Deriving inherent optical properties from water color:
a multiband quasi-analytical algorithm for optically
ZhongPing Lee, Kendall L. Carder, and Robert A. Arnone
For open ocean and coastal waters, a multiband quasi-analytical algorithm is developed to retrieve
absorption and backscattering coefficients, as well as absorption coefficients of phytoplankton pigments
and gelbstoff. This algorithm is based on remote-sensing reflectance models derived from the radiative
transfer equation, and values of total absorption and backscattering coefficients are analytically calcu-
lated from values of remote-sensing reflectance.In the calculation of total absorption coefficient, no
spectral models for pigment and gelbstoff absorption coefficients are used.
coefficients are spectrally decomposed from the derived total absorption coefficient in a separate calcu-
lation. The algorithm is easy to understand and simple to implement.
past and current satellite sensors, as well as to data from hyperspectral sensors.
empirical relationships involved in the algorithm, and they are for less important properties, which
implies that the concept and details of the algorithm could be applied to many data for oceanic obser-
vations. The algorithm is applied to simulated data and field data, both non-case1, to test its perfor-
mance, and the results are quite promising. More independent tests with field-measured data are
desired to validate and improve this algorithm.© 2002 Optical Society of America
010.4450, 290.5850, 280.0280.
Actually those absorption
It can be applied to data from
There are only limited
Absorption and backscattering coefficients are inher-
ent optical properties.1
welling light from the Sun and sky, they determine
the appearance of water color, which is normally
measured by the water-leaving radiance or remote-
sensing reflectance2?ratio of water-leaving radiance
to above-surface downwelling irradiance?.
ent optical properties are directly linked to the con-
stituents in the water, their values are used to
determine the type of water, subsurface light inten-
sity, solar heat flux with depth, turbidity, pigment
concentration, and sediment loading to name a few
these optical properties remotely have been under
to accurately retrieve
investigation for several decades, and algorithms
from empirical to full-spectral optimization have
Empirical algorithms2,7–22apply simple or multiple
regressions between the property of interest and the
ratios ?or values? of irradiance reflectance7,22or
remote-sensing reflectance2,11?rrs, see Table 1 for
symbols and definitions used in this paper?.
not require a full understanding of the relationship
between rrsand the properties.
ture of regression, however, these kinds of algorithm
are generally only appropriate to waters with char-
acteristics similar to those used in the algorithm de-
velopment.Their applicability then can be quite
limited and can result in significant errors.
importantly, because of the wide variation of optical
properties found for global waters, one empirical
function cannot fit all waters, unless the waters are
restricted to case1conditions7where all optical prop-
erties co-vary with chlorophyll concentrations.
biggest advantages of this kind of algorithm are the
simplicity and rapidity in data processing, which are
important for the retrieval of information from large
data sets such as satellite images.
The semianalytical algorithms,23–30including the
spectral optimization approach,23,27–29are based on
Because of the na-
When this research was performed, Z. Lee and K. L. Carder were
with the College of Marine Science, University of South Florida, St.
Petersburg, Florida 33701.Z. P. Lee ?firstname.lastname@example.org? and
R. A. Arnone are now with the U.S. Naval Research Laboratory,
Code 7333, Stennis Space Center, Mississippi 39529.
Received 21 February 2002; revised manuscript received 11
© 2002 Optical Society of America
20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS5755
solutions to the radiative transfer equation.
algorithms can be applied to different water types,
and retrieval accuracy is often much better than
those of empirical algorithms.23,31
of these algorithms, however, relies on accurate spec-
tral models for the absorption coefficients of each
individual constituent presented in the water, such
as pigments, gelbstoff ?also called colored dissolved
organic matter ?CDOM??, or suspended sediments.
The accuracy of these models will affect the accuracy
of remote-sensing retrievals.
its applicability in the processing of large data sets,
such as satellite images.
In this study, for waters of open ocean and coastal
areas, we develop a multiband quasi-analytical algo-
rithm ?QAA? for retrieving the absorption and back-
reflectance of optically deep waters.
the derived total absorption coefficient is spectrally
decomposed into the contributions of phytoplankton
pigments and gelbstoff. The algorithm is based on
the relationship between rrsand the inherent optical
properties of water derived from the radiative trans-
fer equation.In the derivation of the total absorp-
tion coefficient, there are no spectral models involved
for the absorption coefficients of pigments and gelb-
stoff. Instead, the derived total absorption coeffi-
cient is further decomposed spectrally into the
absorption coefficients of pigments and gelbstoff
when necessary.The algorithm is applied to both
simulated and field-measured non-case1data to test
its performance. We show that its accuracy is sim-
ilar to that of optimization, but calculation efficiency
is similar to that of empirical algorithms.
This QAA can be quickly applied to data from past
In addition, the opti-
and current ocean-color satellite sensors, such as the
Coastal Zone Color Scanner ?CZCS?, the sea-viewing
wide field-of-view sensor
?MODIS?, as these sensors have only a few spectral
bands in the visible domain.
also be applied to hyperspectral airborne sensors or
future hyperspectral satellite sensors.
This algorithm can
2.Multiband Quasi-Analytical Algorithm
Figure 1, the schematic flow chart, presents the con-
cept of the level-by-level derivation and the QAA.
The left side of Fig. 1 lays out the levels from remote-
sensing reflectance to the concentrations of phyto-
plankton pigments or CDOM.
the QAA used to derive the particle backscattering
is to calculate optical properties first32and in a level-
by-level scheme instead of solving all in one stroke,
such as the optimization approach23,27–29or the linear
This way the accuracy for re-
turns on a higher level ?level 1, for example? has little
or no dependence on how to handle the properties on
the lower levels ?level 2 or 3, for example?, as it should
be in water-color inversion.
level 1 data from level 0 data and level 2 data from
level 1 data are presented here.
ment and gelbstoff absorption coefficients to their
corresponding concentrations can be found in Carder
et al.26and Roesler and Perry.27
In general, on the basis of theoretical analyses22
fer equation, rrsis a function of the absorption and
The right side shows
The methods to derive
Table 1.Symbols and Definitions
Absorption coefficient of the total, aw? a?? ag
Absorption coefficient of phytoplankton pigments
Absorption coefficient of gelbstoff and detritus
Absorption coefficient of pure seawater
Total absorption coefficient at ? from the QAA-555
Total absorption coefficient at ? from the QAA-640
Backscattering coefficient of suspended particles
Backscattering coefficient of pure seawater
Backscattering coefficient of the total, bbw? bbp
Model parameters for phytoplankton specific-absorption coefficient at 440 nm
Spectral power for particle backscattering coefficient
Above-surface remote-sensing reflectance
Below-surface remote-sensing reflectance
Spectral slope for gelbstoff absorption coefficient
Ratio of backscattering coefficient to the sum of absorption and backscattering coefficients,
bb??a ? bb?
