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The Quality of Precise GPS Orbit Predictions for ’GPS-Meteorology’Jan Dousa Geodetic Observatory Pecny of the Research Institute of Geodesy, Topography and Cartography
EGU 5th General Assembly — Vienna, Austria, April 13 – 18, 2008.
Motivation
The quality of the orbits predicted for real-time plays a crutial role in the ’GPS
meteorology’ – precise troposphere delay estimation for the numerical weather
prediction. Two approaches are commonly used: a) precise point positioning
(PPP) using undifference observables and b) network solution using double-
difference observables, both very different in the requirements for the orbit
accuracy.
Since 2000, the International GNSS Service (IGS) provides the ultra-rapid
orbits, which are updated every 6 hours today. In (near) real-time, the use
of 3-10h prediction is thus necessary before getting new IGS product.
Is a quality of current orbit prediction sufficient to ’GPS-meteorology’ applica-
tion ? We monitor the quality of the orbit prediction performance and relevance
of the accuracy code at � � � �� � �� � � � �� � � �� � � � � → � �� �� �� �� � .
The effect of ephemeris errors on ZTD (in PPP)
Linearized equation for a carrier phase observable Lik (scaled to a distance) can
be written as
Lik = |~Ri
k0| + ∆Di
k + ∆C ik + ∆Si
k + λN ik + εi
k (1)
where |~Rik| = | ~X i − ~Xk| is a geometrical distance (in vacuum) between receiver k
and satellite i, ∆Dik is a sum of the distance dependent biases (receiver and
satellite position corrections, delays due to the ionosphere and troposphere),
∆C ik is a sum of the clock related biases (satellite and receiver clock biases,
relativistic corrections), ∆Sik is a sum of the satellite and station dependent
biases (phase center offsets and variations, multipath), λ is wavelength, N ik is
an initial ambiguity of the full cycles in the range and εik is a noise.
We focus on a simplified model considering only a satellite position bias (∆ ~X i)
and a tropospheric path delay (∆T ik) terms from a sum of distance dependent
biases
∆Dik =
~Rik0
|~Rik0|
∆ ~X i −~Ri
k0
|~Rik0|
∆ ~Xk + ∆I ik + ∆T i
k (2)
Other biases are considered accurately provided in advance, modeled or ne-
glected in (near) real-time analysis. Additionally, the station coordinates are
usually kept fixed on a long-term estimated position (we assume ∆ ~Xk = 0) and
the ionosphere bias (∆I ik) can be eliminated for its significant first order effect.
The precise satellite orbits are best estimated from a global network, while
preferably kept fixed in a regional analysis, thus we consider ∆ ~X i as a priori
introduced error (δ ~X i). In a simplest way, the troposphere parameters are
estimated as the time-dependent zenith total delays (ZTD) above each station
of the network
∆T ik = mf(zi
k) · ZTDk ≈1
cos(zik)
· ZTDk (3)
where a zenith dependent mapping function mf(zik) we approximated by cos(zi
k).
Because of their different magnitude, we are interested to express the orbit
errors in a satellite coordinate system (radial, along-track and cross-track;
RAC). Using a transformation
δ ~X i = Rz(λi) · Ry(ϕ
i) · δ ~X iRAC (4)
we distinct only two components: radial and in orbit tangential plane (along-
track + cross-track). Hence, we do not need to consider the satellite track
orientation and we will investigate only the marginal errors. The equation for
our simplified model is
Lik = |~Ri
k0| + ~ei
k · Rz(λi) · Ry(ϕ
i) δ ~X iRAC +
1
cos(zik)
· ZTDk + mf(zik) · ZTDk + λ · N i
k + εik (5)
where ~eik represents a unit vector pointing from station k to satellite i. The
error from the orbit prediction (δ ~X iRAC) is usually significantly larger than the
carrier phase observable noise εik. The orbit errors changes rather slowly (in
hours) and its 3D representation is projected into the pseudorange (1D). A
significant portion of this error can be mapped into the estimated ZTD if not
previously absorbed by the ambiguities (or clock corrections in PPP).
