6ev O'Nly. i *Af. -0-St 4 THE REFLECTION OF RADIO WAVES FROM AN IRREGULAR IONOSPHERE MASSACHUSETTS INSTITUTE OF TECHNOLOGY M. L. V.

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1 i DOCU.EIT R~)6 R0 - RESEARCH LAB(Ai'RY CF EILCTROICS tkassachusetts 1It'S'IJE OF TCVlDDLOGY - -0-St 4 THE REFLECTION OF RADIO WAVES FROM AN IRREGULAR IONOSPHERE M. L. V. PITTEWAY TECHNICAL REPORT 382 NOVEMBER 8, 1960 i *Af 6ev O'Nly MASSACHUSETTS INSTITUTE OF TECHNOLOGY RESEARCH LABORATORY OF ELECTRONICS CAMBRIDGE, MASSACHUSETTS

2 The Reserch Lbortory of Electronics is n interdeprtmentl lbortory of the Deprtment of Electricl Engineering nd the Deprtment of Physics. The reserch reported in this document ws mde possible in prt by support extended the Msschusetts Institute of Technology, Reserch Lbortory of Electronics, jointly by the U. S. Army (Signl Corps), the U.S. Nvy (Office of Nvl Reserch), nd the U.S. Air Force (Office of Scientific Reserch, Air Reserch nd Development Commnd), under Signl Corps Contrct DA sc-78108, Deprtment of the Army Tsk nd Project

3 MASSACHUSETTS INSTITUTE OF TECHNOLOGY RESEARCH LABORATORY OF ELECTRONICS Technicl Report 382 November 8, 1960 THE REFLECTION OF RADIO WAVES FROM AN IRREGULAR IONOSPHERE M. L. V. Pittewy Abstrct The work of recent pper on the reflection of rdio wves from n irregulr ionosphere is extended to tke ccount of strong scttering. Solutions of the wve eqution for horizontlly strtified ionosphere re used s strting point, nd the equtions governing scttering for simple two-dimensionl model re written in coupled form. A ry theory of scttering is exmined from wve theory viewpoint, pplied to scttering by n irregulr lyer of free electrons. Limited results of numericl work re exhibited in curves. This report is chiefly concerned with the physicl interprettion of the mthemtics outlined nd finding useful pproximtions to help future work. I

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5 - TABLE OF CONTENTS 1. Nottion 1 2. Introduction 2 3. A Two-Dimensionl Scttering Problem 4 4. Diffrction by Thin Lyer of Irregulrities 8 5. Coupled Equtions Ry Theory The Reflection Coefficient Mtrix Numericl Work Discussion 24 Acknowledgment 25 References 26 iii

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7 1. NOTATION All electric nd mgnetic quntities in this pper re in rtionlized units. Importnt quntities used re: x, y, z Crtesin coordintes; z is mesured verticlly upwrds. 0 ngulr wve frequency; time fctor exp(iwt) is omitted throughout. e, m chrge nd mss of the electron E electric permittivity of free spce used only to p number of electrons per unit volume define X nd Z. v collision frequency of the electrons X pe 2 /EomW 2 z Z ~/0 F,/W U p.u X(Z) A(x, z) E(x, z) d 1 - iz (1-X/U)1/2, effectively the refrctive index X/U for plne strtified ionosphere function representing irregulrities in two-dimensionl model electric wve-field vector periodicity of irregulrities S /d, where is n integer 0 sin1 S C ( 1 fj(z) term in the Fourier series for A(x, z) E (Z) term in the Fourier series for the y component of E(x, z) N A, B X ' finite summtion limit independent solutions of the wve eqution for strtified ionosphere, depending upon the ngle of incidence 0 coefficients ssocited with A nd B, respectively W the Wronskin A B' - B A'.dSj integrl of fj(z), tken through lyer of irregulrities AQ R complex phse devition produced by diffrction grting squre mtrix of reflection coefficients z grdient prmeter for liner strtified ionosphere b z men height of lyer of irregulrities intensity prmeter of the irregulrities hlf-width of the irregulr lyer The "wek sctter integrl" refers to formul 15 of reference 10; this gives the ngulr spectrum below the ionosphere of wek scttered wve. The unit of length is defined s the wvelength of the incident rdition in free spce, nd thus depends upon the frequency. A prime denotes differentition with respect to z. The dummy suffix nottion is not used in this report. 1

8 2. INTRODUCTION The theory of rdio wve propgtion is often pplied to horizontlly strtified ionosphere, in which the ionosphere is supposed to vry only in the verticl direction. When the ionosphere vries slowly (so tht the ionospheric prmeters chnge only by smll frction in ech wvelength), the process of reflection cn be ignored, nd ry theory pproximtions re vlid. The ry theory pproximtions brek down where the refrctive index in the ionosphere is smll, for there the wvelength becomes very lrge, nd, no mtter how slowly the ionosphere vries, reflection will occur. The ionosphere is not, in fct, plne-strtified, but contins moving irregulrities tht re responsible for scttering the energy of reflected rdio wves (1). The theory must therefore be extended to tke ccount of horizontl vritions. Previous discussions of this type of scttering hve usully been mde in terms of trnsmission through, rther thn reflection from, n irregulr region; this is becuse the work hs been bsed on ry theory diffrction (2, 3, 4), nd is not dequte when reflection occurs. Becuse of this difficulty, the irregulrities re often specified by the effect they produce, rther thn by ctul vritions in the ionospheric prmeters (5, 6). It hs been suggested (7, 8, 9) (Booker nd Gordon 1950, Booker 1955, Jones 1958) tht the reflection level might be of prticulr importnce for scttering. To study this possibility, we must solve the differentil wve equtions, nd with horizontl vritions these will depend upon three independent vribles. If we ssume tht the scttered wve is wek, it is possible to write down useful solution of the wve eqution (10). This is done by dopting solutions of the wve eqution for horizontlly strtified ionosphere s strting point, nd then introducing the irregulrities s smll perturbtion. It is the purpose of this report to show tht solutions of the wve eqution for strtified ionosphere re still useful strting point, even when the scttering is not wek. There seem to be two wys in which strong scttering cn occur: (i) Wek irregulrities producing strong sctter. (ii) Strong irregulrities. In cse (i), we suppose tht wek irregulrities produce strong scttering grdully fter propgtion through thick region; for ny thin slb tken from this region, the scttering will be wek. The differentil wve equtions cn be written in coupled form, where the coupling term is wek nd represents the scttering t ech ionospheric height. In Section 5 of this pper, the coupled equtions re derived for simplified two-dimensionl problem tht is outlined in Section 3. In cse (ii), ech problem requires seprte tretment. Exmples re cylindricl meteor trils (11) nd the thin lyer of irregulrities tht re discussed in Section 4. In Section 6, it is estblished tht the full wve theory tretment given here grees with ry theory under the conditions when the ltter is vlid, nd there is simple method of extending the conventionl ry theory tretment to tke ccount of devition 2

