Reports of the International
Ocean-Colour Coordinating GroupAn Affiliated Program of the Scientific Committee on Oceanic Research (SCOR)An Associate Member of the Committee on Earth Observation Satellites (CEOS)

IOCCG Report Number 5, 2006

Remote Sensing of Inherent Optical Properties:
Fundamentals, Tests of Algorithms, and Applications

Editor:ZhongPing Lee (Naval Research Laboratory, Stennis Space Center, USA)

Report of an IOCCG working group on ocean-colour algorithms, chaired by
ZhongPing Lee and based on contributions from (in alphabetical order):Robert Arnone, Marcel Babin, Andrew H. Barnard, Emmanuel Boss,
Jennifer P. Cannizzaro, Kendall L. Carder, F. Robert Chen, Emmanuel Devred,
Roland Doerffer, KePing Du, Frank Hoge, Oleg V. Kopelevich,ZhongPing Lee, Hubert Loisel, Paul E. Lyon, Stéphane Maritorena,
Trevor Platt, Antoine Poteau, Collin Roesler, Shubha Sathyendranath,
Helmut Schiller, Dave Siegel, Akihiko Tanaka, J. Ronald V. Zaneveld

Series Editor: Venetia Stuart

Correct citation for this publication:
IOCCG (2006). Remote Sensing of Inherent Optical Properties: Fundamentals,
Tests of Algorithms, and Applications. Lee, Z.-P. (ed.), Reports of the International
Ocean-Colour Coordinating Group, No. 5, IOCCG, Dartmouth, Canada.

The International Ocean-Colour Coordinating Group (IOCCG) is an internationalgroup of experts in the field of satellite ocean colour, acting as a liaison andcommunication channel between users, managers and agencies in the oceancolour arena.
The IOCCG is sponsored by NASA (National Aeronautics and Space Administration), NOAA (National Oceanic and Atmospheric Administration), ESA (European
Space Agency), JAXA (Japan Aerospace Exploration Agency), CNES (Centre National d’Etudes Spatiales), CSA (Canadian Space Agency), JRC (Joint ResearchCentre, EC), GKSS Research Centre (Geesthacht, Germany), BIO (Bedford Institute of Oceanography, Canada) and SCOR (Scientific Committee on Oceanic Research).

http://www.ioccg.org

Published by the International Ocean-Colour Coordinating Group,
P.O. Box 1006, Dartmouth, Nova Scotia, B2Y 4A2, Canada.
ISSN: 1098-6030 ISBN: 978-1-896246-56-7©IOCCG 2006
Printed by GKSS Research Centre, Max-Planck Str., Geesthacht, Germany.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1 Why are Inherent Optical Properties Needed in Ocean-Colour Remote
Sensing?
3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The Forward Problem of Ocean Optics . . . . . . . . . . . . . . . . . . 5
1.3 Inherent Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 The Inverse Problem of Ocean Optics . . . . . . . . . . . . . . . . . . . 8
1.5 The Dependence of the Remote Sensing Reflectance on the IOPs . . 10
2 Synthetic and In Situ Data Sets for Algorithm Testing13
2.1 In situ Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Synthetic Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Uncertainties in the Products of Ocean-Colour Remote Sensing
3.1 Sources of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Uncertainties in in situ measurements (LwN , Rrs , C, IOP) . . .
3.1.2 Uncertainties in satellite measurements (LwN ) . . . . . . . . . .
3.1.3 Uncertainties and assumptions in the functional relationship that links LwN and IOP and in the inversion procedureused to derive the products . . . . . . . . . . . . . . . . . . . . .
3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19
20
20
20

4 Simple Algorithms for Absorption Coefficients
4.1 One-Step Spectral Ratio Algorithm . . . . . . . . . . . . . . . . . . . . .
4.1.1 General description . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Results and discussion when applied to the IOCCG data sets
4.2 Spectral Curvature Algorithm . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Results and discussion when applied to the IOCCG data sets
4.3 Spectral-Ratio Algorithm with Chlorophyll Concentration as an Intermediate Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Results and discussion when applied to the IOCCG data sets
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
27
27
27
29
29
30

i

21
24

31
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Remote Sensing of Inherent Optical Properties

5 Inversion of IOP based on Rrs and Remotely Retrieved Kd
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Output and Input Parameters . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Assumptions and Description . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Comparison with synthetic data . . . . . . . . . . . . . . . . . .
5.4.2 Comparison with in situ data . . . . . . . . . . . . . . . . . . . .
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
35
36
36
37
38
39
40

6 The MERIS Neural Network Algorithm
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Description of the MERIS Case 2 Water Algorithm . . . . . . . . . . .
6.3 aNN Results with the IOCCG Data Sets . . . . . . . . . . . . . . . . . .

43
43
43
45

7 The Linear Matrix Inversion Algorithm
7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Inputs of LMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Basic Assumptions of LMI . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Algorithm mathematical description . . . . . . . . . . . . . . .
7.4.2 IOP spectral models . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1 Synthetic data set . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.2 In situ data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.1 Overall results of the linear matrix inversion algorithm . . .
7.6.2 Algorithm weaknesses . . . . . . . . . . . . . . . . . . . . . . . .
7.6.3 Algorithm strengths . . . . . . . . . . . . . . . . . . . . . . . . .

49
49
49
49
50
50
51
52
52
53
54
54
54
55

8 Over Constrained Linear Matrix Inversion with Statistical Selection
8.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Results and Discussion with IOCCG Data Sets . . . . . . . . . . . . . .
8.2.1 Simulated data set . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.2 In situ data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57
57
59
59
60
61

9 MODIS Semi-Analytic Algorithm for IOP
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Algorithm Description . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 Remote-sensing reflectance model . . . . . . . . . . . . . . . . .
9.2.2 Backscattering coefficients . . . . . . . . . . . . . . . . . . . . .
9.2.3 Absorption coefficients . . . . . . . . . . . . . . . . . . . . . . . .
9.2.4 Model inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63
63
63
63
64
65
65

CONTENTS• iii

9.2.5 Empirical portion of Carder− MODIS . . . . . . . . . . . . . . . .
9.2.6 Blending semi-analytic and empirical IOP values . . . . . . . .
9.3 Algorithm Performance with the IOCCG Data Sets . . . . . . . . . . .
9.3.1 Synthetic data set . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2 In situ data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66
67
68
68
70
71

10 The Quasi-Analytical Algorithm10.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Derive Total Absorption and Backscattering Coefficients . . . . . . .
10.3 Decomposition of the Total Absorption Coefficient . . . . . . . . . .
10.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73
73
73
75
76
79

11 The GSM Semi-Analytical Bio-Optical Model
11.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81
81
83
85

12 Inversion Based on a Semi-Analytical Reflectance Model
12.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 The Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3.1 Retrieval of IOPs from the simulated data set . . . . . . . . . .
12.3.2 Retrieval of IOPs from the in situ data set . . . . . . . . . . . .
12.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87
87
89
90
90
92
93

13 Examples of IOP Applications95
13.1 Water Composition and Water-Mass Classification . . . . . . . . . . . 95
13.2 Dissolved and Particulate Organic Carbon . . . . . . . . . . . . . . . . 97
13.3 Diffuse Attenuation Coefficient of Downwelling Irradiance . . . . . . 99
13.4 Oceanic Primary Production . . . . . . . . . . . . . . . . . . . . . . . . . 100
13.5 Chlorophyll Concentration from Remotely Derived Pigment Absorption Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
13.6 Monitoring Coastal Ocean Processes using IOPs and Numerical Circulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
13.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
14 Summary and Conclusions

105

References

111

Acronyms and Abbreviations

123

CONTENTS• 1

Preface
Initially, remote sensing of ocean colour focused primarily on the retrieval of
the concentration of chlorophyll-a in the global oceans. Subsequent studies,
however, have also emphasized the importance of understanding and retrieving,
via remote sensing of ocean colour, inherent optical properties (IOPs), namely,
the scattering and absorption characteristics of water and its constituents (thedissolved and suspended material). Variations in IOPs are clear indications ofchanges in water mass or water constituents.
In the past decade, significant progress has been achieved on remote sensing algorithms for IOPs and applications of IOPs in oceanographic studies. This
report summarizes the progress to date, thus serving to emphasize the importance of IOPs in ocean optics and in ocean-colour remote sensing. It outlines
the fundamental relationships between water-leaving radiance and IOPs (Chapter 1), establishes a data base for algorithm testing and evaluation (Chapter 2),
and provides a discussion of sources of uncertainty (Chapter 3).
The bulk of the report (Chapters 4 to 12) reviews the characteristics of a variety of algorithms commonly used in remote sensing practices and assesses theirperformance when applied to synthetic and in situ data sets. Sufficient details areincluded to allow for easy comparison between the various algorithms and utilisation of the algorithms by interested researchers. Although the ocean-colour
community has accomplished a great deal by developing many algorithms for
ocean-colour remote sensing, very few broad-range tests, validations, or intercomparisons have been available hitherto. This report provides initial results inthis regard, but it should be pointed out that algorithm development is an ongoing process, and we have by no means attempted to include all the algorithms
developed, or under development, by the ocean colour community. The reportends with examples of IOP applications in oceanographic studies (Chapter 13)
and a summary and conclusions chapter (Chapter 14).
On a more general level, the material discussed in the report illustrates therich and quantitative information latent in data on visible spectral radiometry(VSR) of the ocean. The information retrieved from ocean-colour remote sensing
can contribute to our understanding of the planetary carbon cycle and climate
research, as well as other biological and biogeochemical processes in the oceans,and has many other applications including management of marine resources.
This report may not have become a reality without the support of the IOCCG
Committee, and the diligent work of the "Algorithm Working Group". In particular, Z.P.L. wishes to extend his appreciation to Dr. Trevor Platt for his guidance
throughout the duration of this project, and to the series editor, Dr. Venetia
Stuart, for her encouragement and assistance in completing this monograph.
The printing of this report was sponsored and carried out by the GKSS ResearchCentre (Geesthacht, Germany), which is gratefully acknowledged.

Chapter 1
Why are Inherent Optical Properties Needed in
Ocean-Colour Remote Sensing?
Ronald Zaneveld, Andrew Barnard and ZhongPing Lee

1.1 Introduction
In this volume we are interested in the determination of useful oceanographic
parameters from the radiance measured by a satellite-based sensor. The measured radiance originates from sunlight that passes through the atmosphere, isreflected, absorbed, and scattered by constituents in the ocean, and is transmitted back through the atmosphere to the satellite-based sensor. Solar photons
that reach the sea surface are redistributed from those that reached the top of
the atmosphere. Absorption of the aerosols and gases changes the intensity of
the radiance, while scattering changes the intensity as well as the directionality,
resulting in diffuse light that is a function of wavelength. The directional slope
spectrum of the waves at the sea surface, together with the radiance distribution, determine the reflected radiance. White caps, bubbles and surface slicks
also affect the redistribution of light entering the ocean, in addition to the waves.
The processes of scattering and absorption by dissolved and suspended materials in the ocean affect the spectrum and radiance distribution (light field) of
the light emerging from the ocean – the so called water-leaving radiance. The
scattering and absorption characteristics of ocean water and its constituents aredescribed by the inherent optical properties (IOPs) (Preisendorfer, 1976). Notethat the IOPs do not depend on the radiance distribution. If we can remove
the atmosphere and surface effects successfully, the best we can hope for frominversions of the water-leaving radiance are the scattering and absorption characteristics of the dissolved and suspended materials.
While the spectral quality and quantity of the water-leaving radiance is largely
determined by the inherent optical properties, conventionally the modificationof the radiance has been used to determine oceanic constituents directly. Typically the desired parameter has been the chlorophyll-a concentration, C. Usuallyalgorithm development searches for a combination of radiance signals at several
wavelengths to find some ratio, or other combination, that relates empirically to
3

4

•

Remote Sensing of Inherent Optical Properties

the desired parameter. The coefficients contained in these algorithms are generally derived by pooling data collected at various spatial and temporal scales.
This globally and seasonally inclusive approach, which removes “noise” associated with the data sets, diminishes important spatial and temporal features of
the global oceans. This approach assumes that the ocean is a black box, andthat little is to be gained by examining how the black box works, presumably
because the black box is too difficult to be understood. With such a perception,
most algorithm development (even today) uses the black box approach (see Figure 1.1). However, a great deal is known about the inherent optical properties
and their influence on the water-leaving radiance, as is detailed below.
radiance
distribution
and spectrum

Chlorophyll,
production,
particle concentration

Figure 1.1 Diagram of inverse radiative transfer elements using the “blackbox” approach.

In the past (CZCS), present (e.g., SeaWiFS, MODIS), and future (VIIRS-NPOESS)
missions the emphasis of ocean-colour remote sensing has been on the derivation of the concentration of chlorophyll-a (Hooker et al., 1992; Yoder et al., 2001).
This is partly because values of chlorophyll-a play a central role in conventionalalgorithms for primary production or light attenuation coefficients. Also, it isbecause in the earlier days of ocean optics studies, chlorophyll-a concentration(as an index to describe a water body) could be routinely measured at sea. Only inthe recent decade with the advancement of instrumentation (such as the transmissometer and ac-9), have we been able to look further at the fundamentals
and to envisage different inversion schemes. In essence, water colour is determined by inherent optical properties, and chlorophyll is just one of the active
components that determine the IOPs. Therefore C can be determined only witha larger uncertainty from ocean-colour remote sensing than the inherent opticalproperties themselves.
Since no amount of study will modify nature, and the global link between
IOPs and C cannot be improved substantially, no real progress has been madein the accuracy of the determination of C from space in the last two decades.
On the other hand, it is now assumed, as in VIIRS, that ocean colour can be
operational. This should not be interpreted to mean that no further progress
can be made in deriving useful information from remotely sensed radiance. Bystarting at the product end (the need to determine C, production, etc.) the real
inversion signal, IOP, is ignored. Fundamentally, a better approach would be to
ask: “What can water-leaving radiance really give us, and with what accuracy?”
Such an approach, based on physics, would examine how water colour is related
to the IOPs and then, secondarily, how the IOPs are related to the biogeochemical

Why are Inherent Optical Properties Needed in Ocean-Colour Remote Sensing?

• 5parameters of the suspended and dissolved constituents, and finally what theseparameters can tell us about processes. Such an approach, as shown below inmore detail, would enhance our understanding about the remotely sensed signal,
optimize its utilization, and eventually provide improved and reliable products
related to the biogeochemistry of the oceans.

1.2

The Forward Problem of Ocean Optics

The process of forward radiative transfer can be summarized by Figure 1.2. In
ocean-colour remote sensing, the forward radiative transfer problem is to predict the spectral distribution of water-leaving radiance based on a quantitativedescription of all the absorption and scattering characteristics of the opticalcomponents in the ocean. A recent review of radiative transfer can be foundin Zaneveld et al. (2005b). The inverse problem is the determination of usefuloceanic particulate and dissolved parameters when the spectral characteristicsof the water-leaving radiance are known.

particle size, index of refraction,
distributions, and properties ofdissolved materials
Figure 1.2

IOP

radiance
distribution
and spectrum

Diagram of forward radiative transfer elements.

The forward problem is governed by the Equation of Radiative Transfer (ERT).
Without internal sources such as fluorescence or Raman scattering, the ERT is
given by:

4π
Z∇.L (~
x, λ, θ, φ) = −c (~x, λ) L (~x, λ, θ, φ)+~, λ, θ, φ, θ 0 , φ0 L x~, λ, θ 0 , φ0 dω0 .
β x

0

(1.1)
~ is the position vector (x, y, z), θ
The radiance is L, units are W m−2 sr−1 , x
is the zenith angle, φ is the azimuth angle, c is the beam attenuation coefficient(in units of m−1 ). β(~x, θ, φ, θ 0 , φ0 ) is the volume scattering function (VSF), with−1
−1
units of m sr . Many books have been written regarding solutions to the ERT(e.g., Chandrasekhar, 1960; Preisendorfer, 1976).
The most common approach in oceanography is to assume that horizontal
gradients in radiance and IOPs are much smaller than vertical ones, so that hor-6•

Remote Sensing of Inherent Optical Properties

izontal structure is ignored. This leads to:cos(θ)dL(z, λ, θ, φ)/dz = −c(z)L(z, λ, θ, φ)4πZ+β(z, λ, θ, φ, θ 0 , φ0 )L(z, λ, θ 0 , φ0 )dω0 .

(1.2)

0

This is the ERT for the so-called plane parallel assumption without internalsources and is widely applied. Numerical solutions to this equation can be foundin Mobley (1995) (Hydrolight) and Thomas and Stamnes (1999).
There is a large literature on radiative transfer in the ocean and atmosphere.
This body of work is based on deriving radiance distributions when the IOPs are
known. Typically, for oceanographic applications, the IOPs used are based on
knowledge or speculation of the relationship between particulate and dissolved
materials and the IOPs. Again there is a large and developing literature relating
particulate properties such as particle concentration, size distributions, index
of refraction distributions, and shape to IOP (for a recent review see Twardowski
et al., 2005). The forward problem is thus logically broken into two parts: therelationship between biogeochemical parameters and IOPs, and the relationshipbetween the IOPs and the radiance distribution.

1.3

Inherent Optical Properties

Much has been written on inherent optical properties and their wavelength dependencies, examples of which can be found in the books by Shifrin (1988), Kirk
(1994), and Mobley (1994). We will briefly summarize here.
The beam attenuation coefficient (c) is a sum of the coefficients of absorption(a) and scattering (b),c = a + b.

(1.3)

The total scattering coefficient can be divided into forward, bf , and backward,
bb , components:
b = bb + bf ,
and

πZ/2

bf = 2π(1.4)Zπβ(θ) sin(θ)dθ and bb = 2π0β(θ) sin(θ)dθ.

(1.5)

π /2

The theoretical aspects of light scattering are treated extensively in van deHulst (1981). For the various semi-analytical and analytical remote sensing algorithms, we now have defined the two key IOPs relevant to the remote sensingWhy are Inherent Optical Properties Needed in Ocean-Colour Remote Sensing?

• 7reflectance, a and bb . These IOPs are often separated into operationally definedcomponents such as the dissolved and particulate fractions, and water:a = aw + aph + ad + ag ,(1.6)

bb = bbw + bbp ,(1.7)

bf = bfw + bfp ,

(1.8)

and

which applies to Equation 1.3 as:c = a + bf + bb .

(1.9)

The subscripts "g", "p", and "w" represent dissolved (historically called gelbstoff or gilvin), particulate matter, and water, respectively. Subscripts "ph" and"d" represent the algal and non-algal components of the particles, respectively.
Operationally, the dissolved fraction typically comprises all substances that passthrough a 0.2 µm filter. The non-algal component is comprised of non-living
particulate organic material, living particles such as bacteria, inorganic minerals, and bubbles. The relative contributions of these different particle groups toparticulate backscattering are poorly known, but recent progress has been made(Stramski et al., 2001).
Substituting all of the above into the ERT (Equation 1.2) gives:cos(θ)dL(z, λ, θ, φ)/dz = −[aw + ag + aph + ad + bw + bp ](z, λ)L(z, λ, θ, φ)4πZ[βw (z, λ, θ, φ, θ 0 , φ0 ) + βp (z, λ, θ, φ, θ 0 , φ0 )]L(z, λ, θ 0 , φ0 )dω0 .

+0(1.10)Of the IOP parameters in the ERT, only aph relates more or less directly tothe concentration of chlorophyll-a, C (depending on the presence of ancillary
pigments and their proportionality to chlorophyll-a). The other parameters only
relate very indirectly and weakly to C. In so-called Case 1 waters (Morel, 1988),
it is assumed that all non-water components vary closely with C. This has beenshown to be questionable (Mobley et al., 2004; Lee and Hu, 2006), especially incoastal waters. It is thus clear, that in nature, deriving the radiance based on
knowledge of C only, will often lead to incorrect results.
When solving a forward radiative transfer problem, one determines the inherent optical properties in some manner. This can be done by measurement ormodelling. Modelling often involves electromagnetic theory, as this allows one

8•

Remote Sensing of Inherent Optical Properties

to derive IOPs based on the particle size distribution, index of refraction distribution and shape distribution. Clearly, requiring all of the particulate properties
above to be closely related to C, is unreasonable, though in practice has quite
frequently been done (e.g., Morel, 1988; Haltrin, 1999). For the purposes of thedirect inversion of remote sensing to obtain the concentration of chlorophyll-a,these relations were, of course, a necessity. This encourages inattentive people
to believe that all IOPs are in fact a function of chlorophyll only, when this is far
from the truth.

1.4

The Inverse Problem of Ocean Optics

The inversion problem is to determine the biogeochemical parameters from the
upwelling radiance spectrum, i.e. the normalized water-leaving radiance. Zaneveld (1973) has shown that the radiance distribution and its derivative can, intheory, be inverted to obtain the volume scattering function and beam attenuation coefficient, i.e. the inherent optical properties. This has not been done inpractice. An important point is, however, that the entire radiance distribution
and its depth derivative must be known to obtain the IOPs. In remote sensing weonly know the radiance at the surface in a few directions. We therefore cannot
expect to be able to accurately invert for all of the IOPs. A corollary is that we areunable to invert accurately for the complete suite of biogeochemical parameterswhich determine the IOPs.
Inversion for either IOPs or biogeochemical parameters is thus inexact andmust, perforce, depend on approximations. Based on the discussion above, it
is clear that inversion is also a two-step process, explicitly or implicitly: thederivation of IOPs from the radiance, and then biogeochemical parameters fromthe IOPs. Both of these are inexact procedures, especially for the separation ofparticulate and dissolved materials. Due to the extremely complex nature ofthese materials, their full details cannot be expected to be inverted from the
IOPs. Nonetheless, one would logically expect inversion of the water leaving
radiance spectrum to follow an inverse approach to that of Figure 1.2.
Historically, starting with the CZCS, remote sensing inversions have been
focused on the direct derivation of the chlorophyll concentration from waterleaving radiance (e.g. Figure 1.1). This was based on the early recognition thatchlorophyll-laden waters are “greener” than chlorophyll poor waters. While thiswas a reasonable starting point, it is also unfortunate in that this is still theoceanographic parameter chosen for performance criteria of future satellite sensors such as NPOESS–VIIRS. This is unfortunate not because chlorophyll is of nointerest to scientists and managers, but because chlorophyll is only indirectly,
or not at all, related to many of the IOPs that determine radiance, as shown in the
IOP section and Figure 1.3. Using chlorophyll as the primary product therefore

Why are Inherent Optical Properties Needed in Ocean-Colour Remote Sensing?

radiance
distribution
and spectrum

dissolved
matterinorganic

particulatesorganic

IOP
water

• 9

nonliving
phytoplankton
bacteria,
viruses

accessory
pigments

cChlorophyll

Figure 1.3 Diagram of inverse radiative transfer elements. Many further
parameters are derived from these constituents, such as DOC, POC andproductivity.

minimizes the information that can be gained from optical remote sensing. Eventhe chlorophyll concentration itself could be determined with greater accuracy
if there was a full understanding of all the optical processes that connect the
remotely sensed radiance to the IOPs, and the IOPs to chlorophyll.
It is of course possible to find empirical relationships between radiance ratios and C, for example, but the uncertainties in such relationships cannot be
predicted and analyzed. Furthermore such relationships cannot be justified orderived a priori using radiative transfer. Because of this, most of the information
contained in remotely sensed radiance is ignored or overlooked. This approachthus limits the use and applicability of optical remote sensing.
How can one obtain the maximum information from remote sensing? Thisrequires going back to the approach in Figure 1.3. We must recognize that theradiance spectrum depends physically on the IOPs and solar input. Thus, if we
focus on the derivation of the IOPs to the maximum allowed by the geometric restrictions of radiative transfer, we have not diminished the information given tous. Once the IOPs are in hand we can ask the second question: “What particulate
and dissolved properties can be derived from the remotely determined IOPs?”
Such an IOP-based inversion maximizes the information gained from remotesensing, whereas the black box approach minimizes it. Recently, good progresshas been made in the inversion of IOPs from the upwelled radiance spectrum(Roesler and Perry, 1995; Hoge and Lyon, 1996; Lee et al., 1996b; Garver andSiegel, 1997; Carder et al., 1999; Maritorena et al., 2002; Lee et al., 2002; Roesler
and Boss 2003).
Semi-analytical approaches to remote sensing inversion (Gordon et al., 1988;
Morel, 1988) use an IOP approach in that they use the relationship between theoceanic constituents and the IOPs upon which to base the inversion. Such semianalytical approaches are based on simple approximations of the remote sensing
reflectance such as Equation 1.11. To complete this discussion we present below
the fundamental relationship that links remote-sensing reflectance (or waterleaving radiance) and the IOPs.

10•

Remote Sensing of Inherent Optical Properties

1.5 The Dependence of the Remote Sensing Reflectance on
the IOPs
Most remote sensing inversions are based on this simple relationship:
rrs = Lu (0− )/Ed (0− ) = gbb,a(1.11)

where Lu (0− ) and Ed (0− ) are the upwelling radiance and downwelling irradiance just below the sea surface, respectively. A similar relationship developed
for subsurface irradiance reflectance was first derived by Gordon et al. (1975)
and Morel and Prieur (1977) based on modeling the results from radiative transfer calculations. They found that the water reflectance is proportional to thebackscattering coefficient and inversely proportional to the absorption coefficient. The proportionality factor g (also called f /Q in the literature, in units ofsr−1 ), which generally varies over the range 0.084 − 0.15 sr−1 for nadir-viewed
radiance (Morel and Gentili, 1993; Lee et al., 2004), depends on how the backscattered light relates to the backscattering coefficient, and therefore to the detailsof the volume scattering function in the backward direction and the radiancedistribution. Most of the directional effects of radiative transfer are thus contained in the factor g, and this factor has been studied in detail (for example
Gordon et al., 1988; Morel and Gentili, 1993; Lee et al., 2004). Equation 1.11
is the starting point for many inversion algorithms, which remains inaccurate
if the dependence of g on the shape of the volume scattering function and theradiance distribution is ignored.
Based on the derivations of Zaneveld (1982; 1995), a theoretical relationshipof the dependence of the remote sensing reflectance on the IOPs can be obtained
from the ERT (in the form of Equation 1.11) for the nadir radiance, Lu , for whichcos(θ) = −1, and for which we can define a vertical attenuation coefficient ku :
f (z)

b
1
2π bb (z)rrs (z) = Lu (z)/Ed (z) =,µ̄d (z) ku (z) + c(z) − fL bf (z)whereµ̄d (z) =fb (z) =2π
R πR/20

Ed (z),Eod (z)(1.12)(1.13)!,β(z, π , 0, θ 0 , ϕ0 )L(z, θ 0 , ϕ0 ) sin(θ 0 )dθ 0 dϕ0

0





bb (z)
2π Eod (z),(1.14a)fL (z) =

2π
R Rπ

!,
β(z, π , 0, θ 0 , ϕ0 )L(z, θ 0 , ϕ0 ) sin(θ 0 )dθ 0 dϕ0(bf (z)Lu (z)) .

