M. P. Shatz and G. H. Polychronopoulos

An Algorithm for the Evaluation of RadarPropagation in the Spherical Earth

Diffraction Region

We develop an efficient method for computing the Airy function and apply this methodto the problem of calculating the radar propagation factor for diffraction around asmooth sphere. Even for high-altitude antennas, our calculations give accurate resultswell into the visible region, and thus extend the useful range ofthe Spherical Earth withKnife Edges (SERE) radar propagation model.

The design and analysis ofradar systems thattransmit over oceans or smooth terrain - forexample, airborne radars that spot distant in­coming missiles, or coastal radars that fmd drugsmugglers - reqUires accurate evaluation ofradar propagation over a smooth sphere. Analy­sis of smooth sphere diffraction is equally im­portant for general radar applications; bycombining several approximations, compositemodels can evaluate signal propagation overany type of terrain. The Spherical Earth with

MultipathRegion

Fig. I-The algorithm described in this paper extends thearea ofvalidity of the diffraction calculaton into the multipathregion (not to scale).

The Lincoln Labomtory Jownal, Volume 1, Number 2 (1988)

Knife Edges (SEKE) model [1], for instance,combines four approximations in its analysis ofradar propagation over Irregular terrain. (Thebox, "SEKE," describes this radar-propagationmodel in greater detail.)

Figure 1 shows three regions for propagationover a sphere: the multipath region (in yellow],well above the horizon; the diffraction region (inred) below the horizon; and an intermediate re­gion (in orange). In the multipath region a calcu­1ation based on interference between the directand reflected rays can be done with accuracy. Inthe diffraction region a small number ofieadingterms ofan infinite series solution involving Airyfunctions can be used to approximate the propa­gation factor. In the intermediate region thisseries can also be used but more terms and highcomputational accuracy for the Airy Functionare reqUired.

When SEKE was originally developed, it wasdesigned for low altitudes, and its method ofcalculating the Airy functions was sufficientlyaccurate. However, for high-altitude antennasor targets greater accuracy is necessary. Thenew method that is described in this paperprovides the accuracy required for high-altitudeapplications for all reasonable target and an­tenna heights into the visible (multipath) region.

Spherical Earth Diffraction

An accurate model of diffraction about asmooth spherical earth must account for refrac­tion in air. The index of refraction in air, n, is

145

-

Shatz et aI. - An Algorithmjor the Evaluation ojRadar Propagationin the Spherical Earth Diffraction Region

SEKE

The SEKE propagation modelis used to compute the fieldstrengths of radio waves propa­gating from ground-based radarto low-flying aircraft. over irregu­lar terrain. The model was devel­oped at Lincoln Laboratory by Dr.Serpil Ayasli.

The SEKE model combinesseveral propagation models:smooth sphere diffraction. mul­tiple knife-edge diffraction. reflec­tion from a smooth sphere. and aray-tracing computation ofreflec­tions offan irregular surface. Themodel determines. depending onthe terrain. whether to use a dif­fraction calculation. a reflectioncalculation. or a weighted averageofthe two methods. Ifa diffractioncalculation is used. the modelchooses the knife edge. thesmooth sphere. or a weightedaverage of the two methods.Again. the decision depends uponthe terrain. The reflection modelsare implicit in the two versions of

SEKE: SEKE 1 uses the smoothsphere assumption: SEKE 2 usesthe ray-tracing method.

The multiple knife-edge diffrac­tion approach starts by finding theknife edges in the terrain data thatmost block the direct ray. Themultiple knife-edge approach thenimplements Deygout's method [IIfor computing the propagationfactor.

For smooth sphere diffraction,an effective sphere is fitted to theterrain between the target and theantenna. The strength ofthe field isthen determined by the Airy-func­tion method deSCribed in thispaper.