9 effects without troublesome ry-trcing procedures. A convenient method of numericl solution is outlined in Section 7, where the equtions re trnsformed into Rictti form with mtrix vrible. Some curves re drwn in Section 8, but these re limited by the long digitl computer times required for the integrtions. In the finl section, the methods of extension to more generl scttering problem re outlined, but the detiled mthemtics is not given. The use of vrious pproximtions to id further work is discussed with view to hndling stochstic model ionosphere

10 3. A TWO-DIMENSIONAL SCATTERING PROBLEM In this section, we consider simplified two-dimensionl scttering problem. The equtions re trnsformed by Fourier nlysis to introduce the ngulr spectrum, nd this is discussed in terms of ry theory. The irregulrities produce scttering between the different ngles. The propgtion of n electromgnetic wve through the ionosphere is governed by Mxwell's equtions, together with the constitutive reltions for the ionosphere defining the properties of the medium. To simplify the presenttion of the mthemtics, we ignore the effect of the erth's mgnetic field. For monochromtic incident wve of fixed frequency, the properties of the ionosphere re represented by n isotropic complex refrctive index, given by 2 X =1 -- (1) in which X is rel, nd is proportionl to the density of the free ionospheric electrons; U = l-iz; Z is rel nd represents dmping by collision of the electrons with neutrl ir molecules. In generl, is function of three Crtesin coordintes x, y nd z; it is convenient to define the unit of length s the wvelength in free spce, so tht lengths re mesured s multiples of wvelengths, nd we set the z xis verticlly upwrds. As first step, we suppose tht p. is function of x nd z only, nd tht plne wves incident to the ionosphere from below hve their wve normls confined to the x-z plne, so tht 8/y is zero everywhere nd the problem is reduced to two dimensions. Mxwell's equtions then split conveniently into two prts, giving the "electric" nd "mgnetic" polriztions. In the electric polriztion equtions, the electric wve-field vector E is directed everywhere prllel to the y xis, nd the displced electrons give no induced spce chrges. With the mgnetic polriztion, E lies everywhere in the x-z plne; induced spce chrges led to more complicted equtions, which re not considered in detil here. For the electric polriztion, the electric wve-field vector E stisfies the differentil eqution 2 2E + 2E E = ( + + 4n CL' E = (2) x z This is form of wve eqution normlly ssocited with sclr wves, but it is understood tht E is directed s vector long the y xis. The refrctive index p. depends upon the ionospheric prmeter X/U, nd it is convenient to tke this in two seprte prts X = x(z) + (x, z) (3) 4 ( I

11 where X(z) represents the strtified ionosphere upon which the irregulrities A(x, z) re superimposed. In reference 10, A ws introduced in the equtions s smll perturbtion, nd it ws ssumed tht the corresponding chnge in E, representing the scttered wve, ws lso smll. Here, A my be smll, but we wnt to find out wht hppens when the scttered wve is no longer wek. Let us suppose tht A is periodic in the x direction with period d; in the limit, d my be supposed to be lrge to represent completely irregulr ionosphere, but working with d finite leds to equtions with discrete terms nd simplifies questions of convergence nd the reversing of limits. Accordingly, we write A(x,z) = fj(z) exp i (4) where ech fj(z) is n rbitrry function of height. Now suppose tht plne wve is incident normlly onto the ionosphere from below. One of the boundry conditions is tht the solution of the wve eqution must lso be periodic in the x direction with period d, nd so we write where oo E(x, z)= E(z) exp(-2isx) (5) =-oo S (6) d In free spce below the ionosphere, Eq. 5 represents n ngulr spectrum of plne wves; these wves hve wve normls inclined to the verticl t ngles 0, where S = sin 0 ; nd we see from Eq. 6 tht these re just the directions of the spectr tht would emerge from diffrction grting of periodicity d. This rises n interesting point; we hve periodic irregulrities, superimposed upon strtified medium, scttering in exctly the sme directions s if they were in free spce. Now, inside the ionosphere the wvelength is lrger, nd the ngles through which the wve is scttered re correspondingly incresed; but it hppens tht this is corrected s the wve propgtes down to free spce (Fig. 1). Although we hve been considering norml incidence, Eq. 5 is eqully vlid for oblique incidence, provided tht we suppose tht the incident wve hs its wve norml in one of the directions S (In the limit d - oo, the ngulr spectrum becomes continuous, nd incidence t ny ngle is permitted. ) And s different solutions of the wve eqution, Eq. 2, cn be dded to give new solutions, the incident wve need not even be plne, but cn be defined by set of plne wves incident in the different directions S We cn now obtin the differentil eqution stisfied by the trnsformed vribles E (z), bering in mind tht we re exchnging the x dependence of the equtions for 5

