M. P. Shatz and G. H. Polychronopoulos An Algorithm for the Evaluation of Radar Propagation in the Spherical Earth Diffraction Region We develop an efficient method for computing the Airy function and apply this method to the problem of calculating the radar propagation factor for diffraction around a smooth sphere. Even for high-altitude antennas, our calculations give accurate results well into the visible region, and thus extend the useful range ofthe Spherical Earth with Knife Edges (SERE) radar propagation model. The design and analysis ofradarsystemsthat Knife Edges (SEKE) model [1], for instance, transmit over oceans or smooth terrain - for combines four approximations in its analysis of example, airborne radars that spot distant in­ radar propagation over Irregular terrain. (The comingmissiles, orcoastal radars that fmd drug box, "SEKE," describes this radar-propagation smugglers - reqUires accurate evaluation of model in greater detail.) radar propagation over a smooth sphere. Analy­ Figure 1 shows three regions for propagation sis of smooth sphere diffraction is equally im­ over a sphere: the multipath region (in yellow], portant for general radar applications; by well above the horizon; the diffraction region (in combining several approximations, composite red) below the horizon; and an intermediate re­ models can evaluate signal propagation over gion (in orange). In the multipath region a calcu­ any type of terrain. The Spherical Earth with 1ation based on interference between the direct and reflected rays can be done with accuracy. In the diffraction region a small number ofieading terms ofan infinite series solution involvingAiry functions canbeused to approximate the propa­ gation factor. In the intermediate region this series can also be used butmore terms and high computational accuracy for the Airy Function Multipath Region are reqUired. When SEKE was originally developed, it was designed for low altitudes, and its method of calculating the Airy functions was sufficiently accurate. However, for high-altitude antennas or targets greater accuracy is necessary. The new method that is described in this paper provides the accuracy required for high-altitude applications for all reasonable target and an­ tenna heights into the visible (multipath) region. Spherical Earth Diffraction An accurate model of diffraction about a Fig. I-The algorithm described in this paper extends the area ofvalidity ofthe diffraction calculaton into the multipath smooth spherical earth must account for refrac­ region (not to scale). tion in air. The index of refraction in air, n, is The Lincoln Labomtory Jownal, Volume 1, Number 2 (1988) 145 - Shatz et aI. - An Algorithmjor the Evaluation ojRadar Propagation in the Spherical Earth Diffraction Region SEKE The SEKE propagation model SEKE: SEKE 1 uses the smooth path from the radar to the ground is used to compute the field sphere assumption: SEKE 2 uses and then to the target is within strengths of radio waves propa­ the ray-tracing method. one-halfa radarwavelength ofthe gating from ground-based radar The multiple knife-edge diffrac­ length of the path from the radar to low-flying aircraft. over irregu­ tion approach starts by finding the to the specular point and then to '. lar terrain. The model was devel­ knife edges in the terrain data that the target.) Finally. a third effec­ oped at Lincoln Laboratory by Dr. most block the direct ray. The tive sphere is fitted to the first Serpil Ayasli. multiple knife-edge approach then Fresnel zone ofthe specular point The SEKE model combines implements Deygout's method [II on the second sphere, and the several propagation models: for computing the propagation propagation factor is computed. smooth sphere diffraction. mul­ factor. The ray-tracing program finds tiple knife-edge diffraction. reflec­ For smooth sphere diffraction, those points on the terrain that tion from a smooth sphere. and a an effective sphere is fitted to the give specular reflections. It com­ ray-tracingcomputation ofreflec­ terrain between the target and the putes the amplitude and phase of tions offan irregular surface. The antenna. The strength ofthe field is each specularpointusing the un­ model determines. depending on then determined by the Airy-func­ shadowed points in the first the terrain. whether to use a dif­ tion method deSCribed in this Fresnel zone. fraction calculation. a reflection paper. The SEKE software (written in calculation. ora weighted average Smooth sphere reflection. like FORfRAN) can be obtained from ofthe two methods. Ifa diffraction smooth sphere diffraction, requires MIT Technology Licensing Office. calculation is used. the model fitting an effective sphere to the ter­ E32-300. 