ESII Call No. AL Copy No, TRANSMISSION LOSS PREDICTIONS FOR TROPOSPUERIC COMMUNICATION CIRCUITS VOLUME I

Transcription

1 S ft.? &_ = Q t «rt,p t UJ > UJ NBS i. =J I Y- II. \ ESD ACCESSION iflst ESII Call N. AL 469 Cpy N, / ~ ' /- cys. * - *-! r ^ecunical tite 9I. // TRANSMISSION LOSS PREDICTIONS FOR TROPOSPUERIC COMMUNICATION CIRCUITS ^ VOLUME I w.. LüO L. i <: I CC^Y (- ';, D.'LüINi, l.'li P.L.RICE, A.G. LONGLEY, K.A.NORTON, AND A. P. BARSIS U. S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS AD^7ff2

2 THE NATIONAL BUREAU OF STANDARDS The Natinal Bureau f Standards is a principal fcal pint in the Federal Gvernent fr assuring axiu applicatin f the physical and engineering sciences t the advanceent f technlgy in industry and cerce. Its respnsibilities include develpent and aintenance f the natinal standards f easureent, and the prvisins f eans fr aking easureents cnsistent with thse standards; deterinatin f physical cnstants and prperties f aterials; develpent f ethds fr testing aterials, echaniss, and structures, and aking such tests as ay be necessary, particularly fr gvernent agencies; cperatin in the establishent f standard practices fr incrpratin in cdes and specificatins; advisry service t gvernent agencies n scientific and technical prbles; inventin and develpent f device«t serve special needs f the Gvernent; assistance t industry, business, and cnsuers in the develpent and acceptance f cercial standards and siplified trade practice recendatins; adinistratin f prgras in cperatin with United States business grups and standards rganizatins fr the develpent f internatinal standards f practice; and aintenance f a clearinghuse fr the cllectin and disseinatin f scientific, technical, and engineering infratin. The scpe f the Bureau's activities is suggested in the fllwing listing f its fur Institutes and their rganizatinal units. Institute fr Basic Standards. Electricity. Metrlgy. Heat. Radiatin Physics. Mechanics. Applied Matheatics. Atic Physic». Physical Cheistry. Labratry Astrphysics.* Radi Standards Labratry: Radi Standards Physics; Radi Standards Engineering.** Office f Standard Reference Data. Institute fr Materials Research. Analytical Cheistry. Plyers. Metallurgy. Inrganic Materials. Reactr Radiatins. Crygenics.** Office f Standard Reference Materials. Central Radi Prpagatin Labratry.** Insphere Research and Prpagatin. Trpsphere and Spat* Telecunicatins. Radi Systes. Upper Atsphere and Space Physics. Institute fr Applied Technlgy. Textiles and Apparel Technlgy Center. Building Research. Industrial Equipent. Infratin Technlgy. Perfrance Test Develpent. Instruentatin. Transprt Systes. Office f Technical Services. Office f Weights and Measures. Office f Engineering Standards. Office f Industrial Services. * NBS Crup, Jint Institute (r Labratry Astrphysics at the University f Clrad. * * Lcated at Bulder, Clrad.

3 NATIONAL BUREAU OF STANDARDS Technical 92te 0 Issued May 7, 965 TRANSMISSION LOSS PREDICTIONS FOR TROPOSPHERIC COMMUNICATION CIRCUITS VOLUME I P. L. Rice, A. G. Lngley, K. A. Nrtn, and A. P. Barsis Central Radi Prpagatin Labratry Natinal Bureau f Standards Bulder, Clrad NBS Technical Ntes are designed t suppleent the Bureau's regular publicatins prgra. They prvide a eans fr aking available scientific data that are f transient r liited interest. Technical Ntes ay be listed r referred t in the pen literature. Fr sale by the Superintendent l Dcuents, U. S. Gvernent Printing Office Washingtn, D.C Price: $.00

4 FOREWORD A shrt histry f the develpent f the predictin ethds in this Technical Nte will perit the reader t cpare the with earlier prcedures. Se f these ethds were first reprted by Nrtn, Rice and Vgler [955J. Further develpent f frward scatter predictins and a better understanding f the refractive index structure f the atsphere led t changes reprted in an early unpublished NBS reprt and in NBS Technical Nte 5 [Rice, Lngley and Nrtn, 959]. The ethds f Technical Nte 5 served as a basis fr part f anther unpublished NBS reprt which was incrprated in Air Frce Technical Order T.O. 3Z-0- in 96. A preliinary draft f the current technical nte was subitted as a U.S. Study Grup V cntributin t the CCIR in 962. Technical Nte 0 uses the etric syste thrughut. Fr st cputatins bth a graphical ethd and frulas suitable fr a digital cputer are presented. These include siple and cprehensive frulas fr cputing diffractin ver sth earth and ver irregular terrain, as well as ethds fr estiating diffractin ver an islated runded bstacle. New epirical graphs are included fr estiating lng-ter variability fr several cliatic regins, based n data that have been ade available t NBS. Fr paths in a cntinental teperate cliate, these predictins are practically the sae as thse published in 96. The reader will find that a nuber f graphs have been siplified and that any f the calculatins are re readily adaptable t cputer prgraing. The new aterial n tie availability and service prbability in several cliatic regins shuld prve valuable fr areas ther than the U.S.A. Nte: This Technical Nte cnsists f tw vlues as indicated in the Table f Cntents. ii

5 TABLE OF CONTENTS Vlue PAGE NO.. INTRODUCTION - 2. THE CONCEPTS OF SYSTEM LOSS, TRANSMISSION LOSS, PATH ANTENNA GAIN, AND PATH ANTENNA POWER GAIN 2-2. Syste Lss and Transissin Lss 2- Z,Z Available Pwer fr the Receiving Antenna Antenna Directive Gain and Pwer Gain Plarizatin Cupling Lss and Multipath Cupling Lss Path Lss, Basic Transissin Lss, Path Antenna Gain, and Attenuatin Relative t Free Space Prpagatin Lss and Field Strength ATMOSPHERIC ABSORPTION 3-3. Absrptin by Water Vapr and Oxygen Sky-Nise Teperature Attenuatin by Rain Attenuatin in Cluds DETERMINATION OF AN EFFECTIVE EARTH'S RADIUS 4-5. TRANSMISSION LOSS PREDICTION METHODS FOR WITHIN-THE - HORIZON PATHS 5-5. Line-f-Sight Prpagatin Over a Sth r Unifrly Rugh Spherical Earth A curve-fit t terrain The terrain rughness factr,, 5-6 h 5.2 Line-f-Sight Prpagatin Over Irregular and Cluttered Terrain DETERMINATION OF ANGULAR DISTANCE FOR TRANSHORIZON PATHS 6-6. Pltting a Great Circle Path Pltting a Terrain Prfile and Deterining the Lcatin f Radi Hrizn Obstacles Calculatin f Effective Antenna Heights fr Transhrizn Paths Calculatin f the Angular Distance, DIFFRACTION OVER A SINGLE ISOLATED OBSTACLE 7-7. Single Knife Edge, N Grund Reflectins Single Knife Edge with Grund Reflectins 7-3 iii

6 PAGE NO. 7.3 Islated Runded Obstacle, N Grund Reflectins Islated Runded Obstacle with Grund Reflectins DIFFRACTION OVER SMOOTH EARTH AND OVER IRREGULAR TERRAIN 8-8. Diffractin Attenuatin Over a Sth Earth Diffractin Over Irregular Terrain 8-J 8.2. Diffractin ver paths where d s d 8-4 st sr Fr hrizntal plarizatin Single-Hrizn Paths, Obstacle nt Islated FORWARD SCATTER 9-9. The Attenuatin Functin, F(9d) The Frequency Gain Functin, H The Scattering Efficiency Crrectin, F Expected Values f Frward Scatter Multipath Cupling Lss Cbinatin f Diffractin and Scatter Transissin Lss LONG-TERM POWER FADING 0-0. The Effective Distance, d 0-7 e 0.2 The Functins V(50, d ) and Y(p, d ) 0-8 e e 0.3 Cntinental Teperate Cliate Maritie Teperate Cliate Other Cliate Variability fr Knife-Edge Diffractin Paths 0-3. REFERENCES - 2. LIST OF SYMBOLS AND ABBREVIATIONS 2- iv

7 ANNEX I: I..2 ANNEX H: n. i II. 2 II. 3 H.4 H. 5 II. 6 II. 7 II. 8 ANNEX III: HI. IU.2 HI. 3 HI. 4 HI. 5 HI. 6 HI. 7 HI. 8 TABLE OF CONTENTS Vlue 2 PAGE NO. AVAILABLE DATA AND STANDARD CURVES - Available Data as a Functin f Path Length - Standard Pint-t-Pint Transissin Curves -2 BEAM ORIENTATION, POLARIZATION, AND MULTIPATH COUPLING LOSS U-l Representatin f Cplex Vectr Fields H-l Principal and Crss-Plarizatin Cpnents H-4 Unit Cplex Plarizatin Vectrs U-6 Pwer Flux Densities U-8 Plarizatin Efficiency U-0 Multipath Cupling Lss H-2 Idealized Theretical Antenna Patterns H-5 Cnclusins -23 FORMULAS, COMPUTER METHODS, AND SAMPLE CALCU- LATIONS HI- Line-f-Sight IH-2 Diffractin Over a Single Islated Obstacle HI-5 Diffractin Over a Single Islated Obstacle with Grund Reflectins HI-7 Paraeters K and b fr Sth Earth Diffractin IH-23 Frward Scatter IH-24 Transissin Lss with Antenna Beas Elevated r Directed Out f the Great Circle Plane IH-3 7 Lng-Ter Pwer Fading IH-44 HI. 7. HI. 7.2 Diurnal and seasnal variability in a cntinental teperate cliate IH-45 T ix distributins IH-50 Exaples HI-69 III. 8. Line f sight predictins HI-69 III III. 8.3 Transissin lss predictin fr a runded islated bstacle IH-73 Predicted transissin lss fr a transhrizn path... HI-77

8 PAGE NO. ANNEX IV: FORWARD SCATTER IV ' ANNEX V: PHASE INTERFERENCE FADING AND SERVICE PROBABILITY. V-l V. The Tw Cpnents f Fading V.2 The Nakagai-Rice Distributin V " 5 V.3 Nise-Liited Service V-ll V.4 Interference-Liited Service V-3 V. 5 The Jint Effect f Several Surces f Interference Present Siultaneusly V-7 V. 6 The Syste Equatin fr Nise-Liited Service V-8 V.7 The Tie Availability f Interference-Liited Service V-20 V.8 The Estiatin f Predictin Errr V-2 V.9 The Calculatin f Service Prbability Q fr a Given Tie Availability p V-23 V. 0 Optiu Use f the Radi Frequency Spectru V-29 vi

9 TRANSMISSION LOSS PREDICTIONS FOR TROPOSPHERIC COMMUNICATION CIRCUITS P. L. Rice, A. G. Lngley, K. A. Nrtn, and A. P. Bar sis. INTRODUCTION This reprt presents cprehensive ethds fr predicting cuulative distributins f transissin lss fr a wide range f radi frequencies ver any type f terrain and in several cliatic regins. Such quantitative estiates f prpagatin characteristics help t deterine hw well prpsed radi systes will eet requireents fr satisfactry service, free fr harful interference. Thus they shuld prvide an iprtant step tward re efficient use f the radi frequency spectru. The need fr cprehensive and accurate predictin ethds is clearly denstrated when easured transissin lss data fr a large nuber f radi paths are shwn as a functin f path length. In figures I. t I. 4 f annex I, lng-ter edian values f attenuatin relative t free space fr re than 750 radi paths are pltted versus distance. The extreely wide scatter f these data is due ainly t path-t-path differences in terrain prfiles and effective antenna heights. Values recrded fr a lng perid f tie ver a single path shw cparable ranges, seties exceeding 00 decibels. Such treendus path-tpath and tie variatins ust be carefully cnsidered, particularly in cases f pssible interference between c-channel r adjacent-channel systes. The detailed pint-t-pint predictin ethds described here depend n prpagatin path geetry, atspheric refractivity near the surface f the earth, and specified characteristics f antenna directivity. They have been tested against easureents in the radi frequency range 40 t 0,000 MHt (egacycles per secnd). Extensin f the ethds t higher frequencies requires estiates f attenuatin due t absrptin and scattering f radi energy by varius cnstituents f the atsphere. Predictins f lng-ter edian reference values f transissin lss are based n current radi prpagatin thery. A large saple f radi data was used t develp the epirical predictins f reginal, seasnal, and diurnal changes in lng-ter edians. Estiates f lng-ter fading relative t bserved edians are given fr several cliatic regins and perids f tie, including se regins where few bservatins are available. Predictins f transissin lss fr paths within the radi hrizn are based n geetricptics ray thery. Fr paths with a cn hrizn, Fresnel-Kirchff knife-eage diffractin thery is applied and extended t predict diffractin attenuatin ver islated runded bstacles. Fr duble hrizn paths that extend nly slightly beynd the hrizn, a dificatin f the Van der Pl-Breer ethd fr cputing field intensity in the far diffractin regin is -

10 used. Fr lnger paths, extending well beynd the radi hrizn, predictins are based n frward scatter thery. Radi data were used t estiate the efficiency f scattering at varius heights in the atsphere. Where se dubt exists as t which prpagatin echanis predinates, transissin lss is calculated by tw ethds and the results are cbined. Annex I includes a set f "standard" curves f basic transissin lss and curves shwing attenuatin belw free space fr earth space cunicatins, prepared using the ethds described in the reprt. Such curves, and the edians f data shwn n figures I. t.4, ay serve fr general qualitative analysis, but clearly d nt take accunt f particular terrain prfiles r cliatic effects that ay be encuntered ver a given path. Annex II suppleents the discussin f transissin lss and directive antenna gains given in sectin 2. This annex cntains a discussin f antenna bea rientatin, plarizatin, and ultipath cupling lss. Annex III cntains infratin required fr unusual paths, including exact frulas fr cputing line-f-sight transissin lss with grund reflectins, as well as dificatins f the frulas fr antenna beas which are elevated, r directed ut f the great circle plane. Saple calculatins and analytic expressins suitable fr use n a digital cputer are als included. Annex IV reviews trpspheric prpagatin thery with particular attentin t the echaniss f frward scatter fr atspheric turbulence, fr layers, r fr sall randly riented surfaces. References t se f the wrk in this field are included. Annex V presents a discussin f "phase interference fading" as cntrasted t "lngter pwer fading", prvides a ethd fr cputing the prbability f btaining adequate service in the presence f nise and/r interfering signals, and includes a brief suary f ways t achieve ptiu use f the radi frequency spectru. Figures are placed at the end f each sectin, and thse which are nt vertical shuld be turned cunter-clckwise. (The Ordinate labels wuld be upside dwn if the usual cnventin were fllwed.) Previus Technical Ntes in this series, nubered 95 t 03, describe trpspheric prpagatin phenena and siting prbles [Kirby, Rice, and Malney, 96], certain eterlgical phenena and their influence n trpspheric prpagatin [ Duttn, 96; Duttn and Thayer, 96], synptic radi eterlgy [ Bean, Hrn, and Riggs, 962], techniques fr easuring the refractive index f the atsphere [ McGavin, 962], deterinatin f syste paraeters [ Flran and Tary, 962], perfrance predictins fr cunicatin links [ Barsis, Nrtn, Rice, and Elder, 96], and equipent characteristics [ Barghau?en, et al, 963]. -2

11 2. THE CONCEPTS OF SYSTEM LOSS, TRANSMISSION LOSS, PATH ANTENNA GAIN, AND PATH ANTENNA POWER GAIN Definitins have been given in CCIR Recendatin 34 fr syste lss, L, trans- issin lss, L, prpagatin lss, L, basic transissin lss, L, path antenna gain. G, and path antenna pwer gain, G This sectin restates se f the definitins, in- P PP trduces a definitin f "path lss", L, illustrates the use f these ters and cncepts, and describes ethds f easureent [Nrtn, 953, 959, Wait 959]. The ntatin used here differs slightly fr that used in Recendatin 34 and in Reprt 2 [CCIR 963a. b]. Fr the frequency range cnsidered in this reprt syste lss, transissin lss, and prpa- gatin lss can be cnsidered equal with negligible errr in alst all cases, because antenna gains and antenna circuit resistances are essentially thse encuntered in free space. 2. Syste Lss and Transissin Lss The syste lss f a radi circuit cnsisting f a transitting antenna, receiving an- tenna, and the intervening prpagatin ediu is defined as the diensinless rati, Pl/P'i where p' is the radi frequency pwer input t the terinals f the transitting antenna and p" is the resultant radi frequency signal pwer available at the terinals f the receiving antenna. The syste lss is usually expressed in decibels: L = 0 lg (p' t /p a ) = P t-p a db (2.D Thrughut this reprt lgariths are t the base 0 unless therwise stated. The inclusin f grund and dielectric lsses and antenna circuit lsses in L prvides a quantity which can be directly and accurately easured. In prpagatin studies, hwever, it is cnvenient t deal with related quantities such as transissin lss and basic transissin lss which can be derived nly fr theretical estiates f radiated pwer and available pwer fr varius hypthetical situatins. In this reprt, capital letters are ften used t dente the ratis, expressed in db, dbu, r dbw, f the crrespnding quantities designated with lwer-case type. Fr instance, in (2. ), P' = 0 lg pj in dbw crrespnds t p' in watts. Transissin lss is defined as the diensinless rati r p /p, where p is the t r a t ttal pwer radiated fr the transitting antenna in a given band f radi frequencies, and p is the resultant radi frequency signal pwer which wuld be available fr an equivalent lss-free antenna. The transissin lss is usually expressed in decibels: L = 0 lg (p./p ) = P - P «L - L - L db (2.2) id. Id. s ci cr L = 0 lg i. L =0 lg i (2.3) et et er 6 er 2-

12 where / and / as defined in the next subsectin are pwer radiatin and receptin et er effeciencies fr the transitting and receiving antennas, respectively. With the frequencies and antenna heights usually cnsidered fr trpspheric cunicatin circuits, these efficiencies are nearly unity and the difference between L and L is negligible. With an- tennas a fractin f a wavelength abve grund, as they usually are at lwer frequencies, and especially when hrizntal plarizatin is used, L and h are nt negligible, but are influenced substantially by the presence f the grund and ther nearby prtins f the an- tenna envirnent. Fr transitter utput t receiver input, the fllwing sybls are used: Transitter Output Pwer Pwer Input t Antenna Ttal Radiated Pwer Available Pwer at Lss-Free Receiving Antenna Available Pwer at Actual Receiving Antenna Available Pwer at Receiver Input P P< P!r IT It shuld be nted that L and L. are cnceptually different. Since P and it IT ft P' represent the pwer bserved at the transitter and at the transitting antenna, respec- tively, L includes bth transissin line and isatch lsses. Since P' and P * t air represent available pwer at the receiving antenna and at the receiver, isatch lsses ust be accunted fr separately,since L antenna and the receiver. includes nly the transissin line lss between the 2-2

13 2. 2 Available Pwer fr the Receiving Antenna The abve definitins f syste lss and transissin lss depend upn the cncept f available pwer, the pwer that wuld be delivered t the receiving antenna lad if its ipedance were cnjugately atched t the receiving antenna ipedance. Fr a given radi frequency v in hertz, let z., z, and z represent the ipedances f the lad, the actual lssy antenna in its actual envirnent, and an equivalent lss-free antenna, respectively: z. = r, + ix, (2.4a) iv iv iv ' ' z' = r' + ix' (2.4b) V V V z = r + i x (2.4c) V V V where r and x in (2.4) represent resistance and reactance, respectively, Let p. represent the pwer delivered t the receiving ante.-na lad and write p' and p, respective] fr the available pwer at the terinals f the actual receiving antenna and at the terinals f the equivalent lss-free receiving antenna. If v' is the actual pen-circuit r..s. vltage at the antenna terinals, then,2 v r, v Iv z' + z, * When the lad ipedance cnjugately atches the antenna ipedance, s that z = z' r r. = r' and x. = -x', (2.5) shws that the pwer p. delivered t the lad is equal t Iv v iv v r r iv ^ the pwer r p' available fr the actual antenna: av v 2 Kv T? < 2-6» V Nte that the available pwer fr an antenna depends nly upn the characteristics f the antenna, its pen-circuit vltage v, and the resistance r', and is independent f the lad 2-3

14 ipedance. Cparing (2.5) and (2.6), we define a isatch lss factr p' ( r' + r, Y+f*' + x. V I.^=A_^ tv J V " ^i_ (2. 7 ) > P /w 4 r' r, such that the pwer delivered t a lad equals p' H. When the lad ipedance cnjugately atches the antenna ipedance, I has its iniu value f unity, and p = p". Fr any ther lad ipedance, sewhat less than the available pwer is delivered t the lad. The pwer available fr the equivalent lss-free antenna is 2 p.v=rr (2-8) where v is the pen circuit vltage fr the equivalent lss-free antenna. Cparing (2,6) and (2.8), it shuld be nted that the available pwer p' at the terinals f the actual lssy receiving antenna is less than the available pwer p a I p' fr a lss-free antenna at the sae lcatin as the actual antenna:. 2 p r' v I s-ii^il (2.9) erv r p',2 af r v^ v v The pen circuit vltage v' fr the actual lssy antenna will ften be the sae as the pen circuit vltage v fr the equivalent lss-free antenna, but each receiving antenna circuit ust be cnsidered individually. Siilarly, fr the transitting antenna, the rati f the ttal pwer p' delivered t the antenna at a frequency v is I ties the ttal pwer p radiated at the frequency v. '.t-»p;,/p t, (2-0) The cncept r f available pwer fr a transitter is nt a useful ne, and f fr the transetw itting antenna is best defined as the abve rati. Hwever, the agnitude f this rati can be btained by calculatin r easureent by treating the transitting antenna as a receiving antenna and then deterining i t be the rati f the available received pwers fr the equivalent lss-free and the actual antennas, respectively. General discussins f i are given by Crichlw et al [ 955] and in a reprt pre- pared under CCIR Reslutin N. [Geneva 963c]. The lss factr I was successfully 2-4

15 deterined in ne case by easuring the pwer p radiated fr a lss-free target trans- itting antenna and calculating the transissin lss between the target transitting antenna and the receiving antenna. There appears t be n way f directly easuring either I r I withut calculating se quantity such as the radiatin resistance r the transissin lss. ipedance, In the case f receptin with a unidirectinal rhbic terinated in its characteristic I culd theretically be greater than 2 [Harper, 94], since nearly half the received pwer is dissipated in the terinating ipedance and se is dissipated in the grund. Measureents were ade by Christiansen [ 947] n single and ultiple wire units and arrays f rhbics. with frequency and was typically less than 3 db. The rati f pwer lst in the terinatin t the input pwer varied Fr the frequency band v, t v it is cnvenient t define the effective lss factrs L and L as fllws: er et r (d p Idv) d» L =0 lg B er "i r> y db (2.) (d P^/d*) d* L =0 lg et *i r V (d V. p^/do d* (2.2) <dp tk /d*) df >t The liits v and v n the integrals (2.) and (2.2) are chsen t include es- / sentially all f the wanted signal dulatin side bands, but v is chsen t be sufficiently large and v sufficiently sall t exclude any appreciable harnic r ther unwanted radia- tin eanating fr the wanted signal transitting antenna. 2-5

16 2. 3 Antenna Directive Gain and Pwer Gain A transitting antenna has a directive gain g (r) in the directin f a unit vectr f if: () it radiates a ttal f p watts thrugh the surface f any large sphere with the antenna at its center, and (2) it radiates g p /(4ir) watts per steradian in the directin r. The sae antenna has a pwer gain g'(r) in the directin r if: () the pwer input t the antenna terinals is p = I p, and (2) it radiates g'p'/(4ir) watts per steradian in the directin r. The antenna pwer gain g' is saller than the directive gain g siply as a result f the lss factr t It fllws that et G t (f) = G>(?) + L et (2.3a) expressed in decibels abve the gain f an istrpic radiatr. gain G'(r) is less than the antenna directive gain G (f) by the aunt L Nte that the antenna pwer db, where the pwer radiatin efficiency /i is independent f the directin f. et The gain f an antenna is the sae whether it is used fr transitting r receiving. Fr a receiving antenna, the directive gain G (P) and pwer gain G'(f) are related by G (f) = G'(?) + L. (2.3b) r r er The reainder f this reprt will deal with directive gains, since the pwer gains ay be deterined siply by subtracting L r L. The axiu value f a directive gain G(P) is designated siply as G. As nted in Annex II, it is setie» useful t divide the directive gain int principal and crss-plarizatin cpnents. An idealized antenna in free space with a half-pwer sei-beawidth 6 expressed in radians, and with a circular bea crss-sectin, ay be assued t radiate x f its pwer istrpically thrugh an area equal t TT6 percent n the surface f a large sphere f unit radius, and t radiate (00-x) percent f its pwer istrpically thrugh the reainder f the sphere. In this case the pwer radiated in the directin f the ain bea is equal t 2 2 xp /(00it6 ) watts and the axiu gain g is, by definitin, equal t 4ITX/(00TT6 ). One ay assue a bea slid angle efficiency x = 56 percent fr parablic reflectrs with lodb 2 tapered illuinatin, and btain g= 2.24/6. The axiu free space gain G in decibels relative t an istrpic radiatr is then G = 0 lg g = lg 6 db (2. 4) 2-6

17 If aziuthal and vertical beawidths 26 and 26 are different: w z 6 6 (2.5) w z / The abve analysis is useful in cnnectin with easured antenna radiatin patterns. Fr antennas such as hrns r parablic reflectrs which have a clearly definable physical aperture, the cncept f antenna aperture efficiency is useful. Fr exaple, the free space axiu gain f a parablic dish with a 56 percent aperture efficiency and a di- aeter D is the rati f 56 percent f its area t the effective absrbing area f an is- trpic radiatr: G = 0 lg I" 0, 5b?.P /4 ]» 20 lg D + 20 lg f db (2> 6) where D and \ are in eter«and f is the radi frequency in egahertz, MHz. Equatins (2. 4) and (2. 6) are useful fr deterining the gains f actual antennas nly when their bea slid angle efficiencies r aperture efficiencies are knwn, and these can be deterined accurately nly by easureent. With a diple feed, fr instance, and 0 < D/\ < 25, experients have shwn the fllwing epirical frula t be superir t (2. 6): G = 23.3 lg D lg f db (2.7) where D is expressed in eters and f in MHz. Czzens [ 962] has published a ngraph fr deterining parablidal axiu gain as a functin f feed pattern and angular aperture. Discussins f a variety f cnlyused antennas are given in recent bks [ Jasik, 96; Thurel, 960]. Much re is knwn abut the aplitude, phase, and plarizatin respnse f available antennas in the directins f axiu radiatin r receptin than in ther directins. Mst f the theretical and develpental wrk has cncentrated n iniizing the transissin lss between antennas and n studies f the respnse f an arbitrary antenna t a standard plane wave. An increasing aunt f attentin, hwever, is being devted t axiizing the transissin lss between antennas in rder t reject unwanted signals. Fr this purpse it is iprtant t be able t specify, seties in statistical ters, the directivity, phase, and plarizatin respnse f an antenna in every directin fr which ultipath cpnents f each unwanted signal ay be expected. Appendix is devted t this subject. Fr the frequencies f interest in this reprt, antenna radiatin resistances r v at any radi frequency v hertz are usually assued independent f their envirnent, r else the iediate envirnent is cnsidered part f the antenna, as in the case f an antenna unted n an airplane r space vehicle. 2-7

18 2.4 Plarizatin Cupling Lss and Multipath Cupling Lss It is seties necessary t iniize the respnse f a receiving antenna t unwanted signals cing fr a single surce by way f different paths. This requires attentin t the aplitudes, plarizatins, and relative phases f a nuber f waves arriving fr different directins. In any theretical del, the phases f principal and crssplarizatin cpnents f each wave, as well as the relative phase respnse f the receiving antenna t each cpnent, ust be cnsidered. Cplex vltages are added at the antenna terinals t ake prper allwance fr this aplitude and phase infratin. In Annex II it is shwn hw cplex vectrs e and e ay be used t represent transitting and receiving antenna radiatin and receptin patterns which will cntain aplitude, plarizatin, and phase infratin [Kales, 95] fr a given free-space wavelength, \. A bar is used under the sybl fr a cplex vectr e = e + i e, where i = -V - and p c e, e are real vectrs which ay be assciated with principal and crss-plarized cp c pnents f a unifr elliptically plarized plane wave. Calculating the pwer transfer between tw antennas in free space, cplex plarizatin vectrs p(f) and p (-?) are deterined fr each antenna as if it were the transitter and the ther were the receiver. Each antenna ust be in the far field r radiatin field f the ther: <f) = 7/0. (-f) = 7/ g_ (2.8) e = e+i.e,e=e tie (2.9) p c r pr cr fe*l 2 = e 2 + e E, [e* 2 = e ' ' p c < r pr e cr (2.20) The sense f plarizatin f the field e is right-handed r left-handed depending n whether the axial rati f the plarizatin ellipse, a, is psitive r negative: a = e /e (2.2) x c p The plarizatin is circular if le I = le and linear if e =0, where e = e e is in the r ' p c' c p p p principal plarizatin directin defined by the unit vectr e. The available pwer p ay be written as p a = s(t) a e (-f) (. j 2 watts (2.22) 2-8

19 (7) * 7 /(2n > watts/k (2.23) a e (-f) = g r (-f) [ X 2 /(4n)] k 2 (2. 24) where s(r) is the ttal ean pwer flux density at the receiving antenna, a (-f) is the effective absrbing area f the receiving antenna in the directin -f, and fi. 6 is the plarizatin efficiency fr a transfer f energy in free space and at a single radi frequency. The crrespnding plarizatin cupling lss is L cp * -0 lg (. r 2 db (2.25) In ters f the axial ratis a and a defined by (II. 5) and (E. 7) and the acute angle 4> between principal plarizatin vectrs e and e, the plarizatin efficiency ay be written as cs il* (a a + ) + sin 4> (a + a ) P X M r 2 = P X ^ 2 2 (a x+ l)(a xr+ l) " (2.26) This is the sae as (H. 29). Annex II explains hw these definitins and relatinships are extended t the general case where antennas are nt in free space. There is a axiu transfer f pwer between tw antennas if the plarizatin ellipse f the receiving antenna has the sae sense, eccentricity, and principal plarizatin directin as the plarizatin ellipse f the incident radi wave. The receiving antenna is cpletely "blind" t the incident wave if the sense f plarizatin is ppsite, the eccentricity is the sae, and the principal plarizatin directin is rthgnal t that f the incident wave. In thery this situatin wuld result in the cplete rejectin f an unwanted signal prpagating in a directin -P. Sall values f g (-F) culd at the sae tie discriinate against unwanted signals cing fr ther directins. When re than ne plane wave is incident upn a receiving antenna fr a single surce, there ay be a "ultipath cupling lss" which includes bea rientatin, plarizatin cupling, and phase isatch lsses. A statistical average f phase incherence effects, such as that described in subsectin 9.4, is called "antenna-t-ediu cupling lss." Multipath cupling lss is the sae as the 'lss in path antenna gain, " L., defined in the next subsectin. Precise expressins fr L, ay als be derived fr the relatinships in annex II. 2-9

20 2.5 Path Lss, Basic Transissin Lss, Path Antenna Gain, and Attenuatin Relative t Free Space Observatins f transissin lss are ften nralized t values f "path lss" bysubtracting the su f the axiu free space gains f the antennas, G + G, fr the transissin lss, L. Path lss is defined as L = L - G - G db. (2.27) t r Path lss shuld nt be cnfused with basic transissin lss. Basic transissin lss, L, is the syste lss fr a situatin where the actual antennas are replaced at the sae lcatins by hypthetical antennas which are: () Istrpie, s that G (f) = 0 db and G (-f) = 0 db fr all iprtant prpaga- tin directins, f. (2) Lss-free, s that L = 0 db and L =0 db. et er (3) Free f plarizatin and ultipath cupling lss, s that L =0 db. cp Crrespnding t this sae situatin, the path antenna gain, G, is defined as the change in the transissin lss if hypthetical lss-free istrpic antennas with n ulti- path cupling lss were used at the sae lcatins as the actual antennas. The transissin lss between any tw antennas is defined by (2. 2): L = P - P db t a where P dbw is the ttal pwer radiated fr the transitting antenna and P dbw is the crrespnding available pwer fr a lss-free receiving antenna which is therwise equiva- lent t the actual receiving antenna. Replace bth antennas by lss-free istrpic antennas at the sae lcatins, with n cupling lss between the and having the sae radiatin resistances as the actual antennas, and let P, represent the resulting available pwer at the terinals f the hypthetical ab istrpic receiving antenna. Then the basic transissin lss L., the path antenna gain G i and the path antenna pwer gain G, are given by P PP \- P t - P ab =L + G p db (2-28) G p = P a- P ab =L b- L db (2-29a > G = P' - P, = L - L db (2. 29b) pp a ab T) s 2-0

21 In free space, fr instance: P a =P t + G t (f) + G r (-F)-L cp +20 g(^i r )dbw ( 2.30a) P ab = P t + 20 lg C-^_) dbw,2.30b) A special sybl, L.., is used t dente the crrespnding basic transissin lss in free space: L = 20 lg (-^-) = lg f + 20 lg r db (2. 3) where the antenna separatin r is expressed in kileters and the free space wavelength X. equals /f kileters fr a radi frequency f in egahertz. When lw gain antennas are used, as n aircraft, the frequency dependence in (2. 3) indicates that the service range fr UHF equipent can be ade equal t that in the VHF band nly by using additinal pwer in direct prprtin t the square f the frequency. Fixed pint-t-pint cunicatins links usually eply high-gain antennas at each terinal, and fr a given antenna size re gain is realized at UHF than at VHF, thus re than cpensating fr the additinal free space lss at UHF indicated in (2.3). Cparing (2. 28), (2. 29), and (2. 30), it is seen that the path antenna gain in free space, G, is G = G (f)+ G (-f) - L db (2.32) pi t r cp Fr st wanted prpagatin paths, this is well apprxiated by G + G, the su f the axiu antenna gains. Fr unwanted prpagatin paths it is ften desirable t iniize G. This can be achieved nt nly by aking G (?) and G (-f) sall, but als by using different plarizatins fr receiving and transitting antennas s as t axiize L In free space the transissin lss is L =L bf " G pf db < 2 ' 33 > The cncepts f basic transissin lss and path antenna gain are als useful fr nralizing the results f prpagatin studies fr paths which are nt in free space. Defining an "equiva- lent free-space transissin lss", L,, as L f =L bf -G p, (2.34) 2-

