DATE DUE. (o-n-^a r. feb 8 1 ^7Q. h-i^fiom. UCl 1 s h;^ JAl< ia--»o-"3; CA VtOBO PRINTED INU.S.A. AJft-v.'t>-

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1 NISI PUBLICATIONS.

2 DATE DUE UCl s h; feb 8 7Q ia--»-"3; (-n-a r JAl< 2 7 h-ifi AJft-v.t>- CA VtOBO PRINTED INU.S.A.

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7 j 4 itdd ecunlccil flte 92. TRANSMISSION LOSS PREDICTIONS FOR TROPOSPHERIC COMMUNICATION CIRCUITS VOLUME I p. L. RICE, A.G. LONGLEY, K.A.NORTON, AND A. P. BARSIS U. S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS

8 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 devices 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 Bureaus 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 Physics. Physical Cheistry. Labratry Astrphysics.* Radi Standards Labratry: Radi Standards Physics; Radi Standards Engineering.** OflSce f Standard Reference Data. Institute fr Materials Research. Analytical Cheistry. Plyers. Metallurgy. Inrganic Materials. Reactr Radiatins. Crygenics.** OflBce f Standard Reference Materials. Central Radi Prpagatin Labratry.** Insphere Research and Prpagatin. Trpsphere and Space 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 Grup, Jint Institute fr Labratry Astrphysics at the University f Clrad. ** Lcated at Bulder, Clrad.

9 NATIONAL BUREAU OF STANDARDS technical 92te fof Issued May 7, 965 TRANSMISSION LOSS PREDICTIONS FOR TROPOSPHERIC COMMUNICATION CIRCUITS VOLUME P. L. Rice, A. G. Lngley, K. A. Nrtn, and A. P. Brsis Central Radi Prpagatin Labratry Natinal Bureau f Standards Bulder, Clrad NBS Technical Ntes are designed t suppleent the Bureaus 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 f Dcuents, U. S. Gvernent Printing Office Washingtn, D.C Price: $.00

10 Natinal Bureau f Standards FOREWORD AUG ,SSg QOlOOJJSlSd. 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 [955]. 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 xinpublished 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 nnaterial 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.

11 - 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 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 EARTHS RADIUS 4-5. TRANSMISSION LOSS PREDICTION METHODS FOR WITHIN- THE HORIZON PATHS 5-5. Line-f-Sight Prpagatin Over a Sth r Unifrly Rugly Spherical Earth.., A curve-fit t terrain The terrain rughness factr, ", 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 6-3 6, 3 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

12 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 8-8.Z Diffractin Over Irregular Terrain 8-i 8.Z, Diffractin ver paths where d =d st sr ,2 Fr hrizntal plarizatin., Single-Hrizn Paths, Obstacle nt Islated FORWARD SCATTER 9-9. The Attenuatin Functin, F(ed) The Frequency Gain Functin, H The Scattering Efficiency Crrectin, F Expected Values f Frward Scatter Multipath Cupling Lss Cnnbinatin f Diffractin and Scatter Transissin Lss LONG-TERM POWER FADING 0-0. The Effective Distance, d e The Functins V(50, d ) and Y(p, d ) e e Cntinental Teperate Cliate Maritie Teperate Cliate Other Cliates Variability fr Knife-Edge Diffractin Paths 0-3. REFERENCES LIST OF SYMBOLS AND ABBREVIATIONS 2- IV

13 48 ANNEX I: I..2 ANNEX n: n. n. 2 II. 3 n. 4 n.5 n. 6 II. 7 n.8 ANNEX in: III. III. 2 ni.3 ni.4 ni TABLE OF CONTENTS Vlue 2 AVAILABLE DATA AND STANDARD CURVES Available Data as a Functin f Path Length. Standard Pint-t-Pint Transissin Curves BEAM ORIENTATION, POLARIZATION, AND MULTIPATH COUPLING LOSS Representatin f Cplex Vectr Fields... Principal and Crss-Plarizatin Cpnents Unit Cplex Plarizatin Vectrs... Pwer Flux Densities Plarizatin Efficiency Multipath Cupling Lss... Idealized Theretical Antenna Patterns... Cnclusins FORMULAS, COMPUTER METHODS, AND SAMPLE CALCU- LATIONS Line-f-Sight Diffractin Over a Single Islated Obstacle Diffractin Over a Single Islated Obstacle with Grund Reflectins. Paraeters K and b fr Sth Earth Diffractin Frward Scatter Transissin Lss with Antenna Beas Elevated r Directed Out f the Great Circle Plane, Lng-Ter Pwer Fading in. 7. Diurnal and seasnal variability in a cntinental teperate cliate in. 7.2 T ix distributins.. Exaples III. 8. Line f ight predictins PAGE NO. I-l I-l -2 n-6 II- II- II- II n in-5 in- 7 III- 23 in-24 in-37 in in- 50 in-69 Ill- 69 III. 8.2 Transinissin lss predictin fr a runded islated bstacle III- 73 III. 8.3 Predicted transissin lss fr a transhrizn path. III- 77

14 ANNEX IV: FORWARD SCATTER PAGE NO. ANNEX V: PHASE INTERFERENCE FADING AND SERVICE PROBABILITY. V- V, The Tw Cpnents f Fading V.2 The Nakagai-Rice Distributin V.3 Nise -Liited Service V- 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-ZO 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-Z9 IV- V-3 V-5 VI

15 TRANSMISSION LOSS PREDICTIONS FOR TROPOSPHERIG COMMUNICATION CIRCUITS P. L. Rice, A. G. Lngley, K. A, Nrtn, and A. P. Barsis. 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 MHz (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 -

16 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, f ways t achieve ptiu use f the radi frequency spectru. and includes a brief suary Figures are placed at the end f each sectin, and thse which are nt vertical shuld be turned cunter-clckwise. (The rdinate 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 [Buttn, 96; Buttn and Thayer, 96], synptic radi eterlgy [Bean, Hrn, and Riggs, 962], techniques fr easuring the refractive index f the atsphere [ Mc Gavin, 962], deterinatin f syste paraeters [ Flran and Tary, 962], perfrnnance predictins fr cunicatin links [Barsis, Nrtn, Rice, and Elder, I96I], and equipnnient characteristics [ Barghaufen, et al, 963]. -2

17 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, issin lss, L, prpagatin lss, L, basic transissin lss, L, path transs 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 prpagatin 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 Transis sin Lss The syste lss f a radi circuit cnsisting f a transitting antenna, receiving antenna, and the intervening prpagatin ediu is defined as the diensinless rati, Plp> 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 a antenna. The syste lss is usually expressed in decibels: L := 0 lg (pp) P - P db (2.) s t a t a 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 p! in dbw crrespnds t p in watts. Transissin lss is defined as the diensinless rati p, p, where p is the 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) t a t a s et er L = 0 lg i. L =0 lg i (2.3) et et er er 2-

18 where i and IjH 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 antennas a fractin f a wavelength abve grund, as they usually are at lwer frequencies, and especially when hrizntal plarizatin is used, L and L are nt negligible, but are influenced substantially by the presence f the grund and ther nearby prtins f the antenna envirnent. Fr transitter utput t receiver input, the fllwing sybls are used: Transitter Pwer Ttal Available Pwer Available Pwer Available Pwer Output Input t Radiated at Lss-Free at Actual at Pwer Antenna Pwer Receiving Antenna Receiving Antenna Receiver Input ir air It shuld be nted that L and L are cnceptually r different. J Since P it r and P _ represent the pwer bserved at the transitter and at the transitting antenna, respectively, L includes bth transissin line and isatch lsses. Since P and P «t represent available pwer at the receiving antenna and at the receiver, isatch lsses ust be accunted fr separately, since L includes nly the transissin line lss between the antenna and the receiver. 2-2

19 I 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: c "iv "iv + iv (2.4a) z = V r V f :ix V (2.4b) Z : V : r V + ix V (2.4c) where r and x in (2.4) represent resistance and reactance, respectively. Let p. represent the pwer delivered t the receiving antei.na lad and write p and p, respectivt 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 Piv=77- z + z V iv (-J * 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 iv the pwer p available fr the actual antenna: av v. Pv-iT 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

20 r ipedance. Cparing (2,5) and (2,6), we define a isatch lss factr P I + r Y+ x + X, lxji ). "Z (2 7) iv 4 r r V iv such that the pwer delivered t a lad equals p i, When the lad ipedance cniuav v - gately atches the antenna ipedance, i has its iniu value f unity, and p = v iv 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 V P ~ (2.8) a V 4 r V 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 = H p av erv av fr a lss-free antenna at the sae lcatin as the actual antenna: -2..=-:f=-i 2, p r* V (2.9) erw p i2 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 ties the ttal pwer p radiated at the frequency v:. etv = P p. tw tv (2.0) The cncept f available pwer fr a transitter is nt a useful ne, and I fr the transitting 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 prepared under CCIR Reslutin N. [Geneva 963c]. The lss factr i was successfully 2-4

21 deterined in ne case by easuring the pwer p radiated fr a lss-free target transitting 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 Z r withut calculating se quantity such as the radiatin resistance r the transissin lss. In the case f receptin with a unidirectinal rhbic terinated in its characteristic ipedance, 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. The rati f pwer lst in the terinatin t the input pwer varied with frequency and was typically less than 3 db. Fr the frequency band v t v it is cnvenient t define the effective lss factrs L and L as fllws: er at p (d P,,dv) dv T - 0 lg er db (2.) (d p;dv) dv i p V (d Pdv) dv T _ 0 lg et j-j (2.2) (d Pt,dv) dv "i The liits v and v n the integrals (2.) and (2.2) are chsen t include essentially 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 radiatin eanating fr the wanted signal transitting antenna. 2-5

22 The 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 {4Tr) 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 gp(4tr) 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 i. It fllws that et G(r) = G(f ) + L (2.3a) expressed in decibels abve the gain f an istrpic radiatr. Nte that the antenna pwer gain G(f) is less than the antenna directive gain G (r) by the aunt L db, where the t t et pwer radiatin efficiency i is independent f the directin r. 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 (r) and pwer gain G(f) are related by r r G{f) = G(r) + L, {2.3b) The reainder f this reprt will deal with directive gains, since the pwer gains av be deterined siply by subtracting L r L. The axiu value f a directive gain G(f ) is designated siply as G. As nted in Annex II, it is seties useful t divide the directive gain int principal and crss-plarizatin cpnents. An idealized antenna in free space with a half- pwer sei-beawidth 5 expressed in radians, and with a circular bea crss- sectin, ay be assued t radiate x percent f its pwer istrpically thrugh an area equal t Tr5 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 4Trx(l OOtt 6 ). One ay assue a bea slid angle efficiency x = 56 percent fr parablic reflectrs with lodb 2 tapered illuinatin, and btain g = relative t an istrpic radiatr is then axiu free space gain G in decibels G = 0 lg g = lg 6 db (2. 4) 2-6

23 If aziuthal and vertical beawidths 2 5 and 2 5 are different: v z 5 5 (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 diaeter D is the rati f 56 percent f its area t the effective absrbing area f an istrpic radiatr: G = 0 lg Q. 56ttD4 X4u = 20 lg D + 20 lg f db (2.6) where D and X. are in eters 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 < DX. < 25, experients have shwn the fllwing eipirical friula 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, I960]. Much re is knwn abut the aiplitude, 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 II is devted t this subject. Fr the frequencies f interest in this reprt, antenna radiatin resistances r at V 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

24 r watts 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 nnake 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 = j - I 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(r) and p (-r) 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: p(r)7, J-r) =7g (2.8) p c e = e+ie,e=e +ie (2.9) pr cr = e + e, p c = e + e (2.20) pr cr 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 e = e and linear if e =0, where e = e e p c c P P P is in the principal plarizatin directin defined by the unit vectr e. The available pwer p ay be written as p = s(r) a (-r) p p (2.22) 2-i

25 is _ r 2 2 s{r) = e l{zr ) watts k (2.23) ag(-?) - g(-?) [ X(4Tr)] k (2. 24) where s(r) is the ttal ean pwer flux density at the receiving antenna, a (-r) is the effective absrbing area f the receiving antenna in the directin -f, and Ip p the plarizatin efficiency fr a transfer f energy in free space and at a single radi frequency. The crrespnding plarizatin cupling lss is Lp -0 lg g". pp db (2.25) In ters f the axial ratis a and a defined by (II. 5) and (. 7) and the acute angle 4* between principal plarizatin vectrs e and e, the plarizatin efficiency ay be written as lp.pl= cs ij* (a a + ) + sin 4 (a + a ) P P ",, " " (2.26) (a-hl)(a+l) This is the sae as (.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 rnaxiu 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 -f. Sall values f g (-r) 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 gp in annex II. 2-9

26 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 t r db. (2.27) 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: () Istrpic, s that G (r) = db and G (-f) = db fr all iprtant prpagatin directins, r. (2) Lss-free, s that L = db and L =0 db. et er (3) Free f plarizatin and ultipath cupling lss, s that L = 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 ultipath 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 equivalent 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, and the path antenna pwer gain G, are given by p " * pp L = P, - P b t ab p = L + G db (2. 28) G = P - P, = L, - L db (2. 29a) p a ab b G = P - P 3 L - L db (2. 29b) pp a ab b s 2-0

27 : In free space, fr instance: a = t t r<- " cp + S (~4 ) <2. 30a) P, = P + 20 lg f ab t V 4iTr " dbw ((2. 3( A special sybl, L, is used t dente the crrespnding basic transissin lss in free s pa c e L = 20 lg (J ) = 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 (2.3l). 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) pf t r cp Fr st wanted prpagatin paths, this is well apprxiated by G + G, t the su f the r axiu antenna gains. Fr unwanted prpagatin paths it is ften desirable t iniize G.. This can be achieved nt nly by aking G (r) and G (-?) sall, but als by using different plarizatins fr receiving and transitting antennas s as t axiize In free space the transissin lss is L 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 "equivalent free- space transissin lss", L, as 2-

28 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 "-f = -f (=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 G = L, (50) - L (50) (2,36) p b where L, (50) and L (50) are lng-ter edian values f L, and L. b b Multipath cupling lss, r the "lss in path antenna gain", L, is defined as the difference between basic transissin lss 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 The lss in path antenna gain will therefre, in general, include cpnents f bea rientatin lss and plarizatin cupling loss as vell 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

29 . and 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. Repeating the definitins f r and r used in subsectin 2.2, and intrducing the new paraeter r : r = antenna radiatin resistance, t.r r t, r = resistance cpnent f antenna input ipedance, r, r - antenna radiatin resistance in free space, ft, fr where subscripts t and r refer t the transitting antenna and receiving antenna, respectively. Next define L= 0 lg (rr), L = 0 lg (rvr) (2.38) L= 0 lg (r;r), L = 0 lg (rvr) (2.39) L=0 g(rr) = L-L (2.40a) = 0 lg (rr) = L - L (2. 40b) [Actually, (2.) and (2.2) define L, L 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 = L - L - L = L - L - L db f2 p s ft fr rt rr V-J-; 4n Basic prpagatin lss is Basic prpagatin lss in free space is the sae as the basic transissin lss in free space, Lj, defined by (2. 3). The syste lss L 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 Lp, and the field strength E are derived quantities, which in general require a theretical calculatin f L andr L as well as a theretical estiate f the lss in path antenna gain L 2-3

30 The fllwing paragraphs explain why the cncepts f effective radiated pwel-, E.R. P. and an equivalent plane wave field strength are nt recnnended fr reprting prpagatin data. equal t A half-wave antenna radiating a ttal f p watts prduces a free space field intensity s =.64p (4-n-r ) watts k (2.43) 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 s (r)=p g {r)(47rr ) watts k (2.44) as explained in annex II. In (2.44), r = rr and g (r) is the principal plarizatin direc- P 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 as calculated using a field intensity eter r standard signal surce with s as easured using the test antenna: P E.R. P. =P +0 g(s s ) = P +G (rj 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 radiait it tin efficiency i are knwn: et P, =P+L, =P +L + L, dbw (2.46) it t it t et it 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

31 L and L, Fr cnverting reprted values f E in dbu t estiates f P r estiates f the available pwer P at the input t a receiver, the fllwing relatinships ay be useful: P = E + L, + L, - G + L,, - 20 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 transissin lss Ll ay be derived fr the fllwing equatins, L = L + L -G + G - G (r J + 20 lg f - E, db (2.50) s et fr p t pt ikw L = L + G - G (r) + 20 lgf - E, pb rt t pt ikw L, = L +G - G J? J + 20 lg f - E, b rr t pt db (2. 5) db (2.52) 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 envirnent, L = L =0 db, L, = L, 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 assued 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,, and L, et er ft fr ay be very large. Prpagatin curves at HF and lwer frequencies ay be given in ters f L r I_ s that it is nt necessary t specify L and L 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 K, 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, it t pt t L = L. = db, and L, = L, in (2.45) - (2.47), where L, is given by (2.3): it ft pb bf bf I E = lg d dbukw (2.53) p pb 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

32 ; db "equivalent inverse distance field strength", E, is the sae as E. If the antenna radia- I tin resistances r and r are equal t the free space radiatin resistances r and t r r, then (2.52) prvides the fllwing relatinship between E, and L with ir kw b G (rj = G : pt t E = lgf - L, dbukw (2. 54) ft Cnsider a shrt vertical electric diple abve a perfectly-cnducting infinite plane, with E.R. P. = 30 dbw, G:=.76db, and L = 3. db Fr (2. 45) P = dbw, since t rr t G A.) db. Then fr (2. 52) the equivalent inverse distance field is E, = E+L, +L = lg d dbukw (2. 55) I rt rr * crrespnding t e v at ne kileter, r e = 86.4 v at ne ile. In this situatin, the relatinship between E and L is given by (2.52) as E, = lgf - L, dbukw (2.56) ikw * b The freging suggests the fllwing general expressins fr the equivalent free space field strength E and the equivalent inverse distance field E : 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, - L,, = L - L, = E - E db (2. 59) b bf f I Alternatively, attenuatin relative t free space, A, ight have been defined (as it seties is) as basic prpagatin lss relative t that in free space: A = L, L, r = A.- L -- L = E t pb bf rt rr (2.60) 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