0 π /2(1.14b)

Why are Inherent Optical Properties Needed in Ocean-Colour Remote Sensing?

• 11

Equation 1.12 is an exact expression, as it is only a rewrite of the ERT. All of
the details of the radiative transfer process are compressed into the parametersµ̄d (z), fb (z), fL (z), and ku (z).
The simple relationship in Equation 1.11 is thus clearly an approximation.
Based on Equation 1.12, Zaneveld (1995) has derived the following (also approximate) dependence of g:g≈fb,
2π µ̄d (0− )(1 + 1/µ̄∞ )(1.15)

where µ̄∞ is the asymptotic average cosine, which in turn can be described as afunction of b/c (Zaneveld, 1989; Berwald et al., 1995).
We thus find that the remote sensing reflectance can be expressed directlyin terms of IOPs. This is logical, in that reflectance is a measure of water-leaving
radiance, while radiance is determined by the ERT. The difficulty is that all of
the directional effects of radiative transfer are involved. The entire shape of thevolume scattering function thus matters. Therefore any time we use inversion
formulas such as equations 1.11 and 1.15 approximations, uncertainties are
introduced.
A further problem in the interpretation of remotely-derived properties isthe vertical structure of the IOPs. Recently Zaneveld et al. (2005a) derived thedependence of the reflectance at the surface on the vertical structure of opticalparameters from first principles. It was shown that the depth dependence isa function of the derivative of the round trip attenuation of the downwellingand backscattered light. With some approximation it can be shown that thebackscattering to absorption ratio follows the same vertical integration rule. For
backscattering and absorption separately, and for chlorophyll, it can be shownthat there is no general formula that allows one to integrate the vertical structure
and arrive at the remotely sensed parameter. Only in the special case of “opticalhomogeneity” where the ratio of the backscattering and absorption coefficients
does not vary with depth, can the vertical structure be ignored.
What we learn from the above discussion is that in remote sensing inversion,
the directional and vertical details are initially buried in various model parameters. Later, when higher accuracies of inversion are required, this necessitates
the reinsertion of information such as the directional effects, as evidenced byMorel and Gentili (1993; 1996) on the bi-directional reflectance. What has notbeen done, is to start with an expression such as Equation 1.12, which contains
the full ERT, and use this as a basis for the derivation of IOPs, and hence particulate and dissolved properties. This is an approach to the question: “Whatinformation about the oceanic environment can optical remote sensing provide
us?” The multiple connections in Figure 1.3 can then be explored, and such an
approach would allow the maximum information content of the remotely-sensed
data to be obtained.

Chapter 2
Synthetic and In Situ Data Sets for Algorithm
Testing
Stephane Maritorena, ZhongPing Lee, KePing Du, Hubert Loisel, Roland
Doerffer, Collin Roesler, Paul Lyon, Akihiko Tanaka, Marcel Babin andOleg V. Kopelevich

In algorithm testing and evaluation, we are frequently limited by the availabilityof adequate data sets. In many studies, individual groups have measured datafrom limited areas. Those data sets, which are important for the initial development of algorithms, usually lack the dynamic range, and therefore make it
difficult to evaluate an algorithm’s performance in broader scales. To fill thisgap and to have a common ground for algorithm testing, two independent data
sets were compiled and adopted by the “Algorithm Working Group”. One of thedata sets was compiled from global field measurements, where uncertainties
among measured properties are common (see Chapter 3). The other data set
was simulated using the widely accepted numerical code, Hydrolight (Mobley,1995), with input IOPs generated based on extensive measurements made in thefield. This synthetic data set can perhaps be viewed as results from controlled
experiments, where errors from measurement procedures are minimal. This
chapter summarizes the characteristics of both the synthetic and in situ data
sets.

2.1

In situ Data Set

The in situ data set is an extraction from NASA’s SeaWiFS Bio-optical Archive
and Storage System (SeaBASS) (Hooker et al., 1994; http://seabass.gsfc.nasa.
gov/) and contains chlorophyll-a concentration, above-surface remote sensing
reflectance (Rrs , which is the ratio of water-leaving radiance, Lw , to downwelling
irradiance just above the surface, Ed (0+ )) at the first five SeaWiFS bands (412,443, 490, 510 and 555 nm), along with the detrital (ad ), gelbstoff (ag ), and phytoplankton (aph ) absorption coefficients. Detrital (ad ) and gelbstoff (ag ) absorption coefficients were summed to form a single term (Carder et al., 1991) asadg (acdm in Maritorena et al., 2002) and total absorption (a) was calculated by
13

14

•

Remote Sensing of Inherent Optical Properties

adding pure water values (aw ) (Pope and Fry, 1997) to aph and adg at each wavelength. The chlorophyll-a, remote sensing reflectance, and absorption data wereconsidered a match (i.e. coming from a unique station) when all measurements
were made within a 12-hour window and within 0.05 degrees in both latitude
and longitude. Absorption data come from hyperspectral spectrophotometricmeasurements but only the SeaWiFS bands were used, for consistency, with theremote sensing reflectance data.
Methods to measure Rrs , ad , ag and aph are summarized in NASA’s technicalmemorandum (Mueller and Austin, 1992). Generally, phytoplankton absorption
coefficients were obtained by spectrophotometric measurements after filtrationof a water sample through a GF/F filter. Detrital absorption coefficients wereobtained after a methanol extraction of the pigments on the GF/F filter. For
all measurements related to filter pad, there is a need to correct for pathlength
amplification (“beta-factor” correction) (Mitchell and Kiefer, 1988; Bricaud andStramski, 1990). Errors will be introduced when an incorrect “beta-factor” isused (Cleveland and Weidemann, 1993; Allali et al., 1995).
Gelbstoff absorption coefficients were obtained by measuring the absorbanceof the filtrate with a spectrophotometric cell (usually ∼ 10 cm in length). Remote sensing reflectance data were obtained by either in-water or above-surface
radiometric measurements (Mueller et al., 2002). Backscattering measurements
were too rare to be included here. As always, errors (sometimes quite large) areassociated with each of the measured components.
Data were filtered by applying quality control procedures to the remotesensing reflectance and absorption data. For Rrs , these procedures consistedof comparisons with the SeaBAM data set (O’Reilly et al., 1998) and the synthetic data set described in Section 2.2. For a given chlorophyll range, data withRrs (λ) values either 10% higher than the maximum or 10% lower than the minimum value found in the SeaBAM or synthetic data sets were eliminated. While
this procedure removed extra noise in the data set, it may have also removed
some extreme cases such as CDOM or sediment dominated waters. Of the 1,235
original data points, 177 points were eliminated during this step.
For the absorption components, the following controlling factors were applied:
1.0 < adg (412)/adg (443) < 2.0,
1.0 < adg (443)/adg (490) < 3.0,
0.5 < aph (412)/aph (443) < 1.1,
0.1 < aph (490)/aph (443) < 1.0,
and another 402 points were removed during this step. The final in situ data
set contains only 656 stations with a complete set of chlorophyll concentration,
Rrs , and component absorption data. Most of the data come from locations that

Synthetic and In Situ Data Sets for Algorithm Testing• 15

Figure 2.1 Data location of the in situ data set. The origin of the data andthe number of stations (in parentheses) by experiment are also indicated.

are relatively close to the coast and some of the data are from high latitudes.
Figure 2.1 presents a summary of the origin and location of the in situ data set.

2.2

Synthetic Data Set

This data set (a total of 500 points) contains both inherent (IOP) and apparent(AOP) optical properties. IOPs, required as inputs for Hydrolight (Mobley, 1995),
are simulated with optical and bio-optical parameters and models. Detailed
descriptions regarding the simulation of IOPs and AOPs can be found at:
http://www.ioccg.org/groups/OCAG_data.html.
The absorption coefficient of the bulk water was simulated using a four-term
model (Prieur and Sathyendranath, 1981; Roesler et al., 1989; Carder et al., 1991;
Bukata et al., 1995; Fischer and Fell, 1999; Doerffer et al., 2002), with contributions from water molecules, phytoplankton, detritus, and gelbstoff. Values ofaw (λ) were taken from Pope and Fry (1997). Values of aph (λ) were modelled
as aph (440) multiplying the spectral shape of phytoplankton absorption coefficient (a+ph (λ) ≡ aph (λ)/aph (440)), with aph (440) expressed as a function of thechlorophyll concentration (the specific absorption coefficients at 440 nm weretaken from Bricaud et al., 1995; 1998). Oligotrophic and eutrophic waters exhibit different spectral shapes of phytoplankton absorption spectra (Hoepffner
and Sathyendranath, 1992; Stuart et al., 1998). To represent this natural variation, at least to the first order, an a+ph (λ) data bank (600 spectra) was composedfrom the extensive measurements of Bricaud et al. (1995; 1998) and Carder et al.
(1999). This a+ph (λ) data bank is divided into nine groups separated by the measured aph (440) values. Figure 2.2 presents examples of a+ph (λ) from the nine

16•

Remote Sensing of Inherent Optical Properties

Figure 2.2 Examples of aph spectral shape for the nine aph groups (separated by values of aph (440) [m−1 ]). Numbers in parenthesis are the range
of C [mg m−3 ] for those groups.

groups. In the simulation of aph (λ) spectra, a+ph (λ) is selected randomly within
the group in which the aph (440) value falls. Using this process, the variability of
aph spectral shapes is retained in the modelled aph (λ) spectra, and at the sametime the a+ph (λ) of eutrophic waters will not be used to generate aph (λ) of blue
oceanic waters, or vice versa.
Absorption spectra of both detritus and gelbstoff were described as exponentially decreasing functions with wavelength (Bricaud et al., 1981; Roesler etal., 1989). The spectral slopes were treated as random variables but constrainedby ranges commonly observed in the field. The absorption coefficients at 440
nm also varied randomly, but this randomness was constrained such that theranges were wider for higher C values and narrower for lower C values.
The total scattering coefficient was simulated by a three-term model (Bukata
et al., 1995), with contributions from water molecules, phytoplankton, and inorganic particles. Two different particle phase functions were used to represent
the scattering distribution of phytoplankton and inorganic particles. For bothparticulates, the scattering coefficients at 550 nm and the spectral exponents
were varied randomly (but within commonly observed ranges).
With the above modelled absorption and scattering (backscattering) coefficients, Hydrolight was used for the calculation of radiance distribution andthen the AOPs, which include the nadir-viewed above-surface remote-sensing
reflectance (Rrs ), nadir-viewed subsurface remote-sensing reflectance (rrs ), andSynthetic and In Situ Data Sets for Algorithm Testing

• 17

1.5
syntheticin situ

R rs (410)/ R rs (440)

R rs (440) [sr -1 ]

0.016

0.012

0.008

1.2

0.9

0.004
syntheticin situ

0.6
(b)

(a)

0.000
0.01

0.03

0.1

0.3

1
-1

a(440) [m ]

3

0.3

0.5

1

3

5

R rs (490)/ R rs (555)

Figure 2.3 Comparison between in situ and synthetic data sets. (a) Ranges
and variations of Rrs (440) and a(440). (b) Ranges and variations of
Rrs (410)/Rrs (440) and Rrs (490)/Rrs (555).

subsurface irradiance reflectance (R). In the Hydrolight runs, solar input was
simulated with the Gregg and Carder (1990) model with marine aerosols, andthe sky was assumed to be cloud free. A wind speed of 5 m s−1 was applied,
and the water body was assumed to be homogeneous. Spectral bands were set
from 400 to 800 nm, with a spacing of 10 nm. Inelastic scattering (i.e. Raman
scattering, chlorophyll-a fluorescence) was excluded.
For consistency, the synthetic data set was compared with the in situ data
set. For the two data sets Figure 2.3a shows the range and variation ofRrs (440) versus a(440), and Figure 2.3b shows the range and variation ofRrs (410)/Rrs (440) versus Rrs (490)/Rrs (555). We used Rrs (412)/Rrs (443) instead of Rrs (410)/Rrs (440) for the in situ data set, although the effects of thesesmall wavelength differences are considered negligible. For both data sets,a(440) is in the range of ∼ 0.02 − 3.1 m−1 , while Rrs (490)/Rrs (555) is in therange of ∼ 0.3 − 5.2 m−1 . Clearly, the two data sets agree with one anotherin variation and coverage, although for Rrs (490)/Rrs (555) ratios around ∼ 1.0,
some in situ data points have higher Rrs (410)/Rrs (440) ratios.
Although this synthetic IOP-AOP data set may not cover all possible variations that occur in natural waters, it nevertheless covers a wide range of variations encountered in the field, since the models and parameters used in thesimulation process are based on extensive field measurements. In the following
chapters (Chapter 4 – Chapter 12), a series of algorithms currently used for the
retrieval of IOPs from Rrs (λ) are applied to both the synthetic and the in situ
data sets, and the retrieved absorption and/or backscattering coefficients arecompared with known (synthetic) or measured (in situ) values, respectively. To
evaluate the performance of each algorithm, regression results (Type II, Laws,
1997) and Root-Mean-Square-Error (RMSE) are calculated and tabulated for each
property, in log space. RMSE is defined as:18•

Remote Sensing of Inherent Optical Properties
i2 1/2

N h
P
model
true
− log IOPi
log IOPi


 i=1



RMSE = 
,

N −2




(2.1)

stands for the ith property derived from Rrs (λ), IOPtrue
for the ith
where IOPmodeliiproperty known either from simulation or from in situ measurements, and N is
the number of valid retrievals. It is necessary to point out that a slightly different
set of IOPs may be derived from the same Rrs (λ) due to architecture differences.
Also, for the same IOP product, not all algorithms may derive valid retrievals fora given Rrs (λ) spectrum, due to the different settings of the algorithms. Such
non-valid retrievals are then excluded in the performance analysis, and result ina smaller data set, and likely better statistical results.

Chapter 3

Uncertainties in the Products of Ocean-Colour
Remote Sensing

Emmanuel Boss and Stephane Maritorena

Data products retrieved from the inversion of in situ or remotely sensed oceancolour data are generally distributed or reported without estimates of their uncertainties. The accuracy of inversion products such as chlorophyll-a or IOPs
is frequently evaluated by comparison with in situ measurements, but theseanalyses are not always sufficient to determine the level of uncertainty of an
ocean-colour product. This is particularly true for remote sensing data wherematch-up analyses (McClain et al., 2000; http://seabass.gsfc.nasa.gov/matchup_
results.html) can only be performed for an infinitesimal fraction of a sensor’s
records. Although very useful, these analyses cannot provide reliable estimatesof how ocean-colour uncertainties vary with time and/or space. Moreover, because the uncertainties of the input data (for example the normalized waterleaving radiance, LwN ) vary in space and time, the uncertainties of the output
products cannot be reported simply as a single global value unless it is intended
to provide general bounds. Some ocean-colour products are also used as input toother models (for example, to calculate primary production or to assimilate phytoplankton carbon into ecosystem models) for which uncertainty budgets cannot
be properly established without knowledge of the uncertainties associated with
the input data. It is thus important that the variations of the uncertainty in
LwN and in the products derived from them are documented in time and space.
This section discusses the various types of uncertainties present in ocean-colour
data or products and emphasizes recent approaches that allow uncertainties ofsatellite ocean-colour products to be estimated on a pixel-by-pixel basis.
19

20

•

Remote Sensing of Inherent Optical Properties

3.1 Sources of Uncertainty
3.1.1 Uncertainties in in situ measurements (LwN , Rrs , C, IOP)
In situ data are used for algorithm development and for validation of algorithms
and data products. While in situ measurements are frequently considered as “thereference” to which other data (e.g. satellite data) are compared, they containsignificant levels of uncertainties caused by various experimental and environmental factors. Calibration, dark signal, data processing, deployment strategy,
sea and sky states all introduce uncertainties in the radiometric measurements(Siegel et al., 1995; Hooker and Maritorena, 2000; Hooker et al., 2001). Close
compliancy to establish measurement protocols (e.g. Mueller and Austin, 1995
and follow up) along with regular and rigorous calibrations and good characterization of instruments are key to the minimization of uncertainties in the in situ
measurements. Measurements of biogeochemical variables have their own setof difficulties and resulting uncertainties (Mitchell et al., 2000; Van Heukelem etal., 2002; Claustre et al., 2004). Most of the data sets that are publicly available(e.g., SeaBASS) do not contain information regarding the estimated uncertainties of the various variables they contain (e.g., the differences between the triplicate chlorophyll measurements and the uncertainties in the radiometer reading,
based on its variability through the sampling period and its calibration history).
It is frequently assumed that the uncertainties of in situ data are small andin any case much smaller than the uncertainties arising from the natural spatial/temporal variability of a given variable.
Another uncertainty arises from the fact that the match-up field data usuallycharacterize an area of around 1–10 m while the satellite spatial scale is often
100–1,000 m. This environmental mismatch in scales introduces an uncertainty
that is often hard to quantify. Also, satellite measurements represent a watercolumn weighted average (Gordon and Clark, 1980; Sathyendranath and Platt,
1989; Zaneveld et al., 2005a), while in situ measurements usually come fromdiscrete depths. Therefore, for vertically inhomogeneous waters, uncertainties
arise when the two are compared with each other. Some sampling platforms
such as on-line sampling from steaming vessels, undulating vehicles, gliders,and autonomous underwater vehicles (AUVs) are likely to be fruitful approaches
in quantifying these uncertainties.

3.1.2

Uncertainties in satellite measurements (LwN )Various sources of random and systematic error contribute to disagreements between measured normalized water-leaving radiances and their actual values. Uncertainties in LwN are introduced through a variety of factors such as pre-launchcharacterization of the sensor, atmospheric and bi-directional corrections, andUncertainties in the Products of Ocean-Colour Remote Sensing• 21

uncertainties in the monitoring of the changes in the sensor’s performance. Errors in geo-location, contamination with light emanating from adjacent pixels
or other factors like white caps can also add to this uncertainty. The calibration/validation activities of each ocean-colour mission are designed to assess
and minimize the magnitude of this uncertainty (and remove any bias). Prelaunch and on-orbit characterization of the sensors (e.g., measurements of reflected Sun and/or Moon light) along with vicarious calibrations (e.g., the MOBYbuoy) and match-up analyses are the major procedures used to quantify uncertainties of normalized water-leaving radiances.
The calibration/validation activities and the reduction of the uncertainties
in the derived LwN should be one of the primary tasks of space agencies providing the ocean-colour data and much effort must be invested in minimizing
it for various missions. In the remainder of this chapter we will therefore assume the uncertainty in the LwN is known and documented, although at present
uncertainties in atmosphere correction still dominates errors in LwN of coastal
waters.

3.1.3

Uncertainties and assumptions in the functional relationshipthat links LwN and IOP and in the inversion procedure used toderive the products

Uncertainties in the products derived from the inversion of LwN , however, do
not benefit from the same level of effort. In what follows we will address these
uncertainties with reference to the type of algorithm designed to produce them,distinguishing between empirical and semi-analytical inversion algorithms. The
approaches used in some recent works to provide ocean-colour product uncertainties are also described.
3.1.3.1

Obtaining uncertainties in products based on empirical algorithms

Empirical algorithms are developed from data sets where in situ radiometry and a
to-be-derived product (e.g., chlorophyll-a, POC) have been collected at the same
spot of the ocean and within a narrow period of time. A regression is mostoften performed to obtain the ‘best-fit’ function between the two variables andto define the formulation that relates the two quantities. The type of regression
used to relate two variables is relevant to the uncertainty discussion because
regression methods work under different assumptions about uncertainties inthe data involved. Type-I regressions (Laws, 1997) are the most frequently used
and are based on the assumption that only the dependent variable (i.e. y, theproduct) has an uncertainty, while the independent variable (i.e. x, the inputdata) is error free. In Type-I regressions, the individual uncertainties in the input
data are not taken into account and it is generally assumed that the relative error

22•

Remote Sensing of Inherent Optical Properties

in the variable is constant. Conversely, Type-II regressions (Press et al., 1992;
Laws, 1997) assume that both variables have uncertainties and are thus better
adapted for ocean colour where substantial uncertainties frequently exist in thevariables involved (e.g., reflectance ratio, chlorophyll).
An empirical algorithm is as good as the data it is based on, and on how
representative the data are of the environment or bio-optical provinces where
the algorithm is to be applied. In situ data sets are often geographically andseasonally biased due to constraints in the timing and location of oceanic cruises(Claustre and Maritorena, 2003).
In general, it is crucial that data sets used in the development (or validation)of an ocean-colour algorithm have complete information about the location andtime at which the data were collected and about their quality (i.e. associateduncertainties). The geographical and temporal extent of a data set determines
the water types where the algorithm can be applied, whereas uncertainties in
products require information on uncertainties in the input data.
For empirical algorithms, the dispersion of the y-axis data (i.e. the product)around the “mean” relationship of the resulting algorithm provides, to somedegree, information about the uncertainties that can be expected at any given
x-axis value (i.e. the input data). However, this only represents the uncertainties
associated with the data set used in the regression and cannot be generalized
unless the data set fully encompasses all the natural variability that exists forthe water types included. Ideally, to evaluate the uncertainties of an empiricalalgorithm one needs a different data set than that with which the algorithm wasdeveloped; the statistics of the differences between the inverted products andthe measured products in this independent data set can then be used to evaluate
the uncertainties in the product. Additionally, an uncertainty propagation analysis to evaluate the effect of the uncertainties in the LwN on the output has tobe carried out to establish whether or not this uncertainty is a significant source
for uncertainty in the product (e.g., to what extent a 5% relative uncertainty in
LwN at 440 and 555 nm affects the IOPs retrieved).
In the case of neural network (NN) based algorithms, uncertainties should bedetermined from a rigorous statistical approach. Aires et al. (2004) provided an
example of such an approach to products derived from remote sensing (otherthan ocean colour). They use a Bayesian technique to evaluate the uncertainties
in the NN parameters which are then used to compute the uncertainties in the
outputs.
Another way to determine whether the measured reflectance spectrum is
within the domain of the bio-optical models used to simulate reflectance spectra, which in turn were used to train a neural network, has been developed for the
Medium Resolution Imaging Spectrometer (MERIS) (Doerffer and Schiller, 2000;
Krasnopolsky and Schiller, 2003). For this purpose one network is trained to determine concentrations from the eight MERIS bands together with the solar andUncertainties in the Products of Ocean-Colour Remote Sensing• 23

viewing zenith angles and the azimuth difference between viewing and sun direction (see Chapter 6). A second, forward, network is trained with the same dataset, which takes the derived concentrations as input and produces reflectances.
The deviation, calculated as the Chi2 (Sokal and Rohlf, 1981), over all eight bands
between the measured and the computed spectrum, is then used as an indicator
to see if the measured spectrum is within the training range, and thus within thescope of the algorithm. In the case of the MERIS ground segment, a flag is raised
whenever the Chi2 deviation exceeds a certain threshold. However, the Chi2
value can also be used as an uncertainty measure. Furthermore, a technique has
been developed (Schiller and Doerffer, 2005), which combines the neural networks with an optimization procedure, to estimate the uncertainty of a producton a pixel-by-pixel basis.
3.1.3.2

Obtaining uncertainties in products based on semi-analytical models

Semi-analytical models or algorithms are based on the premise of a known relationship (derived from the radiative-transfer theory) between LwN (or a functionof it) and IOPs (generally the absorption, a, and the backscattering, bb , coefficients). These models contain some level of empiricism in the way IOPs areparameterized (i.e. how their variations and spectral shapes are formulated)
and they also use simplified assumptions for some of their components (seeChapter 1). The inversion of semi-analytical models generally allows the simultaneous retrieval of several variables contained in the IOP terms. Like empiricalalgorithms, semi-analytical models are affected by uncertainties in LwN but they
are also influenced by uncertainties associated with the chosen relationship between LwN and IOPs, and uncertainties resulting from the assumptions used intheir formulation.
Sensitivity analyses are frequently used to assess how assumptions used todescribe the component terms of a model affect retrievals (Roesler and Perry,
1995; Hoge and Lyon, 1996; Garver and Siegel, 1997). Although very useful,
this approach does not allow the determination of a product’s uncertainty on acase-by-case (or pixel-by-pixel) basis, but rather provides a general uncertainty
estimate. To our knowledge, only two methods have recently been used withocean-colour data that can estimate the uncertainties of products retrieved by
the inversion of a semi-analytical model on a case-by-case basis. The first one(Maritorena and Siegel, 2005) is a non-linear adaptation of the calculation ofconfidence intervals in linear regressions. Essentially, this method is based on
the projection of the residuals between the observed and reconstructed (fromthe inverted variables) LwN in the solution (i.e. retrieved variables) (Bates andWatts, 1988).
A recent study (Wang et al., 2005) suggests another approach to computeuncertainties of the retrieved variables. In this approach, each of the variables

24•

Remote Sensing of Inherent Optical Properties

to be retrieved has a predefined set of spectral shapes and the model is inverted
for each of the possible combinations of these spectral shapes resulting in an extensive set of possible solutions. These results are then filtered to keep only the“realistic” (e.g., positive) solutions that can closely reproduce the input LwN spectrum (within a pre-described difference from the LwN based on the uncertainties
in LwN and the uncertainties in the theoretical relationship between LwN and
IOP). The final value for each inversion product and its associated uncertainty
is then obtained from the statistics (median and percentiles) on the acceptable
solution subset. The key steps in this approach are the choice of the acceptancecriteria for the solutions (e.g., what is the acceptable difference between observed LwN and that reconstructed from retrieved IOP) and the choice of range
in possible shapes for the spectrum of each individual IOP. The two methods
described above do not produce the same kind of uncertainties, and thus they
are not directly comparable. Both approaches have benefits and limitations. For
example, the Maritorena and Siegel (2005) approach always returns a value forthe confidence interval of the retrieved product because the calculations do notdepend on spectral criteria but on the sum of the residuals (weighted by thespectral uncertainties of the input data, if they are known). On the other end,
this approach does not take into account the uncertainties caused by the model
assumptions. In the Wang et al. (2005) approach, uncertainties in the model anddata are included in the spectral agreement criteria but the inversion may fail tofind any solution that satisfies this criteria. Although it uses an efficient linear
matrix inversion technique (Hoge and Lyon, 1996), the Wang et al. (2005) method
is also more computationally demanding (computational demands increase withnumbers of possible combinations of different shapes of IOPs).

