1. Introduction
Soil moisture plays an essential role in hydrological processes in the atmosphere and underground, including water cycles, vegetation growth, and dust emission. Many remote sensing techniques for detecting surface soil moisture have been exploited, and the microwave remote sensing technique in particular is considered the most practical method. Representative missions for exploring regional and global soil moisture employ satellite products with microwave sensors, such as the Soil Moisture Active Passive (SMAP; Entekhabi et al. 2010), the Soil Moisture and Ocean Salinity (SMOS; Kerr et al. 2010), and the Global Change Observation Mission–Water (GCOM-W1; Imaoka et al. 2010). Soil moisture data generated from microwave sensors are available at daily intervals at most locations because the microwave signal from Earth’s surface is not attenuated by clouds. Data developed in such missions have been widely tested in a number of hydrological studies. Successful methods have been developed, in particular for sparse vegetation, to estimate surface soil moisture, which is based on radar (e.g., Zribi et al. 2007) and on products from passive microwave satellite sensors (e.g., Kaihotsu et al. 2013; H. Lu et al. 2009). Thermal infrared measurements, such as the surface temperature of Earth, have a spatial resolution of several tens of meters at most. Surface temperature measurements with a spatial resolution in kilometers are available on a daily basis or several times a day at any location unless clouds or fog intercept the thermal infrared signals emitted from Earth’s surface.
Thermal inertia denotes the stability of temporal change of soil temperature and is defined as the square root of the product of the volumetric heat capacity and the thermal conductivity. Both the volumetric heat capacity and the thermal conductivity of water are several times those of dry soil, which allows the thermal inertia of soil to increase as the water content increases. These facts led to the idea of using thermal inertia as a proxy for soil water content. Thermal inertia retrieval from satellite surface temperature data was developed using the daily maximum and minimum temperatures, which are favorable for setting measurement timings of sun-synchronous polar orbital satellites, and optimal uses of these satellite surface temperatures were examined in earlier works. Among these, Price (1977) described a pioneering and comprehensive work of soil moisture retrieval using thermal inertia that considered not only the temperature and radiation budget of Earth’s surface but also turbulent heat fluxes. Apparent thermal inertia is a simplified thermal inertia–like parameter that considers the daily surface temperature range and the albedo of the insolation. Some studies have shown that apparent thermal inertia correlates well with surface soil moisture (Chang et al. 2012; Minacapilli et al. 2012; Qin et al. 2013). Various types of models have been developed involving satellite overpass timings to obtain the diurnal fluctuation range of land surface temperatures more accurately and efficiently (Xue and Cracknell 1995; Sobrino and El Kharraz 1999; Verstraeten et al. 2006; Maltese et al. 2013).
This study involves estimating the spatial distribution and intraseasonal changes of thermal inertia over an area of an approximately 2002 km2 scale located in a semiarid region of Mongolia that combines a typical and a dry steppe. A comprehensive model based on radiative, turbulent, and conductive heat exchanges on Earth’s surface with a vegetation canopy was employed to retrieve the thermal inertia of the surface soil layer. The model is classified into the two-source heat balance (TSEB) model, which is based on the force–restore method combined with a TSEB model (Kondo and Watanabe 1992), which possibly allows for the retrieval of model parameters, including thermal inertia, and estimating surface heat fluxes. A lot of specific types of models have been proposed, with emphases on particular parameters and variables relevant to the land surface processes according to models. For sparse vegetation in a semiarid climate, TSEB models have been also successful in retrieving land surface parameters, such as evaporation efficiency, and estimating surface turbulent fluxes (e.g., Boulet et al. 2015; Song et al. 2016). Using comprehensive TSEB models for retrieving thermal inertia and estimating soil moisture have not been sufficiently examined since the earlier stages of thermal inertia studies (Price 1977; van de Griend et al. 1985) because the parameter retrieval can be very time consuming, especially for the calculation of a wide space and a long time range. Matsushima (2007) and Matsushima et al. (2012) have improved and verified the method for retrieving thermal inertia using a TSEB model, but there remain significant fluctuations with retrieved thermal inertia values with respect to each grid. This fluctuation may be attributed to various reasons, such as few frequencies of satellite surface temperature and significant errors in parameters and variables (surface and air temperatures, insolation, albedo, and in particular, errors in satellite surface temperature data due to cloudiness). Matsushima et al. (2012) examined the sensitivity of thermal inertia retrieval in terms of surface temperature frequency and errors in physical values among these possibilities. This study examined effects of the aerodynamic conductance of turbulent heat fluxes in terms of the values almost shifting either in daytime or nighttime.
