Frontiers | Approach for Propagating Radiometric Data Uncertainties Through NASA Ocean Color Algorithms

Frontiers | Approach for Propagating Radiometric Data Uncertainties Through NASA Ocean Color Algorithms
METHODS article
Front. Earth Sci.
, 18 July 2019
Sec. Atmospheric Science
Volume 7 - 2019 |
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METHODS article
Front. Earth Sci.
, 18 July 2019
Sec. Atmospheric Science
Volume 7 - 2019 |
Approach for Propagating Radiometric Data Uncertainties Through NASA Ocean Color Algorithms
Lachlan I. W. McKinna
1,2
Ivona Cetinić
2,3
Alison P. Chase
P. Jeremy Werdell
1.
Go2Q Pty Ltd., Buderim, QLD, Australia
2.
Ocean Ecology Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD, United States
3.
GESTAR/Universities Space Research Association, Columbia, MD, United States
4.
School of Marine Sciences, University of Maine, Orono, ME, United States
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Abstract
Spectroradiometric satellite observations of the ocean are commonly referred to as “ocean color” remote sensing. NASA has continuously collected, processed, and distributed ocean color datasets since the launch of the Sea-viewing Wide-field-of-view Sensor (SeaWiFS) in 1997. While numerous ocean color algorithms have been developed in the past two decades that derive geophysical data products from sensor-observed radiometry, few papers have clearly demonstrated how to estimate measurement uncertainty in derived data products. As the uptake of ocean color data products continues to grow with the launch of new and advanced sensors, it is critical that pixel-by-pixel data product uncertainties are estimated during routine data processing. Knowledge of uncertainties can be used when studying long-term climate records, or to assist in the development and performance appraisal of bio-optical algorithms. In this methods paper we provide a comprehensive overview of how to formulate first-order first-moment (FOFM) calculus for propagating radiometric uncertainties through a selection of bio-optical models. We demonstrate FOFM uncertainty formulations for the following NASA ocean color data products: chlorophyll-a pigment concentration (
Chl
), the diffuse attenuation coefficient at 490 nm (
,490
), particulate organic carbon (
POC
), normalized fluorescent line height (
nflh
), and inherent optical properties (IOPs). Using a quality-controlled
in situ
hyperspectral remote sensing reflectance (
rs,i
) dataset, we show how computationally inexpensive, yet algebraically complex, FOFM calculations may be evaluated for correctness using the more computationally expensive Monte Carlo approach. We compare bio-optical product uncertainties derived using our test
rs
dataset assuming spectrally-flat, uncorrelated relative uncertainties of 1, 5, and 10%. We also consider spectrally dependent, uncorrelated relative uncertainties in
rs
. The importance of considering spectral covariances in
rs
, where practicable, in the FOFM methodology is highlighted with an example SeaWiFS image. We also present a brief case study of two
POC
algorithms to illustrate how FOFM formulations may be used to construct measurement uncertainty budgets for ecologically-relevant data products. Such knowledge, even if rudimentary, may provide useful information to end-users when selecting data products or when developing their own algorithms.
Introduction
NASA has continually collected, processed, archived, and distributed global ocean color data since the launch of the Sea-viewing Wide Field-of-View Sensor (SeaWiFS) in 1997. This two decades-long multi-sensor data climatology continues to provide unprecedented synoptic-scale insight into near-surface oceanographic processes. Some of the satellite-derived variables, such as chlorophyll-a pigment concentration
Chl
(mg m
−3
), are considered as Essential Climate Variables (ECV) and are widely used by the oceanographic community to study phytoplankton ecology, marine biogeochemistry, and ecosystem responses to climate change (IOCCG,
2008
; McClain,
2009
; Franz et al.,
2017
).
Following formal definitions outlined in the Guide to Uncertainty in Measurement (JCGM,
2008
), we can outline the objective of ocean color remote sensing as, to
measure
oceanographic quantities or
measurands
. We note that the
measurement procedure
involves a number of mathematical steps and assumptions that derive the
measurand
from sensor-observed top-of-atmosphere radiances. Thus, a derived ocean color data product is a
result of measurement
and should always be treated as an estimate of the
measurand
which has inherent
uncertainty
Quantifying uncertainty in derived ocean color data products (i.e., measurands) is highly valuable, allowing end-users to: assess if datasets are fit-for-purpose, assess if observed temporal change is greater than uncertainty, assimilate uncertainties into climate models, and assess consistency among sensors (Maritorena et al.,
2010
; Gould et al.,
2014
). Additionally, a thorough understanding of uncertainty sources within a model may help guide the decisions of scientists when developing new satellite algorithms.
The measurement uncertainty (
measurement
), in an ocean color data product,
, can be expressed as the following:
where
model
) represents uncertainties in
due to inherent inaccuracies/limitations in the algorithm (e.g., model coefficients), and
data
) represents uncertainties in
due to uncertainties in sensor-observed radiometry (data). In this paper we focus on
data
), that is, the propagation of radiometric uncertainties through bio-optical algorithms. For brevity, we shorten
data
) to
) throughout this paper unless otherwise stated.
For the ocean color community, much of our understanding of measurement uncertainty in derived data products is sourced from validation exercises using
in situ
datasets (Bailey and Werdell,
2006
; Antoine et al.,
2008
; Melin,
2010
; Mélin et al.,
2016
) or from Monte Carlo-type simulations (Wang et al.,
2005
). We note that advanced statistical methodologies have also emerged for predicting uncertainties in derived ocean color products (Moore et al.,
2009
; Salama et al.,
2009
; Jay et al.,
2018
). While validation studies remain critical for appraising the absolute skill of an ocean color algorithm, such datasets themselves have their own measurement uncertainty associated with
in situ
observations (including uncertainties associated with subpixel temporal/spatial/environmental variability). Monte Carlo-type analyses are particularly useful for understanding measurement uncertainty, however, these approaches can be computationally expensive and are impracticable to implement within pixel-by-pixel ocean color processing.
More recently, analytical first-order first moment (FOFM) methods have been proposed that can directly propagate radiometric uncertainty through an ocean color algorithm to estimate derived data product uncertainty (Neukermans et al.,
2009
; Salama et al.,
2009
2011
; Lee et al.,
2010
; Maritorena et al.,
2010
; Lamquin et al.,
2013
; Qi et al.,
2017
). These approaches are based on the
law of propagation of uncertainty
according to JCGM (
2008
). A FOFM methodology benefits from being computationally efficient, thereby allowing it to be implemented in pixel-by-pixel ocean color data processing software (Lamquin et al.,
2013
). In addition, FOFM calculations can be used to estimate the relative contribution of individual sources to total measurement uncertainty.
Work presented here is the first comprehensive examination of methods that can be used to estimate uncertainties in NASA's standard bio-optical data products. In this study we aim to demonstrate the feasibility of using a FOFM uncertainty framework to approximate ocean color data uncertainty in derived data products. The FOFM method, which itself is an analytical approximation, is first appraised by comparing FOFM-derived uncertainties with Monte Carlo-derived uncertainties. We demonstrate how this approach can be used as a method to check the correctness of FOFM calculations. Second, using FOFM propagation theory, we estimate uncertainty in derived ocean color products given spectrally-flat, uncorrelated relative uncertainties of 1, 5, and 10% in spectral remote-sensing reflectances,
rs,i
(sr
−1
). We also consider spectrally-dependent, uncorrelated relative uncertainties in
rs,i
published by Hu et al. (
2013
). Third, we consider how inclusion of covariances affect uncertainty estimates. A sample SeaWiFS scene of the Hawaiian Islands is used in this case study. Finally, we demonstrate how the FOFM approach may be used to estimate measurement uncertainty budgets. In our case study we consider two algorithms for estimating particulate organic carbon (
POC
; mg m
−3
), a key metric used to understand oceanic biomass and the carbon cycle.
In this work, we utilize a high quality
in situ
hyperspectral
rs,i
dataset that can be spectrally subsampled to match the spectral characteristics of most existing and future ocean color sensors. This includes NASA's Plankton, Aerosol, Cloud, ocean Ecosystem (PACE) mission that is currently under development and will carry the first dedicated hyperspectral ocean color sensor.
Data and Methods
Bio-optical Algorithms and Data Products
The NASA Ocean Biology Data Archive and Active Distribution Center (OB.DAAC) distribute a number of derived marine data products in two separate data suites: (i) the standard ocean color (OC) data product suite and, (ii) the inherent optical properties (IOP) product suite. The OC suite comprises established (legacy) ocean color data products that were developed during the SeaWiFS (1997–2010) and Moderate Resolution Imaging Spectroradiometer aboard Aqua (MODISA 2002–present) missions. The IOP suite comprises spectral component absorption and backscattering coefficients derived using the default configuration of the Generalized Inherent Optical Properties (GIOP) algorithm framework (Werdell et al.,
2013
). A selection of the OC suite and IOP suite products were used in this study (
Table 1
). More comprehensive detail of the bio-optical algorithms used to derive these data products and their associated uncertainties are given in Appendices A–E (
Supplementary Material
). We note that in this study the GIOP used a spectral subset of our
rs
evaluation dataset (described in section Evaluation
rs
Dataset) spanning 412–655 nm.
