2.2.2. Geomagnetic Field and the Consolidated VTEC Databases

The directly proportional relationship between the FRA and the VTEC is determined by Earth's electromagnetic field. Thus, a geomagnetic field dataset is needed. This dataset corresponds to the 12th Generation International Geomagnetic Reference Field (IGRF) [13], which is calculated by the International Association of Geomagnetism and Aeronomy (IAGA). It can be found in [17] and it has a geographical resolution of 5◦ in longitude and 2.5◦ in latitude for the entire globe as well as a daily temporal resolution.

SMOS level 1 data includes the consolidated VTEC. It is built by the SMOS Data Processing Ground Segment (DPGS) [14], and it can be obtained in the SMOS dissemination service website [15]. The data also has a geographical resolution of 5◦ in longitude and 2.5◦ in latitude for the entire globe but a temporal resolution of 2 h.

#### 2.2.3. SMOS Level 2 VTEC (DTBXY Product)

The SMOS level 2 processor computes the VTEC with a methodology that uses the third Stokes parameter and an external database of VTEC. The VTEC value is calculated for the zone of the snapshot with highest sensitivity of TB to VTEC, which corresponds to pixels in the area around ξ = 0, η = 0.2 (ξ = 0 ± 0.025, η = 0.2 ± 0.025). Then, that value is used for the entire EAF-FoV. This VTEC is called A3TEC [11].

This A3TEC dataset is provided in the product called AUX\_DTBXY (Delta TB) [15]. The latitude and the VTEC value per overpass are found in fields 32 and 34 respectively [22].

#### 2.2.4. GPS VTEC

This third VTEC source consists on VTEC maps with a temporal resolution of 2 hours and a geographical resolution of 5◦ in longitude and 2.5◦ in latitude calculated from daily sets of GPS differential code bias values [23]. The IGS Ionosphere Working Group (Iono-WG) calculates the VTEC, which is named IGS Global Total Electron Content (IGSTEC). It is stored in IONEX format, and it can be downloaded from the official server (CDDISA at GSFC/NASA).

#### *2.3. FRA End-to-End Simulator*

In order to evaluate different approaches to retrieve VTEC maps, a first version of a FRA simulator was presented in [10]. This simulator was refined until it worked properly for the assessment. The improved FRA simulator is explained in this section.

For each snapshot, the ξ and η coordinates defined in the antenna frame (director cosine plane) are translated to Earth's surface coordinates. The incidence angle θ and the geometric rotation angle ϕ are computed using standard geometry [16]. Then, a simple geophysical model is used to simulate the Earth's emissions. Ocean TB are built by assuming a Fresnel model with a typical salinity value of 35 psu and a typical sea surface temperature of 294 K. Open ocean TB images per polarization, assuming a uniform ocean, are shown in the top row of Figure 4 (left: X-pol, middle: Y-pol, right: third Stokes parameter).

**Figure 4.** Typical open ocean Fresnel brightness temperature snapshots per polarization (left: X-pol, middle: Y-pol, right: third Stokes parameter). Top: Fresnel modeled brightness temperature (TB), middle: taking into account the FRA, bottom: adding the effect of noise in addition to the FRA.

To add the Faraday rotation for each pixel, Equation (1) is used together with both the VTEC and the geomagnetic databases. This Faraday rotation angle derived from external datasets shall be referred to as "database FRA" from now on in this paper. Once calculated, it is added to the geometrical angle (ϕ) to obtain the total rotation per pixel when simulating Earth's emissions. Open ocean TB images taking into account the FRA are shown in Figure 4 (middle row).

Then, the effect of measurement noise is included in the simulator. To do so, first the radiometric sensitivity per polarization is calculated. The radiometric sensitivity corresponds to the smallest radiometric temperature that the instrument can detect. It is defined in the antenna reference frame as follows [24]:

$$
\Delta T\_{\rm sens} = \Delta S \frac{T\_{\rm sys}}{\sqrt{B \tau\_{\rm \varepsilon}}} \frac{\Omega\_{\rm a}}{\left| F\_n(\xi, \eta) \right|^2} \sqrt{1 - \xi^2 - \eta^2} a\_w \sqrt{N\_v} \tag{4}
$$

where <sup>Δ</sup>*<sup>S</sup>* corresponds to the elementary area in the (ξ, <sup>η</sup>) grid defined by <sup>√</sup> 3*d*2/2; *d* refers to the distance between antennas normalized to the wavelength (d = 0.875 in SMOS); *Tsys* corresponds to the addition of the average antenna temperature (*TA*) and the average receiver noise temperature at the antenna plane (*TR*), both being different per polarization (see typical values for the open ocean in Table 1); *B* is the receiver noise equivalent bandwidth that in MIRAS equals 19 MHz; τ*<sup>e</sup>* is the effective finite integration time (τ*<sup>e</sup>* = τ*<sup>i</sup>* ∗ *Q*, where τ*<sup>i</sup>* corresponds to the integration time that is 1.2 s for pure X and Y epochs and 0.4 s for mixed epochs and *Q* = 0.552 for SMOS 1 bit/2 level digital correlator [25]); Ω*<sup>a</sup>* is the antenna equivalent solid angle that equals 1.4; *Fn*(ξ, η is the antenna pattern measured on the ground; α*<sup>w</sup>* is the used window factor that in this case is a Blackman window (α*<sup>w</sup>* = 0.45); and *Nv* is the total number of visibilities samples that in MIRAS is 2791.

**Table 1.** Average antenna temperature, *TA*, and average noise receiver temperature, *TR*, at the antenna plane (typical values for open ocean).


The effect of noise is added to the TB with a normal distribution of zero mean and the standard deviation equal to the radiometric sensitivity of each pixel. Figure 4 (bottom row) shows typical open ocean TB images with FRA, including the effect of noise.

MIRAS measures sequentially for each polarization at different instants in time. When processing, one single instant is used for all polarizations. This introduces an error when translating the brightness temperatures from the *x* − *y* polarization (antenna frame) to *h* − *v* polarization (ground plane) that is unavoidable in the processing.

VTEC maps can be calculated with the FRA estimated using the TB. These maps can then be used in the correction of the FRA per pixel in the field of view (FoV) of every snapshot of the trace.

#### *2.4. VTEC Retrieval from Radiometric Data*

From the brightness temperature snapshots, the FRA can be retrieved by applying Equation (3). An indetermination emerges when both the numerator and the denominator tend to 0 (*TB xx* ≈ *TByy* and 2*e*(*TB xy*) ≈ 0); that occurs at low incidence angles [26]. To avoid it, pixels with incidence angles lower than 25◦ are discarded. Figure 5a shows a database VTEC snapshot from a descending SMOS overpass over the Pacific Ocean on March 20th, 2014. Figure 5b shows the retrieved VTEC snapshot where some pixels are affected by the indetermination of Equation (3). Figure 5c shows the retrieval once pixels causing the indetermination of Equation (3) are rejected with the chosen threshold.

Once the FRA is retrieved, the VTEC is calculated using Equation (1). This equation presents an indetermination when the geomagnetic field is orthogonal to the wave propagation direction, which occurs close to the Equator, in a zone where the FRA vanishes. To avoid this indetermination, pixels accomplishing Θ*<sup>B</sup>* ≈ π/2 are rejected by using an appropriate threshold established empirically (cos Θ*<sup>B</sup>* < 0.27). Figure 5d shows another database VTEC snapshot from a descending SMOS overpass over the Pacific Ocean on the same date, March 20th, 2014, in order to compare it with its retrieval

(Figure 5e). The error introduced by the indetermination is noticeable. Figure 5f shows the retrieval once pixels causing the indetermination of Equation (1) are rejected with the chosen threshold (pixels causing indetermination of Equation (3) were rejected previously). It is important to remark that this threshold is only used when the geomagnetic field is orthogonal to the signal path.

**Figure 5.** Vertical total electron content (VTEC) snapshots: (**a**) database VTEC, (**b**) its retrieved VTEC snapshot with pixels affected by the indetermination of Equation 3, (**c**) its retrieved VTEC filtering affected pixels, (**d**) another database VTEC snapshot, (**e**) its retrieved VTEC snapshot with pixels affected by the indetermination of Equation (1), (**f**) its retrieved VTEC filtering affected pixels.

After retrieving VTEC snapshots, the geolocation is done at an altitude of 450 km in an ETOPO5 grid (1/12◦).

Figure 6 shows the VTEC maps of a descending SMOS overpass in the Pacific Ocean (March 20th, 2014) processed with the simulator (taking into account the effect of the noise). In order to have a reference, the database VTEC can be seen in the left (Figure 6a). Because both the geographical (5◦ in longitude and 2.5◦ in latitude) and temporal resolution (2 h) of the "Consolidated TEC" is coarse, a spatiotemporal interpolation is done in order to obtain the VTEC in that SMOS overpass in an ETOPO5 grid. In Figure 6b, the retrieved VTEC is shown, where it is noticeable how the effect of noise introduces errors in the retrieval.

To reduce the effect of noise and artifacts in the retrieved VTEC from SMOS data, spatiotemporal filtering techniques are required [9]. The sizes of both filters were optimized with the simulator that takes into account the effect of noise. To do so, the size of the spatial filter was set to 0.179 in the director cosine plane (10 times the minimum Δξ = 0.0179) in TB snapshots and the length of the temporal filter was varied from 15 to 83 snapshots with a step of 4 snapshots. Figure 7a shows the root mean square error (RMSE) of the deviations with respect to the database VTEC of this optimization. The temporal filter is an averaging triangular window considering the current snapshot with the highest weight. Its size was then fixed to 43 snapshots (optimum value from Figure 7a) and the size of the spatial filter was optimized by varying its radius from 0.1253 to 0.2506 with a step of 0.0179. The RMSE of this optimization is shown in Figure 7b, where it can be seen that the optimum size of the spatial filter corresponds to a radius of 0.1969 in the cosine plane. Finally, a fine tuning was done for the size of both filters to select which ones to use. The optimum temporal filter corresponds to 43 snapshots, again, and the spatial filter to 0.189 in the ξ − η plane (Figure 7c). It is important to remark that the temporal filter is applied to TB snapshots, and the spatial filter to VTEC snapshots at the antenna

frame. The spatial filter was also tested over the ground instead of at the antenna reference frame. To do so, the window size of the spatial filter was calculated to be equivalent to the spatial filter at the antenna, which corresponds to a radius of approximately 190 km over the ground. There was not a clear improvement in the retrieval, but there was an important difference in the execution time, with the calculation over the ground being much slower. Therefore, the spatial filter was applied at the antenna reference frame.

**Figure 6.** VTEC of a descendent orbit over the Pacific Ocean, March 20th, 2014: (**a**) database VTEC and (**b**) simulated VTEC retrieval.

**Figure 7.** Root mean square error of the retrieved VTEC with respect to the database VTEC when optimizing (**a**) the size of the temporal filter with a coarse binning, (**b**) the size of the spatial filter with a coarse binning, setting an optimum temporal filter size, and (**c**) the size of the spatial filter with a fine binning, setting the temporal filter with the most optimum temporal filter size.

Hence, the methodology consists of:

