A Comparative Study of Estimating Hourly Images of MODIS Land Surface Temperature Using Diurnal Temperature Cycle Models in Arid Regions

Thermal monitoring of different regions is usually limited to meteorological data in ground stations. Meteorological networks are limited in arid and semi-arid areas, where monitoring climatic conditions is not possible. The aim of this study is to estimate the land surface temperature (LST) hourly for Yazd-Ardakan plain by modeling the diurnal temperature cycle (DTC) using LST imagery of moderate resolution imaging spectroradiometer (MODIS). First, MODIS imagery are reconstructed using the multi-channel singular spectrum analysis, and the complete time series without missing values are created. Then, six DTC models are compared. The accuracy of DTC models is examined by ground LST measurements, air temperature, humidity, and wind speed. In addition, the results of examining the root mean square error (RMSE) images obtained from cross-validation based on MODIS LST imagery show that DTC2 has the highest error, where 73% of the area has RMSE greater than $3^{\circ} \text{C}$ . In DTC1 and DTC2, 64% and 5.8% of the study region has RMSE less than $2^{\circ} \text{C}$ . In general, DTC1, DTC6, DTC5, DTC4, DTC3, and DTC2 models have shown the highest to lowest accuracy in modeling the LST diurnal cycle. In addition, the difference between LST in mountain and plain lands is greater at the time of maximum temperature than at other hours of the day and night. The findings of this research are crucial in studies concerning climate change and land environmental monitoring in arid/semi-arid regions.


I. INTRODUCTION
Land surface temperature (LST), an important parameter in the Earth's climate system, affects the exchange of matter and energy between the Earth and the atmosphere on a regional and global scale [1], [2], [3], [4], [5].LST diurnal cycle changes are related to atmospheric conditions, solar radiation, soil thermal regime and vegetation cover [6], [7], [8], [9].LST diurnal cycle has applications in urban thermal environment monitoring, thermal inertia, energy balance, evaporation, transpiration, and others [10], [11], [12], [13], [14], [15], The associate editor coordinating the review of this manuscript and approving it for publication was Jon Atli Benediktsson .[16].In many models associated with energy and temperature exchange, hourly surface temperature data are required [11].Certain remote sensing satellites provide a unique tool to measure the radiation of the thermal wavelengths of the electromagnetic curve, which can be used to calculate LST with different time resolution [17].However, except for Geostationary satellites that have the ability to measure hourly temperature, the rest of the satellites that have a thermal band do not have this feature [18], [19], [20].
The moderate resolution imaging spectroradiometer (MODIS) sensor was launched in 1999 aboard the Terra satellite and in 2002 aboard the Aqua satellite [21], [22].Both platforms have been acquiring data in 36 spectral bands [23].
In thermal remote sensing, a method has been presented for modeling the LST diurnal cycle [32].Some studies have already been conducted to estimate the hourly temperature by modeling the diurnal temperature cycle (DTC) [33], [34], [35], [36], [37].The results of studies by Göttsche and Olesen showed that by modeling DTC using METEOSAT data, it is possible to estimate the temperature with a 30-minute sequence [38].Coops et al. estimated LST in the morning from 2000 to 2002, when the Aqua satellite was not yet launched [39].Their results showed that LST obtained from the Terra satellite in the afternoon can be used to predict more than 90% of LST in the morning.
Duan et al. evaluated six models of LST diurnal cycle using satellite images in cloudless sky [40].They considered two time periods: one for the whole day, i.e., from sunrise to sunrise (Period A) and another from 09:00 AM to 03:00 AM of the next day in local time (Period B).The results of period A showed that JNG06 and GOT09 models performed best with root mean square error (RMSE) of 0.  [45].All models had higher accuracy in summer than other seasons, while poorer performance was produced in winter.The INA08-ts model showed the best performance in all seasons.Sharifnezhadazizi et al. analyzed the global DTC using MODIS observations [46].Preliminary evaluation of LST interpolated with hourly ground observations showed an error of less than 1 • C.
Sekertekin et al. modeled diurnal LST in arid regions using artificial neural network and Landsat 8 image time series [47].Their results showed that the difference between the estimated surface temperature and the ground data was 1 • C in winter and 2.49 • C in summer, and the accuracy of this model decreases with increasing temperature.Hu et al. improved the monthly LST estimation of the MODIS sensor using the diurnal cycle model of LST [48].This method has made it possible to estimate the average and maximum temperature of 24 hours in a diurnal cycle in every month.Validation of the mean temperature based on DTC estimated the mean difference compared to the ground data of 0.  Xing et al. estimated the daily average LST on a global scale using pairs of observations from the MODIS sensor during day and night [49].Their RMSE results showed less than 1.06 • C. Sismanidis et al. quantified the daily and annual dynamics of LST obtained from satellites and stated that the Earth's diurnal and annual cycles drive the changes of LST in space and time, but their combined effects are transferred differently from place to place as a function of weather, local weather conditions, and surface characteristics [50].Lu and Zhou presented a 4-parameter model to estimate the LST diurnal cycle using MODIS LST [51].They showed that their model can record the changes of LST during a day very well.Jia et al. estimated the hourly product of LST with a spatial resolution of 2 km using advanced baseline imager (ABI) data [52].Compared to the value obtained from the hourly data, the daily average of LST showed RMSE of 1.13 • C and the correlation coefficient of 0.99.Hong et al. generated, validated, and analyzed a global dataset of daily mean LST without missing data using MODIS products [53].Validation using terrestrial data showed that the mean absolute errors (MAEs) in the IADTC framework were 1.4 • C for the SURFRAD data and 1.1 • C for the FLUXNET data.
Thermal monitoring of different regions is usually restricted to meteorological data in ground stations [5], [54], [55].There are few meteorological networks in arid and semi-arid areas, and it is not possible to monitor climatic conditions, especially on a regional and local scale, in these environments [56].Therefore, the aim of the current research is to estimate the LST images on an hourly basis using the DTC modeling.Deriving LST images with hourly sequence has wide applications in monitoring energy balance and soil thermal properties.The rest of this study is organized as follows.Section II describes the study area, employed data sets, and DTC models from MODIS LST measurements.Results and discussion are presented in Section III, and conclusions, recommendations, and future directions are provided in Section IV.