3.2

Summary

While some preliminary uncertainty estimates for ocean-colour products areavailable through match-up analyses, uncertainties are generally not provided
on a per data point basis. This has caused many users to use ocean-colour products as a qualitative descriptor of patterns rather than a quantitative variable.
Others use these products in biogeochemical models (e.g., computing primary
productivity) without being able to propagate uncertainties.
For some ocean-colour missions, such as for MERIS, a sophisticated flagging
system has been developed. It computes, on a pixel-by-pixel basis, indicators forthe reliability of a product by regarding different possible error sources including sun glint, failure in the atmospheric correction, high turbidity in the water,etc. A flag for each possible problem is raised if the uncertainty value exceeds acertain threshold. By this method, the user gets a warning and has to decide if
he can accept this pixel for further computations.

Uncertainties in the Products of Ocean-Colour Remote Sensing• 25

We have reviewed briefly some of the uncertainties present in ocean-colour
data, and have presented different approaches to establish uncertainties in products of ocean-colour remote sensing for either empirical or semi-analytical algorithms. The procedures described above are not complicated and their full
application benefits from the knowledge of uncertainties in the input data. Useof such approaches will help the ocean-colour community establish quantitative
confidence in the remote-sensing products.

Chapter 4
Simple Algorithms for Absorption Coefficients
ZhongPing Lee, Stephane Maritorena, Andrew Barnard

4.1 One-Step Spectral Ratio Algorithm
4.1.1 General description
In a similar fashion to the empirical approach of deriving chlorophyll-a concentration from ocean-colour data, the simplest way to derive absorption coefficients from Rrs (λ) is by empirical relationships. This kind of approach does
not require knowledge of the fundamental relationships between Rrs and IOPs,
but requires an adequate data set to develop the empirical coefficients. For thederivation of the total absorption coefficient at 440 nm, based on limited measurements (63 data points), Lee et al. (1998b) developed an empirical spectralratio algorithm from the spectral ratios of Rrs (λ). To obtain a better fit between
measured and algorithm-derived values, the algorithm uses quadratic polynomials with two spectral ratios:22
log (a(440)) = A0 + A1 ρ25 + A2 ρ25+ B1 ρ35 + B2 ρ35
,

(4.1)

where ρ25 and ρ35 areρ25 = log
Rrs (440)
Rrs (490), ρ35 = log.
Rrs (555)
Rrs (555)(4.2)

Values of A0,1,2 and B1,2 in Equation 4.1, derived by least-square fitting, are-0.674, -0.531, -0.745, -1.469, and 2.375, respectively (Lee et al., 1998b).

4.1.2

Results and discussion when applied to the IOCCG data sets

With Rrs values at 440, 490 and 555 nm (or nearby wavelengths) as inputs, values of a(440) were calculated from Equations 4.1 and 4.2. Figure 4.1 comparesthe derived and known a(440) values for the synthetic and the in situ data sets,
respectively. For the synthetic data set, this empirical algorithm systematicallyoverestimated a(440) for most of the data, though good correlation of determination (R 2 = 0.976) was achieved between the algorithm derived, and known
27

28

•

Remote Sensing of Inherent Optical Properties

3
1:1

1
0.3
0.1
N = 500
n = 500
2
R = 0.976
RMSE = 0.140

0.03
1:1

0.03

0.1

0.3

1

Known a(440) [m-1

Derived a(443) [m -1 ]Derived a(440) [m -1 ]

3

1

0.3
0.1
N = 656n = 6562
R = 0.817
RMSE = 0.202

0.03

3

0.03

]

0.1

0.3

1

Measured a(443) [m

-1

3

]

Figure 4.1 Comparison between algorithm-derived and known IOP, using
algorithm results from the empirical approach of Lee et al. (1998b). Resultsof a(440) from the synthetic data set (left) and a(443) from the in situ dataset (right).

a(440) values (see Table 4.1). These results suggest that the empirical coefficients, derived by forcing Equation 4.1 derived a(440) to match a limited numberof a(440) from field measurements, were biased by data from those measurements. It is likely that when more high-quality data are available, the coefficientsin Equation 4.1 could be fine tuned and the estimation of a(440) from Rrs (λ) bysimple ratios could be improved.
Table 4.1 RMSE and regression (Type II) results for the synthetic data set(θ0 = 30◦ ). N is the number of data tested, while n is the number of valid
retrievals by the relevant algorithm.
intercept

slope

R2

RMSE

500

0.050

0.939

0.976

0.140

0.091

500

-0.466

0.538

0.932

0.356

-0.151Nnaa(440), L98

500a(440), B99b

500

bias

a(490), B99

500

500

-0.488

0.574

0.948

0.281

-0.119a(410), MM01c

500

500

-0.368

0.747

0.976

0.295

-0.221a(440), MM01

500

500

-0.299

0.792

0.976

0.224

-0.156a(490), MM01

500

500

-0.256

0.815

0.965

0.169

-0.096a

Lee et al. (1998b)b

Barnard et al. (1999)cMorel and Maritorena (2001)

For the in situ data set, the R 2 value is 0.817 and the RMSE is 0.202 (see Table4.2), indicating that algorithm-derived a(443) are quite consistent with a(443)values from water samples. The larger differences are likely due to uncertainties
associated with both Rrs (λ) and a(443) in the in situ measurements (see also
Chapter 3 for discussions regarding uncertainties).

Simple Algorithms for Absorption Coefficients• 29

Table 4.2 RMSE and regression (Type II) results for the in situ data set. N
is the number of data tested, while n is the number of valid retrievals by
the relevant algorithm.
Nnintercept

slope

R2

RMSE

bias

a(443), L98

656

656

0.140

1.081

0.817

0.202

0.061

a(443), B99

656

642

0.085

1.039

0.643

0.272

0.047

a(490), B99

656

642

0.152

1.080

0.626

0.255

0.062

a(412), MM01

656

656

-0.228

0.911

0.817

0.237

-0.147a(443), MM01

656

656

-0.158

0.954

0.821

0.210

-0.113a(490), MM01

656

656

-0.117

0.949

0.808

0.171

-0.0594.2 Spectral Curvature Algorithm
4.2.1 General description
A simplistic 3-wavelength ratio method to test in situ measurements of remotesensing reflectance and the absorption coefficient for closure was developed byBarnard et al. (1999). The purpose of the method was to minimize the influenceof parameters of the radiative transfer equation that are difficult to determine insitu, e.g., backscattering. This method uses two ratios with three different wavelengths (λ1 = 440, λ2 = 490, and λ3 = 555 nm) of Rrs to minimize the spectral
dependence of the backscattering coefficient, as well as the angular dependenceof the underwater light field. Based on the semi-analytical relationship between
Rrs and bb /a (Morel and Gentili, 1993) one can derive the following relationshipusing ratios of three different wavelengths:,

Rrs (λ2 )g(λ1 )g(λ3 ) bb (λ1 )bb (λ3 ) [a(λ2 )]2'.
Rrs (λ3 )[g(λ2 )]2
[bb (λ2 )]2 a(λ1 )a(λ3 )(4.3)
As the spectral behaviour of the g parameter is nearly linear over these wavelengths, only a small error is induced by assuming that the triple ratio of g isequal to 1.0. The triple wavelength ratio of the backscattering coefficient inEquation 4.3, evaluated over typical oceanic conditions where the backscattering ranges from particle dominated to water dominated, and where the spectral
dependency of particle backscattering ranges from 0 to 2, varies from 0.93 to
1.02. Thus by choosing a constant value equal to 0.975 for the bb ratio term inEquation 4.3, a maximum error of 4.5% is made for most oceanic conditions.
Substitution of the assumed constant values of g triple ratio (= 1.0) and thebackscattering triple ratio (= 0.975) into Equation 4.3 results in a model that canbe used to compare directly in situ (and modelled) measurements of Rrs (λ) anda(λ).
Rrs (λ1 )Rrs3 (λ1 , λ2 , λ3 ) =Rrs (λ2 )30•

Remote Sensing of Inherent Optical Properties

Rrs3 (λ1 , λ2 , λ3 ) ' 0.975[a(λ2 )]2.
a(λ1 )a(λ3 )(4.4)

If functional relationships exist between the absorption coefficients at the selected three wavelengths, such that the absorption at λ1 and λ3 can be definedsolely in terms of the absorption at λ2 , the above formulation can be used toinvert the remotely-sensed reflectance to determine the spectral absorption at
the selected three wavelengths.
While any functional form for the spectral absorption coefficient can be utilized, Barnard et al. (1999) has shown that the absorption at 440 (originally it
was 443 nm) and 555 nm is significantly, linearly correlated to the absorption
at 490 nm, such that;a(440) = f1 [a(490)] = γ1 [a(490)] + γ2 ,a(555) = f2 [a(490)] = γ3 [a(490)] + γ4 ,(4.5)

where values of γ1−4 are 1.561, -0.012, 0.319, and 0.067, respectively.
Substitution of these functional forms into Equation 4.4 thus allows for theabsorption coefficient at 490 nm (and then at 440 and 555 nm) to be derived
solely from Rrs (λ),
h

i0.5

−(γ1 γ4 + γ2 γ3 ) − (γ1 γ4 + γ2 γ3 )2 − 4 γ1 γ3 − 0.975(γ2 γ4 )Rrs3
a(490) =2 γ1 γ3 − 0.975
Rrs3
(4.6)

4.2.2

Results and discussion when applied to the IOCCG data sets

With Rrs (λ) values at 440, 490 and 555 nm, a(490) and a(440) are calculated
from Rrs3 based on Equations 4.5 and 4.6. Figure 4.2 (also see Tables 4.1 and
4.2) compares model-derived a(440) versus known a(440) values. For the synthetic data, the model-derived values are systematically higher in the lower end(a(440) < 0.05 m−1 ) and systematically lower in the higher end (a(440) > 0.3
m−1 ), indicating a mismatch between data used for algorithm development anddata used for test. However, when the algorithm was applied to the in situ dataset, no such systematic bias was found, although there were 14 points for which
no valid results were obtained.

Simple Algorithms for Absorption Coefficients

3

3

1:1

Derived a(443) [m -1 ]Derived a(440) [m -1 ]

• 31

1
0.3
0.1
N = 500
n = 500
2
R = 0.932
RMSE = 0.356

0.03

0.03

0.1

0.3

1

Known a(440) [m

-1

1:1

1
0.3
0.1
N = 656n = 6422
R = 0.643
RMSE = 0.272

0.03

3

0.03

]

0.1

0.3

Measured a(443) [m

1
-1

3

]

Figure 4.2 Model-derived versus known a(440) (or a(443)) values using
results from the spectral-curvature algorithm (Barnard et al., 1999). Results
for the synthetic data set (left) and the in situ data set (right).

4.3

Spectral-Ratio Algorithm with Chlorophyll Concentration as an Intermediate Link

4.3.1

General description

Using chlorophyll-a concentration (C) derived from the spectral-ratio of Rrs (λ),
and a relationship between Kd and C, along with an analytical function that
expresses a as a function of Kd and R, the value of a can be derived from thespectral ratios of Rrs (Morel and Maritorena, 2001). Specifically, values of C
are first derived from the current operational chlorophyll algorithm for SeaWiFS(OC4v4) (SeaWiFS, 2000),2

34C = 10(a0 +a1 ρ+a2 ρ +a3 ρ +a4 ρ ) ,(4.7)

where ρ = log[max(Rrs (440, 490, 510))/Rrs (555)], and values of a0−4 are 0.366,-3.067, 1.93, 0.649, and -1.532, respectively.
Kd (λ) can be calculated from Equation 4.7 derived C, (Morel, 1988; Moreland Maritorena, 2001):
Kd (λ) = Kw (λ) + χ(λ)C e(λ) ,(4.8)

with the values of Kw (λ), χ(λ) and e(λ) known from statistical analysis of fieldmeasurements (see Table 2 of Morel and Maritorena, 2001).
Semi-analytically, there is (Morel, 1988; Morel and Maritorena, 2001)a(λ) =

0.9Kd (λ)[1 − R(λ)],
1 + 2.25R(λ)

(4.9)

n2w
QRrs (λ).
t2(4.10)

andR(λ) ≈

32•

Remote Sensing of Inherent Optical Properties

3
1:1

1
0.3
0.1
N = 500
n = 500
2
R = 0.976
RMSE = 0.224

0.03

0.03

0.1

0.3

Known a(440) [m

1
-1

]

3

Derived a(443) [m -1 ]Derived a(440) [m -1 ]

3

1:1

1
0.3
0.1
N = 656n = 6562
R = 0.821
RMSE = 0.210

0.03
0.01
0.01

0.03

0.1

0.3

Measured a(443) [m

1
-1

3

]

Figure 4.3 Comparison between algorithm-derived and known IOP, using
results from the spectral-ratio algorithm with C as an intermediate link
(Morel and Maritorena, 2001). Results of a(440) from the synthetic dataset (left) and a(443) from the in situ data set (right).

Here t is the sea-air transmittance, nw is the index of refraction of the water, andQ (sr) accounts the conversion of irradiance to radiance. The quantity Qn2w /t 2
represents the conversion between radiance reflectance to irradiance reflectanceand the air-sea interface effect, and approximates 6.8 for the remote sensing
domain (Morel and Gentili, 1993). Since the value of R is generally less than 0.1
and only plays a secondary role in Equation 4.9, the variation in Qn2w /t 2 does
not greatly affect the value of a(λ) derived by Equation 4.9. Therefore, for agiven Rrs (λ), absorption coefficients can be calculated following Equations 4.7
to 4.10.

4.3.2

Results and discussion when applied to the IOCCG data sets

With Rrs (λ) values at 440, 490, 510 and 555 nm, values of a(410), a(440) anda(490) were calculated from Equations 4.7 to 4.10. The comparison of thesederived values versus known (or measured) values is presented in Tables 4.1
and 4.2. Figure 4.3 shows model-derived a(440) with known a(440). Apparently, this empirical procedure performed very well, especially for the synthetic
data set with a(440) less than 0.2 m−1 . However, this empirical procedure (withthe present coefficients) underestimates absorption when the value of a(440)is greater than 0.2 m−1 , which may be attributed to the data range used to derive the parameters in Equation 4.7 and Equation 4.8. The difference between
model-derived a(443) and known a(443) is larger when applied to the in situ
data set, as was found for the other two algorithms. In addition, a(443) appears
to be underestimated. Even so, an R 2 value of 0.821 and an RMSE of 0.210 wereachieved.

Simple Algorithms for Absorption Coefficients• 33

4.4 Conclusions
Empirical and semi-empirical algorithms are easy to use and straightforward fordata processing. However, since the coefficients used in empirical algorithms arederived from data sets that do not necessarily represent all natural variations,the performance of such algorithms is always subject to compatibility between
the waters under study and the waters from which data were obtained for algorithm development. It is critical to examine this compatibility if robust results
are desired, and if this kind of algorithm is to be applied to an extensive area.

Chapter 5
Inversion of IOP based on Rrs and Remotely
Retrieved Kd
Hubert Loisel and Antoine Poteau

5.1 Background
Based on Monte Carlo and Hydrolight simulations we developed an inverse algorithm to retrieve the total absorption, scattering, and backscattering coefficientsfrom the irradiance reflectance just beneath the surface (R(0− )), and the mean
vertical diffuse attenuation coefficient over the first optical depth, <Kd >1 (Loisel
and Stramski, 2000). Note that while the particulate backscattering coefficient,
bbp , is directly obtained from bb by removing the effect of pure water, our algorithm does not intend to decompose a into its different components such as
the absorption by phytoplankton or by coloured dissolved organic matter. Thistask can be done in a second step (e.g., Chapter 10), by assuming some spectral models for pigments and gelbstoff absorption coefficients, such as thosealready available in the literature (Kopelevich and Burenkov, 1977; Kirk, 1994;
Bricaud et al., 1995), and using least-square fitting methods or equivalent. The
major motivation for the development of our algorithm was the assessment of
total IOP from basic radiometric measurements by means of a simple and fast
approach that does not require any assumption about the spectral shapes of a,b, and bb . A detailed review of the methods used for solving the hydrologicoptics inverse problem was recently performed by Gordon (2002). One of thedifferences among these methods concerns the input parameters they use. The
choice of R(0− ) and <Kd >1 for our algorithm was motivated by the fact that they
can both be estimated from satellite measurements of ocean colour. WhereasR(0− ) is linked to the above surface remote sensing reflectance (Rrs ) in a fairly
straight forward manner (Mobley, 1994), the retrieval of <Kd >1 from space isbased on empirical relationships (Mueller, 2000; Loisel et al., 2001b). Therefore,
our algorithm does not require any spectral assumptions about IOPs, but does
require spectral relationships between <Kd >1 and Rrs , in the frame of remotesensing application. However, while <Kd >1 is still empirically determined from
Rrs , one can imagine a more sophisticated method to retrieve <Kd >1 from space
35

36

•

Remote Sensing of Inherent Optical Properties(such as an iterative scheme based on analytical relationships between <Kd >1
and space retrieved IOP).
Here we test an improved version of our algorithm (Loisel et al., in prep) with
the IOCCG data sets (synthetic and in situ). After a brief overview of the model,
we examine closure between data from both synthetic and in situ IOPs, and theretrieval of these IOPs using our model. Finally, these results are discussed, andcompared to previous validation studies performed with the Loisel and Stramski(2000) algorithm in various oceanic waters.

5.2

Output and Input Parameters

Output parameters of the model are a, b and bb averaged over the first attenuation layer. Because the retrieval of b is highly sensitive to the variations in
the particle phase function, only a and bb can be retrieved reasonably well from
Rrs . For this reason, we will specifically focus on the retrieval of a and bb at
410, 440, 490, and 550 nm. These wavelengths are common to almost all oceancolour sensors (with some slight spectral shifts depending on the sensor).
Input parameters of the model are R(0− , λ), < Kd (λ)>1 , and the sun zenith
angle (θ0 ). Within the context of ocean colour remote-sensing applications, only
Rrs (λ) is available; both R(0− , λ) and <Kd (λ)>1 have to be determined. The exact procedure to assess R(0− , λ) from Rrs (λ) is given in Loisel and Morel (2001).
This step accounts for the process of reflection and refraction of light at the
air-water interface, and of the bi-directional effect as described in Morel andGentili (1993). To estimate <Kd (λ)>1 from Rrs (λ), we originally used an empirical relationship between <Kd (490)>1 and Rrs (490)/Rrs (555), such as the one
developed by Mueller (2000). <Kd (λ)>1 was then estimated empirically from
<Kd (490)>1 . To estimate a(410), a(440), a(490), bb (440), bb (490), and bb (550),
we therefore need Rrs (410), Rrs (440), Rrs (490), and Rrs (550). Moreover, because of the strong influence of the incident light field at the air-sea interface
on the Rrs -IOP relationships, our model also accounts for the change of θ0 .

5.3

Assumptions and Description

The radiative transfer simulations used for the development of our model wererun for an infinitely deep ocean (no bottom reflected light) with an opticallyhomogenous water column, a nearly flat sea surface with no wind, and the absence of inelastic scattering processes. The phase function was derived froma weighted sum of the molecular scattering and the particle scattering phasefunctions proposed by Mobley et al. (1993). An iterative scheme was also developed for removing the Raman contribution, which is always present in theInversion of IOP based on Rrs and Remotely Retrieved Kd• 37

natural environment, from R(0− ). This correction will not be applied here forconsistency with the other models presented in this report.
Our model is based on the following set of equations between a(λ), bb (λ)
and <Kd (λ)>1 , R(0− , λ) which can be applied to any wavelength:µw < Kd >1
a= h
i0.5 ,
R(0− )
2
1 + (2.54 − 6.5µw + 19.89µw) 1−R(0− )(5.1)

bb =< Kd >1 10α [R(0− )]δ .

(5.2)

The α and δ functions were given by:

α = (−0.83 + 5.34η − 12.26η2 ) + µw (1.013 − 4.124η + 8.088η2 ), (5.3)δ = (0.871 + 0.4η − 1.83η2 ).

(5.4)

Where η is the ratio of the molecular scattering to the total scattering (= bw /b),and µw is the cosine of the refracted solar beam angle just beneath the surface.
Some modifications of the original version of the model are performed fora better retrieval of a and bb in the context of ocean-colour remote sensing
applications (Loisel et al., in prep). These modifications are listed briefly below:i) The model accounts directly for Rrs instead of R(0− ).
ii) We developed a new way to account for the effect of η on the derivationof a and bb from remote sensing (new parameterisations coupled with an
iterative procedure). Note that the dependence of η on the assessment of
a was not taken into account in the previous version of our model.
iii) We performed some slight modifications within the a parameterisation toaccount for some more realistic η − b/a combinations at any given wavelength used by ocean-colour sensors.
iv) We used new formulations and parameterisations to determine <Kd (λ)>1
from ratios of remote sensing reflectance:q = Rrs (440)/Rrs (550).
(ν1 (λ) log(q)+ν2 (λ))/(ν3 (λ)+log(q))< Kd (λ) >1 = 10

(5.5)
.

(5.6)

ν1−3 are empirical parameters and are provided in Table 5.1 for the SeaWiFS
bands.

5.4

Results

Using Rrs (λ) values at 410, 440, 490 and 550 nm, IOPs retrieved from the above
steps were compared with known (synthetic) or measured (in situ) values.

38•

Remote Sensing of Inherent Optical Properties

Table 5.1 Parameters for deriving <Kd (λ)>1 from spectral ratio of remotesensing reflectance (Equation 5.6).
v1

v2

v3

410

-4.7636-2.1269

3.1752440-4.6216-2.3587

3.1235490-3.6636-2.31162.5648550-2.0152-1.5296

1.7751

5.4.1 Comparison with synthetic data
Figure 5.1 as well as Table 5.2 present the performance of our model using Rrs (λ)
and the sun angle as inputs to the calculation of the IOPs. In this case, the sunangle is at 30◦ . For the absorption coefficient at 410, 440, and 490 nm, the slopeof the linear regression is very close to 1.0, and the coefficient of determinationis very high (R2 ≥ 0.94). The RMSE values are 0.12, 0.119, 0.136, and 0.138 fora(410), a(440), a(490), and a(550), respectively. Note that the performance ofthe model is slightly degraded at 550 nm compared to other wavelengths.
Table 5.2 RMSE and regression (Type II) results between the derived andthe known values of IOP for synthetic data (for θ0 = 30◦ ). Rrs (λ) at 410,
440, 490 and 550 nm are used as inputs for the derivation of IOP. N is thenumber of data tested, while n is the number of valid retrievals.
intercept

slope

R2

RMSE

500

0.029

0.977

0.973

0.120

0.043

500

-0.007

0.990

0.966

0.119

-0.001500

500-0.017

0.980

0.939

0.136

0.000

500

500

-0.067

0.927

0.818

0.138

-0.002bb (440)500

500-0.173

0.902

0.924

0.123

0.003

bb (490)500

500-0.114

0.935

0.917

0.140

0.007

bb (550)500

500-0.028

0.973

0.934

0.138

0.023

N

n

a(410)

500a(440)

500

a(490)
a(550)

bias

The retrieval of the absorption coefficient at 550 nm is challenging, as it isstrongly dominated by absorption by pure sea water, and because variationsof Rrs are mostly due to the backscattering coefficient in this spectral domain.
The RMSE values for bb (λ) at 440, 490, and 550 nm are very similar to thoseof a(λ), but the slopes differ slightly from 1.0 (0.902, 0.935 and 0.973 at 440,
490, and 550 nm, respectively). Note that the RMSE values for bb are almost
similar for every wavelength. Also, the total absorption and backscattering coefficients are retrieved with the same precision when the sun angle is fixed at60◦ (not shown here). Most of the a(λ) and bb (λ) errors appear at the high endof the data range, where the retrieval of <Kd (λ)>1 from Rrs (λ) is generally much
more doubtful. For instance, by restricting the data set to the a(440) values

Inversion of IOP based on Rrs and Remotely Retrieved Kd

• 39

10

1
0.3
0.1
N = 500
n = 500
2
R = 0.973
RMSE = 0.120

0.03
0.01
0.01

0.03

0.1

0.3

1

Known a(410) [m

3
-1

1:1

3

Derived a(440) [m -1 ]Derived a(410) [m -1 ]

1:1

3

1

0.3

0.1

0.01
0.01

10

N = 500
n = 500
2
R = 0.966
RMSE = 0.119

0.03

0.03

0.1

0.3

Known a(440) [m

]

1
-1

3

]

0.1

0.03

0.01
N = 500
n = 500
2
R = 0.924
RMSE = 0.123

0.003
0.003

0.01

0.1

0.03

Known b b (440) [mDerived b b (490) [m -1 ]Derived b b (440) [m -1 ]

0.3
1:1

-1

]

1:1

0.1

0.03
0.01
N = 500
n = 500
2
R = 0.917
RMSE = 0.140

0.003
0.003

0.01

0.03

0.1

0.3

Known b b (490) [m -1 ]Figure 5.1 Comparison of the derived absorption and backscattering coefficients using the synthetic data set, for a sun zenith angle at 30◦ , andat different wavelengths. Rrs (λ) at 410, 440, 490 and 550 nm are used as
inputs for the derivation of a and bb .

lower than 0.3 m−1 , which includes most oceanic waters, the RMSE drops by afactor of 2 (from 0.119 to 0.058). The same remark holds at other wavelengths.
When measured <Kd (λ)>1 is explicitly taken into account as an input parameter,
the performance of the model is greatly enhanced. For example, the RMSE fora(410), a(440), a(490), a(550), and bb (490) are 0.0445, 0.0102, 0.0131, 0.0101,
and 0.0324, respectively (not shown here).