5756APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002
ment of rrsis a measure of the ratio u of the back-
scattering coefficient to the sum of absorption and
backscattering coefficients, with an error of ?2% to
?10%. This accuracy depends on the accuracy of
one knowing the particle phase function of the water
under study and the accuracy of one modeling rrsas
a function of u. Applying existing rrsmodels, values
of u can be calculated analytically from values of rrs.
Because u is just a ratio of the backscattering coeffi-
cient to the sum of absorption and backscattering
coefficients, then knowledge of the absorption coeffi-
cient enables us to calculate the backscattering coef-
ficient, or vice versa.
To illustrate the derivation of u from rrs, the Gor-
don et al.33formula is used here:
rrs??? ? g0u??? ? g1?u????2, (1)
a ? bb
Here a is the total absorption coefficient and bbis the
total backscattering coefficient.
expressed as2,7bb? bbw? bbp, with bbwthe back-
scattering coefficients for water molecules.
the backscattering coefficients for nonmolecules, col-
lectively called the backscattering coefficients of sus-
pended particles, which actually may include the
contributions of virus39or bubbles40in natural wa-
For nadir-viewed rrs, Gordon et al.33found that
? is the wavelength
g0? 0.0949 and g1? 0.0794 for oceanic case1wa-
ters. Recently Lee et al.34suggested that g0of
0.084 and g1 of 0.17 work better for higher-
scattering coastal waters.
g0and g1may vary with particle phase function,38
which is not known remotely.
however, need to be predetermined as in any semi-
aimed at applying the QAA to both coastal and
open-ocean waters, we used the averaged g0and g1
values of Gordon et al.33and Lee et al.,34which are
g0? 0.0895 and g1? 0.1247.
From Eq. ?1?,
Actually the values of
Values of g0and g1,
u??? ??g0? ??g0?2? 4g1rrs????1?2
Other formulas or procedures for the derivation of u
can be applied on the basis of existing models.34–38
As long as u??? is accurately derived from rrs???, the
following steps are the same.
As u is a simple ratio of bbto ?a ? bb?, knowing a
will lead to
1 ? u,(4)
or knowing bbwill lead to
a ??1 ? u?bb
Fig. 1.Concept and schematic flow chart of the level-by-level ocean-color remote sensing and the QAA.
20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS 5757
For each wavelength, the total absorption coeffi-
cient can be expressed as
a??? ? aw??? ? ?a???, (6)
where awis the absorption coefficient of pure water
and ?a is the contribution that is due to dissolved and
wavelengths ??550 nm?, ?a??? is quite small, with
a??? dominated by the values of aw???, especially for
oligotrophic and mesotrophic waters.
sents examples of a??? variations.
ples, a 20-fold a?440? variation corresponds to a factor
of 2 variation for a?555? and just a fractional varia-
tion for a?640?. These facts suggest that, if a refer-
ence wavelength ??0? is found where rrs??0? from
elastic scattering can be accurately measured and
a??0? can be well estimated, then bbat ?0can be
calculated from Eq. ?4?. As bbis a simple sum of bbw
and bbp?Refs. 2 and 7? and the value of bbw??? is
already known,41then the bbpvalue at ?0is calcu-
We note that, at longer
Figure 2 pre-
In these exam-
The wavelength dependence of bbp??? is normally
bbp??? ? bbp??0??
This suggests that, if the power value Y is known or
can be estimated from remote-sensing measure-
ments, then bbpat any wavelength can be calculated.
If we place this calculated bbp??? value along with the
bbw??? value into Eq. ?5?, then the total absorption
coefficient at that wavelength can be calculated ana-
lytically from rrs???.
As shown above ?Eqs. ?1?–?7??, there are no spectral
models involved for the absorption coefficients of pig-
ments or gelbstoff.As a matter of fact, the simple
math functions such as the log-normal shape used by
Lee et al.28and the Gaussian shape used by Hoge and
Lyon25cannot accurately simulate all the spectral
shapes of the pigment absorption coefficient observed
in the field.The simplification of the QAA here re-
duced the potential errors from inaccurate spectral
models of pigment and gelbstoff absorption coeffi-
cients in the retrieval of the total absorption coeffi-
Using 555 nm as the reference wavelength ??0?, in
Table 2 we detail the steps of applying the QAA.
This ?0can be changed to shorter or longer wave-
lengths, such as 640 nm that is shown in Subsection
4.A. for high-absorbing waters, to obtain better mea-
surement of rrs??0? and a better estimate of a??0?.
For such cases, small adjustments are required ac-
cordingly, but the analytical calculation scheme in
Fig. 1 and Table 2 will remain the same.
also summarizes the optical properties involved, the
mathematical formula for each calculation, the order
of importance of each property, and the character of
each step.Explanations and comments are pro-
vided in Subsection 2.B.
Fig. 2.Examples of a??? variations at different wavelengths.
Table 2. Steps of the QAA to Derive Absorption and Backscattering Coefficients from Remote-Sensing Reflectance with 555 nm as the
?Rrs??0.52 ? 1.7Rrs?
??g0? ??g0?2? 4g1rrs????1?2
?0.0596 ? 0.2?a?440?i? 0.01?, a?440?i? exp??2.0
? 1.4? ? 0.2?2?, ? ? ln?rrs?440??rrs?555??
1 ? u?555?
? 2.2?1 ? 1.2 exp??0.9rrs?440?
??1 ? u?????bbw??? ? bbp????
