2.1.2. Antenna Losses

In SMOS, the NIR antenna losses are divided between the antenna patch (*L*1) and the feeding circuits in the innermost part of the antenna (*L*2) as shown in Figure 3. They are at different physical temperatures. The innermost part of the antenna (*Tp*6) is within the thermal control, whereas the antenna patch is more exposed to the temperature fluctuations of outer space (*Tp*7).

**Figure 3.** Schematic diagram of the structure of the V-channel of the NIR. L stands for loss, Tp for physical temperature, TA for antenna temperature and Tcas for the Calibration Subsystem (CAS) noise temperature [6].

NIR antenna patch losses have been the most challenging problem in SMOS calibration as both *L*<sup>1</sup> and *L*<sup>2</sup> are outside the radiometer's internal calibration loop. A wrong characterisation of the antenna losses introduces variations in the measurements. These variations are related to the variations in the physical temperature of the antenna.

The basic equation for the antenna temperature retrieval at NIR is

$$T\_A = -L\_1 L\_2 T\_{NA} \eta + L\_1 L\_2 [L\_{NC} L\_A L\_{DA} (T\_{II} - T\_{t2}) - T\_{t1}]\_\prime \tag{1}$$

where *TA* is the antenna temperature as measured by the NIR, *L*<sup>1</sup> is the antenna patch loss, *L*<sup>2</sup> is the intermediate layer antenna loss, η is the NIR pulse length, *Lnc*, *LA* and *LDA* are losses of different sections of the cables connecting the antenna to the receiver, *TU* is the load noise temperature and *Tt*<sup>1</sup> and *Tt*<sup>2</sup> are

$$T\_{t1} = \frac{L\_1 - 1}{L\_1 L\_2} T\_{p\mathcal{T}} + \frac{L\_2 - 1}{L\_2} T\_{p6\prime} \tag{2}$$

$$T\_{t2} = \frac{(L\_{\rm NC} - 1)}{L\_{\rm NC} L\_{A} L\_{DA}} T\_{p3} + \frac{(L\_A - 1)}{L\_A L\_{DA}} T\_{\rm Cap} + \frac{(L\_{DA} - 1)}{L\_{DA}} T\_{p1I\prime} \tag{3}$$

and *TNA* is the value measured during calibration, and corresponds to

$$T\_{NA} = \frac{-T\_{A,\text{cal}} + L\_1 L\_2 \left| L\_{NC} L\_A L\_{DA} \left( T\_{1L,\text{cal}} - T\_{t2,\text{cal}} \right) - T\_{t1,\text{cal}} \right|}{\eta L\_1 L\_2} \tag{4}$$

where "*X,cal*" indicates the value of parameter "*X*" obtained during the calibration against the cold sky. Now, if we analyse the equation as a function of the *L*<sup>1</sup> uncertainty using error propagation, we get

$$\frac{\Delta T\_A}{\Delta L\_1} = \frac{\partial T\_A}{\partial L\_1} + \frac{\partial T\_A}{\partial T\_{t1}} \frac{\partial T\_{t1}}{\partial L\_1} + \frac{\partial T\_A}{\partial T\_{NA}} \frac{\partial T\_{NA}}{\partial L\_1},\tag{5}$$

*Remote Sens.* **2020**, *12*, 1645

and finally

$$
\Delta T\_A = \frac{\Delta L\_1}{L\_1} \left| \frac{L\_1 L\_2 \left[ T L\_D - T\_{t1} \right] - T\_A}{L\_1 L\_2 \left[ T L\_{D, \text{cal}} - T\_{t1, \text{cal}} \right] - T\_{A, \text{cal}}} \left( T\_{p\mathcal{T}, \text{cal}} - T\_{A, \text{cal}} \right) - \left( T\_{p\mathcal{T}} - T\_A \right) \right| \tag{6}
$$

where

$$TL\_D = L\_{\rm NC} L\_{\rm A} L\_{\rm DA} (T\_{\rm II} - T\_{\rm t2}),\tag{7}$$

This equation, as given, is difficult to interpret, but by making some realistic numerical simulations, we realised that in a scenario where the calibration was obtained at a *Tp*<sup>7</sup> of 295 K, errors in the *L*<sup>1</sup> antenna patch loss would propagate to *TA* at a different rate depending on the *Tp*<sup>7</sup> during measurement. Figure 4 shows how an error in the antenna losses characterisation will introduce an error in the antenna temperature that will be a function of the temperature of the antenna patch (*Tp*7).

**Figure 4.** Expected error in the antenna temperature as a function of the error in the antenna losses (*L*1), for simulated scenarios with different temperatures, when the calibration of the NIR was obtained at a temperature of *Tp*<sup>7</sup> = 295 K.

Therefore, errors in the antenna loss characterisation should be correlated with the variations of the antenna physical temperature, which is exactly what has been observed in SMOS.

Initially, just after launch, the on-ground characterisation values for *L*<sup>1</sup> and *L*<sup>2</sup> were used. Later, during SMOS's first mission reprocessing, an antenna thermal model was introduced to correct for variations observed during NIR external calibration manoeuvres. However, the antenna thermal model was quickly abandoned, as the instrument became more stable following the initial months in orbit. For the second mission reprocessing, the team derived a method to calibrate the antenna losses in orbit [8]. Antenna losses were measured every 15 days, and, since the values were stable, the average value was used for the entire reprocessing. This correction was key to improve the stability of the data in the second mission reprocessing. The calibration procedure could only measure the antenna losses for whole of the antenna patch and the inner part of the antenna (*L*<sup>1</sup> and *L*<sup>2</sup> losses respectively). But the antenna patch and the innermost part of the antenna in SMOS suffer different temperature excursions. Introducing the correct split in the total antenna loss between *L*<sup>1</sup> and *L*<sup>2</sup> is key for obtaining good instrument stability. This split was obtained by assessing the brightness temperature variations over the ocean against an ocean forward model for Stokes-1 measurements and applying the same antenna loss value at the H and V polarisations.

In the third mission reprocessing, it became evident that a different split was necessary for H and V polarisation, as the antenna has different patch for each polarisation. The exercise was then repeated for each of the two polarisations [9]. Figure 5 shows the variations in the brightness temperature biases over ocean as a function of the physical temperature differences between the antenna patch and the innermost part of the antenna, when the *L*<sup>1</sup> antenna loss has been set to 0 dB. The plots show a clear slope in the data, which can account for the antenna losses. NIR-CA H pol *L*<sup>1</sup> antenna loss was set to 0.27 dB, and V polarisation *L*<sup>1</sup> was set to 0.14 dB. *L*<sup>2</sup> values were set to the difference between the total loss as measured by calibration and the corresponding *L*<sup>1</sup> values (*L*<sup>2</sup> equals 0.19 for H and 0.30 for V polarisation for NIR-CA; the other two NIR units are not used to derive the antenna temperature, but their values can be found in [9]).

**Figure 5.** Variations of brightness temperature biases over the ocean as a function of temperature variations between the antenna patch and the innermost part of the antenna, for H polarisation (**left**) and V polarisation (**right**).
