- Email: [email protected]
On the determination of a new Australian geoid
Phys. Chern. Enrrh (Aj, Vol. 24, No. 1, pp. 61-66, 1999 0 1999 Elsevier Science Ltd
Pergamon
All rights reserved 1464-1895/99/$-see front matter PII: S1464-1895(98)00011-8
On the Determination
of a New Australian Geoid
K. Zhang Institute of Engineering NG7 2RD, U.K. Received
Surveying
and Space Geodesy,
21 May 1998; revised 8 September
University
1998; accepted
Abstract. A new, high precision, high accuracy and high resolution gravimetric geoid of Australia has been produced using most updated data, theory and methodology. This paper presents a concise report of the new Australian geoid determination. Some aspects of the new geoid computation, such as data validation, geoid determination strategies and computational procedures, are described. The relative precision of the new geoid is better than 5 cm for average baseline length of 4km-40km and 18 cm for average baseline length of 120km when compared with three local GPS/levelling networks. 0 1999 Elsevier Science Ltd. All rights reserved.
of Nottingham,
15 September
University
Park,
1998
The first gravimetric geoid for Australia was computed by Mather (1969) using free-air gravity anomalies in Stokes’s integral. From 1970s to early 198Os, a number of interim Austrahan geoids were calculated by Grushinsky and Sazhina (1971), Fryer (1972), and Allman and Veenstra (1984) respectively. Although the overall precision of the geoid has been improved from time to time, the resolution and accuracy could not meet the needs of precise surveying and geodetic applications. Gilhland (1989) and Kearsley (1986) independently computed the Australian geoid based on geopotential coefficients and gravity observations. The methods used by Kearsley were used to produce subsequently AUSGEOID91, which was the first continent-wide gravimetric geoid routinely used by surveying authorities in Australia. These geoid undulation values were in terms of WGS84 and were computed on a 1O’xlO’grid using the OSU89A global geopotential coefficients (Rapp and Pavlis, 1990) and the 1980 Australian gravity data base from the Australian Geological Survey Organisation (AGSO) (Steed and Holtznagel, 1994).
1 Introduction The Global Positioning System (GPS) has found broad applications in surveying and geodesy as it can quickly, easily and accurately determine the ellipsoidal height of a point. Consequently, the combination of a geold with GPS measurements can provide a cheap alternative to geodetic levelling, and especially for determining elevations at difficult-to-access sites. However, if the geoid undulation is not known accurately, the elevation of the point in the local system can not be derived accurately. Therefore, the need for precise geoid determination methodologies is now more pressing than ever. A precise geoid will then allow the surveyor to use GPS to its “full capacity”.
A new continent-wide gravimetric geoid over Australia, AUSGEOID93, was produced by the Australian Surveying and Land Information Group (AUSLIG) in 1993 using the ring integration software developed by Kearsley (1986). The OSU91A geopotential model (Rapp er al. 1991) and the 1980 AGSO gravity data were used for AUSGEOID93 computation.
Since 1960s. there are a number of geodesists have contributed to the determination of the Australian geoid. The first Australian geoid was determined by Fischer and Slutsky (1967) from astro-geodetic deflections of the vertical at about 600 stations over Australia. This geoid was relative to the Australian National Spheroid, and used primarily to enable reductions of mean sea level distance to this spheroid.
AUSGEOID93 has been proven superior to a11 previous geoids in Western Australia by Featherstone and Alexander (1996). Although a number of advances were made with AUSGEOID93, the precision is not as high as it could be. For example, AUSGEOID93 does not include detailed terrain information, which is critical for the short wavelength components of the geoid. In addition to the
Correspondence to: K. Zhang 61
62
K. Zhang: On the Determination of a New Australian Geoid
release of more gravity observations by AGSO and the spot height database by AUSLIG, recent development of satellite techniques has provided us with refined information about the earth’s gravity field, such as new global geopotential models and satellite altimetry (SA) measurements over the oceans. Currently, for high accuracy scientific and engineering applications, the precision of AUSGEOID93 is not sufficient, especially in mountainous regions (Featherstone ef al. 1997). This paper presents a new gravimetric geoid of Australia, its determination, data used, strategies and methodologies. The methodologies developed in this paper are also pertinent to other part of the world for a gravimetric geoid determination over a large area.
