Download Evidence for wavelength dependence of the scattering phase

Evidence for wavelength dependence of the
scattering phase function and its implication for
modeling 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 backscattering ratios in the blue (470 nm) than in the red (676 nm) wave band. Attempts to obtain optical
closure by use of radiance transfer modeling were least successful for stations at which backscattering
ratios are most strongly wavelength dependent. Significantly improved radiance transfer simulation
results were obtained with a modified scattering correction algorithm for AC-9 absorption measurements
that took wavelength dependency in the scattering phase function into account. © 2005 Optical Society
of America
OCIS codes: 000.0000, 010.4450, 290.5820.
1. Introduction
Using commercially available software such as Hydrolight1 to combine radiance transfer computations
with in situ measurements from modern optical instrumentation, it should be possible to construct realistic mathematical models of underwater light
fields and to predict remote-sensing reflectance accurately. Recent results indicate unexpected variability
in the degree to which this aim can be achieved.2–5
The variability has been attributed to poor quality
control of either the inherent optical property measurements used to constrain the models or the radiometric measurements used for model validation.4,5
However, since the degree of closure obtained varies
from one station to another in a given cruise and even
between wavelengths for a given station, it is likely
that the source of variation lies in the optical characteristics of the water body in which the measurements are being made rather than in the
measurement procedures. In this paper, we consider
whether the difficulties encountered in obtaining consistent closure between models and measurements in
shelf seas can be attributed to wavelength depen-
The authors are with Department of Physics, University of
Strathclyde, 107 Rottenrow, Glasgow, G4 0NG, Scotland. D McKee’s e-mail address is [email protected].
Received 23 December 2003; revised manuscript received 7 September 2004; accepted 21 September 2004.
0003-6935/05/010126-10$15.00/0
© 2005 Optical Society of America
126
APPLIED OPTICS 兾 Vol. 44, No. 1 兾 1 January 2005
dence of the scattering phase function. The methodology employed was developed for modeling
coccolithophore blooms,3 but it can be applied to other
waters where standard inherent optical property
(IOP) measurements are difficult to reconcile with in
situ radiometric profiles.
2. Theory
The IOPs of seawater are commonly measured with
an AC-9 dual-tube spectrophotometer (WET Labs,
Philomath, Oregon) and a Hydroscat backscattering
meter (HOBI Labs, Inc., Tucson, Arizona). The AC-9
provides coefficients of attenuation (cn) and absorption (an) for materials other than water, and the Hydroscat provides total backscattering coefficients (bb).
Data from both instruments are subject to correction
procedures that make assumptions about the optical
properties of the medium in which the measurements
are made. These assumptions may be valid for oceanic waters, but they are largely untested for shelf
seas. In this paper, we consider possible modification
of the correction procedure for the AC-9 because the
manufacturer has published a critical analysis of the
geometric and optical principles involved. It is likely
that Hydroscat correction procedures should also be
examined critically in coastal waters, but we only
consider whether the errors involved invalidate the
main thrust of the argument.
A.
AC-9 Correction
It has been reported3 that removal of the assumption
of a wavelength-independent scattering phase func-
tion from the analysis by Zaneveld et al.6 leads to a
modified version of the scattering correction algorithm for reflecting tube absorption measurements of
the form
an共␭兲 ⫽ ai共␭兲 ⫺ ai共␭r兲
ci共␭兲 ⫺ ai共␭兲
ci共␭r兲 ⫺ ai共␭r兲
F共␭, ␭r兲, (1)
where an is the corrected estimate of the absorption
coefficient for materials other than water, ai and ci
are the uncorrected instrument measurements of absorption and attenuation, ␭ denotes the wavelength
at which the corrected value is calculated, and ␭r is
the reference wavelength (715 nm) at which an is
assumed to be zero. Equation 1 differs from the original expression by the presence of an extra factor
F共␭, ␭r兲 that can be written in full as
F共␭, ␭r兲 ⫽
冋
ka共␭兲
1 ⫺ ka共␭兲 ⫺ kc共␭兲
冒
ka共␭r兲
1 ⫺ ka共␭r兲 ⫺ kc共␭r兲
册
Fig. 1. Sigma-corrected total backscattering values show increasing divergence from uncorrected total backscattering as the turbidity of the water increases.
