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Arctic+ Salinity: Algorithm Theoretical Baseline Document

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Abstract and Figures

The main purpose of this Algorithm Theoretical Baseline Document (ATBD) is to provide a detailed definition of the processes and algorithms developed to retrieve Arctic sea surface salinity from the SMOS measures in the context of the Arctic+ salinity project. The data retrieved using this ATBD is available at http://dx.doi.org/10.20350/digitalCSIC/12620
Content may be subject to copyright.
Algorithm Theoretical Baseline Document
Customer: ESA
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Filename: Arctic+Salinity_D1.3_ATBD_v2r0.tex
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Signatures
Name Signature Date
Author Justino Martínez
BEC 14/04/2020
Reviewed by
Carolina Gabarró
BEC 20/04/2020
Antonio Turiel
BEC 20/04/2020
Rafael Catany
ARGANS 20/04/2020
Approved by Justino Martínez
BEC 21/04/2020
Carolina Gabarró
BEC 21/04/2020
Rafael Catany
ARGANS 21/04/2020
Authorized by
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Amendment Record Sheet
Document change record
Date/Issue Description Section/Figure
November 2019 / v1r3 d0computation revised Section 2.1
November 2019 / v1r3 Changed climatology period Section 2.3.1.3, figures 19,
20 and 21 and table 2
November 2019 / v1r3 Computed distance to the ice
Changed L3 filtering
Sections 2.4.0.1 and 2.5
figures 34 and 33
November 2019 /v1r3 Deliver to ESA New document
December 2019 / v1r4 Introduction of key tables Added section 4
December 2019 / v1r4 Justification of not use NS, Gkj Section 2
December 2019 / v1r4 Use of BEC roughness model Section 2.2
December 2019 / v1r4 Algorithm scheme clarification Section 2and figure 1
December 2019 / v1r4 Better explained figure Section 2.3.1.2 and figure 14
December 2019 / v1r4 Better explained
temporal correction Section 2.5 and figure 27
December 2019 / v1r4 Source code moved to appendix Added Appendix A
December 2019 /v1r4 Deliver to ESA New document
April 2020 / v2r0 Added WOA 2018
bibliographic reference and
changes in figures
Section 2.3.1.4
April 2020 / v2r0 Changed period forSMOS-based
climatology computation Section 2.3.1.3
April 2020 / v2r0 Introduced a spatial
correction in L2B
Sections 2.5 and 4,
new sections 2.6 and 2.7, modified
figure 1
May 2020 / v2r0 Deliver to ESA New document
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Table of Content
1 Introduction 15
1.1 Structure of this document ............................... 15
2 Overview and generation of the Arctic+ Salinity product 17
2.1 Generation of the level 1C ............................... 18
2.1.1 The importance of the grid projection ..................... 22
2.1.1.1 Former Arctic Grid Projection: BEC Arctic v2.0 product ..... 22
2.1.1.2 Current Arctic Grid Projection: Arctic+ SSS product ....... 23
2.2 Generation of the level 1D ............................... 24
2.3 Debiasing ........................................ 26
2.3.1 New Debiasing strategy ............................ 28
2.3.1.1 Homogeneous EAF-FOV discretization .............. 28
2.3.1.2 Debiasing brightness temperatures ................ 29
2.3.1.3 SMOS-based climatology computation .............. 32
2.3.1.3.1 Derived statistical quantities ............... 33
2.3.1.4 Annual IF S reference ........................ 36
2.4 Inversion ........................................ 43
2.4.0.1 L2A filtering ............................. 44
2.4.0.2 New L2B generation and its filtering ................ 46
2.4.1 Error propagation ................................ 46
2.4.1.1 L2B computation ........................... 47
2.5 L2B temporal correction ................................ 50
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2.6 L2B spatial correction ................................. 53
2.7 L3 maps creation .................................... 58
3 Dielectric constant study 62
4 Key aspects of processing 66
5 References 68
Appendix Source C code 72
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List of figures
1 General algorithm SSS retrieval. As part of the algorithm it has been used the
debiased non-Bayesian strategy defined by [Olmedo et al., 2017] but the debias-
ing is performed in TBinstead of in SSS as described in 2.3.1. Note that TBsea
is necessary to compute the reflectivity of the sea surface which it is necessary
to perform the TBcorrections and dielectric model must be used in the L1D-BOA
computation. ...................................... 17
2 L1B to L1C TOA algorithm overview ......................... 19
3 (Left) Tessellation of the fundamental hexagon using N hexagons centered on an-
tenna grid points. Example with N = 16 ×16. (Right) Detail of the hexagon cen-
tered in an antenna grid point. Each hexagon is composed by six equilateral trian-
gles with h=dG/2,b=dG/3and dG= 2/(dR3N)where dR= 0.875 is the dis-
tance between receivers in SMOS wavelength units (from [Martínez et al., 2018])
.............................................. 20
4 64 ×64 SMOS hexagonal field of view. Purple line indicate the Earth limit;
beyond this limit the FOV points contain sky TB. Blue and black lines encircle
the AF-FOV and EAF-FOV respectively. Points in the horizontal yellow lines are
known as belt whereas suspenders are the points included in the vertical yellow
lines. Belt ans suspenders are points of transition between free-alias zone and
zones affected by Earth-sky aliases or between zones affected by different Earth-
sky alias, therefore the measures are expected to be somewhat degraded over
there. .......................................... 20
5 FOV projected over the Earth when the SMOS antenna points towards Svalbard
island. SMOS horizon (purple line) extends almost 3000 km beyond the bore-
sight (point 0,0). Green line shows the projection of the hexagonal FOV limits
over the Earth surface. The circles indicate the center of 100×100 km regular
cells in plane coordinates (X-Y) using a Lambert azimuthal equal area projection
centered at 90N. The TBcorresponding to each X-Y cell is computed by retro-
projecting its central grid point on the antenna plane and performing a weighted
average in (ξ,η) following and inverse-square law (compare with figure 4). Only
grid points inside EAF-FOV (black line) will be taken into consideration to retrieve
salinity. ......................................... 21
6 Ascending orbit starting at UTC 2018-08-26 06:35:45 using a Lambert Azimuthal
Equal Area projection with origin (0N, 17.18W) indicated by a purple circle . . 22
7 Snapshot from orbit shown in figure 6 corresponding to UTC 2018-08-26 07:28:54. 23
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8 Brightness temperature obtained for UTC 2018-08-26 07:28:54 processed using
a LAEA with origin in the center of the orbit and reprojected to the EASE-Grid
2.0 North (left) and the same snapshot natively processed in the EASE-Grid 2.0
North (right). Black dots indicate the EASE-Grid 2.0 North grid cells center . . . 24
9 Contributions to the measured TB.......................... 25
10 L1C TOA to L1D BOA algorithm overview ...................... 27
11 Non-homogeneous division of the EAF-FOV performed in BEC Arctic v2.0 prod-
uct to compute the SMOS climatology ........................ 28
12 Homogeneous EAF-FOV grid division used. Left: EAF-FOV grid from the 64 ×
64 FOV grid. Right: Sets of 7 elements (denoted by ξη) that will accumulate
measures in order to compute the SMOS-based climatology ............ 29
13 First Stokes value provided by Meissner & Wentz dielectric model for different
values of salinity and temperature. Note how the system moves away from linear
behavior as the salinity and temperature decrease. ................. 30
14 Example of three measures of half first Stokes (FS/2) obtained for a grid point
affected by land-sea contamination. The grid point is located at North Sea coastal
waters (53.86N, 6.70E), approximately at 35 km of the German coast. The
measures (thick lines) have been obtained from successive snapshots (i.e. for
different incidence angles). Thin lines indicate half first Stokes value provided by
Meissner & Wentz dielectric model for different values of salinity, FS(SSS)/2. . . 31
15 The SMOS climatological representative for ascending passes at different FOV
positions ......................................... 35
16 The SMOS climatological representative for descending passes at different FOV
positions ......................................... 36
17 Schema of the area weighted average procedure. Orange circles indicate the
center of each dashed cell of 0.25×0.25 degres (source grid). Blue squares are
the center of the 25×25 km cells (destination grid). ................. 38
18 Half first Stokes provided by WOA2018 and Meissner & Wentz dielectric model
for different points of the FOV ............................. 39
19 Correction to be applied to measured half first Stokes parameter in 4 difference
FOV positions for ascending passes (equation 20) ................. 40
20 Correction to be applied to measured half first Stokes parameter in 4 difference
FOV positions for descending passes (equation 20) ................ 41
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21 Mean half first Stokes correction in the whole Arctic (half first Stokes from WOA
2018 minus SMOS half first Stokes representant). Left: ascending case. Right:
descending case. .................................... 42
22 Mean ascending minus descending half first Stokes correction in the whole Arctic 42
23 Snapshot from ascending L2A orbit over the Kara Sea on August 2, 2015. Left:
salinity values retrieved using the non-Bayesian procedre described in section
2.4. Right: the corresponding error of each retrieval as has been described in
section 2.4.1 and expressed in equation 26. ..................... 48
24 Ascending L2B orbit over the Kara Sea on August 2, 2015. Left: salinity values
computed from L2A snapshots using expressions 27 and 28. Ob’ and Yenisey
rivers discharge are clear at this level as well as the freshwater accumulation
in the Baydaratskaya bay. Right: the corresponding error of salinity value from
equation 29. ....................................... 49
25 9-day L3 map centered on August 2, 2015 combining ascending and descending
orbits. Left: salinity values computed from L2B orbits using expression 30. Right:
the corresponding error of salinity value from equation 32. ............. 49
26 Argo profilers distribution during the period 2011-2018 (dots). Blue line delimits
the bathymetric curve corresponding to 1000 m. Note the lack of Argo profilers
in the Bering, Beaufort, East Siberian, Laptev, Kara, Barents and North seas and
also in Hudson and Baffin bays. ............................ 50
27 Scheme of the iterative procedure used to correct the temporal salinity bias on
level 2. .......................................... 52
28 Weighted average of all ascending L2B orbits in the period 2013-2019 minus
WOA18-A5B7 ...................................... 54
29 Weighted average of all descending L2B orbits in the period 2013-2019 minus
WOA18-A5B7 ...................................... 55
30 Difference between weighted average of all ascenidng and descending L2B orbits
in the period 2013-2019 ................................ 56
31 Skewness of the half first Stokes distribution. Left: ascending. Right: descending.
The corresponding position in the antenna grid is marked with the red dot. . . . . 57
32 Scheme of the procedure used to apply the spatial correction on level 2. Left-
hand side generates the auxiliary files (one for ascending and one for descend-
ing) that are used to correct all L2B orbits as described in th right-side scheme . 57
33 Salinity error derived from the radiometric error. Arctic+ v3.1 map of the period
August 11-19, 2012. .................................. 58
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34 9-day L3 SSS Arctic+ v3.1 map corresponding to the period August 11-19, 2012.
Note the greater coverage and detail of the gradients to that obtained from the
BEC Arctic v2.0 product (figure 35). A section of this map containing from the
Barents Sea to the East Siberian Sea together with Sea Ice concentration it is
shown in figure 36. ................................... 59
35 9-day objectively analized BEC Arctic v2.0 map [Olmedo et al., 2018] correspond-
ing to the period August 11-19, 2012. Compare it with the new Arctic+ v3.1 (figure
34).A section of this map containing from the Barents Sea to the East Siberian
Sea together with Sea Ice concentration it is shown in figure 37. ......... 60
36 Detail of the map shown in figure 34 together with the minimum Sea Ice concen-
tration provided by OSI SAF for the period August 11-19, 2012. The right color
bar indicates Sea Ice concentration whereas the left color bar indicates salinity. . 61
37 Detail of the BEC Arctic v2.0 map shown in figure 35 together with the minimum
Sea Ice concentration provided by OSI SAF for the period August 11-19, 2012.
The right color bar indicates Sea Ice concentration whereas the left color bar
indicates salinity. .................................... 61
38 Half first stokes value obtained for Meissner & Wentz model (blue line), Klein &
Swift model (black line) and George Washington University model (yellow line) fro
different values of SST: SST=0C (upper left), SST=5C (upper right), SST=10C
(bottom left) and SST=15C (bottom right). ..................... 62
39 Retrieved SSS minus Argo salinity as a function of SST for MW and KS dielectric
models. Up: mean value. Down standard deviation ................. 63
40 Ascending SMOS-climatology representant (up) and standard deviation (down)
for Greenland Sea as a funtion of incidence angle and cross track distance. Left
column corresponds to KS values. Right column shows difference between MW
and KS cases ...................................... 64
41 Ascending SMOS-climatology excess kurtosis (up) and skewness (down) for Green-
land Sea as a funtion of incidence angle and cross track distance. Left column
corresponds to KS values. Right column shows difference between MW and KS
cases .......................................... 65
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List of tables
1 List of acronyms used in this document ....................... 13
2 Statistics for the first Stokes bias correction in the antenna grid .......... 42
3 The relevance takes values from 1 to 5; 1 for the most relevant and 5 for the least
relevant ......................................... 67
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Listings
1 Statistical quantities computation. ........................... 72
2 Area intersection computation. ............................ 77
3 Meissner & Wentz dielectric model. .......................... 78
4 Minimization function. ................................. 80
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Acronyms
AF-FOV Alias free field of view
ARF Antena Reference Frame
ATBD Algorithm theoretical basis document
BEC Barcelona Expert Center
BOA Bottom Of the Atmosphere
CCI Climate Change Initiative
CFI Customer Furnished Item
EAF-FOV Extended alias free field of view
ECMWF European Centre for Medium-Range Weather Forecasts
ESA European Space Agency
FOV Field of view
L2OS Level 2 Ocean Salinity
LAEA Lambert Azimuthal Equal Area
LSC Land-sea contamination
OSI-SAF Ocean and Sea Ice Satellite Application Facility
PHC Polar science center Hydrographic annual Climatology
RFI Radio frequency interferences
SMOS Soil Moisture and Ocean Salinity
SSS Sea Surface Salinity
SST Sea Surface Temperature
TB Brightness Temperature
TOA Top of the atmosphere
WOA World Ocean Atlas
Table 1: List of acronyms used in this document
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1 Introduction
The main purpose of this Algorithm Theoretical Baseline Document (ATBD) is to provide a
detailed definition of the processes and algorithms developed to retrieve Arctic sea surface
salinity from the SMOS measures in the context of the Arctic+ salinity project.
This new processing chain that we have implemented and described in this ATBD includes
improvements respect to the previous one [Olmedo et al., 2018] devoted to obtain more de-
fined salinity gradients and improve freshwater fluxes and currents description. Extreme melting
episodes like 2012 and 2019 Greenland melt [Bennartz R. et al., 2013] are indicators of the im-
portance to monitor changes in the Arctic freshwater system. Improving the sea surface salinity
(SSS) Arctic maps is the best option to attain this objective.
1.1 Structure of this document
The document is structured as follows: Next section introduces an overview of the sea surface
salinity (SSS) retrieval algorithm and explains how the processing chain operates, departing
from L1B up to the L3 salinity maps generation. The changes introduced have been important
and the processing chain has been completely recoded affecting to the debiasing, inversion and
salinity products generation.
The new grid implementation is explained in section 2.1.1. The objective of adopting a regional
Arctic grid instead of an orbit-centered grid as in the previous processing chain, is to avoid
spatial interpolations to not to lose information about salinity gradients captured by SMOS.
Debiasing methodology described in [Olmedo et al., 2017] has been changed to be applied to
brightness temperature (TB) instead doing so at salinity level. At least in the regime of low salin-
ity values, the salinity retrieval is a non-linear process. Therefore, freshwater masses should be
better described using this new methodology. Section 2.3.1 contains information about what im-
provements are expected using the new debiasing algorithm implementation together with the
new antenna binning used to perform SMOS-based climatology computation. The new binning
improves the homogeneity of the adquired measures obteining better statistics for the SMOS
representant computation.
The generation of L3 maps has been also deeply changed (section 2.4). This change comes
partially from the fact that, by including the improvements described above, it is unnecessary to
resort to interpolation methods like objective analysis. But this is not the only change in the L3
generation, the new L3 are generated from a new product that has not been created previouly
in the non-bayesian retrieval scheme: the geo-located L2B salinity orbits derived from L1. This
product is corrected to mitigate the known time-dependent biases and it is the basis to create
the level 3 product. In order to account for the error in the retrieved salinity at L2 and L3, a
propagation error of the radiometric uncertainty is introduced in the inversion procedure.
Section 3, is devoted to expose a brief summary of L-band dielectric constant models in SMOS
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mission work (under development). This study supports the choice of Meissner & Wentz model
[Meissner and Wentz, 2004] as dielectric model to perform the salinity retrieval in the Arctic
instead of Klein and Swift model [Klein and Swift, 1977] or the proposed at George Washington
University by Zhou, Lang and collaborators [Zhou et al., 2017].
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2 Overview and generation of the Arctic+ Salinity product
The general SSS retrieval algorithm is summarized in the figure 1. The processing chain
starts from L1B SMOS product v6.21 distributed by ESA (https://earth.esa.int/web/guest/
missions/esa-operational-eo-missions/smos). This algorithm differs of the nominal SMOS
data processing because it is based in the debiased non-Bayesian (DNB) [Olmedo et al., 2017].
In this work the debiasing is performed in TBinstead in salinity as it is proposed in the referred
paper (see 2.3.1 for detailed information). The general sea surface salinity retrieval algorithm
can be divided in five blocks:
Computation of brightness temperature (TB) at the top of the atmosphere (TOA).
Computation of the measured TBat the bottom of the atmosphere (BOA).
Correction of TBbiases. This bias is assumed as different for each Earth position and
SMOS antenna grid point but constant in time.
After the bias correction is applied, proceed with inversion to retrieve Sea Surface Salinity.
The retrieval is perfored using a non-Bayesian approach as described in section 2.4
Create SSS maps from individual orbits containing retrieved SSS values
Figure 1: General algorithm SSS retrieval. As part of the algorithm it has been used the debiased non-Bayesian strategy defined by
[Olmedo et al., 2017] but the debiasing is performed in TBinstead of in SSS as described in 2.3.1. Note that TBsea is necessary
to compute the reflectivity of the sea surface which it is necessary to perform the TBcorrections and dielectric model must be used
in the L1D-BOA computation.
New techniques to improve the quality of SMOS L1 brightness temperatures have not been
considered in the context of this project because i) we had not a proper reference to assess
the cumulative improvement of those improvements, so the first step is to have a proper L2
chain adapted for the Arctic region (the main goal of this project) and ii) there is no a precise
assessment of the potential improvement of those techniques in the case of the polar regions,
so the use of those techniques could finally lead to a degradation of quality while deviating
significant effort and resources to a goal that is not deemd prioritary right now. More particularly:
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The use of nodal sampling (NS) has been reported not only to decrease the impact of the
tails of Radio Frequency Interferences (RFI) but to considerably reduce the radiometric
noise [González-Gambau et al., 2015,González-Gambau et al., 2016]. However, it has
also been reported to produce a contamination close to coastal and ice edges pixels. This
problem can be very disturbing in the case of the Arctic, as the most dynamics areas (and
those of greater oceanographic interest) happen close to the sea ice border; therefore, use
of NS could lead to a degradation of quality precisely at the zones of maximum interest.
While new versions of NS deal with these problems, the assessment of them has not been
completed yet and therefore their use in this project is deemed as speculative and risky,
therefore discarded.
The use of the modification of the correlation eficiency by means of a small percentage of
change of the non-zero baseline elements of the Gkj matrix [Corbella et al., 2015] in order
to reduce land-sea and sea ice-sea contamination has also been deemed as potentially
risky: in the last approved official reprocessing of SMOS L1, the adaptation of Gkj matrix
was discarded because they are calibrated using NIR and not ALL-LICEF and the quality
of L2 Soil Moisture data was degraded. Besides, the modification of Gkj is known to
slightly increase the range of value of the Ocean Target Transformation (OTT), what is
seen also as a degradation. Therefore, with the evidence available at this time the Gkj
modification should be discarded.
2.1 Generation of the level 1C
Computation of TBstarts from the ESA Level 1B (L1B) product. L1B product contains TB
Fourier components arranged in a time-ordered way according to the integration time. The
brightness temperature is computed at TOA level from ESA L1B product in a similar way as
the standard SMOS L1 processor does (https://earth.esa.int/documents/10174/1854456/
SMOS_L1c-Data-Processing-Models). The used processing chain is summarized in figure 2.
As in the standard SMOS L1 processor, a Blackman window is used to reduce the Gibbs-
like contamination [Anterrieu et al., 2002] and the image is reconstructed by applying an In-
verse Fourier Transformation the resulting TBFourier coefficients. The TBimage is recon-
structed in the Antenna Reference Frame (ARF) and referenced to the antenna coordinates
(ξ,η)=(sin θcos φ,sin θsin φ), where θis the angle from the normal to the instrument plane
(0θπ/2) and φis the angle in the instrumental plane (0φπ/2) [McMullan et al., 2008]
The antenna hexagonal grid in which the TBimage is reconstructed (see figure 3for details),
contains 64 ×64 points instead of 128 ×128 used by standard SMOS L1 processor. This
resolution in the antenna level (4096 grid points) is enough to provide the TBvalues because
the number of visibilities from which snapshots are derived by a linear transformation is 2791.
Therefore, 64 ×64 is the minimum number of points in the antenna grid that provide the best
possible resolution because the hexagonal grid must be constructed as 2n×2ngrid and n5
undersamples the image.
After L1B is processed, measured values of TBare expressed in the 64 ×64 hexagonal antenna
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Figure 2: L1B to L1C TOA algorithm overview
grid. However, due to the SMOS orientation with respect to the Earth, the SMOS field of view
(FOV) contains TBvalues corresponding to the sky (i.e. points located above the Earth horizon).
In addition, due to the interferometric character of the instrument, some parts of the FOV are
affected by aliases: aliases between different parts of the Earth disc and aliases between the
Earth and the sky. The central part of the hexagonal FOV is free of aliases and it is known as
Alias Free Field Of View (AF-FOV). The called Extended alias free field of view (EAF-FOV) is
composed by AF-FOV and the zones containing aliases between Earth and sky TB. The zones
of the FOV containing aliases between different zones of the Earth are discarded (see figure 4).
Additional information can be found in [Corbella et al., 2005].
The geolocation of antenna grid points is performed during the L1C generation using ESA Earth
Explorer Mission CFI propagation libraries version 3.7.4 [ESA, 2014]. The geographic coordi-
nates (longitude and latitude) are transformed to plane coordinates by means of the Lambert
Azimuthal Equal Area map projection (LAEA) [Snyder, 1987]. This coordinate reference sys-
tem is recommended by the European Environment Agency for spatial and statistical analysis
[European Environment Agency, 2003].
Once all the geolocation magnitudes have been computed and the measured TB(ξi, ηi)are
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Figure 3: (Left) Tessellation of the fundamental hexagon using N hexagons centered on antenna grid points. Example with N =
16 ×16. (Right) Detail of the hexagon centered in an antenna grid point. Each hexagon is composed by six equilateral triangles
with h=dG/2,b=dG/3and dG= 2/(dR3N)where dR= 0.875 is the distance between receivers in SMOS wavelength units
(from [Martínez et al., 2018])
Figure 4: 64 ×64 SMOS hexagonal field of view. Purple line indicate the Earth limit; beyond this limit the FOV points contain sky
TB. Blue and black lines encircle the AF-FOV and EAF-FOV respectively. Points in the horizontal yellow lines are known as belt
whereas suspenders are the points included in the vertical yellow lines. Belt ans suspenders are points of transition between free-
alias zone and zones affected by Earth-sky aliases or between zones affected by different Earth-sky alias, therefore the measures
are expected to be somewhat degraded over there.
known in all 64 ×64 FOV points (i∈ {1..4096}), the Earth grid is generated and the points of
this grid are retroprojected up to SMOS antenna coordinate reference system for each SMOS
snapshot. Next, for any retroprojected Earth point at antenna level (ξk,ηk) the brightness tem-
perature at the corresponding geographical point (xk,yk) is computed performing a weighted
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Figure 5: FOV projected over the Earth when the SMOS antenna points towards Svalbard island. SMOS horizon (purple line)
extends almost 3000 km beyond the boresight (point 0,0). Green line shows the projection of the hexagonal FOV limits over the
Earth surface. The circles indicate the center of 100×100 km regular cells in plane coordinates (X-Y) using a Lambert azimuthal
equal area projection centered at 90N. The TBcorresponding to each X-Y cell is computed by retroprojecting its central grid point
on the antenna plane and performing a weighted average in (ξ,η) following and inverse-square law (compare with figure 4). Only
grid points inside EAF-FOV (black line) will be taken into consideration to retrieve salinity.
average in (ξ,η) following and inverse-square law:
TB(xk, yk) = TB(ξk, ηk) = Pd<d0(TB(ξi, ηi)/d2)
Pd<d0(1/d2)(1)
where the sums extend over all the antenna ξiηigrid points (i∈ {1..4096}) and
d2= (ξkξi)2+ (ηkηi)2.(2)
In order to ensure a TBvalue in all the grid points, d0must accomplish the inequality d2
0
h2+ (b/2)2= (10/36)d2
Gd0.0.011 (see figure 3for details). The processing chain uses a
value of d0= 0.012 in antenna units.
The effects of the ionosphere on the polarized Tbare computed in this part of the processing
chain. The ionosphere produces a rotation between the TBpolarizations leaving unaltered the
first Stokes parameter (Tx
B+Ty
B)
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2.1.1 The importance of the grid projection
Figure 6: Ascending orbit starting
at UTC 2018-08-26 06:35:45 using a
Lambert Azimuthal Equal Area projec-
tion with origin (0N, 17.18W) indi-
cated by a purple circle
One of the handicaps of the previous SMOS SSS Arctic prod-
uct produced at BEC (BEC Arctic v2.0 hereafter) is the soft-
eness of the salinity gradients. As compared with SMAP
products [Tang et al., 2019], the gradients shown in SMOS TB
seem to be not adequately translated to salinity. We suspect
that this is caused by the objective analysis made to mitigate
the noise in the simple averaging of level 3 products. There-
fore, any spatial interpolation should be avoided as much as
possible. This can be achieved by means of the definition of a
global grid for the Arctic in the earlier stages of the processing
chain.
2.1.1.1 Former Arctic Grid Projection: BEC Arctic v2.0 prod-
uct
BEC Arctic v2.0 [Olmedo et al., 2018] was created from orbits
processed using a local Lambert Azimuthal Equal Area (LAEA)
projection. The origin of the projection for each orbit was lo-
cated in the equator of the Earth in latitude and in the center
of each orbit in longitude. This means that the brightness tem-
perature (TB) was retrieved in a different grid for each L1C
orbit. A complete orbit projected using LAEA with origin corre-
sponding to its own center (0N, 17.18W) is shown in figure
6. A brightness temperature snapshot belonging to this orbit
and located in the north of Greenland is shown in figure 7. The
selected resolution is 25 km and the center of the orbit (purple
dot in figure 6) defines the center of a cell, i.e. it is a bore-
centered grid. The grid used to perform the geolocation and
SSS retrieval processes for each orbit was created departing
from the projection origin by constructing a set of points (the
center of the cells) regularly separated from each other 25 km in x and y directions. Therefore,
the retrieved SSS were obtained in the projection of their own orbit.
The L3 SSS Arctic product v2.0 was distributed in EASE-Grid 1.0 (EPGS:3408) which corre-
sponds to a Lambert Azimuthal Equal Area projection centered at (90N, 0E) using as ellipsoid
the International 1924 Authalic Sphere (a=b=6371228). The adoption of a regional grid for the
Arctic is essential to generate L3 and L4 products and consequently, in the previous version
of the SSS Arctic product, it was necessary to spatially interpolate every retrieved SSS to this
global grid.
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Figure 7: Snapshot from orbit shown in figure 6corresponding to UTC 2018-08-26 07:28:54.
2.1.1.2 Current Arctic Grid Projection: Arctic+ SSS product
The EASE-Grid North v1.0 has been superseded by WGS 84 / NSIDC EASE-Grid 2.0 North
(EPGS:6931) [Brodzik et al., 2012,Brodzik et al., 2014,Brodzik et al., 2018] which introduces
important improvements. One of the improvements is to be easier for users to import data from
version 2.0 into standard software packages and it minimizes common reprojection errors that
have been encountered with the original EASE-Grid definition.
Based on the experience in previous versions of the Arctic SSS products, two points have been
improved concerning the projection used in the processing chain:
1) Use the same projection origin for all orbits in order to avoid spatial interpolations as much
as possible.
2) Introduce the improved version of EASE-Grid North projection
Therefore, in order to tackle these two points the EASE-Grid 2.0 North projection has been
adopted even in the first stages of the processing chain. This approach produces L1C orbits
expressed in the same regional grid, avoiding spatial interpolation in later L3 salinity maps
creation. As EASE-Grid 1.0 North, this projection corresponds to a Lambert Azimuthal Equal
Area but it uses as ellipsoid the WGS84 and the projection center (90N, 0E) does not define
the center of a cell but the intersection of the central four grid cells. The number of rows and
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Figure 8: Brightness temperature obtained for UTC 2018-08-26 07:28:54 processed using a LAEA with origin in the center of the
orbit and reprojected to the EASE-Grid 2.0 North (left) and the same snapshot natively processed in the EASE-Grid 2.0 North
(right). Black dots indicate the EASE-Grid 2.0 North grid cells center
columns are even (720x720 because the adopted resolution is 25km) and the grid is constructed
from the projection origin establishing the center of four cells at x,y ±12.5 km and every 25 km
from there (black squares in figure 8).
Despite reprojection can be done without interpolation, previous version of TB(or SSS) values
were obtained using different orbit-centered projections definition (figure 7) and could not be
projected over the center of the grid cells of a regional projection (figure 8left). In this case it
was necessary to resort interpolation methods to create L3 maps. Adopting the EASE-Grid 2.0
North in the early stages of the processing chain, TBand SSS can be obtained in this regional
projection making spatial interpolation unnecessary (figure 8right).
2.2 Generation of the level 1D
The TBtransformation from TOA to BOA is performed in a similar way as the operational SMOS
level 2 processor chain does (more details are described in [SSS, 2016]). The measured TB
by SMOS is the result of different contributions [Zine et al., 2008] (figure 9): The galactic and
Sun glitter are attenuated by atmosphere, scattered by sea surface and attenuated again by the
atmosphere before arrive to the top of the atmosphere. The sea emission, composed by the
roughness of the sea surface and the flat sea emission, is also attenuated by the atmosphere.
Even the atmosphere itself contributes to the total L-band emission received by SMOS in two
ways: a direct contribution towards the top of the atmosphere and a contribution that is reflected
by the sea and attenuated in its travel up to SMOS.
To retrieve salinity from measured TBit is necessary to quantify all these contributions and
obtain the contribution of the flat sea emission because flat sea emission depends on the
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sea surface temperature (SST) and SSS. The galactic and Sun glitter contributions are com-
puted following [Tenerelli et al., 2008] and [Reul et al., 2007] respectively. The roughness model
used by the official ESA L2 processing chain (known as model 1). Nevertheless the inver-
sion algorithm developed at BEC uses the roughness model (called model 3) described in
[Guimbard et al., 2012]. The BEC roughness model is based on an empirical approach with
SMOS data and, if necessary, it could be adapted to SMOS data adquired in Arctic conditions.
But this is out of scope of this project.
Figure 9: Contributions to the measured TB
The computation of all L-band contributions implies the use of a wide variety of auxiliary in-
formation concerning atmospheric conditions, galactic emission, etc. Additionally, the auxiliary
information must be interpolated to the geographical coordinates and grid used to obtain the
measured TBbefore its use. Therefore, a higher or similar resolution as the used to process the
measured data is desirable for ancillary data. Auxiliary information is also used to filter data in
zones in which physical conditions are not supported by correction models. The used auxiliary
information is the following:
Atmospheric correction
Air temperature 2m over the surface
Atmospheric pressure at surface
Total column water vapor content
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Sun glitter correction
Wind speed module at 10m of the surface
Zonal and meridional component of neutral wind speed at 10 m of the surface
Galactic correction
Wind speed module at 10m of the surface
Zonal and meridional component of neutral wind speed at 10 m of the surface
Surface roughness correction
Wind speed module at 10m of the surface
Filtering purposes
Rain rate
Sea ice concentration
This information is provided to the official SMOS processing chain by the European Centre
for Medium-Range Weather Forecasts (ECMWF) collocated in time and space for each SMOS
orbit. The grid used to provide this data is the used by the official SMOS processing chain
(Icosahedral Snyder Equal Area or ISEA 4H9, see [Matos et al., 2004]) and is converted to the
plane coordinates using nearest-neighbor interpolation.
The Atmospheric correction is based on the modeling of the absorption coefficient of the oxygen
and the water vapor content following a numerical fitting. Nevertheless, the rest of contributions
are based on lookup tables (LUT) used in official processing chain.
Once all the contributions are computed, the measured TBcorresponding to the flat sea contri-
bution (Tmeas
BF S ) can be obtained.
2.3 Debiasing
Debiasing procedure is an important change with respect to the standard processing used in
the ESA SMOS L2OS chain. This non-Bayesian methodology has been developed at Barcelona
Expert Center (BEC) and introduced in [Olmedo et al., 2017] and used to produce the BEC Arc-
tic v2.0 product [Olmedo et al., 2018]. The minimization of the difference between the measured
first Stokes parameter and the modeled one follows a non-Bayesian scheme [Olmedo et al., 2017],
i.e. SSS is retrieved for each incidence angle.
In the SSS retrieval scheme used in [Olmedo et al., 2018] all the SSS retrieved under the same
geographical location, measuring incidence angle, distance to the center of the swath (cross
track distance) and satellite flight direction are accumulated in a SSS distribution. The used
antenna grid consists in a division of the cross track distance in bins of 50 km and a division
of the incidence angles in 5bins (figure 11). The mean around the mode is estimated for
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Figure 10: L1C TOA to L1D BOA algorithm overview
all SSS distributions. These mean values compose the SMOS climatology. This allows to
characterize the systematic errors for each Earth location, measuring antenna position and flight
direction. In this case no Ocean Target Transformation (OTT) is applied since the systematic
errors are already accounted for with the new methodology (see [Tenerelli and Reul, 2010] for
details about OTT). The poor quality measurements are detected by filtering criteria based on
statistical properties of the obtained SSS distributions. By subtracting the corresponding SMOS
climatological value to each SSS value local biases are mitigated, especially those that are
persistent in time as the produced by land sea contamination and permanent Radio Frequency
Interferences (RFI). The computation of absolute SSS from the anomaly is performed by adding
to each level 3 map an annual SSS reference. This annual SSS reference is the same for all
L3 SSS maps and usually World Atlas Ocean 2013 (WOA) climatology [Zweng et al., 2013] is
used but the Polar science center Hydrographic annual Climatology (PHC) [Steele et al., 2001]
has also been used in Arctic region [Olmedo et al., 2018].
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Figure 11: Non-homogeneous division of the EAF-FOV performed in BEC Arctic v2.0 product to compute the SMOS climatology
2.3.1 New Debiasing strategy
In the development of this new version of the SSS Arctic product important changes concern-
ing to debiasing have been implemented as compared to the previous procedure described in
[Olmedo et al., 2017]: improve of the antenna grid division and perform the debiasing not in
SSS but in TB. Why is it advisable to introduce these changes? Statistics improvememnt and
better bias correction at low salinity ranges. Therefore, it is advisable to correct these short-
comings of this part of the processing chain. The approach has been performed by introducing
an homogeneous EAF-FOV discretization and taking into consideration that the salinity retrieval
departing from TBis not a linear process and the debiasing should be addressed at TBlevel
instead of at salinity level.
2.3.1.1 Homogeneous EAF-FOV discretization
As it is shown in figure 11 the EAF-FOV division formely adopted is not regular, especially
outside of the AF-FOV. This introduces different statistic representativeness for different points
of the antenna when SMOS climatology is computed. Therefore a more homogeneous EAF-
FOV discretization is advisable.
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The hexagonal grid in which FOV is divided is a result of the interferometric nature of SMOS
instrument. This geometry hinderes the division of the FOV in homogeneus sections because
these sections must have hexagonal geometry. On the other hand, each section should include
a large number of measures in order to compute a reliable SMOS-based climatology. This
implies that in order to generate homogeneous distributions of the measures, the 64 ×64
points of the FOV should be also homogeneously grouped to obtain a lower number of antenna
points and increase the statistics of each group. The more effective and simple way to attain this
objective is to group the antenna points contained in EAF-FOV in sets of 7 points (a central point
and their 6 closest neighbors) and accumulate the measures for each 7 points dataset in the
same histogram. This grouping is shown in the right part of figure 12. By using this EAF-FOV
discretization, we pretend to better describe the statistics of the measured TBthan using the
shown in figure 11.
Figure 12: Homogeneous EAF-FOV grid division used. Left: EAF-FOV grid from the 64 ×64 FOV grid. Right: Sets of 7 elements
(denoted by ξη) that will accumulate measures in order to compute the SMOS-based climatology
In order to implement this new feature in the processing chain the computation of the closest
neighbors in the antenna FOV has been performed. This information is computationally ex-
pensive, so it has been stored in a new auxiliary file and it is ingested in the processing chain
developed at BEC.
2.3.1.2 Debiasing brightness temperatures
During the retrieval procedure, the systematic errors in TBare propagated to the SSS. Never-
theless, the SSS retrieval from the measured flat sea TBis not a linear process. Effectively, the
known L-band dielectric models show a non-linear relationship between the measured TBand
the retrieved salinity, especially for low salinity regimes as is shown in figure 13. Therefore, it is
expected that a debiasing of TBprovide better results than the applied to the SSS.
The melting processes as well as rivers discharges are important in the Arctic. So it is possible
to find salinity values as low as to be out from the linear regime (figure 13). Therefore, at least
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in the Arctic, it seems important to adopt a TBdebiasing procedure instead of SSS debiasing.
Figure 13: First Stokes value provided by Meissner & Wentz dielectric model for different values of salinity and temperature. Note
how the system moves away from linear behavior as the salinity and temperature decrease.
Nevertheless, the presence of geophysical low salinity values is not the only reason to adopt this
new debiasing method. The aim of the debiasing is to mitigate systematic biases as the land-
sea contamination (LSC) or persistent RFI. Coastal zones affected by land-sea contamination
can have biases as much as 20 K in TB. Nevertheless it can easily double this quantity in
coastal zones affected by RFI.
Figure 14 shows three measures of half first Stokes parameter (FS/2) obtained at the same grid
point for different incidence angles. Measures are indicated by thick lines (purple and green
affected by LSC) whereas half first Stokes value provided by Meissner & Wentz dielectric model
at the same incidence angles are indicated by thin lines (purple, green and black for each
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measure). Performing the minimization of the differences of both quantities (half first Stokes
measured and modelled) the solution found for purple and green measures are close to SSS=0
psu (the maximum (TBh+TBv)/2for MW model is around this value for usual SST range) with
a value for the cost function (expression 24) of about 7.33 K2for green measure and 4.25 K2for
purple measure. This collapse of the minimization for different values of TBto the same SSS
value recommends to compute the debiasing in TBinstead of in SSS. The situation is different
in the linear regime (black measure providing a cost funcion close to zero at 24.6 psu). In this
case the debiasing in TBshould provide a similar correction than in the debiasing in SSS. This
similar behavior can be expected for small corrections but not for the larger ones because the
dielectric model can only be considered as linear in SSS for small variations of salinity.
Figure 14: Example of three measures of half first Stokes (FS/2) obtained for a grid point affected by land-sea contamination. The
grid point is located at North Sea coastal waters (53.86N, 6.70E), approximately at 35 km of the German coast. The measures
(thick lines) have been obtained from successive snapshots (i.e. for different incidence angles). Thin lines indicate half first
Stokes value provided by Meissner & Wentz dielectric model for different values of salinity, FS(SSS)/2.
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2.3.1.3 SMOS-based climatology computation
The SMOS-based climatology for the Arctic has been computed using the SMOS data from the
2013-2019 period (both included). The year 2011 is discarded because the data corresponding
to this year is too degraded to be considered as typical in the SMOS timeline. This is due
to the large presence of RFI and a payload anomaly (starting at December 30, 2010) which
lead to incorrect temperature readings in one of the segments of the payload antenna arm B.
Canada started to refurbish their equipment in autumn 2011, while Greenland switched off their
transmitters in March 2011. The year 2012 has been also discarded by the abnormal presence
of RFI in Norway and in Ireland and in the United Kingdom.
The SMOS-based climatology computation is performed separately for ascending and descend-
ing orbits and two netCDF files are generated for each satellite direction; one containing the es-
sential statistical information for all valid points and other storing the histograms corresponding
to the measured flat sea first Stokes parameter divided by 2 (Imeas
F S (xy, ξ η)). The histograms
are created accumulating valid measures in bins of 1 K for each 25 km EASE-Grid 2.0 North
grid point (x, y)and FOV division (denoted by ξη, see section 2.3.1.1). Only latitudes beyond
50N are considered. The accumulative Imeas
F S values for each 25×25 km cell include the values
obtained in the closest neighbor cells (i.e. an square of 75×75 km around the central grid point)
in order to increase statistical accuracy and to obtain less noisy results.
A measure is considered as valid if it accomplishes
75K< Imeas
F S <165K (3)
On the other hand two outliers detection are performed. The first one comes from L1D product
in which a linear regression is performed from all the TB(θ)measures for a given geographical
point obtained for different incidence angles. The outliers are detected from the difference
between this linear regression and each individual measure. The second outlier detection is
carried out for each Imeas
F S (xy, ξ η)distribution. In all cases, a measure classified as outlier
according Tukey rule [Tukey, 1977]:
>Q3+ 1.5×IQR
<Q11.5×IQR (4)
is discarded. In this expression, stand for the difference between linear regression and each
individual measure in the case of outliers coming fromk L1D product or Imeas
F S (xy, ξ η)in the case
of outliers from histograms. Q1and Q3are the lower and the upper quartile of respectively
and the interquartile range Q3Q1is denoted as IQR. Also sea ice concentration (SIC) from
ECMWF is taken in to account to discard a valid measure: any measure obtained for SIC>0.3
is not considered in the distribution.
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2.3.1.3.1 Derived statistical quantities
Frequencies
As mentioned above, for a given geographical grid point (x, y)i, the statistical absolute frequency
(histogram) and sum of Imeas
F S (xyi, ξηj)values are obtained by a simple sum of their closer
neighbors for the same FOV point ξηj. The neighbor set includes those contained in a square of
75×75 km from the central grid point (x, y)i. Therefore, for a given grid point iand FOV position
jthe statistical absolute frequency fiin the Imeas
F S interval kis given by the expression
fij
k=X
n
fnj
k(5)
where fij
kis the absolute frequency corresponding to the k Imeas
F S bin (also known as Iclass k)
including nneighbors of the grid point i.
In a similar way, the Isum for a given grid point and Iclass is given by
Iij
k=X
n
Inj
k(6)
Mean
The mean of Imeas
F S for each grid point iand FOV position jis computed by means of the
quantities given by equations 5and 6
< Iij >=PkIij
k
Pkfij
k
(7)
Extending the sum to all Iclases k
Quartiles
The median (second quartile Q2) and interquartile range (IQR=Q3Q1) are computed by
means of interpolation equation
Qij
n=Iij
LBk+ p
100 PN1
l=0 fij
lPk1
l=0 fij
l
fij
k!Iij
k(8)
where pis the percentile, Iij
LBkis the lower boundary of the class containing the n-th quartile (the
k-th class), Iij
kis its width, Nis the number of classes (bins) and fij
lis the statistical absolute
frequency corresponding to the l-th class. In our study Iij
LB ∈ {65,66,67, ..., 199}and Iij
k=
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1.0Kfor all classes. The class kcontainig the n-th quartile is the first one that accomplishes
k
X
l=0
fij
lp
100
N1
X
l=0
fij
l(9)
Second, third and fourth central moments
The second moment (the variance) is expressed here by means of its square root, the standard
deviation σ, which is computed for each grid point iusing the expression
σij =sX
k
ρij
k(< Iij >k< Iij >)2(10)
where the sum extends over all Irange ((75 : 165) K) in steps of 1K, the relative frequency ρij
k
is given by:
ρij
k=fij
k
Pk0fij
k0
(11)
and
< Iij >k=Iij
k
fij
k
(12)
The third and fourth central moments (skewness γand kurtosis Kurt respectively) are given by
expressions
γij =1
σ3
ij X
k
ρij
k(< Iij >k< Ii>)3(13)
Kurtij =1
σ4
ij X
k
ρij
k(< Iij >k< SSSi>)4(14)
Mode and mean around the mode: the climatological representative
The representative value for SMOS flat sea jaf fisrt Stokes at a given grid point, orbit direction
and FOV position (i.e. the SMOS IFS climatological value) is computed as the mean around one
standard deviation from the mode. Although the mode is the value that appears most often and
could be used as the representative value itself, it is not well determined because of the limited
available binning of the IFS histograms (1K in our case). Then, the adoption of our extended
definition for the representative value is better behaved in terms of accuracy.
In order to estimate the mode a 7-point wide rectangular window is applied 3 times to smooth
each histogram. This results in a 19-points pseudo-Gaussian smoothing window which re-
sulting coefficients are in the ratio 1:3:6:10:15:21:28:33:36:37:36:33:28:21:15:10:6:3:1 reducing
high frequency noise and preserving the peak positions, improving the mode estimation. Never-
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theless the lack of sampling reduces the accuracy of its computation. Hence, the representative
value is estimated from the IF S distribution by averaging the range of one standard deviation
(±σ) around the mode found by applying the described haystack smoothing window. The mean
around the mode (Iij) is computed in a range of one standard deviation around the mode. This
means that for a given grid point i(with given overpass direction and FOV position) we will
consider the classes included totally or partially in the range
Iij
mode σij Iij
kIij
mode +σij (15)
For those classes, the sums of IFS are added and divided by the sum of the statistical absolute
frequencies
Iij
F S =Pl=M+m
l=MmIij
l
Pl=M+m
l=Mmfij
l
(16)
where mis the smallest integer greater or equal than the quotient between the standard devia-
tion and the width of the class (1K)
m=pσij
Iij k
q(17)
Figure 15: The SMOS climatological representative for ascending passes at different FOV positions
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Figure 16: The SMOS climatological representative for descending passes at different FOV positions
The C code used to compute these statistical quantities is shown in appendix A, listing 1.
2.3.1.4 Annual IFS reference
As has been mentioned the debiasing procedure consists into replace the SMOS-based clima-
tology by an annual reference. The selected one in our case is the WOA 2018 [Zweng et al., 2018].
In particular, the dataset composed by the statistical mean of salinity on 0.25grid for 2005-2017
years. This dataset has been used because it is the closest period to the SMOS one (2011-
2019). As far as we know, no brightness temperature exists from World Ocean Atlas, therefore
it is necessary to compute it departing from WOA 2018 SSS and SST.
The emissivity can be computed from SSS and SST by means of the adoption of a dielectric
model. The dielectric model used is the Meissnet & Wentz one [Meissner and Wentz, 2004].
Combining this model with Fresnel equations and the emissivity it is possible to compute the
flat sea brightness temperature for horizontal and vertical polarizations and compute Iij
F Sref for
each geographic point as a funcion of the incidence angle (i.e as a function of the FOV position).
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Details about the procedure are provided in the section 2.4.
Nevertheless, to obtain the correction to be applied at measured half first Stokes parameter,
it is necessary to obtain the annual IF S reference in the same projection and grid resolution
as the provided by SMOS-based climatology. The annual WOA 2018 values are provided in
a common longitude latitude projection EPSG:4326 (WGS84 - World Geodetic System 1984,
used in GPS projection see https://epsg.io/4326) with a resolution og 0.25 ×0.25 degrees
whereas SMOS-based climatology is obtained in the projection described in section 2.1.1.2
(WGS 84 / NSIDC EASE-Grid 2.0 North, EPGS:6931) with a resolution of 25×25 km. The
change in the projection is performed using the PROJ coordinate transformation software library
[PROJ contributors, 2019]. Finally, the regridging from the original projected grid to a regular
2525 km is computed by means of the area weighted average. Figure 17 shows an scheme
of superposition of source grid (areas surrounded by dashed lines) and destination grid (areas
surrounded by solid lines). Denoting by Wthe area of a destination grid cell and withe area of
intersection between the original cells and the destination one the following inequality holds:
WX
i
wi(18)
The equality only is accomplished if the original cells completely fill the destination one. In this
case any quantity (SSS and SST for WOA 2018) defined in the original grid can be expressed
in the destination one by means of the weighted average
SSSj=PiSSSiwi
Piwi
SSTj=PiSSTiwi
Piwi
(19)
where j stands for the index of the destination grid.
The code of the common area between cells can be easily found from the doOverlap function
available at https://www.geeksforgeeks.org/find-two-rectangles-overlap/ (appendix A,
listing 2)
Note that annual IF S reference has no differences between ascending and descending passes
(figure 18).
Once both reference IF Sref fields (SMOS representant from figures 15 and 16 and WOA refer-
ence from figure 18) are provided in the same projection and grid (EASE-Grid 2.0 North, see
section 2.1.1.2 for additional information) the debiasing procedure can be applied. The obtained
difference between the WOA IF Sref and the provided by equation 16:
Iij =Iij
F Sref Pl=M+m
l=MmIij
l
Pl=M+m
l=Mmfij
l
(20)
is computed for both passes (figures 19 and 20) and it is added to the measured Imeas
F S (xy, ξ η).
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Figure 17: Schema of the area weighted average procedure. Orange circles indicate the center of each dashed cell of 0.25×0.25
degres (source grid). Blue squares are the center of the 25×25 km cells (destination grid).
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Figure 18: Half first Stokes provided by WOA2018 and Meissner & Wentz dielectric model for different points of the FOV
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Figure 19: Correction to be applied to measured half first Stokes parameter in 4 difference FOV positions for ascending passes
(equation 20)
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Figure 20: Correction to be applied to measured half first Stokes parameter in 4 difference FOV positions for descending passes
(equation 20)
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Figure 21: Mean half first Stokes correction in the whole Arctic (half first Stokes from WOA 2018 minus SMOS half first Stokes
representant). Left: ascending case. Right: descending case.
Orbit passes Correction range Mean STD
Ascending -24.00:-3.16 -8.75 2.80
Descending -21.42:-4.00 -9.20 2.63
Table 2: Statistics for the first Stokes bias correction in the antenna grid
Figure 22: Mean ascending minus descending half first Stokes correction in the whole Arctic
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2.4 Inversion
Once the systematic errors of the measured flat sea Tmeas
BF S are corrected, the SSS retrieval can
be performed for a given known dielectric model. The emitted flat sea radiation is characterized
by its TBand for SST. In the L-band regime, the Rayleigh-Jeans law is a good approximation
to describe its spectral radiance. Therefore, the ocean polarized TBand molecular SST are
linearly related by horizontal (h) and vertical (v) emissivity
e[h,v]=TB[h,v]/SST . (21)
The flat sea emissivity, i.e. the effectiveness in emitting energy as thermal radiation, is governed
by Fresnel reflection law and is a function of the incidence angle of the radiation θand the
dielectric coefficient ε:
eh= 1 "cos θ(εsin2θ)1/2
cos θ+ (εsin2θ)1/2#2
;ev= 1 "εcos θ(εsin2θ)1/2
εcos θ+ (εsin2θ)1/2#2
(22)
Salinity takes part in the formulation because the dielectric coefficient for the sea water should
be expressed, besides a function of SST and measurement frequency ω, in terms of its con-
ductivity and hence in terms of the salinity. Consequently, a reliable dielectric constant model
ε(SS T, SSS, ω)is necessary to describe with enough accuracy the relation between the mea-
sured brightness temperature and the salinity.
In the L-band regime two constant dielectric models have been adopted to retrieve salinity
from remote sensing measurements: Klein and Swift model (KS) [Klein and Swift, 1977] used
by the operational SMOS L2 Ocean Salinity processor and Meissner and Wentz model (MW)
[Meissner and Wentz, 2004,Meissner et al., 2018] used in Aquarius and SMAP salinity proces-
sors. Nevertheless, these are not the unique dielectric constant models in L-band. Recently a
new model, specifically designed for 1.413 GHz, has been developed at George Washington
University by Zhou, Lang and collaborators [Zhou et al., 2017] (designated as GW). All these
dielectric models are based on the Debye equation [Debye, 1970] with a conductivity term.
KS model for a frequency of 1.43 GHz was adjusted using a discrete set of measures at 5C,
10C, 20C an 30C and should be valid in the range of 4-35 psu. MW model interpolates the
dielectric constant as a function of salinity between 0–40 psu and provides accurate values
for the ocean surface emissivities between 2C and 29C. GW model has been computed
specifically for a frequency of 1.413 GHz and adjusts the measures obtained in 30, 33, 35 and
38 psu and a temperature range of 0-35C sampled every 5C. The temperature range used in
KS model seems to be too short to account of the whole SST range, that is about 2C and
35C.
Bias between the retrieved SSS and the one provide by Argo profilers for SST values less than
5C, are larger for KS model than for the MW model [Zhou et al., 2017,Dinnat et al., 2019].
On the other hand GW model has limitations to retrieve SSS below 20 psu (see section 3).
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Therefore, the MW model is used to retrieve SSS in the Arctic. This model is described in
[Meissner and Wentz, 2004] and is implemented in C as it is shown in appendix A, listing 3.
Equations 21 and 22 together with the dielectric model form a closed system of equations.
In order to avoid possible errors in the evaluation of the quantities involved in the ionospheric
rotation of the measured TB, as well as to reduce the degrees of freedom of the system, the
quantity involved is not the TBF S but the first Stokes divided by 2:
IF S =1
2(TBF S h+TBF S v)(23)
since, as has been mentioned, the first Stokes parameter is unaltered by the presence of the
ionosphere.
Therefore, for a given value of SSS and providing the value for SST and incidence angle at
which the measure has been taken (θ), it is possible (in virtue of equations 21 and 22) to find a
modelled value of the half first Stokes for the flat sea (Imod
F S ). Due to the fact that the measured
one (Imeas
F S ) is known, it is possible to obtain the salinity value that minimizes the cost function
F=kImod
F S Imeas
F S k2.(24)
In order to find the SSS value that minimizes Fwe use the well known Newton-Raphson method
[Press et al., 1992]. The implementetation code in C is shown in appendix A, listing 4.
This is a non-Bayesian retrieval scheme because for each measure of brightness temperature a
value of salinity is obtained. This means that the values of SSS recoverred (and their associated
errors ij) depend not only on coordinates over the Earth surface (x,y) but also on the FOV
position (ξ,η) in which the measure was taken. Therefore for each L1C orbit we produce a level
2A orbit (see figure 1). Hence, in the same way that L1C orbits are composed by snapshots of
brightness temperature, L2A orbits are composed by snapshots of salinity (see figure 23).
2.4.0.1 L2A filtering
Not all retrieved salinity values can be considered as valid and some filtering should be applied
on them. In fact, previously to the retrieval procedure, the following TBvalues are not included
in the inversion:
Values of TBobtained too close to the edge of the EAF-FOV or too close to the belts
and suspenders (see figure 4). Points closer than 0.025 antenna units to these zones are
discarded.
Points affected by Sun tails or reflected Sun circle are also discarded. The width for the
Sun tails when its signal is captured in front or back of the instrument is taken as 0.001
antenna units. The Sun radius is assumed as 0.04 antenna units when it is in front of the
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instrument and 0.02 antenna units if it is captured by backlobes when it is in the back of
the instrument.
For each Earth surface point (x,y) and FOV position (ξ,η)TBvalues considered as outliers
during SMOS-based climatology computation (section 2.3.1.3) are discarded as well.
Also TB(x, y, ξ, η)values are discarded according to the SMOS-based climatology if this one is
considered as a moderately nonnormal distribution (kurtsis and skewness conditions according
[West et al., 1995]), statistics have been computed with a low number of measures or standard
deviation is too high close to the coast (suspicion of residual land sea contamination):
Minimum number of measures to create the SMOS-based climatology is taken as 100.
Maximum kurtosis absolute value accepted is 7.
Absolute value of skewness must be less than 2.
For points located less than 100km to the coast, only those having a standard deviation
less than 8 K are accepted.
The TBvalues flaged as good according to above filtering rules are considered as good candi-
dates to proceed with the inversion, therefore they are debiased and introduced in the minimiza-
tion function. It is considered that the minimization has converged if all the following conditions
hold:
The change in salinity values between two consecutive iterations is less than 0.001.
The percentage of variation in the cost function between consecutive steps is less than 1.
The above two conditions are accomplished during 5 consecutive iterations to aboid oscil-
latory solutions.
The above condition is acomplished in less than 150 iterations.
For each L2A salinity value found SSS(x, y, ξ, η)only those lying in the range 0 < SSS < 55
are considered as acceptable values. This range of acceptable salinity ensures that the salinity
distribution is not cropped increasing the relliability of the next level product (L2B). On the other
hand, mensual WOA 2018 is interpolated to the day of the year of the orbit to be corrected using
Akima periodic interpolation [Akima, 1970]. It is expected that retrieved SSS values can not get
too far from the value found for WOA 2018. With this in mind a limit to the difference between
each retrieved salinity value and the provided by interpolated WOA 2018 is imposed. For salinity
values higher than 25 psu the limit imposed is 7 psu. Conversely, a limit of 21 psu is imposed
for salinities less than 25 psu. We have adopted this strategy to avoid restrictions imposed by
WOA 2018 close to the river discharges or in suddenly and unexpected melting episodes where
WOA 2018 could provide salinities larger than 35 psu.
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Concerning to the sea ice concentration (SIC), some preliminary tests point out that this filtering
and retrieval procedure provides a good agreement with the OSIS-SAF Sea Ice Climate Change
Initiative product OSI-450 and OSI-430-b [Lavergne et al., 2019] (i.e it generally fails for CCI
SIC > 0). Nevertheless, to ensure the minimum ice-sea contamination all points having SIC >
0 according Sea Ice CCI product are discarded and they are not included in the minimization
process. The distance to the ice edge (defined by the line SIC=0) is also stored with the same
purpose: minimize ice-sea contamination by avoiding the points too close to the ice in the L3
generation.
2.4.0.2 New L2B generation and its filtering
The generation of L2B level prior to the generation of L3 maps is an improvement introduced in
the context of the Arctic+ salinity project. The L2B orbits are generated from L2A snapshots by
weighted averaging all the measures obtained for a given grid point (section 2.4.1) producing
salinity values independent of the FOV position. An outlier detection is performed in a similar-
way as it was applied in TB: a linear regression is performed from all the ascending sorted SSS
values of the same orbit obtained for a given geographical point. The outliers are detected,
according Tukey rule [Tukey, 1977] (expression 4), from the difference between this linear re-
gression and each individual measure. L2B SSS values are only computed for those grid points
containing more that 12 unfiltered L2A retrieved SSS values.
2.4.1 Error propagation
The propagation of the radiometric error from TBup to salinity is made by performing three
minimizations as described in section 2.4:
Imeas
F S =1
2(TBF S hσh+TBF S vσv)SSSij(Imeas
F S )
Imeas
F S =1
2(TBF S h+TBF S v)SSSij(Imeas
F S )
Imeas+
F S =1
2(TBF S h+σh+TBF S v+σv)SSSij(Imeas+
F S )
(25)
where σ[h,v]is the radiometric accuracy of TB[h,v],istands for the geographical grid point and j
for the FOV position. The salinity error due to the radiometric error is taken as
ij =1
2|SSS(Imeas+
F S )SSS(Imeas
F S )|(26)
The next step is to produce level 2B orbits and level 3 maps depending only of the geographical
coordinates departing from L2A values (individual retrievals of SSS) and propagate their error
up to these upper levels.
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2.4.1.1 L2B computation
As has been mentioned, new L2B orbits are generated from L2A snapshots by weighted av-
eraging all the measures obtained for a given grid point. Assuming the weight function as the
inverse of the squared error of each L2A measure, we ensure that the measures comming from
TBhaving a high radiometric error will have a small influence in the obteined value for SSS at
L2B level.
SSSi=PjSSSij wij
Pjwij
(27)
where the weight function is given by
wij =1
2
ij
(28)
and the jindex stands for all the measures obtained for the grid point i. Therefore, the error of
each L2B salinity value is given by expression
i=1
qPj1/2
ij
.(29)
The error is propagated in the same way from L2B measures to L3 maps:
SSSi=PnSSSinwin
Pnwin
(30)
where the weight function is given by
win =1
2
in
(31)
and the nindex stands for all L2B orbits having salinity value for the grid point iin the L3 period.
The error of each L3 salinity value is given by expression
i=1
qPn1/2
in
.(32)
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Figure 23: Snapshot from ascending L2A orbit over the Kara Sea on August 2, 2015. Left: salinity values retrieved using the
non-Bayesian procedre described in section 2.4. Right: the corresponding error of each retrieval as has been described in section
2.4.1 and expressed in equation 26.
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Figure 24: Ascending L2B orbit over the Kara Sea on August 2, 2015. Left: salinity values computed from L2A snapshots using
expressions 27 and 28. Ob’ and Yenisey rivers discharge are clear at this level as well as the freshwater accumulation in the
Baydaratskaya bay. Right: the corresponding error of salinity value from equation 29.
Figure 25: 9-day L3 map centered on August 2, 2015 combining ascending and descending orbits. Left: salinity values computed
from L2B orbits using expression 30. Right: the corresponding error of salinity value from equation 32.
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2.5 L2B temporal correction
As is pointed out in [Olmedo et al., 2017] the official SMOS L2OS processor correct TBby the
introduction of the Ocean Target Transformation (OTT) [Tenerelli and Reul, 2010]. The OTT is
re-computed daily in version v622. This procedure was implemented to mitigate some global
time-dependent biases cited by [Martín-Neira et al., 2016]. The TBdebiased procedure used
here (section 2.3.1) only accounts for spatial bias. Therefore, a temporal correction should be
implemented.
Previous correction implemented in [Olmedo et al., 2018] operates over L3 maps. The method
uses the Argo profiles [Argo, 2018] available the period covered by each 9-day map and com-
putes the median of the differences between the collocated L3 SSS values and the provided by
Argo profilers. This median is substracted to the SSS map.
Unfortunatelly, the Arctic is a zone with scarce salinity measures and Argo profilers are con-
centrated, due to the bathymetry, in the Atlanctic zone providing a biased sample ot the mean
Arctic SSS value as it is shown in figure 26. Additionally, we need to perform a time-correction
on L2A orbits, not on L3 maps, and the amount of Argo data available for each orbit is clearly
insufficient to perform such correction.
Figure 26: Argo profilers distribution during the period 2011-2018 (dots). Blue line delimits the bathymetric curve corresponding
to 1000 m. Note the lack of Argo profilers in the Bering, Beaufort, East Siberian, Laptev, Kara, Barents and North seas and also in
Hudson and Baffin bays.
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In the absence of SSS in-situ measures, the temporal correction has been performed using
Global Ocean Forecasting System (GOFS) 3.1 (HYCOM + NCODA). In particular the resolution
used corresponds to GLBv0.08 grid having a resolution of 0.08 degrees in longitude and 0.04
degrees in latitude for polar regions [Cummings, 2005,Cummings and Smedstad, 2013]. The
data can be downloaded from https://www.hycom.org/data/glbv0pt08.
After the retrieval of the salinity values (L2A computation) is performed, a first assessment of a
correction is introduced (-12 psu) for each L2A measure. This first assessment only intends to
reduce the number of the necessary iterations. After applying the filtering process described in
section 2.4.0.1 a new value for all L2B points is obtained and mean difference between each
L2B salinity value and the provided by HYCOM is added to each L2A SSS value of the orbit.
This loop is repeated until difference between two accumulated consecutive corrections is less
than 0.01 psu. The iterative loop is shown in the figure 27
Only orbits providing at least 50 common grid points with HYCOM are considered. Due to the
fact that HYCOM provides too salty values in the river mouths, only L2B points having a retrieved
salinity value above 25 psu and an error below 2.5 psu are considered to compute the temporal
correction.
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Figure 27: Scheme of the iterative procedure used to correct the temporal salinity bias on level 2.
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2.6 L2B spatial correction
As it has been described in section 2.3.1.2, the debiasing process is performed at brightness
temperature level before the salinity retrieval. The debiasing method is based on the substitution
of the SMOS-based TBclimatology by the obtained from the WOA 2018 [Zweng et al., 2018].
The WOA dataset used as reference field is the one generated from measurements of the 2005-
2017 period (known as A5B7). Therefore, the average salinity obtained for the period used to
compute the SMOS-based climatology (years 2013-2019) should have a spatial distribution very
close to the reference used to carry out the debiasing (A5B7).
However, when all L2B orbits from the years 2013 to 2019 are weighted averaged the result
is not the expected (figures 28 and 29 should show values close to zero). The average of L2B
orbits is accomplished as it is described in section 2.4.1 using equations 30-32. These averages
have been made separately for ascending and descending passes, finding differences between
them (figure 30).
The cause of the differences between ascending and descending passes is mainly linked to
the different performance of the ascending and descending debiasing. Different undesirable
process can affect the same location depending on the orientation of the orbit. So, depending
on the coast orientation a given point can be affected by land-sea contamination differently in
ascending passes or in descending passes. This means a different correction in both passes,
raising the imprecisions between both. A similar matter takes place with ice-sea contamination
and in the presence of RFI.
The cause of the emergence of differences between the used reference (WOA 2018) and the
weighted average of all the L2B orbits of the period is more subtle. The half first Stokes dis-
tributions provided by SMOS are generally positive skewned (figure 31). This means that its
representant at each geographical point (the mean around the mode as is described in sec-
tion 2.3.1.3.1) generally does not coincide with the mean of the distribution. On the other
hand, the WOA 2018 salinity is obtained through an objective analysis scheme using a cor-
rection factor given by a weighted average of the in-situ measurements in a given limited region
[Zweng et al., 2018, sec. 3.2] assuming, therefore, Gaussianity in this region. The substitution
of the SMOS-based climatology by the TBreference obtained from WOA 2018 salinity intro-
duces innaccuracies due to the skewness of the former. The nonlinearity of the inversion pro-
cess leads to unpredictable inexactitudes in the spatial distribution of the salinity. Nevertheless,
the spatial difference can be estimated by means of the field described above: the difference
between salinity provided by averaging the L2B salinities and the provided by WOA 2018. In this
way, part of the problem can be corrected by subtracting the dame spatial difference (shown in
figures 28 and 29) to every L2B orbit. The algorithm describing this process is ahown in figure
32.
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Figure 28: Weighted average of all ascending L2B orbits in the period 2013-2019 minus WOA18-A5B7
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Figure 29: Weighted average of all descending L2B orbits in the period 2013-2019 minus WOA18-A5B7
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Figure 30: Difference between weighted average of all ascenidng and descending L2B orbits in the period 2013-2019
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Figure 31: Skewness of the half first Stokes distribution. Left: ascending. Right: descending. The corresponding position in the
antenna grid is marked with the red dot.
Figure 32: Scheme of the procedure used to apply the spatial correction on level 2. Left-hand side generates the auxiliary files
(one for ascending and one for descending) that are used to correct all L2B orbits as described in th right-side scheme
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2.7 L3 maps creation
L3 maps are generated daily for 3-days periods, 9-days periods and 18-days periods and they
are obtained by temporal averaging of L2B salinity values. Each salinity value provided at a
fixed geographical point has been obtained by a weighted averaging according to the error of
the salinity measure as is described in section 2.4.1 by equations 30-32. Finally, to minimize ice-
sea contamination and land-sea contamination all L2B points closer to 35 km to the ice edge or
to the coastline are not considered in the L3 maps creation. Figures 34 and 33 show an example
of the resulting 9-days L3 salinity maps and the error derived from radiometric uncertainty. It
is wroth noting the greater coverage and detail of the gradients of Arctic+ v3.1 product to that
obtained from the BEC Arctic v2.0 product (figures 34-37).
Figure 33: Salinity error derived from the radiometric error. Arctic+ v3.1 map of the period August 11-19, 2012.
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Figure 34: 9-day L3 SSS Arctic+ v3.1 map corresponding to the period August 11-19, 2012. Note the greater coverage and detail
of the gradients to that obtained from the BEC Arctic v2.0 product (figure 35). A section of this map containing from the Barents
Sea to the East Siberian Sea together with Sea Ice concentration it is shown in figure 36.
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Figure 35: 9-day objectively analized BEC Arctic v2.0 map [Olmedo et al., 2018] corresponding to the period August 11-19, 2012.
Compare it with the new Arctic+ v3.1 (figure 34).A section of this map containing from the Barents Sea to the East Siberian Sea
together with Sea Ice concentration it is shown in figure 37.
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Figure 36: Detail of the map shown in figure 34 together with the minimum Sea Ice concentration provided by OSI SAF for the
period August 11-19, 2012. The right color bar indicates Sea Ice concentration whereas the left color bar indicates salinity.
Figure 37: Detail of the BEC Arctic v2.0 map shown in figure 35 together with the minimum Sea Ice concentration provided by OSI
SAF for the period August 11-19, 2012. The right color bar indicates Sea Ice concentration whereas the left color bar indicates
salinity.
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3 Dielectric constant study
As has been mentioned in the inversion section (2.4), three dielectric models have been con-
sidered as candidates to be used in the Arctic sea surface salinity retrieval: Klein and Swift
model (KS) [Klein and Swift, 1977] used by the operational SMOS L2 Ocean Salinity proces-
sor, Meissner and Wentz model (MW) [Meissner and Wentz, 2004,Meissner et al., 2018] used
in Aquarius and SMAP salinity processors and the recently developed, and still undergoing im-
provements, by the George Washington University (GW) team by Zhou, Lang and collaborators
[Zhou et al., 2017].
Figure 38: Half first stokes value obtained for Meissner & Wentz model (blue line), Klein & Swift model (black line) and George
Washington University model (yellow line) fro different values of SST: SST=0C (upper left), SST=5C (upper right), SST=10C
(bottom left) and SST=15C (bottom right).
During the inversion implementation described in this ATBD an important handicap was discover
in GW dielectric model to recover freshwaters. Sea surface salinity values below about 20 psu
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can not be described by this dielectric model. Figure 38 shows the half first Stokes parameter
obtained for different SST values. The maximum value obtained around 20-25 psu for GW
model indicates that no solutions below this value could be found using this model. Hence,
where river runoff and melting process are so important. The cause probably is linked with the
limited SSS samples used to develop it (30, 33, 35 and 38 psu) providing a bad extrapolation
for low values of salinity.
Figure 39: Retrieved SSS minus Argo salinity as a function of SST for MW and KS dielectric models. Up: mean value. Down
standard deviation
Shortcomings in KS model against MW at low SST are known [Meissner and Wentz, 2004]
and recently they have been reported in the Aquarius mission context [Zhou et al., 2017] for
individual salinity orbits (level 2). Nevertheless, there is a lack of systematic verification at
the space-time resolutions usually used in SMOS L3 salinity maps. To that goal, we have
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generated L3 salinity maps using the MW and KS dielectric constant models mentioned above.
The study has been carried out by comparing with Argo profilers [Argo, 2018] against three
years (2011-2013) of 9-day L3 salinity global maps using the classic non-bayesian debiased
retrieval [Olmedo et al., 2017]. MW model shows a better behaviour as compared with in-situ
measurements providing maps with a lower bias and standard deviation (figure 39).
Climatologic representantStandard deviation
Figure 40: Ascending SMOS-climatology representant (up) and standard deviation (down) for Greenland Sea as a funtion of
incidence angle and cross track distance. Left column corresponds to KS values. Right column shows difference between MW and
KS cases
The comparison between dielectric models it is also performed at SMOS-based climatology
level. Here the climatology obtained is the classical one ((i.e. salinity representant). The
SMOS-based climatology as a function of incidence angle and cross track distance (the classical
antenna FOV division) it is shown for Greenland Sea (in particfular the area delimited between
68N, 75N and 15W, 7E). The mean values of the SMOS climatology (the representant), stan-
dard deviation, kurtosis and skewness are computed for each antenna FOV division for MW and
KS models. The result in ascending passes is shown in figures 40 and 41. Comparison between
both dielectric models show similar results for descending orbits.
Upper right plot of figure 40 indicates that MW model systematically retrieves lower salinity val-
ues than KS. On the other hand statistical quantities involving salinity distributions like standard
deviation, excess kurtosis and skewness (figure 41) are very similar between both models.
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Excess Kurtosis
Skewness
Figure 41: Ascending SMOS-climatology excess kurtosis (up) and skewness (down) for Greenland Sea as a funtion of incidence
angle and cross track distance. Left column corresponds to KS values. Right column shows difference between MW and KS cases
In conclussion, GW model has been discarded to be used in the Arctic due to its inability to
perform a good salinity retrieval at typical salinity ranges that can be found in stuaries and in
zones with strong melting processes. The comparison with in-situ measurements recommend
the use of MW model instead of KS especially in cold waters.
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4 Key aspects of processing
Processing step Level of improvement
Gradients pattern Salinity values
Grid projection High Low
Dielectric model Low Moderate
EAF-FOV discretization Moderate Low
Debiased in TBLow High
Annual IF S reference Moderate Moderate
L2A filtering Low Moderate
L2B filtering Low Moderate
Temporal correction Low High (global value)
Spatial correction Low High
L3 weighted average Low Low
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Processing step Relevance Changes from BEC Arctic 2.0 Computational bur-
den
Grid projection 1Introduced in 3.0 Low
Dielectric model 3Unchanged Low
EAF-FOV discretization 4Changed in 3.0 Moderate
Debiased in TB1Introduced in 3.0 High
Annual IF S reference 4WOA2018 is used in 3.0 Low
L2A filtering 3Changed in 3.0 Low
L2B filtering 3 Introduced in 3.0 Low
Temporal correction 2ARGO profilers are not used in 3.0
Correction in L2A in 3.0 Low
Spatial correction 2Introduced in 3.1 Low
L3 weighted average 5Introduced in 3.0 Low
Table 3: The relevance takes values from 1 to 5; 1 for the most relevant and 5 for the least relevant
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Appendix A Source C code
1// ................................................................
2/ / FUNCTION : computeSMOSClima
3/ / DESCRIPTION : C omput es SMOS c l i m a t o l o g y and o t h e r s t a t i s t i c a l q u a n t i t i e s
4/ / :
5/ / DATA OUT :
6/ / th eH is : t he H isto gram d at a acc um ula ted from L1D
7/ / t h eC l im a : s t r u c t u r e t h a t w i l l s t o r e d at a
8/ / :
9/ / RETURN : su cce ss or f a i l e d
10 / / :
11 / / NOTE : The ske wnes s and k u rt o s is s t a t i s t i c s ap pea r t o be v er y d epe nde nt
12 / / : on t h e s amp le si z e
13 / / : Read h t t p s : / / www. s p c f o r e x c e l . com / kn ow le dg e / b a si c statistics /areskewnessand
k u r t o s i s u s e f u l statistics
14 / / NOTE a bout zin dex :
15 / / Gha semi , A . , & Z a he d i as l , S . ( 20 1 2 ) . No r m a l i t y t e s t s f o r s t a t i s t i c a l a n a l y s i s :
16 / / A g ui d e f o r nons t a t i s t i c i a n s . I n t e r n a ti o n a l Jo u rn al o f E nd o cr in ol og y a nd
17 / / M et abo li sm , 1 0 , 486 489 . h t t p s : / / www. n c b i . n lm . ni h . gov / pmc / a r t i c l e s / PMC3693611/
18 / /
19 / / A l so n o te th e f o l l o w i n g : M aye rs ( 20 1 3 , p . 5 3) su gg es te d t h a t a c u t o f f o f
20 / / 1 . 9 6 s h o u ld be us ed f o r sam pl es sm a l l e r th a n 5 0 , a c u t o f f o f 2 . 5 8 f o r
21 / / sa mp les f ro m 51 t o 1 00 , and a c u t o f f o f 3 . 29 f o r s amp le s l a r g e r th an 10 0
22 / / when us ed i n c o n j u n c t i o n w i t h th e ex a m in a t io n o f h i s to g r a ms .
23 / /
24 / / Ma yers , A . (2 0 13 ) . I n t r o d u c t i o n t o s t a t i s t i c s a nd SPSS i n ps y ch ol o gy .
25 / / H a rl ow : Pea rs on E d u c at i o n L i m i t e d .
26 / / h t t p s : / / www. n c b i . n lm . ni h . gov / pmc / a r t i c l e s / PMC3591587/
27 / / Ro be rt T reve tha n
28 / / Samp le s i z e S tr a t eg y f o r S & K v al ue s C r i t e r i a
29 / / < 5 0 C o nv er t t o z by d i v i d i n g by st d e r r o r
30 / / I f z > | 1 . 9 6 | , d at a a re n o t n o rm a ll y d i s t r i b u t e d
31 / / 50 t o ~17 5 C on ve rt t o z by d i v i d i n g b y s t d e r r or
32 / / I f z > | 2 . 5 8 | , d at a a re n o t n o rm a ll y d i s t r i b u t e d
33 / / ~17 5 t o 300 C on v er t t o z b y d i v i d i n g by s td e r r o r
34 / / I f z > | 3 . 2 9 | , d at a a re n o t n o rm a ll y d i s t r i b u t e d
35 / / 300+ D o n t c on v e rt t o z v a lue s .
36 / /
37 / / E i t h er an a bs o l ut e s kew ness v a l ue l a r g e r th an 2 o r a n a b s ol u t e k u r t o s i s ( p r op e r )
38 / / l a r g e r t h an 7 may be u sed as v a l u es o f sk ewn ess and k u r t o s i s w i t h o u t f o r
39 / / de t e rm i n in g s u b s t a n t i a l no nn o r m a l i t y .
40 / / P rop ose d by West e t a l (1 9 95 ) W est SG, Fi nc h J F , Cu rr an PJ . S t r u c t u r a l e qu a ti o n
41 / / m ode ls w i t h n on no rm al v a r i a b l e s : pr ob le ms an d r em ed ie s . In : H oy le RH, e d i t o r .
42 / / S t r u c t u r a l e q ua t i on mo de li ng : Co nce pt s , i s su e s and a p p l i c a t i o n s .
43 / / Newb ery P ark , CA: S age ; 1 99 5. pp . 56 75 .
44 / /
45 // ................................................................
46 int compute SMOSClima ( s_H isMa p myHi s , s _Cl ima myClima ) {
47
48 int i , j , k , l Mode , l , l l ;
49 int lm in , lmax ;
50 int counter ;
51 long MaxFreq ; / / The f r eq u e nc y at wh ic h t h e mode i s a t t a i n e d
52 float MeanModeFreq ;
53 float MeanMode ;
54
55 float MeanSum, Mean Bin ;
56 float moment2 , moment3 , moment4 ;
57 float rh o , d e v i a t i o n , d e v i a t i o n 2 ;
58 float l b , pr eAccF req , Acc Freq , N Freq2 ;
59
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60 int stdbin ;
61
62 / / P seu do ga uss ia n i s f a s t e r t ha t p pa sse s o f a wwindow
63 int w= 7; / / w i d t h o f r e c t a n g u l a r wi ndo w
64 int p=3 ; / / nu mber o f pa ss es
65 int p oi n t sG a u ss i a n= 1+p (w1) ; / / 19
66 int po ints G2 ;
67 un si gn ed l on g i n t in de x ;
68 un s ig n ed s h o r t i n t f r e q ;
69 float t b ;
70
71 float his to_ smo ot h ;
72 float sta nd a rdE rr o rKu rt , st an dar dE rro rS ke w , n f ;
73
74 float pseud oGauss ian [ 19]={1.0 , 3.0 , 6 .0 , 1 0. 0 , 1 5. 0 , 21. 0 , 2 8. 0 , 3 3. 0 , 3 6. 0 , 3 7. 0 , 3 6. 0 , 3 3. 0 ,
2 8 .0 , 21 . 0 , 15 . 0 , 1 0 . 0 , 6 . 0 , 3 . 0 , 1 . 0 } ;
75
76 i f ( p oi n ts G au s si a n %2==0) {
77 po in t sG 2= po i nt sG au s si an / 2 ;
78 }els e {
79 po in ts G2 = ( po in ts G au ss ia n 1) / 2 ;
80 }
81
82
83 f o r ( i =0; i <myHis>dimL on ; i + +) {
84 f o r ( j =0; j <myHis>di mLat ; j + +) {
85 f o r ( k = 0 ; k< myH is>d im A nt ; k + + ) {
86 ind ex = ( u nsi gn ed l on g i n t ) ( myHis>d imBin ) (un si gn ed l on g i n t ) ( my His>dim Ant ) (u ns ign ed
l on g i n t ) ( m yHis>di mLa t ) (un si gn ed l on g i n t ) ( i ) ;
87 ind ex += ( u ns ign ed lo ng i n t ) ( myHis>di mBi n ) (un si gne d l on g i n t ) ( m yHis>di mAn t ) (un si gn ed
l on g i n t ) ( j ) ;
88 ind ex += ( u ns ign ed lo ng i n t ) ( myHis>di mBi n ) (un si gne d l on g i n t ) ( k ) ;
89 / / Number o f m eas ur es
90 myClima>Nmeas [ i ] [ j ] [ k ] = 0 ;
91 for ( l = 0; l < myHis>d imB in ; l ++ ) {
92 myClima>Nmeas[ i ] [ j ] [ k ]+=myHis>F re qV ec to r [ i nd e x +( u ns ign ed lo ng i n t ) ( l ) ] ;
93 }
94
95 / / Not eno ugh measures
96 i f ( myCli ma>Nmeas [ i ] [ j ] [ k ]< MIN_MEAS_STAT) {
97 myClima>Nmeas [ i ] [ j ] [ k ] = 0 ;
98 myClima>Mean [ i ] [ j ] [ k ] = ( float )NOVALUE ;
99 myClima>Mode [ i ] [ j ] [ k ] = ( float )NOVALUE ;
100 myClima>MeanMode[ i ] [ j ] [ k ] = ( float) NOVALUE;
101 myClima>S td [ i ] [ j ] [ k ] = ( float) NOVALUE;
102 myClima> K u r t o s i s [ i ] [ j ] [ k ] = ( float )NOVALUE;
103 myClima>Skew ness [ i ] [ j ] [ k ] = ( float) NOVALUE;
104 myClima>Me dia n [ i ] [ j ] [ k ] = ( float) NOVALUE;
105 myClima>Z K u r to s i s [ i ] [ j ] [ k ] = ( float )NOVALUE;
106 myClima>ZSkew ness [ i ] [ j ] [ k ] = ( float) NOVALUE;
107 continue ;
108 }
109 / / T hi s v a lu es o f s td f o r k u r t o si s and sk ewn ess a re o n ly v a l i d f o r n or mal d i s t r i b u t i o n s
110 / / T he re fo r e , d i s t r i b u t i o n i s ass umed as n orm al and we ch ec k zs co r e t o kn ow i f
111 / / th e a ss um p ti on i s c o r r e c t .
112 n f = ( float ) m yClim a>Nmeas[ i ] [ j ] [ k ] ;
113 st a n da r d Er r o rS k e w= s q r t ( 6 . 0 n f ( n f 1) / ( ( n f 2.0)( nf 1) ( n f 3) ) ) ;
114 s t a n d a r d E r r o r K u r t = st a n da r dE r r or S k ew s q r t (4 . 0 ( n f n f 1) / ( ( n f 3.0)( n f +5 ) ) ) ;
115
116 / / ===============================================================
117 / / S mo ot hi ng o f t h e h i s t o gr a m . . .
118 / / A 7po i n t s ( w) wi de r e c t a n g u l a r wind ow a p p l i e d 3 t i m es ( p ) . . .
119 / / E q u i v a l e n t t o a pse udoGaussian with pwp+1 coefficients
120 // 1:3:6:10:15:21:28:33:36:37:36:33:28:21:15:10:6:3:1
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Algorithm Theoretical Baseline Document
Ref: AO/1-9158/18/I-BG
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121 / / The pseudoga u s si a n can be a p p l i e d fr o m e l em e nt 1+ p o i n ts G a u ss i a n / 2
122 / / t o t h e Hi s >dimBin[1+ p o i nt s G au s s i an / 2 ]
123 / / Ex tremes w ei ghte d w it h l e f t o r r i g h t n ei gb or s ( t i p i c a l l y TB < 66 K o r
124 / / l a r g e t h an 1 99 K i f th e r a ng e i s 60 : 20 0 )
125 / / ===============================================================
126
127 for ( l = 0; l < myHis>dim Bi n ; l + +) {
128 myHis>h is to Sm oo th [ i nd ex + ( un si gn ed l on g i n t ) ( l ) ] = 0 . 0 ;
129 cou nt er =0 ;
130 f o r ( l l = l p oi nts G2 ; l l < l + po in ts G2 ; l l ++ ) {
131 i f ( l l > =0 && l l <myH is>dim Bin ) {
132 myHis>h is to Sm oo th [ i nd ex + ( un si gn ed l on g i n t ) ( l ) ] + = ps eu do Ga us sia n [ co u n te r ] myHis
>F r eq Ve ct or [ i nd e x +( u ns ign ed lo ng i n t ) ( l l ) ] ;
133 }
134 c ou nt er + +;
135 }
136 }
137
138 / / ========= Mode (computed usi ng smoothed his tog r am )
139 Max Freq = 0 ;
140 lMo de = 0;
141 MeanSum= 0 ;
142 for ( l = 0; l < myHis>d imB in ; l ++ ) {
143 f r e q =my His>Fr e qV ec to r [ i nd e x +( u ns ign ed lo ng i n t ) l ] ;
144 t b =my His>TB Vec to r [ i n de x +( u ns ig ned lo n g i n t ) l ] ;
145 i f ( f re q > 0) {
146 his to_ smo ot h =myHis>hi st oS mo ot h [ i nd ex + ( un si gn ed l on g i n t ) ( l ) ] ;
147 i f ( h is to _s mo o th >M axFre q ) {
148 MaxFreq= hi st o_sm oot h ;
149 myClima>Mode [ i ] [ j ] [ k ] = t b / ( float) f r e q ;
150 lMo de= l ;
151 }
152 MeanSum+= t b ; / / sum o f a l l TB
153 }
154 }
155 i f ( MeanSum= =0 ) {
156 myClima>Nmeas [ i ] [ j ] [ k ] = 0 ;
157 myClima>Mean [ i ] [ j ] [ k ] = ( float )NOVALUE ;
158 myClima>Mode [ i ] [ j ] [ k ] = ( float )NOVALUE ;
159 myClima>MeanMode[ i ] [ j ] [ k ] = ( float) NOVALUE;
160 myClima>S td [ i ] [ j ] [ k ] = ( float) NOVALUE;
161 myClima> K u r t o s i s [ i ] [ j ] [ k ] = ( float )NOVALUE;
162 myClima>Skew ness [ i ] [ j ] [ k ] = ( float) NOVALUE;
163 myClima>Me dia n [ i ] [ j ] [ k ] = ( float) NOVALUE;
164 myClima>Z K u r to s i s [ i ] [ j ] [ k ] = ( float )NOVALUE;
165 myClima>ZSkew ness [ i ] [ j ] [ k ] = ( float) NOVALUE;
166 continue ;
167 }
168
169 / / ========= Mean
170 / / b y c o n s t r u c t i o n my Clim a>Nmeas[ i ] [ j ] [ k ] = sum_l (myHis>Fr eq [ i ] [ j ] [ k ] [ l ] )
171 myClima>Mean [ i ] [ j ] [ k ]= MeanSum/ myClima>Nmeas[ i ] [ j ] [ k ] ;
172
173 / / C o mp u ta t io n o f t he mom ents
174 moment2 = 0 . 0 ;
175 moment3 = 0 . 0 ;
176 moment4 = 0 . 0 ;
177 for ( l = 0; l < myHis>dim Bi n ; l + +) {
178 f r e q =my His>Fr e qV ec to r [ i nd e x +( u ns ign ed lo ng i n t ) l ] ;
179 t b =my His>TB Vec to r [ i n de x +( u ns ig ned lo n g i n t ) l ] ;
180 i f ( fr eq >0) {
181 Mea nBin = t b / ( float ) ( f r e q ) ;
182 d e v i a t i o n = ( Mea nBinmyClima>Mean[ i ] [ j ] [ k ] ) ; / / D e v i a t i o n f o r t h e bi n l
183 d e v i a t i o n 2 = d e v i a t i o n deviation ;
ARGANS LTD./ICM-CSIC/NERSC. 2020
Arctic+ salinity
Algorithm Theoretical Baseline Document
Ref: AO/1-9158/18/I-BG
Date:14/04/2020
Page: 75 of 82
184
185 / / r h o = f r e q o f b i n / Sum o f f r e q u e n c i e s
186 / / to im p ro ve sp eed and a c cu r ac y we d i v i d e by t he sum o f fr e q u e nc i e s o u t f ro m t h i s
loop
187 rh o =( float) f r e q ;
188 moment2+= r ho deviation2 ;
189 moment3+= r ho deviation2deviation ;