5.4.2

Comparison with in situ data

Figure 5.2 and Table 5.3 show that there is a reasonably good agreement between
the modelled and the measured values of the absorption coefficients, with the
RMSE always being lower than 0.2. In the blue-green spectral domain, the mean
RMSE value is 0.166. Compared to the synthetic data set, the RMSE increases by
a factor of 1.6, 1.6, and 1.24 at 412, 443, and 490 nm, respectively. Note that
RMSE drops from 0.169 to 0.142 by restricting the data set to the a(490) values
lower than 0.3 m−1 . Interestingly, the RMSE at 550 nm is slightly better with the•Remote Sensing of Inherent Optical Properties

3

Derived a(412) [m -1 ]

3

1:1

1
0.3
0.1
N = 656n = 6562
R = 0.847
RMSE = 0.186

0.03
0.01
0.01

0.03

0.1

0.3

1

Measured a(412) [m

-1

1:1

Derived a(443) [m -1 ]

40

1
0.3
0.1

0.01
0.01

3

N = 656n = 6562
R = 0.842
RMSE = 0.198

0.03

0.03

0.1

0.3

1

3

Measured a(443) [m -1 ]

]

1

0.3
0.1
N = 656n = 6562
R = 0.823
RMSE = 0.169

0.03
0.01
0.01

0.03

0.1

0.3

Measured a(490) [mDerived a(555) [m -1 ]Derived a(490) [m -1 ]

1
1:1

1:1

0.3

0.1

0.1

1
-1

N = 656n = 6562
R = 0.670
RMSE = 0.111

0.3

Measured a(555) [m

]

1
-1

]

Figure 5.2 Comparison of the derived and the measured total absorption
coefficients at different wavelengths, using the in situ data set. Rrs (λ) at
412, 443, 490 and 555 nm are used as inputs for the derivation.

in situ data set than with the synthetic data set.
Table 5.3 RMSE and regression (Type II) results between the derived andthe known values of IOP for in situ data. Rrs (λ) at 412, 443, 490 and 555
nm are used as inputs for the derivation of IOP. N is the number of datatested, while n is the number of valid retrievals.
Nnintercept

slope

R2

RMSE

bias

a(412)

656

656-0.052

1.013

0.847

0.186

-0.064a(443)

656

656-0.108

0.997

0.842

0.198

-0.105a(490)

656

656-0.122

0.953

0.823

0.169

-0.069a(555)

656

656-0.126

0.897

0.670

0.111

-0.017

5.5 Conclusions
The retrieval of both a(λ) and bb (λ) is achieved with excellent accuracy in theblue green spectral region when both R(0− ) and <Kd >1 are measured (the mean
RMSE value in this spectral domain is 0.0195 for the absorption coefficient).
When only Rrs is available as an input parameter, the results are obviously degraded, but are still very satisfactory: for the synthetic data set, the mean RMSE
value over the blue-green part of the spectrum for a and bb is 0.128 and 0.134,

Inversion of IOP based on Rrs and Remotely Retrieved Kd• 41

respectively. When comparing with in situ data, we should emphasise that ourmodel is able to predict a with a mean RMSE value of 0.166 over the spectral
domain of interest for ocean-colour related studies. The performance of ourmodel is governed, to a certain extent, by the accuracy of the <Kd >1 assessment
from space. Different approaches have been tested to improve the retrieval of<Kd (λ)>1 from Rrs (λ) (Loisel et al., in prep). Preliminary results for the retrievalof both a and bb are very promising.
The results presented here are consistent with previous comparisons performed in oceanic and coastal waters (Loisel et al., 2001b; Melin et al., 2002;
Dupouy et al., 2003). For example, based on field data collected in waters off
southern California, and in waters surrounding Europe, Loisel et al. (2001b)
showed that the average value and the standard deviation of the relative difference between the measured and the retrieved absorption coefficients from
412 to 555 nm are 26% and 16%, respectively. The new version of the modelsignificantly improves the retrieval of a and bb , especially in the green part ofthe spectrum and at the extreme values (Loisel et al., in prep).

Chapter 6
The MERIS Neural Network Algorithm
Roland Doerffer and Helmut Schiller

6.1 Introduction
In this chapter we present the results of the MERIS Case 2 water algorithm for
the IOCCG algorithm inter-comparison. This algorithm is an artificial neuralnetwork (aNN) inversion procedure (Doerffer et al., 2002; Doerffer and Schiller,
2000; Schiller and Doerffer, 2005; Doerffer and Schiller, 2006), which is used in
the ground segment processor of MERIS. This instrument is operated on board
the Earth observation satellite ENVISAT of the European Space Agency (ESA),
which was launched on 1 March 2002. The aNN algorithm was selected becauseof its capability to invert directional water-leaving radiance reflectance directly
into absorption and scattering coefficients or concentrations of different constituents present in coastal waters, with high efficiency for mass production.
Due to the fixed architecture of the aNN, only the simulated IOCCG data set
could be processed for inter-comparison.

6.2

Description of the MERIS Case 2 Water Algorithm

The MERIS Case 2 water algorithm is a neural network, which takes the log of
the above-surface remote-sensing reflectance (Rrs , which is the directional water leaving radiance divided by the downwelling irradiance) of eight of the fifteen MERIS bands (i.e. after atmospheric correction, Bands 1-7 and Band 9) as
well as three angles (solar zenith, viewing zenith, azimuth difference) as input
and provides the log of the following three optical coefficients as output: pigment absorption (aph (442)), absorption of gelbstoff and bleached suspended
matter (adg (442)), and scattering coefficient of all particles (bp (442)), all at 442
nm (MERIS Band 2). The optical coefficients are then used to compute the concentrations of chlorophyll-a and total suspended matter dry weight. Together withthe gelbstoff absorption, these are the three Case 2 water products of MERIS.
However, by using the inverse of the conversion factors it is also easy to go backone step and compute the three IOPs from the concentrations again, as well as
43

44

•

Remote Sensing of Inherent Optical Properties

the total absorption and scattering.
The neural network is trained with simulated Rrs spectra. About 30,000 spectra are used to cover a large range from Case 1 and Case 2 waters as well as
different observation and solar angles (see Table 6.1). The simulation of Rrs (λ)
is performed using Hydrolight radiative transfer model. The model is set up in
the following way:
❖ No bottom reflection❖ Homogenous vertical distribution of water constituents❖ No inelastic scattering❖ Waves according to a wind speed of 3 m s−1
Standard clear atmosphere with oceanic aerosol and different solar zenith angles(0 - 80◦ from zenith) was used to simulate incoming solar light. The detector
captures the directional, water-leaving radiance and downwelling irradiance justabove the surface, for computing the directional Rrs . For the comparison here,
only the nadir Rrs was used. The part controlling the success of the simulationand training of the aNN is the bio-optical model. For the MERIS aNN algorithm, it
is based on measurements of the IOPs, i.e. absorption and scattering. These data
are mainly from European waters, dominated by measurements in the North Sea.
Table 6.1 Variability and range of the optical properties used for the simulation of water-leaving radiance reflectance spectra that were used to train
the aNN.
Component/Property

Value Range

Gelbstoff absorption wavelength exponent [nm−1 ]0.014 ± 0.002

Bleached particle absorption wavelength exponent [nm−1 ]0.008 ± 0.005

Particle scattering wavelength exponent

0.4 ± 0.2

White particle scattering wavelength exponent

0.0

Phytoplankton pigment absorption spectra

Gelbstoff absorption (ag ) at 442 nm [m−1 ]random selection from > 200
absorption spectra, normalized at 442 nm (MERIS Band 2)
0.005 - 5.0

Particle scattering (bp ) at 442 nm [m−1 ]0.005 - 30.0

White particle scattering (bpw ) at 442 nm [m−1 ]0.005 - 30.0
−1

Phytoplankton pigment absorption (aph ) at 442 nm [m]0.001 - 2.0

Minimum particle scattering at 442 nm [m−1 ]0.25 aph (442)

Bleached particle absorption

0.1 bp (442) + δ† 0.03 bp (442)Sun zenith angle [degree]

0 - 80Viewing zenith angle [degree]

0 - 50Difference between sun and viewing azimuth angle [degree]

0 - 180†– δ is a random value in the range of 0-1

The bio-optical models used here represent mean conditions and variabilities(see Table 6.1). For each case of the simulations, the optical properties are varied

The MERIS Neural Network Algorithm• 45

randomly according to the standard deviations of the measured absorption andscattering spectra. For the absorption of gelbstoff (ag ) and bleached particles(ad ) as well as for the total particle scattering (bp ), the wavelength exponent
is varied according to the measured standard deviations. For the absorption
spectra of phytoplankton pigments (aph (λ)), one out of 223 different measured
spectra is selected randomly for each simulation. The absorption and scattering
coefficients at 442 nm are randomly selected from the range using the log scale(see Table 6.1), while the viewing and sun angles are selected randomly using thelinear scale. The simulated spectra are furthermore randomly degraded using
an estimated error of the instrument and the atmospheric correction. Any Rrs
spectrum that is out of the training range is detected using a forward neuralnetwork. This network takes the optical coefficients from the first backward
network as input to compute a Rrs spectrum. This spectrum is then comparedwith the measured one. If the Chi2 deviations of all eight bands are above acertain level (Doerffer and Schiller, 2000), the spectrum is classified as out of
the training range. However, this test was not used for the IOCCG data sets.
Uncertainties in atmospheric correction over water with low water-leaving
radiance sometimes result in negative reflectance. These incorrect values caneasily be excluded from the neural network by introducing a cut-off. In the
MERIS aNN algorithm, this cut-off was set to a Rrs value of 0.000955 sr−1 . All
reflectance values below this threshold are clipped to this value. The neuralnetwork is trained in the same way.

6.3

aNN Results with the IOCCG Data Sets

The aNN algorithm we have tested here has five hidden layers with 45, 16, 12,
8 and 5 neurons respectively. It is the algorithm which is presently used for
reprocessing all the MERIS data (Doerffer and Schiller, 2006). aph (442), adg (442)and bbp (442) (which is assumed to be 1.5% of bp (442)) were retrieved by applying
this algorithm to the IOCCG data sets. Note that, due to wavelength mismatch,
the aNN algorithm (designed specifically for MERIS) was not applied to the insitu part of the IOCCG data sets. Before applying the aNN algorithm, the reflectance spectra for the MERIS bands were linearly interpolated from the dataset, which has a 10 nm spacing. Also the optical properties of the test data set
were interpolated for 442 nm. It should be pointed out that although the IOCCG
synthetic data set was also simulated with Hydrolight radiative transfer code, it
was nevertheless computed totally independently from the data used for training the aNN. The two data sets are completely independent and are based on
independent bio-optical models, which explains at least part of the deviations.
Figure 6.1 compares the derived properties (for data of 30◦ solar zenith angle)
with their corresponding known values, while Table 6.2 summarizes results from46•

Remote Sensing of Inherent Optical Properties

1:1

0.3
0.1
0.03
0.01

N = 500
n = 500
2
R = 0.943
RMSE = 0.271

0.003
0.003

0.01

0.03

0.1

0.3

1

3

Derived a dg (442) [m -1 ]Derived a ph (442) [m -1 ]

1

1:1

1
0.3
0.1
0.03
N = 500
n = 500
2
R = 0.959
RMSE = 0.230

0.01
0.003
0.003

0.01

1:1

3
1
0.3
0.1

N = 500
n = 500
2
R = 0.994
RMSE = 0.052

0.03
0.01
0.01

0.03

0.1

0.3

0.03

0.1

0.3

1

Known a dg (442) [mDerived b bp (442) [m -1 ]Derived a(442) [m -1 ]Known a ph (442) [m -1 ]

3

]

0.1
0.03
0.01
0.003
N = 500
n = 500
2
R = 0.980
RMSE = 0.082

0.001
1:1

0.001

3

1

-1

Known a(442) [m -1 ]

0.003

0.01

0.03

Known b bp (442) [m

-1

0.1

]

Figure 6.1 Comparison between aNN derived IOPs and the known IOPs,
for the IOCCG synthetic data set (the Sun at 30◦ from zenith). aNN used
Rrs values at 412, 442, 490, 510, 560, 617, 665 and 708 nm to retrieve theIOPs.

statistical analyses. For the entire range of total absorption and backscattering
coefficients the RMSE values are 0.052 and 0.082 (see Table 6.2), respectively,
with slope values nearly 1.0. Similar results were also obtained for the synthetic
data set with the Sun at 60◦ from zenith (not shown here).
Table 6.2 RMSE and regression (Type II) results for the synthetic data set(30o solar zenith angle). IOPs were retrieved with Rrs values at 412, 442,
490, 510, 560, 617, 665 and 708 nm. N is the number of data tested, while
n is the number of valid retrievals.
NnIntercept

slope

R2

RMSE

bias
-0.174adg (442)

500

500

-0.210

0.959

0.959

0.230

aph (442)

500

500

0.407

1.163

0.943

0.271

0.202

a(442)

500

500-0.009

1.006

0.994

0.052

-0.013bbp (442)

500

500

-0.038

0.993

0.980

0.082

-0.024

These results indicate that the aNN algorithm accurately retrieved those optical properties that determine the remote sensing reflectance. When the total
absorption is decomposed into the components of water, gelbstoff and phytoplankton pigments, the scatter is much larger (RMSE values are 0.230 and 0.271
for adg (442) and aph (442), respectively) and the relationships deviate from lin-

The MERIS Neural Network Algorithm• 47

earity in the middle concentration range. The scatter is obviously due to thefact that the bio-optical models used for the IOCCG synthetic data set are different from those used for training the aNN. This is presumably also true for the
maximum difference in the middle of the data range. Since this is also normallythe case in nature, it indicates that total absorption and total backscattering
are more robust variables, which should be derived from reflectance spectra in
addition to other IOPs or concentrations of different water constituents.

Chapter 7
The Linear Matrix Inversion Algorithm
Paul Lyon and Frank Hoge

7.1 Background
The Linear Matrix Inversion (LMI) algorithm was developed by Hoge and Lyon
(1996). This algorithm uses remote-sensing reflectance at three wavelengths tosimultaneously derive three major unknowns algebraically. Due to its linear matrix nature, it is efficient in processing satellite images. In the past decade, thisalgorithm has been applied to data taken from many regions around the world(Hoge and Lyon, 1996; Hoge and Lyon, 1999; Hoge et al., 2001). Nevertheless,since some of the parameters used in the algorithm were developed based on
measurements made mainly from the Mid Atlantic Bight and off the East Coastof the United States, further refinement and improvement is expected in the
coming years.

7.2

Inputs of LMI

The algorithm uses remotely sensed reflectance, Rrs , propagated through theair/ water interface, into semi-analytic reflectance model developed by Gordon
et al. (1988). The present version of the algorithm that is optimized for use
with satellite data uses only three inputs, Rrs (412), Rrs (490) and Rrs (555). The
algorithm also has four empirical parameters, described below, that determine
the spectral shapes of the individual IOP spectrum.

7.3

Basic Assumptions of LMIThere are three assumptions that are fundamental to this inversion technique.
First, it is assumed that the semi-analytic equation, shown in Equation 7.1, is agood description of the relationship between the IOPs and the reflectance overa wide range of environments. Second, it is assumed that globally, the optically significant varying IOPs are absorption coefficients of phytoplankton and
49

50

•

Remote Sensing of Inherent Optical Properties

CDOM (including detritus), and backscattering coefficient of all particles (scattering constituents other than water). Thus, an algorithm based on properly formulated spectral models of these three principal IOPs may be applied to manydifferent water masses. And, third, it is assumed that, using a proper combination of wavelengths, the three major IOPs can be resolved if the following
conditions are met. The wavelengths used should maximize the mathematicaldifferences between the spectral shapes of the three IOPs. And, within each
IOP, a stable spectral dependence must be maintained or the modulations of
the IOP spectral shapes need to be empirically adjusted to reflect their naturalvariability.

7.4

Approach

7.4.1

Algorithm mathematical description

As stated above, the algorithm is based on the reflectance model developed byGordon et al. (1988),
rrs = g1



2
bb
bb
.
+ g2
bb + abb + a(7.1)

Here rrs is the subsurface remote-sensing reflectance, which can be easily calculated from the remote-sensing reflectance (Rrs ) provided by any sensor. g1 =0.0949 and g2 = 0.0794 are model parameters for rrs (Gordon et al., 1988). bb
and a are the total backscattering coefficients and total absorption coefficient,
respectively. Defining u as
bb
u≡
,
(7.2)
bb + awe get a quadratic equation with u as the variable,
g2 u2 + g1 u − rrs = 0,

(7.3)

which can easily be solved for u using the quadratic formula. Since bb is a sum of
bbw and bbp , a is a sum of aw , aph , and adg , and aw and bbw are known constants,
a linear system with aph , adg and bbp as variables can then be constructed byre-arranging u (Hoge and Lyon, 1996):
aph (λ) + adg (λ) + bbp (λ)ν(λ) = −aw (λ) − bbw (λ)ν(λ),
with
ν ≡1−

1
.
u

(7.4)

(7.5)

For an exact solution, three different wavelengths are used to form a systemof three equations with three unknowns. After spectrally modeling the threeThe Linear Matrix Inversion Algorithm• 51

IOP variables with values at a reference wavelength (λr = 410 nm) the equation
becomes,aph (λ)aph (λr ) + adg (λ)adg (λr ) + bbp (λ)bbp (λr )ν(λ) = −aw (λ) − bbw (λ)ν(λ).
(7.6)



Here aph (λ), adg (λ) and bbp (λ) represent the normalized optical properties at
λr (see section 7.4.2). Equation 7.6 can now be used to construct the linear
matrix that could be inverted to derive the IOPs consistent with the input Rrs
and the spectral models,





1
1
ν(λ
)
a
(λ
)
a
(λ
)
+
b
(λ
)ν(λ
)
r
r
r 
bw

  ph r 
 w r
 





 aph (λ2 ) a




=
−
(λ
)
b
(λ
)ν(λ
)
a
(λ
)
a
(λ
)
+
b
(λ
)ν(λ
2
2
2
2
2)  .
bwdg
bp
  dg r 
 w 2



a
(λ
)
a
(λ
)
b
(λ
)ν(λ
)
b
(λ
)
a
(λ
)
+
b
(λ
)ν(λ
)
3
3
3
3
r
w
3
3
3
bp
bw
ph
dg
bp
(7.7)
Note that the inverse matrix must be computed for each data point since input
data, Rrs (λ), is on both sides of the equation (contained in ν(λ)). Any standard method of solving this system of equations can be used. The Hoge/Lyon
inversion algorithm uses lower/upper deconvolution (Hoge and Lyon, 1999).

7.4.2

IOP spectral models

To mathematically solve Equation 7.7, spectral models are required for the three
IOP variables. It is important to select wavelengths where each IOP tends toco-vary among wavelengths (Hoge and Lyon, 1996). Based on many different
published phytoplankton absorption spectra, phytoplankton absorption coefficients at 412, 490 and 555 nm are found to co-vary well. A Gaussian function
centered at 443 nm, with a full-width at half max (FWHM) of 70 nm (σ in Equation 7.8 below) is used to model aph (λ). No improvement in the retrieved IOPs
was found when the FWHM parameter σ was empirically varied.
 2

aph (λ) = aph (λr )e

λr +886(λ−λr )−λ2
2σ 2



.

(7.8)

The combined absorption coefficient of detritus and gelbstoff, adg (λ), ismodelled with an exponential decay function (Bricaud et al., 1981; Roesler etal., 1989; Carder et al., 1991):
adg (λ) = adg (λr )e−S(λ−λr ) .

(7.9)

S is the spectral slope and is set to 0.018 (nm−1 ) for all inversions discussed
within this chapter.
The total particulate backscattering coefficient, bbp (λ) is modelled as a
power-law function of wavelength, Yλrbbp (λ) = bbp (λr ),(7.10)λ52•

Remote Sensing of Inherent Optical Properties

with exponent Y empirically estimated as follows,
Y = m1

Rrs (490)+ m2 .
Rrs (555)(7.11)

Sensitivity studies have found that the magnitude of parameter Y affects
the bbp and the adg retrievals more than the aph retrievals (Hoge and Lyon,
1996). The empirical parameters, m1 (0.8) and m2 (0.2) have been optimized
for use with global satellite data, such that errors in the derived IOPs caused bythis equation are minimized in a global sense. These parameters can also haveregional values to achieve better regional results.

7.5

Results

7.5.1

Synthetic data set

Figure 7.1 shows the agreement between known and derived IOPs. There is alogarithmic offset in the aph (410) retrievals and several outliers, yet the agreement is evident as shown in Table 7.1 by the correlation of determination (R 2 )of 0.877 (n = 484) and the slope close to 1.0, and an RMSE of 0.222. Note thatthe statistics presented in Table 7.1 are affected by the outliers that lie below
the one-to-one line, so that the larger population above the line could still becorrected by using an offset in log space. This infers, of course, that regionalor specific tuning of this and any algorithm may improve its performance forsimilar settings.
Table 7.1 RMSE and regression (Type II) results of the synthetic data set(θ0 = 30◦ ). IOPs were retrieved with Rrs values at 410, 490 and 550 nm
as inputs. N is the number of data tested, while n is the number of valid
retrievals.
Nnintercept

slope

R2

RMSE

bias

aph (410)

500

484

0.053

0.989

0.877

0.222

0.068

aph (490)

500

484

0.114

0.997

0.891

0.23

0.118

adg (410)

500

484

0.069

1.052

0.958

0.161

0.03

adg (490)

500

484

-0.032

1.051

0.921

0.236

-0.095a(410)

500

484

0.067

1.036

0.964

0.14

0.045
0.005

a(490)

500

484

0.012

1.007

0.942

0.133

bbp (410)

500

484

0.043

1.019

0.922

0.15

0.007

bbp (490)

500

484

0.008

1.018

0.936

0.149

-0.027

The agreement between input and output adg (410) is better than that ofaph (410), with an R 2 of 0.958 and slope about 1.1, and RMSE of 0.16 (Table 7.1).
Much better results are achieved for the total absorption and particle backscattering coefficients, with both R 2 and slope values close to 1.0 and RMSE of 0.14
and 0.15, respectively.

The Linear Matrix Inversion Algorithm

• 53

30

0.3
0.1
0.03
0.01
0.003

N = 500n = 4842
R = 0.877
RMSE = 0.222

1:1

0.003

0.01

0.03

0.1

Derived a dg (410) [m -1 ]Derived a ph (410) [m -1 ]1Known a ph (410) [m

1:1

3
1
0.3
0.1
0.03

N = 500n = 4842
R = 0.958
RMSE = 0.161

0.01
0.003
0.003 0.01 0.03

0.3
-1

10

0.1

0.3

1

Known a dg (410) [m

]

-1

3

10

]

1

3
1
0.3
0.1
N = 500n = 4842
R = 0.964
RMSE = 0.140

0.03
0.01
0.01

1:1

0.03

0.1

0.3

Known a(410) [m

3

1
-1

]

Derived b bp (410) [m -1 ]Derived a(410) [m -1 ]

10
0.3
1:1

0.1
0.03
0.01
0.003

N = 500n = 4842
R = 0.922
RMSE = 0.150

0.001
0.001

0.003

0.01

0.03

Known bbp (410) [m

0.1
-1

0.3

]

Figure 7.1 Comparison between retrieved and simulated IOPs for a sun
zenith angle of 30◦ . IOPs were retrieved using Rrs values at 410, 490 and
550 nm.

The smaller number (n) in the statistics analysis (Table 7.1 and Table 7.2)
represents all the data points that the inversion successfully processed (all IOPs
with values greater than zero, or where the output a(410) < 10.0 m−1 ). Data
points with negative IOP retrievals were excluded from the statistics and the figures, as they are physically unrealistic values that are filtered out automatically.
In normal operation of the linear matrix inversion, retrievals where adg (410) >1.0 m −1 or aph (410) > 1.0 m −1 are considered suspect. To allow readers tocompare the results of this algorithm with those of other techniques discussed
in this report, however, inversions with a(410) with values up to 10.0 m−1 areincluded in the figures and tables.

7.5.2

In situ data set

For the in situ data set, the regression statistics are provided in Table 7.2. Retrieved aph (412), adg (412), a(412) and a(490) are compared with their measured
values respectively in Figure 7.2. There were no in situ bbp data for comparison.
Apparently the retrievals of aph and adg scattered much more than that of
the simulated data set. This might be due to the measurement uncertainties
that are common in field-measured data. Also, large portions of the data weretaken in coastal waters, and real in situ properties may not follow the limited

54•

Remote Sensing of Inherent Optical Properties

Table 7.2 RMSE and regression (Type II) results of the in situ data set.
IOPs were retrieved with Rrs values at 412, 490 and 555 nm as inputs. N is
the number of data tested, while n is the number of valid retrievals.
Nnintercept

slope

R2

RMSE

bias

aph (412)

656

642

0.336

1.208

0.654

0.332

0.02

aph (490)

656

642

0.454

1.231

0.686

0.325

0.078

adg (412)

656

642

-0.142

1.007

0.653

0.325

-0.149adg (490)

656

642

-0.315

0.98

0.599

0.427

-0.284a(412)

656

642-0.082

0.96

0.872

0.163

-0.045a(490)

656

642-0.035

0.994

0.804

0.168

-0.028

combinations of spectral shapes used in the simulated data set.
Again, better results are obtained for the total absorption coefficients. Thissuggests that it is easier to retrieve the total absorption using this techniquethan it is to resolve the separate components of the total absorption.

7.6

Discussion

7.6.1

Overall results of the linear matrix inversion algorithm

As described above, the retrievals of the total absorption are quite good for bothin situ and simulated data sets. The separation of the absorption into contributions from phytoplankton and dissolved organic matter are less accurate, but
still retrieved well. The spectral model parameters used in the linear inversionof both simulated and in situ data sets preformed well in spite of the fact thatthe true spectral shapes at the wavelengths used in the inversions varied overdramatic ranges, as shown in Figure 7.1. These results demonstrate that an exact solution derived from a 3-by-3 inversion, can be optimized to retrieve IOPs
at a reference wavelength. The linear inversion method has both weaknesses
and strengths associated with its use, which are briefly described below.