,
(2)
where ka共␭兲 is the fraction of scattered light that is not
collected by the absorption sensor and kc共␭兲 is the
fraction of scattered light that is collected by the attenuation sensor. Zaneveld et al.6 assumed ka共␭兲
⫽ ka共␭r兲 and kc共␭兲 ⫽ kc共␭r兲, and consequently F共␭, ␭r兲
⫽ 1, but this assumption does not hold if the scattering phase function varies significantly with wavelength. Unfortunately, none of the terms on the righthand side of Eq. 2 are readily obtained from in situ
optical measurements, and F共␭, ␭r兲 must be determined by a computationally intensive iterative procedure.3 The magnitude of the error that results from
ignoring wavelength dependence depends on the
product of F共␭, ␭r兲 and ai共␭r兲, and it is greatest in
turbid waters, where ai共␭r兲 values tend to be highest.
In waters in which wavelength dependency was observed, the ratio of the total backscattering coefficient
to the total scattering coefficient (bb兾b) generally decreased with increasing wavelength. From Eqs. 1 and
2, the effect is to produce an overestimate of an and
consequently an underestimate of bn.
B.
Hydroscat Correction
The sigma correction procedure for Hydroscat data
can be written as
bb ⫽ ␴ bbu,
2
␴ ⫽ k0 ⫹ k1Kbb ⫹ k2Kbb
,
(3a)
(3b)
where bbu is the uncorrected signal and k0,1,2 are calibration coefficients supplied by the manufacturer.
Kbb is a measure of the attenuation of the signal by
water within the sensor’s measurement geometry
and is calculated for each wavelength with7
Kbb ⫽ a ⫹ 0.75b.
(3c)
Figure 1 shows the effect of sigma correction on
measurements of backscattering for surface waters
at all stations occupied in the Irish Sea. Significant
divergence between bb and bbu occurs as the level of
turbidity increases, and so backscattering measurements become increasingly sensitive to correction errors in strongly attenuating shelf seas. The
Hydroscat-2 data presented in this paper have been
corrected with values of total absorption and scattering derived from AC-9 measurements in which the
standard scattering correction algorithm was used.
Since positive errors in a and negative errors in b
partially cancel in Eq. 3c, and k0 ⬎ k1 ⬎ k2 in Eq. 3b,
the sigma correction procedure is rather insensitive
to the artifacts in the AC-9 data discussed in the
previous section. In the extreme case of a being overestimated by 50% and b being underestimated by 5%,
bb would be calculated to be 2.5% above its true value
for this data set.
C. Wavelength Dependence of the Scattering Phase
Function
The wavelength dependence of the scattering phase
function can be characterized in terms of available
IOP measurements by
B⫽
共bb 兾b兲470
.
共bb 兾b兲676
(4)
B is subject to errors in the correction procedures
applied to measurements of both total backscattering
(bb) and total scattering (b). The values of a, b and
sigma-corrected bb at 676 nm are assumed to be relatively unaffected by correction errors because this
wave band is close to the AC-9 reference channel.
Consequently our worst-case example of a 5% underestimate in b and a 2.5% overestimate in bb at 470 nm
produces bb兾b and B values that are overestimated by
approximately 8%.
1 January 2005 兾 Vol. 44, No. 1 兾 APPLIED OPTICS
127
D. Possible Influence of Fluorescence on bb676
Much of our discussion here rests on the observation
that measured values of bb兾b470 tended to be higher
than those for bb兾b676 and that their ratio [B in Eq.
