M. P. Shatz and G. H. Polychronopoulos

An Algorithm for the Evaluation of Radar Propagation in the Spherical Earth 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. The integral representation for Ai(z) modifying them to handle complex arguments is is derived from an expreSSion for the modified not straightforward. Hankel function Kv,(z) [Ref. 7, Eq. 6.627]

The Lincoln Laboratory Journal. Volume 1. Number 2 (1988) 147 Shatz et aI. - An AlgorithmJor the Evaluation ojRadar Propagation in the Spherical Earth Diffraction Region

Properties of the Airy Function

The Airy function. first intro­ Thefunctions Ai(ze±27TfJ'3) also we have duced by Airy [I], is used in solv­ satisfy the .jZ d 2 w + _1_ dw ing diffraction problems. and also d 2 y dz2 Vi dz in mathematically analogous ap­ dz2 = zy. Airy plications. was. incidentally. W [z312 + _1-) =0 known equally well for his bril­ Since a second-order differential 4z3/2 . liance and his arrogance. For a equation has only two linearly in­ brief biography of this extraordi­ dependent solutions. the solu­ Changing the independent vari­ ~ (2z312 nary mathematician. see the box. tions Ai(z), Aile27T fJ'3 z), Aile-27T fJ'3 z). able to = l/3 , we have "Sir (1801­ musthave a linearrelationamong d 2 w 1 dw [ 1) 1892).~ them. The power series shows ~2 + T d~ - w 1 + 9~2 =O. The Airy function is defined by that the relation is

Ai(z) = e-7T fJ'3 Ai(e27TfJ'3 z) + So w isa modified Ai(z) = 1r1 f"" cos(t3/3 + zt) dt of order 1/3. w(O =Kl/3 (~I;-lrv'3 is o e7TfJ'3 Aile-27T fJ'3 z). the function with the correct a­ TheAiry function satisfiesthe dif­ symptotic behavior. so ferential equation If we start from the equation,

2 d y Ai(z) _1- ViK [~ z312). cf2Ailz) () = I/3 --=zAiz dz2 -zy =0 1rJ3 3 dz2 and let and most of its important pro­ 1. G.B. Airy. "On the Intensity of perties can be derived from this y= Vi w{z) Light in the Neighborhood of a equation. Caustic,~ Trans. Camb. Phil. Soc. The differential equation has 6, 379 (1838). no singular points in the finite z plane and an irregular singular pointatinfinity. TheAiry function is thus regular in the finite z plane. The possible asymptotic behaviors of the solutions to the Im(z) differential equation are

. e±213 z3/2 Al(Z) - C 4 . VZ The Airy integral vanishes expo­ nentially as z increases on the Re(z) positive real axis, so

e-2/3 z3/2 Ai(z) - c . Yz It canbeshownthattheconstant. C, must be ~.TheAiryfunc­ tion. Ai(z), decreases exponen­ tially as Iz I - 00 if Iarg z 1,< 17"13 and increases exponentially if 17"13 < larg z I < 17", and it oscillates on the negative real axis (Fig. A). TheAiry Fig. A - When evaluated in the complex plane, the Airy function function has an infinite number gives the regions of exponential decay (blue), exponential growth of zeroes, all on the real axis. (red), and oscillation (yellow).

148 The Lincoln Laboratory Journal. Volume 1, Number 2 (1988) Shatz et aI. - An AlgorithmJor the Evaluation ojRadar Propagation in the Spherical Earth Diffraction Region

Sir George Biddell Airy (1801-1892)

Airy. the son ofa fanner, was a it up tothe level ofthe Germanob­ Airy, in 1827. was the first to snobbish, self-seekingyoungster, servatories, which had been forg­ attempt to correct astigmatism in who preferred his well-to-do ing ahead of those in Great Brit­ the human eye (his own) by use of uncle to hisfather. At the age of12 ain. Airy organized data that had a cylindrical eyeglass lens. He he persuaded that uncle to carry been put aside to gather dust. He contributed to the study of inter­ him off and bring him up. He was a conceited, envious, small­ ference fiinges and to the mathe­ entered Cambridge in 1819 and minded man and ranthe observa­ matical theory of rainbows. The graduated in 1823 at the head of tory like a petty tyrant, but he , the central spot oflight hisclassinmathematics. He went made it hum. in the diffraction pattern of a on to teach both mathematics A strange fatality haunted point light source, is named for and astronomy at the university. Airy. causing him to be remem­ him. He advanced himself with bered for his failures. He played ruthless intensity and, although the role ofthevillain ofthe piece in [Excerpted from I. Asimov, everyone disliked him. he was the failure ofJ.C. Adams to carry Asimov's Biographical Encyclope­ successful in his aims. In 1835 he through the discoveryofNeptune. dia oj &ience and Technology, was appointed seventh astrono­ In fact, Airy is far better known as Doubleday, Garden City, NY, mer royal. a post he was to hold the man who muffed the discov­ 1982 and from Encyclopaedia for over 45 years. ery of that planet than for any of Britannica, Fifteenth Edition, He modernized the Greenwich the actual accomplishments for 1986.) Observatory. eqUippingitwith ex­ which he was deservedly cellent instruments and bringing knighted in 1872.

