2. Thermal inertia retrieval from energy balance models

Thermal inertia retrieval from an energy balance model of the Earth's surface began with a geological context in which different rocks or minerals respond differently to the incident solar radiation. Then, researchers' interests moved to the thermal inertia change according to soil moisture, which was coincident with the use of data obtained by sun-synchronous polar orbiting satellites that gave diurnal cycle of LSTs. Retrieval thermal inertia solely according to the soil moisture of the Earth's surface was developed during the last five decades. Most of the proposed methods employed the Fourier series of the LST diurnal variation, which was incorporated in a heat balance model of the Earth's surface.

A comprehensive model for retrieving thermal inertia using the energy balance model for the Earth's surface boundary conditions and the Fourier series for the diurnal change of the surface temperature was proposed by Price [3–5]. Price [6] made the terms of the turbulent heat flux (sensible and latent heat) simpler than previous studies to focus on retrieving thermal inertia. These studies used satellite LST measurements twice a day as the daily maximum and minimum LSTs, corresponding to daytime and nighttime, respectively, and substituted the LSTs into the first component (24-h period) of the Fourier series to calculate thermal inertia. Namely, they approximated the time-differential term of Eq. (7) as <sup>∂</sup>Tð Þ <sup>0</sup>;<sup>t</sup> <sup>∂</sup><sup>t</sup> ! <sup>Δ</sup>Tð Þ <sup>0</sup>;<sup>t</sup> <sup>Δ</sup><sup>t</sup> , where ΔTð Þ 0; t is the difference between daily maximum and minimum LSTs, and Δt is the time difference of the two measurements.

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

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.

inertia itself and its applications. Section 6 presents conclusions.

12 Soil Moisture

incorporated in a heat balance model of the Earth's surface.

difference of the two measurements.

2. Thermal inertia retrieval from energy balance models

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

Thermal inertia retrieval from an energy balance model of the Earth's surface began with a geological context in which different rocks or minerals respond differently to the incident solar radiation. Then, researchers' interests moved to the thermal inertia change according to soil moisture, which was coincident with the use of data obtained by sun-synchronous polar orbiting satellites that gave diurnal cycle of LSTs. Retrieval thermal inertia solely according to the soil moisture of the Earth's surface was developed during the last five decades. Most of the proposed methods employed the Fourier series of the LST diurnal variation, which was

A comprehensive model for retrieving thermal inertia using the energy balance model for the Earth's surface boundary conditions and the Fourier series for the diurnal change of the surface temperature was proposed by Price [3–5]. Price [6] made the terms of the turbulent heat flux (sensible and latent heat) simpler than previous studies to focus on retrieving thermal inertia. These studies used satellite LST measurements twice a day as the daily maximum and minimum LSTs, corresponding to daytime and nighttime, respectively, and substituted the LSTs into the first component (24-h period) of the Fourier series to calculate thermal inertia.

ΔTð Þ 0; t is the difference between daily maximum and minimum LSTs, and Δt is the time

<sup>∂</sup><sup>t</sup> ! <sup>Δ</sup>Tð Þ <sup>0</sup>;<sup>t</sup>

<sup>Δ</sup><sup>t</sup> , where

Namely, they approximated the time-differential term of Eq. (7) as <sup>∂</sup>Tð Þ <sup>0</sup>;<sup>t</sup>

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 compared with the XC model.
