# Improving the Spatial Bias Correction Algorithm in SMOS Image Reconstruction Processor: Validation of Soil Moisture Retrievals With In Situ Data

Ali Khazaal, Philippe Richaume, François Cabot, Eric Anterrieu, Arnaud

Mialon, Yann H. Kerr

► To cite this version:

Ali Khazaal, Philippe Richaume, François Cabot, Eric Anterrieu, Arnaud Mialon, et al. Improving the Spatial Bias Correction Algorithm in SMOS Image Reconstruction Processor: Validation of Soil Moisture Retrievals With In Situ Data. IEEE Transactions on Geoscience and Remote Sensing, 2019, 57 (1), pp.277-290. 10.1109/TGRS.2018.2853619. hal-02619873

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# Improving the Spatial Bias Correction Algorithm in SMOS Image Reconstruction Processor: Validation of Soil Moisture Retrievals With *In Situ* Data

Ali Khazâal<sup>1,2</sup>, Philippe Richaume, François Cabot, Eric Anterrieu<sup>1,2</sup>, Member, IEEE,
Arnaud Mialon, and Yann H. Kerr, Fellow, IEEE

**Abstract**— SMOS is a space mission led by the European Space Agency and designed to provide global maps of Soil Moisture and Ocean salinity, two important geophysical parameters for understanding the water cycle variations and climate change. The SMOS payload is a 2-D interferometer operating at L-band that consists of 69 elementary antennas located along a Y-shaped structure. Important spatial biases persist in the retrieved brightness temperature (BT) images mainly due to the phenomenon of aliasing inside the field of view of SMOS but also due to the Gibbs oscillations near land/ocean transitions. To minimize these biases, a differential image reconstruction algorithm is used in the operational processor that reduces the contrast of the image to be retrieved. To do that, the contribution of a constant artificial temperature map is removed from the measurements prior to reconstruction and then added back after the reconstruction. In this paper, we show that strong residual biases are still present in the retrieved images. To reduce them, we propose to improve the bias correction algorithm by using a more realistic artificial temperature scene based on separating the land and ocean regions and assigning a constant temperature over land and a Fresnel BT model over the ocean. The artificial scene is also improved by means of representing each pixel by its water fraction percentage to smooth the land/ocean transitions. The improved algorithm is validated over the ocean by comparing the retrieved temperatures to a forward geophysical model but also over land by comparing the retrieved soil moisture to in situ measurements.

**Index Terms**— Gibbs oscillations, inverse problems, remote sensing, spatial biases, synthetic aperture imaging.

## I. INTRODUCTION

THE Soil Moisture and Ocean Salinity (SMOS) space mission is a European Space Agency (ESA) mission launched on November 2, 2009, in collaboration with the Centre National d'Etudes Spatiales, France, and the Centro para el Desarrollo Tecnológico Industrial, Spain. It has provided, since 2010, global maps of soil moisture (SM) over

land and sea surface salinity (SSS) over ocean [1], [2]. These two geophysical parameters are crucial for understanding and predicting our planet water cycle evolution. The required accuracy for SM retrieval is about 0.04-m³/m³ volumetric soil moisture with a three-day revisit time and a spatial resolution of 40 Km. For SSS, the required accuracy is 0.1 practical salinity unit (psu) averaged over 10 to 30 days for a spatial resolution of 200 km. The instrument on board SMOS is a 2-D interferometric radiometer operating at L-band at the central frequency of 1.413 GHz [3]. It is a Y-shaped structure with three arms and a central part (hub) carrying a total of 69 equally spaced antennas. The measurements provided by SMOS are called visibilities and are obtained by cross correlating the signals received by every possible pairs of antennas/receivers. These visibilities are then used to reconstruct the brightness temperatures (BTs) that are then used to retrieve SM and SSS.

At L-band, the sensitivity of BT to SSS is very low and ranges from 0.8 to 0.2 K/psu depending on the sea surface temperature, the incidence angle, and the polarization [4]. Therefore, it is crucial to reduce the biases in the retrieved BT maps in order to meet the SMOS requirements. The origin of these biases has been studied in [5] and found to be scene dependent and coming essentially from both the aliasing phenomenon in the field of view (FOV) of SMOS and from the Gibbs oscillations near land/ocean transitions. To correct these biases, the SMOS level 1 operational processor (L1op) uses a differential image reconstruction algorithm in order to reduce the abrupt transitions in the retrieved images [5], [6]. The idea is to introduce an artificial scene as close as possible to the observed one and to subtract its visibilities from the measurements suppressing the transitions and, thus, retrieving a differential BT. The scene is added back afterward. For operational purposes, the L1op currently uses a basic approach (the so-called Gibbs 1) that considers the artificial scene as constant over the entire earth's surface.

In this paper, we will show that important residual biases still persist in the retrieved images. To reduce them, we propose to improve the image reconstruction algorithm by using a more complex bias correction algorithm (so-called Gibbs 2). The main idea is to calculate a realistic artificial scene by separating the land and ocean masses and assigning a constant temperature value over land and a Fresnel BT model over

Manuscript received February 21, 2018; revised May 9, 2018; accepted June 15, 2018. This work was supported by ESA. (Corresponding author: Ali Khazâal.)

The authors are with the Centre d'Études Spatiales de la BIOsphère, Université de Toulouse, CNRS, CNES, IRD, INRA, 31400 Toulouse, France (e-mail: ali.khazaal@cesbio.cnes.fr; philippe.richaume@cesbio.cnes.fr; francois.cabot@cesbio.cnes.fr; eric.anterrieu@cesbio.cnes.fr; arnaud.mialon@cesbio.cnes.fr; yann.kerr@cesbio.cnes.fr).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2018.2853619
<figure><img src="image_2.png" /><figcaption>Fig. 1. (a) Star-shaped frequency coverage H of SMOS in the Fourier domain. (b) FOV of SMOS. The cyan circle corresponds to the entire FOV and the blue ellipsoid corresponds to the earth, as seen by the instrument. The region between the earth's horizon and the unit circle corresponds to the sky. The dot circles correspond to the six aliases. The red hexagon corresponds to the reconstructed FOV limited by the spacing between the antennas. The dark red region corresponds to the EAFFOV.</figcaption></figure>

ocean. To smooth out even more the land/ocean transition, we also propose to represent each pixel by its percentage of water fraction. A comparison between the two algorithms is shown first at BT level over the ocean by comparing the retrieved BT to a forward geophysical model and, then, at soil moisture level over land by comparing the retrieved SM to in situ measurements.

## II. ORIGIN OF SPATIAL BIASES IN SMOS IMAGES

SMOS BT suffers from important spatial biases that exceed the accuracy requirements for SM and SSS retrievals. The origin of these biases has been identified as scene dependent and is mainly due to two phenomena: Gibbs oscillations and aliasing [5]. For the first phenomenon, we recall that each correlation between two receivers $A_k$ and $A_l$, is associated with a spatial frequency ($u_{kl}$, $v_{kl}$) proportional to the baseline distance $d_{kl}$ between $A_k$ and $A_l$. For the 69 antennas of SMOS, the spatial frequencies ($u, v$) = {($u_{kl}$, $v_{kl}$); $k, l$ = 1...69} are confined to a limited star-shaped frequency coverage $H$ defined over a hexagonally sampled grid, as shown in Fig. 1 [7]. The sharp cutoff frequency beyond the star (i.e., for high frequencies) leads to the Gibbs oscillations near the land/ocean transitions in the retrieved images.

Regarding the aliasing, the spacing used between the SMOS antennas ($d$ = 0.875λ, where λ = 21 cm is the wavelength of observation) exceeds the limit required to eliminate the aliasing [the limit is (λ/√3)] according to the Shannon-Nyquist sampling criterion leading to earth and sky aliasing in the FOV, as shown in Fig. 1(b). In fact, the FOV seen by each of the 69 antennas is the entire half-space (i.e., unit circle) in front of the satellite composed by the earth ellipsoid and the sky. This FOV is defined in the director cosine coordinates: $\xi$ = $\sin\theta$ cos $\phi$, $\eta$ = $\sin\theta$ sin $\phi$, with ($\theta$, $\phi$) as the traditional spherical coordinates. The spacing chosen between the antennas leads to a reconstructed FOV that has the form of a hexagon inside the unit circle and suffers from earth and sky aliasing. The regions with only sky aliasing are kept since the sky temperature is well known and its impact can be accounted for. However, the regions with earth aliasing are discarded since the earth temperature is unknown, and thus, the retrieved

temperatures are in this case erroneous. As a consequence, only an extended alias-free FOV region (EAFFOV) is used to retrieve SM and SSS. A consequence of this phenomenon is that the aliased regions do contribute to the SMOS visibilities but cannot be considered in the image reconstruction algorithm leading to a scene dependent bias if their impact is not properly removed.

