5.1 Discrete Dipole Approximation (DDA) Method

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so that (13.5.8)

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The expression of A ij in (13.5.8) will still preserve the translational invariant property so that the FFT can still be taken when the matrix equation is solved by iterative method. After the solution is obtained, we have the solution of the reduced dipole moment Pi for every cell. The electric field at every cell E i is given by Ei = (

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Note that we only have to include those cells which have Epi =1= E. The matrix equation (13.5.1) is of dimension 3N x 3N, where N is the number of small cylindrical subcells. The cylindrical subcells may have different lengths and radii in tree scattering. The factor 3 arises from the x, y, and z components of the polarization vector. After the matrix equation (13.5.1) is solved, the far-field, scattered field in the direction ks is (13.5.10) Based on (13.5.10), the far-field scattering amplitude matrix can be computed readily. The time-averaged power absorbed is equal to

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where E~i is the imaginary part of moment of each cell, we have

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for the ith cell. In terms of dipole

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We have verified the validity of the discrete dipole approximation method by calculating the backscattering coefficients from a vertical cylinder of length l = lA, radius a = 0.05>', and permittivity E = (3 + iO.5)E [Chen, 1994]. p

13 ELECTROMAGNETIC WAVES SCATTERING BY VEGETATION

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The method is further validated by comparing its results with those of another method of moment code for a body of revolution based on the surface integral formulation. The results agree well with each other. Energy conservation is checked by comparing the extinction coefficient obtained by the optical theorem and that by the sum of the total scattering coefficient and the absorption coefficient. The details of the validity of our numerical code can be found in [Chen, 1994]. 5.2 Scattering by Simple Trees In the following study, the trees to be used as our scattering objects are grown by using the stochastic L-systems discussed in Section 2. The input file to the L-systems is the input file B given in Section 2.3. Figure 13.2.3 shows one of the generated trees. Here, we assume that the unit used in the above process is one wavelength. Three hundred trees are generated. The maximum height of these trees is 2.47>'. The shadow of each tree can be put in a circle of minimum diameter 0.632>'. We define a shadow cylinder for each tree as one which has a height of the maximum height of the tree and minimum diameter to cover the shadow of the tree. We define the local fractional volume as the total volume of all the branches of one tree divided by the volume of its shadow cylinder. The averaged local fractional volume for these 300 trees is Ivl = 2.89%. In each realization of our calculation, ten of these trees are put into a pixel of the size of 3>' x 3>'. The positions of the trees are random, but the shadow cylinders of the trees won't overlap each other. The fractional area fa, defined as the sum of the shadow areas of all these ten shadow cylinders divided by the area of the pixel, is 0.346. Thus the fractional volume occupied by the branches in the forest is f = fvda = 1.0%. Each pixel is assumed to have a reflective boundary of permittivity (16 + i4)E o . The following three scattering mechanisms are considered in the presence of the reflective boundary (Fig. 13.3.1a-c). The first term represents the scattering from the incident direction by a scatterer into the scattered direction. The second term represents the scattering of the reflected wave by a scatterer into the scattered direction. The third term represents the scattering from the incident direction by a scatterer, and the wave is then reflected by the boundary before going into the scattered direction. The backscattering coefficients calculated using the discrete dipole approximation (J'~~) can be expressed as follows:

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