Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms

Estimation of phytoplankton pigment concentration in the Gulf ofAqaba (Eilat) by in situ and remote sensing single-wavelengthalgorithms

L. SOKOLETSKY{, Z. DUBINSKY{, M. SHOSHANY{ and

N. STAMBLER{{Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan 52900, Israel;e-mail: [email protected]{Department of Geography, Bar-Ilan University, Ramat-Gan 52900, Israel

(Received 19 April 2001; in final form 25 November 2002 )

Abstract. Bio-optical relationships between inherent and apparent opticalproperties, and between optical properties and phytoplankton pigmentconcentration (C) averaged in a discrete layer, were developed. Theserelationships were derived from analysis of data collected during theperiod 1996–1998 in the Gulf of Aqaba (Eilat), a ‘Case 1’ type waterbody. Parameterization of these relationships was accomplished by com-bining Gershun’s equation, radiative transfer theory for average cosine ofunderwater light field, and a set of different bio-optical models. An analysisof the asymptotic light field was carried out. Semi-analytical single-wavelength(at l~443 nm) algorithms for in situ and remote sensing (RS) estimation ofmean pigment concentration were developed, and evaluated by sensitivity anderror analysis. The advantages of RS single-wavelength algorithms incomparison with current two- and multi-wavelengths RS algorithms arediscussed.

1. Introduction

Estimation of phytoplankton in marine and freshwater bodies is an important

goal in ecosystem research and monitoring, and environmental quality control. In

recent years, due to the availability of new airborne video cameras and satellite

sensors such as SeaWiFS or MODIS, new remote sensing (RS) algorithms were

developed, allowing estimation of phytoplankton pigment concentration (C) (e.g.

Gordon et al. 1988, Morel 1988, Hoge 1994, Morel and Gentili 1996, Gitelson et al.

1996, Fraser et al. 1997, O’Reilly et al. 1998, Avard et al. 2000, Gross et al. 2000).

For this purpose C is considered as the sum of chlorophyll a (Chl a) and degraded

algal pigments, pheophytins (pheo). These algorithms are typically based on four

radiation characterizations: (a) spectral water-leaving radiance Lw (l); (b)

normalised spectral water-leaving radiance: Lwn (l)~Lw(l)/[(cos h0)ta(l)], where

h0 is solar zenith angle, and ta(l) is spectral transmittance of the atmosphere; (c)

International Journal of Remote SensingISSN 0143-1161 print/ISSN 1366-5901 online # 2003 Taylor & Francis Ltd

http://www.tandf.co.uk/journalsDOI: 10.1080/0143116031000069807

INT. J. REMOTE SENSING, 20 DECEMBER, 2003,

VOL. 24, NO. 24, 5049–5073

above-water remote-sensed reflectance Rrs (l)~Lw(l)/Ed(l, 0z), where Ed(l, 0z)

is downwelling irradiance just above sea surface; or (d) underwater remote-sensed

reflectance Rrsw (l)~Lu(l, 02)/Ed(l, 02), where Lu(l, 02) and Ed(l, 02) are

upwelling radiance and downwelling irradiance just below sea surface, respectively.

Most of the existing algorithms have power or polynomial form, utilizing these

characteristics at different wavelength bands as independent variables. Empirical

data for the verification of these algorithms were collected from basin-specific

seasonal measurements or from generalization of regional data sets. In both cases a

delicate approach to utilization of existing algorithms for other water basins and

seasons is necessary.Generalisation of in situ and RS models deriving phytoplankton concen-

tration for water bodies under a wide range of geographical conditions is

of primary importance. The aim of this study was to simplify these models by

utilising a single wavelength band, which could potentially contribute to their

geographical generalisation. The spectral band we selected was the 443 nm

band, which was found to be the most suitable for this purpose by numerous

investigators (e.g. Gordon et al. 1988, Morel 1988, Tilzer et al. 1994, Bricaud et al.

1995, Waters 1995, Garver and Siegel 1997, Stramska and Dickey 1998, Berwald

et al. 1998, Antoine and Morel 1999). Use of the blue range of the spectrum, in

addition to other optical ranges for in situ estimation of pigment biomass, has been

discussed by Gordon and Morel (1983), Smith et al. (1991) and Bartlett et al.

(1998). Garver and Siegel (1997), using only one wavelength (441 nm), obtained

excellent correspondence between Chl a measured in situ and Chl a retrieved

from Rrsw.

The main goal of the present work is to establish new bio-optical relationships

based on state-of-the-art underwater investigations, and to parameterise such

relationships for the Gulf of Aqaba (Eilat). From a practical point of view, the

relationships developed have been used for the solution of the inverse bio-optical

problem, namely, estimation of layer-averaged pigment concentration.

2. Study area

According to most ‘chlorophyllous definitions’ of the trophic state of water

basins (e.g. Shifrin 1988, Morel and Berthon 1989, Dera 1995, Antoine et al. 1996,

Vinogradov et al. 1997), the waters of the Gulf of Aqaba exhibit meso-oligotrophic

rather than oligotrophic-type characteristics (Levanon-Spanier et al. 1979). The

primary productivity rates measured in the Gulf of Aqaba (Levanon-Spanier et al.

1979, Lindell and Post 1995, Iluz 1998) also indicate the seasonal meso-

oligotrophicity of the Gulf.