oo x-ll2e-xK (x) rrJ,K (Y) Substituting ~ = (2/3) i3 /2 into Eq. 4 we find that ___.,,--v_ dx = v '0 X + ~ ~1/2 [cos(lIrr)] (3) fo 1 r foo p(x) 1 Ai(z) = - rr-1/2z-1I4e-" --dx for larg ~ 1< rr and Re(lI) < 2' 2 0 l+x/~

Using for Iz I> °and larg " < rr. The moments, )1k' ofp(xL needed for the Gauss­ rrV3 . 1(3X)2/3] K 1I3(x) = ( ~ xr3 Al 2 ' ian quadrature. can be evaluated in closed form

=foo k _. r(3k+1/2) P- k 0 x p(x)dx- 54kk!r(k+1/2) (5) we can solve Eq. 3 for Ai(z), for k = 0, 1, 2, .... 1 foo p(x) Ai(z) = :2 rr- l12 z-1/4 e-2/3z3/2 3x dx The weights. w. , and abscissas, X., canbe found l I o 1 + 2z3 /2 for the N-point Gaussian quadrature. The Airy for largzl<2rr/3 and Izl>O, (4) function. when evaluated by the Gaussian , quadrature method, depends on N where pIx) is a non-negative exponentially de­ Ai(z N) =_1_ rr-1I2z-1/4e-' ~ Wi(N) creasing function, '2 ~ 1 + Xi(N)/~ 1=0

l12 11/6 2/3 X 3X)2/3] 2 p(x) = rr- 2- 3-213 x- e- Ai I( 2 . where, = """3 z3/2 and Iz I> 0, larg, 1 < rr. (6)

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Setting N = 10 was found to provide adequate values of z, because the Airy function increases accuracy. Table 1 gives a list of weights Wi and exponentially with z, even though the large abscissas Xi for a la-term Gaussian quadrature exponent is offset by the exp(J3a nx/2l factor of integration of the Airy function. Eq. 1in the region ofinterest.Therefore, a scaled Thus for small z, we use the power series; for Airy function is useful large z (I arg(z) I ~ 2nj3). we use the quadrature method. A connection formula [Ref. 5, Eq. M(w) = Ai(w) exp (~ w 3/2). 10.4.7] transforms a point from the sector I arg(z) I ~ 2n/3, where the integral representa­ Equation 1 can therefore be rewritten as tion is notvalid, to a weighted sumoftwo linearly ~ rri rri independent points outside this sector Al(an + ye /3) Al(an + ze /3) F(x,y,z) = 2J7rX ~ erri/3 Ai'(a) erri/3 Al/(a) . ~l n n

Ai(z) = erril3 Ai(ze-2rril3) + e-rri/3 Ai(ze2rril3).

The dividing line between the region that should be solved with the power series and the region (7) that should be solved by the quadrature method was determined empirically. Figure 2 shows Equation 7 is evaluated by computing addi­ where the regions on the complex plane lie that tional terms until two successive terms contrib­ are solved, respectively, by the power series ute less than 0.0005. Then, to verify the compu­ expansion, the Gaussian quadrature method, tation, one more check is performed. If anyone and the connection formula. of the terms of the series of Eq. 1 contributes more than 10,000 per term, we do not return a Use of Airy Function in Computing value for the propagation factor using Fock's Spherical Earth Diffraction series, since the accuracy ofthe value found for the Airy function is at least one in 10,000. The The spherical earth diffraction can be com­ maximum contribution of 10,000 per term was puted by substituting the values of the Airy chosen because, when the Airy function is function computed by this procedure into Eq. 1. evaluated in Eqs. 2, 5, or 6, the accuracy pro­ There may be floating-point overflows for large vided is at least four significant digits.

Table 1 - Abscissas and Weights for the 10-Term Gaussian Quadrature Integration for the Airy Function

Abscissas Weights w. I

1 8.05943 59215 34400 x 10.3 8.12311 4134235980 x 10-1 2 2.010034600905718 x 10- 1 1.439979241604145 x 10-1 , 3 6.418885840366331 x 10-1 3.6440415851 09798 x 10- 2 4 1.34479 70831 39945 x 10° 6.48895 66012 64211 x 10-3 5 2.331065231384954 x 10° 7.15550 10754 31907 x 10.4 6 3.63401 35043 78772 x 10° 4.43509 66599 59217 x 10.5 7 5.30709 43079 15284 x 10° 1.371239148976848 x 10-6 8 7.441601846833691 x 10° 1.75840 56386 19854 x 10-8 9 1.021488548060315 x 10' 6.6367686881 75870 x 10-'1 10 1.4083081071 97337 x 10' 2.677084371247434 x 10-'4

150 The Lincoln Laboratory Journal. Volume 1. Number 2 (1988) Shatz et aI. - An AlgorithmJor the Evaluation ojRadar Propagation in the Spherical Earth Diffraction Region

o Integral Representation the new diffraction algorithm for all values of y _ Power Series and z less than 5000 and found that it provides o Connection sufficient accuracy for all x such that SERE would use the diffraction calculation. These Im(z) values of y and z correspond to heights of 375 kIn at 170 MHz (VHF) or 17 km at 17 GHz (Ku band).