(6)
2 Methodology Compared to other nations, Australia has a relatively large area and is geologically complex. Australian continent is one of the oldest continents and is abundance of mineral deposit. In addition, both the distribution and accuracy of the gravity and terrain data are heterogeneous @hang, 1997). The marine gravity observations along coast are less accurate, whereas the coastal areas are densely inhabited where demand a high precision geoid. This section of the paper will outline the methodologies and data used, evaluation of the SA derived gravity information and their wmbination with marine gravity and determination of an optimal reference field.
where R and a are the mean radius of the spherical Earth and the semi-major axis of the reference ellipsoid respectively; Ag is the freeair gravity anomaly on the geoid, da is the surface integration element; GM and p are the geocentric gravitational constant and the constant density of the Earth’s mass respectively; S(ul) is the Stokes function; r, 0 and il are the spherical polar coordinates of the computation point; F, (co&) are the fully normalised associated Legendre functions; sFm and S,, are the fully normalised spherical harmonic ccefficients of disturbing potential and Histhe height. One dimensional spherical FFT technique (Haagmans, et
2.1 Computational procedures The precise determination of local geoid undulation by gravimetric means has been a dominant research topic of geodesy in the last ten years. Significant developments have taken place in both the theoretical aspects and the practical improvement of the geoid’s precision. Combining a geopotential model with local gravity data and digital terrain elevation data in geoid computation has become a wmmonly used methodology (e.g. For&erg, 1993; Sideris et al. 1992). The Earth geopotential model is generally of long and medium wavelength in nature. The contribution of local gravity data is generally of medium and short wavelength nature. The local topographic effect is relatively small, typically of centimetre to decimetre level and is of short wavelength in nature. The Stokes integral, by applying the so-called removerestore procedure, can be expressed as: N = NY+NGGM+SNm~ (1) with,
al. 1993) is applied to both the computation of the medium
wmponents of the geoid (Ndg) and terrain reduction (&Vh, and AgDm ) since FFT is more efficient for data handling, particularly for a large area. Simulation tests using OSU91A gravity field model as a “ground-truth” data (Tziavos, 1996) indicate that cm-level geoid can be achieved provided the windowing and edge effects, zero padding and removal of the bias in the residual gravity anomalies are properly considered @rang, 1997). In addition, terrain effects computed by Zhang and Featherstone (1997) are also incorporated into this computation (i.e. eq. 5 and eq. 7). 2.2 The data sets and their validation Geoid computation can be considered as a filter with a multiple input and single output. Therefore, to accurately determine the gravimetric geoid of Australia, the data availability, their validation, refinement and preparation are crucial. Since the quality of the data will directly affect the quality of subsequent geoid determination, any physically and geometrically systematic and gross errors should be removed. This procedure gives a homogeneous data source for subsequent geoid determination.
K. Zhang:
On the Determination
Several data sets are used for the new geoid determination of Australia, gravity observations both in land and ocean areas, SA derived gravity anomalies, spot heights, global geopotential model and geometric geoid from GPS/levelling networks. Validation and evaluation of these data sets are described hereafter. 2.2.1 Gravity data The gravity data set used in this research is the AGSO 1992 gravity database (526,091 land and 111,396 marine gravity observations). The spatial resolution of these data is approximately one observation per 7 km in Tasmania and South Australia and one observation per 11 km elsewhere (Gilliland, 1987). However, the resolution is uneven and increases dramatically where detailed gravity surveys have been conducted in areas of geophysical or geological interest. The approaches used to detect erroneous data were based on the strategies described by Featherstone et al. (1997). The marine and land gravity data distribution is shown in Figure 1. The estimated precision of the observed gravity with respect to the Australian gravity network is +0.2-0.5 mGa1 and the precision of the station elevations is better than f 6-10 m (Mather et al. 1976; Dooley and Barlow 1976). This implies that the accuracy of the free-air anomalies is better than +_2-3 mGa1. The accuracy is slightly worse for marine gravity measurements, as these were inevitably made on an accelerating platform. Mather et al. (1976) estimate that the accuracy is f 2-6 mGal for the AGSO marine gravity measurements and their resolution is about 15-30km per observation on average.