Table 1
Product name
Product suite
Symbol
Units
References
Chlorophyll-a pigment concentration
OC
Chl
mg m
−3
O'Reilly et al.,
1998
; Hu et al.,
2012b
Chlorophyll-a derived from band ratio
Chl
BR
mg m
−3
O'Reilly et al.,
1998
Chlorophyll-a derived from line height
Chl
LH
mg m
−3
Hu et al.,
2012b
Diffuse attenuation coefficient at 490 nm
OC
, 490
−1
Mueller,
2000
Particulate organic carbon
OC
POC
mg m
−3
Stramski et al.,
2008a
Normalized fluorescent line height
OC
nflh
mW cm
−2
μm
−1
sr
−1
Behrenfeld et al.,
2009
Absorption coefficient of total non-water components 443 nm
IOP
nw,443
−1
Werdell et al.,
2013
Absorption coefficient of phytoplankton at 443 nm
IOP
ϕ,
443
−1
Werdell et al.,
2013
Absorption coefficient of colored dissolved and detrital matter at 443 nm
IOP
dg,443
−1
Werdell et al.,
2013
Particulate backscattering coefficient at 443 nm
IOP
bp,443
−1
Werdell et al.,
2013
Bio-optical ocean color data products.
Note that NASA's standard Chl product is a dynamic blend of Chl
BR
and Chl
LH
Modeling Bio-Optical Data Product Uncertainty
In this study we used the analytical law of propagation of uncertainty (JCGM,
2008
) to propagate radiometric uncertainties through models used to derive bio-optical quantities. We follow the notation conventions outlined by JCGM (
2008
) where the uncertainty of a measured quantity,
, is denoted as
) and is the positive square root of the variance,
). We note that
is derived from a model,
, of
input quantities,
. Following (JCGM,
2008
), for uncorrelated input quantities,
) can be calculated as:
where,
) is the 1-σ uncertainty in the input quantity
. For our notation of spectral properties used in ocean color remote sensing, subscripts
correspond to wavelength. In this study, partial derivatives of target parameters were calculated analytically, however, these could also be computed numerically. For the situation where uncertainties of input quantities are correlated, Equation 2 is extended to:
where u(
) = u(
) denotes the estimated error covariance associated with the quantities
and
. Comprehensive details of partial derivative calculations for each bio-optical algorithm in
Table 1
are given in Appendices A–E (
Supplementary Material
).
Monte Carlo (MC) methods are routinely used to perform sensitivity analyses as well as quantify model output uncertainties (Refsgaard et al.,
2007
). In this study, we have utilized a MC approach to appraise FOFM calculations. As the partial derivative calculus within FOFM uncertainty estimates can be complex, we have used MC-to-FOFM comparisons as a means of checking calculations.
The MC estimates of uncertainties in this study were computed as follows:
A given bio-optical model,
, that derives an output
, that is considered a function of
spectral remote sensing reflectance bands,
rs,i
, is run 5,000 times.
Upon each iteration, each
rs,i
is perturbed by a factor Δ
which is randomly sampled from a Gaussian distribution Δ
(0,
rs,i
)), in which the mean is zero and the standard deviation, u(
rs,i
), is known or assumed. No spectral correlations are assumed.
The MC simulation then generates a probability density function (
PDF
) for
. From the
PDF
, the mean value, ŷ and the standard deviation, σ
, can be computed.
We note that the MC method captures non-linear effects and thus we cannot always expect direct agreement between
and FOFM-derived
). Indeed, even if a bio-optical model contains weak non-linearities and MC model input uncertainties are normally distributed, the number of MC iterations still needs to be suitably large for
to agree with
).
Evaluation
rs
Dataset
To evaluate our FOFM uncertainty method, we used a dataset of high quality hyperspectral
rs,i
spectra (
= 1124). Hyperspectral radiometric measurements were collected
in situ
during three different expeditions, representing a range of oligotrophic to mesotrophic waters: the SABOR experiment in the Gulf of Maine/North Atlantic/Mid-Atlantic coast (July–August 2014); AE1319 in the North Atlantic and Labrador Sea (August–September 2013); and NH1418 in the Equatorial Pacific (September–October 2014). A HyperOCR system (Sea-Bird Scientific) deployed on a tethered profiler in “buoy mode” was used to collect upwelling radiance,
u,i
(W m
−2
μm
−1
sr
−1
), and downwelling irradiance,
d,i
(W m
−2
), spectra during deployments lasting ~5 min. During sample collection, the instrument was allowed to drift far enough from the boat to avoid the ship's shadow.
The spectra were dark and tilt-corrected, and the upper and lower 25th percentile of the
d,i
spectra were removed from both
d,i
and
u,i
. The mean of the remaining spectra was used in subsequent analysis, providing one spectrum per deployment, and with uncertainties calculated as the standard deviation of the same spectra used to calculate the mean (N.B. uncertainties here represent only the experimental portion of the uncertainties, and calibration bias has not been accounted for). The
u,i
measurements were extrapolated to and across the air-water interface to obtain the water-leaving radiance,
w,i
(W m
−2
sr
−1
), which were then used to calculate remote-sensing reflectance (
rs,i
), defined as:
The spectra were additionally corrected for Raman scattering following methods in Westberry et al. (
2013
), which was necessary to compensate for the scattering that water molecules themselves can contribute to
w,i
, especially at the blue wavelengths in very clear waters (McKinna et al.,
2016
). Finally, the
rs
spectra were normalized to remove the angular effect of the sun position in the sky relative to nadir, following methods in Lee et al. (
2011
). For a more detailed description of the
rs,i
calculations and processing, see Data and Methods section in Chase et al. (
2017
). All hyperspectral
rs,i
used in this study are shown in
Figure 1
Figure 1
Finally, each hyperspectral
rs
spectrum was spectrally sub-sampled. The resulting multiband
rs,i
dataset had sixteen 10 nm-wide spectral bands centered on: 412, 425, 443, 460, 475, 490, 510, 532, 555, 583, 617, 640, 655, 665, 678, and 710 nm. This multispectral subset spanned the visible domain and included bands from both past and present NASA sensors (e.g., SeaWiFS and MODIS).
Radiometric Uncertainties
Spectrally Flat
rs
Uncertainties
For NASA ocean color bio-optical algorithms, model input quantities are typically remote sensing reflectances,
rs,i
(sr
−1
), which are derived from measured top-of-atmosphere radiances,
t,i
(W m
−2
μm
−1
sr
−1
), via atmospheric correction (AC) algorithms. Historically, a desirable science requirements for NASA ocean color missions has been
rs,i
with relative uncertainty of 5% (spectrally flat) or less (Hooker et al.,
1992
; Hooker and McClain,
2000
; McClain et al.,
2004
; PACE Science Definition Team,
2018
). Whilst not directly representative of a true sensor (see section Spectrally-Dependent
rs
Uncertainties), treating relative uncertainties in
rs,i
as spectrally flat is still useful under circumstances where detailed knowledge of sensor performance characteristics is limited, such as during pre-launch scoping studies, to provide rudimentary uncertainty estimates. In this study we first consider 5% relative uncertainty in
rs,i
to compare FOFM-to-MC calculations. We next use the FOFM method consider how spectrally flat relative uncertainties in
rs
of 1, 5, and 10% impact estimated OC and IOP uncertainties. Note, we treat spectrally flat relative uncertainties in
rs
of 1, 5, and 10% as spectrally uncorrelated.
Spectrally-Dependent
rs
Uncertainties
We note that on-orbit uncertainties in
t,i
and
rs,i
have previously been quantified for NASA's SeaWiFS and MODISA missions (Eplee et al.,
2007
; Hu et al.,
2012a
2013
; Angal et al.,
2015
). Whilst historically 5% has been the desired accuracy goal for
rs
in the blue-green spectral range, work by Hu et al. (
2013
) reported that relative uncertainties of
rs,i
for SeaWiFS and MODISA increase monotonically with wavelength, and that
rs,i
relative uncertainty also varies as a function of
Chl
, or water-column optical complexity. To extend this study beyond spectrally flat relative uncertainties, we utilized the relative uncertainties for MODISA
rs,i
estimated for the North Atlantic Ocean (see Table 2 of Hu et al.,
2013
). To estimate relative uncertainty for a given
rs,i
spectra, we followed three steps: (i) linearly interpolate tabulated relative uncertainties to match the spectral resolution of our
in situ R
rs,i
dataset, (ii) estimate
Chl
concentration using NASA's standard OC algorithm, and (iii) linearly interpolate the spectrally tabulated relative uncertainties to estimate relative uncertainty for observed
rs,i
based on the respective
Chl
concentration. Note, where estimated
Chl
exceeded 0.2 mg m
−3
[beyond values reported by Hu et al. (
2013
)] we linearly extrapolated tabulated relative uncertainties.
Figure 2
shows the spectral relative uncertainties in
rs,i
sensu
Hu et al. (
2013
)] used in this study and how they vary with
Chl
concentration. Note, spectrally-dependent relative uncertainties in
rs
computed as a function of
Chl
were treated as spectrally uncorrelated.
