2.1. Description of Atmospheric Correction
The goal of atmospheric correction is to estimate and then remove the atmospheric path radiance contribution. The water-leaving radiance is at most 10–20% of the total top-of-atmosphere (TOA) radiance in the visible bands (VIS) over open ocean waters, and it can reach 50% in the red bands over turbid waters. Therefore, atmospheric correction is a critical step for remotely sensed data [
11]. For the ocean–atmosphere system, after pre-correcting for gas absorption, whitecaps, and sun glitter on the sea surface, the radiance (
) measured by the remote sensor can be decomposed as follows [
11,
23,
24]
where
is the wavelength.
is the radiance due to Rayleigh scattering,
is the contribution of the aerosols to scattering and the scattering between aerosols and air molecules,
is the diffuse transmittance of the atmosphere from the surface to the sensor,
is the viewing direction,
is the solar irradiance at the mean Earth–Sun distance, and
is the remote-sensing reflectance.
Assuming that the corrections for Rayleigh scattering [
25,
26,
27], whitecaps [
28,
29,
30], gas absorption (O3, NO2) [
31,
32], and sun glitter [
26] have all been made, then the left-hand side of Equation (1) becomes
The term is the aerosol path radiance and the term is the Rayleigh-corrected radiance . Equation (2) thus becomes
The purpose of atmospheric correction is to estimate , the combined aerosol and aerosol-Rayleigh reflectance and to subtract it from the left-hand side of Equation (2).
Over open ocean waters, the
estimation is based on the hypothesis of a black ocean in the near-infrared (NIR) bands [
23]. Over turbid waters, this hypothesis is no longer valid; thus, the contribution of the ocean to the NIR must be estimated to accurately estimate
in the visible (VIS) bands [
11,
12,
33].
In this study, we focused on the standard OLCI radiometric product obtained with the standard OLCI atmospheric correction algorithm and the MODIS-AQUA and VIIRS radiometric products obtained with the NASA standard atmospheric correction algorithm [
24]. The latter algorithm has been extensively validated worldwide [
13,
14,
34,
35]. The hypotheses for the standard OLCI atmospheric correction [
36,
37] are similar to those of NASA MODIS/VIIRS atmospheric correction [
23,
24,
38,
39].
The standard OLCI atmospheric correction algorithm first retrieves the suspended particulate matter (SPM) by assuming the water reflectance (
) and the single scattering aerosol reflectance (
) in the NIR bands based on the black ocean hypothesis. Then, the initial SPM is used to get an initial estimate of
and
in the NIR bands via an empirical relationship [
37]. Finally, the
in the NIR bands is used to get
in the VIS bands by accounting for the multiple scattering of air molecules and aerosols [
36] using Equation (3).
To compare the OLCI product with the MODIS-AQUA and VIIRS
, we first present a brief description of the standard NASA atmospheric correction algorithm. This algorithm is described in Bailey et al. (2010) [
24]. First, the black-pixel assumption is used for both NIR bands to obtain the first initial
estimation [
23]. Second, Equation (3) is used to obtain the initial estimate of
in the NIR bands. Third, the initial
is used to obtain an initial estimate of the chlorophyll concentration by using a bio-optical model. Fourth, this chlorophyll concentration is used to obtain the absorption (
) and backscatter coefficients (
) at the NIR bands via an empirical relationship [
24,
38]. Fifth,
and
in the NIR bands are then used to obtain
in the NIR bands and these quantities are used to remove the non-zero
contribution to
from the NIR bands. Finally, this process is repeated until
convergence to obtain the
in all bands.
To summarize, the general flow of these algorithms is as follows:
Step 1. Estimate
(or equivalently
) at the NIR reference wavelengths using the iterative model [
24,
38,
40] so that the non-zero water-leaving radiance can be removed from the TOA signal, leaving only the aerosol reflectance as the contribution to
.
Step 2. The aerosol reflectance at the NIR bands is used to estimate the aerosol properties and extrapolate aerosol reflectance to the VIS bands. Then the aerosol reflectance can be removed from the TOA signal, leaving the (or equivalently )) in the VIS bands.
In the iterative model, the difference between the real and the estimated is called the error of the iterative model, and it stems from the error that occurs when estimating and is later transferred to in the VIS bands (), leading to inaccurate in the VIS bands ().
The aerosol lookup tables (LUTs) used in the atmospheric correction are obtained from simulations of the radiative transfer and take a given number of aerosol models into account [
23,
33,
36]. These estimated aerosol LUTs are ideal, but they may differ from the values observed over the ocean, especially over coastal waters. The difference between the real
and the estimated
, which we term the error of the aerosols LUTs, also has an impact on the final
estimation.
