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Pre-operational Sentinel-3 Snow and Ice (SICE) Products
Algorithm Theoretical Basis Document
Version 3.1
September 22, 2020
A. A. Kokhanovsky (1), J. Box (2), B. Vandecrux (2)
(1) VITROCISET Belgium SPRL, Bratustrasse 7, 64293 Darmstadt, Germany
(2) Geological Survey of Denmark and Greenland (GEUS)
Øster Voldgade 10, 1350 Copenhagen, Denmark
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© 2020 by the author(s). Distributed under a Creative Commons CC BY license.
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Abstract: This document describes the theoretical basis of the algorithms used to determine
properties of snow and ice from the measurements of the Ocean and Land Color Instrument (OLCI)
onboard Sentinel-3 satellites within the Pre-operational Sentinel-3 snow and ice products (SICE)
project: http://snow.geus.dk/. The code used for the SICE retrieval and its documentation can be
found at https://github.com/GEUS-SICE/pySICE. The algorithms were developed after the work
from Kokhanovsky et al. (2018, 2019, 2020).
Keywords: snow; albedo; remote sensing; OLCI; Sentinel-3
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1. Introduction .................................................................................................................... 4
1.1. Ocean and Land and Colour Instrument .................................................................. 4
1.2. Generated Products .................................................................................................. 5
1.3. Summary of assumptions ......................................................................................... 6
2. Snow and ice property retrievals .................................................................................... 6
2.1. Definitions ............................................................................................................... 6
2.1.1. Geometry of the system .................................................................................... 6
2.1.2. Reflectance, spherical and plane albedos ......................................................... 7
2.2. Retrieval overview ................................................................................................... 8
2.3. Atmospheric correction ........................................................................................... 9
2.3.1. Correction of the OLCI TOA reflectance for molecular absorption ................ 9
2.3.2. Molecular and aerosol scattering of light and effects on the top-of the
atmosphere reflectance ................................................................................................. 11
2.4. Retrieval of the surface characteristics .................................................................. 15
2.4.1. Clean snow ..................................................................................................... 15
2.4.2. Polluted snow and ice ..................................................................................... 16
2.5. Broadband albedo calculation ................................................................................ 19
2.5.1. General case ................................................................................................... 19
2.5.1. Approximation used for clean snow ............................................................... 20
2.5.2. Approximation used for polluted snow and ice .............................................. 21
3. NDSI and NDBI ........................................................................................................... 23
4. Appendix: Data tables used in the retrieval .................................................................. 23
References ............................................................................................................................ 24
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1. Introduction
This document describes the theoretical basis of the algorithms used to determine properties
of snow and ice from the measurements of the Ocean and Land Color Instrument (OLCI)
onboard Sentinel-3 satellites within the Pre-operational Sentinel-3 snow and ice products
(SICE) project: http://snow.geus.dk/. The code used for the SICE retrieval and its
documentation can be found at https://github.com/GEUS-SICE/pySICE. The algorithms
were developed after the work from Kokhanovsky et al. (2018, 2019, 2020a).
Snow is composed of ice crystals in contact with each other and surrounded by air. Snow can
include impurities such as dust, soot, algae (e.g., Skiles et al., 2018), here referred to as
‘pollution’. Snow can also contain liquid water. The volume concentration of snow grains is
usually around 1/3 with 2/3 of the snow volume occupied by air (Proksch et al., 2016). The
concentration of pollutants is often low, that is, below 100 ng/g especially in polar regions
(Doherty et al., 2010).
The algorithms described here are dedicated to the retrieval of snow optical properties such
as snow spectral and broadband albedo and also snow microstructure (snow specific surface
area and effective optical grain size). We propose a snow mask based on the Normalized
Difference Snow Index (NDSI) and a technique to retrieve the concentration of pollutants in
snow, which is possible only for the cases with relatively heavy (above 1ppmv) pollution
load (Warren, 2013).