12 d Fig. 1. Scttering by diffrction grting buried in medium of refrctive index [±. sin n = nx L/d (diffrction grting lw) L = Xo/p. (ry theory) nd sin 0n = [L sin n (Snell's lw) Therefore sin 0 n = n 0 /d s in free spce summtions over the discrete vrible. But first we must consider question of convergence. There re two points which ffect the convergence of the series in Eq. 5. First, for ny resonble physicl model of the ionosphere, the functions f(z) must become unimportnt when j is lrge; otherwise, A(x, z) will hve discontinuities. Second, for given vlue of d, there will be vlues of S greter then unity, nd this mens tht lrge vlues of correspond to evnescent wves. Even if A were to hve discontinuities, the exponentil behvior of these evnescent wves ensures tht we need only work with finite number of terms in the summtions to obtin n rbitrry degree of ccurcy, nd t lter stge in the work the infinite limits re replced by finite summtion over the rnge -N N. Accordingly, in deriving the equtions, we ssume tht the orders of differentition nd summtion my be freely interchnged. From Eq. 5, we obtin 2 E -4r 2 E S E (z) exp(-2ri S x) x =- oo nd (7) 2 E 00 2 X E (z) exp(-2tri Sx) z =-oo where the primes denote differentition with respect to z. When we put these into Eq. 2, nd use Eq. 6 to rerrnge the order of summtion, we hve 2 O E =- - o l X Ej exp(-2i S x) = 0 (8) Eqution 8 is n identity, nd the Fourier series on the left-hnd side cn be equted to 6

13 zero for ll vlues of x. It follows tht ech seprte Fourier coefficient must vnish; nd thus we hve E + 4 (C -X E = 42 f(j-) Ej (9) where C 2 1 S 2- we. This is the eqution we require. When we consider plne horizontlly strtified ionosphere, A(x, z) is everywhere zero, nd the right-hnd side of Eq. 9 vnishes; the left-hnd side, equted to zero, cn be recognized s the wve eqution for strtified ionosphere, nd for ech ngle of incidence the corresponding E cn be determined independently. The right-hnd side of Eq. 9 thus represents coupling between the different ngles of propgtion; ech Fourier component fj(z) of the irregulrities produces coupling between ngles given by the ordinry diffrction grting lws. A Boundry Vlue Problem With n infinite plne wve incident on the ionosphere from below, the boundry conditions for Eq. 9 re: (i) Only upgoing wves high in the ionosphere, s there is no wve incident from bove. (ii) The upgoing wve below the ionosphere is just the incident wve; the mplitudes nd phses of the downgoing wves re to be determined. (iii) E(x, z) is periodic in the x direction with period d; this hs lredy been introduced in the nlysis nd is not required gin. Boundry conditions (i) nd (ii) re not esy to hndle becuse they specify conditions on the electric wve-field which re to be pplied bove nd below the ionosphere simultneously. This proves to be one of the most difficult spects of the problem

14 I _ 4. DIFFRACTION BY A THIN LAYER OF IRREGULARITIES Eqution 9 of Section 3 governs the reflection of rdio wves from two-dimensionl irregulr ionosphere. In reference 10, more generl form of this eqution is solved for the cse in which the scttering effect of the irregulrities is wek by using solutions of the differentil wve eqution for strtified ionosphere s strting point. This suggests tht these strtified solutions might be used to dvntge here, even though the scttering my not be wek. To introduce this pproch, we consider in this section the scttering problem when strong irregulrities re confined to discrete level, z =, in the ionosphere. problem is not purely cdemic, s the method of solution cn be used in conjunction (This with ry theory pproximtions to hndle the reflection level in the ionosphere, s discussed in Section 9.) At ll heights other thn z =, the ionosphere will be plnestrtified, nd the solutions for the strtified cse will pply. The problem cn be solved by mtching these solutions t the level z =. It should be emphsized tht this thin lyer of irregulrities is not t ll like the diffrcting screen tht is considered, for exmple, by Rtcliffe (1) - point tht is discussed t some length in Section 6; this is becuse the thin lyer of irregulrities considered here will give considerble bck sctter, wheres the diffrcting screen of ry theory is supposed to be thicker thn one wvelength, so tht there is no bck sctter. At ech ngle of incidence 0 there will be two independent solutions of the secondorder differentil wve eqution for the strtified ionosphere, which we denote by A nd B; both of these re required. The ctul solutions we use re to be mtched t the level z =, nd must be chosen to stisfy the boundry conditions bove nd below the ionosphere. In order tht these boundry conditions my be introduced nturlly into the equtions, let us choose A to be the solution of the strtified wve eqution for plne wve of unit mplitude incident on the ionosphere from below, nd B for unit downgoing wve below the ionosphere. Then A gives no wve incident on the ionosphere from bove nd will stisfy the upper boundry condition; B gives no wve incident on the ionosphere from below nd cn be used to represent the downgoing scttered wve. Now we cn write the required solution of the differentil wve eqution, Eq. 9, in the form: E (z) =X A for z > l (10) E (z) = B + L A for z < where nd,u re constnts to be determined by mtching solutions t z =. L re determined by the incident wve; in prticulr, for plne wve incident normlly: The L = 1 if = 0 = 0 otherwise (11) This is illustrted by Fig. 2. 8

15 LI _ When mtching the solutions t z =, between the derivtives of A nd B: it is helpful to use the Wronskin reltion A B' - B =W (constnt) (12) The definitions of A nd B given bove determine W, which my be evluted in free spce below the ionosphere to give W = 4TiC (13) (For the evnescent terms when S> 1, C defined by (1-S2)2 is imginry; it is con venient to choose C to be negtive complex. Tken together with the omitted time fctor exp(iwt), exp(-2ric z) will then represent n upgoing wve whether C is rel or imginry, nd seprte nlysis for terms with S > 1 is not necessry.) The thin lyer of irregulrities my be introduced into the equtions by dopting delt functions for the f(z), nd these re defined by the limiting process of Fig. 3. _- AI z I I XoA o *XA XIA I / X 2 A 2 o +b \ ib b b ' B-I / FIBzl I11 B A INCIDENT. -R -+A REFLECTED I fj (z) fj(z Fig. 2. Boundry conditions for diffrc- Fig. 3. The function f.(z) defining thin tion by thin lyer of irregu- in the lrities, illustrted for plne lyer of irregulrities; in the normlly incident wve. limit b - 0, ech fj(z) becomes delt function. We suppose tht the lyer becomes thin (so tht its verticl extent is smll compred with the wvelength) but its totl electron content is mintined constnt. By pplying Mxwell's equtions t the two boundries z = nd z = + b, it is esy to show tht when b is smll: (i) E(x, z) is continuous right through the lyer. (ii) E' (x, z) - which is proportionl to the x component of the mgnetic wvefield - hs discontinuity of 4rr 2 s(x) E(x, ), where 9

16 +b rcc jx 1(x) = A(x, z) dz = Qj exp 2rri (14) j=-oc I d (14) Using (i), we mtch the solutions for ll x t z = to obtin: X A() = 11B() + L A() for ll. (15) Similrly, we obtin from (ii): X A'() = B () + LA' () + 4 X A() (16) (j-) j By using the Wronskin reltion, Eq. 12, the unknown my be eliminted between Eq. 15 nd Eq. 16: -ri A X (j-)[jbj + LjAj] = 0 t z = (17) jc -z To solve this eqution, we first rewrite the limits of the summtion -N ~ j ~ N, s discussed in Section 3. Then solution of (2N+1) liner simultneous equtions in the F' gives the different components of the scttered wve below the ionosphere, s required. (In prctice, N might be incresed until sufficient ccurcy ws shown by the convergence of the.) In the specil cse in which dj is smll nd there is plne normlly incident wve, the solution of Eq. 17 is trivil: - I (-) Ao()A() (18) This is in greement with the wek sctter integrl of reference 10. We now go on to consider more relistic problem. 10

17 5. COUPLED EQUATIONS We hve seen tht solutions of the differentil wve eqution for horizontlly strtified ionosphere re useful when strong irregulrities re confined to single ionospheric lyer. Now we use similr pproch for the cse in which scttering is produced by irregulrities spred over considerble verticl rnge. The equtions of this section re derived with view to hndling wek irregulrities producing strong scttering grdully fter propgtion through considerble depth of ionosphere. It so hppens tht with the electric polriztion the coupled equtions re still vlid when the irregulrities themselves re strong, nd for tht reson no pproximtion ppers in the nlysis. We gin use A nd B to denote two independent solutions t ech ngle for the strtified ionosphere. For the ske of generlity, no restriction upon the choice of A nd B need be mde t this stge. For the solution of the trnsformed wve eqution Eq. (9), we write E = XA + B (19) Here, k nd re no longer constnts, but vry with height. To complete the definition of k nd present, we write in such wy tht they re constnt when no irregulrities re E' = A' + B' (20) The differentil equtions stisfied by X nd L re obtined by using Eq. 19 nd Eq. 20 with Eq. 9, nd remembering tht A nd B re themselves solutions of Eq. 9 when the right-hnd side is zero nd there re no irregulrities. The equtions obtined my be solved for X' nd p' by using the Wronskin reltion, Eq. 12, to give 42 o0 W B f(j-) (AjX + Bjj) j-0o = 2 cc W A f(j-) (Ajhj+ Bj j) J=-O (21) nd this is the coupled form of the wve eqution. The right-hnd sides of Eq. 21 represent coupling produced by the irregulrities between the different nd. When the irregulrities re wek, the derivtives of k nd Pu re smll; nd t levels where the ionosphere is strtified, k nd re constnt - help in numericl work with confined lyers of irregulrities. Unfortuntely, the X nd re not in one sense slowly vrying, for the products B Aj, BBj, AA j, nd A Bj chnge sign with the wve oscilltions s z is vried, but this does not relly mtter. Ner the reflection levels in the ionosphere, the wvelength is lrge; low down, where ry theory is vlid, the rpid oscilltions cn be ignored (see Section 6). 11 I ---

18 When the scttered wve is wek, the solution of Eq. 21 by integrtion is trivil, nd gin shows greement with the work of reference 10. An nlytic solution cn lso be written for model in which ll of the fj vnish except one. (This is n rtificil model, for we re tking sinusoidl vrition of X nd Z together to give A(x, z)= f.(z) exp(2rri j-).) For plne incident wve, the nd for ech successive re given in terms of successive integrls; this is becuse the rtificil model gives mechnism for scttering of the energy through the ngles of propgtion in one direction only, nd the issue is not confused by loss of energy in the min wve. (The sinusoidl vrition of Z llows this to be consistent with conservtion of energy.) For model with two or more fj(z) present, the solution is not so esy, nd pproximtions (Section 6) or numericl methods (Section 7) re necessry. 12 _

19 6. RAY THEORY In this section, we show tht the coupled wve equtions of Section 5 cn be pplied to model ionosphere for which ry theory tretment is vlid, nd greement with ry theory is then indicted. A simple numericl method enbles us to tke into ccount the devitions in the ry pth, which re neglected by the conventionl tretment. For vlid ry theory, we must consider slowly vrying ionosphere in which reflection cn be ignored, so tht the refrctive index never pproches zero. Further, it is necessry tht the verticl extent of the irregulrities should everywhere be lrger thn the wvelength, so tht bck sctter cn be ignored. This is the pproch tht should, nd does, led to the diffrction screen considered by Rtcliffe (1). It is convenient to dopt the definitions of A nd B used in Section 4, so tht Eq. 13 pplies. Since there is to be no reflection (we my imgine tht we re studying highfrequency trnsmission problem - perhps stellr scintilltions), A will be simply n upgoing wve of unit mplitude penetrting the ionosphere, nd B sponding downgoing wve, ech given by the usul W. K. B. pproximtion; will be the corre B exp 2rri q dz 21/ z (21) where q2 = C - X(). When the irregulrities hve lrge verticl extent, the products B B nd A Aj on the right-hnd side of Eq. 21 cn be ignored, for the rpid oscilltions of these terms prevent coherent contribution to the chnging X nd ; physiclly, this mens tht we re ignoring bck sctter. Eqution 21 becomes X' =-B C f(. A.X. C j j=-oo f(j- ) J 0o (22) Bj f(j- ) jllj j=-oc Upgoing nd downgoing propgtion hve seprted in Eq. 22, nd ech cn be considered seprtely. This is very convenient; it mens tht the difficulty of the boundry conditions (Sec. 3) hs been removed. eqution in X. ech X t z = 0. For instnce, consider the upgoing propgtion Below the ionosphere the incident wve is known, nd this specifies Now we cn integrte step by step upwrds, determining the X t ech successive height, until we emerge bove the ionosphere with the trnsmitted solution. The troublesome ' known only bove the ionosphere, re not involved. To proceed, we now consider the effect of screen of irregulrities - sufficient thickness to prevent bck sctter - gin with plced in free spce. With no bckground 13 _. II_