77 MassachusettsAve.. chooses the knife edge. the rain between the target and the an­ Cambridge. MA 02139. smooth sphere. or a weighted tenna. The specular point is then 1. J. Deygout. ~Multiple Knife­ average of the two methods. found and a new effective sphere is Edge Diffraction of Microwaves, ~ Again. the decision depends upon fitted to the first Fresnel zone. (The IEEETrans. AntennasPropag. AP­ the terrain. The reflection models first Fresnel zone is the region on 34,480 (1966). are implicit in the two versions of the ground where the length of the approximately given by This approximation - that diffraction around P 0.37e the earth with its atmosphere is eqUivalent to n = 1 + 7.76' 1O-5--y + "'T2' diffraction over a smooth sphere of radius Relf in a vacuum - is used throughout this paper. where Pis the pressure in mbar, Tthe tempera­ The propagation factor. F, is defined as ture in K, and e the partial pressure ofwater in mbar. Usually n decreases with increasingalti­ tude, which causes radio waves to bend to­ F= I:~I. ward the earth. A standard assumption [2], which works for normal conditions, is that where Et is the electric field at the target and Eo the rate ofchange in n with height is a constant is the electric field at the range of the target on the axis of the antenna beam (in free space). F C = -3.9 • 1O-8m -l. Using this assumption, re­ fraction can be accounted for by treating the depends on the target height ht• the antenna height h • and the range r, via the normalized radio waves as though they were in a vacuum a parameters x, y, and z R , above a sphere of radius elf given by - R earth 4 r h a h t ReR' - 1 C R -3 Rearth· x =T'()' y =~. z = h ' 'dJ + earth o 146 The Lincoln Laboratory Journal. Volume 1. Number 2 (1988) Shatz et aI. - An AlgOrithmJor the Evaluation ojRadar Propagation in the Spherical Earth Diffraction Region where ha. the normalization factor for height is Algorithm for Evaluating the Com­ plex Airy Function = _1_ ( a,\2 )1/3 flo - 2 rr2 ' The Airy function is entire (no singularities in the finite complex plane), so it can be expressed and ro' the normalization factor for range is with a convergent power series representation given by [Ref. 5, Eq. 10.4.2]. This series converges very fast for small z. When a large value of z is used, fo -= ( a2,\rr )1/3. the series converges very slowly. and inaccura­ cies due to large cancellations occur. The power series expansion for Ai(z) is Here A is the wavelength of the electromagnetic wave, and is the effective radius ofthe sphere. a Ai(z) = a . h(z) - f3 . g(z) , (2) The general solution to the calculation ofthe propagation factor F for a spherical earth can where ex and f3 are constants be expressed as an infinite series that contains . 3-2/ 3 Airy functions of a complex argument [3] and is a = Al(O) + r(2/3) = 0.3550280538. dependent on x. y, and z co 3-1/ 3 f3 = -Ai'(O) = fO/3) = 0.2588194037, F(x,y,z) = 2.J rrx ~ in(y)Jn(z) exp[ +(v'S + ilanx]. n=1 Ai(a + uerri/3) (1) and in( u) Ai'(a ) where == err''; n h(z) = ~ 3k(_I) z3k - 1 + ~ 3 k (3k)! - k=0 where Ai'(w) is the first derivative of the Airy function and an is the nth zero of the Airy -z31 + --z61·4 + 1· 4' 7 3! 6! 9! z9 + ••• , function, (an < 0). The Airy function (see the box, "Properties of the Airy Function"). Ai(wJ, is de­ fined as ~ k(2) z3k+1 2 glz) = ~ 3 '"3 k (3k + III = z + 4! z4 + 3 Al(W).= -- 1 fco cos (3t + wt) dt. k=0 rr 0 2 . 5 7 + 2· 5 . 8 10 7! z 1O! z + •••• The series in Eq. 1 converges for all (positive) values of x, y, and z. However. the series is not numerically useful in most of the multipath re­ The above expressions for the power series gion (r much less than the distance to the show that each sum grows exponentially on the horizon for a specified ht and heJ, because, in real axis as zbecomes larger, but the difference that region. the largest terms of the series are decreases exponentially as z grows larger. Thus very much larger than the sum of the series. for large z the powerseries becomesnumerically Fortunately, other approximations are valid in unstable (very sensitive to computer round-off that region [3]. With an accurate computationof errors). the Airy function. this series provides numeri­ Schulten et al. [6] give a method for the cally useful results far enough into the multi­ computation of the complex Airy function for path region to meet the needs of models that large z. This method uses an integral represen­ match diffraction and multipath calculations, tation for the Airy function that can be eval­ eg. SEKE [1]. Although algorithms for the evalu­ uated by a. Gaussian quadrature using only a ation of Ai(w) for real ware readily available [4], few terms.