determination of the station coordinates using gps …...we used approximate coordinates of the...
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Determination of the Station Coordinates Using GPS
Satellite Single Frequency Doppler Signal
Iu. Babyk, V. Choliy
Kiev National Taras Shevchenko University, Physical Faculty, Kiev, Ukraine.
V. Taradiy
International Center AMEI, Terskol, Russia.
Abstract. In this work we used doppler data from GPS satellites to determine
coordinates of Terskol observatory. Our goals were to find out whether it is possible
to use these data for determination of coordinates of ground-based stations and
how precise are the coordinates.
Introduction
Each GPS satellite carries an atomic clock to provide timing information for the signals
transmitted by the satellites. Internal clock correction is provided for each satellite clock. Each
GPS satellite transmits two spread spectrum, L-band carrier signals - L1 signal with carrier
frequency f1 = 1575, 42MHz and an L2 signal with carrier frequency f2 = 1227, 6MHz.
Two- or single- frequency receivers are used all around the world to calculate the positions
from analysis of the carrier frequencies modulations.
The data from GPS satellites are transmitted via three kinds of signals: the phase of the
carrier wave P ; pseudo-range of the satellite from the station L; doppler shift of the carrier
frequency D.
In general, everything works fine, but during the periods of increased solar activity P and
L signals may be damaged, while doppler data are mostly not affected [1]. In this way, GPS
satellites become analogous to DORIS (Doppler Orbitography and Radiopositioning Integrated
by Satellite) system and may be processed in analogous way [2], [3]
Task
The Terskol observatory in Russia (3100 m above sea level) was have chosen as the object
of observation, where our detector (Accutime 2000 receiver) was temporary setup. Terskol
mountain at which this observatory is located, is situated on the south slope of the former
volcano Elbrus, so we expected significant oscillations of the coordinates there due to possible
seismicity. The Accutime 2000 class receivers are used for timing purposes mostly and are not
intended to measure the coordinates.
In our work we tried to process GPS doppler signal in a way it is done in DORIS to:
• see whether it is possible to get coordinates when phase and pseudorange signals are
absent;
• get an experience for further combined processing of GPS and DORIS;
• ascertain whether it is possible to calculate the speed of coordinates changes from these
data;
• to see if coordinates of our station are changing, and if so - to find out how are they
changing.
We use basic algorithm for doppler pocessor from [4].
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WDS'08 Proceedings of Contributed Papers, Part II, 192–195, 2008. ISBN 978-80-7378-066-1 © MATFYZPRESS
BABYK ET. AL.: DETERMINATION OF THE STATION COORDINATES
Algorithm
We can calculate the position of the receiver on the Earth surface in Earth Fixed Earth
Centered coordinate system (ECEF) using its a priory longitude λ0, lattitude φ0 and distance
from center R⊕:
~r = (S1, S2, S3) = R⊕(cos φ0 cos λ0, cos φ0 sinλ0, sin φ0). (1)
In analoguous way, using satellite ephemerides, transmitted by the satellite, or downloaded
from the Internet one can calculate the ECEF position of the satellite too:
~ρ = ~ρ(ephemerides parameters). (2)
It is obvious that distance from the receiver to the satellite is:
| ~R |=| ~ρ − ~r | . (3)
Let us rewrite the last equation in the way:
RTHE
= R(φ0, λ0, ephemerides parameters, some extra parameters). (4)
We call R the ”theory”. It means that R comprise all constants, values and algorithms
necessary to calculate theoretical value of satellite-receiver distance RTHE
. Differencing of that
equation give us the theoretical value of the radial velocity of the satellite in receiver-centered
coordinate system ˙RTHE
.
From the other side, the doppler data from the satellie is observed carier frequency doppler
shift ∆f which is:
ROBS
= −c∆f
L1
(5)
The only source of differencies between ROBS
and RTHE
is incorrect or not precise or not
up to date values of the parameters used. To correct the worst values we used least squares
minimisation of theory minus observation squared differencies over all the observations.
This is our first attempt and that is why we worked only with the receiver coordinates and
frequency shift of the receiver clock. That last parameter (B) is the one the theory is extremely
sensitive to.
Using software, developed by authors, we processed doppler data from GPS satellites and
their calculated ephemerides:
ROBS
= RTHE
(φ0, λ0, B0) +∂R
∂φ(φ − φ0) +
∂R
∂λ(λ − λ0) +
∂R
∂B(B − B0); (6)
where
ROBS
observed range rate (Doppler),
RTHE
computed range rate based on current estimate of receiver location,
φ0 initial lattitude,
λ0 initial longitude,
B0 initial estimate of range-rate bias.
On the first iteration, values of φ0 and λ0 are generally available to whithin a few hundred
kilometers based on a priori knowledge. Also, on the first iteration, the initial guess of the bias
B0 is taken be zero. The new values of φ, λ and B obtained after the first iteration are then
used as a priori φ0, λ0 and B0 for the next iteration. This process is repeated until the solution
has converged. The foregoing equations can be rewritten in terms of the three variables ∆φ,
∆λ and ∆B. That is
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BABYK ET. AL.: DETERMINATION OF THE STATION COORDINATES
Figure 1. Time evolution of lattitude and longitude values.
∆R =∂R
∂φ∆φ +
∂R
∂λ∆λ +
∂R
∂B∆B. (7)
where
∆R = Ro − Rc
or residuals which will be minimized as the fit improves,
∆φ = φ − φ0 computed adjustment to a priori geodetic latitude,
∆λ = λ − λ0 computed adjustment to longitude,
∆B = B − B0 computed adjustment to range-rate bias.
All derivatives in the previous equations may be found by direct differencing of the theory
equation:
∂R
∂φ=
∂R
∂S1
∂S1
∂φ+
∂R
∂S2
∂S2
∂φ+
∂R
∂S3
∂S3
∂φ; (8)
∂R
∂λ=
∂R
∂S1
∂S1
∂λ+
∂R
∂S2
∂S2
∂λ; (9)
∂R
∂B= 1.0. (10)
We used approximate coordinates of the station, which were substituted in the algorithm,
and with the help of the least square method we determined corrections to our values. Every
second our receiver produces data for 3–8 equations (7) depending on the amount of visible
satellites (one equation per satellite). Every 10 second we recalculate the parameters and put
them in Fig. 1 and 2. Influence of the ionosphere and troposphere were not taken in account.
Last figure on Fig.1 is the Gdop value. It is Geometric dillution of precision and show us
how much the solution is dilluted due to poor geometry of the satellites in the sky.
Conclusion
Raw Doppler data from GPS satellites is suitable for coordinate purposes. Results strongly
depends on Gdop value. Unfortunately, precision of our results are quite poor, despite of the
fact that repeatibility of the are good. It is mostly due to theory shortcomings. For example,
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BABYK ET. AL.: DETERMINATION OF THE STATION COORDINATES
Figure 2. This image represents raw coordinate point of Terskol observatory.
we found very strong correlation between B parameter and height of the receiver above sea level
but failed to imcorporate them. There are no ionosphere and troposphere models included in
theory. It may cause additional and possibly huge errors.
Acknowledgment. This work is partly supported by RFFI grant 08-02-00458.
References
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Schmid P.E., Lynn J.J Satellite Doppler-Data Processing Using a Microcomputer IEEE Trans., v.Ge-16,
n.4, 1978.
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