1.1. Ocean and Land and Colour Instrument
Ocean and Land and Colour Instrument (OLCI) is a 21 band spectrometer that measures solar
radiation reflected by the Earth’s atmosphere and surface with a ground spatial resolution of
300 m (see Table 1). The OLCI swath width is 1270 km. OLCI is installed on both Sentinel-
3A and Sentinel-3B satellite platforms operated by the European Space Agency (ESA) in
service to the European Union Copernicus Programme. The Sentinel-3 A and B orbit at 802
km altitude, 98.6 orbital inclination and a 10:00 UTC sun-synchronous equatorial crossing
time.
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Table 1. Band characteristics of the Sentinel-3 Ocean and Land Colour Instrument
(OLCI)
1
Band
λ centre
(nm)
Width
(nm)
Band
λ centre
(nm)
Width
(nm)
Band
λ centre
(nm)
Width
(nm)
1
400
15
8
665
10
15
767.5
2.5
2
412.5
10
9
673.75
7.5
16
778.75
15
3
442.5
10
10
681.25
7.5
17
865
20
4
490
10
11
708.75
10
18
885
10
5
510
10
12
753.75
7.5
19
900
10
6
560
10
13
761.25
2.5
20
940
20
7
620
10
14
764.375
3.75
21
1 020
40
1.2. Generated Products
The products of the SICE algorithms are listed in Table 2. Most of retrievals are based on the
measurements at 865 and 1020nm, where the influence of atmospheric light scattering and
absorption processes on top-of-atmosphere signal as detected on a satellite over polar regions
is weak.
Table 2. SICE: Snow and ice products
Snow product name
Related Section
Units
Expected
range
Maximum
acceptable
uncertainty
in modelling
Optimum
uncertainty
1
Snow mask
NDSI and NDBI
-
0, 1
<10%
5 %
2
3
4
Spectral spherical snow
albedo
Spectral planar snow albedo
Spectral surface reflectance
Retrieval of the
surface
characteristics
-
0 -1.0
<10%
5 %
5
Broadband snow albedo
(planar and spherical, for
three spectral ranges)
Broadband
albedo
calculation
-
0-1.0
<15%*
<5%*
6
Snow Specific Surface Area
Clean snow
m2 kg-1
20-200
<15%
5 %
7
Snow grain diameter
Clean snow
mm
0.02-0.2 mm
<15%
5 %
8
Type and concentration of
pollutants
Pollutant
characteristics
ppmv
(10-6)
0.1-10.0
-
-
9
10
Normalized difference snow
index
Normalized difference bare
ice index
NDSI and NDBI
-
-
-
-
* Source: GCOS (WMO, 2011)
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https://sentinel.esa.int/web/sentinel/user-guides/sentinel-3-olci/resolutions/radiometric
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1.3. Summary of assumptions
The simplified snow layer model represents snow as a:
1. Horizontally homogeneous plane parallel turbid medium;
2. Vertically homogeneous layer;
3. Semi-infinite layer. Therefore, there is no need to account for the reflective properties
of underlying surface.
4. Close packed effects are ignored.
5. Geometrical optics can be used to derive local optical snow characteristics.
6. Impurities (dust, soot, etc.) are located external to ice grains.
7. The single light scattering angular pattern is spectrally neutral in the spectral range
given in Table 1.
8. Only pixels completely covered by snow are considered, i.e., pixels with ice and/or
partially snow pixels are ignored.
9. The effects of slopes and snow roughness are not accounted for.
The output is provided if the OLCI reflectance at 1020nm is larger than 0.1 and the derived
diameter of grains is larger than 0.1mm.
2. Snow and ice property retrievals
2.1. Definitions
2.1.1. Geometry of the system
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Figure 1. Definition of the solar zenith angle , azimuth angle , viewing zenith angle
and relative azimuth angle . Illustration adapted from Hudson et al. (2006).