5758APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002
Step 0 converts above-surface remote-sensing reflec-
tance spectra Rrsto below-surface spectra rrsbecause
satellites and many other sensors measure remote-
sensing reflectance from above the surface.
surement is made below the surface, this step can be
skipped. For the Rrsto rrsconversion,33,44
Derivation of Total Absorption and Backscattering
T ? ?QRrs
where T ? t?t??n2with t?the radiance transmit-
tance from below to above the surface and t?the
irradiance transmittance from above to below the
surface, and n is the refractive index of water.
the water-to-air internal reflection coefficient.
the ratio of upwelling irradiance to upwelling radi-
ance evaluated below the surface.
viewing sensor and the remote-sensing domain,35Q,
?in the range of 1% at the high end31? for most oceanic
and coastal waters, the variation of the Q value has
only little influence on the conversion between Rrs
and rrs. From calculated HYDROLIGHT45Rrsand rrs
values, it is found that T ? 0.52 and ?Q ? 1.7 for
optically deep waters and a nadir-viewing sensor.
Knowing rrs, values of u can be quickly calculated,
for example, with Eq. ?3? as shown in step 1.
Step 2 estimates a?555? empirically.
on the sensor’s configurations and sensitivities to
changes of water properties, there could be many
ways to perform this estimation.
use here the Austin and Petzold8approach, with pa-
rameters adjusted for the absorption coefficient in-
stead of the diffuse attenuation coefficient.
initial estimation of a?440?ihere is solely for the em-
pirical estimation of ?a?555? as a?440? is sensitive to
the change of water properties.
on the basis of an earlier study11but is adapted to
bands at 440 and 555 nm as in Mueller and Trees.10
a simple empirical algorithm such as this one may
not accurately estimate a?440?ifor non-case1waters;
in turn ?a?555? may not be accurate either.
ever, as ?a?555? is small compared with a?555? for
most oceanic waters ?see Fig. 2?, the errors of ?a?555?
will have a smaller impact on the accuracy of a?555?.
When the errors for a?555? are no longer tolerable,
such as near the shore or for river plume waters, ?0
has to be shifted to a longer wavelength by this ap-
Step 3 calculates bbp?555? from rrs?555? and a?555?
on the basis of Eq. ?4?.
Step 4 estimates the wavelength dependence ?val-
ue of Y? of the particle backscattering coefficient.
value for Y is required if we want to calculate particle
backscattering coefficients from one wavelength to
another wavelength by Eq. ?7?.
searchers set Y values based on the location of the
water sample,13,29such as 0 for coastal waters and 2.0
for open-ocean waters.Here we used the empirical
For a nadir-
As an example, we
algorithm of Lee et al.46to estimate the Y value, but
adapted it for bands at 440 and 555 nm.
Step 5 computes the particle backscattering coeffi-
cients at other wavelengths given the values of Y and
bbp?555? by use of Eq. ?7?.
Step 6 completes the calculation for a??? given the
values of u??? ?step 1? and bbp??? ?step 5? based on Eq.
As shown from step 1 to step 6, there are two semi-
analytical expressions ?Eqs. ?1? and ?7?? and two em-
pirical formulas ?steps 2 and 4? used for the entire
process.Certainly the accuracy of the final calcu-
lated a??? relies on the accuracy of each individual
step.The semianalytical expressions are currently
widely accepted and used, their improvements are
out of the scope of this study, and Eqs. ?1? and ?7?
could be simply replaced by better expressions when
available. The empirical formulas used either pro-
vide estimates at the reference wavelength ?a?555??
or estimates of less important quantities ?values of Y,
for example?.As shown in Table 2, these properties
have only second-order importance.
The order of importance for a property is based on
its range of variation and its influence on the final
output.Values of rrs, for example, vary widely and
have a great influence on the final results, so they are
of first-order importance.
ever, vary over a much narrower range except near
shore ?see Fig. 2? and have only a small influence on
the final results, so a?555? is of second-order impor-
tance.Although values of Y vary over a range of
0–2.0 or so, they have a relatively small influence on
the final results because this value is used in a power
law on the ratio of wavelengths for the particle back-
scattering coefficient.For example, for the expres-
sion ?555?440?Y, a change of Y from 0 to 2.0 merely
changes the expression from 1.0 to 1.59.
Y value is 1.0 but an estimate of 2.0 is used, this will
make the calculated bbp?440? 21% higher than it
should be.On the other hand, for the same true Y
value of 1.0 but an estimate of 0.0 is used, this will
make the calculated bbp?440? 26% lower than it
should be.These errors will be transferred to the
calculated total absorption coefficient at 440 nm, but,
as shown, the errors are in a limited range.
The quantities with second-order importance, how-
ever, do affect the end products, and further improve-
ments to the end products can be achieved if the
secondary quantities are better estimated with re-
gional and seasonal information, or with improved
Values of a?555?, how-
If the true
The data processing can be stopped here if the inter-
est of remote sensing is on the total absorption coef-
ficient or the particle backscattering coefficient.
many remote-sensing applications, however, it is also
desired to know the absorption coefficients for phyto-
plankton pigment ?a????? and gelbstoff ?ag???? be-
concentrations of chlorophyll26or CDOM,27respec-
Decomposition of the Total Absorption Coefficient
can beconverted to
20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS 5759
It is much more challenging to separate a???? and
ag??? from the total absorption coefficient as the total
absorption is at least a sum of pure water, phyto-
plankton pigment, and gelbstoff.
partition the total absorption if the a??? value is
known only at one wavelength, except perhaps for
For non-case 1 situations, to solve for
the two unknowns, at least the a??? value and the
spectral relations of a???? and ag??? at two or more
bands are required. Table 3 extends the calcula-
tions for this purpose. As in other semianalytical
algorithms,31there is no separation of the absorption
coefficient of detritus from that of gelbstoff, so the
derived ag?440? here is actually the sum of detritus
and gelbstoff absorption coefficients.
veloped a simple empirical algorithm for that sepa-
Basically, the approach here assumes that a??? val-
ues at both 410 and 440 nm are calculated by the
steps in Table 2. For the decomposition, two more
values must be known:
? has been either related to
chlorophyll concentration48or pigment absorption at
As chlorophyll concentration or
pigment absorption are still unknowns, the value of ?
cannot be derived by use of such approaches.
the value of ? is estimated in step 7 by use of the
spectral ratio of rrs?440??rrs?555? based on the field
data of Lee et al.11
The value of ? is calculated in
step 8 when we assume a spectral slope of 0.015
Note that the values of ? and ? may vary
based on the nature of waters under study, such as
pigment composition,50humic versus fulvic acids,51
and abundance of detritus.52
S ?involved in the calculation of ?? will improve the
separation of the absorption coefficients of pigment
When the values of a?410?, a?440?, ?, and ? are
?a?410? ? aw?410? ? ?a??440? ? ?ag?440?,
a?440? ? aw?440? ? a??440? ? ag?440?.