7.6.2

Algorithm weaknesses

One set of weakness in this algorithm is related to the parameterisation of the
IOP spectral shapes. For example, the empirically adjusted bbp spectral modeland the fixed spectral models for aph and adg will not properly represent all
combinations of water constituents, especially when contributions from optically significant constituents not well described by the three IOP basis vectors
are present. The fixed spectral shape of aph limits the accuracy of IOP retrievals
since true phytoplankton absorption spectra vary dramatically (e.g., Hoepffner
and Sathyendranath, 1991), however, the selection of covarying wavelengths reduces the impact of the high variability of other wavelengths. Also, the adg

The Linear Matrix Inversion Algorithm

• 55

3

Derived a dg (412) [m -1 ]Derived a ph (412) [m -1 ]

3
1:1

1
0.3
0.1
0.03
0.01

N = 656n = 6422
R = 0.654
RMSE = 0.332

0.003
0.001
0.001 0.003

0.01

0.03

0.1

0.3

Measured a ph (412) [m

-1

1

1
0.3
0.1

0.03
N = 656n = 6422
R = 0.653
RMSE = 0.325

0.01
1:1
0.003
0.003

3

]

0.01

0.03

0.1

Measured0.3

1adg (412) [m

-1

3

]

1:1

1
0.3
0.1
N = 656n = 6422
R = 0.872
RMSE = 0.163

0.03
0.01
0.01

0.03

0.1

Measured

0.3

3

1

a(412) [m-1]

Derived a(490) [m -1 ]Derived a(412) [m -1 ]

3
3

1:1

1

0.3
0.1
N = 656n = 6422
R = 0.804
RMSE = 0.168

0.03

0.03

0.1

Measured

0.3

a(490) [m

1
-1

3

]

Figure 7.2 Comparison between retrieved and in situ IOPs. IOPs wereretrieved using Rrs values at 410, 490 and 555 nm.

spectral slope coefficient, S, should be varied with type of water mass, but at
this point, no parameterization of S with an a priori data value has improved
the retrievals of adg .
As shown in Figures 7.1 and 7.2, the algorithm has been optimized to retrieve values at 412nm. The IOPs derived at 412 nm can be translated to any
other wavelength through the IOP spectral models but the accuracy of the values
derived at the other wavelengths will be driven by how well the spectral models
reflect the true characteristics of the in-water constituents, which is true of any
spectral model-based algorithm. Methods developed by Wang et al. (2005) could
be implemented to help describe the range of equally valid retrievals of IOPs.
The need for the 412 nm band to separate the CDOM absorption from thephytoplankton absorption exposes the algorithm to potentially large errors in
input Rrs (412), caused by the fact that in coastal regions accurate atmosphericcorrection at the shorter wavelengths is very difficult to achieve. This is a fundamental problem for all semi-analytical algorithms that attempt to use Rrs (412)
to separate phytoplankton and CDOM absorption coefficients.

7.6.3

Algorithm strengths

There are also several advantages gained by using the exact linear inversion
approach. The most important feature is that the algorithm limits errors in IOP
spectral models by using wavelengths where each IOP tends to co-vary. This

56•

Remote Sensing of Inherent Optical Properties

approach is selected to optimize the algorithm for application to global data
sets. Hyper-spectral data was tested and it was found that the best agreement
between the retrieved IOPs and truth data was achieved by using the fewest
number of co-varying wavelengths possible. Therefore, the SeaWiFS bands, 412,
490 and 555 nm, or the closest to those bands on other sensors, are used in thisalgorithm. By using this simplified approach, we sacrifice deriving information
about more constituents in the water but minimize the errors caused by poor
spectral models to describe highly variable portions of the IOP spectra, and great
variability of water mass types on a global scale.
The inversion is computationally fast and no iteration is needed. Large data
set processing is limited more by the rate of data to be read and written to adisk, than by the computation of the IOP outputs.
Inputs from several different satellites that are contemporaneous and geographically coincident can be used in the inversion to produce a multi-satellite
blended product. In this case, the same three wavelengths (or similar wavebands) are used from each satellite to realize the benefit of averaging out theasymmetrical errors in Rrs in an over determined linear inversion, while still
maintaining the inter-wavelength co-variance for each IOP.
With fewer spectral model parameters to adjust, the algorithm is easy to tune
given known or expected values. This allows for tuning of the algorithm to specific regions where characteristics of constituents in the water are constrained
temporally and spatially, so regionally optimized versions of the algorithm canbe developed, without changing the core mathematical implementation.

Chapter 8Over Constrained Linear Matrix Inversion with
Statistical Selection
Emmanuel Boss and Collin Roesler

8.1 General Description
Semi-analytic inversions of remotely-sensed reflectance have been available
since 1995 (Roesler and Perry, 1995). However, a procedure that provides anuncertainty of the inverted parameter for each individual spectrum based on
uncertainties in the remote-sensing data and the model has only recently beendevised (Wang et al., 2005).
We use the same model philosophy as in Wang et al. (2005) with a slight
modification (we use a single phytoplankton absorption spectrum). We assume
a known relationship between rrs and the absorption and backscattering coefficients (Gordon et al., 1988):
rrs (λ) =2Lu (λ, 0− )bb (λ)bb (λ)=
0.0949+0.0794.
Ed (λ, 0− )a(λ) + bb (λ)a(λ) + bb (λ)(8.1)

The quadratic form is important for high rrs (λ) values (Garver and Siegel, 1997).
Gordon et al. (1988) estimated that the model errors in Equation 8.1 are less than10%.
The total absorption coefficient is partitioned as follows:a(λ) = aw (λ) + aph (λ) + adg (λ),(8.2)

where the subscripts “w”, “ph”, and “dg” designate sea water, phytoplankton,
and the combined contribution of CDOM and detrital material. The spectral
absorption coefficient for sea water, aw (λ), is computed for given salinity andtemperature based on Pope and Fry (1997) and Pegau et al. (1997).
The spectral absorption coefficient of phytoplankton is assumed to be:aph (λ) = aph (λ0 )a+ph (λ),(8.3)

where a+ph (λ) is an average of normalized phytoplankton absorption spectra(Roesler and Perry, 1995) and λ0 is commonly set as 440 nm.
57

58

•

Remote Sensing of Inherent Optical Properties

The spectral absorption coefficient of the combined absorption by CDOM anddetritus is:adg (λ) = adg (λ0 ) exp(−S(λ − λ0 )),(8.4)
where S is the spectral slope of the combined absorption coefficient. This function has been found to be an adequate representation of measured CDOM anddetritus absorption coefficient with S ranging between 0.008 to 0.023 nm−1 (e.g.,
Roesler et al., 1989).
The total backscattering coefficient, bb (λ), is approximated bybb (λ) = bbw (λ) + bbp (λ).

(8.5)

The spectral backscattering coefficients of sea water (bbw (λ)) are computed fora given salinity based on the interpolation of the data of Morel (1974) as in Bossand Pegau (2001).
The spectral particle backscattering coefficient is assumed to obey:
bbp (λ) = bbp (λ0 )(λ/λ0 )−Y .

(8.6)

This formulation is consistent with many previous studies, though without inwater validation.
To account for variability in space and time of the spectral shapes of the
IOPs we perform the rrs inversion allowing the shape parameters (spectral slope
S and spectral slope Y ) to vary within most of their observed range of variability(0.01 ≤ S ≤ 0.02, 0 ≤ Y ≤ 2). For each parameter we use 11 different values withequal intervals between their maximum and minimum, resulting in 112 = 121different inversion computations for each rrs .
It can be shown that with known spectral shapes, Equation 8.1 can be solved
to obtain bbp (λ0 ), adg (λ0 ), and aph (λ0 ) using a linear matrix inversion technique(Hoge and Lyon, 1996). When the number of wavelengths exceeds the numberof unknowns (3 in our case), this solution is the best solution in a least-square
sense (Press, 1992).
From all the solutions to Equation 8.1 we select the solution for whichadg (440) and aph (440) > −0.005 m−1 and bbp (440) > −0.0001 m−1 (slightly
negative values are accepted to compensate for finite uncertainties in measurements and calibrations). We further restrict ourselves to the solutions whose
reconstructed rrs (calculated by substituting the solutions into Equation 8.1)
obeys:|rrs,reconstucted (λ) − rrs,known (λ)|/rrs,known (λ) < 0.1 or 0.2 for every λ.
These criteria can result in cases where no solution could be found for a given
rrs . The choice of the criteria should be driven by knowledge of uncertainties
in observed rrs as well as the assumed spectral shapes (in particular that ofphytoplankton).

Over Constrained Linear Matrix Inversion with Statistical Selection• 59

We thus present the results from the two different solution selection criteria
in the tables, but only the criteria of 0.1 in the plots. We provide uncertainties
for the solutions on the plots based on the distance between the 84th and 16th
percentile of the obtained solutions (∼ ± one standard deviation for a normal
distribution).
Given the application to remote sensing we used only the Rrs values at 410,
440, 490, 510 and 550 nm (or nearby for the in situ data set).

Figure 8.1 Comparison of inverted and the simulated data set (Sun at
30o from zenith) for aph (440), adg (440), a(440), and bbp (440) for the 10%
criteria (statistics in Table 8.1). Vertical lines denote the 90% confidence
intervals in the solutions. Rrs values at 410, 440, 490, 510 and 550 nm
were used as inputs for IOP retrieval.

8.2

Results and Discussion with IOCCG Data Sets

8.2.1

Simulated data set

Over the large dynamic range of the data set the inversion fares rather well forboth 10 and 20% criteria (Figure 8.1, Tables 8.1 and 8.2). Not surprisingly thestringent criteria provide less but better solutions (in terms of RMSE error andbias). The agreement between derived and known IOPs can be further improved

60•

Remote Sensing of Inherent Optical Properties

Table 8.1 RMSE and regression (Type II) results for the synthesized data
set. Statistics of comparison of the median of all possible inversion solutions with a 10% agreement criterion. Rrs values at 410, 440, 490, 510
and 550 nm were used as inputs for IOP retrieval. N is the number of datatested, while n is the number of valid retrieval.
Nnintercept

slope

R2

RMSE

bias

a(440)

500

408-0.001

0.995

0.966

0.106

0.003

bbp (440)

500

408

-0.055

0.972

0.935

0.125

0.003

adg (440)

500

408

-0.021

0.982

0.956

0.141

-0.001aph (440)

500

408

0.160

1.113

0.927

0.159

-0.002

Table 8.2 RMSE and regression (Type II) results for the synthesized data
set. Statistics of comparison of the median of all possible inversion solutions with a 20% agreement criterion. Rrs values at 410, 440, 490, 510
and 550 nm were used as inputs for IOP retrieval. N is the number of datatested, while n is the number of valid retrieval.
Nnintercept

slope

R2

RMSE

bias

a(440)

500

438-0.074

0.938

0.946

0.145

-0.025aph (440)

500

438

-0.025

1.014

0.878

0.201

-0.044adg (440)

500

438

-0.082

0.942

0.944

0.169

-0.023bbp (440)

500

438

-0.186

0.925

0.898

0.168

-0.034by choosing other wavelengths (e.g. 410 nm for adg and 550 nm for bbp ) andby adding more wavelengths (e.g. Wang et al. (2005) added a 670 nm channel
and the successful retrieval increased from 408 to 472 with the 10% criteria).
It is encouraging that the uncertainty estimates for both adg (440) and bbp (440)intersect the 1:1 line suggesting the constraint criteria is working well.

8.2.2

In situ data set

Large uncertainties in inverted parameters (in particular aph ) suggest that someof these data have many possible solutions and thus large uncertainties for agiven Rrs (λ). Some data points are way off the line, possibly due to large sunangles and/or poor measurements (Figure 8.2 and Tables 8.3 and 8.4).
In Wang et al. (2005) we used a more complicated phytoplankton absorption
formulation which increased the computation by a factor larger than 10. We
found that this complexity did not improve the match ups significantly and thus
decided here to use a single phytoplankton absorption function. It can easily bedemonstrated that a different choice of wavelengths for inversions or a different
choice of wavelength for the parameter can significantly improve/degrade theretrieval. Thus, if we are after adg , inverting a near UV wavelength provides thebest inversion; while for bb , it is in the NIR that the inversion does best; as long

Derived a ph (443) [m -1 ]1
0.3

N = 656n = 5042
R = 0.613
RMSE = 0.285

1:1

0.1
0.03
0.01
0.003
0.001
0.001 0.003

0.01

Measured

0.03

0.1

a ph (443) [m0.3-1

Derived a dg (443) [m -1 ]Over Constrained Linear Matrix Inversion with Statistical Selection

0.1
0.03
N = 656n = 5042
R = 0.705
RMSE = 0.259

0.01
0.003
0.003

1

1:1

0.3

0.01

0.03

Measured

]

0.1

0.3

a dg (443) [m

1
-1

]

3
1:1

1

0.3
0.1
N = 656n = 5042
R = 0.872
RMSE = 0.146

0.03
0.03

0.1

Measured

0.3

1a(412) [m

-1

3

]

Derived a(443) [m -1 ]3Derived a(412) [m -1 ]

1

• 61

1:1

1

0.3
0.1
N = 656n = 5042
R = 0.849
RMSE = 0.150

0.03
0.03

0.1

Measured0.3

1a(443) [m

-1

3

]

Figure 8.2 Comparison of inverted and the in situ data set (Sun at 30ofrom zenith) for aph (440), adg (440), a(440), and bbp (440) for the 10% criteria(statistics in Table 8.3). Vertical lines denote the 90% confidence intervals
in the solutions. Rrs values at 410, 440, 490, 510 and 555 nm were used as
inputs for IOP retrieval.

as adequate Rrs at those wavelengths could be available.

8.3

Summary

The inversion method presented here was designed to provide uncertainty estimates of inversion products and is dependent on the reality of the assumptionsof the model. For example, it is well known that Equation 8.6 is likely not a good
representation of particulate spectral backscattering, yet it is the only simple
model currently available. Much work is still needed to understand spectral
IOPs, and such work will, without a doubt, improve our ability to retrieve inwater parameters from remote sensing.

62•

Remote Sensing of Inherent Optical Properties

Table 8.3 RMSE and regression (Type II) results for the in situ data set.
Statistics of comparison of the median of all possible inversion solutions
with a 10% agreement criterion. Rrs values at 412, 443, 490, 510 and 555
nm were used as inputs for IOP retrieval. N is the number of data tested,while n is the number of valid retrievals.
Nnintercept

slope

R2

RMSE

bias

a(412)

656

504-0.022

0.942

0.872

0.146

0.029

a(443)

656

504-0.029

0.969

0.849

0.150

0.001

adg (443)

656

504

-0.018

1.043

0.705

0.259

-0.072aph (443)

656

504

0.068

1.031

0.613

0.285

0.024

Table 8.4 RMSE and regression (Type II) results for the in situ data set.
Statistics of comparison of the median of all possible inversion solutions
with a 20% agreement criterion. Rrs values at 412, 443, 490, 510 and 555
nm were used as inputs for IOP retrieval. N is the number of data tested,while n is the number of valid retrievals.
Nnintercept

slope

R2

RMSE

bias

a(412)

656

629-0.036

0.939

0.867

0.157

0.019

a(443)

656

629-0.039

0.977

0.842

0.165

-0.017adg (443)

656

629

0.086

1.069

0.63

0.298

-0.013aph (443)

656

629

-0.057

1.014

0.714

0.266

-0.075

Chapter 9
MODIS Semi-Analytic Algorithm for IOP
Kendall Carder, Jennifer Cannizzaro, Robert Chen and ZhongPing Lee

9.1 Introduction
The Moderate-Resolution Imaging Spectrometer (MODIS) semi-analytic algorithm (Carder− MODIS here after) (Carder et al., 1999; Carder et al., 2004) derives
chlorophyll-a concentrations and inherent optical properties (aph (λ), adg (λ) andbbp (λ)) from remote-sensing reflectance spectrum (Rrs (λ)). This algorithm iscomposed with an algebraic portion and an empirical portion. The algebraic
portion is for waters with low absorption (mostly oceanic waters) while theempirical portion is for waters with high absorption (mostly coastal waters).
One of the main characteristics of this algorithm is that it responds to thelarge global variability observed in both chlorophyll-specific absorption coefficients (a∗ph (λ)), as well as gelbstoff-to-phytoplankton absorption ratios. Thisalgorithm utilizes differences between measured sea-surface temperatures andknown nitrate-depletion temperatures (NDT) ( Kamykowski and Zentara, 1986;
Kamykowski, 1987) to select the most appropriate a∗ph (λ) for a given bio-opticaldomain. The algorithm was first developed and evaluated using high-light, tropical/subtropical and summer temperate field data (Carder et al., 1999) and later
expanded to include parameters appropriate for low-light, polar data (Carder etal., 2004).

9.2

Algorithm Description

9.2.1

Remote-sensing reflectance model

By making several approximations, the Rrs (λ) used in Carder− MODIS algorithm
is simplified to (Carder et al., 1999)
Rrs (λ) ≈ constantbb (λ)
,
a(λ)

(9.1)

where the “constant" is unchanging with respect to wavelength and solar zenith
angle. The value of the constant is not relevant to the algorithm since, as will
63

64

•

Remote Sensing of Inherent Optical Properties

be shown later, the algorithm (for absorption and chlorophyll-a concentration)
uses spectral ratios of Rrs (λ) and the constant term factors out.
Further, both bb (λ) and a(λ) are partitioned into several separate terms.
Each term is described empirically and is written in a general fashion as a function of variables and empirically derived parameters. Since sea-surface temperatures were not provided in the IOCCG data sets, the unpackaged parameters
regarding a∗ph (λ) derived from high-light, tropical/subtropical and summer temperate waters were employed (Carder et al., 1999) (see Table 9.1). While a∗ph (λ)
is extremely important for deriving chlorophyll-a concentrations accurately, retrievals of aph (λ) and adg (λ) are less sensitive to differences in a∗ph (λ).
Table 9.1 Parameters for the MODIS semi-analytical algorithm for regions
without packaged pigments
λ

a0

a1

412

2.20

0.75

443

3.59

0.80

488

2.27

0.59

551

0.42

-0.22

a2

a3

X0

X1

Y0

Y1

S (nm−1 )0.5

0.0112-0.00182

2.058

-1.13

2.57

0.02259.2.2 Backscattering coefficients
The total backscattering coefficient, bb (λ), can be expanded asbb (λ) = bbw (λ) + bbp (λ),(9.2)

with bbp (λ) modelled as (Carder et al., 1999)

bbp (λ) = X

551
λ

Y
.

(9.3)

bbw (λ) is constant (Morel, 1974). X is the particulate backscattering at 551
nm, and Y describes the spectral shape of the particle backscattering spectrum.
Values for X and Y were determined empirically by model inversion (Carder etal., 1999) and are described as
X = X0 + X1 Rrs (551),(9.4)

Rrs (443),
Rrs (488)(9.5)

Y = Y0 + Y1

where X0,1 and Y0,1 are empirically derived constants (Carder et al., 1999) andare provided in Table 9.1.
When absorption due to water molecules does not dominate the total absorption coefficient at 551 nm, algorithms that utilize wavelengths longer than

MODIS Semi-Analytic Algorithm for IOP• 65

551 nm, that take advantage of the larger inflection in the pure water absorption
spectra between 570-610 nm (Pope and Fry, 1997), are required. Using measurements of Rrs (λ) and bbp (λ) collected from the West Florida Shelf, the equationbbp (551) = 10(0.933−0.134 log(Rrs (551))+1.029 log(Rrs (667))) − 0.000966,(9.6)(n=154, r2 = 0.96, RMSE = 0.160) was derived for MODIS-like wavelengths. Thisfunction was used when the Carder− MODIS algorithm was applied to the IOCCG
synthetic data set. Since remote-sensing data with wavelengths longer than 555
nm were not available for the IOCCG in situ data set, however, bbp (551) values
were then estimated using Equation 9.4.

9.2.3

Absorption coefficients

The total absorption coefficient, a(λ), can be expanded asa(λ) = aw (λ) + aph (λ) + adg (λ),(9.7)

with values of aw (λ) taken from Pope and Fry (1997).
The shape of the aph (λ) spectrum for a given water mass changes due to thepigment-package effect and changes in pigment composition. For the MODIS
wavebands centered at 412, 443, 488, and 551 nm, a hyperbolic tangent function was chosen to empirically model the ratio of aph (λ)/aph (675) in order toensure that this ratio approaches an asymptote at very high or very low valuesof aph (675) (Carder et al., 1999),




aph (λ) = a0 (λ) exp a1 (λ) tanh a2 (λ)ln aph (675)/a3 (λ)
aph (675),(9.8)

with values of a0−3 (λ) provided in Table 9.1.
The cumulative effects of detritus and gelbstoff absorption, adg (λ), are expressed asadg (λ) = adg (400) exp(−S(λ − 400)),(9.9)
where S is the spectral slope, and a value of 0.0225 nm−1 provided optimal
retrieval results for the Carder− MODIS algorithm to calculate chlorophyll-a concentrations (Carder et al., 1999). It is larger than the mean ocean value of about
0.015 nm−1 , likely compensating in part for uncertainties in other parts of themodel.

9.2.4

Model inversion

Via Equations 9.1 – 9.9, Rrs (λ) is reduced to a function of three unknowns (“constant" term, aph (675), and adg (400)) along with model constants for X0,1 , Y0,1 ,a0−3 (λ), and S (Table 9.1). To algebraically solve for the values of the two desired

66•

Remote Sensing of Inherent Optical Properties

unknowns (aph (675) and adg (400)), spectral ratios of 412/443 and 443/551 forRrs (λ) as shownb (412)Rrs (412)
Rrs (443)= bbb (443) a(443)a(412) ,Rrs (443)
Rrs (551)

b (443) a(551)= bbb (551) a(443) ,(9.10)

provided the best separation of the two absorption contributions. Details on thecomputational method of solving these equations are discussed in Carder et al.
(1999).

9.2.5

Empirical portion of Carder− MODIS

For waters with high concentrations of gelbstoff and chlorophyll, Rrs (412) andRrs (443) values are small, and therefore the above semi-analytical approach cannot perform properly due to low signal-to-noise ratios. Thus the semi-analytic
approach is designed to return values only when modelled aph (675) values areless than 0.025 m−1 , which is equivalent to a chlorophyll-a concentration of
about 1.5 mg m−3 . Otherwise, the following empirical algorithms derived fromthe West Florida Shelf (1999-2001) and Bayboro Harbor (St. Petersburg, Florida)field data (n = 319) are used.
For aph (443), there is22

aph (443)emp = 10(−1.164−1.2095ρ35 −1.566ρ35 −1.708ρ45 +19.502ρ45 ) ,(9.11)

where ρij is the log-transformed ratio of Rrs (λi ) to Rrs (λj ) and the subscripts iand j are wavebands #1-6 that represent MODIS wavebands 412, 443, 488, 531,
551, and 667 nm, respectively. Since this equation requires the MODIS Rrs (531)
waveband and the SeaWiFS waveband Rrs (510) was provided instead with the
IOCCG in situ data, a modified SeaWiFS algorithm was also developed22

aph (443)emp = 10(−1.189−1.33ρ35 −2.151ρ35 −0.775ρ45S +7.592ρ45S ) ,and applied to the IOCCG in situ data set.
log(Rrs (510)/Rrs (555)).
The empirical algorithm for adg (443) is2(9.12)

Here ρ45S is equal to2adg (443)emp = 10(−1.144−0.738ρ15 −1.386ρ15 −0.644ρ25 +2.451ρ25 ) ,(9.13)

and was applied to the IOCCG in situ data set. Since adding a ρ65 term reduced
the RMSE error by 40% for calculating adg (443) for the West Florida Shelf andBayboro Harbor data, the derived equationadg (443)emp = 10(0.043−0.185ρ25 −1.081ρ35 +1.234ρ65 ) ,(9.14)

MODIS Semi-Analytic Algorithm for IOP• 67

was applied to the synthetic data set where Rrs (670) data (considered equal toRrs (667)) were available.
Empirical retrievals of a(λ) at 412, 443, and 488 nm also improved for theWest Florida Shelf and Bayboro Harbor data set when a red reflectance waveband
was included. Thus, the empirical expression derived from field data and applied
to the synthetic data set takes the forma(λ)emp = 10(c0 (λ)+c1 (λ) log(Rrs (443))+c2 (λ) log(Rrs (488))+c3 (λ) log(Rrs (667))) ,(9.15)

where c0−3 (λ) are empirically derived parameters (Table 9.2a). Note that whilereflectance ratios are used to calculate aph (443)emp and adg (443)emp , reflectancevalues are used to calculate a(λ)emp in Equation 9.15. For the IOCCG in situ
data set that does not have a red reflectance waveband, an empirical expression
similar to that of Lee et al. (1998b)22a(λ)emp = 10(t0 (λ)+t1 (λ)ρ25 +t2 (λ)ρ25 +t3 (λ)ρ35 +t4 (λ)ρ35 ) ,(9.16)

was developed with t0−4 (λ) (Table 9.2b) also derived from the West Florida Shelf
and Bayboro Harbor data.
Table 9.2a Wavelength-dependent parameters for the high-absorption
empirical a(λ) algorithm (Equation 9.15) that requires Rrs (670).
c0 (λ)c1 (λ)c2 (λ)c3 (λ)

a(412)-0.349-1.041

0.171

0.754

a(443)-0.1660.068-1.284

1.077

a(488)-0.1670.478-1.6391.075

Table 9.2b Wavelength-dependent parameters for the high-absorption
empirical a(λ) algorithm (Equation 9.16) that does not require Rrs (670).
t0 (λ)t1 (λ)t2 (λ)t3 (λ)t4 (λ)

a(412)-0.640-0.718-0.650-1.3652.369

a(443)-0.837-0.860-0.791-1.1622.855

a(488)-0.947-0.343-0.721-1.6332.741

9.2.6 Blending semi-analytic and empirical IOP values
In order to provide a smooth transition in modelled IOP values when the algorithm switches from the semi-analytical to the empirical method, a weighted
average of the modelled values returned by both algorithms is used near thetransition border (Carder et al., 1999). When the semi-analytical portion returns
an aph (675) value between 0.015 and 0.025 m−1 , IOP values are calculated as
IOP = w(IOP )sa + (1 − w)(IOP )emp ,(9.17)

68•

Remote Sensing of Inherent Optical Properties

where (IOP)sa is the semi-analytically-derived value, (IOP)emp is the empiricallyderived value, and w is the weighting factor equal to (0.025 − aph (675))/0.010.
Semi-analytical and empirical IOP values are used when modelled aph (675) values are less than 0.015 m−1 and greater than 0.025 m−1 , respectively. Note that
this transition range can vary with pigment packaging (e.g., see Carder et al.,2004).

9.3

Algorithm Performance with the IOCCG Data Sets

The Carder− MODIS algorithm requires Rrs (λ) data at a minimum of five wavebands: 412, 443, 488, 531 and 551 nm. Further inclusion of the Rrs (667) improves retrievals of adg (443)emp (Equation 9.14) and a(λ)emp (Equation 9.15)
values. Since the synthetic Rrs (λ) data was generated in 10 nm increments from
400–800 nm, reflectance values at 410, 440, 490, 530, 550 and 670 nm wereconsidered similar enough to the MODIS wavebands and were input into the algorithm. For the IOCCG in situ data set only Rrs (λ) data at 412, 443, 490, 510,
and 555 nm were input into the equations.

9.3.1

Synthetic data set

Using the Carder− MODIS algorithm, the inherent optical properties a(410),a(440), a(490), aph (440), adg (440) and bbp (550) were derived from the synthetic Rrs (λ) data (Figure 9.1). Statistical analyses were performed on logtransformed data and include the slope, intercept, correlation of determination(R 2 ) and the root-mean-square error (RMSE) (Table 9.3).
Table 9.3 RMSE and regression (Type II) results of the synthetic data set(θ0 = 30◦ ). Rrs (λ) values at 410, 440, 490, 530, 550 and 670 nm were used
as inputs. N is the number of data tested, while n is the number of valid
retrievals.
Nnintercept

slope

R2

RMSE

bias

a(410)

500

500

0.015

0.990

0.990

0.071

0.020

a(440)

500

500

0.030

1.030

0.993

0.059

0.010

a(490)

500

500

0.079

1.082

0.993

0.065

0.008
-0.008bbp (550)

500

500

-0.012

0.998

0.995

0.042

aph (440)

500

500

-0.046

0.908

0.963

0.141

0.071

adg (440)

500

500

0.084

1.098

0.978

0.135

-0.004

Particulate backscattering coefficients at 550 nm retrieved using Equation
9.6 from Rrs (550) and Rrs (670) are very accurate (RMSE = 0.042). Total absorption coefficients at 410, 440 and 490 nm were also retrieved accurately forthe synthetic data set with RMSE errors equal to 0.071, 0.059 and 0.065, re-MODIS Semi-Analytic Algorithm for IOP

• 69

1

3

0.03
N = 500
n = 500
2
R = 0.963
RMSE = 0.141

0.01

0.003
0.003

0.01

0.03

0.1

Known a ph (440) [m

0.3
-1

0.3
0.1
N = 500
n = 500
2
R = 0.993
RMSE = 0.059

0.03

0.1

0.3

Known a(440) [m

3

1
-1

0.1
0.03
N = 500
n = 500
2
R = 0.978
RMSE = 0.135

0.01

0.01

0.03

]

0.1

0.3

1

3

Known a dg (440) [m -1 ]

1

0.01
0.01

0.3

]

0.1

1:1

0.03

1:1

1

0.003
0.003

1

3

Derived a(440) [m -1 ]Derived a dg (440) [m -1 ]0.1

Derived b bp (550) [m -1 ]Derived a ph (440) [m -1 ]

1:1
0.3

0.03
0.01
0.003
N = 500
n = 500
2
R = 0.995
RMSE = 0.042

0.001
1:1

0.00030.0003

0.001

0.003

0.01

0.03

Known b bp (550) [m

-1

0.1

]

Figure 9.1 Relationships between known and retrieved IOPs using theCarder− MODIS algorithm (synthetic data set with Sun at 30◦ from zenith),with Rrs (λ) at 410, 440, 490, 530, 550 and 670 nm used as inputs. Symbols:
semi-analytic (o) and empirical (∆).

70•

Remote Sensing of Inherent Optical Properties

spectively. RMSE values for aph (440) (0.141) and adg (440) (0.135) are slightly
more than double the error calculated for a(440) since phytoplankton and detritus/gelbstoff exhibit overlapping absorption spectra making it difficult to separate them.

9.3.2

In situ data set

The results of the Carder− MODIS algorithm when applied to the IOCCG in situ
data set were not as good as the results observed for the synthetic data set
because errors in field Rrs and IOP data, not present in the synthetic data, aresignificant in the in situ data.
Total absorption coefficients at 412, 443, and 488 nm derived from the insitu Rrs (λ) yielded RMSE errors of 0.197, 0.205 and 0.206, respectively (Figure
9.2, Table 9.4). Errors for aph (443) and adg (443) were only slightly higher thana(443) and were 0.195 and 0.279, respectively. While the semi-analytic aph (443)values derived from synthetic Rrs (λ) data were overestimated, values derived
from the in situ Rrs (λ) data were more centered about the one-to-one line. This
may indicate that perhaps the underlying aph (λ) functions used to generate
the synthetic data for oligotrophic waters are not quite representative of the
distribution of the naturally occurring aph (λ) data, or at least Equation 9.8 ismore consistent with the aph (λ) functionality of the in situ data set than withthat of the synthetic data set.
Table 9.4 RMSE and regression (Type II) results of the in situ data set.
Rrs (λ) values at 412, 443, 490, 510 and 555 were used as inputs. N is thenumber of data tested, while n is the number of valid retrievals.
Nnintercept

slope

R2

RMSE

bias

a(412)

656

656

0.098

1.066

0.826

0.197

0.039

a(443)

656

656

0.030

1.111

0.831

0.205

-0.078a(488)

656

656

0.131

1.173

0.789

0.206

-0.063aph(443)656

656-0.052

0.986

0.827

0.195

-0.032adg(443)656

656-0.041

1.082

0.771

0.279

-0.144Large errors that occur in empirically derived a(412), a(443), and aph (443)values and that appear as linear horizontal rows of data in Figure 9.2 at ∼0.23,
0.15, and 0.07 m−1 , respectively, can be traced to a single investigator for alarge multi-year, coastal data set. Removal of these points would improve theperformance of the empirical portion of our algorithm. Furthermore, empirical
retrievals of a(λ) and adg (440) may also be improved for this data set if Rrs (λ)
data were available for wavelengths longer than 555 nm.

MODIS Semi-Analytic Algorithm for IOP

3Derived a dg (443) [m -1 ]Derived a ph (443) [m -1 ]

1

• 71

0.3
0.1
0.03
0.01
0.003

N = 656n = 6562
R = 0.827
RMSE = 0.195

1:1

0.003

0.01

0.03

0.1

0.3

Measured a ph (443) [m

-1

1:1

1
0.3
0.1
0.03

0.003
0.003

1

N = 656n = 6562
R = 0.771
RMSE = 0.279

0.01

0.01

0.03

0.1

0.3

1

Measured a dg (443) [m

]

-1

3

]

3

1
0.3
0.1
N = 656n = 6562
R = 0.826
RMSE = 0.197

0.03
0.01
0.01

1:1

0.03

0.1

0.3

Measured a(412) [m

1
-1

3

]

Derived a(443) [m -1 ]Derived a(412) [m -1 ]

3
1

0.3
0.1
N = 656n = 6562
R = 0.831
RMSE = 0.205

0.03
1:1

0.01
0.01

0.03

0.1

1

0.3

Measured a(443) [m

-1

3

]

Figure 9.2 Relationships between measured and retrieved IOPs using theCarder− MODIS algorithm (in situ data set), with Rrs (λ) at 412, 443, 490, 510
and 555 nm used as inputs. Symbols: semi-analytic (o) and empirical (∆).

9.4 Conclusions
The Carder− MODIS algorithm (Carder et al., 1999) calculated bbp (550) anda(λ) values very accurately for the synthetic data set. Values for aph (443)
and adg (443) were calculated less accurately because phytoplankton and detritus/gelbstoff exhibit overlapping absorption making it more difficult to separate
them using Rrs (λ). Retrieval errors tripled for a(λ) and doubled for adg (443)when the algorithm was applied to the in situ data set as compared to the synthetic data set. The fact that the partitioned values fell within the same error
range as the total-absorption values suggests that much of the error imputed tothe algorithms for the in situ data set may be attributable to errors or inconsistencies among the measured data sets, whereas the synthetic data set had nomeasurement noise.
IOP retrieval errors calculated for the in situ data set may improve if Rrs (667)
data were available. Significant error reductions were observed for empiricallyderived backscattering and total absorption coefficients when red reflectancedata were used for our high-absorption Florida data set and for the synthetic
data set. Note, however, that while Rrs (667) can be used for “perfect" synthetic
data, accurate measurements of Rrs (667) from space are much more subject toerror due to smaller signal-to-noise ratios. A waveband near 610–620 nm would
perhaps be a better compromise than the use of 667 nm for satellites. The

72•

Remote Sensing of Inherent Optical Properties

Medium Resolution Imaging Spectrometer (MERIS) has such a waveband around
620 nm.
Finally, the expansion of available global data sets in the past 10 years andthe broad range of data synthesized in the numerical data set have provided
examples of how various older algorithms may be improved, and we are gratefulfor being included in this challenging algorithm inter-comparison.

Chapter 10
The Quasi-Analytical Algorithm
ZhongPing Lee, Kendall Carder and Robert Arnone

10.1

General Description

The Quasi-Analytical Algorithm (QAA) was developed by Lee et al. (2002) to derive inherent optical properties of optically deep waters. QAA separates the inversion process into two consecutive sections. The first section is the derivationof coefficients of total absorption and backscattering. In this section, there isno involvement of spectral models for the absorption coefficient of phytoplankton pigments and gelbstoff. The second section, which utilizes the derived total
absorption coefficient from the first section, decomposes the total absorption
coefficient into its major components.

10.2

Derive Total Absorption and Backscattering Coefficients

In this part, QAA follows the generally accepted relationship between remotesensing reflectance and bb /(a + bb ), and the fact that water absorption coefficients dominate most of the longer wavelengths. Here bb is the total backscattering coefficient and a is the total absorption coefficient. QAA starts with thecalculation of a at a reference wavelength (λ0 , 555 or 640 nm), with the assumption that remote-sensing reflectance at this wavelength is well measured from aremote-sensing platform.
The total absorption coefficient at λ0 is expressed asa(λ0 ) = aw (λ0 ) + ∆a(λ0 ),(10.1)

where aw (λ0 ) is the contribution from water molecules (Pope and Fry, 1997), and∆a(λ0 ) represents the contribution from dissolved and suspended constituents.
For this a(λ0 ), errors in its estimation are limited as long as aw (λ0 ) makes up a
big portion (at least one third of the total).
Lee et al. (2002) proposed two λ0 for dealing with IOP inversion: 555 nm foroceanic and most coastal waters and 640 nm for waters with high absorption
73

74

•

Remote Sensing of Inherent Optical Properties

coefficients (a(440) >∼ 0.5 m−1 ). For each λ0 , there could be many ways toestimate a(λ0 ). In the exercise reported here, when 555 nm is selected as λ0 ,a(555) is estimated using the Morel-Maritorena approach (Morel and Maritorena,2001) as described in Chapter 4. That is, Kd (555) is estimated first with ρ =log(max(Rrs (440, 490, 510))/Rrs (555)),
2

3

4

Kd (555) = 0.0605 + 10−1.163−1.969ρ+1.239ρ +0.417ρ −0.984ρ ,and thena(555) =0.9Kd (555)(1 − 6.8Rrs (555)).
1 + 15.3Rrs (555)(10.2)(10.3)For sensors such as MODIS where no 510 nm band exists, a slight adjustment
regarding Equation 10.2 is sufficient for the implementation of QAA.
When 640 nm is selected as λ0 , a(640) is estimated as in Lee et al. (2002),
i.e.,


rrs (640) 1.1,(10.4)a(640) = 0.31 + 0.07
rrs (440)
where rrs is the subsurface remote-sensing reflectance corresponding to the Rrs
measured above the surface.
To estimate a(640) requires measurements of remote-sensing reflectance at
640 nm, a band which does not exist in many satellite sensors (such as SeaWiFS).
To overcome this limitation, Rrs (640) is simulated with measurements made at490, 555 and 670 nm, as described in Lee et al. (2005b),
Rrs (640) = 0.01Rrs (555) + 1.4Rrs (670) − 0.0005Rrs (670)/Rrs (490).

(10.5)

Note that in Lee et al. (2005b) it is Rrs (667) for SeaWiFS spectral bands. This empirical formula was aimed to more or less correct the chlorophyll-a fluorescence
contained in Rrs (670).
rrs (λ) is calculated from Rrs (λ) throughrrs (λ) = Rrs (λ)/(0.52 + 1.7Rrs (λ)),(10.6)

where 0.52 and 1.7 are empirical values derived from data simulated by Hydrolight (Lee et al., 1999). Because rrs (λ) can be modelled as a polynomial functionof bb /(a + bb ) (Gordon et al., 1988; Lee et al., 1998a), bb /(a + bb ) (represented
as symbol u) at λ can be calculated algebraically from rrs (λ) (Hoge and Lyon,
1996; Lee et al., 2002),p
bb (λ)−0.0895 + 0.008 + 0.499rrs (λ)u(λ) ≡=.
a(λ) + bb (λ)
0.249(10.7)The spectral bb (λ) is modelled with the widely used expression (Smith andBaker, 1981; Gordon and Morel, 1983),
 Y
λ0bb (λ) = bbw (λ) + bbp (λ0 ),(10.8)
λThe Quasi-Analytical Algorithm• 75

where bbw and bbp are the backscattering coefficients of pure seawater and suspended particles, respectively. Values of bbw (λ) are provided in Morel (1974).
When a(λ0 ), u(λ0 ), and bbw (λ0 ) are known, bbp (λ0 ) in Equation 10.8 can beeasily derived. The values of bb (λ) at other wavelengths are then calculated when
the wavelength exponent (Y ) is estimated from Lee et al. (2002)
rrs (440)
Y = 2.2 1 − 1.2 exp −0.9.
(10.9)
rrs (555)
Finally, applying bb (λ) back to u(λ) (derived from rrs (λ), Equation 10.7), thetotal absorption coefficient at wavelength λ, a(λ), is calculated algebraically,a(λ) =(1 − u(λ))bb (λ).
u(λ)(10.10)To obtain smooth satellite IOP products where both 555 nm and 640 nm could
be used as reference wavelengths, the final a(λ) product is a combination of theabsorption coefficients derived using 555 nm as reference wavelength (a(λ)[555] )and 640 nm as reference wavelength (a(λ)[640] ), as follows:
a(λ) = a(λ)[555] ,for a(440)[555] < 0.3,






[555]


 a(λ) = 1 − a(440)0.2 −0.3 a(λ)[555] +a(440)[555] −0.3
a(λ)[640] ,for 0.3 ≤ a(440)[555] ≤ 0.5,

0.2




 a(λ) = a(λ)[640] ,for a(440)[555] > 0.5.
(10.11)Further, final bbp (λ) is recalculated using u(λ) and a(λ) based on Equations
10.7 and 10.8.

10.3

Decomposition of the Total Absorption Coefficient

Decomposition of a(λ) used the a(410) and a(440) values derived from theabove steps. In the process, two more parameters are estimated first. One is the
spectral ratio of aph (410)/aph (440) (represented by symbol ζ), while the otheris the spectral ratio of adg (410)/adg (440) (represented by symbol ξ). The valueof ζ is estimated using the spectral ratio of rrs (440)/rrs (555) based on the fielddata (Lee et al., 1998b):
ζ = aph (410)/aph (440) = 0.71 +0.06.
0.8 + rrs (440)/rrs (555)(10.12)The value of ξ is calculated after the spectral slope S (used to describe thespectral shape of adg (λ)) is selected (0.015 nm−1 is used in this exercise):ξ = adg (410)/adg (440) = exp(S(440 − 410)).

(10.13)

76•

Remote Sensing of Inherent Optical Properties

When the values of a(410), a(440), ζ and ξ are known, aph (440) and adg (440)are calculated algebraically, adg (440) = (a(410)−ζa(440)) − (aw (410)−ζaw (440)) ,ξ−ζ
ξ−ζ
(10.14)
 a (440) = a(440) − a (440) − a (440).
ph

10.4

w

dg

Results and Discussion

The above steps to retrieve IOPs from Rrs (λ) are applied to the IOCCG data
sets. For the synthetic data set, Rrs values at 410, 440, 490, 510, 555, and670 nm were used as indicated (Rrs (555) is a simple average of Rrs (550) andRrs (560)). For the in situ data set, however, only Rrs values at the first five
wavelengths were used as Rrs (670) is not available. The retrieved IOPs includea(λ), bbp (λ), aph (λ), and adg (λ) of those wavelengths. To provide a generalidea of the algorithm performance, some retrieved properties were comparedwith known values. Analysis results are presented in Tables 10.1 and 10.2 andthe figures that follow. Performance (not presented) for the synthetic data with
the Sun at 60◦ from zenith is similar to that with the Sun at 30◦ .
Table 10.1 RMSE and regression (Type II) results of the synthetic data set(θ0 = 30◦ ). IOPs were retrieved with Rrs values at 410, 440, 490, 510, 555
and 670 nm. N is the number of data tested, while n is the number of valid
retrievals.
Nnintercept

slope

R2

RMSE

bias

a(410)

500

500-0.001

0.971

0.993

0.061

0.016

a(440)

500

500-0.004

0.971

0.992

0.060

0.016

a(490)

500

500-0.041

0.959

0.989

0.062

-0.005bbp (440)

500

500

-0.099

0.945

0.981

0.081

0.008

bbp (555)

500

500

-0.063

0.983

0.986

0.079

-0.029adg (410)

500

476

-0.010

0.961

0.991

0.076

0.019
0.006

adg (440)

500

476

-0.032

0.960

0.985

0.093

aph (440)

500

476

0.044

1.008

0.930

0.160

0.033

aph (490)

500

476

0.013

1.010

0.825

0.257

-0.002For the synthetic data set, total absorption and backscattering coefficients areaccurately retrieved over the entire data range using the QAA algorithm (slope
and R 2 values are near 1.0 and RMSE values are 0.06 - 0.08). The performanceof the QAA for the in situ data set is not as good as that of the synthetic dataset (see Figure 10.2 and Table 10.2), which is not surprising considering theunavoidable errors and uncertainties (see Chapter 3) in the measurement ofboth Rrs (λ) and IOPs. The natural water environment is also far more complex
than that simulated with computer code. Nevertheless, for such an inclusive
data set, the RMSE values for a(λ) are ∼0.17.

The Quasi-Analytical Algorithm• 77

Table 10.2 RMSE and regression (Type II) results of the in situ data set.
IOPs were retrieved with Rrs values at 412, 443, 490, 510 and 555 nm. N is
the number of data tested, while n is the number of valid retrievals.
Nnintercept

slope

R2

RMSE

bias

a(412)

656

656-0.089

0.963

0.868

0.168

-0.055a(443)

656

656-0.081

0.969

0.840

0.175

-0.051a(490)

656

656

0.001

1.020

0.792

0.174

-0.021adg (412)

656

630

-0.092

0.986

0.820

0.209

-0.077adg (443)

656

630

-0.087

0.989

0.794

0.221

-0.072aph (443)

656

630

0.033

1.067

0.593

0.321

-0.062aph (490)

656

630

0.498

1.310

0.686

0.334

-0.007For both synthetic and in situ data sets, the retrieval of adg (λ) is only slightly
worse than the retrieval of total absorption coefficients, but more errors arefound in the derived aph (λ) (see Tables 10.1 and 10.2 and Figures 10.1 and10.2). This is, in part, because gelbstoff (including detritus) likely contributes
more to the total absorption coefficient at 410 and/or 440 nm.
3
1:1

0.3
0.1
0.03
N = 500n = 4762
R = 0.930
RMSE = 0.160

0.01
0.003
0.003

0.01

0.03

0.1

Derived a dg (440) [m -1 ]Derived a ph (440) [m -1 ]

1

1:1

1
0.3
0.1
0.03
N = 500n = 4762
R = 0.985
RMSE = 0.093

0.01
0.003

0.3

0.003

0.01

Known a ph (440) [m -1 ]

0.03

0.1

0.3

1

3

Known a dg (440) [m -1 ]

1:1

1
0.3
0.1
N = 500
n = 500
2
R = 0.992
RMSE = 0.060

0.03
0.01
0.01

0.03

0.1

0.3

1

Known a(440) [m -1 ]3Derived b bp (440) [m -1 ]Derived a(440) [m -1 ]

0.3
3

1:1

0.1
0.03
0.01
0.003

N = 500
n = 500
2
R = 0.981
RMSE = 0.081

0.001
0.001

0.003

0.01

0.03

0.1

0.3

Known b bp (440) [m -1 ]Figure 10.1 Comparison between QAA-derived IOPs and known IOPs, forthe synthetic data set (sun at 30◦ from zenith). IOPs were derived with
Rrs values at 410, 440, 490, 510, 555 and 670 nm as inputs (see text for
details).

Also, in the explicit decomposition of total a(λ) to aph (λ) and adg (λ), valuesof ζ and ξ are not known precisely, but have to be estimated. Errors in theseestimations will be propagated to the derived values of adg (440) and aph (440).

•

Remote Sensing of Inherent Optical Properties

Derived a ph (443) [m -1 ]1
0.3

N = 656n = 630
2
R = 0.593
RMSE = 0.321

Derived a dg (443) [m -1 ]

78

0.1
0.03
0.01
0.003

1:1

0.003

0.01

0.03

Measured

0.1

0.3

a ph (443) [m

-1

0.1
0.03
N = 656n = 630
2
R = 0.794
RMSE = 0.221

0.01
0.003
0.003

1

1:1

]

0.01

0.03

Measured

0.1

0.3

1

a dg (443) [m

-1

]

3

1
0.3
0.1
N = 656n = 6562
R = 0.868
RMSE = 0.168

0.03
1:1

0.01
0.01

0.03

0.1

Measured

0.3

a(412) [m

1
-1

]

3

Derived a(443) [m -1 ]3Derived a(412) [m -1 ]

1
0.3

1
0.3
0.1
N = 656n = 6562
R = 0.840
RMSE = 0.175

0.03
1:1

0.01
0.01

0.03

0.1

Measured

0.3

a(443) [m

1
-1

3

]

Figure 10.2 Comparison between QAA-derived IOPs and known IOPs, for
the in situ data set. IOPs were retrieved with Rrs values at 412, 443, 490,
510 and 555 nm as inputs.

Note that the value of ξ (directly related to the spectral slope of adg (λ)), as
observed in the field and represented in the synthetic data set, varies widelybased on the nature of waters under study (e.g. humic versus fulvic acids, Carder
et al., 1989), abundance of detritus (Roesler et al., 1989), but the present version
of QAA uses a fixed spectral slope for all cases. Also, QAA currently uses only
one spectral constraint regarding aph (λ) (ratio of aph (410)/aph (440)) in theprocess of decomposing a(λ). Due to errors in Rrs (λ) measurements as wellas errors in the selection of parameter S, negative aph (440) or aph (490) values
appeared (4.8% in synthetic data and 4.0% in the in situ data). Such retrievals
were then flagged and removed in the statistical analyses. This kind of obvious
error can be remedied by replacing with empirical estimates (Lee et al., 1998b),
or by adding more spectral constraints in the derivation of aph (λ) (e.g., the full
spectral models of aph (λ) used in other algorithms), although this will introduce
model uncertainties.
When Rrs (640) was not used in the derivation process (i.e. 555 nm alone asreference wavelength), the performance of QAA with the synthetic data set wasslightly degraded. For instance, the slope and R 2 values for a(440) changed
from 0.971 and 0.992 to 0.907 and 0.978, respectively, and RMSE changed from0.06 to 0.109. The slope and R 2 values for adg (440) (and aph (440)) became
0.905 and 0.969 (0.927 and 0.925, n = 477), respectively. As pointed out in
Lee et al. (2002), the degradation happened to waters with large a(440) (andThe Quasi-Analytical Algorithm• 79

then a(555)) values (mostly turbid coastal waters) where aw (555) makes up
less than 1/3 of the total absorption coefficient. For such cases, there will begreater errors in the estimated a(555) and subsequently in other IOPs. If it islimited to waters with a(440) < 0.5 m−1 (where aw (555) makes up at least ∼ 1/3
of a(555)), however, the performance of QAA with 555 nm as λ0 is significantly
better. The slope and R 2 values are close to unity (n = 334) and RMSE values
are ∼ 0.05 for both total absorption and backscattering coefficients; and the
RMSE are 0.09 and 0.15 for adg (440) and aph (440), respectively. These results
demonstrate the importance of having a red band in the vicinity of 620–640 nm
for remote sensing of coastal waters and the applicability of QAA to satellitedata, especially for oceanic waters.

10.5

Conclusions

The QAA is an algorithm based on the fundamental relationships of ocean optics,
and generally follows the inversion concept described in Chapter 1. When applying QAA to the IOCCG data sets (both synthetic and in situ), the retrieved IOPs
matched known or measured IOPs very well (in particular, absorption coefficientof CDOM and the total, and particle backscattering coefficient). As is the casefor many other inversion algorithms, QAA is mathematically simple and physically transparent. These characteristics make the algorithm easily adaptable tovarious multi-spectral or hyperspectral sensors, and it is also computationallyefficient for processing satellite imagery.

Chapter 11
The GSM Semi-Analytical Bio-Optical Model
Stéphane Maritorena and Dave Siegel

11.1

General Description

The GSM (for Garver-Siegel-Maritorena) semi-analytical ocean colour model was
initially developed by Garver and Siegel (1997) and later updated by Maritorena
et al. (2002). The GSM model is based on the quadratic relationship between
the remote-sensing reflectance (Rrs ) and the absorption and backscattering coefficients from Gordon et al. (1988),
i

2
t2 X
bb (λ)
Rrs (λ) = 2,
gibb (λ) + a(λ)
nw i=1(11.1)

where g1 (= 0.0949) and g2 (= 0.0794) are geometrical factors. The absorption
coefficient (a(λ)) is decomposed into seawater absorption, aw (λ), phytoplankton absorption, aph (λ), and the combined absorption of coloured detrital anddissolved material (CDM), adg (λ) (considered together as a single term becauseof their similar spectral shapes (Carder et al., 1991; Nelson et al., 1998; Nelson and Siegel, 2002). The backscattering coefficient (bb (λ)) is partitioned into
terms due to seawater, bbw (λ), and suspended particulates, bbp (λ). The nonwater absorption and scattering terms are parameterized as a known shape withan unknown magnitude,aph (λ) = Ca∗ph (λ),(11.2)adg (λ) = adg (λ) exp(−S(λ − λ0 )), Y
λ0bbp (λ) = bbp (λ0 ),λ(11.3)(11.4)

where a∗ph (λ) is the chlorophyll-a specific absorption coefficient, S is the spectral decay constant for CDM absorption (Bricaud et al., 1981), Y is the powerlaw exponent for particulate backscattering coefficient, and λ0 is a scaling wavelength (443 nm). For aph (λ), adg (λ), and bbp (λ), the unknown magnitudes are
the chlorophyll-a concentration (C), the detritus/gelbstoff absorption coefficient(adg (443)), and the particulate backscatter coefficient (bbp (443), respectively. In
81

82

•

Remote Sensing of Inherent Optical Properties

application of Equations 11.1–11.4, aw (λ), bbw (λ), nw , t, and gi are taken fromthe literature whereas the values of Y , S, and a∗ph (λ) were determined by “tuning"the model against a large in situ data set (Maritorena et al., 2002) (provided inTable 11.1). The unknowns in Equations. 11.1–11.4, C, bbp (443), and adg (443),
are retrieved by applying a nonlinear least-square technique to fit Equation 11.1
Rrs (λ) data (or normalized water-leaving radiance) collected at four or more
wavelengths. Confidence intervals for the retrieved variables are also generated
during the inversion (See Maritorena and Siegel (2005) and Chapter 3).
Table 11.1

Parameters for GSM Rrs (λ) inversion.

