JOURNAL OF RESEARCH of the National Bureau of Standards- D. Radio Propagation Vol. 66D, No.3, May- June 1962 On the Role of the Process of Reflection In Radio Wave Propagationl F. du Castel, P. Misme, A. Spizzichino, and 1. Voge
Contribution from Centre National d'Etudes des Telecommunications, France
(Received August 21 , 196J ; revi sed December 9, ] 96 1)
Nature offers numerous examples of irregular stratification of t he medium for t he p ro p:t gatio n of radio waves. A st udy of t he process of refl ection in such a mcdium di sting ui shes between specular reflection and d iffu se refl ection. The phenomenon of trans-ho ri zo n tropospheric propagation offers an ex ample of t he appli cation of such a process, necessar." for t he interpretation of experimental results. Other examples a re t hose of ionospheric propagation (sporadic-E laycr) a nd propagation over a n irregula r ground surfact' (phenom enon of a lbedo).
1. Introduction This paper discusses the role of partial r efi ec tion in the propagation o/" radio waves in stratified media. This work drffers considerably from other works by different authors a bout the same problem [Epstein , 1930; F einstein, 1951 and 1952; Wait, 1952; Carroll et aI. , 1955 ; Friis et aI., 1957 ; Smyth et aI. , 1957] . The authors wer e led to make such a study in th e co urse of seeking an interpretation of experimen tal 5 results obtained in trans-horizon tropospheric propa gation for which the classical theory of turbulent scattering did not appear to give a satisfactory ex planation. The process of r efl ection in irregular media points to evidence for the existence of two forms, coherent or specular reftection, and random or diffuse refl ection. These two phenomena, added to the pheonomenon of scattering, permit one to ac FI GUR E 1. Phenomenon of paTt'ial Tejlec tion. count for the ensemble of experimental results ob tained in trans-horizon tropospher ic propagation . The existence of stratified irregular media, or of A plane wave incident n,t an inclination angle a is layers, is far from being limited to the case of the partially refl ected by the surface, S, having a reiiec troposphere. Such layers should, in fact, appear tion coeffi cien t, p, assumed constant at every point each time that a field of force along a dominant of the surface of the irregularities, s (fi g. 1). direction is exer ted on the medium, creating strati fi cation in a perpendicular direction. This is the 2.1. Elementary Reflected Power case for the troposphere or the ionosph ere when subjected to the gravitational field, as well as the Each element of a surface s has a certain inclina magnetosphere in a geomagnetic field. In all these tion , so that it reflects an incident ray in a direction cases, the conclusions reached by a study of the O= a+ .6 relative to it. The element ha a certain process of refl ection should be found to be applicable. radius of curvature, R, and an elemen tary eff ective reflecting cross section, u, corresponding to the 2. Study of the Phenomenon of Reflection in direction 0 with which it can be associa ted (fig. 2). an Irregular Medium Let us take a local system of cylindrical coordi nates, a, cf> , r, such that r is perpendi cular to the bi Consider a small stratum of thickness, H , with a sectrix of angle e and normal to the surfn,ce at a horizontal surface, S, and with an average length, L. center point, P , of the element. To a point M (a,cf» This volume contains irreg ularities having a mean of the tangent plane r=O, ther e corresponds an alti sm-face, s, of average length, l , distributed over th e tude r= a2/8R. At this point M , the radiation is thickness, H. dephased by 2Kr r elative to the center point P, where l/rranslated fr om the French by P . Guzman-Rivas, "Boulder, Colo. K = 2/A a nd A= 'A /(s in 0/2) is the space wavelength. 273 A limiting condition for the element of effective These approximations should not obscure the surface, (J , will be given, for example, by a dephasing fact that the power is reflected into a solid angle of less than 7r/4. Therefore, for the limiting points, around the direction 2a, which can be characterized r=A/4, and a = ~RA / 2, the value of the eff ective by a vertical angle 4> and a horizontal angle f . These reflecting surface, (J , is angles are related to the curvature of the surface within the latter's limits, and can be written: (J = RA/2. (1) The validity of this expression evidently assumes (5) that (J < s, so that (2) if one takes into account small angles 4> and f . The corresponding reflected power, op, in the direction 0 can then be calculated as a function of 2.2. Total Reflected Power the power incident per unit of surface, po, by a reasoning analogous to that for antenna theory. The stratum with a total surface S and having a The expression for the gain of an antenna with dimension L consists of a series of surface irregulari surface (J , in the direction 0, would be ties with mean dimensions s distributed over a thickness H . With each of these is associated a reflected power op along a given direction , and a op ~: (J (si n 0/2) p2 , phase term characteristic of its position. Po(J sin 0/2 1\ Let us take Cartesian coordinate axes x, y, z from p being the amplitude of the reflection coefficient, a central point 0 in the surface S (fig. 3), where x so that and yare horizontal and z is vertical. Associated with each surface