Page 1

Deriving inherent optical properties from water color:

a multiband quasi-analytical algorithm for optically

deep waters

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

OCIS codes:

010.4450, 290.5850, 280.0280.

Actually those absorption

It can be applied to data from

There are only limited

1.

Absorption and backscattering coefficients are inher-

entopticalproperties.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

applications.2–6

Methods

these optical properties remotely have been under

Introduction

Combined withdown-

As inher-

toaccurately retrieve

investigation for several decades, and algorithms

from empirical to full-spectral optimization have

been proposed.2,7–31

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

They do

Because of the na-

More

The

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 ?zplee@nrlssc.navy.mil? 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

June 2002.

0003-6935?02?275755-18$15.00?0

© 2002 Optical Society of America

20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS5755

Page 2

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.

mizationprocedureistime-consuming,31whichlimits

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-

scatteringcoefficients

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

These

The performance

In addition, the opti-

fromremote-sensing

Furthermore,

and current ocean-color satellite sensors, such as the

Coastal Zone Color Scanner ?CZCS?, the sea-viewing

widefield-of-viewsensor

moderate-resolutionimaging

?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.

?SeaWiFS?,

spectroradiometer

and the

This algorithm can

2.Multiband Quasi-Analytical Algorithm

A.

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

andtotalabsorptioncoefficients.

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

matrix inversion.25

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

andnumericalsimulations33–38oftheradiativetrans-

fer equation, rrsis a function of the absorption and

backscattering coefficients.

General Concept

The right side shows

Ourapproachhere

The methods to derive

Converting pig-

Specifically, measure-

Table 1.Symbols and Definitions

Symbol DescriptionUnit

a

a?

ag

aw

a????555?

a????640?

bbp

bbw

bb

A, B

?C?

Y

qn

qn

Rrs

rrs

S

u

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

Pigment concentration

Spectral power for particle backscattering coefficient

Algorithm-derived value

True value

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?

Reference wavelength

a??410??a??440?

ag?410??ag?440?

m?1

m?1

m?1

m?1

m?1

m?1

m?1

m?1

m?1

mg?m3

der

true

sr?1

sr?1

nm?1

?0

?

?

nm

5756APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002

Page 3

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)

with

u ?

bb

a ? bb

.(2)

Here a is the total absorption coefficient and bbis the

total backscattering coefficient.

?whichmaynotbeshownforbrevity?.

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-

ters.

For nadir-viewed rrs, Gordon et al.33found that

? is the wavelength

bbisnormally

bbpis

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-

analyticalalgorithm.Without

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,

preferenceand

u??? ??g0? ??g0?2? 4g1rrs????1?2

2g1

. (3)

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

bb?

ua

1 ? u,(4)

or knowing bbwill lead to

a ??1 ? u?bb

u

. (5)

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 OPTICS5757

Page 4

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

suspended constituents.

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-

lated.

We note that, at longer

Figure 2 pre-

In these exam-

The wavelength dependence of bbp??? is normally

expressed as2,22,42,43

bbp??? ? bbp??0??

?0

??

Y

. (7)

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-

cient.

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.

Table 2

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

Reference Wavelength

StepPropertyMath Formula

Order of

Importance Approach

0

rrs

?Rrs??0.52 ? 1.7Rrs?

??g0? ??g0?2? 4g1rrs????1?2

2g1

?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??

u?555?a?555?

1 ? u?555?

? 2.2?1 ? 1.2 exp??0.9rrs?440?

? bbp?555??

??1 ? u?????bbw??? ? bbp????

u???

1st Semianalytical

1

u???

1stSemianalytical

2

a?555?

2ndEmpirical

3

bbp?555?

?

? bbw?555?

1st Analytical

4

Y

rrs?555???

2nd Empirical

5

bbp???

555

??

Y

1stSemianalytical

6

a???

1st Analytical

5758 APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002

Page 5

B.

Coefficients

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

If mea-

rrs?

Rrs

T ? ?QRrs

,(8)

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,

ingeneral,rangesbetween3and6.35

?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

AspointedoutaboveanddiscussedinSubsection4.A,

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-

proach.

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

? is

Q is

For a nadir-

AsRrsissmall

Depending

As an example, we

The

a?440?iis calculated

How-

A

Historically, re-

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.

?5?.

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

algorithms.

Values of a?555?, how-

If the true

C.

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-

causethese properties

concentrations of chlorophyll26or CDOM,27respec-

Decomposition of the Total Absorption Coefficient

For

canbeconvertedto

20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS 5759

Page 6

tively.

tinues.

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

case1waters.7

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-

ration.

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:

???ag?410??ag?440??.

? has been either related to

chlorophyll concentration48or pigment absorption at

a wavelength.23–29

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

nm?1.49

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

improvedalgorithmstocalculate?andtheparameter

S ?involved in the calculation of ?? will improve the

separation of the absorption coefficients of pigment

and gelbstoff.

When the values of a?410?, a?440?, ?, and ? are

known,2,22,26

?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

Lee47has de-

???a??410??a??440?? and

Here,

Local information or

(9)

Solving this set of simple algebraic equations pro-

vides

?

ag?440???a?410???a?440????aw?410???aw?440??

???

,

a??440??a?440??aw?440??ag?440?.