22 nte that G in (2. 34) is nt equal t G + G unless this is true fr the actual prpagatin path. It is ften cnvenient t investigate the "attenuatin relative t free space", A, r the basic transissin lss relative t that in free space, defined here as A S S" Sf = L - L f db ^>35 ' This definitin, with (2. 34), akes A independent f the path antenna gain, G. Where P terrain has little effect n line-f-sight prpagatin, it is seties desirable t study A rather than the transissin lss, L. Althugh G varies with tie, it is custary t suppress this variatin [Hartan, 963] and t estiate nly the quantity p where L, (50) and L (50) are lng-ter edian values f L and L. b D Multipath cupling lss, r the 'lss in path antenna gain", L, is defined as the difference between basic transissin ls* and path lss, which is equal t the su f the axiu gains f the transitting and receiving antennas inus the path antenna gain: L = L. - L = G + G - G db 2 37) gp b t r p «'. J '> The lss in path antenna gain will therefre, in general, include cpnents f bea rientatin lss and plarizatin cupling lss as well as any aperture-t-ediu cupling lss that ay result fr scattering by the trpsphere, by rugh r irregular terrain, r by terrain clutter such as vegetatin, buildings, bridges, r pwer lines. 2-2

23 2. 6 Prpagatin Lss and Field Strength This subsectin defines ters that are st useful at radi frequencies lwer than thse where trpspheric prpagatin effects are dinant. paraeter r : Repeating the definitins f r and r' used in subsectin 2.2, and intrducing the new r = antenna radiatin resistance, t. r r" = resistance cpnent f antenna input ipedance, t, r r, r = antenna radiatin resistance in free space, ft, fr where subscripts t and r refer t the transitting antenna and receiving antenna, respec- tively. Next define L = 0 lg (r'/r), L «0 lg r' /r J (2.38) et t t er r r L ft = 0 lg (r /r ft ), L fp» 0 lg (rvr (r ) (2.39) L rt = 0 lg (r t /r ft ) = L ft - L^ (2.40a) L =0 lg (r /r ) = L, - L (2.40b) rr r fr fr er [Actually, (2. ) and (2. 2) define L fit and L er while (2. 38) defines r { and r^ given r' ( and r. r' [963a] as Prpagatin lss first defined by Wait [959] is defined by the CCIR L p = L s - L ft - L fr = L - L rt. - L rr db \*.-»i; (2.4) Basic prpagatin lss is L pb = L p +G p {l - 4Z) Basic prpagatin lss in free space is the sae as the basic transissin lss in free space, L bf, defined by (2. 3). The syste lss L g defined by (2. ) is a easurable quantity, while transissin lss L, path lss L., basic transissin lss L, attenuatin relative t free space A, prpagatin lss L, and the field strength E are derived quantities, which in general require a theretical calculatin f L ^ and/r L fj_ as well as a there'tical estiate f the lss in path antenna gain L KP 2-3

24 The fllwing paragraphs explain why the cncepts f effective radiated pwer, E. R. P. and an equivalent plane wave field strength are nt recended fr reprting prpagatin data. equal t A half-wave antenna radiating a ttal f p watts prduces a free space field intensity 2 2 s =.64p/(4trr ) watts/k (2.43) t at a distance r kileters in its equatrial plane, where the directive gain is equal t its axiu value.64, r 2.5 db. The field is linearly plarized in the directin f the antenna. In general, the field intensity s at a pint r in free space and assciated with the principal plarizatin fr an antenna is 2 2 s (r) = p g (r)/(4irr ) watts/k (2.44) IT " r as explained in annex II. In (2.44), r = rr and g (r) is the principal plarizatin direc- tive gain in the directin r. A siilar relatin hlds fr the field intensity s ( r ) assciated with the crss-plarized cpnent f the field. Effective radiated pwer is assciated with a prescribed plarizatin fr a test antenna and is deterined by cparing s signal surce with s as easured using the test antenna : as calculated using a field intensity eter r standard E.R.P. = P + 0 lg(s /s ) = P + G (r,) dbw (2-45) t p t pt ' ' where r in free space is the directin twards the receiving antenna and in general is the initial directin f the st iprtant prpagatin path t the receiver. This abiguity in definitin, with the difficulties which seties arise in attepting t separate characteristics f an antenna fr thse f its envirnent, ake the effective radiated pwer E.R. P. an inferir paraeter, cpared with the ttal radiated pwer P, which can be re readily easured. The fllwing equatin, with P deterined fr (2.45), ay be used t cnvert reprted values f E.R. P. t estiates f the transitter pwer utput P when transissin line and isatch lsses L and the pwer radiatin efficiency /i are knwn: P it= P t' +L!t =P t + L et + L it dbw (2-46) The electragnetic field discussed in annex II is a cplex vectr functin in space and tie, and infratin abut aplitude, plarizatin, and phase is required t describe it. A real antenna respnds t the ttal field surrunding it, rather than t E, which crrespnds t the r..s. aplitude f the usual "equivalent" electragnetic field, defined at a single pint and fr a specified plarizatin. 2-4

25 Fr cnverting reprted values f E in dbu t estiates f P. r estiates f it the available pwer P at the input t a receiver, the fllwing relatinships ay be use- ful: P^ = E + L. + L, - G + L lgf dbw (2.47) it it ft t pb P, = E-L - L. +G -L - 20 lgf dbw (2-48) ir ir fr r gp P. = P' - L. = P - L - L. dbw (2.49) ir a ir a er ir ' ' In ters f reprted values f field strength E in dbu per kilwatt f effective radiated pwer, estiates f the syste lss, L, basic prpagatin lss L., r basic s pb transissin lss L. ay be derived fr the fllwing equatins, L = L + L r -G + G - G Jr.) + 20 lg f - E, db (2.50) S et fr p t pt ikw ' ' L. = L + G - G Jr.) + 20 lgf - E. db (2. 5) pb rt t pt B ikw * ' L= L + G - G Jr.) + 20 lgf - E. db (2.52) b rr t pt ikw ' ' prvided that estiates are available fr all f the ters in these equatins. Fr an antenna whse radiatin resistance is unaffected by the prxiity f its en- virnent, L = L =0 db, L, = L, and L, = L In ther cases, especially iprrt rr ft et fr er tant fr frequencies less than 30 MHz with antenna heights cnly used, it is ften as- sued that L = L = 3.0 db, L, = L db, and L = L db, crrespnding rt rr ft et fr er t the assuptin f shrt vertical electric diples abve a perfectly-cnducting infinite plane. At lw and very lw frequencies, L, L. L r, and L ay be very large Prpagatin n et er ft fr curves at HF and lwer frequencies ay be given in ters f L r L, s that it is nt p pb necessary t specify L, and L ' F ' et er Naturally, it is better t easure L directly than t calculate it using (2.50). It ay be seen that the careful definitin f L, L, L., r L is sipler and re direct s p than the definitin f L, L, A, r E. The equivalent free-space field strength E in dbu fr ne kilwatt f effective radiated pwer is btained by substituting P = P = E.R. P. = 30 dbw, G (r ) = G = 2.5 db, L = L f = 0 db, and L = U, in (2.45) - (2.47), where L is given by (2.3): E = lg d dbu/kw (2.53) where r in (2.3) has been replaced by d in (2.53). Thus e is illivlts per eter at ne kileter r 39.4 illivlts per eter at ne ile. In free space, the 2-5

26 "equivalent inverse distance field strength", E, is the sae as E. If the antenna radia- tin resistances r and r are equal t the free space radiatin resistances r, and t r ft r, then (2.52) prvides the fllwing relatinship between E and L with V ; i )=G t : E = lg f - ^ dbu/kw (2.54) Cnsider a shrt vertical electric diple abve a perfectly-cnducting infinite plane, with E.R.P. = 30 dbw, G =. 76 db, and L = 3.0 dh Fr (2. 45) P = dbw, since t rr t G (f ) =. 76 db. Then fr (2. 52) the equivalent inverse distance field is E T = E + L t + L = lgd dbu/kw (2.55) rt rr crrespnding t e = 300 v/ at ne kileter, r e = 86.4 v/ra at ne ile. In this situatin, the relatinship between E.^ and L, is given by (2.52) as E x lgf - L dbu/kw (2.56) The freging suggests the fllwing general expressins fr the equivalent free space field strength E and the equivalent inverse distance field E : I E = (P - L + G» - 20 lg d dbu (2. 57) t rt t E = E + L + L dbu (2. 58) I rt rr Nte that L in (2.57) is nt zer unless the radiatin resistance f the transitting antenna in its actual envirnent is equal t its free space radiatin resistance. The definitin f "attenuatin relative t free space" given by (2.35) as the basic transissin lss relative t that in free space, ay be restated as A=L.-LL.= L-L=E-E db (2. 59) Alternatively, attenuatin relative t free space, A, ight have been defined (as it se- ties is) as basic prpagatin lss relative t that in free space: A = L. - L. * A - L - L = E - E db (2.60) t pb f rt rr Fr frequencies and antenna heights where these definitins differ by as uch as 6 db, cautin shuld be used in reprting data. Fr st paths using frequencies abve 2-6

27 50 MHz, L + L is negligible, but cautin shuld again be used if the lss in path rt rr antenna gain L is nt negligible. It is then iprtant nt t cnfuse the "equivalent" free space lss L f given by (2.34) with the lss in free space given by (2.33). 2-7

28 3. ATMOSPHERIC ABSORPTION At frequencies abve 2 GHz attenuatin f radi waves due t absrptin r scattering by cnstituents f the atsphere, and by particles in the atsphere, ay seriusly affect icrwave relay links, cunicatin via satellites, and radi and radar astrny. At frequencies belw GHz the ttal radi wave absrptin by xygen and water vapr fr prpagatin paths f 000 kileters r less will nt exceed 2 decibels. Absrptin by rainfall begins t be barely nticeable at frequencies fr 2 t 3 GHz, but ay be quite appreciable at higher frequencies. Fr frequencies up t 00 GHz, and fr bth ptical and transhrizn paths, this sectin prvides estiates f the lng-ter edian attenuatin A f radi waves by xygen and water vapr, the attenuatin A due t rainfall, and the rder f agnitude f absrptin by cluds f a given water cntent. The estiates are based n wrk reprted by Artan and Grdn [ 954], Bean and Abbtt [ 957 ], Bussey [ 950 ], Crawfrd and Hgg [ 9 56 ], Gunn and East [ 9 54 ], Hathaway and Evans [ 9 59 ], Hgg and Mufrd [ I960 ], Hgg and Seplak [ 96 ], Lane and Saxtn [ 952 ], Laws and Parsns [ 943 ], Perlat and Vge [ 953 ], Straitn and Tlbert [ I960 ], Tlbert and Straitn [ 957 ], and Van Vleck [947a, b; 95). 3. Absrptin by Water Vapr and Oxygen Water vapr absrptin has a resnant peak at a frequency f GHz, and xygen absrptin peaks at a nuber f frequencies fr 53 t 66 GHz and at 20 GHz. Figure 3., derived fr a critical appraisal f the abve references, shws the differential absrptin V ' and v in decibels per kileter fr bth xygen and water vapr, as deterw ined fr standard cnditins f teperature and pressure and fr a surface value f abslute huidity equal t 0 gras per cubic eter. These values are cnsistent with thse prepared fr the Xth Plenary Assebly f the CCIR by U. S. Study Grup IV [ 963d ] except that the water vapr density is there taken t be 7.5 g/. Fr the range f abslute huidity likely t ccur in the atsphere, the water vapr absrptin in db/k is apprxiately prprtinal t the water vapr density. The ttal atspheric absrptin A decibels fr a path f length r kileters is cnly expressed in ne f tw ways, either as the integral f the differential absrptin y(r) dr: A = \ v(r) dr db (3. ) a J 0 r in ters f an absrptin cefficient T(r) expressed in reciprcal kileters: A = -0 lg exp - \ T(r) dr = \ I"(r) dr db (3.2) a L J J 0 3-

29 The arguent f the lgarith in (3. 2) is the aunt f radiwave energy that is nt absrbed in traversing the path. The ttal gaseus absrptin A ver a line-f-sight path f length r kileters is r A = \ dr [ Y (h) + v (h)) db (3.3) a J w where h is the height abve sea level at a distance r fr the lwer terinal, easured alng a ray path between terinals. Fr radar returns, the ttal absrptin is 2A db. Cnsidering xygen absrptin and water vapr absrptin separately, (3.3) ay be written A=vr+yr db (3.4) a e w ew where r and r are effective distances btained by integrating y /v and v /\ e ew w w ver the ray path. The effective distances r and r are pltted versus r and frequency fr elee ew vatin angles 6=0, 0.0, 0.02, 0.05, 0., 0.2, 0.5,, and W2 radians in figures Figure 3. 5 shws the relatinship between r and the sea level arc distance, d, fr these values f 8. A ay be estiated fr figures I. 2 t I. 26 f annex I, where attenuatin relative a t free space, A, is pltted versus f, 9, and r, ignring effects f diffractin by terrain. Fr nnptical paths, the ray fr each antenna t its hrirn akes an angle 9 t r 9 with the hrizntal at the hrizn, as illustrated in figure 6. f sectin 6. The r hrizn rays intersect at distances d and d fr the transitting and receiving terinals. The ttal absrptin A is the su f values A and A a at ar A = A + A a at ar where A = A (f, 9, d,). A = A <f, 9, d,) at a t ar a r 2 (3.5) Fr prpagatin ver a sth earth, 9 =9 =0, and A a 2 A (f, 0, d/2). Fr transt r a a hrizn paths and the frequency range GHz, figure 3. 6 shws A pltted versus distance ver a sth earth between 0 eter antenna heights. 3-2

30 3.2 Sky-Nise Teperature The nninized atsphere is a surce f radi nise, with the sae prperties as a reradiatr that it has as an absrber. The effective sky-nise teperature T ay be de- terined by integrating the gas teperature T ultiplied by the differential fractin f re- radiated pwer that is nt absrbed in passing thrugh the atsphere t the antenna: T^'KJ-j" T(r)r(r) expf-j^ r(r')dr'" dr (3.6) 0 where the absrptin cefficient T(r) in reciprcal kileters is defined by (3.2) stance, assuing Fr in- T(r) = ( h) *K fr h =s 2 k, and T(r) = 20*K fr h a 2 k, figure 3.7 shws the sky-nise teperature due t xygen and water vapr fr varius angles f elevatin and fr frequencies between 0. and 00 GHz. In estiating antenna teperatures, the antenna pattern and radiatin fr the earth's surface ust als be cnsidered. 3-3

31 3.3 Attenuatin by Rain The attenuatin f radi waves by suspended water drplets and rain ften exceeds the cbined xygen and water vapr absrptin. Water drplets in fg r rain will scatter radi waves in all directins whether the drps are sall cpared t the wavelength r cparable t the wavelength. In the latter case, raindrps trap and absrb se f the radi wave energy; accrdingly, rain attenuatin is uch re serius at illieter wavelengths than at centieter wavelengths. In practice it has been cnvenient t express rain attenuatin as a functin f the precipitatin rate, R, which depends n bth the liquid water cntent and the fall velcity f the drps, the latter in turn depending n the size f the drps. There is little evidence that rain with a knwn rate f fall has a unique drp-size distributin, and the prble f estiating the attenuatin f radi waves by the varius frs f precipitatin is quite difficult. Ttal absrptin A due t rainfall ver a path f length r can be estiated by integrating the differential rain absrptin y (r)dr alng the direct path between tw intervisible antennas, r alng hrizn rays in the case f transhrizn prpagatin: r A = \ y (r)dr decibels (3. 7) r J r Fitting an arbitrary atheatical functin epirically t theretical results given by Hathaway and Evans [ 959] and Ryde and Ryde [ 945], the rate f absrptin by rain v be expressed in ters f the rainfall rate R in illieters per hur as ay v = KR" db/k (3.8) fr frequencies abve 2 GHz. The functins K(f ) and «(f) are pltted in figures 3.8 and 3.9, where f is the radi frequency in GHz. K= [3(f G -2) 2-2(f G -2)] x 0-4 (3.9a) a = [ (f G - 2) 7 ] f (f G - 3.5) exp( f Q ) ] (3.9b) fr An exainatin f the variatin f rainfall rate with height suggests a relatin f the R/R = exp(-0.2 h 2 ) (3.0) 3-4

32 where R rs is the surface rainfall rate. Then A = y r db, (3.) r 'rs er r C 2 v = KR db/k, r = \ drexp(-0.2ah ) k (3.2) 'rs rs er J r where v i» tne surface value f the rate f absrptin by rain, and r is an "effective 'rs er rainbearins distance". Figures shw r versus r fr several values f 8 er and. The curves shwn were cputed using (3.2). A "standard" lng-ter cuulative distributin f rain absrptin is estiated, using se statistics fr Ohi analyzed by Bussey [950], wh relates the cuulative dis- tributin f instantaneus path average rainfall rates fr 25, 50, and 00-kileter paths, respectively, with the cuulative distributins fr a single rain gauge f half-hur, ne-hur, and tw-hur ean rainfall rates, recrded fr a year. abut 0 centieters. The ttal annual rainfall in Ohi is Rainfall statistics vary cnsiderably fr regin t regin, seties fr year t year, and ften with the directin f a path (with r acrss prevailing winds). Fr instance, in Nrth Aerica, east-west systes see particularly vulnerable, as they lie alng the path f frequent heavy shwers. Fr very lng paths, the cuulative distributin f instantaneus path average rain- fall rates, R, depends n hw R relatin f rainfall with distance alng the path. instantaneus path average rainfall rate R varies with elevatin abve the surface and upn the cr- Figure 3. 4 prvides estiates f the exceeded fr 0.0, 0,,, and 5 percent f the year as a functin f r and nralized t a ttal annual rainfall f 00 c. T btain A er r fr (3. ),replace R in (3.2)with R fr figure 3. 4, ultiplied by the rati f the ttal annual rainfall and 00 c. These estiates are an extraplatin f the results given by Bussey [ 950] and are intended t allw fr the average variatin f R given by (3. 0)and allwed fr in the definitin f r fall rate R with distance alng the surface, as analyzed by Bussey. with height, as, and fr the crrelatin f surface rain- 3-5

33 3.4 Attenuatin in Cluds Clud drplets are regarded here as thse water r ice particles having radii saller than 00 icrns r 0.0 c. Althugh a rigrus apprach t the prble f attenuatin by cluds ust cnsider drp-size distributin, it is re practical t speak f the water cntent f cluds rather than the drp-size distributin. Reliable easureents f bth paraeters are scarce, but it is pssible t ake reasnable estiates f the water cntent, M, f a clud fr a knwledge f the vertical extent f the clud and the gradients f pressure, teperature, and ixing rati, which is the rati f the ass f water vapr t the ass f dry air in which it is ixed. The absrptin within a clud can be written as A = K, M db c (3.3) where A is the ttal absrptin attenuatin within the clud, K is an attenuatin cefficient, values fr which are given in table 3., and M is the liquid water cntent f the clud, easured in gras per cubic eter. The aunt f precipitable water, M, in a given pressure layer can be btained by evaluating the average ixing rati in the layer, ultiplying by the pressure difference, and dividing by the gravity. Using this ethd f btaining M and the values f K fr table 3., it is pssible t get a fairly reliable estiate f the absrptin f radi energy by a clud. Several iprtant facts are denstrated by table 3.. The increase in attenuatin with increasing frequency is clearly shwn. The values change by abut an rder f agnitude fr 0 t 30 GHz. Clud attenuatin can be safely neglected belw 6 GHz. The data presented here als shw that attenuatin increases with decreasing teperature. These relatins are a reflectin f the dependence f the refractive index n bth wavelength and teperature. The different dielectric prperties f water and ice are illustrated by the difference in attenuatin. Ice cluds give attenuatins abut tw rders f agnitude saller than water cluds f the sae water cntent. TABLE 3. One-Way Attenuatin Cefficient, K, in db/k/g/ 3 Teperature ( C) Frequency, GHz, Water Clud 20 0 { L (extraplated) (extraplated) Ice X 0" X 0' X 0" -3 2, 46 X 0" 3 3 r... l X 0' 2. X 0" Clud.46 X 0' B. 9 X 0" X 0' l. 45 X 0".0 X 0" 5. 63X 0" 4 3-6

34 : SURFACE VALUES X AND Yw OF ABSORPTION?0 BY OXYGEN AND WATER VAPOR PRESSURE 760Hg TEMPERATURE 20»C WATER VAPOR DENSITY IOg/ i 0.5 It 0.2 Ui u S.i X 0.05 J * i g 0.02 CO 5 0.0! u I l_>b I FREQUENCY IN GHz Figur«3. 3-7

35 EFFECTIVE DISTANCES r e AND r ew FOR ABSORPTION BY OXYGEN AND WATER VAFOR 6^ « MO 2SOJ J RAY PATH r0 IN KILOMETERS BOO UJ *3 % %-QOI i 20 ^,-002 z 3 a (00 80 H 40 a ^ y, _.-.- / V' --- ^ K -"'"!n J ^J i / u 5» LiJ - > F REOUE ^c f i G ca 2 ) l» i 0!' 0 If 0 Hi t Y 2( 0 'A rh 2 '0 IN 2 0 Kl 2 JO M TE RS 2 *3 I 00 5 a 3 (0 i X Fl 3U re 2 5-8

36 EFFECTIVE DISTANCES r e AND r ew FOR ABSORPTION BY OXYGEN AND WATER VAPOR 200 en 80 (r UJ 60 HI *_l 40 * : t 20 in 00 g 80 t 60»- ;,; 40 u. u. LU 20 0 e 0 c 0 GcAj e.i fl 0.Q2 / / feg J^').^\ y^lxorfti' >»i»», " ~a x ^ - i ygjjg wyifjgzi _ ai. Ql Gc/ L-L.._i.. u_. r : ' r *^h q_ r - /tätf^-' _r~?&&?- 5S «ti*-"i" " ' >**»* " SKJ Tl. i i FREQUENCY IN GcAl. pf 2 ri' i ^ Tl T II II II lilol g v z «-< 80.(»0-u<: y ; in S UJ > FR EQUENCY IN Gc/«- 0 -ICX T a O.I IOO I u. yf> ; 3I-IOO )? 0 4 ) 6 ) 8 0 K It RAY PATH r 0 IN KILOMETERS Figure

37 JU EFFECTIVE DISTANCES r e AND r ew FOR ABSORPTION BY OXYGEN AND WATER VAPOR 0 O «0.5, I, ir/2 i r i 28 fl. -OS e 0.i?6 W» «M FREQUENCY IN GCA 6 i 4 do.igc/*5 [05 I? JO.I GC/S '* i02 l ^ 0 s ^0.3 E M i " ^ -T""" v it; 0 k» Q - Ä A»in t 5 L.. l«n ) 2 < 0 2 \ i i t i O32M36M S485O52 RAY PATH r 0 IN KILOMETERS 0 i I r 0 / S-0.9 *< ir/2 > ^ J "5 l-lot )l l v Hi J.l-K»( I * \ I I 3 I 2 3 t 5 r i i i RAY PATH r 0 IN KILOMETERS Figure

38 SEA LEVEL ARC DISTANCE d, IN KILOMETERS 3J > 5 I i 5 x en CO Ü a II < S3 Ü CO OJ c CO CO T> OJ r Ui - < r~ J> si CO H J> H O 3-

39 OXYGEN AND WATER VAPOR ABSORPTION IN DECIBELS g I vn w \\ \\ H > -i g 2 -z CT z C _ (/> O -5 - > 5 3) 5 8 s *> O > l u c 0,000 c -< z E z S *3 P* > CD en O g t) H z ] \ 3-2

40 SKY NOISE TEMPERATURE DUE TO RERADIATION BY OXYGEN AND WATER VAPOR c Q a' cr Q < X 8 <r UJ Q. : CO 2 5 FREQUENCY IN GHz Figure

41 RAINFALL ABSORPTION COEFFICIENT K vs FREQUENCY 7= KR? db/k, WHERE R r IS THE RAINFALL RATE IN MILLIMETERS/HOUR 0 00 FREQUENCY f G IN GHz Figure

42 X) J> z ~n J> 7 i - > r Z x > r 7) II OD cr - 3 ^ ^ X T T> ^ O 7] Z rn ^ <L xi -i Ml * * (7) xi t/> H Q <r lw X T u Xi <_ JJ O c -< 3-5

43 EFFECTIVE DISTANCE r er IN KILOMETERS S 8 S X XI s a > \ \SSSs x \ \ %\ v V - - b \\ w \ ^ V \ l\ ^ ^ v Ik \VV\\ \\\\\ \ u\ \ ft ^ l> \\ ii V i * ö * < g > O p <r "* t» CD t/) O XI 5 -^ u_ P e

44 ' I EFFECTIVE DISTANCE r er IN KILOMETERS cn S ts> & «&S en cn cr> cn c: cn c3 c5 en S\^ Iv < J> -< g r O -H C/5 8 v n \w ^ w\ KM \YA W u\ n\ iiv \ K? \w \ ~- i i, J.., O GO H J> c?> 2T O II Q rf^ Ui ^ T O r\i IE,U > z > CD CO ;c Tl n s $> ' > rl" P 5 ' _ - P - P Q OJ 00 w p jr < > ö 3-7

45 EFFECTIVE DISTANCE r er IN KILOMETERS r K 3] > K K s D J> - I tn -n cr> ct c - CTl ^ c=> OJ r * r H rr- X 00 K ~i CJ en g g e \v WV k M t ^\ w \\\ V <5 II en D CO H > Z 3 X CD CO TJ * ii h P i? g gö p i>j K) Ö (D OD 3-8

46 EFFECTIVE OISTANCE Ife, IN KILOMETERS K b «V N > e - r 8 0, ) ^ S? - ; *. 3 5 I I r «l ^ V i \ g CO 3?^ 33 J> -, </> y r t OD "* < < tiff If 5 r r <L 8090 k 8894 k.987 k.256 k.494 k ill :> 5 W \\\\\ l\\ \ i \ \ ' ' i\ > > CD CO O XI ix r - D 8 3-9

47 PATH AVERAGE RAINFALL RATE, R" r, VS EFFECTIVE RAINBEARING DISTANCE, r er (TOTAL ANNUAL RAINFALL, 00c) IUU 50 tr 3 i e 3 z?0 PERCENT OF THE YEAR THAT R, IS EXCEEDCO 0 ^k "^. 8r - - "" * ^ 5,. iufl ^' tr I Z s I K?.0 05.% isj 02 ni EFFECTIVE RAINBEARING DISTANCE, r^,in KILOMETERS ' Figurt

48 4. DETERMINATION OF AN EFFECTIVE EARTH'S RADIUS The bending f a radi ray as it passes thrugh the atsphere is largely deterined by the gradient f the refractive index near the earth's surface. In rder t represent radi rays as straight lines, at least within the first kileter abve the surface, an "effective earth's radius" is defined as a functin f the refractivity gradient, AN, r f the surface refractivity value N, N = (n - ) X 0 (4.) s s where n is the atspheric refractive index at the surface f the earth, s In the United States the fllwing epirical relatinship has been established between the ean N and the ean refractivity gradient AN in the first kileter abve the surface: AN/k = exp( N ) (4.2) Siilar values have been established in West Gerany and in the United Kingd, [CCIR 963 e] In this paper values f N are used t characterize average atspheric cnditins during perids f iniu field strength. In the nrtnern teperate zne, field strengths and values f N reach iniu values during winter afternns. Thrughut the wrld, reginal changes in expected values f transissin lss depend n iniu nthly ean values f a related quantity, N, which represents surface refractivity reduced t sea level: N = N exp( h ) (4.3) s s where h is the elevatin f the surface abve ean sea level, in kileters, and the s refractivity N is read fr the ap shwn in figure 4. and taken fr Bean, Hrn, and Ozanich [ 960]. Mst f the refractin f a radi ray takes place at lw elevatins, s it is apprpriate t deterine N and h fr lcatins crrespnding t the lwest elevatin f the radi rays s st iprtant t the geetry f a prpagatin path. As a practical atter fr within-the- hrizn paths, h is defined as the grund elevatin iediately belw the lwer antenna terinal, and N is deterined at the sae lcatin. Fr beynd-the-hrizn paths, h and N are deterined at the radi hrizns alng the great circle path between the antennas, and N is the average f the tw values calculated fr (4.3). An exceptin t this latter s rule ccurs if an antenna is re than 50 eters belw its radi hrizn; in such a case, h and N shuld be deterined at the antenna lcatin, s The effective earth's radius, a, is given by the fllwing expressin: a=a [ exp( N )]"' (4.4) 4-

49 where a is the actual radius f the earth, and is taken t be 6370 kileters. Figure.2 shws the effective earth's radius, a, pltted versus N. The ttal bending f a radi ray which is elevated re than radians (45*) abve the hrizn and which passes all the way thrugh the earth's atsphere is less than half a illiradian. Fr studies f earthsatellite cunicatin ray bending is iprtant at lw angles. At higher angles it ay ften be neglected and the actual earth's radius is then used in geetrical calculatins. Large values f AN and N are ften assciated with atspheric ducting, which is usually iprtant fr part f the tie ver st paths, especially in aritie cliates. The average ccurrence f strng layer reflectins, superrefractin, ducting, and ther fcusing and defcusing effects f the atsphere is taken int accunt in the epirical tie variability functins t be discussed in sectin 0. Additinal aterial n ducting will be fund in papers by Andersn and Gssard [ 953a, b], Bean [ 959], Bker [ 946], Bker and Walkinshaw [ 946], Clew and Bruce-Claytn [ 963], Duttn [ 96], Fk, Vainshtein, and Belkina [ 958], Friend [ 945], Hay and Unwin ( 952], Ikegai [ 959], Kitchen, Jy, and Richards [ 958], Nura and Takaku [ 955], One and Nishikri [ 957], Pekeris [ 947], Sch'uneann [ 957], and Unwin [ 953]. 4-2

50 <P -J <7> <-» g g 8 6«z58gfi833 c 8 * B 8 s * * 8 8 S I S S SI g s» a SsSS S S 8 s 8 s s 8 <" ö«> *s * 4-3

51 600 EFFECTIVE EARTH'S RADIUS, Q, VERSUS SURFACE REFRACTIVITY, N s 0800 CO Q: UÜ i- UJ CO 5 <t cr c x \- cr < LU > LÜ LL LL Figure

52 5. TRANSMISSION LOSS PREDICTION METHODS FOR WITHIN-THE-HORIZON PATHS Grund wave prpagatin ver a sth spherical earth f unifr grund cnductivity and dielectric cnstant, and with a hgeneus atsphere, has been studied extensively. Se f the results were presented in CCIR Atlases [ 955, 959]. Recent wrk by Bachynski [ 959, I960, 963], Wait [ 963], Furutsu [ 963], and thers cnsiders irregularities f electrical grund cnstants and f terrain. A distinctin is ade here between the rughness f terrain which deterines the prprtin between specular and diffuse reflectin f radi waves, and large scale irregularities whse average effect is accunted fr by fitting a straight line r curve t the terrain. A cprehensive discussin f the scattering f electragnetic waves fr rugh surfaces is given in a recent bk by Beckann and Spizzichin [ 963]. Studies f reflectin fr irregular terrain as well as absrptin, diffractin, and scattering by trees, hills, and an-ade bstacles have been ade by Beckann [ 957], Bit [ 957a, b], Kalinin [ 957, 958], Kühn [958], McGavin and Malney [ 9 59], McPetrie and Frd [ 946], McPetrie and Saxtn ( 942], Saxtn and Lane [ 955], Sherwd and Ginztn [ 955], and any ther wrkers. Exaples f studies f reflectin fr an cean surface ay be fund in papers by Beard, Katz and Spetner [ 956], and Beard [96]. If tw antennas are intervisible ver the effective earth defined in sectin 4, geetric ptics is rdinarily used t estiate the attenuatin A relative t free space, prvided that the great circle path terrain visible t bth antennas will supprt a substantial aunt f reflectin and that it is reasnable t fit a straight line r a cnvex curve f radius a t this prtin f the terrain. Reflectins fr hillsides r bstacles ff the great circle path between tw antennas seties cntribute a significant aunt t the received signal. Discriinatin against such ff-path reflectins ay reduce ultipath fading prbles, r in ther cases antenna beas ay be directed away fr the great circle path in rder t increase the signal level by taking advantage f ff-path reflectin r knife-edge diffractin. Fr shrt perids f tie, ver se paths, atspheric fcusing r defcusing will lead t sewhat saller r uch greater valueb f line-f-sight attenuatin than the lng-ter edian values predicted fr the average path by the ethds f this sectin. 5. Line-f-Sight Prpagatin Over a Sth r Unifrly Rugh Spherical Earth The siplest ray ptics frulas assue that the field at a receiving antenna is ade up t tw cpnents, ne assciated with a direct ray having a path length r, and the ther assciated with a ray reflected fr a pint n the surface, with equal grazing angles *\>. The reflected ray has a path length r + r. The field arriving at the receiver via the direct ray differs fr the field arriving via the reflected ray by a phase angle which is a functin f the 5-

53 path length difference, Ar = r + r - r, illustrated in figure 5.. The reflected ray fie'd is als dified by an effective reflectin cefficient R and assciated phase lag (IT - c), which depend n the cnductivity, perittivity, rughness, and curvature f the reflecting surface, as well as upn the rati f the prducts f antenna gain patterns in the directins f direct and reflected ray paths. Let g and g represent the directive gain fr each antenna in the directin f i z the ther, assuing antenna plarizatins t be atched. Siilar factrs g and g are defined fr each antenna in the directin f the pint f grund reflectin. reflectin cefficient R is then e / g g \ -0.6 <7 sin \\\ The effective where the divergence factr surface, and ay be apprxiated as D allws fr the divergence f energy reflected fr a curved -[ 2d d l + ' a d tan + (5.2) An expressin fr the divergence factr, D, based n geetric ptics was derived by Riblet and Barker, [948]. The ter R represents the agnitude f the theretical cefficient, R exp[-i(ir -c) ], fr reflectin f a plane wave fr a sth plane surface f a given cnductivity and dielectric cnstant. In st cases c ay be set equal t zer and R is very nearly unity. A ntable exceptin fr vertical plarizatin ver sea water is discussed in annex III. Values f R and cvsi i are shwn n figures III. t III. 8 fr bth vertical and hrizntal plarizatin ver gd, average, and pr grund, and ver sea water. The grazing angle 4 and the ther geetrical paraeters d, d, d, and a are shwn n figure 5.. The terrain rughness factr, a, defined in sectin 5..2, and the radi wave length, \, are expressed in the sae units. The expnent (a sin +)/* is Rayleigh's criterin f rughness. If the prduct DRexp(-0.6 <r, sin^/x) is less than si sinij/, and is less than 0.5, grund reflectin ay be assued t be entirely diffuse and R is then expressed as R r _fr f Ii _ 8in+ ' e L g. g > J where terrain factrs D, R and a are Ignred, "''he factr g g / g" g in (5.3) akes h ri n 0 02 R apprach zer when narrw-bea antennas are used t discriinate against grund reflectins. (5.3) 5-2

54 Fr a single grund reflectin, the attenuatin relative t free space ay be b- tained fr the general frula A-G p - -0 lg {g i g z l + R e 2-2 R e c 8 (i^-c)] } db (5.4) where the path antenna gain G ay nt be equal t the su f the axiu antenna gains. At frequencies abve GHz an estiate f lsses due t atspheric absrptin as shwn in (3.4) shuld be added t A as cputed by (5.4) r (5.5). Over a sth perfectly-cnducting surface, R = and c = 0, Asuing als that free space antenna gains are realized, s that G = 0 lg(g g ), the attenuatin relative t free space is A = lg sin 2 (IT Ar/\) db (5. 5) Exact frulas fr cputing Ar are given in annex III. The apprpriate apprxiatins given in (5.9) t (5.3) suffice fr st practical applicatins. If Ar is less than 0. 2X, (5.4) ay underestiate the attenuatin and ne f the ethds f sectin 5.2 shuld be used. Sectin 5.. shws exact frulas fr antenna heights h and h' abve a plane earth, r abve a plane tangent t the earth at the pint f reflectin. The grazing angle + is then defined by tan+ = h /d = h^/d 2 (5.6) where heights and distances are in kileters and d. and d are distances fr each antenna t the pint f specular reflectin: d +d 2 =d, dj =d(l +h 2 /h' )", d 2 = d(l +h' /h 2 )" (5.7) The distances d and d ay be apprxiated fr a spherical earth by substituting antenna heights h and h? abve the earth fr the heights h.' and h' in (5.7). Then these heights ay be calculated as 2,,,.,,.,2 h \ = h l - d /(2a), h 2 = h 2 - d 2 /(2a) (5. 8) lr an earth f effective radius a, and substituted in (5. 7) t btain iprved estiates f d l "2 btained. 5-3

55 The path length difference between direct and grund reflected rays is Ar = JdS^+h!,) 2 - Jd 2 + (h; -h' 2 ) 2 2h;h^/d (5.9) where the apprxiatin in (5.9) is valid fr sall grazing angles. Referring t (5.5) the greatest distance, d, fr which A is zer, (assuing that R = and that free space gains are realized) ccurs when Ar = X./6. Fr (5.9) Ar a 2h'h'/d; therefre: d = 2 h'. h'/x. (5. 0a) I c This equatin ay be slved graphically, r by iteratin, chsing a series f values fr d, slving (5.8) fr h, h', and testing the equality in (5.0a). Fr the special case f equal antenna heights ver a spherical earth f radius a, the distance d ay be btained as fllws: Ar = \/6 = 2 r..2 h - d /(8a) a z 2h 2 /d - hd /(2a) +d 3 /(32a 2 ) (5.0b) ' where d = d = d/2, h = h, = h, and h' = h - d 2 /(8a) L I C O the angle if Fr this special case where h p h ver a sth spherical earth f radius a, ay be defined as tan + = 2h/d - d/(4a) (5.a) and Ar = d(sec+ - i) ) = dl dt. 'i' + tan +- (5.b) Let 0, represent the angle f elevatin f the direct ray r relative t the hrizntal at the lwer antenna, h, assue that h < < h?, h < < 9a + / 2, and that the grazing angle, IJJ, is sall; then, ver a spherical earth f effective radius a, L N L2 Ar ^ 2h x sin^ S hj ^ 0 h + 4 hj/^a) + 9 h "! (5.2) whether 9 is psitive r negative. Fr 9 = 0, d =5 2h /(34J). 5-4

56 Tw very useful apprxiatins fr Ar are Ar = 2 <\> dd /d S 2h sin + kileters (5.3) and the crrespnding expressins fr the path length difference in electrical radians and in electrical degrees are 2irAr/\ = 4.97 f hjh' /d = 4.97 f + d.^/d a 42 f h sin+ radians (5. 4a) 360Ar/\ = f h' h' /d = f + 2 d^/d S 2402 f h^ sin+ degrees (5. 4b) where f is the radi frequency in MHz, and all heights and distances are in kileters. The last apprxiatin in (5.3) shuld be used nly if h, is sall and less than h.,/20, 2 as it invlves neglecting d /(2a) relative t h in (5.8) and assuing that d S d. As nted fllwing (5. 5), ray ptics frulas are liited t grazing angles such that Ar > 0.06 \. With this criterin, and assuing R =, the attenuatin A is 5 db fr the crrespnding iniu grazing angle where antennas are barely intervisible. I J et «V 0.03 \ d/(d,d,) radians k c, A cparisn with the CCIR Atlas f sth-earth diffractin curves shws that the attenuatin relative t free space varies fr 0 t 20 decibels fr a zer angular distance (8 = 0, + = 0) except fr extreely lw antennas. Figure 5. la shws hw rays will bend abve an earth f actual radius a = 6370 kil eters, while figure 5. lb shws the sae rays drawn as straight lines abve an earth f ef- fective radius a. Antenna heights abve sea level, h and h, are usually slightly greater than the effective antenna heights h' and h', defined in 5... This difference arises fr tw circustances: the sth curve ay be a curve-fit t the terrain instead f representing sea level, and straight rays abve an effective earth verestiate the ray bending at high ele- vatins. This latter crrectin is insignificant unless d is large. 5.. A Curve-Fit t Terrain A sth curve is fitted t terrain visible fr bth antennas. It is used t define an- tenna heights h' and h^, as well as t deterine a single reflectin pint where the angle f incidence f a ray r is equal t the angle f reflectin f a ray r? in figure 5.. This curve is als required t btain the deviatin, v, f terrain heights used in cputing R in (5.). Experience has shwn that bth h' and h' shuld exceed 0.6 \ fr the fllwing frulas t be applicable. Fr ther predictin ethds, see subsectin

57 First, a straight line is fitted by least squares t equidistant heights h (x ) abve sea level, and x /(2a) is then subtracted t allw fr the sea level curvature /a illustrated in figure 6.4. The fllwing equatin describes a straight line h(x) fitted t 2 equidistant values f h.(x.) fr terrain between x. = x and x^ - x 2Q kileters fr the transitting antenna. The pints x and x are chsen t exclude terrain adjacent t either antenna which is nt visible fr the ther: h(x) = h + (x - x) (5.5a) 20 2 V h(i-0) H >y,, x=^l^. =_fe (5.5b) l U0 77 < X 20-V 20 Sth dified terrain values given by y(x) = h(x) - x 2 /(2a) (5. 6) will then define a curve f radius a which is extraplated t include all values f x fr x = 0 t x = d, the psitins f the antennas. The heights f the antennas abve this curve are h'j = h tg - h(0), hi, = h rs - h(d) (5. 7) If h' r h' is greater than ne kileter, a crrectin ter, Ah, defined by (6. 2) and shwn n figure 6.7 is used t reduce the value given by (5.7). Where terrain is s irregular that it cannt be reasnably well apprxiated by a single curve, <r is large and R = 0, nt because the terrain is very rugh, but because h e it is irreaular. In such a situatin,ethd 3 f sectin 5.2 ay be useful The terrain rughness factr, 9- The terrain rughness factr v in (5.) is the rt-ean-square deviatin f dified terrain elevatins, y., relative t the sth curve defined by (5.6), within the liits f the first Fresnel zne in the hrizntal reflecting plane. The utline f a first Fresnel zne ellipse is deterined by the cnditin that r ll +r 2 = r l +r 2 +V/2 where r. + r,. is the length f a ray path crrespnding t reflectin fr a pint n the edge f the Fresnel zne, and r + r is the length f the reflected ray fr which angles f incidence and reflectin are equal, Nrtn and Oberg [ 947 ] give general frulas fr deterining a first Fresnel zne ellipse in the reflecting plane. Frulas are given in annex III fr calculating distances x and x^ fr the transitter t the tw pints where the first Fresnel ellipse cuts the great circle plane. A saple calculatin f <r is given in Exaple f Annex III. h 5-6

58 5.2 Line-f-Sight Prpagatin ver Irregular and Cluttered Terrain Where ray ptics frula«described in sectin 5. are nt applicable, a satisfactry estiate f'line-f-sight transissin lss ay seties be ade by ne f the fllwing ethds:, If a slight change in the psitin f either antenna results in a situatin where ray ptics frulas ay be used, then A ay be estiated by extraplatin r interplatin. A useful set f calculatins fr 9=0 is given by Db and Pryce [ 947]. 2, Instead f a single curve fit t terrain as in 5., in se cases the ethd ay be extended t ultiple curve fits and ultiple reflectins fr these curves. 3, If terrain is s irregular it cannt be reasnably well apprxiated by a single curve, the line-f-sight knife-edge frulas f sectin 7 ay be applicable. 4, Interplatin between curves in an atlas, r standard prpagatin curves such as thse given in appendix I, ay prvide a satisfactry estiate. 5, Epirical curves drawn thrugh data apprpriate fr the prble f interest ay be useful. Fr exaple, the dashed curves f figures. -.3 shw hw values f attenuatin relative t free space vary with distance and frequency fr a large saple f recrdings f televisin signals ver rand paths. The data shwn in figures L, -.4 crrespnd t a re careful selectin f receiving lcatins and t a greater variety f terrain and cliatic cnditins. The effects f refractin, diffractin, and absrptin by trees, hills, and an-ade bstacles are ften iprtant, especially if a receiving installatin is lw r is surrunded by bstacles. Absrptin f radi energy is prbably the least iprtant f these three factrs except in cases where the nly path fr radi energy is directly thrugh se building aterial r where a radi path extends fr a lng distance thrugh tress. Studies ade at 3000 MHz indicate that stne buildings and grups f trees s dense that the sky cannt be seen thrugh the shuld be regarded as paque bjects arund which diffractin takes place [McPetrie and Frd, 946 ]. At 3000 MHz the lss thrugh a 23- centieter thick dry brick wall was 2 db and increased t 46 db when the wall was thrughly saked with water. A lss f.5 db thrugh a dry sash windw, and 3 db lss thrugh a wet ne were usual values. The nly bjects encuntered which shwed a lss f less than 0 db at 3000 MHz were thin screens f leafless branches, the trunk f a single tree at a distance exceeding 30 eters, wd-fraed windws, tile r slate rfs, and the sides f light wden huts. Field strengths btained when a thick belt f leafless trees is between transitter and receiver are within abut 6 db f thse cputed assuing Fresnel diffractin ver an bstacle slightly lwer than the trees. Lss thrugh a thin screen f sall trees will rarely exceed 6 db if the transitting antenna can be seen thrugh their trunks. If sky can be seen thrugh the trees, 5 db is the greatest expected lss. 5-7

59 The fllwing epirical relatinship fr the rate f attenuatin in wds has been given by Saxtn and Lane [ 955 ] : A w = d (0.244 lg f ) decibels, (f > 00 MHz) (5.8) where A w is the absrptin in decibels thrugh d eters f trees in full leaf at a frequency f egahertz. The situatin with a high and a lw antenna in which the lw antenna is lcated a sall distance fr and at a lwer height than a thick stand f trees is quite different fr the situatin in which bth antennas ay be lcated in the wds. Recent studies at apprxiately 500 MHz shw the depressin f signal strengths belw sth earth values as a functin f clearing depth, defined as the distance fr the lwer antenna t the edge f the wds [Head, I960 ]. Expressing this epirical relatin in ters f a frula: A c =52-2 lg d decibels (5.9) where ii c is the depressin f the field strength level belw sth earth values and d is the clearing depth in eters. A particularly interesting applicatin f se f the sth-earth frulas given in this sectin is the wrk f Lewin [ 962] and thers in the design f space-diversity cnfiguratins t verce phase interference fading ver line-f-sight paths. Diffractin thery ay be used t establish an ptiu antenna height fr prtectin against lng-ter pwer fading, chsing fr instance the iniu height at which the attenuatin belw free space is 20 db fr a hrizntally unifr atsphere with the axiu psitive gradient f refractivity expected t be encuntered. Then the frulas f this sectin will deterine the ptiu diversity spacing required t prvide fr at least ne path a siilar 20 db prtectin against ultipath fr direct and grund-reflected cpnents thrughut the entire range f refractivity gradients expected. In general, the refractive index gradient will vary ver wider ranges n ver-water paths [Ikegai, 964]. 5-8

60 GEOMETRY FOR WITHIN-THE-HORIZON PATHS Figure

61 6. DETERMINATION OF ANGULAR DISTANCE FOR TRANSHORIZON PATHS The angular distance, 8, is the angle between radi hrizn rays in the great circle plane defined by the antenna lcatins. This iprtant paraeter is used in diffractin thery as well as in frward scatter thery. Angular distance depends upn the terrain prfile, as illustrated in figure 6., and upn the bending f radi rays in the atsphere. Figure 6. assues a linear dependence n height f the atspheric refractive index, n, which iplies a nearly cnstant rate f ray refractin. If heights t be cnsidered are less than ne kileter abve the earth's surface, the assuptin f a cnstant effective earth's radius, a, akes an adequate allwance fr ray bending. Atspheric refractivity N = (n - ) X 0 re than ne kileter abve the earth's surface, hwever, is assued t decay expnentially with height [Bean and Thayer, 959 ]. This requires crrectins t the effective earth's radius frulas, as indicated in the fllwing subsectins. T calculate 6, ne ust first plt the great circle path and deterine the radi hrizns. 6. Pltting a Great Circle Path Fr distances less than 70 kileters, the great circle path can be apprxiated by a rhub line, which is a line intersecting all eridians at the sae angle. Fr greater distances, the rganizatin f a ap study is illustrated n figure 6.2. Here, a rhub line is first pltted n an index ap t shw the bundaries f available detailed tpgraphic sheets. Segents f the actual great circle path are later pltted n these detailed aps. The spherical triangle used fr the cputatin f pints n a great circle path is shwn n figure 6.3, where PAB is a spherical triangle, with A and B the antenna terinals, and P the nrth r suth ple. B has a greater latitude than A, and P is in the sae heisphere. The triangle shwn is fr the nrthern heisphere but ay readily be inverted t apply t the suthern heisphere. B is any pint alng the great circle path fr A t B, and the triangle PAB' is the ne actually slved. The latitudes f the pints are dented by *, *, and», while C and C are the differences in lngitude between A and B and A and B', respectively. Z and Z are the crrespnding great circle path lengths. The fllwing frulas are practical fr hand cputatins as well as fr autatic digital cputers. Equatins (6.) t (6.4) have been taken, in this fr, fr a well-knwn reference bk [I. T. and T., 956 ], where they appear n pages The initial bearing (X fr terinal A, and Y fr terinal B) are easured fr true nrth, and are calculated as fllws: _ -. \ // $_ + $» tan Y - X. C r - = ct T sin r K-^)] 6-

62 Y + X C *B " *A N tan - : ct - / I sin r ) (6.2) 2 2 V, 2 Y+X JL Y-X, Y + X Y - X v,.,> 5 + r = Y, and 5 - ^ = X (6. 3) The great circle distance, Z, is given by * - 4 Z B A tan T- = tan 2 2 [("" T ii )/("» ^)] " «T cnvert the angle Z btained in degrees fr (6.4) t units f length, the fllwing is used, based n a ean sea level earth's radius f 6370 k: d^ =.8 Z (6.5) The fllwing frulas shw hw t calculate either the latitude r the lngitude f a pint n the great circle path, when the ther crdinate is given. The given crdinates crrespnd t the edges f detailed aps, and t interediate pints usually abut 7.5 inutes apart, s that straight lines between pints will adequately apprxiate a great circle path. Fr predinantly east-west paths, calculate the latitude 4 fr a given lngitude difference C: cs Y = sin X sin C sin «- cs X cs C (6.6) cs 4 = sin X cs 4 /sin Y' (6. 7) Fr predinantly nrth-suth paths, calculate the lngitude difference C fr a given latitude *,: sin Y' = sin X cs 4 /cs 4 (6. 8) C Y - X \ ( B' A ct -T- = tan 2 2 V cs M-*-^^)] Where the bearing f a path is clse t 45 degrees, either ethd ay be used. 6-2

63 6. 2 Pltting a Terrain Prfile and Deterining the Lcatin f Radi Hrizn Obstacles This subsectin explains hw t deterine the sea level arc distance, d. fr an Lt, r antenna t its radi hrizn bstacle, and the height, h f this bstacle abve ean sea Lt, r level. The hrizn bstacles are represented by the pints (d, h ) and (d, h. ) in Lit Lit LdV Lr the great circle plane cntaining the antennas. These pints ay be deterined by the tps f high buildings, wds, r hills, r ay be entirely deterined by the bulge f the earth itself. All f the predictins f this paper replace the earth by a cylinder whse eleents are per- pendicular t the great circle plane and whse crss-sectin is in general irregular and deterined by the antenna and hrizn lcatins in the great circle plane. When the difference in elevatins f antenna and hrizn greatly exceeds ne kileter, ray tracing is necessary t deterine the lcatin f radi hrizns accurately [Bean and Thayer, 959 ]. Elevatins h. f the terrain are read fr tpgraphic aps and tabulated versus their distances x. fr the transitting antenna. The recrded elevatins shuld include thse f successive high and lw pints alng the path. The terrain prfile is pltted n linear graph paper by difying the terrain elevatins t include the effect f the average curvature f the radi ray path and f the earth's surface. The dified elevatin y. f any pint h. at a distance x. fr the transitter alng a great circle path is its height abve a plane which is hrizntal at the transitting antenna lcatin: y. = h. - x 2 /(2a) (6.0) where the effective earth's radius, a, in kileters is calculated using (4. 4), r is read fr figure 4.2 as a functin f N. The surface refractivity, N, is btained fr (4. 3), s s where N is estiated fr the ap n figure 4.. A plt f y. versus x. n linear graph paper is the desired terrain prfile. Figure 6.4 shws the prfile fr a line-f-sight path. The slid curve near the btt f the figure indicates the shape f a surface f cnstant elevatin (h = 0 k). Prfiles fr a path with ne hrizn cn t bth antennas and fr a path with tw radi hrizns are shwn in figures 6.5 and 6.6. The vertical scales f these three figures are exaggerated in rder t prvide a sufficiently detailed representatin f terrain irregularities. Pltting terrain elevatins vertically instead f radially fr the earth's center leads t negligible errrs where vertical changes are sall relative t distances alng the prfile. On a cartesian plt f y. versus x., as illustrated in figures 6.4, 6.5, and 6.6, the ray fr each antenna t its hrizn is a straight line, prvided the difference in antenna and hrizn elevatins is less than ne kileter. case are indicated in the next subsectin. Prcedures t be fllwed where this is nt the 6-3

64 6. 3 Calculatin f Effective Antenna Heights fr Transhrizn Paths If an antenna is lcated n anther structure, r n a steep cliff r untainside, the height f this structure, cliff, r untain abve the surrunding terrain shuld be included as part f the antenna height. T btain the effective height f the transitting antenna, the average height abve sea level h f the central 80 per cent f the terrain between the transitter and its hrizn is deterined. The fllwing frula ay be used t cpute h fr 3 evenly spaced terrain elevatins h. fr i = 0,, 2,... 30, where h _ is the height abve sea level f the grund belw the transitting antenna, and, h = h : 27 V 2T I \i> \-\s-\ i=3 fr V \0! < 6 - Ua) therwise V h ts- h to (6 - Ub) where h is the height f the transitting antenna abve ean sea level. The height h is siilarly defined. If h r h as defined abve is less than ne kileter, h = h rh = h. t r te t re r Fr antennas higher than ne kileter, a crrectin Ah, read fr figure 6.7, is used t reduce h r h t the value h r h : t r te re h te. h t - Ah(h t. N s ), h re = h r - Ah(h r, N g ) (6. 2) The crrectin Ah was btained by ray tracing ethds described by Bean and Thayer [ 959 ]. Fr a given effective earth's radius, the effective antenna height h crrespnding t a given hrizn distance d is saller than the actual antenna height, h. Over a sth Lt t spherical earth with h < k and h < k, the fllwing apprxiate relatinship exists between effective antenna heights and hrizn distances: \e - 4t /(2a) ' h re = d L /(2a) < 6 ' 3a > If the straight line distance r between antennas is substantially different fr the sea level arc distance d, as in cunicatin between an earth terinal and a satellite, the effective antenna heights ust satisfy the exact relatin: h te = a[ sec(d Lt /a) - ], h^ = a[ sec(d Lr /a) - ] (6. 3b) 6-4

65 The angular distance, 6.4 Calculatin f the Angular Distance, 0 8, is the angle between hrizn rays in the great circle plane, and is the iniu diffractin angle r scattering angle unless antenna beas are elevated. Calculatins fr cases where the antenna beas are elevated are given in annex. In calculating the angular distance, ne first calculates the angles 0 and 0 by et r which hrizn rays are elevated r depressed relative t the hrizntal at each antenna, as shwn n figure 6.. In this reprt, all heights and distances are easured in kileters, and angles are in radians unless therwise specified. When the prduct d8 is less than 2, 0 = 8 = d/a (6. 4) et er % where "a" in (6. 4) is the effective earth's radius defined in sectin 4. The hrizn ray elevatin angles 6 and 8 ay be easured with surveying instruents in the field, r deterined directly fr a terrain prfile plt such as that f figure 6.5 r 6.6, but are usually cputed using the fllwing equatins: h T - h d_ h T - h d. 8 t = L * t8 - ", 8 = Lr, rs - 3L (6. 5) et d T 2a er d T 2a s ' Lt Lr where h, h are heights f hrizn bstacles, and h, h are antenna elevatins, Lt Lr ts rs all abve ean sea level. As a general rule, the lcatin (h, d ) r (h, d ) f a Lit Lt Lr Lr hrizn bstacle is deterined fr the terrain prfile by using (6.5) t test all pssible hrizn lcatins. The crrect hrizn pint is the ne fr which the hrizn elevatin angle 8 r 8 is a axiu. When the trial values are negative, the axiu is the value et er nearest zer. Fr a sth earth, 8 = - -JTK 7ä fr h < k et, er te, re te.re At the hrizn lcatin, the angular elevatin f a hrizn ray, 8. r 8, is t r greater than the hrizn elevatin angle 8 r 8 e. = e. + d T./ a 8 t et Lt - nr = 8 er +d T Lr / a (6.6)» ' If the earth is sth, 8 and 8 are zer, and 8 = D /a, where t r s D S =d - d Lt- d Lr (6 ' 7 > Figure 6.8, valid nly fr 8 +8 = 0, is a graph f 8 versus D fr varius values f ' t r «- g surface refractivity, N. 6-5

66 In the general case f irregular terrain, the ancles and ß shwn in figure 6. are calculated using the fllwing frulas: h - h a = ± tg. " (6.8a) 2a et d h - h ß = rs t8. r 2a er d (6.8b) These angles are psitive fr beynd-hrizn paths. T allw fr the effects f a nn-linear refractivity gradient, r and ß are dified by crrectins A a and Aß t give ' " r the angles a and 6 whse su is the angular distance, 9, and whse rati defines a path asyetry factr 6: r = a + Aa (6. 9a) ß r = ß r + Aß r (6. \ 9b) ' I 9 = a + P, s = a /p (6. 9c) * ' The crrectins Aa and Aß are functins f the angles 8 and 8, (6. 6), r t r and f the distances d and d fr each hrizn bstacle t the crssver f hrizn st sr rays. These distances are apprxiated as d = dp /8 - d T. d = da /8 - d T (6. 20) st r Lt sr Lr * ' ' The su f d and d is the distance D st sr s between hrizn bstacles, defined by (6.7). Over a sth earth d =d = D /2. st sr s Figure 6.9, drawn fr N = 30, shws Aa as a functin f 8 and d, s t st Siilarly, Aß is read fr the figure as a functin f 9 and d. ' " r sr Fr values f N ther than 30, the values as read fr the figure are ultiplied by C(N ) : Aa (N ) - C(N ) Aa (30), Aß <N ) C(N ) Aß (30) (6.2 a ) s s ss ' C (N g ) = (.3 N^ - 60 N g ) x 0" 5 (6.2b) Fr instance, C(250) = 0.66, C(30) =.0, C(350) r. 38, and C(400) =.84. Figure 6. 0 shws C(N ) pltted versus N. 6-6

67 Fr sall 9^ f n crrectin A^ r A^ is required fr values f d, less than 00 K. When bth \a and AD arc n f li!w». 8t * r } and \p are negligible: which is the sae as (6. 4). 9 = 8 = a + ß (6.22) If 9 r 0 is negative, cpute t r d = d - la 9 I r d' = d - Ia8 I, (6.23) st st ' t sr sr! r v ' substitute d' fr d r d' fr d. and read figure 6.9, using 8 =0. st 6t sr sr' t, r If either 9 r 9 is greater than 0. radian and less than 0.9 radian, t r deterine A r r Aß fr 9 =0. radian and add the additinal crrectin ter t N (9.97 -cte ) ( - exp( d )] X 0 radians 8 t, r' L r st, r' ' The bending f radi rays elevated re than 0.9 radian abve the hrizn and passing all the way thrugh the atsphere is less than radian, and ay be neglected. Other geetrical paraeters required fr the calculatin f expected transissin lss are defined in the sectins where they are used. Many f the graphs in this and subsequent sectins assue that s = a /ß S, where f and ß are defined by (6. 9a) and (6. 9b). It is therefre cnvenient, since the transissin lss is independent f the actual directin f transissin, t dente as the transitting antenna whichever antenna will ake s less than r equal t unity. Alternatively, s ay be replaced by /s and the subscripts t and r ay be interchanged in se f the frulas and graphs, as nted in later sectins. 6-7

68 PATH GEOMETRY SCATTER VOLUME DISTANCES ARE MEASURED IN KILOMETERS ALONG A GREAT CIRCLE ARC. De 3x> = "a" + ^t + #r = a + ^et + 0 ( er Figure

69 6-9

70 (POLE CORRESPONDING TO HEMISPHERE OF B) SPHERICAL TRIANGLE FOR GREAT CIRCLE PATH COMPUTATIONS Figure

71 MODIFIED TERRAIN PROFILE FOR A LINE-OF-SIGHT PATH DISTANCE, X, IN KILOMETERS Figure

72 MODIFIED TERRAIN ELEVATION, y, IN KILOMETERS n/k O g 3D CO H T> x H 30 O 3] I - -n 30 CO I "0 6-2

73 MODIFIED TERRAIN ELEVATION,y, IN KILOMETERS I» CT CO H > -I JO t Tl O s 33 O c GD r i x M O - 3> 6-3

74 REDUCTION OF ANTENNA HEIGHT FOR VERY HIGH ANTENNAS 000 MO? in UJ 2 I- ui > 3 i z 05 < ? i i, a.t: /A- - ttl -4ii / /III / i //// '/// y iu f U-- -(- /// Iff / //// \ h = h,-ah -«-~~Vv V «Lf^yS HEIGHT, h fl IN KILOMETERS Figure

75 ANGULAR DISTANCE, 9, IN RADIANS CD en z CD H c x rsi t O in S O i> (/> O > en h CD I " ^ 33 CO i t > CO 6-5

76 CORRECTION TERMS A 0, Aft, FOR N $ = 30 2( ,, IN MILLIRADIANS Figure

77 THE COEFFICIENT C(N S ) 32 I:::::!:::::!:::::::::::: Aa 0 (N«) = C(N») Aa 0 (30D C(N») = (l.3ng'-60n*) xio" S 2.4 n^^^h^^^^^h 2.0 i iniihiniiiiirff «\r * *»» <_>.6 a <«*> > > >> < > * > > ««««a'4 «*» * «> >«' iiiiiiiiiiii «::::::::::::::::::::::::::::::::::::'.::::::::::::::::::::::::::::::::::::::::::::::: > IIIIIIHII»! Ill «a«* «*» * «««««««a» * i! ) I«* **>' iihiiiiiiiiimiiiiiaimii tiiiiiil * * > * :ll!ll:ll-:!:l:illll N c Figure

78 7. DIFFRACTION OVER A SINGLE ISOLATED OBSTACLE A prpagatin path with a cn hrizn fr bth terinals ay be cnsidered as having a single diffracting knife edge. In se cases, reflectin fr terrain ay be neglected as discussed in sectin 7. I; in ther cases, grund reflectins ust be cnsidered as shwn in sectin 7.2 and appendix III. In actual situatins, the cn hrizn ay be a untain ridge r siilar bstacle, and such paths are seties referred t as "bstacle gain paths",[ Barsis andkirby, 96; Dicksn, Egli, Herbstreit and Wickizer, 953; Furutsu, 956, 959, 963; Kirby, Dugherty and McQuate, 955; Rider, 953; Ugai, 963]. A ridge r untain peak ay nt prvide an ideal knife edge. The thery f "runded bstacles" is discussed by Bachynski [ 960], Dugherty and Malney [ 964], Neugebauer and Bachynski [ 960], Rice [954], Wait [958, 959], and Wait and Cnda [ 959]. Furutsu [ 963] and Millingtn, Hewitt,and Iirzi [ 962a] have recently develped tractable expressins fr ultiple knife-edge diffractin. In se cases, ver relatively sth terrain r ver the sea, the cn hrizn ay be the bulge f the earth rather than an islated ridge. This situatin is discussed in sectin Single Knife Edge, N Grund Reflectins A single diffracting knife edge where reflectins fr terrain ay be neglected is illustrated in figure 7., where the wedge represents the knife edge. The diffractin lss A(v, 0) is shwn n figure 7. as a functin f the paraeter v, fr Schclling, Burrws, and Ferrell [933] and is defined as v = ± 2\TÄr~7\ = ± 'JJ^S tanä'~iinß~]t\~ (7. la) r in ters f frequency in MHz: v = ± 2,583 e^tt^jt (7.b) where the distances are all in kileters and the angles in radians. The distance Ar a r, + r, - r = 9 2 d,d,/(2d) 2 2 is discussed in sectin 5, and the distances d. and d fr the knife edge t the trans- itter and receiver, respectively, are shwn n figure 7.. The radi wavelength, X, is in the sae units as the ttal path distance, d. The angles a, ß, and 8 are defined in sectin 6. In this case, h. = h., and si-ice d = d = 0, n crrectins A r r Lt Lr' st sr Aß are required. Fr the line-f-sight situatin, shwn in figure 7. and discussed in sectin 5.2, the angles and ß are bth negative, and the paraeter v is negative. Fr transhrizn paths, a and ß are bth psitive and v is psitive. 7-

79 If v is greater than 3, A(v, 0) ay be expressed by: A(v,0) lg v db (7.2) The basic transissin lss, L, fr a knife-edge diffractin path is given by adding A(v, 0) t the free space lss; L^ = L^ + A(v, 0) db (7.3) where L is given by (2. 3). Fr frequencies abve abut GHz, an estiate f the lss bf due t absrptin (3. ), shuld be added t (7.3) and (7.4). If the angles a and ß are sall, the basic transissin lss ver a knife-edge diffractin path ay be written as : U= 30 lgd + 30 lgf + 0 lg r + 0 lg p" db (7.4) which, hwever, is accurate nly if v > 3, d >>\, and (d/\) tan a tan ß > 4. Fr any paths, the diffractin lss is greater than the theretical lss shwn in (7.2), (7.3), and (7.4), because the bstacle is nt a true knife edge, and because f ther pssible terrain effects. abut 0 t 20 db. Fr a nuber f paths studied, the additinal lss was The prble f ultiple knife-edge diffractin is nt discussed here, but fr the duble knife-edge case, where diffractin ccurs ver tw ridges, a siple technique ay be used. The path is cnsidered as thugh it were tw siple knife-edge paths, (a) trans- itter -first ridge-secnd ridge, and (b) first ridge-secnd ridge -receiver. The diffrac- tin attenuatin A(v, 0) is cputed fr each f these paths, and the results added t btain the diffractin attenuatin ver the whle path. When the paraeter v is psitive and rather sall fr bth parts f the path, this ethd gives excellent results. Methds fr apprx- iating theretical values f ultiple knife-edge diffractin have been develped by Wilkersn [964]. 7-2

80 7.2 Single Knife Edge with Grund Reflectins Theretically, received fields ay be increased by as uch as 2 db due t enhanceent, r deep nulls ay ccur due t cancellatin f the signal by grund reflectins. Reflectin ay ccur n either r bth sides f the diffracting edge. When an islated knife edge frs a cn hrizn fr the transitter and receiver, the diffractin lss ay be estiated as: A = A(v,0) - G(hj) - G(h 2 ) db (7.5) where hj» B 2 (K.b*) (f 2^)* h^ a 5.74 (f 2^)* h^ h, = B 2 (K.b«) (f 2 /a,) 7 h a 5.74 (f 2 /a,)> h 2 2' re 2 re (7.6a) 2 2 (7.6b) a i = V^V' a 2 - d L/t 2h re> The paraeters b*. K, and B(K, b*) are defined in subsectin 8.. The knife-edge attenuatin A(v, 0) is shwn n figure 7., and the functin G(h) intrduced by Nrtn, Rice and Vgler [955] is shwn n figure 7.2. Effective antenna heights h, h, and the distances d, tc re l.t d are defined in sectin 6. In these and ther frulas, f is the radi frequency in MHz. Lr The functin G(h ) represents the effects f reflectin between the bstacle and the transitter and receiver, respectively. These ters shuld be used when re than half f the terrain between an antenna and its hrizn cuts a first Fresnel zne ellipse which has the antenna and its hrizn as fcii and lies in the great circle plane. Definite criteria are nt available, but in general, if terrain near the iddle distance between a transitting antenna and its hrizn is clser t the ray than 0.5(X.d ) kileters, G(h~ ) shuld be Lit used. The sae criterin, depending n d, deterines when G(h,) shuld be used. Lr 2 When details f terrain are nt knwn, an allwance fr terrain effects, Gfh,,), shuld be used if 0.5(\d Ti, ) > I h T t, - h /2, where all distances and heights are in Lt, Lr ' Lt, Lr ts.rs 6 kileters. When the reflecting surface between the diffracting knife-edge and either r bth an- tennas is re than the depth f a first Fresnel zne belw the radi ray, and where ge- etric ptics is applicable, the fur ray knife-edge thery described in annex III ay be used t cpute diffractin attenuatin. that reflecting planes ay be deterined rather accurately. This ethd is used when details f terrain are knwn s Using the fur ray thery, the received field ay Include three reflected cpnents, with assciated reflectin cefficients and ray path differences, in additin t the direct ray cpnent. 7-3

81 7. 3 Islated Runded Obstacle, N Grund Reflectins Dugherty and Malney [ 964 ] describe the diffractin attenuatin relative t free space fr an islated, perfectly cnducting, runded knife edge. The runded bstacle is cnsidered t be islated fr the surrunding terrain when k h[2/(kr)] J > > where k = 2ir/\, r is the radius f curvature f the runded bstacle, and h is the saller f the tw values [(d + r ) - r ] and [(d + r ) - r ]. The diffractin lss fr an islated runded bstacle and irregular terrain shwn i.i figure 7. 3 is defined as: A(v.p) = A(v,0) + A(0,p)+U(vp) db (7.7) where v is the usual diensinless paraeter defined by (7. ) and p is a diensinless index f curvature fr the crest radius, r in kileter«, f the runded knife edge: vp =.746 0(fr)5 < 7-8) 4 '/a p = r 3 f 6 [d/( ri r 2 )] (7.9) where, f is the radi frequency in MHz, d is the path distance in kileters, and r,r, shwn in figure 7.3 are the distances in kileters fr the transitter and receiver, respectively t the runded bstacle. Fr all practical applicatins, r, r, ay be replaced by d d. Where the runded bstacle is the brad crest f a hill, the radius f curvature, r, fr a syetrical path is: r = D /8 (7. 0) s where D = d - d. - d. is the distance between transitter and receiver hrizns in s Lt Lr kileters, and 9 is the angular distance in radians (6. 9). Where the rati a /ß ^, the radius f curvature is defined in ters f the harnic ean f radii a and a defined t r in the next sectin, (8.9), and shwn in figure 8. 7: r = 2 D d d S St 8r d st sr ) (7.) In (7.7), the ter A(v, 0) is the diffractin lss fr the ideal knife edge (r = 0), and is read fr figure 7.. The ter A(0, p) is the agnitude f the intercept values (v = 0) fr variuß values f p and is shwn n figure 7.4. The last ter U(v p) is d functin f the prduct, vp, and is shwn n figure

82 Arbitrary atheatical expressins, given in annex III, have been fitted t the curves f figures 7., 7.3, 7.4, and 7.5 fr use in prgraing the ethd fr a digital cputer. The diffractin lss A(v, p) as given by (7.7) is applicable fr either hrizntally r vertically plarized radi waves ver irregular terrain prvided that the fllwing cnditins are et: (a) the distances d, d, d, and r are uch larger than X, (b) the extent f the bstacle transverse t the path is at least as great as the width f a first Fresnel zne: ^T^TTTT^TdT, (c) the cpnents a and ß f the angle 9 are less than 0.75 radians, and l_ (d) the radius f curvature is large enugh s that (IT r/\? > >. In applying this ethd t cputatin f diffractin lss ver irregular terrain, se variance f bserved fr predicted values is t be expected. One iprtant surce f errr is in estiating the radius f curvature f the runded bstacle, because the crests f hills r ridges are rarely sth. t be greater at UHF than at VHF. Differences between theretical and bserved values are apt 7-5

83 7. 4 Islated Runded Obstacle with Grund Reflectins If a runded bstacle has a sall radius and is far fr the antennas, (7.7) ay neglect iprtant effects f diffractin r reflectin by terrain features between each antenna and its hrizn. Such terrain fregrund effects ay be allwed fr, n the average, by adding a ter, 0 exp(-2.3p) t (7.7). The effect f this ter ranges fr 0 db fr p =0 t db fr p =. When se infratin is available abut fregrund terrain, the G(h. 7) ters discussed in sectin 7.2 ay be used if re than half f the terrain between an antenna and its hrizn cuts a first Fresnel zne in the great circle plane: A = A(v,p) - CKh^ - G{h" 2 ) db (7.2) where A(v,p) is defined by (7.7), h, h by (7.6), and the functins G(h?) are shwn n figure 7.2. When details f terrain are knwn, and the reflecting surfaces between the runded bstacle and either r bth antennas are re than the depth f a first Fresnel zne belw the radi ray, the geetric ptics fur-ray thery described in annex cable. ay be appli- In this case, the phase lag f the diffracted field with reference t the free space field ust be cnsidered in additin t the ray path differences f the reflected cpnents. The phase lag *(v, p) f the diffracted field is defined as 2 *(v, p) = 90 v + dxv, 0) + 4X0» p) + (Hvp) degrees (7.3a) where the functins 4>(v,0), 4>(0, p), and <Hvp) are shwn n figures 7., 7.4, and 7.5, respectively. Fr an ideal knife-edge, p = 0, the phase lag f the diffracted field is 2 *(v,0) = 90v +(tkv,0) fr v > 0 (7.3b) *nd *(v, 0) = 4>(v, 0) fr v is 0 (7. 3c) 7-6

84 KNIFE EDGE DIFFRACTION LOSS, A(v,0) 40 </> S Q > Figur«7. 7-7

85 G(h )2 ) t 7-8

86 DIFFRACTION LOSS, A(v,/>), FOR A ROUNDED OBSTACLE aaaaaalaaaaal»*»-» *» flai«>* «.»»» «a ««*»! «nil in'tk i. «)» * *!. * i < *][ * > ->aaiaaaat*ak a i tun ii ( i > IIIIIII. i i a. HI)iMMiM)iliMM>" > IIIIIII la a IM M. iaa ^ «iiiiiia«lflli«iim>imiii«ilinitii itii.iii a : < iiimiiiiii iiiimiiiiiimiii < a a. «a» iiiiiiiiiaiialt>ia>ai aaai!,<! n> iiitllllm ** ' a in a a «{ >iiie«l^ia>tila>amli ill>t IN Mttti * { ««^ ( ^ i aaafaaaa.i MMtll IIIMMIH IIIIMIIIII II! > >. figure

87 INTERCEPT MAGNITUDE AND PHASE FOR DIFFRACTION OVER A ROUNDED OBSTACLE Figure /0

88 UNIVERSAL DIFFRACTION CURVE FOR A ROUNDED OBSTACLE IV 0 iiiiiiiihiiiiiiiiniiiiiffl It iiiiiiiiin iiiiiiii IM?0 LL) 30 U(V _4-^4---l^-UJ 20 ) ;~t ~"lt~"t ~T~ F r~]~~t" in UJ LlJ a: 40 < ~r~[f i v.p) ' " :jj j f~f ~-j «r~j 60 r._i.i_w 4~] N 3 iiiiiiiiiiiiiiii'iiiiiiihii Figure

89 . 8. DIFFRACTION OVER SMOOTH EARTH AND OVER IRREGULAR TERRAIN Diffractin attenuatin ver an islated ridge r hill has been discussed in sectin 7. The fllwing ethds are used t cpute attenuatin ver the bulge f the earth and ver irregular terrain. The ethds are applicable t the far diffractin regin, where the diffracted field intensity ay be deterined by the first ter f the Van der Pl-Breer residue series [Breer, 949]. This regin extends fr near the radi hrizn t well beynd the hrizn. A criterin is given t deterine the iniu distance fr which the ethd ay be used. In se situatins the first ter f the series prvides a valid apprxiatin t the diffracted field even at pints slightly within line-f-sight [Vgler. 964]. A siplified graphical ethd fr deterining grund wave attenuatin ver a spherical hgeneus earth in this far diffractin regin was recently de "i^ped by Vgler [ 964], based n a paper by Nrtn [ 94]. The ethd described in sectin 8, is applicable t either hrizntal r vertical plarizatin, and takes accunt f the effective earth's radius, grund cnstants, and radi frequency. In sectin 8.2, a dificatin f the ethd fr cputing diffractin attenuatin ver irregular terrain is described, and sectin 8.3 cnsiders the special case f a cn hrizn which is nt an islated bstacle. Fr frequencies abve 000 MHz, the attenuatin due t gaseus absrptin shuld be added t the diffractin lss. See (3. ) and figure Diffractin Attenuatin ver a Sth Earth The attenuatin relative t free space ay be expressed with fur ters; ne cntains the distance dependence, tw represent the dependence n antenna heights, and the furth ne depends n electragnetic grund cnstants, the earth's radius, and the radi frequency: A = G(x Q ) - F(Xj) - F(x 2 ) - C^K.b") db (8.) where x = d B, x =d B, x =d B (8.2a) 0 Lt c. Lr i 2 i B = f J C B(K,b ), C = (8497/a) 3 (8.2b) The distances d, d, d, and the effective earth's radius, a, have been defined Lt l_,r in sectins 4 and 6, and f is the radi frequency in egacycles per secnd. The paraeters K and b depend n plarizatin f the radi wave and the relative dielectric cnstant, c, and cnductivity, <r, f the grund. Figures 8. and 8.2 shw curves f K and b versus frequency fr cbinatins f < and a crre- spnding t pr, average, and gd grund, and t sea water. Figure 8. shws K fr a = 8497 k. Fr ther values f effective earth's radius, K(a) = C K(8497) (8.3) 8-

90 General frulas fr K and b fr bth hrizntal and vertical plarizatin are given in sectin.4 f annex III. The paraeter B(K, b") in (8.2b) is shwn as a functin f K and b in figure 8.3. The liiting value B =.607 fr K 0 ay be used fr st cases f hrizntal plarizatin. The paraeter C.(K,b ) in (8.) is shwn in figure 8.4. The functin G(x ) in (8.) is shwn n figures 8.5 and 8.6, and is defined as G(x 0 ) = x Q - 0 lg x Q (8.4) and the height functins F(x y) are pltted in figures 8.5 and 8.6 versus K and b. Fr large values f x r x-, F(x) is apprxiately equal t G(x). Because this ethd is based n nly the first ter f the residue series, it is liited t the fllwing distances t insure that A is accurate within apprxiately.5 db; x - x (Ax ) - x 2 (Ax 2 ) > 335, fr B =.607, (K-0.0) (8.5a) x Q - x (Ax x ) - x 2 (Ax 2 ) > 5, fr B = 0.700, (K > 0) (8. 5b) Fr values f B lying between these tw liits, linear interplatin between the A(x) curves f figure 8.6, and the tw iniu values in (8.5) gives a fair apprxiatin f the range f validity f (8.). Using linear interplatin: x 0 -x A(x.B).x 2 A(x 2 B) >x ln (8.6) where x = (.607-B) (8.7a) an A(x,B) = A(x,.607) +.03(.607-B) [A(x, 0.700) - A(x,.607)] (8.7b) A(x, 0.700) and A(x,.607) are the values read fr the upper and lwer curves f Ax in figure The basic diffractin transissin lss, L,, is btained by adding the attenuatin A t the free space lss L_. defined by (2. 3), including an allwance fr atspheric absrptin when required. 8-2

91 8.2 Diffractin ver Irregular Terrain T cpute diffractin attenuatin ver irregular terrain, the single effective earth's radius, a, used in (8.2) is replaced by fur different radii as shwn in figure 8.7. The radii a and a, f the terrain between the antennas and their hrizns, and the radii a t and a f the terrain between radi hrizns and the crssver pint f hrizn rays are defined by a l ed L /(2h f>' a 2 = d L /(2h re> < 8 ' 8) a t = D s d st /(9d sr>' a r = D s d.r /(8d.t> (8 * 9) The distances D, d, d, d., d., the effective antenna heights h and s st sr Lt Lr B te h, and the angular distance 0 are defined in sectin 6. re Fur values f C are cputed fr (8.2b) with C, C, C, and C crre r i t t r spnding t a, a?, a, and a, respectively. These are used in (8.3) t btain values f K.. fr the crrespnding earth's radii, and B, are then read fr l,z,t,r l,<,t,r figure 8.3 crrespnding t each value f K. The diffractin attenuatin relative t free space is then: A = G(x 0 ) - F( Xl ). F(x 2 ) - Cj (K 2) + A. (8. 0) where A is the atspheric absrptin defined by (3.), and is negligible fr frequencies a less than GHz. and C. (K ) is the weighted average f C.(K.,b) and C. (K.,b) read fr figure 8.4: CjtKj 2) = [x C (K )+x 2 C (K 2 )]/(x l +x 2 ) (8.) x.=b.c f 3 d,. x, =B,C f d T (8.2) i Lt Lr l ' x = (B C 2 d + B C 2 d ) f +x, +x, (8.3) \ t t st r r sr/ 2 C = (8497/a,,, ) J, K., t =C K(8497) i,2,t, r l,2,t, r l,2,t, r i, 02, t, r B l,2,t,r = B^l,2,t,r' b > This ethd is applicable t cputatin f diffractin attenuatin ver irregular terrain fr bth vertical and hrizntal plarizatin fr transhrizn paths. The ethd ay be sewhat siplified fr tw special cases: diffractin ver paths where däd, and fr st paths when hrizntal plarizatin is used. 8-3

92 8. 2. Diffractin ver paths where d S d st sr Fr paths where the distances d and d are equal, the paraeter x ay be defined in ters f D and the crrespnding earth's radius a : X 0 = 2 I J B + x, +x s C f s D s +X l +X 2 (8 ' U) D s = 2d st = 2d sr' a s = V e ' C s = < 8497/a s )i < 8 ' 5a > K =C K(8497), B =B(K,b) (8.5b) S OS 8 8 where Xj and x.^ are defined by (8. 2). The diffractin attenuatin is then cputed using (8. 0) Fr hrizntal plarizatin Fr hrizntally plarized radi waves, at frequencies abve 00 MHz, and with K(a) s 0.00, the paraeter B(K,b) appraches a cnstant value, B».607, and C (K,b) = db. Assuing B =.607 and C. = 20.03, the diffractin attenuatin ay be cputed as fllws: A = G(x Q ) - F( Xl ) - F(x 2 ) db (8. 6a) Xj = 669 f J d Lt / ai 2, x 2 = 669 f T d Lr /a 2 2 (8.6b) i. z x n = 669 f J 6 D. +x,+x, (8.6c) 0 str 2 x ' where D. = (d d ) J (d ]' + d T )/ (d 4 + d )* str st sr' \ st sr J/ st sr' The paraeter D is shwn in figure 8.8 as a functin f d and d str st sr Fr aths where d = d, using hrizntal plarizatin, the paraeter x r st sr u siplifies t i? I x Q = 669 f 3 (8 D s ) 3 + x { + x 2 (8. 6d) 8-4

93 8. 3 Single-Hrizn Paths, Obstacle nt Islated In se cases, ver rather regular terrain r ver the sea, a cn hrizn ay be the bulge f the earth rather than an islated ridge r untain. Fr such paths, the path distance, d, is just the su f d and d, and in this case, the ethd described Lt Lr in sectin 8.2 is siplified t ne with nly tw earth's radii instead f fur. The para- eters Xj and x 2 are defined by (8.2), and x =x +x. The diffractin attenuatin is then cputed using (8. 0). The diffractin lss predicted by this ethd agrees very well with bserved values ver a nuber f paths in the United Kingd and the United States where the cn hrizn is nt islated. Fr transhrizn paths f shrt t ediu length, when it is nt knwn whether diffractin r scatter is the dinant prpagatin echanis, bth diffractin and scatter lss shuld be cputed. The next sectin shws hw t cpute scatter lss, and hw t cbine the tw cputed values when they are nearly equal. 8-5

94 CARRIER FREQUENCY IN MHz g g g «. «- g - g X -n <a Q en c ii <T> 8 OD tß O -J O > < Q O a> a> O q q in q <* O>00 >. M H M H N M 2* 2«P«^* 9 VJ <J _S O ra. ^ \ (^«- -" \ \ i= ^^ *>"' j-r"^" i-*"' _,. * ^ i - '"' -...«-»""" - -^ V -- -»"»",-- -->-» j «- ** "" V * ' \ f" -' \ *s* A,.-'-^ J.-" ^ti 9 "» J =-.^a -" ' \ \ -' \ * ^ S \ «*> -*-' \ \ *",,-" r \.'.-- *"»- *,' -' \ \ JB t» <r> *"" > -' O S O y * 8 s X 2. ' q z ' S 6 * / r / > y ' y V A («* K / n a» > / " 3) «(AIR) «cr IN hs/eter HORIZONTAL VERTICAL * «gs H 3 3) J> n n H < > > O c II 03 * 3 8-6

95 b IN DEGREES S I s f CD s O en c z < s a s _ L 4 Hi in -ff '\ / IC,- jtt * \\ " HE \ ^"'S q * 3 \ >\ - A - - i ^ /- ^ -0^ -is i ^ I - > s>»o wo Af \ \ % -f X / 7*"s V -% % -} /- \/ *r SJ_/ \-%2 \, ^»_ "^2!>^z: ^ L_ P5g ^N \\ /A / sg-- I \ \> // / j C i \ ^, yr?\ I i s ' v \\ i / l\ i 5 J I * stg C^IJL.J :s?g? : t <->~8 \ If/ P "SS II ( 'S.'*-\\i' \ K> E i»t "I i r I'?*rn V?77 3. S *.$ * 3 * Jl/f ^ 3_ f :]:_:! ::i :::JL_:: ::: :x ' i/, I i/ w i *r-4- SI J 7. - > - -Z - ^ >* * - : t : \ - * s v es S 7 > \ t i ^ ^. ^ sl- V ' - ss_ C -J ii 4--/ i- J 5 3) > 9 - Ti 33 CO 5 3) 2 r > > 33 ^ 3 X) 8-7

96 THE PARAMETER B(K,b) Figur«

97 C,(K, b") jtr K> O OJ 7 CO \ M \ S <*^_ " ^v s A H X > -i 3) n cr xrtt I 5' A? * 8-9

98 THE FUNCTIONS F(x,,x 2 ) FOR KsO.I AND G(x Q ). FOR LARGE x: F(x)~G(x) ;. : u UJ O x 0,x,, OR %2 IN KILOMETERS Figur«

99 _. 30 THE FUNCTIONS F(x,), F(x 2 ) AND G(x Q ) FOR THE RANGE OSK* I < 20 Ü.4» 0,t(/6373,a,,,:EFFECTIVE EARTH RADIUS ~ 0 0 br- 90 If- ISO FOf LARGE x:fu)~g<i) )«00575U-IO 0«x 0 (j! -20 :..- y K-l $g ;. S.' Q> ^ -rtt- K '0.5,;.:,.(, K K-0 2 _» ( 0 -w 60 :. K-0 0 s v ^ y, _ --*-***' Jrf ' x/ W- i>j 'A,^r AT j,'jr J HÖ ERROR IN At 5 db r J_i J» r335,k<oi 70 -^ 4 IM MIHl«tMl*tf -M SO =s: K-0 OOI»ST/. I L-l-I ^."" =ft I /// I 6?' yy/ l^nkih-f- J.J>^rJ> t! ß / ' ' "* "T II ' "^ " l~~t ; -IfO i - K«0 =H=.:: y / f. / / n ; -i r 2 ) ftp - -. J>T. - -! 3 t 2C r-' 0 0 SO 0 IC 00 2 '/3»O.I.Z'^C Wtt <*0,l.2<l"") Figure

100 GEOMETRY FOR DIFFRACTION OVER IRREGULAR TERRAIN Figure

101 J iu Illllllllllllllllfllllll lllllllllllilllllllllllllllllmlllllllllllllllllllllllllllllllilllllllllllllllllllllllllllllllllllllllllll IIIMIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIilllllllllllll Illllllllllllllllllll O H X OD OD r 3) 8-3

102 9. FORWARD SCATTER This sectin gives ethds fr calculating reference values f lng-ter edian radi transissin lss ver paths that extend well beynd the hrizn. A ethd is given fr c- bining diffractin and scatter transissin lss estiates where this is apprpriate. ethds f this sectin prvide a reference edian value f basic transissin lss. pirical estiates f the^edian values and lng-ter variability f transissin lss fr sev- eral cliatic regins and perids f tie are given in sectin 0 and annex III. The E- Fr lng trpspheric paths the prpagatin echanis is usually frward scatter, especially during ties f day and seasns f the year when ducts and elevated layers are rare. Often, fr ther perids f tie, as scattering beces re cherent it is re prperly called reflectin. The exainatin f transissin lss variatin ver a particular path during se perid fr which detailed infratin abut layer heights, tilts, and inten- sities is available can be very illuinating; see fr instance Jsephsn and Eklund [ 958]. Seties n distinctin can be ade between "frward scatter" fr a turbulent atsphere and "incherent reflectins" fr patchy elevated layers. The first viewpint is develped in papers by Pekeris [ 947], Bker and Grdn [ 950a, b], Megaw [ 950, 954, 957], Millingtn [ 958], Staras [ 952, 955], Ta[l957], Tritsky [ 956, 957a], Villars and Weisskpf [ 955], Vge [ 953, 955], and Wheeln [ 957, 959], while the secnd viewpint is ephasized in papers by Beckann [ 957, I960, 96a, b], ducastel, Mise, and Vge [958], Friis, Crawfrd and Hgg ( 957], Starkey, Turner, Badce, and Kitchen [ 9 58], and Vge [ 956, I960]. The general predictin ethds described here are fr the st part cn- sistent with either viewpint, and agree with lng-ter edian values fr all available data. A brief discussin f frward scatter is given in Annex IV. The reference value, L,, f lng-ter edian basic transissin lss due t frbsr ward scatter is L, = 30 lg f - 20 lg d + F(0d) - F +H +A db (9.) bsr a Fr st applicatins the first three ters f (9. ) are sufficient fr calculating L, ' g bsr In (9. ) f is the radi frequency in MHz, and d is the ean sea level arc distance in kil- eters. The attenuatin functin F(9d), the scattering efficiency ter F, and the frequency gain functin H, are discussed in the fllwing subsectins. Atspheric absrptin, A, a defined by (3. ) and shwn n figure 3. 6, ay be neglected at lwer frequencies, but ay be re than 2 db ver a lng path at 000 MHz, and beces increasingly iprtant with in- creasing frequency. Fr grund-based scatter links the sea level arc distance, d, and the straight line distance, r, between antennas are apprxiately equal. T estiate transissin lss between the earth and a satellite, where r is uch greater than d, a ter 20 lg(r /d) 9-

103 shuld be added t the reference value L,. Annex III cntains a discussin f transissin bsr lss expected when antenna beas are elevated abve the hrizn, r directed away fr the great circle plane deterined by the antenna lcatins. The edian frward scatter transissin lss, L, is the basic transissin lss, sr L,, inus the path antenna gain, G. Sectin 9.4 shws hw t estiate the lss in path bsr p antenna gain, L, when there is a lss in antenna gain due t scatter. Sectin 9. 5 shws gp hw t cbine diffractin and scatter lsses. Fllwing Arns [956], the scattering f dif- fracted fields and the diffractin f scatter fields are ignred. 9. The Attenuatin Functin F(9d) The attenuatin functin F(8d) depends upn the st iprtant features f the prpagatin path and upn the surface refractivity, N. The functin includes a sall epirical adjustent t data available in the frequency range fr 00 t 000 MHz. Fr st land-based scatter links figure 9. ay be used, where F(9d) is pltted versus the prduct 9d fr N = 400, 350, 30 and 250. The path distance, d, is in kileters and the angular distance, 8, in radians. fr all values f s. Fr values f 9d «0 the curves f figure 9. are valid Fr values f 9d greater than 0 the curves ay be used fr values f s fr 0.7 t unity. Fr s greater than use /s in reading the graph. Fr highly asyetrical paths with 8d > 0, figures III. t III. 4 f annex III are used t btain F(9d). Annex III als cntains analytical functins fitted t the curves F(9d) fr 0.7 s fr all values f the prduct 9d and fr N = 250, 30, 350, and 400. Using the expressins fr the functin F(9d) with N issin lss is Fr 9d * 0: =30, the reference edian basic trans- L a lg f + 30 lg lg d d (9.2a) Fr 0 < 9d s 50i L u bsr a lg f + 35 lg lg d d (9.2b) Reference values ay be cputed in a siilar anner fr ther values f N. s The apprxiatins in (9.2) d nt ake any allwance fr the frequency gain functin, H. Fr usual cases f transissin at frequencies abve 400 MHz the apprxiatins in (9.2) give gd results. Fr the higher frequencies an estiate f atspheric absrptin shuld be added. Fr lwer frequencies, r lw antenna heights, grund-reflected energy tends t cancel the direct ray and the apprxiatin in (9.2) will underestiate the transissin lss. 9-2

104 9.2 The Frequency Gain Functin, H It is assued that if antennas are sufficiently high, reflectin f energy by the grund dubles the pwer incident n scatterers visible t bth antennas, and again dubles the pwer scattered t the receiver. As the frequency is reduced, effective antenna heights h /X. and h /\ in wavelengths bece saller, and grund-reflected energy tends t cancel direct-ray energy at the lwer part f the cn vlue, where scattering efficiency is greatest. frequency gain functin H lss. in (9. ) is an estiate f the crrespnding increase in transissin This functin first decreases rapidly with increasing distance and then appraches a cnstant value. Fr h /K > 4 a/d and h /K > 4a/d, H is negligible. The upper liit te re f H as h and h apprach zer is H => 6 + A db, where A :s the diffractin atten te re uatin ver a sth earth, relative t free space, at 9=0. Fr frequencies up t 0 GHz. A ay be estiated fr the CCIR Atlas f Grund Wave Prpagatin Curves [955, 959]. H shuld rarely exceed 25 db except fr very lw antennas. The frequency gain functin, lengths, path asyetry, and the paraeter n The H, depends n effective antenna heights in ters f wave shwn n figure 9.2 and defined as n = h [l+( N x 0~ N xi" )exp(-3.8 h x 0~ )] (9.3a) SOS I h = sd6/(l + s) k. (9.3b) The paraeters r. and r are defined as r = 4ir0h l\, r, = 4IT6 h /X. (9.4a) te 2 re where 0 is the angular distance in radians, and the effective antenna heights h, h are in the sae units as the radi wave length, \. In ters f frequency r and r ay be written r =4.929fh, r =4.928fh (9.4b) te 2 re where 6 is in radians, f in MHz, and h, h are in kileters. te re Fr the great ajrity f transhrizn paths, s is within the range 0.7sss. The effect f very sall values f s, with a «p, ay be seen in figures. 5 t III. 9, which have been cputed fr the special case where effective transitting and receiving antenna heights are equal. 9-3

105 a) Fr T greater than r equal t : Read H (r,) and H (r,) fr figure 9.3; then H I is H =[H (r ) + H (r )]/2+AH (9.5) l O C where AH = 6(0.6 - lg n )lg s lg q 8 = a JP O n O = r? /sr i I If Ti > 5 the value f H fr ti = 5 is used. The crrectin ter AH is zer fr ss T = 4, s *, r q = and reaches a axiu value, AH =3.6 db, fr highly asyetrical paths when r\ =. The value f AH ay be cputed as shwn r read fr the ngra, figure 9.4. A straight line between values f s and q n their respective scales intersects the vertical line arked. This pint f intersectin when cnnected by a straight line t the apprpriate value f n intersects the AH scale at the desired value. The fllwing liits shuld be applied in deterining AH : If s > 0 r q 2: 0, use s = 0 r q = 0. If s 0. r q s 0., use s = 0. r q = 0.. If AH a[h (r.) + H (r,)]/2, use H = H (r,) + H (r,). l c l c If AH wuld ake H negative, use H =0. b) Fr r less than : s First btain H fr n = as described abve, then read H fr TI =0 fr figure s 0 8 The desired value is fund by interplatin: H ( ns < ) - H (n 8 = 0) JH^ = ) - H^ = 0)] (9.6) The case rj = 0 crrespnds t the assuptin f a cnstant atspheric refractive index. A special case, h = h, r = r, ccurs frequently in systes design. Fr this case H has been pltted versus r in figures III. 5 t III. 9 fr n =, 2, 3, 4, 5 and fr s = 0., 0.25, 0. 5, 0.75 and. Fr given values f n and s, H is read directly fr the graphs using linear interplatin. N crrectin ter is required. Fr n < the value f H (n = ) is read fr figure 9.3 with r = r and H (T\ =0) is read fr figure 9.5 OS <_. s as befre. 9-4

106 9.3 The Scattering Efficiency Crrectin, F The crrectin ter F in (9. ) allws fr the reductin f scattering efficiency at great heights in the atsphere: F =.086(r, /h )(h - h - h - h ) db (9.7) O S O O L.t Lir where n and h are defined by (9. 3) and h, is defined as 's ' h x = sd 8 e/(l + s) 2, D B =d - d Lt- d Lr (9.8) The heights f the hrizn bstacles, h, h and the hrizn distances d d are defined ijt l_«r ^t I r in sectin 6. All heights and distances are expressed in kileters. The crrectin ter F large that h exceeds h, by re than 3 kileters, exceeds Z decibels nly fr distances and antenna heights s 9-5

107 9.4 Expected Values f Frward Scatter Multipath Cupling Lss Methds fr calculating expected values f frward scatter ultipath cupling lss are given in several papers, by Rice and Daniel [ 955], Bker and de Bettencurt [ 955], Staras [ 957], and Hartan and Wilkersn [ 959]. This reprt uses the st general ethd available depending n the paper by Hartan and Wilkersn [ 959]. As explained in sectin 2, the path antenna gain is G = G + G - L db (9.9) P t r gp where G and G are free space antenna gains in decibels relative t an istrpic radiatr. The influence f antenna and prpagatin path characteristics in deterining the lss in path antenna gain r ultipath cupling lss L separately. are interdependent and cannt be cnsidered This sectin shws hw t estiate nly that cpnent f the lss in path antenna gain which is due t phase incherence f the frward scattered fields. This quantity is readily ap- prxiated fr figure 9.6 as a functin f n, defined by (9. 5), and the rati 9/U, where U = 26 is the effective half-pwer antenna beawidth. If the antenna beawidths are equal, U = U, and if s =, values f L fr figure 9.6 are exact. When antenna beawidths t r gp are nt equal the lss in gain ay be apprxiated using Q = sfntt. The relatin between the free-space antenna gain G in decibels relative t an is- trpic radiatr and the half pwer beawidth ft = 26 was given by (2. 4) as: G = lg 6 = lg 2 db where 6 and il are in radians. Assuing 56% aperture efficiencies fr bth antennas, 6/n a Ö(n t ft r )"' / * a exp [ (G fg )J (9.0) where 8 is the angular distance in radians and G, G are the free space gains in decibels. Sectin 2 shws that the gain fr parablic dishes with 56% aperture efficiency ay be cputed as (2. 6) : G = 20 lg D + 20 lg f db where D is the diaeter f the dish in eters and f is the frequency in MHz. Fr diple-fed parablic antennas where 0 < D/\ < 25. an epirical crrectin gives the fllwing equatin fr the antenna gain (2. 7) : G = 23.3 lg D lg f db 9-6

108 The general ethd fr calculating L requires the fllwing paraeters: v= n /2, u = 6 /6 (9.) 8 r t Fr su 2, n * a I b. Fr s u s, n = ß /6_ (9. 2a) t r n = (n v)/f(v) (9.2b) f(v) = [ v] [ exp(-0. 56v)l" ' (9.3) where n, s, a and 0 have been defined, 6 and 6 are the effective half-pwer sei- O t r beawidths f the transitting and receiving antennas, respectively, and f(v) as defined by (9. 3) is shwn n figure Figure 9.8 shws L versus n fr varius values f the prduct su Fr su < read gp figure 9.8 fr /(su) instead f su Cbinatin f Diffractin and Scatter Transissin Lss Fr transissin paths extending nly very slightly beynd line-f-sight, diffractin will be the dinant echanis in st cases and scattering ay be neglected. Cnversely, fr lng paths, the diffracted field ay be hundreds f decibels weaker than the scattered field, and thus the diffractin echanis can be neglected. In interediate cases, bth ech- aniss have t be cnsidered and the results cbined in the fllwing anner: Figure 9.9 shws a functin, R(0. 5), which depends n the difference between the dif- fractin and scatter transissin lss. Calculate this difference (L, - L, ) in decibels, dr sr deterine R(0. 5) fr figure 9.9 and then deterine the resulting reference value f hurly edian transissin lss, L, fr the relatin cr L cr = L dr- R( - 5) <9 - U) If the difference between the diffracted and the scattered transissin lss values exceeds 5 db, the resulting value f L will be equal t L, if it is saller than L, r t L cr ^ dr sr sr if this is the saller value. In general, fr st paths having an angular distance greater than 0.02 radians the diffractin calculatins ay be itted; in this case, L = L. cr sr 9-7

109 THE ATTENUATION FUNCTION, F(0d) d IS IN KILOMETERS AND 0 IS IN RADIANS (0 75SSSI) Q CD CD d F'gur«9. 9-8

110 THE PARAMETER 77 s (h Q ), USED TO COMPUTE H ( ^,=0.5696h 0 t ( NgXlO NfKl0 6 )e i * h * t0 J h 0 = Sd«/(i + S) k MJ, = h 0 [ ^s _L r-^i r I II h 0 IN KILOMETERS Figure

111 THE FREQUENCY GAIN FUNCTION, H 0 Figur«

112 CO J-* ^^, -_ ^_, _ ^_. CM -t C/ 0> -«4 O CO CD J L J I I I ' ' I ' I I I I I I I I I I I III llll LIHINIIIIIMI J I I L I I ' ' ' > I ^n J» t>j rj llllllllllllllllllljlll -2 t _ N w * yi lllllllllllilllllllhll L 5 ß II t> T M CD r» >«. O CO J> UD CO -i CO a O t n CO O a 2 O CD > CD O H t z t> X J 02 9-

113 THE PARAMETER H Q FOR 77,. =0 (0.7SSS ) 0 5 L-, 0 \f 5 a: <J a 20 V?5 I 0 5! -jvy fsl UHJJ In u /y^ j ^ - > *tjf ^ s s <s>f 35 ' //&y r ^03 /*< /<*$ / 40 Ä / y > '/ \IJ* > ; pf M ' ^y. <&jt.<^. J* / > w Si i/^ )0I 00? I 2 Figur«

114 LOSS IN ANTENNA GAIN, L gp assuing equal free space gains G, and G r at the terinals f a syetrical path ü t =ß r,s«! 60 Sif5^rJ]i5?^Tji7T~T"l PflHQillHI JURIS ffll igfflnbsoh «SS HU «IPS MS,tn^twaBE^r-" U.I U.d U.O I 0 z Rati ö/ß f angular distance 9 t half-pwer antenna beawidth, l Figure

115 f(l/) 7 O c <» X r % 's > > rs> r n rvj CM <_~ \ ^ ^ V >. ^ v >, \^ s. > V v _^ V \ V ^. \ V \ ^ > V V ^ V \ V v 5 v V 5 <X> C*> S I 9 ä X) 9-4

