequately describe the scatteringproperties of natural particles with a range ofmorphologies and shapes that might diverge moreor less strongly from sphericity. However, Mie calcu-lations serve to illustrate the fact that deviation from
Fig. 8. Particulate backscattering spectramodeled usingMie the-ory with real refractive index n ¼ 1:15 and a power-law size dis-tribution with slope ξ ¼ 3. (a) The imaginary refractive index ismodeled to increase exponentially with decreasing wavelengthin a manner consistent with observed mineral absorption charac-teristics. (b) The effect of mineral absorption is to depress bbp=bp inthe blue (dashed curve) relative to the case with zero absorption(solid curve).
a power-law size distribution can introduce signifi-cant wavelength dependence in bbp=bp. In reality,the observed spectral variability in bbp=bp could bedue to the combined effects of both particle absorp-tion and deviation from a power-law size distribu-tion. Furthermore, it is possible that naturalparticle size distributions may include subpopula-tions with different refractive indices. Risović [27]has examined the potential impact on the scattering
properties of natural waters of multimodal, non-power-law PSDs with variable refractive indicesfor each component, and observed a degree of spec-tral variability in bbp=bp as a result of invoking amore complex particle size/refractive index model.The complexity of modeling such a system is beyondthe scope of this paper, but it seems clear that ourobservation of wavelength-dependent bbp=bp isphysically realistic if deviations from power-law sizedistribution and absorption effects are considered.
H. Inherent Optical Property Measurement Uncertainties
The ability to make in situ measurements of IOPs isa relatively recent advance and one that has yet toreach full maturity. A key step in this process is de-veloping an understanding of random and systematicmeasurement uncertainties, assessing their magni-tude, and working toward improved correction proce-dures. In this study we have seen evidence of bothsystematic errors (e.g., the offset between bbp andbp) and random measurement uncertainties. Wehave observed the impact that such measurementuncertainties can have on estimates of particulatebackscattering ratio, particularly when the point-by-point approach is taken. It is essential thatfurther efforts are made to understand measurementuncertainties in other optical water types. For exam-ple, the random uncertainty ranges observed for thetidally dynamic Bristol Channel may be larger thanin oceanic waters where small-scale spatial variabil-ity may be less marked. This could explain why it ispossible to make high-quality measurements in clearwaters with essentially similar equipment to thatused in this study [6,18,28].
I. Point-by-Point Versus Regression Approaches
This paper compared estimates of bbp=bp from point-by-point and regression analysis approaches andfound that the point-by-point approach was highlysensitive to the effect of both random and systematicmeasurement uncertainties, particularly when scat-tering signals were low. Equation (1) gives a clearindication of why this is so. In this study, we wereparticularly fortunate to have measurements forwaters with a strongly dominant class of particlesgenerating a broad range of scattering magnitudes.This made it possible to successfully use geometricmean regression to determine an alternative esti-mate of bbp=bp for the population, which carriesgreater confidence because regression analysisweights random uncertainties against the wholesignal range rather than individual signals, andthe regression slope is independent of systematic off-set errors. However, it should be noted that this ap-proach does not resolve potential errors associatedwith the backscattering slope calibration and is onlyappropriate when a single population dominates anda reasonably broad range of signal magnitudesrelative to the measurement uncertainty canbe found.
Fig. 9. Particulate backscattering spectra modeled by Mie theoryfor power-law size distributions with constant values of real andimaginary refractive index (dashed and dotted curves) exhibitno significant spectral variability. Introducing a separate mode(lognormal in this case) of particles is enough to introduce signifi-cant wavelength dependence in the bbp=bp spectrum (squares).bbp=bp from regression analysis of field data (circles) is shownfor an order of magnitude comparison, but there are insufficientdata available to perform a full theoretical validation of the fieldvalues of bbp=bp.
