1. Introduction: thermal inertia of unsaturated soil

Thermal inertia P is one of the parameters used to characterize the thermal properties of soil and is defined as the square root of the product of the volumetric heat capacity C and thermal conductivity λ, which is given as

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

$$P = \sqrt{\mathcal{C}\lambda}.\tag{1}$$

periodic cycles of soil temperature, under the boundary conditions that the temperature change is sinusoidal at the surface and constant at an infinite depth. The basic solution is as

d

if the surface boundary condition is Tð Þ¼ 0; t A � exp ½ � iωt . In Eq. (3), A is the amplitude of a periodic change with an angular velocity ω, d is the scale depth at which the amplitude is e�<sup>1</sup> of

r

d

G zð Þ¼� ; <sup>t</sup> <sup>λ</sup> <sup>∂</sup>T zð Þ ; <sup>t</sup>

where the vertical profile of soil temperature is obviously required to calculate the soil heat flux. However, when the soil temperature solution (3) or (5) is applied to the soil heat flux Eq. (6) and then z is set to zero, it only uses the time series of surface temperature as follows:

is used. Eq. (7) is known as the force-restore method (FRM) for calculating the surface soil heat flux [2]. In Eq. (7), the numerator of the parameter is defined as thermal inertia P, which is

<sup>C</sup><sup>λ</sup> <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi

The above derivation describes thermal inertia being effective for quantifying soil thermal properties when only the LST is known, which leads to the analysis of the land surface processes using satellite LSTs. The above procedure leads to the studies proposed by Matsu-

ffiffiffiffiffiffi Cλ 2ω <sup>r</sup> <sup>∂</sup>Tð Þ <sup>0</sup>; <sup>t</sup>

∂T zð Þ ; t

<sup>P</sup> <sup>¼</sup> ffiffiffiffiffiffi

� � � cos <sup>ω</sup><sup>t</sup> � <sup>z</sup>

<sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>ω</sup>Tð Þ <sup>0</sup>; <sup>t</sup>

ffiffiffiffiffiffiffi 2λ ωC

d ¼

T zð Þ¼ ; <sup>t</sup> <sup>A</sup>exp � <sup>z</sup>

The conductive heat flux in soil at depth z and time t, G zð Þ ; t , is defined as

Gð Þ¼ 0; t

In deriving Eq. (7), the following relation derived from Eq. (3),

shima and co-researchers, which are described in Section 2.

given as

� � � exp <sup>i</sup> <sup>ω</sup><sup>t</sup> � <sup>z</sup>

d

d

h i � � , (3)

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11

Thermal Inertia-Based Method for Estimating Soil Moisture

�<sup>1</sup> <sup>p</sup> . The scale depth <sup>d</sup> is formulated as

: (4)

h i, (5)

<sup>∂</sup><sup>z</sup> , (6)

� �: (7)

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>i</sup>ωT zð Þ ; <sup>t</sup> , (8)

crλ p : (9)

T zð Þ¼ ; <sup>t</sup> <sup>A</sup>exp � <sup>z</sup>

follows using the complex number expression:

the surface value, and i is the imaginary number ffiffiffiffiffiffi

if the surface boundary condition is Tð Þ¼ 0; t A � cos ωt.

Eq. (3) is rewritten as follows using the real number expression:

Thermal inertia appears in the formulation of the ground heat flux when it is formulated with a unique variable known as the land surface temperature (LST), and one does not have to consider the vertical profile of the soil temperature.

In terms of the relation between soil thermal properties and soil moisture, both the volumetric heat capacity C, which is the product of specific heat c and the bulk density r of the soil, and the thermal conductivity λ increase as the soil moisture increases. Accordingly, thermal inertia P also increases as soil moisture increases. Therefore, soil moisture can be estimated inversely if the thermal inertia value is known (Figure 1). The volumetric heat capacity is a moderate linear function of soil moisture [1]. By contrast, thermal conductivity has a strong nonlinearity with soil moisture, making it difficult to parameterize thermal conductivity and hence thermal inertia.

Thermal inertia is effective when the time series of the surface temperature is available, but the vertical profile of the soil temperature is not available. The differential equation for heat diffusion is given as

$$\frac{\partial T(z,t)}{\partial t} = \frac{\lambda}{C} \frac{\partial^2 T(z,t)}{\partial z^2},\tag{2}$$

where T zð Þ ; t is the soil temperature at depth z and time t, which is the difference from a constant value at an infinite depth. This equation is solved to reproduce the daily and yearly

Figure 1. Schematic of the relationship between volumetric heat capacity, thermal conductivity, and thermal inertia in terms of soil moisture.

periodic cycles of soil temperature, under the boundary conditions that the temperature change is sinusoidal at the surface and constant at an infinite depth. The basic solution is as follows using the complex number expression:

$$T(z,t) = A \exp\left(-\frac{z}{d}\right) \cdot \exp\left[i\left(\omega t - \frac{z}{d}\right)\right],\tag{3}$$

if the surface boundary condition is Tð Þ¼ 0; t A � exp ½ � iωt . In Eq. (3), A is the amplitude of a periodic change with an angular velocity ω, d is the scale depth at which the amplitude is e�<sup>1</sup> of the surface value, and i is the imaginary number ffiffiffiffiffiffi �<sup>1</sup> <sup>p</sup> . The scale depth <sup>d</sup> is formulated as

$$d = \sqrt{\frac{2\lambda}{a\mathcal{C}}}.\tag{4}$$

Eq. (3) is rewritten as follows using the real number expression:

$$T(z,t) = A \exp\left(-\frac{z}{d}\right) \cdot \cos\left[\omega t - \frac{z}{d}\right],\tag{5}$$

if the surface boundary condition is Tð Þ¼ 0; t A � cos ωt.

<sup>P</sup> <sup>¼</sup> ffiffiffiffiffiffi

Thermal inertia appears in the formulation of the ground heat flux when it is formulated with a unique variable known as the land surface temperature (LST), and one does not have to

In terms of the relation between soil thermal properties and soil moisture, both the volumetric heat capacity C, which is the product of specific heat c and the bulk density r of the soil, and the thermal conductivity λ increase as the soil moisture increases. Accordingly, thermal inertia P also increases as soil moisture increases. Therefore, soil moisture can be estimated inversely if the thermal inertia value is known (Figure 1). The volumetric heat capacity is a moderate linear function of soil moisture [1]. By contrast, thermal conductivity has a strong nonlinearity with soil moisture, making it difficult to parameterize thermal conductivity and hence thermal

Thermal inertia is effective when the time series of the surface temperature is available, but the vertical profile of the soil temperature is not available. The differential equation for heat

where T zð Þ ; t is the soil temperature at depth z and time t, which is the difference from a constant value at an infinite depth. This equation is solved to reproduce the daily and yearly

Figure 1. Schematic of the relationship between volumetric heat capacity, thermal conductivity, and thermal inertia in

∂T zð Þ ; t <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>λ</sup> C ∂2 T zð Þ ; t

consider the vertical profile of the soil temperature.

inertia.

10 Soil Moisture

diffusion is given as

terms of soil moisture.

<sup>C</sup><sup>λ</sup> <sup>p</sup> : (1)

<sup>∂</sup>z<sup>2</sup> , (2)

The conductive heat flux in soil at depth z and time t, G zð Þ ; t , is defined as

$$G(z,t) = -\lambda \frac{\partial T(z,t)}{\partial z},\tag{6}$$

where the vertical profile of soil temperature is obviously required to calculate the soil heat flux. However, when the soil temperature solution (3) or (5) is applied to the soil heat flux Eq. (6) and then z is set to zero, it only uses the time series of surface temperature as follows:

$$G(0,t) = \sqrt{\frac{\overline{\mathbf{C}\lambda}}{2\omega}} \left[ \frac{\partial T(0,t)}{\partial t} + \omega T(0,t) \right]. \tag{7}$$

In deriving Eq. (7), the following relation derived from Eq. (3),

$$\frac{\partial T(z,t)}{\partial t} = \mathrm{i}\omega T(z,t),\tag{8}$$

is used. Eq. (7) is known as the force-restore method (FRM) for calculating the surface soil heat flux [2]. In Eq. (7), the numerator of the parameter is defined as thermal inertia P, which is given as

$$P = \sqrt{\mathbb{C}\lambda} = \sqrt{\mathfrak{c}\rho\lambda}. \tag{9}$$

The above derivation describes thermal inertia being effective for quantifying soil thermal properties when only the LST is known, which leads to the analysis of the land surface processes using satellite LSTs. The above procedure leads to the studies proposed by Matsushima and co-researchers, which are described in Section 2.