A. STUDY REGION
The study region is Yazd-Ardakan plain, located in Yazd province in the center of Iran (Fig. 1).This region has desert conditions and a hot and arid climate.Rainfall is very little and irregular, with a long-term annual average of 60 mm.This region is selected for the DTC modeling due to its desert  nature and the sharp increase in air and land temperatures in summer.

B. DATASETS AND PREPROCESSING
The MODIS LST is estimated using a generalized split window algorithm.This algorithm is similar to the split window method presented by [57] and [58] for AVHRR data based on the day and night physics developed in [59].In this method, LST is derived using classification-based emissivity [60].These two polar orbiting satellites carry out imaging with daytime resolution of about 10:30 AM/PM local solar time for Terra and 1:30 AM/PM for Aqua [61], [62], freely available as MOD11A1 and MYD11A1 [63].Herein, MODIS LST images are employed with a sequence of four times per day in all days of 2020 that includes a total of 1460 images.All these images are reconstructed using the multi-channel singular spectrum analysis (MSSA) to create a time series without missing data and outliers for modeling the LST cycle.The reconstruction of these images using MSSA is described in detail in [64].The MSSA is a robust time series analysis model that simultaneously accounts for the temporal and spatial correlations between multiple time series.More details on how MSSA is applied to reconstruct LST images can be found in [64] and [65].The six DTC models mentioned below (DTC1-DTC6) are compared.Ground data as well as MODIS images are employed for validation.After selecting the optimal method, LST is modeled hourly in every day and night.Annual LST time series is created with hourly sequence (8760 images for one year).The steps of modeling the diurnal cycle of LST using MODIS imagery are illustrated in Fig. 2. Note that the labels 'A' to 'H' in red in Fig. 2 are the same as the labels of the subsections in Section III to ease understanding.

