# A Practical Split-Window Algorithm for Retrieving Land Surface Temperature from Landsat-8 Data and a Case Study of an Urban Area in China

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## Abstract

**:**

## 1. Introduction

## 2. The SWA for Landsat-8 TIRS

#### 2.1. Theoretical Derivation of the SWA

_{n}(T) denotes the object’s spectral radiance, T its absolute temperature, k the Boltzmann constant, h the Planck constant, and c the speed of light. The units of B

_{n}(T) are Wm

^{−2}sr

^{−1}µm

^{−1}. The general radiance transfer equation [38] for the remote sensing of LST can be formulated as follows in Equation (2):

_{s}is the LST, T

_{i}is the brightness temperature in channel i, ε

_{i}is the ground emissivity and τ

_{i}is the atmospheric transmittance in band i. B

_{i}(T

_{s}) is the ground radiance, and I

_{i}↓ and I

_{i}↑ are the downward and upward path radiances, respectively. I

_{i}↓ and I

_{i}↑ can be expressed by Equations (3) and (4), respectively [27]:

_{a}is the average temperature of the upward radiance of atmosphere. Every term of the radiance transfer equation includes the Planck function. According to Qin’s [39] analysis, few differences result from using T

_{a}instead of ${{T}_{a}}^{\downarrow}$. Therefore, Equation (2) can be written as follows:

_{i}(T

_{i}) can be calculated by substituting the brightness temperature ${T}_{i}$ into the Planck function. The terms B

_{i}(T

_{s}) and B

_{i}(T

_{a}) need to be simplified. In this paper, considering the scope of the application and the desired retrieval accuracy, we applied different simplified fitting methods for B

_{i}(T

_{s}) and B

_{i}(T

_{a}). Because B

_{i}(T

_{s}) has a greater impact on the retrieval precision and it contains inversion parameters, we decided to nonlinearly fit it with a quadratic function, whereas B

_{i}(T

_{a}), which has less impact on the retrieval accuracy, can be linearly fit. Then, Equation (6) can be written as follows:

_{s}, the result can be expressed as

#### 2.2. Determination of Coefficients

#### 2.2.1. Determination of the Fitting Parameters for ${B}_{i}({T}_{s})$ and ${B}_{i}({T}_{a})$

_{i}(T

_{s}) via a quadratic fitting method and B

_{i}(T

_{a}) via a linear simplification approach. To strengthen the universality of the model, the temperature range is set from 180 K–363 K (−90 °C–90 °C) for Landsat-8 thermal infrared bands 10 and 11. The selection of the temperature range is based on a full consideration of the existence of extreme surface temperature regions. The extreme minimum temperature was reported to have reached −71 °C in Oymyakon village in Russia, and the surface temperature reached as high as 82.3 °C in several desert areas in the summer. Based on the TIR spectral response function of Landsat-8, the effective wavelengths [40] and wavenumbers of these TIRS are calculated to be 10.9034 μm and 917.1417608 cm

^{−1}for the minimum temperature, respectively, and 12.0028 μm and 833.1387464 cm

^{−1}for the maximum temperature, respectively. Then, the thermal radiation values can be obtained using Equation (1). The corresponding Planck curve is shown in Figure 1. The quadratic fitting function is shown in Table 1, the linear regression formula is listed in Table 2, and the fitting parameters B

_{i}(T

_{s}) and B

_{i}(T

_{a}) are listed in Table 3.

TIRS | Quadratic Fitting Expression | R^{2} | SEE |
---|---|---|---|

band10 | ${B}_{10}({T}_{s})=0.0006678{{T}_{s}}^{2}-0.2333226{T}_{s}+21.1666266$ | 0.999979 | 0.0309 |

band11 | ${B}_{11}({T}_{s})=0.0006188{{T}_{s}}^{2}-0.1990475{T}_{s}+16.7224278$ | 0.999997 | 0.0117 |

