GEOID DETERMINATION AND LEVELLING BY GPS: SOME EXPERIMENTS ON A TEST NETWORK
ABSTRACT
Henri DUQUENNE, Zhiheng JIANG, Cecile LEMAruE Laboratoire de Recherche en Geodesie Institut Geographique National, France
An increasing demand for local or national geoids results of the development of GPS positioning. In order to determine the precision that could be reached in heights determination with different data and methods, a test network covering 560 km2 was established in 1993-1994. 63 points were observed with GPS and precise levelling in a hilly area located in the South of France. Gravity measurements and a digital terrain model were also available. Two methods are compared. The first one is based on simple interpolation, the second one uses a gravimetric geoid and greatly improves the precision.
THE TEST NETWORK OF MANOSQUE
The geodetic test network used for these studies was established around Manosque, in the South of the French Alps. It covers 28x20 km2. The heights are between 260 and 790 m, and the Luberon Mountains (more than 1000 m high) are located at less than 15 km. The geodetic measurements were performed by students of the Ecole Nationale des Sciences Geographiques and of the Ecole Superieure des Geometres et Topographes in 1993 and 1994. The network consists of82 GPS points. 17 points were measured during 45 minutes with Ashtech and Leica two frequencies receivers, and constitute a basic network linked to ITRF reference frame. The GPS baselines were computed with Ashtech GPPS and Leica SKI programs and the co-ordinates were adjusted by IGN's SSCMIX software (see Boucher and Willis, 1988). Their precision is about 2 cm. 65 points were measured by rapid static method, each point during 15 minutes. The co-ordinates were adjusted using GEOLAB. The precision is 2-3 cm. The normal heights of 63 GPS points were measured in the national IGN69 height reference system, using precise spirit levelling or precise trigonometric motorised levelling techniques. Hence the precision of the heights can be evaluated to 1 cm in a local sense. Figure 1 shows the location of the levelled GPS points. An interesting characteristic of this network is its density, about 1 point per 9 km2, which permits a good experimental evaluation of the precision for GPS levelling and geoid determination.
GRAPHIC SYSTEM PEG A SUS A PRODUCT OF rue IAPG UNJ'vERSIlY FAf MUNICH 31·MAY-94 1201
Fig. 1. Distribution of the levelled GPS points of the test network of Manos que.
SIMPLE INTERPOLATION OF THE HEIGHT REFERENCE SURFACE
According to whether the type of height is orthometric or normal, the basic equation for GPS levelling is: H=h-N (1)
or: H*=h-' ~)
where H is the orthometric height, H* the normal height, h the ellipsoidal height, Nand , the geoidal undulation and the height anomaly (see Heiskanen and Moritz, 1967). The main problem is to calculate N or ,. A method extensively used by surveyors consists of interpolation of N or , on non levelled GPS points, between some well distributed levelled GPS points. The precision of this method is limited by the precision of GPS h coordinates, by the precision of the normal heights of the levelled GPS points, and by the high frequency undulations of the geoid or the quasi-geoid. In order to test this kind of method, five computations were achieved using various numbers of levelled GPS points, and comparing the interpolated height anomalies with the observed ones. The precision is estimated by the standard deviation of these residuals. The computations were carried out with the programs GEOGRID and GEOIP of GRA VSOFT software package. Height anomalies were first gridded using weighted mean interpolation, then interpolated using bicubic spline interpolation. Although other interpolation methods might be proposed which could slightly improve the results, they were not yet been applied to the test area. Table 1 shows the worsening in precision when the number of levelled GPS points decreases. When the density of these points is less than 1 per 36 km2 (that is 1 point every 6 km), the standard deviation becomes greater than 5 cm, and is not at all compatible with
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Table 1. Apparent precision of heights predicted by simple interpolation.
Number of levelled Standard deviation of Points with high residual GPS points predicted heights (m)
Fig. 2. Quasi-geoid realized by GPS and levelling. 60 points were used. (Same area as in fig. 1)
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50.2
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N
(m)
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Fig. 3. Quasi-geoid realized by GPS and levelling. 5 points (the black circles) were used.