Smooth sphere reflection. likesmooth sphere diffraction, requiresfitting an effective sphere to the ter­rain between the target and the an­tenna. The specular point is thenfound and a new effective sphere isfitted to the first Fresnel zone. (Thefirst Fresnel zone is the region onthe ground where the length of the

path from the radar to the groundand then to the target is withinone-halfa radar wavelength ofthelength of the path from the radarto the specular point and then tothe target.) Finally. a third effec­tive sphere is fitted to the firstFresnel zone of the specular pointon the second sphere, and thepropagation factor is computed.

The ray-tracing program findsthose points on the terrain thatgive specular reflections. It com­putes the amplitude and phase ofeach specular point using the un­shadowed points in the firstFresnel zone.

The SEKE software (written inFORfRAN) can be obtained fromMIT Technology Licensing Office.E32-300. 77 Massachusetts Ave..Cambridge. MA 02139.1. J. Deygout. ~Multiple Knife­Edge Diffraction of Microwaves, ~IEEE Trans. Antennas Propag. AP­34,480 (1966).

'.

approximately given by

P 0.37en = 1 + 7.76' 1O-5--y + "'T2'

where Pis the pressure in mbar, Tthe tempera­ture in K, and e the partial pressure of water inmbar. Usually n decreases with increasing alti­tude, which causes radio waves to bend to­ward the earth. A standard assumption [2],which works for normal conditions, is thatthe rate of change in n with height is a constantC = -3.9 • 1O-8m -l. Using this assumption, re­fraction can be accounted for by treating theradio waves as though they were in a vacuumabove a sphere of radius Relf , given by

R- R earth 4

eR' - 1 C R -3 Rearth·'dJ + earth

146

This approximation - that diffraction aroundthe earth with its atmosphere is eqUivalent todiffraction over a smooth sphere of radius Relfin a vacuum - is used throughout this paper.

The propagation factor. F, is defined as

F= I:~I.

where Et is the electric field at the target and Eois the electric field at the range of the target onthe axis of the antenna beam (in free space). Fdepends on the target height ht• the antennaheight ha • and the range r, via the normalizedparameters x, y, and z

r h a h tx =T'()' y =~. z = h o '

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

Shatz et aI. - An AlgOrithmJor the Evaluation ojRadar Propagationin the Spherical Earth Diffraction Region

Algorithm for Evaluating the Com­plex Airy Function

The Airy function is entire (no singularities inthe finite complex plane), so it can be expressedwith a convergent power series representation[Ref. 5, Eq. 10.4.2]. This series converges veryfast for small z. When a large value of z is used,the series converges very slowly. and inaccura­cies due to large cancellations occur. The powerseries expansion for Ai(z) is

where ha. the normalization factor for height is

= _1_ ( a,\2 )1/3flo - 2 rr2 '

and ro' the normalization factor for range isgiven by

= ( a2,\ )1/3fo - rr .

Here A is the wavelength of the electromagneticwave, and a is the effective radius ofthe sphere.

The general solution to the calculation of thepropagation factor F for a spherical earth canbe expressed as an infinite series that containsAiry functions of a complex argument [3] and isdependent on x. y, and z

Ai(z) = a . h(z) - f3 . g(z) ,

where ex and f3 are constants

. 3-2/3a = Al(O) + r(2/3) = 0.3550280538.

(2)

coF(x,y,z) = 2.J rrx ~ in(y)Jn(z) exp[+(v'S+ ilanx].

n=1

3-1/ 3f3 = -Ai'(O) = fO/3) = 0.2588194037,

. 1 fco ( t3)Al(W) =-- cos 3 + wt dt.

rr 0

where Ai'(w) is the first derivative of the Airyfunction and an is the nth zero of the Airyfunction, (an < 0). The Airy function (see the box,"Properties of the Airy Function"). Ai(wJ, is de­fined as

