os 11 139 2015

os 11 139 2015
Ocean Sci., 11, 139–158, 2015
www.ocean-sci.net/11/139/2015/
doi:10.5194/os-11-139-2015
© Author(s) 2015. CC Attribution 3.0 License.
Using empirical orthogonal functions derived from remote-sensing
reflectance for the prediction of phytoplankton pigment
concentrations
A. Bracher1,2 , M. H. Taylor3 , B. Taylor1 , T. Dinter1,2 , R. Röttgers4 , and F. Steinmetz5
1 Alfred
Wegener Institute Helmholtz Centre for Polar and Marine Research, Bussestraße 24,
27570 Bremerhaven, Germany
2 Institute of Environmental Physics, University of Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany
3 Leibniz Center for Tropical Marine Ecology, Fahrenheitstraße 6, 28359 Bremen, Germany
4 Helmholtz Zentrum Geesthacht Center of Materials and Coastal Research, Max-Planck-Str.,
21502 Geesthacht, Germany
5 HYGEOS, Euratechnologies, 165 Avenue de Bretagne, 59000 Lille, France
Correspondence to: A. Bracher ([email protected])
Received: 30 July 2014 – Published in Ocean Sci. Discuss.: 11 September 2014
Revised: 29 November 2014 – Accepted: 6 January 2015 – Published: 3 February 2015
Abstract. The composition and abundance of algal pigments
provide information on phytoplankton community characteristics such as photoacclimation, overall biomass and taxonomic composition. In particular, pigments play a major
role in photoprotection and in the light-driven part of photosynthesis. Most phytoplankton pigments can be measured
by high-performance liquid chromatography (HPLC) techniques applied to filtered water samples. This method, as well
as other laboratory analyses, is time consuming and therefore limits the number of samples that can be processed in
a given time. In order to receive information on phytoplankton pigment composition with a higher temporal and spatial
resolution, we have developed a method to assess pigment
concentrations from continuous optical measurements. The
method applies an empirical orthogonal function (EOF) analysis to remote-sensing reflectance data derived from shipbased hyperspectral underwater radiometry and from multispectral satellite data (using the Medium Resolution Imaging Spectrometer – MERIS – Polymer product developed
by Steinmetz et al., 2011) measured in the Atlantic Ocean.
Subsequently we developed multiple linear regression models with measured (collocated) pigment concentrations as
the response variable and EOF loadings as predictor variables. The model results show that surface concentrations of
a suite of pigments and pigment groups can be well predicted
from the ship-based reflectance measurements, even when
only a multispectral resolution is chosen (i.e., eight bands,
similar to those used by MERIS). Based on the MERIS reflectance data, concentrations of total and monovinyl chlorophyll a and the groups of photoprotective and photosynthetic
carotenoids can be predicted with high quality. As a demonstration of the utility of the approach, the fitted model based
on satellite reflectance data as input was applied to 1 month
of MERIS Polymer data to predict the concentration of those
pigment groups for the whole eastern tropical Atlantic area.
Bootstrapping explorations of cross-validation error indicate
that the method can produce reliable predictions with relatively small data sets (e.g., < 50 collocated values of reflectance and pigment concentration). The method allows for
the derivation of time series from continuous reflectance data
of various pigment groups at various regions, which can be
used to study variability and change of phytoplankton composition and photophysiology.
1
Introduction
Optical measurements taken from various platforms have
been successfully used to determine the total chlorophyll a
(TChl a) concentration (e.g., see the summary by McClain 2009). Those measurements can be taken continuously,
Published by Copernicus Publications on behalf of the European Geosciences Union.
140
A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
thereby allowing for the estimation of TChl a concentration
at a much higher temporal and spatial resolution than possible from chemical measurements in the laboratory, e.g.,
by high-performance liquid chromatography (HPLC) analysis of discrete water samples. Chl a is the major pigment
in all phytoplankton species and is often used as an indicator of phytoplankton biomass. When pigments are measured by HPLC, TChl a is defined as the sum of monovinyl
Chl a (MVChl a), divinyl Chl a (DVChl a) and chlorophyllide a (which is mainly formed as an artifact of the former two during the extraction process and therefore included
in the calculation). DVChl a exists only in the prokaryotic
genus Prochlorococcus, while MVChl a is the Chl a pigment
for all other phytoplankton (other cyanobacteria and eukaryotes). Besides Chl a, there are many other pigments in phytoplankton that are either involved in light harvesting, such
as chlorophyll b (Chl b), chlorophyll c (Chl c) and photosynthetic carotenoids (PSC), or in protecting Chl a and other
sensitive pigments from photodamage, such as photoprotective carotenoids (PPC). Some pigments only occur in certain phytoplankton groups and thus are indicator pigments
for their identification, e.g., peridinin in dinoflagellates (e.g.,
Letelier et al., 1993; Vidussi et al., 2001).
When analyzing biogeochemical fluxes in the oceans,
however, it is inadequate to consider phytoplankton as a single variable (i.e., TChl a) because various groups have different roles in the biogeochemical processes (such as carbon fixation and export, nitrogen fixation, and silicon uptake). TChl a is far from being a sole function of phytoplankton biomass and varies, as other phytoplankton pigments
do, with taxonomic composition and mean physiological algal assemblage state in response to several factors such as
light, temperature and nutrients (Behrenfeld and Boss, 2006).
Thus, knowledge of a wider array of phytoplankton pigment
concentrations provides insight into phytoplankton composition, overall light absorption and physiological state. Phytoplankton absorption bears the imprints of different types
of pigments and can be measured by optical measurements.
However, different phytoplankton pigments may correlate in
parts of their spectrum, making individual pigment detection
difficult.
Several recent studies have investigated the potential of using continuous optical data to derive surface concentrations
of pigments other than TChl a, with the advantage of being able to supply estimates over larger spatial and temporal scales than obtained with in situ water sampling. Chase
et al. (2013) decomposed a large global data set of hyperspectral particulate absorption measurements into Gaussian
function components and assessed the magnitude of specific
Gaussian functions in relation to the absorption by specific
pigments or pigment groups. The method provided robust
results for obtaining concentrations of TChl a, TChl b (sum
of different types of Chl b), TChl c (sum of different types
of Chl c), PSC, PPC and phycoerythrin (PE). Organelli et
al. (2013) used a multivariate approach applied to fourthOcean Sci., 11, 139–158, 2015
derivative spectra of phytoplankton or particulate absorption
(aph and ap , respectively) data to retrieve TChl a, the total
concentrations of seven diagnostic pigments and three phytoplankton size classes. However, ap and aph are inherent optical properties (IOP) which cannot be directly determined
from satellite ocean-color measurements (after successful atmospheric correction), such as the apparent optical properties
(AOP). The estimation of IOP from AOP is based on a certain inversion model (e.g., the Quasi-Analytical Algorithm
by Lee et al., 2002), which introduces additional uncertainty.
The water-leaving reflectance (ρw ) is related not only to
phytoplankton absorption but also to the scattering and absorption of water and other water constituents and to changes
in the radiance distribution in response to environmental conditions such as observation geometry, surface waves and atmospheric conditions.
Pan et al. (2010) developed empirical algorithms based
on reflectance ratios to approximate key phytoplankton pigment concentrations. The band-ratio algorithms were developed from underwater radiometric measurements collocated
to pigment data taken in northeastern US coastal waters and
were successful in deriving the concentration of TChl a,
TChl b, TChl c and nine different carotenoids. However, such
band-ratio algorithms require a very large database (> 400
collocations with satellite data) from a certain region to derive robust results. Pan et al. (2013) later described that the algorithm had to be adapted by modifying the pigment-specific
coefficients based on a regionally specific data set.
Craig et al. (2012) developed local models to estimate
TChl a and aph at different wavelengths from hyperspectral
in situ measurements of remote-sensing reflectance, Rrs (λ),
in an optically complex water body. The models were based
on empirical orthogonal functions (EOF) analysis of normalized Rrs (λ) spectra and a subsequent linear fitting of measured TChl a concentration and aph (λ) as response variables
to EOF loadings as predictor variables. Taylor et al. (2013)
showed that the method could be used similarly to derive PE
concentrations from underwater upwelling radiance spectra,
Lu (λ), which enabled continuous profile predictions of PE
concentrations.
The present study aims to use the spectral information contained in reflectance data to derive the optical signature of
different pigments by an automatic and generic technique.
The EOF analysis is applied to Rrs and to ρwN (i.e., normalized ρw just above surface) data measured in the field and
by satellite sensors, respectively, in the Atlantic Ocean. The
dominant EOF loadings were subsequently assessed as predictors in a multiple linear regression for the concentration
of phytoplankton pigments and pigment groups as response
variables. The prediction error of each model is evaluated by
a permuted cross-validation routine, which is used to estimate the critical sample sizes necessary for reliable prediction. In addition, we demonstrate the approach’s utility in estimating the large-scale distribution and photophysiology of
the phytoplankton assemblage.
