Global Positioning System Reference
In-Depth Information
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The variance of the random variable
r
follows from the law of variance-covariance
propagation:
r
a
2
cos
2
b
2
sin
2
σ
=
ψ +
ψ
(4.311)
Th
e variance (4.311) is geometrically related to the standard ellipse. Let the ellipse
be
projected onto the direction
ψ
. The point of tangency is denoted by
P
0
. Because
th
e equation of the ellipse is
z
1
z
2
b
2
a
2
+
=
1
(4.312)
the slope of the tangent is
z
2
a
2
z
1
b
2
dz
1
dz
2
=−
[14
=−
tan
ψ
(4.313)
Se
e Figure 4.7 regarding the relation of the slope of the tangent and the angle
ψ
. The
Lin
—
0.4
——
No
PgE
se
cond part of (4.313) yields
z
01
a
2
z
02
b
2
sin
ψ −
cos
ψ =
0
(4.314)
This equation relates the coordinates of the point of tangency
P
0
to the slope of the
tangent. The length
p
of the projection of the ellipse is, according to Figure 4.7,
p
=
z
01
cos
ψ +
z
02
sin
ψ
(4.315)
[14
Ne
xt, (4.314) is squared and then multiplied with
a
2
b
2
, and the result is added to the
sq
uare of (4.315), giving
p
2
a
2
cos
2
b
2
sin
2
=
ψ +
ψ
(4.316)
By
comparing this expression with (4.311), it follows that
p
; i.e., the standard
de
viation in a certain direction is equal to the projection of the standard ellipse onto
th
at direction. Therefore, the standard ellipse is not a standard deviation curve. Figure
4.
8 shows the continuous standard deviation curve. We see that for narrow ellipses
th
ere are only small segments of the standard deviations that are close to the length of
the semiminor axis. The standard deviation increases rapidly as the direction
σ
r
=
moves
away from the minor axis. Therefore, an extremely narrow ellipse is not desirable if
the overall accuracy for the station position is important.
As a by-product of the property discussed, we see that the standard deviations of
the parameter
x
1
and
x
2
ψ
σ
x
1
=σ
0
√
q
x
1
(4.317)
σ
x
2
=σ
0
√
q
x
2
(4.318)