The series in Eq. 1 converges for all (positive)values of x, y, and z. However. the series is notnumerically useful in most of the multipath re­gion (r much less than the distance to thehorizon for a specified h t and heJ, because, inthat region. the largest terms of the series arevery much larger than the sum of the series.Fortunately, other approximations are valid inthat region [3]. With an accurate computation ofthe Airy function. this series provides numeri­cally useful results far enough into the multi­path region to meet the needs of models thatmatch diffraction and multipath calculations,eg. SEKE [1]. Although algorithms for the evalu­ation of Ai(w) for real ware readily available [4],modifying them to handle complex arguments isnot straightforward.

and

1· 4' 79! z9 + ••• ,

2 . 5 7 + 2· 5 . 8 107! z 1O! z + ••••

h(z) = ~ 3k(_I) z3k - 1 +~ 3 k (3k)! -k=0

1 1·4-z3 + --z6 +3! 6!

~ (2) z3k+1 2glz) = ~ 3k '"3 k (3k + III = z + 4! z4 +

k=0

The above expressions for the power seriesshow that each sum grows exponentially on thereal axis as zbecomes larger, but the differencedecreases exponentially as z grows larger. Thusfor large z the power series becomes numericallyunstable (very sensitive to computer round-offerrors).

Schulten et al. [6] give a method for thecomputation of the complex Airy function forlarge z. This method uses an integral represen­tation for the Airy function that can be eval­uated by a. Gaussian quadrature using only afew terms. The integral representation for Ai(z)is derived from an expreSSion for the modifiedHankel function Kv,(z) [Ref. 7, Eq. 6.627]

(1)Ai(a + uerr i/3)where in( u) == err''; Ai'(an)

The Lincoln Laboratory Journal. Volume 1. Number 2 (1988) 147

Shatz et aI. - An AlgorithmJor the Evaluation ojRadar Propagationin the Spherical Earth Diffraction Region

Properties of the Airy Function

Fig. A - When evaluated in the complex plane, the Airy functiongives the regions of exponential decay (blue), exponential growth(red), and oscillation (yellow).

The Airy function. first intro­duced by Airy [I], is used in solv­ing diffraction problems. and alsoin mathematically analogous ap­plications. Airy was. incidentally.known equally well for his bril­liance and his arrogance. For abrief biography of this extraordi­nary mathematician. see the box."Sir George Biddell Airy (1801­1892).~

The Airy function is defined by

1 f""Ai(z) = 1r cos(t3/3 + zt) dto

The Airy function satisfies the dif­ferential equation

cf2Ailz) ( )--=zAizdz2

and most of its important pro­perties can be derived from thisequation.

The differential equation hasno singular points in the finite zplane and an irregular singularpoint at infinity. The Airy functionis thus regular in the finite zplane. The possible asymptoticbehaviors of the solutions to thedifferential equation are

. e±213 z3/2Al(Z) - C 4 .

VZThe Airy integral vanishes expo­nentially as z increases on thepositive real axis, so

e-2/3 z3/2Ai(z) - c .

YzIt can be shown that the constant.C, must be ~.The Airy func­tion. Ai(z), decreases exponen­tially as Iz I - 00 if Iarg z 1,< 17"13 andincreases exponentially if 17"13 <larg z I < 17", and it oscillates on thenegative real axis (Fig. A). The Airyfunction has an infinite numberof zeroes, all on the real axis.

The functions Ai(ze±27TfJ'3) alsosatisfy the differential equation

d 2 ydz2 = zy.

Since a second-order differentialequation has only two linearly in­dependent solutions. the solu­tions Ai(z), Aile27TfJ'3 z), Aile-27TfJ'3 z).must have a linear relation amongthem. The power series showsthat the relation is

Ai(z) = e-7TfJ'3 Ai(e27TfJ'3 z) +

e7TfJ'3 Aile-27TfJ'3 z).

If we start from the equation,

d 2ydz2 -zy =0

and let

y= Vi w{z)

Im(z)

we have

.jZ d 2 w + _1_ dwdz2 Vi dz

W [z312 + _1-) =04z3/2 .

Changing the independent vari­able to ~ =(2z312l/3 , we have

d 2w 1 dw [ 1)~2 + T d~ - w 1 + 9~2 =O.

So w is a modified Bessel functionof order 1/3. w(O =Kl/3 (~I;-lrv'3 isthe function with the correct a­symptotic behavior. so

Ai(z) =_1- Vi K I/3 [~ z312).1rJ3 3

1. G.B. Airy. "On the Intensity ofLight in the Neighborhood of aCaustic,~ Trans. Camb. Phil. Soc.6, 379 (1838).

Re(z)

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

Shatz et aI. - An AlgorithmJor the Evaluation ojRadar Propagationin the Spherical Earth Diffraction Region

Sir George Biddell Airy(1801-1892)

Airy. the son ofa fanner, was asnobbish, self-seekingyoungster,who preferred his well-to-douncle to his father. At the age of 12he persuaded that uncle to carryhim off and bring him up. Heentered Cambridge in 1819 andgraduated in 1823 at the head ofhis class in mathematics. He wenton to teach both mathematicsand astronomy at the university.

He advanced himself withruthless intensity and, althougheveryone disliked him. he wassuccessful in his aims. In 1835 hewas appointed seventh astrono­mer royal. a post he was to holdfor over 45 years.

He modernized the GreenwichObservatory. eqUipping it with ex­cellent instruments and bringing

it up to the level ofthe German ob­servatories, which had been forg­ing ahead of those in Great Brit­ain. Airy organized data that hadbeen put aside to gather dust. Hewas a conceited, envious, small­minded man and ran the observa­tory like a petty tyrant, but hemade it hum.

A strange fatality hauntedAiry. causing him to be remem­bered for his failures. He playedthe role ofthe villain ofthe piece inthe failure of J.C. Adams to carrythrough the discovery ofNeptune.In fact, Airy is far better known asthe man who muffed the discov­ery of that planet than for any ofthe actual accomplishments forwhich he was deservedlyknighted in 1872.

Airy, in 1827. was the first toattempt to correct astigmatism inthe human eye (his own) by use ofa cylindrical eyeglass lens. Hecontributed to the study of inter­ference fiinges and to the mathe­matical theory of rainbows. TheAiry disk, the central spot of lightin the diffraction pattern of apoint light source, is named forhim.

[Excerpted from I. Asimov,Asimov's Biographical Encyclope­dia oj &ience and Technology,Doubleday, Garden City, NY,1982 and from EncyclopaediaBritannica, Fifteenth Edition,1986.)

Using

foo x-ll2e-xK (x) rrJ,K (Y)

___.,,--v_ dx = v '0

X + ~ ~1/2 [cos(lIrr)]o

1for larg ~ 1< rr and Re(lI) < 2'

(3)

Substituting ~= (2/3) i3/2 into Eq. 4 we find that

1 r foo p(x)Ai(z) = - rr-1/2z-1I4e-" -- dx

2 0 l+x/~

for Iz I> °and larg " < rr.

(5)

,

rrV3 . 1(3X)2/3]K 1I3(x) = ( ~ xr3 Al 2 '

we can solve Eq. 3 for Ai(z) ,

1 foo p(x)Ai(z) = :2 rr- l12 z-1/4 e-2/3z3/2 3x dx

o 1 + 2z 3 /2

for largzl<2rr/3 and Izl>O, (4)

where pIx) is a non-negative exponentially de­creasing function,

I( 3X)2/3]p(x) = rr- l12 2-11/6 3-213 x-2/3 e-X Ai 2 .

The Uncoln Laboratory JournaL Volume 1, Number 2 (1988)

The moments, )1k' of p(xL needed for the Gauss­ian 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)

for k = 0, 1, 2, ....