element (J is a phase term kr, (3) where r is the distance to the point O. Separating horizontal and vertical characteristics, one can write, t being the horizontal distance from 0 of the The surface s can be considered as consisting of a element 0': combination of elements of effective surfaces (J, each reflecting an energy op in a direction O. It can r= t + 2z(t) sin a, (6) be assumed that the surface s is small and that its where: mean radius of curvature R is large, with the radii of curvature of the surfaces (J approximated by the t = x sin a sin 4> +Y sin f , (7) average radius R. In the expression for K , the reflection angles {3 can be approximated by the angle with 4> and f characterizing the direction of observa a, so that 8= 2a. One obtains for the radius of tion (3 by a vertical angle and a horizontal angle curvature R, if H is the maximum heigh t of the around a, and z(t) describing the position of the irregulari ty: elements (J. It amounts to restoring them entirely R = l2/8H. to a horizontal plane, but conserving for each el ement its proper phase term 2Kz(t). Defining: Hence, we obtain an equation for op:
7r l4 'Y (t) = ej2Kz(t ) , (8) oP = PoP2 16H (4) we obtain a function describing the distribution of and, for condition (2): irregularities. H > A/16. a . Specular Reflection To calculate the total power reflected in a direction 0, two cases come into consideration, depending on whether the elementary terms involved are in phase or not. R
E
FIGURE 2. R eflection by an irregulan:ty of a sUlface. FIGU RE 3. R eflection by an irregular surface. 274 For terms in phase, the reHected power, P s, is Diffuse reflection is therefore written as: obtained by considering elementary energies, i. e., by squaring the phase terms: (18)
i kl (9) P s=op DIs e -y (l)([2t J. c. Solid Angle of Reflection
Since the terms are in phase, one must have kt= 27r Coming back to the physical meaning of this or c/> = 1/; = 0. Reflection in phase occurs only along the mathematical analysis, a certain number of elements direction of specular reflection (3 = IX. Therefore: are distributed at random in a stratum, each element reflecting a certain power into a solld angle, w, around the direction of spec ular refl ection (fi g. 1). (10) The distribution of these clemenLs i analyzed by their spectral distribution, r (k) ; i.e., with the space wavelength, A, in the direction of propagation, Since -y et ) is a cissoiclal functioll, for an approximate and the wavelength, A, in the transverse direction. value of its integral over 8, one can take: Elements whose phases are random give a diffuse reflection in an angle equal to the angle w. ElemenLs sIn u -y =--, (11) in phase give a more intense reflection in a much u smaller angle W' . where: The preponderance of one phenomenon 01' the other u = 2KH (12) depends on the size of the irreg ularities. By fixine; a limit, u = 7r/4, Rayleigh 's criterion [01' an in egular and H is the maximum height of the irregularities surface is found: z(t). H ence, for specular reflection, H > A/16, this corresponding to condition (2 ). (13) The solid angles involved have limits fixed by the phase variation. In the vertical plane, LIte b. Diffuse Reflection phase term, within limits of the surface, is Kl sin C/> , for diffuse reflection, where the surface s is involved, For terms whose phases differ, it is necessary to and KL sin c/> for specular reflection. resort to a reasoning analogous to that ",{hich is used The phase varies from 7r for small angles equal to : in the theory of scattering, and to take in to account the random character of the distribution of refiecLing i:I> s=A/L (19) elements. In considcring two elements correspond ing to t and t + 0, the reflected power will be: In the horizontal plane, the phase is Kl sin 1/;. For a variation of 7r : Pd= OP~s Jrse jkl -y ( t ) cl2t~ s Jrse ik(lHl -y (t + o)cl2 (t + 0). (14 ) 1/;s= A/L 1/;a= A/l. (20) This expression can be transformed into: These limits are stricter than the geometrical limits of formula (5) if H > A/8. We note also that reflection creates a lobar structure, as in the case Pd = oP~Is eik5 GL -y(t)-y (t + o)cl2t Jd 20. (15 ) of antennas, and that the results involve only the main lobe. In this form, the correlation function of -y (t ) is Several remarks can b e made concerning the shown as: approxin1ations used. Introduction of the approx imation 'Y instead of -y et) in (11 ) is tantamount to (16) considering the correlation function )/ (0) in (16 ) equal to 1 when o
275 This equation shows the number of reflecting dimensions II and hi (corresponding to a mean clements which enter into the second power in surface 8 1), for irregularities of the second order coherent reflection. (mechanical movements), as in figure 5. Limits must be considered in the real surface of reflection (fig. 4). For specular reflection, the hypothesis of a phase phenomenon introduces a Fresnel's ellipse when we no longer have a plane wave. If D is the emission-reception distance, its dimensions are:
(22)
Only that zone of 8 in the Fresnel's ellipse (viz, 8 s) will be involved. In th e case of diffuse reflection , the maximum in clination (small) T of the elements will be involved. This corresponds to a width, Ld = D sin CXT, where T~H/ l. The possible transverse limit for 8 d then is:
(23)
Hence, the reflected power (2 1) should be written in the form: FIGURE 4. SwJnce of reflection.