(10)

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?.

a????

Unlike

3.

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-

sociatedwith themeasurement

avoided. Then the differences between retrievals

and inputs are solely due to the algorithm.

Data to Test the Quasi-Analytical Algorithm

Simulated data have

processes are

A.

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-

Simulated Data

The follow-

It

However, because of the

Without losing

Table 3.Steps to Decompose the Total Absorption to Phytoplankton and Gelbstoff Components, with Bands at 410 and 440 nm

Step PropertyMath Formula

Order of

ImportanceApproach

7

? ? a??410??a??440?

? 0.71 ?

0.06

0.8 ? rrs?440??rrs?555?

2ndEmpirical

8

? ? ag?410??ag?440?

? exp?S?440–410??

2ndSemianalytical

9

ag?440?

??a?410? ? ?a?440??

? ? ?

? a?440? ? ag?440? ? aw?440?

–?aw?410? ? ?aw?440??

? ? ?

1stAnalytical

10

a??440?

1stAnalytical

5760 APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002

Page 7

cients are needed to create rrsby use of Eqs. ?1? and

?2?2,22,42,52:

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

coastal waters:

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.

Furthermore,

(12)

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-

scattering values.

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:

555

??

Y

,(13)

Note

A ? 0.03 ? 0.03?1,

p1? 0.3 ?

3.7?2a??440?

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

observations.37,48,55,57

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

and Y.

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

chlorophyll algorithms.16

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

Fig-

Figure 3?d? shows how ?C?

The scatter of the data

B.

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

Field Data

For each station,

For

20 September 2002 ? Vol. 41, No. 27 ? APPLIED OPTICS5761

Page 8

of Gordon62to make it comparable to those derived

from color inversions.

4.

Table4aswellasFigs.4–12presenttheperformance

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

640 nm.

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:

RMSElog10???n?1

N

?log10?qn

der? ? log10?qn

N

true??2

?

1?2

.

(16)

Then the linear percentage error is

ε ? 10RMSElog10? 1. (17)

Fig. 3.

pigment concentrations.

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

Data Approach

a?440?

bbp?555?

a??440?

ag?440?

a?440? ? 0.3 m?1

Two-band empiricala

Optimization

QAA-555

QAA-640

Two-band empiricala

Optimization

QAA-555

QAA-640

0.436

0.059

0.083

0.079

0.426

0.061

0.143

0.076

0.333

0.096

0.094

0.136

0.402

0.176

0.166

0.123

0.031

0.067

0.069

0.143

0.131

0.134

Full range 0.01 ? a?440? ? 2.0 m?1

0.039

0.186

0.073

0.138

0.175

0.130

aLee et al.11empirical formula.

5762APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002

Page 9

This kind of analysis puts equal emphasis on under-

estimates as well as on overestimates.

A.

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.

derivations,becauseoftheirempiricalnatureandthe

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-

Simulated Data

Apparently these two

However,

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-

ations.

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

For

Not surpris-

Methods to bet-

These two algorithms

The two-

This could be

The

This is

Fig. 4.

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

Page 10

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

ag?440? values.

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

theQAA-640performsbetterthantheQAA-555,with

similar performance at the lower-absorption end.

This does not mean, however, that QAA-640 is al-

Assumingthereisabandat640nm,

The

To esti-

rrs?640?

rrs?440??

1.1

,(18)

ε values are listed

Fig. 5.

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

Fig. 6.

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

5764APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002

Page 11

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

If

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??

rrs?640?

rrs?555??

1.7

? 0.03?.

(19)

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.

catesthepossibleareasforfutureimprovementofthe

QAA.

Af-

Clearly

This example indi-

Fig. 7.

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 OPTICS5765

Page 12

B.

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

atmospheric corrections.

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

absorption coefficient.

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

Figure 9

For both tests, al-

Not surpris-

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-

vations.

As in

C.

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

twosetsofresultsareinexcellentagreement,andthe

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?.

Field Data

For the four

The SeaWiFS

Fig. 8.

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

Page 13

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.

Ap-

Note that the

More independent tests

D.

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.

assumedthattheoptimizationapproachprovidesone

of the best returns in ocean-color retrieval.31

Comparison with Optimization Results when Applied

It is currently

If the

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

SargassoSea,and near

Chlorophyll-a concentrations ranged from 0.05 to

50.0 mg?m3. Only optically deep waters were used.

Locationsandmeasurementsforthesedataweredoc-

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.

KeyWest

?Florida?.

Fig. 9.

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 OPTICS5767

Page 14

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.

Assumingthata????555?isthederivedtotalabsorption

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

of QAA.

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

The mis-

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

Fig. 10.

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??? ? a????640?,

for a?440??640?? 0.2,

0.3 ? a?440??640?

0.1

?a????555???1 ?0.3 ? a?440??640?

0.1

?a????640?,

for 0.2 ? a?440??640?? 0.3,

for a?440??640?? 0.3.

(20)

5768APPLIED OPTICS ? Vol. 41, No. 27 ? 20 September 2002

Page 15

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

coefficient.38

For sensors not viewing at

5.

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-

Summary

The algo-

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

The

The

Because few

Fig. 11.

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-

Fig. 12.

optimization-derived a?440? for earlier collected field data ?see text

for details?.

?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