X i
iA
R
ZiA
iR
iψA
XA
RadialTangent
RA
For a priori orbit errors compensated mostly by ZTDs
we can write
~eik · Rz(λ
i) · Ry(ϕi) · δ ~X i
RAC +1
cos(zik)
· δZTDk ≈ 0 (6)
Either receiver position nor satellite position or velic-
ity have to be known if we express the impact of the
satellite radial and tangential errors only as a function
of zenith distance to the satellite. Following the sketch in figure we can express
~eBA · Rz(λ
i) · Ry(ϕi) · δ ~X i
RAC = cos(ΨiA) · δX i
Rad + sin(ΨiA) · δX i
Tan (7)
where
ΨiA = arcsin(sin(zi
A) ∗ RA/Ri) (8)
and we derive the plot for the impact of the radial and tangential orbit errors
to the range δLiA and their potential mapping into the ZTDA.
0.000.100.200.300.400.500.600.700.800.901.00
0 10 20 30 40 50 60 70 80 90
impa
ct (t
o ra
nge,
to Z
TD)
zenith distance
Zero-diff: Impact of radial/tangent orbit error
Jan
Dou
ša -
ZTD
erro
rs.g
pt 2
008-
04-2
5Radial error to δLAi
Tangent error to δLAi
Radial error to δZTDATangent error to δZTDA
Maximal impact from the radial error is in
zenith (impact 1.0). For satellite in horizon
it only slightly decreases to 0.97 for error in
δLiA and to 0.0 for δZTDA. The impact of the
tangential errors is much smaller with max-
imum 0.13 at zik = 45deg and minimum 0.0 when
tangential error is perpendicular to ∆ ~X ik. For
example, 10 cm tangential or 1 cm radial error
in orbit can cause max. 1.3 cm or 1.0 cm in ZTD, respectively.
The effect in the difference observables
Commonly used double-difference phase carier observations are written
Lijkl = Li
kl − Ljkl = (Li
k − Lil) − (Lj
k − Ljl ). (9)
This approach cancels a significant portion of the common biases at two re-
ceivers or two satellites. Some biases are cancelled perfectly (e.g. satellite
clocks), others are more or less significantly reduced depending on baseline
length (e.g. satellite position errors, troposphere path delays).
We investigate here an orbit error impact from a single satellite and thus we
can use solely single-difference observations. According to (5) and (9) they
are written as
Likl = |~Ri
k| − |~Ril| + (~ei
k − ~eil) · δ
~X i + mf(zik) · ZTDk − mf(zi
l) · ZTDl (10)
Any orbit error is simply projected into the single-difference observation by a
difference in the unit vectors (eik − ei
l). If it is compensated by the difference of
the estimated ZTDs (also in pseudorange projection), then
(~eik − ~ei
l) · Rz(λi) · Ry(ϕ
i)δ ~X iRAC + mf(z
ik) · δZTDk − mf(zi
l) · δZTDl ≈ 0 (11)
We need to know the baseline length, the zenith distance of the satellite at
one of the stations and the direction of the satellite with respect to baseline.
This is a bit more complicated case to generalize and we will thus study its
two marginal cases which both meets in a zenith above a mid of the baseline:
•� �� �� � � �� � � � � - satellite is in the same azimuth like the second station•� �� �� � � � � � � - zenith distances to the satellite are equal at both stations
X i
iA
R
ZiA
iR
XA
ZiB
iRB
iBψ
iψA
XB
RadialTangent
ABS
RA RB
We do not need again to know a satellite velocity vector
if we distiguish the ephemeris errors only in radial and
tangential direction. According to (7), (8) and (10)
we get a relation for the impact
(~eiA − ~ei
A) · Rz(λi) · Ry(ϕ
i) · δ ~X iRAC = (12)
(cos(ΨiA) · δX i
Rad + sin(ΨiA) · δX i
Tan) − (cos(ΨiB) · δX i
Rad + sin(ΨiB) · δX i
Tan)
which is evaluated for two cases above and baseline SAB = 1000km in the plots.