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A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
2
141
Material and methods
Two sets of optical and pigment data from the Atlantic Ocean
were used in the analysis. The first model setup used a data
set which included only optical measurements taken in situ
(as depth profiles) and collocated surface pigment data collected during three transatlantic RV Polarstern cruises in
2008 and 2010. These data enabled us to study the difference in EOF methods between hyper- and multispectral resolution. In the following, we call this data set “field data
set”. For a second data set, the “satellite-based data set”, we
considered water reflectance measurements from the satellite
sensor Medium Resolution Imaging Spectrometer (MERIS),
collocated to pigment data from various researchers in the
tropical Atlantic Ocean. These data enabled us to study the
generic application of the method.
2.1
Field data set
Samples for the field data set were collected during three
RV Polarstern cruises: the expeditions ANTXXIV/4 in
April/May 2008 and ANTXXVI/4 in April/May 2010 followed a south-to-north transect through the Atlantic Ocean
from Punta Arenas (Chile) to Bremerhaven (Germany); ANTXXV/1 in November 2008 followed a north-to-south transect through the eastern Atlantic Ocean from Bremerhaven
to Cape Town (South Africa) (see Fig. 1; for more details
see Table S1, upper panel in the Supplement). Sampling was
generally conducted at 12:00 local time and involved conductivity temperature density (CTD) casts with water samplers, below-water radiance and irradiance measurements
and above-water irradiance measurements. Water samples
from surface water (< 10 m) for pigment analysis and for PE
analysis were filtered on GF/F filters and on 0.4 µm polycarbonate filters, respectively. Filters were immediately shockfrozen in liquid nitrogen and stored at −80 ◦ C until further
analysis at the laboratories of the Alfred-Wegener-Institute
Helmholtz Centre for Polar and Marine Research (AWI).
2.1.1
Pigment data
The composition of pigments that were soluble in organic
solvents was analyzed by HPLC following the method
by Barlow et al. (1997) and adjusted to our temperaturecontrolled instruments (a Waters 600 controller combined
with a Waters 2998 photodiode array detector, a Water717plus auto sampler and a LC Microsorb C8 HPLC column) as detailed in Taylor et al. (2011). We determined the
list of pigments shown in Table 1 of Taylor et al. (2011)
and applied the method by Aiken et al. (2009) for quality
control of the pigment data. HPLC data for ANTXXV/1, as
opposed to the other two cruises, were already published
in Taylor et al. (2011) and are available from PANGAEA
(doi.pangaea.de/10.1594/PANGAEA.819070). The relative
concentration of PE was taken from the data set published
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Figure 1. Position of pigment samples used in this study. Red: field
data set; black: samples which are collocated to satellite-based but
not to field reflectance data; circles: samples which are collocated to
field but not to satellite-based reflectance data; stars, diamonds and
squares: collocations to MERIS Polymer data based on the 1 × 1,
3 × 3 and 5 × 5 pixel criteria, respectively.
for all three cruises in PANGAEA (doi.pangaea.de/10.1594/
PANGAEA.819624) and analyzed in Taylor et al. (2013). As
outlined in Taylor et al. (2013), the PE concentration is expressed as a relative value, while all other pigments concentrations are directly measured values.
2.1.2
Reflectance data field data set
For all three cruises as AOP input data, we used Rrs (λ) data
obtained from profiles of radiance and irradiance from 320
to 950 nm, with an optical resolution of 3.3 nm and a spectral
accuracy of 0.3 nm, measured with hyperspectral radiometers
(RAMSES, TriOS GmbH, Germany) at the same time and
place as pigment data of Sect. 2.1.1. Rrs data of ANTXXV/1
were already published in Taylor et al. (2011) and are available from PANGAEA (doi.pangaea.de/10.1594/PANGAEA.
819506). For the other two cruises we applied the same technique and instrumentation as in Taylor et al. (2011) to derive the Rrs spectrum at each station. To test the influence of
Ocean Sci., 11, 139–158, 2015
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A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
spectral range and resolution of AOPs, the hyperspectral field
Rrs (λ) data were used within the range of 350 to 700 nm and
380 to 700 nm and reduced to the multispectral bands (412,
443, 490, 510, 560, 620, 665 and 681 nm) of MERIS by taking the integral over all wavebands within one band (±10 nm
around the center wavelength except when 681 nm ± 7.5 nm
was used).
2.2
Satellite-based data set
For this data set, pigment concentrations had been determined from the sea surface (< 10 m) with HPLC by several investigators within the area of 35◦ N–10◦ S and 42◦ W–
3◦ E during the MERIS/ENVISAT mission lifetime (2002–
2012; for more details on the data set see Supplement Table S1, lower panel). A large part of those data are publicly available from the SEABASS and BODC databases.
The remaining pigment data are from the field data set
within this area, including additional data from stations
where no radiometric measurements had been taken and
from four other cruises: pigment data from the RV Maria
S. Merian cruise MSM-18/3 were analyzed by AWI as described above in Sect. 2.1.1; data from two RV Polarstern
cruises (ANTXXIII/1 and ANTXXIV/1) were analyzed by
HZG following Zapata et al. (2000); data from the Bonus
Good Hope (BGH) cruise, conducted by the Laboratoire
d’Océanographie de Villefranche, were acquired as outlined
in Speich et al. (2008) and analyzed following the method by
Ras et al. (2008).
AOP input data is from the MERIS Polymer level 2
ρwN (λ) product given for the same eight wavebands as listed
in Sect. 2.1.2. The Polymer algorithm (for details see Steinmetz et al., 2011) provides a powerful atmospheric correction. It is an iterative spectral matching method over the
whole available sensor spectrum and uses two decoupled
models. First, the water reflectance is modeled using two parameters: the Chl a concentration and the particle backscattering coefficient. Second, the reflectance of the atmosphere,
including aerosols and contamination by sun glint, is simplified by using an analytical expression that can account for
multiple interactions between molecular and aerosol scatterings (and glitter) without referring to a specific aerosol
model. Hence, it allows for the retrieval of large amounts
of MERIS observations in sun glint, thin clouds or heavy
aerosol plumes; these contaminated conditions could not be
treated correctly by standard atmospheric correction schemes
extrapolating from the near infrared. MERIS Polymer products thus improve the spatial coverage by almost a factor of
2 and have proven successful for retrieving MERIS Ocean
Colour products: Polymer was selected as the MERIS processor for atmospheric correction for the Ocean Colour Climate Change Initiative after an extensive validation and intercomparison with other atmospheric correction algorithms in
which each algorithm’s uncertainty was assessed (Müller and
Krasemann, 2012). However, additional uncertainties probaOcean Sci., 11, 139–158, 2015
bly result from the difference in spatial resolution between
satellite (1 km by 1 km) and ship-based (20 cm by 20 cm)
sampled data.
Matchups between pigment data and MERIS Polymer
ρwN (λ) and TChl a products were determined according to
the MERMAID (MERIS MAtchup In-situ Database) as 1×1
(within the MERIS pixel), 3×3 and 5×5 pixels, respectively,
measured on the same day around the field observation (see
Barker et al., 2008). For the 3 × 3 and 5 × 5 MERIS pixel
match-ups, the mean ρwN (λ) and TChl a concentrations from
the MERIS products were calculated. Then the 1 × 1, mean
3×3 and mean 5×5 MERIS ρwN (λ) matchup data were used
for deriving predicted (modeled) pigment concentrations, as
outlined in Sect. 2.3. The mean MERIS Polymer TChl a data
were validated with the in situ TChl a data of the satellitebased data set. The R 2 , percent bias (PB), mean percent difference (MPD) and root mean square error (RMSE) between
the two collocated data sets were calculated as outlined in
Werdell et al. (2013) and used to determine pigment prediction full-fit statistics (see Sect. 2.3.2).
2.3
Statistical methods to retrieve pigment
concentrations from reflectance
Figure 1 presents the distribution of collocated pigment and
reflectance measurements for both field and satellite-based
data sets that were used separately as input for the EOF prediction analysis. The field data set covered 53 collocated reflectance and pigment data points (Fig. 1, red points). We
used three setups of the field Rrs (λ) spectra for the development of pigment-specific models:
1. Rrs (λ) data in hyperspectral
“hyper_Rrs ”) from 350 to 700 nm,
(1 nm
resolved,
2. “hyper_Rrs ” from 380 to 700 nm and
3. Rrs (λ) data in MERIS band resolution (“band_Rrs ”).
The three satellite-based data sets consisted of 139, 155
and 160 collocated reflectance and pigment data points from
2002 to 2012 for the 1 × 1 (Fig. 1, stars), 3 × 3 (Fig. 1, diamonds) and 5 × 5 (Fig. 1, squares) pixel collocation criteria,
respectively, covering all months except January, March and
December (details on the spatial and temporal distribution of
collocations are given in the Supplement Table S1). Eighteen
collocations of the field data matched the 1×1 pixel satellitebased data set (Fig. 1, red stars), but no additional field data
matched the two other (3 × 3 and 5 × 5 pixel) satellite-based
data sets.