4] was usually greater than unity. However, the Hydroscat 676-nm channel covers a relatively broad
wave band (20 nm FWHM) that partly overlaps with
the fluorescence emission spectrum of chlorophyll-a
in vivo. It must therefore be presumed that Hydroscat bb676 measurements contain a fluorescence
component, though no quantitative study has been
published. The effect of fluorescence would be to raise
the bb676 signal and to generate erroneously high
apparent values of bb兾b676. As a result, actual B
values would be even greater than those measured
here. The observed wavelength dependence in the
backscattering ratio is therefore qualitatively robust
in the presence of fluorescence artifacts in bb676.
The magnitude of fluorescence augmentation of the
backscattering signal in the present data set is difficult
to measure directly. However, its likely significance
can be judged from an analysis of seawater composition. Chlorophyll-a concentrations measured by acetone extraction were generally low, with 95% of the
stations having values of less than 2 mg m⫺3. The
bb676 signal was poorly correlated with chlorophyll
concentration (r2 ⫽ 0.29) for stations with significant
chlorophyll (1–7 mg m⫺3) and low mineral (⬍5 g m⫺3)
concentrations but was highly correlated with suspended mineral concentration (r2 ⫽ 0.92) for all stations. It therefore appears that backscattering in this
data set is dominated by the suspended mineral component and that chlorophyll fluorescence is likely to
be of minor significance.
E.
Fig. 2. Backscattering ratios are generally greater in the blue
(470 nm) than in the red (676 nm) for most stations in the Irish and
Celtic Seas.
the upward radiance (Lu) is also underestimated.
For coccolithophore blooms, we were able to show
that the use of Eq. 1 with computationally derived
values of F共␭, ␭r兲 for correcting AC-9 data produced
improved fits between radiance transfer models and
in situ measurements in turbid waters. We extend
these observations to a larger geographical area (the
Irish and Celtic Seas) whose optics are not dominated
by coccolithophores.
3. Materials and Methods
Data were obtained during a series of cruises in the
Irish and Celtic Seas between May 2001 and July
Predicted Effects on Radiance Transfer Calculations
The qualitative effect of wavelength-dependent scattering phase functions on attempts to achieve optical
closure can be predicted from established relations
between IOPs and apparent optical properties. To
reasonable approximations, the radiance reflectance
(RL) is given by8
RL ⫽
f bb
Q a
,
(5)
and the diffuse attenuation coefficient for downward
irradiance (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 measurements, and since
RL ⫽
128
Lu
Ed
,
APPLIED OPTICS 兾 Vol. 44, No. 1 兾 1 January 2005
(7)
Fig. 3. Ratio of backscattering ratios at 470 and 676 nm, B, shows
significant variability for surface waters of the Irish and Celtic
Seas. Less than 5% of stations have a value of B between 0.9 and
1.1, where the scattering phase function might be considered wavelength independent. Most stations have stronger backscattering
ratios in the blue than in the red. It can be concluded that the
scattering phase function is generally wavelength dependent for
these waters.
2002. A total of 120 stations were occupied covering a
wide range of water types.
A.
Fig. 4. Significant degree of variability is exhibited in the relation
between backscattering ratios at 470 and 676 nm for the top 10 m
of four sample stations. Lines (gradients between 1.0 and 1.8)
indicate the strength of the wavelength dependency in the backscattering ratio, B.
Inherent Optical Property Measurements
A 25-cm path-length WET Labs AC-9 was used to
measure the absorption coefficient (an) and beam
attenuation coefficient (cn) of materials other than
water at nine wavelengths (10 nm FWHM) across
the visible spectrum. Optical blanks for the AC-9
were regularly measured with ultrapure Millipore
water treated with ultraviolet light, and calibration
of the two optical channels remained within the
manufacturer=s specifications of ⫾0.005 m⫺1. Absorption and attenuation signals at 715 nm were
corrected for temperature-dependent water absorption,10 and the data were averaged over 1-m depth
intervals. We assumed that the standard correction
algorithm always suffices for AC-9 measurements at
676 nm, since this is close to the 715-nm reference
wavelength. Total absorption (a) and attenuation (c)
coefficients were obtained by addition of partial coefficients for water obtained from the literature.11,12
Scattering coefficients were obtained from b ⫽ c
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.