20 strtified ionosphere, A nd B become A = exp(-2wi C z) B = exp(2tri C Z) (23) Before we obtin greement with ry theory, yet nother pproximtion is necessry. We suppose tht ll pprecible energy is restricted to smll rnge of ngles, nd tke ll C to be pproximtely one. (This lso voids the prtil reflection tht would otherwise be inevitble t very oblique ngles. ) Then the eqution in X becomes 00 ), = i f(j-). (24) From the viewpoint of ry theory we need to know the phse devition tht the thick From the viewpoint of ry theory we need to know the phse devition tht the thick screen would impose upon normlly incident plne wve. This is obtined by integrting the phse chnge long lines of constnt x, so tht devitions of the phse pth from the norml re ignored. is everywhere smll, this is given by Provided tht the devition of the refrctive index from unity (= ir f A(x, z) dz (25) If there is collisionl dmping, A(x) will be complex, nd the lyer will impress devitions of both phse nd mplitude upon the incident wve. With no dmping, A will be rel, nd the lyer will ct s phse diffrcting screen. For sinusoidl vrition of electron density with no dmping, we cn write in terms of Bessel functions, solution of Eq. 24 tht grees with Rtcliffe' s phse screen (1). More generlly, it is possible to solve Eq. 24 by mens of simple Fourier trnsform (the reverse of the nlysis of Sec. 3); the X re given by the Fourier trnsform of exp(ia4q(x)), nd this grees with the formul for Frunhofer diffrction by screen. If we discrd the C -l pproximtion, the equtions re not so esily solved, even without the strtified ionosphere, except by numericl integrtion. But Eq. 22 does give simple method of extending the ry theory tretments to tke ccount of trcing the devitions in the ry pths, for the pproximtion C 1 is exctly equivlent to ignoring ny devition of the ry pth from the verticl. Ry trcing in two-dimensionl irregulr medium is not prticulrly esy, nd so Eq. 22 might prove useful. 14 I I

21 7. THE REFLECTION COEFFICIENT MATRIX We hve lredy mentioned tht the boundry conditions of the scttering problem re not esily hndled numericlly, unless we cn ignore reflection nd bck sctter s in Section 6. It is with this spect of the problem tht we re concerned here. Suppose tht we ttempt to solve the coupled wve eqution, Eq 21, for given model by numericl step-by-step integrtion. For simplicity, A nd B my once gin be given the useful definitions of Section 4. We know the incident wve below the ionosphere, nd hence the X; the u below the ionosphere re unknown nd re to be determined. Above the ionosphere (or sufficiently high in n overdense ionosphere), we know tht the p must ll be zero, for there is no wve incident from bove, but the X re unknown. Now suppose tht we strt step-by-step integrtion with some rbitrry vlues chosen for the X bove the ionosphere, replcing the infinite summtion limits in 21 by -N j N. When we hve integrted downwrds we shll hve solution of Eq. 21, but this is unlikely to mtch the incident plne wve specifying the lower boundry condition. In generl, the solution we hve obtined will correspond to (2N+1) plne wves incident in the (2N+1) different directions 0. To obtin the prticulr solution we wnt, it is necessry to perform (2N+1) seprte downwrd integrtions, ech strting with different rtios between the k bove the ionosphere. Then, since the differentil equtions re liner nd homogeneous, we cn combine the (2N+1) independent solutions to mtch the specified incident wve. The sitution is nlogous to the first method developed by Budden (12) to obtin reflection coefficients for horizontlly strtified but nisotropic ionosphere. The two mgneto-ionic components of tht pper require two seprte integrtions; here we hve (2N+ 1) components. Unfortuntely, if we ttempt the (2N+1) seprte integrtions, we meet nother difficulty tht is fmilir in the numericl solution of differentil equtions of this type. As the integrtion proceeds, it hppens tht ner the reflection level in the ionosphere the independence of the (2N+l) solutions is grdully lost, nd then the combintion below the ionosphere is inccurte or impossible. Briefly, this is becuse wves t oblique ngles re reflected lower in the ionosphere thn the verticl component; between the different reflection levels, the prt of the solutions due to the reflected wves grows exponentilly in mgnitude, nd the propgting wves re swmped nd lost in rounding off errors. This swmping difficulty cn be resolved by trnsforming to Rictti type eqution with mtrix vrible, corresponding to the second method used by Budden (12). First we rewrite Eq. 21 in mtrix form: k' = -4Zr W B F(AX+BI) At = 4r 2 W AF(A k +BI) (26)