For this latter purpose, data processing con-
It is impossible to
Local information or
Solving this set of simple algebraic equations pro-
And, if values of a???, ag?440?, and S are known, the
a???? spectrum can then be easily calculated:
? a??? ? aw??? ? ag?440? exp??S?? ? 440??.
earlier approaches,23,27–29the derivation of a???? here
requires no prior knowledge of what kind of phyto-
plankton pigments might be in the water or of a
spectral model for a???? at all wavelengths, although
we do need to know a??410??a??440?.
To test the performance of the QAA, we applied it to
both simulated and field-measured data sets, neither
of which are case1dependent.
no involvement in the measurements; the errors as-
avoided.Then the differences between retrievals
and inputs are solely due to the algorithm.
Data to Test the Quasi-Analytical Algorithm
Simulated data have
To create a valid data set, we varied the pigment
concentration ?C?, as in Sathyendranath et al.,22and
other bio-optical parameters in a way that generally
mimics those found in the natural field.
ing provides details about the data simulation.
would be better to use a numerical simulation tech-
nique such as Monte Carlo33,35or HYDROLIGHT45to cre-
ate a simulated data set.
wide variation of water properties in the field, it is
quite time-consuming to create a large case2data7set
with such numerical techniques.
generality regarding the QAA, we used Eq. ?1? to
replace the tedious radiative transfer computations.
Later the QAA is applied to rrsvalues with added
noise. These rrsvalues are then closer to the data
from field measurements or from satellite sensors.
Values of absorption and backscattering coeffi-
However, because of the
Table 3. Steps to Decompose the Total Absorption to Phytoplankton and Gelbstoff Components, with Bands at 410 and 440 nm
? ? a??410??a??440?
? 0.71 ?
0.8 ? rrs?440??rrs?555?
? ? ag?410??ag?440?
??a?410? ? ?a?440??
? ? ?
? a?440? ? ag?440? ? aw?440?
–?aw?410? ? ?aw?440??
? ? ?
5760APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002
cients are needed to create rrsby use of Eqs. ?1? and
a??? ? aw??? ? a???? ? ag???,
bb??? ? bbw??? ? bbp???. (11)
Values for aw??? and bbw??? are already known.41,53
We used the following bio-optical models33,48,54,55to
create optical data sets that simulate oceanic and
a??440? ? ?A?C??B??C?,
ag?440? ? p1a??440?,
bbp?555? ? ?0.002 ? 0.02?0.5
? 0.25 log??C????p2?C?0.62.
a???? ? ?a0??? ? a1???ln?a??440???a??440?,
ag??? ? ag?440?exp??S?? ? 440??,
bbp??? ? bbp?555??
where values for a0??? and a1??? are known.56
that here ?C? is used only as a free parameter for
designation of a wide range of absorption and back-
For case1waters,33,54p1? 0.5, p2? 0.3, Y ? 1.0;
and average A and B values are 0.0403 and 0.332
?Ref. 48? so that all optical properties co-vary with ?C?
values, and only one fixed rrs??? spectrum will be
created for a ?C? value.It is found in the field, how-
ever, that different rrs??? spectra exist for the same
?C? values.To accommodate such observations, we
kept B ? 0.332 and perturbed the other case1param-
eters in the following way:
A ? 0.03 ? 0.03?1,
p1? 0.3 ?
0.02 ? a??440?,
p2? 0.1 ? 0.8?3,
Y ? 0.1 ?1.5 ? ?4
1 ? ?C?
S ? 0.013 ? 0.004?5, (14)
where ?1, ?2, ?3, ?4, and ?5are random values
between 0 and 1. These kinds of perturbation make
A, p1, p2, Y, and S random values for each ?C? value,
but fall in a range generally consistent with field
In general, A will range be-
tween 0.03 and 0.06, p1between 0.3 and 4.0, p2be-
tween 0.1 and 0.9, Y between 0.1 and 2.5, and S
between 0.013 and 0.017 nm?1.
tent with field observations, the range for p1is nar-
rower for low ?C? values ?open ocean? and wider for
high ?C? values ?coastal?, and Y decreases with in-
creasing ?C? values,22but in a random way for both p1
Also, to be consis-
Values of g0and g1are also needed if we want to
create rrs??? by Eq. ?1?.As discussed above, values of
g0and g1are not constant; they may vary with phase
function and scattering properties.38
not known yet how they vary, we perturbed the g0
and g1values in the following way:
Because it is
g0? 0.084 ? 0.011?6,
g1? 0.0794 ? 0.0906?7. (15)
?6and ?7are random values between 0 and 1; thus
g0will be randomly between 0.084 and 0.095 and g1
randomly between 0.0794 and 0.17, within the re-
ported values of Gordon et al.33and Lee et al.34
Therefore, for a ?C? value, there will be a range of
rrs??? spectra simulated that are not just a function of
?C?, but are also a function of the seven random val-
ues ??1–7?. Figure 3?a? shows examples of rrs???
spectra for ?C? ? 1.0 mg?m3, with rrs??? curves widely
spread for the same ?C? value.