1 January 2005 兾 Vol. 44, No. 1 兾 APPLIED OPTICS
129
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 small
overestimate of modeled upward radiance for ST19 (d) may be attributable to limitations in the performance of sigma correction of
backscattering data for very turbid water such as this.
⫺ a. Values of absorption and scattering at 470 nm,
which are used at various points in the paper, were
obtained by interpolation between readings at 440
and 488 nm. Sets of modified AC-9 data were generated with Eq. 1 with values of F共␭, ␭r兲 ranging between 1 and 1.5. Total backscattering (bb) was
measured at 470 and 676 nm by use of a Hydroscat-2
(HOBI Labs) and was corrected with calibration constants supplied by the manufacturer.
monitored at the start and the end of each cruise by
use of a 100-W standard lamp, and the radiance sensors were checked by use of the same lamp to illuminate a Spectralon reflectance target. All sensors
remained within factory specifications. Signals from
the SPMR were processed with ProSoft, a MATLAB
module supplied by the manufacturers. Data processing steps included application of calibration constants and averaging over 1-m depth intervals.
B.
C.
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 SeaWiFS Profiling Multi-Channel Radiometer (SPMR).
The SPMR was deployed at a distance of at least 20 m
from the ship to avoid shadowing. A reference radiometer measuring surface irradiance (Es) in the same
wave bands was mounted on the superstructure of
the ship. The stability of the irradiance sensors was
130
APPLIED OPTICS 兾 Vol. 44, No. 1 兾 1 January 2005
Radiance Transfer Modeling
Radiance transfer calculations were carried out with
Hydrolight (Sequoia Scientific, Bellevue, Washington).1 Surface irradiance data for each station were
obtained from the SPMR deck reference. The choice of
wavelength for this study was limited by the fact that
we had access only to a two-channel Hydroscat backscattering meter operating at 470 and 676 nm. The
analysis was limited to the blue wave band to avoid the
complicated effects of inelastic processes on the radio-
Fig. 7. IOPs of stations ST12 and ST16 indicate (a) similar levels of absorption at 488 nm, (b) significantly higher levels of scattering at 488
nm, (c) backscattering at 470 nm, and (d) backscattering ratio at 470 nm for ST12. All IOPs were generated with standard correction
algorithms.
metric measurements. A direct comparison was made
between Hydrolight computations at 488 nm and
SPMR profiles at the same wavelength. The IOP inputs were bb values at 470 nm measured with the
Hydroscat 2 and a and b values at 488 nm measured
with the AC-9. It is assumed that the 18-nm wavelength discrepancy between the backscattering data
and the other IOPs has a negligible effect on the
results. We performed separate runs with AC-9 input
files generated by use of different values of F共␭, ␭r兲 for
the modified scattering correction algorithm [Eq. 1].
Fournier–Forand13,14 scattering phase functions for
each station and depth were chosen using sigmacorrected Hydroscat-2 values for the backscattering
coefficient. The match between the Hydrolight output
and the in situ measurements from the SPMR was
evaluated by use of the average standard percentage
error ε, defined as
␧⫽
兺n ⱍ共xin
⫺ xmod兲 兾 xinⱍ ⫻ 100
n
,
(8)
where n is the number of observations, xin refers to in
situ 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兾b at 470 nm, plotted against bb兾b
at 676 nm for surface waters at 120 stations in the
Irish Sea, derived from Hydroscat and AC-9 data
corrected with the standard procedures. For most stations the points lie above the 1:1 line, indicating that
backscattering ratios are higher at shorter wavelengths. Figure 3 shows the distribution of B for the
data in Fig. 2. For 95% of the stations, B ⬎ 1.1, and
for 50% B ⬎ 1.3. These percentages are not changed
significantly by the possible existence of an 8% overestimate in B values for the most wavelengthdependent stations. The implications of this result for
the correction of AC-9 measurements and the ability
to obtain optical closure were therefore investigated.