Figure 2
Spectrally-Correlated
rs
Uncertainties
Our initial analyses treated
rs
spectral uncertainties as uncorrelated, which in practice is an oversimplification. Indeed, AC algorithms utilize near-infrared bands to make assumptions about the contribution of atmospheric aerosols to
(Gordon and Wang,
1994
; Bailey et al.,
2010
). Thus,
rs,i
uncertainties are inherently spectrally correlated. While much work has been done to characterize radiometric uncertainties of NASA sensors used for ocean color (Eplee et al.,
2007
; Hu et al.,
2012a
2013
), few studies have quantified off-diagonal elements of the variance-covariance matrices for top-of-atmosphere radiance,
Lt
, and remote sensing reflectances,
Rrs
, respectively. We note that while beyond the scope of this work, parallel efforts are underway by the research community to derive pixel-by-pixel estimates of
rs,i
) by propagating radiometric uncertainties through ocean color atmospheric correction algorithms (Gillis et al.,
2018
).
Recently, Lamquin et al. (
2013
) demonstrated a methodology to estimate
Lt
for MERIS data and propagate these through ESA's clear water branch AC algorithm and into bio-optical data products. Critically, Lamquin et al. (
2013
) demonstrated that ignoring covariances can lead to overestimated data product uncertainties. In this study, using a similar methodology to Lamquin et al. (
2013
), we estimate
Lt
for SeaWiFS and then using a numerical approximation estimate
Rrs
. A full description of this method can be found in Appendix F (
Supplementary Material
). We note that while our estimates of
Rrs
are somewhat rudimentary, they are still useful for demonstrating the importance of including covariance terms in FOFM-based uncertainty estimates.
Satellite Data Processing
A SeaWiFS image of Hawaii captured on 1 December 2000 was used to demonstrate the FOFM methodology when applied to ocean color imagery. SeaWiFS Level-1 data was downloaded from NASA's Ocean Biology Distributed Active Archive Center (NASA OB.DAAC) Level 1 and 2 Browser website (
. Data were then processed from Level 1 to Level 2 using NASA Ocean Color Science Software (OCSSW). These processing steps include radiometric calibration, geolocation, and atmospheric correction. A prototype version of OCSSW code was used to compute
Chl
) using FOFM methodology where
rs,i
) was estimated using an empirical methodology described in Appendix F (
Supplementary Material
).
Results
Appraisal of Methodology
The MC methodology, while computationally expensive, was expected to give robust estimates of measurand uncertainties. Thus, MC outputs provided a benchmark to which the FOFM uncertainty estimates could be compared with for correctness. Direct calculations of FOFM uncertainties,
), were compared with MC output uncertainties, σ
. To compare MC and FOFM calculations we used 5% spectrally flat relative uncertainty in
rs
and computed the following comparison statistics: bias and Type II linear regression slope. When computing these statistics for the purpose of FOFM-to-MC comparisons, we assume that MC-estimated uncertainties were quasi-truth. We note that variables were log-transformed for these calculations following Seegers et al. (
2018
). Bias was computed as:
where
= 1124 is the number of input spectra. Given that bias was computed using log-transformed variables, it becomes interpretable as multiplicative metrics (Seegers et al.,
2018
). We note that our bias calculations assume estimated OC and IOP uncertainties follow log-normal distributions; a property that is demonstrated later in the paper (e.g.,
Figures 4
).
The MC and FOFM estimation of derived product uncertainties were in good agreement for the following OC products:
,490
POC
, and
nflh
. This was indicated by slope and bias and statistics (
Table 2
) having values of, or near to, 1.0. However, regression statistics indicated
Chl
uncertainties derived using the FOFM method did not completely agree with the MC method (
Table 2
). To assess this discrepancy more closely, uncertainties in each component of the
Chl
algorithm were inspected, namely the band ratio (
BR
), line height (
LH
), and blended components. Regression statistics indicated that FOFM estimates of
Chl
blend
product uncertainties did not agree well with MC values and were typically biased low by 27%, visualized further by the color-coded scatter plot in
Figure 3A
Table 2
Derived product uncertainty
Product
Bias
Slope
Chl
(all)
0.95
0.96
Chl
BR
1.00
1.00
Chl
LH
0.99
1.00
Chl
blended
0.73
0.72
,490
0.99
1.00
POC
0.99
1.00
nflh
0.99
1.00
nw,443
0.99
1.00
ϕ,
443
0.98
1.00
dg,443
0.98
1.00
bp,443
0.99
0.98
Log-normal statistics comparing Monte Carlo (MC) and first-order first-moment (FOFM) uncertainty calculations for
rs
with spectrally flat, uncorrelated 5% relative uncertainty.
Blended LH and BR Chl product span 0.134–0.165 mg m
-3
Figure 3
Derived uncertainties for IOP products generally agreed with MC simulations. Specifically,
Table 2
shows FOFM estimates of uncertainties with respect to MC estimates for
nw
,443
ϕ, 443
dg
,443
, and
bp
,443
were biased low by 1%, low by 2%, low by 2% and, high by 2%, respectively. Slight disagreement between MC and FOFM estimates of
bp
,443
) can be visualized in
Figure 3
when
bp
,443
) > 2.0 ×10
−3
−1
. In addition, MC and FOFM estimates of
ϕ,443
) showed slight disagreement when
ϕ,443
) > 1.0 ×10
−2
−1
These results demonstrate that while FOFM uncertainty calculations are computationally inexpensive, they serve as approximations only, especially in the case of
Chl
. Indeed, while FOFM-derived uncertainties can be expected to agree with MC-derived values for simple functions that vary linearly, it may not be unusual for FOFM-derived uncertainties to differ from MC-derived values; particularly when analyzing complicated non-linear problems (Putko et al.,
2001
; Mekid and Vaja,
2008
). For example, with the IOPs we found slight differences in the order of 1% between MC and FOFM uncertainty estimates. For such mathematical functions, higher order methods such as Second Order First Moment (SOFM) methods may be useful, however, the added mathematical complexity may be prohibitive.
Uncertainties Estimated From
in situ
Radiometric Data
OC Product Uncertainties
Using the multispectral
rs
evaluation dataset, uncertainties in derived OC products associated with 5% spectrally-flat relative, uncorrelated uncertainty in
rs
were computed.
Figure 4
shows histograms of derived OC products, absolute uncertainties, and relative uncertainties. MC computations are summarized in
Table 3
, while FOFM computations are provided for comparative purposes in
Table 4
Figure 4
Table 3
Derived value
Absolute uncertainty
Relative uncertainty (%)
Product
Range
Median
Range
Median
Range
Median
Chl
(mg m
−3
3.96 × 10
−2
−1.27
0.110
2.56 × 10
−5
-0.231
7.00 × 10
−3
1.73–18.2
9.74
,490
(m
−1
2.01 × 10
−2
−0.131
2.91 x10
−2
1.19 × 10
−3
−1.36 × 10
−2
2.68 × 10
−3
5.92–10.5
8.94
POC
(mg m
−3
18.8–203.4
33.1
1.37–14.6
2.44
7.11–7.60
7.37
nflh
(mW cm
−2
μm
−1
sr
−1
5.25 × 10
−6
−2.74 × 10
−2
2.20 × 10
−3
3.18 × 10
−4
-4.47 × 10
−3
9.86 × 10
−4
14.8–1.7 × 10
41.9
OC products and associated uncertainties derived via MC method with 5%, uncorrelated relative uncertainty in
rs
Table 4
Derived value
Absolute uncertainty
Relative uncertainty (%)
Product
Range
Median
Range
Median
Range
Median
Chl
(mg m
−3
3.96 × 10
−2
−1.28
0.110
3.89 × 10
−5
-0.230
6.70 × 10
−3
0.26–18.7
9.67
,490
(m
−1
2.01 × 10
−2
−0.131
2.91 × 10
−2
1.18 × 10
−3
−1.33 × 10
−2
2.68 × 10
−3
5.86–10.2
8.91
POC
(mg m
−3
18.8–203.4
33.1
1.37–14.9
2.42
7.31
7.31
nflh
(mW cm
−2
μm
−1
sr
−1
2.05 × 10
−6
−2.73 × 10
−2
2.19E × 10
−3
3.21 × 10
−4
−4.43 × 10
−3
9.87 × 10
−4
15.1–3.24 × 10
42.1
OC products and associated uncertainties derived via FOFM method with 5%, uncorrelated relative uncertainty in
rs
Relative uncertainties in POC computed using FOFM method were constant over the dynamic range
The range of derived
Chl
confirmed that the dataset spans oligotrophic (0.04 mg m
−3
) to mesotrophic conditions (1.28 mg m
−3
) with a median value of 0.11 mg m
−3
. Values of
Chl
) span four orders of magnitude and have median values of 7.00 ×10
−3
and 6.70 ×10
−3
mg m
−3
for the MC and FOFM methods, respectively. The relative uncertainties for
Chl
span a single order of magnitude and have median values of 9.74 and 9.67% for the MC and FOFM methods, respectively. Although the histogram of derived
Chl
in
Figure 4
appears log-normally distributed, two distinct peaks are present; a low peak (ranging from 0 to 0.5 mg m
−3
) and a high peak (centered on 1.1 mg m
−3
). Since bio-optical properties are log-normally distributed in the ocean (Campbell,
1995
), the peaks observed in the distributions of derived bio-optical variables are probably due to the limited size of the hyperspectral
rs
dataset (
= 1124), that does not uniformly span the entire range of oceanic conditions (see Figure 1A in Chase et al.,
2017
).
The range of derived
d,490
spans an order of magnitude with a median value of 0.0291 m
−1
. The values of
d,490
) also span an order of magnitude with median values of 2.68 ×10
−3
−1
for both MC and FOFM calculations. The relative uncertainties for
d,490
span a single order of magnitude and have a median value of 8.94 and 8.91% for MC and FOFM calculations, respectively. The range of derived
POC
spans two orders of magnitude with a median value of 33.1 mg m
−3
. The values of
POC
) span an order of magnitude and have median values of 2.44 and 2.42 mg m
−3
for MC and FOFM calculations, respectively. The relative uncertainties in
POC
have a value of 7.37 and 7.31% for MC and FOFM calculations, respectively. We note that the relative uncertainty in
POC
as computed by FOFM method exhibits no spread. For uncorrelated, spectrally flat relative uncertainties,
POC
)/
POC
is a function of
rs,443
)/
rs,443
and
rs,555
)/
rs,555
. Thus, when
rs,443
)/
rs,443
and
rs,555
)/
rs,555
are fixed (e.g., at 5%),
POC
)/
POC
is fixed. In practice, this will not always hold true, particularly when relative uncertainties in
rs
are variable and spectrally dependent. We note that in
Figure 4
the MC-derived relative uncertainties for
POC
are normally distributed over a narrow range centered on 7.37%.
The range of
nflh
spans three orders of magnitude with an MC-estimated median value of 2.20 ×10
−3
mW cm
−2
μm
−1
sr
−1
. We note that direct calculations of
nflh
resulted in a median value of 2.19 ×10
−3
mW cm
−2
μm
−1
sr
−1
. The values of
nflh
) span an order of magnitude with median values of 9.86 ×10
−4
and 9.87 ×10
−4
mW cm
−2
μm
−1
sr
−1
for MC and FOFM calculations, respectively. The median relative uncertainty in
nflh
was 41.9 and 42.1% for MC and FOFM calculations, respectively (
Figure 4
). We note that the range of relative errors for
nflh
is very large (for MC calculations: 14.8–1.7 ×10
%), and these should be interpreted with a caution. Low values of
nflh
, in the order of 1 ×10
−6
mW cm
−2
μm
−1
sr
−1
, were derived from the evaluation dataset which in most likelihood would be beyond the detection limit of existing ocean color sensors. Further, while the absolute uncertainties associated with these low
nflh
values may also be small in magnitude, they can still manifest as large relative uncertainties.
IOP Product Uncertainties
Using the radiometric evaluation dataset, uncertainties in derived IOP products associated with 5% relative, uncorrelated uncertainty in
rs,i
were computed following the methodology in Appendix E (
Supplementary Material
).
Figure 5
shows histograms of derived IOP products, absolute uncertainties, and relative uncertainties. MC computations are summarized in
Table 5
while FOFM computations are provided for comparative purposes in
Table 6
Figure 5
Table 5
Derived value
Absolute uncertainty
Relative uncertainty (%)
Product
Range
Median
Range
Median
Range
Median
nw
(443) (m
−1
9.40 × 10
−3
−0.127
0.0185
1.79 × 10
−3
−1.13x10
−2
2.31 × 10
−3
8.16–19.4
12.6
(443) (m
−1
5.80 × 10
−3
−9.43 x10
−2
9.60 × 10
−3
1.63 × 10
−3
−9.68 × 10
−3
2.04 × 10
−3
10.0–29.2
21.4
dg
(443) (m
−1
3.50 × 10
−3
−3.72 x10
−2
8.71 × 10
−3
6.66 × 10
−4
−5.90 × 10
−3
1.07 × 10
−3
7.92–19.9
14.5
bp
(443) (m
−1
4.18 × 10
−4
−4.00 × 10
−3
1.08 × 10
−3
8.98 × 10
−5
−2.25E × 10
−4
1.34 × 10
−4
5.57–34.1
13.8
IOP products and associated uncertainties derived using MC method with 5%, uncorrelated relative uncertainty in
rs
Table 6
Derived value
Absolute uncertainty
Relative uncertainty (%)
Product
Range
Median
Range
Median
Range
Median
nw
,443
(m
−1
9.42 × 10
−3
−0.127
0.0185
1.79 × 10
−3
-1.03 × 10
−2
2.26 × 10
−3
8.12–19.1
12.2
ϕ,
443
(m
−1
5.86 × 10
−3
−9.45 × 10
−2
9.63E-3
1.64 × 10
−4
−8.73 × 10
−3
2.00 × 10
−3
9.02–28.6
20.8
dg
,443
(m
−1
3.51 × 10
−3
−3.70 × 10
−2
8.73E-3
6.51 × 10
−4
−5.63 × 10
−3
1.05 × 10
−3
7.93–18.9
14.1
bp
,443
(m
−1
4.16 × 10
−4
−4.01x10
−3
1.00E-3
9.00 × 10
−5
−2.11 × 10
−4
1.33 × 10
−4
5.25–34.1
13.9
IOP products and associated uncertainties derived using FOFM method with 5%, uncorrelated relative uncertainty in
rs
The range of derived
nw,443
spans two orders of magnitude with a median value of 0.0185 m
−1
. Values of
nw,443
) span an order of magnitude with median values of 2.31 ×10
−3
and 2.26 ×10
−3
−1
for MC and FOFM methods, respectively. The median relative uncertainty in
nw
,443
spans a single order of magnitude and has median values of 12.6 and 12.2% for MC and FOFM methods, respectively. The range of
ϕ, 443
dg,443
, and
bp,443
all span a single order of magnitude with median values of 9.6 ×10
−3
, 8.71 ×10
−3
, and 1.08 ×10
−3
−1
, respectively. Absolute uncertainties in IOPs all span two orders of magnitude apart from
ϕ,
443
) which spanned a single order of magnitude. Highest relative uncertainties of all GIOP-derived products are for
ϕ,443
(~20%), whereas
nw,440
dg,440
, and
bp,440
have relative uncertainties of similar magnitude that are all <15%.
Summary of MC and FOFM Comparisons
FOFM and MC estimates of OC and IOP uncertainties were generally in good agreement. This provides confidence that our FOFM analytical formulations were correct. However, FOFM-to-MC comparisons of
Chl
and IOP uncertainties, whilst similar in magnitude, exhibited a degree of scatter around the one-to-one line. We expect that these differences may be due to the MC method's ability to handle non-linearity and discontinuities in the models more robustly than the FOFM approach. For example, the
Chl
model has several complex features such: switching between
Chl
BR
and Chl
LH
, the
Chl
BR
model's selection of maximum band ratios, and the blending of
Chl
BR
and Chl
LH
, which may not be fully captured by the FOFM method.
We thus found FOFM-to-MC comparisons to be useful as a “quick acceptability checking” of FOFM calculations. In practice, however, one should not always assume the two methods will closely agree as the MC model may handle non-linearities and discontinuities more robustly than the FOFM method. The FOFM and MC calculations also indicate that for normally distributed radiometric input uncertainties, the estimated output uncertainties for OC and IOP were log-normally distributed (as per
Figures 4
). Such highly dynamic and variable nature of uncertainties in ocean color data products highlights the need for these estimates to be done on a pixel-by-pixel basis, rather than a single scene-wide estimate, further justifying the need for simplified, computationally inexpensive approach (i.e., FOFM).
We note that our FOFM uncertainty formulation for the GIOP currently does not consider uncertainty in spectral shape models [i.e.,
) and
)]. Indeed, we believe that this may be why there were some noticeable differences when comparing FOFM and MC methods, for example: when
bp,443
) > 2.00 ×10
−4
−1
Figure 3H
). In a cursory study, we re-ran both FOFM and MC calculations with the shape models parametrized as spectral constants (i.e., having no uncertainties). This resulted in improved FOFM-to-MC comparisons (results not shown) and further highlighted how spectral shape uncertainties impact our FOFM uncertainty estimates. As part of future work, we thus plan to extend our current GIOP FOFM uncertainty formulation to include the spectral shape uncertainties. Additionally, we note that
) and
), computed as functions of
Chl
and a red-green
rs,i
ratio, respectively, are also correlated. Thus, an improved GIOP FOFM uncertainty formulation should also consider covariances between spectral shape models.
GIOP Model Misfit Uncertainties
In this analysis we used our high-quality evaluation
rs
dataset to approximate GIOP model misfit uncertainties. Our assumptions in this exercise were: (i) the uncertainties in our
rs
dataset are small, and (ii) the least squares residual of the optimal solution (model misfit) are thus due to an imperfect model.
In this analysis we first computed the error-covariance matrix,
rrs
, for each
rs
observation as follows: (i) employ the Levenberg-Marquardt non-linear least squares optimization to iteratively find an optimal solution for the free variables
dg
, and
which correspond to
Chl
concentration,
dg,440
, and
bp,440
, respectively (see Appendix E in
Supplementary Material
for further detail). We note that in the standard implementation of the GIOP, the cost function (Chi-squared) is unweighted. (ii) feed the optimal set of
dg
, and
back in the forward reflectance model to compute a best-fit spectral sub-surface remote sensing reflectance,
. (iii) calculate the spectral residual, ε
rrs,i
, between
and sensor-observed subsurface remote sensing reflectance. (iv) set the diagonal elements of
rrs
as the square of ε
rrs,i
Next, by substituting
rrs
for
rrs
in Equation E13 the parameter error-covariance matrix,
, can be computed as:
Where
is the Jacobian matrix of the forward model (see Appendix E in
Supplementary Material
for derivation). Finally, the estimates of parameter uncertainties due to model misfit were calculated as the square root of the diagonal elements of
. The model-misfit uncertainties are summarized in
Table 7
and compared to parameter uncertainties due to Hu spectrally-dependent radiometric uncertainties (as per
Table 6
).
