## 1 Introduction
Polar orbiting satellites, such as NASA's Terra (1999-2020) and Aqua (2002-2020) missions, provide global LST maps at 1 km every 1-2 day using the Moderate-resolution Imaging Spectroradiometer (MODIS) sensor [14]. Geostationary satellites (i.e., those with the same orbital period as the Earth's rotation) are able to provide continental coverage at higher temporal resolution. The Spinning Enhanced Visible and InfraRed Imager (SEVIRI) sensor from the Meteosat Second Generation (MSG) mission (2002-2021) of ESA & European Organization for the Exploitation of Meteorological Satellites (EUMETSAT) provides continental LST maps at 3 km every 15 min [15]. However, the effective temporal resolution of both polar and geostationary LST observations can bemuch lower than the theoretical value since optical observations, and particularly TIR observations, are masked in presence of cloud cover. SM and LST are closely inter-related and their link is central to land-surface variables and processes such as soil emission, evapotranspiration (ET) and thermal inertia. Soil emission is influenced by the SM and soil physical temperature. Owing to this relationship, LST is a necessary input of any microwave SM retrieval algorithm [16; 17; 18]. LST and SM are also related through ET, which is a fundamental variable in hydrological and water/energy balance studies [19; 20; 21]. According to the latter, there are two different ET regimes: the energy-limited and the water-limited. In the energy-limited regime, SM is above a certain SM value (critical SM) and ET is mainly controlled by the available energy (not by the soil water content). In the water-limited regime, in contrast, SM is below the critical SM and acts as the main driver of the ET process. The critical SM is known to be bounded by the soil wilting point (WP), which is the lower limit of the plant available water, and the soil field capacity (FC), defined as the upper limit of SM available to plants. It typically lies between the 50% to 80% of FC [22]. The thermal inertia describes the resistance of a material to temperature variations and is proportional to the material's thermal conductivity, density and specific heat capacity. In general, a decrease in SM produces an increase in LST diurnal range, and the maximum LST under moist conditions occurs later than under dry conditions (with a time lag within the day) [23]. This is because the specific heat capacity of water is typically higher than that of bare soil and, consequently, dry soils respond faster than wet soils to temperature variations. Several studies have been focused on the indirect estimation of SM using TIR remote sensing [24]. Because the surface brightness temperature depends on its physical temperature and also on the surface water content and vegetation, most of these approaches are based on the so-called triangle or trapezoidal method of the LST/vegetation index space [25; 26; 27; 28]. Usually, it is used with an ET model, e.g., the Atmosphere-Land Exchange Inverse (ALEXI) model [29], or the thermal inertia [30]. An intercomparison study showed that SM estimates from microwave and TIR provide complementary information [31]. Microwave retrievals are optimal for measuring accurate SM under all-weather conditions, but the spatial resolution obtained is limited (around tens of km). TIR retrievals, in turn, provide high resolutions (from tens of meters to several km), but are sensitive to atmospheric effects. To benefit from the advantages of both SM retrieval techniques, there are several SM downscaling or disaggregation algorithms based on the synergy of coarse resolution microwave and high resolution TIR observations [32; 33; 34]. A recent review describes these relationships and the synergistic use of passive microwave and optical Visible/InfraRed (VIS/IR) for multi-scale SM estimation [35]. Most radiometer-based and TIR-based SM retrieval algorithms employ the instantaneous LST in the inversion.