In the past only a limited number of phytoplankton concentration studies have

been conducted in the Gulf of Aqaba, and these have been limited to single dates

and locations (Levanon-Spanier et al. 1979, Dubinsky et al. 1990, Stambler 1992,

Lindell and Post 1995, Iluz 1998, Badran and Foster 1998). This does not allow

development of statistically reliable bio-optical algorithms for the region under

consideration. The present study is based on an extensive oceanographic survey

consisting of about 70 cruises that were conducted in the Gulf between January

1996 and December 1998.

The chosen site for the measurement of optical, hydro-physical and other

5050 L. Sokoletsky et al.

characteristics was station A1 (29‡ 28’N, 34‡ 56’E), which is situated at the northern

tip of the Gulf about 5 km off the coast. Bottom depth at this station is

approximately 700 m. This station was determined as a good representative of the

offshore waters in the Gulf.

Trophic conditions in the region under consideration were estimated according

to pigment concentration averaged in the penetration and euphotic layers (DZp and

DZe, respectively). These layers are defined as layers from the just below sea surface

(Z~0–) to the depth at which photosynthetically active radiation, EPAR (e.g.

downwelling irradiance averaged within the spectral range from 400 to 700 nm) is

reduced to 1/e (Z~Zp) or 1% (Z~Ze) of its sub-surface value, respectively. Thus,

trophic conditions were classified as oligotrophic during the May–October stratified

period (when observed penetration-layer averaged concentrations vCpw were

<0.12–0.25 mg m{3 and euphotic-layer averaged concentrations vCew were

<0.30–0.46 mg m{3), and mesotrophic during the November–April mixing period

(vCpw <0.33–0.72 mg m{3 and vCew <0.37–0.68 mg m{3). Concomitantly, Zp

and Ze varied from 13.6 and 73.7 m, respectively on 4 March 1996 up to 28.9 and

114.2 m, respectively on 15 June 1998. These observations and classification are in

agreement with the early observations of Levanon-Spanier et al. (1979).

3. Data acquisition and pre-processingData of Chl a and pheo concentrations were derived from in situ measurements

conducted at depths Z from 0– to y600 m, and radiometry data — from 0– to

y100–120 m, in accordance with the depth of the euphotic layer. The number of

vertical profiles of C was 52. Samples were taken usually every 20 metres and after

that measurements were repeated several times.

The fresh samples were analysed within a few hours after collection.

Determination of C concentrations in the water column was performed by the

standard fluorometric method (Holm-Hansen et al. 1965, Schanz et al. 1997).

Statistical analysis of repeated concentration measurements give a mean coefficient

of variance CV~15.8% for Chl a for initial dataset (N~45) and CV~9.8% for a

dataset remaining after rejecting obviously erroneous measurements (N~24). With

the aim of diminishing of stochastic (space and temporal) error and estimation of

intermediate values of concentrations, quadratic or cubic smoothing of raw data

were applied.

The spectra of the downwelling irradiance Ed(l, Z), and upwelling radiance

Lu(l, Z) were acquired with a submersible Profiling Reflectance Radiometer (Model

PRR-600 from Biospherical Instruments Ltd) at seven spectral bands (412, 443,

490, 510, 555, 665, and 694 nm) with 10-nm bandwidths. All spectra were measured

every several centimetres and in two directions (down and up). The least significant

bit that the spectroradiometer could resolve corresponded to a spectral irradiance

value of <0.01 mW m{2 cm{1. Initial number of measured optical profiles

(including repeated measurements) was 56. However, after rejection of erroneous

measurements and incomplete optical or concentration data, 24 pairs (concen-

trationszoptics) of vertical profiles and 30 pairs of bio-optical data just below air-

sea interface (Z~0–), were saved for further data processing.For more exact assessment of the downwelling irradiance just below sea surface

Ed(l, 0–), additional meteorological data of incoming global solar irradiance at

Estimation of phytoplankton pigment concentration 5051

Eilat during 1990–1998 and estimates obtained from the regionally-tuned algorithm

at l~443 nm (Sokoletsky et al. 2000), were used. In order to reduce the effects of

noise in the data, the third order polynomials were used (henceforth, the symbol l

will be omitted):

In Ed Zð Þ½ �~{ a0za1Zza2Z2za3Z

3� �

, ð1Þwhere coefficients a0, a1, a2, and a3 were determined by comparison of mea-

sured values of ln[Ed(Z)] to their model values (simultaneously within all range of

depths) by means of non-linear least-squares method (NLSM). Equation 1 was

found to be significant in most cases (the coefficient of determination R2 was

greater than 0.99). A similar approach was used for smoothing of Lu(l, Z) data.