P _ 247r sin2a 2[8; 2+ 8d (1- 2) J . r-POP ", 2 (J 82 'Y 8 · 'Y (24)
3. Process of Reflection in Trans-Horizon Tropospheric Propagation
3.1. Characteristics of the Propagation Media As a consequence of gravity, the atmosphere has a general tendency to become horizontally stratified. H A consideration of movements in t he atmosphere shows the possibility of two processes, a laminar flow and a turbulent flow [Misme, 1958] . The limit between these two processes is clearly marked and depends on atmospheric stability. In a zone of laminar flow, a stratification of the atmosphere appears which can be characterized by layers of a stable nature, with highly variable hori zontal as well as vertical dimensions. In a zone of turbulent flow , elementary unstable blobs appear, which are more or less organized depending upon the thickness of the zone. A highly plausible representation of the atmos phere would show a spatial superposition of laminar- flow zones (or stable strata) and turbulent-flow zones (or turbulent strata). 1 Stable strata can be characterized by a dis continuity in t he gradient of refractive index, corresponding to a variation on of the index in a very small t hickness, e. The structure of the stratum is subject to the influence of turbulent flow in neighboring strata and to mechanical vertical movements of the atmosphere. The corresponding irregularities could be expressed b~T horizontal and vertical mean dimensions land h, correspon ding to a mean surface 8, for irregularities N of the first order (turbulent-flow) and by average FIGU RE 5. Schematic of the troposphere. 276 The stratulll hfts a total thickn ess, H, comprising b. First Order Irregularities ~),ll such irregulftrities. H could be consid er ed the Sft me t1.S hi with ft total dimension L , corresponding Jrr egularities s(l ,h), comprised in a surface s, give to a s urface S. rise to a specular r efl ection , PI S, and a difl'use r efl ec The orders of magnitude of these difl'erent charac t ion , Pld: teristic parameters of the strftti fi cation, as far as they can be deduced from experimental observations (29) are the following: '
Discontinuity BN~1O - 2N e~ 10 cm (30) Dimensions L~lO km H~lO m Irregularities l~lO m h~ll1l wh ere ,), = sin u /u a nd u = 47rall/ X. The eff ective l l~ l km hl '"'-' 10 m reflectin g surface, (1 , defin ed by (1 ), can b e wri tten: A simplified schematic of t he stratificfttion is tha t wherein irregularities with mean dimensions l alld h (31) ~), r e distributed through a stratum of dimensions L The mean surface of the irregularities s would th en and H. S uch a simple model is consider ed in the b ' , preceding stud y. e: it should be noted t hitt observations of t lte upper s=v (32) surfaces of cloud layers gi ve evidence frequently of t be exi stence of irregula ri ties s uch as those r ecorded h er e The total s urf ~1ce of d ifru se r efl ection IS equal Lo S l, [Misme, 1960] . this b ein g small : (33) 3.2. Propagation by Reflection in Tropospheric Strata The total surface of sp ecular r efl ection is limited tr a, nsverscly in general by the s ma,H dia meter of In the case of trans-horizon troposph eric propaga Fresnel's ellipse, L ; = ~XD , if D is th e total length of tion, angles of inclination a a re always small so that th e path : one may ta ke the a ngle for the sin e. (34) a. Reflection Coefficient
For a disco ntinuity of the index dN, sm all com The expression for th e r efl ected power can be O"iven pared to a n an gle of inclination a , th e coeffi cient of in r el atio ~ to t ~l e ~orr es pondin g free-space pow e~, Po , r eflection of a n clemen t, according to Fresnel's lIl stead of the lI1 cldent power per unit of urface, Po. formula, can b e \Hitten: Then: (35) (25) if D is average distan ce, close to actual d istance, d, For a discontin uity thickness e, th e coeffi cient of such that: r efl ection is: _ J, edN jZK z (36) p- 2---;> e o a- where dl and d2 are the distances from the r eflecting or, upon introducing th e index gradien t g(z)= dN/dz: element ~o the extremes of th e path close to d/2. EquatlOns for reflected powers ar e, ther efore:
(26) (37)
A limited development permits one to write:
