5. Applications and discussion for future exploration

determined by the United States Department of Agriculture (USDA), and parameter b is a

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,

λ ¼ λdry þ λ<sup>∗</sup> � λdry

P ¼ Pdry þ P<sup>∗</sup> � Pdry

formulation of Kp is given as the parameterization proposed by Lu et al. [31] in [39], as

To calculate the thermal inertia value, parameters Pdry and P<sup>∗</sup> have to be determined, and the

Kp <sup>¼</sup> exp <sup>γ</sup> <sup>1</sup> � <sup>S</sup><sup>γ</sup>�<sup>δ</sup>

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

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.

r

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 parame-

Kp (17)

Kp (18)

, (19)

predictive parameter of the clay ratio in soil [37].

terization as follows:

18 Soil Moisture

4.2. Analogous to Johansen's thermal conductivity model

ratio, improving the accuracy of thermal inertia retrieval.

silt, and clay. The generalized form of the parameterization is given as
