PPP algorithms - The Ohio State University
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Transcript PPP algorithms - The Ohio State University
Part VI
Precise Point Positioning Supported by
Local Ionospheric Modeling
GS894G
Presentation Outline
Research objectives
The Benefits of PPP
MPGPS™ Software
Methodology
Experiments and test results
Summary and Conclusions
Research Objectives
Develop precise point positioning (PPP) methodology and
algorithms for surveying and navigation applications
Take advantage of the existing IGS products (precise
orbits and clock corrections)
Provide local ionospheric maps (LIM) and tropospheric
total zenith delays (TZD) from permanent GPS stations to
support single-frequency PPP
Evaluate the quality of single and dual-frequency static
and kinematic PPP in post processing
MPGPS™ - Multi Purpose GPS software
Developed at The Ohio State University (OSU)
Positioning Modules
Long-range instantaneous (single epoch) RTK GPS
Rapid-static
Static
Multi-station DGPS
Precise point positioning (PPP)
Atmospheric Modules
Ionosphere modeling and mapping
Troposphere modeling
Positioning Solutions
Single-baseline
Multi-baseline (network)
Stand-alone
The Benefits of PPP
Single receiver operation (low-cost)
Can be applied anywhere and anytime under different
dynamics (remote areas, space applications, etc)
Not limited by baseline length as relative techniques
Independence on GPS reference stations
Can be applied for static and kinematic platforms
Methodology
Error Sources in PPP
Errors affecting the GPS observations
Satellite orbit and clock corrections, (provided by IGS)
•
accuracy < 5 cm and <0.1 ns (3 cm)
Relativistic effects (included in the IGS orbits, except for the periodic
relativity, which is modeled in MPGPS™)
•
Receiver and satellite antenna phase center offsets (provided by IGS
or NGS)
•
up to 2 ns (0.6 m), accuracy 0.1 ns (3 cm)
Receiver DCB (GPS receiver calibration in MPGPS™ or IGS)
•
satellites - up to 1.023 m, receiver up to - 0.2 m
Satellite P1P2 and P1-C1 differential code biases (DCBs) (provided
by IGS)
•
periodic relativity - up to 30 ns (~9 m)
up to 20 ns (6 m), accuracy 0.1 ns (3 cm)
Phase wind-up
•
up to 1 cycle (~0.2 m) of carrier phase data
Methodology
Error Sources in PPP
Errors affecting the GPS observations (cont.)
Ionospheric refraction
•
Ranges from <1 m to >100 m
Tropospheric refraction
•
TZD = ~ 2.3 m (for standard atmosphere)
Errors affecting the station coordinates
Atmospheric loading
•
Ocean loading
•
corrections : horizontal < 2 cm, vertical < 5 cm
Solid Earth tides
•
correction: vertical < 1 cm
correction: horizontal < 5 cm, vertical < 30 cm
Earth Rotation Parameters, i.e., pole position and UT1-UTC
(included in the IGS orbits)
Methodology
Adjustment Model
GLS – Generalized Least Squares adjustment
All parameters in the mathematical model are considered
pseudo-observations with a priori information (σ = 0 ÷ )
F ( LbF , LbX ) 0
Two groups of parameters (pseudo-observations) of interest:
LbF
LbX
BFVF BX VX WF 0
instantaneous parameters (e.g., ionospheric delays)
- accumulated parameters (e.g., ambiguities)
-
Flexibility, easy implementation of:
stochastic constraints
fixed constraints
weighted parameters
filters
Methodology
PPP Functional Model
Lki ik c(ti t k ) ik Ti I ik Bik
0
Pi k ik c(ti t k ) ik Ti I ik c(b k bi ) 0
Lki , Pi k
- undifferenced carrier phase and code observations (in meters)
ik
- geometric distance (satellite-receiver)
Bik
- constant bias, where Bik ( Nik Nik ) c(d k di )
- integer carrier phase ambiguity and non-zero initial fractional phase
Nik , Nik
ti , t k
- receiver and satellite clock offsets
Ti
- tropospheric total zenith delay (TZD)
ik
- troposphere mapping function
I ik
- slant ionospheric delay
bi , bk ; di , d k
- receiver and satellite code and phase hardware delays
- corresponding carrier wavelength
c
- speed of light
- random error or residual
Methodology
PPP - Functional Model Unknowns
Permanent GPS station solution for local ionosphere
maps (LIM)
Single-frequency positioning solution
receiver clock
tropospheric TZD
slant ionospheric delays
bias parameters (non-integer ambiguities and hardware delays)
rover coordinates
receiver clock
bias parameters
Dual-frequency (ionosphere-free) positioning solution
rover coordinates
receiver clock
tropospheric TZD
bias parameters
Methodology
Local Ionospheric Model (LIM)
