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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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

Aiframovich E., Demyanov V., Ishin A., Smolkin G. Total failures of GPS-GLONASS functioning caused

by the powerful solar radio emission // EGU 2008-A-00177.

Hoffman-Wellenhof B., Lichtenegger H., Collins J. Global Positioning System – Theory and Practice,

NY, 1995, 377 page.

Mohinder S. Grewal L., Weill R., Andrews A. Global Positioning Systems,Inertial Navigation, and Inte-

gration, NY, 2001, 409 page.

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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