3.2. Model methods using multiple satellite data

Earth observing satellites are divided into polar-orbiting and geostationary. Both types have thermal-infrared bands to estimate LST. The temporal resolution of the geostationary satellites is superior to that of the polar-orbiting ones. By contrast, the spatial resolution and temperature accuracy of the geostationary satellites have been improving but are not yet superior to those of the polar-orbiting one [22, 23]. Therefore, in this section, models using polar-orbiting satellites are described and discussed.

Most of the models proposed so far for estimating thermal inertia were based on the model proposed by Xue and Cracknell [13] (XC model). This model is based on solving the thermal diffusion equation using the Fourier series expansion, which is described in Section 1. This model uses the first 24-h period and the second 12-h period Fourier harmonics of the sinusoidal components to reproduce the diurnal variation of LSTs. However, the two components were not always enough to reproduce an actual diurnal LST change. In these cases, the phase differences from the diurnal change of the insolation of respective components have to be adjusted. Measured values in thermal-infrared bands were used to calculate the LSTs. The LSTs are almost the daily maximum and minimum, which are suitable for accurately estimating thermal inertia. Various improvements were proposed based on the XC model. Among these improvements, a method using four satellite LSTs during a diurnal cycle irrespective of the daily maximum and minimum successfully retrieved thermal inertia [9]. Schemes for the phase and amplitude adjustments were also introduced. The phase adjustment was applied to the time-difference between the actual timing of the maximum and minimum LSTs and the overpass timings of the satellite measurements. This adjustment scheme was to have the satellite overpass timing approach the daily maximum or minimum LSTs using a cosine function for the phase difference when their estimated times were given [24]. On the other hand, an adjustment that decayed the LST amplitude after sunset was applied during nighttime due to the LST change being smaller than that in the daytime to avoid overestimating nighttime cooling [25]. These adjustments allowed arbitrary and many timings of the satellite LSTs to be incorporated into the models for thermal inertia retrieval. The formulations for estimating thermal inertia were analytical but relatively complicated to describe and grasp. Moreover, various approximations were adopted to avoid the implicit formulations that required iterative calculations to retrieve thermal inertia. The input data required are according to individual models but are almost limited to the satellite LSTs, and some parameters with regard to the insolation are also required to specialize in retrieving thermal inertia from other parameters regarding the land surface processes. Models that simply address turbulent heat flux were proposed assuming that turbulent heat flux is proportional to the temperature difference between surface and atmosphere [26], or was ingeniously neglected [27]. Maltese et al. [10] proposed an XC-based model that only used the first Fourier component, which was enough to accurately retrieve thermal inertia using three LSTs during a diurnal cycle.

used instead of the temperature at an infinite depth), and Gð Þ0 is soil heat flux at the surface averaged over the time span Δt. Matsushima et al. [20] estimated the thermal inertia of asphalt pavement based on Eq. (10) using data from Tð Þ0 measured by a portable radiative thermometer approximately 1 m above the surface as well as a copper-constantan thermal couple on the surface and Gð Þ0 measured by a heat flux plate on the asphalt surface over 30 min under a summer daytime clear sky, assuming that the daily average thermocouple temperature was the temperature at an infinite depth. The results of the thermal inertia were 1140 J m�<sup>2</sup> K�<sup>1</sup> s

and the thermocouple, respectively. The values agree well with the standard value of

capacity and thermal conductivity, calculated using the definition of thermal inertia Eq. (1)). The above result shows that thermal inertia can be estimated at a local scale using simple measurement equipment and also shows the feasibility of thermal inertia estimation using

Earth observing satellites are divided into polar-orbiting and geostationary. Both types have thermal-infrared bands to estimate LST. The temporal resolution of the geostationary satellites is superior to that of the polar-orbiting ones. By contrast, the spatial resolution and temperature accuracy of the geostationary satellites have been improving but are not yet superior to those of the polar-orbiting one [22, 23]. Therefore, in this section, models using polar-orbiting

Most of the models proposed so far for estimating thermal inertia were based on the model proposed by Xue and Cracknell [13] (XC model). This model is based on solving the thermal diffusion equation using the Fourier series expansion, which is described in Section 1. This model uses the first 24-h period and the second 12-h period Fourier harmonics of the sinusoidal components to reproduce the diurnal variation of LSTs. However, the two components were not always enough to reproduce an actual diurnal LST change. In these cases, the phase differences from the diurnal change of the insolation of respective components have to be adjusted. Measured values in thermal-infrared bands were used to calculate the LSTs. The LSTs are almost the daily maximum and minimum, which are suitable for accurately estimating thermal inertia. Various improvements were proposed based on the XC model. Among these improvements, a method using four satellite LSTs during a diurnal cycle irrespective of the daily maximum and minimum successfully retrieved thermal inertia [9]. Schemes for the phase and amplitude adjustments were also introduced. The phase adjustment was applied to the time-difference between the actual timing of the maximum and minimum LSTs and the overpass timings of the satellite measurements. This adjustment scheme was to have the satellite overpass timing approach the daily maximum or minimum LSTs using a cosine function for the phase difference when their estimated times were given [24]. On the other hand, an adjustment that decayed the LST amplitude after sunset was applied during nighttime due to the LST change being smaller than that in the daytime to avoid overestimating nighttime cooling [25]. These adjustments allowed arbitrary and many timings of the satellite

�1/2 using the LSTs measured by the portable radiative thermometer

�1/2 in the literature [21] (actually, this value is based on the volumetric heat

and 1350 m�<sup>2</sup> K�<sup>1</sup> s

airborne and satellite thermal-infrared thermometry.

3.2. Model methods using multiple satellite data

satellites are described and discussed.

1220 m�<sup>2</sup> K�<sup>1</sup> s

14 Soil Moisture

�1/2

In contrast to the XC-based models, a series of models proposed by Matsushima and coresearchers (the most recent one is [16], hereafter referred to as M2018) was essentially based on the sinusoidal solution of the thermal diffusion equation. However, they adopted the FRM to avoid complicated analytical formulations for estimating thermal inertia while maintaining linearity. The FRM requires time integration to calculate the surface temperature as shown in Eq. (7). The characteristic frequency ω was set to the diurnal change (= 2π=86400(s<sup>1</sup> )) in M2018. Hence, the FRM appeared to be able to only reproduce the sinusoidal changes whose frequencies were near the characteristic frequency. However, shorter period changes were reproduced according to the input data changes shown in Figure 4 of M2018 because the force term (the time derivative term) was more temporally sensitive than the restore term (the product of the frequency and temperature difference from the daily average). The boundary conditions at the surface, the left side of Eq. (7), are also required. The boundary conditions are converted to the budget of net radiation, and sensible and latent heat at the surface using the heat balance equation. This requirement for the surface boundary conditions is as same as that of the XC-based models. M2018 was not specialized at retrieving thermal inertia, but other parameters with regard to the sensible and the latent heat flux, including the diurnal time series of insolation, air temperature, specific humidity, and wind speed, were required as input data. Also, the surface albedo and leaf area index were used as parameters, and the LSTs for the model optimization are described below. The above types of data were required, but all were readily available from satellite and meteorological data archives through the Internet, that is, a special observation was not needed. Instead of requiring many types of input data, the M2018 formulation was relatively simple and did not require phase and amplitude adjustments. The shift in values of the bulk transfer coefficients for the sensible and latent heat flux in the daytime and nighttime was required only to improve the accuracy of thermal inertia retrieval, which was approximately equivalent to the amplitude adjustment in the XC-based model by Schädlich et al. [25]. The time integration did not require an implicit scheme, but an optimization algorithm was required to retrieve thermal inertia and the other parameters at the same time. The optimization algorithm (the downhill simplex method that was employed in M2018) took time to retrieve parameters; hence, the M2018 had no advantage for the worldwide spatial scale and the temporal scale of several decades. In M2018, daily values of thermal inertia-derived soil moisture were estimated with a 3-km spatial resolution at 2<sup>∘</sup> <sup>2</sup><sup>∘</sup> in latitude and longitude using the 1-km spatial resolution of Moderate Resolution Imaging


<sup>λ</sup> <sup>¼</sup> <sup>X</sup> N

been expanded, and the details are provided in a review by Dong et al. [32].

water content) when the soil (mineral) density and water density are known.

ship between M2018 thermal inertia and soil moisture, which is formulated as

Cs ¼ Cs,<sup>∗</sup>

s ¼ θ<sup>∗</sup>

Substituting Eq. (13) into Eq. (14), after several calculations, yields

and

where

Cs <sup>¼</sup> <sup>2</sup> P ffiffiffiffiffiffiffiffi

> θ∗ maxð Þ θ; θ<sup>w</sup>

where τ ¼ 2π=ω, and subscripts \* and w denote the saturation and wilting points, respectively.

Cs,<sup>∗</sup> 2

The constant s depends on the parameters b, θ∗, and Cs,∗, of the Clapp and Hornberger parameterization [36]. In particular, parameter b is related to 11 categories of soil types

ffiffiffi τ π � � r ð Þ 2ln 10=<sup>b</sup>

� �ð Þ 2ln 10=<sup>b</sup>

i¼0

KiXiλi=

where N is the number of types of granules and particles that make up the soil, including the minerals, organic matter, water, and air inside the soil. Each component has a thermal conductivity λ<sup>i</sup> and a volume fraction Xi, and Ki is a weighting factor that is the ratio of the average temperature gradient in the granules of the i-th component to the average temperature gradient in the medium. See Appendix for details on the formulation of Ki. Minacapilli et al. [30] combined the linear relation of the volumetric heat capacity proposed by de Vries [29] and an empirical parameterization for thermal conductivity proposed by Lu et al. [31] to derive thermal inertia. The proposed models for the thermal conductivity of unsaturated soil have

Ma and Xue [33] proposed an empirical parameterization that often appears in the literature. This parameterization calculates thermal inertia for a given soil moisture (gravimetric soil

Noilhan and Planton [34] derived the relationship between thermal inertia and soil moisture in another way. This method was basically a combination of thermal inertia, the relationship between soil moisture and matric potential of soil, and a parameterization of the thermal conductivity as a function of the matric potential proposed by McCumber and Pielke [35]. In their paper, they showed the relationship between soil moisture and soil thermal coefficient Cs, which was defined in their paper, but was able to be rearranged to according to the relation-

X N

KiXi, (12)

http://dx.doi.org/10.5772/intechopen.80252

17

Thermal Inertia-Based Method for Estimating Soil Moisture

<sup>τ</sup>=<sup>π</sup> <sup>p</sup> , (13)

<sup>θ</sup> <sup>¼</sup> sPð Þ 2ln 10=<sup>b</sup> (15)

, (14)

(16)

i¼0

Table 1. Comparison of the differences between the XC-based models and M2018 model.

Spectroradiometer (MODIS) LST (MOD11\_L2 and MYD11\_L2). Thermal inertia retrieval at a 2-km resolution was, therefore, possible in principle [28].

Comparisons of the differences between the XC-based models and M2018 model are provided in Table 1.