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\ 50 . 2
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the usual specifications for current levelling work. In the last column of the table, points with residual larger than 2 times the standard deviation are listed. The results depend upon the levelled GPS points used, any attempt to detect outliers would be doubtful. As a matter of fact, residuals contain information not only on measurement errors but also on geoidal undulation.
Figures 2 and 3 illustrate two realizations of the quasi-geoid by GPS and levelling only, the first one based on 60 levelled GPS points, the second using 5 points only. Although the results are roughly compatible with the realizations of the geoid (see fig. 6), they are clearly affected by errors in GPS measurements and, to a lower extend, in normal height. Some "holes" or "bump" appearing in figure 2 canlt be explained otherwise. However, the simple interpolation method does not allow to distinguish between GPS measurement errors and local geoidal undulations.
GRAVIMETRIC GEOID COMPUTATION
The local geoid computations were carried out using GRA VSOFT, the global geopotential model OSU91A complete to degree and order 360, a gravity data set and a digital terrain model. Classical Stokes integration combined with remove-restore technique were employed. An integration radius of 55 km was adopted, wich is insufficient to eliminate errors in the geopotential model. However, these errors were cancelled by using levelled GPS points.
The gravity data consisted of 795 points selected in a data set supplied by the Bureau Gravimetrique International. Heights of the gravity points were checked with the digital terrain model, the tolerance was fixed to 40 m and about 1 % of the gravity data were rejected. Two zones were defined, see figure 4. In the inner zone that extends 15 km beyond the geodetic network, a minimum distance of 2 km between two gravity points was chosen. In the outer zone, as far as 55 km beyond the area of interest, the distance was fixed to 6 km. The points are well distributed in the west part of the area, but are on the roads and in the valleys in the east part, in the high mountainous area near Italy.
The digital terrain model was extracted from a national model described in (Duquenne, 1992). The grid spacing is 4.5" in latitude and 6" in longitude, about 140x140 m2. The DTM extends beyond the gravity data set as showed on figure 4, allowing to take into account the terrain effect on all gravity values.
The scheme of computation, inspired by (Sideris and Forsberg, 1990), (Balmino et al. 1992) and (Tscherning et aI, 1990) comprised the following steps:
Preparation of three DTM: the first one with fine grid spacing (4.5'i X6"), the second one with coarse grid spacing (45 I x60"), the third one filtered by moving-mean method
Checking of the heights and selection of the gravity data
Gravimetric reductions: atmospheric correction, computation of residual terrain effect, free-air reduction. The fine DTM was used in a cap of radius 25 km, and the coarse DTM between 25 and 55 km
Computation of modelled gravity from OSU91A
Computation and gridding of residual free-air anomaly
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Stokes integration in a cap of 0.55°
Computation ofthe modelled geoid from OSU91A
Computation of residual terrain effect on the geoid
Computation of geoidal undulation.
Figure 5 and 6 show the obtained geoid. The hollow in the west part fits the valley of the Durance River, which is 150-200 m lower than the remaining terrain.
Fig. 5. Gravimetric geoid in the area of Manos que (m)
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Fig. 6. Gravimetric geoid in the area of Manos que.
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43.92°
N (m)
50.6
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COMPARISON BETWEEN (QUASI-)GEOID REALIZATIONS
When (quasi-)geoids issued from GPS and levelling are compared with the gravimetric geoid, some problems appear. Most of them are now well described in the literature, see for instance (Forsberg and Madsen 1990).
Geodetic reference system The co-ordinates of the GPS points and the geoidal undulation derived from gravity do not refer to the same geodetic reference system. For the test network of Manosque, the first system is close to ITRF, as the network was linked to 2 control points the coordinates of which were expressed in ITRF. The second system is the one underlying to the global geopotential model. It is realized by a set of pursuit stations of which the coordinates are fixed. The geoidal undulation may be corrected by t1Nref ' the height of the
ellipsoid of the geopotential model above the other ellipsoid. A simplified but sufficient expression of t1Nref is:
t1Nref = AX" cOStpCOSA + ilY cos tp sin A + LlZ sin tp+ (ailf + fila) sin 2 tp- ila +ailk (3)
where the parameters for datum change have been introduced. If the two geodetic systems are well identified and the translation (AX", ilY, LlZ) and change in scale ilk well known, equation (3) may be applied. But the parameters are generally unknown and are to be computed using some levelled GPS points. It is often proposed to form observation equations with (1) or (2) and (3), and to compute the unknown parameters by a leastsquare adjustment. Although this solution is efficient, the computed parameters are not significant. On the one hand, some of them are correlated with others, on the other hand the undulation of the geoid computed from the geopotential model are contaminated by other errors, see below. For a small area as the test network of Manosque, this error appears as a constant bias and a tilt of the ellipsoids, with respect to each other.
Type of heights The ideal reference surface for orthometric heights is the geoid, while the one for normal heights is the quasi-geoid. If the height reference system uses normal heights, and geoidal undulations are derived from gravity, the difference in height type must be taken into account. Let LlH be the difference H* - H, G the constant of gravitation, g the gravity, r the normal gravity on the spheropotential surface, ilgB the Bouguer anomaly, p the mean density of the crust, J and Jo the mean local curvature of the ellipsoid and of the geoid. The value of LlH is (combining several formulae of physical geodesy, see (Heiskanen and Moritz 1967»:
LlH=N-r;= LlgB H+(J-Jo)H2 (4) r
where: ilgB = g- r- 27rpGH (5)
Here r must be computed with the same normal gravity formula as used to compute normal heights (1930 formula for France). It is well known that the Bouguer anomaly varies more slowly than the surface anomaly g - r and can be easily interpolated if dense gravity data are available, see fig. 7. The second term of the right member of equation (4) is more tricky to calculate, a geoid with very fine resolution is necessary to derive its
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43.73° '--_______ ~ ___ ~ ____ ~ ___ ___'
5.75° 6.15°
Fig. 7 Bouguer anomaly in the geodetic network area (mgal).
curvature. The term was evaluated for the test area and seems to be less than 10-6 m. On the contrary the first term is not negligible and the correction Ml varies from -6 to -1 cm. Hence it was taken into account.
Height reference system A height reference system is defined by a fundamental benchmark and a height type (orthometic or normal). The height of the fundamental benchmark is fixed from see level measurements, regardless of the gravimetric geoid and providing an unknown constant difference between the geoid and the height reference surface. The reference system is realized by a network affected by systematic errors. In France for instance, the systematic error of the IGN69 network appears to be a tilt of the reference surface of less than 0.45 m per 1000 krn, and oriented from south to north, see (Levallois and Maillard, 1971) and (Kasser, 1989).
E"ors in geopotential model Global potential models are affected by truncation errors (the coefficients of the expansions in spherical harmonics are known to a maximum degree, for instance 360) and errors in the coefficients. Using Stokes integral of gravity anomalies leads to minimize the short wavelength errors of the global model, but the long wavelength errors persist. For a small area as the test network of Manosque, this remaining error appears as a constant bias and a tilt of the computed geoid with respect to the true geoid.
E"ors in residual N These errors occur because of the errors in gravity data, the lack of data (gravity, DTM ... ), the limited cap of Stokes' integration. They have short wavelength components that cannot be eliminated but made smaller by enlarging inner zone.
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RESULTS FOR GRAVIMETRIC-AIDED GPS LEVELLING
The difference between the (quasi-)geoid computed from gravity data and the height reference surface realized by the reference levelling network can be minimized by fitting both surfaces using some levelled GPS points. For the small test region, this was made by linear regression. Decreasing number of levelled GPS points were used to compute the coefficients a, b, C of the regression, with:
N + H - h = a +b(rp- rpo) +c(A- ..1,0) cosrp (6) as observation equation. Then residuals of all points and their standard deviation were computed (as for the simple interpolation method). In a first series of computation, two points had high residuals (more than two times the standard deviation, i.e. 8 em) whatever the number of levelled GPS points used for the regression, see table 2. These points were considered as possible blunders (remember that the field-operators were students) and a second series of computations was achieved, see table 3.
The estimated precision is less than 3 em, close to the one of GPS co-ordinates, and does not vary with the number of levelled GPS points used for regression. In the conditions of the test, the precision seems to be limited by the one of the GPS.
Table 2. Points with high residual before rejection.
Number of levelled Points with hig~ residual GPS points
Table 3. Precision of heights predicted by gravity-aided GPS levelling after rejection.
Number of levelled Standard deviation of Points with high residual GPS points predicted heights (m)
30 0,028 m 7057 14 0,028 m 7057 10 0,028 m 5 0,028 m 7039 - 7057
FINAL REMARKS AND CONCLUSIONS
The tests demonstrate clearly the superiority of the gravimetric-aided method against the simple interpolation for levelling by GPS. The former method is more exact, in the sense that predicted heights are affected only by measurement errors, not by undulation of the geoid. These undulations are in fact obtained from the gravity. data (for the medium
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frequency part) and from the DTM (for the high frequency part). A detection of blunders is conceivable. The advantage of the method would be more obvious in a high mountainous area. For the authors, the goal is now to complete an operational quasi-geoid for whole France, controlled by the 1000 levelled GPS points of the future first order geodetic network. Nevertheless, some exciting problems persist, as listed below:
Efficiency of the terrain corrections and precision of GPS in mountainous areas
Decreasing of the precision in coastal zones, due to the sparse gravity data at sea
Data acquisition from foreign countries
Unification of the geodetic and heights references systems used
Blunder detection, especially when linking levelling benchmarks to GPS antennas
Acknowledgements We wish to thank Prof. c.c. Tscherning and Dr. R. Forsberg for kindly providing the GRA VSOFT package. Gravity data was supplied by the Bureau Gravimetrique International: we are grateful to Prof. G. Balmino and his team.
REFERENCES
Balmino, G., Balma, G., Sarrailh, M. and Toustou, D. (1992). Geoide gravimetrique franyais, etat d'avancement et programme de travail au GRGSIBGI. Toulouse, France.
Boucher, C. and Willis, P. (1988). Analyse d'un logiciel de combinaison de jeux de coordonnees tridimensionnelles. Internal documentation IT/G nO 58, IGN, SaintMande, France.
Duquenne, H. (1992). The new digital terrain model of France. Presented at the First Continental Workshop on the geoid in Europe. Prague, Czech Republic.
Heiskanen, W. A. and Moritz, H. (1967). Physical Geodesy. Reprint Institute of Physical Geodesy, Technical University, Graz, Austria, reprint 1981.
Forsberg, R. and Madsen, F. (1990). High-precision geoid heights for GPS levelling. In: Proceedings of the Second International Symposium on Precise Positioning with the Global Positioning System. Ottawa, Canada.
Kasser, M. (1989). Un nivellement de tres haute precision: la traversee Marseille Dunkerque. c.R. Acad. Sci. Paris, t. 309, serie II, p. 695-700.
Levallois, 1.-1. and Maillard, 1. (1971). Le nouveau reseau de nivellement de premier ordre du territoire franyais. Presente it la XVo Ass. Gen. de l'AIG, Moscou, URSS.
Sideris, M. G. and Forsberg, R. (1990). Review of Geoid Prediction Methods in Mountaneous Regions. In Determination of the Geoid, Present and Future, lAG Symp. n° 106, Milan, Italy.
Tscherning, C.c., Forsberg, R. and Knudsen, P. (1992). The GRAVSOFT Package for Geoid Determination. Presented at the First Continental Workshop on the geoid in Europe. Prague, Czech Republic.