0.00
0.01
0.02
0 10 20 30 40 50 60 70 80 90
impa
ct (t
o ra
nge,
to Z
TD)
zenith distance
Single-diff: Impact of radial orbit error
Jan
Dou
ša -
ZTD
erro
rs.g
pt 2
008-
04-2
5
Baseline SAB = 1000 km
δLABi [case: equal azimuths]
δLABi [case: equal zeniths]
δZTDA = - δZTDB [case: equal azimuths]δZTDA = - δZTDB [case: equal zeniths]
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 10 20 30 40 50 60 70 80 90
impa
ct (t
o ra
nge,
to Z
TD)
zenith distance
Single-diff: Impact of tangential orbit error
Jan
Dou
ša -
ZTD
erro
rs.g
pt 2
008-
04-2
5
Baseline SAB = 1000 km
δLABi [case: equal zeniths]
δLABi [case: equal azimuths]
δZTDA = - δZTDB [case: equal zeniths]δZTDA = - δZTDB [case: equal azimuths]
In case of equal zeniths (ziA = zi
B, RiA = Ri
B, ΨiA = Ψi
B) the impact is always cancelled
out for the radial orbit error. Tangential orbit error has maximal impact when
the satellite is above baseline and the error is parallel with baseline (±0.027 for
ZTDA, ZTDB). It is cancelled out when the error is perpendicular to baseline and
reduced to the horizon. In case of equal azimuths the impact of the radial orbit
error is maximal at ziA = 38 deg (±0.0023 for ZTDA, ZTDB respectively). Tangential
error impact is the largest again above the baseline (the same as in equal zenith
case) and slightly faster reduces to the horizon. The impact is reduced with
decreasing the baseline length (approx. half for baseline 500km).
Simulation in network analysisBOGO
BRUS
BUCU
CHIZ
GLSVGOPE
HELG
HERS
LLIV
MATE
ONSA
OROS
POTS
SULP
VIS0
ZIMM
We used a network solution processed with
the Bernese GPS software to simulate the
Radial/Along-track/Cross-track (RAC) or-
bit errors. The ZTDs were estimated us-
ing ’star’ and ’circle’ networks with the longest baseline of
1300km (’star’). The precise IGS final orbits were used for data
pre-processing, ambiguity fixing, for estimating the reference
coordinates and ZTDs. The synthetic biases (1cm−100cm) were
introduced into the IGS final orbits successively for G01, G03,
G05 and G25 satellite in the RAC components independently.
Two ZTD solutions were provided and compared to the ref-
erence ZTDs – ambiguity fixed (top Figs) and ambiguity free
(bottom Figs). The ZTD map differences are plotted for
radial, along-track and cross-track components in 3-hour in-
terval when satellite is above the region (Figs below).