22 where X nd t re the column vectors of order (2N+1) of the Xk nd ; W, A nd B re digonl mtrices formed, respectively, from W, A nd B; F is squre stripe mtrix, of which the generl term F j = f(j-) Next we define R by writing = RX (27) Differentiting Eq. 27 with respect to z, we obtin W1' = R' X + RX' (28) Using Eqs. 26, 27, nd 28, we cn eliminte ', X', nd p. successively; nd remembering tht digonl mtrices commute, we obtin R' X = 4 2 (A+RB) W 1 F(A+BR) (29) In free spce below the ionosphere, R is mtrix vrible tht represents generlized solution of the scttering problem; for, given n incident pttern of wves described by X, the downgoing scttered wves re given by forming = RX. Now the X cn be chosen with (2N+1) degrees of freedom, so tht the definition of R, Eq. 27, is complete, nd we cn drop from Eq. 29 to obtin the differentil eqution stisfied by R. Further, using Eq. 13 in mtrix form, we hve W = 4ri C (30) where C is the digonl mtrix of the C, nd so we hve R' = rri(a+rb) C 1 F(A+BR) (31) The boundry conditions re now esily introduced, for we know tht whtever vlues the X tke, FL is zero bove the ionosphere, so R = 0 t the strt of n integrtion. Below the ionosphere, once we clculte R we cn determine the scttered wve for ny Below the ionosphere, once we clculte R we cn determine the scttered wve for ny Fig. 4. Derivtion of the reflection coefficient mtrix form of the coupled equtions by repeted ppliction of the wek sctter integrl. The irregulrities, but not the bckground ionosphere upon which they re superimposed, re divided into slbs. Ech slb is thin enough to be treted s wek scttering lyer, nd successive slbs re dded from below. incident wve X, nd in fct we now hve one single integrtion of the (2N+1) 2 vribles in R, insted of (2N+1) seprte integrtions in (2N+1) vribles. Eqution 31 hs n interesting physicl interprettion, which leds to n lterntive method of derivtion. We know tht the work of reference 10 is not vlid here, becuse 16 rl -

23 the scttered wve is not necessrily wek. But we cn imgine tht the irregulrities re divided into seprte slbs s in Fig. 4, ech superimposed upon the bckground strtified ionosphere. The wek scttering pproximtion cn be pplied to ech slb seprtely, nd the results re combined by dding successive slbs downwrds through the ionosphere. This leds to Eq. 31 if we tke very thin slbs for the ddition process. But if we cn tke slbs tht re thick compred with the wvelength, yet still hve the wek sctter pproximtion vlid, we obtin Eq. 31 smoothed of rpid oscilltions, nd then R is in every sense slowly vrying. An lterntive form of the R eqution cn be derived by dividing the bckground ionosphere, s well s the irregulrities, into the seprte slbs. But in this eqution the vribles re in no sense slowly vrying, nd even if there were no irregulrities R would still be function of height. Symmetry nd the Conservtion of Energy It is obvious tht the presence of C on the right-hnd side of Eq. 31 destroys the symmetry of the eqution, nd in fct R is not symmetric. It might be thought tht the theorem of ionospheric reciprocity, vlid when the effect of the erth's mgnetic field is ignored (13), suggests tht R should be symmetric, for there must be reciprocl reltion between Rjk nd Rkj. But cre is necessry when pplying the theorem of reciprocity to infinite plne wves; strictly, the theorem pplies only to reciprocl trnsmissions between two finite rdio erils. Creful considertion shows tht R C, -1 rther thn R itself, should be symmetric, nd if Eq. 31 is rewritten with R C s vrible, the symmetry is t once obvious. Further, if the ionospheric irregulrities re tken to be symmetric bout the line x = 0, it turns out tht R C bout its triling digonl. is symmetric lso Since this reduces the number of vribles in R from (ZN+1)2 to (N+1)2, gret sving of computtion time cn be chieved. Cre is lso necessry when pplying the principle of conservtion of energy to infinite plne wves. Agin, the cosines of the ngles re involved, just s in the wellknown prdox tht the Fresnel coefficients of reflection nd trnsmission t plne boundry do not show conservtion of energy unless the concentrtion due to diffrction of the wve norml is considered. in R to stisfy In fct, conservtion of energy requires the terms Rjk(Cj/Ck) /2 < 1 for ll rel ngles Cj, C. k (If there is no collisionl dmping, nd ll evnescent terms re ignored, conservtion of energy lso requires tht CilRi2 1 C = 1 for ll i) When using the symmetricl form of the equtions, cre is required for terms t -1 grzing incidence where C = 0, for t this point C will not exist. A convenient trick 17 -I------I-- --

24 is lwys to use nonintegrl vlues for d, so tht this troublesome point is voided; but, even so, scling difficulties my rise. The trouble is tht the A nd B re lible to lose their independence t grzing ngles, unless they re specilly modified; the difficulty cn be resolved either by considering zero C seprtely or by hndling the limiting process when one C tends to zero. It cn be shown tht the symmetric vrible (R+I) C - 1, where I is the unit mtrix, hs terms tht ll remin finite with zero C for ny physicl model of the ionosphere. 18

25 8. NUMERICAL WORK To test the method, some numericl integrtions of the R mtrix eqution developed in Section 7 were tried on the IBM 704 computer t the Computtion Center, M.I. T. These tril integrtions used very simple ionospheric model. Collisions re ignored, so Z = 0. We suppose the electron density vrible X to be given by X = Xo(z) + 2f(z) cos(z2x/d) (32) Thus the "irregulrities" re represented by sinusoidl vrition in the x direction. We notice tht fl(z) = f_l(z) = f(z), nd tht ll other fj(z) vnish. Further Eq. 32 hs symmetry bout the line x = 0, so we cn use symmetric form of the R eqution in which the mtrix vrible is symmetric bout both leding nd triling digonls. Z z - i i I 1.0 X o (z) U b f (z) Fig. 5. A simple model ionosphere, for which the clcultions of section 8 re mde. There is no collisionl dmping; X(z) represents liner plne strtified ionosphere, nd f(z) superimposed prbolic lyer of sinusoidl "irregulrities." The functions X(z) nd f(z) re drwn in Fig. 5. strtified ionosphere: We use liner model for the Xo(z) = z/z o =0 for z 0 for z < 0 This model is convenient becuse the solutions of the strtified wve eqution re Airy integrl functions; the hndling of these is comprtively strightforwrd, s the prmeter C ffects only the scle of the Airy functions. The function f(z) is given prbolic form: f(z) = b[1-(z-)2/ 2 ] = 0 for - r z + - otherwise 19

26 It is interesting to consider such lyer becuse the effect of verticl spred in the irregulrities cn be studied by vrying ; nd integrtion is necessry only in the rnge of the lyer. Prtil reflection t the shrp grdient boundries t z = ± - my obscure the issue, but if, for exmple, Gussin f(z) hd been used, much longer integrtion through the til of the lyer would be required. Returning to the strtified ionosphere, we see tht for plne wve incident from below (leding to A), the corresponding solution of the Airy integrl eqution is denoted by A i (ref. 14). Since, without dmping, the ionospheric reflection will be totl, it is possible to represent A by the rel function A i (provided tht the phses of the incident wves in the different directions 0 re suitbly defined). If we use the previous definitions of the B, we obtin complex functions in which both mplitude nd rgument vry with height. It is tempting, therefore, to discrd this definition, nd to use the second stndrd solution of the Airy eqution, denoted by B i. The Wronskin reltion is chnged, nd ll the W re rel; R becomes purely rel vrible (Eq. 29 with X dropped). Unfortuntely, this introduces into the eqution complictions tht do not occur if the previous complex definition of the B is used; for, to obtin the desired scttered Cn x /- Fig. 6. The ngulr spectrumof scttered wves for plne normlly incident wve, plotted s //z function of b, the intensity of the irregulri- RI. / ties; z o = 100, d = 6.5, = 90, nd - = 1/8. R 2 The broken lines re clculted by using the wek sctter pproximtion, nd the devi ;i-c I-,,- c +F'rr t,-c h irrr,-llr]otiie. hprnmp b LI. LLj 1I. I11,L, Ll I 1. I..,. i.- _.I.,.. 5 strong re pprent. I) Ro - R 2 0 wves from this new rel vrible R mtrix inversion is necessry. This inversion is impossible fter long integrtions becuse the swmping difficulty mentioned in 20

27 Section 7 gin limits the method, nd in fct the rel R my hve singulrities. Nonetheless, the curves plotted here were ll obtined by mens of the rel R vrible. The reson for this is tht the integrtions tke considerble mount of computer time. To work with R complex would be too slow nd expensive. The curves of Figs. 6, 7, nd 8 were limited prtly by vilble mchine time nd prtly by filure of the rel R method. Since inccurcy must come before complete filure, the right-hnd end of ech curve must be viewed with suspicion. However, the grphs do exhibit some interesting fetures. The curves re ll drwn to give the scttered wve for norml incidence (lthough Inm 0 MI The work of Fig. 6 is repeted for lyer of irregulrities plced low in the ionosphere t = 10. Notice tht the scttering is no longer enhnced by the proximity of the reflection levels, nd thus the b scle is com- pressed. b Fig R z O I -90 R 0 R R 3 -IC)R results for oblique incidence re inherent in the solutions obtined). In the curves, Ro gives the wve returned normlly for unit incident wve, R 1 gives the wve returned t 01 or 01', R 2 t 02 or 0 2' nd so on. Both phse nd mplitude re plotted s function of b, the intensity of the irregulrities. In ll of the curves, zo = 100 (100 freespce wvelengths between the bottom of the ionosphere nd reflection t norml incidence), nd d = 6.5 (6 ngles other thn 0 = 0 for propgtion in free spce). Figure 6 is drwn with - = 1/8 ( thin lyer), nd = 90 (the lyer plced high in the ionosphere). At this height, the wves t ngles 03, or more, hve been reflected, nd so R 3, R 4,..., re smll nd re not shown. A vlue of bout 5 for N proved 21 _I_ -

28 dequte. The dotted lines re forms predicted by the wek sctter integrl of reference 10. As b is incresed, it is interesting to see the wek sctter pproximtion grdully brek down. There is good greement until R 1 reches 0. 1 (bout 2 per cent of the energy of the min wve scttered into the first-order sidebnds t 01 nd 01)' then RO drops, which shows filure of the Born pproximtion. Simultneously, R 2 is produced by scttering of energy from R 1, nd the phses wnder from the constnt vlues of wek scttering. Figure 7 repets Fig. 6, except tht the irregulrities re plced low in the ionosphere t = 10. The scttering effect is weker, nd the b scle is compressed. More oblique terms re involved (physiclly, this mens tht diffrction below the irregulr lyer is no longer sufficient to prevent wide ngles of sctter), nd for lrger vlues of b it ws necessry to use N = 7 or 8 for stisfctory results. 0 :o Fig. 8. The work of Fig. 6 is repeted for thicker lyer of irregulrities, with = b imu R2\ cr 90 R z 0 9. I. Figure 8 repets Fig. 6, with the lyer gin high t = 90, but now o = 1. 0, so tht the lyer is much thicker. The integrtions took longer with this thick lyer, but the curves re not much different, possibly becuse the wvelength is sufficiently lrge t z = 90 for bck sctter to still be considerble. It ws hoped tht results might be obtined to check the comprison with ry theory in Section 6, perhps with Ad equl to 5 Or 10. Unfortuntely, very long 22

29 numericl integrtion is required to do this, especilly since the complex R must be used, nd so the correspondence with ry theory must rest on the nlytic comprison lone. It cn be checked tht the plotted results show conservtion of energy; but it hppens tht conservtion of energy is implicit in using the rel vrible X., only the sorting-out process, not the step-by-step integrtion. nd thus this checks The Airy Functions The Airy functions were clculted by first forming nd storing tble by Tylor integrtion process. Outside the rnge of this tble, symptotic pproximtions were used. The methods re described by Miller (14). Even with floting-point computer, it is necessry to scle the quntities relting to the evnescent wves; for, if b is lrge, energy my be temporrily stored in highly evnescent terms. Otherwise, the progrmming is strightforwrd I -

