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+ (P - We)E a + (1 - P)E r + iaW; (We S Wt)
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where Ew , Ea , Ei, and Er are the complex dielectric constants of water, air, ice, and rock, respectively, P is the porosity of dry soil, and We is the moisture content (cm3 /cm3 ). The transition moisture Wt, conductive corrections a and , are parameters that are determined by measurement. They depend on the soil type. For Zaneis loam soil, W t = 0.22, a = 8.0, and, = 0.4. In the calculation, the other parameters are taken to be P = 0.5, Ea = (1 + iO), Ei = (3.2 + iO.1), and Er = (5.5 + iO.2). The dielectric constant of water Ew is calculated by using Debye formula [Ulaby, 1975] as given by
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where f is frequency (Hz), EwO is static dielectric constant of pure water, Ewoo is the high frequency (or optical) limit of Ew , and T w is the relaxation
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time (s). They were determined by measurement [Lane and Saxton, 1952; Stogryn, 1970; Klein and Swift, 1977J and given by
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= 4.9
fwo(T) = 88.045 - 0.4147 T + 6.295
1O-4T 2 + 1.075
1O- 5T 3 (5.4.4) (5.4.5)
(T) = 1.1109 x 10- 10 - 3.824 x 1O- 12 T + 6.938 x 1O- 14 T 2 - 5.096 X 1O- 16 T 3
where T is water temperature in C. Because fwO and T w are functions of the water temperature, the dielectric constant of water is also temperaturedependent besides being frequency-dependent. Therefore, the dielectric constant of soil is also temperature-dependent. In Fig. 5.4.1, we plot the dielectric constant of Zaneis loam soil at temperature of 295 K as a function of frequency for moisture content of 5% and 30%. There is only a small change in the dielectric constant as a function of frequency for dry soil (5% moisture). The large change for the wet soil (30% moisture) is due to the dominant contribution from water and shows a similar pattern as that of water. In Fig. 5.4.2, the dielectric constants are plotted as a function of soil moisture content. We see that both the real and imaginary parts increase as the moisture content increases. The imaginary part of dielectric constant is related to the attenuation of wave propagation. The attenuation is usually characterized by the optical depth defined as 1/ (2kE") with the imaginary part of dielectric constant E". In Fig. 5.4.3, we plot the optical depth of the soil as a function of moisture content for frequencies of 0.25 GHz, 1.42 GHz, 5.0 GHz, and 10.69 GHz. We see optical depth decreases as the moisture content increases. For low frequency, the optical depth is larger, which means that a low-frequency wave can penetrate into the deeper region of soil. At a frequency of 0.25 GHz, the optical depth ranges from 30 cm to 90 cm, depending on moisture content. At high frequency, optical depth is smaller and has stronger dependence on the moisture content. At 10.69 GHz, the optical depth is only on the order of millimeters, and it changes from 1.8 cm to 0.31 mm as moisture content is changed from 2% to 35%. To examine the characteristics of microwave emission from soils, We first study the soil of uniform profile with temperature of 295 K. In Fig. 5.4.4, the brightness temperature is plotted as a function of moisture content at an observation angle of 45 . It shows that the brightness temperature decreases as the moisture content increases. This can be explained by the increase in reflection and decrease in the emission as the value of dielectric constant of soil increases and as the moisture content increases. The difference between