The following pigments were included in the construction
of all different statistical models (further explained below):
MVChl a, DVChl a, chlorophyllide a, MVChl b, DVChl b,
Chl c1/2 (Chl c type 1 and 2), Chl c3 (Chl c type 3), Hex
(19-hexanoyl-fucoxanthin), But (19-butanoyl-fucoxanthin),
Allo (alloxanthin), Diadino (diadinoxanthin), Diato (diatoxanthin), Fuco (fucoxanthin), Lut (lutein), Peri (peridinin),
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A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
Viola (violaxanthin) and Zea (zeaxanthin). In addition, the
concentrations of different pigment groups were considered,
such as the sum concentration of α- and β-carotene (Caro);
MVChl a, DVChl a and chlorophyllide a (TChl a); MVChl b
and DVChl b (TChl b); all pheopigments (TPheo); according
to Hooker et al. (2005) and Roy et al. (2011), the photosynthetically active carotenoids (PSC: Fuco, But, Hex, Peri), and
the photoprotective carotenoids (PPC: Allo, Diadino, Diato,
Zea, Caro). PE, although measured by a different technique,
was also included since it also had a major impact on the optical Lu (λ) data of the field data set (see Taylor et al., 2013).
Neo and PE were only considered in the models using the
field data set because, for some pigment samples of the data
set collocated to the satellite data, these pigments have not
been analyzed.
Figure 2 gives an overview describing the various steps of
the development and validation of our EOF method to predict
various pigments and pigment groups’ concentrations, which
are described in detail in the following subsections.
2.3.1
Empirical orthogonal function analysis
Following Taylor et al. (2013), the spectral data were subjected to an EOF analysis, also known as a principal component analysis, in order to reduce the high dimensionality
of the data and derive the dominant signals (“modes”) that
best describe variance within the data set. In addition to dimension reduction of spectral data, the use of EOF modes
in statistical model building also avoids problems associated with multicollinearity amongst the original predictor
variables. All calculations in the following were done with
the statistical computing software R (R Development Core
Team, 2013).
Spectral data were contained in a data matrix X with dimensions M, sample rows, by N, reflectance band columns.
Spectral samples were collocated to the respective pigment data set Y with dimensions M by P , pigment
columns (pigments and pigment groups included are outlined above). While hyper_Rrs data consisted of 350–700 nm
(N = 351) or 380–700 nm (N = 321) bands, band_Rrs and
the satellite_ρwN data consisted of the eight MERIS visual
wavebands (N = 8). As in Taylor et al. (2013), spectral data
sets X were standardized for each sample row by first subtracting the mean spectral value (centering) followed by division by the spectral standard deviation (scaling), which focused the analysis on the spectral shape rather than the magnitude. The standardized matrix X was then subjected to singular value decomposition (SVD) in order to derive EOF
modes:
X
X = U6VT , xij =
uik σk vkj ,
(1)
k=l,N
where V is a N × N matrix containing the EOFs (spectral
pattern), U is an M × N matrix containing the principal components (PCs), 6 is an N × N matrix containing the sinwww.ocean-sci.net/11/139/2015/
143
gular values on the diagonal and k is the EOF mode index
(length N ). Only EOFs ≤ min (M, N ) will carry information.
This notation differs slightly from that presented in Taylor et
al. (2013), where a covariance matrix of the data set was subjected to Eigen decomposition with subsequent projection of
data onto EOFs to derive PCs. The results of both approaches
are similar except that U derived via SVD is unitary, and 6
contains standard deviation rather than variance. The SVD
method is presented here due to its more straightforward notation: EOFs and PCs are determined in a single step whereas
the alternate Eigen decomposition is a three-step calculation
(Fig. 2, the upper part of the panel on the left summarizes
these steps).
2.3.2
Log transformed general linear model
A general linear model was used to predict log-transformed
pigment concentrations of each pigment, yp , based on a
subset of PCs, U, as covariates (Fig. 2, the lower part of
the panel on the left summarizes these steps). The linear
model uses log-transformed pigment concentrations. Since
only positive, non-zero values are permissible with this transformation, a small value was added to all concentrations
(0.00001 mg m−3 ) to allow for the inclusion of samples
where pigment concentrations were essentially zero or below
the detection limit. A truncated subset of PCs was used as defined by the magnitude of their standard deviation. PCs with
standard deviations of ≤ 0.0001 times the standard deviation
of the first component were omitted. The resulting multiple
regression had the form
log(yp ) = a + b1 u1 + b2 u2 + · · · + bn un ,
(2)
where log(yp ) is the natural log-transformed concentration
of pigment p, u1,2,...n are the leading n PC scores from U, a
is the intercept and b1,2,...n are the regression coefficients. A
bidirectional stepwise routine was used to search for smaller
multiple regression models based on fewer predictor terms.
Best linear models were selected through minimization of
the Akaike information criterion (AIC). Once the best linear
model was determined, the relative importance of included
terms was defined by the change in AIC (1AIC) following
each term’s removal.
Since the range of concentration varies greatly among the
different pigments, we calculated mainly relative error statistics. According to Werdell et al. (2013), the coefficient of
determination (R 2 ), the RMSE, the slope (S) and the intercept (a) of the linear regression are based on the log-scaled
predicted (log(yp )) as opposed to the log-scaled observed
(log(yo )) pigment concentration data, while the MPD, the PB
and the median percent difference (MDPD) are based on the
non-log-transformed pigment concentrations. The following
Ocean Sci., 11, 139–158, 2015
144
A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
Figure 2. Schematic overview of the steps used in model building and prediction. Multiple linear regression models are fit to log-transformed
pigment concentrations, yp , as the response variable and EOFs derived from a spectral (reflectance) data set, X, as predictor variables. Model
building (left) is used for “full-fit” models to all data samples (M) or to a training subset of samples for cross-validation (Sect. 2.3.3).
Prediction (right) is used for the assessment of the model error on a validation subset of samples (I) for cross-validation (Sect. 2.3.3) or in the
extrapolation of model predictions to an new data set of reflectance spectra, as was done for the larger area of the tropical eastern Atlantic
region in this study (Sect. 2.3.4).
equations for these statistics were used:
RMSE = root of
MPD =
N
2
1 X
log(ypi ) − log(yoi ) ,
N i=1
N 100 X
ypi − yoi /(yoi )
N i=1
[%] ,
N
100 X
ypi − yoi /(yoi ) [%] ,
N i=1
MDPD = Median of ypi − yoi /(yoi ) × 100
i = 1, N [%] .
PB =
of points used for validation:
(3)
(4)
(5)
tp = n × d, with d = 0.1, 0.15, 0.2, . . ., 0.9,
(7)
vp = n (1 − d).
(8)
Since the number of permutations for data splitting affects the overall computing time, the procedure was run for
500 permutations, similar to the recommendation of Craig et
al. (2012). Such a high number of permutations rules out the
model error being assessed based on a spatially or temporally
biased data set.
Each cross-validation procedure was as follows:
(6)
1. For 500 permutations, do steps 2–8.
2.3.3
Model prediction error
In addition to the statistics performed for each pigment linear model (Sect. 2.3.2), we performed a cross-validation of
the linear model fitting in order to better test the robustness of
the models’ prediction error. Data were split into two groups:
the first part of the data was used for model fitting (Fig. 2, left
panel), while the second part was used for prediction validation (Fig. 2, right panel). According to Craig et al. (2012),
we assessed the number of observations required to achieve
adequate predictions by the pigment linear models using the
variable jack-knife procedure of Wu (1986). So the proportion used for data splitting for the cross-validation procedure
was varied as follows, where n is the total number of samples, tp is the number of training points and vp is the number
Ocean Sci., 11, 139–158, 2015
2. Randomly select n × d of collocated samples to include
in training sets Xtrain and Ytrain for spectra and pigment
data, respectively. Remaining n(1 − d) of samples are
allocated to the validation sets Xvalid and Yvalid .
3. Standardize Xtrain and perform EOF following Eq. (1)
to obtain Utrain , 6 train and Vtrain .
4. For each pigment concentration ypvalid of Yvalid , do steps
5–9.
5. Fit linear model to log-transformed pigment concentrations using selected Utrain as in Eq. (2):
+ b2 utrain
+ · · · + bn utrain
(9)
log yptrain = a + b1 utrain
n .
1
2
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A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
6. Perform bidirectional stepwise search for smaller linear
model.
3
7. Standardize validation set and project Xvalid onto the
−1
EOFs Vtrain and the inverse of singular values 6 train
to derive their PCs Uvalid :
3.1
−1
Uvalid = Xvalid · Vtrain · 6 train .
(10)
Use selected PCs of Uvalid as variables in Eq. (10) in order to predict pigment concentrations for the validation
data set:
.
+ · · · + bn uvalid
+ b2 uvalid
log ypvalid = a + b1 uvalid
n
2
1
(11)
8. Record pairs of observed and predicted validation pigment concentrations yo and ypvalid in a new object for all
permutations for later calculation of prediction error.