The angles describing the solar and satellite positions around the point observation are
presented in Figure 1. From these we derive the cosine of the solar zenith angle , the cosine
of the viewing zenith angle and the scattering angle :
(2.1.1)
2.1.2. Reflectance, spherical and plane albedos
The top-of-atmosphere reflectance is defined as:
(2.1.2)
where, is the intensity of reflected light, is the solar flux at the top-of-atmosphere. Many
satellite instruments simultaneously measure both and and allow the derivation of the
top-of-atmosphere reflectance. In the absence of cloud, the bottom-of atmosphere
reflectance or snow reflectance is defined by Eq. (2.1.2) when applied at the bottom of
the atmosphere.
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The reflectance depends on atmospheric effects due to molecular and aerosol scattering and
absorption of solar radiation. For retrieval of surface optical properties, these effects must be
removed.
The plane albedo is defined as the integration of bottom-of atmosphere reflectance R
across all viewing azimuth and zenith angles:
(2.1.3)
The spherical albedo is found by integration of R over all incident angles :
(2.1.4)
2.2. Retrieval overview
We first convert the top of the atmosphere radiance to reflectance using the SNAP Rad2Refl
module. The top of the atmosphere reflectances are then corrected for ozone and
molecular scattering. Retrievals are thereafter approached in two ways, depending on
whether they are considered as clean or polluted snow. The test differentiating these two
snow conditions is based on the theory presented in Section Clean snow: given the pixel’s
illumination and viewing geometry, we calculate the theoretical RTOA at band 1 (λ = 400 nm)
that this pixel would have if it had a surface reflectance of 1. If the observed RTOA is higher
than this virtual RTOA, the pixel is considered as clean snow. Otherwise it is considered as
polluted snow.
Clean snow retrieval approach
If the pixel is classified as clean snow, we derive snow spectral albedo in the spectral range
0.3-2.4 micrometres using the two-parameter analytical equation as described by
Kokhanovsky et al. (2019) under assumption that atmospheric effects can be ignored at OLCI
channels 16 and 21 (λ = 865 nm and λ = 1020 nm). This simple approach has produced good
performance when comparing the retrieved snow spectral albedo and broadband albedo to
ground observations (Kokhanovsky et al., 2019).
Polluted snow retrieval approach
The atmospheric correction for the polluted snow is treated in two ways depending on the
OLCI reflectance at band 21 (λ = 1020 nm).
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Case 1: Polluted pixels with RTOA at band 21 over 0.4
If OLCI reflectance at channel 21 is above 0.4, we assume that scattering and absorption of
light by surface impurities and atmosphere can be ignored at the wavelengths 865 and
1020nm but not below 865 nm. Therefore, reflectances of bands above 16 are being retrieved
based on the OLCI measurements at bands 16 and 21 (λ = 865 nm and λ = 1020 nm) using
the analytical equation for the spectral surface albedo presented by Kokhanovsky et al.
(2019). For wavelengths below 865 nm, we account for gaseous absorption and light
scattering by aerosol in the framework of the theory described in Section Atmospheric
correction.
Case 2: Polluted pixels with RTOA at band 21 under 0.4
In the case of polluted and dark (RTOA at band 21 under 0.4) snow and ice pixels, scattering
within the atmosphere affect all OLCI channels. We can no longer assume that pollutants
and other effects have small influence on OLCI reflectance above 865nm. Here we need an
additional assumption on the intrinsic optical properties of these dark surfaces before the
retrieval is performed.
In both cases, the albedo inside O2 and water vapor absorption bands is derived using the
linear interpolation of results for neighbouring channels.
