2.1.3. PMS Sensitivities

The sensitivity of the power measurement system (PMS) gain to physical temperature variations was first characterised by the receiver supplier and later verified on-ground at instrument level during the test in thermal vacuum conditions at the Large Space Simulator (LSS) at ESTEC [10], and then again during special calibration events in the SMOS commissioning phase [11]. The latter values have been used until now for adjusting the temperature sensitivity. However, a recent analysis of the variations of the PMS gain through the years showed that the pre-launch PMS sensitivities provide for a more natural behaviour of the PMS gain's aging with time. Figure 6 shows PMS behaviour for receiver LICEF C10 (LICEF stands for lightweight cost-effective front end). As it is seen, using the pre-launch sensitivities provides the best cancellation of the PMS gain oscillations due to physical temperature swings. Similar results are seen in other receivers. For the third mission reprocessing, PMS sensitivities' pre-launch values were used again.

#### 2.1.4. PMS Heater Correction

A known problem in the SMOS instrument, which was detected during the thermal tests at the LSS chamber, was PMS offsets jumps following the instrument heater switching from on to off and vice versa. A correction was introduced early in the mission to mitigate this effect, which consisted of a delayed voltage offset with respect to the heater status transition, but the problems were still noticeable, particularly for a few receivers [11].

**Figure 6.** SMOS PMS gain for receiver LCF\_C10 over the years. The blue line indicates the value as measured during the calibration event at the physical temperature during the calibration event. The cyan line and black line show the PMS gain transported to 21 degrees Celsius using the second mission reprocessing and pre-launch PMS sensitivities, respectively.

A more careful analysis showed that the jumps related to the heater status do not correspond to a simple delayed offset, but that the behaviour follows a double exponential [9]. Figure 7 shows the PMS voltages for a calibration event, where a constant noise from an internal warm load source was introduced at the receivers.

**Figure 7.** PMS voltages for LICEF C-03 when a constant noise source is introduced. In green, the status of the heater is depicted. Red crosses for the PMS voltage indicate that the heater is on and blue crosses show when the heater is off.

Based on this analysis, the correction applied, Δ*V*, was set to

$$
\Delta V\_{\rm ON} = a\_{\rm ONi} (V\_i - V\_{\rm max}) \left( 1 - e^{\frac{-t}{\tau\_{\rm ON} \cdot 1i}} \right) + \beta\_{\rm ONi} (V\_i - V\_{\rm max}) \left( 1 - e^{\frac{-t}{\tau\_{\rm ON} \cdot 2i}} \right) \tag{8}
$$

$$
\Delta V\_{\rm OFF} = \alpha\_{\rm OFFi} (V\_{\rm i} - V\_{\rm max}) e^{\frac{-l}{\tau\_{\rm OFF}}} + \beta\_{\rm OFFi} (V\_{\rm i} - V\_{\rm max}) e^{\frac{-l}{\tau\_{\rm OFF}}},\tag{9}
$$

where *Vi* is the current PMS value without correction for each of the 72 receivers, in volts. *Vmax* is the maximum measurable PMS, set to a value of 2.5V. α, β, τ*ON* and τ*OFF* are fixed constants for each receiver that empirically determine the double exponential behaviour, and *t* is the number of epochs since the corresponding transition of the heater status (on to off, or vice versa).

The validation of this correction was performed by means of a relative comparison of the antenna temperature of one receiver to the average of all receivers. Figure 8 shows the behaviour for L1OP v620 (delay heater correction) and for L1OP v724 (double exponential heater correction). The new correction clearly reduces the obvious PMS offset jumps due to the heater status.

**Figure 8.** Difference between the antenna temperature measurement of receiver LCF\_C\_03, with respect to the average of all receivers, when applying the v620 delayed offset heater correction (**a**) and when applying the new v724 double exponential heater correction (**b**). The colour of the points in both plots indicates the status of the heater: red when the heater is on and blue when it is off.

## 2.1.5. Antenna Patch Thermistor Correction

Another aspect that was discovered during the second mission reprocessing analysis was an increased bias immediately following the Sun's eclipse by the Earth, relative to the instrument. This effect clearly pointed to another problem related to temperature variations. While the instrument backend is kept under thermal control [McMullan et al., 2008], the antenna patches suffer large thermal excursions. Those changes are monitored by three thermistors placed at the screw of each NIR antenna patch (*Tp*<sup>7</sup> in Figure 3) and are used in the NIR radiometric equation presented in Section 2.1.2.