The vertical attenuation coefficient Kd (Z) for downwelling irradiance, which is

determined by

Kd Zð Þ~{Lln Ed Zð Þ½ �

LZð2Þ

was then derived from equations 1 and 2 as follows:

Kd Zð Þ~a1z2a2Zz3a3Z2: ð3Þ

4. Average cosine of the underwater light field: RTT implementation

The average cosine of underwater light field (henceforth, ‘average cosine’) �mm is

an important geometrical characteristic of the underwater light regime, and serves

as a factor relating an inherent optical property (IOP), a, with the apparent optical

property (AOP), Kd (Plass et al. 1981, Stavn 1988, Kirk 1994, McCormic 1995,

Pelevin and Rostovtseva 2001) as follows:

a~�mmKd: ð4Þ

Equation 4 can be considered as a form of Gershun’s equation (Gershun 1939),

and was tested by Plass et al. (1981), who showed that the mean accuracy of this

equation is in the range of 0.5%, and the maximum deviation for clear oceanic

waters is 1.8%. Later processing of numerous experimental ocean data by Pelevin

and Rostovtseva (2001) slightly increased the boundary of accuracy to 4%.

In turn, �mm (at given vertical depth Z) can be represented in the approximate

form (Berwald et al. 1995):

�mm~�mm?z �mm0{�mm?ð Þ exp {PzZð Þ, ð5Þwhere �mm0 and �mm? are average cosines at Z~0– and Z~‘ (i.e. at infinitely-thick

oceanic layer), respectively; Pz is defined from equation 5 as the rate (at any vertical

depth Z) of exponential decrease of the quantity �mm{�mm?ð Þ= �mm0{�mm?ð Þ. Taking into

account the independence of �mm0 and �mm? of the depth, Pz has a physical meaning as

the rate of decrease of the average cosine �mm.

It is reasonable to assume that if the sun is in the zenith, then �mm0 and �mm? are

IOPs, i.e. optical properties independent of the ambient light field. Indeed, Berwald

et al. (1995), by means of HYDROLIGHT radiative transfer numerical model and

following least-squares fitting, were able to get equations relating these properties to


another IOP, single-scattering albedo v0:

�mm0~mw 0:000421v0

1{v0

� �2

{0:0274v0

1{v0

� �z1

" #, ð6Þ

�mm?~mw {1:59 v0ð Þ4z1:71 v0ð Þ3

{0:467 v0ð Þ2{0:347v0z1

h i, ð7Þ

v0~b= azbð Þ, ð8Þwhere mw is cosine of solar zenith angle at Z~0–; b is a beam scattering coefficient.

Pz is often taken as either a constant (McCormick 1985, Zaneveld 1989) or a

weak function of v0 (Berwald et al. 1995), however, authors of the latter paper

noted a significant deviation of real behaviour Pz within the first few optical depths

from such representations. In the recent work of Kirk (1999), by straightforward

application of radiative transfer theory, expression for the rate of exponential

decreasing (Pd) of �mm with the average distance (d), traversed by photons, has been

derived:

Pd~b 1{�mmsð Þ, ð9Þwhere �mms is an average cosine of single scattering (asymmetry parameter). For

conversion from Pd to Pz, we substitute d by Z=�mmd (e.g. Kirk 1991), where �mmd is the

downwelling average cosine of underwater light field, obtaining the expression for

Pz:

Pz~b 1{�mmsð Þ

�mmd

: ð10Þ

The relation between �mmd and �mm is described by the equation (Aas 1987, Morel and

Gentili 1991):

�mmd~�mm 1z2Rð Þ

1{R, ð11Þ

where R is reflectance, defined as the ratio of upwelling irradiance Eu to

downwelling irradiance Ed. Returning to equation 5 and taking into account

equations 10 and 11, we get a implicit expression for �mm:

�mm~�mm?z �mm0{�mm?ð Þ exp {bZ 1{�mmsð Þ 1{Rð Þ

�mm 1z2Rð Þ

� �: ð12Þ

Solution of this equation can be obtained in explicit form by sequential substitution

of �mm into equation 12; by doing so we get solution in infinite form:

�mm~�mm?z �mm0{�mm?ð Þ exp {q

�mm?z �mm0{�mm?ð Þ exp { qP

� �" #

, ð13Þ

where q is certain generalised parameter, equalling

q~bZ 1{�mmsð Þ 1{Rð Þ

1z2R: ð14Þ

Now we can derive some important relationships among the optical properties

of the underwater light field �mm Zð Þ, Pz(Z) and Kd(Z) themselves. Indeed, as far as the

optical properties a, b, R and �mms are concerned, these are only C-dependent and

Z-independent. Therefore, from equations 4, 10 and 11, for any two depths Z1 and


Z2, follows the similarity relationship:

Pz Z1ð ÞPz Z2ð Þ~

�mm Z2ð Þ�mm Z1ð Þ~

Kd Z1ð ÞKd Z2ð Þ , ð15aÞ

or in its differential form:

LlnPz Zð ÞLZ

��~ L�mm Zð Þ

LZ

��~ LlnKd Zð Þ

LZ

��: ð15bÞ

Figure 1 illustrates the computation [based on equations (6), (7) and (13)] of the

normalised average cosine �mm=mw as a function of q and v0.

The form of the relationship between �mm=mw and q (at v0~const) is similar to the

forms of relationships �mm=mw vs. Z or vs. optical depth t~Zc, represented in several

earlier papers (e.g. Preisendorfer 1959, Zaneveld 1989, Sathyendranath and Platt

1991, Gordon et al. 1993, Berwald et al. 1995, 1998, McCormick 1995). At first

�mm=mw strongly decreases as the parameter q increases; however, with further increase

of q, it reaches an almost constant value, i.e. the underwater light field becomes

asymptotic. A more detailed study of the asymptotic light field will be undertaken

below, in a separate section.