of a New Australian
Geoid
63
2.2.2 Geometric geoid from GPSLevelling Independent geometric geoids from several GPS and spirit levelling networks are used in this study to evaluate the accuracy of the new geoid. These GPS/levelling networks are the Australian National and Fiducial Networks (ANNIAFN) (59 stations, -500km per point on average) and three local networks in Australian Capital Territory (86 points, -3km per point), Victoria (18 points, -120km per point) and Western Australia (21 points, -40km per point). They are also used as part of the evaluation process to determine a best-fitted geopotentail model for Australia. The precision of these networks has been examined through analysing their various error sources including GPS related errors (e.g. ionosphere, troposphere, satellite orbit and multipath), tide correction (e.g. ocean and atmosphere loadings) and Australian height datum errors as demonstrated by Zhang (1997). 2.2.3 Topographic data The Australian landscape is strikingly distinctive, with a variety of landforms ranging from extensive plains and plateaus to great expanses of desert. Australia is the lowest, flattest and smallest continent in the world. The highest mountain in Australia is 2228 m (Mt. Kosciusko) and lowest elevation on land is -15 m (Lake Eyre). There are a total of 5,726,698 records of spot heights to form a digital terrain model @TM) of Australia. On average, the density of the spot height is approximately one point per 1.3 km’ which corresponds to a resolution of 1.lkm (full wavelength). The terrain height data have been
20'
25'
a0
,a5
Fig.1 LMdandmMinegrWitydIltallsed
in the gravimeuic gwid determination of AustraIia (linear projcclion).
64
K. Zhang: On the Determination of a New Australian
formatted and transformed to WGS84 where necessary. Any erroneous records have been corrected or removed. In order to minimise the aliasing effect in the terrain reduction, a grid size of l’xl’, which is about 1.8 km resolution, was chosen to produce the digital terrain model. The minimum curvature spline in tension method (Wessel and Smith, 1995) was used to produce the l’xl’ DTM. This DTM is then used to compute terrain corrections and terrain indirect effect in the remove-restore technique (eq 1 and eq. 3).
2.2.4 Global geopotential model (GGM) The selection and verification of an optimal geopotential model is also studied to give the best local reference field through comparisons with the geometric geoid and the land, marine and SA derived gravity anomalies. More than ten sets of high degree and order GGMs from the USA, Germany, Canada and the International Geoid Service (IGeS), Italy are tested. It is demonstrated that both OSU91A (Rapp et al. 1991) and EGM96 (Lemoine et al. 1997) fit the Australian gravity field very well. However, OSU91A is slightly better fitted with land gravity (Zhang, 1997). Because the geoid undulation in land is of most importance and the land gravity measurements are more accurate than marine and SA gravity anomalies, OSU91A is used as the best fitted GGM for Australia. 2.3 Satellite altimetry derived gravity anomalies and their assessment The sparse coverage of marine gravity measurements offshore Australia restricts the gravimetric geoid determination near the coast @hang, 1998). The coast of Australia, however, is densely populated and highly developed and thus demands the most reliable geoid solution. Therefore, recent satellite altimetry results may be a supplementary data source to improve data coverage and quality, and thus improve the geoid near the coast. A wide data coverage around the coast is also very useful to remove the boundary effects in the geoid determination by the fast Fourier transform @hang, 1997). Sandwell et al. (1995) and Sandwell and Smith (1997) have calculated global gravity anomalies through a combination of satellite altimeter profiles from multiple missions, which include data from GEOSAT, ERS-1, SEASAT and TOPEX/Poseidon (frp sire:baltica.ucsd.edu). The accuracy and resolution of the derived field is approaching the best fields derived from declassified GEOSAT Geodetic Mission data south of 30’S (Sandwell ef al., 1995). In order to avoid a multi-satellite crossover adjustment, sea surface topography profiles were first differentiated, then combined to produce grids of east and north vertical deflection. These were, in turn, converted to
