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Figure 12.3.4 Radiation pattern a(<ps) from N cylinders with fractional volume f = 0.005 and size ka = 0.05. The separation of the parallel plates is such that kd = 0.3. (a) N = 16, (b) N = 64, and (c) N = 256.
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Au, W. C., J. A. Kong, and L. Tsang (1994), Absorption enhancement of scattering of electromagnetic waves by dielectric cylinder clusters, Microwave Opt. Technol. Lett.,
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Au, W. C., L. Tsang, R. T. Shin, and J. A. Kong (1996), Collective scattering and absorption in microwave interaction with vegetation canopies, Progress in Electromag. Res., 14, 182-231, EMW Publishers, Cambridge, Massachusetts. Chew, W. C. (1990), Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York. Chiu, T. and K Sarabandi (2000), Electromagnetic scattering from short branching vegetation, IEEE Trans. Geosci. Remote Sens., 38(2),911-925. Gu, Q., M. A. Tassoudji, S. Y. Poh, R. T. Shin, and J. A. Kong (1994), Coupled noise analysis for adjacent vias in multilayered digital circuits, IEEE Trans. Circuits and Systems I:
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Fundamental Theory and Applications, 41(12), 796-803.
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Karam, M. A., A. K. Fung, and Y. M. M. Antar (1988), Electromagnetic wave scattering from some vegetation samples, IEEE Trans. Geosci. Remote Sens., 26(6), 799--808. Kong, J. A. (2000), Electromagnetic Wave Theory, EMW Publishing, Cambridge, MA. Kuga, Y and A. Ishimaru (1984), Retroreflectance from a dense distribution of spherical particles, J. Opt. Soc. Am., 1, 831-835. Lin, Y-C. and K. Sarabandi (1995), Electromagnetic scattering model for a tree trunk above a tilted ground plane, IEEE Trans. Geosci. Remote Sens., 33(4), 1063-1070. Sarabandi, K, P. F. Polatin, and F. T. Ulaby (1993), Monte Carlo simulation of scattering from a layer of vertical cylinders, IEEE Trans. Antennas Propagat., 41(4), 465-475. Seker, S. S. and A. Schneider (1988), Electromagnetic scattering from a dielectric cylinder of finite length, IEEE Trans. Antennas Propagat., 36(2), 303-307. Sommerfeld, A. (1962), Partial Differential Equations, Academic Press, New York. Tsang, L., C. H. Chan, J. A. Kong, and J. Joseph (1992), Polarimetric signatures of a canopy of dielectric cylinders based on first and second order vector radiative transfer theory,
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J. Electromag. Waves and Appl., 6(1),19-51.
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Tsang, L., K H. Ding, G. Zhang, C. C. Hsu, and J. A. Kong (1995), Backscattering Enhancement and clustering effects of randomly distributed dielectric cylinders overlying a dielectric half space based on Monte-Carlo simulations, IEEE Trans. Antennas Prop-
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agat., 43(5), 488--499.
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Tsang, L. and A. Ishimaru (1984), Backscattering enhancement of random discrete scatterers, J. Opt. Soc. Am. A, 1, 836-839. Tsang, L.,.1. A. Kong, and R. T. Shin (1985), Theory of Microwave Remote Sensing, WileyInterscience, New York.
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Scattering of Electromagnetic Waves: Numerical Simulations. Leung Tsang, Jin Au Kong, Kung-Hau Ding, Chi On Ao. Copyright 2001 John Wiley & Sons, Inc. ISBNs: 0-471-38800-9 (Hardback); 0-471-22430-8 (Electronic)
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Introduction Plant Modeling by Using L-Systems Lindenmayer Systems Turtle Interpretation of L-Systems Computer Simulations of Stochastic L-Systems and Input Files Scattering from Trees Generated by L-Systems Based on Coherent Addition Approximation Single Scattering by a Particle in the Presence of Reflective Boundary
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2.1 2.2 2.3
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3.1.1 Electric Field and Dyadic Green's Function 3.1.2 Scattering by a Single Particle
Scattering by Trees Coherent Addition Approximation with Attenuation Scattering from Plants Generated by L-Systems Based on Discrete Dipole Approximation Formulation of Discrete Dipole Approximation (DDA) Method Scattering by Simple Trees Scattering by Honda Trees Rice Canopy Scattering Model Model Description Model Simulation References and Additional Readings
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Various theoretical models have been developed to characterize the electromagnetic wave scattering properties of vegetation canopies. In the past, wave scattering from vegetation has been studied extensively with vector radiative transfer theory as described in Volume 1. Classical radiative transfer theory assumes that the particles scatter independently so that the scattering phase functions add. This assumption is based on the random phase of scattering by different particles and is valid if the particle positions are independent and the randomness of relative positions is comparable to or larger than a wavelength. However, such an assumption can be invalid for microwave scattering of certain cases of vegetation canopy where the randomness of their relative positions is less than a wavelength. For example, branches and leaves in a tree occur in clusters, and there are correlations between their relative positions. Scatterers with this kind of cluster structure can demonstrate collective scattering effects. In 7, we have shown by simulations of point scatterers the effects of clustering. Collective scattering effects include correlated scattering and take into account the relative phase of scattered waves from the scatterers and their neighbors. The mutual coherent wave interactions between scatterers are also to be included. In numerical solutions of Maxwell's equations for random media, the positions and characteristics of the scatterers are randomly generated according to prescribed statistics. We have described procedures for generating 2-D and 3-D dense media in s 8 and 9. For the case of vegetation, the scattering by correlated scatterers was first studied by using the coherent addition approximation (CAA). The probability density functions of positions are introduced. However, it is difficult to calculate the probability density functions and the pair distribution functions for natural vegetation. Coherent scattering effects including the relative structures of branches and leaves draw important attention [Yueh et al. 1992; Au et al. 1994, 1996]. To build realistic tree structures, Lindenmayer systems can be used [Prusinkiewicz and Lindenmayer, 1990]. Subsequently coherent scattering models are developed for these Lindenmayer systems. [Chen et al. 1990, 1996; Zhang et al. 1996; Le Toan et al. 1997; Lin and Sarabandi, 1999a,b]. In this chapter we study wave scattering by trees. The trees are grown by using stochastic L-systems. The correlation of scattering by different branches are included by using their relative positions as given by the growth procedure. The advantages of this method are that (1) the structure of trees is controlled by growth procedure and the calculations of the pair distribution functions and probability density function are not needed, (2) the