Journal ofAtmospheric andTerrestrial Physics, 1970, Vol.32,pp.QtU-Q66. Pergamon Press.Printedin Northern lreland
SHORT PAPER
Amplitude and phase scintillations of spherical waves BOHUMIL CHYTIL* Geophysical
Institute, (Received
Czechoslovak 19 March
Academy
of Science, Prague,
1969; in revised form
15 September
Czechoslovakia 1969)
Abstract-The formulae for variances of amplitude and phase fluctuations of a wave after scattering from ionospheric irregularities are derived for the oblique incidence of a spherical wave. 1. INTRODUCTION
THE SCINTILLATION of the characteristics
of an electromagnetic wave after passing through the layer of irregularities and its application to the ionosphere has been widely studied, especially for it plane incident wave. For satellites however, a point source at a finite distance would be more appropriate, and this problem has also been solved but only for perpendicular incidence of a wave (e.g. YEH, 1962, DE WOLF, 1965, TATARSKY, 1967).
Therefore, in this contribution we have derived the expressions for variances of amplitude and phase fluctuations for oblique incidence of a spherical wave. The spatial autocorrelation function of refractive index fluctuation has not been specified. Ot,herwise, the usual simplifications of single scattering have been used. 2. ESTIMATION OF VARIANCES When solving the wave equation of a scattering problem using the perturbation Rytov’s method (DE WOLF, 1965; TATARSKY, 1967), we can obtain the solution of it in the form Y? = yO + v1 + . . . where y,, is the unperturbed wave and y, are scattered contributions to the resultant field. These quantities are complex and written in the form Y = In A + iS, y,, = In A, + iS,, vi = In A, + iS,. Symbols il and S stand for the amplitude and phase respectively. In our simple investigation into fluctuations of observable characteristics of an electromagnetic wave we shall, as usual, confine ourselves to the first scattered contribution, i.e. to the quantity y1 --Y - yO and put y1 = x + i&Y, where 2 = In (A/A,), 6S = S - SO. In order to find the quantity y1 we must solve the following integral (DE WOLF, 1965; TATARSKY, 1967)
Y,(r) -
E
s
exP
cik1’ - PI)%(f)
form ~O(~)/uO(~) = (r/f) . w * Present 15
address:
Integra
WP
uom
Ir - PI
where uO stands for the primary wave. -
&(P)
dxo
dy
&
0
03
(1)
For a point source it will be written in the T)l, where T = IrI and r is a vector from the
Inc., Taborska
9, Prague
961
4, Czechoslovakia.
BOHUMIL CHYTIL
962
origin to the point of observation P(x, y, z). Similarly p = 1~1,where p is a vector from the origin to a point Q(z,, v,,, za) in the scattering medium. The geometry of the problem is simply sketched in Fig. 1. A source of spherical waves is situated at the co-ordinate origin at the height z over the Earth. The layer of irregularities is assumed to be plane, of thickness 1 and placed at the height h over the ground. The most suitable simplification of expression (1) for oblique incidence can be achieved using the approximation by BUDDEN (1965). In principle, we express the co-ordinates of both points P and Q with the use of the co-ordinates P’ and Q’
Fig. 1. The general geometry of the problem.
respectively, (cf. Fig. 1). Naturally, this approximation holds for small distances X, Y only, or for small scattering angles, but this limitation is known and follows also from the assumption of the simple scattering theory. Instead of the exponent in equation (1) we get then approximately
k[lr - PI + (P - 41 - %[-4/(~ - %‘) + A,/%’ - A&l,
(2)
where A, = a, . (X -
X,)2 + u2 . (Y -
A, = al . x,2 + u‘J . Y,2 A,=a,.X2+u2.
2 . a3 .
Y,)2 -
2 . a3 . (X -
X,) . (Y -
x, . Y,,
Y2-2.a3.X.
Y;
and further a, = $k . cos 0, u2 = 1 -
a, = 1 -
sin2 0 . sin2 9,
sin2 0 . cos2 ip,
u3 = sin2 0 . sin 7 . cos fp,
Yo),
963
Amplitude and phase scintillations of spherical waves
In the denominator
we put simply 9.. (p . jr -
p[)-1 N 2 . f.q)’ * (2 -
z*‘) . SW q-1.
For estimation of variances, that is for (x2), (&Y) and a co-variance @X3), we shall use two auxiliary quantities (yIyI*) and (yla), (where * denotes a complex conjugate), because it holds (TATARSKY, 1967) W> (34
= 4E(nvtl*>& Re 0,$% (8S2) (xdS) = 4 Im (y12).
@b)
The brackets ( > form the space mean values of random variables. In the rest of our paper we shall simply write yl = y, omitting the index 1. The details of the calculrttion of quantities {yy*) and (yz), which is more lengthy than difficult are not quoted here. The second set of points in the integrated region is denoted by &“(x,,“, y,,“, x0”). Having performed the formal mean values of the proper products and substituted (%’ - 5”) = t, @Jo - y,“) = 24,XOB= 0, y,” = w, we integrated along the variables v and W. In the next step, the auto~orrelation function of refractive index ~uctuation was replaced by the corresponding two-~mensional spectral density function according to the following relation (AE(~‘)A+“))
= B(x,’
-
zOn,y,’ -
=
SJ-
y,,“, zO’ -
F(‘Qt ‘% 20 -
X0”) exp
Having done so, the remaining integrations were performed with the result (yy*>
-
(I% set 0) ‘jj&
d%j+(‘%
zO”) (i(~&
-f@
KS, 20) -
SC% @)zj/dKI dK,SjF(f$
K$U))
drcl dKZ.
(4)
with respect to the variables t and 7~
Z/)
exp (--i&,b,K, (Y2> -
+
KZ,Z; -
+ ~~~~~~~dxO’ dzo”,
(5)
+ i/?b,K,} dzo’ dzo”,
(6)
Zen)
exp { -iab,K, where
li,
=
b, =
KIX
+
22 -
K&f_/
+
(Z@’ f
.Z
t&Xl @ (‘cl
ZoN),
COS
9
+
b, = (zo’ -
Kz
sin p7),
go”),
b, = zo’(z - 2,‘) + z~n(x - Zen), a = (2kZCOS30)-l,
j3= l/z.
Both expressions can be simplified. Let us turn first to the internal integral of the function (yy*). We may use the assumption that the characteristic dimension of irregularities E. is much larger than the wave-length a. As ib holds that IK(Zo’ zo”)l G 1 (TATARSKY, 1967) and therefore also IK,b,b,/kxl G A/Z, < 1, we cm take approximately exp ( --iab,b,K,) M 1.
BOI~~MIL CIIYTIL
964
Let us introduce two new variables 4 = (zO’ - so”), 7 = j&z,’ + zO”). The fiction F(KI, K~,E), being an even fiction, the iteration of the variable 7 gives us the result &I I, =
ss
F exp (i/lb&,)
dz,’ dzO” = 2
a
s0
‘(1 - E) . I? s cos (,9&E) d4‘.
As the contribution of the spectral density function is considerable only in the region C 6 I?(TBTARSKY,1967), we can approximately write I, -
21 mF . cos (/X,5) s0
dE = 23iZ.
+(q,
Kz>
B&L
where # is the TV-dimensional spectral density function of refractive index fluctuation. Substituting (7) into expression (5), we have obtained the resulting formula for the quantity (yy*) (yy*>
-
&71.(k
see
@)21
$(KI,
K2>
@K,)
SKI
F-9
dK2.
SJ’
The ~a~angement of the expression for (y2> has been done in a similar way. Substituting again (zO’ - zO”) = 5 and (zO’ + z@“) = 2~ in the double integral dtz Is =
ss d
F . exp (--irxb,KI + i,!?b,K,} dz0) dzO”,
we transformed the boundaries to (0, t) and put approxima~ly exp ( --idCl~~f2) for the same reason as above. After integration of variable 73we got I, = drr . exp ( --Gp2z2). ( cXJ-~/~ . ‘% s
M 1,
9 43, E) cm (B&6)
O x ACHE) -
fa%?) + ~~~~) -
a.P2)>,
(9)
where C(z) and S(x) are the Fresnel integrals, the definition of which is C(z) + is@) = and further p = (~I~2)1/2, Pl =P@
--?a-
Ef,
Pa
= PoEZ-
qfa+ Z) -I- El.
The contribution of the function F is negligibIe for E > 1. Then if we, in our approximation, neglect E against the quantity z in the arguments p, and p,, we may again use the relation (7) and ultimately obtain -+7r(k: set 0)2
[7&z cos3 0/2&]I” . +(K,, K2, BK,) IS X exp ( -ipzx2){c(pzI) - c(pz,) + iS(pzI) - i8(pz2)) dKI dKZ, (10) where ZI = x - 2fa,x2 = z - 2(b + I). W> -
Amplitude and phase scintillations of spherical waves
965
Now, from the equations (3), (8) and (lo), the final expressions for variances are easily obtained. They read
-
n(j&
set
@)2
. l
.
+(K1,
K2,
/?K,)[l
F
R,H,]
dK1 dfr2,
(11)
(dS2) (Zd8) -
?~(+kset @)2
where we put
. l
.
+(K~,
ss
K~,
@K2)
. &Hz
dK1 dK2,
(12)
K, = [kz cosa 0/2K,P]‘, HI
=
H2 =
CC@~l)
%9z2)1 . cm3 (p)2
-
MP1)
-
Colz2)l
+
* sin (pz)” -
[fw,)
-
4&l
* sin (PZ)“,
[5@z1)
-
S(pz2)]
. CO8 ($x)2.
The above formulae are general in the sense that, once the autocorrelation function of refractive index fluctuation has been chosen, the last step of calculation is not difficult to perform. 3. SPECIAL CASES Under special assumptions our expressions turn to those which have been derived earlier by several authors. 3.1 Perpendicular
incidence
Most often the case of the perpendicular calculated. Then it holds 0 = 0, from which (1) If both the source and the observer irregular medium, i.e. for d = h = 0, z = I, same as those derived by TATARSKY (1967). (2) If the validity of Gaussian correlation the corresponding spectral density function performing the integration, we would obtain has been published by DE WOLF (1965).
incidence of a wave on the layer is a, = a, = 1, a3 = 0, and also K, = 0. are assumed to be immersed in the the expressions (11) and (12) are the function is assumed,then substituting to the equations (11) and (12) and after rearrangement the result which
3.2 Plane wave limit The simplest way to find the case of a plane incident wave is to perform the limit z -+ co in the equations (5) and (6). Integrating in a similar way as above, we have easily found the final expressions in the following form
ss
(x2> - 77(&kseC @)2z. @S2) x (1 (XSS) -
?T(&kSW
0)’
. i
.
$?%(K,,
(qz)-’ . sin(qz) . cos [q(2d + Z)]} dK1 dK2,
7
K2, y) x
where y = tan @ ( K1
co9
p
+
4(K1, K2, 7).
(41)-l . sin (pl) . sin [9(2d + Z)] dK1 dK2,
~~ sin pl), q = K,
WC3
0/2k.
BOEUMIL
966
These expressions
are identical
CHYTIL
with those derived
by the author
elsewhere
(CHYTIL, 1968). 4. CONCLUsION
The derived formulae for variances allow us to compare the behavior of variances for different plausible autocorrelation functions of refractive index fluctuation in the general case of an obliquely incident spherical wave. In order to find the error that we admit when receiving the signal from an artificial satellite but using the formulae for a plane wave, which is very often done, a suitable numerical analysis would have to be done. REFERENCES BUDDEN K. G. CHYTIL B. DE WOLF D. A. TATAR~KY V. I.
1965 1968 1965 1967
YEli
1962
K. C.
J. Atmoqh. Terr. Phys. 27, 155. J. Atmosph. Terr. Phys. 30, 1687. Trans. IEE APl.3, 48. Prop. of Waves in Turbulent Atmosphere (in Russiau), Chap. 49. Moscow. J. Res. nutn. Bur. Stand. 6f3D, 621.