Thus, Step 1 of the algorithms depends on the accuracy of the iterative model used to estimate the aerosol properties in the NIR bands, while Step 2 depends on the accuracy of the aerosol LUTs, which are used to extrapolate the aerosol reflectance to the VIS bands. Note that the iterative model and the aerosol LUTs algorithms differ between ESA and NASA, although their general principles are similar.
2.2. Error Budget
To improve the product accuracy for ocean color satellites, it is necessary to not only determine the total error but also to find the most algorithm component that contributes the most to the total error. The error budget decomposes the total error into sub-error categories and then compares the sub-errors to obtain the maximal error contribution. Therefore, calculating the error budget is an essential part of the validation.
If the measurement result is determined by a functional relation from other quantities, rather than directly measured, that function or model should represent not only the physical/bio-optical laws but also the measurement process. In particular, it should include all the quantities that may have a significant impact on the uncertainty of the measurement result [
41]. Here, we introduce an error budget model that represents both the physical/bio-optical laws and the measurement processes and apply it to estimate the error budget of the
-derived product [
42].
If the result (C) includes two independent parts, such as A and B. The error of C is
If the function is given as
then the error budget is
where
is the result;
,
,
, and
are function quantities;
is the error of the result; and
,
,
, and
are the error of quantities [
42].
If the function is given as
with B, being a constant. Then the error budget is
where
is the result,
is the quantity,
is the error of the result, and
is the error of the quantity [
42].
2.2.1. Total Error of the Satellite Product
With the in-situ data and match-up procedures, the total difference in
between the in-situ data and the satellite product can be obtained in the VIS bands, for example, the absolute percentage difference (APD, Equation (25)) and the bias (Equation (24)). The in-situ data and the satellite product are independent, according to Equation (4); therefore, the difference between in-situ data and the satellite product can be obtained by
where
is the difference between the in-situ data and a satellite-derived product,
is the error of the in-situ data (also named the uncertainty on the measurement of the in-situ data),
is the error of the satellite-derived product. Then, we can calculate the error of the satellite-derived product in the VIS and NIR bands.
Because the error of in-situ data is independent of the satellite measurement, the total error of the satellite-derived product can be computed as
where
is the difference between the in situ and satellite-derived
,
is the error of the in situ
, and
is the error of satellite-derived
.
2.2.2. Decomposition of the Total Error
For NASA and OLCI atmospheric correction algorithms, can be derived using the equation
For a given band, when the observation geometry and aerosol type are known, then
is also known, and can be considered a constant [
43]. Then, the error of
can be derived using Equations (5)–(8), leading to
Thus, the error of includes two parts: the error of , which is called the error of the Rayleigh-corrected radiance and the error of , which includes the error of the iterative model and the aerosol LUTs.
2.2.3. Error of the Iterative Model
For the NASA and OLCI AC algorithms,
is estimated using an iterative model, and the error of
is the difference between the
from the iterative model and the
from the in-situ data. Thus, the error of
is
where
is the error of
at NIR bands,
, the
from the iterative model in the NIR bands, and
, the true/in situ
in the NIR bands.
Because the error budget relies on the calculation/iterative process, by determination the , the error budget of the error of the iterative model is calculated as follows:
(a) The error on
is passed to
using Equation (11), and the error of
due to the iterative model (
) is
(b) Considering the laws of aerosol radiance in atmospheric correction algorithms [
23,
33,
44],
can be derived as
where
is the Ångström exponent or Ångström coefficient, which is derived from aerosol LUTs using the aerosol optical thickness
.
For a given band and a known aerosol type, is constant. Then, is passed to , and the error of from the iterative model () can be derived using Equations (4)–(7)
(c)
is passed to
using Equation (10), and the error of
from the iterative model (
) is
where
is the error of the iterative model in the VIS bands.
Because
is not included in the L2 standard products, it can be obtained from [
45]
where
,
, and
are the Rayleigh, aerosol, and ozone optical thicknesses, respectively;
is the aerosol single scattering albedo; and
and
are the Rayleigh and aerosol forward scattering probabilities, respectively. The
value was computed using Bodhaine et al. (1999) [
27]. The value of
was taken as a constant equal to 0.008. The value
was also taken as a constant equal to 0.5, while
was allowed to spectrally vary from 0.89 at 412 nm to 0.86 at 670 nm. The value
was extracted from the OLCI L2 radiometric product, and from the orbitons for VIIRS and MODIS. The value of
was extracted from the L2 standard products.
2.2.4. Error of the Aerosol LUTs and the Rayleigh-Corrected Radiance
Given the total error of the satellite-derived product (Equation (10)) and the error of the iterative model (Equation (17)), the error of the aerosol LUTs and the error of the Rayleigh-corrected radiance can be calculated using Equation (4). The error of the aerosol LUTs and the Rayleigh-corrected radiance is
where
is the error of the aerosol LUTs, and
is the error of Rayleigh-corrected radiance. The error of
includes the errors of both
and
.