0.03–30 mg?m3, 480 rrsspectra with wavelengths at
410, 440, 490, 555, and 640 nm were simulated.
ure 3?b? shows how the simulated a?440? and bbp?555?
values relate to the ?C? values of this data set,
whereas Fig. 3?c? shows the ag?440??a??440? varia-
tions.These wide variations clearly indicate that
this data set is not case1.
and a?440? of this data set relate to the spectral ratio
of rrs?490??rrs?555?, a ratio often used in empirical
points indicates that it is hard to develop accurate
empirical algorithms for ?C? or a?440? simply on the
basis of this ratio.
For a ?C? range of
Figure 3?d? shows how ?C?
The scatter of the data
The data collected for waters around Baja California
were used to see how the QAA-derived total absorp-
tion coefficient compared with AC9- ?WET Labs, Inc.?
measured values.This data set was collected in
October 1999 during the Marine Optical Character-
ization Experiment 5, with a chlorophyll-a concentra-
tion range of 0.2–9.4 mg?m3.
remote-sensing reflectance was measured with a
hand-held spectroradiometer from above the sea sur-
face.The total absorption coefficients were mea-
sured through a vertically profiling of an AC9
instrument.The measurement method for remote-
sensing reflectance can be found in Lee et al.11,46and
Mueller et al.58
The vertical profiles of the total absorption coeffi-
cients ?the Optical Group at Oregon State University?
were obtained by an AC9 instrument attached to the
Slow Descent Rate Optical Platform ?WET Labs,
Inc.?. The profiling system was deployed with a
winch wire, with descent rates typically of the order
of 25–50 cm?s, providing submeter resolution.
details regarding the measurement and calibration of
the AC9 instrument as well as the Slow Descent Rate
Optical Platform see Pegau et al.59and Ref. 60.
Because remote sensing is a measure of the upper
water column,61the absorption values from the ver-
tical profiles were optically averaged by the approach
For each station,
20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS5761
of Gordon62to make it comparable to those derived
from color inversions.
of the QAA. To be comparable with a few other in-
version algorithms, Table 4 summarizes the percent-
age errors for a?440?, bbp?555?, a??440?, and ag?440?.
A comparison at other wavelengths is presented ei-
ther in Figs. 4–12 or in the text.
In Table 4, the two-band empirical algorithm is
from Lee et al.,11which uses a spectral ratio of
rrs?490??rrs?555?. The spectral optimization algo-
rithm is the same as used in Lee et al.34,63by use of
Results and Discussion
available rrsat wavelengths 410, 440, 490, 555, and
The percentage error listed in Table 4 is calculated
in the following way.First we calculated RMSElog10
?root-mean-square error in log10scale? of quantity qn
between the derived and the true values:
der? ? log10?qn
Then the linear percentage error is
ε ? 10RMSElog10? 1.(17)
spectral ratio of remote-sensing reflectance at 490 and 555 nm.
Characteristics of the simulated data.
?c? Values of ag?440??a??440? versus values of a??440?.
?a? Possible rrs??? curves for ?C? ? 1.0 mg?m3.
?b? Values of a?440? and bbp?555? versus
?d? Pigment concentration and a?440? versus the
Table 4. Percentage Error ? between Derived and True Values of the Simulated Data
a?440? ? 0.3 m?1
Full range 0.01 ? a?440? ? 2.0 m?1
aLee et al.11empirical formula.
5762APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002
This kind of analysis puts equal emphasis on under-
estimates as well as on overestimates.
Figure 4?a? presents the empirically derived a?555?
values ?from step 2, open circles? versus true values,
and Fig. 4?b? presents empirically derived Y values
?step 4? versus true values.
algorithms used, did not produce accurate returns ?ε
values are 16% for a?555? and 35% for Y?.
as discussed in Section 2, these two properties are of
second-order importance, so the inaccuracy here does
not affect the end results much.
the a?555? errors ?underestimate? were at a?555?
greater than ?0.2 m?1?which is a factor of 3 of the
aw?555? value?, which indicates less accuracy in the
estimation of a?555? for higher-absorbing waters, as
discussed in Section 2.
Figure 5 compares the optical properties derived
from the QAA with 555 nm ?QAA-555? as the refer-
Apparently these two
Actually, most of
ence wavelength versus their true values, with Fig.
5?a? for the total absorption coefficient at 410, 440,
and 490 nm; Fig. 5?b? for the particle backscattering
coefficient at 555 nm; Fig. 5?c? for the pigment ab-
sorption coefficient at 410, 440, and 490 nm; and Fig.
5?d? for the gelbstoff absorption coefficient at 440 nm.
Separation into a???? and ag??? was achieved after we
added the 410-nm band ?Table 3? and used an S value
of 0.015 nm?1. In the process, it is noticed that
?a?555? in step 2 is derived from the coarse first
estimate of a?440?i, which involves some empirical
parameters that may not well represent the waters
under study.To overcome the uncertainties, the
steps from step 2 to step 6 are repeated once, with
a?555? estimated ?step 2? with a?440? values from the
first round of calculations.
found when additional iteration was applied, and ap-
parently the results did not converge with more iter-
Clearly the derived a??? and a???? match their true
values well for the three wavelengths shown, espe-
cially for the clearer waters ?a?440? ? 0.3 m?1?.
the entire data set ?0.01 ? a?440? ? 2.0 m?1?, the
linear percentage error ?ε value? is 14.3% for a???,
18.6% for bbp?555?, 17.7% for a????, and 17.5% for
ag?440? values. For clearer waters ?a?440? ? 0.3
m?1, which encompass most oceanic waters54?, the ε
values for a???, bbp?555?, a????, and ag?440? are 8.2%,
6.7%, 10.6%, and 13.1%, respectively.
ingly, additional error is introduced when we further
partition a??? into a???? and ag??? values because the
parameters for a??410??a??440? and ag?410??ag?440?
used in the algorithm ?Table 3? may not match their
input values ?S was randomly in a range of 0.013–
0.017 nm?1in the data simulation?.
ter estimate the S and ? values are needed if we want
to improve the a???? and ag??? separation.
Figure 6 compares algorithm-derived a?440? values
versus true a?440? values.
process data at similar speeds ?they take seconds to
process a standard SeaWiFS image?, but the error of
the QAA is a factor of 3 smaller than that of the
empirical algorithm for the entire a?440? range ?a
factor of 5 smaller for a?440? ? 0.3 m?1?.
band empirical algorithm11overestimated the a?440?
values, especially at the lower end.
because ?1? the data set used for the algorithm devel-
opment did not cover enough lower a?440? points or
?2? the a?440? values used for the algorithm develop-
ment, which were derived from the downwelling dif-
fuse attenuation coefficient,11might still contain
errors associated with the spectral variation of the
average cosine for downwelling irradiance.19
comparison here, however, further emphasizes that
performance of empirical algorithms depends on the
consistency and accuracy of the data set used in the
algorithm development, but the QAA presented here
reduced that dependence.
Most of the a??? error for the QAA-555 appears at
the higher end of the data, which further leads to
larger errors in a???? and ag??? ?see Fig. 5?.
for waters with higher chlorophyll or gelbstoff con-
No improvement was
Methods to bet-
These two algorithms
This could be
values ?x axis?.
values ?x axis?.
?a? Empirically estimated a?555? values versus true a?555?
?b? Empirically estimated Y values versus true Y
20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS 5763
centrations, where the a?555? values depart more
from the values of pure water, and the empirically
estimated a?555? ?step 2? are subject to larger errors
when the water is less clear ?see Fig. 4?a??.
For such higher-absorbing waters ?a?440? ? 0.3
m?1?, the reference wavelength needs to be shifted to
a place longer than 555 nm, where pure-water ab-
sorption is still dominant even for larger a??440? and
this band is used as the reference wavelength.
analytical steps listed in Table 2 remain the same,
but the estimation of a?555? will be changed to the
estimation of a?640?. Absorption of pure water at
640 nm is a factor of 5 larger than that at 555 nm, and
the absorption of pigment and gelbstoff have limited
contributions at 640 nm for most oceanic and coastal
waters. As a result, the variation of a?640? is even
narrower than that of a?555? ?see Fig. 2?.
mate a?640?, step 2 is changed to
a?640? ? 0.31 ? 0.07?
with 0.31 in Eq. ?18? the absorption coefficient of pure
water at 640 nm.53
The backscattering coefficient at
640 nm ?step 3? is then calculated with known u?640?
and a?640? values.
a??? is further calculated at step
6.With bands at 410 and 440 nm, a??440? and
ag?440? values are also calculated with the steps
listed in Table 3.
Figure 7 presents the same comparisons as Fig. 5,
but with retrievals by the QAA with 640 nm as the
reference wavelength ?QAA-640?.
in Table 4. Clearly, for the higher-absorbing waters
similar performance at the lower-absorption end.
This does not mean, however, that QAA-640 is al-
ε values are listed
the simulated data.
Comparison of the QAA-555 derived values of ?a? a???, ?b? bbp?555?, ?c? a????, and ?d? ag?440? versus their true values ?x axes? of
true a?440? values of the simulated data.
algorithm is from Lee et al.11
Comparison of algorithm-retrieved a?440? values versus
The two-band empirical
5764 APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002
ways better than QAA-555.
times that of aw?555?, rrs?640? will be ?5 times
smaller than rrs?555?, which makes it more difficult to
accurately measure rrs?640? than to measure rrs?555?
for open-ocean waters, where rrs?640? is small.
Therefore more errors could be introduced when
QAA-640 is applied to open-ocean waters.
The errors are generally smaller for the four optical
properties when we use a spectral optimization ap-
proach34?see Table 4?. The spectral optimization
used rrsvalues at all available wavelengths ?QAA
used only two bands for the derivation of the total
absorption coefficient at 440 nm; more bands could be
included to fine-tune the estimation of a??0??, which
improved retrieval accuracy at the higher-absorption
end. Also, it is the same model for the pigment ab-
sorption coefficient used in the optimization and in
the simulation, which helps to match the spectral
shapes of the pigment absorption coefficient.
these shapes do not match each other closely, how-
ever, extra errors will be introduced to the optimiza-
tion retrievals.Also, it must be kept in mind that
optimization processing is time-consuming,31and
currently it may take hours to process, for example, a
standard SeaWiFS image.
When the QAA uses 640 nm as the reference wave-
length, however, the difference between the accuracy
of the QAA and the accuracy of the optimization is
small. Also, if we ignore these high-end values but
Because aw?640? is ?5
focus on the data with a?440? less than 0.3 m?1, then
the ε value for a??? is 8.2% by the QAA-555 and 5.9%
by the optimization. It is encouraging that the
QAA-555 provides equivalent results to those from
the optimization algorithm for these clearer waters,
suggesting that the QAA-555 could be used to process
satellite data of such waters with results comparable
to optimization processing.
Furthermore, if there is a band at 640 nm, the
estimation for a?555? can be greatly improved, at
least for this data set, with
a?555? ? 0.0596 ? 0.56??
The a?555? values fromapproximation ?19? are shown
in Fig. 4?a? ?solid circles, 3.9% percentage error?.
ter we replace the formula in step 2 with approxima-
tion ?19? and derive a???, bbp???, a????, and ag???
values again using the steps listed in Tables 2 and 3,
Fig. 8 compares them to their true values.
all four properties are retrieved better than those
shown in Fig. 5, with linear percentage errors of
6.5%, 7.3%, 12.7%, and 12.2% for a???, bbp?555?,
a????, and ag?440?, respectively.
This example indi-
the simulated data.
Comparison of the QAA-640 derived values of ?a? a???, ?b? bbp?555?, ?c? a????, and ?d? ag?440? versus their true values ?x axes? of
20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS 5765
In the above, we used error-free data ?perfect rrs? to
test the QAA.In the real world, rrsvalues will not
be as good as the above because of measurement and
forms for erroneous data, two tests were carried out.
In test 1 we added ?10% random noise to the simu-
lated rrsvalues at each wavelength and ran the QAA-
555. In test 2 we increased all rrsvalues by 20% and
again ran the QAA-555.
viewed as errors to rrsintroduced by imperfect sensor
calibration or atmospheric correction.
shows the results of test 1, and Fig. 10 shows the
results of test 2. As above for perfect data, steps 2–6
in Table 2 were repeated once.
though the rrsvalues were perturbed, the QAA-555
still returned quite good a??? estimates ?ε values are
19.5% for test 1 and 14.6% for test 2?.
ingly, the random noise has great influence on the
decomposition of the total absorption into phyto-
plankton pigment ?ε ? 57.0% for a????? and gelbstoff
?ε ? 37.3% for ag?440??, whereas the 20% increase has
more influence on the backscattering coefficient ?ε ?
