# Estimation of Land Surface Temperature under Cloudy Skies Using Combined Diurnal Solar Radiation and Surface Temperature Evolution

^{*}

## Abstract

**:**

_{cloud}) from diurnal NSSR and surface temperatures is proposed. Validation is performed against in situ measurements that were obtained at the ChangWu ecosystem experimental station in China. The results show that the root-mean-square error (RMSE) between the actual and estimated LSTs is as large as 1.23 K for cloudy data. A sensitivity analysis to the errors in the estimated LST under clear skies (T

_{clear}) and in the estimated NSSR reveals that the RMSE of the obtained T

_{cloud}is less than 1.5 K after adding a 0.5 K bias to the actual T

_{clear}and 10 percent NSSR errors to the actual NSSR. T

_{cloud}is estimated by the proposed method using T

_{clear}and NSSR products of MSG-SEVIRI for southern Europe. The results indicate that the new algorithm is practical for retrieving the LST under cloudy sky conditions, although some uncertainty exists. Notably, the approach can only be used during the daytime due to the assumption of the variation in LST caused by variations in insolation. Further, if there are less than six T

_{clear}observations on any given day, the method cannot be used.

## 1. Introduction

_{cloud}) are urgently needed. Because of the absorption of surface emission by clouds, T

_{cloud}cannot be calculated directly from remotely sensed thermal-infrared information. Presently, only microwave remote sensing can be used to obtain T

_{cloud}because it is able to penetrate clouds. Jia et al. [7] retrieved LST data based on passive-microwave remotely sensed data and achieved an accuracy of 3 K relative to the MODIS LST product. However, these methods of using microwave observations are limited because microwave remote sensing is very sensitive to surface roughness and surface moisture.

_{clear}) pixels within 100–300 Km or within two days. The method is limited if the clear and cloudy pixels are not homogeneous and the atmospheric conditions are non-uniform. To overcome this weakness, Lu et al. [8] calculated T

_{cloud}, which is interpolated from temporal-based neighboring-pixel T

_{clear}observations and is compared using the spatial-based neighboring-pixel method. The result shows that the temporal “neighboring-pixel” method is better than a spatial approach, and the absolute error is within 1.5 K. However, one disadvantage of this approach is inevitable. Specifically, T

_{cloud}is interpolated from T

_{clear}of temporally neighboring pixels, while the difference in net solar shortwave radiation (NSSR) in the proposed method is obtained from spatially neighboring pixels.

## 2. Method

^{2}∙s

^{−1}) and T(z,t) represents the soil temperature at a distance (z) below the surface at time (t). Under a specific set of initial and boundary conditions, the model gives the surface-subsurface temperature profiles as a function of time. Initial conditions are specified 24 h or more before the time to weaken the dependence of the model results on initial values. The lower boundary condition is that the temperature is constant at a depth of 50 cm and the upper boundary condition is the energy balance equation which is listed as Equation (2) [9].

^{−}

^{2}), L

_{net}is the net longwave radiation from ground and air (W∙m

^{−}

^{2}), H is sensible heat transfer between ground and air (W∙m

^{−}

^{2}), LE is the latent heat transfer between ground and air (W∙m

^{−}

^{2}) and G is heat conducted in the soil or rock unit (W∙m

^{−}

^{2}).

_{d}is the time at which the surface temperature (z = 0) reaches its maximum and w is the angular diurnal frequency of surface temperature in a period (which is nearly π/DD (DD is the duration of daytime)).

^{−}

^{1}∙K

^{−}

^{1}), P is the thermal inertia (W∙s

^{1/2}∙m

^{−2}∙k

^{−1}) and the other parameters are the same as those in Equation (3).

_{n}is the net radiation (W∙m

^{−}

^{2}), ε

_{s}is the surface emissivity, L

_{atm}

_{↓}is the atmospheric downward radiation, λ is the latitude, δ

_{s}is the solar declination angle, NSSR is the net shortwave solar radiation (W∙m

^{−}

^{2}), w

_{1}is the angular diurnal frequency of solar radiation in a period that is nearly π/DD (DD is the duration of daytime), t

_{s}is the time (local time 12 h in general) at which the NSSR is maximized (NSSR

_{max}).