TIRS | Linear Regression Results | R^{2} | SEE |
---|---|---|---|

band10 | ${B}_{10}({T}_{a})=0.1312942{T}_{a}-26.7808503$ | 0.9465 | 1.6404 |

band11 | ${B}_{11}({T}_{a})=0.1387986{T}_{a}-27.7043284$ | 0.9584 | 1.5193 |

TIRS | Fitting Parameters | |
---|---|---|

Quadratic Fitting Parameters | Linear Regression Parameters | |

${B}_{i}({T}_{s})={a}_{i}{{T}_{s}}^{2}+{b}_{i}{T}_{s}+{c}_{i}$ | ${B}_{i}({T}_{a})={k}_{i}{T}_{a}+{d}_{i}$ | |

band10 | a_{10} = 0.0006678 b_{10} = −0.2333226 c_{10} = 21.1666266 | k_{10} = 0.1312942 d_{10} = −26.7808503 |

band11 | a_{11} = 0.0006188 b_{11} = −0.1990475 c_{11} = 16.7224278 | k_{11} = 0.1387986 d_{11} = −27.7043284 |

#### 2.2.2. Determination of Atmospheric Transmittance

^{−2}[41] and 0.10 g·cm

^{−2}, the corresponding atmospheric transmittance evaluation errors are less than 0.031 and 0.016, respectively.

**Table 4.**The water vapour of the atmosphere and the corresponding atmospheric transmittance as simulated by MORTRAN 4.0.

Water Vapour Content (g·cm^{−2}) | TIR10-${\tau}_{10}$ | TIR11-${\tau}_{11}$ |
---|---|---|

0.5 | 0.93542 | 0.89660 |

0.6 | 0.92903 | 0.88448 |

0.7 | 0.92217 | 0.87220 |

0.8 | 0.91483 | 0.85967 |

0.9 | 0.90700 | 0.84686 |

1.0 | 0.89869 | 0.83372 |

1.1 | 0.88990 | 0.82021 |

1.2 | 0.88064 | 0.80637 |

1.3 | 0.87093 | 0.79215 |

1.4 | 0.86076 | 0.77758 |

1.5 | 0.85015 | 0.76266 |

1.6 | 0.83913 | 0.74742 |

1.7 | 0.82769 | 0.73187 |

1.8 | 0.81588 | 0.71603 |

1.9 | 0.80370 | 0.69993 |

2.0 | 0.79117 | 0.68360 |

2.1 | 0.77830 | 0.66706 |

2.2 | 0.76514 | 0.65034 |

2.3 | 0.75168 | 0.63347 |

2.4 | 0.73798 | 0.61649 |

2.5 | 0.72401 | 0.59941 |

2.6 | 0.70983 | 0.58229 |

2.7 | 0.69546 | 0.56512 |

2.8 | 0.68092 | 0.54797 |

2.9 | 0.66622 | 0.53084 |

3.0 | 0.65140 | 0.51378 |

**Table 5.**The cubic fit regression functions of the atmospheric transmittance and water vapour content range of 0.5–3.0 g·cm

^{−2}.

TIRS | Cubic Regression Results | R^{2} | SEE |
---|---|---|---|

band10 | ${\tau}_{10}=0.9570356-0.0277340w-0.0333734{w}^{2}+0.0028800{w}^{3}$ | 0.999999 | 0.0001 |

band11 | ${\tau}_{11}=0.9456728-0.0857755w-0.0290912{w}^{2}+0.0032169{w}^{3}$ | 0.999995 | 0.0003 |

^{−2}, $\alpha =0.020$, $\beta =0.651$, and ${r}_{19}$ and ${r}_{2}$ denote the reflectance of MODIS band19 and band2, respectively.

#### 2.2.3. Determination of the Emissivity of the Ground

_{imix}is the emissivity of the mixed pixels for band10 and band11, ε

_{iv}is the emissivity of the fully vegetated pixels for band10 and band11, ε

_{in}is the emissivity of the non-water and non-vegetated pixels for band10 and band11, and F = 0.55 is a shape factor considering the geometrical distribution [48]. The parameter p

_{v}is the scaled NDVI value, in which NDVI

_{min}and NDVI

_{max}are the NDVI values for non-vegetated and fully vegetated land covers, respectively. Please note that, in recent years, colourful steel plates mainly composed of electro-galvanized steels have been installed in architectural fields, with blue-coloured plates being especially common. As a result, regions with blue roof plates in our study area are relatively prevalent. Because the corresponding NDVI of these blue colour steels is relatively large, generally above 0.20 and sometimes even as high as 0.53, we separated these areas from the 0.20–0.50 range of NDVI to produce amplitude ratio values. According to the ASTER spectral database, the Roofing Materials--Galvanized Steel Metal category features wavelengths of 10.9009 μm and 12.0108 μm and corresponding reflectances of 0.041 and 0.038. Therefore, we calculated the Galvanized Steel Metal emissivity values in band10 and band11 to be 0.959 and 0.962, respectively.