2.3. Atmospheric correction
2.3.1. Correction of the OLCI TOA reflectance for molecular absorption
The top-of-the-atmosphere reflectance is corrected for ozone absorption using the
ozone transmittance function :
(2.3.1)
Where the transmittance function is defined as in Rozanov and Rozanov (2010):
(2.3.2)
Where
is the air mass factor and is the ozone vertical optical depth (VOD) at the
wavelength , defined as:
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(2.3.3)
Here is the ozone absorption cross-section at the height z and the wavelength
is the concentration of the ozone at the height z. Equation (2.3.3) depends on the
vertical profile of and but in the absence of such information, we use the
simplification by Kokhanovsky et al., (2020b) where the ozone optical depth is expressed as
the product the total column ozone and an reference ozone absorption cross-section
:
(2.3.4)
The value of was estimated by Kokhanovsky et al., (2020b) as follows. A reference
vertical profile of pressure, temperature and ozone concentration was extracted from the 2D
chemistry-climate model from Sinnhuber et al. (2009) at 75oN and for the month of August.
For these reference profiles, was calculated using the parametrization of ozone
cross-section from Serdyuchenko et al. (2014), as shown in Figure 2, and integrated vertically
to calculate the reference optical depth (Eq. (2.3.3)). Eventually, the reference optical
depth was normalized by the total ozone of the reference profile,
= 405 DU =
8.6728e-3 kg m-2, to derive the reference ozone absorption cross-section:
(2.3.5)
Eventually, the transmittance can be calculated for each pixel using Equation (2.3.2) and
(2.3.4):
(2.3.6)
Where
is the total column ozone concentration provided by the European Centre for
Medium-Range Weather Forecast (ECMWF) within the OLCI files.
We note that one should account for the instrument spectral response function because the
measurements are usually performed not at a single wavelength but in narrow spectral range
Therefore, the value of will differ for different instruments even if measured
at the same central wavelength.
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Figure 2. Optical depth of the total ozone column as a function of wavelength. The OLCI
bands are highlighted in blue.
2.3.2. Molecular and aerosol scattering of light and effects on the top-of the
atmosphere reflectance
The background atmospheric aerosol in Arctic is usually characterized by the low values of
aerosol optical thickness and values of single scattering albedo close to one. Therefore, one
can neglect light absorption by aerosol and assume that the atmosphere-underlying surface
reflectance (due to molecular and aerosol scattering and reflectance from underlying surface)
can be presented in the following way:
(2.3.7)
where the surface spherical albedo and snow reflectance are the quantities we want to
quantify in this retrieval. But before that, three characteristics of the atmosphere need to be
quantified: the atmospheric reflectance , the spherical albedo of atmosphere , and the
total atmospheric transmittance from the top-of-atmosphere to the surface and back to the
satellite .
Before the atmospheric reflectance can be derived, several characteristics of the atmosphere
should be described: the molecular and aerosol optical depth, which describe how opaque the
atmosphere is at a given wavelength, the phase function and its derivatives: the asymmetry
parameter and backscattering fraction. In this section we describe how we derive these
characteristics and eventually present the atmospheric reflectance calculation in Section:
Calculation of the atmospheric reflectance, transmittance and spherical albedo.
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Molecular and aerosol optical depth
The atmospheric reflectance depends on the atmospheric optical thickness, which can be
presented in the following form:
(2.3.8)
The molecular optical depth can be approximated as
(2.3.9)
where
, p is the site pressure, and the
wavelength is in microns. We calculate the site pressure using the following equation:
. Here the height of the underlying surface provided in OLCI files and H = 7.64
km is the scale height.
It follows for the aerosol optical depth:
(2.3.10)
where . The pair represents the Angström parameters. Currently, we use
the fixed values of in our retrievals. Due to low aerosol load in Arctic,
this assumption does not lead to the substantial errors.
Phase function, asymmetry parameter and backscatter of the atmosphere
The phase function of a media define the light intensity scattered by the media at a given
wavelength and towards the scattering angle . The phase function is normalized so that
integrating for all gives one. The asymmetry parameter of the media is then defined
as the intensity-weighted average cosine of the scattering angle (Hansen and Travis, 1974).