2−1a∗ph (λ) [m mg ]S[nm−1 ]Y

412

0.006650.02061.0337443

0.05582490

0.02055510

0.01910555

0.01015

The results presented below were obtained using the set of model parametersdescribed in Maritorena et al. (2002). In this version, model parameters were optimized using an in situ data set that consisted mostly of offshore oceanic Case
1 waters with very few stations from eutrophic waters. In order to streamline
the tuning process and to limit the number of unknowns to retrieve, the parameterisation of the original GSM model includes some simplifying assumptions.
In particular, several parameters are held constant in the model while they actually vary in nature. For example, a∗ph (λ) is expressed as a constant mean spectrum while a more sophisticated function could account for photoadaptationor community structure shifts (e.g., Bricaud et al., 1998). Similarly, particulate
backscattering is modelled using a simple function with a fixed spectral dependence (through exponent Y in Equation 11.4) while such wavelength dependence
tends to disappear in turbid waters. The slope of the spectral decrease in adg
absorption, S, is also held constant in the model whereas it actually depends
on a complex system involving land/sea interactions, the productivity and stateof the phytoplankton communities, the microbial loop and photochemistry (seealso discussion in Maritorena et al., 2002). Since these parameters were optimized from a large global Case 1 in situ data set they are generally well suited forsuch conditions and for the original GSM retrievals (C, bbp (443), and adg (443)).
However, in waters where optical characteristics differ strongly from those used
to tune the model, coastal Case 2 or phytoplankton rich waters in particular,
the model performance can be significantly degraded. Although not presented
here, other tuned versions of the model have been developed that are more appropriate for specific situations (e.g., Santa Barbara channel coastal waters).
The GSM model was initially designed for use with SeaWiFS data, andThe GSM Semi-Analytical Bio-Optical Model• 83

chlorophyll-a concentration is one of its three originally retrieved variables (Maritorena et al., 2002). For consistency with some of the other models presented
here, additional calculations were implemented in order to generate total andphytoplankton absorption coefficients at 440 nm as well (considered negligible
difference from that at 443 nm). The total absorption coefficient was calculated
by solving Equation 11.1 for a(440) using the input Rrs (440) values and theretrieved bbp (440). The phytoplankton absorption coefficient was then calculated by subtracting aw (440) and the retrieved adg (440) value from a(440).
The original GSM retrieved variables have to satisfy the following criteria to beconsidered valid:
0 < C < 100.0 mg m−3 ,0 < adg (443) < 2.0 m−1 ,and, 0.0001 < bbp (443) < 0.1 m−1 .

11.2

Results

Taking the Rrs values at 410, 440, 490, 510, and 555 nm, the variables obtainedby inversion of the model were compared to the known or in situ data using
simple regression analyses. Type II regressions on log-transformed data were
performed for each of the retrieved variables. The statistical parameters presented in Table 11.2 include: the slope and intercept of the regression, R 2 , RMSE
error, bias and the number of valid retrievals.
Table 11.2a RMSE and regression (Type II) results of the synthetic dataset (θ0 = 30◦ ). IOPs were retrieved with Rrs values at 410, 440, 490, 510 and
555 nm as inputs. N is the number of data tested, while n is the numberof valid retrievals.
Nnintercept

slope

R2

RMSE

bias

a(443)

500

479

0.032

1.068

0.974

0.115

-0.017adg (443)

500

479

0.036

1.053

0.965

0.145

-0.013aph (443)

500

479

0.162

1.171

0.957

0.173

-0.060bbp (443)

500

479

0.198

1.133

0.957

0.152

-0.062

Table 11.2b RMSE and regression (Type II) results of the in situ data set.
IOPs were retrieved with Rrs values at 412, 443, 490, 510 and 555 nm as
inputs. N is the number of data tested, while n is the number of valid
retrievals.
Nnintercept

slope

R2

RMSE

bias

a(443)

656

646-0.034

1.097

0.838

0.223

-0.129adg (443)

656

646

0.003

1.084

0.798

0.246

-0.103aph (443)

656

646

0.029

1.175

0.737

0.350

-0.221

84•

Remote Sensing of Inherent Optical Properties

Results of the inversion using the synthetic data set are presented in Figure11.1. Overall, the retrievals for the four variables presented show good statistical results with small negative biases and high R 2 values. Slopes are greaterthan 1.0 for all variables and retrievals tend to slightly underestimate synthetic
values at the low end and slightly overestimate at the high end. In general,
dispersion tends to increase when absorption or backscattering reaches highvalues because, as explained above, this version of the model is not ideal insuch conditions. Also, no valid retrievals were achieved for a small portion ofboth synthetic and in situ data sets.

3

Derived a dg (443) [mDerived a ph (443) [m

-1

-1

]

]

1
0.3
0.1
0.03
0.01

N = 500n = 4792
R = 0.957
RMSE = 0.173

1:1

0.003

0.003

0.01

0.03

0.1

Known a ph (443) [m

0.3
-1

0.3
0.1
0.03
0.01
0.003

N = 500n = 4792
R = 0.965
RMSE = 0.145

1:1

0.001
0.001 0.003 0.01 0.03

1

]

0.1

0.3

Known a dg (443) [m

1
-1

3

]

0.3

1

0.3
0.1
N = 500n = 4792
R = 0.974
RMSE = 0.115

0.03
1:1
0.01
0.01

0.03

0.1

0.3

Known a(443) [m -1 ]1

3

Derived b bp (443) [m -1 ]3Derived a(443) [m -1 ]

1

0.1
0.03
0.01
0.003
N = 500n = 479
R2 = 0.957
RMSE = 0.152

0.001
1:1

0.00030.0003 0.001 0.003

0.01

0.03

Known b bp (443) [m

0.1
-1

0.3

]

Figure 11.1 Comparison of the modelled and known IOPs for the synthetic data set (sun at 30◦ from zenith) using the GSM model with Rrs (λ)
at 410, 440, 490, 510 and 555 nm as inputs.

Figure 11.2 presents the GSM retrievals when applied to the in situ data
set. As expected, the statistical results are slightly degraded. The dispersion
is higher than with the synthetic data and R 2 values are lower. This is likely aconsequence of the noise and uncertainties associated with in situ AOP and IOP
measurements. The slopes show the same trends as with the synthetic data but
are slightly higher. In general, the GSM retrievals tend to be slightly lower than
the in situ data.

The GSM Semi-Analytical Bio-Optical Model

• 85

3

0.3
0.1
0.03
0.01
0.003

N = 656n = 6462
R = 0.737
RMSE = 0.350

1:1

0.001
0.001 0.003

0.01

0.03

0.1

0.3

Measured a ph (443) [m-1

Derived a dg (443) [m -1 ]Derived a ph (443) [m

-1

]

1
1

0.3
0.1

0.03
N = 656n = 6462
R = 0.798
RMSE = 0.246

0.01
1:1
0.003
0.003

1

0.01

0.03

0.1

0.3

1

Measured a dg (443) [m

]

-1

3

]

Derived a(443) [m -1 ]

3

1

0.3

0.1
N = 656n = 6462
R = 0.838
RMSE = 0.223

0.03
1:1
0.01
0.01

0.03

0.1

0.3

Measrued a(443) [m

1
-1

3

]

Figure 11.2 Comparison of the modelled and in situ IOPs using the GSM
model with Rrs (λ) at 412, 443, 490, 510 and 555 nm as inputs.

11.3

Conclusions

The GSM model is a simple semi-analytical ocean-colour model originally designed for use with SeaWiFS and MODIS-like satellite data over non-coastal waters. While both the synthetic and in situ data sets used here have a strong“coastal" component, the model performed well but as expected, its performance
was lower in highly absorbing or backscattering situations. Other versions of themodel exist or are being developed for specific coastal waters or to implement
new features (e.g., band-independent or “Trichodesmium" versions, Westberry
et al., 2005).

Chapter 12
Inversion Based on a Semi-Analytical ReflectanceModel
Emmanuel Devred, Shubha Sathyendranath and Trevor Platt

The algorithm presented here is based on the theoretical reflectance model developed by Sathyendranath and Platt (1997; 1998). They used the assumptionof quasi-single scattering to express the reflectance in the ocean as a functionof the diffuse attenuation coefficient, Kd , which was in turn expressed as a function of IOPs. This model has since been implemented for remote sensing applications in the North West Atlantic (Sathyendranath et al., 2001; Devred et al.,2005) and coastal waters off Vancouver Island (Sathyendranath et al., 2004). Although the model was designed primarily for application in Case 1 waters, themathematical formulation (Sathyendranath and Platt, 1997) accounts for multiple orders of scattering, and the computer programme used in the analysis
presented here incorporates scattering events up to the fifth order. Thus themodel is easily adapted to more turbid Case 2 waters, such as coastal areas.
Moreover, some assumptions made to develop the model (e.g., value of 1.0 for
the ratio of backscattering to upward-scattering coefficients) may be satisfied
in turbid waters in the presence of multiple scattering. This model has been
widely used for various applications ranging from chlorophyll-a concentration
retrieval to primary production computations. Here it is coupled with a nonlinear, least-square fitting method to retrieve IOPs (absorption and backscattering
coefficients) of marine components (phytoplankton and detrital material, dissolved and particulate) from remote-sensing reflectance, Rrs .

12.1

Theoretical BackgroundSub-surface irradiance reflectance (R(0− , λ)) is expressed as the ratio of upwelling irradiance (Eu ) to downwelling irradiance (Ed ) just below the surface.
Sathyendranath and Platt (1997) showed that R(0− , λ) for a homogeneous watercolumn can be expressed as:
R(0− , λ) =sbb (λ),µd (Kd (λ) + κ(λ))
87(12.1)

88•

Remote Sensing of Inherent Optical Properties

where Kd (λ) and κ(λ) are respectively the diffuse attenuation coefficients (m−1 )for downwelling and upwelling irradiance (note that κ(λ) defines the rate ofattenuation of upwelling light as it travels to the surface, and that this is different
from the attenuation coefficient for upwelling light with increasing depth), µd is
the average cosine for the downwelling irradiance, s is a shape factor defined as
the ratio of upward-scattering coefficient bu (m−1 ) to backscattering coefficientbb (m−1 ). The parameter s takes the value of 1.0 in very oligotrophic waters
where molecular scattering is dominant. The average cosine for downwelling
irradiance (µd ) just beneath the sea surface can be written as the sum of a direct
and a diffuse component (Equation 12.2).
The cosine for the direct component is equal to cos(θs ) where θs is the subsurface solar zenith angle and mean cosine for the diffuse component is 0.83
(Sathyendranath and Platt, 1988). Thus, the mean cosine for the total downwelling irradiance at the sea surface is given by:µd =Edd cos(θs ) 0.83Eds
+
,
Ed
Ed

(12.2)

where Ed , Edd and Eds correspond respectively to the total, direct and diffuse
solar radiation at the sea surface (Gregg and Carder, 1990). Further details regarding the assumptions and approximations in the ocean-colour model usedhere are available in Sathyendranath and Platt (1997).
Sathyendranath and Platt (1988) have expressed the diffuse attenuation coefficient as (wavelength argument is omitted here for clarity),
Kd =a + bb,µd(12.3)

κ=a + bb,µu(12.4)and similarly,

where µu corresponds to the mean cosine for the upwelling light, which is approximated as 0.5.
From Equations 12.1–12.4, the reflectance R can be expressed as a function ofbackscattering and absorption coefficients of the marine components at a given
wavelength. This model has also been extended to deal with stratified waters andinelastic (Raman) scattering (Sathyendranath and Platt, 1998). However, thesefeatures of the model are not exploited here to facilitate comparison with theother models in this report. The model has been used to provide a theoreticalunderpinning for empirical algorithms for retrieval of chlorophyll-a from oceancolour data (Sathyendranath et al., 2001), and to develop improved algorithms
for chlorophyll-a retrieval for Case 1 waters of the North West Atlantic (Devred
et al., 2005). Here, we examine the use of a nonlinear optimization technique toretrieve optical properties of the IOCCG data sets that include both Case 1 andCase 2 waters.

Inversion Based on a Semi-Analytical Reflectance Model

12.2

• 89

The Approach

To retrieve the inherent optical properties from remote-sensing reflectance, weapplied a classical nonlinear least-square fitting method to Equation 12.1. At agiven wavelength, the reflectance at the sea surface is a function of five unknown
parameters: R = f (aph , ag , ad , bb,ph , bb,p ) where subscripts “ph", “g", “d" and “p"stand respectively for chlorophyll (phytoplankton), yellow substances (also referred to as CDOM), detritus and other particulate material, when the absorption
and backscattering coefficients in the model are expressed as the sums of theircomponents. Note that absorption (aw ) and backscattering (bbw ) by pure seawater can be computed at a given wavelength (see respectively Pope and Fry, 1997;
Morel, 1974) and do not appear as unknown parameters in the above equation.
Further, based on historical measurements and bio-optical models (Bricaud etal., 1981; Ulloa et al., 1994; Bricaud et al., 1995; Loisel and Morel, 1998; Ciotti etal., 2002; Bricaud et al., 2004; Devred et al., 2006; ), the spectral dependencies
of some components are described as follows:adg (λ) = adg (440) exp[−S(λ − λ0 )],(12.5)

for combined absorption coefficients of yellow substances (ag ) and detritus (ad )at λ, where S, the exponential decrease of absorption with wavelength, is set tothe average value of 0.014 nm−1 ; andλbbp (λ) = bbp (440)440− log(C),(12.6)

for particulate backscattering (organic and mineral). Note that the wavelength
dependence is also a function of chlorophyll concentration (C) as in Sathyendranath et al. (2001).
The model of Sathyendranath et al. (2001) was used to describe phytoplankton absorption,aph (λ) = U(λ)(1 − exp(−F C)) + a∗
2 (λ)C.

(12.7)

Values of U(λ), F and a∗
2 (λ) are provided in Table 12.1, whereas details on interpretation of these parameters can be found in Devred et al. (2006). The threeparameters of the model were determined by fitting the model to the databasefrom the Bedford Institute of Oceanography.
With the above prescriptions on the spectral dependencies of the opticalproperties of some of the components, and by combining the absorptions by
detritus and yellow substances into a single component (Equation 12.5), thenumber of unknown parameters in Equation 12.1 is reduced to four (namely,
aph (440), adg (440), bbp (440) and C). When remote-sensing reflectance at 410,
440, 490, 510, 555 and 670 nm are available from ocean-colour sensors (for example, SeaWiFS, MODIS and MERIS, which are the most commonly-used), we get90•Remote Sensing of Inherent Optical Properties

Table 12.1

Parameters for aph (λ) model.

wavelength

U (λ)

2
−1
a∗
2 (λ) [m mg ]F

412

0.03690.02431.582

490

0.03380.0129510

0.01800.0114555

0.00360.0070670

0.00890.0172a system of six equations with four unknowns. This facilitates the convergenceon the solution for the four unknowns. Note that parameters bbp (λ) and aph (λ)
are related to C through Equations 12.6–12.7.
To apply our approach to the IOCCG data sets (both synthetic and in situ),
the reflectance R was estimated from remote-sensing reflectance, Rrs , using:
R=n2w
QRrs .
t2(12.8)

Here n2w /t 2 (≈ 1.89) accounts for the air-sea interface effects, and Q converts
radiance to irradiance. It is known that the factor Q varies with solar zenith angle, sea-surface roughness (wind-induced) and substances present in the water.
Here, the dependence of Q on solar zenith angle was computed using the modelof Åas and Højerselev (1999) and an empirical function (Devred et al., 2005) wasused to compute the dependence of Q on chlorophyll content (Morel and Gentili,1993).

12.3

Results and Discussion

We used remote-sensing reflectance at 410, 440, 490, 510, 555, and 670 nm (notethat Rrs (670) is not available for the in situ data set) of the IOCCG data sets toderive total, phytoplankton, and detrital (dissolved and particulate) absorption
coefficients, and particulate backscattering coefficient at 440 nm.

12.3.1Retrieval of IOPs from the simulated data set

The interest in inverting synthesized data lies in the control of all environmental
variables such as the sea surface state, solar zenith angle and optical properties.
It is then possible to assess accurately the performance of the reflectance modeland the fitting method to retrieve inherent optical properties.
Figure 12.1 shows derived versus synthesized total, phytoplankton and detritus absorption coefficients and particulate backscattering at 440 nm for datawith a solar zenith angle of 30◦ . The retrieved data are consistent with the simulated data. For each of the derived IOPs, the optimization method failed toInversion Based on a Semi-Analytical Reflectance Model

0.3

N = 500n = 4922
R = 0.827
RMSE = 0.288

3

Derived a dg (440) [m -1 ]Derived a ph (440) [m -1 ]

1

0.1
0.03
0.01
0.003

1:1

0.001
0.001

0.003

0.01

0.03

Known aph (440) [m

0.1
-1

1

0.1
0.03
0.01
0.003

]

0.1

0.3

Known a dg (440) [mN = 500
n = 500
2
R = 0.960
RMSE = 0.166

0.3
0.1
0.03

0.1

1
-1

3

]

N = 500
n = 500
2
R = 0.938
RMSE = 0.160

0.03
0.01
0.003
0.001
1:1

1:1

0.01
0.01

1:1

0.001
0.001 0.003 0.01 0.03

0.3

Derived b bp (440) [m -1 ]Derived a(440) [m -1 ]1

N = 500n = 4922
R = 0.873
RMSE = 0.348

0.3

0.3
3

• 91

0.0003
0.03

0.1

1

0.3

Known a(440) [m

-1

]

3

0.0003 0.001 0.003

0.01

0.1

0.03

Known b bp (440) [m

-1

0.3

]

Figure 12.1 Comparison between retrieved and simulated IOPs for a solar
zenith angle of 30◦ . IOPs were retrieved using Rrs values at 410, 440, 490,
510, 555 and 670 nm.

retrieve the parameters in eight cases for absorption of phytoplankton and yellow substances (with detritus). Linear regression (Type II) on log-transformed
data (omitting the cases where convergence was not obtained) gave slopes close
to 1.0 (except for phytoplankton absorption with a slope of 1.16) with a small
negative bias for all variables (Table 12.2). Note that similar results were alsoachieved with the synthesized data of 60◦ solar zenith angle (not shown). This
demonstrates that the assumptions made on the spectral dependence of the
IOPs are suitable for ocean-colour inversion.
Matching of phytoplankton absorption coefficients presents the poorest
agreement (although still acceptable) with a slope of 1.156 and a bias of -0.053.
It also exhibits the lowest correlation coefficient with a value of R 2 = 0.827. One
observes an increase in discrepancy in the retrieved data (Figure 12.1) as aph (440)increases. This is probably due to the phytoplankton absorption model used in
our algorithm. It is noteworthy that the retrieved total absorption at 440 nm
shows a better agreement than does the retrieved phytoplankton absorption. At
low backscattering coefficients (bbp (440) < 0.002 m−1 ) our algorithm showed a92•

Remote Sensing of Inherent Optical Properties

Table 12.2 RMSE and regression (Type II) results for the synthetic dataset (Sun at 30◦ ). IOPs were retrieved using Rrs values at 410, 440, 490, 510,
555 and 670 nm. N is the number of data tested, while n is the number ofvalid retrievals.
Nnintercept

slope

R2

RMSE

bias

a(440)

500

500-0.122

0.974

0.960

0.166

-0.104bbp (440)

500

500

-0.093

0.981

0.938

0.160

-0.056aph (440)

500

492

0.145

1.156

0.827

0.288

-0.053adg (440)

500

492

-0.119

1.090

0.873

0.348

-0.200systematic underestimation of the retrieved backscattering. It probably resultsfrom the formulation of the spectral dependence of the backscattering coefficients as a function of chlorophyll concentration. This approach may not beappropriate at low chlorophyll concentrations, and therefore for low backscattering coefficients. We will explore this problem further.

12.3.2Retrieval of IOPs from the in situ data set

Inversion of in situ measurements becomes more challenging not only because
the parameters defined in the previous section (IOPs, sun angle, vertical profile)show random and/or systematic variability in their natural environment, butalso because external variables (for example, measurement errors) add perturbation to the entire system (defined here as the reflectance/IOP pairs). We cantherefore expect a higher variability when retrieving the IOPs as confirmed in
Figure 12.2. Only results for absorption coefficients are shown in Figure 12.2
because backscattering measurements were not available.
Table 12.3 RMSE and regression (Type II) results for the in situ data set.
IOPs were retrieved using Rrs values at 410, 440, 490, 510 and 555 nm. N
is the number of data tested, while n is the number of valid retrievals.
Nnintercept

slope

R2

RMSE

bias

a(443)

656

656

0.011

1.048

0.762

0.218

-0.036aph (443)

656

656

0.654

1.537

0.648

0.442

-0.110adg (443)

656

491

0.416

1.312

0.380

0.470

-0.003The standard deviation has increased for all of the matching pairs (Table
12.3). Retrieved phytoplankton absorption coefficients show the highest discrepancy with the in situ data (slope of 1.537 and bias of -0.110), perhaps because of the performance of the absorption model. Previous works (Burenkov etal., 2001; Reynolds et al., 2001; Sathyendranath et al., 2001; Gohin et al., 2002;
Devred et al., 2005) showed that local bio-optical models should be preferred
to global ones. This type of approach would likely decrease the discrepancy

Inversion Based on a Semi-Analytical Reflectance Model

10

0.3

N = 656n = 6562
R = 0.648
RMSE = 0.442

1:1

Derived adg (443) [m -1 ]Derived a ph (443) [m -1 ]

3
1

• 93

0.1
0.03
0.01
0.003
0.001
0.001 0.003

1

N = 656n = 4912
R = 0.380
RMSE = 0.470

1:1

0.1

0.01

0.001
0.01

0.03

0.1

0.3

Measured a ph (443) [m

-1

1

0.01

3

0.1

1

10

Measured a dg (443) [m

]

-1

]

Derived a(443) [m -1 ]

3

1:1

1
0.3
0.1
N = 656n = 6562
R = 0.762
RMSE = 0.218

0.03
0.01
0.01

0.03

0.1

0.3

3

1

Measured a(443) [m-1]

Figure 12.2 Comparison between retrieved and in situ IOPs. IOPs wereretrieved using Rrs values at 412, 443, 490, 510 and 555 nm.

between the retrieved and in situ data. Comparison between retrieved and insitu absorption coefficients of yellow substances and detritus is also less consistent (slope of 1.312 and bias of -0.003) than the previous case (synthesizeddata set). In Figure 12.2, cases where the fitting procedure for adg (440) failed toconverge are shown as filled triangles. These points were not retained in the statistical analysis (resulting in a smaller number of samples). The total absorption
coefficients show a good agreement with a slope of 1.048 and a bias of -0.036.
This is not inconsistent as phytoplankton absorption seemed to be slightly overestimated while yellow substances are underestimated. These effects cancelledeach other, resulting in a better agreement when comparing the total absorption
coefficients.

12.4

Conclusion

The reflectance model of Sathyendranath and Platt (1997), although based on
the quasi-single scattering assumption, proved to be robust when applied to agreat variety of optical marine environments: cases ranging from low to high94•

Remote Sensing of Inherent Optical Properties

albedo (scattering to absorption coefficients ratio).
Retrieval of inherent optical properties shows better accuracy when performed on the synthesized data set at low solar zenith angle. A small decreasein the accuracy was observed as solar zenith angle increased (not presented).
A greater discrepancy occurred when retrieving phytoplankton absorption at
440 nm than for other IOPs. This may be explained by the phytoplankton absorption model used in our algorithm. However, retrieval of the total absorption coefficient seems not affected by this feature. Our algorithm underestimated backscattering coefficients at small values, perhaps a limitation of our
bio-optical model adapted from Loisel and Morel (1998).
For the in situ data set, our algorithm yielded consistent results, althougha greater variability around the 1:1 line was observed than that observed when
inverting the synthesized data set. We showed that an underestimation (overestimation) of retrieved phytoplankton absorption lead to an overestimation (underestimation) of retrieved yellow substances absorption (not knowing which
one is the cause). This will be further analysed to improve the performanceof our algorithm. However, our algorithm yielded results comparable to othermodels reported here.

Chapter 13
Examples of IOP Applications
Robert Arnone, Hubert Loisel, Kendall Carder, Emmanuel Boss,
Stephane Maritorena and ZhongPing Lee

The IOPs retrieved from ocean colour provide innovative tools and opportunities
for oceanographic studies, as their values can be used directly or indirectly to
study biological and biogeochemical processes in the oceans (Gould and Arnone,
1997; Bissett et al., 2001; Coble et al., 2004; Hu et al., 2004, 2005). For instance,earlier studies (Kirk, 1984; Sathyendranath and Platt, 1988) have shown thatthe diffuse attenuation coefficients of the water can be adequately estimatedfrom water’s inherent optical properties. Recent studies (Stramski et al., 1999;
Loisel et al., 2001a; Balch et al., 2005) have shown that particulate carbon can bewell estimated from particle backscattering coefficient. Further, a new generation of biological models (Bissett et al., 2005; Penta et al., 2005) now integrate
explicitly two or more species of plankton, as well as dissolved (DOC) and particulate organic carbon (POC), whereas IOPs play important roles in observing andmonitoring blooms of red tides (Cullen et al., 1997; Cannizzaro et al., 2006). As
confidence in the IOP products continues to grow, our understanding of how IOP
properties are linked to ocean processes expands. This research is moving theocean community beyond the traditional applications centered on the oceanic
chlorophyll-a. In this chapter, we present some examples of IOP applications inthis regard.

