4.1. Combination of C � θ and λ � θ relations

One of the principal methods for deriving the relationship between thermal inertia P and soil moisture (volumetric water content in most cases) θ uses the definition of thermal inertia <sup>P</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi <sup>c</sup>r<sup>λ</sup> <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffi <sup>C</sup><sup>λ</sup> <sup>p</sup> . Specifically, the effective models for using soil moisture to determine the volumetric heat capacity and thermal conductivity, respectively, proposed by de Vries [29] were used in several thermal inertia models [7, 8]. The volumetric heat capacity is formulated as

$$\mathbf{C} = \mathbf{C}\_w \boldsymbol{\theta} + \mathbf{C}\_m (1 - \boldsymbol{\theta}\_\*),\tag{11}$$

where Cw and Cm are the volumetric heat capacities of water and minerals, respectively, and θ<sup>∗</sup> is the soil moisture at saturation, in other words, the porosity of the soil, and the thermal conductivity λ is formulated as

Thermal Inertia-Based Method for Estimating Soil Moisture http://dx.doi.org/10.5772/intechopen.80252 17

$$\lambda = \sum\_{i=0}^{N} \mathcal{K}\_i \mathcal{X}\_i \lambda\_i / \sum\_{i=0}^{N} \mathcal{K}\_i \mathcal{X}\_i \tag{12}$$

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

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

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

$$\mathcal{C}\_s = \frac{2}{\mathcal{P}\sqrt{\pi/\pi}},\tag{13}$$

and

Spectroradiometer (MODIS) LST (MOD11\_L2 and MYD11\_L2). Thermal inertia retrieval at a

s surface constant at infinite depth

Analytical solution Model optimization (with other parameters

Needed in most models Adjusted through time integration

based on force-restore model (FRM)

Time integration of the respective surface temperatures of the two sources

LSTs insolation air temperature, specific humidity, wind speed albedo, and the leaf

Arbitrarily determined (at least two—one in the daytime and the other in the nighttime—

area index of the surface

regarding turbulent heat)

incorporating input time series

are suitable) [19]

Model Fourier series expansion Two-source energy balance (TSEB) model

Comparisons of the differences between the XC-based models and M2018 model are provided

One of the principal methods for deriving the relationship between thermal inertia P and soil moisture (volumetric water content in most cases) θ uses the definition of thermal inertia

volumetric heat capacity and thermal conductivity, respectively, proposed by de Vries [29] were used in several thermal inertia models [7, 8]. The volumetric heat capacity is formulated as

where Cw and Cm are the volumetric heat capacities of water and minerals, respectively, and θ<sup>∗</sup> is the soil moisture at saturation, in other words, the porosity of the soil, and the thermal

<sup>C</sup><sup>λ</sup> <sup>p</sup> . Specifically, the effective models for using soil moisture to determine the

C ¼ Cwθ þ Cmð Þ 1 � θ<sup>∗</sup> , (11)

2-km resolution was, therefore, possible in principle [28].

4.1. Combination of C � θ and λ � θ relations

4. How soil moisture is derived from thermal inertia

Items XC-based models M2018 [16]

[10] or both first and second components [11–14]

LSTs parameters in terms of insolation parameters in terms of turbulent heat flux (according to the

Two (without timing adjustment for maximum and minimum [11–14]); two (with timing adjustment

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

Sinusoidal function at the Earth<sup>0</sup>

Solution Fourier series components (first component only

[24]); three or more [9, 10]

according to models)

models)

Basic equation Differential equation of heat diffusion

in Table 1.

Boundary conditions

16 Soil Moisture

Input data and parameters

Thermal inertia retrieval

Number of diurnal LST measurements

Phase adjustment

of LST

<sup>P</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi

<sup>c</sup>r<sup>λ</sup> <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffi

conductivity λ is formulated as

$$\mathbf{C}\_{s} = \mathbf{C}\_{s,\*} \left[ \frac{\boldsymbol{\theta}\_{\*}}{\max(\boldsymbol{\theta}, \boldsymbol{\theta}\_{w})} \right]^{(2\ln 10/b)}\text{\textit{\textit{\textit{1}}}}\text{\textit{\textit{1}}}\tag{14}$$

where τ ¼ 2π=ω, and subscripts \* and w denote the saturation and wilting points, respectively. Substituting Eq. (13) into Eq. (14), after several calculations, yields

$$
\Theta = \text{sP}^{(2\ln 10/b)} \tag{15}
$$

where

$$s = \theta\_\* \left(\frac{\mathbb{C}\_{s,\*}}{2} \sqrt{\frac{\pi}{\pi}}\right)^{(2\ln 10/b)}\tag{16}$$

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 determined by the United States Department of Agriculture (USDA), and parameter b is a predictive parameter of the clay ratio in soil [37].

#### 4.2. Analogous to Johansen's thermal conductivity model

Johansen [38] proposed a model for determining thermal conductivity as a function of soil moisture. The concept of the model is that thermal conductivity is formulated as a universal function of soil moisture and that the function shape is determined by parameters of the formulation. The parameters are determined according to the soil type, such as sand, loam, silt, and clay. The generalized form of the parameterization is given as

$$
\lambda = \lambda\_{dry} + (\lambda\_\* - \lambda\_{dry})\mathbf{K}\_p \tag{17}
$$

5. Applications and discussion for future exploration

flux, including evapotranspiration.

approximately 1 week after a rainfall.

5.2. Assimilation with microwave-based data

needs to be explored.

5.1. Advantages and disadvantages compared to microwave-based methods

Thermal inertia-derived soil moisture can be estimated by combining methods as described in Sections 3 and 4. An advantage of the thermal inertia method that uses satellite data is that the spatial resolution is a couple of kilometers, which is much more precise than that of the microwave-based method, which has the spatial resolution of several tens of kilometers. However, there are also disadvantages, such as the precision and accuracy of thermal inertia retrieval being affected by the sky conditions, especially clouds, which are the weakest point in using the thermal-infrared bands. A recent study showed that the microwave brightness temperatures complemented the thermal-infrared derived LST, but instead of this, the spatial resolution of the thermal-infrared LST had to be sacrificed [43]. Another disadvantage is that the thermal inertia of a surface covered with dense vegetation is difficult to retrieve. Soil moisture retrieval using the microwave bands also has the same problem. Thermal inertia retrieval over a surface covered with sparse vegetation has been achieved in many studies in which M2018 is categorized in the two-source energy balance (TSEB) concept [44, 45]. In M2018, the vegetation canopy is modeled according to its surface temperature, the three parameters that should be optimized, and the leaf area index, which is given as satellite data. The effectiveness of the TSEB model is not only to retrieve thermal inertia but also possibly to accurately calculate heat flux with regard to the surface heat balance. The denser the vegetation, the less accurate the thermal inertia retrieval. It should be noted that the thermal inertiaderived soil moisture is calculated through a simple TSEB model as well as the surface heat

Thermal Inertia-Based Method for Estimating Soil Moisture

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

19

According to the above advantages and disadvantages, soil moisture derivation for a surface is more effective in arid and semi-arid regions where clear sky conditions overwhelm other conditions and where there are spatial soil moisture contrasts, for example, between an oasis and other land cover, as well as significant temporal changes, for example, from just after to

The optimization scheme for the thermal inertia retrieval is crucial to save calculation time. M2018 uses the downhill simplex method [46], which is generally suitable for optimizing less than approximately five parameters. It takes approximately 20–40 s to retrieve seven parameters including the thermal inertia of one grid in M2018 using a workstation. The downhill simplex method has the advantage of not diverging in the optimizing process, but the algorithm is not simple and requires a long time to complete. A more efficient optimization scheme

Data assimilation procedures are downscaled schemes of microwave-based soil moisture, which has a scale of several tens of kilometers, to one to a couple of kilometers. These schemes have recently been improved [47–50] using visible, near-infrared, and thermal-infrared satellite data, which have more precise spatial resolution than microwaves. These procedures can be

where the subscript dry denotes zero soil moisture, and Kp is the universal Kersten function. The formulation is defined as a function of soil moisture from zero to the saturation point (porosity). Then, the problem is reduced to determine the specific formulation and its parameter values. The specific form of the Kersten function is a power function, and the curve shape depends on the power according to the soil type, which has strong nonlinearity in most cases.

Murray and Verhoef [39] applied the above Johansen type model to thermal inertia parameterization as follows:

$$P = P\_{dry} + \left(P\_{\ast} - P\_{dry}\right)K\_p \tag{18}$$

To calculate the thermal inertia value, parameters Pdry and P<sup>∗</sup> have to be determined, and the formulation of Kp is given as the parameterization proposed by Lu et al. [31] in [39], as

$$K\_p = \exp\left[\gamma \left(1 - S\_r^{\prime - \delta}\right)\right],\tag{19}$$

where γ and δ are the coefficients for optimization according to the soil type, and Sr is the soil moisture normalized by saturation. Lu et al. [40] proposed a similar parameterization as that in [39]. Minacapilli et al. [41] tested the performance of the Murray and Verhoef model [39] and extended the Johansen model concept to the apparent thermal inertia. Recently, Lu et al. [42] showed that Pdry was parameterized as a function of porosity, in other words, the soil clay ratio, improving the accuracy of thermal inertia retrieval.

Again, thermal inertia is the square root of the product of volumetric heat capacity and thermal conductivity, and volumetric heat capacity increases modestly according to soil moisture. By contrast, thermal conductivity has strong nonlinearity compared to soil moisture. Therefore, thermal inertia can be formulated as a nonlinear function, and even, the square root operates the product Cλ. Lu et al. [40] applied the formulation to thermal inertia and determined the parameter values according to three soil types. The minimum and maximum thermal inertia values range over soil moisture from zero to saturation. If some amount of error is added to the retrieved value of thermal inertia from a model calculation or laboratory experiment, the value may be less than the minimum, and hence, soil moisture cannot be calculated due to Kp being negative.