Note, that recent experimental data (IOPs, AOPs and vertical profiles of C)

gathered by Sosik et al. (1998) in northwest Atlantic coastal waters, revealed a clear

vertical increase of Kd(443) even in the case of oligotrophic (C v0.5 mg m{3) and

homogeneous waters.

Objective analysis of above equations and general background (see e.g. Kirk

1994) show that the underwater light regime is governed by various components

Figure 1. Vertical profiles (along axis q) of the average cosine of underwater light field atzenith sun �mm computed for the set of single-scattering albedo v0 (thin curves) andpercentage criterion for asymptotic property x (thick curves). v0 accepts values of0.2, 0.3, …, 0.9, 0.95 (on the right to the left); x increases from 5% on the bottom to50% on the top by 5% steps.


of the complex atmosphere–water column system (including air/water interface),

primarily by:

1. sun position, affecting mw and R;2. atmospheric conditions (cloudiness, pollution, winds, etc) affecting mw;

3. vertical depth Z, affecting Pz and �mm;

4. concentration of solid and dissolved matter in the water, affecting all IOPs

and AOPs, including �mm0, �mm?, �mms and �mm.

5. Bio-optical modelling

The following initial bio-optical relationships (all taken at wavelength of

443 nm) were used in the present work:

. Total absorption a (m{1) vs. C (e.g. Prieur and Sathyendranath 1981, Morel

1991, Bricaud et al. 1995, Ciotti et al. 1999):

a~aw, diszaaCba, ð16Þ

where aw;dis is absorption coefficient for water and dissolved matter, aa and ba

are positive coefficients;. Total backscattering bb (m{1) vs. C (e.g. Gordon et al. 1988, Sathyendranath

and Platt 1988):

bb~bbwzabbCbbb, ð17Þ

where bbw is the water backscattering coefficient, given as 0.00239 m{1 by

Morel (1974), abb and bbb are positive coefficients;

. The ratio of particle backscattering (bbp~bb2bbw) to particle scattering

(bp~b2bw) (Ulloa et al. 1994, Sathyendranath et al. 2001) is:

bbp

bp~0:0078{0:0042 log10 C, ð18Þ

where bw is the water scattering coefficient, given as 0.00478 m{1 by Morel

(1974);

. Reflectance R vs. bb/a and cosine (ma) of solar zenith angle in air (Morel and

Gentili 1991) is:

R~½0:6279{0:2227gb{0:0513 gbð Þ2{

0:3119{0:2465gbð Þma�bb

a

� �,

ð19Þ

where gb is the contribution of molecular backscattering into total

backscattering, i.e.:

gb~bbw=bb; ð20Þ

. Average single-scattering cosine �mms vs. C, derived by approximation of the

data from table 1 of Kirk (1991) (see also Walker 1994, p. 58, Gordon and

Table 1. Values of IOP parameters, estimated at l~443 nm.

aw;dis (m{1) aa ba abb bbb

0.0186 0.0297 0.788 0.00175 0.253


Boynton 1997, Boynton and Gordon 2000) is:

�mms~0:975 exp {2:594bb=bð Þ; ð21Þ. Underwater remote-sensed reflectance Rrsw (sr{1) vs. bb/a (Kirk 1994):

Rrsw~0:083bb=a: ð22Þ

It should be noted that equations 16–22 illustrate the assumption about the

depth-independency of a, b, bb, R, Rrsw and �mms.

6. Parameterization of bio-optical equations

Attempts to estimate the parameters of the equations listed above by solving

them for a large range of the depths lead to absurd results. Such a situation is

characteristic for the solution of large non-linear systems of equations. For this

reason, we used a special computer algorithm, which can be divided into three

steps:

1. Values a and bb at l~443 nm and Z~0–) were computed from measured

values of Kd, Rrsw, C and mw (‘pseudodata’). The last parameter was

computed by Snell’s law. The computation was executed by exact solution of

the system of equations 4, 6, 8, 18 and 22 for each sample (N~30).

2. Pseudodata obtained from (1) were approached by means of three-

parametric power bio-optical models (equations 16 and 17) by NLSM.3. The remaining inherent optical properties (b, c, gb, v0, �mms, �mm0, and �mm?), the

quasi-inherent optical property (Rrsw) and the apparent optical properties (R,

�mm? Pz, �mm, �mmd, and Kd) were computed by equations listed in Sections 4 and 5.

The input parameters for modelling (C and Z) were taken from the following

ranges, near to real measurement conditions: (0ƒCƒ1 mg m{3 and

0ƒZƒ100 m).

As can be seen from table 1 and figures 2–4, the bio-optical models (at

l~443 nm) found are close to ones encountered in literature, describing Case 1

waters. For instance, an a vs. C model close to ours can be found in Morel (1988),

Cleveland (1995), Lee et al. (1998); b vs. C in Gordon et al. (1993) and Haltrin

(1999); bb vs. C in Sathyendranath and Platt (1988) and Morel and Maritorena

(2001); v0 vs. C in Berwald et al. (1995), Ciotti et al. (1999) and Morel and

Maritorena (2001); Kd(0–) vs. C in Platt et al. (1994). It should be noted, however,

that our models at Cv0.1 mg m{3 are expected to be less accurate than at

0.1ƒCƒ1 mg m{3 since they are based mainly on data within the second, higher

range of pigment concentrations.

The total accuracy of models found was estimated from comparison of

computed optical properties with directly-observed optical properties [Kd(0–) and

Rrsw], or with optical pseudodata (a and bb); it was represented (table 2) by values

of normalised (to average values of the ‘true’ optical property) root-mean-squared

error (NRMSE), coefficient of determination (R2) and significance level (p).

Developed and parameterised above, bio-optical models define characteristics of

the underwater light field R, q, Pz, �mm=mw and mwKd as functions of ma, C and Z.

However, in contrast to C and Z, impact of the sun position on these characteristics

is not as large. We verified this impact by doing all computations for the two


extreme values of the cosine (ma) of solar zenith angle, namely, ma~0.6 (this is the

minimal value for the entire period of observations) and for ma~1 (its maximal

theoretical value) for the chosen range of input parameters. The normalised

(to the average values of selected optical property, computed for both values of ma)

root-mean-squared differences (NRMSD) estimated for �mm=mw and mwKd are

less than 0.3% (table 3) and, therefore, these important optical characteristics may

be assumed to be independent of the sun’s position over a broad range of its

variations.

The generalised parameter q is a quantity proportional to the depth Z

(equation 14), with a slope increasing with pigment concentration (figure 5). This is

Figure 2. Relationships (the main model) between different optical properties and totalpigment concentration. Note that Kd(0–) and R(0–) are computed for zenith sun.


Figure 3. Relationships (observed data, the main and simplified models) between Kd(0–) atsun in zenith and sub-surface pigment concentration C0.

Figure 4. As figure 3, but for water-leaving radiance Rrsw.

Table 2. Accuracy of bio-optical models at l~443 nm and Z~0– (N~30).

Optical property NRMSE (%) R2 p

a 11.7 0.793 4.41610{11

bb 12.2 0.243 5.65610{3

Kd 11.4 0.813 1.05610{11

Rrsw 16.8 0.443 5.98610{5


explained due to the larger positive contribution of beam scattering b than that of

the negative contribution of the factor (1{�mms) to parameter q.

Vertical profiles of Pz, �mm and Kd (at zenith sun) plotted for the C values in the

chosen range, are shown in figures 6–8, respectively.

7. An asymptotic light field

Below sufficient depth in a homogeneous ocean, the angular shape of the

radiance distribution and the rates of decay of the magnitude of radiances with

depth are constant and depend only on the IOP of the water. This is referred to as

the asymptotic light field (e.g. Preisendorfer 1959, Berwald et al. 1998). Taking into

account that Pz, �mm and Kd vary with depth in the same manner (see the similarity

relationship in equation 15), we can define an asymptotic light field more generally

as that range of depths at which the magnitudes of Pz, �mm and Kd become constants,

depending only on the IOP of the water.The criterion for asymptotic property (x) giving the upper boundary of the

asymptotic light field (Zx) was chosen analogously to the criterion applied by

Table 3. Normalized root-mean-squared difference (NRMSD, in %) for characteristics ofthe underwater light field computed between ma~0.6 and ma~1.

R q Pz �mm=mw mwKd

16.7 2.3 0.25 0.22 0.25

Figure 5. Vertical profiles (along axis of depths) of generalized parameter q (thin curves)computed for C~0, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7 and 1.0 mg m{3 on the leftto the right. x (thick curves) increases from 5% on the bottom to 40% on the top by5% steps.


Gordon et al. (1993), namely,

Pz ?ð Þ{Pz Zxð ÞPz ?ð Þ ~

�mm Zxð Þ{�mm?�mm?

~Kd ?ð Þ{Kd Zxð Þ

Kd ?ð Þ vx

for all Z > Zx:

ð23Þ

Similar expressions can be written for qxwq(Zx).

Figure 7. Vertical profiles (along axis of depths) of the average cosine �mm of underwater lightfield (at zenith sun), computed for C~0, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7 and1.0 mg m{3 on the right to the left (thin curves). x (thick curves) varies as in figure 5.

Figure 6. Vertical profiles (along axis of depths) of the rate of exponential decrease Pz ofaverage cosine of underwater light field (thin curves), computed for C~0, 0.01, 0.05,0.1, 0.2, …,1.0 mg m{3 on the left to the right. x (thick curves) varies as in figure 5.


The plots illustrating asymptotic light ‘fields’ corresponding to different values

of x (in %) are presented in figures 1 and 5–7. These plots clearly demonstrate the

significant variability of AOPs (Pz, �mm and Kd) along the vertical axes q (figure 1) or

Z (figures 5–7). For example, a general model, based only on RTT (figure 1), yields

more than 50% variability of these AOPs as the single-scattering albedo v0 varied

over the range 0.63 to 0.83, and the generalised parameter q varied from 0 to ‘. In

more realistic cases when v0 ranges from 0.79 to 0.83 (this corresponds to the

pigment concentration C ranging from 0.1 to 1 mg m{3) and vertical depths vary

from 0 to 100 m, the above-mentioned AOPs vary from 24 to 36%, depending on C

(figures 5–7). The onset of the asymptotic light field Zx (in m) is a function of the

percentage criterion x, and pigment concentration C. However, for the following

ranges of parameters: 5%ƒxƒ35% and 0.1ƒCƒ1 mg m{3, the uncertainty due to

C is less than 14%, and Zx can be estimated from the approximated equation 24,

derived from a general in situ model (with R2~0.999):

Zx~156:5{10:1xz0:295x2{0:00355x3: ð24Þ

8. Simplified bio-optical models

Although above-presented in situ and remote sensing bio-optical models allow

the possibility of solving the direct problem (estimation of Kd and Rrsw from mw, C

and Z), the solution of the inverse problem (estimation of C from Kd (or Rrsw), mw

and Z) may, in some cases, lead to absurd results. With the aim of removing

such situations and simplifying the solution of the inverse problem, the

following simplified models were developed and used in further investigations (at

l~443 nm, 0ƒCƒ1 mg m{3 and 0ƒZƒ100 m for Kd and Z~0– for Rrsw):

Figure 8. Vertical profiles of Kd computed by two models for the following set of pigmentconcentrations C (from the left to the right): 0, 0.01, 0.05, 0.1, 0.2, …, 1.0 mg m{3.The main model is shown by the thick curves, and the simplified model by the thincurves.


Kd Zð Þ~Kw, dis Zð ÞzaK Zð Þ C Zð Þ½ �bK , ð25Þwhere Kw;dis(Z) is downwelling attenuation coefficient for water and dissolved

matter:

Kw, dis Zð Þ~ aw, dis

mw�mmw0

� �1zk1Zzk2Z

2� �

, ð26Þ

aK(Z) is a positive coefficient:

aK Zð Þ~ aa

mw�mmw0

� �1za1Zza2Z

2� �

, ð27Þ

Rrsw~0:015{0:0241C0z0:0290 C0ð Þ2{0:0125 C0ð Þ3

, ð28Þwhere bK was assumed equal to ba~0.788; C0 is sub-surface pigment concentration;

�mmw0 is sub-surface average cosine �mm0 at mw~1 (i.e. at zenith sun), computed by

equations 6, 8 and 16–18 and averaged for the overall range of C. The remaining

values of parameters for our simplified in situ (Kd vs. C) model were estimated from

comparison of this model with the main in situ model by the least-squares method

(table 4).A simplified RS model (equation 28) was derived, based on observed values of

C0 and Rrsw (N~30). The plots for which both types of models (main and

simplified) were used simultaneously (figures 3 and 8 for Kd and figure 4 for Rrsw),

confirm the high accuracy of our simplified models.

9. Sensitivity and error analysis

Sensitivity analysis for the single-wavelength Kd(0–) and Rrsw vs. C models,

derived above, was conducted by using an error amplification factor H, deter-

mined as a ratio of relative error of output value (C for our application) to

relative error of input value (F) at very small values of the F error. It is easy to

show that H can be expressed by the following equation (see, for instance, Shifrin

1988, p. 212):

H C vs:F Cð Þ½ �~ F Cð ÞC LF Cð Þ=LC½ � , ð29Þ

where F(C) is presented by Kd(0–) or Rrsw.

From equations 25, 28 and 29 it follows:

H C vs:Kdð Þ~ Kd

bK Kd{Kw, disð Þ , ð30Þ

H C0 vs:Rrswð Þ~ Rrsw

C0 {0:0241z0:0579C0{0:0374 C0ð Þ2h i : ð31Þ

The plot of error amplification factor for C vs. Kd at two depths, 0– and 100 m

(figure 9) demonstrate a sharp increase of error H as C approaches 0, and slow

Table 4. Parameters of equations 24–26.

�mmw0 bk k1 (m{1) k2 (m{1) a1 (m{1) a2 (m{1)

0.888 0.788 21.51610{4 1.51610{5 1.04610{2 24.85610{5


asymptotic behavior as C tends to infinity (limit values of H is 1/bK~1.270). In

order to obtain magnitudes of errors, resulting from the seasonal variations in

pigment concentrations, dependences H as functions of mean values of observed

C(Z), averaged separately for the mixing period (November–April) and for

the stratified period (May–October), were also plotted in figure 9. As could be

expected, during a mixing period (when the pigment concentrations were

C0~0.52¡0.21 mg m{3 and C100~0.45¡0.12 mg m{3) the magnitude of H was

usually lower than during a stratified period (when C0~0.15¡0.05 mg m{3 and

C100~0.60¡0.30 mg m{3).

Values of the error amplification factor H for C0 vs. Rrsw (figure 10) are negative

for all values of C0, in the range under consideration, but their absolute values are

close to values for H(C0 vs. Kd). The relationship between H(C0 vs. Rrsw) and C0

has a more complex character than that between H(C0 vs. Kd) and C0, and is much

lower related to season. Indeed, values of H(C0 vs. Rrsw) computed for the mean

sub-surface pigment concentrations (¡ standard deviations) measured during both

periods, are {4:05z0:7{2:4 and {4:9z0:9

{1:85 for the mixing and stratified period,

respectively. Note that absolute values of H(C0 vs. Rrsw) are even somewhat lower

than those (about 6), obtained by Vasilkov et al. (2001) for Case 1 and Case 2

waters, using bio-optical simulations for the 5-wavelength RS algorithm.

If we accept that the error of retrieving Rrsw at 443 nm is within ¡5% (see e.g.