C. DIURNAL TEMPERATURE CYCLE MODELS
To estimate LST with hourly sequence, a model consisting of a harmonic and an exponential term is fitted to the Earth's DTC model, which describe the effect of the sun and the reduction of LST at night, respectively [66], [67], [68].Modeling estimates the parameters that describe DTC and can also be useful for interpolating missing data due to technical or cloud problems [69], [70].The parameters depend on all modeled temperatures and are therefore hardly affected by outliers [71].

1) DTC1
A two-part, semi-empirical DTC model is developed in [38].The LST cycle is modeled using the cosine function to predict the evolution of LST during the day and based on the thermal diffusion equation and an exponential function to describe the temperature decrease at night, assuming that natural surfaces follow Newton's law of cooling [72].In the present study, this model is called DTC1 and is defined as where k is the temperature decrease constant given by the following equation, All the parameters of these equations are also ilustrated in [40, Figure 1].Here, t denotes the time, T 0 is the residual surface temperature around sunrise, T a is the range of surface temperature in one day and night, w is the width in half of the period (considered as the length of the day), t m is the time in which the surface temperature reaches its maximum in 24 hours, t s is the starting time of the temperature decrease after sunset, δ T is the temperature difference between the lowest temperature and T 0 .In this model, clear sky conditions are assumed (non-cloudy pixels) without any significant change in wind speed and temperature drop after the thermal sunset defined by t s .
The components of the time of sunrise and sunset, the length of the day and the local noon (when the sun has the most amount of radiation) are calculated using the geographical location of the study region and the day of the year for each day of the year to estimate t s , t m , and w.The length of the day, w, also called the maximum possible sunshine hours, is defined as the time between sunrise and sunset [73]: where δ is the solar declination angle in degrees and φ is the latitude of the region in degrees.A suitable approximate equation for calculating the solar declination using the day of the year, D n , is presented by Cooper [74] as follows δ = 23.45sin (360/365)(284 The time of sunrise, t sr , and the time of sunset, t ss , can be calculated with the location's latitude (φ degrees) and solar declination (δ degrees) as The local solar time, when the sun is at the highest level of the sky, is used to calculate the time of maximum temperature, t m .The solar time, defined based on the daily apparent movement of the sun around the Earth, is the hourly angle Here, the coefficient 4 minutes is because the Earth rotates one degree every 4 minutes, E 0 T stands for equation of time, and δ T UTC is the difference between the local time and the universal coordinated time (UTC) in hours, which is 3.5 hr in the study region.The E 0 T is an empirical equation that corrects for Earth's orbital eccentricity and Earth's axial tilt, approximately equal to E 0 T = 9.87 sin(2B) − 7.53 cos(B) − 1.5 sin(B), (9) where B = (360/365)(d − 81) and d is the number of days that have passed since the beginning of the year.

3) DTC3
Van den Bergh et al. found that the width of the best fitted cosine term of the DTC is different on the upslope (morning) than on the downslope (afternoon) [76].As a result, they proposed a DTC model in which an additional term is introduced based on previous models.The whole DTC is represented by three functions, which represent two curve functions during the day, one for the upslope and the other for the downslope, while the third term describes the shape of the nocturnal curve.This model, DTC3, is defined as: where w 2 = 3.4 t m and k is calculated as 4) DTC4 Hu et al. proposed a different way for calculating the day and night LST parameters [48].This method, named DTC6 hereafter, uses the same equations for calculating T day and T night as DTC5, but k, T 0 , T a , and δ T are calculated by the following equations.
In these equations, T sr,d is LST at the time of sunrise on the first day, and T sr,d+1 is LST at the time of sunrise of the next day whose images are estimated using the mean and range.Also, T max is the maximum day and night LST at time t m .

