Global Positioning System Reference
In-Depth Information
llm
−
=
∑∑
( )
mn
Nuv
,
a uv
(5)
mn
mn
==
00
where
a
mn
are the polynomial coefficients for
m
,
n
= 0 to
l
, which is the order of polynomial.
u
and
v
represent the normalized coordinates, which are obtained by centring and scaling the
geodetic coordinates ϕ and λ
.
In the numerical tests of this study, the normalized
coordinates were obtained by
u
=
k
(ϕ − ϕ
o
) and
v
=
k
(λ − λ
o
) where ϕ
o
and λ
o
are the mean
latitude and longitude of the local area, and the scaling factor is
k
= 100/
ρ
°.
In Equation 5, the unknown polynomial coefficients are determined with least squares
adjustment solution. According to this, the geoid height (
Ni
) and its correction (
Vi
) at a
reference benchmark having (
u
,
v
) normalized coordinates as a function of unknown
polynomial coefficients is:
NVa auav
au
+= + +
+
i
i
00
10
11
2
2
+
auv av
+
20
21
22
3
2
2
3
+
au
+
auv auv
+
+
av
(6)
30
31
32
33
4
3
2
2
3
4
+
au
+
auv auv
+
+
auv
+
av
40
41
42
43
44
...
and the correction equations for all reference geoid benchmarks in matrix form is:
⎡
NV
⎤⎡⎤⎡
1
u
v
...
⎤⎡ ⎤
a
1
1
1
1
00
⎢
⎥⎢⎥⎢
⎥⎢ ⎥
NV
1
u
v
...
a
⎢
⎥⎢⎥⎢
⎥⎢ ⎥
2
2
2
2
10
⎢
⎥⎢⎥⎢
⎥⎢ ⎥
.
+
.
=
.
.
.
...
.
(7a)
⎢
⎥⎢⎥⎢
⎥⎢ ⎥
.
.
.
.
.
...
.
⎢
⎥⎢⎥⎢
⎥⎢ ⎥
⎢
⎥⎢⎥⎢
⎥⎢ ⎥
NV
1
u
v
...
a
⎣
⎦⎣⎦⎣
⎦⎣ ⎦
i
i
i
i
mn
NV AX
+ =
(7b)
and the unknown polynomial coefficients (
a
mn
elements of the
X
vector, see Equations 7a
and 7b):
( )
T T
−
XAAA
=
A
(8)
and the cofactor matrix of
X
XX
QAA
−
( )
1
T
=
(9)
are calculated. In the equations
A
is coefficients matrix and
ℓ
is the vector of observations
that the elements of the vector are the geoid heights (
N
GNSS/levelling
).
One of the main issues of modelling with polynomials is deciding the optimum degree of
the expansion, which is critical for accuracy of the approximation as well and its decision
mostly bases on trial and error (Erol, 2009). Whilst the use of a low-degree polynomial
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