2
For each permutation,
the R based on the log-scaled
predicted (log ypvalid ) versus the log-scaled measured
(log(yp )) were derived and finally, over all permutations, the
mean value (R 2 cv) was calculated. In accordance with statistics in Sect. 2.3.2, the prediction error was described in terms
of the absolute squared difference based on log-transformed
pigment concentrations, (log(ypvalid )−log(yo )) 2 , and relative
difference based on non-log-transformed pigment concentrations, (ypvalid − yo )/yo . Mean and median relative difference (MPDcv and MDPDcv, respectively) and the root mean
square absolute difference (RMSEcv) over all permutation
were determined as follows:
N h
i
100 X
[%] ,
MPDcv =
(ypvalid − yo )/yo
(12)
N i=1
v
u
N h
i2
u1 X
log ypvalid − log(yo ) ,
(13)
RMSEcv = t
N i=1


pred
valid
yi,p − yi,p
MDPDcv = median value of 
× 100 ,
valid
yi,p
i = 1, N [%].
2.3.4
(14)
Pigment concentration predictions with MERIS
reflectance data
In order to predict pigment concentration from MERIS
ρwN (λ) for a whole month of data in November 2008, for
which we did not have corresponding pigment measurements, the following method was applied: we projected standardized MERIS ρwN (λ) data onto the EOF loading (V) to
derive their principal components (U), which were subsequently used for the prediction with the fitted linear model
(as in Sect. 2.3.3, step 7, Eq. 11, Fig. 2, right panel), where
b1,2,...n are taken from the EOF model developed with the
1 × 1 MERIS Polymer ρwN (λ) matchups (following Fig. 2,
left panel).
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145
Results and discussion
Characteristics of input data sets
Figure 3 shows the original and standardized spectra of the
field and satellite-based data sets. Considering the conversion of Rrs (λ) to ρwN (λ) data by a factor of π , the magnitude
and shape of the original and standardized spectra are similar for the band-resolved data sets, except that the standardized satellite_ρwN data set contains only one spectrum with
maximum reflectance in the green at 560 nm, while the standardized field data set contains four spectra with maxima at
510 nm.
The composition and range of pigments (as detailed with
maximum, minimum, mean and standard deviation in Supplement Table S2) show, for all pigments, that the collocations to the field data set contain higher maxima and minima than the collocations to the satellite-based data set (except for Fuco, for which it is equal and for Zea, for which it
is inverted). For most pigments, mean values are very similar for both data sets. However, standard deviations for the
field data set are 2 to 3 times higher than the mean for all
pigments. In the satellite data set, the standard deviation is
of a similar magnitude to the mean value. The higher concentration of total pigments in the field data set may explain
the small differences in the shape of the reflectance spectra
of the two (field versus satellite-based) data sets. However,
DVChl b, MVChl b, TChl b, Allo, Diato, Lut, Neo, Peri, Viola and TPheo had values of 0 mg m−3 in more than 20 % of
all stations in both data sets. Also, Chl c3 had a concentration of 0 mg m−3 in one sample collocated to the field and in
over 30 % of samples collocated to the satellite-based data
set. Several pigments had concentrations of 0 mg m−3 only
occasionally (< 10 %) in samples collocated to the satellitebased data set (Caro, Chl c1/2 , But, Hex, Zea, DVChl a, Diadino and Fuco) and in the field data sets (DVChl a, Diadino
and Fuco). All other pigments not listed here had detectable
concentrations in all samples.
3.2
EOF analysis – shape of modes and relevance for
predictions
Following the EOF truncation criteria outlined in Sect. 2.3.2,
the decomposition of the standardized spectra resulted in
nine modes (EOF-1 to EOF-9) for the hyper_Rrs and seven
modes for the band_Rrs and satellite_ρwN data sets (the first
four modes are presented in Fig. 4). EOF modes for the three
satellite_ρwN data sets were nearly identical. For simplicity
we only show (Fig. 4) and discuss the EOF modes of the
1×1 pixel collocation data set. For all data sets, the first three
modes explain over 99.8 % of the variance for all three data
sets, with EOF-1 explaining between 94.5 % and 96 % of the
variance (Table 1).
The shapes of the first three EOF modes are very similar
among all three reflectance data sets. They are nearly idenOcean Sci., 11, 139–158, 2015
146
A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
Figure 3. (a)–(c) Original (in sr−1 ) and (d)–(f) standardized (subtracted mean and divided by standard deviation) reflectance spectra of
hyper_Rrs (a) and (d), band_Rrs (b) and (e) and satellite_ρwN (from MERIS Polymer) data within the 1 × 1 pixel collocation boxes (c) and
(f).
Table 1. Percent of total variance explained (Expl. variation; upper panel) and cumulative proportion (Cum. proport.; lower panel) by the
significant EOFs derived from field Rrs spectra in hyperspectral resolution (hyper_Rrs ) and multispectral resolution (band_Rrs ) and from
satellite_ρwN (from MERIS Polymer) using the 1 × 1 pixel collocation criterion.
% Expl. variation
hyper_Rrs
band_Rrs
satellite_ρwN
% Cum. proport.
hyper_Rrs
band_Rrs
satellite_ρwN
EOF-1
EOF-2
EOF-3
EOF-4
EOF-5
EOF-6
EOF-7
EOF-8
EOF-9
95.0
94.5
95.9
4.1
5.0
3.9
0.7
0.4
0.1
0.1
< 0.1
< 0.1
< 0.1
< 0.1
< 0.1
< 0.1
< 0.1
< 0.1
< 0.1
< 0.1
< 0.1
< 0.1
< 0.1
EOF-1
EOF-2
EOF-3
EOF-4
EOF-5
EOF-6
EOF-7
EOF-8
EOF-9
95.0
94.5
95.9
99.1
99.5
99.7
99.8
99.9
99.9
99.9
99.95
99.95
99.965
99.990
99.985
99.981
99.999
99.996
99.990
100.0
100.0
99.994
99.996
tical for the band_ Rrs and the satellite_ρwN data sets but
show smoother shapes and peaks for hyper_Rrs for the first
two modes. Due to the limited number of wavelengths for
the two multispectral data sets, EOFs show evidence of a
shift in peak location, starting with EOF-3 (peak at 412 and
443 nm for EOF-3 and EOF-4, respectively), as compared
to hyper_Rrs (peak at 360 and 410 nm for EOF-3 and EOF4, respectively). This is likely due to the increased spectral
resolution of the hyperspectral data, which allows for more
precision in identifying spectral regions of higher variance.
For EOF-4, the satellite_ρwN mode is much flatter beyond
500 nm and shows no trough between 600 and 650 nm compared to the EOF-4 for the other two data sets. Not much
similarity is seen among the EOF-5 modes of the different
spectra data sets; for EOF-6, the two field data sets are simi-
Ocean Sci., 11, 139–158, 2015
lar in the overall shape, but peak locations are shifted towards
longer wavelengths for the satellite data set. EOF-7 and EOF8 show very similar shapes for hyper_Rrs and deviate from
EOF-7 in the band data sets, while EOF-9 from hyper_Rrs
looks much more like the later ones.
The EOF analyses identify dominant modes of variance,
which can be interpreted as imprints of changes in the optical properties of water constituents in the water column. For
this study, only reflectance spectra taken in high TChl a waters with measurable mineral fraction (identified as cluster V
for the ANTXXV/1 data in Taylor et al., 2011) show any resemblance to spectral shapes obtained in the case 2 waters of
Lubac and Loisel (2007, e.g., class 5) and Craig et al. (2012).
The remaining spectra (typical case 1 water) show characteristics not observed in those studies. This difference explains
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A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
147
Figure 4. First four EOF modes (EOF-1 to EOF-4) derived from field Rrs data set in hyperspectral resolution (hyper_Rrs , solid lines) and
in multispectral band resolution (band_Rrs , dashed lines) and from using satellite_ρwN (from MERIS Polymer, dotted line) data within the
1 × 1 pixel collocation box.
the minor variations in the shape and loading of EOFs between their and our data sets. In the following, we focus the
discussion on our hyper_Rrs data set results with specific
comparison to the study by Craig et al. (2012), which was
also based on hyperspectral Rrs data.
Our first three EOF modes correspond to the ones derived
for the hyperspectral case 2 reflectance data set of Craig et
al. (2012). As pointed out in their study, EOF-1 is likely the
signature of bulk oscillations in phytoplankton biomass concentration (including its effect on backscattering). However,
our EOF-1 already explains much more of the variance than
in Craig et al. (2012), where it only accounted for 72.4 %
and showed much more structure and a weaker exponential
decrease from 400 to 550 nm. EOF-2 superficially resembles the overall changes in the total absorption over broad
band structures. It strongly decreases from 350 to 510 nm
and increases again above 570 nm, which is connected to
total pigment and water absorption, respectively. There is a
peak around 683 nm which can be linked to MVChl a and
DVChl a fluorescence. While this peak is present in EOF-1
in the Craig et al. (2012) data set, it is not in the EOF-1 of our
data set likely because of the lower TChl a concentrations.