Recently, land data assimilation systems have been successfully used to downscale microwave-based satellite soil moisture data (e.g., Yang et al. 2009; Sawada et al. 2015) to simulate overall land surface interactions and soil moisture from the surface to the root zone and to compensate for the weakness whereby the spatial resolution is not as precise compared to the visible and infrared products. The thermal inertia method directly uses thermal-based satellite surface temperatures for estimating soil moisture. It is important to examine differences in ability of estimating soil moisture between the thermal-based method and the microwave-based method. This study examined correlations of overall data of both methods in the study period, and more specifically, agreements of daily changes of soil moisture in dry-down periods after antecedent rainfall. In this analysis, a parameterization for converting thermal inertia to volumetric soil water content (VSWC) was used. This parameterization is based on the land surface model developed by Noilhan and Planton (1989) and is applied based on a relevant study by the authors (Matsushima et al. 2017).
The objectives of this study are twofold: 1) to clarify the effects of the aerodynamic conductance of turbulent heat fluxes in terms of the values almost shifting either in daytime or nighttime and 2) to clarify the differences in the ability of estimating soil moisture between the thermal inertia method and the microwave-based method. A time series of spatial distribution of thermal inertia–derived soil moisture over the study area in a relatively fine resolution (3 km), which was finer than the results shown in Matsushima (2007) (6 km), is included here.
2. Materials and methods
a. Model and parameter retrieval
1) Model
2) Optimization and parameter retrieval
The Runge–Kutta method was employed for the integration of Eq. (1) with time intervals of 60 s (30 s when a divergence of the integration occurred using a 60-s interval). The time integral started at 2300 local time (LT) on the previous day after a lead time of 6000 s to begin the calculation and ended at 0400 LT on the next day.
3) Modification of the bulk transfer coefficients
The criterion for identifying daytime and nighttime was whether the net radiation was positive or negative. The above separation was effective for the calculation of turbulent heat fluxes at nighttime because the surface temperatures at nighttime were calculated more accurately compared to previous studies, in which the sensible and latent heat fluxes were overestimated compared to the observations when the daytime value of 
b. Study area and in situ soil moisture data
The spatial and temporal ranges of the present study were 45°–47°N and 106°–108°E and from May to September 2012. The spatial range mostly consists of flat terrain at an altitude of between 1300 and 1600 m above mean sea level, where all of the in situ soil moisture stations described in section 2d were located (Kaihotsu et al. 2013). The topography in the area is almost flat, with an average inclination between 1/500 and 3/5000 (Kaihotsu et al. 2004). The area was covered with pasture grass and sparse shrubs (Kaihotsu et al. 2013) with an LAI of at most 0.5 in 2012. The soil texture is almost sandy in most of the area and sandy with silt/humus in certain areas (Kaihotsu et al. 2004). Precipitation during this study at the routine meteorological station at Mandalgobi (45.75°N, 106.27°E) was 159 mm during the study period, which was a little higher than the long-term average (135 mm) of total precipitation from May through September (1944–99) at that station. The average air temperature was 16.1°C during the study period, somewhat higher than the long-term average of 14.9°C. This study selected a growing season over sparse vegetation as a study period and area for analyses, which is because one objective is to show how the thermal inertia–derived soil moisture and the AMSR2 soil moisture agree with the in situ soil moisture. It is suitable for better analyses of the present study that there are conditions of a large amount of periodic rainfall, large soil moisture fluctuations that resulted from the rainfall, and less vegetation on the ground surface. The growth season of 2012 in the study area meets the above conditions best compared to the other years for which the same analyses are possible (2002 and following years).