116 60 LOSS IN PATH ANTENNA GAIN vs ft ' r~ 56 CO _J Ld CD O ÜJ Q < <L S M s M = / / i=, ~7 // >/u.=2 U / = 4 ~~MJ LJJ I /// A / / / / / / / CO CO Q. 6 2 / // /i / 8 *f I A n Figure

117 R(0.5) IN DECIBELS i 00 OJ r CO _i_ i 03 O a CD > CO CD f\j Lcr = MEDIAN BASIC TRANSMISSION LOSS CORRESPONDING TO THE RESULTANT FIELD. L, = BASIC TRANSMISSION LOSS CORRESPONDING TO THE DIFFRACTED FIELD, dr Lsr = MEDIAN BASIC TRANSMISSION LOSS CORRESPONDING TO THE RAYLEIGH DISTRIBUTED SCATTERED FIELD. _ ffi C CO 33 < r I" I c * H P en > XI Zu CO c x g > Sen?"^ L_ 3J O c > O i e < g > CD Q c cr> 8 9-6

118 0. LONG-TERM POWER FADING The variability f trpspheric radi transissin lss depends upn changes in the atsphere and upn cplex interrelatinships between varius prpagatin echaniss. Shrt-ter variability r "phase interference fading," assciated with siultaneusly ccuring des f prpagatin, is discussed in annex V. The effects f this type f fading expected within an hur are allwed fr by deterining an hurly edian ra-carrier-t-rsnise rati which defines the grade f service that will be prvided. Lng-ter pwer fading, identified with the variability f hurly edian values f transissin lss, is usually due t slw changes in average atspheric refractin, in the degree f atspheric stratificatin, r in the intensity f refractive index turbulence. An estiate f the lng-ter pwer fading t be expected ver a given path is iprtant t insure adequate service ver the path. The pssibility that unusually high interfering fields ay ccur fr an appreciable percentage f tie places restrictins n services perating n the sae r adjacent frequencies. The basis fr the ainly epirical predictins f lng-ter variability given here needs t be well understd in rder t appreciate their value as well as their liitatins. An increase in atspheric refractin increases lng distance diffractin r frward scatter fields but ay lead t ultipath fading prbles ver shrt paths. Increased turbulence f the atsphere ay result in either an increase r a decrease f radi transissin lss depending n the geetry f a particular path and n the dinance f varius prpagatin echaniss. Increased stratificatin favrs prpagatin by reflectin fr elevated layers and seties the guiding f energy by ducts r layers. Such stratificatin usually increases lng-distance fields but ay be assciated with prlnged fadeuts at shrt distances. Just beynd radi line-f-sight, fading rate and fading range depend in a very cplex anner n the relative iprtance f varius prpagatin echaniss. During perids f layering and ducting in the atsphere, transissin lss shws a tendency t g int relatively deep fades, with duratins fr less than a inute t re than an hur. Ordinarily a diffractin signal fades slwly if at all, and the fades are f relatively shrt duratin and very deep. A trpspheric frward scatter signal, n the ther hand, exhibits the rapid and severe fading characteristic f the Rayleigh distributin. An interediate type f fading results when the scattered pwer is nearly equal t pwer intrduced by se echanis such as diffractin, fr which the variatin in tie is usually very slw. Aircraft reflectins intrduce rapid, intense, and relatively regular fading. Meter bursts and se types f inspheric prpagatin add spikes t a paper chart recrd. Space-wave fadeuts [Bean, 954] ay represent pwer fading due t defcusing f radi energy in se regins f space, (radi hles) accpanied by a fcusing effect and 0-

119 Signal enhanceent in ther regins [Dherty, 952; Price, 948], r ay crrespnd t phase interference fading phenena. In teperate cntinental cliates, space-wave fadeuts are likely t ccur priarily at night and st frequently during the suer nths; they are re frequent at UHF than at VHF, and their ccurrence can be crrelated with the ccurrence f grund-dified refractive index prfiles [Barsis and Jhnsn, 962]. Such fading predinates in gegraphic areas where layers and ducts ccur frequently. Ordinary space diversity des nt appear t be helpful in vercing this type f fading. During perids f unifr refractive-index lapse rates, prlnged fadeuts are uch less intense r d nt exist. Seties thse that d exist are caused by ultipath reflectins which arrive in such a phase and aplitude relatinship that a slight change in the lapse rate will cause a large change in the resultant field. The latter type can be verce in st instances by either relcating the terinal antennas r by the use f space diversity. General discussins f the tie fading f VHF and UHF radi fields will be fund in reprts by Bullingtn [950], du Castel [957a], Chernv [ 956],Grsskpf [ 958], Krasil'nikv [ 949], Tritski [957b], and Ugai [ 96]. Silveran [ 957], discusses se f the thery f the shrt-ter fading f scatter signals, Breer [ 957] discusses signal distrtin due t trpspheric scatter, while Beckann [ 96b] cnsiders related deplarizatin phenena. The bserved crrelatin f radi data with varius eterlgical paraeters is discussed by Bean [ 956, 96], Bean and Cahn [ 9 57], du Castel and Mise [ 957], Jsephsn and Blquist [ 958], Mise [958, 960a, b, c, 96], Mler and Hlden [960], and Ryde [946], Meterlgical paraeters such as surface refractivity and the height gradient f refractive index have been fund re useful as a basis fr predicting reginal changes than fr predicting diurnal r seasnal variatins. In this reprt eterlgical infratin has been used t distinguish between cliatic regins, while radi data are depended n t predict lng-ter variability abut the cputed lng-ter edian value in each f these regins. The basic data used in develping these estiates f lng-ter pwer fading were recrded in varius parts f the wrld ver re than a thusandprpagatin paths. Path distances extend fr within line-f-sight t abut 000 kileters, and frequencies range fr 40 MHz t 0 GHz. As re data are cllected, particularly in regins where little infratin is currently available, these estiates shuld be re-exained and revised. Allwances shuld seties be ade fr predictable lng-ter variatins in antenna gain, interference due t reflectins fr aircraft r satellites, and variatins in equipent perfrance. Micrwave attenuatin due t rainfall, discussed in sectin 3, shuld be allwed fr in estiating 0-2

120 the variability f transissin lss at frequencies abve 5 GHz. f xygen and water vapr absrptin ay be iprtant abve 5 GHz. The lng-ter variability It is ften desirable t specify rather precisely the cnditins fr which an estiate f pwer fading characteristics is desired. Fr instance, the average frequency dependence f lng-ter variability ver a given type f prfile depends critically n the relative d- inance f varius prpagatin echaniss, and this in turn depends n cliate, seasn, tie f day, and average terrain characteristics. Cliatic regins ay be defined in several different ways: () by gegraphic areas n a ap, (2) by averarge eterlgical cnditins, (3) by the predinance f varius prpagatin echaniss r (4) by averages f available data. In varius s-called "cliates," at different ties f day r seasns f the year, different prpagatin echaniss ay be dinant. Fr exaple, in a cntinental teperate cliate the characteristics f a. re- ceived signal ver a given path ay be quite different in the early rning hurs f May than during the afternn hurs in February. Prbably the st serius bstacle t iprveent f ethds fr predicting the characteristics f trpspheric prpagatin is the lack f adequate paraeters fr des- cribing layers and ducts. It is iprtant t learn hw t describe atspheric stratifica- tin well enugh t predict the intensity f fcusing, defcusing, and reflectin as well as the percentage f tie such phenena are likely t be iprtant fr a given regin, tie f day r seasn, and radi frequency. When it beces pssible t describe the actual inh- geneus, stratified, and turbulent atsphere re adequately, it shuld als be fund wrthwhile t 'ix" predicted cuulative distributins f transissin lss fr a variety f prpagatin echaniss. Based n ur current knwledge f eterlgical cnditins and their effects n radi prpagatin, the Internatinal Radi Cnsultative Cittee[ CCIR 963f ] has defined several "cliates." A large aunt f data is available fr cntinental teperate and aritie teperate cliates. Other cliatic regins, where few data are available, are discussed in annex III. The divisin int cliates is sewhat arbitrary, based n present knwledge f radi eterlgy, and is nt necessarily the sae as eterlgical cliates [ Haurwitz and Austin, 944]. Three iprtant effects f the atsphere n radi prpagatin have been cnsid- ered in defining the varius cliates. These are: bending f the radi rays, the effects f atspheric turbulence, and the degree and stability f atspheric stratificatin. aunt a radi ray is bent and the intensity f atspheric turbulence are usually crrelated The 0-3

121 with the surface refractivity, N. The intensity and stability f varius types f stratificatin are ften nt well crrelated with N. Rather stable and extensive layers f dry war air ver relatively cl ist air tend t fr ducts fr VHF and UHF cunicatin. The sharp isture discntinuity bends the radi rays, which then tend t fllw the dry-ist air interface. The phenenn f super-refractin, assciated with the ccurrence f radi ducts clse t the surface f the earth, is essentially a fine weather phenenn. Inland, during fine weather, ducting is st nticeable at night. Over the sea super-refractin is st arked where the war dry air f an adjacent land-ass is able t extend ut ver a cparatively cl sea. Rugh terrain and high winds bth tend t increase ixing in the atsphere and cnsequently reduce super-refractin. Areas f divergence, usually favrable fr elevated duct fratin, appear t be st persistent ver cean areas fr 0 t 40 nrth and suth latitude, especially during winter nths. [ Mler and Hlden, I960; Randall, 964]. Elevated ducts are usually less iprtant fr trpspheric prpagatin than thse clse t the surface. Wrld aps f iniu nthly ean N, figure 4., and f the annual range f nthly ean N, figure III. 3, ay be useful in deciding which cliate r cliates are applicable in a given regin. The bundaries between varius cliatic regins are nt well defined. In se cases it ay be necessary t interplate between the curves fr tw cliates giving additinal weight t the ne whse ccurrence is cnsidered re likely. Se iprtant characteristics f the cliatic regins fr which estiates f tie variability are shwn, are nted belw:. Cntinental Teperate characterized by an annual ean N f abut 3 20 N- units with an annual range f nthly ean N f 20 t 40 N-units. A cntinental cliate in a large land ass shws extrees f teperature in a "teperate" tne, such as 30 * t 60 nrth r suth latitude. Prnunced diurnal and seasnal changes in prpagatin are expected t ccur. On the east cast f the United States the annual range f N ay be as uch as 40 t 50 N-units due t cntrasting effects f arctic r trpical aritie air asses which ay ve int the area fr the nrth r fr the suth. 2. Maritie Teperate Overland characterized by an annual ean N f abut 320 N-units with a rather sall annual range f nthly ean N f 20 t 30 N-units. Such cliatic regins are usually lcated fr 20 * t 50 * nrth r suth latitude, near the sea, where prevailing winds, unbstructed by untains, carry ist aritie air inland. These cnditins are typical f the United Kingd, the west casts f Nrth Aerica and Eurpe and the nrthern castal areas f Africa. 0-4

122 Althugh the islands f Japan lie within this range f latitude the cliate differs in shwing a uch greater annual range f nthly ean N, abut 60 N-units, the prevailing winds have traversed a large land ass, and the terrain is rugged. One wuld therefre nt expect t find radi prpagatin cnditins siilar t thse in the United Kingd althugh the annual ean N is 30 t 320 N-units in each lcatin. Cliate is prbably re apprpriate than cliate 2 in this area. Ducting ay be very iprtant in castal and ver- sea areas f Japan. 3. Maritie Teperate Oversea castal and versea areas with the sae general characteristics as thse fr cliate 2. The distinctin ade is that a radi path with bth hrizns n the sea is cnsidered t be an versea path; therwise cliate 2 is used. Ducting is rather cn between the United Kingd and the Eurpean Cntinent, and in suer alng the west cast f the United States. 4. Maritie Subtrpical Overland characterized by an annual ean N f abut 370 N-units with an annual range f nthly ean N f 30 t 60 N-units. Such cliates ay be fund fr abut 0* t 30 nrth and suth latitude, usually n lwlands near the sea with definite rainy and dry seasns. fr a cnsiderable part f the year. Where the land area is dry radi-ducts ay be present 5. Maritie Subtrpical Oversea cnditins bserved in castal areas with the sae range f latitude as cliate 4. Typical f this cliate is the nrthwest cast f Africa. 6. Desert, Sahara characterized by an annual ean N f abut 280 N-units with s year-rund seiarid cnditins. The annual range f nthly ean N ay be fr 20 t 80 N-units. 7. Equatrial aritie trpical cliate with an annual ean N f abut 360 N-units and annual range f 0 t 30 N-units. Such cliates ay be bserved fr 20 * N t 20 * S latitude and are characterized by ntnus heavy rains and high average suer teperatures. Africa. Typical equatrial cliates ccur alng the Ivry Cast and in the Cng f 8. Cntinental Subtrpical typified by the Sudan and nsn cliates, with an an- nual ean N f abut 320 N-units and an annual range f 60 t 00 N-units. This is a ht s cliate with seasnal extrees f winter drught and suer rainfall, usually lcated fr 20 t 40" N latitude. 0-5

123 A cntinental plar cliate, fr which n curves are shwn, ay als be defined. Teperatures are lw t derate all year rund. The annual ean N is abut 30 N- units with an annual range f nthly ean N f 0 t 40 N-units. Under plar cnditins, 8 which ay ccur in iddle latitudes as well as in plar regins, radi prpagatin wuld be expected t shw sewhat less variability than in a cntinental teperate cliate. ter edian values f transissin lss are expected t agree with the reference values L. cr Lng- It is difficult t predict the percentage f tie that high fields due t ducting cndi- tins ay be expected t ccur. listed by Bker [ 946] are: Se f the better-knwn aritie areas f super-refractin suer nths; a) British Isles, Atlantic casts f France, Spain and Prtugal and all year; the Mediterrean Sea b) Red Sea,Gulf f Aden, Persian Gulf c) west and suth casts f Australia, New Suth Wales and New Zealand d) Pacific cast f United States and Canada, Atlantic cast nrth f Chesapeake Bay e) casts f China and Japan f) plar regins, althugh se sub-refractin ay als be expected a) west and suthwest cast f Africa, especially arked in suer b) west cast f India and the Bay f Bengal except during the suth- west nsn c) nrthern part f the Arabian Sea,especially during the Indian ht seasn d) nrth and nrthwest casts f Australia except during the nrth- west nsn It is apparent that the st intense super-refractin is encuntered in a trpical (nt equa- trial) cliate, in trade wind areas ver the ceans, and in st f the principal deserts f the wrld. High untain areas r plateaus in a cntinental cliate are characterized by lw values f N and year-rund seiarid cnditins. The central part f Australia with its s ht dry desert cliate and an annual range f N as uch as 50 t 70 N-units ay be inter- ediate between cliates and 6. Predictin f lng-ter edian reference values f transissin lss, by the ethds f sectins 3 t 9, takes advantage f thery in allwing fr the effects f path geetry and radi ray refractin in a standard atsphere. Meterlgical infratin is used t dis- tinguish between cliatic regins. Median values f data available in each f these regins are related t the lng-ter reference value by eans f a paraeter V(50, d ) which is a functin f an "effective distance, " d, defined belw. Lng-ter fading abut the edian fr each cliatic regin is pltted in a series f figures as a functin f d. Fr regins where a large aunt f data is available, curves are presented that shw frequency-related effects. cliate. ) (Seasnal and diurnal changes are given in annex III fr a cntinental teperate 0-6

124 0. The Effective Distance, d e Epirical estiates f lng-ter pwer fading depend n an effective distance, d, which has been fund superir t ther paraeters such as path length, angular distance, distance between actual hrizns, r distance between theretical hrizns ver a sth earth. The effective distance ake allwance fr effective antenna heights and se allw- ance fr frequency. Define 9 as the angular distance where diffractin and frward scatter transissin si lss are apprxiately equal ver a sth earth f effective radius a = 9000 kileters, and define d as Then: si si A d = 65(00/f) 3 k (0. ) si The value f d is cpared with the sth-earth distance, d, between radi hrizns: si s d = d- 3 stzrt~ - 3N/"ZTJ k (0.2) s te re where the effective antenna heights h and h are expressed in eters, the path length d te re in kileters and the radi frequency f in MHz. It has been bserved that the lng-ter variability f hurly edians is greatest n the average fr values f d nly slightly greater than d. The effective distance d is arbitrarily defined as: d = 30/ l+(d -d )/dl k, fr d s d (0.3a) e l si s ' s si d = 30 + d -d k, frd >d (0.3b) e s si s si 0-7

125 0.2 The Functins V(50. d ) and Y(p. d ) e e The predicted edian lng-ter transissin lss fr a given cliatic regin L (50), n characterized by a subscript n, is related t the calculated lng-ter reference value L, cr by eans f the functin V(50, d ) L (50) = L - V (50. d ) db (0.4) n er n e where L (50) is the predicted transissin lss exceeded by 50 percent f all-hurly edians n in a given cliatic regin. V (50, d ) is shwn n figure 0. fr several cliates as a functin f the effective distance d. Fr the special case f frward scatter during winter afternns in a teperate cntinental cliate, V(50) = 0 and L(50) = L. In all ther cases, the calculated lng-ter reference value L, the particular cliatic regin r tie perid cnsidered. shuld be adjusted t the edian L, (50) fr The functin F(8d) in the scatter predictin f a lng-ter reference edian cntains an epirical adjustent t data. ter V(50, d ) prvides a further adjustent t data fr all prpagatin echaniss and fr different cliatic regins and perids f tie. In general, the transissin lss exceeded (00-p) percent f the tie is The L (p) = L (50) - Y (p, d ) db (0.5) n n n e where Y (p, d ) is the variability f L (p) relative t its lng-ter edian value L. (50). n e n n Fr a specified cliatic regin and a given effective distance, the cuulative distributin f transissin lss ay be btained fr (0. 5). issin lss is ften nearly lg-nrally distributed. In a cntinental teperate cliate trans- The standard deviatin ay be as uch as twenty decibels fr shrt transhrizn paths where the echaniss f diffractin and frward scatter are abut equally iprtant. When a prpagatin path in a aritie teperate cliate is ver water, a lg-nral distributin ay be expected fr L(50) t M99.9), DUt cnsiderably higher fields are expected fr sall percentages f tie when prnunced superrefractin and ducting are present. 0-8

126 0. 3 Cntinental Teperate Cliate Data fr the U.S.A., West Gerany, and France prvide the basis fr predicting lng-ter pwer fading in a cntinental teperate cliate. Mre than half a illin hurly edian values f basic transissin lss recrded ver se tw hundred paths were used in develping these estiates. Figure 0.2 shws basic estiates Y (p) f variability in a cntinental teperate cliate. Curves are drawn fr 0 and 90 percent f all hurs f the day fr suer, winter and all year fr a "typical" year. In the nrthern teperate zne, "suer" extends fr May thrugh Octber and "winter" fr Nveber thrugh April. A 'frequency factr" g(p, f) shwn in figure 0.3 adjusts the predicted variability t allw fr frequency-related effects: Y(p) = Y (p,d e )g(p,f) (0.6) The functin g(p, f) shws a arked increase in variability as frequency is increased abve 00 MHz t a axiu at 400 t 500 MHz. Variability then decreases till values at r 2 GHz are siilar t thse expected at 00 MHz. The epirical curves g(p, f) shuld nt be regarded as an estiate f the dependence f lng-ter variability n frequency, but represent nly an average f any effects, se f which are frequency-sensitive. The apparent frequency dependence is a functin f the relative dinance f varius prpagatin echaniss, and this in turn depends n cliate, tie f day, seasn, and the particular types f terrain prfiles fr which data are available. Fr exaple, a heavily frested lw altitude path will usually shw greater variability than that bserved ver a treeless high altitude prairie, and this effect is frequency sensitive. An allwance fr the year-t-year variability is als included in g(p,f). Data suarized by Williasn et al [960] shw that L(50) varies re fr year t year than Y(p). Assuing a nral distributin f L within each year and f L(50) fr year t year, L wuld be nrally distributed with a edian equal t L(50) fr a "typical" year. Y(p) is then increased by a cnstant factr, which has been included in g(p, f). Estiates f Y(0) and Y(90) are btained fr figures 0.2, 0.3 and fr equatin 0.6). These estiates are used t btain a predicted cuulative distributin using the fllwing ratis: Y(0.0) = 3.33 Y(0) Y(99.99) = 2.90 Y(90) Y(0.) =2.73Y(0) Y(99.9) =2.4Y(90) (0.7) Y(l) =2.00Y(0) Y(99) =.82Y(90) 0-9

127 Fr exaple, r assue f = 00 MHz, d =0 k, and a predicted reference edian e basic transissin lss, L L = 80 db, s that V(50,d ) =,9 db, (figure 0.). Y (0, d ) bcr e e = 8 db, and Y (90, d ) = db, (figure 0.2), g(0, f) = g(90, f) =.05 (figure 0.3). Then e Y(0) =. 05 Y (0) = 8. 5 db, and Y(90) =. 05 Y (90) = -6. db. Using the ratis given abve: Y(0.0) = 28.3, Y(0.) = Y(l) = 7.0, Y(0) = 8.5, Y(99.99) = -7.7, Y(99.9) = -4.7, Y(99) = -., Y(90) = -6.. The edian value is L (50)= L - V(50) = 78. db and the predicted distributin f basic transissin lss is; L(0.0) = L(0.) = 5.9, L(l) = 6., L(0) = 69.6, L(50) = 78., L(90) = 84.2, L(99) = 89.2, U99.9) = 92.8 ^d M99.99) = 95.8 db. These values are pltted as a functin f tie availability, p, n figure 0.4 and shw a cplete predicted cuulative distributin f basic transissin lss. Fr antennas elevated abve the hrizn, as in grund-t-air r earth-t-space cunicatin, less variability is expected. This is allwed fr by a factr f(9, ) discussed in annex III. Fr transhrizn paths f(9 ) is unity and des nt affect the distributin. Fr line-f- sight paths f(9, ) is nearly unity unless the angle f elevatin exceeds 0. 5 radians. Allwance ust seties be ade fr ther surces f pwer fading such as attenuatin due t rainfall r interference due t reflectins fr aircraft that ay nt be adequately represented in available data. Fr exaple, at icrwave frequencies the distributin f water vapr, xygen, rain, snw, cluds and fg is iprtant in predicting lngter pwer fading. Let Y, Y Y represent estiates crrespnding t each f these surces f variability, and let p, be the crrelatin between variatins due t surces i and j. Then the ttal variability is apprxiated as: Y 2 (P) - Y x 2 (p) + 2 Y.Y.p.. (0.8) = i.j = l i<j where Y(p) is psitive fr p < 50 percent, zer fr p = 50 percent, and negative fr p > 50 percent. Sectin 3 shws hw t estiate Y (p) and Y (p) fr atspheric absrptin by a r xygen and water vapr, and fr rain absrptin respectively. Let p be the crrelatin la between variatins Y f available data and variatins Y due t icrwave absrptin by 0-0

128 xygen and water vapr. Let p be the crrelatin between Y and Y. Assuing that l r r p a =, p^ = 0.5, and p^ = 0, Y 2 (p) = (Y + Y ) 2 + Y Z + YY (0.9) a r r This ethd was used t allw fr the effects f rainfall at frequencies abve 5 GHz, fr 99 and percent f all hurs in figures I. 6 t I. f annex I. Figures 0. 5 t 0. 0 shw variability, Y(p) abut the lng-ter edian value as a functin f d fr perid f recrd data in the fllwing frequency grups; 40-88, 88-08, , , , and > 000 MHz. The curves n the figures shw predicted values f Y(p) fr all hurs f the year at the edian frequency in each grup. These edians are: 47., 98.7, 92.8, 7, 700, and 500 MHz fr data recrded in a cntinental teperate cliate. Equatin (0.6) and figures 0.2 and 0.3 were used t btain the curves in figures 0. 5 t An analytic functin fitted t the curves f V(50, d ) and Y (p, d ) is given in annex III. Diurnal and seasnal variatins are als discussed and functins listed t predict variability fr several ties f day and seasns. 0-

129 0.4 Maritie Teperate Cliate Studies ade in the United Kingd have shwn appreciable differences between prpagatin ver land and ver sea, particularly at higher frequencies. Data fr aritie teperate regins were therefre classified as verland and versea, where versea paths are categrized as having the castal bundaries within their radi hrizns. tend ver a ixture f land and sea are included with the verland paths. The data were divided int frequency grups as fllws: Bands I and II (40-00 MHz) Band III ( MHz) Bands IV and V ( MHz). Paths that ex- Lng-ter variability f the data fr each path abut its lng-ter edian value is shwn as a functin f effective distance in figures 0. t Curves were drawn thrugh edians f data fr each percentage f tie p = 0.0, 0.,, 0, 90, 99, 99.9, Figures 0. t 0. 6 shw that it is nt practical t use a frula like (0.6) fr the aritie teperate cliate, because the frequency factr g(p, f) = Y/Y is nt independent f d, as it is in the e case f the cntinental teperate cliate. The iprtance f trpspheric ducting in a ari- tie cliate is ainly respnsible fr this difference. grups. These figures denstrate greater variability versea than verland in all frequency The very high fields nted at UHF fr sall percentages f tie are due t per- sistent layers and ducts that guide the radi energy. In cases f prpagatin fr great dis- tances ver water the fields apprach free space values fr sall percentages f tie. Curves have been drawn fr thse distance ranges where data peritted reasnable estiates. Each curve is slid where it is well supprted by data, and is dashed fr the reainder f its length. 0-2

130 0. 5 Other Cliates A liited aunt f data available fr ther cliatic regins has been studied, [CCIR 963f ]. Curves shwing predicted variability in several cliatic regins are shwn in annex III, figures III. 25 t III. 29. At ties it ay be necessary t predict radi perfrance in an area where few if any easureents have been ade. In such a case, estiates f variability are based n whatever is knwn abut the eterlgical cnditins in the area, and their effects n radi prpagatin, tgether with results f studies in ther cliatic regins. If a sall aunt f radi data is available, this ay be cpared with predicted cuulative distributins f transissin lss crrespnding t sewhat siilar eterlgical cnditins. In this way estiates fr relatively unknwn areas ay be extraplated fr what is knwn. 0.6 Variability fr Knife-Edge Diffractin Paths The variability f hurly edians fr knife-edge diffractin paths can be estiated by cnsidering the path as cnsisting f tw line-f-sight paths in tande. The diffracting knifeedge then cnstitutes a cn terinal fr bth line-f-sight paths. The variability f hurly edian transissin lss fr each f the paths is cputed by the ethds f this sectin and characterized by the variability functins Vj(p) = VjISO) + Yj(p) V 2 (p) = V 2 (50) + Y 2 (p) db db During any particular hur, the ttal variability functin V fr the diffractin path wuld be expected t be the su f V plus V. T btain the cuulative distributin f all values f V applicable t the ttal path a cnvlutin f the individual variables V and V ay be eplyed ( Davenprt and Rt, 958]. Assuing that V and V are statistically independent variables, their cnvlutin is the cuulative distributin f the variable V = V + V. The cuulative distributin f V ay be btained by selecting n equally-spaced percentage values fr the individual distri- butins f V (p) and V (p) calculating all pssible sus V =V.+ V. and fring the cuulative distributin f all values V, btained in this anner. k Anther ethd f cnvlutin that gives gd results requires the calculatin and 2 rdering f nly n, instead f n, values f V. As befre V (p) and V (p) are btained fr n equally spaced percentages. Then ne set is randly rdered cpared t the ther s that the n sus V = V. + V are randly rdered. The cuulative distributin f these sus then prvides the desired cnvlutin f V. and V. If the distributin f V - V is desired this is the cnvlutin f V and -V. 0-3

131 V(50,d e ) IN DECIBELS CD 5 3 s -z. F x en / A / V* r «A N N ^ / I S l/l A 5 N ft \ / - s t tr \ / // w \ / \» / // I J, / { ' s! en!' > / /! i _, i i «! I i i A. CONTINENTAL TEMPERATE 2. MARITIME TEMPERATE OVERLAI* 3. MARITIME TEMPERATE OVERSE 4. MARITIME SUBTROPICAL OVERL CD -^ O) <J» MARITIME SUBTROPICAL OVERSEA DESERT. SAHARA EQUATORIAL CONTINENTAL SUBTROPICAL r Z J> -i i i i -i i i i i i -n c z H 5 < 2 w Q. CO O a) > 3) z C/5 0-4

132 Y 0 (90,d e ) indb Y 0 ( i,d e ) IN db < z g -I -n > % c O 2 3) Q z i' C") H i H. 3i <f =r H > T O H $ 2 JJ 3 i > H ^ <, > r =r > H -i a i; < 0-5

133 POWER FADING ADJUSTMENT FACTOR g(p,f) BASED ON U.S. OVERLAND DATA FREQUENCY IN MHz FREQUENCY IN MHz Figur«

134 EXAMPLE OF A CUMULATIVE DISTRIBUTION L b (p) VERSUS p HU 50 J 60 V) V) _l z S 70 < e» TIME AVAILABILITY, p Figur«

135 Y(p)= VARIABILITY ABOUT THE LONG-TERM MEDIAN. IN DECIBELS O - < -n en H > c 7 Q I'l Ü a. fo d s XI en 8 c3i <=3 n ii O-i N> rv> LM en <r> yy f *?*k Pi / // / / / / < 0 0 /» & "^ ^\ G r v 5 J"» v r c / \»H 0 r^> \ \ O 0 f«\ M 0 M M - / "» ' *i t i L«\ D 0 I H / CD \. / <_w \ I M / / C=> \ / / S \\ cr«cr> CT> v i a c; <-<-< < 2 cr> ) C3 ~i / J 2_ -g «i Q -M M U J 0 O W W T) <-< i -< I - ~ rn <=» c 52 22?=! «O CO a - < i.3 ;* < r Die «S "> "* ifiuj in ii ii ii» -n - rj r 2 '* «- c *< < < Bi ip ID. D O O O Y(OOI), YI99.99)» YlOl). Y(99.9) Y(l), Y(99) Y(IO), Y(90) «EDIAN FREQUENCY 47 MHi > j s i > O z r- 2 r H S tr> a) 5 > <: H -a pi > CD OD i O cr> 0 If t 0 IOIO 0 loiolo _ 9 0-8

136 Y(p)«VARIABILITY ABOUT THE LONG-TERM MEDIAN, IN DECIBELS i 8 -n c~ en C3 H < CO -n H " > X» c H «- (t> O a cr» > a) > 5 a 40 ff *JL?'i /f "V "^R^s X*J V S C ^^V? M <A0l Ä " N s * v /fr i-i "b ' V A»^. : 'i - r" T X - ' \ " t I.V a O p. l t «Ill» 0 a? " In* - <, 4» + T t & / l f '* / ; \ j t-v i 0) > * A 0i /» 7 H : ' / \ J \ T^ \\ i / i / r 7 \ i i \ - j L. R L. 9 n h H » * 3 ; 5 - P C! z y) -<^<< ( > C """' 2- p gr 3 - r w w s = 2 O r i c n w w 2 5 i c It O g -g r S c z. H 2 ^ H J> r H 2 O MI 2 H x> 2 H u * r s 3) > -n H t> ni OB D s X z 4 0 I - (0 (0 IC O -n - r r ;< c 3 in A < 'S» r ~ < K _^ K c I» U) IOKO O - - = 0-9

137 Y(p)-VARIABILITY ABOUT THE LONG-TERM MEDIAN, IN DECIBELS i r < g c n 2 2 S O <D CO en» /* w, /" 7I O xjf ft> 7^ r> & <* D \ *>»0 /.00 0 ^i *l 0 * ^ M / p / > 0 i» L x VJ 8 O». v - CD 0 *^ \ J a MB J " O / " r \ \ r» + / <=> A \ \. 0 O I / \ s " \ X <J r <^ \ \ ' % t n a c» 0 -c* \ CD en CD <-n CD CT>» a 0 3 t*.c" CD M > 0 " O C X n - y 5-PP ZS 2 ~ =» " " 0 -N N W J < -J 30>H * ^ ^ ;i x CD O ~~ 0 0 i^ X * r ( OHHD 0 C :> t t t (0 0 U> MEDIAN FREQUENCY 92.8 MHi Y(O.OI), YI99.99)» Y(0.l), Y(99.9) Y(l), Y(99) 0 Y(I0), Y(90) OS > CD J CD CD CD CD i < i S! H H ; xi > 2 ~i T) O r ^ r> ZC H,'l > u ^ N (Tl O 2 T IN r. CD H n H 3S - r r r $> a> *> t ;S q <-<-<-< ix> J CD tx> is t (0 0 to S 0 to t t t b L

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147 . REFERENCES The references given belw include nly selected papers referred t in the text f this reprt. A cprehensive survey f wrk in the field f trpspheric prpagatin, and an extensive bibligraphy will be fund in the fllwing reprt: Shkarfsky, I. P. (March 958), Trpspheric scatter prpagatin, Res. Rpt. N , RCA Victr C., Ltd. Res. Labs, Mntreal, Canada. Fur recent bibligraphies are: Abbtt, R. L. (Nv. 960), Bibligraphy f trpspheric radi wave scattering, NBS Tech. Nte N. 80. Abbtt, R. L., and E. R, Westwater (Dec. 96), Bibligraphy f icrwave theral eissins by atspheric gases, Private Cunicatin. Nupen, Wilhel (964), Bibligraphy n prpagatin f radi waves thrugh the trpsphere, NBS Tech. Nte N Dugherty, H. T. (Aug. 964), Bibligraphy f fading n icrwave line-f-sight trpspheric prpagatin paths and assciated subjects, NBS Tech. Nte N Andersn, L. J., and E. E. Gssard (Oct. 953a), The effect f the ceanic duct n icrwave prpagatin. A. Gephys. Unin Trans. 34, N. 5, Andersn, L. J., and E. E. Gssard (Jan. 953b), effect n UHF, Prc. IRE 4_. N., Predictin f the ncturnal duct and its Arns, L. D. (Oct. 956), An analysis f radi-wave scattering in the diffractin regin, Crnell University E. E. Reprt 32. Artan, J. O., and J. P. Grdn (Dec. 954), Absrptin f icrwaves by xygen in the illieter wavelength regin, Phys. Rev. 96, N. 5, Bachynski, M. P. (959), Micrwave prpagatin ver rugh surfaces, RCA Review 20, N. 2, Bachynski, M. P. (July-Aug. I960), Prpagatin at blique incidence ver cylindrical bstacles, J. Res. NBS 64D (Radi Prp. ), N. 4, Bachynski, M. P. (March 963), Scale del investigatins f electragnetic wave prpagatin ver natural bstacles, RCA Review 24, N., Barghausen, A. F., F. O. Giraud, R. E. McGavin, S. Murahata, and R. W. Wilber (Jan. 963), Equipent characteristics and their relatin t syste perfrance fr trpspheric cunicatin circuits, NBS Tech. Nte 03. Barsis, A. P., and M. E. Jhnsn (Nv. - Dec. 962), Prlnged space-wave fade-uts in trpspheric prpagatin, J. Res. NBS 66D (Radi Prp. ), N. 6, Barsis, A. P., and R. S. Kirby (Sept. - Oct. 96), VHF and UHF signal characteristics bserved n a lng knife-edge diffractin path, J. Res. NBS 65D (Radi Prp. ), N. 5, Barsis, A. P., K. A. Nrtn, P. L. Rice, and P. H. Elder (Aug. 96), Perfrance predictins fr single trpspheric cunicatin links and fr several links in tande, NBS Tech. Nte 02. (See als IRE Transactins n Cunicatin Syste«CS-0, N., 2-22, March 962). -

148 Batchelr, G. K. (947), Klgrff's thery f lcally Istrpie turbulence, Prc. Cab. Phil. Sc Batchelr, G. K. (953), Press). The thery f hgeneus turbulence, (Cabridge University Bean, B. R. (May 954), Prlnged space-wave fadeuts at,046 Mc bserved in Cheyenne Muntain prpagatin prgra, Prc. IRE 42, N. 5, Bean, B. R. (956), Se eterlgical effects n scattered VHF radi waves, IRE Trans. C. Syst., CS4(), Bean, B. R. (July-Aug. 959), Cliatlgy f grund-based radi ducts, J. Res. NBS 63D (Radi Prp.), N., Bean, B. R. (96), Cncerning the bi-expnential nature f the trpspheric radi refractive index, Beitrage zur Physik der Atsphäre 34, N. /2, 8-9. Bean, B. R., and R. L. Abbtt (957), Oxygen and water vapr absrptin f radi waves in the atsphere, Gefisica Pura e Applicata - Milan 37, Bean, B. R., and B. A. Cahn (Nv. 957), The use f surface bservatins t predict the ttal atspheric bending f radi rays at sall elevatin angles, Prc. IRE 45, N., Bean, B. R., J. D. Hrn, and A. M. Ozanich, Jr. (Nv. I960), Cliatic charts and data f the radi refractive index fr the United States and the wrld, NBS Mngraph N. 22. Bean, B. R., J. D. Hrn, and L. P. Riggs (Oct. 962), Tech. Nte 98. Synptic radi eterlgy, NBS Bean, B. R., and G. D. Thayer (May 959), index, Prc. IRE 47. N. 5, Mdels f the atspheric radi refractive Beard, C. I. (Septeber 96, Cherent and incherent scattering f icrwaves fr the cean, IRE Trans. Ant. Prp. AP-9, Beard, C. I., I. Katz, and L. M. Spetner (April 9 56), Phenenlgical vectr del f icrwave reflectin fr the cean, IRE Trans. Ant. Prp. AP-4, N. 2, Beckann, P. (957), A new apprach t the prble f reflectin fr a rugh surface, Acta, Tech. Cesksl. Akad. 2, 3-355; see als pp , (959). Beckann, P. (I960), A generalized Rayleigh distributin and its applicatin t trpspheric prpagatin, Electragnetic Wave Prpagatin, (Sypsiu, Liege, 958), (Acadeic Press, Lndn, ). Beckann, P. (96a), The statistical distributin f the aplitude and phase f a ultiply scattered field. Inst. Rad. Eng. andelec, Czechslvak Akad. Sei., Paper N. 8. See als NBS Jur. Res. 66D, (Radi Prpagatin), pp , 962. Beckann, P. (96b), The deplarizatin f electragnetic waves scattered fr rugh surfaces, Inst. Rad. Eng. and Elect., Czechslvak Akad. Sei., Paper N. 9. Beckann, P. (Septeber 964), Rayleigh distributin and its generalizatin, NBS Jur. Res. 68D, (Radi Science), N. 9, pp Beckann, P. and A. Spizzichin (963), The scattering f Electragnetic waves fr rugh surfaces, Internatinal Series f Mngraphs n Electragnetic Waves, Vl. 4, (Pergan Press, New Yrk, N. Y.). -2

149 Bit, M. A. (Dec. 957a), Se new aspects f the reflectin f electragnetic waves n a rugh surface, J. Appl. Phys. 28, N. 2, Bit, M. A. (Nv. 957b), Reflectin n a rugh surface fr an acustic pint surce, J. Acust..Sc. A. 2% N., Bker, H. G. (946), Eleents f radi eterlgy: Hw weather and cliate cause unrthdx radar visin beynd the geetrical hrizn, J. Inst. Elec. Engrs. (Lndn) 93. Pt. in-a, N., Bker, H. G., and J. T. de Bettencurt (Mar. 955), Thery f radi transissin by trpspheric scattering using very narrw beas, Prc. IRE 43, N. 3, Bker, H. G., and W. E. Grdn (Sept. 950a), Outline f a thery f radi scattering in the trpsphere, J. Gephys. Res. j>5. N. 3, ; see als Prc. IRE 38, N. 4, 40, (April, 950b). Bker, H. G., and W. Walkinshaw (April 946), The de thery f trpspheric refractin and its relatin t waveguides and diffractin, Reprt n Cnference n Meterlgical Factrs in Radi Wave Prpagatin (The Phys. Sc., and the Ryal Met. Sc., Lndn), Bray, W. J., F. Hpkins, A. Kitchen, and J. A. Saxtn (Jan. 955), Review f lngdistance radi-wave prpagatin abve 30 Mc/s, Prc. IEE, Paper N. 782R, Pt. B, 02, Breer, H. (949), Terrestrial radi waves; thery f prpagatin, (Elsevier Publishing C., Asterda and New Yrk, N. Y. ). Breer, H. (Sept. 957), Distrtin in trpspheric scatter, Phillips Telec. Rev. 28, N. 3, Breer, H. (May 959), On the thery f the fading prperties f a fluctuating signal ipsed n a cnstant signal, NBS Circular 599. Bugnl, O. S. (July 958), Multiple scattering f electragnetic radiatin and the transprt equatin f diffusin, IRE Trans. Ant. Prp. AP-6, N. 3, 30. Bullingtn, K. (Jan. 950), Radi prpagatin variatins at VHF and UHF, Prc. IRE 38, N., Bullingtn, K. (Oct. 955), Characteristics f beynd-the-hrizn radi transissin, Prc. IRE 43, N. 0, Bussey, H. E. (July 950), Micrwave attenuatin statistics estiated fr rainfall and water vapr statistics, Prc. IRE 38, N. 7, CCIR (955), Atlas f grund wave prpagatin curves fr frequencies between 30 Mc/s and 300 Mc/s, ITU, Geneva. CCIR (959), Atlas f grund wave prpagatin curves fr frequencies between 30 and 0,000 Mc/s (Vertical plarizatin nly; prepared by the Radi Research Labratries, Ministry f Pstal Services, Tky, Japan, January 958), ITU, Geneva. CCIR (963a), The cncept f transissin lss in studies f radi systes, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. Ill, Recendatin 34,

150 CCIR (963b), Transissin lss in studies f radi systes, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. Ill, Reprt 2, CCIR (963c), Optiu use f the radi spectru, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. in, Reslutin,. CCIR (963d), Line frequencies r bands f interest t radiastrny and related sciences, in the Gc/s range arising fr natural phenena, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. IV, Reprt 223, CCIR(963e), Reference atspheres, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. II, Reprt, 23, CCIR (963f), Estiatin f trpspheric-wave transissin lss, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. II, Reprt 244, CCIR (963g), Prpagatin curves fr VHF/UHF bradcasting in the African Cntinent, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. HI, Reprt 240, CCIR (963h), VHF and UHF prpagatin curves fr the frequency range fr 40 Mc/s t 000 Mc/s - Bradcasting and bile services, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. II, Recendatin 370, CCIR (963i), Cunicatin satellite systes-frequency sharing between cunicatin satellites systes and terrestrial services, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. IV, Reprt 209, CCIR (963j), Influence f the atsphere n wave prpagatin, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. II, Reprt 233, CCIR (963k), Prpagatin data required fr radi relay systes, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. II, Reprt 242, CCIR (963), Fading f signals prpagated by the insphere, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. II, Reprt 266, CCIR (963), Ters and definitins, Dcuents f the Xth Plenary Assebly, ITU, Geneva, Vl. I, Reprt 32, CCIR (964), Optiu use f the radi frequency spectru, Dcuent being prepared fr the Xlth Plenary Assebly, in accrdance with Reslutin f the Xth Plenary Assebly, ITU, Geneva, Vl. ID,. Chernv, L. A. (Jan. -June 955), Crrelatin f aplitude and phase fluctuatins fr wave prpagatin in a ediu with rand irregularities, Akust. Zh., 89; translatin in Sviet Phys. - Acust., N. -2, Christiansen, W. N. (947), Rhbic antenna arrays, A. W.A.Tech. Rev. [Aal. Wireless Australia ] T_, N. 4, Clew, D. B., and E. H. Bruce-Claytn (Jan. 963), Lng range VHF air/grund cunicatins, Brit. IRE J. 25, N., Czzens, D. E. (June 962), Ngraph fr deterining parablidal gain as a functin f feed pattern and angular aperture, Micrwave J. V, N. 6, Crawfrd, A. B., and D. C. Hgg (July 956), Measureent f atspheric attenuatin at illieter wavelengths, Bell Syst. Tech. J. 35,

151 Crawfrd, A. B., D. C. Hgg, and W. H. Kuer (Sept. 959), Studies in trpspheric prpagatin beynd the hrizn, Bell Syst. Tech. J. 38_, N. 5, Crichlw, W. Q., D. F. Sith, R. N. Mrtn, and W. R. Crliss (Aug. 955), Wrldwide radi nise levels expected in the frequency band 0 Kc t 00 Mc, NBS Circular 557. Crysdale, J. H. (July 958), Cparisn f se experiental terrain diffractin lsses with predictins based n Rice's thery fr diffractin by a parablic cylinder, IRE Trans. Ant. Prp. AP-6, N. 3, Crysdale, J. H., J. W. B. Day, W. S. Ck, M. E. Psutka, and P. E. Rbillard (April 9 57), An experiental investigatin f the diffractin f electragnetic waves by a dinating ridge, IRE Trans. Ant. Prp. AP-5, N. 2, Davenprt, W. B., and W. L. Rt (958), An intrductin t the thery r ranae- signals and nise, McGraw-Hill Bk C., Inc., New Yrk, Chapter 3 dejager, C. (952), The spectru f turbulence in the earth's upper atsphere, Me. Sc. Ry. des Sei., Liege 2, Dicksn, F. H., J. J. Egli, J. W. Herbstreit, and G. S. Wickizer (Aug. 953), Large reductins f VHF transissin lss and fading by the presence f a untain bstacle in beynd-line-f-sight paths, Prc. IRE 4_, N. 8, See als subsequent crrespndence by Crysdale and rebuttal by Dicksn, et al., in Prc. IRE 43, N. 5, (May 955). Dherty, L. H. (Sept. 952), Geetrical ptics and the field at a caustic with applicatins t radi wave prpagatin between aircraft, Crnell University Schl f Electrical Engineering Research Reprt EE-38. Dlukhanv, M. P. (957), Investigatins int the prpagatin f radi waves ver the earth's surface in the USSR., Radi Engr. and Electrnics (USSR) 2, N., Db, C, and M. H. L. Pryce (Sept. 947), The calculatin f field strengths ver a spherical earth, IEE 94, Part III, N. 3, Dugherty, H. T., and L. J. Malney, (Feb. 964) The applicatin f diffractin by cnvex surfaces t irregular terrain situatins, J. Res. NBS 68D (Radi Science), N. 2, ducastel, F. (May 957a), Different types f fluctuatins f trpspheric fields and their physical interpretatin, L'Onde Electrique 22«N. 362, ducastel, F. (Nv. 957b), The use f ultra shrt waves fr lng distance telephne links in Africa (Results f Tests in the Caerns), L'Onde Electrique 37, N. 368, ducastel, F. (Nv.-Dec. 960), Experiental results fr transhrizn trpspheric prpagatin, Ann des Telrfc. _5, N. -2, ducastel, F., and P. Mise (Nv. 957), Eleents f radi cliatlgy, L'Onde Electrique _37. N. 368, ducastel, F., P. Mise, and J. Vge (March 958), Reflectin f an electragnetic wave fr an atspheric layer with variable index f refractin, C. R. Acad., Sei. Fr. 246, N. 2, ducastel, F., P. Mise, A. Spizzichin, and J. Vge (962), On the rle f the prcess f reflectin in radi wave prpagatin, J. Res. NBS 66D (Ra..4 Yt^-I» -I - 3, Duttn, E. J. (June 96), On the cliatlgy f grund-based radi ducts and assciated fading regins, NBS Tech. Nte

152 Duttn, E. J., and G. D. Thayer (Oct. 96), Techniques fr cputing refractin f radi waves in the trpsphere, NBS Tech. Nte 97. Fengler, G. (964), Untersuchungen der elektragnetischen Wellenausbreitung i 500 MHZ-bereich über land unter besnderer berücksichtigung der eterlgischen, Berichte des Instituts für Radieterlgie und Maritie Meterlgie an der Universität Haburg, Reprt N. 8. Fengler, G., J. Jeske, and G. Stilke, Radieterlgical papers II, Berichte des Instituts für Radieterlgie und Maritie Meterlgie an der Universität Haburg, Reprt N. 9. Flr an, E. F., and J. J. Tary (Jan. 962), Required signal-t-nise ratis, RF signal pwer, and bandwidth fr ultichannel radi cunicatins systes, NBS Tech. Nte 00. Fk, V. A., L. A. Vainshtein, and M. G. Belkina (958), Radiwave prpagatin in surface trpspheric ducts, Radi Eng. Electrn. (USSR), 3, N. 2, -27. Friend, A. W. (June 945), A suary and interpretatin f ultra high frequency wave prpagatin data cllected by the late Rss A. Hull, Prc. IRE 33, 358. Friis, H. T., A. B. Crawfrd, and D. C. Hgg (May 957), A reflectin thery fr prpagatin beynd the hrizn, Bell Syst. Tech. J. 36, N. 3, Furutsu, K. (956), On the ultiple diffractin f electragnetic waves by spherical untains, J. Radi Res. Labs., Tky 3, 33. Furutsu, K (959), Wave prpagatin ver an irregular terrain, I, n, HI, J. Radi Res. Labs., Tky 4, 35, 349(957), and 6, 7(959). Furutsu, K. (Jan. - Feb. 963), On the thery f radi wave prpagatin ver inhgeneus earth, J. Res. NBS 67D (Radi Prp.), N., Grsskpf, J. (June 956), On the existing cnditin f research in the real f trpspherically scattered radiatin, Nachrtech. Z. 9, N. 6, Grsskpf, J. (Nv. 958), Se rearks n the analysis f fading in the eter and decieter range, Nachrtech. Z., N., Gunn, K. L. S., and T. W. R. East (Oct. 954), The icrwave prperties f precipitatin particles, Quart. J. Ry. Meterl. Sc. (Lndn) 80, Harper, A. E. (94), Rhbic antenna design, (D. van Nstrand C., Princetn, N.J.). Hartan, W. J. (May 963), Path antenna gain and cents n "Prperties f 400 Mcps lng-distance trpspheric circuits," Prc. IEEE 5, N. 5, Hartan, W. J., and R. E. Wilkersn (Nv. - Dec. 959), Path antenna gain in an expnential atsphere, J. Res. NBS 63D (Radi Prp. ), N. 3, Hathaway, S. D., and H. W. Evans (Jan. 959), Radi attenuatin at kmc and se iplicatins affecting relay syste engineering, Bell Syst. Tech. J. 38, N., Haurwitz, B. and J. M. Austin [ 944], Cliatlgy, McGraw-Hill C. Inc., New Yrk. Hay, H. G., and R. S. Unwin (Dec. 952), Trpspheric wave prpagatin in a duct f nnunifr height, Phys. Sc. Lndn Prc. 65, N. 396b,

153 Head, H. T. (June I960), The influence f trees n televisin field strengths at ultra-high frequencies, Prc. IRE 48, N. 6, Heisenberg, W. (Dec. 948), On the thery f statistical and istrpic turbulence, Prc. Ry. Sc. Lndn A 95, Herbstreit, J. W., and P. L. Rice (Sept. 959), Survey f Central Radi Prpagatin Labratry research in trpspheric prpagatin, , NBS Tech. Nte N. 26. Hirai, Masaichi (May 96a), Multipath prperties f trpspheric prpagatin f very shrt radi waves beynd the hrizn. Jur. Radi Res. Lab., Japan d, N. 37, Hirai, Masaichi (Sept. 96b), Diversity effects in spaced-antenna receptin f trpspheric scatter waves, Jur. Radi Res. Lab., Japan J, Hitchcck, R. J., and P. A. C. Mrris (July 96), The HF band: Is a new lk required? Wireless Wrld, 375. T 378. Hgg, D. C, and W. W. Mufrd (March 960), The effective nise teperature f the sky. Micrwave J., Hgg, D. C., and R. A. Seplak (Sept. 96), The effect f rain and water vapr n sky nise at centieter wavelengths, Bell Syst. Tech. J. 4, N. 5, Ikegai, F. (July 959), Influence f an atpsheric duct n icrwave fading, IRE Trans. Ant. Prp. AP-7. N. 3, Ikegai, F. (May-June 964), Radieterlgical effects in prpagatin ver the sea and islands, Rev. Elect. Cun. Lab., Tl"/,. rt-2, 5-H Internatinal Telephne and Telegraph Crpratin (956), Reference data fr radi engineers, Furth Editin, (ITT, New Yrk). Janes, H. B., and P. I. Wells (Oct. 955), Se trpspheric scatter prpagatin easureents near the radi hrizn, Prc. IRE 43, N. 0, Jasik, H. (96), Antenna Engineering Handbk, (McGraw Hill). Jhnsn, M. A. (958), A review f trpspheric scatter prpagatin thery and its applicatin t experient, Prc. IEE 05B, Suppl. 8, Jsephsn, B., and A. Blquist (April 958), The influence f isture in the grund, teperature and terrain n grund wave prpagatin in the VHF band, IRE Trans. Ant. Prp. AP-6, N. 2, Jsephsn, B., and G. Carlsn (April 958), Distance dependence, fading characteristics and pulse distrtin f 3000 Mc trans-hrizn signals, IRE Trans. Ant. Prp. AP-6, N. 2, Jsephsn, B., and F. Eklund (April 958), Se icrwave prpagatin experiences fr a just-belw-hrizn path, IRE Trans. Ant. Prp. AP-6, N. 2, Jwett, J. K. S. (Jan. 958), The easureent and predictin f VHF trpspheric field strengths at distances beynd the hrizn, Prc. IEE 05B, Supp. 8, 9-96, and 22-26, Paper N. 2500R. < Jy, W. R. R. (Jan. 958a), The lng-range prpagatin f radi waves at 0 c wavelength, Prc. IEE 05B, Supp. 8, 53-57, Paper N. 2522R. Jy, W. R. R. (958b), Radi prpagatin far beynd the hrizn at abut 3.2 c wavelength, Prc. IEE 05B, Supp. 8, and 84-88, Paper N. 2528R. U-7

154 Kales, M. L. (May 95), Elliptically plarized waves and antenna, Prc. IRE 39, N - 5, Kalinin, A. I. (957), Apprxiate ethds f cputing the field strength f ultra shrt waves with cnsideratin f terrain relief, Radi Eng, _2, N. 4, 3.26, Raditekhn. i Elektrn., N. 4, Kalinin, Iu. K (958), Perturbatin f plane radi wave by inhgeneities f the earth's surface, Raditech, and Elecktrn ^( , Translatin in Radi Engineering and Electrnics _3. N - 4 i Kerr, D. E. (964), Prpagatin f shrt radi waves, MIT Radiatin Labratry Series 3, (Bstn Technical Publishers, Inc., Lexingtn, Mass.). Kirby, R. S., H. T. Dugherty, and P. L. McQuate (Oct. 955), Obstacle gain easureents ver Pike's Peak at 60 t 046 Mc/a, Prc. IRE 43, N. 0, Kirby, R. S., P. L. Rice, and L. J. Malney (Oct. 96), Characteristics f pint-t-pint trpspheric prpagatin and siting cnsideratins, NBS Tech. Nte N. 95. Kitchen, F. A., and I. J. Richnd (March 957), Se characteristics f lng distance scatter transissins (tw parts), British C, and Electr. 4, N. 2, (Feb. 957); 4, N. 3, 46-48, (March 957). Kitchen, F. A., E. G. Richards, and I. J. Richnd (Jan. 958), Se investigatins f etre-wave radi prpagatin in the transhrizn regin, Prc. IEE 05B, Supp. 8, 06-6, Paper N. 2509R. Kitchen, F. A., W. R. R. Jy, and E. G. Richards (Aug. 958), Influence f the seiperanent lw-level cean duct n centietre wave scatter prpagatin beynd the hrizn. Nature 82, N. 4632, Klgrff, A. N. (94), Dissipatin f energy in lcally istrpic turbulence, Cptes Rendus (Dklady) de l'acadaie des Sciences de l'ussr 3_2, N., 6-8. Krasil'nikv, V. A. (949), The effect f variatins f the cefficient f refractin in the atsphere upn the prpagatin f ultra-shrt waves, Izvest. Akad. Nauk.S. S. S. R. Sev. Gegraf i Gefiz _3, N., (in Russian). Kühn, V. (Feb. and May 958), Prpagatin investigatin f the effect f varius types f terrain in frequency bands I, II, and III, Tech. C. Lab. fr Cercial Radi and Telev., BRF, DDR. Lane, J. A., and J. A. Saxtn (July 952), Dielectric dispersin in pure plar liquids at very high radi frequencies, Prc. Ry. Sc. A23, Laws, J. O., and D. A. Parsns (Apr. 943), The relatin f raindrp-size t intensity, Trans. Aer. Gephys. Unin 24, Lewin, L. (July 962), Diversity receptin and autatic phase crrectin, Prc. IEE 09, Part B, N. 46, McGavin, R. E. (May 962), A survey f the techniques fr easuring the radi refractive index, NBS Tech. Nte 99. McGavin, R.E., and L. J. Malney (Sept. - Oct. 959), Study at 046 Mc/s f the reflectin cefficient f irregular terrain at grazing angles, J. Res. NBS 63D (Radi Prp. ), N. 2, McPetrie, J. S., and J. A. Saxtn (Sept. 942), Nature 50, 292. Diffractin f ultra-shrt radi waves, -8

155 McPetrie, J. S., and L. H. Frd (946), Se experients n the prpagatin ver land f radiatin f 9. 2 c wavelength, especially n the effect f bstacles, Prc. EE 93, Pt. 3-A, Ns. -4, Megaw, E. C. S. (Dec. 950), Scattering f electragnetic waves by atspheric turbulence, Nature 66, Megaw, E. C. S. (April 954), Interpretatin f stellar scintillatin, Quart. J. Ry. Met. Sc. 80, Megaw, E. C. S. (Sept. 957), Fundaental radi scatter prpagatin thery, Prc. IEE, Pt. C 04, N. 6, 4-455, see als Mngraph 236R, May 957. Merkulv, V. V. (957), On the thery f prpagatin f electragnetic waves in edia with rand inhgeneities in the index f refractin, Sviet Physics: Tech. Phys. 2, , J. Electr-Tech. Phys. 27, N. 5, 05. Millingtn, G. (May 958), Trpspheric scatter prpagatin, Electrnic Eng. 30, N. 363, Millingtn, G., R. Hewitt, and F. S. Iirzi (Sept. 962a), Duble knife-edge diffractin in field strength predictins, Prc. IEE 09, Part C, N. 6, See als IEE Mngraph N. 507E (Mar. 962). Millingtn, G., R. Hewitt, and F. S. Iirzi (Sept. 962b), The Fresnel surface integral, Prc. IEE 09, Part C, See als IEE Mngraph N. 508E (Mar. 962). Millingtn, G., and G. A. Isted (July 950), Grund wave prpagatin ver an inhgeneus, sth earth, Part 2: Experiental evidence and practical iplicatins, Prc. IEE, Part III 97, N. 48, 209. Mise, P. (July 958), The crrelatin between the electric field at a great distance and a new radieterlgical paraeter, Trans. IRE Trans. Ant. Prp. AP-6, N. 3, Mise, P. (March-April 960a), The equivalent gradient direct easureents and theretical calculatins, Ann. des Tele"c. _5, Ns. 3-4, Mise, P. (Aug. 960b), Cents n "Mdels f the atspheric radi refractive index," Prc. IRE 48, N. 8, Mise, P. (Nv.-Dec. 960c), Se aspects f radieterlgy and radicliatlgy, Ann. des Tele'c. _5, N. -2, Mise, P. (May-June 96), The influence f the equivalent gradient and atspheric stability n transhrizn paths in the Sahara and the Cng, Ann. des Telec. 6^, Ns. 5-6, 0-6. Mler, W. F., and D. B. Hlden (Jan. - Feb. I960), Trpspheric scatter prpagatin and atspheric circulatins, J. Res. NBS 64D (Radi Prp. ), N., Nakagai, M. (Oct. 940), Study n the resultant aplitude f any vibratins whse phases and aplitudes are rand, Nippn Elec. C. Eng. 2, Natinal Bureau f Standards (954), Applied Matheatics Series 32, and csine integrals fr arguents fr Table f the sine Natinal Bureau f Standards (June 964), Matheatical Functins. Applied Matheatics Series 55, Handbk f -9

156 Neugebauer, H. E. J., and M. P. Bachynski (July-Aug. I960), Diffractin by sth cnical bstacles, J. Res. NBS 64D (Radi Prp.), N. 4, Newtn, R. G., and T. F. Rgers (Nv. 953), Dependence f ttal icrwave atspheric absrptin n prpagatin path elevatin, Air Frce Cabridge Reprt AFCRC Tech A. Nura, Y. A., and K. Takaku (Aug. 955), On the prpagatin f electragnetic waves in an inhgeneus atsphere, J. Phys. Sc. Japan 0, N. 8, Nrtn, K. A. (Dec. 94), The calculatin f grund-wave field intensity ver a finitely cnducting spherical earth. Prc. IRE 29, N. 2, Nrtn, K. A. (950), Addendu t Reference E t the reprt f Ad Hc Cittee f the F.C.C. fr the Evaluatin f the Radi Prpagatin Factrs Cncerning the TV and FM Bradcasting Services in the Frequency Range Between 50 and 250 Mc/s. (See Nrtn and Fine reference belw. ) Nrtn, K. A. (Jan, 953), Transissin lss in radi prpagatin, Prc. IRE 4, N., Nrtn, K. A. (July-Aug. 959), Syste lss in radi wave prpagatin, J. Res. NBS 63D (Radi Prp. ), N., Nrtn, K. A. (July I960), Carrier frequency dependence f the basic transissin lss in trpspheric frward scatter prpagatin, J. Gephys. Res. 65, N. 7, Nrtn, K. A. (962), Efficient use f the radi spectru, NBS Tech. Nte 58. Nrtn, K. A., and E. C. Barrws (964), Observed vertical wavenuber spectra f refractivity near the grund, t be published. Nrtn, K. A., and H. Fine (Aug., 949), A study f ethds fr the efficient allcatin f radi frequencies t bracasting services perating in the range abve 50 Mc, Reference E t the Reprt f Ad Hc Cittee f the F.C.C. fr the Evaluatin f the Radi Prpagatin Factrs Cncerning the TV and FM Bradcasting Services in the Frequency Range Between 50 and 250 Mc/s. Nrtn, K. A., and A. C. Oberg (Jan. 947), Prc. IRE 35, N., The axiu range f a radar set, Nrtn, K. A., P. L. Rice, H. B. Janes, and A. P. Barsis (Oct. 955), The rate f fading in prpagatin thrugh a turbulent atsphere, Prc. IRE 43, N. 0, Nrtn, K. A., P. L. Rice, and L. E. Vgler (Oct. 955), The use f angular distance in estiating transissin lss and fading range fr prpagatin thrugh a turbulent atsphere ver irregular terrain, Prc. IRE 43, N. 0, Nrtn, K. A., H. Staras, and M. Blu (Feb. 952), A statistical apprach t the prble f ultiple radi interference t FM and televisin service, IRE Trans. Ant. Prp. AP-, Nrtn, K. A., L. E. Vgler, W. V. Mansfield, and P. J. Shrt (Oct. 955), The prbability distributin f the aplitude f a cnstant vectr plus a Rayleigh-distributed vectr, Prc. IRE 43, N. 0,

157 Obukhv, A. M. (94), On the distributin f energy in the spectru f turbulent flw, Bull. Acad. Sei. USSR Ge. and Gephys. Ser. 4-5, 453. See als Cptes Rendus (Dklady) L'Acadeie des Sciences de 'USSR 32, N., 9 (94). Obukhv, A. M. (953), On the effect f inhgeneities f the atsphere n sund and light prpagatin, Bull. Acad. Sei. USSR, Geg. and Gephys. Ser. 2, 55. One, M., M. Hirai, and S. Niwa (April 958), Results f experients f lng distance verland prpagatin f ultra-shrt waves, J. Radi Res. Lab. (Tky) 5, N. 20, One, M.. and K. Nishikri (Oct. 957), Micrwave prpagatin ver the sea beynd the line f sight, Radi Res. Lab. J. 4, N. 8, Pearcey, T. (956), Table f the Fresnel integral, (Cabridge Univ. Press, New Yrk, N. Y.). Pekeris, C. L. (Feb. 947), Nte n scattering f radiatin in an inhgeneus ediu, Phys. Rev. 7, N. 3, Perlat, A., and J. Vge (Dec. 953), Attenuatin f centieter and illieter waves by the atsphere, Ann. des Tele'c. 8, N. 2, Price, W. L. (July 948), Radi shadw effects prduced in the atsphere by inversins, Prc. Phys. Sc. Lndn 6_, N. 343, Randall, D. L. (964), A suary f trpspheric radi duct eterlgy at V.H.F. and UHF as bserved n a trip arund the wrld, Wrld Cnference n Radi Meterlgy Bulder, Clrad, Septeber 4-8. Rayleigh, Lrd (Aug. 880), On the resultant f a large nuber f vibratins f the sae pitch and f arbitrary phase, Phil. Mag. 0, Riblet, H. J., and C. B. Barker (948), A general divergence frula, J. Appl. Phys. % 63. Rice, P. L., and F. T. Daniel (Apr. 955), Radi transissin lss vs. distance and antenna height at 00 Mc, Trans. IRE Ant. Prp. AP-3, N. 2, Rice, P. L., and J. W. Herbstreit (964), Trpspheric prpagatin, (t be published in Vl. 20 f Advances in Electrnics, Acadeic Press). Rice, S. O. (Jan. 945), Matheatical analysis f rand nise, Bell. Syste Tech. J. 24, Rice, S. O. (954), Diffractin f plane radi waves by a parablic cylinder, Bell Syste Tech. J. 33, Rider, G, C. (953), Se VHF experients upn the diffractin effect f hills, The Marcni Rev. 6, N. 09, 96-06, 2nd quarter. Rwden, R. A., L. F. Taghl, and J. W. Stark (958), A survey f trpspheric wave prpagatin easureents by the BBC, , Prc. IEE 05B, Suppl. 8, and 22-26, Paper N. 257R. Ryde, J. W. (946), The attenuatin and radar eches prduced at centietre wavelengths by varius eterlgical phenena, Cnference n eterlgical factrs in radiwave prpagatin, Phys. Sc. (Lndn) and Ryal Meterlgical Sciety, Ryde, J. W. and D. Ryde (945), Attenuatin f centieter waves by rain, hail, fg, and cluds, General Electric C., Webley, England. -

158 Saxtn, J. A. (Sept. 95), The prpagatin f etre radi waves beynd the nral hrizn. Part, Prc. IEE 98, Part, N. 55, Saxtn, J. A., and J. A. Lane (May 955), VHF and UHF receptin-effects f trees and ther bstacles, Wireless Wrld 6L Saxtn, J. A., J. A. Lane, R. W. Meadws, and P. A. Matthews (Feb. 964), Layer structure f the trpsphere, Prc. IEE, N. 2, Schelkunff, S. A. and H. T. Friis (952), Antennas, thery and practice, Wiley and Sns, New Yrk City. Schelling, J. C, C. R. Burrws, and E. B. Ferrell (Mar. 933), prpagatin, Prc. IRE 2, N. 3, Ultra-shrtwave Schüneann, R. (Sept. 957), Mechanis f ultra shrt wave prpagatin ver great distances, Hchfreq. u. Electrak. 66, N. 2, Sherwd, E. M., and W. L. Ginztn (July 955), Reflectin cefficients f irregular terrain at 0 c., Prc. IRE 43, N. 7, Shkarfsky, I. P. (Mar. 958), Trpspheric scatter prpagatin, Res. Rpt. N , RCA Victr C., Ltd. Res. Labs., Mntreal, Canada. Siddiqui, M. M. (March-April 962), Se prbles cnnected with Rayleigh distributins, J. Res. NBS 66D (Radi Prpagatin), N. 2, pp Silveran, R. A. (Apr. 957), Fading f radi waves scattered by dielectric turbulence, J. Appl. Phys. ^8, N. 4, Als New Yrk Univ. Inst. f Math. Sei., Electragnetic Res. Divisin, Res. Rept. EM 0 (Jan. 957). Staras, H. (Oct. 952), Scattering f electragnetic energy in a randly inhgeneus atsphere, J. Appl. Phys. 23, N. 0, Staras, H. (Oct. 955), Frward scattering f radi waves by anistrpic turbulence, Prc. IRE43, N. 0, Staras, H. (April 957), Antenna-t-ediu cupling lss, IRE Trans. Ant. Prp. AP-5, N. 2, Starkey, B. J., W. R. Turner, S. R. Badce, and G, F. Kitchen (Jan. 958), The effects f atspheric discntinuity layers up t and including the trppause n beynd-thehrizn prpagatin phenena, Prc. IEE 05B, Suppl. 8, and 22-26, Paper N. 2486R. Stkes, G. G. (922), Matheatical and physical papers, Vl. Ill, On the cpsitin and reslutin f streas f plarized light fr different surces, (Cabridge University Press, Lndn), Straitn, A. W., and C. W. Tlbert (May 960), Analies in the absrptin f radi waves by atspheric gases, Prc. IRE 4_8, N. 5, Suttn, O. G. (955), Atspheric turbulence (Jhn Wiley and C.). Ta, K. (Jan. 957), On the relatinship between the scattering f radi waves and the statistical thery f turbulence, J. Radi Res. Lab. (Tky) 4, N. 5, T.A.S.O. (March 959), Engineering aspects f televisin allcatins, Reprt f the televisin allcatins study rganizatin. -2

159 Taylr, G. I. (922), Diffusin by cntinuus veents, Prc. Lndn Math. Sc. II 20, 96. Thurel, L. (I960), The antennas, translated by H. de Laistre Banting (Jhn Wiley and Sns, Inc., New Yrk, N. Y. ). Tlbert, C. W., and A. W. Straitn (Apr. 957), Experiental easureent f the absrptin f illieter radi waves ver extended ranges, IRE Trans. Ant. Prp. AP-5, N Tritski, V. (May 956), The prpagatin f ultra-shrt waves at great distances beynd the hrizn. Radi Technika, N. 5, Tritski, V. N. (Jan. 957a), Abut the influence f the fr f the structure functin f nn-hgeneus dielectric pereability f air n lng distance trpspheric prpagatin f ultra shrt waves, Radi Eng. 2, Tritski, V. N. (957b), Fading f ultra-shrt waves in radi relay systes. iü.- Electrsviaz Ugai, S. (May-June 96), Characteristics f fading due t ducts and quantitative estiatin f fading, Rev. Elect. C. Lab., Japan 9, N. 5-6, Ugai, S., S. Ayagi, and S. Nakahara (May 963), Micrwave transissin acrss a untain by using diffractin gratings, Electrnics and Cunicatins in Japan 46, N. 5, 7-7. Unwin, R. S. (Nv. 953), Ultra-shrt-wave fie Id-strength in a grund-based radi duct, Nature 72, N. 4384, Van Vieck, J. H. (Apr. 947a), The absrptin f icrwaves by xygen, Phys. Rev. 7, N. 7, Van Vleck, J. H. (Apr. 947b), The absrptin f icrwaves by uncndensed water vapr, Phys. Rev. 7, N. 7, Van Vleck, J. H. (95), Thery f absrptin by uncndensed gases, Prpagatin f Shrt Radi Waves, (McGraw-Hill Bk C., New Yrk, N. Y. ), Villars, F., and V. F. Weisskpf (Oct. 955), On the scattering f radi waves by turbulent fluctuatins f the atsphere, Prc. IRE 43, N. 0, Vge, J. (Mar. 953), The trpsphere and wave prpagatin (Suary f Prceedings f Cissin II, 0th General Assebly URSI (952),) L'Onde Electrnique 33, N. 32, Vge, J. (955), Radielectricity and the trpsphere, parti, theries f prpagatin t lng distances by eans f atspheric turbulence, L'Onde Electrique 35, Vge, J. (Nv. 956), Useful bandwidth in scatter transissin, Prc. IRE 44, N., Vge, J. (Nv.-Dec. I960), Theries f transhrizn trpspheric prpagatin, Ann. des Tellc. _5. N., Vgler, L. E. (July 964), Calculatin f grundwave attenuatin in the far diffractin regin, Radi Sei. J. Res. NBS/USNC-URSI 68D, N. 7,

160 Vgler, L. E., and J. L. Nble (Sept. - Oct. 963), Curves f grund prxiity lss fr diple antennas (a digest), J. Res. NBS 67D (Radi Prp.), N. 5, VvedenBkii, B. A., and A. G. Arenberg (957), Lng distance trpspheric prpagatin f ultra-shrt waves, Radi Eng. 2, N., 3-3; Radi Eng. 2, N. 2, 0-25, Vvedenskii, B. A., and A. V. Sklv (957), Investigatin f trpspheric prpagatin f eter, decieter, and centieter radi waves in the USSR, Radi Eng. and Elect. (USSR) 2, N., Vyskvskii, D. M. (957a), Calculatin f ultiple scattering in the diffusin prpagatin f ultra-«hrt waves in the trpsphere, Radi Eng. and Elect. (USSR) 2, N. 6, Vyskvskii, D. M. (957b), Geetrical characteristics f the scattering f radi waves by turbulent inhgeneities in the trpsphere, Telecunicatins (USSR) 9, -20. ~ Vyskvskii, D. M. (958), Diffused prpagatin f ultra-shrt waves in the trpsphere with high-directivity antennas, Telecunicatins (USSR) 5, Wait, J. R. (958), On the thery f prpagatin f electragnetic waves alng a curved surface, Can. J. Phys. 36, N., 9-7. Wait, J. R. (959), Electragnetic radiatin fr cylindrical structures (Pergan Press, New Yrk, N. Y. ). Wait, J. R. (April 9 59), Transissin f Pwer in Radi Prpagatin, Electrnic and Radi Engineer, Vl. 36, Series N. 4, pp Wait, J. R. (962), Electragnetic waves in stratified edia. Internatinal Series f Mngraphs n Electragnetic Waves 3, (Pergan Press, New Yrk, N. Y. ) Wait, J. R. (Nv. 963), Oblique prpagatin f grund waves acrss a castline, part I, J. Res. NBS 67D (Radi Prp.), N. 6, Wait, J. R., and A. M. Cnda (Sept. - Oct. 959), Diffractin f electragnetic waves by sth bstacles fr grazing angles, J. Res. NBS 63D (Radi Prp. ), N. 2, Wait, J. R., and C. M. Jacksn (Nv. 963), Oblique prpagatin f grund waves acrss a castline, part II, J. Res. NBS 67D (Radi Prp.), N. 6, Wheeln, A. D. (June 957), Relatin f radi easureents t the spectru f trpspheric dielectric fluctuatins, J. Appl. Phys. 28, Wheeln, A. D. (Sept.-Oct. 959), Radi-wave scattering by trpspheric irregularities, J. Res. NBS 63D (Radi Prp. ), N. 2, ; als, J. Ats. and Terr. Phys. _5. Ns 3, 4, (Oct. 959). Wilkersn, R. (964), Multiple knife-edge diffractin, (private cunicatin). Williasn, D. A., V. L. Fuller, A. G. Lngl-y, and P. L. Rice (Mar. I960), A suary f VHF and UHF trpspheric transissin lss data and their lng-ter variability, NBS Tech. Nte

161 2. LIST OF SYMBOLS AND ABBREVIATIONS In the fllwing list the English alphabet precedes the Greek alphabet, and lwer-case letters precede upper-case letters. fr quantities expressed in decibels, fr exaple p is transitter pwer in decibels abve ne watt. As a general rule, upper-case letters have been used is transitter pwer in watts, and P Seties a sybl ay be used in quite different cntexts, in which case it is listed fr each separate cntext. Subscripts are used t dify the eaning f sybls. The rder is:. Sybl withut a subscript. h 2. Sybl with a subscript, (letter subscripts in alphabetical h rder fllwed by nuber subscripts in nuerical rder). h 3. Sybl as a special functin. h(x) 4. Abbreviatins. ht. Fllwing each definitin an equatin nuber r sectin nuber is given t shw the ter in its prper cntext. Where applicable, reference is ade t a figure. Thrughut the reprt, lgariths are t the base 0 unless therwise nted. a Effective earth's radius, allwing fr average radi ray bending near the surface f the earth, (4.4) figure a An equivalent earth's radius which is the harnic ean f the radii a and a, e t r (7.0). a The "effective absrbing area" f an antenna, (2. 24). a The effective absrbing area fr the n en tenna fr a single surce, (.34). discrete plane wave incident n an an- a, a The effective absrbing area f the receiving antenna fr each f tw waves, ei ei (.86). a The fractin f energy absrbed alng a ray path, r scattered ut f it, (II. 26). a, a The fractin f energy, a abve, fr the and n ultipath cpnents p pn p fr a single surce, where and n take n integral values fr t N, (.39). a The radius f a circular arc that is tangent t the receiving antenna hrizn ray at the hrirn, and that erges sthly with the crrespnding arc thrugh the transitting antenna hricn, (8.9) figure a Effective earth's radius factr crrespnding t D, (8.5). a Radiwave scattering crss-sectin f a single scatterer r grup f scatterers, s (IV. 3). 2-

162 \ Radius f a circular arc that is tangent t the transitting hrizn ray at the hrizn, and that erges sthly with the crrespnding arc thrugh the re- ceiving antenna hrizn, (8,9) figure a Radiwave scattering crss-sectin per unit vlue, (IV. 4). v Radiwave scattering crss-sectin fr refractivity turbulence, (IV. 2). v a, a, a Radiwave scattering crss-sectins per unit vlue fr large, ediu, and VI VZ V3 sall layers, (IV. 5) t (IV. 7). a The axial rati f the plarizatin ellipse f a plane wave, (II. 5). X th a, a, a Axial ratis f the plarizatin ellipse f the n, first, and secnd plane xn xl X2 r wave fr a single surce, (.3 5) and (.8 5). a The axial rati f the plarizatin ellipse assciated with the receiving pattern, (.7). a, a, a Axial ratis f the plarizatin ellipse assciated with the receiving patxrn xn xrj r tern fr the n, first, and secnd plane wave fr a single surce, (.35) and (.82). a The actual earth's radius, usually taken t be 6370 kileters, (4.4). a Radius f the circular arc that is tangent t the transitting antenna hrizn ray at the hrizn, and that passes thrugh a pint h itting antenna, (8.8) figure 8.7. kileters belw the trans- a Radius f the circular arc that is tangent t the receiving antenna hrizn ray at the hrizn, and that passes thrugh a pint h antenna, (8.8) figure 8.7. kileters belw the receiving a, a Psitive r negative aplitudes f real and iaginary cpnents f a cplex vectr: a = a + ia, a = a + a, (II. 52). a, ä The real vectr a = aä, where ä is a unit vectr. a., a Real vectrs defining real and iaginary cpnents f a cplex vectr: a = Tj + ix.,, (.4 5). a A cplex vectr: a = a, + i a, (II. 45). a. A cplex vectr defined in ters f the unit vectr syste x, x, x, (.62). a (-f) The effective absrbing area f a receiving antenna in the directin (-?), (2. 22) and (2.24). A An antenna terinal, figure 6.3. A Attenuatin relative t free space, expressed in decibels, defined as the basic transissin lss relative t that in free space, (2.35). See A. A The lng-ter edian attenuatin f radi waves due t atspheric absrptin by xygen and water vapr, sectin

163 A.A Fr transhrizn paths, A = A + A, the su f the absrptin fr the ar at a at ar K transitter t the crssver f hrizn rays and the absrptin fr the crss- ver f hrizn rays t the receiver, sectin 3. A Ttal absrptin attenuatin within a clud, (3.3). A The hurly edian attenuatin relative t free space, annex I. r A Ttal absrptin due t rainfall ver a given path, (3. 7). A Attenuatin relative t free space, defined as basic prpagatin lss relative t that in free space, (2.47). See A. A Rate f attenuatin thrugh wds in full leaf, (5. 8). A_ Diffractin attenuatin relative t free space at an angular distance 9=0 ver a sth earth, sectin A, A Antenna terinals, figure 6.. A(v, 0) Attenuatin relative t free space as a functin f the paraeter v, (7. 2) figure 7.. A(v, p) Diffractin attenuatin relative t free space fr an islated perfectly cnducting runded bstacle, (7.7), figure 7.3. A(0, p) The diffractin lss fr 6 = 0 ver an bstacle f radius r, (7. 7) figure 7.4. A(v.) Attenuatin relative t free space fr each f several rays paraeter v., where j =, 2, 3, 4, (III. 34). as a functin f the A (p) The tie availability f hurly edian values A Figures :2-:26 shw A (p) pltted against the straight-line distance, r, fr values f p ranging fr 0.0 t percent. D The diensins f an atspheric layer r feuillet in any directin perpendicu- lar t K, (IV. 9). b Effective bandwidth f a receiver in cycles per secnd, (V.7). b* The paraeter b, a functin f grund cnstants, carrier frequency, and plari- zatin, expressed in degrees, figure 8.2, and equatins (III. 40) and (III. 4). b. The paraeter b fr hrizntal plarizatin defined by (III. 40). h b The paraeter b fr vertical plarizatin, (III. 4). B Effective bandwidth, b, expressed in decibels abve ne cycle per secnd, (V.8). B An antenna terinal, figure 6.3. B The paraeter B(K, b) crrespnding t the effective earth's radius a, (8. 5). s s B Values f the paraeter BfK, b) that crrespnd t values f K, (8. 3). B Defined by (8. 2) as the prduct f several factrs, cbined fr cnvenience in cnsidering diffractin. 2-3

164 B' Any pint alng the great circle path fr A t B, figure 6.3. B(K, b*) A paraeter pltted in figure 8.3 as a functin f K and b', (8. 2). c Free space velcity f radi waves, C 2' C 3 c = ± 0. 3 k/sec. c A paraeter shwing the phase change ir - c assciated with the cplex plane wave reflectin cefficient R exp {-M* -c)] crrespnding t reflectin fr an infinite sth plane surface, (5.4) figures III. thrugh III. 8. c, c Values f c fr hrizntal and vertical plarisatin, respectively, (III. 3) and h v (III. 4) figures III. thrugh III. 8. c Plarizatin efficiency f the pwer transfer fr transitter t receiver, P (IV. 3). c Ci c. J c c, r c, Ol C The phase changes assciated with the cplex reflectin cefficients R. R, (III. 32). Difference in lngitude between A and B, (6. ) and (6. 2). Csine integral, (III. 5). Fresnel integral, (III. 33), where j =, 2, 3, 4. A paraeter which relates K t K(8497), (8.2). C Values f C crrespnding t effective earth's radii a, a, and a, t r s t (8.3). Values f C crrespnding t effective earth's radii a and a, (8.3). Difference in the lngitude f the pints A and B', (6.6) t (6.9). Fresnel csine integrals, (IV.8). A paraeter used in calculating diffractin attenuatin, (8. ) figure 8.4. C (K, b ), C (K, b') The paraeter C (K, b*) crrespnding t K and K, als written C^Kj) and C (K ), (8.). C~ (K ) The weighted average f values f C (K, b) and C (K, b), (8.). Ci(r) Csine integral as a functin f r, (III. 5). Ci(r ), Ci(r ) Csine integral as a functin f r, r, (III. 50). CCIR Internatinal Radi Cnsultative Cittee. CRPL Central Radi Prpagatin Labratry, Natinal Bureau f Standards, U.S.A. C. W. Cntinuus wave. d Great circle prpagatin path distance, easured at sea level alng the great circle path deterined by tw antenna lcatins, A and A, figure 6.. d c Clearing depth in eters, defined as the distance fr the edge f wds t the d C, OS C 02 C(u), C(v) Cj(K. b') lwer antenna alng a prpagatin path, (5. 9). Effective prpagatin path distance, a functin f d, f, h, and h, sectin c te re 0., (0.3). d Lp Great circle distance fr the receiving antenna t its hrizn, figure 6.. d Lt Great circle distance fr the transitting antenna t its hrizn, figure

165 dq A differential aplitude reflectin cefficient fr a trpspheric layer, (IV. 5). d r Distance used in calculating grund reflectins in knife edge diffractin; d r defined by (III. 29). d Distance between the receiving antenna hrizn and the crssver f hrizn rays as easured at sea level, (6. 20). d Distance between the transitting antenna hrizn and the crssver f hrizn rays as easured at sea level, (6. 20). d', d' If 0 r 6 is negative, d' r d' is cputed and substituted fr d sr st r t sr st r sr r d in reading figure 6.9, (6.23). d A factr used t nralise effective antenna heights in cputing d, (0.?.). s e d The theretical distance where diffractin and scatter fields are apprxiately si equal ver a sth earth, (0. ). d The greatest distance fr which the attenuatin relative t free space is zer, (5.0). d, d Distance fr the transitting, r the receiving antenna, t the crssver f hrizn rays, easured at sea level, figure 6.. d, d Great circle distance fr ne antenna f a pair t the pint f reflectin f a reflected ray, figure 5.. d,d,d., d Distances used in cputing diffractin attenuatin with grund reflec- tins, (III. 3) figure IH. 9. db Decibels = 0 lg... (pwer rati) r 20 lg (vltage rati). In this reprt, all lgariths are t the base 0 unless therwise stated. dbu Decibels abve ne icrvlt per eter. dbw Decibels abve ne watt. D Divergence cefficient, a factr used t allw fr the divergence f energy due t reflectin fr a cnvex surface, (5. 2). D Diaeter f a parablic reflectr in eters, (2. 6). D Great circle distance between transitting and receiving hrizns, (6. 7), fig- ure 6.. D A functin f d, d used in cputing diffractin lss, (8. 6), figure 8.8. str st sr e The psitive r negative aplitude f the crss-plarized vectr cpnent e c c f a cplex plarizatin vectr e, sectins 2.4 and II. 2. e The psitive r negative aplitude f the crss-plarized vectr cpnent e f a receiving antenna respnse pattern, (II. 6). e. The psitive r negative aplitude f the real vectr e. assciated with a c- plex plane wave s/2 - (e + ie.) exp (IT), where e and e. are tie-invariant and exp (IT) is a tie phasr, (II. 8b). is 2-5

166 e The psitive r negative aplitude f the principal plarizatin cpnent e P _ P f a cplex plarizatin vectr e, sectins 2.4 and II. 2. e The psitive r negative aplitude f the principal plarizatin cpnent e f a receiving antenna respnse pattern, (.6). e The psitive r negative aplitude f the real vectr cpnent e e e I r 2 ". % e, c e. C < P < P e cr, e pr e cr, e pr e. I > r V? 2 V % assciated with a cplex plane wave slz~(e + ie.) exp (ii - ), where e and e. are tier l r l invariant and exp (IT) is a tie phasr, (II. 8a) Equivalent free space field strength, (II. 5), (2.44). Equivalent inverse distance field strength, (2.45). The psitive r negative real aplitudes f real and iaginary cpnents f the cplex plarizatin vectr e, (II. 0). The psitive aplitudes f real vectrs e and e assciated with the 6 and cpnents f a cplex plane wave, (II. 7) figure II.. Real vectrs assciated with crss and principal plarizatin cpnents f a unifr elliptically plarized plane wave, annex II,sectin II. 2. Directins f crss and principal plarizatin, chsen s that their vectr prduct e x e is a unit vectr in the directin f prpagatin, (II. 4). P c Crss and principal plarizatin field cpnents f a receiving antenna re- spnse pattern, (II. 6). Directins f crss and principal plarizatin cpnents f a receiving an- tenna respnse pattern, (II. 8), (II. 20). The real vectr assciated with the iaginary cpnent f the tie-invariant part f a cplex plane wave \IT~ (e + ie.) exp (i"r), (II. 8b). The real vectr assciated with the real cpnent f the tie-invariant part f a cplex plane wave \[Z~ (e + ie.) exp (ii - ), (II. 8a). Real vectr cpnents f a cplex plarizatin vectr e which has been re- slved int cdnents which are rthgnal in bth space and tie, (II. 0). Real vectrs assciated with the 0 and <J> cpnents f a cplex plane wave v/2 - [e exp (it ) + e exp (it )] exp (ii - ), where nly the phasr exp (it) depends n tie, (II. 7) figure II. e A unit vectr e y r perpendicular t e and r, (II.3 b) figure II e A unit vectr (r x x )/sin 9 perpendicular t r and x, (II. 3a) figure II.. 9 " e, e A bar is used under the sybl t indicate a cplex vectr: e = e + ie, r ' r p c 7 = 7 + ie, (2. 9). -_* -r pr cr * e The cplex cnjugate f e: e = e - ie. e, e The agnitudes f the cplex vectrs e_ and e, (II. 22). Ie I, I e I, I e I, I e The acnitudes f the crss and principal plarizatin c c er p pr pnents e, e, e, and e, sectins 2.4. II. 2. II. 3. c er p pr 2-6

167 E Field strength in dbu, (2.43). E The equivalent free space field strength in dbu, (2.44). E The equivalent inverse distance field, (2.45). E. R. P. Effective radiated pwer, E. R. P. = P t + G (r,) dbw. E Field strength in dbu per kilwatt effective radiated pwer, see sectin 2. lkw Radi wave frequency in egahertz (egacycles per secnd). f Diffractin lss fr each f several distinct rays ver an islated bstacle, where j j =, 2.3,4, (IU.32-UI.35). f.. M., Hz Radi wave frequency»» in egahertz. e i Operating nise factr f a receiving syste, (V. 7). p f., iy f-, f. Diffractin lss fr each f fur distinct rays ver an islated bstacle, (UI.32). f(r ), f(r ) Functins f the nralized antenna heights r and r, (IU. 50). f(v.) A functin identically equal t f. fr v = v., (III. 33) figure IU. 0. f(9.) A factr used t reduce estiates f variability fr antenna beas elevated h abve the hrizn plane, (III. 64) figure IU. 22. See 9, and 0,. h b f( v) A functin used in cputing path antenna gain, defined by (9. 3) figure F The crrectin ter F allws fr the reductin f scattering efficiency at great heights in the atsphere, (9. ) and (9. 7). F. Scattering efficiency crrectin ter fr the i 0 (UI.63). lbe f an antenna pattern, F Operating nise factr f a receiving syste expressed in decibels, (V.8). p F(x ), F(x ) Functins used in cputing diffractin attenuatin, (8. ) and figures 8. 5 and 8.6. F(6d) The attenuatin functin used in calculating edian basic transissin lss fr scatter paths, (9. ) figures 9., and HI. t III. 4. F(9.d) This functin is the sae as F(9d) with the effective angular distance 9 ei ei substituted fr the angular distance, 9, annex III, (HI. 57). FM Frequency dulatin, g Grade f service. A specified grade f service guarantees a crrespnding degree f fidelity f the infratin delivered t the receiver utput, annex V, sectin V. 5. g Maxiu free space directive gain, r directivity, the rati f the available ean pwer flux density and e /n II. fr a lss-free antenna, sectin 2.3, annex g A high gain antenna radiates g watts per unit area in every directin nt acb b cunted fr by the ain bea r by ne f the side lbes f an antenna, annex III, sectin III. 6. g The directive gain g fr a transitting antenna, annex IU, sectin III. 6. bt b 2-7

168 g c «er 8 P V g pt The crss-plarizatin cpnent f the directive gain, (II, 24). The crss-plarizatin cpnent f the directive gain f a receiver, (II. 9). Principal plarizatin directive gain, (II. 24). Principal plarizatin directive gains fr the receiving and transitting antennas, respectively, (.26). g r' ««rn ' g tn g r, g * 8. g e' «g(f) g(p. *) g rz g z g(0, f), g(90, f) Maxiu free space directive gains fr the receiving and transitting antennas respectively, sectin 2.3. The directive gains g and g fr the n f a series f plane waves, (II. 33) and (II. 34). Directive gain factrs defined fr each antenna in the directin f the pint f grund reflectin, (5.). The axiu value f the perating gain f a receiving syste, (V. 7). The directive gain fr ne antenna in the directin f the ther, sectin 5.. The directive gain f the transitting and receiving antennas, each in the di- rectin f the ther, assuing atched antenna plarizatins, (5. ). Directive gains assciated with the field cpnents e, e, (II. 4). 6 (J> A frequency crrectin factr shwn in figure III. 30, (III. 66). A frequency factr used t adjust predicted lng-ter variability t allw fr frequency-related effects, (0.6) figure 0.3. The frequency factr g(p, f) used t adjust Y^IO) and Y Q (90) pre- dicted values, (0.6) figure 0.3. g(r) Directive gain in the directin r, (.56). g(-r) Directive gain in the directin (-r), annex II. g (r). g (r) C P Crss plarizatin and principal plarizatin directive gains in the directin r, (.6). g (-r) Free space directive gain f the receiving antenna in the directin (-r). (2.32). a (r ). a (r ) Directive gains assciated with direct and grund-reflected rays, respec- lively, (II. 84). g (T) Free space directive gain f the transitting antenna in the directin r, see als g'(r), sectin g' Pwer gain f a transitting antenna when the pwer input t the antenna ter- inals is p' watts, sectin 2.3. g (r) Pwer gain f a transitting antenna in the directin r, see subsectin g Gras, g/ Gras per eter. G The axiu free space directive gain relative t an istrpic radiatr, (2. 4). G Decibel equivalent f g, G = 0 lg g, annex III sectin III. 6. b b b b G Decibel equivalent f g fr a transitting antenna, annex III sectin III. 6. bt b G The hurly edian perating signal gain f a receiving syste, (V.8). s 2-8

169 G Path antenna gain, the change In transissin las r prpagatin lss if P hypthetical lss-free istrpic antennas with n rientatin, plarizatin, r ultipath cupling lss were used at the sae lcatins at the actual antennas, (2,29). G, Path antenna gain in free space, (2. 32). P f G Lng-ter edian value f G, (2.36). p p G Path antenna pwer gain, (2. 29). PP G, G Free space gains f the receiving and transitting antenna, respectively, in decibels relative t an istrpic radiatr, (2.37). th G., G. Gains f the i lbe f receiving and transitting antennas, respectively, (.57). G Maxiu value, expressed in decibels, f the perating gain f the receiving syste fr C. W. frequencies in the receiver pass band, (V. 2). G(h) Residual height gain functin, figure 7.. G(h ) The functin G(h) fr the transitting r receiving antenna. G(h ),G(h ) The functin G(h) fr the transitting and receiving antennas, respectively, (7.5). G(r) Directive gain f an antenna in the directin r. is G, sectin The axiu value f G(r) G (r), G (-r) Directive gain, in decibels, f a receiving antenna in the directins r <-r), (2.32). Q r (r li2 J - 0 gg r (r- >2 ). G' (r) Pwer gain, in decibels, f a receiving antenna, (2. 3). G (r) Directive gain, in decibels, f a transitting antenna, (2.3). G'(r) Pwer gain, in decibels, f a transitting antenna, G'(r) = G (r) - L, (2. 3). i iv et G(x ) A functin used in cputing diffractin attenuatin, (8. ) figures 8. 5 and GHz Radi frequency in gigacycles per secnd. h Height abve the surface f the grund as used in (3. 0), (3. 2). h Height referred t sea level. h Height fr elevated beas that is equivalent t h fr hrizn rays, (.63). h Equidistant heights f terrain abve sea level, (5. 5), (6. 0). h Height f the receiver hrizn bstacle abve sea level, (6. 5). I_,r h Height f the transitter hrizn bstacle abve sea level, (6. 5). Lit h Height f the intersectin f hrizn rays abve a straight line between the an tennas, deterined using an effective earth's radius, a, (9. 3b) and figure 6.. h, h Height f the receiving r transitting antenna abve grund, assuing a sth earth. A sth earth is assued in the curves f figures I. 5 and I. 7 t I. 26. h, h The height h r h is defined as the height f the receiving r transitting antenna abve the average height f the central 80% f the terrain between the antenna and its hrizn, r abve grund, whichever gives the larger value, (6. ). and 2-9

170 h, h Effective height f the receiving r transitting antenna abve grund. Fr re te h, h leas than ne kileter h = h, h = h. Fr higher antennas a r t re r te t crrectin Ah is used, (6. 2). h, h Height f a knife edge abve a reflecting plane n the receiver r transitter r t side f the knife edge, (III. 37). h, h Height f the receiving antenna r transitting antenna abve sea level, fig- ure 6., used in (6. ), (6. 5). h Elevatin f the surface f the grund abve ean sea level, (4. 3). s h. The heights abve sea level f evenly spaced terrain elevatins between the transitter and its hrizn, (6. ). h The height abve sea level f the grund belw the transitting antenna, (6. ). h The height f the hrizn bstacle h = h., (6. ). t*> * tj Lt h Height f the crssver f hrizn rays abve a straight line between the trans- itter and receiver hrizn bstacles, (9. 7) figure 6.. h, h Heights f antenna terinals and 2 abve the surface f the earth, figure 5.. h, h' Heights f antenna terinals and 2 abve a plane tangent t a sth earth at the bunce pint f a reflected ray, (5.8). h Average height abve sea level, (5. 5). h Average height f the transitting antenna abve the central 80% f terrain between the transitter and its hrizn, (6. ). h, h Nralized heights f the transitting and receiving antennas, (7.6). h(r) A functin f r shwn in figures III. 20 and III. 2. h(r ), h(r ) A functin f r r r defined by (III. 50) and shwn n figures III. 20 and HI. 2. h(x) A straight line fitted by least squares t equidistant heights abve sea level. h(0), h(d) ( 5. 5). Height abve sea level f a sth curve fitted t terrain visible t bth an- tennas, and extraplated t the transitter at h(0) (5.7). and the receiver at h(d). h.(x.) A series f equidistant heights abve sea level f terrain visible t bth antennas sectin 5.. H The frequency gain functin, discussed in sectin th H. The frequency gain functin fr the i bea intersectin in a scattering plane, (HI. 57). H (r,), H (r ) The frequency gain functin, H, as a functin f r and r, respec 2 2 tively, (9. 5). 2-0

171 H (n < ), H (r =) Value f the frequency gain functin, H, where the paraeter n is less than r equal t ne, respectively, (9.6). H (n =0) The frequency gain functin when n = 0 which crrespnds t the assup- tin f a cnstant atspheric refractive index, figure Hz Abbreviatin fr hertz = cycle per secnd. i i = \CT, (2. 9) annex II. I Current in r.. s. aperes where = 0,, 2. I-, I., I Current in r.. s. aperes crrespnding t three eleentary diples in three utually perpendicular directins, (.46). j Represents a series f subscripts, 2, 3, 4, as used in equatins (III. 27) t (UI. 35). k Prpagatin cnstant, k= 2ir/K, (II. ). k Bltzann's cnstant, -23 k= x 0 jules per degree, (V.7). kt b Jhnsn's nise pwer that wuld be available in the bandwidth b cycles per secnd at a reference abslute teperature T (V.7).. k Abbreviatin fr kileter. kw Abbreviatin fr kilwatt. K A frequency-dependent cefficient, (3.8). K A paraeter used in cputing diffractin attenuatin, = degrees Kelvin, K is a functin f the effective earth's radius, carrier frequency, grund cnstants, and plariza- tin, figure 8, and annex III. 4. K The decibel rati f the rt-su-square f Rayleigh cpnents f a received signal relative t a cnstant r pwer-fading cpnent, annex V, sectin V. 2 and figure V.l. K The diffractin paraeter K fr hrizntal plarizatin, annex III. 4. h K An arbitrary cnstant in the systes equatin, (V. 22). K The rati K fr an unwanted signal, annex V. u K The diffractin paraeter K fr vertical plarizatin, annex III. 4. K A frequency and teperature-dependent attenuatin cefficient fr absrptin within a clud, (3. 3) and table 3.. K, K, K, K, K Values f the diliractin paraeter K fr crrespnding earth's 2 r s t v f 6 radii a,, a,, a, a, a, (8.8) t (8. 3). 2 r s t K(a), K(8497) The diffractin paraeter K fr an effective earth's radius a, and fr a = 8497 k. K(f_ u ) A frequency-dependent cefficient used in cputing the rate f absrptin by G Hz rain, (3.9a) and figure 3.8. K(N ) A functin f the surface refractivity, N, used in cputing F(0d), (III. 46). s s 2-

172 K (N ), K (N ), K (N ) Functins f the surface refractivity, N, used in cputing * F(ed), (III. 48). I Used as a subscript t indicate a lad, fr exaple, z pedance f a lad at a radi frequency v, (2.4)., represents the i- i A range f eddy sizes r layers', the radi wave scattered frward is st af- fected by a particular range f "eddy sizes, " i, r by layers f an average thickness /2, that are visible t bth antennas, (IV. ). I The effective lss factr fr a receiving antenna, r the reciprcal f the pwer receiving efficiency, (2.3). / The effective lss factr fr a receiving antenna at a frequency v hertz, defined as the rati p /p', (2.9) av r av t The effective lss factr fr a transitting antenna, r the reciprcal f its pwer radiatin efficiency, (2.3). I The effective lss factr fr a transitting antenna at a radi frequency hertz, (2. 0). I A isatch lss factr defined by (2. 7). I Scale f turbulence, (IV, 9). L Transissin lss expressed in decibels, (2. 2). L, Basic transissin lss, (2.28) and (2. 29). b L,. Basic transissin lss fr-a diffractin path, (7.3), (7.4). bd L Basic transissin lss in free space, (2.3). bf L Hurly edian basic transissin lss, b L, Reference value f lng-ter edian basic transissin lss due t frward bsr scatter, (9. ). L Calculated value f transissin lss. c L Plarizatin cupling lss, (2.25). L Reference value f hurly edian transissin lss when diffractin and cr scatter lsses are cbined, (9. 4). L, Reference value f hurly edian transissin lss due t diffractin, (9. 4). dr L Effective lss factr fr a receiving antenna, expressed in decibels, (2. ). L The effective lss factr, L, at a radi frequency v hertz, erv er n ' (2.). L, L The effective lss factr fr a transitting antenna, expressed in decibels, et etv (2.3) and (2. ). L An "equivalent free-space transissin lss, " (2.34). L., L The decibel rati f the resistance cpnent f antenna input ipedance t the free space antenna radiatin resistance fr the receiving and transitting antennas, respectively, (2.39). L Lss in antenna gain fr the i scattering subvlue, (III. 57). 2-2

173 L Lss in path antenna gain, defined as the difference between basic transis- 8P sin lss L, and path lss L, r as the difference between the su f the b axiu gains f the transitting and receiving antennas and the path antenna gain: L = L L - L = G + G - G db, (2.37). gp b t r p A L. Transissin lss assciated with the i pwer cntributin, (III. 55) and i (III. 57). L Transissin line and atching netwrk lsses at the receiver. Ir L Transissin line and atching netwrk lsses at the transitter, (V. 20). L Hurly edian transissin lss, (V.20). L The transissin lss exceeded (00-p) percent f the tie with a prbability Q, (V.43). L Path lss, defined as transissin lss inus the su f the axiu free space gains f the antennas: L = L - G -G,(2. 27). L Prpagatin lss, (2.4). P L Basic prpagatin lss, (2.42). L The rati f the actual radiatin resistance f the receiving antenna t its rarr diatin resistance in free space, (2.40). L The rati f the actual radiatin resistance f the transitting antenna t its rt radiatin resistance in free space, (2.40). L The syste lss expressed in decibels, defined by (2. ). grund and dielectric lsses and antenna circuit lsses. Syste lss includes L Reference value f edian frward scatter transissin lss, used with L, sr dr t btain the reference value L, (9. 4). cr L Median transissin lss f an unwanted signal, annex V.4. urn L.i L... L... L A series f hurly edian values f transissin lss arranged in rder fr the sallest t the largest value, annex III, subsectin L(p) Transissin lss exceeded (00-p) percent f the tie, (III. 68). L(50) The lng-ter edian value f transissin lss, sectin 0.3. L(0.0), L(0. ),... L(99.99) Transissin lss exceeded (00-p) percent f the tie where p = 0.0, 0., , sectin 0.3. L, (50) Lng-ter edian value f basic transissin lss, sectin b L (p) Tie availability f hurly edian basic transissin lss, annex I, figures b I. 7 t I. 7. L, (50) Lng-ter edian value f L,, (2.36). b b L.(p) Instantaneus values f transissin lss nt exceeded p percent f the tie, (V.5). 2-3