## III. IMAGE RECONSTRUCTION ALGORITHM

SMOS provides cross-correlation measurements or visibilities, $V$, between each pair of antennas/receivers that are used to retrieve the BT images $T$ of the earth's surface. It is a fully polarimetric instrument as described in [8]. Each antenna operates in two linear orthogonal polarizations and is equipped with two ports: a horizontal port $X$ perpendicular to the orbital plane and a second port $Y$ tilted by an angle of 32° from the horizontal [9]. Three sets of visibilities are measured every 2.4 s, namely $V^X$, $V^Y$, and $V^{XY}$ in three different polarizations $X$, $Y$, and $XY$, respectively. These measurements are used to retrieve four BT maps, namely $T^X$, $T^Y$, and $T^{XY} = T^3 + j T^4$, where $2T^3$ and $2T^4$ correspond, respectively, to the third and fourth Stokes parameters [10]. For simplification reasons, the image reconstruction algorithm is recalled here without referring to the different polarizations. Indeed, the relationship between the visibilities, $V$, defined in the data space $F$, and the BT to retrieve $T$, defined in the object space $E$, is linear and given by

$$V(u, v) = G(u, v, \xi, \eta) \times T(\xi, \eta) \quad (1)$$

where $(u, v) = \{(u_{kl}, v_{kl})$; $k, l$ = 1...69} are the frequencies inside the star-shaped frequency coverage $H$ and $G(u, v, \xi, \eta)$ is the modeling operator describing the relationship between the temperature to retrieve $T$ and the visibilities $V$.

It has been shown, in [11], that (1) is ill-posed with an infinite number of solutions for $T$. A physical regularization was also proposed in [11], which takes advantage of the limited frequency coverage inside the star $H$ to do the reconstruction in the Fourier domain. Equation (1) is in this case equivalent to

$$V = J \times \hat{T} \quad (2)$$

where $\hat{T}$ is the Fourier components of $T$ inside $H$ and $J = G_h U^* Z$ is a new operator defined from the data space $F$ into the Fourier domain $H$, with $G_h$ is the subset of $G$ inside the hexagon, $U$ is the Fourier transform operator, and $Z$ is the zero-padding operator beyond $H$. This new linear problem is well posed, thanks to the redundancies of SMOS, where different visibilities are associated with the same frequency. Thus, the number of visibilities is always larger than the number of spatial frequencies inside $H$, and the $J$ matrix is in this case well-conditioned. As a consequence, a unique solution $\hat{T}_r$ is obtained (subscript $r$ means retrieved)

$$\hat{T}_r = J^+ V \quad (3)$$

where $J^+$ is the Moore-Penrose pseudoinverse of $J$. Finally, the retrieved temperature $T_r$ is obtained by

$$T_r = U^* Z \hat{T}_r = U^* Z J^+ V. \quad (4)$$
A preliminary correction of the Gibbs oscillations due to the sharp frequency cutoff in the frequency coverage is to apply a Blackman apodization window function W to $T_r$ [12]

$$ \mathcal{T}_r = \mathbf{U}^* \hat{\mathbf{W}} \mathbf{U} \mathcal{T}_r \quad (5) $$

where $\hat{\mathbf{W}}$ is the Fourier transform of W.

## IV. SPATIAL BIAS CORRECTION ALGORITHM

As explained in Section II, the spatial biases are mainly due to the aliasing in the reconstructed FOV but also to the Gibbs oscillations due to the sharp transition beyond the limited frequency coverage. To reduce the impact of the aliased region, their contribution to the visibilities should be removed prior to inversion. Also, to reduce the Gibbs oscillations, the sharp land/ocean transition should be smoothen as much as possible [13]. One way to do both things is to apply a differential image reconstruction algorithm to the SMOS data. The idea is to discard from the visibilities V, the contribution $\tilde{V}$ of an artificial scene $\tilde{T}$, as close as possible to the observed one T. By that, we obtain a differential system to solve $\delta V = \mathbf{G}\delta T$ with $\delta V = V - \tilde{V}$ and $\delta T = T - \tilde{T}$, where $\tilde{V} = \mathbf{G}\tilde{T}$. The above-described image reconstruction algorithm is applied to this differential system where (2) becomes

$$ \delta V = \mathbf{J}^+ \times \delta \tilde{T} \quad (6) $$

and yielding to the following solution:

$$ \delta T_r = \mathbf{U}^* \mathbf{Z} \mathbf{J}^+ (V - \tilde{V}). \quad (7) $$

Finally, the part of $\tilde{T}$ inside the hexagon, $\tilde{T}_h$, is added back to the solution thus obtained so that (4) becomes

$$ T_r = \mathbf{U}^* \mathbf{Z} \mathbf{J}^+ (V - \tilde{V}) + \tilde{T}_h. \quad (8) $$

The remaining point now is how to calculate or simulate $\tilde{T}$ for each snapshot. For that, two approaches are presented here: 1) a basic approach (namely, Gibbs 1) that is currently implemented in the L1op (version 620) and 2) a more complex approach (namely, Gibbs 2) but yielding to a more realistic artificial scene.

### A. Gibbs 1 Algorithm

To calculate $\tilde{T}$, the Gibbs 1 algorithm considers a constant BT over the entire earth ($t_E$) calculated by setting the artificial visibility associated with the zero frequency, $\tilde{V}(0, 0) = \mathbf{G}(0, 0, \xi, \eta) \times t_E$, as equal to the measured one $V(0, 0)$

$$ t_E = \frac{V(0, 0)}{\sum_{\xi, \eta} \mathbf{G}(0, 0, \xi, \eta)}. \quad (9) $$

This solution has the advantage of being simple and operational. However, important residuals still persist in the retrieved images as we will show hereafter, hence, the necessity of improving the algorithm.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Fresnel model $\mathcal{F}$ over ocean: variation of the BT (in Kelvin) with the incidence angle $\theta_i$. For this model, the sea surface temperature and the SSS are set to 293 °K and 35 psu, respectively.</figcaption></figure>

### B. Gibbs 2 Algorithm

To improve the current bias correction algorithm, we propose to use a realistic artificial scene instead of the constant BT. Ideally, we should separate the land pixels from the ocean ones and use a geophysical BT model over each region.

Over land, it is rather difficult to obtain a simplified BT model that can be ingested into an operational processor. For that, we propose to use a constant BT $t_L$ over the land pixels.

Over ocean, we propose to use a so-called Fresnel model ($\mathcal{F}$) close to the one used in the operational level 2 (L2) ocean salinity retrieval algorithm [14]. For this model, we calculate the dielectric constant $\epsilon$ using the model defined by Klein and Swift [15]. Then, the Fresnel reflection coefficients are calculated at the earth's surface plane in H and V polarizations using the following equations:

$$ R_H = \frac{\cos \theta_i - \sqrt{\epsilon - \sin^2 \theta_i}}{\cos \theta_i + \sqrt{\epsilon - \sin^2 \theta_i}}, \quad R_V = \frac{\epsilon \cos \theta_i - \sqrt{\epsilon - \sin^2 \theta_i}}{\epsilon \cos \theta_i + \sqrt{\epsilon - \sin^2 \theta_i}} $$

where $\theta_i$ is the incidence angle. The surface BTs are then calculated as:

$$ \mathcal{F}_H = \mathbf{sst} \times (1 - |R_H|^2), \quad \mathcal{F}_V = \mathbf{sst} \times (1 - |R_V|^2), $$

where $\mathbf{sst}$ is the sea surface temperature fixed at 293 °K. Fig. 2 shows the BT of this model ($\mathcal{F}_H, \mathcal{F}_V$) as a function of the incidence angle $\theta_i$ in H and V polarizations. Finally, a geometric rotation angle is applied in order to calculate the BT of this model ($\mathcal{F}_X, \mathcal{F}_Y, \mathcal{F}_3$) at antenna level in X, Y, and XY polarizations [8]. We indicate that the model is equal to 0 for the imaginary part in XY polarization: ($\mathcal{F}_4 = 0$).

Once the ocean model $\mathcal{F}$ is calculated, we propose to multiply it by a constant value $t_O$ since the average BT over the ocean differs as the satellite moves. This constant can be considered as a scaling or an amplification factor.

Furthermore, in order to smoothen the land/ocean transition in the artificial scene, we also propose to characterize each pixel over the earth by its percentage of water content computed from the United States Geological Survey 1-km land/sea mask. For that, we will use a global water fraction mask (WF), as shown in Fig. 3, that has a range value going from 0 describing a 100% land pixel up to 200 for a 100% ocean pixel.
<figure><img src="image_4.png" /><figcaption>Fig. 3. Global water fraction content as used in the L1OP: “0” corresponds to 100% land (or ice) pixels and “200” corresponds to 100% ocean pixels.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Example of the artificial scene used in (Left) Gibbs 1 and (Right) Gibbs 2 bias correction algorithms.</figcaption></figure>

Using the proposed improvements, the artificial scene $\tilde{T}$ is modeled as follows:

$$ \tilde{T} = \frac{t_L \times [200 - \mathbf{WF}(\xi, \eta)] + t_O \times \mathbf{WF}(\xi, \eta) \times \mathcal{F}(\xi, \eta)}{200} \quad (10) $$

The two constants $(t_L, t_O)$ are estimated by minimizing the difference between the measured and the artificial visibilities

$$ \min_{t_L, t_O} \|V - \tilde{V}(t_L, t_O)\|_F^2. \quad (11) $$

Replacing $\tilde{V}$ by $G\tilde{T}$ and $\tilde{T}$ by its expression in (10), we obtain the following optimization problem

$$ \min_{t_L, t_O} \|V - (t_L V_L + t_O V_O)\|_F^2 \quad (12) $$

where $V_L = ((G \times (200 - WF))/200)$ is the contribution of a 1 K scene over the land pixels and $V_O = ((G \times WF \times F)/200)$ is the contribution of the Fresnel model over the ocean pixels. This least-square optimization problem has an explicit solution, as shown in [16, p. 237]

$$ [t_L \quad t_O] = (V_L \quad V_O)^+ \times V. \quad (13) $$

Finally, Fig. 4 shows an example of the artificial scene obtained using both algorithms. We clearly notice that the artificial scene in the Gibbs 2 algorithm is much more realistic than the constant value used in the Gibbs 1 algorithm. In Sections V and VI, we will compare SMOS operational data obtained using the Gibbs 1 algorithm to data processed using the proposed Gibbs 2 algorithm.

<caption>TABLE I</caption>

MEAN AND RMS (OTT/$\sigma_{OTT}$) OF THE OTT USING THE GIBBS 1 AND THE GIBBS 2 BIAS CORRECTION ALGORITHMS FOR THE L1A DESCENDING ORBIT OF MAY 2, 2011. COLOR HIGHLIGHTS THE DIFFERENCES IN THE STATISTICS BETWEEN GIBBS 1 AND GIBBS 2: RED FOR WORSE, GREEN FOR BETTER, AND BLACK FOR NOT SIGNIFICANTLY DIFFERENT

<table><thead><tr><th></th><th>Gibbs 1</th><th>Gibbs 2</th></tr></thead><tbody><tr><th>T<sup>x</sup></th><td>1.9 / 2.3</td><td>1.8 / 2.2</td></tr><tr><th>T<sup>y</sup></th><td><span style="color:red;">0.3</span> / 2.8</td><td><span style="color:red;">1.3</span> / 1.9</td></tr><tr><th>T<sup>3</sup></th><td><span style="color:red;">0.6</span> / 2.5</td><td><span style="color:green;">0.2</span> / 1.9</td></tr><tr><th>T<sup>4</sup></th><td>0.3 / 1.5</td><td>0.3 / 1.2</td></tr></tbody></table>

## V. RESULTS AT BT LEVEL

All the results presented hereafter are obtained using the SMOS real interferometric measurements (L1A data) taken from the operational data processing ground segment (DPGS) of ESA. They are then processed using the above-described two Gibbs algorithms and the results are shown in full-polarization mode. The two algorithms are here compared at BT level through two tests: 1) the Ocean Target Transformation (OTT) [17] and 2) the global averaged retrieved BT images over the entire ocean regions (Dwell lines).

### A. OTT

The OTT is calculated in the SMOS L2 ocean salinity processor to correct the retrieved BT maps from the residual biases [2]. It is obtained over the open ocean where the FOV of SMOS covers 100% of ocean for N images taken for a latitude range between 5 °N and 45 °S. Over this area, the OTT is defined as the average of the difference between the retrieved BT maps, $T_{r,i}$ ($i$ = 1 to N), and the BT of a reference ocean model, $T_{m,i}$

$$ \text{OTT} = \frac{\sum (T_{r,i} - T_{m,i})}{N}. $$

The ocean model used here is similar to the one used in the operational L2 ocean salinity retrieval algorithm and described in [14]. For this model, the specular emission is computed using the dielectric model of [15] and the SSS is based upon the World Ocean Atlas 2009 [18].

Fig. 5 shows the OTT obtained, using both algorithms, for an L1A orbit over the Pacific ocean taken at May 2, 2011 at 02:40:43 h (UTC time). Table I shows its mean OTT and its root mean square (rms) $\sigma_{OTT}$ in both cases. From
<figure><img src="image_6.png" /><figcaption>Fig. 5. OTT of the descending orbit of May 2, 2011, taken at 02:04:43 h (UTC time) using (a) Gibbs 1 and (b) Gibbs 2 algorithms.</figcaption></figure>

these results, we notice an improvement of the residual error
near the borders of the EAFFOV using the Gibbs 2 algorithm
mainly for $T^X$ and $T^Y$. We also notice that the distribution
of the residual error is more homogeneous using the Gibbs 2
algorithm as compared to Gibbs 1 mainly for $T^Y$, $T^3$, and $T^4$.
This is confirmed by the values of Table I, where we see a
decrease of 0.9, 0.6, and 0.3 K of the rms, respectively, for
$T^Y$, $T^3$, and $T^4$.

B. Dwell Lines

As SMOS progresses in its orbit, a given fixed point on
earth defined by its latitude and longitude appears in several
images within the EAFFOV at different locations in the (ξ, η)
grid and for different incidence angles going from 0° to 60°.
We refer to these locations as dwell lines, and they are nearly
parallel to the subsatellite track in the instrument FOV.

To globally compare the two Gibbs algorithms, we calculate
now the dwell lines averaged difference BT maps over the
ocean between the retrieved BT by SMOS and the forward
ocean BT model described previously. For that, 10 days of
L1A ascending orbits from May 1 to 10, 2011 are processed
using the two Gibbs algorithms. A single OTT (one for
each algorithm) is used to correct the retrieved maps $T_r$
in order to reduce, furthermore, the residual bias. For each
image, the retrieved and corrected BT, $T_r$ - OTT, is compared
with the BT of the ocean forward model, $T_m$, to obtain the
error of reconstruction $\Delta T_r = T_r - \text{OTT} - T_m$. Then, $\Delta T_r$
is projected on the earth's surface over the discrete global
grid (DGG) latitude and longitude pixels of the Icosahedral
Snyder Equal Area grid, which is used in ESA processing at
DPGS. Finally, the several BT differences $\Delta T_r$ of each dwell
line and corresponding to a specific DGG pixel are averaged.

Figs. 6 and 7 show the global dwell lines averaged error
of reconstruction for Gibbs 1 and Gibbs 2, respectively. First,
we must indicate that the ocean model BTs are not valid over
the ocean regions covered by ice (near Antarctica, Alaska,

Canada, and the Arctic ocean). Therefore, the corresponding
$\Delta T_r$ values are not valid, and they are not considered in the
analysis of the results. Also, the retrieved BTs for some ocean
regions affected by nearby radio frequency interference (RFI)
over land (mainly close to southeast Asia) are also not valid
and not considered in the analysis of the results. Apart from
that, Fig. 6 shows important spatial biases in all polarizations
using the Gibbs 1 algorithm, which are reduced using Gibbs 2,
as shown in Fig. 7. This reduction of the bias is mainly seen
in $T_X$ and $T_Y$ over the Pacific ocean around the American
continent and over the Atlantic ocean between the American
and the African continents and also around Australia. The
phenomenon of land-sea contamination near the coasts is
reduced in $T_X$, $T_Y$, and $T_4$ but not totally corrected, which is
quite normal since it is due to a combination of problems, such
as calibration of the instrument, uncertainties in the antenna
patterns, floor error noise, and image reconstruction errors.
Therefore, improving the image reconstruction algorithm will
not solve this problem completely but will certainly reduce it.
We also precise that we do not dispose off an ocean model
for $T_4$, and that what, we retrieve is the residual radiometric
noise. Therefore, the closer $T_4$ is to zero, as for Gibbs 2, the
better.

To illustrate even more the comparison between the two
algorithms, we show in Fig. 8 the probability density
function (pdf) of the maps in Figs. 6 and 7 after filtering the
regions covered by ice and those contaminated by RFI. The
pdf curves show clearly the advantage of Gibbs 2 with respect
to Gibbs 1. We notice that the pdf curve for Gibbs 2 has a more
symmetric shape and is much closer to a normal distribution
than Gibbs 1. Table II shows the percentage of ocean pixels
having a bias in the range of [-1 1] and [-0.5 0.5] K, which
confirms that Gibbs 2 outperforms Gibbs 1.

VI. VALIDATION AT SOIL MOISTURE LEVEL

After comparing the Gibbs 2 algorithm to the operational
one (Gibbs 1) at BT level (L1), it is quite natural now to com-
<figure><img src="image_7.png" /><figcaption>Fig. 6. Global-dwell lines averaged error of reconstruction $\Delta T_r$ (in Kelvin) between the SMOS retrieved temperatures corrected by the OTT ($T_r - OTT$) and the ocean model ($T_m$) for the period of May 1 to 10, 2011, using the Gibbs 1 bias correction algorithm in X (top-left), Y (top-right), and XY (bottom-left for the real part and bottom-right for the imaginary part) polarizations.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Global dwell-lines averaged error of reconstruction $\Delta T_r$ (in Kelvin) between the SMOS retrieved temperatures corrected by the OTT ($T_r - OTT$) and the ocean model ($T_m$) for the period of May 1 to 10, 2011, using the Gibbs 2 bias correction algorithm in X (top-left), Y (top-right), and XY (bottom-left for the real part and bottom-right for the imaginary part) polarizations.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Comparison of the pdf of the dwell lines $\Delta T_r$ for the period of May 1 to 10, 2011 between the Gibbs 1 (blue curve) and the Gibbs 2 (red curve) bias correction algorithms. (Left to Right) pdf in X, Y, and XY polarizations (real and imaginary parts), respectively.</figcaption></figure>

pare both algorithms at SM level (L2) since any changes on the L1 data will have a significant impact on the L2 retrievals. In this section, we describe briefly the L2 SM operational

algorithm, and then, we compare the SM retrievals using the operational (OP) and the Gibbs 2 (G2) algorithms to in situ SM measurements over Australia.
<caption>TABLE II</caption>

COMPARISON OF THE PERCENTAGE OF PIXELS OVER OCEAN WITH A BIAS RANGE OF [-1] K AND [-0.5 0.5] K USING THE GIBBS 1 AND THE GIBBS 2 BIAS CORRECTION ALGORITHMS OF THE DWELL LINES $\Delta T_r$ FOR THE PERIOD OF MAY 1 TO 10, 2011. COLOR HIGHLIGHTS THE DIFFERENCES IN THE STATISTICS BETWEEN GIBBS 1 AND GIBBS 2: RED FOR WORSE, GREEN FOR BETTER, AND BLACK FOR NOT SIGNIFICANTLY DIFFERENT

<table><thead><tr><th rowspan="2"></th><th colspan="2">bias range [-1 1](K)</th><th colspan="2">bias range [-0.5 0.5](K)</th></tr><tr><th>Gibbs 1</th><th>Gibbs 2</th><th>Gibbs 1</th><th>Gibbs 2</th></tr></thead><tbody><tr><th>ΔT<sup>x</sup></th><td>77,9</td><td>80,7</td><td>53,5</td><td>57,6</td></tr><tr><th>ΔT<sup>y</sup></th><td>73,8</td><td>83,6</td><td>51,9</td><td>63,8</td></tr><tr><th>ΔT<sup>3</sup></th><td>86,4</td><td>85,7</td><td>71,5</td><td>69,9</td></tr><tr><th>ΔT<sup>4</sup></th><td>94,8</td><td>94,9</td><td>83,7</td><td>85,5</td></tr></tbody></table>

## A. SMOS L2 Soil Moisture Retrieval Algorithm

The SMOS L2 SM retrieval algorithm is extensively detailed in [19] and [20]. It uses, as inputs, SMOS L1C products (i.e., retrieved BT) that consist in profiles of several BTs observed at different incidence angles and polarizations available at given grid points (DGG) on the earth. These BT come from the sequence of consecutive snapshots that include these DGGs. The retrieval part of the algorithm consists in fitting modeled incidence angle profiles of BTs to those observed by SMOS and account for the possible mixture of various type of emitters: soil covered by low vegetation or forest, water bodies, and other specific surfaces, such as frozen soils, barren surfaces, or snow-covered areas.

The BT models are mainly the function of the two retrieved parameters, the SM and the vegetation opacity (Tau) but also to other fixed parameters, such as soil and vegetation temperatures and single-scattering albedo. The optimization of SM and vegetation opacity is carried on following the Bayesian approach aiming to minimize the sum of squared differences, J, between the simulated model and the radiometer measurements normalized by the measurements uncertainty along the collection of N incidence angles and polarization observed by SMOS. It is given by

$$J = \frac{1}{N - N_R} \sum_{k}^{N} (r_k^{p_k})^2 \quad (14)$$

with $N_R$ is the number of retrieved parameters, and $r_k^{p_k}$ is defined for the observed and modeled BTs at incidence angle $\theta_k$, polarization $p_k$ (X or Y), and radiometric accuracy $RA_{\text{SMOS}}^{p_k}$ by

$$r_k^{p_k} = \frac{\text{BT}_{\text{SMOS}}^{p_k}(\theta_k) - \text{BT}_{\text{model}}^{p_k}(\theta_k; \text{SM, tau...})}{\text{RA}_{\text{SMOS}}^{p_k}(\theta_k)}. \quad (15)$$

In (14), J is actually a $\chi^2$ normalized by the number of degrees of freedom $N - N_R$. In perfect conditions, when the retrieved model and the observations are in good agreement within the SMOS observation uncertainties ($RA_{\text{SMOS}}^{p_k}$), J is expected to follow a normalized chi-square distribution with $N - 2$ degrees of freedom as we are retrieving two parameters, SM and Tau ($N_R = 2$). When N is high enough ($> 50$), which

is mostly the case in the L2 retrievals, this normalized $\chi^2$ tends to a normal distribution $N[1, (2/\sqrt{N-2})]$.

When J is lower than $1 + (4/\sqrt{N-2})$, then the goodness of fit with 95% confidence is reached and implies that the residuals $r_k^{p_k}$ are expected to follow a standard normal distribution $N(0, 1)$. In particular, the mean residual $\bar{r}$ along the N samples shall be close to 0. This condition still holds when considering separately the two half subpopulations of X and Y polarizations residuals, and we expect $\bar{r} \approx \bar{r}^X \approx \bar{r}^Y \approx 0$ though with less accuracy for the two latter as they are computed from a smaller population size (the half).

When J is significantly higher than $1 + (4/\sqrt{N-2})$, then the goodness of fit with 95% confidence fails. A disagreement between the model and the observations can be suspected with a possible origin coming from systematic biases and/or because of too optimistic observation uncertainties ($RA_{\text{SMOS}}^{p_k}$).

## B. Results

We have shown earlier that the Gibbs 2 algorithm reduces spatial biases at BT level over ocean. Its impact over land on L2 SM retrieval is here analyzed from a radiometry change perspective. We will investigate how it changes the mean residuals $\bar{r}$, the mean residuals per X and Y polarizations ($\bar{r}^X$ and $\bar{r}^Y$), and the goodness of fit, compared to the operational L2 SM products version 620 (OP) that uses the Gibbs 1 algorithm. For this purpose, one year (2011) of L1A products over Australia was processed using the Gibbs 2 algorithm and the associated L1C was generated and fed to the L2 SM processor. Australia was chosen as it is one of the cleanest environments in term of RFIs and its homogeneous surface corresponds to a nominal case for the retrieval algorithms, hence minimizing the modeling errors. These two important qualities allowed us a clear analysis of the radiometry change impact on the L2 retrievals. Besides, the Australian OZNET in situ measurement network located close to Melbourne offers the opportunity to conduct the analysis up to SM retrievals and compare G2 and OP SM against in situ SM time series of 17 validation sites taken from Yanco and Kyeamba stations [21], [22]. The results of the analysis of G2 versus OP are presented from two perspectives: radiometry versus model agreement and retrieved SM versus ground measurements.

Since the beginning of the SMOS mission, the normalized $\chi^2$ of the retrievals is significantly higher than the theoretical expectation ($< 1 + (4/\sqrt{N-2})$), as shown in Fig. 9 (left column) that displays a typical example of OP SMOS profile of BTs (in colors) and the associated retrieved model (in black) and the residuals. While $\bar{r} = 1/2(\bar{r}^X + \bar{r}^Y)$ is close to 0, it is the result of the two populations $\bar{r}^X$ and $\bar{r}^Y$ having nonnegligible magnitudes but of opposite sign that compensate each other. It indicates a quite important polarization disagreement between the model and the observations, which contributes to the reported high normalized $\chi^2$ value of 2.06. In this example, $N = 76$, the $\chi^2$ value is expected to be less than $\approx 1.47(= 1 + (4/\sqrt{76-2}))$ with 95% of probability. The origin could be a modeling error but obviously not (or not only) as, for the same retrieval, G2 significantly reduces $\chi^2$
<figure><img src="image_10.png" /><figcaption>Fig. 9. (Top) SMOS observed BT incidence angle profile in X (o) and Y (x) polarizations; the radiometric accuracy as error bars and the model fit (horizontal black line) at the end of the optimization. (Bottom) The associated $r_k^X$ and $r_k^Y$ normalized residuals with $\bar{r}, \bar{r}^X$, and $\bar{r}^Y$ are shown as black, blue, and red horizontal lines, respectively. Left column: the results obtained using the OP version. Right column: the results obtained using G2.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10. Mean residuals $\bar{r}$ (black dots), $\bar{r}^X$ (blue dots), and $\bar{r}^Y$ (red dots) of retrievals obtained at the DGGID 8173743 for the year 2011 expressed as function of the Xswath in time from (Left) standard OP 620 L1C and from (Right) Gibbs 2 L1C for (Top) descending and (Bottom) ascending products.</figcaption></figure>

from 2.06 down to 1.41 and the biases in ((X + Y/2), X, Y) polarizations from (−0.15, 0.5, −0.8) to (−0.02, 0.06, −0.1),

as shown in Fig. 9. Note that, in this example, G2 passes the criteria but marginally, suggesting either possible residual
<table><caption>TABLE III</caption><caption>RESIDUALS AND χ<sup>2</sup> MEAN STATISTICS PER SWATH FOR Descending Orbits USING 400-km BANDWIDTH FOR THE LEFT BAND (L600 km, L200 km), THE MIDDLE ONE (L200 km, R200 km), AND THE RIGHT ONE (R200 km, R600 km). COLORS HIGHLIGHT THE OP AND G2 STATISTICS WHEN THEY ARE SIGNIFICATIVELY DIFFERENT: WORSE IN RED, BETTER IN GREEN, AND EQUIVALENT IN BLACK. THE TWO SCORE ROWS REPORT THE NUMBERS OF BETTER, EQUIVALENT, AND WORSE STATISTICS FOR OP AND G2 WITH RESPECT TO THE OVERALL NUMBER. THE LAST ROW REPRESENTS THE RELATIVE χ<sup>2</sup> COMPUTED AS 100(1 - mean(χ<sup>2</sup><sub>OP</sub>/χ<sup>2</sup><sub>G2</sub>))</caption><thead><tr><th></th><th colspan="6">Descending Orbits</th></tr><tr><th></th><th colspan="3">Left Swath</th><th colspan="3">Mid Swath</th><th colspan="3">Right Swath</th></tr><tr><th>Polarization</th><th>All</th><th>X</th><th>Y</th><th>All</th><th>X</th><th>Y</th><th>All</th><th>X</th><th>Y</th></tr></thead><tbody><tr><td>mean <span style="text-decoration: overline;">r</span><sub>OP</sub></td><td>-0.09</td><td>0.00</td><td>-0.18</td><td>-0.03</td><td>0.01</td><td>-0.07</td><td><span style="text-decoration: overline;">0.01</span></td><td>0.69</td><td>-0.68</td></tr><tr><td>mean <span style="text-decoration: overline;">r</span><sub>G2</sub></td><td><span style="text-decoration: overline;">0.01</span></td><td>0.42</td><td>-0.41</td><td>-0.04</td><td><span style="text-decoration: overline;">-0.00</span></td><td>-0.07</td><td>0.03</td><td>0.09</td><td>-0.04</td></tr><tr><td>qmean σ<sub>OP</sub></td><td>1.47</td><td>1.58</td><td>1.34</td><td>1.47</td><td>1.49</td><td>1.42</td><td>1.64</td><td>1.40</td><td>1.52</td></tr><tr><td>qmean σ<sub>G2</sub></td><td>1.45</td><td><span style="text-decoration: overline;">1.32</span></td><td>1.42</td><td>1.44</td><td>1.45</td><td>1.42</td><td><span style="text-decoration: overline;">1.42</span></td><td>1.46</td><td>1.36</td></tr><tr><td>mean χ<sup>2</sup><sub>OP</sub></td><td colspan="3">1.65</td><td colspan="3">1.99</td><td colspan="3">2.26</td></tr><tr><td>mean χ<sup>2</sup><sub>G2</sub></td><td colspan="3">1.86</td><td colspan="3">2.00</td><td colspan="3">1.79</td></tr><tr><td>Score OP</td><td><span style="text-decoration: overline;">4/7</span></td><td>1/7</td><td>2/7</td><td><span style="text-decoration: overline;">0/7</span></td><td>6/7</td><td>1/7</td><td><span style="text-decoration: overline;">1/7</span></td><td>1/7</td><td>5/7</td></tr><tr><td>Score G2</td><td>2/7</td><td>1/7</td><td>4/7</td><td>1/7</td><td>6/7</td><td>0/7</td><td><span style="text-decoration: overline;">5/7</span></td><td>1/7</td><td>1/7</td></tr><tr><td>Relative χ<sup>2</sup></td><td colspan="3">9.3% ± σ</td><td colspan="3">0.4% ± σ</td><td colspan="3">-30.1% ± σ</td></tr><tr><td></td><td colspan="3">= 19.0%</td><td colspan="3">= 9.9%</td><td colspan="3">=31.6%</td></tr></tbody></table>

<table><caption>TABLE IV</caption><thead><tr><th colspan="10">Descending Orbits</th></tr><tr><th></th><th colspan="3">Left Swath</th><th colspan="3">Mid Swath</th><th colspan="3">Right Swath</th></tr><tr><th>Polarization</th><th>All</th><th>X</th><th>Y</th><th>All</th><th>X</th><th>Y</th><th>All</th><th>X</th><th>Y</th></tr></thead><tbody><tr><td>mean <span style="text-decoration: overline;">r</span><sub>OP</sub></td><td>-0.12</td><td>0.31</td><td>-0.54</td><td>-0.01</td><td>0.32</td><td>-0.33</td><td>0.05</td><td>0.51</td><td>-0.42</td></tr><tr><td>mean <span style="text-decoration: overline;">r</span><sub>G2</sub></td><td><span style="text-decoration: overline;">0.02</span></td><td>0.25</td><td>-0.21</td><td><span style="text-decoration: overline;">-0.00</span></td><td>0.00</td><td>-0.01</td><td><span style="text-decoration: overline;">-0.01</span></td><td>0.07</td><td><span style="text-decoration: overline;">-0.08</span></td></tr><tr><td>qmean σ<sub>OP</sub></td><td>1.55</td><td>1.48</td><td>1.45</td><td>1.44</td><td>1.41</td><td>1.38</td><td>1.54</td><td>1.37</td><td>1.52</td></tr><tr><td>qmean σ<sub>G2</sub></td><td><span style="text-decoration: overline;">1.38</span></td><td>1.39</td><td>1.32</td><td>1.43</td><td>1.39</td><td>1.45</td><td><span style="text-decoration: overline;">1.37</span></td><td>1.34</td><td>1.39</td></tr><tr><td>mean χ<sup>2</sup><sub>OP</sub></td><td colspan="3">1.82</td><td colspan="3">1.91</td><td colspan="3">1.91</td></tr><tr><td>mean χ<sup>2</sup><sub>G2</sub></td><td colspan="3">1.68</td><td colspan="3">1.93</td><td colspan="3">1.72</td></tr><tr><td>Score OP</td><td><span style="text-decoration: overline;">0/7</span></td><td>0/7</td><td>7/7</td><td><span style="text-decoration: overline;">1/7</span></td><td>3/7</td><td>4/7</td><td><span style="text-decoration: overline;">0/7</span></td><td>1/7</td><td>6/7</td></tr><tr><td>Score G2</td><td>7/7</td><td>0/7</td><td>0/7</td><td><span style="text-decoration: overline;">4/7</span></td><td>3/7</td><td>1/7</td><td><span style="text-decoration: overline;">6/7</span></td><td>1/7</td><td>0/7</td></tr><tr><td>Relative χ<sup>2</sup></td><td colspan="3">-9.3% ± σ</td><td colspan="3">1.1% ± σ</td><td colspan="3">-14.1% ± σ</td></tr><tr><td></td><td colspan="3">= 22.6%</td><td colspan="3">= 11.3%</td><td colspan="3">=20.3%</td></tr></tbody></table>

<table><caption>RESIDUALS AND χ<sup>2</sup> MEAN STATISTICS PER SWATH FOR Ascending Orbits USING 400-km BANDWIDTH FOR THE LEFT BAND (L600 km, L200 km), THE MIDDLE ONE (L200 km, R200 km), AND THE RIGHT ONE (R200 km, R600 km). COLORS HIGHLIGHT THE OP AND G2 STATISTICS WHEN THEY ARE SIGNIFICATIVELY DIFFERENT: WORSE IN RED, BETTER IN GREEN, AND EQUIVALENT IN BLACK. THE TWO SCORE ROWS REPORT THE NUMBERS OF BETTER, EQUIVALENT, AND WORSE STATISTICS FOR OP AND G2 WITH RESPECT TO THE OVERALL NUMBER. THE LAST ROW REPRESENTS THE RELATIVE χ<sup>2</sup> COMPUTED AS 100(1 - mean(χ<sup>2</sup><sub>OP</sub>/χ<sup>2</sup><sub>G2</sub>))</caption><thead><tr><th></th><th colspan="6">Ascending Orbits</th></tr><tr><th></th><th colspan="3">Left Swath</th><th colspan="3">Mid Swath</th><th colspan="3">Right Swath</th></tr><tr><th>Polarization</th><th>All</th><th>X</th><th>Y</th><th>All</th><th>X</th><th>Y</th><th>All</th><th>X</th><th>Y</th></tr></thead><tbody><tr><td>mean <span style="text-decoration: overline;">r</span><sub>OP</sub></td><td>-0.12</td><td>0.31</td><td>-0.54</td><td>-0.01</td><td>0.32</td><td>-0.33</td><td>0.05</td><td>0.51</td><td>-0.42</td></tr><tr><td>mean <span style="text-decoration: overline;">r</span><sub>G2</sub></td><td><span style="text-decoration: overline;">0.02</span></td><td>0.25</td><td>-0.21</td><td><span style="text-decoration: overline;">-0.00</span></td><td>0.00</td><td>-0.01</td><td><span style="text-decoration: overline;">-0.01</span></td><td>0.07</td><td><span style="text-decoration: overline;">-0.08</span></td></tr><tr><td>qmean σ<sub>OP</sub></td><td>1.55</td><td>1.48</td><td>1.45</td><td>1.44</td><td>1.41</td><td>1.38</td><td>1.54</td><td>1.37</td><td>1.52</td></tr><tr><td>qmean σ<sub>G2</sub></td><td><span style="text-decoration: overline;">1.38</span></td><td>1.39</td><td>1.32</td><td>1.43</td><td>1.39</td><td>1.45</td><td><span style="text-decoration: overline;">1.37</span></td><td>1.34</td><td>1.39</td></tr><tr><td>mean χ<sup>2</sup><sub>OP</sub></td><td colspan="3">1.82</td><td colspan="3">1.91</td><td colspan="3">1.91</td></tr><tr><td>mean χ<sup>2</sup><sub>G2</sub></td><td colspan="3">1.68</td><td colspan="3">1.93</td><td colspan="3">1.72</td></tr><tr><td>Score OP</td><td><span style="text-decoration: overline;">0/7</span></td><td>0/7</td><td>7/7</td><td><span style="text-decoration: overline;">1/7</span></td><td>3/7</td><td>4/7</td><td><span style="text-decoration: overline;">0/7</span></td><td>1/7</td><td>6/7</td></tr><tr><td>Score G2</td><td>7/7</td><td>0/7</td><td>0/7</td><td><span style="text-decoration: overline;">4/7</span></td><td>3/7</td><td>1/7</td><td><span style="text-decoration: overline;">6/7</span></td><td>1/7</td><td>0/7</td></tr><tr><td>Relative χ<sup>2</sup></td><td colspan="3">-9.3% ± σ</td><td colspan="3">1.1% ± σ</td><td colspan="3">-14.1% ± σ</td></tr><tr><td></td><td colspan="3">= 22.6%</td><td colspan="3">= 11.3%</td><td colspan="3">=20.3%</td></tr></tbody></table>

modeling errors or other image reconstruction errors different from the noise floor error corrected by the Gibbs 2.

The residuals ($\bar{r}$, $\bar{r}^X$, $\bar{r}^Y$) and normalized $\chi^2$ analysis have been also extended to the retrievals of the 17 sites for 2011. Fig. 10 shows the example of the Yanco Y1 site (DGG 8173743, Longitude 145.8°E, Latitude 34.7°S), where the mean residual triplets (mR, mR<sub>X</sub>, mR<sub>Y</sub>) are shown, respectively, in black, blue, and red dots for all the successful retrievals obtained with either OP on the left or G2 on the right. These residuals are displayed as a function of the cross-track distance of the DGG to SMOS satellite subtracks (XSwath) within each L1C swath that observed the point. The perfect case would have displayed no dependence with the XSwath, no difference between black, blue and red dots, just a white noise mixture of them centered along the 0 y-axis line. However, this is not the case, and as shown in Fig. 10, the overall shape of the cloud of points has distinct features, especially for the operational products. These shapes are also observed for the 16 other DGGs with varying magnitudes and some variations but with similar aspects.

When analyzing the results of Fig. 10, the agreement between model and SMOS G2 radiometry is generally increased: the biases in X and Y polarizations are consequently reduced, especially for high XSwath. They become also more symmetrical ($\bar{r}^X = -\bar{r}^Y$). In the region close to the satellite subtrack, G2 and OP are more similar. There is a systematic exception for left-hand swath position (LXSwath > 200 km) of descending orbits (top right plot of Fig. 10), which is also observed for all others DGGs. In this area of the FOV, the G2 algorithm displays equivalently high or even higher X and Y polarization biases than OP even though the symmetrization of $\bar{r}^X$ and $\bar{r}^Y$ is, in general, still improved.

The overall numerical statistics of all the retrievals for the 17 validation sites are given in Tables III and IV for descending and ascending passes, respectively. The results are partitioned per orbit passes and per three 400-km wide portion of the swath (left, middle, and right) as suggested by Fig. 10. For the two outer bands of ascending orbits, almost all the statistics are improved with G2, whereas for descending orbits, the left part of the swath is degraded. The right part of the descending swath shows improvements similar to that of ascending orbits. The central part of the swath, as already mentioned, is more balanced between G2 and OP, especially for descending orbit with mostly equivalent statistics. Still, the ascending pass is more in favor of G2. Globally, the G2 radiometry change improves the agreement between the models and the observations. The degradation observed for the left side of the descending is not well understood but might be linked to a different land/sea mass in front of the instrument, letting the synthetic scene less suited to reduce the noise floor error. It might be an indication to also consider, similar to what is done over oceans, a more realistic scene over land. But such an approach is more difficult to implement than a constant temperature.

When compared with SM ground measurements, G2 has a positive to very positive impact depending on the orbit pass as shown in the Taylor diagrams [23] presented in Figs. 11 and 12 for the Yanco Y9 station. These diagrams are very convenient to summarize the standard deviation of the error statistics (STDE) between several estimates against one reference; in our case, SM OP and SM G2 minus the in situ SM. It represents, in polar coordinates, the decomposition of the STDE into the standard deviation of the time series (module), their correlations with the reference (angle) and
<figure><img src="image_12.png" /><figcaption>Fig. 11. (Top) Time series of retrieved SM for OP (red circle), G2 (blue square), and *in situ* Yanco Y9 station (black dots) for descending orbits. (Bottom right) Corresponding Taylor diagram comparing retrieved SM for OP (1) and G2 (2) against *in situ* station Y9. (Bottom left) Numerical statistics.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12. (Top) Time series of retrieved SM for OP (red circle), G2 (blue square), and *in situ* Yanco Y9 station (black dots) for ascending orbits. (Bottom right) Corresponding Taylor diagram comparing retrieved SM for OP (1) and G2 (2) against *in situ* station Y9. (Bottom left) Numerical statistics.</figcaption></figure>

biases (mean differences) as a color bar. The reference is always located on the x axis (correlation of 1 with itself) and 0 bias. The green concentric circles centered on the reference

give the STDE ranges. Closer the position of a series to the reference, better the agreement. For this Y9 case, the gain in correlation provided by G2 for the descending orbit is very
## Concatenated Retrieval OP and G2 versus CalVal Sites Insitu Time Series - Descending orbits

<figure><img src="image_14.png" /><figcaption>Fig. 13. Taylor diagram comparing retrieved SM time series for OP (1) and G2 (2) against all Yanco and Kyeamba concatenated *in situ* time series for descending orbits.</figcaption></figure>

## Concatenated Retrieval OP and G2 versus CalVal Sites Insitu Time Series - Ascending orbits

<figure><img src="image_15.png" /><figcaption>Fig. 14. Taylor diagram comparing retrieved SM time series for OP (1) and G2 (2) against all Yanco and Kyeamba concatenated *in situ* time series for ascending orbits.</figcaption></figure>

clear, whereas for ascending orbits, G2 and OP are more similar though slightly in favor of G2.

Finally, Tables V and VI give the statistics for all the validation sites. The RMSE is almost systematically improved for the descending orbits and similar or slightly improved for the ascending orbits in particularly for the correlation.

The Taylor diagrams shown in Figs. 13 and 14 made from the concatenation of all retrieval series against all station series provide a visual confirmation. Looking more in detail into Tables V and VI, the improvement comes mainly from improved STDE and correlations (highest green score numbers for G2) rather than biases that appear mostly similar (highest
<table><caption>TABLE V</caption><caption>Descending Orbits. OP/G2 RETRIEVAL STATISTICS AGAINST GROUND MEASUREMENTS TIME SERIES OF YANCO STATIONS AND KYEAMBA STATIONS FROM OZNET NETWORK: SAMPLE MEAN (μ), SAMPLE STANDARD DEVIATION (σ), SAMPLE RMSE OF THE DIFFERENCE OF SM BETWEEN SMOS AND In Situ DATA, PEARSON SAMPLE CORRELATION (ρ), SAMPLE CORRELATION P-VALUE (pv), AND THE NUMBER (N) OF MEASUREMENTS COLLOCATED IN TIME (30' WINDOW) OF SMOS AND In Situ SAMPLES. COLOR HIGHLIGHTS THE DIFFERENCES BETWEEN OP AND G2 STATISTICS: WORSE IN RED, BETTER IN GREEN, AND EQUIVALENT IN BLACK. THE LAST SCORE ROW REPORTS THE NUMBER OF IMPROVED OP/ EQUIVALENT OP-G2/IMPROVED G2 STATISTICS</caption></table>

<table><caption>TABLE VI</caption><thead><tr><th rowspan="2">station</th><th colspan="6">Descending Orbits<br>V620 operational / V620 Gibbs 2</th></tr><tr><th>μ</th><th>σ</th><th>RMSE</th><th>ρ</th><th>pv</th><th>N</th></tr></thead><tbody><tr><td>K6</td><td>0.04/0.04</td><td>0.08/0.06</td><td>0.09/0.07</td><td>0.50/0.52</td><td>0.00/0.00</td><td>100/100</td></tr><tr><td>K7</td><td>0.07/0.07</td><td>0.10/0.06</td><td>0.12/0.09</td><td>0.49/0.74</td><td>0.00/0.00</td><td>165/167</td></tr><tr><td>K8</td><td>0.01/0.02</td><td>0.08/0.07</td><td>0.08/0.08</td><td>0.33/0.30</td><td>0.00/0.01</td><td>80/80</td></tr><tr><td>K13</td><td>0.04/0.04</td><td>0.08/0.05</td><td>0.09/0.07</td><td>0.19/0.51</td><td>0.22/0.00</td><td>43/44</td></tr><tr><td>K14</td><td>-0.04/-0.02</td><td>0.07/0.09</td><td>0.08/0.09</td><td>0.58/0.52</td><td>0.00/0.00</td><td>59/55</td></tr><tr><td>Y1</td><td>0.04/0.04</td><td>0.06/0.04</td><td>0.07/0.06</td><td>0.69/0.79</td><td>0.00/0.00</td><td>61/63</td></tr><tr><td>Y2</td><td>-0.01/-0.01</td><td>0.06/0.05</td><td>0.06/0.05</td><td>0.60/0.72</td><td>0.00/0.00</td><td>147/147</td></tr><tr><td>Y4</td><td>-0.03/-0.04</td><td>0.06/0.03</td><td>0.07/0.05</td><td>0.44/0.89</td><td>0.33/0.01</td><td>7/7</td></tr><tr><td>Y5</td><td>0.06/0.05</td><td>0.07/0.06</td><td>0.10/0.08</td><td>0.53/0.63</td><td>0.00/0.00</td><td>142/143</td></tr><tr><td>Y6</td><td>0.01/0.02</td><td>0.08/0.06</td><td>0.08/0.06</td><td>0.37/0.72</td><td>0.08/0.00</td><td>24/24</td></tr><tr><td>Y7</td><td>0.05/0.05</td><td>0.06/0.04</td><td>0.08/0.07</td><td>0.55/0.71</td><td>0.00/0.00</td><td>105/106</td></tr><tr><td>Y8</td><td>0.09/0.08</td><td>0.08/0.05</td><td>0.12/0.10</td><td>0.41/0.64</td><td>0.01/0.00</td><td>37/37</td></tr><tr><td>Y9</td><td>0.03/0.03</td><td>0.08/0.06</td><td>0.08/0.06</td><td>0.56/0.72</td><td>0.00/0.00</td><td>158/156</td></tr><tr><td>Y10</td><td>-0.02/-0.03</td><td>0.07/0.05</td><td>0.07/0.06</td><td>0.65/0.79</td><td>0.00/0.00</td><td>94/89</td></tr><tr><td>Y11</td><td>0.06/0.06</td><td>0.07/0.06</td><td>0.09/0.09</td><td>0.56/0.61</td><td>0.00/0.00</td><td>154/147</td></tr><tr><td>Y12</td><td>0.04/0.04</td><td>0.08/0.07</td><td>0.09/0.08</td><td>0.69/0.80</td><td>0.00/0.00</td><td>51/51</td></tr><tr><td>Y13</td><td>0.06/0.06</td><td>0.08/0.07</td><td>0.10/0.09</td><td>0.57/0.62</td><td>0.00/0.00</td><td>100/105</td></tr><tr><td>Score</td><td>4/10/3</td><td>1/0/16</td><td>1/2/14</td><td>2/0/15</td><td>0/14/3</td><td></td></tr></tbody></table>

<table><caption>Ascending Orbits. OP/G2 RETRIEVAL STATISTICS AGAINST GROUND MEASUREMENTS TIME SERIES OF YANCO STATIONS AND KYEAMBA STATIONS FROM OZNET NETWORK: SAMPLE MEAN (μ), SAMPLE STANDARD DEVIATION (σ), SAMPLE RMSE OF THE DIFFERENCE OF SM BETWEEN SMOS AND In Situ DATA, PEARSON SAMPLE CORRELATION (ρ), SAMPLE CORRELATION P-VALUE (pv), AND THE NUMBER (N) OF MEASUREMENTS COLLOCATED IN TIME (30' WINDOW) OF SMOS AND In Situ SAMPLES. COLOR HIGHLIGHTS THE DIFFERENCES BETWEEN OP AND G2 STATISTICS: WORSE IN RED, BETTER IN GREEN, AND EQUIVALENT IN BLACK. THE LAST SCORE ROW REPORTS THE NUMBER OF IMPROVED OP/EQUIVALENT OP-G2/IMPROVED G2 STATISTICS</caption></table>

<table><thead><tr><th rowspan="2">station</th><th colspan="6">Ascending Orbits<br>V620 operational / V620 Gibbs 2</th></tr><tr><th>μ</th><th>σ</th><th>RMSE</th><th>ρ</th><th>pv</th><th>N</th></tr></thead><tbody><tr><td>K6</td><td>0.03/0.02</td><td>0.06/0.06</td><td>0.07/0.06</td><td>0.52/0.51</td><td>0.00/0.00</td><td>99/99</td></tr><tr><td>K7</td><td>0.07/0.05</td><td>0.07/0.07</td><td>0.10/0.09</td><td>0.65/0.66</td><td>0.00/0.00</td><td>170/170</td></tr><tr><td>K8</td><td>0.01/0.01</td><td>0.07/0.06</td><td>0.07/0.06</td><td>0.20/0.42</td><td>0.07/0.00</td><td>80/80</td></tr><tr><td>K13</td><td>0.03/0.03</td><td>0.04/0.04</td><td>0.05/0.05</td><td>0.48/0.55</td><td>0.01/0.00</td><td>26/26</td></tr><tr><td>K14</td><td>-0.01/-0.02</td><td>0.10/0.10</td><td>0.10/0.10</td><td>0.45/0.47</td><td>0.00/0.00</td><td>62/62</td></tr><tr><td>Y1</td><td>0.07/0.07</td><td>0.04/0.04</td><td>0.08/0.08</td><td>0.81/0.81</td><td>0.00/0.00</td><td>64/64</td></tr><tr><td>Y2</td><td>0.00/0.00</td><td>0.05/0.05</td><td>0.05/0.05</td><td>0.64/0.67</td><td>0.00/0.00</td><td>144/144</td></tr><tr><td>Y4</td><td>-0.04/-0.03</td><td>0.03/0.03</td><td>0.05/0.04</td><td>0.84/0.83</td><td>0.01/0.01</td><td>8/8</td></tr><tr><td>Y5</td><td>0.07/0.07</td><td>0.05/0.05</td><td>0.09/0.09</td><td>0.71/0.75</td><td>0.00/0.00</td><td>147/147</td></tr><tr><td>Y6</td><td>0.02/0.02</td><td>0.05/0.05</td><td>0.06/0.05</td><td>0.70/0.73</td><td>0.00/0.00</td><td>23/23</td></tr><tr><td>Y7</td><td>0.05/0.05</td><td>0.04/0.04</td><td>0.07/0.06</td><td>0.76/0.73</td><td>0.00/0.00</td><td>102/102</td></tr><tr><td>Y8</td><td>0.09/0.09</td><td>0.04/0.03</td><td>0.10/0.09</td><td>0.93/0.91</td><td>0.00/0.00</td><td>37/37</td></tr><tr><td>Y9</td><td>0.03/0.02</td><td>0.06/0.05</td><td>0.07/0.06</td><td>0.77/0.79</td><td>0.00/0.00</td><td>156/156</td></tr><tr><td>Y10</td><td>-0.01/-0.02</td><td>0.05/0.05</td><td>0.05/0.05</td><td>0.83/0.83</td><td>0.00/0.00</td><td>97/98</td></tr><tr><td>Y11</td><td>0.07/0.07</td><td>0.06/0.06</td><td>0.09/0.09</td><td>0.67/0.69</td><td>0.00/0.00</td><td>151/151</td></tr><tr><td>Y12</td><td>0.04/0.04</td><td>0.06/0.05</td><td>0.07/0.06</td><td>0.86/0.89</td><td>0.00/0.00</td><td>53/53</td></tr><tr><td>Y13</td><td>0.06/0.06</td><td>0.07/0.08</td><td>0.10/0.10</td><td>0.55/0.53</td><td>0.00/0.00</td><td>101/102</td></tr><tr><td>Score</td><td>2/11/4</td><td>1/12/4</td><td>0/8/9</td><td>5/2/10</td><td>0/16/1</td><td></td></tr></tbody></table>

black score numbers for G2 and OP). It is worth noting that the degradation of the left-hand swath of descending orbits using G2 in term of residuals is unnoticed on the retrieved G2 SM. We have also seen that even though $\chi^2$ for this particular case is higher in G2 than OP, the symmetry of the associated residuals $\bar{\tau}^X$ and $\bar{\tau}^Y$ is still improved. The magnitude of these residuals impacts the retrieved opacity, but still the gain in symmetry seems a more determinant factor improving the SM retrievals.

## VII. CONCLUSION

Important spatial biases still appear in the SMOS retrieved images and are mainly due to the contribution of the aliased regions to the interferometric measurements and the Gibbs oscillations near the land/ocean transition. Their impact is reduced by means of introducing an artificial scene and solving a differential system. For operational purposes, the artificial scene currently implemented in the SMOS L1op is a basic approach (Gibbs 1) with a constant temperature over the earth. In this paper, we propose to use a more realistic approach (Gibbs 2) in order to obtain an artificial scene much closer to the observed one. The main idea is to separate the land and ocean regions and assign an estimated constant temperature over the land pixels and a scaled Fresnel BT model

over the ocean ones. Furthermore, we have also proposed to characterize each pixel over the earth by its percentage of water content in order to smoothen the land/ocean transition in the artificial scene. A least-square optimization approach with explicit solution is used to estimate the constant over land and the scaling factor over ocean.

The validation of the Gibbs 2 algorithm and its comparison to Gibbs 1 were shown at the level of retrieved temperatures (OTT, dwell lines) but also at the level of retrieved SM and comparing them to in situ measurements.

At BT level, we have shown that the OTT is more homogeneous using Gibbs 2 with a decrease of the rms up to 0.9 K. Globally over ocean, the error of reconstruction using Gibbs 2 also decreases with respect to Gibbs 1, as shown over the dwell lines. The land-sea contamination phenomenon is slightly reduced but not totally corrected since it is due to a combination of problems, such as calibration of the instrument, uncertainties in the antenna patterns, floor error noise, and image reconstruction errors.

From the perspective of the SM retrievals, the use of the Gibbs 2 algorithm is very beneficial, where the radiometry is generally improved. Better agreements between models and observations are found for ascending passes and right-hand swath for descending orbits but appear slightly worse for left-
hand swath of descending orbits. Compared to 17 Australian stations, Gibbs 2 improves the correlation and STDE of the retrieved SM against ground measurement, very significantly for the descending orbits and moderately for ascending ones. In the future, it might be worthwhile to consider the possibility of using a more realistic scene over land just, as it is done over ocean, even though the complexity of the processor will be significantly increased.

Finally, we indicate that this algorithm is now under implementation and it will be used in the upcoming version (v720) of the SMOS L1op.

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**Ali Khazâal** was born in Ansarieh, Lebanon, in 1981. He received the Engineer degree in telecommunication and computer science from the Faculty of Engineering, Lebanese University, Beirut, Lebanon, in 2003, the M.S. degree in signal, image, acoustic, and optimization from the Institut National Polytechnique de Toulouse, Toulouse, France, in 2006, and the Ph.D. degree in signal/image processing and optimization from Paul Sabatier University, Toulouse, in 2009. His Ph.D. thesis was on the Image Reconstruction Algorithms for the Soil Moisture and Ocean Salinity (SMOS) Mission.

He was with the Signal, Image and Instrumentation Group, Laboratoire d'Astrophysique de Toulouse-Tarbes, Observatoire Midi-Pyrnes, Paul Sabatier University. Since 2009, he has been a Research Scientist with the Centre d'Études Spatiales de la BIOsphere, Toulouse. His research interests include numerical analysis and signal and image processing with direct application on the European Space Agency's (ESA) SMOS mission.

Dr. Khazâal is a member of the ESA's SMOS Payload Calibration Committee.

<figure><img src="image_16.png" /></figure>

**Philippe Richaume** received the Engineer degree in computer, electronic, and automatic from the École Supérieure d'Informatique, Électronique, Automatique, Paris, France, in 1990, the M.Sc. degree in computer sciences and artificial intelligence from Paul Sabatier University, Toulouse, France, in 1991, and the Ph.D. degree in computer sciences and applied mathematics from the Conservatoire national des arts et métiers, Paris, 1996.

For the last 20 years, he was with various geophysical laboratories, putting to stress advanced computer science and applied mathematics paradigms against real problems, particularly in the remote sensing context. He is currently with the Centre d'Etudes Spatiales de la BIOsphere, Toulouse. His research interests include signal processing, nonlinear modeling, and inverse problem, particularly using artificial neural networks, such as for real-time signal processing controller of a radio receiver dedicated to solar wind plasma line tracking onboard the WIND/WAVES spacecraft, or for direct-inverse modeling of ocean surface wind from ERS 1/2 scatterometer or biophysical parameters, LAI, chlorophyll, and so on, from POLDER optical directional reflectance, or using traditional iterative minimization approaches, such as for soil moisture retrieval from the Soil Moisture and Ocean Salinity brightness temperature.

<figure><img src="image_17.png" /></figure>

**François Cabot** received the Ph.D. degree in optical sciences from the University of Paris-Sud, Orsay, France, in 1995.

From 1995 to 2004, he was with the wide field of view of the Instruments Quality Assessment Department, Centre National d'Etudes Spatiales (CNES), Paris, France, where he was involved in absolute and relative calibration of CNES-operated optical sensors over natural terrestrial targets. In 2004, he joined the Centre d'Etudes Spatiales de la BIOsphere, Toulouse, as a Soil Moisture and Ocean Salinity System Performance Engineer. He has been a Principal Investigator or Co-Investigator for various calibration studies for MSG, Terra, ENVISAT, and ADEOS-II. His research interests include radiative transfer both optical and microwave and remote sensing of terrestrial surfaces.
<figure><img src="image_18.png" /></figure>

**Eric Anterrieu** (M'02) was born in Brive-la-Gaillarde, France, in 1965. He received the Engineer and M.S. degrees in solid-state physics from the Institut National des Sciences Appliquées, Toulouse, France, in 1988, and the M.S. and Ph.D. degrees in image reconstruction in astronomy from Paul Sabatier University, Toulouse, in 1989 and 1992, respectively. His Ph.D. thesis was on the image reconstruction algorithms for multiple aperture interferometry.

He was with the Radio and Optical Aperture Synthesis Group, LATT, UMR 5572, Toulouse, from 1993 to 2000, the Signal and Image Processing Team, CERFACS, URA 1875, Toulouse, from 2000 to 2004, and the Signal, Image and Instrumentation Group, IRAP, UMR 5277, Toulouse, from 2005 to 2016. Since 1993, he has been an Engineer of research in computer science with the Centre National de la Recherche Scientifique, Paris, France. In 2017, he joined the Soil Moisture and Ocean Salinity (SMOS) Team, Centre d'Etudes Spatiales de la BIOsphère, Toulouse. His research interests include numerical analysis and image and signal processing with a particular emphasis on the SMOS mission and the future microwave missions using aperture synthesis.

<figure><img src="image_19.png" /></figure>

**Arnaud Mialon** received the M.S. degree in climate and physics-chemistry of the atmosphere from Université Joseph Fourier, Grenoble, France, in 2002, and the joint Ph.D. degree in ocean-atmosphere-hydrology from Université Joseph Fourier (Grenoble I), Grenoble, and in remote sensing from the Université de Sherbrooke, Sherbrooke, Québec, Canada, in 2005.

In 2006, he joined the Centre d'Etudes Spatiales de la BIOsphère, Toulouse, France. He is part of Soil Moisture and Ocean Salinity satellite mission and especially in the soil moisture retrieval. He is also involved in field campaigns. His research interests include the passive microwave remote sensing of continental surfaces.

<figure><img src="image_20.png" /></figure>

**Yann H. Kerr** (M'88-SM'01-F'13) received the Engineering degree from the École nationale supérieure de l'aéronautique et de l'espace, Toulouse, France, in 1977, the M.Sc. degree in electrical and electronic engineering from University of Glasgow, Glasgow, U.K., in 1981, and the Ph.D. degree from Paul Sabatier University, Toulouse, in 1992.

From 1980 to 1985, he was employed by CNES, Paris, France. In 1985 he joined LERTS, Toulouse, where he was the Director in 1993-1994. He spent 19 months at the Jet Propulsion Laboratory (JPL), Pasadena, CA, USA, in 1987-1988. He has been working at CESBIO, Toulouse, since 1995 (Deputy Director and Director since 2007). His fields of interest are in the theory and techniques for microwave and thermal infrared remote sensing of the Earth, with emphasis on hydrology, water resources management, and vegetation monitoring. He has been involved with many Space missions. He was an EOS principal investigator (interdisciplinary investigations) and Principal Investigator and precursor of the use of the SCAT over land. In 1990, he started to work on the interferometric concept applied to passive microwave earth observation and was subsequently the science lead on the MIRAS project for the European Space Agency (ESA) with Matra Marconi Space and Observatoire Midi Pyrénées, Toulouse. He was also a Coinvestigator on IRIS, OSIRIS, and HYDROS for NASA. He was science advisor for MIMR and Coinvestigator on AMSR. In 1997, he first proposed the natural outcome of the previous MIRAS work with what was to become the SMOS Mission which was eventually selected by ESA in 1999 with him as the SMOS mission Lead-Investigator and Chair of the Science Advisory Group. He is also in charge of the SMOS science activities coordination in France. He has organized all the SMOS Science workshops.