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In Volume I, we have employed the radiative transfer theory to evaluate the scattering of waves by discrete scatterers. The extinction rate and phase matrix are constructed by assuming that the particles scatter independently. Hence, the extinction rate will be linearly proportional to the number of particles per unit volume and the fractional volume of particles f = novo (Fig. 9.1.1). However, physical intuition indicates that the linear relation cannot be correct for arbitrary f. For example, at f = 1, when the entire volume is occupied by scatterers, the medium becomes a homogeneous medium. Hence, in the absence of absorption, scattering should be equal to zero in the limit f = 1 (Fig. 9.1.1). Independent scattering is not valid for materials with an appreciable fractional volume of scatterers. This has been verified by controlled laboratory experiments [Ishimaru and Kuga, 1982; Mandt et al. 1992; West et al. 1994]. When a coherent wave propagates through a scattering medium, it is attenuated by both absorption and scattering. The coherent wave propagation constant is denoted by K, where

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(9.1.3) (9.1.4)

The attenuation K i is a summation of the absorption K a and scattering K s The scattering part of the attenuation is dependent on the particle size.

When an object is created from the master class, code in the master class can assign this property to the appropriate detail collection class as follows:

(9.1.5)

FIGURE 2.3 Bulk enzyme market, 2002.

(9.1.6)

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Equation (915) gives the approximate relation We use and "'e are the absorption coefficient, scattering coefficient, and extinction coefficient, respectively The effective permittivity includes absorption and scattering The classical mixing formulas for effective permittivity applies when scattering attenuation can be ignored [Bottcher, 1952; Maxwell-Garnett, 1904; Polder and van Santern, 1946] Scattering is generally size dependent For example, Rayleigh scattering gives a scattering cross section that is proportional to k4 a 6 Hence, at very low frequencies, scattering attenuation can be neglected In such a limit, the effective permittivity will be the same as the effective permittivity from classical mixture formula Thus classical mixture formula can be regarded as a very low frequency limiting case of the effective permittivity and wavenumber of dense media Random discrete scatterers can be classified as in Table 911.

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Particles are described as tenuous if their dielectric properties are only slightly different from the background medium The characteristics of wave propagation in such media will be quite similar to those of COntinuous random media The properties of K r and K i for the four classes of particles are described in Table 911, where k is used to denote the propagation constant of the background medium Of the four classes of random discrete particles, class D is the subject of dense media 12 Extinction as a Function of Frequency In experiments of scattering by random distribution of particles, the scattering and extinction are usually measured as a function of frequency For example, in satellite passive microwave remote sensing, the brightness temperatures are measured at 10 GHz, 19 GHz, 37 GHz, and 94 GHz In satellite active microwave remote sensing backscattering measurements, the frequencies are 1.

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5 GHz, 5 GHz, and 10 GHz The objectives of these sensors are to extract the snow parameters from these multi-frequency measurements Thus it is important to study the frequency dependence of scattering and.

+ iKa )2 + 2iKs K r "'a = 2Ka, "'s = 2Ks, "'e = "'a + "'s, where "'a, "'s,

(9.1. 7)

extinction by random distribution of particles For a small particle of radius a, it is well known that the dependence of scattering is k4 a 6 for 3-D particles, and k 3 a4 for 2-D scattering The dependence of frequency is strong When the frequency is doubled, scattering increases by 16 times and 8 times respectively for 3-D and 2-D scattering For particle sizes comprable to the wavelength, then the freqeuncy dependence is weaker and becomes independent of frequency at the geometric optics limit For sparse concentration of random distribution of particles, because independent scattering is valid, the frequency dependence follows that of single particles If the particles have a size distribution with some particles smaller than the wavelength and others comparable to the wavelength, then the medium can exhibit varied frequency dependence of scattering.

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