_{min}= (1 − A)S τsin(λ)sin(δ

_{s}), approximately represents the minimum daytime NSSR, and S

_{max}= (1 − A)S τcos(λ)cos(δ

_{s}), approximately represents amplitude of the daytime NSSR, w

_{1}and t

_{s}are the same as those in Equation (5).

_{min}, S

_{max}, w

_{1}and t

_{s}), we defined the model as diurnal solar cycle model (DSC).

_{0}are the two parameters to be defined: approximately represents the minimum temperature (T

_{min}), while T

_{0}represents the amplitude (approximately T

_{max}− T

_{min}, where T

_{max}is the maximum daytime LST). Furthermore, t

_{rs}is the starting time of the attenuation (near sunset), β is the decay coefficient during the nighttime, t

_{d}and w are the same as those in Equation (3).

_{atm↓}− H – LE = aT + b and that the surface longwave radiation function is linearized in the vicinity (T

_{i}), the following function can be derived by simple mathematical manipulation of Equations (4)–(6):

_{1}is the same as w in Equation (9), i.e., w

_{1}= w, and substituting Equation (7) into Equation (9), the daytime surface temperature (T) can be expressed as follows:

_{d}− t

_{s})) with T

_{0}as follows:

_{max}, t

_{s}and w) in Equation (6) and two parameters (T

_{0}and t

_{d}) in Equation (8) can be estimated by fitting Equations (6) and (8) using least square method, then P can be obtained using w × (t

_{d}− t

_{s}), T

_{0}and S

_{max}with Equation (12).

_{d}) often lags behind the time at which the NSSR is maximized (t

_{s}). Meanwhile, the amplitude of the variation in surface temperature is also affected by thermal inertia. When the sky is cloudy, the reduced incoming solar radiation causes a drop in the LST; but, the time in temperature change will be postponed, perhaps lasts t

_{d}− t

_{s}, and the amplitude of the temperature change increases incrementally (from 0 to 1). Assuming that the LST variations are caused by variations in insolation (ΔS), which is related to cloudiness, at the same time, the amplitudes of the LST variations are related to thermal inertia (P). Considering ΔS is less than 500 W∙m

^{−}

^{2}, and P changes from 400 to 4000 W∙s

^{1/2}∙m

^{−}

^{2}∙k

^{−}

^{1}in most situations, Lu et al. [8] also pointed out the ratio of solar radiation change and temperature change is between 30 and 300. So, the amplitudes of the LSTs variations are enlarged by a factor of 10. The method to calculate the daytime LST under cloudy skies combining the diurnal solar radiation and surface temperature is proposed as follows:

_{cloud}= T

_{clear}– 10 × ΔS/P

_{cloud}is the LST under cloudy skies, T

_{clear}is the LST under cloud-free skies (estimated using Equation (8)), and ΔS is the difference in the NSSR values between clear or cloudy skies. Considering the lag between LST variations and insolation variations, ΔS cannot attain a maximum once a cloud appears. Instead, the value reaches a maximum at some time after a cloud appears (t

_{d}− t

_{s}). Moreover, the degree of the influence increases incrementally (from 0–1). Therefore, ΔS is given as follows:

_{actual}is the actual NSSR at t time, S

_{fit}is fitted using Equation (6) which means NSSR under cloud-free skies (NSSR

_{clear}) at t time, and t is time.

_{max}, t

_{s}and w) in Equation (6) and two parameters (T

_{0}and t

_{d}) in Equation (8) can be estimated by fitting Equations (6) and (8) using least square method, then P can be obtained using w × (t

_{d}− t

_{s}), T

_{0}and S

_{max}with Equation (12). Meanwhile, T

_{clear}at any time during the daytime can be estimated using Equation (8) with six parameters (T, T

_{0}, w, t

_{d}, t

_{s}, and β which are inversed using more than six observations with least square method). Similarly, NSSR

_{clear}can be also obtained using Equation (6), and ΔS can be calculated after NSSR

_{clear}minus observed NSSR, in the end, substituting these variables (T

_{clear}, ΔS and P) to Equation (13), T

_{cloud}can be calculated.

## 3. Data

#### 3.1. Data from Field Experiments

#### 3.2. Satellite Data

## 4. Results and Discussions

#### 4.1. LST under a Cloudy Sky

#### 4.1.1. Determining Parameters in the DTC Model

_{clear}observations. However, the six T

_{clear}observations must be distributed throughout the day, i.e., they cannot be concentrated in the morning or afternoon. Figure 2 displays the fitted results for six days under various cloud conditions. The black points denote the measured LST under cloud-free skies, and the black hollow points denote the LST fitted using Equation (8). The root-mean-square error (RMSE) between the measured LST and the predicted LST using Equation (8) are within 2 K. The fit is suitable for the daytime but not for the nighttime. Although the overcast time is different among the six days, the DTC model can be used to describe the daily LST evolution, even when most of the daytime is cloudy (e.g., 8 June in Figure 2).

**Figure 2.**Comparison of the measured daily temperature evolution with that predicted by the DTC model for six days under various cloud conditions.

#### 4.1.2. Determining the Parameters in the DSC Model

_{clear}). Considering that clouds suppress the amount of solar radiation reaching the surface, it is assumed that the fitted NSSR is equal to or slightly higher than the actual NSSR. Therefore, the RMSEs between the measured and predicted NSSR are large; most RMSEs exceed 170 W∙m

^{−2}. However, the fitted curve reflects the changes in the NSSR in one day and suggests that the DSC model described in Equation (6) can predict the diurnal NSSR evolution on clear days. Just as shown in Figure 2 and Figure 3, measured diurnal LST curve is smoother than measured diurnal NSSR curve, it is because LST will display the slow process of change and the time in LST change lag the time in the reduced incoming solar radiation under the influence of thermal inertia.

**Figure 3.**Comparison of the measured daily NSSR evolution with that predicted by the DSC model for six days under various cloud cover conditions.

#### 4.1.3. Estimation of LST Under a Cloudy Sky

_{cloud}is calculated by the proposed method after obtaining T

_{clear}, NSSR

_{clear}and thermal inertia using Equations (6), (8) and (13), and the results for the six days are shown in Figure 4. The black points denote the measured LST under cloud-free skies, the red solid points denote the measured LST under cloudy skies, and the red hollow points denote the estimated LST using the proposed method. The RMSE values of the measured and estimated LST are 1.2, 1.8, 1.05, 1.2, 0.8 and 0.72 K for April 17, June 8, June 15, June 22, August 21 and September 28, respectively. Compared with the other five days, the overcast time on 8 June was much longer, which influenced the fit (Equation (8)) shown in Figure 2. As a result, the absolute errors of the measured LST and the predicted LST using Equation (13) on 8 June are higher than those on the other five days. Compared to the RMSEs between the measured and estimated LSTs under cloud-free conditions (Figure 2), the RMSE between the measured and estimated LSTs is smaller on 28 September. Because errors between the measured and estimated LSTs under cloud-free conditions are primarily produced at night (after midnight) and the RMSE between the measured and estimated daytime LSTs under cloud-free conditions is only 1.10 K, the RMSE is smaller on 28 September. The curve of the LST predicted by Equation (13) is very similar to that of the measured LST under cloudy skies on 17 April, 15 June, 22 June, 21 August and 28 September; therefore, Equation (13) can be used to describe the response of the LST changes to variations in the NSSR. Prior knowledge that the fitted LST is equal to or slightly higher than the actual LST when estimating the six parameters in Equation (8) suggests that a few of the measured LSTs under cloud-free skies are less than the estimated LSTs.

**Figure 4.**Comparison of the measured daily temperature evolution with that predicted by the proposed method for six days under various cloud conditions.