Table 7
Absolute uncertainty (m
−1
Relative uncertainty (%)
Difference between absolute data and absolute model misfit uncertainties
(%)
Product
Range
Median
Range
Median
Median
tw,443
(m
−1
3.88 × 10
−4
−5.71 × 10
−3
4.87 × 10
−4
1.26–5.70
3.15
−77
ϕ,
443
(m
−1
3.67 × 10
−4
−5.25 × 10
−3
4.54 × 10
−4
3.02–9.09
4.68
−77
dg,443
(m
−1
1.07 × 10
−4
−2.26 × 10
−3
1.434 × 10
−4
0.81–7.48
2.86
−86
bp,443
(m
−1
2.94 × 10
−5
−2.17 × 10
−4
5.22 × 10
−5
1.57–9.58
4.52
−61
GIOP model-misfit uncertainties estimated using the evaluation
rs
dataset.
Differences between median absolute model uncertainties in this table and median absolute radiometric (data) uncertainties (column RU: Hu in
Table 9
We found that estimated GIOP model misfit uncertainties were 60–90% smaller than those imparted by radiometric uncertainty. Thus, by combining the two during pixel-by-pixel processing, it would be possible to more completely estimate
measurement
) for IOPs. However, we accept that our FOFM model-data misfit approach is approximate only and does not consider all uncertainties in the GIOP model formulation.
Comparing Product Uncertainties Due to Various Radiometric Input Uncertainties
In order to evaluate the impact of different
rs
uncertainty values on derived product uncertainties, using the FOFM method we: (i) propagated spectrally flat, uncorrelated
rs
relative uncertainties of 1, 5, and 10% through OC and IOP models, and (ii) propagated spectrally-dependent, uncorrelated
rs
) through OC and IOP models by linearly interpolating/extrapolating tabulated data published by Hu et al. (
2013
), referred to as “Hu uncertainties” (see
Figure 2
). Summary results of this analysis are given in
Tables 8
. As expected, introducing spectrally flat, uncorrelated
rs
uncertainties of lower and higher value than the previously evaluated 5%, resulted in respectively, lower and higher uncertainties in data products, while the distribution of uncertainties kept the same shape as for the 5% run (
Figure 6
). For the product uncertainties derived using the “Hu
rs
uncertainties,” both the shape of the distribution and median values changed from the 5% run (
Figure 6
). These results demonstrate the importance of considering spectral dependence in radiometric uncertainties. Notably, considering spectrally flat 5% relative uncertainties in
rs
for a data product such as
nflh
, which utilizes red-end bands, may result in significant underestimation of likely data product uncertainties.
Table 8
Median absolute uncertainties
Median relative uncertainties (%)
RU: 1%
RU: 5%
RU: 10%
RU: Hu
RU: 1%
RU: 5%
RU: 10%
RU: Hu
Product
Chl
(mg m
−3
1.52 × 10
−3
6.70 × 10
−3
1.46 × 10
−2
6.50 × 10
−3
1.96
9.67
19.35
8.29
,490
(m
−1
5.37 × 10
−4
2.68 × 10
−3
5.36 × 10
−3
5.07 × 10
−3
1.78
8.91
17.8
17.3
POC
(mg m
−3
4.84 × 10
−1
2.42
4.84
4.38
1.46
7.31
14.6
13.1
nflh
(mW cm
−2
μm
−1
sr
−1
1.97 × 10
−4
9.87 × 10
−4
1.97 × 10
−3
4.47 × 10
−3
8.41
42.1
84.1
197.6
Median OC data product uncertainties computed as relative uncertainties (RU) in
rs
vary.
Table 9
Median absolute uncertainties
Median relative uncertainties (%)
RU: 1%
RU: 5%
RU: 10%
RU: Hu
RU: 1%
RU: 5%
RU: 10%
RU: Hu
Product
tw,443
(m
−1
4.52 × 10
−4
2.26 × 10
−3
4.52 × 10
−3
2.76 × 10
−3
2.45
12.2
24.5
15.1
ϕ,
443
(m
−1
4.00 × 10
−4
2.00 × 10
−3
4.00 × 10
−3
2.42 × 10
−3
4.15
20.8
41.6
23.8
dg,443
(m
−1
2.11 × 10
−4
1.05 × 10
−3
2.11 × 10
−3
1.33 × 10
−3
2.82
14.1
28.2
15.9
bp,443
(m
−1
2.67 × 10
−5
1.33 × 10
−4
2.67 × 10
−4
1.73 × 10
−4
2.78
13.9
27.9
17.9
Median IOP data product uncertainties computed as relative uncertainties (RU) in
rs
vary.
Figure 6
Spectrally flat relative uncertainty in
rs
(e.g., 5% in the blue-green region) is a commonly used accuracy goal for ocean color missions. However, we know from on-orbit data that sensors such as SeaWiFS and MODIS have largely not achieved their desired accuracy goals over the full spectral range (Hu et al.,
2013
), particularly at red wavelengths. In lieu of any knowledge of a sensor's radiometric uncertainty characteristics (e.g., during design trade studies), one might decide to utilize desired relative radiometric accuracy goals to approximate ocean color data product uncertainties. However, our results have shown spectrally flat (5%) and spectrally-dependent (Hu) relative
rs
uncertainties lead to different estimates of OC and IOP uncertainties. Indeed, for improved uncertainty estimates, we recommend the use of more representative spectrally-dependent
rs
)/
rs
, if known.
Application to Satellite Chlorophyll Image
The potential impact that spectrally-correlated uncertainties in
rs
have upon ocean color data product uncertainties was evaluated using a scene of the southern Hawaiian Islands captured on 1 December 2000 (
Figure 7
). We have estimated on a pixel-by-pixel basis the covariance matrix of remote sensing reflectances,
Rrs
, as per the methodology described in Appendix F (
Supplementary Material
). Estimates of
Chl
) were then calculated both with- and without the off-diagonal terms in
Rrs
to demonstrate the impact of incorporating covariance terms (if known) when estimating uncertainties.
Figure 7
The sample SeaWiFS
Chl
image (
Figure 7A
) shows that the clearest waters occurred southeast of Island of Hawaii (largest island) with two large eddies to the west. Regions of elevated
Chl
concentration are also visible along the northeast coast of the Island of Hawaii, and also adjacent to coastal waters of four islands (Maui, Lanai, Molokai, and Kahoolawe) to the northwest of Hawaii. Derived
Chl
blend
ranges from 1.83 ×10
−3
to 0.498 mg m
−3
with a median of 0.066 mg m
−3
. When the off-diagonal terms in
Rrs
were considered, the estimated values of
Chl
) ranged from 1.30 ×10
−3
to 0.075 mg m
−3
with a scene-wide median of 5.20 ×10
−3
mg m
−3
Figure 7B
) and the relative uncertainties spanned 0.84–38.6% with a median of 7.89% (
Figure 7C
). When the off-diagonal terms in
Rrs
were not considered (i.e., set to zero), estimated values of
Chl
) ranged from 1.30 ×10
−3
to 0.109 mg m
−3
with a scene-wide median of 5.50 ×10
−3
mg m
−3
Figure 7D
) and relative uncertainties spanning 0.85–46.1 % with a median of 8.27% (
Figure 7E
). Note, these image statistics were computed with standard NASA level-2 quality control flags applied to remove the effect of: land, clouds, sun glint, atmospheric correction failure, product failure, and straylight contamination.
These results demonstrate how a FOFM method can be utilized in operational processing code to estimate uncertainties in derived bio-optical data products. The FOFM method was straightforward to implement within l2gen code and did not add any appreciable processing overhead. Whilst our estimation of
Rrs
is rudimentary (Appendix F in
Supplementary Material
), it allowed us to consider the covariance terms in the FOFM derivation of
Chl
). Critically, we demonstrated that the inclusion of off-diagonal covariance terms from
Rrs
led to lower estimates of both
Chl
) and
Chl
)/
Chl
when compared to the same calculations performed with off-diagonal elements of
Rrs
set to zero; a result consistent with findings of Lamquin et al. (
2013
). Additionally, this example was done with an operational processing code, demonstrating the easiness of implementing a FOFM method within day-to-day ocean color processing.
POC Algorithm Case Study
Recall from Equation 1, we broadly defined measurement uncertainty as having two sources: data uncertainty and model uncertainty. Throughout this paper we have focused heavily on deriving data uncertainties (i.e., propagation of radiometric uncertainty) which is useful if one is trying understand how a specific sensor's noise characteristics may impact derived data product uncertainties. However, this information alone does not provide a complete picture of measurement uncertainty; model uncertainty also needs to be considered. We thus wish to demonstrate how with knowledge of model uncertainties one can draw more complete conclusions about biogeochemically-relevant data product uncertainties. As such, we present a case study in which we estimate
POC
measurement uncertainty for two different algorithms: (i) Stramski et al. (
2008a
) and (ii) Rasse et al. (
2017
).