-40
-30
-20
-10
0
10
20
30
40
00 02 04 06 08 10 12 14 16 18 20 22 00
ZTD
diff
eren
ce [m
m]
hours
ZTD difference due to satellite shift in Radial position
Satellite G03 (’Star’ network, 16 stations)- ambiguity fix
+100cm+25cm+10cm
+5cm+1cm -40
-30
-20
-10
0
10
20
30
40
00 02 04 06 08 10 12 14 16 18 20 22 00
ZTD
diff
eren
ce [m
m]
hours
ZTD difference due to satellite shift in Along-track position
Satellite G03 (’Star’ network, 16 stations)- ambiguity fix
+100cm+25cm+10cm+5cm+1cm -40
-30
-20
-10
0
10
20
30
40
00 02 04 06 08 10 12 14 16 18 20 22 00
ZTD
diff
eren
ce [m
m]
hours
ZTD difference due to satellite shift in Cross-track position
Satellite G03 (’Star’ network, 16 stations)- ambiguity fix
+100cm+25cm+10cm
+5cm+1cm
-40
-30
-20
-10
0
10
20
30
40
00 02 04 06 08 10 12 14 16 18 20 22 00
ZTD
diff
eren
ce [m
m]
hours
ZTD difference due to satellite shift in Radial position
Satellite G03 (’Star’ network, 16 stations)- ambiguity free
+100cm+25cm+10cm
+5cm+1cm -40
-30
-20
-10
0
10
20
30
40
00 02 04 06 08 10 12 14 16 18 20 22 00
ZTD
diff
eren
ce [m
m]
hours
ZTD difference due to satellite shift in Along-track position
Satellite G03 (’Star’ network, 16 stations)- ambiguity free
+100cm+25cm+10cm+5cm+1cm -40
-30
-20
-10
0
10
20
30
40
00 02 04 06 08 10 12 14 16 18 20 22 00
ZTD
diff
eren
ce [m
m]
hours
ZTD difference due to satellite shift in Cross-track position
Satellite G03 (’Star’ network, 16 stations)- ambiguity free
+100cm+25cm+10cm
+5cm+1cm
Radial Along-track Cross-track
5
−45 −35 −25 −15 −5 5 15 25 35 45
ZTD−err [mm]
1m radial error star solutionDec 10, 2002 [18:30]
18
19
20 Satellite G03
20 25 30 35
40
−45 −35 −25 −15 −5 5 15 25 35 45
ZTD−err [mm]
1m along−track errorstar solutionDec 10, 2002 [18:30]
18
19
20 Satellite G03
5
10
−45 −35 −25 −15 −5 5 15 25 35 45
ZTD−err [mm]
1m cross−track errorstar solutionDec 10, 2002 [18:30]
18
19
20 Satellite G03
0
−45 −35 −25 −15 −5 5 15 25 35 45
ZTD−err [mm]
1m radial error star solutionDec 10, 2002 [19:30]
18
19
20 Satellite G03
−15
−10
−50510
15
20
25
30
35
−45 −35 −25 −15 −5 5 15 25 35 45
ZTD−err [mm]
1m along−track errorstar solutionDec 10, 2002 [19:30]
18
19
20 Satellite G03
−20
−15
−10
−5
0
5
10
15
−45 −35 −25 −15 −5 5 15 25 35 45
ZTD−err [mm]
1m cross−track errorstar solutionDec 10, 2002 [19:30]
18
19
20 Satellite G03
0
−45 −35 −25 −15 −5 5 15 25 35 45ZTD−err [mm]
1m radial error star solutionDec 10, 2002 [20:30]
18
19
20 Satellite G03
−30
−25
−20
−15
−10
−5
0
5
−45 −35 −25 −15 −5 5 15 25 35 45ZTD−err [mm]
1m along−track errorstar solutionDec 10, 2002 [20:30]
18
19
20 Satellite G03
−15
−10−5
05
10
−45 −35 −25 −15 −5 5 15 25 35 45ZTD−err [mm]
1m cross−track errorstar solutionDec 10, 2002 [20:30]
18
19
20 Satellite G03
Monitoring the quality of the IGS ultra-rapids
The overall accuracy of the precise IGS ultra-rapid orbit product is usually
presented by means of weighted rms. We present a detail evaluation with
respect to each individual satellite and with respect to every hour of 0-24h
interval prediction. The aim is to evaluate independently the individual satellite
orbit quality and assigned accuracy codes.