A. FOUR MODIS LST PRODUCTS
Four products of MODIS LST at 1:00 AM, 10:00 AM, 1:00 PM, and 10:00 PM hours are utilized to model the temperature image of the earth's surface hourly in a one-year time series, i.e., total of 1460 MODIS LST images where each image covers the entire study region.Note that for the Yazd-Ardakan plain, the actual acquisition times of MODIS LST products are very close to 10:00 AM/PM (for Terra) and 1:00 AM/PM (for Aqua).Due to the drastic change in the temperature of the earth's surface, especially in desert areas, it is crucial to obtain the hourly image of LST.In Fig. 3, four MODIS LST products at different hours are illustrated, showing significant variations.
The LST changes are compared in four MODIS products.At 1:00 AM, 30% of the study region, which mostly includes highlands and mountainous areas, has LST less than 1 • C, and at this hour, 95% of the study region has LST less than 6 • C. At 10:00 AM, for 88% of the study region, LST is estimated to be more than 20 • C, which has changed a lot compared to 1:00 AM.Between 10:00 AM and 1:00 PM, the biggest area change is related to temperatures over 30 • C. In the time interval from 10:00 AM to 1:00 PM, 15% of the region of the temperature category 30-35 • C is reduced and 35% is added to the area of the temperature category of more than 35 • C, which indicates a sharp increase in temperature during these hours.At 10:00 PM, which is an average of 4 hours after sunset, the temperature has dropped sharply and in 58% of the study region, it is less than 6 • C. The comparison of LST between 10:00 PM and 1:00 AM shows that the temperature change between these two hours is less compared to LST changes during the day.Also, the changes in LST in one pixel during one year are checked, and the amount of changes in each product is compared with each other (Fig. 4).

B. COMPARISON OF DTC MODELS
The LST imagery for all hours of the day and night in a one-year time series are estimated using the six DTC models.Four MODIS LST products per day, geographical coordinates, and day of the year are used as inputs for the DTC models.A MATLAB script is prepared to calculate the hourly images of the LST obtained from different methods with these inputs.First, the range and average images for each day are calculated using the four MODIS LST products.Using the images of average, range and coefficients that are mentioned in the previous section, LST is calculated hourly.The results of different models in one pixel and in one day of summer are illustrated in Fig. 5.Note that the LST starting time in Fig. 5 is at sunrise.The DTC models consist of two parts of calculating the temperature before and after sunset.In some models, one of these parts is the same and the other part is calculated by a different method.Therefore, the numbers of LST calculated using different methods overlap with each other at some hours.In DTC1, DTC2, DTC4, and DTC6, the LST at sunrise and the first hour of the day is estimated at 27.

C. COMPARISON OF DTC MODELS WITH MODIS LST PRODUCTS AT FOUR DIFFERENT HOURS
For a better and more comprehensive comparison, the MODIS LST images at 1:00 AM, 10:00 AM, 1:00 PM, and 10:00 PM are compared with the LST images obtained by different DTC models (Fig. 6).From Fig. 6, one can observe that LST in mountainous areas is generally lower than LST in plain areas, see Fig. 1.The DTC2 has the most difference from MODIS at 1:00 AM.In DTC2, more than 30% of the region is in the temperature category of higher than 22 • C, while in the MODIS image at 1:00 AM, 4% of the region show a temperature higher than 22 • C. In DTC2, it is assumed that there is a difference between LST at the time of sunrise and LST at the time before sunrise the next day.Therefore, in DTC2, LST at 1:00 AM, when the temperature decreases, is estimated higher than the actual value.The DTC3 and DTC4 show exactly the same results in this watch, that is why they are displayed together.In these two models, the LST region of the category above 22 • C is 23% more than the one in the MODIS LST image.The DTC5 image also has a significant difference with the MODIS LST image and the temperature in this model is estimated higher than the values of MODIS.Therefore, 58% of the region has a temperature of higher than 18 • C, while in the images 38% of the region is in this temperature category.This method also estimates LST at 1:00 AM higher than the actual value.The DTC6 and DTC5 are more similar to the MODIS LST at 1:00 AM.In the MODIS LST image, the largest area corresponds to the category 16 to 18 • C, but in DTC6, the largest area belongs to the category 18 to 20 • C. The DTC1 classes are more similar to the ones in the MODIS LST image.In DTC1, the area of the temperature categories below 10 • C and above 22 • C are estimated more and less than the values in MODIS LST, respectively.In general, DTC2, DTC3, DTC4, DTC5, DTC6 and DTC1, have, respectively, lowest to the highest similarity to MODIS LST at 1:00 AM.
The MODIS LST image at 10:00 AM and the six DTC models are also compared with each other (Fig. 7).The DTC1, DTC2, DTC3, DTC4, and DTC6 have estimated similar values for the temperature at 10:00 AM, and so they are displayed in one single image.Since parameter w in the DTC5 is calculated by a different method, this model estimates different LST than other models at 10:00 AM.The DTC1 to DTC4 and DTC6, LST is estimated to be lower than MODIS LST, so that the area of LST category of above 56 • C is 4.2% in these models but 10.2% for MODIS, while DTC5 has estimated LST higher than other DTC models.In DTC5, the area of the temperature category of 54-56 • C is 19.8% while this area is 16.1% in the MODIS LST image.In general, DTC5 is more different from the MODIS LST image at 10:00 AM, and comparing the area of the LST categories, the results of other models are closer to the MODIS product.
The LST estimated via the different DTC models has the same results at 1:00 PM, and their results are compared with the MODIS product at this time (Fig. 8).From Fig. 8, the LST of the category above 58 • C covers 29% of the study region, while in the MODIS LST image, the area of this category is 22.5%.
Finally, the LST obtained from the different DTC models at 10:00 PM are also compared with the MODIS LST image (Fig. 9).In DTC2 and DTC3, the temperature calculation algorithm is the same in the decreasing slope of the day and night temperature.For this reason, these two methods have similar results and are displayed together.The LST of DTC2 and DTC3 is more different from the MODIS LST than other models.In DTC2 and DTC3, 42% of the study region show a surface temperature of above 24 • C, while in the MODIS image, this area is about 3.5%.The DTC4 has also estimated the LST in the category of above 26 • C by 9% more than MODIS.Therefore, DTC2, DTC3 and DTC4 have the most difference with the MODIS product at 10:00 PM.The DTC5 and DTC6 have also estimated LST higher than the MODIS LST image for some categories.In these models, respectively, 48.5% and 42.5% of the study region has a temperature above 22 • C, while in the MODIS image, this area is 23%.The DTC1 has the most similarity with MODIS.In DTC1 and MODIS LST, 23% and 21.5% of the study region is above 22 • C, respectively.In order words, the LST obtained from DTC1, DTC6, DTC5, DTC4, DTC2,3 show the most to the least similarity with the MODIS product at 10:00 PM.

D. COMPARISON OF DTC MODELS IN MOUNTAIN AND PLAIN LOCATIONS IN ONE DAY
To compare the six DTC models in different lands, two locations are selected in the mountain and plain lands, see Fig. 10(a).One day in summer is chosen to compare the dayto-day changes in LST, Fig. 10(b).At 10:00 AM in the plain land, the temperature is above 50 • C, and in the mountain land, the surface temperature is below 30 • C. Therefore, at the same time in the study region, there is a difference of about 20 • C between different land types.The LST at 1:00 AM, 10:00 AM, 1:00 PM, and 10:00 PM obtained from MODIS is also displayed on the diagram of DTC in Fig. 10(b).
The results show that the maximum LST in mountain and plain lands is 39.2 and 60 • C, respectively.In other words,  there is a difference of 20 • C between the maximum LST in mountain and plain lands.The difference between the LST in mountain and plain lands is greater at the time of the maximum temperature than at other hours of the day and night.At 10:00 AM, the difference between LST in the mountain and the plain lands is 18 • C. The difference in LST in the mountain and plain lands at night has decreased, so that the difference between 1:00 PM and 10:00 PM is estimated to be around 10 • C, i.e., LST at 1:00 PM is 15.3 • C in the mountain land and 25 • C in the plain land.The range of changes in LST in one day and night is estimated to be 20 • C in mountain lands and 30 • C in plain lands.In addition, the average LST during the day and night is 26.2 • C in the mountain lands and 40 • C in the plain lands, showing the difference of 14 • C in the average LST in these lands.In general, the results show that the LST changes in one day and night are more in the plain lands than in the mountain lands, and the maximum temperature difference between the plain and mountain lands is during the peak temperature hours and at the local noon time while the minimum is at night.

E. COMPARISON OF DTC MODELS IN TWO DAYS OF WINTER AND SUMMER IN ONE LOCATION
The DTC models are compared at one place in two times of the cold and warm seasons of the year (Fig. 11).The day length, times of sunrise, sunset, noon, and the MODIS LST images as inputs of the DTC models are different on these two dates, causing significant changes in the estimated LST.In winter, the maximum LST 6 hours after sunrise is 20 • C and in summer 8 hours after the sunrise, it is estimated to be 56.4 • C. Sunrise time is calculated at 6:00 AM in summer and 5:00 AM in winter.Note that the policies related to moving the clock forward and backward are not considered in this phase of the study.The length of the day, which determines the width of the peak and sharp drop in temperature, is estimated to be 10 hours in winter and 13 hours in summer.Therefore, accurate estimation of sunrise, sunset, day length, etc. is very important for modeling the LST cycle.Considering that the study region is located in the hot and arid belt of the Earth and the salient features of desert areas are the large difference in day and night temperatures, the results of this section also show that the difference in the LST between day and night in summer is much greater than in winter, so that the difference in LST at noon and midnight in the warm season of the year is estimated as 33.3 • C and in the cold season as 20 • C. In desert areas, the main factor of soil formation and physical destruction of rocks is the large difference in day and night temperatures.

F. VALIDATION OF DTC MODELS BY GROUND MEASUREMENTS
Validation is the most important part of remote sensing studies, because the modeling is not reliable without evaluating the accuracy of the product.Using a thermometer with the ability to measure temperature with the required sequences, LST is measured in several consecutive days.However, the low spatial resolution of MODIS images and single-point temperature measurement are challenging in ground databased validation [9].Therefore, considering that the study region is located in the arid belt of the Earth and has desert conditions, the areas that have homogeneous land cover in a large area and about a few pixels of the MODIS image are selected for validation.As an example of such validation, a one-day real and modeled (DTC) measurements are shown in Fig. 12. Dry areas with land covers, bare lands, Hamada, barren lands and vast deserts without any cover are very suitable for validating the surface temperature and solving the problem of mixed pixels [9].The validation results based on ground data also show that DTC1, DTC6, DTC5, DTC4, DTC3 and DTC2 have, respectively, the highest to lowest accuracy in modeling the LST diurnal cycle.
Validation is also done by comparing the hourly LST from different DTC models with hourly air temperatures acquired at an automated meteorological station.The LST is not measured in weather stations of the National Meteorological Department and in the automatic weather station used in this research, and only hourly air temperature information is available.Therefore, the hourly temperature estimated using different DTC models is compared with the hourly air temperature, air humidity percentage, and wind speed in two days of the year in summer and winter, see Fig. 13.
The automated weather station used is located on the edge of the Abarkouh desert (Latitude 31 • 15 ′ 9 ′′ N and Longitude 53 • 43 ′ 56 ′′ E), which has high air and LST due to its proximity to the desert.For a better comparison, one day in summer and one day in winter are selected.In general, LST is much higher than the temperature of the air, so that the maximum surface temperature of the selected hot day is 58 • C and the maximum air temperature at the same time is 34.4 • C. In the comparison between LST and air temperature in the summer season,    around 3 to 5 hours after sunset, LST and air temperature are almost the same.The air temperature on the two study days clearly show an inverse relationship with air humidity percentage and wind speed, while the LST pattern did not match with wind speed and humidity percentage.It can be seen that the air temperature decreases with the increase in wind speed and humidity and vice versa.

G. VALIDATION OF DTC MODELS BY MODIS PRODUCTS
The six DTC models for a one-year time series are evaluated by MODIS LST images.Using different DTC models, hourly images of the LST are prepared in one year.For crossvalidation, these images are compared with MODIS LST images at 1:00 AM, 10:00 AM, 1:00 PM, and 10:00 PM.Finally, using this comparison in one year and evaluating 4 images in each day of the year, the RMSE images of each DTC model are prepared and shown in Fig. 14.These images are the result of cross-validation and comparison of 4 modeled LST images using different DTC models and MODIS products at those times.Because the goal is select the best DTC models in estimating the LST at all hours of the day and night, accuracy assessment is not done separately for day and night.
The RMSE results show that DTC2 has the highest error.In this method, 73% of the area is covered by RMSE error greater than 3 • C. Of this value, more than 42% of the area is covered by RMSE greater than 4 • C. In DTC2, it is assumed that the difference between LST before sunrise and the coldest LST at night is zero, i.e., δ T is not considered.DTC3 and DTC4 have also shown little accuracy following this method.In DTC3, 34% and in DTC4, 25% of the study region is covered by the RMSE error greater than 4 • C. DTC5 is more accurate than DTC2, DTC3, and DTC4 and less accurate than the DTC6 and DTC1.In DTC5, 11% of the study region is covered by RMSE error of greater than 4 • C. DTC6 and DTC1 are more accurate than other methods.In DTC6, 6% and DTC1, 4% of the region has RMSE error greater than 4 • C. In DTC1, most of the area (38%) has RMSE error smaller than 1 • C, and in the DTC6, the error category 2-3 • C accounts for the largest area with 43%.In DTC1, 64% of the study region show RMSE error of smaller than 2 • C, while the area of this category in DTC6 is 30%.
In general, the cross-validation results show that DTC1, DTC6, DTC5, DTC4, DTC3 and DTC2, respectively, show the highest to lowest accuracy in modeling the LST diurnal cycle.These results are consistent with the results in [40] and [44].The results of estimating the average daily temperature are also consistent with the results presented in [49].Also, the results of the present study are consistent with the findings of Coops et al. [39] who investigated the differences in morning and afternoon LST estimates from Terra and Aqua satellites in a wide range of categories, places and dates of land cover in Canada.The results show that there is a statistically significant difference between LST in the morning and in the afternoon, which varies from 0.3 to 2.3 • C, depending on the type of land cover, and from 1.2 to 5 • C, depending on the time of the year.

H. HOURLY LST IMAGERY IN ONE YEAR
Using different validation datasets, it is shown that DTC1 has the highest accuracy for modeling LST, and so the hourly LST images in for the year 2020 are produced by this model, illustrated in Fig. 15.An an example, one pixel is selected, and its hourly time series is also displayed in Fig. 15.

IV. CONCLUSION
In this paper, the modeling of the LST cycle was carried out using four MODIS LST images, and the hourly LST images were prepared for one year for Yazd-Ardakan plain.Six DTC models were compared and their LST results were examined by cross-validation using MODIS LST products at 1:00 AM, 10:00 AM, 1:00 PM, and 10:00 PM.The DTC models were also validated by ground measurements.It was shown that DTC1 performs better than other models for LST estimation in the arid regions.Future direction would be to investigate the possibility of modeling the LST diurnal cycle using UAV images.

COMPUTER CODE
The MATLAB code developed for this research is available online at https://github.com/Fahimearabi.
5 • C. The GOT01_0 model performed the worst with an overall RMSE of 1 • C. The results of period B showed that except for the GOT01_0 model, other models had similar results with an overall RMSE of 0.4 • C. Holmes et al. investigated the spatial patterns in the timing of the LST diurnal cycle [41].In fact, the structural differences in the timing of DTC, caused by the choice of measurement device or model framework, were examined.Duan et al. estimated DTC at high spatial and temporal resolution using MODIS products and evaluated with LSTs derived from MSG-SEVIRI [42].Their results achieved RMSE smaller than 1 • C. Results of the study conducted by Malbeteau et al. showed that the unmanned aerial vehicle (UAV)-based diurnal cycle was consistent with ground measurements with an average correlation coefficient and RMSE of 0.99 • C and 0.68 • C, respectively [43].Nie et al. simulated DTC using MODIS imagery [44].Their results showed that RMSE in eight validation points in different land covers was smaller than 0.72 • C.They validated their results by comparing the simulated LST of MODIS products with FY-2F.Chang et al. evaluated 4-parameter DTC models using MODIS LST images and ground measurements

