### Methods in which LSE is assumed to be known beforehand

**1. Single channel methods**

Methods that use one single satellite channel imply a simple inversion of the RTE. As mentioned earlier, the emissivity is assumed to be known, as is information about atmospheric composition. Currently atmospheric profiles from numerical prediction models forecasts and analyses are used as alternatives to radio soundings, which have a poor network density. Nevertheless, this has proven to produce large errors in LST retrieval (Chédin et al, 1985). On the other hand, the use of satellite sounders provides more accurate information. The problem is that there are not many instruments that have a thermal imager with sounders onboard, so satellite sounding data is not widely used. Single channel methods rely on the use of Radiative Transfer Models (RTMs) to simulate radiance at the TOA, so the method is also dependent on the accuracy of the RTM.

Figure 12 gives a schematic view of how single channel methods work: A dataset of brightness temperatures at the top of the atmosphere is simulated with a RTM using prescribed values of (LST, ε_{λ}) and atmospheric profiles. The obtained Lookup Tables (LUTs) are used to estimate LST, which corresponds to the sensor's BT.

**Fig. 12**: A schematic of the general principle of single channel methods for LST retrieval.

**2. Multi-channel methods**

There are many sensors with more than one channel in the TIR suited to retrieving LST from satellites, such as those onboard MODIS, AVHRR, AATSR, MSG, GOES or Himawari. One of the most popular methods uses two channels centered at about 11 μm and 12.0 μm to account for the atmospheric effect. The difference in TOA radiances measured in channels near the TIR and with distinct atmospheric transmissivities gives an estimate of the atmospheric water vapor content. This technique, known as Split Windows (SW) was first proposed to estimate surface temperature over the ocean, where emissivity is homogeneous and can be assumed to be one with little detriment to LST calculations. For land surfaces, however, emissivity is highly variable spatially; it depends on the emissivity distribution within the pixel, the spectral range of measurement, and view angle. Yet, given the success of the SW technique, it has also seen widespread use over land. These algorithms were first developed for specific regions. Later, Wan and Dozier (1996) proposed an algorithm that can be applied to different regions of the globe and under distinct atmospheric conditions, named Generalized Split Windows (GSWs). Over the years several different GSW formulations have been proposed. These types of algorithms rely on the linearization of the RTE with respect to the temperature or the wavelength. A typical linear SW algorithm can be written as:

LST = a_{0} + a_{1}T_{i} + a_{2}(T_{i} - T_{j}) **(eq.12)**

where a_{k} (k=0, 1, and 2) are coefficients that depend primarily on the spectral response function of the two channels (i and j), the two channels' emissivities, water vapor content, and the satellite's viewing zenith angle (VZA).

There are also SW algorithms that rely on non-linear simplifications of the RTE, giving rise to formulations of the type:

LST = c_{0} + c_{1}T_{1} + c_{2}(T_{i} - T_{j}) + c_{3}(T_{i} - T_{j})^{2} **(eq.13)**

where c_{k} (k=0-3) are coefficients pre-determined by regressing this equation with simulated satellite data values for a set of atmospheres and surface parameters.

**Linear or non-linear multi-channel algorithms?**

When more than 2 TIR channels are available, the LST can be estimated from a linear or non-linear combination of the TOA brightness temperatures in those channels using methods similar to the SW algorithms. As an example, the linear multi-channel formulation proposed by Sun and Pinker (2003) uses 3-channels (2 SW and the Middle Infrared (MIR) 3.9 μm) to estimate LST. It derives emissivity from the different surface types that cover a pixel and uses the MIR 3.9 μm channel to improve the atmospheric correction during night.

**3. Multi-angle methods**

Similarly to the SW, multi-angle methods rely on the estimation of atmospheric effects. It is assumed that in a given spectral channel an object is observed from different viewing angles, giving differing absorption values due to the distinct optical paths. It was primarily developed for the ATSR (Advanced Track Scanning Radiometer) onboard ERS -1, the first sensor operating in biangular mode, i.e., capable of providing 2 views of the same scene within about two minutes: at near-nadir (0°-22°) and at forward view under 55°.