Supports PPP in case of single-frequency receiver
Single layer model (SLM) ionosphere approximation
Slant ionospheric delays estimation from dual-frequency GPS
data at the neighboring permanent stations
Slant ionospheric delays conversion to vertical total electron
content (VTEC) at ionosphere pierce points (IPPs)
Kriging interpolation to produce LIM in a form of a grid using
the calculated vertical TEC values at IPPs
Methodology
Local Ionospheric Model (LIM)
SLM – Single Layer Model
z
- zenith angle
H - SLM height
R - Earth radius
1 TECU = 1016 ellectron/m2
= 0.162 m delay/advance
SLM assumes that all free electrons are contained
in a shell of infinitesimal thickness at altitude H
Methodology
PPP Models
Three PPP MPGPS™ models were tested in post
processing mode
Static PPP
– dual-frequency (ionosphere-free)
Static PPP
– single-frequency supported by LIM
Kinematic PPP – single-frequency supported by LIM
Adaptive filter for kinematic solution
Follows the dynamic variations of the system estimates and
stochastic models
Propagates the coordinate and ionosphere residuals together
with their stochastic characteristics
Forward and backward filters
Experiments and test results
Data Source
Four stations, IGS/EPN (EUREF permanent network)
Three stations were used to derive LIM and TZD
(BOR1, GOPE, KRAW)
One station was selected as a rover (WROC)
Two three-hour sessions
01 - 04 UTC (nighttime - lowest TEC level)
13 - 17 UTC (daytime - highest TEC level)
30-second sampling rate (i.e., 360 epochs per session)
Phase-smoothed pseudoranges
Distances between permanent stations ~330 km (average)
Distances to the rover ~130–230 km
Experiments and test results
Test
Area Map
____________________________________________________
N
Rover
- LIM/TZD
- PPP (rover)
Poland
Czech
Republic
Experiments and test results
Satellite
Geometry - Station WROC
____________________________________________________
01-04 UTC nighttime
4-7 satellites
13-17 UTC daytime
4-5 satellites
GDOP
=~1000
GDOP
= ~80
Poor satellite geometry, high GDOP - usually over 5
A short period with very poor geometry occurred in both sessions
Experiments and test results
Example
LIM-derived ionospheric delays
_______________________________________________________
13-17 UTC daytime
highest TEC
01-04 UTC nighttime
lowest TEC
7
7
PRN 15
PRN 16
PRN 18
PRN 23
PRN 31
PRN 14
PRN 11
PRN 3
6
5
PRN 8
PRN 27
PRN 28
PRN 29
PRN 26
PRN 9
6
5
4
[m]
[m]
4
3
3
2
2
1
1
0
0
50
100
150
200
250
300
350
0
0
50
100
Epochs
150
200
Epochs
Station WROC (rover)
250
300
350
Experiments and test results
Static
PPP Analysis – Station WROC
____________________________________________________
Nighttime, dual-frequency
(ionosphere-free LC)
Daytime, dual-frequency
(ionosphere-free LC)
Nighttime, single-frequency
supported by LIM
Daytime, single-frequency
supported by LIM
Experiments and test results
Static PPP Analysis – Station WROC
Ionosphere-free solution
Horizontal - sub-decimeter-level position accuracy
Vertical
Nighttime - convergence after 40 minutes
Daytime - convergence after 25 minutes
- decimeter-level
Single-frequency solution supported by LIM
Good agreement with its ionosphere-free counterpart
Similar accuracies and convergence times
LIM proved to be efficient in removing the ionospheric delays
Experiments and test results
Kinematic
PPP Analysis – Station WROC
____________________________________________________
01-04 UTC (nighttime)
Unfiltered
singlefrequency
supported by
LIM
Filtered
singlefrequency
supported by
LIM
13-17 UTC (daytime)
Experiments and test results
Kinematic PPP Analysis – Station WROC
The unfiltered solutions are very noisy in both
sessions
In the filtered solution the large residuals were
smoothed out after a few iterations (3-4)
The filtered kinematic solutions show similar
accuracies as obtained in the static case
Sub-decimeter horizontal and decimeter-level vertical
position accuracy was achieved
Summary and Conclusions
The sequential GLS adjustment was successfully applied in the
PPP algorithm
Single-frequency static and kinematic PPP solutions, supported
by LIM, are comparable to the ionosphere-free solutions
The results prove a good quality of the obtained LIM
The effectiveness of the adaptive filter was presented in the
kinematic mode, even under unfavorable satellite geometry
This algorithm may be applied in geodetic applications, where
sub-decimeter level accuracy is required