13.1

Water Composition and Water-Mass Classification

The absorption and backscattering coefficients bring some complementary information on the water composition, because of their different sensitivity to thevarious optically significant materials in water. While the absorption coefficientis affected by the presence of both suspended and dissolved material in water,
the backscattering coefficient represents the concentration (to first order) of organic and inorganic suspended particles, and bubbles. The decomposition of thetotal absorption coefficient into its different components, as discussed in Chapter 1, allows the monitoring of phytoplankton and of the remaining absorbing
95

96

•

Remote Sensing of Inherent Optical Properties

Figure 13.1 Comparison between the SeaWiFS-chlorophyll concentrationand the ratio of the particle backscattering to absorption obtained from aninverse algorithm (Loisel and Stramski, 2000) over the North Atlantic (south
Island) in June 1998. As seen, C and bbp (555)/(a(490)-aw (490)) represent
different patterns, with the latter clearly showing different particle populations. These particles have been identified as coccolithophorid, which arecharacterized by a high backscattering efficiency.

materials. Therefore, synoptic satellite observations of a and bb give a valuable
picture of composition of surface waters. For example, the bbp /a ratio may be
used to discriminate different families of particles (Figure 13.1).
New applications have also used the IOP characteristics of the water as atool to fingerprint a water mass and identify the controlling optical processes(Traykovski and Sosik, 2003; Arnone and Parsons, 2004). Besides water absorption, the total absorption is additionally composed of the absorption from
CDOM, detritus and phytoplankton (see Chapter 1). By defining the percent contribution of each of these components, a water mass can be defined by which
component controls the absorption budget. A ternary plot of these three components provides a useful method for fingerprinting water mass and the dominantabsorption process (Gould and Arnone, 2003; Arnone et al., 2004). This method
has been applied to satellite absorption properties derived from semi-analyticalalgorithms for SeaWiFS and MODIS ocean-colour imagery (Figure 13.2a). Thiswater-mass classification can be represented by an RGB image representing percent detritus, phytoplankton and CDOM absorption (Figure 13.2b). These images easily illustrate the controlling biogeochemical processes for monitoring
coastal and offshore water masses. Note that this classification method identifies the dominance of the absorption processes, and not the absolute valuesof the absorption coefficients. This classification method can be used on sequential satellite images of the absorption components to identify changes inabsorption processes and to track water masses based on a specific fingerprintof the absorption components.

Examples of IOP Applications• 97

Figure 13.2 (a) Water-mass classification from the absorption budget.
Ternary plot of the percent absorption attributed to detritus, CDOM andphytoplankton for each pixel to identify the dominant components. (b)RGB image of percent absorption of detritus (red), phytoplankton (green)
and CDOM (blue). Absorption products derived from SeaWiFS were used
to determine the absorption budget. Intensity of the colour indicates thedominant component

13.2

Dissolved and Particulate Organic Carbon

Examination of the temporal variations of absorption and backscattering coefficients and comparison with that of chlorophyll over the global ocean have also
provided important information about the dynamics of marine particles and dissolved organic carbon, because the absorption and backscattering coefficientsare related to different biogeochemical parameters. For instance, the feasibilityof estimating POC (in mg m−3 ), and the coloured detrital and dissolved materials(CDM) (in m−1 ), from the remotely detected bb and a was recently demonstrated(Stramski et al., 1999; Loisel et al., 2001b; Loisel et al., 2002; Siegel et al., 2002;
Balch et al., 2005) (see Figures 13.3 and 13.4). A phase shift between the annual
cycles of bbp and chlorophyll was evidenced, and was attributed to the presenceof a pool of non-pigmented particles originating from the accumulation of deadphytoplankton cells, as well as zooplankton detritus, in the summer stratified
surface layer (Loisel et al., 2002). The decrease of the Chl/POC ratio in living
phytoplankton at high irradiance in summer was also used to explain the lag
between the Chl and bbp maxima (Loisel et al., 2002).
Figure 13.4 shows global distributions of CDM of two seasons in 1998, derived from SeaWiFS data (Siegel et al., 2002). Clearly, there are significant spatial
and temporal variations in global CDM (a part of DOC). Because POC and CDM
represent different pools of carbon stored in oceans, and since CDM plays animportant role in regulating subsurface blue/ultraviolet radiation (Siegel et al.,2002), analysis of their spatial/temporal distributions is important for the understanding of the carbon cycles in oceans.
Behrenfeld et al. (2005), using backscattering and chlorophyll-a derived from98•

Remote Sensing of Inherent Optical Properties

Figure 13.3 Global chlorophyll (SeaWiFS product) and POC distribution in
January 2000 (adapted from Loisel et al., 2002).

Figure 13.4 Global distribution of CDM (in m−1 ) derived from SeaWiFS
data by the GSM algorithm (Chapter 11) (adapted from Siegel et al., 2002).

Rrs as inputs, also developed a novel primary production model based on thephysiological link between phytoplankton growth rate and growth conditions(temperature, nutrients, and light) as reflected in the ratio of chlorophyll tocarbon of phytoplankton. This novel (and debatable) approach is to use thebackscattering coefficient to estimate phytoplankton biomass and assuming alinear relation between total POC and phytoplankton biomass. The observedchange (Figure 13.5) in the ratio of chlorophyll-a to phytoplankton carbon isinterpreted as reflecting a physiological change, rather than a change in theparticulate composition. Net primary production is then computed from theestimated growth rate through a simple multiplication by the phytoplankton
carbon and a function that accounts for its vertical distribution with depth.

Examples of IOP Applications• 99

Figure 13.5 Phytoplankton growth rates for Boreal summer (June to August) and winter (December to February). Adapted from Behrenfeld et al.
(2005).

13.3

Diffuse Attenuation Coefficient of Downwelling Irradiance

The availability of absorption (a) and backscattering coefficients (bb ) also makes
it straightforward to calculate the diffuse attenuation coefficient of downwelling
irradiance, either at a single wavelength (Kd (λ)) or for the broad band (350 700 nm) visible domain (Kvis ). Because both Kd and Kvis are apparent opticalproperties, they are directly linked to the IOPs (Sathyendranath and Platt, 1988;
Gordon, 1989; Lee et al., 2005a,b). Traditionally, estimation of Kd is based on
the spectral ratios of Lw (λ) or Rrs (λ). Such an approach does not reveal thefundamental relationship between AOPs and IOPs, and is found to work only
for waters with limited dynamic range (Mueller, 2000). Figure 13.6(a) shows acomparison between measured Kd (490) and Rrs derived Kd (490), for a wide rangeof Kd (490) ( 0.04 - 4.0 m−1 ) measured from different regions and at different
times, using an algorithm based on a and bb whose values were derived firstfrom Rrs (Lee et al., 2005b). Clearly, excellent agreement is achieved between
the two independent measurements and determinations.
Kvis (wavelength range of 350 - 700 nm) is a parameter needed for models of
oceanic photosynthesis and heat transfer in the upper water column. Kvis varies
significantly from the surface to depth (z), even for vertically homogeneous waters, which is different from the characteristics of Kd . To represent this verticalvariation, earlier studies used multiple exponential terms to describe the vertical propagation of visible solar radiation, with the coefficients of these multiple
terms expressed as empirical functions of chlorophyll (Morel and Antoine, 1994;
Ohlmann and Siegel, 2000). Again, realizing the intrinsic limitations between an
optical property (e.g., Kvis ) and chlorophyll, a model has been developed (Lee etal., 2005a) that can be used to adequately estimate the vertical variation of Kvis

100

•

Remote Sensing of Inherent Optical Properties

Figure 13.6 (a) Measured Kd (490) vs Rrs -derived Kd (490). In the derivationof Kd from Rrs , values of a and bb were derived from Rrs first, and then Kd is
calculated based on these a and bb values (from Lee et al., 2005b). (b) modelled Kvis (z) compared with Kvis (z) from Hydrolight simulations (adaptedfrom Lee et al., 2005a).

when values of a(490) and bb (490) are available. Figure 13.6(b) shows modelled
Kvis compared with Hydrolight-calculated Kvis for different values of a, bb , andz. The average difference between the two sets of Kvis (z) is 2.2%.

13.4

Oceanic Primary Production

Knowing the values of IOPs can also provide some basic information for theestimation of oceanic primary production. Currently, this estimation is done
centred on the values of chlorophyll-a concentration (Platt and Sathyendranath,
1988; Behrenfeld and Falkowski, 1997). When chlorophyll is used as an input
parameter representing the function of phytoplankton, a value regarding thechlorophyll-specific absorption coefficient is also explicitly or implicitly utilized.
Numerous field measurements (Bricaud et al., 1995; Cleveland, 1995; Lutz et al.,
1996; Bricaud et al., 1998) and theoretical studies (Bricaud and Morel, 1986)
have pointed out that this property varies widely from place to place and time
to time, therefore large uncertainties are automatically introduced when thisparameter is involved. Because primary production measures the conversionof solar energy absorbed by phytoplankton to sustenance in the photosynthetic
process (Morel, 1978; Smith et al., 1989), remotely derived or locally measured
phytoplankton absorption and other IOPs can then be utilized directly in this estimation (Zaneveld et al., 1993). One example of taking this approach is demonstrated in Lee et al. (1996a), with Figure 13.7 showing primary production calculated from values of Rrs (along with other auxiliary information) compared withprimary production measured from in situ incubation.

Examples of IOP Applications• 101

Figure 13.7 Daily primary production calculated from Rrs versus thatfrom in situ incubation. Adapted from Lee et al. (1996a)

13.5

Chlorophyll Concentration from Remotely Derived
Pigment Absorption Coefficient

When the absorption coefficient of phytoplankton pigment is derived from ocean
colour, it adds the possibility of deriving the concentration of chlorophyll-a(Carder et al., 1999; Lyon et al., 2004) for different regions of the world, as indicated in Carder et al. (1999). Applying the semi-analytic code (Carder et al., 1999)
to an upwelling site (Smyth et al., 2002) and a river-plume site (Hu et al., 2003),
it provided much more realistic estimations of chlorophyll concentration forboth cases (where the empirical band-ratio approach (OC4) underestimated thehigh chlorophyll concentrations of the upwelling site and overestimated chlorophyll concentration for gelbstoff-rich river-plume regions). MODIS-Terra chlorophyll images from the GES DAAC (Goddard Earth Science Distributed Active
Archive Center) derived by the semi-analytic code (chl_a_3) and empirical-ratio
code (chl_a_2) were composited in 39-km bins for December 2000 and are shownin Figure 13.8. The subtropical gyre regions appear similar for the two images,
but the chlorophyll values are clearly elevated with the semi-analytic code forthe high-latitude and equatorial upwelling regions which have higher pigment
concentrations.

102

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Remote Sensing of Inherent Optical Properties

Figure 13.8 Global composited maps (December 2000) of chlorophyll-aconcentration (mg m−3 ) retrieved using empirical (top) and semi-analytic(bottom) algorithms from MODIS-Terra radiometry (adapted from Carder
et al., 2004).

13.6

Monitoring Coastal Ocean Processes using IOPs andNumerical Circulation Models

IOPs provide an improved capability to understand how physical processes influence the bio-optical processes (Bissett et al., 2001; Arnone and Parsons, 2004).
For instance, ocean colour IOP products from MODIS and SeaWiFS are being
integrated with numerical circulation models. The Navy Coastal Ocean Model(NCOM) is forced by large scale ocean models which currently assimilate sea
surface height from altimetry and sea surface temperature (SST) from AVHRR.
These models are at 32-degree resolution with 41 sigma levels to characterize themesoscale features (http://www7320.nrlssc.navy.mil/global_ncom/). Overlaying
the modelled properties (currents, salinity, surface heights) with optical properties adds continuity to understanding IOP image products. This fusion ofphysical models and IOP imagery enables improved understanding of the distri-Examples of IOP Applications• 103

Figure 13.9 The circulation along the California Coast develops coastal
filaments shown in surface currents and MODIS-Aqua IOP products. Differences in the locations of backscattering at 551 nm (associated with particles) and the total absorption at 443 nm (detritus, CDOM and phytoplankton) indicate varying bio-optical processes within these filaments.

bution of bio-optical processes that are linked with mesoscale ocean circulation
features. Different IOP properties, such as backscattering, CDOM and phytoplankton absorption respond differently to mesoscale processes. Along the U.S.
West coast, filaments associated with the California Current System are driven
by the physical circulation as shown in Figure 13.9 (Shulman et al., 2004; Pentaet al., 2005). The corresponding IOP distribution within these filaments is used
to define the response of bio-optical processes. For example, the influence of thestrong southerly flow off the Channel Islands is characterized by backscatteringand total absorption products. Within this filament, the elevated particles arelocated south of the strong flow (point A) as shown in the bb (551) image (a result of advection), whereas the strongest currents (point B) located close to land
have elevated total absorption as shown in the a(443) image (a result of coastal
upwelling). Divergent and convergent mesoscale fronts are revealed by the IOP
properties observed in satellite imagery (Figure 13.9). Similar differences in the
distribution of backscattering and absorption have been observed by Otero andSiegel (1995).

13.7

Conclusions

Our understanding of how the optical properties of water constituents are related to ocean processes has advanced significantly in the last decade. Use of
IOPs to characterize ocean processes provides improved methods for monitoring and understanding the role of the oceans on a global scale. Because IOPs are

104

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Remote Sensing of Inherent Optical Properties

closely associated with the water leaving radiance measured by satellites and
IOP retrievals are robust and stable as shown in previous chapters, IOP products are critical for monitoring and detecting changes in the ocean’s climatology
and forecasting ocean biogeochemical processes.

Chapter 14
Summary and Conclusions
ZhongPing Lee, Ronald Zaneveld, Stephane Maritorena, Hubert Loisel,
Roland Doerffer, Paul Lyon, Emmanuel Boss, Kendall Carder,
Emmanuel Devred and Robert Arnone

Most algorithms used in ocean colour remote sensing attempt to derive, directly,
the concentrations of water constituents, mainly phytoplankton chlorophyll concentration. In this report, however, we present and discuss algorithms whichhave been developed to derive inherent optical properties (IOPs) from waterleaving radiance, in a one-step or multi-step process. The IOPs are then decomposed into the contributions by different optical components, such as absorption by phytoplankton pigments, and finally the IOPs of different components
are converted into concentrations.
IOPs are the fundamental parameters of hydrological optics. The IOPs, in
combination with radiances from the sun and sky, determine water-leaving radiance, which in turn defines water colour (an apparent optical property). At thesame time, IOPs are also environmental properties. Their variations are directly
related to changes in concentration, size distribution and composition of particulate matter and/or dissolved constituents. IOPs derived from remote sensingof ocean colour provide innovative opportunities for environmental observationand oceanographic studies on time and space scales not achievable with in situ
measurements.
To derive, accurately, various IOPs from water colour, as presented here, is
not a simple task. This report presents some frequently encountered methods
for IOP retrieval. These algorithms have different levels of complexity; some are
explicit about all elements and derivation processes, some are implicit; somehave fewer empirical inputs, while others have more empiricism built into them.
Table 14.1 highlights their similarities and major characteristics.
When presenting and comparing models, it is always useful to rememberthat models, by their very nature, represent some sort of reduction, or simplification. It naturally follows that practically all models will have some limitation
when they attempt to mimic nature. Thus there is often a need to tailor models for specific applications or for specific regions. If models are applied for
purposes for which they were not designed, there is always a risk of poor per105

106

•

Remote Sensing of Inherent Optical Properties

Table 14.1 Algorithm highlights. L98 - Spectral-ratio algorithm (Lee etal. 1998, Chapter 4); B99 - Spectral curvature algorithm (Barnard et al.
1999, Chapter 4); MM01 - Spectral-ratio algorithm (Morel and Maritorena,
2001, Chapter 4); Loisel - Inversion of IOP (Chapter 5); D&S - MERIS Neural Network Algorithm (Chapter 6); Lyon - Linear Matrix Inversion (Chapter7); Boss - Over constrained Linear Matrix Inversion (Chapter 8); Carder MODIS semi-analytical algorithm (Chapter 9); QAA - Quasi-Analytical Algorithm (Chapter 10); GSM - Garver, Siegel and Maritorena semi-analyticalmodel (Chapter 11); SPD - Sathyendranath, Platt and Devred semi-analyticalreflectance model (Chapter 12).
Algorithm

Type

Key features

L98

Empirical

Empirical constants; products at 440 nm only

B99
MM01

Relationships between total absorption coefficients
Semi-empirical

D&S

Bio-optical models; hyperspectral
Kd (λ) from Rrs (λ) empirically

LoiselNeural Network

Neural constants; MERIS only
Spectral models for aph (λ), adg (λ), and bbp (λ)

Lyon
Boss

Algebraic (Linear
Matrix Inversion)

Varying spectral shapes for aph (λ), adg (λ), andbbp (λ); statistical selection of solution; generates
output confidence intervals; applicable to multi- andhyperspectral data

CarderAlgebraic for low
absorption waters(iterative solution);
empirical for otherSpectral models for aph (λ), adg (λ), and bbp (λ); empirical coefficients for different properties

QAA

Algebraic

Separate derivations for the total and individualcomponents; spectral models for adg (λ) and bbp (λ);
retrieve multi- or hyperspectral aph spectrum

GSM
Spectral optimization
SPD

Optimized spectral shapes for aph (λ), adg (λ), andbbp (λ); applicable to multi- and hyperspectral data;
can use input uncertainties and generates output
confidence intervals
Varying spectral shapes for aph (λ), adg (λ), andbbp (λ); applicable to multi- and hyperspectral data

formance. Thus a golden rule in application of algorithms (and algorithms area type of model) is to test them always for the specific application or region
envisaged, before routine use is made of the algorithm. But knowledge of thefeatures of the model would often help in making the initial selection of an algorithm for a particular application. For example, it is often useful to know if aparticular algorithm is non-linear or not; if it is purely empirical or if it is basedon theoretical considerations; if it is multi-variable or not; if it is computationally demanding or not. Such relevant features of the algorithms presented inthis report are shown in Table 14.1 and Figures 14.1 - 14.4. But, as is often thecase with summary tables and figures, they do not represent the whole story,
but merely highlight some emergent properties when all models were made asSummary and Conclusions• 107

Figure 14.1 RMSE values for total absorption coefficient of both synthetic
and in situ data sets, for all algorithms tested (see Table 14.1 for notationof algorithms). The “Lyon" results are for 410 nm, while all other results
are in the vicinity of 442 nm. Numbers in parenthesis indicate percentage
of valid retrievals for each algorithm. Invalid retrievals are excluded fromthe calculation of RMSE and other statistical analyses.

comparable as possible, for the purpose of this report.
The RMSE errors presented in Figures 14.1 - 14.4 not only represent the performance of each algorithm, but show also the deviation of the bio-optical model
behind each algorithm, from that of the bio-optical model used to prepare thesynthetic data set. In addition, the RMSE errors for the in situ data set include
uncertainties associated with field measurements. It should be noted that not
all the algorithms tested used the same number of spectral bands, and somealgorithms used fewer bands than what they can potentially use (especially forthe synthetic data set).
An inversion algorithm works as a mathematical filter analagous to physicalor chemical filters used in the lab or field. In this filtering process, uncertainties
are introduced, explicitly or implicitly, into the desired products. More uncertainties are introduced when fewer parameters are under control. Clearly, theresults of the various algorithms indicate that there remains room for improvement in the derivation of IOPs from ocean colour. As new information becomes
available, it is anticipated that the present algorithms could be revised, or exciting new methods could be developed. It is natural that algorithm development
is always a continuing and evolving process.
Nevertheless, we can safely draw the following conclusions based on the

108

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Remote Sensing of Inherent Optical Properties

Figure 14.2 As Fig. 14.1, but for particle backscattering coefficient (synthetic data set only).

presentations and discussions of the various algorithms:
1. In general, the best properties that can be obtained from ocean-colour data,
regardless of the algorithm used (see Figures 14.1 to 14.4) and as expectedfrom the inversion of radiative transfer (see Figure 1.3), are the spectral
absorption and backscattering coefficients of the total water volume.
2. Using the synthetic data set as a reference (the in situ data set prevents
the separation of algorithm error from measurement error), more reliable
results are obtained for clearer waters (a(440) < ∼0.3 m−1 ). Due to limitations of algorithm architecture and availability of reliable remote-sensing
reflectance at specified wavelengths, less accurate results are generally obtained for more absorbing waters (a(440) > ∼0.3 m−1 ).
3. When decomposing the total absorption coefficient into the componentsof phytoplankton and coloured material, less accurate results (see Figures
14.3 and 14.4) are anticipated due to overlapping of spectral signals andbecause the spectral shapes of the components are not constant.
4. If the chlorophyll-a concentration (C) is desired from ocean colour, more
uncertainties will be introduced because the chlorophyll-specific absorption coefficient is not constant at a given wavelength, nor is the relationshipbetween backscattering and chlorophyll well defined.
5. Because there are more unknown factors that affect the retrieval of C fromocean colour than there are unknown factors that affect the retrieval ofabsorption and backscattering coefficients, we should revisit the issue ofC remaining the primary product of ocean-colour remote sensing, rather
than the IOPs of the bulk water or the optical properties of phytoplankton.

Summary and Conclusions• 109

Figure 14.3 As in Figure 14.1, but for absorption coefficient of detritus
and gelbstoff combined.

6. The robust and stable results of the total absorption and backscattering
coefficients from these various algorithms (again using the synthetic data
set as reference), which were developed independently and are based on
different principles, clearly indicate that these optical properties should betaken as standard products for all ocean-colour satellite missions. These
optical properties, similar to the sea surface temperature, could serve as
climatology data records to study long-term changes of the global oceans.
7. Space-based sensors should be equipped with at least one spectral bandin the region of 620-640 nm. Such a band is very important for coastal
remote sensing (or for more turbid waters), and algorithm performance
would be improved when such a band is included in the process.
8. Algorithms based on the fundamentals of hydrological optics are strongly
advocated. Simple empirical relationships prevent understanding of thebasics and, therefore, limit advancement in ocean-colour remote sensing.
On the other hand, analytical or semi-analytical algorithms enable opportunities to trace back the error sources.
Because inherent optical properties provide important indices for our water
environments and open new doors for oceanographic studies, we should spend
a great deal of effort on the following issues to improve IOP products:❖ Increased high-quality, co-located measurements of remote-sensing reflectance and IOPs.
❖ Improved methods to select model parameters such as the spectral shapesof individual IOPs that include bb (λ), aph (λ) and adg (λ). Separation of the
global ocean into dynamic biogeochemical provinces may provide vital help
in this regard (see IOCCG working group on “Global Ecological Provinces"

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Figure 14.4 As in Figure 14.1, but for the absorption coefficient of phytoplankton pigments.
http://www.ioccg.org/groups/dowell.html for more information).

❖ Better quantification of uncertainties in derived products. An in-depth
analysis of error sources and their propagation are highly desirable in thisregard.
❖ Improved procedure for atmospheric correction. All algorithms tested use
remote-sensing reflectance (Rrs ) as inputs for the calculation of IOPs. Quality of Rrs , which is one of the products derived from atmospheric correction, plays a critical role in the accuracy of retrieved IOPs. Addition of
UV-a bands would assist in the derivation of Rrs from satellite measured
radiance, especially for coastal waters. Also, such bands may increase theability to separate phytoplankton absorption from that of dissolved andnon-pigmented particulate materials.
❖ And, finally, enhance and broaden applications of IOPs for oceanographic
studies, which are the ultimate goal of ocean-colour remote sensing.
It should be pointed out that in this exercise, the water column was assumed
to be homogeneous in terms of its optical properties. Passive optical remotesensing becomes quite a challenge when the optical properties of the upper water column are significantly stratified. Furthermore, we did not touch on issues
related to optically shallow environments in this report (for discussions on thisissue see IOCCG Report 3). To resolve these important issues, we need to effectively combine measurements from satellite with those from other observatory
platforms, such as LIDAR, gliders, and the Network of Coastal Observatories.

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Acronyms and AbbreviationsaNN

Artificial Neural Network

AOP

Apparent Optical Property

AVHRR

Advanced Very High Resolution Radiometer

CDM

Coloured Detrital and Dissolved Material

CDOM

Coloured Dissolved Organic Matter

CZCS

Coastal Zone Colour Scanner

DOC

Dissolved Organic Carbon

ERT

Equation of Radiative Transfer

FWHM

Full-Width at Half Max

GSM

Garver Siegel Maritorena

IOP

Inherent Optical Property

LMI

Linear Matrix Inversion

MERIS

Medium Resolution Imaging Spectrometer

MOBY

Marine Optical Buoy

MODIS

Moderate Resolution Imaging Spectroradiometer

NCOM

Navy Coastal Ocean Model

NDT

Nitrate Depletion Temperature

NN

Neural Network

NPOESS

National Polar-orbiting Operational Environmental Satellite System

POC

Particulate Organic Carbon

QAA

Quasi Analytical Algorithm

RGB

Red Green BlueRMSE

Root Mean Square Error

SeaBASS

SeaWiFS Bio-Optical Archive and Storage System

SeaWiFS

Sea-viewing Wide Field-of-view Sensor

SST

Sea Surface Temperature

VIIRS

Visible Infrared Imager Radiometer Suite

VSF

Volume Scattering Function
123Mathematical Notations

SymbolDescription

Units

a

Absorption coefficient

m−1

ad

Absorption coefficient of non-algal
particlesm−1ag

Absorption coefficient of yellow
substance (gelbstoff)

m−1

adg

Sum of absorption coefficients of
non-algal particles plus yellow substances (gelbstoff)

m−1

aph

Absorption coefficient of phytoplankton pigments

m−1

a∗
ph

Specific absorption coefficient ofphytoplankton pigments (normalized to chl concentration)

m2 (mg chl)−1a+phAbsorption coefficient of phytoplankton pigments normalized at
440 nmaw

Absorption
moleculesb

Scattering coefficient

m−1

bf

Forward scattering coefficient

m−1

bb

Backscattering coefficient

m−1

bbp

Backscattering coefficient of particles

m−1cBeam attenuation coefficientm−1

C

Concentration of chlorophyll-a

mg m−3

Ed

Downwelling irradiance

W m−2

Eod

Downwelling scalar irradiance

W m−2

Eu

Upwelling irradiance

W m−2coefficient

125

of

water

m−1

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Kd

Diffuse attenuation coefficient for
downwelling irradiance

m−1

ku

Diffuse attenuation coefficient for
upwelling radiance

m−1

KVIS

Diffuse attenuation coefficient for
downwelling broad band (350–700nm) irradiance

m−1

L

Radiance

W m−2 sr−1

Lu

Upwelling radiance

W m−2 sr−1

Lw

Water-leaving radiance

W m−2 sr−1

LwN

Normalized water-leaving radiance

W m−2 sr−1

Q

Ratio of upwelling irradiance to upwelling radiance

sr

rrs

Remote sensing reflectance just below the surface

sr−1

R

Irradiance reflectanceRrs

Remote sensing reflectance justabove the surface

sr−1

S

Slope of absorption coefficient ofyellow substance

nm−1

Y

Exponent for particle backscattering
coefficientβVolume scattering function

m−1 sr−1λLight wavelength in free spacenmη

Ratio of molecular scattering to total
scatteringµd

Mean cosine of the downwelling irradianceµu

Mean cosine of the upwelling irradiance