The IGS ultra-rapid orbits are epoch by epoch compared to the IGS rapid
product (3 rotations estimated). From the differences, which are stored in
a database we generate plots of the dependency of the orbit accuracy with
respect to the prediction interval (Fig 1), the evolution of the individual orbits
in time-series (Fig 2), to monitor the real orbit differences together with the
triple of expected error assigned to satellite (3 ∗ 2AccCode), Fig 4. Figure 3 shows
1D RMS from the IGS ultra-rapid comparison to IGS rapids provided by the
IGS ACC, which includes the eclipsing periods identifying prediction problems.
��� "!$#&%(
'
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10
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2005 2006 2007 2008 2009
RM
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m]
IGU x IGR : orbit position differences
Satellite G03
Dou
sa -
plot
_Orb
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.gpt
[2
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fit
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2005 2006 2007 2008 2009
RM
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IGU x IGR : orbit position differences
Satellite G05
Dou
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fit
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2005 2006 2007 2008 2009
RM
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Satellite G09
Dou
sa -
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[2
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3]
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fit
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2005 2006 2007 2008 2009
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Dou
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[2
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3]
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fit
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2005 2006 2007 2008 2009
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3]
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�)� "!*#&%,+
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t [2
008-
04-1
3]
RadAloCroPos
0
2
4
6
8
10
12
14
16
18
20
22
24
0 5 10 15 20
mon
thly
RM
S [c
m]
@ 2
008_
01
prediction [h]
IGU x IGR : orbit prediction
Satellite G05
Dou
sa -
plot
_Orb
Pre
d.gp
t [2
008-
04-1
3]
RadAloCroPos
0
2
4
6
8
10
12
14
16
18
20
22
24
0 5 10 15 20
mon
thly
RM
S [c
m]
@ 2
008_
01
prediction [h]
IGU x IGR : orbit prediction
Satellite G09
Dou
sa -
plot
_Orb
Pre
d.gp
t [2
008-
04-1
3]
RadAloCroPos
0
2
4
6
8
10
12
14
16
18
20
22
24
0 5 10 15 20
mon
thly
RM
S [c
m]
@ 2
008_
01
prediction [h]
IGU x IGR : orbit prediction
Satellite G18
Dou
sa -
plot
_Orb
Pre
d.gp
t [2
008-
04-1
3]
RadAloCroPos
0
2
4
6
8
10
12
14
16
18
20
22
24
0 5 10 15 20
mon
thly
RM
S [c
m]
@ 2
008_
01
prediction [h]
IGU x IGR : orbit prediction
Satellite G20
Dou
sa -
plot
_Orb
Pre
d.gp
t [2
008-
04-1
3]
RadAloCroPos
0
2
4
6
8
10
12
14
16
18
20
22
24
0 5 10 15 20
mon
thly
RM
S [c
m]
@ 2
008_
02
prediction [h]
IGU x IGR : orbit prediction
Satellite G03
Dou
sa -
plot
_Orb
Pre
d.gp
t [2
008-
04-1
3]
RadAloCroPos
0
2
4
6
8
10
12
14
16
18
20
22
24
0 5 10 15 20
mon
thly
RM
S [c
m]
@ 2
008_
02
prediction [h]
IGU x IGR : orbit prediction
Satellite G05
Dou
sa -
plot
_Orb
Pre
d.gp
t [2
008-
04-1
3]
RadAloCroPos
0
2
4
6
8
10
12
14
16
18
20
22
24
0 5 10 15 20
mon
thly
RM
S [c
m]
@ 2
008_
02
prediction [h]
IGU x IGR : orbit prediction
Satellite G09
Dou
sa -
plot
_Orb
Pre
d.gp
t [2
008-
04-1
3]
RadAloCroPos
0
2
4
6
8
10
12
14
16
18
20
22
24
0 5 10 15 20
mon
thly
RM
S [c
m]
@ 2
008_
02
prediction [h]
IGU x IGR : orbit prediction
Satellite G18
Dou
sa -
plot
_Orb
Pre
d.gp
t [2
008-
04-1
3]
RadAloCroPos
0
2
4
6
8
10
12
14
16
18
20
22
24
0 5 10 15 20
mon
thly
RM
S [c
m]
@ 2
008_
02
prediction [h]
IGU x IGR : orbit prediction
Satellite G20
Dou
sa -
plot
_Orb
Pre
d.gp
t [2
008-
04-1
3]
RadAloCroPos
��� "!$#&%.
-
0
20
40
60
80
100
Mar 07 Mar 11 Mar 15 Mar 19 Mar 23 Mar 27 Mar 31diffe
renc
es &
3-s
igm
a ex
pect
ed a
ccur
acy
[cm
] G03 : IGU x IGR orbit differences & expected accuracy
Dou
sa -
plot
_Orb
Acc
.gpt
[2
008-
04-0
4]
Position error [cm]Compared accuracy [cm]
IGU accuracy [cm]temporary unhealthy
permanent unhealthy
0
20
40
60
80
100
Mar 07 Mar 11 Mar 15 Mar 19 Mar 23 Mar 27 Mar 31diffe
renc
es &
3-s
igm
a ex
pect
ed a
ccur
acy
[cm
] G13 : IGU x IGR orbit differences & expected accuracy
Dou
sa -
plot
_Orb
Acc
.gpt
[2
008-
04-0
4]
Position error [cm]Compared accuracy [cm]
IGU accuracy [cm]temporary unhealthy
permanent unhealthy
��� "!$#/%,0
0
20
40
60
80
100
120
140
160
180
200
220
012007
032007
052007
072007
092007
112007
012008
032008
1D o
rbit
RM
S [c
m]
(sat
ellit
e sh
ifted
+20
cm
)
Eclipting satellites and 1D orbit RMS (IGU/6h prediction w.r.t. IGR)
G01
G02
G03
G04
G05
G06
G07
G08
G09
G10
Dou
sa -
plot
_Orb
Ecl
ips.
gpt
[200
8-04
-04]
0
20
40
60
80
100
120
140
160
180
200
220
012007
032007
052007
072007
092007
112007
012008
032008
1D o
rbit
RM
S [c
m]
(sat
ellit
e sh
ifted
+20
cm
)
Eclipting satellites and 1D orbit RMS (IGU/6h prediction w.r.t. IGR)
G11
G12
G13
G14
G15
G16
G17
G18
G19
G20
Dou
sa -
plot
_Orb
Ecl
ips.
gpt
[200
8-04
-04]
1 �2 �� � � 3 4� � � �� � � � � �� � �5 � �� 6 � � � � � �� � �� 7 �� 3 � �� � 8 9: � 9; � 9< ; < �
Summary Radial Tangent
PPP 1 cm 7 cm
Network 217 cm 19 cm
The table shows the re-
quirements for the radial
and tangential orbit posi-
tions to ensure the ZTD contamination less than 1cm when
not absorbed by the ambiguities or clocks. 1000km baselines are
considered in the network (the effects will reduce to one-half
if 500km). Also the ambiguities can absorb a significant por-
tion of the relative ZTD values. General requirements setup
is difficult. The satellite constellation, the network configu-
ration and especially the pre-processing (solving for ambigu-
ities, clocks, coordinates) altogether differentiate the situa-
tion in which the orbit errors can be absorbed into different
model constituents. The most inaccurate orbit component
(along-track) causes the problem in network ZTD solution
when satellite is flying in the baseline direction. The radial
component is usually crucial for PPP ZTD when satellite is
near zenith, while the along-track component cause maximal
error in elevation of 45deg if satellite is flying to or from the
station. It is necessary to distinguish GPS Block IIR and IIA
satellites, due to a different performance in prediction when
eclipsing. The orbit prediction is fastly degraded for the Block
IIA satellites (44%) and can not be often predicted enough
accurately even for a few hours. Also the accuracy code is
often underestimated for old satellites in eclipsing periods.