The weights. w. , and abscissas, X., can be foundl I

for the N-point Gaussian quadrature. The Airyfunction. when evaluated by the Gaussianquadrature method, depends on N

Ai(z N) =_1_ rr-1I2z-1/4e-' ~ Wi(N)'2 ~ 1 + Xi(N)/~

1=0

2where, = """3 z3/2 and Iz I> 0, larg, 1 < rr. (6)

149

Shatz et aI. - An AlgorithmJor the Evaluation ojRadar Propagationin the Spherical Earth Diffraction Region

Setting N = 10 was found to provide adequateaccuracy. Table 1 gives a list of weights Wi andabscissas Xi for a la-term Gaussian quadratureintegration of the Airy function.

Thus for small z, we use the power series; forlarge z (I arg(z) I~ 2nj3). we use the quadraturemethod. A connection formula [Ref. 5, Eq.10.4.7] transforms a point from the sectorIarg(z) I~ 2n/3, where the integral representa­tion is not valid, to a weighted sum oftwo linearlyindependent points outside this sector

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

The dividing line between the region that shouldbe solved with the power series and the regionthat should be solved by the quadrature methodwas determined empirically. Figure 2 showswhere the regions on the complex plane lie thatare solved, respectively, by the power seriesexpansion, the Gaussian quadrature method,and the connection formula.

Use of Airy Function in ComputingSpherical Earth Diffraction

The spherical earth diffraction can be com­puted by substituting the values of the Airyfunction computed by this procedure into Eq. 1.There may be floating-point overflows for large

values of z, because the Airy function increasesexponentially with z, even though the largeexponent is offset by the exp(J3 a nx/2l factor ofEq. 1 in the region of interest.Therefore, a scaledAiry function is useful

M(w) = Ai(w) exp (~ w 3/2).

Equation 1 can therefore be rewritten as

~ Al(an + yerri/3) Al(an + zerri/3)F(x,y,z) = 2J7rX ~ erri/3 Ai'(a) erri/3 Al/(a) .

~l n n

(7)

Equation 7 is evaluated by computing addi­tional terms until two successive terms contrib­ute less than 0.0005. Then, to verify the compu­tation, one more check is performed. If anyoneof the terms of the series of Eq. 1 contributesmore than 10,000 per term, we do not return avalue for the propagation factor using Fock'sseries, since the accuracy of the value found forthe Airy function is at least one in 10,000. Themaximum contribution of 10,000 per term waschosen because, when the Airy function isevaluated in Eqs. 2, 5, or 6, the accuracy pro­vided is at least four significant digits.

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

150

12345678910

Abscissas

8.05943 59215 34400 x 10.3

2.010034600905718 x 10- 1

6.418885840366331 x 10-1

1.34479 70831 39945 x 10°2.331065231384954 x 10°3.63401 35043 78772 x 10°5.30709 43079 15284 x 10°7.441601846833691 x 10°1.021488548060315 x 10'1.4083081071 97337 x 10'

Weightsw.

I

8.12311 4134235980 x 10-1

1.439979241604145 x 10-1

3.6440415851 09798 x 10-2

6.48895 66012 64211 x 10-3

7.15550 10754 31907 x 10.4

4.43509 66599 59217 x 10.5

1.371239148976848 x 10-6

1.75840 56386 19854 x 10-8

6.6367686881 75870 x 10-'12.677084371247434 x 10-'4

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

,

Shatz et aI. - An AlgorithmJor the Evaluation ojRadar Propagationin the Spherical Earth Diffraction Region

10..-------------:---------,

Summary

600550

c:o

.!::!...o

:::c

350 400 450 500

Range (km)

ha = 1000 mht = 8000 m

Frequency = 167 MHz

the new diffraction algorithm for all values of yand z less than 5000 and found that it providessufficient accuracy for all x such that SEREwould use the diffraction calculation. Thesevalues of y and z correspond to heights of 375kIn at 170 MHz (VHF) or 17 km at 17 GHz (Kuband).

Fig. 3 - The blue curve shows the values of the one-waypropagation factor, F 2, obtained from the multipath model.The red curve gives the values of F 2 found by the sphericalearth diffraction model.

The diffraction algorithm increases the accu­racy of the calculated propagation factor in thespherical earth diffraction region for a largerange of normalized heights. This algorithmrequires the application of Fock's series andrequires the accurate evaluation of the complexAiry function.

The complex Airy function was evaluated forI z I close to the origin by using a power seriesexpansion. For large Iz I , the 10-term Gaussianquadrature approximation to an integral repre­sentation was used. The calculated Airy func­tion agrees with published tables of the Airyfunction to within at least four significant digits.Incorporating this program in the SERE propa­gation model gives accurate values for thepropagation factor to heights of 375 kIn at VHFor 17 kIn at Ku band.

NLL. -40

-50

-60

-70 L...-_-L..._.......l'--_...l-_---L__..L-_~

300

-----+-Re(z)

o Integral Representation

_ Power Series

o Connection

We tested our calculation by comparing itsresults with the results of a multipath calcula­tion over a smooth conducting sphere. Theresults are shown in Fig. 3 for a high-altitudecase - the most challenging situation. Thedotted curve shows the results of the multipathcalculation; the solid curve gives the results ofthe diffraction calculation.

It can be seen that there is a region (betweenabout 430 kIn and 450 kIn) where the twocalculations are very close. We can concludethat, for this case, the diffraction calculationwas accurate through the intermediate regioninto the multipath region, for these antennaheights. The diffraction calculation is accurateeven within the second multipath lobe in Fig. 3.

We then examined the range ofx, y, and z overwhich our calculation gives accurate results.SERE uses the diffraction calculation if theminimum clearance of a ray is less than

Fig. 2 - The Airy function can be evaluated with threemethods: a power series expansion, the Gaussian quadra­ture method, and a connection formula. The correct choiceofevaluation method depends on the region of the complexplane under consideration.

Im(z)

Test of the Model

where It is the wavelength and d} (d2

) is theground range from the antenna (target) to thepoint of minimum clearance. We have checked

The Lincoln Laboratory Journal. Volume 1. Number 2 (1988) 151

Shatz et aI. - An Algorithmjor the Evaluation ojRadar Propagationin the Spherical Earth Diffraction Region

References

1. S. Ayasli, "SEKE: A Computer Model for Low-AltitudeRadar Propagation Over Irregular Terrain: IEEE Trans.Antennas Propag. AP-M. 1013 (1986).2. M.L. Meeks. Radar Propagation at Low Altitudes, Artec­House. Dedham. MA (1982).3. V.A. Fock, Electromagnetic Diffraction and PropagationProblems. Pergamon, Oxford, England (1965).

4. Association for Computing Machinery. Transactions ojMathematical Sojtware. Program 498. "The Airy Subrou­tine:5. M. Abramowitz and 1. Stegun. HandbookojMathematicalFunctions, Sect. 10, National Bureau of Standards, U.S.Govt. Printing Office. Washington (1970).6. Z. Schulten, D.G.M. Anderson, R.G. Gordon, "An Algo­rithm for the Evaluation of the Complex Airy Functions," J.Comput. Phys. 31. 60 (1976).7. I.S. Gradshetyn and I.M. Ryzhik. Tables oj Integrals,Series. and Products, Academic Press. New York (1965).

GEORGE H. POLYCHRO­NOPOULOS was a staff as­sociate in the Systems andAnalysis Group. He receiveda BSEE from PrincetonUniversity and an MS fromMIT. and is currently pursu­ing a PhD, also at MIT.

&. George's studies are fo-cused on air defense systems. In his free time. he enjoystennis, soccer, and basketball.

MICHAEL P. SHATZ hasbeen a staff member in theSystems and AnalysisGroup for four years. His re­search at Lincoln labora­tory concentrates on analy­sis of air defense systems.Mike came to Lincoln fromthe California Institute of

Technology. where he received his PhD in physics in 1984.He received a BS in physics from MIT in 1979.

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