1 q q' . gil ] r [ . J' ZK} P= 2a2 l J 2K+ (21<' )2-J (21<.)3 + . .. e Z • with num erical values al s ~6.2 10- 3 and aid ,-,6.2 10- 3• (27) c. Second Order Irregularities For a very smal! thick:n ess, e, e< < A, we will CO Il sider a discontinuity gradient g= BN/e and write: . Over ~ ~otal surface, S, of a stratum, we assume lrregularIties 8 I Ct], H ) superposed upon irregularities s(l,"0 ). The .power of diffuse reflection, Pl d, due to (28) an lrregulanty '~I, c~n b e consider ed as playing thp, role of an elem ent of reflected power. Then th e sum 277 of these powers will cause a total noncoherent reflec Other assumptions would lead to equations a little t ion to appear, which will be the total power of different from the role played by the different pa diffuse reflection from th e surf'ace S, and a total rameters. Moreover, the ch aracteristics and dimen coherent reflection, which can be considered as being sions of a.tmospheric strata and their irregularities th e total power of quasi-specular refl ection of the can be markedly variable in time and space. If, surface S. then, the assumptions which have been made cor The first can be written: respond to average orders of magnitude, the equa tions which result from them should be considered (39) as having limits in their application for a particular case. It is essential to k eep in mind that r efl ection The surface of diffuse reflection will be limited phenom ena in an irregular medium involve two transversely by the maximum inclination, 1', of the components: a reflection with coherent character, elem ents of surface s, i.e. , by a transverse dimension affecting a limited surface of the stratum and appear La=Dar.With the approximation 1' = H/l I , we have ing dominant if the irregularities are dimension ally for S a: small relative to the space wavelength; and a reflec (40) tion with diffuse character, involving a large portion of the surface of the stratum, and all the more Then the equation for diffuse reflected power is: marked as the irregularities are more important. Rayleigh 's criterion gives us the limits within 2 which one or the other component will be pre on LlIH ? /-.. 2 Pra= P ad ---:) - h2 (1- -y-) ~D ' (41 ) o e- a ponderant. One other significant characteristic of the propagation by reflection is th e fine structure of th e phenomenon involving a limited number of with a numerical value ad"-'6.2 10 - 3• For total power of quasi-specular reflection, one reflecting elements. Some numerical examples permit one to account must consider the power Pld as an elemen t of power for the orders of magnitude involved, and the limits corresponding to an effective surface (jl associated of approximation used. \vith SI and no longer to (j associated with s. The expression for reflected power, therefore, will b e: For a distance D = 200 km, th e minimum value for the angle of inclination is about a= 12.5.10- 3 rad; for D= 400 km, a= 25·10- 3 rad. The space wavelengths (j ~ S; 2 P rs = Pl d 2---;'- (1- ,), ). (42) for radiation of wavelength /-.. = 1 m or /-.. = 10 cm are, (j Sia r especti\Tely, for 200 km, A= 800 m or A= 80 m; for 400 km, A= 400 m or A= 40 m. For the eff ective surface (jI , we have: For a discontinuity thiclmess, e=2 or 3 m, we have always e< < A/8, and fo rmula (28) for the (43) reflection coeffi cien t is generally valid. Rayleigh 's criterion for an irregular surface shows, As far as th e total surface of reflection, Ss, is on th e contmry, that it is suffi cient that th e thickness concerned, the transverse dim ension of Fresnel's of th e irregularities of a stratum, H , b e greater than ellipse, L;= ~ /-..D will, as b efore, be involved. 10 m for the surface to appear irregular to radiation H ence: of /-.. = 10 cm. However, the thiclmess has to be greater than 50 or 100 m, according to the distance, (44) for the surface to appear irregular to radiation of /-,, = 1 m . The expression for the power of quasi-specular The dimensions of Fresnel's ellipse vary from reflection will b e: L s=38 km and L;=450 m for D= 200 km and /-,, = 1 m, to L s=8 km and L;=200 III for D= 400 kIll and /-,, = 10 cm. For stratum dimensions on th e order of L = l km, th e assumption that L; Equations (37) and (38 ), or (41) and (45), for The factor, T, which is the ma,ximum inclination of specular and diffuse refl ection powers point to the strata, is introduced by the transverse limits of evidence that there exists a decrease of sp ecular the reflection surface_ reflection with frequency (\ 3) more rapid than the case involving diffuse reflection (\ 2). d. Space Selectivity of Reflected Power It must, however, b e remarked that, while the frequency increases, irregularities of th e m edium The diversity distance is rehted to the measure b eco me more apparent under radiation, to th e detri ments of the useful zone of the a,tmosphere, more ment of specular reflection. On the other hand, exactly, to the angle at which the zone is seen from eq (28) for th e coefficient of reflection ceases the point of em ission. to be valid for frequencies that are too high, and th e In a transverse direction, diversity distan ce OH ensuing terms of the development introduce much would be equal to: higher exponents of \ . (48) h . Effect Due to Distance In the ver tical direction, however, tbe small Upon admitting a proportionality of ex to distance, thickness, II, 01' Lhe stratum. i involved, hen ce: } the expr es ions for reflected power point to an evident and rapid deer'ease of specular reflection with dis lAD tance as compar ed to diffuse reflection . If one, how oV=8 H ' (49) ever, take into accoun t the fact that th e dim ensions of the strata appear to increase with altitude, diffuse increasing with the frequency. reflection will tend to b e dominant at very great distances. e. Frequency Selectivity of Reflecte d Power c. Statistical Properties of the Signal The dim ension s or the useful zone of the atmo - phere involved, in limiting the b a,ndwiclth transmis Fluctuations of signals are due essentially to in sible by reflection, will be the ma,ximum dimensions stantaneous interference b etween signal com ponents. in a horizontal, rather th a, n vertical, plane, by The differences in path lengths can be more signifi 1'ea,son of the sma,ll thickness of the stratum. cant, therefore, th e greater the dimensions of the In this ma,nne1', the equa,tion for the bandwidth atmospheric zon e which a,re involved. It can b e /:'j is in the fo rm: seen that th e limiLs of t he useful surfaces differ for specular reflection (S s transver::;ely limited by (50) Fresnel's ellipse, Ls=-,/\D) and for diffuse reflection (Sa limited transversely by the inclination of refiect For a correlation sufficiently close to 1, when /:'j ing elem ents, La= Dcx r). In thickness, th e atmos is descriptive of the transmissible bandwidth, the pheric zone is limited by the thickness, H , of the value of the numerical coefficient is close to b= 10, stratum. if b..f is in M c/s and D is in 100-km units. The r esult is that the fading range will b e less than with Rayleigh 's law. Upon considering the number, f. Fine Structure of the Field n, of r efl ecting surfaces, equation (2 1) for r efl ected power could b e put in the form: The instantaneous cha,racteristics of the reflected fields differ sensibly from the eh a,ra,cter of the scat (46) tered fields. In a rapid spatial ana,lysis, difl'usion will appea,r This form corresponds to the expression for the only as a large bright zone, depending upon the distribution of a vector sum, whose phase is distrib dimensions of the scattering volume. R efl ection, on uted in a random m anner between two limits, W the contrary, can cause very narrow zone to appear, and -w [B eckmann, 1957], with: corresponding to surfaces of reflecting strata. These will appear as "brilliant points," by virtue of the ,), = sin w/w. reduced openings of the antenna, lobes. Experiences in rapid swinging of a narrow beam of an antenna a,r e, The fading rate is a function of the velocity of in this connec tion, a, confirma,tion of the possible relative displacement of r eflecting elements, each existence of these two pheno mena, of propagation relative to the others, resultin g in average move [Waterm a,ll , 19581. m ents, of velocity u, and vertical movements, of In a rapid frequency analysis, the statistical prop velocity v, of the a,tmosphere. This fading rate can erties of the frequency stru cture will similarly be be expressed by th e number N of crossin gs of the different for scattering and reflection. Besides an median level. An approximate expression as a increase in the maxima due to differences in the 625S29- 62--5 279 ---' transmissible bandwidth, and besides a reduced small percentage of the time, and will be weaker as fading rate in the case of reflection, it is possible to distance increases and frequency rises. This is the relate the spectral characteristics of fluctuations to origin of the high and little fluctuating levels observed the atmospheric zones involved. The existence of in trans-horizon paths. different types of spectral distributions, within the The phenomenon of diffuse reflection and scatter range of experiments in rapid frequency analysis, ing will be essential elements in trans-horizon propa also afford a verification of the existence of several gation as long as distance is sufficiently large or fre propagation phenomena [Landauer, 1960; Biggi et aI., quency high enough to make the effect of the earth 1960]. and its relief neglible compared to the effect of the troposphere. 3.4. The Complex Phenomenon of Trans-Horizon Although the fundamental characteristics of these Tropospheric Propagation two phenomena are quite close, as long as the phe nomena are incoherent, specific properties are at Propagation phenomena by scattering and reflec tached to each, and it will be necessary to resort to tion each play their part in the complex processes of one or the other in order to interpret all the experi propagation in a troposphere consisting of turbulent mental results [du Oastel, 1960]. layers and stable strata. A comparative study of these principal properties The phenomenon of specular or quasi-specular re can be made. R esults based on very general hy flection, which presupposes specific conditions of potheses for each of the phenomena are shown in regularity in the strata, will appear generally only a figure 6. o ,..., b ~ ~ u 0 ~ ~ "- ;" 10 \ ~ ,,\ -'" ~ 20 '\\ "- ~ " , co '\, 30 [\\- ~ '>~ ~ ~. > 0 '\ I ~ f\ " ...J '~ 50 10 2 3102 10 3 3.10 3 10 4 100 200 300500 1000 Fe'quen, . MH z Di stance km c d >- e MHz ~ r ,:-,- --.