*2.1. Three Split-Window Algorithms*

A. Generalized Split-Window Algorithm by Du et al. [14]

The generalized split-window algorithm Du et al. [14] applied to two Landsat 8 TIR bands expressed as:

$$T\_s = b\_0 + (b\_1 + b\_2 \frac{1 - \varepsilon}{\varepsilon} + b\_3 \frac{\Delta \varepsilon}{\varepsilon}) \frac{T\_{10} + T\_{11}}{2} + (b\_4 + b\_5 \frac{1 - \varepsilon}{\varepsilon} + b\_6 \frac{\Delta \varepsilon}{\varepsilon}) \frac{T\_{10} - T\_{11}}{2} + b\_7 (T\_{10} - T\_{11})^2 \tag{1}$$

where *T*<sup>10</sup> and *T*<sup>11</sup> are the brightness temperatures at the top of the atmosphere (TOA) from TIRS Bands 10 and 11, respectively; ε is the average emissivity of the two bands; ∆ε is the band emissivity difference (∆ε = ε<sup>10</sup> − ε11); and *b<sup>k</sup>* (*k* = 0, 1, . . . , 7) refers to the algorithm coefficients, which are obtained directly from simulation dataset on the basis of the thermodynamic initial guess retrieval (TIGR) [28] atmospheric profile database and the MODTRAN 5.2 [29] atmospheric transmittance/radiance code. Du et al. [14] calculated those coefficients independently on column water vapor (CWV) to reduce the effect of atmospheric water vapor. The CWV was divided into five subranges (0 < CWV ≤ 2.5 g/cm<sup>2</sup> , 2 < CWV ≤ 3.5 g/cm<sup>2</sup> , 3 < CWV ≤ 4.5 g/cm<sup>2</sup> , 4 < CWV ≤ 5.5 g/cm<sup>2</sup> , and CWV > 5.0 g/cm<sup>2</sup> ). This method was also applied on Sentinel-3A [30] and improved by dividing the simulation data into temperature subranges. We refined these algorithm coefficients as Zheng et al. [30] did in two ways to make the evaluation more precise. First, the largest difference between the bottom air temperature (*Tair*) and LST used in the simulation dataset was increased from 20 K in the study of Du et al. [14] to 35 K for a barren or desert surface that probably has high surface temperature. Second, the brightness temperature of Band 10 (*T*10) was divided into several subranges, which were used to determine the coefficients together with the CWV subranges. *T*<sup>10</sup> varies with LST and atmospheric conditions; thus, in accordance with the value ranges of *T*<sup>10</sup> under various CWV subranges, *T*<sup>10</sup> was divided into four subranges for 0 ≤ CWV ≤ 2.5 g/cm<sup>2</sup> as *T*<sup>10</sup> < 270 K, 270 K ≤ *T*<sup>10</sup> < 300 K, 300 K ≤ *T*<sup>10</sup> < 330 K, and *T*<sup>10</sup> ≥ 330 K. For the other four CWV subranges, *T*<sup>10</sup> was divided into *T*<sup>10</sup> < 300 K and *T*<sup>10</sup> ≥ 300 K. Table 1 lists the new algorithm coefficients of Equation (1) for different combinations of CWV subranges and *T*<sup>10</sup> subranges and the root-mean-square error (RMSE) of the predicted temperature compared with the value in the simulation dataset.

#### B. Linear Split-Window Algorithm by Rozenstein et al. [13]

Based on the linear relationship between band radiance and temperature in specified temperature ranges, the linear split-window algorithm proposed by Rozenstein et al. [13] to estimate LST from Landsat 8 TIRS image is expressed as:

$$T\_s = A\_0 + A\_1 T\_{10} - A\_2 T\_{11} \tag{2}$$

where, *A*0, *A*<sup>1</sup> and *A*<sup>2</sup> are algorithm coefficients given by following equations derived from thermal radiative transfer equation [31] and linearizing Planck's radiance function:

$$A\_0 = E\_1 a\_{10} + E\_2 a\_{11} \tag{3a}$$

$$A\_1 = 1 + A + E\_1 b\_{10} \tag{3b}$$

$$A\_2 = A + E\_2 b\_{11} \tag{3c}$$


**Table 1.** The coefficients *b<sup>k</sup>* in different column water vapor (CWV) and brightness temperature of Band 10 (*T*10) intervals, and root-mean-square error (RMSE) of the predicted temperature.

In the above equations, *E*1, *E*<sup>2</sup> and *A* are coefficients determined by pixel emissivity and atmospheric transmittance, and written as:

$$E\_1 = D\_{11}(1 - C\_{10} - D\_{10})/E\_{0\prime} \ E\_2 = D\_{10}(1 - C\_{11} - D\_{11})/E\_0$$

$$A = D\_{10}/E\_{0\prime} \text{ and } E\_0 = D\_{11}C\_{10} - D\_{10}C\_{11} \tag{4}$$

$$\text{with } \mathbf{C}\_{l} = \varepsilon\_{l}\tau\_{l} \text{ and } D\_{l} = (1 - \tau\_{l}) \left[ 1 + (1 - \varepsilon\_{l})\tau\_{l} \right] \tag{5}$$

where, ε*<sup>i</sup>* and τ*<sup>i</sup>* are the pixel emissivity and atmospheric transmittance of TIRS Band 10 or 11, respectively. The atmospheric transmittance τ*<sup>i</sup>* is obtained from the negative correlation with CWV by the results of MODTRAN 4.0 simulations [13], which is the only empirical fitting step in this algorithm. Therefore, different from the generalized split-window algorithm, this linear split-window algorithm is not dependent on the temperature simulation data, but the direct features of atmospheric profiles. This makes more stable simulation results of this linear split-window algorithm; because the process of temperature simulation needs more other parameters besides the atmospheric profile, such as the input LST and emissivity, while the process of transmittance simulation only needs the atmospheric profile.
