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a
b
1000
300
995
100
0
990
−100
985
−300
980
975
−500
−10
−5
0
5
10
−10
−5
0
5
10
X
X
Fig. 7.1 Gaussian phase error model. In ( a ) is shown a single member ( thin solid )ofthe
distribution centered at the origin as well as the mean of the distribution ( thick solid ). The vertical
dashed lines denote the location of the inflection points. In ( b ) is shown the variance ( thin solid )
and the third moment ( thick solid )
of a disturbance may lead to non-Gaussian phase uncertainty. The strength of the
shearing of the flow is shown to be the determining factor as to whether the resulting
phase distribution will be Gaussian or non-Gaussian.
Returning to our phase error model of the pressure field of a TC we may write a
Taylor-series approximation of the pressure field at
x 0 , of the form:
p.x 0 I '/ D p.x 0 / C ˇ.x 0 '/ 2 C :::;
(7.32)
where
ˇ ˇ ˇ ˇ x D x 0 >0:
ˇ D 1
2
d 2 p
dx 2
(7.33)
Note that the term at leading-order that depends on the phase of the disturbance
is quadratic because the term proportional to dp
=dx
vanishes at the center of the
TC. Hence, the distribution of
at the mean (center) location of the disturbance is
therefore approximately chi-square 1 , which can be seen in the values of its scalar
moments:
p
h p i D p 0 C ˇ 2 ;
(7.34)
D
.p h p i / 2 E
D 2 4 ;
(7.35)
D
.p h p i / 3 E
D 3 6 :
(7.36)
1 A chi-square distribution with one-degree of freedom is constructed by squaring each random
draw from a Gaussian distribution.
 
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