_{clear}observations are available). The scatter plot shows that most points are distributed near the 1:1 line, and the RMSE is 1.23 K. The histogram indicates that most errors between the measured LST and the estimated LST using the proposed method under cloudy skies are distributed within ±2 K. The maximum error is −6 K, implying that the proposed method can be used to estimate the LST under cloudy skies by combining the diurnal solar radiation with surface temperature under clear skies. The errors are caused by the uncertainty in the algorithm, DTC, the DSC model, and measurement errors, among other factors.

**Figure 5.**Comparison of the measured daytime LST and the daytime LST predicted by the proposed method under cloudy skies at the Chang Wu Ecosystem experimental station in 2012 (

**left**: scatter plot;

**right**: histogram of errors of the measured and estimated LST).

#### 4.2. Error Analysis

#### 4.2.1. Sensitivity to Errors of the Estimated LST Under Cloud-Free Skies

_{cloud}) is retrieved using T

_{clear}observations in the proposed method. In general, T

_{clear}is obtained using infra-thermal band data, and errors are unavoidable due to the uncertainty in the emissivity, unknown atmospheric conditions and the inversed method, etc. Thus, errors in T

_{clear}are fully translated into errors in the estimation of T

_{cloud}. Regarding the errors in T

_{cloud}from T

_{clear}, a sensitivity analysis is performed by adding ±0.25 K and ±0.5 K to the actual T

_{clear}. Figure 6 shows the resulting error histograms. The RMSE values of the measured and estimated T

_{cloud}are 1.34, 1.39, 1.36 and 1.44 K with a mean error of 0.35 K, 0.12 K, −0.57 K and −0.34 K, respectively when adding −0.25, −0.5, 0.25 and 0.5 K biases to the real T

_{clear}. We find that most errors in T

_{cloud}are within ±2.5 K from the histogram of errors. The mean error is positive when T

_{clear}is underestimated, while the mean error is negative when T

_{clear}is overestimated.

#### 4.2.2. Sensitivity to Errors in the Estimated NSSR

_{cloud}. Regarding the errors in T

_{cloud}introduced by NSSR, a sensitivity analysis is performed after adding ±5 and ±10 percent NSSR biases to the actual NSSR, respectively. Figure 7 shows the corresponding histograms of errors. The RMSE values of the measured and estimated T

_{cloud}are 1.484, 1.424, 1.42 and 1.31 K with a mean error of −0.05 K, −0.05 K, −0.13 K and −0.14 K respectively after adding −5, −10, 5 and 10 percent NSSR biases to the actual NSSR.

**Figure 6.**Histogram of errors in the measured T

_{cloud}and in the estimated T

_{cloud}based on adding biases to T

_{clear}((

**A**): add a −0.25 K bias; (

**B**): add a −0.5 K bias; (

**C**): add a 0.25 K bias; and (

**D**): add a 0. 5 K bias).

**Figure 7.**Histogram of errors of the measured T

_{cloud}and the estimated T

_{cloud}after adding biases to the NSSR ((

**A**): add a −5 percent NSSR bias to the actual NSSR; (

**B**): add a −10 percent NSSR bias to the actual NSSR; (

**C**): add a 5 percent NSSR bias to the actual NSSR; and (

**D**): add a 10 percent NSSR bias to the actual NSSR).

## 5. Application to Actual MSG-SEVIRI Satellite Data

**Figure 8.**Comparison of LST products from LSASAF, with the LST estimated by the proposed method on 1 April 2012 at 11:00 UTC (

**left**: estimates from the proposed method;

**right**: the LST products).

## 6. Conclusions

_{clear}) and diurnal NSSR. Considering the errors in the estimated LST under cloudy conditions (T

_{cloud}) when using NSSR and T

_{clear}, a sensitivity analysis regarding the uncertainty of T

_{clear}and NSSR was also performed. The results show that the accuracy of the LST retrieval can be off by 0.11 K, 0.16 K, 0.17 K and 0.21 K, respectively, when adding a −0.25 K, −0.5 K, 0.25 K and 0.5 K bias to the actual T

_{clear}. The effect of the uncertainty in the NSSR on the LST retrieval could be approximately 0.254, 0.194, 0.19 and 0.08 K, respectively, when adding −5, −10, 5 and 10 percent NSSR errors to the real NSSR.