EOF-3 of our data set as compared to the one of Craig et
al. (2012) shows a much steeper decrease with wavelength
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in the blue spectral range. These changes may reflect concomitant changes of absorption by chlorophyll, colored dissolved organic matter and non-algal particles expected to be
co-varying and of much lower concentration in our case 1
waters. Scattering by particles other than phytoplankton was
much higher in the case 2 water of Craig et al. (2012), leading to a less steep slope of this EOF mode. EOF-4 appears
different in relation to the three peaks. Similar to EOF-2 and
EOF-3, these differences are caused by the different composition and overall loading of water constituents of our and
their sampled stations.
In summary, in contrast to more coastal waters where measurable mineral fraction can affect Rrs properties, the total
attenuation is much more affected by total pigment concentration in our open-ocean, case 1 data set. Our data set was
largely composed of samples from waters with lower TChl a
concentration, ranging from 0.005 to 3.553 mg m−3 , while
in the study of Craig et al. (2012) it ranged from 0.584
to 18.02 mg m−3 . EOFs greater than 4 were not presented
in Craig et al. (2012) because they were not used to predict TChl a from Rrs data, as was the case for our TChl a
(and MVChl a) linear model predictions (Sect. 3.3.3). Higher
EOF modes probably reflect the influence of specific pigment
Ocean Sci., 11, 139–158, 2015
148
A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
groups or pigments, as indicated by the results of the 1AIC
values and further discussed in Sect. 3.3.3.
3.3
3.3.1
Pigment prediction by linear models
Field data set linear models
All pigments that were detected in the full set of the field data
samples were well predicted by linear models based on hyperspectral (hyper_Rrs ) or the reduced eight-band (band_Rrs )
resolution spectra. The correlations between predicted and
observed concentrations for these pigments were highly significant (p < 0.0001) and cross-validation statistics reached
reasonable quality with R 2 cv ≥ 0.5, MDPDcv ≤ 45 % and
MPDcv ≤ 60 % (Table 2a, upper part). For some pigments
(TChl a, PSC, MVChl a, Hex, Caro) EOFs based on 380
to 700 nm produced much better linear model results using
hyper_Rrs data than based on 350 to 700 nm (for all statistical parameters see Supplement Table S3; models based on
hyper_Rrs (a) at 350 to 700 nm and (b) at 380 to 700 nm and
(c) on band_Rrs ). Lower quality for one statistical parameter
for both linear models was reached for Zea (R 2 cv 0.35 and
0.28), But (MPDcv 81 and 95 %) and for two parameters for
PE (MDPDcv 65 and 67 %, MPDcv 139 and 156 %).
Plots of observed versus predicted values for the full data
set of well-predicted pigments TChl a, PSC, PPC, Hex and
Zea are shown in Fig. 5. For pigment groups and pigments
with a high range of data (TChl a, PSC and Hex), covering
about 3 orders of magnitude, the intercept is much lower and
the regression closely aligns with the 1 : 1 reference line. The
predicted versus observed regression for Zea was of lower
quality (R 2 < 0.6) likely due to a much lower range of observed concentrations.
For all other pigments, predictions were of low quality (results not shown), demonstrating that the linear model approach does not produce robust predictions for situations
where pigment were not detected (i.e., 0 mg m−3 ) in every
sample (see results for all pigment predictions in Supplement
Table S3). Even pigments that were only occasionally undetected (e.g., DVChla, TChlb, MVChlb) showed increased
error in cross-validation prediction as revealed by MDPDcv
and RMSEcv values far above 100 % and 1, respectively. We
re-ran the predictions for specific pigments where only a few
samples (< 10 %) had concentrations of 0 mg m−3 , as was
the case for DVChl a, Fuco, Diadino and Chl c3 (see Supplement Table S2). In those specific linear model runs we
only included as input data the data points where the specific
pigment concentrations were > 0 mg m−3 . The resulting predictions (Table 2a, lower part; for DVChl a see full-fit results in Fig. 5d) from using the adjusted input data for those
pigments show robust and significant cross-validation results
within the same quality range as for the pigments which were
detected in all data. For other pigments, where non-detection
occurred more frequently (> 20 % of the samples), the reOcean Sci., 11, 139–158, 2015
moval of non-detection samples did not result in robust predictions (results not shown).
Cross-validation results of well-predicted pigments (Table 2a) show that, especially regarding the R 2 cv and RMSEcv values, hyper_ Rrs -based linear models perform either
the same (PSC), slightly better (PPC, Chl c1/2 ) or much better (TChl a, MVChl a, But, Hex, Zea, Caro, PE, DVChl a,
Chl c3 , Diadino, Fuco) than predictions based on eight wavelengths (band_Rrs data set). In particular, RMSEcv is much
improved for several pigment predictions where RMSEcv
reaches high values (> 0.65 mg m−3 ), i.e., for PE, Fuco, But,
Chl c3 , Diadino and Hex. The benefit was less clear when
observing the statistics of MDPDcv and MPDcv in several
pigments (MVChl a, Chl c1/2 , TChl a and PSC predictions).
For these pigments the multispectral resolution appears to be
sufficient for obtaining similarly robust predictions. TChl a
(in line with MVChl a) and PSC dominate the overall phytoplankton pigment composition and absorption. TChl a concentrations have been well retrieved by band-ratio algorithms
as a main phytoplankton biomass indicator (e.g., see Brewin
et al., 2014). For pigments very similar in spectral range,
such as But, Hex and Fuco, the hyperspectral resolution of
the linear models provides much more robust pigment predictions (Table 2a). The hyper_Rrs linear models also produced better predictions for DVChl a, Zea, Diadino and PPC,
where the specific linear models included a much larger set
of EOF modes (see Sect. 3.3.3) which may indicate the importance of higher-resolution spectral details not available in
the band_Rrs data.
3.3.2
Satellite-based data set linear models
Results for the models predicting pigment concentration
from the satellite-based data set were very similar when
using 1 × 1, 3 × 3 or 5 × 5 collocated MERIS ρwN data
(for all statistical parameters see Supplement Table S3:
satellite_ρwN models based on 1×1 (d), 3×3 (e) and 5×5 (f)
collocations). Deviations were within 1 to 3 % for all statistical parameters. R 2 cv values were best in all cases for
well-predicted pigment concentrations in the 1 × 1 collocations, while MPDcv was best in the 3 × 3 collocations. Results clearly show that even models based on 5 × 5 pixel collocations can produce robust results. For simplicity, in the
following we present and discuss the results of the 1 × 1 collocated reflectance data only.
In line with field data linear model results, pigment
groups and pigments, which were detected in every sample
(MVChl a, TChl a, PSC and PPC; the full-fit linear model
results are shown in Fig. 6a–c), are well predicted with similar cross-validation statistic values using the satellite_ρwN
data set (Table 2b, upper part). Also, good predictions for
some pigments (DVChl a, Zea, Diadino, Hex, But, Fuco
and Chl c1/2 ) could be obtained by re-running the linear
model analysis with concentrations of 0 mg m−3 excluded
(Table 2b, lower part). For example, the full-fit linear model
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A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
149
Table 2. Statistics of linear models using EOF modes based on a) field Rrs data in hyperspectral (hyper; normally 350–700 nm; when *
then 380–700 nm) resolution and multispectral (band) resolution and (b) the satellite_ρwN (from MERIS Polymer) using the 1 × 1 pixel
collocation criterion data set. Cross-validation results are presented with 500 permutations for data splitting into 80 % of the data used for
training and 20 % for validation. Only well-retrieved pigment prediction results, with correlations being highly significant at p < 0.0001,
are given. Abbreviations of pigments are explained in Sect. 2.3.1. Pigments listed in the upper part of each table show high-quality results
using the entire data set. In the lower part of each table (listed under “> 0 mg m−3 ”) models are based only on the data set of collocated Rrs
samples where the respective pigment reached concentrations above 0 mg m−3 . Bold: here band-model performs better than hyper-model.
Red signifies only medium quality as specified in the text.
(a)
MDPDcv
MPDcv
R 2 cv
RMSEcv
Band/hyper
N
band
hyper
band
hyper
band
hyper
band
hyper
TChl a*
PSC*
PPC
MVChl a*
Chl c1/2 *
But
Hex*
Zea
Caro*
PE*
53
53
53
53
53
53
53
53
53
53
28
32
28
31
34
44
37
31
34
67
32
34
27
34
36
43
36
28
33
65
42
51
51
45
48
95
57
45
55
156
43
53
49
44
49
81
53
42
50
139
0.54
0.62
0.57
0.54
0.57
0.90
0.69
0.52
0.62
1.28
0.49
0.62
0.56
0.50
0.56
0.82
0.60
0.48
0.54
1.16
0.72
0.75
0.52
0.77
0.79
0.50
0.64
0.28
0.55
0.65
0.77
0.75
0.52
0.79
0.80
0.55
0.70
0.35
0.62
0.69
49
52
52
52
26
40
45
37
24
37
40
33
45
61
92
59
39
58
73
53
0.49
0.86
1.02
0.71
0.44
0.69
0.82
0.61
0.59
0.71
0.65
0.57
0.67
0.73
0.66
0.67
> 0 mg m−3 :
DVChl a
Chl c3 *
Fuco*
Diadino
(b) satellite_ρwN
TChl a
PSC
PPC
MVChl a
N
MDPDcv
MPDcv
RMSEcv
R 2 cv
139
139
139
139
32
42
27
40
47
59
40
55
0.64
0.79
0.49
0.67
0.67
0.70
0.50
0.72
124
127
126
132
128
138
132
40
41
39
40
40
29
45
59
89
55
60
65
46
91
0.73
0.84
0.63
0.68
0.85
0.51
0.84
0.25
0.64
0.74
0.68
0.71
0.40
0.57
> 0 mg m−3 :
DVChl a
Chl c1/2
X19_But
X19_Hex
Fuco
Zea
Diadino
results for DVChl a, Hex and Zea are shown in Fig. 6d–f.
Nevertheless, some of these pigments show only medium
quality for one cross-validation statistical parameter (lower
R 2 cv for DVChl a and Zea, higher MPDcv for Fuco, Chl c1/2
and Diadino).
The full-fit results shown in Fig. 6 show that the models based on the satellite data show much poorer predictions
(e.g., a, R 2 and RMSE) than the field data models for all pigment or pigment groups (except Zea) even though the satellite data models are based on more samples. This may be
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caused by the lower quality of water-leaving reflectance data
obtained from the satellite as opposed to direct radiometric
measurements in the water column. Another explanation may
be that the lower standard deviation of the pigments in the
satellite-based data set leads to less precision of the EOFbased models. The latter may explain why the full-fit results
for predicting Zea concentrations are very similar for the two
model types.
Similar to the field data linear models, no robust predictions were obtained for all other pigments that reached
Ocean Sci., 11, 139–158, 2015
150
A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
Table 3. 1AIC for the robust pigment predictions of the pigment groups TChl a, PSC and PPC and the pigments MVChl a, Zea and DVChl a
by the EOF models based on field Rrs in (a) hyperspectral resolution (hyper_Rrs ) and (b) multispectral resolution (band_Rrs ) and (c) the
satellite_ρwN (from MERIS Polymer) using the 1 × 1 pixel collocation criterion. The pigments listed under “no 0 mg m−3 ” were predicted
using a reduced data set where the respective pigment reached concentrations above 0 mg m−3 . Bold highlights the EOF mode with the
highest 1AIC.
(a) hyper_Rrs
TChl a
PSC
PPC
MVChl a
Zea
EOF-1
EOF-2
EOF-3
EOF-4
16
9
33
16
21
16
8
5
80
78
35
88
6
7
10
21
EOF-1
EOF-2
2
86
80
38
90
5
12
17
EOF-1
EOF-2
7
5
10
4
148
146
62
151
38
13
63
EOF-5
EOF-6
4
0.4
EOF-7
EOF-8
3
2
EOF-9
11
10
7
1
5
9
21
1
20
9
4
7
EOF-3
EOF-4
EOF-5
EOF-6
EOF-7
10
5
20
10
6
15
5
9
9
10
3
6
1
1
4
20
23
EOF-4
EOF-5
no 0 mg m−3 :
DVChl a
(b) band_Rrs
TChl a
PSC
PPC
MVChl a
Zea
1
7
0.3
EOF-6
EOF-7
no 0 mg m−3 :
DVChl a
(c) satellite_ρwN
TChl a
PSC
PPC
MVChl a
EOF-3
1
16
3
5
0.2
3
4
no 0 mg m−3 :
Zea
DVChl a
17
26
> 0 mg m−3 in less than 80 % of all samples concentrations,
even when only data points with specific pigment concentrations > 0 mg m−3 were included (results not shown).
3.3.3
EOF modes relevant for pigment predictions
Table 3 presents the results of EOF significance based on
1AIC from their removal as model terms. For the hyper_Rrs
data set, the prediction linear models used EOF-2 and EOF3 for all pigments. EOF-2 was the most relevant in the respective models for all pigment prediction except for Zea and
DVChl a, for which EOF-3 was the most important, closely
followed by several other EOF modes. For all other wellpredicted pigments, EOF-3 followed EOF-2 in importance,
except for Chl c3 (EOF-4) and PE (EOF-1). Besides PE, only
EOF-1 was included (with medium importance) for the prediction of But, DVChl a and Zea concentrations. Nearly all
linear models using the hyper_Rrs data set to predict pigment
concentrations incorporated the loadings of three to five EOF
Ocean Sci., 11, 139–158, 2015
35
1
2
11
6
modes. In contrast, predictive models for DVChl a, Zea and
PPC incorporated nine, eight and six EOF modes, respectively.
As discussed in Sect. 3.2, EOF-2 reflects the optical imprint of all phytoplankton pigments. The high 1AIC value of
EOF-2 for most pigments’ linear models is probably caused
by the increase in concentration of these specific pigments
and most phytoplankton groups when TChl a increases. In
contrast to that, cyanobacteria and especially its subgroup
Prochlorococcus, containing the marker pigments Zea and
DVChl a, respectively, are the most abundant phytoplankton under low TChl a concentrations. This has manifested
in the abundance-based algorithms to retrieve picoplankton
from TChl a data (Uitz et al., 2006; Hirata et al., 2011) and
may explain why predictions of those marker pigments by
our linear models show lower 1AIC for EOF-2 and require
several different EOF modes in their linear models.
As in Craig et al. (2012), EOF-2 to EOF-4 were relevant
for our hyper_Rrs -based TChl a and MVChl a predictions.
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151
Figure 5. Examples of regressions between observed (obs.) and predicted (pred.) concentrations for pigment groups, (a) TChl a, (b) PSC and
(c) PPC, and specific pigments, (d) DVChl a, (e) Hex and (f) Zea. Observed values have been measured by HPLC (obs.), while predictions
are made using a linear model based on EOF modes derived from field Rrs data in hyperspectral resolution (hyper_Rrs ). For DVChl a, the
model data set was reduced by excluding collocated samples where DVChl a had concentrations of 0 mg m−3 .
EOF models developed by Taylor et al. (2013) to predict PE
concentrations based on Lu data required the first four EOF
modes, while our PE prediction based on Rrs data required
the first three EOFs only. For all other pigments, the higher
EOFs were also necessary for robust predictions.
Similarly to the hyper_Rrs linear models, the two multispectral linear models also showed EOF-2 to be the most
important predictor for specific pigment models except for
DVChl a (both models) and Zea (only band_Rrs ).
3.4
Number of data points to construct robust models
Our presented linear models to predict specific pigment or
pigment group concentration are calibrated for an oceancolor data set of a specific region with coincidental pigment measurements. Results of the variable jack-knife procedure indicate that the minimal number of training points
needed to set up a robust linear model varies among pigments and pigment groups, as revealed by several statistical
error measures: the ratio of R 2 cv to R 2 (R 2 cv /R 2 ), the ratio
of MPDcv to MPD (MPDcv / MPD) and the ratio of RMSE
to RMSEcv. Examples for predicting TChl a, PSC, PPC and
PE are shown in Fig. 7. The ratio R 2 cv / R 2 for PPC in all
linear models (Fig. 7a, d) drops below 0.8 after a threshold
of 50 training data and then decreases exponentially with diminishing data, while other pigments can maintain a high ratio with as few as 30 samples and even 15 samples in the
case of the hyper_Rrs PE linear model. The threshold where
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the slope increases in RMSEcv / RMSE (Fig. 7c, f) is for all
pigments and linear models probably around 20 to 30 training points. MPDcv/MPD ratios below 1.4, which would indicate robust fits, are obtained for all pigments above 50 training points for the satellite_ ρwN (Fig. 7e) and above 30 for
the hyper_Rrs data sets (Fig. 7b). Generally, we observe that
band_Rrs -based models are more sensitive to training sample size as compared to the hyper_ Rrs -based models, especially for TChl a and PE. As a general recommendation, a
requirement of at least 45 to 50 training data points is advised
for most cases, while some pigments (e.g., TChl a) may be
well predicted with as few as 25 training samples when using
models based on hyper_Rrs data. Based on these results, we
are confident that the models presented in Sect. 3.3 are able
to provide robust predictions for both field and the satellitebased data. In the case of PE, the number of samples seems
to have been too small, especially for the multispectral resolution, to provide robust PE predictions.
3.5
Comparison to other approaches deriving pigment
concentration
Our hyper_Rrs TChl a linear model results (R 2 = 0.84,
RMSE = 0.4, R 2 cv = 0.77, RMSEcv = 0.49; Fig. 5 and Table 2) are comparable to results by Craig et al. (2012;
R 2 =0.84, RMSE = 0.3, R 2 cv = 0.76, RMSEcv = 0.21).