-----+-Re(z) 10..------:------,

Fig. 2 - The Airy function can be evaluated with three methods: a powerseries expansion, the Gaussian quadra­ N c: LL. -40 o ture method, and a connection formula. The correct choice .!::! ofevaluation methoddepends on the region ofthe complex ha = 1000 m ...o -50 :::c plane under consideration. ht = 8000 m -60 Frequency = 167 MHz -70 L...-_-L..._...... l'--_...l-_---L__..L-_~ Test of the Model 300 350 400 450 500 550 600 Range (km) We tested our calculation by comparing its results with the results of a multipath calcula­ Fig. 3 - The blue curve shows the values of the one-way tion over a smooth conducting sphere. The 2 propagation factor, F , obtained from the multipath model. results are shown in Fig. 3 for a high-altitude The red curve gives the values ofF 2 found by the spherical case - the most challenging situation. The earth diffraction model. dotted curve shows the results of the multipath calculation; the solid curve gives the results of the diffraction calculation. Summary It can be seen that there is a region (between about 430 kIn and 450 kIn) where the two The diffraction algorithm increases the accu­ calculations are very close. We can conclude racy of the calculated propagation factor in the that, for this case, the diffraction calculation spherical earth diffraction region for a large was accurate through the intermediate region range of normalized heights. This algorithm into the multipath region, for these antenna requires the application of Fock's series and heights. The diffraction calculation is accurate requires the accurate evaluation ofthe complex even within the second multipath lobe in Fig. 3. Airy function. We then examined the range ofx, y, and z over The complex Airy function was evaluated for which our calculation gives accurate results. I z I close to the origin by using a power series SERE uses the diffraction calculation if the expansion. For large I z I, the 10-term Gaussian minimum clearance of a ray is less than quadrature approximation to an integral repre­ sentation was used. The calculated Airy func­ tion agrees with published tables of the Airy function to within at least four significant digits. Incorporating this program in the SERE propa­ It (d ) where is the wavelength and d} 2 is the gation model gives accurate values for the ground range from the antenna (target) to the propagation factor to heights of 375 kIn at VHF point of minimum clearance. We have checked or 17 kIn at Ku band.

The Lincoln Laboratory Journal. Volume 1. Number 2 (1988) 151 Shatz et aI. - An Algorithmjor the Evaluation ojRadar Propagation in the Spherical Earth Diffraction Region

References 4. Association for Computing Machinery. Transactions oj Mathematical Sojtware. Program 498. "The Airy Subrou­ tine: 1. S. Ayasli, "SEKE: A Computer Model for Low-Altitude 5. M. Abramowitz and 1. Stegun. HandbookojMathematical Radar Propagation Over Irregular Terrain: IEEE Trans. Functions, Sect. 10, National Bureau of Standards, U.S. Antennas Propag. AP-M. 1013 (1986). Govt. Printing Office. Washington (1970). 2. M.L. Meeks. Radar Propagation at Low Altitudes, Artec­ 6. Z. Schulten, D.G.M. Anderson, R.G. Gordon, "An Algo­ House. Dedham. MA (1982). rithm for the Evaluation of the Complex Airy Functions," J. 3. V.A. Fock, Electromagnetic Diffraction and Propagation Comput. Phys. 31. 60 (1976). Problems. Pergamon, Oxford, England (1965). 7. I.S. Gradshetyn and I.M. Ryzhik. Tables oj Integrals, Series. and Products, Academic Press. New York (1965).

MICHAEL P. SHATZ has GEORGE H. POLYCHRO­ been a staff member in the NOPOULOS was a staff as­ Systems and Analysis sociate in the Systems and Group for four years. His re­ Analysis Group. He received search at Lincoln labora­ a BSEE from Princeton tory concentrates on analy­ University and an MS from sis of air defense systems. MIT. and is currently pursu­ Mike came to Lincoln from ing a PhD, also at MIT. the California Institute of &. George's studies are fo- Technology. where he received his PhD in physics in 1984. cused on air defense systems. In his free time. he enjoys He received a BS in physics from MIT in 1979. tennis, soccer, and basketball.

152 The Lincoln Laboratory Journal, Volume 1. Number 2 (1988)