32.1% for bbp?555??. The results of these tests fur-
ther indicate that the QAA itself is quite noise toler-
ant to address field data, at least for the total
It is necessary to keep in mind that the above error
analysis by use of the simulated data provides only a
Influence of Nonperfect Data
To see how the QAA per-
This added noise could be
For both tests, al-
general guidance about the performance of the QAA.
This is because ?1? the simulated data cannot cover
all the variations of the natural waters and ?2? the
expressions of Eqs. ?13? and ?14? may not be accurate
in describing the responses of optical properties to
biogeochemical properties of natural waters.
any other algorithms, use of accurate field-measured
data to test, validate, and improve the QAA should be
done if we want to apply this QAA for oceanic obser-
Figure 11?a? compares the QAA-derived a??? versus
the AC9-measured a??? for wavelengths at 410, 440,
490, and 530 nm.In the inversion process, a?555?
was estimated by approximation ?19?.
wavelengths, the ε value is 12.5% and r2is 0.98 ?N ?
80?, with the values from the AC9 slightly larger than
the values from QAA.These values suggest that the
differences are close to the accuracy limitations of
each method itself. These results indicate that the
true total absorption coefficients are near the ones
from each process, although it is hard to know which
set is closer to the truth as each method has its own
sources of errors.
Figure 11?b? compares the QAA-derived a?490? ver-
sus AC9-measured a?490?, along with the current
SeaWiFS algorithm-derived a?490?.
For the four
the simulated data, after improved a?555? estimation.
Comparison of the QAA-555 derived values of ?a? a???, ?b? bbp?555?, ?c? a????, and ?d? ag?440? versus their true values ?x axes? of
5766APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002
a?490? is converted from the SeaWiFS K?490? ?diffuse
attenuation coefficient at 490 nm ?Ref. 10?? after we
applied the model of Morel,54with K?490? derived
empirically with the ratio of rrs?440??rrs?555?.10
parently the SeaWiFS a?490? values are much larger
at the higher end, whereas the QAA a?490? is slightly
smaller than the AC9 a?490? ?the ε value is 9.8%?.
The larger SeaWiFS a?490? at the higher end sug-
gests that this algorithm needs to be adjusted for
some of the turbid coastal waters.
QAA process used rrs?640? values ?in step 2 by ap-
proximation ?19??, but the SeaWiFS process did not.
This field data set, however, is quite small and
covers only a narrow range of natural waters, which
is insufficient to completely validate the accuracy of
the QAA, although the QAA was tested with the wide
range of simulated data.
with field-measured data are desired and required for
the validation and improvement of the QAA.
Note that the
More independent tests
to Other Field Data
As an indirect test and consistency check of the per-
formance of the QAA, it is applied to our earlier col-
lected field data,11which covers a wider range of
waters. The QAA-derived a??? were compared with
those from the spectral optimization.
of the best returns in ocean-color retrieval.31
Comparison with Optimization Results when Applied
It is currently
QAA is valid, its retrievals should be at least close to
that from optimization to data not only from simula-
tion but also from field measurements.
This field data came from a variety of oceanic and
coastal environments with a wide range of water
types, covering waters such as the West Florida
Shelf, the Gulf of Mexico ?including the Mississippi
River plume?, the Bering Sea, the Arabian Sea, the
Sargasso Sea,and near
Chlorophyll-a concentrations ranged from 0.05 to
50.0 mg?m3. Only optically deep waters were used.
umented in Lee et al.11
Figure 12?a? shows derived a?440? values ?in a
range of 0.025–2.0 m?1? fromthe QAA compared with
those by use of full-spectrum optimization ?400–800
nm with 5-nm spacing, 65 effective bands28?.
Clearly, as shown in the simulated data set, the QAA-
555 provides almost identical results to those from
optimization for a?440? of less than 0.3 m?1?ε ?
7.3%?, but underestimates a?440? values when it gets
larger ?ε ? 16.9% for the entire range?.
With 640 nm as the reference wavelength, the al-
gorithm improves the estimates for higher a?440? val-
ues, but performs less accurately at the lower a?440?
values. As discussed in Subsection 4.A, rrs?640? in
the field are small for clearer waters and difficult to
accurately measure.They can be further perturbed
by nonperfect removal of surface-reflected skylight.
simulated data after ?10% random noise was added to the rrsvalues at each band.
Comparison of the QAA-555 derived values of ?a? a???, ?b? bbp?555?, ?c? a????, and ?d? ag?440? versus their true values of the
20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS 5767
Also, the effects to rrsof Raman scattering are larger
for clearer water at longer wavelengths.
These results suggest that a combination between
the results of QAA-555 and that of QAA-640 can im-
prove estimates for the entire range of this data set.
coefficient from the QAA-555 and that a????640?is the
derived total absorption coefficient from the QAA-
640, a combination of the two are
Figure 12?b? shows the combined a??? versus
optimization-derived a???, with an ε value of 10.3%,
suggesting that the two retrievals are consistent with
each other. However, because the results from the
spectral optimization also contain errors,28,34they
cannot be used as standards to validate the accuracy
There is no field-measured backscattering coeffi-
cient for comparison here.
that, when a remote-sensing reflectance model is ap-
It needs to be pointed out
plied in the field, because the actual shape of the
particle phase function is not known, the differences
between the used and required g0and g1values may
be larger than the differences that appeared in the
simulated data ?g0and g1values are fixed in the
inversion, but their values are somewhat random in
the simulation?. This mismatch can contribute as
much as 20% extra error to the retrieval of the par-
ticle backscattering coefficients.35,36,64
match has only limited influence, however, on the
absorption coefficient, as absorption has no angular
dependence. Therefore we should expect larger er-
rors for estimates of bbpwhen applied to field data
than those errors found for the simulated data.