1 January 2005 兾 Vol. 44, No. 1 兾 APPLIED OPTICS
131
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 the
scattering correction of in situ measurements of absorption.
B.
Case Studies
Figure 4 shows backscattering ratios at 470 nm plotted against backscattering ratios at 676 nm for the
top 10 m of four stations that cover a representative
range of B values. These ratios were derived from
IOP data corrected with standard procedures. Stations B4 and ST19 are examples with low wavelength
dependency in the backscattering ratio (B ⬍ 1.2),
whereas stations ST12 and ST16 are examples with
marked wavelength dependency (1.2 ⬍ B ⬍ 1.8).
Equations 5–7 predict that radiance transfer models
in which standard IOP inputs are used will produce
satisfactory outputs for stations B4 and ST19 and
will underestimate both Ed and Lu for ST12 and
ST16.
C. Closure for Stations with B ⬍ 1.2
Figure 5 shows IOP profiles generated with AC-9
data corrected by use of the standard algorithm for
stations B4 and ST19. B4 was a relatively clear station in the middle of the Irish Sea, occupied in April
132
APPLIED OPTICS 兾 Vol. 44, No. 1 兾 1 January 2005
2002, with low values of absorption, scattering, and
backscattering throughout the top 25 m of the water
column. ST19 was a turbid station close to the Welsh
coast, occupied in November 2001, with significantly
higher values of all three IOPs. The backscattering
ratio for ST19 was almost double that at B4. The
IOPs in Fig. 5 were used as inputs for Hydrolight
simulations, and the modeled profiles of Ed and Lu for
the top 10 m of the water column compared with the
in situ measurements obtained from the SPMR radiometer system (Fig. 6). For both these stations the
average standard percentage error was less than 6%
for measurements of Ed. There was a tendency to
overestimate Lu, with values of ␧ ⬃ 11% for B4 and
⬃32% for ST19. However, the apparently poor performance of the simulation for Lu at ST19 corresponded to an average absolute error of less than 5 ⫻
10⫺5 W m⫺2 nm⫺1 sr⫺1 and represented the smallest
absolute error of any of the simulations presented.
Light levels were very low for this station, which
provided a severe test of modeling and measurement
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 be
minimized for stations ST12 and ST16 by varying only the value of F共␭, ␭r兲 used in the scattering correction for the AC-9 measurements.
The optimal value of F共␭, ␭r兲 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共␭, ␭r兲 ⫽ 1.2 for ST12 [(a)
and (b)], and F共␭, ␭r兲 ⫽ 1.3 for ST16 [(c) and (d)].
1 January 2005 兾 Vol. 44, No. 1 兾 APPLIED OPTICS
133
Fig. 11. Absorption values obtained with the modified scattering correction algorithm and F共␭, ␭r兲 ⬎ 1.0 are generally lower than
standard (F共␭, ␭r兲 ⫽ 1.0) values of absorption. Higher levels of scattering at ST12 than at ST16 are responsible for the greater difference
between standard and modified absorption values at ST12 despite the optimal value of F共␭, ␭r兲 being lower for this station.
precision. In both cases the simulations provided a
reasonable degree of optical closure by use of standard AC-9 data.