Geoid
gravity anomalies using a planar fast Fourier transform algorithm (Sandwell, 1992). This approach has been shown to be superior to the use of an inverse Stokes’s formula by Olgiati et al. (1995). To study the possibility of combining SA derived gravity anomalies with local gravity anomalies, the resolution and accuracy of the SA gravity anomalies must be evaluated. The accuracy of SA gravity anomalies is, in the most accurate cases, 3-6 mGa1 (Nerem et al. 1995; hbelos and Ttiavos, 1994). However, the accuracy of the Australian marine gravity observations is 2-6 mGal (Gilliland, 1987) and their resolution is about 15-30km per observation on average. Another consideration is the actual resolution of the SA observations around Australia, which is approximately 20-30 km (Sandwell and Smith, 1997) although the nominal grid size is 2’. For these reasons, a block mean comparison method is chosen instead of point to point comparison. The comparison is made between the SA gravity anomalies around Australia and marine gravity anomalies for different grid sizes, namely 6/x6’, 1O’xlO’and 3O’x30’. The block mean values for each grid are determined by using a moving weighted average strategy (Zhang, 1998). In addition, the SA results are expected to be unreliable near coastal areas because of loss of altimeter lock. Therefore, the comparison is performed about 20 km away Comparisons indicate that the from the coastline. corresponding standard deviation errors of the SA derived mean gravity anomalies are 5.89 mGal for 6’x6’, 6.34 mGa1 for lO’x10’ and 4.20 mGal for 3O’x30’respectively. The strategies used to combine the SA gravity anomalies with marine gravity anomalies are: . in the areas where both gravity anomalies exist, the weight for SA gravity anomalies was determined by their accuracy and the number of observations; . in the areas w&e marine gravity anomalies do not exist, SA gravity anomalies were used; and . in Papua New Guinea Strait area, SA gravity data were not used. In order to test the improvement of the new geoid by using SA data, the new geoids both with and without inclusion of SA data are compared at the 35 AFN/ANN coastal stations (see Table 1). Table 1 Grid dXem~ces between the new gaid (N-J and AFN/ANN geoid (NO=)at 35 coastal stationsboth with and without the SA data (units ill-)
Geoid differences N,, -NGPS N,, with SA)-NGPS
Max 0.85 0.70
Min -1.37 -1.22
Mea
RM
STD
-0.15 -0.06
0.50 0.41
0.48 0.40
From Table 1, it is concluded that SA gravity anomalies can not only improve the coverage and resolution of the
K. Zhans: On the Determination
of a New Australian Geoid
-10”
m 90 80 70 60 50
-20’
40 30 -25”
20 12
-30”
4 -4 -12
-35”
-20 -30
-40’
-40 -50 -45”
gravity
field, it can also improve the quality of the gravity
field and thus geoid. The SA data give an 8 cm improvement over the new geoid. 3 Evaluation
of the new geoid
From Table 2, it is concluded that the relative precision of the new geoid is higher than 5cm for mean baseline length of 4-40km, 18 cm for mean baseline length of 120km. The absolute accuracy of the new geoid is higher than 33cm. This precision is comparable with the precision specrfication
A gravtmetnc geoid of Australia using the techniques and data described above is computed and show11 on Figure 2 using GMT software package (Wessel and Smith, 1995). Comparisons of the new geoid with GPYlevelling &cold are listed in Table 2. In addition to the national-wide GPSllevelling network (AFN/ANN), three regronal or local GPYlevelling networks in Australian Capita1 Territory (ACT), Victoria (Vie) and Western Australia (WA) are used for thus comparison. Note that the errors rn the GPS/levelling results are not taken into account in these compartsons.
-60
of third
order
levelling
(12,/a
) of
Australia.