It ranges from -1 for completely backscattered light to +1 for entirely forward scattered light
and is defined as:
(2.3.11)
In presence of both molecular scattering and aerosol, the phase function can be presented in
the following form:
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(2.3.12)
where
(2.3.13)
is the molecular scattering phase function and is the aerosol phase function. We shall
represent this function as:
(2.3.14)
Therefore, it follows for the asymmetry parameter:
(2.3.15)
The parameter varies with the location, time, aerosol, type, etc. We shall assume that it
can be approximated by the following equation:
With
(2.3.16)
The coefficients in this equation are derived from multiple year AERONET observations
over Greenland.
Another useful quantity is the so-called backscattering fraction, meaning the fraction of light
scattered in the backward direction defined as:
(2.3.17)
Which, using Eq. (2.3.12), translated into:
(2.3.18)
where =0.5 and
(Kokhanovsky et al., 2020b).
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It should be pointed that the system of equations given above enables the calculation of
underlying snow-atmosphere reflectance as a function of the aerosol optical thickness for a
known value of the snow spherical albedo.
Calculation of the atmospheric reflectance, transmittance and spherical albedo
The atmospheric reflectance , caused by coupled aerosol-molecular scattering, can be
presented within the framework of the Sobolev approximation (Sobolev, 1975) as the sum of
the reflectance due to single scattering and the reflectance due to multiple scattering :
(2.3.19)
Both contributions can be expressed as a function of the atmospheric optical depth and
of the atmosphere’s phase function , and its asymmetry parameter .
The single scattering contribution is expressed as:
(2.3.20)
The multiple light scattering contribution is approximated as
(2.3.21)
where
(2.3.22)
The transmission function is approximated as follows:
(2.3.23)
Where B is the backscattering coefficient defined in Eq. (2.3.17) and Eq. (2.3.18).
The atmosphere’s spherical albedo is found using the approximation proposed by
Kokhanovsky et al. (2007):
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(2.3.24)
The coefficients of polynomial expansions of all coefficients (a, b, c, ) with respect to the
value of g are given by Kokhanovsky et al. (2005).
2.4. Retrieval of the surface characteristics
2.4.1. Clean snow
The asymptotic radiative transfer theory relates the reflectance of a medium to its spherical
albedo and consequently allows for the determination of spherical albedo using reflectance
observations for a given observation geometry. See He and Flanner (2020) for a review of
current radiative theories. The framework used here was first developed and verified with
airborne measurements of albedo and reflectance over a bright cloud field with the spherical
albedo in the range 0.8-0.95 (Kokhanovsky et al., 2007). This relationship was later adapted
to snow by Kokhanovsky et al. (2018, 2019, 2020a) to retrieve the snow spherical albedo
from the atmosphere-corrected OLCI reflectance as:
(2.4.1)
where is the theoretical reflectance of snow in the absence of absorption (Kokhanovsky
et al., 2019, Appendix A), is the cosine of the solar zenith angle, is the cosine of the
viewing zenith angle, and is the angular function (Kokhanovsky et al., 2019) defined as:
(2.4.2)
where is either or .
The spherical albedo for clean snow can be presented in the following form (Kokhanovsky
et al, 2018):
(2.4.3)
Where l is the effective absorption length in snow and is the bulk absorption coefficient
of ice calculated from the wavelength and, the imaginary parts of ice refractive index
(Warren and Brandt, 2008) reported in Section Appendix: Data tables used in the retrieval:
(2.4.4)
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The plane albedo can be derived eventually from spherical albedo. Namely, it follows
(Kokhanovsky et al., 2019):
(2.4.5)
The effective absorption length l does not depend on the wavelength in the OLCI spectral
range as demonstrated by Kokhanovsky et al. (2018). The same is true for , the reflectance
of non-absorbing snow layer. Therefore, we can derive both parameters from measurements
of OLCI reflectance at two wavelengths and as in Kokhanovsky et al., (2018) that
satisfy the following two criteria: i) the reflectance at these channels must be sensitive to the
parameters of interest, and ii) these channels must be least influenced by atmospheric
scattering and absorption processes. Consequently, the OLCI channels centred around 865
and 1020 nm are the best candidates for the retrieval.
