Land surface temperature （LST） is an important variable in climate and environmental research，which has been widely used in global climate change^［1-2］，drought monitoring and warning^［3］，urban heat island^［4-6］，urban livable assessment^［7］，satellite precipitation^［8］，hydrological and ecological effects^［9］，crop growth monitoring and assessment^［10］. The development of high precision temperature data sets with the appropriate resolution is of great significance to any kind of ecological environment research worldwide.

Many researchers have developed algorithms based on surface temperature inversion. Major LST inversion algorithms include RTE^［11-12］，which is a direct method for LST retrieval using a single TIRS band，the parameters required for inversion mainly include the atmospheric transmittance，surface reflectivity，atmospheric upwelling radiation and atmospheric downwelling radiation. The process calculation of RTE is complicated，but the accuracy of the obtained result is relatively high. The SC algorithm^［13-15］ includes two calculation methods. The first one is based on the calculation of atmospheric water vapor parameters，the calculation process is simple，but the calculation accuracy is poor. The second method is based on the calculation of atmospheric transmittance，atmospheric upward radiation，atmospheric downward radiation and other parameters. The calculation process of this method is relatively simple and the calculation result has a relatively high precision，so it is also a common method for USGS to produce surface temperature data. The MW algorithm^［16］ is an algorithm developed specifically for Landsat data，which needs to use parameters including surface emissivity，atmospheric transmittance，and average atmospheric temperature. Due to the huge variation of the estimated mean atmospheric temperature，the inversion accuracy of the MW algorithm is relatively poor. The SW algorithm^{［13，17-19］} uses two thermal infrared bands for calculation. Because the eleventh band calibration of Landsat is not stable，the inversion accuracy of this algorithm is the worst compared with the above.

In these LST inversion algorithms，multiple basic parameters need to be input^［20］. Multiple parameters are estimated variables，not standard variables. Each algorithm is tested on the satellite data for which it is applicable，and sometimes not always applicable faced with new satellite data. For example，the split window algorithm is not suitable for Landsat 8 data，because the calibration of the 11th band is not accurate. It is therefore necessary to evaluate land surface temperature inversion for each new type of satellite data^［21］.

The study area was chosen to be covered by both Landsat 8 and Landsat 9 data，shown in Fig. 1. The data track number was 123，032. It mainly covered Beijing. We selected Landsat 8 data with imaging date of November 26，2021，and Landsat 9 data with imaging date of November 22，2021 （test data），which were the two images closest to the USGS release date，four days apart. For four days we don't think anything has changed on the surface. The main variation is the influence of atmospheric aerosols and other parameters on temperature inversion，and the data needs to be normalized in the comparative study.

Accurate measurements from 20 meteorological stations （marked in green in Fig. 1） were selected as the true values. The temperature measuring device at the weather station was located 1.5 meters above the ground and recorded data every hour. It is necessary to convert the measured data from the meteorological station to ground data when using the measured data from the meteorological station.

In the inversion of surface temperature in this study，LSE，atmospheric transmittance，upward radiation，downward radiation，atmospheric water vapor content，atmospheric average temperature，and other parameters required by various inversion algorithms are obtained by USGS using interpolation measurements from various stations around the world. A description of the parameter dataset can be found in the USGS official documentation.

In the data pre-processing stage，the atmospheric correction was mainly carried out on the selected Landsat-8 and Landsat-9 level-1 product data. Then，we combined five commonly used LST inversion algorithms using land surface emissivity，upward radiation，downward radiation，atmospheric water vapor，and average temperature. LST inversion was performed for the corrected atmospheric data. Secondly，we fitted the five temperature inversion results with the in-situ measurement results of weather stations to compare the accuracy of the five algorithms. The sensitivity of each dependent parameter of the inversion algorithm was tested by controlling the parameters with an equal step size. Finally，we classified the study area，and several pixels were randomly selected for data statistics in each category in the study area and we measured the stability of each inversion algorithm on Landsat-8 and Landsat-9 data according to the mean and standard deviation. The overall process is shown in Fig. 2.

Five LST inversion schemes are discussed in this study，which are shown in Table 1.

