An Improved UAV-Based ATI Method Incorporating Solar Radiation for Farm-Scale Bare Soil Moisture Measurement

: The use of UAV-based remote sensing for soil moisture has developed rapidly in recent decades, with advantages such as high spatial resolution, ﬂexible work arrangement, and ease of operation. In bare and low-vegetation-covered soils, the apparent thermal inertia (ATI) method, which adopts thermal infrared data from UAV-based remote sensing, has been widely used for soil moisture estimation at the ﬁeld scale. However, the ATI method may not perform well under inconsistent weather conditions due to inconsistency of the intensity of the soil surface energy input. In this study, an improvement of the ATI method (ATI-R), considering the variation in soil surface energy input, was developed by the incorporation of solar radiation measurements. The performances of the two methods were compared using ﬁeld experiment data during multiple heating processes under various weather conditions. It showed that on consistently sunny days, both ATI-R and ATI methods obtained good correlations with the volumetric water contents (VWC) (R 2ATI-R = 0.775, RMSE ATI-R = 0.023 cm 3 · cm − 3 and R 2ATI = 0.778, RMSE ATI = 0.018 cm 3 · cm − 3 ) on cloudy or a combination of sunny and cloudy days as long as there were signiﬁcant soil-heating processes despite the different energy input intensities; the ATI-R method could perform better than the ATI method (cloudy: R 2ATI-R = 0.565, RMSE ATI-R = 0.024 cm 3 · cm − 3 and R 2ATI = 0.156, RMSE ATI = 0.033 cm 3 · cm − 3 ; combined: R 2ATI-R = 0.673, RMSE ATI-R = 0.028 cm 3 · cm − 3 and R 2ATI = 0.310, RMSE ATI = 0.032 cm 3 · cm − 3 ); and on overcast days, both the ATI-R and ATI methods could not perform satisfactorily (R 2ATI-R = 0.027, RMSE ATI-R = 0.024 cm 3 · cm − 3 and R 2 ATI = 0.027, RMSE ATI = 0.031 cm 3 · cm − 3 ). The results indicate that supplemental solar radiation data could effectively expand applications of the ATI method, especially for inconsistent weather conditions.


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Visible-near-infrared methods have been widely applied; however, they may be easily affected by vegetation and surface roughness, which could lead to unstable monitoring results [6,16]. Microwave methods have better penetration capability than visible-nearinfrared methods; however, the equipment is usually large and is challenging to carry by mini UAVs, being more suitable for manned aircraft and satellite-based remote sensing [15,[17][18][19]. In contrast, thermal infrared cameras are usually light and accurate, and the relationships between soil thermal properties and soil moisture have been fully studied [20][21][22][23][24]; therefore, they seem more suitable for UAV-based soil moisture monitoring. However, thermal infrared-based methods are sensitive to energy inputs and could be easily affected by weather conditions [24,25]. A method to reduce the impact of inconsistent energy input on thermal infrared remote sensing during various weather conditions is the primary focus of this study.
In bare soil or sparse-vegetation-covered surface areas, the thermal inertia method is the most commonly used method for soil moisture estimation [20,22]. The thermal inertia method based on thermal infrared imagery was first proposed in the 1970s by Waston et al. [26]. Pratt improved the thermal inertia model based on application tests by considering more factors, such as thermal conductivity, solar radiance, average humidity of the surface and atmosphere, average air temperature and surface roughness [27]. The main difficulty in applying Pratt's model is that it requires numerous parameters and complex calculations; Price proposed a simplified apparent thermal inertia (ATI) index to simplify the calculation [28], and the ATI index positively correlates with soil moisture.
where α is the surface albedo (dimensionless) and ∆T d is the diurnal (daytime and nighttime) temperature difference ( • C). However, there may exist ATI index application problems, which uses the soil diurnal temperature difference and the surface albedo to characterize the proportion of energy inputted into soil [28], when the surface energy inputs vary among the sampling days [29] due to the method's simplification of the energy input. Even on clear and cloudless days, which are recommended conditions for the application of the ATI method [30], the actual solar radiation intensities still vary significantly, which then introduces inaccuracy into the soil moisture measurements. For multi-process soil moisture measurements, the ∆T d values may be the same in different processes, but the inability to quantify solar radiation makes it impossible to determine whether the corresponding water contents are the same. Therefore, the correlation between soil moisture and the ATI index is difficult to accurately find under multiple heating processes, so the ATI method is unstable in measuring multi-process soil moisture. Palombo et al. used the ATI method to measure soil moisture by recalculating the ATI index for each sampling process and fitting the results to the current day water contents, with a resulting R 2 of approximately 0.62 [31]. This result was acceptable since it considered the diurnal variations in solar radiation, but the requirement to recalculate the ATI index each time would increase computational effort, and it would be difficult to apply the model for the current day to the next day's soil moisture measurements. In contrast, when fitting ATI indices and moisture content from different processes without considering radiation variations, the correlations between ATI indices and moisture content weakened; in some relevant studies, the R 2 fell below 0.6 [32][33][34].
From the perspective of soil physics, adding solar radiation measurement may be a feasible improvement to the ATI method for VWC monitoring under different solar radiation processes. Soil volumetric heat capacity is the quantity of heat that needs to be absorbed or released by the soil for every 1 • C change [35]. Similarly to soil thermal inertia, soil volumetric heat capacity is also affected by moisture, resulting in different rates of soil temperature variations for the same energy input. However, the volumetric heat capacity of the soil is an explicit relationship between the amount of energy and the amount of temperature change. In contrast, the ATI replaces the actual energy with 1−α. The surface Remote Sens. 2023, 15, 3769 3 of 17 energy inputs are different for multiple natural heating processes. Therefore, adding solar radiation measurements to quantify the surface energy input for each process makes it clear that ∆T occurs under each energy input. Such a method can measure soil moisture for multiple processes only if the relationship between the energy input and the ∆T is clarified in the method.
The objectives of this study are to (1) propose an improved ATI method (ATI-R) incorporating the measurement of total solar radiation, (2) examine the feasibility and performance of the ATI-R method under a single heating process and multiple heating processes, and (3) compare the performance of the ATI-R and ATI methods under different weather types.

