Department of Astronomy and Geodesy, Kazan State ...old.kpfu.ru/journals/georesur/bin_files/12_komarovv!153.pdf · R.V. Komarov, R.A. Kascheev, R.V. Zagretdinov Geoid Determination

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R.V. Komarov, R.A. Kascheev, R.V. Zagretdinov Geoid Determination by GPS/Levelling Method...

R.V. Komarov, R.A. Kascheev & R.V. Zagretdinov Department of Astronomy and Geodesy, Kazan State University, Kazan, Russia

[email protected]; [email protected]; [email protected]

Geoid Determination by GPS/Levelling Method

in the Republic of Tatarstan

The building of a geoid heights map using a Global Positioning System (GPS) and levelling data is presented. GPS

measurement was taken at benchmarks of Almetievsky geodynamical polygon and controls of the state geodetic network

of the Republic of Tatarstan. For the first time the local geoid heights by GPS/leveling method in Tatarstan was computed.

Local geoid heights have been compared with global geoid models. Precision of computed heights have also been made.

1. IntroductionDuring last decades the GPS (Global Positioning System)

has been used in many applications of geodesy, geophysics

and surveying. The GPS observations are referred to geodet-

ic ellipsoid WGS-84 and characterized by rectangular or geo-

detic coordinates.

The modern differential GPS-techniques provide the ellip-

soidal heights with unprecedented accuracy up to one cen-

timeter at regional and global scales. On the other hand, many

applications in geodesy, geophysics and surveying requires

physically defined orthometric or normal heights related to

the Earth’s gravity field, typically produced by geometric spirit

levelling. For these kind of applications the high precision

geoid models must be established with an accuracy compara-

ble to the GPS and the levelling measurements accuracy.

2. Geoid/quasigeoid determinationThe geoid, as an equipotential surface of the gravity field

suitably fitting the physical surface of the earth and deter-

mined in geodesy as the basic surface which orthometric or

normal heights is refered to. For determination of the geoid

height the ellipsoidal and orthometric heights are used. The

ellipsoidal height is referred from the surface of any reference

ellipsoid to the point of interest along ellipsoidal normal. The

orthometric height is referred from the geoid to the point of

interest along the curved plumbline. The geoid height or ge-

oid-ellipsoid separation is referred from the surface of any

reference ellipsoid to the geoid along the ellipsoidal normal

(e.g. Heiskanen & Moritz, 1967). The transformation of ellip-

soidal heights to orthometric heights therefore requires that

the geoid height refer to the same reference ellipsoid. In the

case of GPS-derived ellipsoidal heights the geocentric WGS84

ellipsoid are used. Therefore, the geoid model must refer to

this ellipsoid or another that is compatible.

The relationships between ellipsoidal, orthometric and

geoid heights are shown in Fig. 1. The determination of the

geoid height at each point can be calculated using a well-

known formula:

N ≈ h – H, (1)

where N is the geoid height, h is the ellipsoidal height and H

is the orthometric height. The approximate equality in Eq. (1)

results from neglecting the departure of the plumbline from

the ellipsoidal normal, which is also called the deflection of

the vertical. There is also torsion in the plumbline, but the

deflection of the vertical is usually the dominant effect of the

approximation in Eq.(1).

In software for post-processing GPS observations in dif-

ferent global and regional geoid models are used and the

most common of them are EGM96 and OSU91A.We built up

for the territory of our republic global geoid model map height

according to EGM96 (Lemoine et al., 1996) presented in figure

2,and model OSU91A (Rapp et al., 1991) presented in figure 3.

Russian gravimetric geoid RGG2000 computed by Central

Research Institute of Geodesy, Aerial Surveying and Cartog-

raphy in 2000 with resolution 2 by 2 arc-minute grid also can

be used for our republic Fig. 4 (http://zeus.wdcb.ru/wdcb/

gps/rgg/html, 2000). Unfortunately for many geodesy and

surveying applications these models are not accurate enough

on recalculation from ellipsoidal for orthometric heights.

3. GPS surveys and levellingSince 1991 precise levelling has been performed on Alme-

tievsky geodynamical polygon (AGDP) in the area of the Ro-

mashkino oil deposit. The levelling traverse was used to de-

termine the normal heights of the benchmarks and was per-

formed according to II class levelling standards. This corre-

sponds to the maximum allowable misclosures or repeatabili-

ty between levelling runs of ±1√L mm, where L is the direct

distance along the level traverse in kilometers. The observa-

tions on the polygon were carried out using optical levels and

invar staves. This polygon consists of about 355 km double

run levelling paths and about 500 levelling benchmarks.

Two epochs of GPS observations using dual frequency

receivers were made in 2000 and 2001 respectively. On their

bases the local GPS network was established in the South-

Eastern part of the Republic Tatarstan in Russia. The local

GPS network consist of 12 benchmarks of geopolygon. The

coordinates of our benchmarks were calculated relatively

nearest to the IGS stations (ARTU, GLSV). The GPS measure-

Table 1.

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R.V. Komarov, R.A. Kascheev, R.V. Zagretdinov Geoid Determination by GPS/Levelling Method...

ment of the baselines were taken

using TRIMBLE dual frequency

GPS receivers in static mode with

12 hour and 24 hour seance dura-

tions in 2000 and 2001 respective-

ly. Processing of the GPS base-

lines has been performed using

Trimble Geomatics Office software

with precise IGS ephemeris. The

95 % confidence error for the

WGS 84 ellipsoidal heights in the 12 points local scales GPS

network are shown in Table 1.

The errors presented in Table 1 are a result of the post-

processing GPS observation software estimation. Further, we

calculated the geoid heights by formula (1) and built up the map

of the local geoid heights for the Almetievsky geopolygon.

In several districts of the Republic of Tatarstan the GPS

static observations were performed from 2000 to 2002 under

the controls of the state geodetic network. We used these

data to expand the size of the Almetievsky geopolygon geoid

in this territory. The GPS surveys were carried out using sin-

gle and double frequency Trimble and Ashtech receivers.

The normal heights of controls were determined by trigono-

metric and spirit levelling with the maximum allowable misclo-

sures of ±5 – 10√L mm. For determination of WGS-84 coordi-

nates on these controls all GPS measurements were connect-

ed to the nearest IGS stations ARTU and GLSV with the dis-

tances approximately 500 km to 1000 km respectively.

We have used 35 points observed by dual frequency meas-

urements and 38 single frequency with the mean time of these

observations about 10 hour. The GPS vectors processing was

made as described above. For single-frequency measurements

the solution of GPS vectors is a gained float, but we shall try to

estimate an error heights. Using the new calculated heights we

have expanded the local geoid of the Almetievsky geopolygon

on all the territory of the Republic of Tatarstan. Accordingly,

for the first time the height map of local geoid was computed

using the GPS/leveling method in the Republic of Tatarstan.

The interpolating surface and contour map were computed

by the linear Kriging method. The results are shown in Fig. 5.

4. Results, comparison and discussionGeoid heights at a moment can calculated for our Repub-

lic using global geoid models and regional gravimetric geoid

model on our territory. It should be emphasized that the pur-

pose of this article is to give local geoid heights in a first

approximation. Because the GPS data were obtained at differ-

ent times the GPS network not homogeneous and data are

not reduced on one epoch. This also can introduce an error at

calculating geoid heights. It is easy to see from the formula

(1) that the error of geoid height consists of error determina-

tion orthometric height and ellipsoidal height. First, we shall

consider an error of definition of orthometric heights. Ac-

cording to our estimations this error is about 1 cm for bench-

marks of Almetievsky geopolygon and about 10 cm for con-

trols of state geodetic network. Further we discuss errors of

ellipsoidal heights determination. We estimate, that the er-

rors of the ellipsoidal heights determination amounts to be

about 2-5 cm for dual-frequency observations. For an estima-

tion of error single frequency data we have performed GPS

measurements at the same point using dual and single fre-

quency receivers. The results comparison of precision of GPS

baselines and coordinates presented in Table 2.

In this table the single frequency GPS-vectors introduced

in the upper two lines and dual frequency vectors in the bot-

tom two lines. We can see that error definition ellipsoidal

height using single frequency data is about 20 cm.

Earth surfacegeoid

ellipsoid

H h

N

Fig. 1. The relationships be-

tween ellipsoidal, orthomet-

ric and geoid heights.

Table 2.

Fig. 4. Contour map of RGG2000.

Fig. 5. Contour map of local geoid.

Fig. 2. Contour map of EGM96. Undulations values in meters (Figs.

2-8).

Fig. 3. Contour map of OSU91A.

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R.V. Komarov, R.A. Kascheev, R.V. Zagretdinov Geoid Determination by GPS/Levelling Method...

Accordingly we have an error of ellipsoidal height of about

5 cm and 20 cm for dual and single frequency data. Therefore

an error of geoid heights can vary from 5 cm up to 25 cm for

dual-frequency data given on benchmarks of an Almetievsky

geodynamical polygon and controls of a state geodetic net-

work respectively. On controls, the error of geoid heights can

be estimated by 30 cm at the best and 1 m in the worst case for

single-frequency data. In this case the primary factor is the

error of a solution GPS baseline. It is known, that the biggest

error in determination of baselines during processing of a

single-frequency data on major distances is introduced by

ionosphere effects. Unfortunately in Tatarstan at the present

time there are wide distribution single-frequency receivers

and lack of dense GPS network stations. In a first attempt to

compare local geoid with other models our geoid was com-

pared with global models EGM96 and OSU91A.

The differences between local geoid heights and EGM96

model are not more than 20 cm for Almetievsky geopolygon

benchmarks (Fig. 6) and about 1 m for other points. The

OSU91A model differences are about 1 m on geopolygon (Fig.

Fig. 7. Contour map of differences local geoid and OSU91A model

on geopolygon.

Fig. 6. Contour map of differences local geoid and EGM96 model

on geopolygon.

Fig. 8. Contour map of differences local geoid and RGG2000 mod-

el on geopolygon.

Table 3.

7) and 2 m for oth-

er points. In a sec-

ond step the local

geoid was com-

pared with a Rus-

sian gravimetric

geoid. The differ-

ences of geoid

heights not more

then 50 cm on ge-

opolygon (Fig. 8)

which is about 1

m for other points.

Find the re-

sults of compari-

son of our geoid

with above the

tree geiod models

presented in Ta-

ble 3. From the

comparison of ge-

oid height on Almetievsky geopolygon we can also see some

systematic displacement between the above models. In re-

maining points this displacement is not so evident through

bigger errors of geoid heights determination.

Where is first 12 points is benchmarks of Almetievsky

geodynamical polygon, * and ** are controls of state geodetic

network measured dual-frequency and single-frequency re-

ceivers respectively. In Table 3 loc-wgs is the difference be-

tween our local geoid height and the global model EGM96

geoid height, loc-rgg is difference between our local geoid

height and Russian gravimetric model geoid height, and loc-

osu is the difference between of our local geoid height and

global model OSU91A geoid height.

ConclusionFor the first time the local geoid model has been comput-

ed for the Republic of Tatarstan. The error of geoid heights

can vary from 5 cm up to 25 cm for dual-frequency data given

on benchmarks of a Almetievsky geodynamical polygon and

controls of state geodetic network respectively. On controls

the error of geoid heights determination can be estimated by

30 cm at the best and 1 m in the worst case for the single-

frequency data. This paper presents some preliminary results

for geoid heights from Tatarstan. In order to improve the com-

puted geoid, it is necessary to increase the density of points

using dual frequency GPS receivers.

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