LST1→ Radiative transfer equation （RTE） is a method of surface temperature inversion using a single thermal infrared band. This can be given by Eq. （1）：

whereLλ(w·m-2·sr-1·µm-1) is the brightness value of the bandλ. BTs is the blackbody radiance energy. For more detailed parameter description，please read the original literature^{［11-12，22］}.

LST2→ The Single channel （SC） algorithm can be expressed by Eq. （2） and Eq. （3）：

where bγ is equal to 1 320 K for Band 10. ψ1,ψ2,ψ3 are functions of water vapor content （w）. For more detailed parameter description，please read the original literature^［14］.

LST3→ When w in the SC algorithm is greater than 3g·cm-2，Jiménez-Muñoz suggests using Eq. （4） to calculate ψ1,ψ2andψ3. This algorithm is the USGS official method for producing LST datasets.

LST4→ Some researchers refer to the split window （SW） algorithm of MODIS satellite and transfer it to Landsat data，which can be calculated by Eq. （5）：

where T10 and T11 are the brightness temperatures. ∆ε is the LSE difference of Band 10 and Band 11. For more detailed parameter description，please read the original literature^［13］.

LST5→ The Mono Window （MW） algorithm can be expressed by Eq. （6）：

where a=-67.35 and b=0.45 of Band 10，Ta is the mean temperature. C=ε⋅τ and D=1-τ1+1-ε⋅τ. For more detailed parameter description，please read the original literature^［23］.

We fitted the five kinds of inversion LST with the temperature measured in situ by the weather station （from near-surface temperature to LST）. The T-based technique was used to evaluate the fitting data^［23-26］. Root mean square error （RMSE） and average BIAS^［27］ were used as evaluation criteria in this study. The formulas of these metrics are given by：

where TLandsat and TStation are the Landsat-8 and Landsat-9 derived LST and in-situ LST，respectively，and n represents the number of in-situ measurements. In this study，in-situ measurement data of 20 meteorological stations were used，so n=20.

Sensitivity analysis of model parameters is an application of a model output error （fuzzy approximation，large number，statistical or other） that is inversely partitioned and inversely assigned to different sources of uncertainty in the model input^［28］. We consider the inversion parameters of surface temperature in different places as independent variables and separately control the changes of each parameter to observe the influence of the changes of various parameters on the results. First，assume that with the image after the atmospheric correction by one pixel，the DN value is a fixed value. Then，according to experience，the parameters are selected as control variables，and finally the sensitivity of each parameter is analyzed. The selected length and step size of each parameter are shown in Table 2.

To make the results comparable under the same measurement，we normalized the surface temperature. The following equation is utilized：

whereSe is the LST difference calculated for each increase in step size； Tsx+∆x and Tsx refer to the LST calculated for “x+∆x” and “x”，respectively.

The stability of five inversion algorithms was discussed，and the inversion results on different land use types were selected for statistical analysis. The random forest method was used to classify land use in the study area. Since the accuracy of classification directly affected the test results，the overall classification accuracy was required to be higher than 90%. Considering the spatial resolution of Landsat data and the separability and high precision requirements of land cover，as well as the subsequent research on the thermal environment using surface temperature，we divided land use in the study area into seven categories： water，vegetation，dark buildings，bright soil，dark soil，and high reflectivity buildings. Spectral statistics and analysis were performed for each category，and the specific classification sample selection and classification process，please refer to our previous literature^［29］.

Five temperature retrieval algorithms were used to retrieve LST from the Landsat-8 and Landsat-9 data，the results are shown in Fig. 3. It can be seen that the results of each LST inversion algorithm and the graphical trend of the measured data are consistent. This is consistent with our common knowledge that the temperature drops by an average of 6 ℃ for every 1000 meters of elevation. In Fig. 3，the measured data of the high-altitude site is displayed in a concave shape with the adjacent data.

The inversion result values of five temperature inversion algorithms were used to linearly fit the measured temperature values. The fitting results are shown in Fig. 4. It can also be seen from the statistical fitting results that the fitting results of LST1 and LST3 are relatively ideal，and the fitting slope is around 0.7 and R² is around 0.6. The best fitting between the model inversion results and the measured results is the mono-window algorithm，which is ideal both in the fitting slope and R²，and which is the main reason why USGS adopts the MW algorithm. It is worth noting that the fitting slope of LST4 is above 0.9，but R² is around 0.45. This phenomenon may be caused by the lack of sample point data on the one hand. Two groups of pixel points are close to the measured points，but the inversion results of some pixel points differ greatly from the measured results of the station. This shows that the stability of the model is poor and the estimation result is not ideal. In addition，the LST2 method based on atmospheric water vapor content parameters and the mono-window algorithm for surface temperature inversion has a lower fitting slope and R²，which further illustrates the instability of the inversion algorithm based on atmospheric water vapor content parameters.

From the accuracy of the algorithm inversion results and the sensitivity analysis of the parameters in the algorithm，the RTE and SC algorithms calculated using the LSE parameters are better than other algorithms. The MW algorithm yields slightly higher retrieval results than the measured data，and the SW algorithm yields a large difference from the measured data. This may be related to the unstable radiometric calibration in Band 11 of the Landsat-8 TIRS. In addition，the calibration parameters of the 11th band of Landsat-9 are still being tested. It is hoped that USGS will provide more accurate calibration parameters in the future，and calculate atmospheric influence through two thermal infrared bands to obtain a more accurate surface temperature.

The inversion temperature of each algorithm was normalized，and the sensitivity of the parameters in the algorithm was analyzed by controlling variables，and the analysis results of each parameter are shown in Fig. 5.

It can be seen from Fig. 5（a） that the atmospheric transmittance variable has a logarithmic function relationship with the inversion result values of the RTE，SC and MW algorithms. As the atmospheric transmittance variable increases，the retrieved surface temperature values gradually decrease. From the curvature shown in the image，the values of the retrieved results of the RTE algorithm change rapidly with the atmospheric transmittance，while the values of the retrieved results of the SC and MW algorithms change slowly with the atmospheric transmittance. In Fig. 5（b），the inversion temperature values of various LST inversion algorithms gradually decrease with the increase of upward radiation parameters. However，the effect of ascending radiation parameters on the LST1 algorithm presents a logarithmic curve，and the effect on the LST3 algorithm presents a linear relationship. In Fig. 5（c），the inversion temperature value of various LST inversion algorithms gradually decreases with the increase of the descending radiation parameter value. However，the influence of down-radiation parameters on LST1 and LST3 shows a linear relationship. The LST inversion results decrease by 0.3 units with each unit increase of downward radiation. As can be seen from Fig. 5（d），the inversion temperature values of various LST inversion algorithms gradually decrease with the increase of ground object-specific emissivity. The effects of specific emissivity parameters on all inversion algorithms show a linear relationship. For every 0.1 units increase in specific emissivity，the retrieved LST decreases by 1 unit. In Fig. 5（e），the inversion values of LST2 and LST4 gradually increase with the increase of the atmospheric water vapor content. However，the parameters of atmospheric water vapor content show a logarithmic trend to LST2 and a linear relationship to LST4. As can be seen from Fig. 5（f），the inversion value of the LST5 algorithm gradually decreases with the increase of average surface temperature parameters. The influence of average temperature parameters on LST5 shows a linear relationship，and the inversion MW value of average surface temperature decreases by 0.01 units when the average surface temperature increases by 1 unit.

The random forest algorithm was used to classify the land cover in the study area，and then five inversion LSTs were superimposed. The maximum value，minimum value，mean value，and standard deviation of different temperature inversion methods in different types of two images were calculated to indicate the stability of temperature inversion with different temperature inversion methods in different ground covers. We selected 100 pure pixel points in each category and carried out the ground object verification with the data taken by an unmanned aerial vehicle on the imaging day. The statistical results are shown in Table 3-7 and Fig. 6.

As can be seen from Table 3-7 and Fig. 6，the inversion temperature of the five LST inversion algorithms on the water body and vegetation is the most stable，and the percentage error is relatively low among all land cover types. This may be because，on the one hand，the water body and vegetation area are relatively stable，and the temperature change will not be disturbed by too many human factors； on the other hand，the water body and vegetation have relatively high separability. All inversion algorithms for high reflectivity building inversion results are the most unstable，its percentage error is the largest，the possible reason is that high reflectivity buildings are generally factories or large warehouses，its production operation will generate large heat，stop operation will rapidly lose heat，resulting in large fluctuations in statistical data.