Figure 9. Error amplification factor (H) computed for the estimation of C(Z) from Kd(Z).The thick curve was obtained for Z~0– and thin curve for Z~100 m. The symbolsshow H computed for the averaged pigment concentration values measured indifferent seasonal periods; open triangles correspond to mixing period, while closecircles correspond to stratified period. The symbols are shown for the depths from0– to 100 m in 10 m steps.


Vasilkov et al. 2001, Gordon and Wang 1994, Wang and Gordon 1994), then the

error of remote-sensed retrieving of C0 can be estimated as +20% and +24:5% for

mixing and stratified period, respectively. Therefore, it seems obvious that if the

first SeaWiFS accuracy goal of 5% in water-leaving radiances (Mueller and Austin

1992) is achievable, then the second goal of 35% accuracy of Chl a retrieval is also a

realistic minimum for the waters under consideration.

10. Relationships between pigment concentrations averaged in select layers

One of the challenging objectives in biological oceanography is estimation of

underwater layer-averaged biomass or total pigment concentration. Remote-sensing

imagery gives estimations of C or Chl a only from the near-surface layer rather

than from deeper layers. For example, as was shown by Gordon and McCluney

(1975), approximately 90% of the information contained in water-leaving radiance

originates from the near-surface (‘penetration’) layer. However, in Case 1 waters,

main biomass (expressed by chlorophyll concentration) is usually situated below, in

the layer approximately next to the ‘euphotic’ depth, Ze. Thus preliminary in situ

determination of relationships between pigment concentrations averaged in

different layers has paramount value. Such relationships were observed for

different water basins by various investigators (e.g. Morel and Berthon 1989,

Arrigo et al. 1998, Ignatiades 1998).

We assumed a simple linear regression between C0 and layer-averaged concen-

trations, vCpw (for Z varying from 0– to Zp), vC50w (for Z varying from 0– to

50 m) and vCew (for Z varying from 0– to Ze), which were parameterized based

Figure 10. Error amplification factor H for C0 vs. Rrsw. The solid curve is computed forcontinuous values of C0 from 0 to 1 mg m{3, while the symbols show H computed forthe mean values of C0 (¡ standard deviations) observed during two differentseasonal periods.


on in situ measurements as follows (figure 11):

vCp > ~0:895C0z0:061 R2~0:972� �

, ð32Þ

vC50 > ~0:687C0z0:172 R2~0:856� �

, ð33Þ

vCe > ~0:474C0z0:290 R2~0:725� �

: ð34ÞThus, equations 32–34 along with equations of simplified models (Section 8) can

be utilized for in situ or remote sensing single-wavelength estimation of mean

phytoplankton pigment concentration in select layers.

11. In situ/RS estimation of layer-averaged pigment concentrations

The in situ and RS versions of bio-optical relationships represented above allow

development of an algorithm for estimation of layer-averaged pigment concentra-

tions from the measured quantities of mwKd(443) or Rrsw(443). It would be

reasonable to begin from estimation of sub-surface pigment concentration C0, and

then use equations 32–34 to estimate vCDZw.

An analytical solution for C0 estimated in situ was found by inversion of

equation 25 taking into account equations 26–27 at Z~0– as follows:

C0~mw�mmw0Kd 0{ð Þ{aw, dis

aa

� �b{1K

%

0, if mwKd 0{ð Þ¡0:0209 m{1

74:8 mwKd 0{ð Þ{0:0209½ �1:270, otherwise

( ð35Þ

Analogously, an analytical solution for C0 estimated by the RS algorithm was

Figure 11. Relationships between sub-surface pigment concentration C0 and layer-averagedpigment concentrations vCDZw. The observed data are shown by symbols: solid,dotted and dot-and-dash curves are approaches to vCpw, vC50w and vCew vs.C0 relationships, respectively.


developed by inversion of equation 26:

C0~4:440{653:1Rrswz24060 Rrswð Þ2: ð36ÞFigures 12 and 13 show modelled dependences of mwKd(443, 0–) and Rrsw(443),

respectively, as a function of C0, vCpw, vC50w and vCew. Comparison

between modelled (by both algorithms) and measured euphotic layer-averaged

pigment concentrations (figure 14 and table 5) demonstrate sufficiently high and

approximately the same accuracy of in situ and RS algorithms.

12. Discussion and conclusion

New single-wavelength (at l~443 nm) algorithms for in situ and remote sensing

estimation of layer-averaged pigment concentration are developed and parameter-

ized for the Gulf of Aqaba (Eilat). The novelty of these algorithms is joint

consideration of depth-dependent average cosine of underwater light field and semi-

analytical bio-optical relationships. The error analysis outlined in this paper

indicates that a single wavelength band centred on 443 nm may be utilised for C

estimation even in a strongly stratified water column with accuracy, which meets

current goals of the marine biological community. Due to such simplification,

developed bio-optical models could potentially contribute to geographical general-

isation of new local algorithms. Such algorithms can be used for real-time optical

measurements conducted from in situ platforms (e.g. ships, optical moorings,

drifters and profiling floats) as well as from remote platforms (e.g. aircrafts or

satellites). It is also clear that single-wavelength algorithms could be used in further

applications of bio-optical methods for water quality assessment and monitoring.