---,-----.40 o ~ , ."! " 30 ~ 3 o ~ /' 2 20 .""•.. "'" 2 o "" -, , 10 ~ , 1o o co " " " ". r'- \--- -4-- -+- --10 0 0 200 300 400 500 km 200 300 400 500km 200 300 400 500km ,-- -,---,---,- -.10 o ~--L-_~ __~ _---, 20 10 50 90 9 9% FIGURE 6. Phenomenon of trans-horizon tropospheric propagation. Diffu sion (broken line), specular reflection (solid line). Diffuse refleeWon (dot-and-dasb), a, According to frequency; b, According to distance; c, Fading speed; d, DiYcrsity distance; e, Trans missible bandwidth; f, Distribution of fading range. 280 4 . Process of Reflection in Other Propagation (5 1) Media eo being the electron charge; mo, electron mass; EO, the dielectric constant in vacuum. In practical General co nsiderations concerning r eflection pro units, we have approximately: cesses in irregular media can be applied to numerous > instances where the existence of a field of force in a jo(Mc/s) = 10 - 2 ,/N(cm-3) . (52) dominant direction leads to some stratification of the propagation medium. The troposphere is an ex For N o= 3 103 cm-3']0= 0.5 Mc/s. For any frequency ample, but there are others. Following the consid less th~n jo, the reflection is total, and the reflected erations that have been made in section 3, we shall power IS: briefly consider the case involving the ionosphere. (53) Moreover, if an irregular stratification leads to a phenomenon of reflection, the same result will evi At a frequency j higher thanjo, the ionization gmdi dently exist for reflection on an irregular ground ent oN introduces a partial reflection, corresponding surface. We shall similarly deal with this latter case to an index variation on. succinctly, as another example of application of the We have: more general analysis. (54) or, in practical units: 4 .1. Process of Reflection in Ionospheric Propa gation (Sporadic-E Layer) n 2= 1- 10- 4N(cm- 3)/p(Mc/s) (55) so that: The possible existence in the lower ionosphere of on= 5.10- 50N/p. (56) strata of ionization i , at present, an experimental fact [Bowles, 1959; Seddon, 1960]. However, in The change in the index takes place over a thickness, terpretations of the phenomenon are not all in e, to which corresponds a gradient of index change: agreement. For some, this stratification would result from propagation throngh the atmosphere of g= on/e. gravity waves, creating vibrations opposite in phase for points separated by a half-wavelength. Such For example, for oN=104 cm- 3, and e= 10m, g2= vibrations would create strong gradients of hori 2.10- 3 atf= l Mc/s, and g2 = 2. 10 - 7 atj= 10 Mc/s. zontal velocity. In transitional zones between two Consider the term for specular reflection in eq oppositely directed horizontal currents, the electrified (24). We will have, for vertical sounding: , particles which are transported are subject to vertical magnetic forces of opposite sense because of the hOrIzontal component of the earth's magnetic field. Thus, a reinforced ionization will result, correspond ing to sporadic-E layer [Hines, 1960; Whitehead, The surface of specular reflection involved is 1960]. limited by a condition of phase coherence, i.e., a Such strati.fication would not be uniform. Under Fresnel's ellipse, and has the value: the influence mainly of turbulent movements in lower adj acent layers, structural irregularities would appeal' S s='AD/4. (58) [Voge, 1961]. Therefore: Let us consider a model sporadic-E layer, charac terized by a gradient of electronic density, corre 610- 6 P s-- P , . g 2~.4h2 (59) sponding to a variation of density, oN (on the order of 104 electrons pel' cm3) , over a small thickness, e (on the order of 10 m), the average density at the For the term of diffuse reflection in eq (24), \ve have level under consideration being No (on the order of here: 3 X 103 electrons per cm3). Horizontally, the dimension of the stratum is L (60) (on the order of 10,000 m) and vertically H (on the order of 1,000 m). The surface shows irregularities of average dimension, horizontally l (on the order of The surface of diffuse reflection involved is limited by 1,000 m) and vertically h (on the order of 100 m). the inclination of the elementary surfaces of reflec The stratum is at a mean altitude D (on the order of tion, and has a value: 100 km). What is the significance in this case of the reflection Sa= L(DI-I/L). (61) formula of section 2 for vertical transmitting power, Therefore: P e, corresponding to an ionospheric sounding? Ionospheric plasma has a cutoff frequency,jo, such (62) that: 281 With orders of magnitude as follows: h= 100 m, to the ground, at altitude Z. This is, for example, l= 1,000 m, H = l ,OOO m, L = 10,000 m, the numerical the calculation for the earth's albedo under the in values indicated in figure 7 are obtained for reflected fluence of solar radiation [Spizzichino and Beckmann]. power as a function of frequency. Transition The general schematic of the ground surface will be between specular reflection and diffuse reflection analogous to that of models previously adopted, with corresponds to a frequency j, such that: irregularities of dimension l and height 11" distributed over the surface with a thickness H (fig. 8). 'A' = 2Hl/L. (63) Each surface element dS, such that its normal is a bisector of directions of the emitter (at infinity) and Here,]' = 1. 5 M c/s. It should be noted that we h ave receiver, will reflect energy as given b y eq (24): not considered here the term 'Y of section 2, character izing the r espective importance of the two reflection 2 dP =..JL 2 47r sin ex (T2 dS, (65) terms by: T 4711,2 P 'A2 S 'Y = sin u/u where Po is the incident energy per unit of surface; where u = 47rH/'A. The term 'Y would introduce a r is the distance of the surface element to point of frequency of transitionj' ,such that: reception; p is the coefficient of reflection equal to unity. 'A" = 47rh. (64) The effective surface of reflection has a value according to equation (1) : Here,]" = 0.15 Mc/s. In fact, frequency j" involved is less than frequ ency j' because of the dominant (T = 'AF/16h sin a. (66) role played by the different actual reflection surfaces, S . and Sd. Eq (65) 'will then be written: In an ionogram, there will appear an occlusion frequency jo = 0.5 Mc/s corresponding to total reflec _ 1 l2 dS tion, and a limiting frequency jl, corresponding to dPT = 2~6 -,?- 2· (67) o ~- r t he reception threshold, hence dependent upon the sen sitivity of the sounding equipmen t. If it permits It can be remarked that the ratio P/h2 is, in effect, r eception with an attenuation of 60 db, then one proportional to inclination 7 of the surface element would, for example, have j1 = 3 M c/s, according to dS on the ground surface, so that (fig . 8) : figure 7. 7= 4h/l. (68) 4.2. Diffuse Reflection by an Irregular Ground Surface The equation for the reflected energy can then be written, for the case of a unit incident energy: We shall study the case of an irregular ground sur face illuminated by a source at infinity in a direction making an angle a with the ground. We shall cal (69) culate the reflected energy received at a point close PT= 1~r 2L ~~. The integration should be extended over all the real surface of reflection, S, i.e., over all the portion I of the surface of the ground for which the angle o between normal and vertical is less than the angle of inclination of the reflecting elements, 7. In order to calculate this surface S, the system u,v of angular coordinates could be employed, as ro .." defined in figure 9. The equation for the limiting 50 curve of the surface is then: c . __ . _- 0 , , sin v+sin a +> 2 cos u = cos v/cos a + cos a/cos u-tg2r ---'-- I' , cos V cos ex '"::> Pd c I ',<..., , .... , ' GI .., I ,', ...... (70) +> , F I GURE 7. Energy reflected by an irregular spol'adic-E layer, and as a function of the fl'equency. P . = specular reflection; Pd = diffuse reflection. (7 1) 282 L __ F IGURE 8. R eflecti on by i r re:J~da r ground surJace. F I GUR E 10. Di,D'erent examples oj actual sll1jace oj j'eflecti on by an irregular ground smjace. refl ection poin t remains small. However, the actual surface of reflection should be, never theless, limited FIGU RE: 9. Ac t ~tal sUljace oJ j'ejiection by an irregular sll1 jace. to the horizon of the point . A restrictive hypothesis Sys tem of angular coordinates. is equally implied, and the equations for reflected power cease to be valid when a« 7, obligin g one to limit the curves in figure 11 to small values. The equation for th e limiLing curve IS therefore of the form: 5. Conclusions u = U(v, a) (72) The general study of refl ection in irregular medi a and the reflected energy will be: (section 2) and the discussion of applications, com plete for the case of trans-horizon tropospheri c "2 propagation (section 3), but only summarily made P r = (1/167 2) f u (v,a) cot v dv. (73) "1 for other cases (section 4), show the importance of reflection phenomena in numerou s problems of It must be noted that this expression is illdepend propaga tion. ent of the altitude Z of the point of reception . The problems brought up here have been discussed According to the limits VI and V2 introduced by in different French publications. A series of articles conditions (71) , the limits of the surface S can be on the subject, published 1958- 60, have been edited very differ en t. Figure 10 gives some idea of cases [du Castel, Misme, Voge, Spizzichino, 1960]. A which can be encountered, while figure 11 gives the book studying trans-horizon tropospheric propaga behavior of variations of the corresponding refl ected tion was recently published [du Castel, 1961]. energy. The authors, within the framework of their work We note that the calculation assumes a plane at the C.N.E .T ., envision other applications of this ground surface and neglects the earth's curvature. study, namely, to problems of ionosphere and exo The approximation is valid if the altitude of the sphere propagation of decameter waves. 283 RHltctton co tfficient du Castel F . P. Misme, J . Voge, and A. Spizz ichino, R e- ' flexions' p~ )' t i e ll e s dans l'atmosphere et propagation a gra nde distance, Ed. Revue d 'Optiqueo Paris (1960): d u Castel, F ., Propagation troposphcnque et faI sceaux ~.10 0 m hertziens transhorizon, Ed . Chiron, Paris (1961) .. Epstein, P. S., R efl ection of waves in an inhomogen eous a bsorbing m edium, Nat. Acad. Sci., U .S.A. 16, 627- 637 (Oct. 1930) . . . ,------ F ein stein, J ., Tropospheric propaga tIOn beyond the honzon, J . Appl. Phys. 22, No. 10, 1292- 1293 (Oct. 1951). 0)5 ~.100km F einst ein, J ., Gradient reflections from the atmosphere, Trans. 1. R . E . AP, No. 4, 2- 3 (D ec. 1952). F einstein, J., The role of partial reflection in t ropospheric .1 00m propagation b eyond the horizon, Trans. 1. R. E. AP, N o.2, 'C.20' 2- 8 (Mar. 1952); No.3, 101- 111 (Aug. 1952)...... --- Friis, H . 1'., A. B . Crawford, and D. C. Hogg, A r eflectIOn / / t,10 ,m , '--- t heory for propaga tion beyond the horizon, Bell Syst. Tech. /" ...... ,.. Journal 36, No.5, 627- 644 (May 1957). // , I.lOO '". Hines, C. 0 ., Theoret ica l survey of motions in the ionosphere, U. R . S. 1. XIII General Assembly, Doc. III 3, London (1960). . Landauer, vV . E., Experimental swept-frequency tropospherIc scatter link, Trans. 1. R . E. AP, No. 8, 423- 428 (July 1960). '1:.10' M isme, P., Mesures m 6teorologiques, Ann. Telec. 13, No. 7- 8, 209- 219 (July- Aug. 1958). " M isme, P., Interpretations des m esures met6orologlques, Ann. T616c. 13, No. 9- 10, 265- 270 (Sept.- Oct. 1958). Misme, P ., Quelques aspects de la radiomet6orologie et de la l : Alt itude of receiv ing po iot radioclimatologie, Ann. T6lcc. 15, No. 11- 12, 266- 273 (Nov.- D ec. 1960). 't: Inc l i oa i ~ on of i rrtgulari l ie~ Seddon, J. C., Su mmar y of rocket and satellite observations relative t o t he ionosphere, U . R. S. 1. XIII General Assem- bly , Doc. III 10, London (1960). . r , Smith, E. K., The morphology of sporadIC E , U . R . S. 1. XIII 20 30 40 50 60 70 General Assembly, Doc. III 6, J.,ondon (1960) . Smyth, J . B., and L . J. Andersol:, Vimportance des WOc?SSUS de refl 6xion dans les commUlllcatlOns au-dela d e 1 honzon, Onde E lectr. 37, No.5, 485- 490 (May 1957). Spizzichino, A., Differentes composantes du champ re<;m au FIGU RE 11 . Variation of energy reflected by an irregula r dela de l'horizon, Ann. T6lec. 15, No. 5- 6, 122- 136 (M ay- ground surfac e. June 1960). . . Spizzichino, A., and P . Beckmann, Ed., Scattenng of radIO waves by irregular surfaces (Pergamon Press, London, to be 6 . References pu bli shed). VOlTe J . R 6flexion sp eculaire et r6flexion diffuse sur des f~~ill ets atmospheriques, Ann. T 6l6c. 15, No. 1- 2, 38- 47 Beckmann, P., A new approach to the problem of reflection (J an.- Feb. 1960). from it rough surface, Acta T echnica C. S. A. V. 2, 311- Voge, J ., R 6flexion et diffusion en propagation t roposph6rique, 355 (April 1957). Ann.T6lcc. 15, No. 5- 6, 107- 121 (May- June 1960) . Biggi, J ., F . du Castel, J . C. Simon, A. Spizzichino, an~ J . Voge, J., ?,heories de l ~ propagation t r opo s p h ~ riqu e au ~ d e la Vorre E tude exp6rimentale de la structure frequentleUe de l'hOrl zon, Ann. T elec. 15, No. 11- 12, 260- 265 (Nov. inst a;ltan6e de la propagation en milieu heterogt'me, Ann. D ec. 1960). T 616c. 15, No. 11- 12, 274-276 (Nov.-Dec. 1960). Voge J. Note sur Ie rOle des feuillets a surface irreguliere Bowles, K. L ., Symposium on fluid mechanics in t he iono da~ s l~ formation des couches E sporadique, Ann. T6lec. spherc, J . Geophys . R esearch 64, No. 12, 2072 (D e c.195~ ). 16, No. 1- 2, 62- 63 (Jan.- Feb. 1961).. . Carroll, T . J ., and R . :tiL Ring, Prop agatIOn of short radlo 'Wait, J . R., R efl ect ion of electromagne tIC waves obliquely waves in a normally stratified t roposphere, Proc. 1.R.E. from an inhomogeneous m edium, J. Appl. Phys. 23, 1403- 43, No. 10, 1384- 1390 (Oct. 1955). 1404 (D ec. 1952). ... . du Castel, F ., and A. Spizzichino, R 6flexion en milieu inhomo 'Wat erman, A. T ., A rapid beam SW1llg1llg expenment m gene, Ann. T 616c. 14, No. 1- 2, 33- 90 (Jan.-Feb. 1959). trans-horizon propagation, Trans. 1. R. E. AP, No. 6, du Cast el, F., Notes sur les phenomenes de propagation tropo 338- 340 (Oct . 1958). spherique, Ann. T 6lec. 15, No. 5- 6, 137- 142 (May-June Whitehead, J . D., Formation of sporadic E layer in t he t em 1960). pcrate zones, Nature 188, 567 (Nov. 12, 1960). du Castel, F., R esult at s experimentaux en propagation t ropo spherique t ranshorizon, Anll. T elec. 15, N o. 11- 12, 255- 259 (Nov .- D ec. 1960). (Paper 66D3- 195) 284 l