J. Wavelength-Dependent bbp=bp and Particle SizeDistribution
Spectral dependence in the particulate backscatter-ing ratio is inconsistent with a power-law distribu-tion of nonpigmented particles [16], and it is unlikelythat inclusion of realistic wavelength-dependentimaginary refractive indices would be sufficient toproduce a good fit with our field observations [4].However, the introduction of additional modes of par-ticles is sufficient to induce spectral variation inbbp=bp of the magnitude required to accommodateour results. Although we have insufficient data tofully model our observations using Mie theory, it isclear that the scope for understanding spectrally de-pendent bbp=bp is greatly increased if non-power-lawparticle size distributions and absorption effects areconsidered together, as previously suggested byChami et al. [13]. It is equally important to realizethe potential role of submicrometer particles in de-termining the backscattering properties of naturalwaters [29,30]. Peng and Effler [31] recently pre-sented PSDs for minerogenic particles in a reservoirthat deviated strongly from a power-law distribution,and Wells and Goldberg [32] used electron micro-scopy to show non-power-law distributions for colloi-dal particles for open waters, such as the South andNorth Atlantic oceans. Considerably more informa-tion about PSDs, including submicrometer particles,over broader geographic ranges than are currentlyavailable, is urgently required. The impact of non-power-law PSDs on scattering properties, particu-larly bbp=bp, should be studied in more detail as well,potentially using earlier work by Risović [27] as astarting point.
K. Optical Implications of Wavelength-Dependent bbp=bpDevelopments in instrument technology are provid-ing us with new opportunities to understand thecomplexity of natural particle assemblages and theiroptical properties. In a relatively short time we havemoved from being reliant on a handful of in situmea-surements [2] to a situation where the full VSF isbeing measured with specialist equipment in a vari-ety of locations [3,4], and particulate scattering andbackscattering data sets regularly consist of thou-sands of samples [7,8,18]. Variability in the magni-tude of the particulate backscattering ratio haslong been recognized, but it is now becoming clearthat this may be coupled with spectral variabilityin some circumstances. It may be necessary in thefuture to include this variability in radiative transfercalculations, particularly when attempting to obtainclosure with radiometric measurements. Significantspectral variability in bbp=bp, such as observed here,violates a key assumption in the Zaneveld et al. [20]scattering error correction procedure for AC9 mea-surements [10]. It also calls into question the validityof assumptions of power-law size distributions, andthis has potential implications for estimating refrac-tive index from scattering signals [14,15].
4. Conclusions
Our analysis suggests that random measurementuncertainties can play a major role in overstatingvariability in the magnitude of the particulate back-scattering ratio. A key observation in this process isthat the random measurement uncertainty range isconstant, i.e., it does not scale withmeasurement sig-nal and, as a result, signal-to-noise ratio deterioratesas signal levels drop. Random measurement uncer-tainties could have similar impacts on other impor-tant optical parameters involving ratios of IOPs,such as material-specific IOPs. One outcome fromthis work is a new question: how much of the ob-served variability in material-specific IOPs can be at-tributed to random measurement uncertainty? Thiswill form the basis for future work on this topic, butit is already clear that we should not ignore IOPmeasurement uncertainties.
An original approach based on regression analysishas been used to parameterize measurement uncer-tainties and establish probable wavelength depen-dency in the particulate backscattering ratio forthe mineral-rich waters sampled in this case study.Limitations in current sensor technology precludea definitive statement for or against wavelength de-pendence of bbp=bp. Further work is needed todetermine to what extent other natural waters exhi-bit significant wavelength dependence in bbp=bp.However, application of Mie theory to hypotheticalPSDs has shown that wavelength dependency inbbp=bp can be generated by the effect of particleabsorption (nonzero imaginary refractive index) onscattering and backscattering, and/or deviation ofthe PSD from a power-law distribution. One can en-visage scenarios where these conditions could occur,such as resuspension of benthic material and the for-mation of algal blooms. Given the sensitivity ofbbp=bp to submicrometer particles and the scarcityof PSD data available for this size class, there arereasonable grounds for wondering if wavelength-dependent scattering phase functions might be thenorm rather than the exception in the global ocean.
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