Inelastic scattering such as Raman scattering and
gelbstoff fluorescence are not considered in this
study. Their effects can be corrected, however, to
first order with proposed models65–67given knowl-
edge of the absorption and backscattering coefficients
simulated data after a 20% increase to the rrsvalues at each band.
Comparison of the QAA-555 derived values of ?a? a???, ?b? bbp?555?, ?c? a????, and ?d? ag?440? versus their true values of the
a??? ? a????555?,
a??? ? a????640?,
for a?440??640?? 0.2,
0.3 ? a?440??640?
?a????555???1 ?0.3 ? a?440??640?
for 0.2 ? a?440??640?? 0.3,
for a?440??640?? 0.3.
5768 APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002
of the water column.
nadir,68values of T, g0, and g1need to be slightly
adjusted. Without any correction, however, it is
found that most of the errors will be transferred to
the retrieved particle backscattering coefficient, with
only little influence on the retrieved total absorption
For sensors not viewing at
For open-ocean and coastal waters, a multiband QAA
is developed to retrieve total absorption and back-
scattering coefficients, as well as absorption coeffi-
cients of phytoplankton pigments and gelbstoff, from
remote-sensing reflectance spectrum.
rithm is based on relationships between remote-
sensing reflectance and inherent optical properties of
the water derived from the radiative transfer equa-
tion. For the derivation of total absorption and
backscattering coefficients, there are no spectral
models involved for pigment and gelbstoff absorption
coefficients, which reduce model influences on the
derived values.Furthermore, the near-analytical
nature and the explicit step-by-step sequential calcu-
lation of the QAA make it easy to diagnose possible
error sources in water-color inversion.
The algorithm uses multiband rrsvalues as inputs,
and absorption and backscattering coefficients are
calculated analytically from the values of rrs.
algorithm can be applied to sensors with multispec-
tral or hyperspectral bands as long as accurate rrs???
values are available.It can also be easily adapted
for use of subsurface irradiance reflectance values as
inputs for the calculations.
There are only two empirical aspects in the deri-
vation of the total absorption and backscattering co-
efficients in the QAA. One relates to the estimation
of the total absorption coefficient at a reference wave-
length ?a??0?, as shown here a?555? or a?640??; the
other relates to the estimation of the spectral slope
?Y? of the particle backscattering coefficient.
reference wavelength is where rrs??0? from elastic
scattering can be well measured and a??0? can be well
estimated. For a??0?, because of its dominance by
water absorption, errors are limited.
measurements of Y are available, less is known about
how Y varies.Often in practice a smaller value
?close to 0? is assumed for coastal waters, and a larger
measured a??? for field data.
isonoftheQAA-deriveda?490? versusAC9-measureda?490?, along
with SeaWiFS algorithm-derived a?490?.
?a? Comparison of the QAA-derived a??? versus AC9-
?b? For the same field data, compar-
optimization-derived a?440? for earlier collected field data ?see text
?b? Comparison of the QAA-derived a??? versus
optimization-derived a??? ?x axes?.
the results from QAA-555 and QAA-640.
?a? Comparison of the QAA-derived a?440? versus
QAA-a??? is a combination of
20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS 5769
value ?around 2.0? is assumed for open-ocean wa-
Based on a field study, here we use a
rrs?440??rrs?555? ratio to empirically estimate the Y
To test the first part of the QAA ?Table 2?, we
applied it to a simulated data set and a field data set,
neither of which are case1dependent.
ulated data set, the pigment concentration ranged
from 0.03 to 30 mg?m3, with the total absorption
coefficient at 440 nm in a range of 0.01–2.0 m?1.
With the QAA-555, the percentage error for a??? is
14.3% for the entire range and ?8% for a?440? ? 0.3
m?1. To the field data, the percentage difference for
a??? is 12.5% ?r2? 0.98, N ? 80? when compared with
We also applied the QAA to an earlier collected
field data set to test its consistency versus the opti-
mization results. We found that the percentage dif-
ference for a??? is 10.3% for a??? ranging from 0.025 to
2.3 m?1and it is just 6.9% for a?440? ? 0.3 m?1.
These results suggest that the performance of the
QAA is at least close to that of the optimization ap-
proach in the retrieval of total absorption coefficients
For less clear waters, a band around 640 nm as the
reference wavelength is suggested for better retriev-
als. We used 640 nm here for the possible applica-
tion to the MODIS sensor, which has a band at 645
nm. For future sensors such as the medium-
resolution imaging spectrometer ?MERIS? or the
ocean-color imager, a band around 620 nm may work
as well as the 640-nm band.
or coastal waters, it may be necessary to shift ?0to
even longer wavelengths.
It is much more challenging to separate the phyto-
plankton absorption and gelbstoff absorption from
the total absorption coefficient.
information regarding the spectral relations of the
pigment and gelbstoff absorption at a few bands is
required. With bands at 410 and 440 nm, the pig-
ment and gelbstoff absorption coefficients are sepa-
coefficient, after a simple empirical algorithm is em-
ployed for the ratio of a??410??a??440?.
ulated data here, the percentage error was 18% ?10%
for clearer waters? for a???? and 18% ?13% for clearer
waters? for ag?440? when the QAA-555 was applied.
More importantly, the separation of Tables 2 and 3
clearly indicates that the retrieval of the total absorp-
tion coefficients is independent of mathematical mod-
els for the absorption coefficients of phytoplankton
pigments and gelbstoff.
The errors in the derived inherent optical proper-
ties here by the QAA are similar to those found by a
full-spectrum optimization approach, but the compu-
tation time is dramatically reduced ?hours to seconds
makes it suitable to process large data sets such as
For the sim-
For turbid river plume
For the separation,
For the sim-
We are in debt to Andrew Barnard for his AC9 data
and grateful for the comments and suggestions from
Jennifer Cannizzaro and the two anonymous review-
ers. Financial support was provided by the follow-
ing contracts and grants:
97137, NAS5-31716, and NAG5-3446; the U.S. Office
of Naval Research grants N00014-96-I-5013 and
N00014-97-0006; and the National Oceanic and At-
mospheric Administration Coastal Ocean Program
NASA grants NAS5-
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