4 Summaries This research produces the most accurate gravimetrrc geoid which is up-to-date for Australia. The 1992 release of the Australian gravity database is used to include the most-upto date land and marine gravity. The optimal geopotential model, OSU91A for the test areas, has been adopted to give the best reference gravity field. The refined gravity reduction procedures are employed for the new geoid. The terrain effects are evaluated using a detailed DTM and combined into the computation of the new geoid. Sandwell’s satellite altimetry-derived gravity anomalies have been assessed and combined with the Australian gravity database to improve the quantity and coverage of the gravity field offshore Australia. It is demonstrated that a proper inclusion of SA gravity data can give -1Ocm root mean squared improvement of the gravimetric geoid.
66
K. Zhang: On the Determination of a New Australian
The new geoid may provide an alternative and costeffective method to geodetic levelling and for many scientific applications, such as geodetic positioning, geodynamics and geophysical exploration. Relative precision of the new geoid is cm-level, with absolute precision higher than existing geoid and resolution of a few kilometres. Satellite altimetry data collection, analysis, assessment and implementation of marine geoid together with the sea surface topography information and marine and continental gravity observations are important for further refinement of the Australian geoid. Satellite altimetryderived gravity anomalies need properly assessed in terms of error features and accuracy so as to be optimally combined with local gravity data. This is of importance for FIT computation and thus is expected to further improve the accuracy and precision of the geoid along the coastline of Australia. This is particularly useful for Australia, since the coastline is densely populated relative to the interior. Finally, the methodologies developed in this paper are pertinent to other part of the world for a gravimetric geoid determination over a large area, particularly for maritime countries. Acbrowledgement~. The Authorwould
Iike to express his gratitude to Curtin University of Technology for awarding him with two prestigious researchscholarships,namely Ovaseas PostgraduateResearch Scholarship and Curtin UniversityPostgraduateResearch Scholarshipand to Associate professorWE Fcatherso(ne,School of Spatial Sciences. Cutin Universityof Technology for his assistancefor pat of this research. Comments from ProfessorAH D&on, The Universityof N&tin&am and from ProfessorLN. Tziavos. AristotleUniversityof Tbessalonikiarehighly appreciated.
References Allman, IS and Veenstra, C (1984) Geodetic Model of Australia 1982, Technical Report 33, Division of NationalMapping, Canberra.60 pp. Arab&s, D. and Tziavos. I.N. (1994) Gravity and G&d in the Meditermneanfrom a Common Adjwtnwnt of ERS-I and TOPEX Altimeter Data, Proceedings of Intemationol Association of Geodesy Symposium No. 113, Graz. Au&a, Sept., pp.316-385. Do&y. JC and Barlow, BC (1976) Gmvimetry in Australia. 1819-1976, BLURJournal ofAustralian Geology and Geophysics, Vol. 1, No. 4. pp. 261-271. Featherstone,WE and Alexander, K (1996) An Analysis of GPS Height Detamination in W&em Australia, The Australian Surveyor, Vol. 41, No. 1. pp. 29-34. Feathastone~WE, Kearsley,AHW and GiIIiland.JR (1997) Data Preparations for a New AustralianGravimetricG&d, The Australian Surveyor,Vol. 41, No. 2. Piscber, I and Slutsky, M (1967) A PreliminaryGenid chart of Australia,77~ Australian Surveyor, Vol. 21, No. 8, pp. 327-332. Fombug. R (1993) Model& the Fmestmcture of the G&d: Methods,Data Reqiremen~ and SomeResults. Surveys in Geophysics, Vol. 14, pp.403418. Fq%%JO (1972) The.AustralianG&d, nlreAustralian Surveyor. Vol. 24, No. 4. pp. 203-214. GiIliIand.JR (1987) An AustralianGravity Anomaly Data Bank for Geoid Detaminations. The Aurtrolian Surveyor. Vol. 33, No. 7, pp. 57% 581.
Geoid
Giiand, JR (1989) A Gravim&ic Geoid of AustmIia, The Ausfralinn Surveyor, Vol. 34, No. 7, pp. 699-706. Gnwhimky, NP and Sazbina. NB (1971) The GravitationaIReid and the G&d of AustmIia.Journal ofthe Geological Society of Awtralia.Vol. 18, No. 3, pp. 183-199. Haagmans.RRN. de Min. E and van Gcldapn M (1993) Fast Evaluationof Gmvolotion In+als on the Sphere Using ID FFT. and a Comparison with Existing Me&& for Stokes’s Integral. Manurcripta Geodaetica. Vol. 18, pp. 227-241. Keamley.AHW (1986) The Deiumination of PreciseGeoid Iieigbt Di&wtces UsingRing Integration,Bolletino di Geodesia e Scienze .Qini, Vol. 2, pp. 151-174. Lemoine, FG, Smith, DE, Kunz, L. Smith, R, Pavlis, EC, Pa&., NK, Klosko, SM. Cbinn. DS, Tonace MH. Williamson, RG, Cox. CM, Rachlin. KE, Wang, YM, Kenyon, SC, Salman, R, Trimmer, R bPP* RH and Nerem. RS (1997) The developmentof the NASA GSFC and NIMA joint geqotential model, Gravify, Geoid and Marine Geodesy. Intemational Association of Geodesy Symposium, Vo1.117, Berlin, Springer-V&g. pp.461-469. Mat&, RS (1969) The Freeair Geoid of Australia, Geophysical Journal of the Royal Astronomical Society, Vol. 18, pp. 499-516. Mather, RS, Rizos, C and Hirsch, B (1976) An AustralianGravity Data Bank for Sea Surface Topogrqhy Detaminations (AUSGAD 76). Unisurv Report No. G25, Universityof New South Wales, Sydney, pp. 54-84. Nerem, RS, Jekeli, C and Kaula, NM (1995) Gravity Reid Delamination and Charactaitics: Retmspaive and Prospective, Journal of Geophysical Research. Vol. 100, No. B8, pp. 15053-15074. Olgiati, A. Balmino. G., Saw&, M. and Green. CM. (1995) Gravity Anomalies from Satellite Altimcry: Comparison Between Computationvia G&d Heights and via Deflections of the Vatical, Bulletin Gt%d&ique. Vol. 69, pp. 252-260. Rapp. RH and P&is, NK (1990). The Dev&pmatt and Analysis of Gqotauial Coef?icimt Models to Spherical HarmonicDegree 360: OSU89A and OSU89B, Journal of Geophyrical Research, Vol. 95. No. B13, pp. 21855-21911. Rspp, RH, Wang, YM and PavIis, NK (1991) The Ohio State 1991 Gqotential and Sea Surface Topography Harmonic CoefticientModels, Report No. 410, Departmentof Geodetic Science and Surveying,The Ohio StateUniversity,Columbus,Dec., 94 pp. Sandwell. D.T. (1992) Antarctic Marine Gravity Field from Highdaxity SatelliteAltimetry.Geophysical Journal International, No. 109, pp. 437448. Sandwell, D.T.. Yale, M. and Smith, W.H.F. (1995) Gravity Anomaly Profdes from ERS-1. TOPEX and GEOSAT Altimetry, EOS Supplement; Transactions ofthe American Geophysical Union, Vol. 76, No. 17, p. 89. Sandwell, DT and Smith, WHF (1997) Marine Gravity Anomaly from GEOSAT and ERS-1 Satellite Altimetry, Journal of Geophysical Research, Vol. 102, No. B5. pp. 10039-10054. Side& MG. Maim&z. A and Fonbq, R (1992) G&d TestingUsing GPS and LeveIIing,Australian Journal of Geodesy, Photogmmmetry and Surveying, No. 57, pp.62-77. Steed. J and Holtuugel, S (1994) AHD Heights from GPS Using AUSGEOID93, The Austmlinn Suweyor, Vol. 39, No. 1, pp. 21-27. Tziavos. IN (1996) Comparisons of Spectral Techniques for G&d ComputationsOver Large Regions, Journal of Geodesy. Vol. 70, pp. 357-373. Wessel, P and Smith, WHF (1995) New Version of the Generic Mapping Tools Released, EOS; Transactions ofthe Amsrican Geophysical Union. Vol. 76. p. 329. Zhang, KF (1997) An evaluationof FFT geoid de&mination techniquesand their application to height determinationusing GPS in Australia.PhD thesis, School of Spatial Sciences, Curtin University of Technology, 261~. Zhang. KP (1998) Altime& gravity anomalies, their assessmat and combinationwith local gravity field, Finnish Geodetic Institute Report (h pm). Zhang KP and Featbaau~e WE (1997) A preliminaryevaluation of the terraineffects on gmvim&ic g&d determinationin Australia,Gravity. Geoid and Marine Geodesy, International Association of Geodesy Symposium.Vol.1 17, Berlin. Springer-V&g. pp.565-572.
Pergamon
All rights reserved 1464-1895/99/$-see front matter PII: S1464-1895(98)00011-8
On the Determination
of a New Australian Geoid
K. Zhang Institute of Engineering NG7 2RD, U.K. Received
Surveying
and Space Geodesy,
21 May 1998; revised 8 September
University
1998; accepted
Abstract. A new, high precision, high accuracy and high resolution gravimetric geoid of Australia has been produced using most updated data, theory and methodology. This paper presents a concise report of the new Australian geoid determination. Some aspects of the new geoid computation, such as data validation, geoid determination strategies and computational procedures, are described. The relative precision of the new geoid is better than 5 cm for average baseline length of 4km-40km and 18 cm for average baseline length of 120km when compared with three local GPS/levelling networks. 0 1999 Elsevier Science Ltd. All rights reserved.
of Nottingham,
15 September
University
Park,
1998
The first gravimetric geoid for Australia was computed by Mather (1969) using free-air gravity anomalies in Stokes’s integral. From 1970s to early 198Os, a number of interim Austrahan geoids were calculated by Grushinsky and Sazhina (1971), Fryer (1972), and Allman and Veenstra (1984) respectively. Although the overall precision of the geoid has been improved from time to time, the resolution and accuracy could not meet the needs of precise surveying and geodetic applications. Gilhland (1989) and Kearsley (1986) independently computed the Australian geoid based on geopotential coefficients and gravity observations. The methods used by Kearsley were used to produce subsequently AUSGEOID91, which was the first continent-wide gravimetric geoid routinely used by surveying authorities in Australia. These geoid undulation values were in terms of WGS84 and were computed on a 1O’xlO’grid using the OSU89A global geopotential coefficients (Rapp and Pavlis, 1990) and the 1980 Australian gravity data base from the Australian Geological Survey Organisation (AGSO) (Steed and Holtznagel, 1994).
1 Introduction The Global Positioning System (GPS) has found broad applications in surveying and geodesy as it can quickly, easily and accurately determine the ellipsoidal height of a point. Consequently, the combination of a geold with GPS measurements can provide a cheap alternative to geodetic levelling, and especially for determining elevations at difficult-to-access sites. However, if the geoid undulation is not known accurately, the elevation of the point in the local system can not be derived accurately. Therefore, the need for precise geoid determination methodologies is now more pressing than ever. A precise geoid will then allow the surveyor to use GPS to its “full capacity”.
A new continent-wide gravimetric geoid over Australia, AUSGEOID93, was produced by the Australian Surveying and Land Information Group (AUSLIG) in 1993 using the ring integration software developed by Kearsley (1986). The OSU91A geopotential model (Rapp er al. 1991) and the 1980 AGSO gravity data were used for AUSGEOID93 computation.
Since 1960s. there are a number of geodesists have contributed to the determination of the Australian geoid. The first Australian geoid was determined by Fischer and Slutsky (1967) from astro-geodetic deflections of the vertical at about 600 stations over Australia. This geoid was relative to the Australian National Spheroid, and used primarily to enable reductions of mean sea level distance to this spheroid.
AUSGEOID93 has been proven superior to a11 previous geoids in Western Australia by Featherstone and Alexander (1996). Although a number of advances were made with AUSGEOID93, the precision is not as high as it could be. For example, AUSGEOID93 does not include detailed terrain information, which is critical for the short wavelength components of the geoid. In addition to the
Correspondence to: K. Zhang 61
62
K. Zhang: On the Determination of a New Australian Geoid
release of more gravity observations by AGSO and the spot height database by AUSLIG, recent development of satellite techniques has provided us with refined information about the earth’s gravity field, such as new global geopotential models and satellite altimetry (SA) measurements over the oceans. