Geometrical-Optics Model Geometrical optics and the geometrical theory of diffraction( GTD)are high-frequency asymptotic techniques wherein the fields are propagated using such optical concepts as shadowing, ray tubes, and refraction and diffraction. Although conceptually straightforward, optical techniques are limited analytically by the unavail- ability of diffraction coefficients for various geometries and material bodies and numerically by the need to trace rays over complex surfaces. There is a vast literature on geometrical optics and GTD, as may be ascertained by examining the yearly and cumulative indexes of such publications as the Transactions of the IEEE Antennas and Propagation Society. 45. 4 Numerical Issues in Developing a Computer Model Sampling Functions At the core of numerical analysis is the idea of polynomial approximation, an observation made by Arden and Astill [1970] in facetiously using the subtitle "Numerical Analysis or 1001 Applications of Taylors Series. The ea is to approximate quantities of interest in terms of sampling functions, often polynomials, that are en substituted for these quantities in various analytical operations. Thus, integral operators are replaced by finite sums, and differential operators are similarly replaced by generalized finite differences. For example,use f a first-order difference to approximate a derivative of the function F(x)in terms of samples F(x ) and F(x_) leads to dF(x) F(x,-F(x_) (45.2a) h and implies a linear variation for F(x) between x, and x as does use of the trapezoidal rule F(x)dx [F(x+)+F(x) (45.2b) to approximate the integral of F(x), where h=x-x. The central-difference approximation for the second d F(x F(x)-2F(xo)+F(x) similarly implies a quadratic variation for F(x)around xo=x4-h/2=x+ h/2, as does use of Simpson's rule F(x)dx==[F(x)+4F(x0)+F(x.) (45.2d) to approximate the integral. Other kinds of polynomials and function sampling can be used, as discussed in a large volume of literature, some examples of which are Abramowitz and Stegun [1964], Acton [ 1970), and Press et al.[1986]. It is interesting to see that numerical differentiation and integration can be accomplished using the same set of function samples and spacings, differing only in the signs and values of some of the associated eights. Note also that the added degrees of freedom that arise when the function samples can be unevenly spaced, as in Gaussian quadrature, produce a generally more accurate result(for well-behaved functions)for c 2000 by CRC Press LLC© 2000 by CRC Press LLC Geometrical-Optics Model Geometrical optics and the geometrical theory of diffraction (GTD) are high-frequency asymptotic techniques wherein the fields are propagated using such optical concepts as shadowing, ray tubes, and refraction and diffraction. Although conceptually straightforward, optical techniques are limited analytically by the unavail￾ability of diffraction coefficients for various geometries and material bodies and numerically by the need to trace rays over complex surfaces. There is a vast literature on geometrical optics and GTD, as may be ascertained by examining the yearly and cumulative indexes of such publications as the Transactions of the IEEE Antennas and Propagation Society. 45.4 Numerical Issues in Developing a Computer Model Sampling Functions At the core of numerical analysis is the idea of polynomial approximation, an observation made by Arden and Astill [1970] in facetiously using the subtitle “Numerical Analysis or 1001 Applications of Taylor’s Series.” The basic idea is to approximate quantities of interest in terms of sampling functions, often polynomials, that are then substituted for these quantities in various analytical operations. Thus, integral operators are replaced by finite sums, and differential operators are similarly replaced by generalized finite differences. For example, use of a first-order difference to approximate a derivative of the function F(x) in terms of samples F(x+) and F(x–) leads to (45.2a) and implies a linear variation for F(x) between x+ and x– as does use of the trapezoidal rule (45.2b) to approximate the integral of F(x), where h = x+ – x–. The central-difference approximation for the second derivative, (45.2c) similarly implies a quadratic variation for F(x) around x0 = x+ – h/2 = x– + h/2, as does use of Simpson’s rule (45.2d) to approximate the integral. Other kinds of polynomials and function sampling can be used, as discussed in a large volume of literature, some examples of which are Abramowitz and Stegun [1964], Acton [1970], and Press et al. [1986]. It is interesting to see that numerical differentiation and integration can be accomplished using the same set of function samples and spacings, differing only in the signs and values of some of the associated weights. Note also that the added degrees of freedom that arise when the function samples can be unevenly spaced, as in Gaussian quadrature, produce a generally more accurate result (for well-behaved functions) for dF x dx Fx Fx h x xx () ( ) ( ) ª - £ £ + - - + ; F x dx h Fx Fx x x ( ) [ ( ) ( )] ª + - + Ú + - 2 dFx dx Fx Fx Fx h 2 2 0 2 () [( ) ( ) ( ) 2 ª + - - + F x dx h Fx Fx Fx x x ( ) [ ( ) ( ) ( )] ª ++ - + Ú + - 6 4 0