The team considered that the reading of the thermistor did not properly describe the temperature of the antenna patch and proposed a correction [12]. The correction was introduced based on the observed thermal latency during inertial external manoeuvres. During these manoeuvres, the instrument points at the cold sky for several minutes. However, *Tp*<sup>7</sup> thermistor readings take a long time to stabilise to a constant temperature. The thermistor reading was considered to be thermally coupled to the innermost part of the antenna through the antenna screw, inside whose head the thermistor is mounted. As such, the thermistor reading is not fully representative of the antenna patch region. The team decided then to apply a correction to the thermistor reading by assuming that the temperature to which the thermistor stabilises at the end of the external manoeuvre is the actual temperature during the entire inertial manoeuvre. The following correction was derived:

$$T\_{p\mathcal{T}} = T\_{p\mathcal{T}} - \frac{1}{LP} \frac{dT\_{p\mathcal{T}}}{dt} \, , \tag{10}$$

where *T*ˆ *<sup>p</sup>*<sup>7</sup> is the corrected thermistor temperature, *Tp*<sup>7</sup> the actual thermistor reading, and *LP* a constant that was estimated to be −0.0031.

The correction was then used to process the ocean brightness temperature, and the bias with respect to the forward model was re-assessed. The impact of this correction is a clear mitigation of the bias observed during the eclipse period. Figure 9 shows the Y polarisation brightness temperature bias observed over the ocean with and without the *Tp*<sup>7</sup> correction applied. The increased bias in the eclipse is observed around 35N to 60N degrees in latitude during the Northern Hemisphere (NH) winter months in the left plot.

**Figure 9.** Hovmoller plot showing Y pol BT bias over the ocean as a function of time and latitude with (**a**) and without (**b**) the *Tp*<sup>7</sup> correction.

#### *2.2. Changes in Image Reconstruction*

Image reconstruction is a process where the calibrated MIRAS data are turned into radiometric maps that can be projected on the Earth's surface. For the v724 processing baseline, three main improvements were made to this process. They are improvements related to


In the following sub-sections, we describe these updates in detail.

## 2.2.1. Gibbs-2 Algorithm

The so-called Gibbs-2 algorithm is an evolution of the Gibbs-1 algorithm applied to SMOS measurements since its launch. Originally, the Gibbs correction aimed to reduce the Gibbs artefacts that appeared in the image following large BT transitions between land and ocean (or sky and Earth) due to limited coverage in the visibility domain. Soon after, the team realised that the correction not only reduces the Gibbs artefacts but also a floor error induced in the retrieved images due to dissimilarities in the antenna patterns and the aliasing [13,14]. Gibbs-1 correction reduces this so-called floor error by removing a constant brightness temperature (BT) in the reconstruction process, which reduces the visibility values before inversion, and adding it back at the end of the inversion process. In Gibbs-2, the process has evolved to include the use of an artificial scene as close as possible to the observed one. This artificial scene, *Va*, uses a Fresnel model over the ocean and a constant value (250 K) over land. Figure 10 shows an example of the artificial scene as used in the image reconstruction processor. The visibilities of this artificial scene are computed using an SMOS instrument model:

$$V\_a = GT\_{a\nu} \tag{11}$$

where *Ta* represents the modelled BT of the artificial scene, *G* is the instrument model, and *Va* are the visibilities derived from the *Ta* scene.

**Figure 10.** Artificial scene of one SMOS observation of the Iberian peninsula and northern Africa.

Then, the image reconstruction algorithm is applied over the following differential linear problem:

$$V - V\_a = G(T - T\_a),\tag{12}$$

yielding to the following retrieved BT:

$$T\_I = \mathbb{U}^\* Z \mathbb{J}^+ (V - V\_a) + T\_{a\prime} \tag{13}$$

where *U* is the Fourier transform operator, *Z* is the zero-padding operator beyond the SMOS frequency coverage, *J* = *GU*\* *Z* is the image reconstruction operator used in the SMOS processor and *J* <sup>+</sup> is the pseudo-inverse of *J* [15].
