.20 ROCEEDINGS OF 7TH1. IRE

and hence the Z transform is an irrationial function ofp in general. The Z transform miiay be either a rationalor an irrational function of z, depending upon whether(37) results from exparnsion of a rational fraction in z ornot, respectively.

Let us assume that G*(z) can be found as a rationalfraction in z, at least in approximation, as

a5± a z + a2-2z +2 + arnz-sG*(z) y (39)

bo + btz1 + b2Z-2 + - . +b,zn

where one of the coefficients may be prescribed (asunity, for example) at no loss in generality. Note thatwith z=exp (pT), G*(z) becomes the Laplace transformG(p) and hence

G(p) -ao + a1e-pT + -. + a.e-l"IT

bo + biepT + -+ b,enpTIt often occurs that we wish to determine the Z

transform for a waveform defined in terms of a rationalLaplace transform. In this event, we seek to find coefficients alt anid bk for an assuimed degree of complexity(i.e., values of m and n) so that (40) approxilTiates arationial funiction of p. A specific case of initerest isG(p) = pR where R is a positive integer. For this, expandexponentials in (40) itn their power series and cross-multiply as in the usual Pade' approxinmation. Thereresul ts

./°°O PiT(It+RbopR +±f (- 1)>

q= q.(1Iqbl + 2qb2 + -4 n4qb,)

00o Jrprro'ao+ E(- 1)r- (Ias,+ 2ra2 + e +m'ar4 (41)

which, for any R, enables tus to equate the coefficients

of the lower powers of p. Of particular iiiterest are firstand second-order approximnation-is for 1/p. We use IR -

and invert the resulting solutioni to obtainiI 4

,:Z:8I

A Ip 2z±1 T z I

P 2 z +I(42)

where the first-order approximation-r is the well kniown-iZ transform for a unit step fuIction which has Laplacetransform 1/p and where the second-order approximnation is the Z form used in certain types of generalizedanalysis and in theories of integra-tion [4] [6j

V BIBLOGRAPUHY[1] D. F. Tuttle, Jr., "Network Synthesis, John Wilev anid Sotis,

1nc., New York, N. Y., pp. 761 782, 816; 1958.[2] R. 1) Teasdale, "Time domaini approxiniationi by Use of Pade

approximants," 1953 IRE CONVENTION RracoDt, pt 5, pp. 89594,

'31 F. Ba Hli, "A geineral mnethod tor titmie don iinnetwork syn.-thesis,' IRE TRANS. ON CIRCIT TiilnOIOY, vol. CTfl.IP. 21l28;September, 1954.

141 R. Boxer aiid S. rhaler, 'A simplified method for solving liuiearanid nonlinear svsteims," Pizoc. IR1E, vol 44, pp. 89-101 jaiinary, 1956.

151 S. Thdler at-nd R. Boxer, "Anl operational calculus for iLi-nmericalanalysis," 1956 IRE CONVENTION RECORD, pt. 2 pp. 100-1054

[61 R. Boxer, "A note oni numerical transform calculus, " PRoc. I REvol. 45, pp. 1401 1406; October, 1957.

[71 J. T. Tou, "Digital and Sampled-Data Control Systemus," Mc-Graw-Hill Book Co., Inc., New York, N. Y.; 1959.

181 R. D. Middlebrook and R. M Scarlett, "An approxiniiationi toalpha of a junctionl tranisistor," IRE TRANS. o\t EL iCTti0oNDEvica1s, vol. ED-3, pp. 25-29; Ja-nuary, 1956.

[91 R. M. Scarlett, "lHigh Frequeiicy EquLivalent (ircuits foi Jtnnction Triodes,' presenited at- WESCON\ Los Angeles, Calif.;August, 1956.

1101 E. Weber, "Linear Tranisienit Analysis,' John Wiley and SonsInc., New York, N. Y., vol. 2; 1955.

111 J. L. Stewart, "Fundamentals of Signal lIheory," NVeGraw-HillBook Co., I ne., New Yorl, N. Y.; 1960

A Study of Surface Roughness and Its Effect on

the Backscattering Cross Section of Zpheres*

R. E. HIATTt, SENIOR MEMBER, IRE, T. B. A. SENIORt, AND V II. WESTONt

Summary-The effect on the scattering cross section of a per-fectly conducting object produced by surface roughaess whose scaleis only a small fraction of a wavelength is discussed. The object itselfis assumed large compared with the wavelength. The roughness isassumed to be statistically random in nature and it is shown thatthis can be treated by means of an impedance boundary condition.This permits the use of known results to determine the resultingmodification to the cross section of the unperturbed object. Experi-

* Received by the IRE, July 6, 1960; revised manuscript received,August 29, 1960. The work described herein was carried out for theU. S. Air Force ulnder Conitracts AF 30(602)-1808, AF 30(602)-2099,an-d AF 19(604)-5470.

f Radiation Lab)., Unis. of Michigan, Ainn Arbor, Mich.

mental data obtained by measuring the backscattering cross sectionof a large rough sphere at three frequencies, S, X and K band, arepresented. It is found that even for a sphere whose depth of rough-ness is as large as 10-2 X, the measured change in cross section is nonore than about 0.1 db. This is in good agreement with the theoretical prediction.

1. I:NTRODUCTIONURING the last two years it has become apparent that a difference of opinion exists as tothe influence of suiface imperfectionis in i(lnoel

scattering expert-ients. On- the one( lhauicl there are

Decembelr2008

I ,,

I-6iatt, Senior and Weston: A Study of Surface Roughness

those who believe that an rms surface finish good to10-1 X (approximately) is required if the effects of sur-face roughness are to be discounted, and that an in-crease in the roughness to 10-4 X could produce a detect-able change (of order 1 db) in the scattering cross sec-tion. In comparison with this, a tolerance of about10-3 X on the absolute dimensions of the body is re-garded as sufficient.The albove viewpoint is held by several experimen-

talists of considerable reputation, and if the restrictionsare indeed necessary it is questionable whether thescattering cross section of any practical shape can bepredicte(d satisfactorily by means of model experiments.For exarnple, individual rivets would then become im-portant.On the other hand, there are many who do not accept

the necessity for these restrictions, and who feel thatsurface imperfections of as much as 10-2 X will seldom(if ever) affect the scattering cross section in any de-tectable manner. The only exceptions are those caseswhere thie return from the smooth (unperturbed) bodyis either small in magnitude (as in backscattering froman infinite cone nose-on), or the result of a particularphase relationship which is seriously disturbed by thepresence of the roughness. This opinion is shared by theauthors, and as a contribution towards a better under-standing of roughness effects in general, some resultsobtained from a study of a particular type of roughnessare presented here.The type of roughness considered here is one in

which the surface irregularities are distributed at ran-dom, but in a statistically uniform and isotropic man-ner. The surface slopes are assumed small, and theminimurn (effective) radius of curvature (or dimension)of the mean (unperturbed) surface is assumed largecompared with the wavelength. The effects of the sur-face roughness can then be discussed in terms of animpedance boundary condition applied at the meansurface, and this approach is described briefly in Sec-tion II. As an illustration the method has been used todetermine the back scattering cross section of a roughsphere, and the results obtained appear in Section III.The details of the analysis are given in the Appendix.

In ordler to test the theoretical predictions a series ofexperiments has been carried out in which the scatter-ing cross section of a suitably chosen rough sphere hasbeen measured relative to the cross section of a smoothsphere of about the same diameter. Three different fre-quencies were employed thereby simulating three dif-ferent scales of roughness. The results are presented inSection V and confirm -that even for a sphere whoseroughness depth is as large as 10-2 X the change in crosssection is no more than about 0.1 db. This is in reason-able agrieement with the theory.

II. APPROXIMATE BOUNDARY CONDITIONS

Let us consider first an infinite perfectly conductingplane which is perturbed in some manner so as to yielda type of rough surface. Let z = ¢(x, y) be the amplitude

of the perturbation measured from a mean surfacewhich, for convenience, is taken to be the plane z=0.Then if the perturbed surface is defined in a statisticalmanner so that v is effectively a random variable, andif the statistical properties are uniform and isotropic,the boundary conditions at the actual surface z = canbe written as relations connecting the tangential com-ponents of the electric and magnetic fields at the meansurface z=0. The only characteristics of the surfacewhich enter into these equations are the correlationfunction F (and its derivatives) and the standard devia-tion ¢o of the amplitudes. It is assumed that F is afunction of the distance p between neighboring pointson the surface, and falls rapidly to zero for p>>», where Ican be interpreted as the scale of the irregularities (orthe size of a typical "hump"). The details of the analysisare given in Senior,1 who shows that the above resultsare valid providing v and its first derivatives are con-tinuous and the slope of the surface is everywheresmall. In the practical case to be investigated here weshall only be concerned with values of 1 for which kl < 1,where k = 2r/X, and a sufficient condition upon theslope is then to<<l.The boundary conditions on the mean surface are

functions of the anigle at which the field is incident,and in any general application of the conditions thisvariation is a severe handicap. For the present pur-poses, however, only the approximate magnitude of theperturbation effect is required, and it seems reasonableto expect that the accuracy of the boundary conditionwill not be seriously impaired if an average is taken overall directions of the incident field. The boundary condi-tions which result from the averaging process are

Ex = - qZHy (1)

E, = nZHx (2)

where Z= 1/ Y is the intrinsic impedance of free spaceand , is a parameter defined in terms of the surface char-acteristics by

4__ - 00 { +a 2)( /2kp71 ~ Iik +9. - +k FJo( -I4 L J \p 3p '\ \V2/

k dF kp d

\/2 ap -\V2/j(3)

Eqs. (1) and (2) can be written as

E(n- E)n' = nZ' A H, (4)

where n' is a unit vector normal in the outwards direc-tion, and this will be recognized as the usual impedanceboundary condition for a material whose effective sur-

face impedance is 7.

I T. B. A. Senior, "Impedance boundary conditions for statisti-cally rough surfaces," submitted for publication in A ppl. Sci. Res.

1960 2009

PROCEfD[AIGS OF L%I-fE IREImb

A boundary condition- of the form (4) is freq1uentlyapplied at the surface of a medium whose refractive index N is large compared with urnity; i is thenl iunterpreted as a function of the electrical properties of thematerial and is proportional to t/N. A rigorous dlerivation of the bounidary contiditiot-n as applied to such mla-terials is given-i in a paper by Seniior,2 where it is showntthat (4) is also valid for surfaces of varying cuirvatureprovidii-ig

Im NI kd >% 1,

where d is the smnallest radius of curvature (or dimnen-sion) of the surface. If the permneability IA is niot la{Irgecompared with to, a sufficient restrictiotn upoII d is

which are of a higlher order in r asud ii coiisequeiice, itthe fields are capable of expansion in series of ascending(positive) powers of q, the peife(ctly smooth atproxina-tion (corresponding to q 0) cin be iiserted iito theright-hand side ol (4) In geileral, such expansions willbe -valid and lead to solutions which {iie cssentially"perturbations" about the solutions tor I-lie surfacewithout roughness.

Wle shall niow use this fact to (leteir nue the hack-scattered field when a plan-e wave is incident oni a uniiformfly rough sphere for which ka>>1, where a is theimi-eani radius. If the inicident held is polarized with itselectric vector in the x direction, the scatteied electricfield at a distance _R fronm the en-ter of the sphere is

kd7>> ImqIReturn-ing now to the boundary con-idition uat a rot

surface, this caii be gei-ieralized so as to apply to a irsurface which is curved by ana anialysis similar inrespects to that giveri in Senior,2 providing the nim-yium radius of curvature (or diniensioni) of the misnirface satisfies the inequality (5). In additioii,noted that the roughriess paramiieter 7 eniters intoproblem only via the boui-idary con-dition (4), andcordirngly a rough (but perfectly coniducting) surfaccequivalenti to an imperfectly coniductitng (but smiiocsurface so far as its scattering properties are coniceriThis enables us to associate ani effective conductivitwith the rough surface. Inr the particular case kl(3) gives

77 -' I2/r 1Po22 1

and hence for small scale roughniesses,

4 12s -I V-Y mrhos/m.

r kolj

As an example, if kl1 1/5 and k¢(1=t/100,

t0,

xmhos/rn

and at X band this is comiparable to the conductivityof ordinary nietals. In this instantce at least it would notappear that the roughness can have any appreciableeffect.

.11. SCATTERING [IY A RoUGH SPHEREThe inmpedance condition0 (4) is an approximnation to

the exact boundary conditions, and apart from anyerrors initroduced by the averaging over the incidentfield directions, (4) is correct to the first order in 77,Accordingly, in any solutiotn obtained using this conidition there is no physical justification for retaining ternis

2 T. B. A. Senior, "IImpedance bouLndary conditions for iniperfectly conductinig surfaces," suLbmlitted for publication in Appi. Sri.Res.

(5)

tgh.eanall

1ini-cant istheac-

a 'I i-4 I (

2R -a k/a _K (8)

xhere

Ao t- 2

2i(R- a) 2R

(2R- a) 2 21? a

(9)

(1 i)n, (10)

le is and this result is valid if 771| <<(ka) '. Fhe detailed.tth) anialysis is giveni ini the Apperidixied Eq. (8) is of particular interest in showing the varia-ty s tion of the roughtness effect-, as a fuiction of the distance'<I1?R. The doniinant contributiotn to the overall effect is

provided bv the terni Ao and this is indepe ident of R.The first contributioni which is a function of R comiies

(6) froim the term A4 and is reduced in n-iagnitude by the

(large) factor kac ini the deioniinatorm As P -oc , A--s i/2(1+217), but as the receivei- approaches tthesphere A--Im 2 (1 i)77. Sin-ce 177 is smnall cotmlipared withuinity, the ratio of these two termns is approxiinatelr

(7) 4(1+1077, which implies a decrease mui the effect of sti-face roughtness as the receiver moves itnto the niear field.'In practice, hiowever, it is utnlikely that suchla chan-gewould be detected in view of the factor ka by which thetermi A, is divided, and to a first arpproxirnatilon A1 and(iall subsequernt term:ns cani be nieglected. :I'he magni.ituLeof the scattered field is thetn

E, Ia

2R -aI I 2qI (It)

which onily differs fronm the "sn-mooth" result by at m-iosta few per cen-t for the- type of roughness beinig conisid-ered here. Moreover, for kl/<KK 7 is purely imnaginaryanid (11) shows that the cross section is inicreased by thepresence of the small roughiiess. As kl increases, how-ever, the approximate formula (6) ultimiately ceases tloapply, and the impedance assumn--es a resistive palrt- asindicated by (3); the cross sectiont of the spheie miiaythen: be either increased or decreased depeiidimng o-- therelative magniitudes of the real anid imaginary parts of'q. This is discussed at more lengtlh itn Senior.2

2010 De)cember

19Hiatt, Senior and TVeston: A Study of Surface Roughness

IV. AN EXPERIMENTTo test the above conclusions and, at the same time,

to obtain some direct measurements of the effect ofroughness, an experiment was carried out in which theback scattering cross section of a rough sphere wasmeasured against the cross section of a smooth (stand-ard) sphere at a variety of different distances rangingfrom 16 feet to (about) 6 inches. Each sphere was castin aluminum, a hemisplhere at a time, and it was foundthat the casting process could be modified so as to pro-vide a suitable degree of roughness. The first spherewas left in its rough initial state, while the second wasmachined to give a smooth sphere of radius approxi-mately equal to the meani radius of the other. Thedimensions (in cm) were found to be as follows:

Rough Sphere Smooth Spherea 12.857+0.013 12.697±0.010O 0037 ----.037

0.101 ------

where a is the mean radius (the variation is a conse-quence of slight asymmetries), ¢o is the rms amplitudeof the roughness, and 1 is the scale. The measurementof ¢o and 1 was made using a vernier caliper, and al-though there was some variation from point to pointon the sphere, the above values are typical of those ob-tained.The two spheres are shown in Fig. 1 and the close-up

photograph of the rough sphere in Fig. 2 gives someidea of both the type of surface and the degree of rough-ness.

In order to simulate three different degrees of rough-ness, the cross sections of the two spheres were measuredat the frequencies 2.87, 9.7, and 23 kMc, correspondingto the wavelenigths 10.5, 3.1, anid 1.3 cm, respectively.The measurements were made in an indoor anechoicroom 30 feet wide by 60 feet long using conventionialequipment and technique. Particular care was taken toachieve the greatest possible accuracy, and it is be-lieved that the results for the relative cross sections aregood to about 0.2 db.A block diagram of the equipment is shown in Fig. 3.

At X band a cavity stabilized oscillator was employed,and at the S and K band frequencies the stability wasobtained from a crystal oscillator. The receiver was ofthe microwave superheterodyne type using a separatemixer for each frequency band. The models were sup-ported on a styrofoam column resting on a pedestalwhich could be rotated about its axis, and this in turnwas mounted on a trolley to facilitate measurements asa functioni of range. A photograph of the room and partof the equipment is given in Fig. 4.The comparison between the cross sections of the

spheres was carried out in two different ways. In thefirst, the cross sections of the spheres were individuallyrecorded as each was rotated through 360°. This pro-cedure proved adequate at the lowest frequency wherethe roughness produced a negligible effect. At the higherfrequencies point by point data were taken in addition

Fig. 1-Ten-inch metal spheres, rough and smooth surface.

Fig. 2-Surface condition of rough ten-inch sphere.

Fig. 3-Block diagram of equipment.

Fig. 4-Anechoic room and K-band equipment.

1960 2011

PROCEEDINGS OF THE- IREFJn

to the 3600 plots. In obtaininig these further data eightpoints oIn the rough sphere were selected, four on ealchhemnisphere, int such a way that the plane of the junictioInbetweeiu the two hemispheres was niever parallel or

perpendicular to either the direction-i of the illuminiat-in-g beacm or the electric vetctor. The anitennia beacm was

theni "directed" successively at each of these points,anid the average signial recorded as the ranige wAcas variedthrough ±X/4. Trhe cointribution diue to the background(lwas thereby minimi-ized.IThe change in this contributionas a funictioni of ranige could genierally be kept to lessthan onie or two decibels, ani( miucCh of the time it- wN7as

i1o more thani onie.

Initerspersed betweeni these eight r-eadlings, three read-inigs were obtained with the smootlh sphere, and thedifference in averages was then recordled or plotted as

otne poinlt (see, for example, Fig. 7). Poinit by point tlataof this type were obtainied at both X anid K bands, Al-

though at 23 kMc the ii-umiiber- of readinigs wN as furtlherinicreased to 24 byI rotating the sphere tl-hrough + 50at each of the above-mientioned eiglht poinits.

V. REnSULTSAt all the frequencies at which the experimiienital work

was carried out, the values of kl are small coniparedlwith unity, and since ¢o<</, (6) canr be used to calculatethe effective surface i mpedance consequent upon thepresenice of the roughniess. ThuLIS W"' have

X 1O.5 cm,

X = 3.1 cm,

X = 1.3 cm,

= 0.009m,n = 0.03i,

7q= 0.07i.

Usinig (8) nlow, the roughn-ess is founid to increase thebackscattering cross sectioni of the sphere by an amnoutitwhich increases from 2 X l0-3 db at X = 10.5 cm, through2X10-2 db at X=3.1 cim to 10-1 (lb at X=1.3 cnm. nti

addition, however, ther-e is the change in the cross sec-

tion of the rough sphere over the smiioothl (standard)sphere produced by its larger imeani radius. At S band(_-:where the sphere is near the tipper enid of the resornantregion the change is -(0. db; for the X anid K banidfrequetncies the change is 0.1 (lb. These changes plus thetheory outlinied ini Sectionis I I and 11I then-i predict thatthe cross sectioni of the rough sphere will excee(d thecross sectioni of the sm11ooth (staindard) sphere by the fol-

lowing amounits:

X = 10.5 cmi,

X 3.1 cm,

-0.110 db,

0.12 db,X 1t.3 cm, 0.20 db.

Before going oni to compare these with the valuesfound experimentally, it m-nay be of interest to list thedegrees of roughness appropriate to the frequenciesemployed. ff d deniotes the total depth of the roughness(approxiimately 2¢,,) and(i w denotes the total width of a

typical bum-np (rather tlhani the width b)etweenl, 3-dbpoints used in the specificationi of the scalde 1), the various paratmeters are:

X(lui)

10 .3

3.1'I .3

ka

7 625.477 5

d/A7 X 1()-I:2XIO -2

sx 10 -2

3X1() 2

It)

2X10I

In viexw of tlhie large values of kca it is not to b)e, expettedltlhat anyc change in the relative cross ser timis as it f(ilic-tionl of distance will be detectable.

In Fig. 5 the experimtnental restlts at 2.87 k ic are

showin in the formii of 360( Iplots for three (lifferentranges. Eacth plot contains four traces- -one tbor tIhesmiiootlh splhere and(i onie fom- each ol the tlhree oriellta..tions of the rough sphere- -cand il genieral the traces are

mnore or less coincident witlh oine anotheir. By iitspect-tioni of these traces (and othier- similar traces not pire-sente(l iere), it, iS ConltIided that there is 1(1o imeastir-

alble effect- (lue to the r-oLughiness at thlis frequiiencyx In

)aISS11ng- it shotLIld be noted that tte tlhickniess ol' thetraces in Fig. 5 is of or(der 0.1 (db), and atccordinglvmiiore detailed ianlysis wotul(d b)e iecessaryv if tlie pre-

dicted change tIue to the dlifferent sphere radii is t-o be

(letected. Since this change is niot truly at roughness ef-

fect, nio such manlysis was performe(d.

+2 i

0--2 -T

n ~~~~~~~~~~~~~~~~~~~~~Range- 12 feet

'_ I0-2 .4 4 !-,--1- ---I I-4 -+ t

Rarnge - 8 feet

Cy2 A

T- -- I| h~~~~~~~Rnge - 4 fee

I80' 144 108 72 36 o6 36 72 108 144 180°

Fig. 5-----Relative ra(lar (ross sect iou, o-, of s.mooth and(l routgh sphereat X=10ctm. Each of the three multiple tra(ces show a for- thesmooth sphere anl for three dlifferent orientations of the: rouglsphere.

Ihe 360° plots at 9.7 kMc are showni in Uig. 6. Theeffect of the surface roughness is quite apparenth-:ere,but tl-ie peak deviatiotis fromii the sm-iooth splIere ret-lurnlare limliitecd to about 1 (lb, antI the atverage (lifference inithe returnis is even1 less. This is brought- out iior-e clearlyini IFig. 7 in whilcih tie poitit by point nleasureniiiitts aicshownr as a function of the ranige R. For comparlisonwxith thlle Rayleigh distance, t1e IIaximunlii rainge (RP 16feet) is equlivalenit to R 9.4 (a2 X) wh1'iere a is thIie sph)hereradius, anld at the iminitnuinim range (=- 6i itches)RP 0.29(a2/X).

ThI-le results ill [1ig. 7 shiowxx no stat isticallv significaitran-ige depen-denice, though there appears to be a tend-enicv for the r-elative cross section to deicrease with le-creasing ranige. This is in accortdanice with thie theory.Wheni all the poinits in Fig. 7 are averaged regartiless olrange, the cross sectionl of the rough sphere is loutnd tobe 0.12 db above that for the smooth sphere, and while

2012 Dec:ember

Hiatt, Senior and Weston: A Study of Surface Roughness

- 1I11 1r I7I r-I-----Smooth Sphere -:| |=|ughX30°I-4- --

100 144 [08 72 36 00

6 I

36 72

108I

108

Fig. 6 -Relative radar cross section, a-, of smooth and rough sphere at X=3.1 cm.

I

16 14 12 10

I i

I 0

0 0

8 6 4 2 1 0

Range il Feet

Fig. 7-Effect of ranige oni the ratio of of roLugh to smuooth sphereX=3.1 cm. Poinits plotted are the average of 16 or miiore readinigs. 0

rnw

0,2

.2

to

uzp

Cd

2

0I = 10.5 Cm

-2

2-

O%_ 23.1 Cm

-f2 A

02

-2 L.IotS,,moomth Sph.m_

b Rough Sphere Range 16', 16+X/4, 16-X/4

-2 -4Smooth Sphere Range 16', 16 +n/4,16-X/4

I80'180° 144 108 72 36 0° 36 72 108 144 180°

A- 1.3 C a

- 400

Fig. 9-Effect of surface roughness on backscatter patterni of 25 cm

metal sphere. Average bump size abouLt 0.7 mm-nn deep by 3 mmwide.

Fig. 8 Radar cross section, a, of rough aindand smooth sphere at X = 1.3 cm.

the standard deviation of the experimental values issomewhat large (0.40 db), the average is in truly re-

markable agreemeint with the theory. The extent of theagreement is, perhaps, a little fortuitous, but does pro-

vide confirmation of the theoretical approach.The final set of 3600 plots are given in Fig. 8 and are

for 23 kMc. The surface roughness now has a markedeffect, and the peak deviations from the cross sectioni ofthe smooth sphere are as much as 4 db. The multipletraces sh'own result from changing the range by ±X/4and serve to indicate the effect of the background signal.Such traces as these were entirely reproducible, and, ifsufficient care was taken, the sphere could be removedfrom its pedestal and then replaced, with the same

traces obtained once again.At this frequency the bumps on the sphere are about

X/5 wide by X/20 deep and it seems probable that thebumps are here acting singly or in combination of twoor three at a time to produce the individual features inthe traces. The fine structure in the traces is no more

than 2° in width and corresponds to a displacement ofthe sphere's surface of approximately 0.4 cm. It can beseen from Fig. 2 that this is comparable to the width ofthe bumps, and under these circumstances a theorybased on the random addition of the returns from many

small irregularities is clearly inappropriate. It thereforecomes as no surprise that the predicted chanige in cross

section differs from that observed.Information on the average change inl measured

cross section was obtained by the poinit by poinitmethod. Almost 800 readings were averaged regardlessof range and showed that the cross sectioni of the roughsphere exceeded that of the standard by 0.51 db. Thestandard deviation of the points was, however, 1.04 db.

In order to facilitate comparison of the returnis atthe three frequencies, sample recordings of the roughsphere data are given in Fig. 9. The way in which theroughness effect increases with increasing frequency isclearly visible.

VI. THEORETICAL DIscUSSION

The theory outlined in Sectionis II and III is basedon an impedance boundary condition derived in Senior.This condition is accurate to the first order in theroughness effect providing the inclination of the actualsurface to a mean surface is everywhere small, and pro-viding the irregularities are distributed at ranidom butin a statistically uniform and isotropic manner. If theserestrictions are fulfilled, the main effect of the surfaceroughness is to modify slightly in phase aind amplitudethe field scattered by each surface element of thesmooth body.

+2b

1;-2-

0.4 _

i

0.2 -

0 -0.2

°) -0.4 _-

144 1I 0°

1----t- I- ----------fqi-.! ;-4= -1- -- F2:lz;7I.I-I,

Sphoro-,-.-

-.,,...,.-..,I

1960 2013

1404

b

-2_

- 400

2014PROCEEDINGS OF TfiE FRE

If any of the above restrictions are relaxed, theboundary condition may cease to hold. Thus, if theslopes of the irregularities are large, there is the possi-bility of scattering taking place from the sides of theindividual humps, thus produicirig a field in a directionother than that for the smooth body and of a magnitudewhich is no longer negligible. If the surface of an infinitecone were roughened in this mnanner, a contributioncould be expected which was not from the tip. Similarly, if the irregularities are not distribiuted at ranidom,then in certain directions the fields produced by the in-dividual element mav add up in phase, and here againthe boundary condition is not applicable. As an exampleof this, if small concentric grooves are cut in the sidesof an infinite cone a first-order modification to thefield may result, particularly for backscattering in adirection normal to the rings.

If the surface irregularities do not satisfy the aboverestrictions, alternative methods must be employedfor assessing the effect of the surface roughness, andonly for certain special types of irregularitv are appro-priate methods available Thus for one or more iso-lated bumps whose dimensions are small compared withthe wavelength, the total scattered field can be ob-tained by using the Rayleigh scattering formula foreach individual bump and neglecting the interactionwith the field of the smooth body. Since the cross sec-tion in Rayleigh scattering is proportional to the sixthpower of a linear dimension, the percentage change inthe total scattering cross section will be small unless thefield of the unperturbed body is itself small, or the num-ber of bumps is large. If, on the other hand, the surfaceperturbations are of a very regular kind and can be approximated by a series of corrugations, the effect maybe estimated by using the known solutions for scatter-ing by a corrugated sheeti or by a corrugated cylinder.4In this case, the perturbation field may be comparableto the field of the smooth body.

VII. CONCLUSIONSIn embarking on a study of surface roughness and its

effect on radar scattering cross sectioiis, one of the ob-jectives was to consider the degree of surface finishwhich is necessary in model scattering experiments. Aspart of the experimental program the backscatteringcross section of a suitably chosen rough sphere has beenmeasured at S-, X-, and K-band frequencies and com-pared with the cross section of a smooth sphere of ap-proximately the same diameter. It was found that evenwith a surface roughness which would normally be re-garded as completely unacceptable for model work thechange in cross section due to roughness was relatively

3 P. C. Clemmow and V. H. Weston, "Diffraction by an AlmostCircular Cylinder," University of Michigan, Ann Arbor, Rept. No.2871-3-T; September, 1959.

4 T. B. A. Senior, "The scattering of electromagnetic waves by acorrugated sheet," Canad. J. Phys., vol. 37, pp. 787, 1572; July,December, 1959.

srmiall. Thus, at X band the spheie used had a roughnesswhose depth was 0.02 X, b3Ut still the average changeiccross section which could be attributed to the roughnesswas less than 0.1 db, and at S band io change couldi bedetected. As is to be expected, the effect illcreases withircreasing fiequency, and at K banud where the bumpswere 0.05 X in depth the scattering patterns were quiteirregular,On the theoretical side an analysis of the general

problem of roughness has shown that geoimuetrical ir-regularities characterized by s malI surface gradientsand random but statistically uniformn properties anl behandled by the usual type of i mpedanice boundary con-dition. This implies that the rouighness produces, a comparable effect to a change in the conlductivity of theuinperturbed surface. Results obtained with this approachare in good agreement with the experimental data.

APPENDIX

BACKSCATTERING FROM A LARGE ROUGH SPHI,-RE

Followinrg the theory outlined in Section 11 it is as-sumed that the surface roughness produces an effectivesurface impedance iq. The boundary coridition which isapplied at the surface R a is then

E -(fzeE)>i - = i2 A H. (1.2)If the incident field is a plane wave traveling in the

positive z direction with aii electric vector

Ei ixeZt= sct iotnl ^er)} 2n + I

e i"'t ijflMol~sn(i) - tin,,(')

where m0l. and n5in are the spherical vector wave func-tions defined by Strattonj5 the scattered field can bewritten as

2n + IEs - e scot Z in {aImoln(3

ol n(nl+ 1)ibnfneic13) }

and by application of the boundary condition (12) thecoefficients at, and b,, are found to be

Pi, (P) ip[p,(p)I' (13)pjs(p) --- +n ip/,?(p) I'[pj, (p) II + iqpj", (p) 1.A

[ph" (p) I' + itiplu(p)(14)

where p ka and the primes denote differentiation withrespect to the entire argument. The backscattered fieldis then

00

Es --1,2

( -i) 7?('n 2 1) a,h,t(kR)n2.l

ib5 [kRhn(kR) ]'} (15)

IJ. A. Stratton, "Electromagnetic Theory," McGraw-Hill BookCo., Inc., New York, N. Y.; 1941.

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1fiatt, Senior and Weston: A Study of Surface Roughness

To evaluate this expression for large ka it is conven-ient to separate out the portioni appropriate to a smoothsphere (q = 0). If E(O) is the x component of the electricvector in the backscattered field for this case, and ifEl'=xE(n), (15) can be writteni in the form

E = E(0) + h(-i)n(n + 2{hn(kR) [an(7) -a.(0)3?1

+ [[kRhn(kR)]'[bn(J)- bn(), (16)

where the new coefficients are defined by

a.(q) - a.(0) = {PhA(p) - iq [Phn(P)I'}ph,(p)

b,Q(q) - bn(0) = ---t [ph (p) I' + inp1in(p) }I1[phIs(p)s l

If qI issfficietly small,

bn(7)- bn (0) - n{I [phn(p) I'-2171 <<IIqI «P-113

and the coefficients will be replaced by these asymptoticvalues. This has the effect of neglecting the residuesproduced by the first-order poles of an(t1) -a,(0) andb,(-) -b,(0) when the series in (16) is transformed bymeans of a Watsoni transformation into a contour in-tegral plus a residue series. Since these residues cor-respond to the diffracted field, the approximation repre-sented by (17) and (18) is sufficient whenever the re-flected field is dominant.By using the relationis

in (p) n(p)-j (p)h,(p)-p2

[pji(p)P + {1 _.j-+ 1)}pi,(p) =0

(16) can now be simplified to give

E(77) = L- i7 E(O) + tqS

where

n(on + 1) [kRh,(khR)3i_hE(-t) ( + 2) kR{p [phn(p)2]}

Taking first the portion corresponding to a smoothsphere, the analysis in Weston6 shows that

a-eik(R- 2a)

2R - a

r Al(o) A2(0)I +-- + 2+ka (ka)

6 V. H. Weston, "Exact Near-Field and Far-Field Solutioon fcBack Scattering of a Pulse from a Perfectly Conducting SphUniiv. of Michigan, Ann Arbor, Rept. No. 2778-4-T; February,

(19)

(20)

ere,">1959.

for large ka, where

2i(R - a)2(2R - a)2

a(R - a)(2R2- 4Ra + 3a2)A2(0) a

(2R- a)4

(22)

(23)

The remaining task is to sum the series S, anid for thispurpose the Watson transform technique is used. If acontour C is drawn surrounding the poles of cos v7r onithe positive real axis of the v planie, S can be written as

/2 eir'4 e>7'2v(v2 1/4) [VkR H,(kR)'dS- -= - dv./7r 2kRp2J cosv7r{I[NpH(p)]'}2

The contour C may now be deformed into a straight-line path from - oo to + oc and passing through theorigin at an- angle 3 to the positive real v axis, where

(17) -it22<K<0. The odd portiofi of the initegrand then(18) integrates to zero, so that

V 2 e3ir/4wkRp2

ooexp (-iA) v tan vr(V2 - 1/4) [VER H,(kR)j'_ - dv.Jo neiprI2{ [v\p Hv(p) 1} } 2

To evaluate this new integral it is convenient tobreak it inlto two parts by setting

tan vr=cos vt

Taking first the integral SI corresponding to the firstof these two terms, it is permissible to put f3=7r/2.Writing v =-ip we then have

/2 eir/4 F Ie-,) PSI - -hRp2p(pJ + 1/4) c irf

r kRp2J cosh 7rp

[\/2R Hip(kR) ]dpe p

Since the dominant behavior of the integrand is pro-vided by the factor e- P/cosh irp, Si may be approxi-mated as

1 rX e-rpSI ~- 2 ei(R-2a) p(p2 + 1/4) dp

kRp2 o cosh irp

(21) and although this integral can be easily evaluated, forpresenit purposes it is only necessary to note that

rthe 1SI - eik(R-e2) X constant.

kR(ka) 2

1960 2Ot5

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PROCEEDINGS OF THE IRE

'rhe other integral S2 is given by

1/2 ei'r/4S2 =X

7r kRp2

ro exp ( i)) I \/kR HI(kr) I'di'1/4) e- vI

eipr/2 -\[p Hp(p) '} 2

:Hence,

(24)a) ka-'2f

and the backscattered field for the rongh sphere istherefore

and is evaluated by replacing the Hankel functionsby their asymptotic expansions for Iv| <p:.7 We thenhave

[V/P Hv(p)]I'ro{ exp {ip (v 1/2)i

iv2 iA+ I (V2 1/4)

2p 8p

7r

2

I iV4

4p2 24p'

.1 Ao(,q) +ka

(25)A)2(n+

(ka) I

where

and the integral can now be approximated as

eik(R-2a) co exp (-,B) (jiV2/ 1 2S2 v3exp - { - dv

kRp2 JO 2 kR ka,/

__ ik(R 2a){+ /( )$

kR 2 -- 2

\R

J. M. C. Scott, "Ain Asymptotic Series for the Radar ScatteringCross Section of a Spherical Target," (British) Atomic Energy Res.Establ. Rept. T/M 30; 1949.

A o (,) =1 In

2i(R a)' 2R

(2R a-a)2 2R -a

,q(1 e)

This result holds for sufficiently large ka (such that thediffracted field is negligible) and for sufficiently small1|q| (il.e., 1,q1 <<(ka)-13).

ACKNOWLEDGMENT

The authors wish to express their thanks to TheodoreH-Ion for his assistance with both the measuremrnents andthe illustrations.

Correspondence

Doppler Navigation and Tracking*The series of papers by the personnel of

Johns Hopkins in the April Space Electronicsissue, on the use of Doppler techi-siques forsatellite trackling and navigatioin were mostinteresting, and I would like to take thisopportunity to expand oi- the fundamecntalprinciples and background uniderlying theseprojects.

The fundamental principle embodied inthese navigationi and tracking systemns con-

cerns the fact that at any inistant a receiverwill detect a Doppler and rate of change ofDoppler frequency which is utiique for that"target" position1 and velocity. As so aptlyput by Drs. Weiffenbach and Guier,wheni properly utilized, each segmiienit of thereceived Doppler- curve provides useful in-formation about the track of the satel-lite. ." Proper utilization will consist ofsome form of computationial or apparatuscturve fitting procedure. The realization thatit is possible to resolve a moving "target"inu azimuth and range, using CW traiismis-

* Received by the IRE, June 6, 1960.

sion, is most iiiteresting and powerful.Space does not permit a theoretical anal-

ysis here; however, some of our experi-mental data will be of interest. The followiingdiscussion and data makes use of a movingaircraft to ground "target" configuration,as this was the first embodiment makinguse of these funidamental priinciples. A sin-gle target will return- a Doppler frequency,i.e., a Doppler curve which is a direct andanalytic function of the aircraft path andvelocity. Two targets are sufficien-t to es

tablish aircraft position (given altittide aniddirection one target is sufficient, as is thecase for a satellite of approximate knownorbit). A more powerful aid to visualizatioriof these priniciples of resolving a movingtarget in "azimuth" an-id "range" is providedby a Doppler mosaic. Flig. 1 shows such a

mosaic oriented for a satellite with onie-way

propagation. This mosaic was originlally developed for anl aircraft with two-way propa-gation under conditions similar to thoseused for the experimental data (fii13.5kmc, velocity-120 mph, Af-as shown,dAf/dt--Xl0) anid simply relabeled to pro-duce Fig, 1. Two sets of cu-rves are shown,

lines of constant Doppler frequency (radiallines converging at zero) and lines of con-

stant rate of change of Doppler frequency(semi-elliptical curves, all tangent at thecelnter). In three dimensions, the pattern isa figure of revolution around the velocityvector. Since we are concernied only withrelative motion, consider the diagram at-tached to the satellite (or aircraft) with re-

ceiving sites (targets) moving through thepattern from right to left. TI he satellite pathis assumed circular (altitude 400 iniles), aiidthuis a receiving statioIm will follow thecurved path (overhead pass) represented bythe earth's surface. Off-coujrse passes can bevisualized in three dimenlsions as coricentriccircles arouiid the earth's center aiid in.tei-secting the imosaic above or below the planeof the paper.

The resolving power in. determniiingsatellite position is governed by equipmentability to mseasure Doppler frequeincy com-

ponents in the presence of nioise. The important point is that each element or segment of the Doppler curve can be properlyweighted to provide a best "average" matchor fit betweei thie received signal (Doppler

a f 2Rs '= e ik(R--2a) .

2R- a ka(2R

E(X)a

2R ae ik (R- 2a)

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