3. Thermal inertia retrieval according to the spatial and temporal resolutions

### 3.1. A simple method using in-situ field measurements on a local scale

A simple method for estimating thermal inertia using simply measured surface radiative temperatures can be performed based on a finite difference form of Eq. (7), which is given as

$$\overline{G(0)} = \sqrt{\frac{P}{2\omega}} \left[ \frac{\Delta T(0)}{\Delta t} + \omega \overline{T(0)} \right],\tag{10}$$

where ΔTð Þ0 is a significant increase in LST during a relatively short-time span Δt (e.g., 30 min), Tð Þ0 is the temporal average of the LST difference from the soil temperature at infinite depths during the time span (practically, the daily average surface or air temperatures can be 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 �1/2 and 1350 m�<sup>2</sup> K�<sup>1</sup> s �1/2 using the LSTs measured by the portable radiative thermometer and the thermocouple, respectively. The values agree well with the standard value of 1220 m�<sup>2</sup> K�<sup>1</sup> s �1/2 in the literature [21] (actually, this value is based on the volumetric heat 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 airborne and satellite thermal-infrared thermometry.

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

Thermal Inertia-Based Method for Estimating Soil Moisture

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

enough to accurately retrieve thermal inertia using three LSTs during a diurnal cycle.

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>

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

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