_{cloud}was calculated using T

_{clear}and Down-welling Surface Short-wave Radiation Flux (DSSR) and albedo products of MSG2-SEVIRI from the Land Surface Analysis Satellite Applications Facility (LSASAF) using the new method. The new algorithm can be applied to LST data retrieved from a geo-stationary satellite during cloudy conditions, and it provides the ability to reconstruct diurnal LST cycles from geo-stationary satellite observations. These cycles are particularly useful in regions where ground-based meteorological observations are scarce.

_{clear}values in one day are needed to fit the DTC model; therefore, the algorithm cannot be used to estimate the LST when less than six T

_{clear}observations are available. These limitations will be addressed in future research.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Li, Z.-L.; Tang, R.; Wan, Z.; Bi, Y.; Zhou, C.; Tang, B.-H.; Yan, G.; Zhang, X. A review of current methodologies for regional evapotranspiration estimation from remotely sensed data. Sensors
**2009**, 9, 3801–3853. [Google Scholar] [CrossRef] [PubMed] - Scarino, B.; Minnis, P.; Palikonda, R.; Reichle, R.H.; Morstad, D.; Yost, C.; Shan, B.; Liu, Q. Retrieving clear-sky surface skin temperature for numerical weather prediction application from geostationary satellite data. Remote Sens.
**2013**, 5, 342–366. [Google Scholar] [CrossRef] - Li, Z.-L.; Wu, H.; Wang, N.; Qiu, S.; Sobrino, J.A.; Wan, Z.; Tang, B.-H.; Yan, G. Land surface emissivity retrieval from satellite data. Int. J. Remote Sens.
**2013**, 34, 3084–3127. [Google Scholar] [CrossRef] - Li, Z.-L.; Tang, B.-H.; Wu, H.; Ren, H.Z.; Yan, G.; Wan, Z.M.; Trigo, I.F.; Sobrino, J.A. Satellite-derived land surface temperature: Current status and perspectives. Remote Sens. Environ.
**2013**, 131, 14–37. [Google Scholar] [CrossRef] - Fan, W.; Yu, S.; Wu, W. Waterbody identification under semitransparent cloud in MODIS image. J. Atmos. Environ. Opt.
**2007**, 2, 73–77. [Google Scholar] - Jin, M.L. Interpolation of surface radiative temperature measured from polar-orbiting satellites to a diurnal cycle 2. Cloudy-pixel treatment. J. Geophys. Res.
**2000**, 105, 4061–4076. [Google Scholar] [CrossRef] - Jia, Y.-Y.; Tang, B.-H.; Zhang, X.; Li, Z.-L. Estimation of land surface temperature and emissivity from AMSR-E data. In Proceedings of 2007 IEEE International Geoscience and Remote Sensing Symposium, IGARSS 2007, Barcelona, Spain, 23–28 July 2007; pp. 1849–1852.
- Lu, L.; Venus, V.; Skidmore, A.; Wang, T.; Luo, G. Estimating land-surface temperature under clouds using MSG/SEVIRI Observations. Int. J. Appl. Earth Obs.
**2011**, 13, 265–276. [Google Scholar] [CrossRef] - Anne, B.K.; John, P.S.; Ronald, E.A. Sensitivity of thermal inertia: Calculations to variations in environmental factors. Remote Sens. Envion.
**1984**, 16, 211–232. [Google Scholar] [CrossRef] - Jiang, G.-M.; Li, Z.-L.; Nerry, F. Land surface emissivity retrieval from combined mid-infrared and thermal infrared data of MSG-SEVERI. Remote Sens. Environ.
**2006**, 105, 326–340. [Google Scholar] [CrossRef] - Zhao, W.; Li, Z.-L. Sensitivity study of soil moisture on the temporal evolution of surface temperature over bare surfaces. Int. J. Remote Sens.
**2013**, 34, 3314–3331. [Google Scholar] [CrossRef] - Duan, S.-B.; Li, Z.