33 50 MHz, L + L_ is negligible, but cautin shuld again be used if the lss in path antenna gain L, is nt negligible. It is then iprtant nt t cnfuse the "equivalei free space lss L given by (2.34) with the lss in free space given by (2.33). 2-7

34

35 :, - 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, appreciable at higher frequencies. but ay be quite 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 a 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 [ 956 ], Gunn and East [ 954 ], Hathaway and Evans [ 959 ], Hgg and Mufrd [ I960 ] Hgg and Seplak [ 96 ], Lane and Saxtn [ 952 ], Laws and Parsns [ 943 ], Perlat and Vge[l953], Straiten and Tlbert [ I960 ], Tlbert and Straiten [ 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 Y and V in decibels per kileter fr bth xygen and water vapr, as deter w 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 ] 3 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 dbk 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 (r) dr A = Y(r) dr db (3. ) a Jq r in ters f an absrptin cefficient r(r) expressed in reciprcal kileters: A = - 0 lg exp r(r) dr =4.343 r(r) dr db (3.2) 0 - * 3-

36 The arguent f the lgarith in (3. 2) is the aunt f radivave 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 a r A = dr [y (h) + Y (h)] db (3.3) cl O 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. a Cnsidering xygen absrptin and water vapr absrptin separately, (3.3) ay be written A = Y r + y r db (3.4) a e w ew where r and r are effective distances btained by integrating y y and y y e ew v w ver the ray path. The effective distances r and r are pltted versus r and frequency fr elee ew vati n angles 9=0, 0.0, 0.02, 0.05, 0., 0.2, 0.5, L and tt 2 radians in figures Figure 3. 5 shws the relatinship between r and the sea level arc distance, d, fr these values f 9. A a ay be estiated fr figures I. 2 t I. 26 f annex I, where attenuatin relative 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 hrizn akes an angle 9 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 (3.5) where A i A (f,, dj, A = A (f, 9, d) at a t ar a r 2 Fr prpagatin ver a sth earth, 9 =9 =0, and A = 2A (f, 0, d2). Fr transt r a a hrizn paths and the frequency range GHz, figure 3. 6 shvs A pltted versus distance ver a sth earth between 0 eter antenna heights. 32

37 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 des terined by integrating the gas teperature T ultiplied by the differential fractin f reradiated pwer that is nt absrbed in passing thrugh the atsphere t the antenna: T ( K) = T(r)r(r) exp r. r(r)dr[dr (3.6) L - where the absrptin cefficient r(r) in reciprcal kileters is defined by (3. 2) Fr instance, assuing and T(r) = ( h) "K fr h < 2 k, T(r) = 20 K fr h > 2 k, figures.? 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 frnn the earths surface ust als be cnsidered. 3-3

38 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 = V {r)dr decibels (3.7) r Jq 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 y ay be expressed in ters f the rainfall rate R in illieters per hur as V = KR* dbk (3.8) r r fr frequencies abve 2 GHz. The fictins K(f ) and 0!( ) are pltted in figures 3.8 G G and 3.9, where f is the radi frequency in GHz. K= [3(f- 2) - 2(f- 2)] X 0" (3.9a) a - [ (f- 2)] [ (f- 3.5) exp( f) ] (3.9b) fr An exainatin f the variatin f rainfall rate with height suggests a relatin f the g = exp(-0.2 h) (3.0) 3-4

39 where R is the surface rainfall rate. Then rs V r db. (3.) rs er r = KR dbk, r_ = dr exp ( a h ) k (3.2) s rs er where v is the surface value f the rate f absrptin by rain, and r is an "effective r s er rainbearine distance". Figures shw r versus r fr several values f 9 er and a. 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 distributin 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. The ttal annual rainfall in Ohi is abut 0 centieters. 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 rainfall rates, R, depends n hw R varies with elevatin abve the surface and upn the crr r relatin f rainfall with distance alng the path. Figure 3. 4 prvides estiates f the instantaneus path average rainfall rate R exceeded fr 0.0, 0.,, and 5 percent f the r 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 with height, as given by (3. 0)and allwed fr in the definitin f r, and fr the crrelatin f surface rainfall rate R with distance alng the surface, as analyzed by Bussey. 3-5