All LST inversion algorithms for Landsat data have certain errors. Such errors can only be minimized，not eliminated. Since the required parameters in the LST inversion algorithm are not exact values，and some parameters need to be estimated initialized，it is very necessary to explore the disturbance of each parameter on the results. All the algorithms that worked for Landsat-8 also worked for Landsat-9 data. However，unlike Landsat-8 data，the radiometric calibration file of Band 11 of Landsat-9 has not been updated yet. We expect USGS to provide more accurate calibration parameters so that we can use the data of the two channels to constrain each other to obtain higher precision temperature inversion results.

Considering the accuracy and parameter sensitivity of the inversion algorithm，the RTE and SC algorithms using LSE parameters have high accuracy and good algorithm stability. The inversion results of the SC algorithm and SW algorithm based on the calculation of atmospheric water vapor content parameters fit poorly with the measured data. The MW algorithm based on LSE and average atmospheric temperature parameters fit poorly with the measured results. Therefore，we believe that the results obtained by the LSE parameter inversion algorithm are better than those obtained by the atmospheric water vapor content parameter inversion algorithm. By comparing LST1 and LST5，it can be seen that the LST5 algorithm has one more atmospheric average temperature estimation parameter than the LST1 algorithm，leading to a poor fitting effect on inversion results，which may be due to the negative impact of excessive uncertain parameters on results. We strongly recommend LST1 and LST3 algorithms for the LST inversion from Landsat data. Even though it is not possible to find atmospheric profiles （radiosonde data，etc.） in place at any time and in any place，this use （using ACPC to simulate atmospheric profile information） can affect the accuracy of the method，but from our results and the literature^{［5，12，14-16，18，30-32］} ，NASA's ACPC provided an inversion algorithm with high accuracy. Wang^［33］ found that SW had the lowest sensitivity to input parameter errors，but the inversion accuracy was not as high as other algorithms. The relationship between the dependence of the inversion algorithm on the parameters and the accuracy of the result needs to be further explored. In a humid environment，parameter error has little effect on the results. Xu^［34］ compared the inversion results of SC and SW algorithms and concluded that the inversion results of the SC algorithm were significantly better than those of SW，especially when the atmospheric water vapor content was more than 2.5 g·cm-2.

In the comparative analysis of all LST inversion model results，error tracing is very necessary. Through error tracing，we can reverse calculate which parameters the error mainly comes from and which parameters have high sensitivity in model calculation. In the subsequent calculation process，various considerations can be taken to reduce the error accumulation. According to the inversion results of different temperature inversion algorithms on different data and different land cover types，we can see that the inversion percentage error of the same inversion algorithm and the same land cover type on Landsat-9 is smaller than that on Landsat-8，indicating that the data quality of Landsat-9 has been improved.

The algorithm for Landsat-8 can also be applied to Landsat-9 data. The calculation process of the SC （LST3） algorithm is a little simpler than that of RTE （LST1），but there is little difference in accuracy between the two algorithms. The RTE algorithm and SC algorithm based on LSE parameters are superior to other algorithms in terms of both accuracy of results and sensitivity to parameters. The retrieval results of the SC （LST2） algorithm and SW （LST4） algorithm based on the atmospheric water vapor retrieval are higher than the measured temperature. The inversion effect of the MW （LST5） algorithm based on average temperature parameters is not particularly ideal. This phenomenon shows that among all the current surface temperature inversion algorithms，the accuracy of surface temperature inversion based on the single window algorithm is the highest，which is also the algorithm used in the advanced products released by USGS. However，the starting point of the split window algorithm is to eliminate the error caused by atmospheric influence with the help of two thermal infrared channels，so as to obtain higher inversion accuracy. But the actual result is the opposite. This may be due to unstable radiometric calibration of Landsat-8 TIRS Band 11. Calibration parameters for Band 11 of Landsat-9 are still being tested. It is hoped that USGS will provide more accurate calibration parameters in the future and calculate atmospheric effects through two thermal infrared bands to obtain more accurate surface temperatures.

With the same inversion algorithm and the same ground cover type，the inversion percentage error on Landsat-9 is smaller than that on Landsat-8，indicating that the data quality of Landsat-9 has been improved. From the inversion results of different inversion algorithms on the same data，the results of water and vegetation have good stability.