c. Data for model calculation
Data from MODIS, installed on the Terra and Aqua satellites of the Earth Observing System (EOS), were used in this study. The MODIS data were used for the LST (MOD11_L2 and MYD11_L2; NASA 2012c,d), the albedo (MCD43B3; NASA 2012a), and the LAI (MOD15A2; NASA 2012b). The insolation was calculated from the visible, near-infrared, and thermal infrared data of the geostationary satellite MTSAT-2 by employing an algorithm developed by Kawamura et al. (1998). Data from MTSAT-2 were referred to the geocoordinate mapped data archive for the GEWEX Asian Monsoon Experiment (GAME) research area given at a spatial resolution of 0.05° × 0.05° and were acquired at Kochi University, Japan (http://weather.is.kochi-u.ac.jp/). The surface meteorological data (the air temperature, specific humidity, and wind speed) were supplied from the archive of the Integrated Surface Hourly Database (ISD) of the National Centers for Environmental Information (
d. Soil moisture data
Long-range observations of the surface soil moisture have been carried out at many locations in the study area (Kaihotsu et al. 2004, 2009, 2013). The soil moisture data acquisition was first intended to validate the satellite soil moisture data from the Advanced Microwave Scanning Radiometer for EOS (AMSR-E) and AMSR2 [Mongol AMSR-E/AMSR2 Validation Experiment (MAVEX)], which are also used for the evaluation in the SMOS mission (Kaihotsu et al. 2013). The data from VSWC at a depth of 3 cm from 13 stations were applied to the comparison with the retrieved thermal inertia on a daily basis. Figure 1 shows the locations of the in situ soil moisture stations. Soil moisture data from the AMSR2 (available at https://gcom-w1.jaxa.jp/) were also compared to the retrieved thermal inertia results after 3 July. The AMSR2 data were archived from that date forward. The utility of SMOS data were not good because of radio frequency interference (RFI; Kaihotsu et al. 2013), while the AMSR2 data were not influenced by RFI because of the higher frequency.
Locations of in situ soil moisture stations and the spatial domain of the present study.
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
e. Data preprocessing for model calculation
Parameter retrieval was performed in the same manner as described in Matsushima (2007). The optimal parameters were retrieved at each 0.03° × 0.03° spatial resolution, at which other parameters, such as albedo, were given. Each 0.03° × 0.03° grid contained nine grids of 0.01° × 0.01° resolution where the model calculation was performed separately, although the model parameters were mutually used in each 0.03° × 0.03° grid. This approach was employed to prevent the number of degrees of freedom of the calculation from being higher than the number of optimized parameters (seven), and this condition was met when at least one available MODIS LST at each 0.01° × 0.01° grid for a diurnal calculation existed. Parameter retrieval was not performed if the number of available MODIS LSTs in the daytime during the diurnal period was less than nine in the 0.03° × 0.03° grid and if MODIS LSTs were available only in the daytime during the diurnal period even the number of the LST was more than nine after the result of Matsushima et al. (2012) (described in section 2a3). The grid scales described above are illustrated in Fig. 2.
Illustration of grid scales for the data used in this study and the statistical analyses.
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
According to the above-described method for parameter retrieval, data for the diurnal calculation were basically interpolated to grid data of a spatial resolution of 0.01° × 0.01° as follows, except for the insolation. The MODIS LST swath measurements were interpolated to a 0.01° × 0.01° spatial resolution by incorporating the geolocation data (MOD03 and MYD03) into the MODIS Reprojection Tool (MRT) Swath tool. The ISD meteorological data were interpolated to a spatial resolution of 0.01° × 0.01° based on the weighted average of the data observed at the routine meteorological stations within 500 km of a grid where interpolated data should be allocated. The weight was calculated as the sum of the reciprocal of the distance between the grid and the meteorological stations (Matsushima 2007). The downward longwave radiation was calculated at the same spatial resolution. The insolation data of the original resolution (0.05° × 0.05°) was directly used for the diurnal calculation of a 0.01° × 0.01° grid included in the grid of insolation. The albedo and the LAI given as the model parameters were aggregated to a 0.03° × 0.03° spatial resolution from the original 0.01° × 0.01° resolution to match the spatial resolution of the optimized parameters. Figure 2 illustrates the grid scales of the data for the diurnal calculation and parameters.
The data of the input variables of the model, which are in the column vector of the second term on the right side of Eq. (1), were interpolated in the temporal domain except for 
The availability of the MODIS LST occurred eight times a day at most, provided that the MODIS of both Terra and Aqua experienced cloudless conditions all day, but several times a day was the most typical frequency over the period of the present study. No data were available and no parameters were retrieved on days that were overcast throughout the day.
f. Thermal inertia–derived soil moisture
Thermal inertia is defined as the square root of the multiplication of the volumetric heat capacity and the thermal conductivity as presented in section 1. This definition is deeply connected to the parameterization for VSWC when the thermal inertia is given. In particular, a variety of models are proposed for the relationship between the thermal conductivity and the VSWC. Dong et al. (2015) categorizes the models into three types: the mixing model, the empirical model, and the mathematical model. Of these types, the empirical model is appropriate for the present study because the formulation is simple and easy to calculate. The empirical models can be further categorized into a few different types, one of which is an analogy model of the normalized thermal conductivity concept proposed by Johansen (1975), and the other type is a model connecting thermal inertia and thermal conductivity using the relationship between the matric potential and thermal conductivity. The former type includes models proposed by Murray and Verhoef (2007) and S. Lu et al. (2009), and this type of model relies on the change in thermal inertia due to the VSWC being regarded as the same as the change in the thermal conductivity, which has strong nonlinearity. The latter type includes the model proposed by Noilhan and Planton (1989), which employs the parameterization of the matric potential of soil and the thermal conductivity developed by McCumber and Pielke (1981), together with the parameterization of VSWC and the matric potential of soil developed by Clapp and Hornberger (1978). In this study, the Noilhan and Planton (1989) model is used to convert the thermal inertia to the VSWC according to the results of Matsushima et al. (2017), which examined the performance of the above three models using almost the same dataset as the present study. Table 1, adapted from Matsushima et al. (2017), lists the results of retrieved parameters with RMSEs of the calculations to the in situ observations and relevant soil types with respect to the parameter combinations for individual models. The results showed that all three models performed well at the same level of RMSE (0.039 m3 m−3) once the parameters of the respective model were recalibrated based on in situ soil moisture data (recalibrated). However, the soil type corresponding the recalibrated parameters (nearest type) of the Murray and Verhoef model (fine-textured soil) did not match the observed soil type (sandy loam, sandy silt loam; “loam” in Table 1) which had been examined by Yamanaka et al. (2007), while that in the Noilhan and Planton model was “loamy sand.” The Lu model had two soil types to classify (coarse or fine), which may be rough in terms of determining soil types, while the Noilhan and Planton model was able to classify 11 soil types according to the USDA classification using the exponent parameter b from the Clapp and Hornberger model, and the RMSE was smallest in using parameters for “nearest type” with respect to the individual models. Furthermore, the Johansen-type models use the logarithm of the normalized thermal inertia, in which the normalized thermal inertia was defined by the subtraction of the minimum thermal inertia from a retrieved thermal inertia divided by the subtraction of the minimum thermal inertia from the maximum. The normalized thermal inertia value was sometimes negative even when the maximum and minimum thermal inertia were recalibrated according to the in situ soil moisture data, and the logarithm was not applicable when the retrieved thermal inertia value was less than the minimum. Thus, not all cases were applicable to the models. The Noilhan and Planton model was adopted in this study to convert the thermal inertia to the VSWC due to the above reasons.
Applied or recalibrated values of individual parameters with respect to the parameterizations for thermal inertia converting to VSWC (adapted from Matsushima et al. 2017). Symbols are referred to in section 2f and appendix C. The terms “loam,” “recalibrated,” and “nearest type” are referred to section 2f.
3. Results
a. Effect of the nighttime bulk transfer coefficients on thermal inertia retrieval
Figure 3 clearly shows an example of the effect of introducing nighttime 
Comparison of retrieved values of the daily thermal inertia at BTS between V0 and V1 with in situ soil moisture.
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
Averages of daily standard deviations of retrieved thermal inertia at nine grids located at and around each station in terms of versions.
The effect of introducing nighttime values for the bulk transfer coefficients can be summarized as retrieved values of thermal inertia fluctuated less with respect to each grid, and the absolute values decreased in most cases. This can be explained as calculated surface temperatures were not likely to decrease to observed surface temperatures because of the calculated turbulent heat fluxes at nighttime being likely to be larger in magnitude, which resulted in thermal inertia having larger values. Figure 4 shows an example of diurnal changes of surface temperature and turbulent heat fluxes of both versions. Surface temperature during nighttime of V0 was higher than that of V1 and the MODIS temperature, which resulted from the sensible and latent heat fluxes at nighttime having a magnitude of 10–50 W m−2 (negative sign for sensible heat). Figure 5 shows averages and the standard deviations of subtractions of thermal inertia of V1 from that of V0 with respect to each grid over a 1° × 1° square (45°–46°N, 106°–107°E, which contains 1089 grids of 0.03° × 0.03°) on a daily basis. The daily averages of thermal inertia of V1 were mostly less than those of V0. The differences between versions were enhanced in May and September when the nighttime surface temperature was more likely to decrease than in summer.
(a) Diurnal change of calculated surface temperatures of V0 and V1 with MODIS surface temperatures on 14 Jul 2012 at a grid near BTS. (b) As in (a), but for sensible and latent heat fluxes.
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
Averages and the standard deviations of subtractions of thermal inertia of V1 from that of V0 with respect to each grid over a 1° ×1° square (45°–46°N, 106°–107°E, which contains 1089 grids of 0.03° × 0.03°) on a daily basis.
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
b. Thermal inertia–derived soil moisture
The approximate linearity of the thermal inertia and the VSWC at all stations are revealed in Fig. 6, except for station MGS. Figure 6 shows that the relationship between the thermal inertia and the VSWC for MGS is different from those at the other stations. This may have been partly due to the depth of the soil moisture sensors, which may have been deeper than the intended depth (3 cm). One of the authors (Asanuma) measured the depths of the in situ soil moisture sensors in the summer of 2015 and found that the depth at MGS was 13 cm and at the other stations was only 3–7 cm. The sensors were buried at 3 cm in the summer of 2008, and some of the depths may have increased as the sensors became covered with additional soil and/or humus since then. In this study, the in situ soil moisture data at MGS were excluded from the overall statistical analyses. Equation (15) shows that this linearity is determined by the exponent of 
Scatterplot of the thermal inertia vs the VSWC at the 12 stations (811, 813, 814, 815, 816, 817, 818, 820, 821, 5N, BTS, and DRS; squares) and MGS (crosses).
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
Retrieved parameters [
As in Table 3, but for representative values of parameters covering 12 stations (all stations except MGS).
c. Comparison of the retrieved thermal inertia with in situ observations and AMSR2 data
Figure 7 shows examples of the daily time series of retrieved thermal inertia compared to in situ soil moisture (3 cm) and the AMSR2 soil moisture. The thermal inertia estimates are statistically illustrated as the average and the standard deviation of nine values optimized at individual grids in the 0.03° × 0.03° resolution. Namely, the average value is a representation of an area of 0.09° × 0.09°, in which the grid including an in situ soil moisture station was located at the center and was surrounded by eight contiguous grids (see Fig. 2).
Daily time series of retrieved thermal inertia, in situ soil moisture (3 cm), and the AMSR2 soil moisture at (a) 821 (DOY 120–200), (b) 821 (DOY 200–280), (c) BTS (DOY 120–200), and (d) BTS (DOY 200–280).
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
Correlation coefficients between thermal inertia, in situ observation, and AMSR2 for (a) descending and (b) ascending paths at individual soil moisture stations.
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
Fitted values of parameter 
d. Spatial distribution
Figure 9a shows the spatial distribution of estimated thermal inertia–derived VSWC values calculated by Eq. (16) using the representative parameter values (
(a) Spatial distribution of the estimated thermal inertia–derived VSWC (m3 m−3) over the study area. Values of the soil water content are averages over 10 or 11 days for (left) days 1–10, (center) days 11–20, and (right) days 21–30 (or 31) of each month from May through September 2012. Blank grids are included where the values were not calculated because of the lack of retrieved thermal inertia data (less than 5 days).
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
As in (a), but for AMSR2 VSWC (level 2 product averaged over each 0.1 × 0.1 grid) from July through September 2012. For July, the first 10 days are from day 3 to 10 (instead of days 1–10).
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
Daily time series of thermal inertia–derived soil moisture (estimation), in situ soil moisture (3 cm), and the AMSR2 soil moisture at stations (a) 813 (DOY 120–200), (b) 813 (DOY 200–280), (c) 821 (DOY 120–200), and (d) 821 (DOY 200–280). “Representative” parameters of Eq. (16) were used for these calculations.
Citation: Journal of Hydrometeorology 19, 1; 10.1175/JHM-D-17-0040.1
4. Discussion
The range of retrieved thermal inertia was evaluated and appears reasonable. The range of thermal inertia found in this study was approximately 400–2300 J m−2 s−1/2 K−1 with the in situ VSWC ranging between 0.02 and 0.32 m3 m−3, and the values of the Clapp and Hornberger (1978) exponent indicated that the soil type was similar to loamy sand. Murray and Verhoef (2007) examined the thermal inertia of loam or loamy soil, and the resulting range of thermal inertia was 500–2000 J m−2 s−1/2 K−1; the corresponding range of the actual VSWC was 0.04–0.30 m3 m−3. Pratt and Ellyett (1979) theoretically examined the correspondence of the soil type (composition of sand and clay) with the thermal inertia and the VSWC, which resulted in the range of thermal inertia being 600–2000 J m−2 s−1/2 K−1 when the actual VSWC range is approximately 0.02–0.35 m3 m−3 for sandy soil. The above results were consistent with those of the present study.
The error statistics, that is, the RMSE of thermal inertia–derived VSWC, was 0.038 m3 m−3 in the present study, which was also evaluated as reasonable. S. Lu et al. (2009) proposed a Johansen-type model to derive soil moisture from thermal inertia, and they applied the model to data obtained by themselves and other data from reported studies. The RMSEs between the estimated and in situ soil moisture were between 0.02 and 0.05 m3 m−3. The present soil moisture estimates were also comparable to the results reported by Minacapilli et al. (2009), with the RMSE being approximately 0.03 m3 m−3 in an overnight experiment on an agricultural field covering an area of 1002 m2 (42 m2 of grid size). Bandara et al. (2014) used the Joint U.K. Land Environment Simulator (JULES) land surface model with a particle swarm optimization algorithm to retrieve the soil hydraulic parameters, including the Clapp and Hornberger (1978) exponent, and to estimate the daily time series of the soil moisture at the surface and root zone. The RMSE of the surface soil moisture estimation was 0.035–0.036 m3 m−3. Bandara et al. (2015) extended the method to estimate the spatial distribution with a resolution of 1 km in a semiarid grassland, and the resulting RMSE was 0.05–0.08 m3 m−3. Several studies investigated the upscale results of thermal inertia or apparent thermal inertia to match their scales to those of soil moisture results of the microwave passive sensors or an assimilated result from a land surface model. In a study by Qin et al. (2015), the RMSE of surface soil moisture estimation was 0.02–0.03 m3 m−3.
It is revealed in Figs. 7 and 10 that retrieved thermal inertia and thermal inertia–derived VSWC did not always follow the in situ VSWC well when VSWC was higher than approximately 0.15 m3 m−3. This may be partly due to the retrieval procedure and the characteristics of temporal changes of soil moisture of the grassland in Mongolia. VSWC in the study area was mostly small, but a sudden rise occurred after a significant rainfall and a rapid decay followed. Namely, large values of VSWC are limited in a very short period, which is likely to have the parameters of Eq. (16) fit optimally for dry conditions, and not sufficiently follow the large VSWC. To avoid this kind of result, relatively large values of thermal inertia should outweigh small ones in determining parameters of Eq. (16). An optimal method for weighting retrieved values is a future issue.
Yamanaka et al. (2007) explored the soil hydraulic properties of the stations in the same area as the present study and showed that values of the Clapp and Hornberger (1978) exponent [originally a pore-size distribution index 
5. Conclusions
The thermal inertia of soil is one parameter of a surface heat budget model and depends on the water content of soil. Surface thermal inertia was retrieved from a heat budget model incorporating satellite and surface meteorological data. The two objectives of this study are to clarify the effects of the aerodynamic conductance for turbulent heat fluxes in terms of the values almost shifting either in daytime or nighttime, and to clarify the differences in the ability to estimate soil moisture between the thermal inertia method and the microwave-based method. New formulations for the aerodynamic conductance with respect to the atmospheric stability significantly improved the accuracy of the thermal inertia retrieval. Thermal inertia–derived soil moisture correlated well with in situ soil moisture. In particular, dry-down processes of in situ soil moisture after antecedent rainfalls qualitatively agreed with thermal inertia–derived soil moisture but not with AMSR2, which was found after fitting the daily time series of individual soil moistures to an exponentially decaying function. The relationship between thermal inertia and VSWC is almost linear if the soil is sand or loam. The form of the function included Clapp and Hornberger’s (1978) exponent. In applying this function to the data, the approximate linearity was found, and the soil types derived from the exponent b agreed with the past observation results. Based on the above results, the spatiotemporal distribution of the thermal inertia–derived soil moisture was estimated over a grassland in Mongolia. The accuracy of the proposed method was comparable to the results from reported studies, which use both thermal inertia and apparent thermal inertia in a land surface model with data assimilation and/or upscaling. The accuracy of the thermal inertia method proposed in this study still has room to improve after solving problems with regard to the data, in particular, possible errors in satellite surface temperatures and also the method itself for better estimating sporadic increases of soil moisture. More precise resolution of satellite surface temperatures can raise the accuracy. Further validation of the exponent b associated with the soil type should be done with datasets other than the spatial or temporal ranges of this study.
This study was supported by JSPS KAKENHI [Awards 24510017 (Matsushima), 26289159 (Asanuma)] and JAXA GCOM 2nd RA [Award JX-PSPC-337505 (Kaihotsu)]. The Terra-, Aqua-, and combined MODIS products were acquired from the Level-1 and Atmosphere Archive and Distribution System (LAADS) Distributed Active Archive Center (DAAC), located in the Goddard Space Flight Center in Greenbelt, Maryland (https://ladsweb.nascom.nasa.gov/). AMSR2 data were supplied by the GCOM-W1 data providing service, Japan Aerospace Exploration Agency. We are grateful to Dr. G. Davaa and the staff at the Institute of Meteorology, Hydrology and Environment of Mongolia for their assistance in conducting this study. Two anonymous reviewers gave invaluable comments which have further improved the manuscript.
APPENDIX A
On Improved Bulk Transfer Coefficient Formulations Considering Atmospheric Stability with Regard to Linearity
APPENDIX B
Relevance of Parameter b for Soil Heat Transfer
APPENDIX C
Specific Formulations with Respect to the Parameterizations that Appear in Table 1
The Noilhan and Planton (1989) formulation is applied in present study to Eqs. (16) and (17), in which the parameters that should be recalibrated or applied according to soil type are 
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