D. Closure for Stations with B ⬎ 1.2
Figure 7 shows profiles of IOPs for the top 25 m of
stations ST12 and ST16. Both stations were fairly
homogeneous over the top 15 m, with similar absorption coefficients. The scattering and backscattering
coefficients and backscattering ratios were all significantly higher for ST12 than for ST16, but in both
cases backscattering ratios were higher in the blue
than in the red (Fig. 4). Radiance transfer modeling
with AC-9 data corrected by use of the standard algorithm [F共␭, ␭r兲 ⫽ 1.0] produced systematic underestimates of both Ed and Lu (Fig. 8) for both these
stations. The discrepancy was most obvious for ST12
for which the average standard percentage errors
were 38% and 54% for Ed and Lu, respectively. The
errors for ST16 were lower at 13% for Ed and 17% for
Lu. As predicted, the standard [F共␭, ␭r兲 ⫽ 1.0] AC-9
scattering correction algorithm for these stations results in a systematic overestimate of absorption at
blue wavelengths.
Figure 9 shows the effect of varying F共␭, ␭r兲 on the
performance of radiance transfer simulations. For
ST12 the minimum average error for both Ed and Lu
occurred when F共␭, ␭r兲 ⫽ 1.2. For ST16 the optimal
value of F共␭, ␭r兲 was closer to 1.3. It is interesting to
note that the overall magnitude of the error with
F共␭, ␭r兲 ⫽ 1.0 was greater for ST12 than for ST16
even though the optimal value of F共␭, ␭r兲 was less for
ST12 than for ST16. This occurrence was probably
due to a higher residual absorption [ai共␭r兲] for ST12 as
a result of significantly higher scattering at this station. When the optimal values of F共␭, ␭r兲 derived
above were incorporated in the AC-9 correction algorithm (Fig. 10), the average standard percentage errors for the radiance transfer simulations were less
134
APPLIED OPTICS 兾 Vol. 44, No. 1 兾 1 January 2005
than 10% for ST12 and less than 5% for ST16. This
result represents a significant improvement in modeling accuracy over the results from use of standard
AC-9 data.
Figure 11 illustrates the effect of using optimized
F共␭, ␭r兲 values on absorption coefficients for these
two stations. In both cases the modified AC-9 absorption signals were lower than the standard values,
with the average percentage difference varying from
27% for ST12 to 11% for ST16. The resulting scattering coefficients (not shown) were approximately 4%
greater than the standard values for both stations.
5. Discussion
The observations presented here show that it can be
difficult to obtain optical closure in waters where
scattering coefficients are high relative to oceanic values. The nature of the errors observed in radiance
transfer calculations and the changes in AC-9 correction procedures necessary to obtain closure were both
consistent with the hypothesis of wavelength dependence in the scattering phase function. In the Irish
Sea, where the backscattering ratio was generally
found to decrease with increasing wavelength, the
standard AC-9 scattering correction produces overestimates in the absorption coefficient and underestimates in the scattering coefficient at blue and green
wavelengths. These errors can be reduced by introduction of a modified scattering correction algorithm
[Eq. 1] with an additional term F共␭, ␭r兲 whose magnitude can be determined computationally by minimization of the discrepancies between radiance
transfer models and IOP measurements. However,
the significance of the IOP measurement errors will
be determined not only by F共␭, ␭r兲 but also by ai共␭r兲,
the magnitude of the residual scattering signal at 715
nm. Since the latter term generally varies with suspended particle concentration, the greatest errors are
expected in coastal and shelf seas.
One approach to obtaining appropriately corrected
measurements of the absorption and scattering coefficients in these waters would be to extend the
method described here to all wave bands by deploying
spectrally matched instrument packages. It may be
necessary to carry out further Monte Carlo simulation of the reflecting tube design15,16 with appropriate wavelength dependency being prescribed for all
IOPs, including the scattering phase function.17 Alternatively, a number of relations between IOPs and
apparent optical properties have been published that
would allow IOPs to be derived from radiometric
measurements with reduced precision but greater
freedom from systematic errors.18 –20 Whichever
method is adopted, improvement in the accuracy of
IOP measurements is urgently required for validating remote-sensing products and for interpreting
data from moored arrays of optical sensors in coastal
seas.
8.
9.
10.
11.
12.
13.
14.