FIGURE 1 .
FIGURE 1.The Yazd-Ardakan plain in the province of Yazd in Iran.

FIGURE 2 .
FIGURE 2. The workflow of this study.
of the sun relative to the observer's meridian.The local solar time is the sum of local time and the time correction (TC) factor.The local time usually differs from the local solar time because of eccentricity of the Earth's orbit and as a result of human adjustments for summer savings.The TC or net time correction factor (in minutes) calculates the local solar time changes in a given time zone due to longitude changes in the time zone: Duan et al.[42] optimized the LST cycle model presented by Inamdar et al.[77].Duan et al. reduced the width, w, greater than half a period in the cosine term of the thermal diffusion equation.The equation for estimating day and night temperature in this model, named hereafter DTC5, is identical to DTC4, but only the method of estimating w component is different, estimated using the following equation:

FIGURE 4 .
FIGURE 4. Comparison of LST ( • C) in four MODIS products in a one-year time series at the scale of one pixel.

FIGURE 5 .
FIGURE 5. A comparison of the results of the six DTC models ( • C) in one pixel and in one day of summer.
7 • C while in DTC3 and DTC5, it is estimated at 23.7 • C. The LST at sunset is 41.6 • C in DTC1, 37.4 • C in the DTC2 and DTC3, and 38, 34.6, and 43.8 • C in DTC4, DTC5, and DTC6, respectively.In DTC6, LST at sunset is estimated higher than other models, and the decrease in LST is more severe than in other models one hour after sunset.The lowest LST in a day and night is related to the time before sunrise, estimated as 25.5, 28.8, 28.8, 28.5, 27.2, and 26.9 • C for DTC1 to DTC6, respectively.Using MODIS images, the results show that the estimated values of hourly LST are different for different models.The maximum daily temperature is the same in all models, and LST at sunrise in different models differs by 4 • C. It also varies by 9 • C and 3 • C at sunset and before sunrise in different models, respectively.With identical inputs, different models have estimated different LST at different hours.Therefore, it is necessary to validate and use the appropriate model with the highest precision.Note that this comparison is made at the scale of one pixel.

FIGURE 10 .
FIGURE 10.(a) The mountain and plain locations for comparing DTC models at 10:00 AM, and (b) The LST comparison from DTC models and MODIS.

FIGURE 11 .
FIGURE 11.Comparison of DTC and MODIS LST ( • C) at one place in winter (top panel) and summer (bottom panel).

FIGURE 12 .
FIGURE 12. Validation by ground measurements.The sampling intervals for the real and modeled LST are 10 minutes and 1 hour, respectively.The DTC1 is also used here.

FIGURE 13 .
FIGURE 13.Comparison of LST ( • C) resulting from different DTC models and components of air temperature, humidity percentage, and wind speed in a hot day (top panel) and a cold day of the year (bottom panel).

FIGURE 14 .
FIGURE 14. RMSE image ( • C) resulting from the validation of different methods of modeling the LST diurnal cycle: a) DTC1 b) DTC2 c) DTC3 d) DTC4 e) DTC5 and f) DTC6 g) Comparison of the percentage area of the RMSE image categories.

FIGURE 15 .
FIGURE 15.Monitoring LST hourly in a one-year time series: a) schematic image of the location of the investigated place and hourly images of the LST in one year, b) hourly examination of LST in one year, c) a magnified view in a season, and d) a magnified view of the hourly changes of the LST in 11 days.