Examples of such applications include estimation of penetrating in-depth solar

irradiance, phytoplankton primary productivity or modelling of time-spatial

Figure 12. Modelling of relationship between mw Kd (0–) and the pigment concentrationsjust below the surface (C0) as well as averaged over selected layers.


Figure 14. Comparison between modelled (by means of in situ and remote sensing lgorithms)and measured values of pigment concentrations averaged over euphotic layer.

Table 5. Results of statistical verification of in situ and remote sensing algorithms foraveraged (in euphotic layer) pigment concentration; linear regression of estimatedconcentrations vs. measured concentrations is assumed.

Algorithm Slope Intercept NRMSE (%) R2 p

In situ 0.731 0.116 20.0 0.555 2.36610{6

RS 0.989 0.017 24.9 0.559 2.05610{6

Figure 13. As figure 12, but for Rrsw.


structure of plankton community (e.g. Olesen et al. 1999, Kamenir et al. 2000,

Reynolds et al. 2001, Sathyendranath et al. 2001, Sokoletsky et al. 2001).

In spite of its own merits, several aspects of the proposed algorithms remain

questionable and require further investigation. One such question is to what degree

do the equations for average cosine, used in the present work, reflect the real

underwater light situation? An especially important point here is the impact of

absorption on the geometry of underwater light fluxes. For instance, if to use the

value of single-scattering albedo from our bio-optical modelling [v0~0.82 (¡0.01),

see figure 2(a)], then, according to the findings of Berwald et al. (1995), the

contribution of absorption to Pt~Pz/c (and, hence, to Pz) should be about 13%.

The second important question, not discussed in the present work, is the choice

and number of wavelength bands for RS estimation of chlorophyll (for Case 1

waters). The number of bands usually used in the current RS applications, from

two to four (e.g. O’Reilly et al. 1998) is not necessarily optimal. Moreover, there are

examples in which good results were obtained from a single wavelength (441 nm) in

the blue range (Garver and Siegel 1997, O’Reilly et al. 1998). The following are

additional arguments supporting the possibility of using only one wavelength:

1. Spectral remote-sensed reflectances [Rrs(l) and Rrsw(l)], with high accuracy,

proportional to spectral normalised water-leaving radiance Lwn(l) (e.g.

Gordon 1990, Clark 1997, Fraser et al. 1997, Gordon and Voss 1999, Mobley

1999, Loisel et al. 2001), the main feature of which is maximal removal of the

effects of the atmosphere and the solar zenith angle from upwelling signal.

Therefore, use of Rrs(l) and Rrsw(l) also promotes removal of these effects.

2. Remote sensing observations (McClain et al. 1998, Gordon and Voss 1999)

of Lwn(l) accompanied by ground-truth data, demonstrate the minimal error

of Lwn(l) retrieval for lv555 nm. A possible explanation of this may be the

large variability of the Raman scattering contribution to Lw(l) at l~550 nm

(from 18% for pure water to 3% for 1.0 mg Chl m{3) in comparison to

l~440 nm (from 6% for pure water to 2% for 1.0 mg Chl m{3) (Waters

1995).

3. The uncertainties of retrieving Lwn(l) and, hence, Rrs(l) or Rrsw(l) may have

different signs and there is no correlation between the sensor noise at

different wavelengths (Gordon 1990, Clark 2001). Moreover, it can be

assumed that the absolute values of spectral Rrs(l), Rrsw(l) or Lwn(l) errors

are just the same, positive and negative signs of errors equiprobable, and

spectral errors independent from one another. Then, from sensitivity analysis

of model equations 16, 17 and 22 it follows that the mean error of C

estimation from Rrs(443), Rrsw(443) or Lwn(443) would not be greater than

that from any spectral differences or ratios of these parameters. For example,

from the spectral radiance data of Wang and Gordon (1994), it follows that

for modelled range of C: 0.1ƒCƒ1 mg m{3, the mean-square error of C vs.

Lwn(443) algorithm on 3.9% and 5.5% lower than the mean-square error of C

vs. Lwn(443) – Lwn(550) and Lwn(443)/Lwn(550) algorithms, respectively.

These findings may be expanded easily to a higher number of wavelengths by

the propagation errors method. Nevertheless, conclusions about the possibility and

even preference of single-wavelength use have preliminary value only. It seems that

further experimental (particularly in different sites of the Gulf) and theoretical


investigations are necessary for verification of this conclusion, as well as for

refinement of presented one-wavelength algorithms. Undoubtedly, significant

improvements in the bio-optical models may be obtained from additional

microstructure information and particle optics.

Acknowledgments

This study was part of the Ph.D. study of the first author and was conducted

within the framework of the ‘Red-Sea Program’, a joint German, Egyptian,

Palestinian and Israeli program funded by the German Ministry of Science,

Technology and Education (BMBF). The authors cordially thank Professors

J. T. O. Kirk (Kirk Marine Optics), N. J. McCormick (University of Washington),

A. Morel (Universite Pierre et Marie Curie), R. H. Stavn (University of North

Carolina), and K. J. Voss (University of Miami) for valuable suggestions and

comments offered in the course of paper preparation. We would like to thank

anonymous reviewers who provided helpful comments on the manuscript. We also

gratefully acknowledge Ms Sharon Victor for English assistance.

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Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms