**1. Introduction**

Over the past few years, the earth has been undergoing significant climate change and extreme weather events. Even though the water cycle is the most influential process in this situation, it is still relatively poorly understood. For this reason, its understanding remains a priority field under study by different research groups. Earth's water cycle and climate are intrinsically linked to some geophysical variables. Two of those variables are soil moisture and ocean salinity, which change constantly depending on the exchange of water between oceans, atmosphere, and landmasses [1]. Global measurements of both parameters were not available with a suitable temporal and spatial resolution until 2009.

In November 2009, The European Space Agency launched the SMOS (Soil Moisture and Ocean Salinity) mission to observe soil moisture over the earth's landmasses and salinity over the oceans at a global and frequent scale [2,3]. Its unique payload is MIRAS (Microwave Imaging Radiometer by Aperture Synthesis), a two-dimensional Y-shape synthetic aperture radiometer operating in the L-band (1.413 GHz) [4]. After more than 10 years in operation, MIRAS continues to provide good quality full polarimetric brightness temperature (TB) [5] to generate continuous and global maps of both geophysical variables.

MIRAS was designed to measure the radiation emitted from the earth (i.e., the brightness temperatures (TBs)). Each scene is measured with multi-incidence angles, which are taken into account when processing the data to construct an image (snapshot) over the extended alias-free field of view (EAF-FoV). A block of full polarimetric data per scene is obtained every 2.4 s [6].

As microwave radiation from Earth propagates through the ionosphere, the electromagnetic field components are rotated at an angle, called the Faraday rotation angle (FRA), which depends on the vertical total electron content (VTEC) of the ionosphere, the frequency, and the geomagnetic field. At the SMOS operating frequency (1.4135 GHz), the Faraday rotation is not negligible and must be compensated for to get accurate geophysical retrievals. It can be estimated using a classical formulation [7] that makes use of total electron content (TEC) and geomagnetic field data provided by external sources.

The Faraday rotation angle can alternatively be retrieved from the SMOS radiometric data. This is possible thanks to improvements in the image reconstruction algorithms developed in the last few years, particularly regarding the third and fourth Stokes parameters [6]. However, estimating the Faraday rotation from SMOS radiometric data per each pixel in the SMOS field of view is not straightforward because of the presence of spatial errors in SMOS images. Spatial ripples are due to calibration inaccuracies, image reconstruction artifacts, and antenna pattern uncertainties [8] that limit the quality of the retrieval. A previous work showed that the FRA can be dynamically retrieved at boresight per snapshot directly from SMOS full-polarization TB by applying filtering techniques [9]. The results show a good performance, but the FRA at boresight is not representative for the entire SMOS field of view as shown in [10].

The possibility of retrieving the Faraday rotation from SMOS radiometric data opens up the opportunity to estimate the total electron content of the ionosphere by using an inversion procedure from the measured rotation angle in the SMOS field of view. Currently, the SMOS ocean salinity team computes the VTEC over the ocean from the SMOS third Stokes measurements following the procedure detailed in [11]. These VTEC retrievals have been shown to improve the salinity retrievals. This methodology considers only the SMOS field of view region with the highest sensitivity of TB to VTEC, and it calculates the VTEC starting with a first order approximation initiated with the VTEC value from an external database. This value is then assigned to the entire SMOS field of view not taking into account the VTEC spatial variation within it. Therefore, this methodology is dependent not only on an external VTEC database but also on a forward radiative model.

The present work proposes a novel methodology to derive VTEC maps from SMOS radiometric data over the EAF-FoV. This methodology is expected to work independently on the target measured by the instrument. It applies spatiotemporal filtering techniques to overcome the issues involved. The structure of this paper is as follows: Section 2 details the different data sources and the methods used for deriving the VTEC maps from SMOS measurements. The results obtained with the novel methodology are shown and discussed in Section 3. Finally, conclusions are drawn in Section 4.

#### **2. Data and Methods**

## *2.1. Faraday Rotation*

The rotation angle in the polarization of an electromagnetic field is directly proportional to the VTEC of the ionosphere, the geomagnetic field, and the sensor orientation, according to the following equation [7,12]:

$$
\Omega\_f = 1.355 \ast 10^4 \ast f^{-2} \ast B\_0 \ast \cos\Theta\\p \ast \sec\Theta \ast V\\TEC\_\prime \tag{1}
$$

where Ω*<sup>f</sup>* represents the FRA in degrees; *f*, the frequency in GHz (1.4135 GHz); *B*0, the geomagnetic field in Tesla; Θ*B*, the angle between the magnetic field and the wave propagation direction; θ, the angle between the wave propagation direction and the vertical to the surface or so called incidence angle; and VTEC, the vertical total electron content in TEC Units (TECU) [1016*electrons*/*m*2]. Both the geomagnetic field and the VTEC are given at a geodetic altitude of 450 km.

The geomagnetic field is obtained from the data set of the International Geomagnetic Reference Field (IGRF) [13]. In the SMOS Level 2 operational processor, the vertical electron content used to correct the FRA is read from a SMOS auxiliary data field called "consolidated TEC" [14] (referred to hereafter as the VTEC database) for ascending and descending orbits over land, and only for ascending orbits over ocean. For measurements over the ocean in descending orbits, VTEC values are computed from SMOS TB measurements following the methodology detailed in [11], and the so derived VTEC values can be found in the OSDAP2 (Level 2 Ocean Salinity Data Analysis Product) [15].

The Faraday rotation can alternatively be retrieved directly from SMOS radiometric data using a different technique. At each spatial direction, Earth's radiation arrives at the instrument with a rotation equal to the addition of two angles. The first one is the geometric angle (ϕ) and it is given by the third Ludwing polarization definition [16] according to the instrument attitude and orientation with respect to the nominal ground-referenced horizontal (*h*) and vertical (*v*) polarizations. The second one is the FRA (Ω*f*). Assuming that the *h* and *v* polarizations emitted by Earth are uncorrelated, the relationship between the brightness temperatures in full polarization at the ground and antenna levels can be expressed as follows [16]:

$$
\begin{bmatrix} T\_B^{xx} \\ 2T\_B^{xy} \\ T\_B^{yy} \end{bmatrix} = \begin{bmatrix} \cos^2(\varphi + \Omega\_f) & \sin^2(\varphi + \Omega\_f) \\ -\sin 2(\varphi + \Omega\_f) & \sin 2(\varphi + \Omega\_f) \\ \sin^2(\varphi + \Omega\_f) & \cos^2(\varphi + \Omega\_f) \end{bmatrix} \begin{bmatrix} T\_B^{hh} \\ T\_B^{vv} \end{bmatrix} \tag{2}
$$

where the superscripts represent the polarization frames.

From Equation (2), the FRA can be calculated using full-pol radiometric data as follows (equivalent to Equation (22) of [12]):

$$
\Omega\_f = -\varphi - \frac{1}{2} \arctan\left(\frac{2\Re c(T\_B{}^{xy})}{T\_B{}^{xx} - T\_B{}^{yy}}\right) \tag{3}
$$

where 2*e*(*TB xy*) corresponds to the third Stokes parameter at antenna level and *TB xx* and *TByy* to the brightness temperatures in the x and y polarizations, respectively.

In a previous paper [9], the FRA was estimated from MIRAS data using spatiotemporal filtering techniques. A unique FRA value per snapshot (at boresight) was retrieved by averaging the FRA over a circle of radius 0.3 around the boresight in the ξ−η plane, defining ξ and η as the director cosines with respect to the X and Y axes, respectively. Even though this methodology reproduces the natural variation of the Faraday rotation accurately, it is not enough to use one FRA value for all the EAF-FoV due to its spatial variation within the snapshot. Figure 1a shows the FRA over the EAF-FoV of one snapshot over a descending orbit in October 2011. The circle of radius 0.3 is drawn in black. The FRA was calculated using Equation (1) and reading the geomagnetic field and the VTEC from the external datasets [14,17], respectively. The error when the boresight FRA is considered for all pixels over the EAF-FoV is shown in Figure 1b.

**Figure 1.** Faraday rotation angle (FRA) over the extended alias-free field of view (EAF-FoV): (**a**) database FRA, (**b**) systematic FRA error when considering the FRA value at boresight for the entire EAF-FoV.

It is known that the VTEC and therefore the FRA vary with solar activity, which can be assessed using sunspot numbers [18], and in a geographical and temporal way [7,19]. SMOS data is here considered within the 24th sun cycle, which started on January 4th, 2008, with each cycle lasting about 11 years. The sun's peak activity during this cycle was reached on March 2014 [18,19]. It is also important to note that the value of the FRA reaches its highest point during the year in the March equinox due to the sun's illumination geometry over the earth during that season.

The geographical variability of the FRA is presented in Figure 2. Equation (1) and the same mentioned external databases were used again to calculate the FRA at the coordinates of the SMOS boresight over a 3-day period. Two different time frames were used: one with high FRA (March 19th to 21st, 2014) and another with low FRA (January 14th to 16th, 2011). The FRA of both descending (DES) and ascending (ASC) orbits are shown for each period (be aware of the different scales in ASC/DES maps). Additionally, the latitude-time Hovmöller plots of the FRA for the entire mission are shown in Figure 3 for descending and ascending orbits to show the FRA temporal variability. The FRA was calculated over the eastern Pacific Ocean using also the database VTEC. These Hovmöller diagrams confirm the selected periods of high and low FRA.

**Figure 2.** FRA in the Soil Moisture and Ocean Salinity (SMOS) boresight coordinates of 3 days in different periods: (**a**) descending orbits in March 2014 (high sun activity), (**b**) descending orbits in January 2011 (low sun activity), (**c**) ascending orbits in March 2014 (high sun activity), (**d**) ascending orbits in January 2011 (low sun activity).

From these figures, it is perceived that descending orbits present much higher FRA and higher dynamic ranges than ascending ones. In the afternoon local time, the surviving amount of VTEC generated by sunlight during the preceding hours is higher than in the morning local time [19] and because SMOS is in a 6 am—6 pm sun-synchronous orbit, the resulting FRA range is large. Consequently, in the first stage, the analysis was focused on descending orbits. A preliminary analysis was done during the March equinox of 2011 [20], but it was later decided to extend the study to the March equinox of 2014 as well, because, as can be perceived in Figure 3, the highest peak of FRA in the SMOS mission up until now corresponds to that period of time.

**Figure 3.** Latitude–time Hovmöller plots of the boresight FRA for the full mission for: (**a**) descending orbit, (**b**) ascending orbits.

#### *2.2. Data Sources*

#### 2.2.1. SMOS Brightness Temperatures

MIRAS measures the brightness temperature of each scene providing a full-polarimetric block (Tx, Ty, and Txy) every 2.4 s [16]. The MIRAS Testing Software (MTS), developed by the Polytechnic University of Catalonia (UPC), is an independent processor used as a breadboard to test calibration and image reconstruction algorithms before their introduction into the SMOS Level 1 operational processor [21]. The SMOS brightness temperatures used in this work were processed by MTS from level 0 (raw data) up to level 1C (geolocated brightness temperatures).