-L.; Wang, N.; Wu, H.; Tang, B.-H.; Jiang, X.; Zhou, G. Modeling of day-to-day temporal progression of clear-sky land surface temperature. IEEE Geosci. Remote Sens. Lett.
**2013**, 10, 1050–1054. [Google Scholar] [CrossRef] - Cracknell, A.P.; Xue, Y. Review article: Thermal inertia determination from space-a tutorial review. Int. J. Remote Sens.
**1996**, 17, 431–461. [Google Scholar] [CrossRef] - Wan, Z.; Dozier, J. A generalized split-window algorithm for retrieving land-surface temperature from space. IEEE Trans. Geosci. Remote Sens.
**1996**, 34, 892–905. [Google Scholar] [CrossRef] - Wan, Z.; Li, Z.-L. Radiance-based validation of the V5 MODIS land-surface temperature product. Int. J. Remote Sens.
**2008**, 29, 5373–5395. [Google Scholar] [CrossRef] - Zhang, X.; Li, L. A method to estimate land surface temperature from Meteosat Second Generation data using multi-temporal data. Opt. Express.
**2013**, 21, 31907–31918. [Google Scholar] [CrossRef] [PubMed] - Manalo-Smith, N.; Smith, G.L.; Tiwari, S.N.; Staylor, W.F. Analytic forms of bi-directional reflectance functions for application to Earth radiation budget studies. J. Geophys. Res.
**1998**, 103, 19733–19751. [Google Scholar] [CrossRef] - Tang, B.-H.; Li, Z.-L.; Zhang, R. A direct method for estimating net surface shortwave radiation from MODIS data. Remote Sens. Environ.
**2006**, 103, 115–126. [Google Scholar] [CrossRef] - Land Surface Analysis Satellite Applications Facility (LSA SAF). Available online: http://landsaf.meteo.pt (accessed on 1 January 2014).
- Duan, S.-B.; Li, Z.-L.; Wang, N.; Wu, H.; Tang, B.-H. Evaluation of six land-surface diurnal temperature cycle models using clear-sky in situ and satellite data. Remote Sens. Environ.
**2012**, 124, 15–25. [Google Scholar] [CrossRef] - Duan, S.-B.; Li, Z.-L.; Tang, B.-H.; Wu, H.; Tang, R.; Bi, Y.; Zhou, G. Estimation of diurnal cycle of land surface temperature at high temporal and spatial resolution from clear-sky MODIS data. Remote Sens.
**2014**, 6, 3247–3262. [Google Scholar] [CrossRef] - Duan, S.-B.; Li, Z.-L.; Tang, B.-H.; Wu, H.; Tang, R. Direct estimation of land-surface diurnal temperature cycle model parameters from MSG-SEVIRI brightness temperatures under clear sky conditions. Remote Sens. Environ.
**2014**, 150, 34–43. [Google Scholar] [CrossRef] - Gautam, B.; Rafael, L. Bras estimation of net radiation from the MODIS data under all sky conditions: Southern Great Plains case study. Remote Sens. Environ.
**2010**, 114, 1522–1534. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Zhang, X.; Pang, J.; Li, L.
Estimation of Land Surface Temperature under Cloudy Skies Using Combined Diurnal Solar Radiation and Surface Temperature Evolution. *Remote Sens.* **2015**, *7*, 905-921.
https://doi.org/10.3390/rs70100905

**AMA Style**

Zhang X, Pang J, Li L.
Estimation of Land Surface Temperature under Cloudy Skies Using Combined Diurnal Solar Radiation and Surface Temperature Evolution. *Remote Sensing*. 2015; 7(1):905-921.
https://doi.org/10.3390/rs70100905

**Chicago/Turabian Style**

Zhang, Xiaoyu, Jing Pang, and Lingling Li.
2015. "Estimation of Land Surface Temperature under Cloudy Skies Using Combined Diurnal Solar Radiation and Surface Temperature Evolution" *Remote Sensing* 7, no. 1: 905-921.
https://doi.org/10.3390/rs70100905