Materials and Methods
In this study, we proposed the ATI-R method based on the soil thermal properties, which combines the ATI index with the solar radiation measurements. The ATI-R method was evaluated and compared to the ATI method in a field moisture-controlled experiment using UAV-obtained thermal and ground-obtained total solar radiation information.

The ATI-R and ATI Methods
As a three-phase mixture of solid, liquid, and gaseous matter, soil has a volumetric heat capacity composed of the heat capacity of each phase and its corresponding proportion [36].
where X a , X w , and X sj are the volume fractions of soil air, water, and solid-phase components, respectively, and C a , C w , and C sj are the volumetric heat capacities of air, water, and soil solid-phase components, respectively. The relationship between soil volumetric heat capacity (C s ) and soil moisture is similar to that which exists between ATI and soil moisture and is as follows [35]: where C s is the soil volumetric heat capacity (J·cm −3 · • C −1 ), Q is the absorbed energy (J), ∆T S is the soil temperature difference ( • C), and V is the soil natural volume (cm 3 ). C a can usually be ignored when calculating C s , since the volumetric heat capacity of air (1.26 × 10 −3 J·cm −3 · • C −1 ) is much lower than that of water (4.18 J·cm −3 · • C −1 ) and solid-phase components (approximately 1.92 J·cm −3 · • C −1 ) [37]. The soil bulk density can be used instead of the volume fraction of the solid phase for calculating the soil volumetric heat capacity.
where θ is the soil volumetric water content (cm 3 ·cm −3 ), ρ b is the soil bulk density (g·cm −3 ), C w is the water volumetric heat capacity (4.18 J·cm −3 · • C −1 ), and C b is the soil solid-phase heat capacity (approximately 0.71-0.91 J·g −1 · • C −1 ). Substituting Equation (4) into Equation (3), the equation becomes: C w is a constant, and C b can also be considered a constant for a specific type of soil. ρ b varies slowly under undisturbed conditions for the same area. Therefore, ignoring the change in bulk density, the soil volumetric water content is proportional to the ratio of Q to ∆T: Substituting albedo and measurement of solar radiation intensity for Q, Equation (6) can be written as: where R t is the cumulative intensity of the total solar radiation (kJ). Based on the relationship between C s and VWC and Equation (7), an ATI-R index is proposed and defined as: where R t is the cumulative total solar radiation intensity (kJ), and ∆T is the soil temperature increment ( • C). The ATI index was calculated based on Equation (1).
The following equation exists between ATI-R and VWC and ATI and VWC:

Field Experiment Design
A field moisture-controlled experiment was arranged at the Fengqiu State Key Agro-Ecological Experimental Station, Chinese Academy of Sciences (34 • 53 -35 • 14 N, 114 • 14 -114 • 46 E) in Fengqiu, Henan Province, China, in May 2022. The study area is flat, with an average elevation of 67.5 m above sea level; the primary soil type is fluvo-aquic soil, and the main cropping pattern is winter wheat-summer corn rotation. The area has a semi-arid, semi-humid temperate monsoon climate with an average annual precipitation of 615.1 mm, but precipitation resources are unevenly distributed and mainly concentrated in summer (about 45-65% of the annual precipitation).
A bare field was selected as the study area, with a flat surface without noticeable bumps and depressions. The experiment used a completely randomized design with 5 soil moisture levels and 3 replications for each level, totaling 15 plots, each with an area of 6.25 m 2 (2.5 m × 2.5 m), with a protective row of 0.75 m width between each plot to avoid water disturbance between neighboring plots (Figure 1a). The field water-holding capacity of the experimental area was measured to be about 0.320 cm 3 ·cm −3 . Accordingly, we designed 5 water treatments (T0, T1, T2, T3, and T4), where T0 was the no-watering treatment, as this experiment's control check (CK). The remaining 4 treatments (T1, T2, T3, and T4) increased by 10%, 20%, 30%, and 40% of field holding capacity based on the T0 soil moisture content. The 5 watering treatments with watering amounts were 0, 3.2, 6.4, 9.6, and 12.8 mm, respectively. The VWC used for method testing was based on actual measurements of the oven-drying method. Substituting albedo and measurement of solar radiation intensity for Q, Equation (6) can be written as: where Rt is the cumulative intensity of the total solar radiation (kJ). Based on the relationship between Cs and VWC and Equation (7), an ATI-R index is proposed and defined as: where Rt is the cumulative total solar radiation intensity (kJ), and ΔT is the soil temperature increment (°C). The ATI index was calculated based on Equation (1).
The following equation exists between ATI-R and VWC and ATI and VWC:

Field Experiment Design
A field moisture-controlled experiment was arranged at the Fengqiu State Key Agro-Ecological Experimental Station, Chinese Academy of Sciences (34°53′-35°14′N, 114°14′-114°46′E) in Fengqiu, Henan Province, China, in May 2022. The study area is flat, with an average elevation of 67.5 m above sea level; the primary soil type is fluvo-aquic soil, and the main cropping pattern is winter wheat-summer corn rotation. The area has a semiarid, semi-humid temperate monsoon climate with an average annual precipitation of 615.1 mm, but precipitation resources are unevenly distributed and mainly concentrated in summer (about 45-65% of the annual precipitation).
A bare field was selected as the study area, with a flat surface without noticeable bumps and depressions. The experiment used a completely randomized design with 5 soil moisture levels and 3 replications for each level, totaling 15 plots, each with an area of 6.25 m 2 (2.5 m × 2.5 m), with a protective row of 0.75 m width between each plot to avoid water disturbance between neighboring plots (Figure 1a). The field water-holding capacity of the experimental area was measured to be about 0.320 cm 3 ·cm −3 . Accordingly, we designed 5 water treatments (T0, T1, T2, T3, and T4), where T0 was the no-watering treatment, as this experiment's control check (CK). The remaining 4 treatments (T1, T2, T3, and T4) increased by 10%, 20%, 30%, and 40% of field holding capacity based on the T0 soil moisture content. The 5 watering treatments with watering amounts were 0, 3.2, 6.4, 9.6, and 12.8 mm, respectively. The VWC used for method testing was based on actual measurements of the oven-drying method.   The watering amounts with T1,  T2, T3, and T4 were 10% (3.2 mm), 20% (6.4 mm), 30% (9.6 mm), and 40% (12.8 mm) of the field water-holding capacity, respectively, based on the T0 soil moisture.
During May 2nd and 11th, 2022, UAV-based remote sensing images were obtained daily in the experimental plots. In addition, ground-based experimental data, including the total solar radiation intensity and the VWC measurement from the soil samples, were collected synchronously. The corresponding methods of data acquisition were as follows.
The flight altitudes of UAV-based remote sensing were all conducted at 6 m and the soil temperatures and Vis-NIR images were acquired by a thermal infrared camera (Zenmuse H20T, Shenzhen, China, Figure 2b) and a multispectral camera (Rededge-MX, Micasense, Seattle, WA, USA, Figure 2c) mounted on the UAV (M300 RTK, Shenzhen, China, Figure 2a), respectively. The H20T camera operates in the 8-14 µm wavelength range, has a temperature measurement range between −40 to 150 • C (high gain mode) and −40 to 550 • C (low gain mode), and has an image resolution of 640 × 512. The RedEdge-MX has 5 wavelength bands, of blue, green, red, red edge, and near-infrared, respectively, and has an image resolution of 1280 × 960. Two batches of thermal infrared images were taken daily at approximately 5:30 and 14:00 local time. The times representing the lowest and highest soil surface temperatures were determined in advance by soil temperature loggers (IDEASTYLE, Suzhou, China, Figure 2d) buried 1-2 cm beneath the surface. The soil temperature increment was calculated by: where T 1400 and T 0530 are the soil temperatures acquired by the UAV-based thermal infrared camera at 14:00 and 5:30, respectively. Multispectral images were taken between 10:00 and 14:00 to calculate soil α [38]: where ρ band,1 , ρ band,2 , ρ band,3 , and ρ band,4 are the reflectivity in the blue (459-491 nm), green (546-573 nm), red (660-676 nm) and near-infrared (814-871 nm) bands, respectively.
the total solar radiation intensity and the VWC measurement from the soil samples, were collected synchronously. The corresponding methods of data acquisition were as follows.
The flight altitudes of UAV-based remote sensing were all conducted at 6 m and the soil temperatures and Vis-NIR images were acquired by a thermal infrared camera (Zenmuse H20T, Shenzhen, China, Figure 2b) and a multispectral camera (Rededge-MX, Micasense, Seattle, WA, USA, Figure 2c) mounted on the UAV (M300 RTK, Shenzhen, China, Figure 2a), respectively. The H20T camera operates in the 8-14 µm wavelength range, has a temperature measurement range between −40 to 150 °C (high gain mode) and −40 to 550 °C (low gain mode), and has an image resolution of 640 × 512. The RedEdge-MX has 5 wavelength bands, of blue, green, red, red edge, and near-infrared, respectively, and has an image resolution of 1280 × 960. Two batches of thermal infrared images were taken daily at approximately 5:30 and 14:00 local time. The times representing the lowest and highest soil surface temperatures were determined in advance by soil temperature loggers (IDEASTYLE, Suzhou, China, Figure 2d) buried 1-2 cm beneath the surface. The soil temperature increment was calculated by: where T1400 and T0530 are the soil temperatures acquired by the UAV-based thermal infrared camera at 14:00 and 5:30, respectively. Multispectral images were taken between 10:00 and 14:00 to calculate soil α [38]: where ρband,1, ρband,2, ρband,3, and ρband,4 are the reflectivity in the blue (459-491 nm), green (546-573 nm), red (660-676 nm) and near-infrared (814-871 nm) bands, respectively. The total solar radiation intensity in the ATI-R index was obtained by a total radiation sensor (SP-110, Apogee Instruments, Inc., Logan, UT, USA, Figure 2e) placed on a fixed platform located 2 m above the ground surface. The spectral of SP-110 ranges from 360 to 1120 nm with 0.2 mV per W·m −2 of sensitivity. The sensor records the half-hourly average solar irradiance value, and the cumulative total solar radiation intensity is the integration of irradiance over time.
where I0600, I0630, ..., I1400 are the mean values of irradiance (W·m −2 ) from 5:30 to 6:00, 6:00 to 6:30, ..., and 13:30 to 14:00, respectively. t is the measurement time of the sensor, which is 30 min in this study, and S is the area of soil that receives solar radiation, taken as 1 m 2 in this study. Three sampling points were randomly marked within the square area (1.5 m × 1.5 m) in the center of each plot, soils at 0-5 cm were sampled and the oven-drying method was The total solar radiation intensity in the ATI-R index was obtained by a total radiation sensor (SP-110, Apogee Instruments, Inc., Logan, UT, USA, Figure 2e) placed on a fixed platform located 2 m above the ground surface. The spectral of SP-110 ranges from 360 to 1120 nm with 0.2 mV per W·m −2 of sensitivity. The sensor records the half-hourly average solar irradiance value, and the cumulative total solar radiation intensity is the integration of irradiance over time.
R t = (I 0600 + I 0630 + · · · + I 1400 )·t·S (12) where I 0600 , I 0630 , . . ., I 1400 are the mean values of irradiance (W·m −2 ) from 5:30 to 6:00, 6:00 to 6:30, . . ., and 13:30 to 14:00, respectively. t is the measurement time of the sensor, which is 30 min in this study, and S is the area of soil that receives solar radiation, taken as 1 m 2 in this study. Three sampling points were randomly marked within the square area (1.5 m × 1.5 m) in the center of each plot, soils at 0-5 cm were sampled and the oven-drying method was then performed to determine the VWC. Meanwhile, ∆T and α were obtained from thermal infrared images (Equation (10)) and multispectral images (Equation (11) tions corresponding to the marked sampling points to calculate the ATI-R index and ATI index, respectively.
Watering occurred on May 1st and May 6th during the experiment, and rainfall occurred on May 5th and 9th, so we did not conduct the sampling described above on those days.

Methods Evaluation
The R 2 was calculated to analyze the correlation between the ATI-R/ATI index and the VWC. The RMSE and mean absolute error (MAE) were calculated to evaluate the performance of the ATI-R/ATI method for VWC measurement.
where VWC cal , VWC oven , and VWC mean are the VWC calculated by the ATI-R/ATI method, the VWC measured by the oven-drying method, and the mean value of VWC oven , respectively. The six heating processes were divided into three weather types: sunny (May 2nd, May 3rd, May 4th), cloudy (May 7th and May 11th), and overcast (May 8th), based on temperature, solar irradiance, and cumulative solar radiation intensity variations. Sunny days were when the temperature and solar irradiance increased significantly with time, and the cumulative solar radiation intensity was higher than 15 MJ. Cloudy days were when the temperature and solar irradiance increased gradually, and the cumulative solar radiation intensity was 6-15 MJ. An overcast day was when the temperature and irradiance increased non-significantly, and the cumulative radiation intensity was lower than 6 MJ. Figure 4a shows the soil VWCs under different moisture treatments. On the day after watering (May 2nd and 7th), the VWC difference among treatments was approximately 0.03 cm 3 ·cm −3 , consistent with the experimental design. On May 4th, the VWC under the T0 treatment was 0.11 ± 0.01 cm 3 ·cm −3 , whereas the VWC under the T0 treatment was 0.14 ± 0.02 cm 3 ·cm −3 on May 7th due to rainfall that occurred on May 5th. A similar impact on the VWC from a rain event also occurred on May 9th.   Figure 4a shows the soil VWCs under different moisture treatments. On the day after watering (May 2nd and 7th), the VWC difference among treatments was approximately 0.03 cm 3 ·cm −3 , consistent with the experimental design. On May 4th, the VWC under the T0 treatment was 0.11 ± 0.01 cm 3 ·cm −3 , whereas the VWC under the T0 treatment was 0.14 ± 0.02 cm 3 ·cm −3 on May 7th due to rainfall that occurred on May 5th. A similar impact on the VWC from a rain event also occurred on May 9th. Figure 4b shows the soil temperature increments under each treatment during six heating processes. On sunny days, the soil temperature increment is approximately 20-40 °C, and the lower the VWC is, the higher the temperature increment. On cloudy days, the soil temperature increment decreased to 9-29 °C. On the overcast day, the temperature increment was only 3 ℃, or was even a negative value.

Soil VWC and Temperature
The above results indicate that the solar radiation intensity also affects the soil temperature difference besides the VWC. The effect of the VWC on the soil temperature increments is significant when the solar radiation intensity is higher and is less significant when the energy input is lower [39]. For example, on May 3rd (sunny, Rt was 17.86 MJ), the maximum difference in VWC was 0.08 cm 3 ·cm −3 , and the maximum difference in the soil temperature increment was 15 °C. On May 11 (cloudy, Rt was 6.83 MJ), the maximum difference in the VWC was 0.07 cm 3 ·cm −3 , and the maximum difference in the temperature increment was only 3 °C. Therefore, it is necessary to clarify the solar radiation intensity that causes the soil temperature increment.

Comparison of the ATI-R and ATI Methods
The correlations between the ATI-R method and VWC and the ATI method and VWC are shown and discussed separately in three ways: (a) with data from a single heating process; (b) with data from multiple heating processes under the same weather type; and (c) with data from multiple heating processes of inconsistent weather types.  Figure 4b shows the soil temperature increments under each treatment during six heating processes. On sunny days, the soil temperature increment is approximately 20-40 • C, and the lower the VWC is, the higher the temperature increment. On cloudy days, the soil temperature increment decreased to 9-29 • C. On the overcast day, the temperature increment was only 3°C, or was even a negative value.

Single Heating Process
The above results indicate that the solar radiation intensity also affects the soil temperature difference besides the VWC. The effect of the VWC on the soil temperature increments is significant when the solar radiation intensity is higher and is less significant when the energy input is lower [39]. For example, on May 3rd (sunny, R t was 17.86 MJ), the maximum difference in VWC was 0.08 cm 3 ·cm −3 , and the maximum difference in the soil temperature increment was 15 • C. On May 11 (cloudy, R t was 6.83 MJ), the maximum difference in the VWC was 0.07 cm 3 ·cm −3 , and the maximum difference in the temperature increment was only 3 • C. Therefore, it is necessary to clarify the solar radiation intensity that causes the soil temperature increment.

Comparison of the ATI-R and ATI Methods
The correlations between the ATI-R method and VWC and the ATI method and VWC are shown and discussed separately in three ways: (a) with data from a single heating process; (b) with data from multiple heating processes under the same weather type; and (c) with data from multiple heating processes of inconsistent weather types.

Single Heating Process
The correlations between ATI-R and VWC, and ATI and VWC during each heating process were calculated, and the results are shown in Table 1. The R 2 from the linear regression showed that ATI-R and ATI correlated well with VWC under single sunny and cloudy days, with R 2 above 0.700. However, both the ATI-R and ATI methods failed on an overcast day. The reason is that the solar radiation intensity was weak on May 8th (R t = 5.11 MJ, Figure 3c) and caused the soil temperature increment to be low or negative (∆T ranged from −1.5 to 3 • C, Figure 4b). The ∆T is a denominator in the ATI-R and ATI indices (Equations (1) and (8)). Low or negative values of ∆T make the indices extremely large or small, thus weakening the correlation.
Observation of the linear regression equations of the two methods with VWC revealed that the slopes of the equations for the ATI-R method were similar for the different heating processes (except for May 8, which was cloudy). In contrast, the slopes for the ATI method varied considerably. This could lead to the ATI-VWC equations obtained using a single heating process as the dataset, applied to other heating processes, to produce significant errors. The ATI-VWC and ATI-R-VWC linear regression equations for each day were used as predictive models to confirm this assumption. The models were validated with the remaining five heating processes as validation datasets. The results are shown in Figure 5. The dates in the horizontal coordinate indicate that the data of the day was used as the modeling set, and the dates in the vertical coordinate indicate that the data of the day was used as the validation set. nificant errors. The ATI-VWC and ATI-R-VWC linear regression equations for each day were used as predictive models to confirm this assumption. The models were validated with the remaining five heating processes as validation datasets. The results are shown in Figure 5. The dates in the horizontal coordinate indicate that the data of the day was used as the modeling set, and the dates in the vertical coordinate indicate that the data of the day was used as the validation set. Figure 5. (a) R 2 of ATI method validation results with ATI-VWC for a single heating process as the modeling set, (b) R 2 of ATI-R method validation results with ATI-VWC for a single heating process as the modeling set, (c) RMSE of ATI method validation results with ATI for a single heating process as the modeling set, (d) RMSE of ATI-R method validation results with ATI-R for a single heating process as the modeling set, (e) MAE of ATI method validation results with ATI for a single heating process as the modeling set, and (f) MAE of ATI-R method validation results with ATI-R for a single heating process as the modeling set. Figure 5. (a) R 2 of ATI method validation results with ATI-VWC for a single heating process as the modeling set, (b) R 2 of ATI-R method validation results with ATI-VWC for a single heating process as the modeling set, (c) RMSE of ATI method validation results with ATI for a single heating process as the modeling set, (d) RMSE of ATI-R method validation results with ATI-R for a single heating process as the modeling set, (e) MAE of ATI method validation results with ATI for a single heating process as the modeling set, and (f) MAE of ATI-R method validation results with ATI-R for a single heating process as the modeling set. Figure 5a,b show the R 2 between the VWC estimated by the ATI and ATI-R methods and the measured VWC, respectively. It was found that the R 2 of both methods was above 0.70 except for the overcast day, indicating good correlations between the two methods and the VWC on both sunny and cloudy days. At the corresponding positions in Figure 5a,b, the R 2 were the same for both methods. Because ATI-R is ATI multiplied by R t , and R t is equivalent to a constant during a single heating process. This can also be seen in the equations in Table 1, where the two equations for the same day differ only in slope.
To further compare the performance of the two methods, the RMSE and MAE of the two methods are presented in Figure 5c-f, respectively. The ATI method was similar to the ATI-R method in its effectiveness in estimating VWC on sunny days using sunny day data as a predictive model (called the sunny-day model). When predicting VWC on cloudy days with the sunny-day model, the ATI method exhibited more significant bias, with RMSE ATI > 0.100 cm 3 ·cm −3 and RMSE ATI-R ≤ 0.050 cm 3 ·cm −3 , and with MAE ATI > 0.090 cm 3 ·cm −3 and MAE ATI-R < 0.050 cm 3 ·cm −3 . Both methods showed some bias when predicting VWC on sunny days with the cloudy-day model, with RMSE ATI-R ≤ 0.050 cm 3 ·cm −3 and MAE ATI-R < 0.050 cm 3 ·cm −3 , while RMSE ATI and MAE ATI were above 0.050 cm 3 ·cm −3 in some validation sets. The ATI model for May 7th predicted better than ATI-R with VWC on sunny days, probably because of the higher radiation and lower temperature variation on May 7th, which made the soil temperature change a lower amount during the prescribed warming; thus the ATI-R index overestimated VWC on this day.
In addition, the ATI showed a good correlation with the VWC on cloudy days, which differed from the clear and cloudless conditions suggested by existing studies [30]. This phenomenon is because in this study, the soil temperature still increased on cloudy days (May 7th: 13-29 • C; and May 11th: 8-12 • C). Additionally, the multispectral images used to calculate albedo were acquired when the weather conditions were stable, ensuring consistency among images during the same sampling processes [40]. The effect of clouds can be approximated to be the same on soil spectral information because the UAV flies under clouds.
The results above indicate that ATI and ATI-R methods correlated with VWC for a single heating process on sunny and cloudy days. However, both methods failed on overcast days when soil heating was insignificant or negative. Moreover, when weather conditions for relationship exploration were different (radiation differences), the ATI method produced a more significant bias in estimating the VWC for different weather conditions than the ATI-R method.

Multiple Heating Processes under the Same Weather Type
We compared the performance of the ATI-R and ATI methods for the VWC measurement under the same weather type: sunny (May 2nd, May 3rd, and May 4th), cloudy (May 7th and May 11th). Since the ATI-R and ATI methods failed on the overcast day, the overcast data (May 8th) were not repeated in this subsection. The results are shown in Figure 6. Figure 6a,b show the correlation between ATI-R and VWC and the correlation between ATI and VWC under sunny conditions, respectively. The ATI-R and ATI correlated well with the VWC, with R 2 ATI-R = 0.775, RMSE ATI-R = 0.023 cm 3 ·cm −3 , and MAE ATI-R = 0.018 cm 3 ·cm −3 (Figure 6a), and R 2 ATI = 0.778, RMSE ATI = 0.018 cm 3 ·cm −3 , and MAE ATI = 0.013 cm 3 ·cm −3 (Figure 6b), respectively. Under cloudy conditions, R 2 ATI-R = 0.565, RMSE ATI-R = 0.024 cm 3 ·cm −3 and MAE ATI-R = 0.020 cm 3 ·cm −3 (Figure 6c), and R 2 ATI = 0.156, RMSE ATI = 0.033 cm 3 ·cm −3 and MAE ATI = 0.027 cm 3 ·cm −3 (Figure 6d), indicating that the ATI-R method correlated better with the VWC than the ATI method correlated with the VWC (Figure 6c,d).
The results under the same weather type show that the ATI-R and ATI methods had acceptable and similar performances on sunny days; the ATI-R method outperformed the ATI method on the cloudy days. On sunny days, the ATI-R and ATI methods similarly performed well because the cumulative radiation intensities during the three warming processes were similar (R t-May2 = 17.48 MJ, R t-May3 = 17.86 MJ, R t-May4 = 18.41 MJ), and the daily differences in the solar radiation were slight (∆R t ranged from 0.38-0.93 MJ). Such weather was also consistent with clear and cloudless conditions suitable for the ATI method [30]. On cloudy days, the daily variations in solar radiation were significant and unignorable (∆R t was 4.58 MJ), resulting in the different performances of the ATI-R and ATI methods. However, since the ATI-R method adds the solar radiation measurement to identify the variations in R t , the ATI-R method outperforms the ATI method on cloudy days.  Figure 6a,b show the correlation between ATI-R and VWC and the correlation between ATI and VWC under sunny conditions, respectively. The ATI-R and ATI correlated well with the VWC, with R 2 ATI-R = 0.775, RMSEATI-R = 0.023 cm 3 ·cm −3 , and MAEATI-R = 0.018 cm 3 ·cm −3 (Figure 6a), and R 2 ATI = 0.778, RMSEATI = 0.018 cm 3 ·cm −3 , and MAEATI = 0.013 cm 3 ·cm −3 (Figure 6b), respectively. Under cloudy conditions, R 2 ATI-R = 0.565, RMSEATI-R = 0.024 cm 3 ·cm −3 and MAEATI-R = 0.020 cm 3 ·cm −3 (Figure 6c), and R 2 ATI = 0.156, RMSEATI = 0.033 cm 3 ·cm −3 and MAEATI = 0.027 cm 3 ·cm −3 (Figure 6d), indicating that the ATI-R method correlated better with the VWC than the ATI method correlated with the VWC (Figure 6c,d).
The results under the same weather type show that the ATI-R and ATI methods had acceptable and similar performances on sunny days; the ATI-R method outperformed the ATI method on the cloudy days. On sunny days, the ATI-R and ATI methods similarly performed well because the cumulative radiation intensities during the three warming processes were similar (Rt-May2 = 17.48 MJ, Rt-May3 = 17.86 MJ, Rt-May4 = 18.41 MJ), and the daily differences in the solar radiation were slight (ΔRt ranged from 0.38-0.93 MJ). Such weather was also consistent with clear and cloudless conditions suitable for the ATI method [30]. On cloudy days, the daily variations in solar radiation were significant and unignorable (ΔRt was 4.58 MJ), resulting in the different performances of the ATI-R and ATI methods. However, since the ATI-R method adds the solar radiation measurement to identify the variations in Rt, the ATI-R method outperforms the ATI method on cloudy days. Figure 6. (a) Correlation between the ATI-R and the VWC on sunny days, (b) correlation between the ATI and the VWC on sunny days, (c) correlation between the ATI-R and the VWC on cloudy days, (d) correlation between the ATI and the VWC on cloudy days. The red-dashed line is the result of the linear fit. Similar to Section 3.2.1, the sunny-day dataset was used to model and predict VWC for cloudy days, and the cloudy-day dataset was used to model and predict VWC for sunny days. The results are displayed in Table 2. The first row of the data indicates that the ATI-R-VWC model was established with the sunny-day dataset, and the ATI-R of the cloudy-day dataset was brought into the model to calculate a set of VWC estimates. R 2 , RMSE, and MAE between VWC estimates and measured VWC are calculated and used for method evaluation. Table 2 shows that although the correlations between the ATI-R and ATI methods with VWC under sunny conditions are similar, the two methods appear significantly different when used to estimate VWC under cloudy conditions. Similar results are seen in the estimation of sunny-day VWC with the cloudy-day models. The above results show that the ATI-R method can be used to estimate the VWC under different weather conditions with higher accuracy than the ATI method.

Multiple Heating Processes under Different Weather Types
The ATI-R and ATI methods were compared under inconsistent weather types. Inconsistent weather included (a) a combination of sunny and cloudy days (May 2nd, 3rd, 4th, 7th, and 11th), (b) a combination of cloudy and overcast days (May 7th, May 11th, and May 8th), and (c) a combination of sunny and overcast days (May 2nd, 3rd, 4th, and 8th). Figure 7 illustrates the correlation between ATI-R with VWC and the correlation between ATI with VWC under the two-by-two combinations of the three types of weather. On combined sunny and cloudy days, the ATI-R method correlated better with the VWC than the ATI method did, where R 2 ATI-R = 0.67, RMSE ATI-R = 0.028 cm 3 ·cm −3 , MAE ATI-R = 0.018 cm 3 ·cm −3 , and R 2 ATI = 0.31, RMSE ATI = 0.032 cm 3 ·cm −3 , MAE ATI = 0.025 cm 3 ·cm −3 (Figure 7a,b). However, after considering data from the overcast day, neither the ATI-R method nor the ATI index showed correlations with VWC (R 2 lower than 0.01, Figure 7c-f). In addition, vertical lines were found in Figure 7c-f due to the data for the overcast day deviating significantly from those for sunny and cloudy days. The purple dots indicate data from the overcast day, representing ATI-R ranging from −17,000 to 17,000 and ATI ranging from −2.5 to 2.5, respectively. In contrast, the points in Figure 7a,b were from data from sunny and cloudy days; ATI-R ranged from 150 to 700, and ATI ranged from 0.015 to 0.085.
The results under inconsistent weather types showed a good correlation between the ATI-R method and VWC as long as there were significant increases in soil temperature, even if different weather types were mixed in the dataset. The more complex solar radiation intensity under inconsistent weather types (6.84-18.41 MJ), compared to the same weather type , led to daily variations that could not be approximated equally in solar radiation. The lack of radiation measurements made it difficult for the ATI method to explain the conditions under which the soil temperature varied. Therefore, the correlation between the ATI index and VWC was weak under inconsistent weather types. However, both methods failed when introducing data from the overcast day, indicating that the ATI-R/ATI index of the corresponding processes is hardly correlated with the VWC when there is no significant increase in soil temperature (∆T < 3 • C, in this study, Figure 4b) and that the overcast days should not be considered in the multiple soil-heating processes.
In this study, the ATI-R method expands the applicability of the ATI method by adding the total solar radiation measurement and calculating the cumulative solar radiation intensity of the soil-heating process. The ATI-R method was applied to multiple soil-heating processes with significant increases in soil temperature (∆T ≥ 7 • C) even if these processes had substantial differences in solar radiation intensity. The R t of the ATI-R method that was applicable to multiple processes ranged from 6.84 to 18.41 MJ, and the half-hourly mean irradiance ranged from 223.2 to 601.7 W·m −2 . In contrast, the ATI method is only applicable to multiple soil-heating processes with similar radiation intensities. The difference in the R t of the ATI method applicable for multiple processes was small at 0.93 MJ (17.48-18.41 MJ), while the correlation between the ATI method and the VWC was weak when the difference in R t was 4.58 MJ (6.83-11.41 MJ), with R 2 ATI = 0.156 and RMSE ATI = 0.033 cm 3 ·cm −3 . These results indicate that adding a solar radiation measurement sensor allows for quantifying the energy input during each soil-heating process. The energy input from the sun varies in different processes, which leads to different ∆T values for different heating processes, even for soils with the same VWC. Although α can represent the proportion of energy absorbed by the soil [28,41], the actual energy input that causes the soil temperature variation, especially for multiple processes, cannot be replaced by α. Therefore, quantifying the solar radiation intensity during different warming processes is necessary, which makes data from different processes comparable.  The results under inconsistent weather types showed a good correlation between the ATI-R method and VWC as long as there were significant increases in soil temperature, even if different weather types were mixed in the dataset. The more complex solar radiation intensity under inconsistent weather types (6.84-18.41 MJ), compared to the same weather type (17.84-18.41 MJ), led to daily variations that could not be approximated equally in solar radiation. The lack of radiation measurements made it difficult for the ATI method to explain the conditions under which the soil temperature varied. Therefore, the (a) Correlation between the ATI-R method and the VWC on sunny and cloudy days, (b) correlation between the ATI method and the VWC on sunny and cloudy days, (c) correlation between the ATI-R method and the VWC on cloudy and overcast days, (d) correlation between the ATI method and the VWC on cloudy and overcast days, (e) correlation between the ATI-R method and the VWC on sunny and overcast days, and (f) correlation between the ATI method and the VWC on sunny and overcast days. The red-dashed line is the result of the linear fit.
In this study, adding a total solar radiation sensor increases the monitoring cost, which is acceptable because it extends the applicability of the traditional ATI method and improves the accuracy of soil moisture monitoring in multiple processes. Meanwhile, this study was carried out in an agricultural field of flat terrain, where the size of the experimental area matched the effective measurement scale represented by the solar radiation sensor and where factors such as the terrain caused less variation among the plots [42]. Therefore, the ATI-R method proposed in this study is suitable for long-term soil moisture monitoring at the farm-scale in plains areas. With the increase in the monitoring area, a single radiation sensor cannot accurately represent the solar radiation in the study area, and the number of sensors needs to be increased, which undoubtedly will increase the monitoring cost and introduce some errors.
Additionally, the use of the flexible and convenient UAV platform can better meet the needs of the ATI-R method, which is also one of the factors in improving the accuracy of soil moisture measurement. The ATI-R method must obtain the soil temperature at the beginning and end of a specific soil-heating process to calculate the soil temperature increment. The UAV-carried thermal infrared camera can capture the infrared soil image at the corresponding time required to obtain the soil temperature [43]. As a result, the soil temperature information obtained by the UAV and the solar radiation information obtained by the radiation sensor are highly temporally matched, making the ATI-R index calculation more accurate. Soil albedo is another critical parameter used to calculate the ATI-R index. In this study, to ensure the accuracy of the α calculation, UAV-based multispectral images were obtained under stable weather conditions such as cloud cover and wind speed [44]. Relatively stable weather conditions can ensure comparability between α in the same sampling process, and a flexible UAV platform can meet this sampling requirement. Furthermore, resampling can occur even if weather conditions change significantly during sampling, since the UAV-based visible-near-infrared (VNIR) images and thermal infrared (TIR) images are all surface measurements. Therefore, we validate the method on bare soil, which is more intuitive and accurate to reflect the correlation between soil temperature and moisture. In the future, if vegetation needs to be considered, the ATI-R method can extract soil pixels by supervised classification before modeling under low-vegetation cover [45]. In contrast, the ATI-R method may also be limited under medium and high-vegetation cover due to the lack of a physically meaningful and clear vegetation canopy-land surface temperature conversion relationship.

Conclusions
This study proposed an ATI-R method for soil moisture estimation under multiple processes with different weather conditions based on thermal infrared UAV-based remote sensing. The ATI-R method quantifies the surface energy input during different soil-heating processes by adding solar radiation measurement during the heating processes. A moisture control experiment was designed in flat topographic farmland in Henan Province, China. The performance of the ATI-R method for VWC estimation was verified by comparing the ATI method. The main findings are as follows: (1) ATI-R and ATI methods both had acceptable correlations with VWC during single heating process. However, using the single-day expression, the ATI-R was more accurate in estimating VWC under conditions of significant differences in solar radiation. Both methods failed on cloudy days with insignificant soil heating. (2) Under multiple heating processes with similar weather, ATI-R and ATI correlated well with VWC on sunny days with similar solar radiation. At the same time, ATI-R correlated better with VWC on cloudy days with differences in solar radiation than ATI. ATI-R obtained more accurate estimates when estimating VWC on cloudy days with a model with sunny-day data. (3) The correlation between ATI-R and VWC was significantly better than that between ATI and VWC when the combination of sunny and cloudy days was considered. In contrast, the correlation between the two methods and VWC was almost zero, as long as cloudy days with insignificant surface heating (∆T ≤ 3 • C, in this study, Figure 4b) were considered.
Comparison results of the ATI-R method with the ATI method showed that the ATI-R method incorporating solar radiation measurements broadened the weather applicability conditions of the method and improved the accuracy of multi-process VWC measurements. Moreover, the ATI-R method can be used to estimate VWC under different weather conditions (except overcast), which provides a new idea for farm-scale soil moisture monitoring. The ATI-R method is currently only aimed at VWC measurements on bare soil in plains. Although the possibility of applying it to low-vegetation cover scenarios is suggested, more consideration should be given to VWC monitoring in vegetated areas, including mediumand high-vegetation cover.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.