Enhancing SMOS brightness temperatures over the ocean using the nodal sampling image reconstruction technique

Article (PDF Available)inRemote Sensing of Environment 180:205-220 · July 2016with 115 Reads
DOI: 10.1016/j.rse.2015.12.032
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Abstract
Abrupt changes in the Soil Moisture and Ocean Salinity (SMOS) brightness temperatures, such as those produced by land/sea/ice transitions and Radio-Frequency Interference (RFI) sources, produce artificial rippling patterns (i.e. the so-called Gibbs-like contamination) that propagate through the SMOS-reconstructed image. A nodal sampling technique, focused on the reduction of this kind of contamination by sampling at the points where the perturbation cancels, was introduced by González-Gambau et al. (2015). In this work we show that the quality of nodal sampling can be largely improved by refining the determination of the nodal grid. In addition, we have carried out an extensive validation of the resulting data over the ocean. Nodal sampling reduces sidelobe levels and ripples in the reconstructed images leading to brightness temperatures in better agreement with the theoretically modeled ones. Validation of the salinity retrievals against close-to-surface Argo salinity observations shows that nodal sampling leads to improved salinity retrievals in open ocean, while close to the coast land–sea contamination seems to deteriorate the quality. Besides, spectral analysis shows that nodal sampled salinities become closer to what is geophysically expected without loss of effective spatial resolution.
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Enhancing SMOS brightness temperatures
over the ocean using the nodal sampling
image reconstruction technique
Ver´onica Gonz´alez-Gambau
a,c,
, Estrella Olmedo
a,c
,
Antonio Turiel
a,c
, Justino Mart´ınez
a,c
,
Joaquim Ballabrera-Poy
a,c
, Marcos Portabella
a,c
and
Mar´ıa Piles
b,c
a
Department of Physical Oceanography, Institute of Marine Sciences, CSIC
Passeig Maritim de la Barceloneta, 37-49. 08003 Barcelona. Spain
b
Department of Signal Theory and Communications, Universitat Polit`ecnica de
Catalunya, Jordi Girona 1-3, E-08034 Barcelona, Spain
c
SMOS Barcelona Expert Center, Pg. Mar´ıtim 37-49, E-08003, Barcelona, Spain
Abstract
Abrupt changes in the Soil Moisture and Ocean Salinity (SMOS) brightness tem-
peratures, such as those produced by land/sea/ice transitions and Radio-Frequency
Interference (RFI) sources, produce artificial rippling patterns (i.e. the so-called
Gibbs-like contamination) that propagate through the SMOS-reconstructed image.
A nodal sampling technique focused on the reduction of this kind of contamination
by sampling the signal at the points where the perturbation cancels was introduced
by Gonzalez-Gambau et al. (2015). This work presents an improvement over that
technique by refining the nodal grid determination. A comprehensive assessment of
this new approach over the ocean is presented. The new brightness temperatures are
compared to the ones provided by the current SMOS image reconstruction baseline
and by the original nodal sampling method. The new technique reduces sidelobe lev-
els and ripples in the reconstructed images, being the new brightness temperatures
in better agreement with the theoretically modeled ones. A validation of the salinity
retrievals against near-surface Argo salinity observations shows that the new method
Preprint submitted to Elsevier Science August 3rd, 2015
leads to improved salinity retrievals in open ocean. However, in coastal areas, where
systematic biases due to land-sea contamination are also present, improvements de-
pend on the retrieval algorithm used. This indicates that further research on salinity
retrieval methods is necessary. Research that would benefit from the nodal sampling
technique.
Key words: Soil Moisture and Ocean Salinity (SMOS), interferometry,
radiometry, image reconstruction, nodal sampling, nodal points, sidelobes, Radio
Frequency Interferences (RFI), salinity retrievals
1 Introduction1
The Soil Moisture and Ocean Salinity (SMOS) mission is the first Earth Obser-2
vation (EO) mission devoted to the remote sensing of both soil moisture over3
the continental surfaces and sea surface salinity (SSS) over the oceans (Barre4
et al., 2008; Font et al., 2010; Kerr et al., 2010). Its unique payload, MIRAS,5
is a completely new type of instrument: a two-dimensional synthetic aperture6
radiometer with multiple incidence angles and full polarimetric capabilities7
(Martin-Neira and Goutoule, 1997; McMullan et al., 2008).8
The operating principle of MIRAS is based on the measurement of the complex9
cross-correlations of the signals collected by each pair of receivers, providing10
the basic measurements of a two-dimensional interferometer: the samples of11
the visibility function (Corbella et al., 2004). Errors in the visibility samples12
need to be corrected by means of several calibration procedures (Brown et al.,13
2008) before applying an image reconstruction algorithm to obtain the bright-14
ness temperatures (TB) (Anterrieu and Khazaal, 2008; Camps et al., 2008;15
Corbella et al., 2009).16
To whom correspondence should be addressed
Email address: vgonzalez@icm.csic.es (Ver´onica Gonz´alez-Gambau).
2
Due to the finite extent of the instrument and the fixed locations of the anten-17
nas, visibilities are measured only at selected points in a star-shaped subarray18
over a hexagonal grid. The incomplete sampling at high spatial frequencies19
generates Gibbs-like contamination, i.e., sidelobes and ripples when sharp20
transitions are present in the brightness temperature scenes. In the SMOS21
operational image reconstruction strategy, a Blackman window is applied to22
the Fourier components of the brightness temperatures in order to decrease23
these artifacts. This windowing considerably reduces the amplitude of the rip-24
ples and improves the radiometric sensitivity while meets the spatial resolution25
mission requirement (Anterrieu et al., 2002; Bar´a et al., 1998).26
Radio-Frequency Interferences (RFI) at L-band have been a serious problem27
for SMOS ever since the beginning of the mission. According to international28
regulations, such emissions are illegal, but they are quite frequent. Owing to29
intense efforts conducted by ESA, many RFI sources have been switched off30
(Oliva et al., 2012). However, there is still significant contamination of SMOS31
TB by RFI, which render the retrieval of ocean salinity in some coastal areas32
impossible, and also have a strong impact on the quality of the soil moisture33
retrievals. Hence, many efforts have been dedicated to developing appropriate34
RFI detection and mitigation strategies (Camps et al., 2011; Castro et al.,35
2012; Daganzo-Eusebio et al., 2013; Park et al., 2015; Soldo et al., 2014).36
The presence of any RFI source not only impacts the area directly affected37
by the interference, but also causes strong sidelobe levels and ripples that38
contaminate all over the brightness temperature image. It must be noted,39
however, that Gibbs-like contamination is not restricted to RFI hot spots, but40
it is present wherever any sharp transition in SMOS brightness temperature41
scenes exist, such as land/sea/ice transitions and even the Earth surface/sky42
transition. This means that any image has some small but non-negligible de-43
gree of contamination because, even when no RFI (or land/sea/ice transition)44
3
appears, the Earth horizon is always present.45
This work is being motivated by the fact that, even after applying a Blackman46
window, the sidelobes and ripples caused by strong point sources are still47
significant (see as an example the image in Fig. 1). The effect of these ripples48
is not irrelevant in the case of the ocean salinity retrieval: the sensitivity of the49
L-band brightness temperature to salinity is relatively low and therefore, any50
perturbation of few Kelvins may cause large deviations on the retrieved SSS51
(Yueh et al., 2001). As such, the development of specific methods to reduce52
the impact of these perturbations over ocean scenes is a crucial task.53
Nodal sampling, first presented in (Gonz´alez-Gambau et al., 2015), is a novel54
image reconstruction algorithm focused on the reduction of Gibbs-like contam-55
ination. The first approach to nodal sampling showed a more than significant56
reduction of Gibb-like contamination (mainly ripples and sidelobes). But even57
implying a large gain in quality, there were some remaining issues in the for-58
mulation of nodal sampling, especially regarding the size of the RFI sources59
originating the largest sidelobes. The aim of this paper is twofold: on one60
hand, to discuss the issue of the poor description of the area directly affected61
by the presence of RFI sources, first, and then introduce an improvement in62
the nodal sampling algorithm that leads to a general quality enhancement;63
on the other hand, we want to illustrate how the use of nodal sampling will64
contribute to the improvement of salinity retrievals.65
The paper is structurated as follows. The improvements in the nodal grid de-66
termination and the analysis of the system performance using this algorithm67
are presented in section 2. The latest definition of the nodal sampling algo-68
rithm is compared at brightness temperature level using ocean scenes to the69
initial approach as well as to the currently SMOS operational image recon-70
struction strategy. The comparison of the brightness temperatures to those71
theoretically modeled is also used as a metric to evaluate the performance of72
4
the method (section 3). Section 4 is devoted to the impact assessment of the73
nodal sampling application on the SSS retrievals, using in-situ SSS measure-74
ments as reference.75
2 Nodal sampling image reconstruction tecnique76
2.1 Nodal sampling: conceptual basis and current approach77
The nodal sampling algorithm was recently developed by (Gonz´alez-Gambau78
et al., 2014, 2015). This method is based on sampling TB images at the nodal79
points, i.e., those points at which the oscillating interference causes the min-80
imum distortion of the geophysical signal. The underlying hypothesis of the81
nodal sampling is that the geophysical signal of interest varies relatively slowly,82
at the scale of the spatial resolution of the instrument. At those points where83
there is no perturbation, the subsampled signal will take a value which is the84
result of the interpolation among the points nearby, which introduces some85
error. However, if the geophysical signal present a slow variation this error will86
be rather small. This approximation is especially well suited over the ocean,87
which presents structures of typically tens to hundreds of kilometers.88
The nodal sampling algorithm encompasses three steps: (i) the spatial over-89
sampling of the TB image, (ii) the selection of the nodal grid in the oversam-90
pled TB image and (iii) the reconstruction of a corrected image at the original91
spatial sampling. It is important noting that no apodization window is applied92
to the Fourier coefficients before the image reconstruction. Brightness temper-93
ature images are oversampled in such a way that the Fourier coefficients at94
the known spatial frequencies are kept the same. This oversampling leads to95
a better definition of the oscillating structures and, as such, the points where96
5
the perturbation cancels can be more accurately determined. Nodal points are97
characterized by the local minima of the Laplacian at the areas defined by the98
pixels at the original spatial sampling. Finally, a new image is constructed by99
taking the values of the oversampled image at the selected nodal points, such100
that each pixel of the new image is given by one specific pixel of the associated101
oversampled pixels. Hence, the corrected image has the same spatial sampling102
than the original one.103
The key point for the practical application of nodal sampling is to determine104
the set of nodal points, something for which there is no straightforward pro-105
cedure. In (Gonz´alez-Gambau et al., 2015) it was noticed that nodal points106
are referred to the local minima of the Laplacian, but the determination of107
those minima requires a simultaneous change of the sampled points and the108
values of the Laplacian. The iterative procedure for the determination of the109
nodal grid introduced by (Gonz´alez-Gambau et al., 2015) has shown a signif-110
icant improvement of the brightness temperature quality: over ocean scenes,111
previous works yield an average error reduction of approximately 0.7 K. This112
improvement should lead to an error reduction in the SSS estimates of about113
1.4 psu, which is quite significant taking into account that the requirements114
of the mission are to fulfill an accuracy of 1 psu at Level 2 (see (scientific115
requirements, 2015)). However, some limitations in the determination of the116
nodal grid were already evident in previous works. As it will be next discussed,117
the observed problems have to do with the particular choice of the nodal grid118
and this choice can be improved with the convenient strategy.119
2.2 Refinement of the nodal grid determination120
The initial nodal sampling approach leads to unexplained changes in the spa-121
tial extent of some RFI sources, that become either wider or significantly122
6
reduced than that of the currently operational (i.e., nominal) image recon-123
struction algorithm. We have identified that part of this increase/decrease of124
the spatial extent of some RFI sources is linked to the condition imposed in the125
spatial locations of the nodal points. In the initial definition of the algorithm126
in (Gonz´alez-Gambau et al., 2015) (hereafter referred to as NSv1), for each127
pixel of the original image, the associated nodal point is chosen as one of the128
oversampled points corresponding to that pixel, that is, a subpixel belonging129
to the area covered by the pixel at original sampling.130
In this paper, a modification of the nodal sampling strategy is proposed to131
better track the nodal lines. In particular, the search condition for nodal points132
has been relaxed by allowing some nodal points to cross the boundaries defined133
by the original pixel. Once a first guess of the nodal points has been estimated134
by finding the local minima of the Laplacian in the oversampled image, an135
iterative refinement of the nodal points selection is performed in order to136
reduce the Laplacian in the original grid (see (Gonz´alez-Gambau et al., 2015)137
for more details). The new algorithm (hereafter referred to as NSv2) proceeds138
as follows (see scheme in Fig. 2):139
(1) The original TB image is spatially oversampled using an oversampling140
factor β = 9. The resulting pixels are referred as subpixels. In a first141
step, candidates to nodal points are searched within the subpixels which142
belong to the area of each original pixel. A point will be considered as a143
candidate if the Laplacian of the generated image (obtained by subsam-144
pling the image with the new candidate at the location under study and145
the previous candidates at any other location) is minimum at that loca-146
tion. The process is then repeated at all locations of the original image.147
As all the points are not updated at the same time, it can not be granted148
that any new point verify to be the one leading to the minimum Lapla-149
cian when its surrounding is kept and the point is updated. Therefore,150
7
several iterations are required to attain a close-to-equilibrium state. This151
step is iterated a prestablished number i of times. After the i iterations,152
those candidates which are strictly inside the area belonging to the origi-153
nal pixel are kept fixed. Hence, only those points lying on the boundaries154
of this area can be updated in the successive steps.155
(2) The area where the nodal points are searched for is extended by 1 subpixel156
in all directions (see Fig. 2). The search of the nodal points (step 1) is157
then repeated.158
(3) The algorithm is iterated until all the candidates are fixed. The resulting159
grid is the nodal grid.160
It is worth remarking that the new search condition is more mathematically161
coherent, since the local minimum is now searched in an open set (not in a162
close one as the previous definition of the algorithm does).163
One of the parameters to be analyzed in the algorithm performance is the164
number of subpixels (in all directions) where the search domain is extended165
beyond the area corresponding to the original pixel. Nodal sampling is a non-166
linear method and thus, it does not produce the same effect in all the points167
over the image. From our experiences, it has been observed that the search168
domain is usually extended beyond the original boundaries to not more than169
3 subpixels in the oversampled image. This ensures that the spatial resolution,170
for those pixels where the search domain is extended, is not degraded more171
than a third (accounting that β = 9) with respect to the NSv1 (nodal points172
are always selected inside the original pixel boundaries). Only in the case173
of very strongly contaminated pixels, the search domain is extended up to 5174
subpixels. In general terms, about half of the candidates lie within the original175
boundaries (and therefore they are fixed from step 1), and a quarter lie exactly176
at the boundaries. The remaining quarter is mainly at 1 subpixel of the pixel177
boundaries, with a very small percentage lying beyond.178
8
Three nodal subgrids are shown in Fig. 3- the regular hexagonal subgrid which179
leads to the same values of brightness temperatures as the original (nominal)180
image, the resulting subgrid from the nodal sampling algorithm proposed in181
(Gonz´alez-Gambau et al., 2015) and that from the new version. White stars182
indicate the points which belong to each subgrid. This image corresponds to183
a zoom-in view of the oversampled TB image in Fig. 4 over the RFI located184
around the position (ξ, η) = (0.3, 0.2). As it can be seen, the nodal points are185
more concentrated in between the peaks and the valleys (where the Laplacian186
minima are located) in the new version of the NS algorithm (Fig. 3c) than in187
the previous one (Fig. 3b).188
Brightness temperature images have been reconstructed taking the values of189
the oversampled image at the three nodal subgrids (see Fig. 4). As expected,190
the nominal processing shows the RFI-induced ripples everywhere in the im-191
age (Fig. 4a). NSv1 reduces very much the Gibbs contamination, although192
remaining sidelobes are still present (Fig. 4b). In contrast, the NSv2 algo-193
rithm (Fig. 4c) is able to further reduce the remaining sidelobes in the NSv1194
TB image. In addition, some spurious spikes which remain in the NSv1 TB195
image have been also reduced using the new algorithm.196
An example illustrating the differences on the image reconstruction techniques197
is given in Fig. 5, which shows a scene over the Atlantic Ocean strongly con-198
taminated by a RFI source produced by a ship. A clear reduction in the level199
of sidelobes and ripples is observed using nodal sampling, in its both versions.200
In the case of the NSv2, the spatial extent of the RFI source is significantly201
narrower than that using NSv1. This effect can be better appreciated by look-202
ing at the masks of the brightness temperature images in Fig. 6. The mask203
is set for those pixels with negative TB values or values higher than 350 K,204
which correspond to non-natural emissions. This illustrates that NSv2 reduces205
significantly the sidelobes and could lead to more localized (narrower) sources.206
9
2.3 Analysis of the radiometric resolution207
One of the parameters used to define a radiometer system performance is208
the radiometric resolution, defined as the temporal standard deviation of the209
brightness temperatures over a flat and stable scene (Corbella et al., 2000; Tor-210
res et al., 1997). The SMOS radiometric resolution can be estimated using the211
brightness temperature difference between consecutive snapshots in the same212
polarization in order to reduce the system drift and the scene variation (Wu,213
2014). Using this methodology, the radiometric resolution was empirically es-214
timated during Commissioning Phase (Corbella et al., 2011) and compared215
to the theoretical one, computed using the following formula (Camps et al.,216
1998)217
T
B
(ξ, η) =
3d
2
2
T
sys
q
Bτ
eff
a
t(ξ, η)
q
1 ξ
2
η
2
α
w
N (1)218
where d is the distance between the antennas in wavelengths; T
sys
is the aver-219
age system temperature used for the visibility denormalization; B is the noise220
equivalent bandwidth; τ
eff
is the effective integration time for the one-bit cor-221
relator (Corbella et al., 2000); α
w
is a coefficient that depends on the apodiza-222
tion window used in the TB reconstruction (1 for a rectangular window, 0.45223
for a Blackman window);
a
is the antenna equivalent solid angle and t(ξ, η)224
is the normalized antenna power pattern (both measured on ground); and N225
is the total number of visibilities.226
In this paper, this analysis has been performed both for nominal and NSv2227
processings using ocean scenes over the South Central Pacific in June 2014.228
Figure 7a shows the radiometric resolution estimated from the nominal TBs229
of one descending overpass for dual X-polarization epochs (i.e, epochs with230
all arms in the same polarization status) (Martin-Neira et al., 2002). The cut231
along ξ = 0 in Fig. 7b, shows the agreement between the estimation and the232
10
theoretical computation of the radiometric resolution (with an average 6 %233
error in the EAF-FOV between them). All the overpasses over that zone during234
June 2014 have been analyzed, showing a stable performance over time. The235
radiometric resolution when using NSv2 is presented in Fig. 7c. In this case,236
the standard deviation of the random errors is half the one of the nominal237
processing (on average in the EAF-FOV), as it can be observed in the cuts238
along ξ = 0 (Fig.7b).239
It is worth noting that this improvement in the radiometric resolution could240
as well be obtained by using a spatial low pass filter, but in this case, the241
geophysical structures in SSS are degraded with respect to the nominal SSS,242
as we show in section 4.3. This reduction in the expected random errors of the243
brightness temperatures is quite significant. Note that to achieve an equiva-244
lent reduction from the hardware point of view, the integration time of the245
measurement should be increased by a factor of four.246
In the case of the nodal sampling, the input data are the Fourier components247
of the brightness temperatures without applying any apodization window. The248
α
w
parameter in Eq. (1) is not known for the case of applying the nodal sam-249
pling and has been obtained by least-squares fitting the estimated radiometric250
resolution to the theoretical values. For verification purposes, this method has251
been first applied to the nominal case. An α
w
value of 0.46 has been obtained,252
as expected for a Blackman window (Camps et al., 1998). The same procedure253
has been then applied to find the α
w
value for the nodal sampling technique.254
In this case, α
w
= 0.23, which is consistent with the reduction factor observed255
in the radiometric resolution. This parameter is then used to compute the256
theoretical radiometric resolution in eq. 1 for each snapshot, which is used to257
weight the observational (i.e., TB) term of the Bayesian-based salinity inver-258
sion cost function (Gabarro et al., 2009), as it will be explained later in section259
4.260
11
3 Impact of nodal sampling on brightness temperatures over the261
ocean262
One of the metrics used in this work for the quality assessment of the bright-263
ness temperatures over ocean scenes is the comparison of the SMOS TB mea-264
sured to the theoretically modeled ones, using geophysical priors as input265
variables. The modeled TB have been derived from the forward model or the266
Geophysical Model Function (GMF) presented in (Guimbard et al., 2012), us-267
ing geophysical priors for SSS (climatology), SST and wind speed (data from268
ECMWF) (Zine et al., 2008). The difference between measured and modelled269
brightness temperatures is computed and the statistics of this new variable are270
assessed. Prior data do not correspond to the actual values of the geophysical271
parameters but they are typically close to them. Comparisons against mod-272
eled brightness temperatures have been widely used as they can help in the273
discrimination of errors larger than those associated to the models.274
The analysis is performed over a very stable zone in the South Central Pacific275
Ocean where the SMOS operational OTT (Ocean Target Transformation) is276
computed (Meirold-Mautner et al., 2009; Tenerelli and Reul, 2010). The OTT277
is used to correct for residual antenna-frame systematic errors and is computed278
as the temporal median of the TB difference between the measured and the279
modelled TBs as a function of the spatial direction. The OTT is computed for280
each TB dataset (nominal, NSv1 and NSv2) using 10 overpasses. OTTs in Fig.281
8 correspond to the ascending node, dual epoch and Y-polarization measure-282
ments. The statistics computed for all the points in the Alias Free (AF) and283
Extended Alias Free (EAF) FOV are also annotated in the plots. Similar spa-284
tial patterns can be recognized in the three cases, although these are smoother285
for NSv1 and NSv2 than for the nominal processing. The reduction of the spa-286
tial bias in the AF is around 23.4% for NSv2 and 22.5% for NSv1 with respect287
12
to the nominal OTT. Similar results in terms of the spatial bias reduction are288
obtained for X and Y-polarizations and for ascending/descending overpasses289
(see Table 1).290
In order to assess the error reduction on brightness temperatures when using291
NS, the temporal standard deviation of the TB difference has been computed292
at each point in the EAF-FOV. The distributions of the standard deviation293
values (shown in Fig. 9) reveal that for NSv2 the retrieved brightness tempera-294
tures are in a better agreement to those modelled than for NSv1 and especially295
the nominal processing. The average reduction in the brightness temperature296
errors (w.r.t. the nominal processing) is 1.43 K (0.84 K) in X-polarization and297
1.36 K (0.76 K) in Y-polarization for NSv2 (NSv1). Notice also that the tails298
of the error distributions have been substantially reduced. With this reduc-299
tion, errors are close to the minimum expected when auxiliary information of300
geophysical parameters is used in SSS retrievals. Results confirm that NSv2301
performs better than the previous approach. For this reason, from this point302
onwards, only NSv2 will be analyzed and compared to the nominal processing.303
The analysis discussed above have been carried out over the OTT (clean and304
stable) zone and for only a few orbits. In order to extend this study, in both305
the spatial and temporal domains, a 9-day 0.25
resolution global map of the306
First Stokes brightness temperature difference (measurements minus model)307
has been generated from both the nominal and NSv2 TBs at Bottom of the308
Atmosphere (BOA). The maps are obtained after averaging all the pixels in the309
AF-FOV. The corresponding OTT has been subtracted to both nominal and310
NSv2 TB datasets, respectively, before the computation of the difference with311
the modeled brightness temperatures. The residual differences between the312
OTT-corrected brightness temperatures and the modeled TBs present similar313
structures both for the nominal and the nodal sampling datasets, although314
the latter are smoother (see Fig. 10). These residual patterns are consistent315
13
with those shown in (Corbella et al., 2015) and their origin can be attributed316
to two different effects: the floor error in the image reconstruction (Anterrieu317
et al., 2015; Corbella et al., 2014) and the land-sea contamination (LSC).318
Regarding the standard deviation of the TB differences, the NSv2 (Fig. 11b)319
clearly shows lower values (1.68 K on average) than the nominal (2.62 K on320
average) reconstruction (Fig. 11a). This reduction can be appreciated not only321
in open ocean (where the average reduction is around 36% at a global scale)322
but also in strongly RFI-contaminated coastal areas such as in the North-East323
Atlantic Ocean, the Arabian Sea, the Bay of Bengal and the China Seas.324
4 Impact of nodal sampling on salinity retrievals325
The impact of improved TB on SSS retrievals will be assessed in this sec-326
tion. Two methods for inverting SSS from TB data have been used: (i) a327
non-Bayesian approach, based on the retrieval of a SSS value per each indi-328
vidual TB measurement and then estimating the most probable value from329
the SSS distributions and (ii) a Bayesian approach similar to that used in the330
operational Level 2 processor (L2OS, 2014) retrieving the SSS consensus of all331
the TB measurements. Salinity maps are generated from both nominal and332
NSv2 TBs using both retrieval methodologies. The geophysical consistency of333
SSS retrievals and the inter-comparison/validation against in-situ data (Argo334
buoys) are also analyzed in this section.335
4.1 Non-Bayesian SSS retrieval336
This inversion scheme is based on retrieving a SSS value for each TB mea-337
surement and accumulate them for each geographical point of the map. Note338
that SMOS has multiple incidence angle capabilities and therefore multiple339
14
TB measurements for each satellite overpass at a certain grid point on the340
ground. All the retrieved SSS values are accumulated for each grid point over341
the selected period, e.g., 9-day average for the 9-day SSS product. To perform342
this simple single-TB retrieval, all the unknown parameters defined in the343
GMF, excepting the SSS, are fixed using auxiliary data (SST, wind speed and344
Total Electron Content). As such, the SSS is retrieved for each point and TB345
measurement at a given incidence angle (θ) by minimizing the difference be-346
tween the measured (T
meas
B
) and the modeled (T
mod
B
) brightness temperatures347
by means of the Newton’s method348
f(SSS) = (T
meas
B
(θ) T
mod
B
(θ; SSS))
2
(2)349
This direct approach does not require an error model and is applied without350
previous filtering of the brightness temperatures. Therefore, the analysis of351
the SSS distributions directly provides a quality indicator of the brightness352
temperatures used in the salinity retrievals.353
All the retrievals at different incidence angles for the same geographical point354
in a 0.25
spatial grid during 9 days have been accumulated in order to get355
significant statistics. SSS values outside the range [0-50] psu have been filtered356
out. Two examples of the a posteriori SSS distributions are shown in Fig. 12357
for two representative grid points: (i) over the South Pacific Ocean (clean358
zone) and (ii) over the Indian Ocean close to the Arabian Sea (very affected359
by land-sea and RFI contaminations). The histograms show the relative fre-360
quency of the retrieved SSS values using bins of 0.5 psu for the nominal (in361
green) and the nodal sampled TBs (in blue). The SSS estimates from Argo362
buoys (cyan vertical line) and climatology from WOA13 (red vertical line) are363
also displayed for reference. The NSv2 SSS distributions evidence a noticeable364
reduction in the standard deviation with respect to the nominal distributions365
(36.7% for the Pacific grid point and 19.7% for the Indian grid point).366
15
At a global scale (Fig. 13), the standard deviations of the a posteriori SSS dis-367
tributions are 7.01 psu for the nominal processing (Fig. 13a) and 5.25 psu on368
average for the NSv2 approach (Fig. 13b). This reduction in the standard devi-369
ation of the distributions (approximately 25%) implies that the nodal sampling370
image reconstruction requires a smaller number of accumulated measurements371
to retrieve a representative SSS value with the same accuracy as that of the372
nominal approach.373
From these SSS distributions, several statistical descriptors (mean, median374
and the mean around the mode) have been explored to find which one of375
them provides the most appropriate estimator for the SSS retrievals. The mean376
around the mode of the distribution considering an interval of ±σ is used to377
obtain the retrieved SSS values. The retrieved salinities are presented in Fig.378
14 for both the nominal (Fig. 14a) and the NSv2 (Fig. 14b) TB datasets. Note379
that although the general patterns are similar, regional differences are quite380
evident, notably in the Southern Ocean. The corresponding anomaly maps381
(w.r.t. the World Ocean Atlas 2013 or WOA13 climatology data) are shown382
in Figs. 14c and 14d. The analysis of these maps reveals that NSv2 SSS are in383
better agreement with the expected mean SSS values than the nominal SSS,384
which show a positive bias (i.e., Fig 14d presents more unbiased SSS areas385
than Fig. 14c). This could be explained by the large standard deviation of386
SSS retrievals in the case of the nominal TB.387
4.2 Bayesian SSS retrieval388
A similar approach to that used in the SMOS operational Level 2 processor389
has been also used to produce SSS maps from the nominal and the NSv2390
TBs. At each grid point all the measurements provided by one overpass of the391
satellite are used for computing a single SSS value. The Bayesian approach392
16
accounts for both TB measurement uncertainties and a priori knowledge on393
the geophysical parameters. The Bayesian-based cost function is defined as:394
f(SSS) =
N
m
1
X
j=0
(T
meas
B
(θ
j
) T
mod
B
(θ
j
; SSS))
2
σ
2
T B
(θ
j
)
+
N
p
1
X
j=0
(P
j
P
prior
j
)
2
σ
2
P
j
(3)395
where N
m
is the number of measurements available for the current grid point;396
N
p
is the number of geophysical parameters to be retrieved; σ
T B
is the standard397
deviation of the error associated to the measured and modelled brightness398
temperatures (which can be deduced from Eq. (1)); P
j
is the j-th parameter399
to be retrieved; P
prior
j
is its a priori value, and σ
2
P
j
is an estimation of its error400
variance.401
The cost function used in this work is a simplification of that defined in402
(Gabarro et al., 2009). As in the non-Bayesian approach, all the auxiliary403
parameters (SST, wind speed and TEC) have been fixed, so the second term404
of the right hand side of equation (3) is zero. The semi-empirical roughness405
model defined in (Guimbard et al., 2012) has been used to model the rough-406
ness component of the GMF. As such, only one geophysical parameter, i.e.,407
the SSS, is retrieved by minimizing the cost function using the Levenberg408
Marquardt’s method. Only dual polarization measurements have been used to409
retrieve SSS maps. Previous to the inversion, several filters have been applied410
to the TB measurements to disregard any measurement411
(1) that is outside the [0,300] K range,412
(2) that when compared to the modeled TB, the absolute value differs more413
than the minimum between 5 K and 3 times the theoretical radiometric414
resolution.415
Binned maps have been produced by a weighted average of the retrieved salin-416
ities, considering only those values within ±300 km from the center of the417
sub-satellite track (see (SMOS-BEC , 2015) for more details on this product).418
17
No post-filtering has been applied to the retrieved salinities. The SSS retrieved419
from both TB datasets are shown in Fig. 15. From the maps of the anomalies420
in Figs. 15c (nominal) and 15d (NSv2) it can be observed that both maps ex-421
hibit similar large-scale structures and the anomalies are less noisy in the case422
of the nodal sampled SSS. The land-sea contamination effect can be clearly423
observed in both maps, being similar in terms of their spatial extension but the424
bias seems to be higher in the case of the nodal sampling processing. Notice425
that the land-sea transition induces two simultaneous effects. One is a Gibbs426
effect, which is expected from the incomplete sampling of Fourier frequencies;427
nodal sampling is expected to remove this contribution. The other effect is428
still under investigation and seems to be related to correlator efficiency errors429
(Corbella et al., 2015). This second effect has long range and is not oscillatory:430
it tends to decrease the average gradient making the transitions smoother and431
much broader than they actually are. As this second effect has a moderate432
gradient nodal sampling cannot remove it, and the removal of the oscillatory433
Gibbs contamination makes it more apparent in nodal sampling images.434
4.3 Geophysical consistency of the SSS retrievals435
One of the questions about the performance of nodal sampling is if this tech-436
nique induces some degradation of the spatial resolution. A visual inspection437
of Figs. 4-5 shows a significant widening of some structures, but at the same438
time other structures seem to be better resolved. Note that the nodal sam-439
pling is a non-linear technique and therefore, it acts differently in the different440
parts of the snapshot. Therefore, a simple assessment of its effect on the final441
resolution of SSS products is not straightforward.442
In order to assess the impact on the effective spatial resolution of SSS products443
when nodal sampling is used or not, we have applied a standard oceanographic444
18
methodology: a spectral analysis (incluir ref) of the two 9-day SSS maps (3-12445
June 2014), both retrieved with the Bayesian approach, has been performed.446
For this test, we have focused on a zonal box extending from 180
W to 120
447
W in longitude and from 5
S to 5
N in latitude. This choice of zonal box is448
justified because the region is dominated by the presence of a rich variety of449
mesoscale structures, so leading to well-defined spectral characteristics.450
The power spectra of the nodal sampled (red) and nominal (green) SSS are451
presented in Fig. 16. The approximate slopes of the spectra are also shown452
for providing a rough reference. As shown in the figure, both NS and nomi-453
nal present peaks and valleys at the same low spatial frequencies (i.e., below454
1
(
)
1
). This means that both have the same resonance spatial frequencies,455
that is, they represent the same large-scale oceanographic structures (e.g.,456
Tropical Instability Waves, Equatorial Current, filaments, eddies, etc.). Both457
products present a similar cutoff frequency of around 1.5
(
)
1
, from which458
the amplitude of the noise floor is dominant (that is, the spectrum becomes a459
horizontal line). This cutoff frequency represents the minimum resolved spa-460
tial scale and hence informs us about the effective spatial resolution of the461
products, which in this case is of about 0.75
. Hence, the application of nodal462
sampling is not reducing the effective spatial resolution of SSS products.463
However, it is clear from the figure that the spectrum of the nominal product464
can be approximately fitted by k
1.1
, that is, by a power law of slope -1.1.465
A power law behaviour in the spectrum is expected for any ocean scalar at466
mesoscale and larger scales because of the effect of quasi-geostrophic turbu-467
lence (incluir ref). However, the spectral slope is expected to be in the range468
from -5/3 to -3 (incluir ref). A spectral slope close to -1 is an evidence of the469
nominal product being dominated by correlated noise, even at large scales. On470
the contrary, the approximate spectral slope for NS is -1.5, which is more geo-471
physically consistent. This means that the oceanographic structures described472
19
by NS SSS have an energy content closer to reality.473
Spectral analysis also puts into evidence the advantage of applying a non-linear474
method as NS for the removal of Gibbs-like contamination. For instance, when475
a standard spatial filtering is applied to the data, the analysis of its spectrum476
reveals a certain loss of geophysical consistency. To illustrate the differences477
in the performance, SSS maps applying a spatial filtering to the Fourier coef-478
ficients of the TB have been generated. In this case, we have used a circular479
window in such a way that only the spatial frequencies in the inner circle of480
the star-shape spatial coverage are used in the brightness temperature inver-481
sion (Corbella et al., 2012). In this case, the improvement in the comparison482
with modeled TB is on average 0.2 K with respect to the nominal case. Re-483
garding SSS, when the zonal spectrum is computed (see Fig. 17) we see that484
some spectral peaks and valleys are lost. This implies a significant change in485
some geophysical structures, what is clearly an artifact. Besides, as in the case486
of the nominal SSS, the circular SSS are dominated by correlated noise (the487
approximate slope in this case is k
1.2
). Finally, the cutoff frequency is clearly488
lower than those of nominal and NS products, hence evidencing a worse ef-489
fective resolution in the case of this particular spatial averaging. We conclude490
that nodal sampling is more suitable from a geophysical perspective than a491
linear spatial filtering.492
4.4 Validation with in-situ measurements493
The validation of the SSS maps against in-situ measurements is carried out494
using Argo floats (see http://www.argo.ucsd.edu). The analysis is split in two495
different ocean regions: (i) coastal regions (i.e., wihtin 800 km off the coast) and496
(ii) open ocean (all the ocean but coastal regions). This is done to differentiate497
the nodal sampling effect over land-sea contaminated regions and areas where498
20
no significant impact from this kind of contamination is expected. All the499
available Argo SSS estimates within the same 9-day period are considered500
in the validation. Each Argo SSS estimate is compared with the SSS value501
provided by its corresponding L3 grid. Therefore, may happen that several502
Argo estimates are compared with the same L3 grid. In this case, all the503
differences are considered in the full statistics. Since one month of nominal504
and NSv2 data (June 2014) is used for validation purposes, seven 9-day maps505
(generated every 3 days) have been generated. The statistics (the average of506
the statistics for each individual 9-day map) have been computed in the study507
regions indicated in Table 2.508
Figure 18 displays the statistics (mean and standard deviation) of the dif-509
ferences between the 9-day, 0.25
SMOS measurements and the Argo surface510
salinities for the nominal and the NSv2 processings. In open ocean (top row),511
the standard deviation of the SSS differences for the NSv2 processing is lower512
than that for the nominal processing in all the study regions. For the NSv2513
processing, the standard deviation of the differences of SSS retrievals and Argo514
are quite similar for both (Bayesian and non-Bayesian) retrieval methods, al-515
though slightly lower for the Bayesian approach. The difference between the516
NSv2 and the nominal SSS quality is much larger for the non-Bayesian re-517
trieval than for the Bayesian approach. This may be due to the fact that the518
representative SSS for the 9-day accumulated distributions is not well defined519
in the nominal case. Hence, accumulated measurements over a larger period520
would be needed to define a proper SSS using this retrieval methodology. Glob-521
ally, the biases are very similar for both the NSv2 and the nominal processing.522
Slight differences appear in North-Pacific (better in nominal) and Southern523
Ocean (better in NSv2), which require further investigation.524
Regarding the matchups located no farther than 800 km from the coast (bot-525
tom row), both the bias and the standard deviation of the SSS differences526
21
are higher than in open ocean, as expected due to the RFI and land-sea con-527
tamination effects. The standard deviation of the SSS differences for the non-528
Bayesian approach is lower for NSv2, except in the North Atlantic zone. This529
isolated effect needs to be further investigated. However, this is not the case530
for the Bayesian retrieval, where the standard deviation seems to be generally531
higher for NSv2 than for the nominal processing. This could be due to the pre-532
filtering criteria used in the Bayesian retrieval, which might be masking the533
improvements on the NSv2 TB quality with respect to that of the nominal TB.534
In terms of error biases, the matchups close to the coast display a larger bias535
in the case of NSv2 than the nominal case. We have not at this moment a com-536
plete explanation of this larger deviation close to the coast when the Bayesian537
method is used on NS images, something that does not happen when they are538
processed with the non-Bayesian method. We guess that the removal of the539
oscillatory Gibbs makes brightness temperatures to be systematically biased540
in the NS case whereas in the nominal case the Gibbs contamination makes541
them to get sometimes closer to the right value. Since the TB differences are542
averaged together in the cost function this would explain a stronger systematic543
bias on SSS retrieved from NSv2 than from nominal TBs. An exception is the544
Southern Ocean, where NSv2 performs better and seems to partially overcome545
the limitations of the SSS retrievals in areas of high winds and low SST values,546
where the low sensitivity of the brightness temperatures to salinity changes547
produce large errors in the SSS retrievals.548
5 Conclusions549
In this paper, a modification of the nodal sampling technique has been pro-550
posed to improve the quality of the reconstructed TBs. This improvement con-551
cerns to the fine tuning of the nodal grid determination. This refined technique552
(NSv2) has led to a significant quality improvement of the brightness temper-553
22
atures with respect to those of the previous nodal sampling method (NSv1)554
(Gonz´alez-Gambau et al., 2015) and notably those of the current SMOS image555
reconstruction baseline (nominal). At snapshot level, NSv2 improves the TB556
reconstruction in terms of both the sidelobe levels and ripples. In addition to557
that, the spatial extent of the RFI sources, sometimes wider in NSv1 than in558
the nominal images, is clearly reduced using NSv2.559
Since NSv2 leads to a decrease of the expected errors, a new radiometric560
resolution has been empirically estimated over the ocean. The results show561
that NSv2 TB present half the radiometric resolution of the nominal TBs,562
which is an important improvement in terms of the system performance. A key563
point for the general application of the nodal sampling technique is to assess564
the effective spatial resolution of nodal sampling images. As nodal sampling565
is a non-linear method (its effect is not simply applying a convolution kernel566
to the image, as some parts of the image behave in a different way than other567
parts) the concept of resolution should be rather considered as an effective568
quantity. The introduction of power spectral analysis has allowed us to show569
that the application of nodal sampling does not degrade the effective spatial570
resolution of SSS products; on the contrary, the spectral slope of NS products571
becomes much closer to what should be geophysically expected.572
The validation of SMOS brightness temperatures over ocean scenes shows that573
NSv2 TBs are in better agreement with the modeled TBs than the nominal574
TBs. Over clean and stable zones, such as the zone in the Pacific Ocean where575
the SMOS OTT is computed, the average error of the NSv2 TBs is about 1.3576
K lower than that of the nominal TBs. At global scale, the error reduction577
is of about 1 K. This improvement in the brightness temperatures is a good578
indicator that the quality of geophysical retrievals could be improved using579
this technique.580
Salinity maps have been produced for both the NSv2 and the nominal process-581
23
ing using two retrieval methods. The non-Bayesian approach allows a more582
direct analysis of the quality of the TB used in the retrieval. It is shown that583
the NSv2 SSS distributions are narrower and therefore better defined than the584
nominal SSS distributions. When compared to the climatology, the NSv2 SSS585
anomalies show similar large-scale structures but less noisy than the nominal586
SSS anomalies.587
From the validation with Argo SSS estimates, it is found that NSv2 improves588
the accuracy of the SSS retrievals in open ocean with respect to the nomi-589
nal SSS product, regardless of the retrieval methodology. The validation has590
been carried out separately for coastal areas (i.e., within the first 800 km off591
the coast), where besides RFI, the effect of the land-sea contamination is also592
present. In these regions, the advantage of the higher quality brightness tem-593
peratures provided by NSv2 is only present in the case of the non-Bayesian594
approach. In this regard, two issues need to be further investigated: (i) the ef-595
fect of the systematic bias induced by the land-sea contamination and (ii) the596
Bayesian-based retrieval which has been used, notably regarding the quality597
control.598
Future improvements at the SMOS image reconstruction level are aimed at599
assessing the capabilities of the nodal sampling to provide valid retrievals600
in strongly RFI-contaminated areas once the land-sea contamination effects601
have been corrected. To do that, the corrections currently proposed by the602
SMOS Level 1 team will be applied prior to the image reconstruction by nodal603
sampling.604
In the context of salinity retrievals, our team is investigating improvements of605
the Bayesian cost function using nodal sampled TB to retrieve, not only SSS,606
but also the wind speed, similarly to the Level2 operational processor. This will607
imply to revisit the main mathematical hypothesis for applying the Bayesian608
approach (unbiased and independent Gaussian errors) to the SMOS case. In609
24
this line, this work allows analyzing the impact that brightness temperature610
biases have on the SSS retrieval when the TB standard deviations are reduced,611
as in the nodal sampling processing.612
Acknowledgments613
This work has been funded by the Spanish Ministry of Economy through the614
National R+D Plan by means of MIDAS-7 project AYA2012-39356-C05-03615
and previous grants. The validation part of this work has been done with the616
support of the FP7-SPACE E-AIMS project (grant agreement 312642). The617
Argo data were collected and made freely available by the International Argo618
Program and the national programs that contribute to it (http://www.argo.ucsd.edu,619
http://argo.jcommops.org). The Argo Program is part of the Global Ocean620
Observing System.621
Authors would like to thank to Francesc Torres Torres and Nuria Duffo Ubeda622
from the Universitat Polit´ecnica de Catalunya for their useful insights in the623
system performance assessment.624
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Table 1
Spatial bias reduction (in percent) of the ascending and descending OTT in the
AF-FOV using both versions of NS with respect to the nominal reconstruction.
NSv1 NSv2
X-pol Y-pol X-pol Y-pol
ASC 19.1 22.5 19.7 23.4
DES 15.3 17.1 16.5 18.5
Table 2
Definition of the study regions where the statistics of the comparison between SSS
retrievals and Argo estimates are provided.
Zone Description Latitude Longitude
Global Tropics and mid-latitudes [60S-60N] All
Tropics Tropics [30S-30N] All
N. Atlantic North-Atlantic region [15N-37N] [50W,0]
N. Pacific A region of the North Pacific [45N-60N] [170E,140W]
S. Ocean Southern Ocean [60S-40S] All
31
Fig. 1. SMOS brightness temperatures over southern Europe using the nominal
processing (after applying a Blackman window). The sidelobes and ripples caused
by the RFI sources can be appreciated also over the Mediterranean Sea, very far
away from the sources locations.
Fig. 2. Scheme to illustrate how the NSv2 algorithm works.
32
(a) Original (b) NSv1 (c) NSv2
Fig. 3. Zoom-in view of the oversampled brightness temperature image (without
Blackman window) in Fig. 4 over the RFI source located around the position
(ξ, η) = (0.3, 0.2). Subgrids where sampling the oversampled image are indicated
by the white stars: (a) the original regular hexagonal subgrid (which leads to the
same values of brightness temperatures as the original image), (b) nodal subgrid
from NSv1 and (c) selection of nodal points from NSv2.
(a) Nominal (b) NSv1 (c) NSv2
Fig. 4. Reconstructed brightness temperatures in the fundamental hexagon of a
snapshot over North Africa using the three nodal subgrids in Fig. 3. The reduction
of the sidelobe levels and the ripples can be clearly appreciated when using nodal
sampling.
33
(a) Nominal (b) NSv1 (c) NSv2
Fig. 5. Brightness temperatures in the fundamental hexagon of a scene over the
Atlantic Ocean strongly contaminated by an RFI produced by a ship reconstructed
using the (a) nominal, (b) NSv1 and (c) NSv2 image reconstruction approaches.
(a) Nominal (b) NSv1 (c) NSv2
Fig. 6. Masks of pixels with brightness temperatures corresponding to non-natural
emission (negative TB values and higher than 350 K) for images in Fig. 5. A clear
reduction of the RFI source spatial extension is observed in NSv2 TB with respect
to NSv1 TB.
34
(a) Nominal (b) Cut along ξ = 0 (c) NSv2
Fig. 7. Left: Radiometric resolution [K] estimated using ocean scenes over the South
Central Pacific Ocean for nominal processing. Middle: comparison of estimated ra-
diometric resolution for nominal TBs (dashed blue line) with the theoretically com-
puted ones (dashed red line) along the cut ξ = 0 and estimated radiometric resolu-
tion for NSv2 TBs (solid line). A reduction factor of 2 is achieved on average in the
EAF-FOV using NSv2 with respect to the nominal. Right: Radiometric resolution
[K] estimated for NSv2 processing.
(a) Nominal (b) NSv1 (c) NSv2
Fig. 8. OTT (median of the difference between TB measurements and the theoreti-
cally modeled TB for the EAF-FOV) for dual epochs in Y-polarization. 10 ascending
passes over the South Central Pacific Ocean have been used to compute the OTT
(Meirold-Mautner et al., 2009; Tenerelli and Reul, 2010). A reduction of the spatial
bias is observed both for NSv1 and NSv2 with respect to the nominal case.
35
(a) X-polarization (b) Y-polarization
Fig. 9. Distributions of the standard deviation of the difference between the TB
measurements and the model for nominal (red), NSv1 (blue) and NSv2 (black)
approaches. A mean error reduction of approximately 0.8 K is obtained using NSv1
technique and 1.4 K using NSv2 with respect to the SMOS nominal processing. It
also must be pointed out the significant reduction in the tails of the distributions
with NSv2 approach.
36
(a) Nominal
(b) Nodal Sampling
Fig. 10. 9-day 0.25
spatial resolution map of the First Stokes TB difference (OT-
T-corrected SMOS TB minus TB model) for (a) nominal and (b) nodal sampling
image reconstruction approaches.
37
(a) Nominal
(b) Nodal Sampling
Fig. 11. 9-day 0.25
spatial resolution map of the standard deviation of the First
Stokes TB difference (OTT-corrected SMOS TB minus TB model) for (a) nominal
and (b) nodal sampling image reconstruction approaches.
38
(a)
(b)
Fig. 12. A posteriori SSS distributions per a given grid point over (a) the South
Pacific Ocean (clean zone) and (b) Indian Ocean close to the Arabian Sea (strongly
contaminated by RFI and LSC).
39
(a) Nominal
(b) Nodal Sampling
Fig. 13. Map of the standard deviation of the SSS distributions using (a) nominal
and (b) NSv2 TBs. SSS retrievals have been accumulated during 9 days over a 0.25
spatial grid.
40
(a) SSS nominal (b) SSS nodal sampling
(c) Anomaly nominal (d) Anomaly nodal sampling
Fig. 14. Map of SSS using a non-Bayesian retrieval with (a) nominal and (b) NSv2
brightness temperatures. Map of the anomalies between SSS retrievals and clima-
tology from WOA13 using (c) nominal and (d) NSv2 brightness temperatures. SSS
retrievals have been accumulated during 9 days over a 0.25
spatial grid.
41
(a) SSS nominal (b) SSS nodal sampling
(c) Anomaly nominal (d) Anomaly nodal sampling
Fig. 15. Map of the SSS retrievals using the Bayesian approach from (a) nominal
and (b) NSv2 brightness temperatures. Map of the anomalies between SSS retrievals
using the Bayesian approach and climatology from WOA13 using (c) nominal and
(d) NSv2 brightness temperatures. Binned maps have been generated using SSS
retrievals during 9 days over a 0.25
spatial grid.
42
0.001
0.01
1
psu
2
Degree
-1
Zonal spectra: NS (red) vs nominal (green)
Fig. 16. Zonal spectra for the nominal (green) and nodal sampled (red) SSS maps
obtained by the Bayesian approach. The approximate slope for the nominal spectra
(magenta) corresponds to k
1.1
and the one for the nodal sampled SSS spectra
(blue) corresponds to k
1.5
, which is more consistent with the geophysical signal.
43
0.001
0.01
1
psu
2
Degree
-1
Zonal spectra: NS (red) vs circular window (green)
Fig. 17. Zonal spectra for the circular window (green) and nodal sampled (red) SSS
maps obtained by the Bayesian approach. The approximate slope for the circular
SSS spectra (magenta) corresponds to k
1.2
and the one for the nodal sampled SSS
spectra (blue) corresponds to k
1.5
.
44
(a) Open ocean, mean (b) Open ocean, standard deviation
(c) Coastal regions, mean (d) Coastal regions, standard deviation
Fig. 18. Statistics of the comparison of SSS retrievals using both Bayesian and
non-Bayesian (indicated in the legend as mode) approaches with Argo SSS esti-
mates. Mean and standard deviation are shown for the 800 km closest to the shore
(top) and from 800 km offshore on (bottom), separately.
45
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