Our motivation here is to solely demonstrate how one might develop algorithm uncertainty budgets (data and model uncertainty as per Equation 1) using a FOFM framework. Our calculations, however, are limited by: (i) the representativeness of our
in situ R
rs
dataset which does not encompass all optical water-types found in the World's oceans, (ii) our spectral
rs
) values which are estimated from data published by Hu et al. (
2013
) for a MODIS-like sensor without co-variance terms, and (iii) our knowledge of model uncertainties, such as coefficients uncertainties, which is limited to those reported in literature and/or our best-guess estimates. We hence caution the reader should not use our reported numbers as a basis for algorithm selection.
POC Measurement Uncertainty Estimates
In this exercise, we performed rudimentary calculations to estimate measurement uncertainty budgets for two
POC
algorithms: (i) NASA's standard
POC
algorithm (Stramski et al.,
2008a
) and (ii) the IOP-based model of Rasse et al. (
2017
). Conveniently for this exercise, both
POC
models have a power law formulation:
where
in Stramski et al. (
2008a
) is a blue-to-green reflectance ratio (
rs
,443
rs,555
, as per Appendix C in
Supplementary Material
) and the coefficients
poc
and
poc
have the values of 203.2 and −1.034, respectively. For the approach of Rasse et al. (
2017
is
bp
470
and the coefficients
poc
and
poc
have the values of 141,253 and 1.18, respectively. Note, in this case study we use GIOP-derived estimates of
bp
470
as inputs to the Rasse et al. (
2017
) model.
First, let us consider the model uncertainty component due to imperfect model coefficients. For both
POC
algorithms, with the coefficients
poc
and
poc
and their assigned uncertainties of
model
poc
) and
model
poc
), respectively, we can estimate the model variance for
POC
as:
In the third term on the right-hand side of Equation 8, we set
model
) = 0 and
model
) =
model
bp,470
) for Stramski et al. (
2008a
) and Rasse et al. (
2017
), respectively. We have also assumed the covariance of the coefficients
poc
and
poc
, which are determined by regression fit, is zero. For the Rasse et al. (
2017
) model, the reported model coefficient uncertainties
model
poc
) and
model
poc
) are 45,534 and 0.046, respectively. For the Stramski et al. (
2008a
) model, values of
model
poc
) and
model
poc
) were not reported. We did, however, estimate these model uncertainties by reanalyzing the original published dataset (Stramski et al.,
2008b
) and considering the likely uncertainty introduced by not accounting for the effect of filter pad absorption of
POC
(Novak et al.,
2018
). Following this cursory analysis (results not shown), we estimated
model
poc
) and
model
poc
) for the Stramski et al. (
2008a
) model to be ~2.20 and 0.015, respectively.
Next, we considered the data uncertainty component. The Stramski et al. (
2008a
) model's data uncertainty FOFM calculus was formulated in Appendix C (
Supplementary Material
). For the Rasse et al. (
2017
) model, we first estimated
data
bp
470
). To do so,
bp
470
was calculated from GIOP-derived
bp
440
as:
The variance in
bp
470
due to data uncertainty was then estimated as:
For this exercise, we used GIOP-derived values of
data
bp
470
) and
(γ). The correlation between derived values of
bp
,547 and γ was used to estimate the covariance term
bp
547
, γ) as −1.64 ×10
−6
−1
nm
−1
. Using, the GUM methodology the variance in the Rasse et al. (
2017
POC
model due to data uncertainty was then estimated as:
We finally estimated the measurement uncertainty budgets for both
POC
models using our
rs
evaluation dataset and with Hu spectrally-dependent, uncorrelated radiometric uncertainties (results are shown in
Table 10
).
Table 10
Algorithm
Median derived value (mg m
−3
Median absolute uncertainty in mg m
−3
(median relative uncertainty in %)
Data
Model
Measurement
Stramski et al.,
2008a
33.1
4.40 (13.1)
0.94 (2.85)
4.50 (16.6)
Rasse et al.,
2017
37.8
6.96 (18.4)
17.30 (45.8)
18.6 (49.2)
Simplified random uncertainty budgets for two POC models.
Median absolute uncertainties and median relative uncertainties were computed using our R
rs
evaluation dataset with Hu spectrally-dependent, uncorrelated radiometric uncertainties and basic knowledge of model coefficient uncertainty. We note that these data are intended to illustrate how one might formulate measurement uncertainty budgets. These data are not intended for algorithm comparison purposes
In our rudimentary measurement uncertainty budget for the Stramski et al. (
2008a
POC
algorithm, we found the contribution of data (radiometric) uncertainty was larger than model uncertainty. Conversely, for the Rasse et al. (
2017
POC
algorithm, the contribution of model uncertainty was larger than data uncertainty. Whilst these
POC
algorithm uncertainty budgets may not be fully representative due to the assumptions we partook here, the exercise nonetheless demonstrates an important point: data and model uncertainties should both be considered if one wishes to use uncertainties as a means of benchmarking/comparing ocean color algorithms.
From an algorithm development perspective one can also use FOFM method to explore the relative contribution of individual uncertainty sources to the combined measurement uncertainty. We have graphically displayed the estimated component uncertainty contribution for each
POC
algorithm using pie charts (
Figure 8
). Such information may assist algorithm designers identify and minimize uncertainty sources within a model.
Figure 8
Summary of POC Case Study
Our brief example demonstrates the benefits of using the FOFM method for analytically estimating measurement uncertainty in
POC
. From an ecological perspective, this is particularly useful if one is trying to understand the variability in observed patterns, and distinguish real change from variation in uncertainty. Additionally, it allows for sensitivity analysis, thereby providing a guideline for improving model parameterization. The case study demonstrates how an uncertainty budget can provide additional information to end-users regarding data product quality, potentially informing algorithm selection, and/or guiding new algorithm development. Although ocean color algorithms are typically benchmarked based upon validation matchup metrics (Seegers et al.,
2018
), we expect model selection and development may be better guided by considering how data and model uncertainties manifest in derived data products.
This case study highlights a challenge if one wishes to compare/benchmark legacy ocean color algorithms based on their measurement uncertainty; one must have reasonable and complete knowledge of both data and model uncertainties to do so. Whilst we have demonstrated that it is possible to estimate and propagate random radiometric uncertainties using the FOFM framework, estimating model uncertainties remains a challenge. This is because model component uncertainties (e.g., model coefficient uncertainties) of legacy ocean color algorithms were not routinely reported. To address this, re-analysis of the structure of legacy ocean color algorithms using high quality bio-optical datasets, such as NASA's bio-Optical Marine Algorithm Dataset (NOMAD; Werdell and Bailey,
2005
), may be necessary. Without such knowledge, it remains a challenge to formulate complete measurement uncertainty budgets for legacy ocean color algorithms.
Conclusions
In this paper we demonstrated a FOFM-based method for estimating uncertainties in a selection of NASA OC and IOP products, namely:
Chl, K
d,490
POC, nflh, a
nw,440
ϕ, 440
dg,440
, and
bp,440
, due to sensor-observed radiometric uncertainty. Using a high quality hyperspectral
rs
dataset subsampled to our target wavelengths, we first appraised the FOFM methodology by comparing FOFM-derived uncertainty estimates with uncertainties estimated from MC simulations with an assumed relative spectrally flat, uncorrelated uncertainty in
rs
of 5%. Our analyses showed that OC and IOP uncertainties estimated using the FOFM method generally agreed with MC simulations. Collectively, the FOFM-to-MC comparisons provided a basis for checking the correctness of the FOFM formulations, which are often algebraically complex. Further, we demonstrated that the FOFM formulation, which is computationally inexpensive, can be applied in routine pixel-by-pixel data processing for estimating uncertainties in derived ocean color data products.
This paper has primarily focused on propagating radiometric uncertainties through bio-optical models (
data
) in Equation 1). In practice, the combined measurement uncertainty in derived ocean color data products is expected to be larger once model uncertainties are included. In this study, we have broadly assumed that coefficients within the bio-optical algorithms themselves are errorless, which is not the case. Indeed, most coefficients in bio-optical algorithms have been derived empirically using
in situ
oceanographic datasets, which themselves have inherent uncertainties due to measurement method and environmental variability. The GIOP, for example, makes assumptions about spectral shapes of IOPs, utilizes an approximate forward reflectance model (Gordon et al.,
1988
), and employs a model to convert
rs
to
rs,i
(Lee et al.,
2002
). Thus, there are a number of GIOP model components whose uncertainties, if characterized, may improve the overall estimate of IOP measurement uncertainty. Our case study of
POC
algorithms also highlighted how the addition of model (e.g., coefficient) uncertainties can further inform end-users, and may potentially guide algorithm development and/or selection.
Although this work represents a first step toward implementing pixel-by-pixel uncertainty estimates in NASA operational ocean color processing code, we recognize that continued effort is required. For example, strategies for quantifying uncertainties in look-up-table (LUT) based models, such as the two-band particulate inorganic carbon (PIC) algorithm (Balch et al.,
2005
) and bidirectional reflectance distribution function (BRDF) correction (Morel et al.,
2002
), are needed. Globally, there are a multitude of ocean color algorithms maintained by various researchers and/or institutes and formulating uncertainty estimates must be a collective effort. While the community continues to innovate new bio-optical algorithms, we strongly encourage model developers to characterize uncertainties as a matter of routine.