Considerable effort has been made to estimate the thermal inertia of the Earth's surface mainly using LST data from satellites. Most of this effort has been concentrated on retrieving daily values of thermal inertia due to the availability of daily maximum and minimum LSTs observed from polar orbiting or geostationary satellites. Models using these types of satellite LSTs are based on the Earth's surface energy balance principle, which includes not only the radiation budget but also turbulent heat flux. A Fourier series expansion was introduced to solve Eq. (1) under the above Earth's surface boundary conditions using the solution of the real number expression, Eq. (5). Models have been improved from those using only the two daily extreme LSTs [3–8] to those using LSTs that are irrespective of time in a diurnal change [9, 10] and other significant studies that follow a series of important proposals by Xue and Cracknell [11–14], which are also described in Section 2 when compared with studies by Matsushima and co-researchers.

Based on a series of studies performed by Price [3–6], Xue and Cracknell [11–14] proposed improved methods, which showed that data from satellites were good enough to accurately retrieve thermal inertia as well as using the time of the maximum LST. These models used the first- and second-order harmonics of the diurnal change (24- and 12-h periods) to fit the LST change considering the phase differences of both components to insolation. Thermal inertia was obtained from analytical but relatively complicated formulations. Based on the series of models proposed by Xue and Cracknell (hereinafter the XC model), several improved methods were proposed in terms of the timing of satellite measurements, actual timing of the diurnal maximum and minimum LSTs, and difference in LST change between daytime and nighttime.

Thermal Inertia-Based Method for Estimating Soil Moisture

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Other than the above methods, Matsushima [15] applied the FRM to the surface heat balance model to retrieve thermal inertia. The FRM is also based on a sinusoidal boundary condition at the surface and the heat diffusion equation, which is essentially the same as the models based on the XC model. The Matsushima model [15] employed an FRM that was designed not only to mostly respond to the diurnal change but also to more rapid changes according to the temporal resolution of input the variables (insolation, air temperature, etc.). A change of the LST over a period of approximately a few hours was fairly reproduced by the FRM that had a characteristic period of 24 h, as illustrated in [16]. Similar results were found in other studies [2, 17], or higher-frequency nonsinusoidal forcing did not significantly affect the LST prediction [18]. This means that the FRM can reproduce temporal changes that have a wide range of LST frequencies via its relatively simple formulation. Using this method, the timing of satellite LST measurements was arbitrary in principle, irrespective of the daily maximum and minimum, but was more accurate for thermal inertia retrieval that the LSTs measured both in the daytime and in the nighttime, as shown in [19]. The accuracy of thermal inertia retrieval is improved if the coefficients of the atmospheric turbulent heat flux are set differently in the daytime and nighttime, as illustrated in [16]. The details are described in Section 3.2 when

3. Thermal inertia retrieval according to the spatial and temporal

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

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

<sup>Δ</sup><sup>t</sup> <sup>þ</sup> <sup>ω</sup>Tð Þ<sup>0</sup>

� �, (10)

ffiffiffiffiffiffi P 2ω <sup>r</sup> <sup>Δ</sup>Tð Þ<sup>0</sup>

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

Gð Þ¼ 0

The details of the above schemes are described in Section 3.2.

compared with the XC model.

resolutions

This chapter reviews former and state-of-the-art methods for estimating soil moisture by exploring the relationship between thermal inertia and soil moisture. Section 2 reviews past developments of methods for thermal inertia retrieval from land surface models. Section 3 describes how thermal inertia is experimentally observed, and how it is retrieved from land surface models in terms of the Xue and Cracknell-based models and the Matsushima models. Section 4 describes several semi-empirical parameterizations of thermal inertia in terms of soil moisture. Section 5 describes applications of thermal inertia for analyzing hydrometeorological phenomena around the Earth's surface and also discusses further exploration of thermal inertia itself and its applications. Section 6 presents conclusions.