Craig et al. (2012) used measurements only from a single location, sampled about weekly over the course of 1 year, while
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A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
Figure 6. Examples of regressions between observed (obs.) and predicted (pred.) concentrations for pigment groups, (a) TChl a, (b) PSC and
(c) PPC, and specific pigments, (d) DVChl a, (e) Hex and (f) Zea: observed values have been measured by HPLC (obs.), while predictions are
made using a linear model based on EOF modes derived from satellite_ρwN (from MERIS Polymer) data within the 1 × 1 pixel collocation
box. For DVChl a, Hex and Zea, the model data set was reduced by excluding collocated samples where DVChl a had concentrations of
0 mg m−3 .
our field data set was from a much larger region (covering
95◦ in latitude and 85◦ in longitude) and sampled during two
seasons in 2008 and 2010 only. In their study the same linear
model setup was used with collocated in situ reflectance and
TChl a data sampled at Compass Buoy station in the Bedford
Basin near Halifax as input data. The number of collocations
used for training to obtain robust results for TChl a predictions was also similar for both studies, with more than 25
recommended for our hyper_Rrs linear model and more than
15 in the Craig et al. (2012) study.
Chase et al. (2013) used Gaussian functions to derive different chlorophyll types, PSC and PPC concentrations from
a large global data set of hyperspectral particulate absorption measurements. Their validation results showed MDPD
values between predicted and observed concentrations of 30
and 36 %, 40 and 53 %, 49 % and 51 % for TChl a, TChl c,
PSC and PPC, respectively. Our linear models show similar (TChl a 27–32 %) or even much better MDPDcv values (Chl c1/2 : 34–41 %, PSC: 32–43 %, PPC: 24–28 %). We
believe that this further indicates the robustness of our approach, especially given that we use a more indirect measure
of pigments, AOP (reflectance), as opposed to the IOPs used
in their study.
Pan et al. (2010) developed pigment specific band-ratio
algorithms with collocated in situ Rrs (λ) and pigment measurements from the northeastern coast of the United States.
Ocean Sci., 11, 139–158, 2015
Those algorithms are based on deriving pigment-specific coefficients for third-order polynomial functions using the band
ratio of either 490–550 nm or 490–670 nm (for SeaWiFS;
for MODIS changed accordingly to MODIS bands 488 and
547 nm). Validation of results with collocated satellite (SeaWiFS and MODIS) reflectance data and pigment concentrations showed very good-quality predictions for several pigments (TChl a, TChl c, Caro, Fuco, Diadino and Zea) using
SeaWiFS bands (MPD from 36 to 48 %, RMSE from 0.23
to 0.29, and R 2 from 0.65 to 0.90; similar results were also
obtained using MODIS bands). This method was modified
to the northern South China Sea using globally derived relationships and locally identified links between pigment concentration and sea surface temperature (Pan et al., 2013) with
similar validation results as in Pan et al. (2010). Compared
to our linear model results, the quality of pigment concentration prediction is similar: while our results for MPDcv and
R 2 cv are slightly worse (42–50 % and 0.61–0.80, respectively), our results for RMSEcv (0.48–0.61 mg m−3 , except
Fuco: 0.82 mg m−3 ) are much better.
PE is not well predicted by both our linear models based
on the field data set. Still, hyper_Rrs linear model crossvalidation measures are much better than the PE band_Rrs
linear model. In Taylor et al. (2013), PE concentrations were
predicted from the same underwater light measurements but
using Lu instead of Rrs data and the model was based on pig-
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A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
153
Figure 7. R 2 cv / R 2 (a, d), MRDcv / MRD (b, e) and RMSEcv / RMSE (c, f) as a function of number of training points (tp) for the linear
models. Shown are results for specific models for TChl a, PSC, PPC and PE using reflectance data from the field (a–c) in hyperspectral
(hyper_Rrs , solid lines) and multispectral band (band_Rrs , dotted lines) resolution and from satellite MERIS Polymer within the 1 × 1
pixel collocation box (satellite_ρwN , (d–f)). The number of total samples points was n = 53 for hyper_Rrs and band_ Rrs and n = 139 for
satellite_ρwN . Cross validation is based on 500 permutations using tp for training and as number of validation points (vp): vp= n−tp.
ment concentrations at surface and deeper depths. No crossvalidation was performed within their study. Our results for
R 2 cv (0.69) are even better than their results for using the
data from all three cruises for predictions (R 2 of 0.58). The
data set of Taylor et al. (2013) was nearly 3 times larger
than our field data set and a log-link generalized linear model
(GLM) was used instead of a log-transformed linear model.
For the latter we tested both settings for our pigment linear
models. Cross-validation revealed a similar prediction error
for PE using the log-link GLM instead of the log-transformed
linear model, but the error increased when GLM was used for
other pigment predictions.
As for TChl a predictions from the satellite_ρwN linear
model, validation results of the MERIS Polymer TChl a
product collocations with in situ TChl a from the satellitebased data set showed marginal differences for the 1×1, 3×3
or 5×5 pixel collocations (Table 4, upper panel). The TChl a
Polymer product obtained 3 % higher MPD and similar R 2 ,
RMSE and PB values (of about 0.74, 0.51 and 10 % on average, respectively) to the TChl a linear model predictions.
In the global validation by Brewin et al. (2015), the OC4V6
(Ocean-Chlorophyll-4 algorithm version 6; O’Reilly et al.,
2000) was selected from amongst various TChl a satellite
products as the best TChl a algorithm. This algorithm is used
to produce the MERIS Polymer TChl a from atmosphericcorrected MERIS Polymer data. Global validation by Brewin
et al. (2015), with 1039 collocations and retrievals of TChl a
directly from in situ ρwN (λ) data, showed an R 2 of 0.87 and
a RMSE of 0.29 for OC4V6 based on non-log-transformed
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concentrations (which compares to our RMSE values on logscale shown in Table 4 of Bracher et al., 2014). We conclude
that both MERIS Polymer TChl a products, the level 2 and
linear models, show high quality within the eastern Atlantic
Ocean although they are retrieved from satellite data and not
in situ ρwN data.
The comparison with other methods of retrieving pigment
concentrations from reflectance data shows that our method,
based on a linear model using EOFs from reflectance data,
gives robust results for pigment groups and pigments that are
always present in the region investigated. To test our EOF
methods for independent data sets using the method established by a certain testing data set, we have used the crossvalidation technique. The technique allows the re-sampling
of all data for 500 different subsets (i.e., run by 500 permutations) into testing and validation data sets.
The advantage of our approach is that it allows for the estimation of several pigments and pigment groups using either reflectance data measured directly in the ocean water or
obtained from a satellite ocean-color sensor. For the eastern
tropical Atlantic Ocean data set, these additional pigments
(other than TChl a) include PPC, PSC, DVChl a and MChl
a. Additional pigments may also be accurately predicted with
this approach; however, the results suggest that the prediction
error increases for pigments that are found in lower concentrations or with a high number of samples below the detection limit (i.e., referred to in statistics as “censoring”). This
poor performance may be in part due to the fact that pigments
found in small concentrations are likely to have a limited efOcean Sci., 11, 139–158, 2015
154
A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
Table 4. TChl a validation statistics for MERIS Polymer TChl a (left panel) and TChl a obtained from the full-fit linear model on satellite_
ρwN (from MERIS Polymer data, EOF full-fit model, right panel) with different collocation criteria (either 1 × 1 or the mean of 3 × 3 or 5 × 5
pixel values) for the MERIS Polymer data compared to the in situ (from HPLC) value.
MERIS
Polymer
EOF full-fit model
Collocation
N
R2
RMSE
PB
MPD
R2
RMSE
PB
MPD
1 × 1 pixel
3 × 3 pixel
5 × 5 pixel
139
155
160
0.74
0.75
0.73
0.51
0.50
0.51
6
10
11
39
39
40
0.72
0.74
0.74
0.54
0.52
0.52
10
10
10
37
36
36
fect on spectral shape, but further modeling work may also
need to focus on better approaches for the treatment of censored values. Generally, we can also see from the field data
linear models that using a coherent in situ data set, where
all pigments have been measured by the same method and
instrumentation, may be better suited for the modeling approach due to the homogeneous error across the range of pigment concentrations. An advantage of our linear method to
pigment-specific band algorithms is that we require a much
smaller data set for establishing the prediction (about 50 as
opposed to several hundreds) of collocated pigment and reflectance data.
3.6
Application of linear model to study large-scale
pigment distributions
For demonstrating the application of our linear model, we
used the satellite_ρwN specific pigment’s full-fit models for
TChl a, MVChl a, PSC and PPC and ran these specific models using November 2008 MERIS Polymer ρwN level 2 data
to retrieve those pigments for an example time period on a
larger spatial scale. By subtracting the MVChl a value from
TChl a we also derived concentrations of DVChl a. Figure 8 shows the monthly averages for those various pigment
groups and pigments. Also, the MERIS Polymer TChl a concentration for the same time and region is shown.
