Dominio IV: Desarrollo de la profesionalidad y la identidad docente
RESULTADOS DE LA INVESTIGACION
4.3 PRUEBA DE HIPÓTESIS
4.3.1 Prueba de hipótesis
As already discussed, the irradiance p a ttern of scintillations at a d istan t receiver will be a random function of both space an d time. The expressions developed so far hav e generally been valid for the lim iting case of point receivers. H ow ever, if such apertures are larger than the spatial scale of the irradiance fluctuations, then these w ill be averaged over the receiver an d will consequently be less than for the lim iting case of a point receiver. An a p e rtu re av erag in g factor A , is defined as th e ratio of the norm alised variance of the signal from a receiver w ith ap ertu re diam eter D to th at from a receiver w ith an infinitesim ally sm all aperture, norm alised by the square of the average signal.
In th e w eak turbulence^® reg im e the g en eral form of the a p e rtu re averaging factor for a circular ap ertu re of d iam eter D is given by. Fried 1967:
JtD 0 B;(0)
/ ^ \ f ^2 > COS'Y 1_ P. 1 - P
[d J D \ pdp (3.75)
w here Bj(p) is the covariance function if irradiance, Bj(0) is the variance of irradiance, and the quantity w ithin the square brackets is proportional to the area of overlap of tw o circles of diam eter D w hose centres are located a d istan ce p ap art. The covariance fu nction in h om ogenous, isotropic turbulence is given by Lawrence and Strohbehn 1970:
B , ( p ) = 1 6 n V ( K) Kdkj Jo (pKs) sin‘
0 0
K ^ ( L - z ) s
2k dz (3.76)
w here k is the optical w avenum ber, 0 ^ ( K ) is the spectrum of refractive index fluctuations, L is the path length, Jq is the zero order Bessel function
of the first kind and s is a geom etry factor w hich is unity for a plane w ave ^®Transverse coherence length of the field in the receiver plane is m uch larger than the Fresnel zone size. The standard deviation of the irradiance fluctuations are m uch less than their respective m eans and will have a lognomnal distribution.
Chapter 3 : The Stochastic Wave Equation,
and z/L for a spherical wave.
For spherical w ave propagation, s = z/L. C onsidering a sm all inner scale of tu rb u le n c e , [Iq « '^(L/k)], th e K o lm o g o ro v s p e c tru m can be u se d , 0„(K)=O,O33C/K-^i/^ and: 0 0 V /
KhjL-z)
2kL
dz (3.77) and for p = 0: Bi(0)=^A91C/FI^ L::/6 (3.78)S u b stitu tio n of these expressions in to th e g en eral a p e rtu re av erag in g function yields for large D [kD^I4L » 1]:
A = 1 + 0.214
-1
(3.79)
For a very large inner scale [Iq» V(L/fc)], the Tatarskii spectrum can be used,
0„(K)=O.O33C„2x-^^/'^exp(-O.O285K2/j2) and the covariance becomes:
Bj(p) = (0.033)4ji^C„^r^ exp[-0.02S5K‘^t l)d kj k ^ ^ y ^ L - z f d z
an d the variance is:
Bi(0)=L2SC„%-7/3L3,
The ap ertu re averaging function m ay then be stated as:
(3.80) (3.81) A = 1 + 0.109(d 1 < 4 y -1 (3.82) — 66 —
C h u rn s id e 1992, also calcu lates A u sin g th e H ill sp ectru m an d no significant difference betw een the Tatarskii sp ectru m values and the Hill sp e ctru m values w ere observed. For sp h erical w a v e p ro p ag atio n , the s m a ll/la rg e inner scale approxim ations are eq u al fo r /o = 1.5 tim es the Fresnel zone. Therefore, the sm all inner scale ap p ro x im atio n should be u se d for Iq <1.5 VfL/U an d th e large in n er scale a p p ro x im atio n for Iq
>1.5V(Llk).