30 9. DISCUSSION In this finl section, we re concerned with extensions to more generl problems nd with pplictions of the theory. It seems tht results of prcticl worth will depend upon the use of vrious pproximtions. The Mgnetic Polriztion All of the work given here cn be duplicted for the mgnetic polriztion (Sec. 3) when wve is incident on the two-dimensionl model ionosphere with its electric wvefield vector directed everywhere in perpendiculr direction to the y xis. It is necessry to decide whether strong scttering is produced by wek irregulrities or by irregulrities tht re, themselves, strong. With wek irregulrities, the derivtion of the coupled equtions is lmost trivil, either by following the nlysis of Section 5 or - better - by pplying the wek sctter integrl to the successive slbs of Section 8. When, with the mgnetic polriztion, the irregulrities themselves re strong, it 2 2 is necessry to extend the work of Section 3 nd to nlyze 1/p, s well s l itself, into Fourier components. It is dvisble to work with Ex nd the y component of the mgnetic wve-field (rther thn ny derivtive of the wve-fields) s vribles, so s to void troublesome derivtives of p.. In order for the series nd other limiting processes to converge, it is necessry to hve some collisionl dmping so tht Z 0; otherwise the equtions hve singulrities t X = 1. use lrge vlue of N where X (Even so, it might be necessry to = 1. ) The need for these extensions cn be ttributed to the effect of induced spce chrges, nd when these cn be neglected (s, for instnce, when ry theory is vlid) there is no difference between the two polriztions. of l, Corresponding equtions exist in the limit d - oc for completely generl vrition the summtions of this report being replced by integrls. The effect of the erth' s mgnetic field cn be included without difficulty (except tht the electric nd mgnetic polriztions no longer exist independently). Extension to three-dimensionl problem requires the use of double summtions, R becoming four-suffix mtrix. The Use of Approximtions Owing to the difficult boundry conditions, gret del of computer rithmetic is necessry to complete even simple integrtion of the type considered in Section 8. It would therefore seem tht unless some rel dvnce is mde in the nlysis, it is necessry to pproximte. In Section 6 we sw tht the conventionl pproximtions of ry theory, where vlid, simplify the work considerbly, even when we improve on the usul tretments with terms tht llow for ry pths tht devite from the norml. On the other hnd, reference 10 shows tht solution is possible when the scttered wve is wek. In fct, it is simple to extend the ry theory pproch little further to llow certin mount

31 of bck sctter. Provided tht we cn ssume tht the bck scttered wve is wek, it is not necessry to neglect it ltogether to void the boundry vlue problem, nd so it is possible to hndle problems of sctter communictions when most of the incident wve penetrtes the ionosphere. Brekdown of the ry theory pproximtions is, of course, inevitble t the reflection levels in the ionosphere, nd with oblique components pprecible, this could be wide region. (Ry theory fils t different levels for the different 0 ; in fct, where X = C.) Now the equtions for thin sheet of irregulrities in Section 4 re vlid for region of ionosphere tht is thin compred with the wvelength. At the reflection levels, the wvelength is lrge, nd this mens tht the thin-sheet model might perhps be used to hndle quite thick slb of ionosphere, which could subsequently be mtched to the ry theory pproximtions tht re vlid lower down. This is promising pproch. The Stochstic Problem The work described in this report is only n pproch to the problem of ionospheric scttering, for we relly wnt to consider model in which the irregulrities re known only in terms of sttisticl prmeters. It hs been shown (15) tht when the wek sctter pproximtion is vlid, the ngulr power spectrum of the scttered wve cn be determined from knowledge of the utocorreltion function of the irregulrities; this is ccomplished by tking n verge over ssemblies. And other workers hve mde considerble mount of progress bsed on ry theory, which, s mentioned bove, is help in the lower prt of the ionosphere. To extend the work to tke ccount of strong scttering by stochstic type of irregulrities situted ner the reflection level is one of the next relevnt theoreticl problems under considertion. Acknowledgment I would like to thnk Dr. K. G. Budden of the Cvendish Lbortory, Cmbridge University, Englnd, for help with the initition of the work described in this report, nd the Director nd Stff of the Reserch Lbortory of Electronics, Msschusetts Institute of Technology, for mking it possible to use the IBM 704 computer of the Computtion Center, M.I.T. I m indebted to Hrkness House, New York, for the Commonwelth Fellowship tht finnced the work. 25

32 References 1. J. A. Rtcliffe, Rep. Progr. Phys. (London: Physicl Society) 19, 188 (1956). 2. J. A. Fejer, Proc. Roy. Soc. (London) A220, 455 (1953). 3. E. N. Brmley, Proc. Roy Soc. (London) A225, 515 (1954). 4. S. A. Bowhill, J. Atmos. Terr. Phys. 8, 129 (1956). 5. H. G. Booker, J. A. Rtcliffe, nd D. H. Shinn, Phil. Trns. Roy. Soc. (London) A242, 579 (1950). 6. B. H. Briggs, G. J. Phillips, nd D. H. Shinn, Proc. Phys. Soc. (London) B63, 106 (1950). 7. H. G. Booker nd W. E. Gordon, Proc. IRE 38, 401 (1950). 8. H. G. Booker, J. Atmos. Terr. Phys. 7, 343 (1955). 9. I. L. Jones, J. Atmos. Terr. Phys. 12, 68 (1958). 10. M. L. V Pittewy, Proc. Roy Soc. (London) A246, 556 (1958). 11. N. Herlofson, Arkiv Fysik, 247 (1951). 12. K. G. Budden, Proc. Roy. Soc. (London)A227, 516 (1955). 13. J. R. Crson, Proc. IRE 17, 952 (1929). 14. J. C. P. Miller, B. A. Mthemticl Tbles, Prt-Vol. B, (Cmbridge University Press, 1946). 15. M. L. V. Pittewy, Proc. Roy. Soc. (London) A254, 86 (1960). 26