174 L.(0. ), L.(0.9) Instantaneus values f transissin lss nt exceeded 0 and 90 percent L «.W L (50) L (p, Q) L ( 8) L n (p) Ln(50) urn L (50) urn Li. tn ni f the tie, (V.4). Hurly edian transissin lss nt exceeded fr p percent f all hurs r exceeded fr (00-p) percent f all hurs, (V.25). Lng-ter edian transissin lss, (V. 2). Hurly edian transissin lss nt exceeded fr p percent f the tie with a prbability Q. Maxiu allwable transissin lss fr a grade g service, (V.27), Transissin lss nt exceeded p percent f the tie in a given cliatic re- gin, (0. 5). Predicted edian lng-ter transissin lss fr a given cliatic regin, characterised by the subscript n, (0.4). Hurly edian transissin lss f an unwanted signal nt exceeded fr p per- cent f all hurs, (V. 33). Lng-ter edian transissin lss fr an unwanted signal, (V.39). Abbreviatin fr liit, as used fr exaple n figure 8.4. A sybl used t designate the slpe f a straight line, (5. 5). A subscript used t identify service liited by nise, annex V. Average refractive index gradient, dn/dz, acrss a layer, (IV. 5). M M,, h v in. h.. v/ M Paraeters used in cputing the agnitudes R and R f the sth plane earth reflectin cefficient R, (III. 0). Average refractive index gradient fr the regin in which a layer is ibedded, (IV. 5). Abbreviatin fr iniu. A unit f cnductance, the reciprcal f resistance which is easured in hs, annex III., figures III. t III. 8. Abbreviatin fr illieter. Millivlts per eter. Liquid water cntent f a clud easured in gras per cubic eter, (3. 3). A ter defined by (IV. 7) used in the pwer reflectin cefficient q 2, (IV. 6). A ter defined by (IV. 22) used in defining a, the scattering crss-sectin v fr refractivity turbulence. 2-4

175 MHz Radi frequency in egahertz. M.U.F. Abbreviatin f axiu usable frequency. n Refractive index f the atsphere, sectin 4. n The rati a lb. r p lb used t cpute n, (9. 2). t r n Atspheric refractive index at the surface f the earth, (4. ). s n, n Refractive indices f adjacent layers f hgeneus edia, (IV. 3). n A paraeter used in calculating path antenna gain, (9. 2). 6 N Atspheric refractivity defined as N = (n-) X 0, sectin 4. N The nuber f layers per unit vlue f a scattering crss-sectin, (IV. 5) t (IV. 7). N Surface refractivity reduced t sea level, (4.3). N The value f N at the surface f the earth, (4. ). s N The nuber f scattering subvlues that ake an appreciable cntributin t the ttal available pwer, (IV. ). p Tie availability, the percentage f tie a given value f transissin lss is nt exceeded, sectin 0. P A functin f the dielectric cnstant and grazing'angle used in cputing the plane wave reflectin cefficient, (III. 8). P a Radi frequency signal pwer that wuld be available fr an equivalent lss- free receiving antenna, (2.2). P a b The available pwer crrespnding t prpagatin between hypthetical is- trpic antennas, (.40). P a j Cntributin t the ttal available pwer fr the i (III. 55) and (IV. ). scattering subvlue, P a Radi frequency signal pwer available at the terinals f the receiving an- tenna, (2. ). P av Available pwer at the terinals f an equivalent lss-free receiving antenna at a radi frequency v, (2.9). P av Available pwer at the terinals f the actual receiving antenna at a radi fre- quency v, (2.6). Pj "Instantaneus" radi frequency signal pwer available at the terinals f an equivalent lss-free antenna, defined as the average pwer fr a single cycle f the radi frequency, annex V. Pj r ' Pf r v Pwer delivered t the receiving antenna lad, at a radi frequency v, (2. 5). 2-5

176 Pwer delivered by the transitter t the transissin line, (2. 5). ^t' **ltw p The edian wanted signal pwer available at the receiver, annex V. p Median value f the ttal nise pwer in watts, (V. 7). n p Operating sensitivity, the edian wanted signal pwer, p, required fr sat- isfactry service in the presence f nise, annex V. p Fixed value f transitter pwer utput, expressed in watts, (V. 26). p Pwer, in watts, radiated fr a wanted statin, (V. 34). p Ttal pwer radiated fr the transitting antenna in a given band f radi frequencies, (2.2). p Ttal pwer radiated at a frequency v, (2. 0). p Radi frequency pwer input t the terinals f the transitting antenna, (2. ). p' Ttal pwer delivered t the transitting antenna at a frequency v, (2. 0). p Pwer radiated fr an unwanted statin, (V. 34). *u p. Instantaneus pwer f an unwanted signal available t a receiving syste, annex V. p Available pwer per unit scattering vlue, (IV. ). p. Available pwer per unit scattering vlue fr the i (IV. 2) p Median unwanted signal pwer, annex V. urn p.(q) Value f "instantaneus" available pwer exceeded fr shrt perid, (V.6). See p. scattering subvlue, 00 q percent f a P (g) The value f p required t prvide service f grade g, annex V. r r Unit cplex plarizatin vectr fr the incident wave, (II. 2). th p Unit cplex plarizatin vectr fr the n incident plane wave, (.3 5). p Unit cplex plarizatin vectr assciated with a receiving pattern, (II. 8). p The cplex plarizatin vectr th the n incident wave, (.35). assciated with a receiving pattern and, The cplex receiving antenna plarizatin vectrs fr each f tw ray paths between transitter and receiver, (.83). (r), (-r)unit cplex plarizatin vectrs fr the transitter, and fr the receiver,, in the directin (-r), (2. 8)., in the directin (r) * A. 7 I. I Plarizatin efficiency fr transfer f energy in free space at a single radi frequency, (2. 22) and (II. 29). P The nrth r suth ple in figure P The available pwer fr a lss-free receiving antenna which is therwise equivalent t the actual receiving antenna, (2. 2). P' The radi frequency signal pwer available at the terinals f the receiving antenna, (2. ). 2-6

177 P Available pwer at the terinals f a hypthetical lss-free istrpic receiving ab antenna, assuing n rientatin, plarizatin, r ultipath cupling lss be- tween transitting and receiving antennas, (2. 28). P P. = 0 lg p., decibels, (V. ). the instantaneus pwer f a wanted radi signal expressed in P Pwer, in dbw, delivered t the receiving antenna lad, (2.5). Ir P Pwer, in dbw, delivered by the transitter t the transissin line, (2. 5). IT P The cpnent f P. which is nt affected by the usually rapid phase interfer i ence fading, st ften identified as the shrt-ter edian f the available pwer Pj, (V. ). P The hurly edian value in dbw f the ttal nise pwer delivered t a receiver n utput: P = 0 lg p, (V.8). n n P Operating sensitivity, assuing a specified type f fading wanted signal and a r specified type f nise, annex V. P A fixed transitter utput pwer, expressed in dbw, (V. 26). P Ttal pwer radiated fr a wanted statin, expressed in dbw, (V.34). P The ttal pwer radiated fr the transitting antenna in dbw : P = 0 lg p, (2.2). P' Radi frequency pwer input t the terinals f the transitting antenna, in dbw, (2. ). P Pi.wer radiated fr an unwanted statin, (V.33). P u The edian unwanted signal pwer available at a receiving antenna fr an unwanted statin radiating p watts, annex V. P,(p) Percentage f tie p that a given value f instantaneus pwer is exceeded, annex V. Pj(q) The percentage, 00 q, f a shrt perid f tie r the prbability q that P. will exceed P.(q) is knwn if the phase interference fading distributin f P. relative t the shrt-ter edian value P is knwn, annex V. i P. t Transitter utput pwer which will prvide at least a grade g service fr p percent f the tie, (V.25). P (p) The hurly edian wanted signal pwer P exceeded fr p percent i all hurs, annex V. P (50) The lng-ter edian f all hurly edian values P, usually identified als as the lng-ter edian f P., (V. 2). P (p) Observed values f P (p). r r P (g) The perating sensitivity f a receiving syste, defined as the iniu r value f P which will prvide a required grade f service a, in the pre r sence f nise alne, (V.9). 2-7

178 P (p) The hurly edian pwer P expected t be available at least p percent urn urn f the tie, annex V. P (50) The hurly edian pwer P expected t be available at least 50 percent urn urn f the tie, annex V. q 00 q is the percentage f a shrt perid f tie r q is the prbability that p will exceed p.(q) fr a given edian value p, which is the sae as the r i l-ji-i/ r prbability that Y. will exceed Y.(q), annex V. q A paraeter used in calculating a plane wave reflectin cefficient, (III. 7) t (III. 4). q The rati q=r /sr used t cpute AH, (9.5). 2 q The pwer reflectin cefficient, q, fr a trpspheric layer is apprxi- ated by (IV. 6). q The plane wave Fresnel reflectin cefficient fr an infinitely extended plane bundary, (IV. 3). Q Service prbability, discussed in subsectin V.8. Q(p) Service prbability crrespnding t the tie availability p, annex V. Q(z ) Service prbability expressed as a functin f the standard nral deviate z, (V.44). r The length in free space f the direct ray path between antennas, figure 5.. r Radius f curvature, (7.9). r Resistance f an antenna, sectin 2. r Magnitude f the vectr r = r r the plar crdinate syste r, 6, 4>, annex II. in the directin r(0, t)>), and a crdinate f r Effective distance fr absrptin by xygen in the atsphere, (3.4) figures 3.2 t 3.4. r Effective rain-bearing distance, (3. ) and (3. 2) figures 3. 0 t r Effective distance fr absrptin by water vapr in the atsphere, (3.4). r figures 3.2 t 3.4., r Antenna radiatin resistance in free space fr the receiving and transitting antennas, respectively, (2.38). r. Resistance f a lad, (2.4). r Rati between the hurly edian wanted signal pwer and the hurly edian perating nise pwer, annex V. r r A specified value f r which ust be exceeded fr at least a specified r perr centage f tie t prvide satisfactry service in the absence f unwanted signals ther than nise, annex V. 2-8

179 r Length f a direct ray between antennas ver an effective earth f radius a, n figure 5.. r, r Antenna radiatin resistance f the receiving and transitting antennas, re- spectively, (2.38). r, r' Resistance cpnent f antenna input ipedance fr the receiving and trans- itting antennas, respectively, (2.38). r Rati between the hurly edian wanted signal pwer and the hurly edian u unwanted signal pwer available at the receiver, annex V, page V-l. r A specified value f r which ust be exceeded fr at least a specified perur u centage f tie t prvide satisfactry service in the presence f a single unwanted signal, annex V. r Resistance f an equivalent lss-free antenna, (2.4). r' Resistance f an actual antenna in its actual envirnent, (2.4). v Paraeters used in cputing the frequency gain functin H, and defined r r r 2 by (9.4). r, r Distances whse su is the path length f a reflected ray, figure 5.. r. r, r, r Distances t and fr the bunce pint f reflected rays, (.28) fig- lb b C. C, ure UI. 9. r, r Straight line distances fr transitting and receiving antennas t a pint n the grund a distance x. fr the transitting antenna, figure 6.4. r The vectr distance between tw antennas, r = rr, (.47). r A unit vectr directed away fr an antenna, annex II, subsectin U.. r, r Directin f the st iprtant prpagatin path fr the transitter t the receiver, r fr the receiver t the transitter. r, e, e. A cartesian unit vectr crdinate syste, annex. 9 <t> r (g) The iniu acceptable signal t nise rati which will prvide service f a given grade g in the absence f unwanted signals ther than nise, annex V. r (g) The prtectin rati, r, required t prvide a specified grade f service, g, annex V sectin V.4. r.. s. Abbreviatin f rt-ean-square. R Lcatin f the receiving antenna, figure 5.. R The agnitude f the theretical cefficient R exp[-i(ir -c)] fr reflectin f a plane wave fr a sth plane surface f a given cnductivity and dielec- tric cnstant, (5. ). 2-9

180 R An "effective" grund reflectin cefficient, (5. ). R Plane earth reflectin cefficient R fr hrizntal plarizatin, (III. 2) and a figures III. t III. 8. R R = 0 lg r decibels, the edian wanted signal t edian nise rati available at the receiver utput, (V.9). R The decibel equivalent f r, R =0 lg r, annex V. rat r r r R Rainfall rate in illieters per hur, (3. 0). R Surface rainfall rate, (3. 0). rs R R = 0 lg r, the rati between the hurly edian wanted signal pwer and u u u the hurly edian unwanted signal pwer available at the receiving antenna terinals, (V. 5). R The rati between the instantaneus wanted signal pwer and the instantanui eus unwanted signal pwer at the receiving antenna terinals, (V. 0). R The decibel equivalent f r, R =0 lg r ur ur ur ur R Plane earth plane wave reflectin cefficient R fr vertical plarizatin, v (III. 2) figures III. t III. 8. R. R Q Vectr distances fr transitter and receiver, respectively, t a pint R. R' R 0 Unit vectrs fr the centers f radiatin f the receiving and transitting antennas, respectively, (IV. ) R i A pint fr which pwer is cherently scattered r reflected, (IV. ). R Cuulative distributin f instantaneus path average rainfall rates, figure R(0. 5) A functin f L, L, (9. 4) figure dr cr R (50) The value f R exceeded at least 50 percent f the tie, (V.29). R (p) The value f R exceeded at least p percent f the tie, (V.24). R (g) The iniu value f R that will prvide a desired grade f service in the presence f nise alne, (V.9). R (p) A specified value f R exceeded at least p percent f the tie, (V.36). R (50) A specified value f R exceeded at least 50 percent f the tie, (V. 36). R (g) Median wanted signal t edian unwanted signal rati required t prvide a R ur <8, P ' grade g service, annex V sectin 4. The re uired rati R cent f the tie, (V.3 5). t prvide service f grade g fr at least p per- R ur <8) The required value f R fr nn-fading wanted and unwanted signals, (V. 4). s Path asyetry factr, s = a /(}, (6.9). 2-20

181 Ttal ean pwer flux density, (.25). Mean pwer flux density assciated with crss-plarizatin cpnents, (II. 23). c s The fractin f the ttal flux deity that cntribute» t the available pwer, (II. 43a). e s Path asyetry factr fr beas elevated abve the hrizn, s = a /ß, e ' ' e e e (UI. 64). s Mean pwer flux density assciated with left-handed plarizatin, (II. 28). s Free space field strength in watts per square kileter, (2.43). s Mean pwer flux density assciated with principal plarizatin cpnents, (II. 23). 8 Mean pwer flux density assciated with right-handed plarizatin, (II. 28). < > The statistical "expected value" f s, (II. 43b). s(r) Ttal ean pwer flux density at the receiving antenna, (2. 23). *» s (r), s (r) Mean pwer flux densities assciated with the crss and principal plariza- tin cpnents f e in the directin r, (II. 23). S. Fresnel integral, (III.33). S(u), S(v) Fresnel sine integrals, (IV. 8). Si(r) Sine integral as a functin f r, (III. 5). t Tie at the transitter, in secnds, (II. ). T Lcatin f transitting antenna, figure 5.. T Reference abslute teperature, T = degrees Kelvin, T(r) Teperature in the trpsphere in degrees Kelvin. T (*K) Effective sky nise teperature in degrees Kelvin. T. A. S. O. Abbreviatin f Televisin Allcatins Study Organizatin. u A subscript used t indicate signal fr an unwanted r interfering statin, annex V. u A paraeter defined by (IV. 9). U(vp) A paraeter used in cputing diffractin ver a runded bstacle, (III. 26) and figure UHF Abbreviatin f ultra high frequency, v A paraeter used in cputing diffractin ver an islated bstacle, (7. ). v A paraeter defined by (IV. 9). ~,_ th V. The r scattering subvlue, (IV. ). v. The paraeter v fr each f j paths ver an islated bstacle, (III. 27). v Cplex pen-circuit r..s. signal vltage fr cherently phased ultipath cpnents, (.32). v The pen-circuit r.. s. vltage fr an equivalent lss-free antenna at a v frequency v, (2.8). The actual pen-circuit r.. s. vltage at the antenna terinals at a fre- quency v, (2. 5). 2-2

182 V(50, d ) A paraeter used with the calculated lng-ter reference value, L, t e cr predict edian lng-ter transissin lss, figure 0. equatins (0.4) Y (50, d ) and (III. 67). The paraeter V(50, d ) fr a given cliatic regin characterized by the subscript n, (0.4) figure 0.. VHF Abbreviatin f very high frequency. w Half the width f a first Fresnel zne, (IV. 7). x A specified value, the discussin preceding (2. 4). x A variable designating distance fr an antenna, figure 6.4. x, x Pints at which a first Fresnel ellipse cuts the great circle plane, III. 8 t III. 23. x., x', x Reactance f a lad, an actual lssy antenna, and an equivalent lss-free lv v v n antenna, respectively, (2.4). x. The i distance fr the transitter alng a great circle path, figure 6.4. x One f three utually perpendicular directins, = 0,, 2, annex II. XQ, x, x Paraeters used t cpute diffractin lss, (8. 2) figures 8. 5 and 8.6. x, x Pints chsen t exclude terrain adjacent t either antenna which is nt visible t the ther in cputing a curve fit, (5. 5). x, x, x Axes f a cartesian unit vectr crdinate syste, (II. 2) figure II.. x The average f distances x and x, (5. 5b). X Initial bearing fr antenna terinal A, easured fr true nrth, figure 6.3. y. Terrain elevatins, dified t accunt fr the curvature f the earth, (6. 0). 2 y(x) Mdified terrain elevatin, y(x) = h(x) - x /(2a), (5.6). Y Initial bearing fr antenna terinal 6.3. B, easured fr true nrth, figure Y A sybl used t describe the characteristics f lng-ter fading, (V. ) and (V.3). Y. The phase-interference fading f a received signal, (V. ) and (V.3). Y Lng-ter fading cpnent f unwanted signal fading, (V. 6). Y. The phase interference cpnent f the fading f an unwanted signal, (V. 0). Y' Bearing fr any pint B' alng the great circle path AB, figure 6.3. Y(p) Lng-ter variability f L r f P in ters f hurly edians, (0. 6) and (V.4). Y.(q) Phase interference fading evaluated fr the particular phase interference fading characteristic f a wanted signal exceeded p percent f the tie, where p = 00 q, (V. 5). \l-ll

183 Y.(q, K) Cuulative distributin functin fr the phase interference fading f a wanted signal, (V. 2). Y.(q, K ) Cuulative distributin functin fr the phase interference fading f an un- u wanted signal, (V.2). Y (p) Variability f hurly edian transissin lss, (V.29). Y (p) Variability f the perating nise factr, Y (p) = F (p) - F (50), (V.32). Y (00 - p) Value f Y (p) exceeded (00 - p) percent f the hurs r nt exceeded p n n percent f the hurs, (V.3). Y (p) Basic estiate f variability in a cntinental teperate cliate, figure 0.2. Y (p, d ) Basic estiate f variability as a functin f effective distance, (0.6) fig- ure Y D (P) Variability f the rati f wanted t unwanted signal, (V.38). R Y (p) Variability f an unwanted signal, (V.39). Y (00 - p) Value f Y (p) exceeded (00 - p) percent f the tie, (V.38). z Thickness f a trpspheric layer, (IV. 6). z. Ipedance f a lad, (2.4). tv z A standard nral deviate, (V.43). z The thickness f a trpspheric layer, (IV. 6). z A standard nral deviate, (V. 50). P z A standard nral deviate crrespnding t the ttal variance 2 a (p) f an estiate f the service criterin, (V. 53). z Ipedance f an equivalent lss-free antenna, (2.4). z' Ipedance f an actual lssy antenna, (2.4). z' * The cnjugate f z', fllwing (2. 5). Z Great circle path length between antenna terinals A and B, figure 6.3. Z The difference between the tw rand variables Y and Y, (V. 6). u Z. The difference between the phase interference fading cpnents Y. and Y., (V. ). ui Z Great circle path distance between an antenna and an arbitrary pint B, figure Z (p) Apprxiate cuulative distributin functin fr the rand variable Z = Y - Y, (V. 7). u Z (50) The edian value f Z (p), by definitin equal t zer. Z.(q,», «) The cuulative distributin functin fr the special case where Y. is Ray- leigh distributed, (V. 3). Z. (q,», <*>) The apprxiate value f Z.(q,»,») see (V. 2) and Table V.2. la i Z. (q, K, K (Apprxiate cuulative distributin functin f the rand variable Z., ta u i (V. 2). 2-23

184 The paraeter is defined in equatin (3.9b) and pltted as a functin f fre- quency n figure 3.9. a, ß The angles between the "btts" f transitting r receiving antenna beas e e r side lbes and a line jining the antennas, (.6). a., 0. Angles a and ß fr the i lbe f an antenna pattern, ei ei e e a, ß When beas are elevated sufficiently that there is n bending f the ray due t e e atspheric refractin = a, ß = ß, (III. 60); when ray bending ust be e e e e cnsidered a and ß are cputed using (III. 6). a, ß The angles a, ß dified by the crrectins Aa, Aß, (6. 9). a., ß. The angles a, ß ade by each f j rays, ver an islated bstacle, (III. 36). j OJ a, ß The angles between a transitter r receiver hrizn ray and a line drawn be- 00 tween the antenna lcatins n an earth f effective radius, a, (6. 8) figure 6.. a, a, &, ß The angles a and ß fr each f fur rays ver an islated bstacle, 0 02 Ol 02 O O (III. 36). Q (f/- u ) T^e functin a in (3.9b) as a functin f frequency in GHz, figure 3.9. Y Differential absrptin in decibels per kileter fr xygen under standard cn- ditins f teperature and pressure, (3.4). Y Rate f absrptin by rain, (3.8). y Surface value f the rate f absrptin by rain, (3. ). Y Differential absrptin in decibels per kileter fr water vapr under standard cnditins f teperature and pressure and fr a surface value f abslute hu- idity f 0g/cc, (3.4). Y(r) Differential atspheric absrptin in db/k fr a path length r, (3. ). Y (r) Differ ntial rain absrptin alng a path r, (3.7). Y (h), Y ( n ) Differential absrptin in db/k fr xygen and water vapr, respectively, as a functin f height, h, (3.3). r(r) Absrptin cefficient as a functin f path distance r, (3.2) and (3.6). 6 A paraeter used in cputing the first Fresnel zne in a reflecting plane, (III. 8). 6 The effective half-pwer sei-beawidth f an antenna, (2. 5) and annex III. 6 The effective half-pwer sei-beawidth f an antenna that is elevated r di- rected ut f the great circle plane, annex III The sei-beawidth f an equivalent bea pattern with a square crss-sectin, 6 = 6Vir/4, annex III. 6. 6, 6 The effective half-pwer sei-beawidth fr the receiving and transitting antennas, respectively, (9.) and (9. 2). 6, 6 Aziuthal eqi equivalent sei-beawidths with square crss-sectin, (III. 58) rw tw figure III , 6 Vertical angle equivalent sei-beawidths with square crss-sectin, (III. 58) rz tz figure III

185 6 Aziuthai sei-beawidth, (2.5). w 6 Aziuthai equivalent sei-beawidth with square crss-sectin, annex III. 6. w 6 Vertical angle sei-beawidth, (2.5). 6 Vertical angle equivalent sei-beawidth, annex III. 6. z Aa, Aß Crrectin ters applied t cpute a, ß (6. 9) figure 6.9. A Depressin f field strength belw sth earth values, (5. 9). Ah A crrectin ter used t cpute the effective height fr high antennas, (6. 2) figure A. The j th value f Ar, where Ar = r, + r, - r, (III. 27) and (III. 29). j Z An The deviatin f refractive index fr its expected value, (IV. 20). Ar The path length difference between a direct ray, Ar = r, + r,- r, (5.4), (5.9) and (7. ). r, and a reflected ray, A A Auxiliary functins used t check the agnitude f errr in the graphical deterina- X l' X 2 tin f diffractin attenuatin, (8. 5) figures 8. 5 and AH A crrectin ter applied t the frequency gain functin, H, (9. 5) and figure 9.4. AN The refractivity gradient fr the surface value, height f ne kileter abve the surface, (4.2). N, t the value f N at a s A., A-, A, A Ray path differences between a direct ray and a ray path ver a single is- lated bstacle with grund reflectins, (III. 28) figure (III. 9). A., A?, A, A. Ray path difference between straight and grund reflected rays n either side f an islated bstacle, (III. 3, (III. 37) figure (III. 3). Aa (N ), Aß (N ) The crrectin ters Aa, Aß fr values f N ther than 30, s S O O 8 (6. 2) figure AQ (30), Aß (30) The crrectin ters Aa, Aß fr N =30, (6. 2) read fr s figure 6.9. Ah(h, N ), Ah(h, N ) The crrectin Ah as a functin f N and f receiver and transr s t s s itter heights h and h, (6. 2) figure <An> The expected value f refractive index, (IV.20). <(An) > The variance f fluctuatins in refractive index, (IV. 9). Rati f the dielectric cnstant f the earth's surface t the dielectric cnstant f air, figures 8. and 8.2, annex III. 4. < A sall increent as defined by (II. 64) and used in (II. 65), (II. 72) t (II. 75). <,,, Angle between the axis f the ain bea and the axis f the first side lbe f an rl tl antenna pattern, figure III

186 <,, e., Aziuth angles f the first and secnd lbes f a transitting antenna relative twl tw2 t the ain bea axis, figure III. 23. «,, c Elevatin angles f the first and secnd lbes f a transitting antenna relative tzl tz2 t the ain bea axis, figure.23. t, The angle that a scattering plane akes with the great circle plane, (III. 60), (III. 6), and figure IU. 22. t A functin f h and N used in cputing F and H, (9.3) and figure 9.2. s s n A functin f h and N used in cputing F. and H. fr scattering fr se es i i antenna beas directed abve the hrizn r away fr the great circle plane, (III. 64). n Characteristic ipedance f free space, n = 4irc. 0 a 20ir hs, (II. 5). 0 The angular distance, 9, is the angle between radi hrizn rays in the great circle plane defined by the antenna lcatins, (6. 9). 9 A plar crdinate, (II. 56). 9 Angle f elevatin f the lwer half pwer pint f an antenna bea abve the b hrizntal, (.62). See 9, and f(9j. h h 9,, 9, Values f 9, fr the receiving and transitting antennas, respectively, (III. 6). tv, 9,., 9,. Values f 9, fr the i bea intersectin, (.59). bri bti b 9 The angle between radi rays elevated abve the hrizn and/ r away fr the great circle plane, (III. 64). 0. The angle 9 at the i intersectin f radi rays elevated abve the hrizn fi and/r away fr the great circle plane, (III. 57). 9, 9 Hrizn elevatin angles at the receiver and transitter, respectively, (6. 5). 9,9,... 9 The angle 9 ei ez en e fr the first, secnd,... n intersectin f radi rays, figure III , Angle f elevatin f a direct ray relative t the hrizntal at the lwer antenna, h (5. 2). See 9 U and f(0j. b h 9, 9 Angle f elevatin f a knife edge relative t the hrizntal at the receiving r transitting antenna, (III. 38). 9. Angle between direct and/r reflected ray ver a knife-edge, where j =, 2, 3, 4 as shwn in figure III Angles defined in (III. 29), where j =, 2, 3, 4, which are added t 9 t deterine 9., J 0=9+9.. J J' 9,, 9,, 9,, 9. Values f 0. fr j=,2,3,4, (.29). lr 2r 3r 4r jr 0 Angle f elevatin abve the hrizntal, figures 3.2 t

187 0 Angle between radi hrizn rays, assuing straight rays abve an earth f ef fective radius, a, figure 6., 0, 6 The angular elevatin f a hrizn ray at the receiver r transitter hrizn, r t (6. 6), figure , 8-, 8,, 8 The angle between rays fr the transitting and receiving antennas ver an islated bstacle with grund reflectins, figure III. 9. Ic A wave nuber directin defined by (IV. ). \ Free space radi wave length, used fr exaple in (2. 6). H The rati 6 /6 used in (9. 2) and figure w A paraeter that is half the value f TI, used in cputing lss in antenna gain, (9. ), (9. 2) and figure 9.7. v Radi frequency in hertz, (2.4) t (2.2). v, v Liits f integratin (2. ) and (2. 2) chsen t include essentially all f the x wanted signal dulatin side bands, it A cnstant, IT a p Crrelatin cefficient between tw rand variables, p Index f curvature fr the crest curvature f a runded bstacle in the great circle path directin, (7.8). p.. The crrelatin between variatins due t surces i and j, (0.8). p The crrelatin between variatins Y and Y, (0.9). a a p The crrelatin between variatins Y and Y, (0.9). r i r r p The nralized crrelatin r cvariance between path-t-path variatins f P (50) and P (50), (V.4 5). urn p K The lng-ter crrelatin between P and F, (V.27). tn B p p r The lng-ter crrelatin between P and P, (V.34). tu urn a Surface cnductivity in hs per eter, figures 8. and 8.2, annex III. 4. a (p) The standard deviatin crrespnding t the variance a (p). 2 a (p) The path-t-path variance f bserved fr predicted p-percentiles C f transc issin lss fr a large nuber f randly different paths with a given set f values fr all paraeters used in the predictin prcess, (V.36) figure V (50) The path-t-path variance f the difference between bserved and predicted lng- ter edian values f transissin lss. The crrespnding standard deviatin is a (50), annex V. 6. c The rt-ean-square deviatin f great circle path terrain elevatins relative h t a sth curve fitted t the terrain, (5. ). 2 - (p) Ttal variance f any estiate f the service criterin fr service liited nly by external nise, (V.43). The crrespnding standard deviatin is <J (p). 2 p a (p) Ttal variance f any estiate f the service criterin fr service liited nly by interference fr a single unwanted surce, (V.45). standard deviatin is CT (p). uc 2-27 The crrespnding

188 er Variance f the estiate R (p, a), (V.45). ur " 20 Z A sybl t represent the suatin f ters, as in (5. 5) where. h. eans the su f all values f h. fr i = 0 t i = 20. l T The aunt a radi ray bends in the atsphere, (III. 62). T Delayed tie f the phasr, exp (it), where T = k(ct-r) tin at free-space radi-wave velcities, waves, (U. ). is the tie f recep- c is the free space velcity f radi- t is the tie at the radi surce, and r is the length f the radi ray, T Tie eleent defined by (II. 70) as T = IOIT cs 6. a a T, T The tie eleent T crrespnding t direct and grund-reflected waves at ai a2 a the receiving antenna, (.79). T A tie-independent phase which is a functin f r, (II. 9), (.3). T. The tie-independent phase fr the n cpnent f an incident wave, annex in II. 6. T, T. The tie-independent phase fr tw cpnents f an incident wave, (.8 5). 2 T Initial phase f the current supprted by ne f eleentary diples, where = 0,, 2, (.46). T, T, T Initial phases f the currents supprted by three eleentary diples, (.46). T Tie-independent phase which is a functin f the ray path, including allwances P fr path length differences and diffractin r reflectin phase shifts, (.3). T t T t T The phase functin T fr the n, first, and secnd plane wave incident pn pi- p2 P n an antenna fr a single surce, (.32) and (.85). T Antenna phase respnse fr the receiving antenna, (II. 6). r th T, T, T The antenna phase respnse, T, fr the n, first, and secnd plane rn n TI r wave incident n the receiving antenna, (.32) and (.8). T Antenna phase respnse fr a transitting antenna, (II. 6). T, T, T The antenna phase respnse T tn ti t2 r t fr the n, first, and secnd plane wave, (II. 32) and (.85). T, T Phases assciated with the electrical field cpnents e, e, (II. 7). 9 4> 0 9 T(9,, d, N ) Bending f a radi ray that takes ff at an initial angle 9, and travels d b s b kileters thrugh an atsphere characterized by a surface refractivity N, s (III. 6). <t> One f the plar crdinates, r, 0,, (II. 56) and figure II.. 9(v, 0) Cpnent f phase lag due t diffractin ver an idealized knife edge, (7. 3) figure 7., and (III. 30). <Mvp) Cpnent f phase lag due t diffractin ver an islated perfectly-cnducting runded bstacle, (7. 3) figure 7. 5 and (III. 30). <j>(0, p) The cpnent f the phase lag f the diffracted field ver an islated perfectly- cnducting runded bstacle fr v = 0, (7. 3) figure 7.4 and (III. 30). 2-28

189 *, * Latitudes f antenna terinals A and B, (6. ) t (6.9) figure 6.3. A B *, Latitude f an arbitrary pint alng the great circle path fr A t B, (6. 7). 4. The phase lag f the diffracted field fr the j ray ver an islated perfectly- cnducting runded bstacle (III. 30a), where j =, 2, $(v, p) The ttal phase lag f the diffracted field ver an islated runded bstacle with reflectins fr terrain, (7. 3). *(v, 0) The ttal phase lag f the diffracted field ver an ideal knife edge with grund reflectins, (7.3). *.(v, p) The phase lag f the diffracted ray ver an islated runded bstacle fr the j J ray, *.(v, p) 3 *., (III. 30)..th *.(v, 0) The phase lag ver an ideal knife edge fr the j ray, (III. 30). *,»,*,* The phase lag» (v, p) fr values f j =, 2, 3, 4, (III. 32). vji The grazing angle f a ray reflected fr a pint n the surface f a sth earth, (5. ) figure 5., r grazing angle at a feuillet, annex IV. 4/ Miniu grazing angle, sectin 5.. I\I The acute angle between principal plarizatin vectrs e and e, (2.26). P P P* I I, i\> The acute angle, +, fr each f tw waves, (.8 5). Pi PJ P v i, <JJ The angle between the plane f the lwer half-pwer pint f an antenna bea and the receiver r transitter hrizn plane, (III. 60). il., d> The angle J/ r ill fr the i lbe f an antenna pattern, (III. 59). ri ti r t i i, 4», Angle f reflectin at the grund f a reflected ray that passes ver a knife- edge, (III. 36) figure III. 9. ft The half-pwer beawidth, il = 26, (9. 0) and figure III. 22. Q, a The half-pwer beawidths f the receiving and transitting antennas, re- spectively, (9. 0). W, SI,, U, U Half-pwer beawidths crrespnding t , fr the receiving r rl t tl and transitting antenna patterns, respectively, figure III *»O «42

190 5- VI*.