As we enter the hyperspectral world of PACE, it is credible to expect an evolutionary leap in remote sensing observation of ocean processes detailing, for example, phytoplankton diversity, physiological preferences, and ecology from space. This, parallel to the increase in computational power of the day-to-day data processing, will allow for more complex algorithms; algorithms which will need detailed evaluation of uncertainty budgets, to understand what is real, and what is hidden under the dashed line.
Statements
Author contributions
LM, IC, and PW: conceptualization, methodology, simulations, and data analysis. AC and IC: hyperspectral dataset. LM, IC, PW, and AC: original draft, reviewing, and editing.
Acknowledgments
Many thanks to NASA Ocean Biology Processing Group staff for their advice during the preparation of this manuscript. The authors are thankful for all the scientists that contributed to collection of this dataset, especially Wayne Slade and Nicole Poulton, and captains and crews of the UNOLS research vessels. We are also very grateful to the two reviewers for their detailed and insightful comments which have greatly improved this work.
Conflict of interest
LM was employed by company Go2Q Pty Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Supplementary material
The Supplementary Material for this article can be found online at:
Footnotes
1.
NASA Goddard Space Flight Center, Ocean Ecology Laboratory, & Ocean Biology Processing Group. Sea-viewing Wide Field-of-view Sensor (SeaWiFS) L1 Data (data/10.5067/ORBVIEW-2/SEAWIFS/L1/DATA/1).
References
Angal
A.
Xiong
X.
Sun
J.
Geng
X.
2015
).
On-orbit noise characterization of MODIS reflective solar bands
J. Appl. Remote Sens.
094092
Google Scholar
Antoine
D.
d'Ortenzio
F.
Hooker
S. B.
Bécu
G.
Gentili
B.
Tailliez
D.
et al
. (
2008
).
Assessment of uncertainty in the ocean reflectance determined by three satellite ocean color sensors (MERIS, SeaWiFS and MODIS-A) at an offshore site in the Mediterranean Sea (BOUSSOLE project)
J. Geophys. Res. Oceans
113
22
10.1029/2007JC004472
CrossRef
Google Scholar
Bailey
S. W.
Franz
B. A.
Werdell
P. J.
2010
).
Estimation of near-infrared water-leaving reflectance for satellite ocean color data processing
Opt. Express
18
7521
7527
10.1364/OE.18.007521
Pubmed Abstract
CrossRef
Google Scholar
Bailey
S. W.
Werdell
P. J.
2006
).
A multi-sensor approach for the on-orbit validation of ocean color satellite data products
Remote Sens. Environ.
102
12
23
10.1016/j.rse.2006.01.015
CrossRef
Google Scholar
Balch
W. M.
Gordon
H. R.
Bowler
B. C.
Drapeau
D. T.
Booth
E. S.
2005
).
Calcium carbonate measurements in the surface global ocean based on Moderate-Resolution Imaging Spectroradiometer data
J. Geophys. Res. Oceans
110
21
10.1029/2004JC002560
CrossRef
Google Scholar
Behrenfeld
M. J.
Westberry
T. K.
Boss
E. S.
O'Malley
R. T.
Siegel
D. A.
Wiggert
J. D.
et al
. (
2009
).
Satellite-detected fluorescence reveals global physiology of ocean phytoplankton
Biogeosciences
779
794
10.5194/bg-6-779-2009
CrossRef
Google Scholar
Campbell
J. W.
1995
).
The lognormal distribution as a model for bio-optical variability in the sea
J. Geophys. Res. Oceans
100
13237
13254
10.1029/95JC00458
CrossRef
Google Scholar
Chase
A. P.
Boss
E.
Cetinić
I.
Slade
W.
2017
).
Estimation of phytoplankton accessory pigments from hyperspectral reflectance spectra: toward a global algorithm
J. Geophys. Res. Oceans
122
9725
9743
10.1002/2017JC012859
CrossRef
Google Scholar
Eplee
J. R. E.
Patt
F. S.
Barnes
R. A.
McClain
C. R.
2007
).
SeaWiFS long-term solar diffuser reflectance and sensor noise analyses
Appl. Opt.
46
762
773
10.1364/AO.46.000762
Pubmed Abstract
CrossRef
Google Scholar
10
Franz
B. A.
Behrenfeld
M. J.
Siegel
D. A.
Signorini
S. R.
2017
).
Global ocean phytoplankton [in: State of the Climate in 2016]
Bull. Amer. Meteor. Soc.
99
S94
S96
10.1175/2018BAMSStateoftheClimate.1
CrossRef
Google Scholar
11
Gillis
D. B.
Bowles
J. H.
Montes
M. J.
Moses
W. J.
2018
).
Propagation of sensor noise in oceanic hyperspectral remote sensing
Opt. Express
26
A818
A831
10.1364/OE.26.00A818
Pubmed Abstract
CrossRef
Google Scholar
12
Gordon
H. R.
Brown
O. B.
Evans
R. H.
Brown
J. W.
Smith
R. C.
Baker
K. S.
et al
. (
1988
).
A semianalytic radiance model of ocean color
J. Geophys. Res. Atmos.
93
10909
10924
10.1029/JD093iD09p10909
CrossRef
Google Scholar
13
Gordon
H. R.
Wang
M.
1994
).
Retrieval of water-leaving radiance and aerosol optical thickness over the oceans with SeaWiFS: a preliminary algorithm
Appl. Opt.
33
443
452
10.1364/AO.33.000443
Pubmed Abstract
CrossRef
Google Scholar
14
Gould
W. G.
McCarthy
S. E.
Coelho
E.
Shulman
I.
Richman
J. G.
2014
).
Combining satellite ocean color and hydrodynamic model uncertainties in bio-optical forecasts
J. Appl. Remote Sens.
083652
10.1117/1.JRS.8.083652
CrossRef
Google Scholar
15
Hooker
S. B.
Esaias
W. E.
Feldman
G. C.
Gregg
W. W.
McClain
C. R.
1992
).
An Overview of SeaWiFS and Ocean-Color, NASA Tech. Memo. 104566
Vol. 1
, eds S. B. Hooker, and E. R. Firestone.
Greenbelt, MD
NASA Goddard Space Flight Center, 24
Google Scholar
16
Hooker
S. B.
McClain
C. R.
2000
).
The calibration and validation of SeaWiFS data
Prog. Oceanogr.
45
427
465
10.1016/S0079-6611(00)00012-4
CrossRef
Google Scholar
17
Hu
C.
Feng
L.
Lee
Z.
2013
).
Uncertainties of SeaWiFS and MODIS remote sensing reflectance: implications from clear water measurements
Remote Sens. Environ.
133
168
182
10.1016/j.rse.2013.02.012
CrossRef
Google Scholar
18
Hu
C.
Feng
L.
Lee
Z.
Davis
C. O.
Mannino
A.
McClain
C. R.
et al
. (
2012a
).
Dynamic range and sensitivity requirements of satellite ocean color sensors: learning from the past
Appl. Opt.
51
6045
6062
10.1364/AO.51.006045
Pubmed Abstract
CrossRef
Google Scholar
19
Hu
C.
Lee
Z.
Franz
B.
2012b
).
Chlorophyll a algorithms for oligotrophic oceans: a novel approach based on three-band reflectance difference
J. Geophys. Res. Oceans
117
C1
),
25
10.1029/2011JC007395
CrossRef
Google Scholar
20
IOCCG
2008
).
Why Ocean Colour? The Societal Benefits of Ocean- Colour Technology
Vol. 7
Dartmouthn, NS
IOCCG
Google Scholar
21
Jay
S.
Guillaume
M.
Chami
M.
Minghelli
A.
Deville
Y.
Lafrance
B.
et al
. (
2018
).
Predicting minimum uncertainties in the inversion of ocean color geophysical parameters based on Cramer-Rao bounds
Opt. Express
26
A1
A18
10.1364/OE.26.0000A1
Pubmed Abstract
CrossRef
Google Scholar
22
JCGM
2008
).
Evaluation of Measurement Data - Guide to the Expression of Uncertainty in Measurement
JCGM
100
2008
Google Scholar
23
Lamquin
N.
Mangin
A.
Mazeran
C.
Bourg
B.
Bruniquel
V.
D'Andon
O. F.
2013
).
OLCI L2 Pixel-by-Pixel Uncertainty Propagation in OLCI Clean Water Branch
ESA ATBD ref. S3-L2-SD-01-C01-ACR-TN
Google Scholar
24
Lee
Z.
Arnone
R.
Hu
C.
Werdell
P. J.
Lubac
B.
2010
).
Uncertainties of optical parameters and their propagations in an analytical ocean color inversion algorithm
Appl. Opt.
49
369
381
10.1364/AO.49.000369
Pubmed Abstract
CrossRef
Google Scholar
25
Lee
Z.
Carder
K. L.
Arnone
R. A.
2002
).
Deriving inherent optical properties from water color: a multiband quasi-analytical algorithm for optically deep waters
Appl. Opt.
41
5755
5772
10.1364/AO.41.005755
Pubmed Abstract
CrossRef
Google Scholar
26
Lee
Z.
Du
K.
Voss
K. J.
Zibordi
G.
Lubac
B.
Arnone
R.
et al
. (
2011
).