**Table 6.**The emissivity values of water, vegetation and non-vegetation for Landsat-8 TIRS band10, and band11.

ε_{water} | ε_{vegetation} | ε_{non-vegetation} | ε_{Galvanized-Steel} | |
---|---|---|---|---|

TIR-band10 | 0.991 | 0.984 | 0.964 | 0.959 |

TIR-band11 | 0.986 | 0.980 | 0.970 | 0.962 |

## 3. Steadiness Analysis and Validation of the SWA

#### 3.1. Steadiness Analysis of the SWA

#### 3.1.1. Steadiness Analysis of Atmospheric Transmittance

_{10}= 283.0–333.0 K, with an interval of 10.0 K, for a T

_{10}–T

_{11}difference of −3–3 K to calculate the LST estimation error, assuming e

_{10}= 0.967 and e

_{11}= 0.971 and undervaluing the atmospheric water vapour content by 0.1 and 0.2 g·cm

^{−2}, as was done by Rozenstein et al. (2014) The water vapour content range was 1.0–4.0 g·cm

^{−2}.

^{−2}and T

_{10}is 3 K lower than T

_{11}. In addition, for constant values of the other parameters, higher T

_{10}values correlate with higher LST errors. The specific error variation is shown in Figure 2A,B. In the case of underestimating the water vapour content by 0.1 g·cm

^{−2}and 0.2 g·cm

^{−2}, the corresponding maximum LST errors are 0.56 K and 1.11 K, respectively, and the RMSEs of the LST error are 0.30 K and 0.59 K, respectively. The contribution of the calculated atmospheric transmittance error to the retrieved LST is not only related to the water vapour content but also correlates with the LSEs of band10 and band11. The ratio of the LSEs of band10 and band11 is invariant in the analysis process. When the water vapour content column is undervalued by 0.1 and 0.2 g·cm

^{−2}, the corresponding maximum LST errors are 0.21 K and 0.41 K, respectively, and the RMSEs of the LST error are 0.10 K and 0.19 K, respectively, as illustrated in Figure 3A,B.

**Figure 2.**(

**A**) The LST evaluated error (K) in the case of undervaluing the atmospheric water vapour content by 0.1 g·cm

^{−2}for a water vapour content in the range of 1.0–4.0 g·cm

^{−2}and an interval of 1.0 g·cm

^{−2}over six gradations of T

_{10}= 283.0–333.0 K and a T

_{10}–T

_{11}value range of −3.0–3.0K. e

_{10}= 0.967, and e

_{11}= 0.971. (

**B**) The LST evaluation error (K) for undervaluing the atmospheric water vapour content by 0.2 g·cm

^{−2}under the same conditions as in (A).

**Figure 3.**The LST evaluation error (K) in the case of undervaluing the atmospheric water vapour content by 0.1 g·cm

^{−2}(

**A**) and 0.2 g·cm

^{−2}(

**B**) for water vapour contents in the range of 1.0–4.0 g·cm

^{−2}at an interval of 1.0 g·cm

^{−2}over six gradations of T

_{10}= 283.0–333.0 K and a T

_{10}–T

_{11}value of 1 K. The LSE variable is independent.

#### 3.1.2. Stability Analysis of LSE

_{10}and T

_{10}–T

_{11}values and has a negative correlation with LSE, as illustrated in Figure 4A,B. To be specific, the maximum LST calculation error, 0.44 K, occurs when T

_{10}is 330 K, the T

_{10}–T

_{11}value is 3 K and the LSE is 0.900 due to undervaluing by 0.005. In the case of underestimating the LSE by 0.001, the maximum LST calculation error is 0.09 K. In Figure 4A,B, the atmospheric water vapour content was kept constant at 1.5 g·cm

^{−2}. The sensitivity of the LSE to the LST estimation error was analysed as the atmospheric water vapour content varied from 1.0–4.0 g·cm

^{−2}, as seen in Figure 5A,B. The LST calculation error has a positive correlation with the T

_{10}value and has a negative correlation with LSE and water vapour content. The maximum LST calculation error, 0.45 K, occurs when T

_{10}is 333 K, the water vapour content is 1.0 g·cm

^{−2}, and LSE is 0.900 due to undervaluing by 0.005.

**Figure 4.**The illustration of the LST calculation error (K) in the case of undervaluing LSE by 0.001 (

**A**) and 0.005 (

**B**) over four gradations of T

_{10}= 270.0–330.0 K with an interval of 20 K. The atmospheric water vapour content was constant at 1.5 g·cm

^{−2}, and the T10–T11 values were 1.0, 2.0, and 3.0 K. The LSE variable is independent.

**Figure 5.**The illustration of the LST calculation error (K) in the case of undervaluing LSE by 0.001 (

**A**); and 0.005 (

**B**) over six gradations of T

_{10}= 283.0–333.0 K with an interval of 10.0 K. The atmospheric water vapour content varies in the range of 1.0–4.0 g·cm

^{−2}with an interval of 1.0 g·cm

^{−2}, and the T10–T11 value is 1.0 K. The LSE variable is independent.

#### 3.2. Evaluation of the Accuracy of the Proposed SWA

_{i}input parameters for the SWA. Then, according to our SWA and the given input parameters, the LST was calculated. Finally, the LST evaluation errors were obtained by calculating the difference between the given and calculated LSTs. The 90 combination scenes are listed in Table 6. The water vapour content values were 1.0, 2.0, or 3.0 g·cm

^{−2}, the range of the assumed LST values was 10–60 °C, with an interval of 10.0 °C, and the range of the LSE values was 0.940–0.980, with an interval of 0.010. The accuracy assessment results, shown in Table 7, are satisfactory, and the RMSE of the LST estimation errors is 0.51 °C, outperforming the result of 0.93 °C from Rozenstein et al. (2014). To assess the general applicability of this SWA to different atmospheric conditions, we also evaluated the accuracy of this SWA under a tropical model and 1976 US standard atmospheric conditions. The assessment results are listed in Table 8 and Table 9. The RMSEs of the LST errors under the tropical model and the 1976 US standard atmospheric conditions are 0.70 °C and 0.63 °C, respectively.

**Table 7.**Estimation errors of LST for different simulated combinations of specified LST, LSE, and water vapour content values based on the Mid-Latitude Summer atmospheric profile.

Water Vapour Content (g·cm^{−2}) | LST (°C) | LSE = 0.980 | LSE = 0.970 | LSE = 0.960 | LSE = 0.950 | LSE = 0.940 |
---|---|---|---|---|---|---|

1 | 10 | 0.3710 | 0.3641 | 0.3561 | 0.3492 | 0.3413 |

20 | 0.3384 | 0.3302 | 0.3220 | 0.3138 | 0.3056 | |

30 | 0.3993 | 0.3914 | 0.3921 | 0.3691 | 0.3700 | |

40 | 0.5339 | 0.5348 | 0.5221 | 0.5232 | 0.5103 | |

50 | 0.7360 | 0.7350 | 0.7268 | 0.7186 | 0.7178 | |

60 | 0.9933 | 0.9870 | 0.9926 | 0.9864 | 0.9731 | |

2 | 10 | 0.3235 | 0.2868 | 0.2550 | 0.2231 | 0.1897 |

20 | 0.2835 | 0.2605 | 0.2238 | 0.2007 | 0.1635 | |

30 | 0.3388 | 0.3250 | 0.2833 | 0.2586 | 0.2450 | |

40 | 0.4810 | 0.4671 | 0.4374 | 0.4075 | 0.3936 | |

50 | 0.6935 | 0.6682 | 0.6520 | 0.6264 | 0.6102 | |

60 | 0.9515 | 0.9278 | 0.9177 | 0.8939 | 0.8611 | |

3 | 10 | 0.1532 | 0.0701 | −0.0164 | −0.1170 | −0.2146 |

20 | 0.1456 | 0.0527 | −0.0281 | −0.1099 | −0.1927 | |

30 | 0.2104 | 0.1390 | 0.0603 | −0.0194 | −0.0933 | |

40 | 0.3684 | 0.2865 | 0.2227 | 0.1388 | 0.0734 | |

50 | 0.5708 | 0.5059 | 0.4403 | 0.3739 | 0.3067 | |

60 | 0.8190 | 0.7716 | 0.7073 | 0.6423 | 0.5933 |

**Table 8.**Estimation errors of LST for different simulated combinations of specified LST, LSE, and water vapour content values based on the Tropical model atmospheric profile, RMSE = 0.70 °C.

Water Vapour Content (g·cm^{−2}) | LST (°C) | LSE = 0.980 | LSE = 0.960 | LSE = 0.940 |
---|---|---|---|---|

1.0 | 20 | 0.3130 | 0.2958 | 0.2766 |

40 | 0.5476 | 0.5326 | 0.5163 | |

60 | 1.0371 | 1.0250 | 1.0122 | |

2.0 | 20 | 0.3078 | 0.2415 | 0.1730 |

40 | 0.5845 | 0.5282 | 0.4723 | |

60 | 1.1176 | 1.0713 | 1.0220 | |

3.0 | 20 | 0.2281 | 0.0416 | −0.1486 |

40 | 0.6080 | 0.4500 | 0.2914 | |

60 | 1.2313 | 1.0958 | 0.9587 |

**Table 9.**Estimation errors of LST for different simulated combinations of specified LST, LSE, and water vapour content values based on the 1976 US Standard atmospheric profile, RMSE = 0.63 °C.

Water Vapour Content (g·cm ^{−2}) | LST (°C) | LSE = 0.980 | LSE = 0.960 | LSE = 0.940 |
---|---|---|---|---|

1.0 | 20 | 0.3893 | 0.3836 | 0.3762 |

40 | 0.5563 | 0.5511 | 0.5439 | |

60 | 0.9996 | 0.9940 | 0.9893 | |

2.0 | 20 | 0.3129 | 0.2704 | 0.2288 |

40 | 0.4576 | 0.4206 | 0.3851 | |

60 | 0.8739 | 0.8440 | 0.8145 | |

3.0 | 20 | 0.2749 | 0.1358 | −0.0071 |

40 | 0.5500 | 0.4310 | 0.3118 | |

60 | 1.0849 | 0.9823 | 0.8787 |

## 4. A Case Study of an Urban Area in China

#### 4.1. Study Area

^{2}and has approximately 4.41 million inhabitants (China, the data of the Sixth Population Census, 2010). The region features a warm continental monsoon climate.

#### 4.2. Data Used

#### 4.3. LST Retrieval

**Figure 7.**The LST Spatial distribution of the study urban area retrieved from Landsat-8: (

**A**) on June 27, 2013, and (

**B**) on June 30, 2014.

## 5. Conclusions

_{10}–T

_{11}change together, as is seen in Figure 2B, especially when the water vapour content is 0.5–1.0 g·cm

^{−2}. Based on the validation results of the simulation data generated by MODTRAN 4.0, the SWA performs very well. The verification result is satisfactory in that the maximum and minimum LST errors are 0.993 K and 0.016 K, respectively, and the LST error RMSE is 0.51 K under Mid-Latitude Summer atmospheric conditions. The LST error RMSEs under Tropical and 1976 US standard atmospheric conditions are 0.70 K and 0.63 K, respectively. The assessment results demonstrate that the SWA is not only steady and accurate but also universal for different atmospheric conditions.

## Acknowledgements

## Author Contributions

## Conflicts of Interest

## References and Notes

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**MDPI and ACS Style**

Jin, M.; Li, J.; Wang, C.; Shang, R.
A Practical Split-Window Algorithm for Retrieving Land Surface Temperature from Landsat-8 Data and a Case Study of an Urban Area in China. *Remote Sens.* **2015**, *7*, 4371-4390.
https://doi.org/10.3390/rs70404371

**AMA Style**

Jin M, Li J, Wang C, Shang R.
A Practical Split-Window Algorithm for Retrieving Land Surface Temperature from Landsat-8 Data and a Case Study of an Urban Area in China. *Remote Sensing*. 2015; 7(4):4371-4390.
https://doi.org/10.3390/rs70404371

**Chicago/Turabian Style**

Jin, Meijun, Junming Li, Caili Wang, and Ruilan Shang.
2015. "A Practical Split-Window Algorithm for Retrieving Land Surface Temperature from Landsat-8 Data and a Case Study of an Urban Area in China" *Remote Sensing* 7, no. 4: 4371-4390.
https://doi.org/10.3390/rs70404371