References
1. C. D. Mobley, “Light and Water: Radiative Transfer in Natural
Waters,” (Academic, San Diego, 1994).
2. A. Cunningham, J. C. Boyle, and P. Wood, “Radiative transfer
modelling of the relationship between seawater composition
and remote sensing reflectance in sea lochs and fjords,” Int. J.
Remote Sensing 23, 3713–3724 (2002).
3. D. McKee, A. Cunningham, and S. Craig, “Semi-empirical correction algorithm for AC-9 measurements in a coccolithophore
bloom,” Appl. Opt. 42, 4369 – 4374 (2003).
4. T. J. Smyth, G. F. Moore, S. B. Groom, P. E. Land, and T.
Tyrrell, “Optical modeling and measurements of a coccolithophore bloom,” Appl. Opt. 41, 7679 –7688 (2002).
5. G. C. Chang, T. D. Dickey, C. D. Mobley, E. Boss, and W. S.
Pegau, “Toward closure of upwelling radiance in coastal waters,” Appl. Opt. 42, 1574 –1582 (2003).
6. J. R. V. Zaneveld, J. C. Kitchen and C. M. Moore, “The scattering error correction of reflecting-tube absorption meters,” in
Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE 2258, 44 –55
(1994).
7. M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A.
15.
16.
17.
18.
19.
20.
H. Barnard, and J. R. V. Zaneveld, “A model for estimating
bulk refractive index from the optical backscattering ratio and
the implications for understanding particle composition in case
I and case II waters,” J. Geophys. Res. 106, C7, 14129 –14142
(2001).
A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters.
II. Bidirectional aspects,” Appl. Opt. 32, 6864 – 6879 (1993).
H. R. Gordon, “Can the Lambert-Beer law be applied to the
diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389 –1409 (1989).
W. S. Pegau, D. Gray, and J. R. V. Zaneveld, “Absorption and
attenuation of visible and near-infrared light in water: dependence on temperature and salinity,” Appl. Opt. 36, 6035– 6046
(1997).
R. M. Pope and E. S. Fry, “Absorption spectrum (380 –700 nm)
of pure water. II. Integrating cavity measurements,” Appl.
Opt. 36, 8710 – 8723 (1997).
R. C. Smith and K. Baker, “Optical properties of the clearest
natural waters (200-800 nm),” Appl. Opt. 20, 177–184 (1981).
G. R. Fournier and J. L. Forand, “Analytic phase function for
ocean water,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE
2258, 194 –201 (1994).
C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function
effects on oceanic light fields,” Appl. Opt. 41, 1035–1050
(2002).
J. T. O. Kirk, “Monte Carlo modeling of the performance of a
reflective tube absorption meter,” Appl. Opt. 31, 6463– 6468
(1992).
J. Piskozub, P. J. Flatau, and J. R. V. Zaneveld, “Monte Carlo
study of the scattering error of a quartz reflective absorption
tube,” J. Atmos. Oceanic Technol. 18, 438 – 445 (2001).
D. Stramski and J. Piskozub, “Estimation of scattering error in
spectrophotometric measurements of light absorption by
aquatic particles from three-dimensional radiative transfer
simulations,” Appl. Opt. 42, 3634 –3646 (2003).
R. A. Leathers, C. S. Roesler, and N. J. McCormick, “Ocean
inherent optical property determination from in-water light
field measurements,” Appl. Opt. 38, 5096 –5103 (1999).
M. Stramska, D. Stramski, B. G. Mitchell, and C. D. Mobley,
“Estimation of the absorption and backscattering coefficients
from in-water radiometric measurements,” Limnol. Oceanogr.
45, 628 – 641 (2000).
D. McKee, A. Cunningham, and S. Craig, “Retrieval of inherent optical properties from in situ radiometric measurements
in case II waters,” Appl. Opt. 42, 2804 –2810 (2003).
1 January 2005 兾 Vol. 44, No. 1 兾 APPLIED OPTICS
135