An inherent-optical-property-centered approach to correct the angular effects in water-leaving radiance
Appl. Opt
50
3155
3167
10.1364/AO.50.003155
Pubmed Abstract
CrossRef
Google Scholar
27
Maritorena
S.
d'Andon
O. H. F.
Mangin
A.
Siegel
D. A.
2010
).
Merged satellite ocean color data products using a bio-optical model: characteristics, benefits and issues
Remote Sens. Environ.
114
1791
1804
10.1016/j.rse.2010.04.002
CrossRef
Google Scholar
28
McClain
C. R.
2009
).
A decade of satellite ocean color observations
Ann. Rev. Mar. Sci.
19
42
10.1146/annurev.marine.010908.163650
Pubmed Abstract
CrossRef
Google Scholar
29
McClain
C. R.
Feldman
G. C.
Hooker
S. B.
2004
).
An overview of the SeaWiFS project and strategies for producing a climate research quality global ocean bio-optical time series
Deep Sea Res. Part II Topical Stud. Oceanogr.
51
42
10.1016/j.dsr2.2003.11.001
CrossRef
Google Scholar
30
McKinna
L. I. W.
Werdell
P. J.
Proctor
C. W.
2016
).
Implementation of an analytical Raman scattering correction for satellite ocean-color processing
Opt. Express
24
A1123
A1137
10.1364/OE.24.0A1123
Pubmed Abstract
CrossRef
Google Scholar
31
Mekid
S.
Vaja
D.
2008
).
Propagation of uncertainty: expressions of second and third order uncertainty with third and fourth moments
Measurement
41
600
609
10.1016/j.measurement.2007.07.004
CrossRef
Google Scholar
32
Melin
F.
2010
).
Global distribution of the random uncertainty associated with satellite-derived Chl a
IEEE Geosci. Remote Sens. Lett.
220
224
10.1109/LGRS.2009.2031825
CrossRef
Google Scholar
33
Mélin
F.
Sclep
G.
Jackson
T.
Sathyendranath
S.
2016
).
Uncertainty estimates of remote sensing reflectance derived from comparison of ocean color satellite data sets
Remote Sens. Environ.
177
107
124
10.1016/j.rse.2016.02.014
CrossRef
Google Scholar
34
Moore
T. S.
Campbell
J. W.
Dowell
M. D.
2009
).
A class-based approach to characterizing and mapping the uncertainty of the MODIS ocean chlorophyll product
Remote Sens. Environ.
113
2424
2430
10.1016/j.rse.2009.07.016
CrossRef
Google Scholar
35
Morel
A.
Antoine
D.
Gentili
B.
2002
).
Bidirectional reflectance of oceanic waters: accounting for Raman emission and varying particle scattering phase function
Appl. Opt.
41
6289
6306
10.1364/AO.41.006289
Pubmed Abstract
CrossRef
Google Scholar
36
Mueller
J. L.
2000
).
“SeaWiFS algorithm for the diffuse attenuation coefficient, K(490), using water-leaving radiances at 490 and 555 nm,”
in eds S. B. Hooker and E. Firestone, R
NASA Technical Memorandum 2000-206829
Vol. 11
Greenbelt, MD
NASA Goddard Space Flight Center
),
51
Google Scholar
37
Neukermans
G.
Ruddick
K.
Bernard
E.
Ramon
D.
Nechad
B.
Deschamps
P.-Y.
2009
).
Mapping total suspended matter from geostationary satellites: a feasibility study with SEVIRI in the Southern North Sea
Opt. Express
17
14029
14052
10.1364/OE.17.014029
Pubmed Abstract
CrossRef
Google Scholar
38
Novak
M. G.
Cetinić
I.
Chaves
J. E.
Mannino
A.
2018
).
The adsorption of dissolved organic carbon onto glass fiber filters and its effect on the measurement of particulate organic carbon: a laboratory and modeling exercise
Limnol. Oceanogr. Methods
16
356
366
10.1002/lom3.10248
Pubmed Abstract
CrossRef
Google Scholar
39
O'Reilly
J. E.
Maritorena
S.
Mitchell
B. G.
Siegel
D. A.
Carder
K. L.
Garver
S. A.
et al
. (
1998
).
Ocean color chlorophyll algorithms for SeaWiFS
J. Geophys. Res. Oceans
103
24937
24953
10.1029/98JC02160
CrossRef
Google Scholar
40
PACE Science Definition Team
2018
).
Pre-Aerosol, Clouds, and ocean Ecosystem (PACE) Mission Science Definition Team Report
Greenbelt, MD
Google Scholar
41
Putko
M. M.
Taylor
I. I. I. A. C.
Newman
P. A.
Green
L. L.
2001
).
Approach for input uncertainty propagation and robust design in CFD using sensitivity derivatives
J. Fluids Eng.
124
60
69
10.1115/1.1446068
CrossRef
Google Scholar
42
Qi
L.
Lee
Z.
Hu
C.
Wang
M.
2017
).
Requirement of minimal signal-to-noise ratios of ocean color sensors and uncertainties of ocean color products
J. Geophys. Res. Oceans
122
2595
2611
10.1002/2016JC012558
CrossRef
Google Scholar
43
Rasse
R.
Dall'Olmo
G.
Graff
J.
Westberry
T. K.
van Dongen-Vogels
V.
Behrenfeld
M. J.
2017
).
Evaluating optical proxies of particulate organic carbon across the surface atlantic ocean
Front. Marine Sci.
18
10.3389/fmars.2017.00367
CrossRef
Google Scholar
44
Refsgaard
J. C.
van der Sluijs
J. P.
Højberg
A. L.
Vanrolleghem
P. A.
2007
).
Uncertainty in the environmental modelling process – a framework and guidance
Environ. Model. Softw.
22
1543
1556
10.1016/j.envsoft.2007.02.004
CrossRef
Google Scholar
45
Salama
M. S.
Dekker
A.
Su
Z.
Mannaerts
C. M.
Verhoef
W.
2009
).
Deriving inherent optical properties and associated inversion-uncertainties in the Dutch Lakes
Hydrol. Earth Syst. Sci.
13
1113
1121
10.5194/hess-13-1113-2009
CrossRef
Google Scholar
46
Salama
M. S.
Mélin
F.
Van der Velde
R.
2011
).
Ensemble uncertainty of inherent optical properties
Opt. Express
19
16772
16783
10.1364/OE.19.016772
Pubmed Abstract
CrossRef
Google Scholar
47
Seegers
B. N.
Stumpf
R. P.
Schaeffer
B. A.
Loftin
K. A.
Werdell
P. J.
2018
).
Performance metrics for the assessment of satellite data products: an ocean color case study
Opt. Express
26
7404
7422
10.1364/OE.26.007404
Pubmed Abstract
CrossRef
Google Scholar
48
Stramski
D.
Reynolds
R. A.
Babin
M.
Kaczmarek
S.
Lewis
M. R.
Rottgers
R.
et al
. (
2008a
).
Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans
Biogeosciences
171
201
10.5194/bg-5-171-2008
CrossRef
Google Scholar
49
Stramski
D.
Reynolds
R. A.
Babin
M.
Kaczmarek
S.
Lewis
M. R.
Röttgers
R.
et al
. (
2008b
).
Concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans. Supplement to: Stramski, D et al. (2008): Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans
Biogeosciences
171
201
10.5194/bg-5-171-2008
CrossRef
Google Scholar
50
Wang
P.
Boss
E. S.
Roesler
C.
2005
).
Uncertainties of inherent optical properties obtained from semianalytical inversions of ocean color
Appl. Opt.
44
4074
4085
10.1364/AO.44.004074
Pubmed Abstract
CrossRef
Google Scholar
51
Werdell
P. J.
Bailey
S. W.
2005
).
An improved
in-situ
bio-optical data set for ocean color algorithm development and satellite data product validation
Remote Sens. Environ.
98
122
140
10.1016/j.rse.2005.07.001
CrossRef
Google Scholar
52
Werdell
P. J.
Franz
B. A.
Bailey
S. W.
Feldman
G. C.
Boss
E.
Brando
V. E.
et al
. (
2013
).
Generalized ocean color inversion model for retrieving marine inherent optical properties
Appl. Opt.
52
2019
2037
10.1364/AO.52.002019
Pubmed Abstract
CrossRef
Google Scholar
53
Westberry
T. K.
Boss
E.
Lee
Z.
2013
).
Influence of Raman scattering on ocean color inversion models
Appl. Opt.
52
5552
5561
10.1364/AO.52.005552
Pubmed Abstract
CrossRef
Google Scholar
Summary
Keywords
ocean color
remote sensing
bio-optics
uncertainties
oceanography
radiometry
biogeochemistry
Citation
McKinna LIW, Cetinić I, Chase AP and Werdell PJ (2019)
Approach for Propagating Radiometric Data Uncertainties Through NASA Ocean Color Algorithms
Front. Earth Sci.
7:176. doi:
10.3389/feart.2019.00176
Received
30 November 2018
Accepted
21 June 2019
Published
18 July 2019
Volume
7 - 2019
Edited by
David Antoine, Curtin University, Australia
Reviewed by
Constant Mazeran, Solvo (Europe), France; Frederic Melin, Joint Research Centre, Italy
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This is an open-access article distributed under the terms of the
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*Correspondence: Lachlan I. W. McKinna
lachlan.mckinna@go2q.com.au
This article was submitted to Atmospheric Science, a section of the journal Frontiers in Earth Science
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