Geoscience Reference
In-Depth Information
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
2ˇ
2
4
;
(7.35)
D
.p
h
p
i
/
3
E
D
8ˇ
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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