In th e strong turbulence^® regim e the covariance fim ction has been found to contain tw o scales, Clifford 1974(b). This com prises of a sm all scale peak of w id th (ctj2 + i ) / 2 of the covariance, w h ere is the total irradiance variance. The long end tail is about the w id th of th e scattering disk size. C ovariance functions are m ore com m only expressed as a series expansions d eriv ed from the asym ptotic theory. The form of the covariance function w ith sm all inner scale is identical for both plane an d spherical waves;
B,{p) = exp
P o j
(3.83)
-3/
w h ere the transverse coherence length po * Po =(o.545fc^LC^) and N3
3.86. The
function is given by, Shishov et al 1974, as:
bi(p) = 0.915Jx xf d x] x^ ^ ]Q
0 0 V L
exp _ x / 3 ( l - dx. (3.84)
The function b2 is very difficult to evaluate b u t it is assum ed to have the
form :
b2{p) = exp
vPoy
(3.85)
^^Defined as w hen the transverse coherence length of the field in the receiver plane is much sm aller than a Fresnel zone.
Chapter 3 : The Stochastic W ave Equation.
the variance of the intensity is given by:
o f = 1 + 3.86
This leads to the following approxim ation for the ap ertu re averaging factor:
A = 2 a f 1 + 0.908 2 a j
’A
-1 (3.86)If th e in n er scale is m uch larg er th an the coherence len g th b u t m uch sm aller than the scattering disk, the covariance function can be expressed in the sam e w ay, b u t using the following for the coherence length:
Po = 0 . 5 4 5 k ^ L C h / ^n^O y
(3.87)
The sam e form is also assum ed for the function b2(p) b u t use the following
for bi(p): hi(p) = 1.05 ykpoy j x { I - x ) dxjK^^exp 0 0 JoiKxpyiK. (3.88)
The variance in this case is given by:
o f =1 + 2.27
V y (3.89)
and the aperture averaging factor by:
_ o f + l 2 a j 1 + -1
o f - 1
2of
1 + 0.534 k D p ^ \ 2L ) %1-1 (3.90)C hurnside 1992, has perform ed a series of experim ents to test the validity of the above expressions, w hich are w ithin about a factor of tw o of the exact calculations. Path lengths of 100, 250, 500 an d 1000 m w ere u sed together w ith apertures of 1, 2.25, 5, 10, 25 and 50 mm. Sim ultaneous m easurem ents of C / and inner scale w ere also m ade over an adjacent path. Each data ru n w as a v erag ed o v er 25 cycles w ith the valu e of the m ean b a ck g ro u n d su b tracted . The experim ental results, sh o w good a g reem en t w ith th e aperture averaging factors of spherical waves in w eak turbulence. O ver the 1000 m path , ie. the strong turbulence regime, the aperture averaging factor does not d ro p of as fast as is predicted by the theory, over the range of the m easurem ents. This is show n b y Figure 3.7. The turbulence strength w as = 4.20 ± 0.33 X 10’^^ The inner scale Iq = 5.98 ± 0.35 m m , w hich
w as larger than the coherence length of = 2.84 ± 0.11 m m . The Fresnel zone size of (L/kŸ^^ w as also larger than p^. The tw o scales predicted by the asym ptotic theory are not seen. This, how ever, is not su rp risin g since the variance needs to be of the o rd er of u n ity plus a sm all p e rtu rb atio n term w hereas the stated variance w as 3.15 ± 0.24.
Chapter 3 : The Stochastic W ave Equation.
- L
<
D /2 p ,
Figure 3.7: A perture averaging factor A vs. ratio of aperture rad iu s D /2 to coherence length
Pq. The po in ts rep resen t d a ta taken at 1000 m , the solid curve is the strong-turbulence
approxim ation, and the dashed curve is the weak turbulence approxim ation.
It is th o u g h t th a t v ery m u ch larg er a p e rtu re s w o u ld p ro v id e b e tte r approxim ations to the asym ptotic theory. Figure 3.8 show s the ap ertu re averaging factor for strong an d w eak turbulence, this is taken from , Fante 1975.
0.8 0 6 cr/ : 100 o 0 4 o 02 0 02 04 0 6 OjB 12 14 1.6 18 2 O'/ (X O ‘^
Figure 3.8: A perture averaging in strong and weak turbulence.