The distributions of TChl a from the EOF model prediction or from the Polymer algorithm are very similar, ranging from 0.00003 to 7.52 mg TChl a m−3 . For this particular
month, the total biomass of phytoplankton shows a strong
phytoplankton bloom (> 2 mg m−3 ) at the Mauritanian upwelling spread in two parts, 190–24◦ N and 14–7◦ N, and
high values (> 0.5 mg m−3 ) at all coastal areas of the African
continent. Enhanced TChl a concentrations > 0.3 mg m−3
are also spreading into the open ocean especially at 5–20◦ N
and 30–40◦ W, along the 0◦ latitude from Africa to South
America, and south of this at 3–10◦ S from 3◦ E to about
25◦ W. MVChl a follows more or less the TChl a distribution, however, only reaching the magnitude indicated by
the TChl a values at the northern bloom. The deviation between TChl a and MVChl a is obvious in the distribution
of DVChl a, which indicates that at the northern part of
the Mauritanian upwelling bloom, Prochlorococcus (the only
Ocean Sci., 11, 139–158, 2015
phytoplankton genus which contains DVChl a) seems to
have contributed to this bloom by only a very minor fraction
(i.e., a few percent), while elsewhere it presents a substantial
background of about 30 % of all phytoplankton.
Our predicted PPC concentrations show values in the same
range as TChl a at the oligotrophic areas and about 50 %
in the enhanced TChl a areas and the southern part of the
bloom. As for DVChl a, in the northern part of the bloom
PPC concentrations are significantly lower and only contribute less than 10 % to the total pigment concentrations.
PSC concentration in the oligotrophic and enhanced TChl a
areas are much lower than PPC or even DVChl a concentrations but reflect the TChl a distribution more or less on the
large scale. Within the northern part of the Mauritanian upwelling PSC concentrations reach values even as high as for
TChl a, while concentrations at the bloom further south contribute to less than 10 % of the total pigment concentrations.
In Taylor et al. (2011) the analysis of pigment and additional
microscopic data clearly showed very high concentrations of
Fuco, a main pigment of PSC, and a high dominance of diatoms within water samples at the northern bloom collected
at the same time period.
From our results, we can conclude that the northern phytoplankton bloom at the Mauritanian upwelling seems to have
been freshly growing with very high photosynthetic activity,
while for most of the other areas a lot of the energy build-up
via photosynthesis was used for photoprotection. We have
no information on photodegradation since no significant prediction linear model could be developed for phaeopigments.
These pigments had only been identified in less than 60 %
of all samples collocated to the field and satellite-based data
sets, and the results show that this pigment group was not
well predicted by the linear model. Based on the biogeography of Longhurst (2006), the oligotrophic areas on our maps
fall in the North Atlantic Subtropical Gyre Province East at
> 25◦ N (the border between the two is the subtropical convergence) and the North Atlantic Tropical Gyre Province at
25◦ N to about 12◦ N. At the eastern corner towards the coast
of these provinces, in the Canary Coastal Province (CNRY),
concentrations of all predicted pigments and pigment groups
may have been increased due to eddy-driven processes that
increase the supply of nutrients. In Taylor et al. (2011), the
two blooms analyzed by field samples at CNRY have been
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A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
155
Figure 8. Monthly mean concentrations (in 0.25◦ grid resolution) for November 2008 of (a) TChl a of the MERIS Polymer algorithm
(TChl a MERIS Polymer) and predicted (b) TChl a, (c) MVChl a, (d) DVChl a, (e) PSC and (f) PPC by the LM, based on the full fit of
satellite ρwN data within the 1 × 1 pixel collocation box and the EOFs of this month’s MERIS Polymer ρwN data.
identified to cluster differently due to their pigment composition. The northern “fresh” bloom with low photoinhibition,
high dominance of phytoplankton and strong photosynthetic
efficiency was related to a major upwelling focusing in the
area south of Cape Blanc (western Sahara) off the coast of
Mauritania. DVChl a was absent in this bloom, which is in
line with our results obtained from the linear model. The
southern part of the CNRY bloom was placed within the
African dust veil where mineral-rich dust fertilizes the ocean.
In the northern bloom stations, the spectral shape and absolute values of particulate non-phytoplankton absorption spectra, presented in Taylor et al. (2011) and coinciding with the
pigment data used in our study, clearly indicated that mineral
particle absorption was very high.
Comparisons of our predictions to pigment data not used
for the development and validation of our EOF model show
consistent results: Partensky et al. (1996) measured TChl a
concentrations of about 1.2 mg m−3 in December 1992 (EU-
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MELI 5 cruise) at a station within a phytoplankton bloom
at 18◦ 290 N and 21◦ 050 W, similar to the range of our predicted values at the southern edge of the northern bloom.
Barlow et al. (2002) measured, within the area of our predictions, concentrations of TChl a, DVChl a, PSC and PPC
during the AMT-3 cruise in October 1996 at 20◦ N and 20◦ W
(0.4, 0.05, 0.175 and above 0.09 mg m−3 , respectively) and
30◦ N and 22◦ W (0.05, 0.01, 0.022 and above 0.04 mg m−3 ,
respectively), similar to our predicted concentrations for the
same pigments just east of the northern bloom of the Mauritanian upwelling and at the North Atlantic Subtropical Gyre
Province East, respectively.
4
Conclusions
We present robust predictions of concentrations of various
pigments and pigment groups from linear models based on
fitting empirical orthogonal function on a set of reflectance
Ocean Sci., 11, 139–158, 2015
156
A. Bracher et al.: Using empirical orthogonal functions derived from remote-sensing reflectance
data to collocated pigment concentrations. Spectral shapes
of the reflectance spectra from the eastern Atlantic and
of their derived EOF modes reflect typical case 1 water
characteristics. In our study, it was shown that EOFs derived from both hyperspectral underwater radiometric measurements and multispectral reflectance data from field or
satellite (MERIS Polymer) enable reliable predictions of
the concentration of nine different pigments/pigment groups
(TChl a, PPC, PSC, MVChl a, Chl c1/2 , But, Hex, Zea, Caro,
PE). A limitation of all predictions was that only those pigments can be predicted that have been identified in every
collocated sample; adding a small value (0.0001 mg m−3 )
to censored samples was not an appropriate solution to this
problem.
The method proves for the first time to be applicable for
predicting concentrations of not only TChl a and PE but
also of other pigments and pigment groups with weaker, but
spectrally unique, features on the underwater light field. Statistical resampling used for cross-validation indicates that
predictions were robust (R 2 cv ≥ 0.5, MDPDcv ≤ 44 % and
MPDcv ≤ 60 %) for all pigments (except for PE, Zea and
But, which deviated for one of these measures) and pigment
groups. Hyperspectral linear models proved to be already stable with less collocated samples for most pigment or pigment
groups used for training (n > 30 to 40) than linear models
based on multispectral reflectance data (n > 50). The linear
models using MERIS Polymer reflectance data as input were
applied to 1 month of satellite data to predict the concentrations of TChl a, PSC, PPC, MVChl a and DVChl a for
the whole eastern tropical Atlantic. For the first time a consistent picture of several phytoplankton pigments indicating
group-specific behavior and photophysiology on a larger spatial scale for this area was shown.
Our presented linear models are generic and can be applied
to even a small, consistently collocated reflectance and pigment data set to enable various specific pigment predictions
from continuous optical measurements. The optical data can
be obtained from radiometric measurements based on various platforms (buoys, gliders, floats or satellite). On a global
scale, TChl a, PSC and PPC are consistently accurately predicted, while other pigments may be better predicted on
smaller spatial scales. Highly temporally resolved time series
data, which – depending on the platform – may even provide
good spatial coverage, can be used to study variability and
change of overall phytoplankton and photophysiological responses to environmental variables. While we established the
linear models for prediction of various pigments in typical
case 1 waters, the method should be tested in the future for
its applicability in case 2 waters as well.
Ocean Sci., 11, 139–158, 2015
The Supplement related to this article is available online
at doi:10.5194/os-11-139-2015-supplement.
Author contributions. A. Bracher designed and ran the experiments
and wrote the manuscript. M. H. Taylor developed the statistical
method and wrote the R code; B. Taylor supplied all field and part
of the additional pigment data, prepared all input data and designed
part of the experiment; T. Dinter contributed to programming; R.
Röttgers contributed to Rrs data and additional pigment data; and
F. Steinmetz supplied MERIS Polymer data and match-ups. All coauthors assisted in writing the manuscript.
Acknowledgements. We thank AWI, Helmholtz Impulse Fond
(HGF Young Investigators Group Phytooptics), Total Foundation
(project PHYTOSCOPE) and the Leibniz Association (project
OCEANET) for funding. We thank Erika Allhusen, Helmke
Hepach, Kerstin Heymann, Laila Bentama, Anja Bernhardt and
Sonja Wiegmann for work in the laboratory and on-board RVs
Polarstern and Merian S. Merian. We thank the crew, principal
investigators and other scientists on ANTXXIII/1, ANTXXIV/1,
ANTXXIV/4, ANTXXV/1, ANTXXVI/4 and MSM 18-3 for support
on board and fruitful discussions afterwards. We thank Oliver
Zielinski and his group at ICBM, University Oldenburg, previously
at the University of Applied Sciences, Bremerhaven, for the use of
their equipment. We thank ESA for MERIS level 1 data. We thank
Amélie Talec for BGH sample collection and Hervé Claustre and
Josephine Ras (all LOV) for BGH pigment sample analysis. This
study contains data supplied by the Natural Environment Research
Council (NERC) via BODC and NASA via SEABASS. We thank
all researchers behind these large pigment data sets.
Edited by: O. Zielinski
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