The and parameters are subsequently defined as follows.
(2.4.6)
And can then be used in Eq. (2.4.1) and (2.4.3) at the wavelength to determine l :
(2.4.7)
The derived value of l can be used to determine the snow spherical/plane albedo and also
snow reflection function (OLCI bottom of atmosphere reflectance) at any OLCI wavelength
using Eqs. (2.1.1)-(2.3.5). The diameter d of ice grains in snow is estimated using the
effective absorption length (Kokhanovsky et al., 2019):
(2.4.8)
where the parameter depends on the type of snow and shape of grains. We assume that
A=0.06 in the retrievals as suggested by Kokhanovsky et al. (2019). The snow specific
surface area is derived as
(2.4.9)
where g cm-3 is the bulk ice density.
2.4.2. Polluted snow and ice
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As in the case of clean snow, we assumed that scattering and absorption of light by
atmosphere and impurities in snowpack can be ignored at the wavelengths 865 and 1020nm.
This makes it possible to derive the parameters , l, d and in the same way as for clean
snow. However, for polluted snow, cannot be derived from Eq. (2.4.3) because the spectral
reflectance in the visible is influenced not just by ice grains but also by various impurities
(soot, dust, algae).
Nevertheless, Eq. (2.3.1) describing ozone absorption, Eq. (2.3.7) describing atmospheric
correction and Eq. (2.4.1) that links observed reflectance to surface reflectance and spherical
albedo can still be used at all channels except those subject to molecular absorption by
oxygen: channels 13-15; and water vapor: channels 19 and 20. For the remaining channels,
1-12, 16-18 and 21, Equations (2.3.1) (2.3.7) and (2.4.1) give the following system:
(2.4.10)
Where is the measured top-of-atmosphere reflectance function, is atmospheric
contribution to the measured signal, is the spherical albedo of the atmosphere, is the
bottom-of-atmosphere surface reflectance, is atmospheric transmittance from the top-of-
atmosphere to the underlying surface and back to the satellite position, is the transmittance
of purely gaseous atmosphere. Given that is measured and that, , , , can
be calculated following the approach detailed above, the system presented in Eq. (2.4.10) has
therefore only one unknown, , which cannot be presented in closed form. We consequently
derive iteratively using Simpson’s rule.
For the channels 13-15, affected by oxygen absorption, and 19-20, affected by water vapor
absorption, the spherical albedos are linearly interpolated between the retrieved spherical
albedo at channels 12 and 16 in the first case and channels 18 and 21 in the second.
For the very dark, polluted pixels (<0.4 at band 21), we assume that the underlying
surface is not snow, the application of the equation (2.4.6) relating the snow albedo to the
snow grain size is not justified. In this case, it is assumed that the value of reflectance for a
non-absorbing surface can be approximated, as discussed by Kokhanovsky et al. (2019), by
the following expression:
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(2.4.11)
where A = 1.247, B = 1.186, C = 5.157, and
(2.4.12)
and θ is the scattering angle in degrees. This formulation of is then used when solving Eq.
(2.4.10).
Pollutant characteristics
The concentration of pollutants in snow is estimated using the approach described below. It
is assumed that the spherical albedo, solved in Eq. (2.4.10), can also be expressed as in
Kokhanovsky et al. (2018):
(2.4.13)
where is spectral absorption coefficient of impurities, is so-called absorption
enhancement parameter for ice grains (Kokhanovsky et al., 2019), is the volumetric
concentration of ice grains in snowpack and is the bulk absorption coefficient of ice
defined in Eq. (2.4.4). We shall assume that Babs=1.6 in the retrieval procedure.
In the visible spectrum, the absorption by ice particle can be neglected () and the
polluted snow spherical albedo can be presented in the following form:
or
(2.4.14)
Let us assume that the impurity absorption coefficient can also be expressed in
following form:
(2.4.15)
where The Angström absorption coefficient can then be derived from Eq.
(2.4.14) and (2.4.15) evaluated at400 nm and 412.5 nm where the spherical
albedo was previously derived:
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(2.4.16)
The Angström coefficient can then be used to characterize the type of pollutant present in the
snow. Since soot has a typical Angström coefficient around one while dust’s Angström
coefficient ranges from 3 to 7, we here assume that the snow is polluted by black carbon if m
< 1.2 and by dust pollution otherwise.
Eq. (2.4.15) evaluated for also gives:
(2.4.17)
Eq. (2.4.13) can be used to derive , the normalized absorption coefficient of
impurities:
(2.4.18)
Once again, neglecting absorption by ice particle from the measurements at the wavelength
400 nm (), the relative volumetric concentration of pollutants in snow can
be derived:
(2.4.19)
where is the volumetric absorption coefficient of impurities.
In the case of soot, can be approximated as in Kokhanovsky et al., (2018):
(2.4.20)
Here, is the enhancement
is the bulk absorption coefficient of soot ,
is the imaginary part of soot refractive index, currently set at a constant .
In the case of dust pollution, we assume: to calculate the relative
volumetric concentration of pollutants in snow .
2.5. Broadband albedo calculation
2.5.1. General case
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The derived spectral albedo is used to integrate the planar and spherical broadband albedo
(BBA) over any wavelength interval [:
(2.5.1)
where is the incident solar flux at the snow surface, is plane (p) or spherical (s)
albedo depending plane or spherical BBA, , is to be calculated. Currently, only
shortwave spherical/plane BBA () is being retrieved but
additional ranges may be added in the future depending on user demand.
Broadband albedo are only weakly sensitive to the variation of . The spectrum of
incident solar flux at the snow surface is therefore assumed to be identical in all pixels and is
approximated by the following analytical equation:
(2.5.2)
where 3.238e+1, -1.6014033e+5, 7.95953e+3, 11.71 , and
2.48. The coefficients have been derived using the code SBDART (Ricchiazzi et al.,
1998) in the spectral range 300-2400 nm at the following assumptions.
Table 3. Assumptions used in SBDART to derive the solar flux at the surface
Parameter
Value
water vapor column
2.085 g m-2
ozone column
0.35 atm-cm
tropospheric ozone
0.0346 atm-cm
aerosol model
rural (Shettle and Fenn, 1979)
vertical optical depth of
boundary layer at 550nm
0.1
altitude
825 m a.s.l.
solar zenith angle
60 degrees
snow albedo at the surface
calculated using spherical
grains of 0.25 mm diameter
2.5.1. Approximation used for clean snow
In the case of clean snow, the exact integration of BBA (Eq. (2.3.14)) is possible because the
spectral reflectance is known for each of OLCI measurement wavelength. However, this
integration is time consuming. To speed up the retrieval process, the shortwave spherical
albedo can be directly expressed as a function of the retrieved grain diameter:
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(2.5.3)
where d is expressed in microns and a = 0.642, b = 0.1044, c = 0.1773, 158.62 , and
= 2448.18 .
For the shortwave broadband plane albedo, we use the same equation as for the spherical
albedo. However, the second order polynomial is used to represent the dependence of the
coefficients a,b,c, ,with respect to the cosine of the solar zenith angle :
,
… etc. The coefficients of parametrization are given
in Table 4.
Table 4. The coefficients of the parametrization for the shortwave plane albedo.
a
0.7389
-0.1783
0.0484
b
0.0853
0.0414
-0.0127
c
0.1384
0.0762
-0.0268
, microns
187.89
-69.2636
40.4821
, microns
2687.25
-405.09
94.5
2.5.2. Approximation used for polluted snow and ice
For polluted snow, the spherical albedo and planar albedo cannot be expressed in a
closed form as it is the solution of the system of equation (2.4.10). Nevertheless, ( and
respectively) are known for each of the wavelength corresponding to OLCI channels. To
circumvent this issue, we build functions of the wavelength that approximate the retrieved
over three intervals:
1) Over 400-709 nm, we approximate spherical and planar albedo by a polynomial
of the second order fitted to the retrieved 400 nm), 560 nm), and
709 nm).
2) Over 709-865 nm, we approximate spherical and planar albedo by a polynomial
of the second order fitted to the retrieved 709 nm), 753 nm), and
865 nm).
3) Over 865-2400 nm, we approximate the spherical and planar albedo with an
exponential function fitted to the retrieved 865 nm), and 1020nm).
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These assumptions make it possible to derive the value of BBA analytically.
First, for all three intervals, the denominator of Eq. (2.5.1) can be calculated as:
,
(2.5.4)
Over the intervals , either equal to 400-709 nm or 709-865 nm, the spherical albedo
can be expressed using its polynomial approximation:
(2.5.5)
Where a, b and c take different values whether they are fitted to derived or . With this
formulation of , the numerator in Eq. (2.5.1) can be expressed in the following form:
(2.5.6)
where
,
(2.5.7)
And
(2.5.8)
For wavelengths in the 865-2400 nm range, we use the exponential approximation of :
(2.5.9)
Where and take different values whether they are fitted to derived or . And in that
case, the numerator in Eq. (2.5.1) can be expressed in the following form:
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(2.5.10)
3. NDSI and NDBI
The normalized difference snow index (NDSI) is calculated as:
(2.5.1)
A pixel is considered snow-covered if NDSI is in the range 0.3-0.4 and R(400nm) is larger
than 0.75. The normalized difference bare ice index (NDBI) is calculated as:
(2.5.2)
The bare ice is classified in two steps. First, dark bare ice is identified where NDBI is less
than 0.65 and R (400nm) is less than 0.75. Then for cases the dark bare ice flag is not set, the
bare ice flag is equal to one (100% bare ice – covered pixel), if NDSI is larger than 0.33. Also
is assumed that the dark dirty bare ice flag is equal to one (100% dark dirty bare ice – covered
pixel), if NDBI is smaller than 0.65 and R (400nm) is smaller than 0.75 and that a land mask
is used.
4. Appendix: Data tables used in the retrieval
Table A 1. The ozone vertical optical thickness (as function of wavelength
in terrestrial atmosphere at the ozone concentration of 405 DU = 8.6728e-3 kg m-2.
(nm)
(-)
(nm)
(-)
(nm)
(-)
400
1.38E-04
665
2.10E-02
767.5
2.73E-03
412.5
3.05E-04
673.75
1.72E-02
778.75
3.26E-03
442.5
1.65E-03
681.25
1.47E-02
865
8.96E-04
490
8.94E-03
708.75
7.98E-03
885
5.19E-04
510
1.75E-02
753.75
3.88E-03
900
6.72E-04
560
4.35E-02
761.25
2.92E-03
940
3.13E-04
620
4.49E-02
764.375
2.79E-03
1020
1.41E-05
Table A 2. The imaginary part of ice refractive index at OLCI bands and wavelength ()
as reported in Warren and Brandt (2008).
Band
(nm)
Band
(nm)
Band
(nm)
Oa1
400
2.37E-11
Oa8
665
1.78E-08
Oa15
767
8.13E-08
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Oa2
412
2.70E-11
Oa9
673
1.95E-08
Oa16
778
9.88E-08
Oa3
442
7.00E-11
Oa10
681
2.10E-08
Oa17
865
2.40E-07
Oa4
490
4.17E-10
Oa11
708
3.30E-08
Oa18
885
3.64E-07
Oa5
510
8.04E-10
Oa12
753
6.23E-08
Oa19
900
4.20E-07
Oa6
560
2.84E-09
Oa13
761
7.10E-08
Oa20
940
5.53E-07
Oa7
620
8.58E-09
Oa14
764
7.68E-08
Oa21
1020
2.25E-06
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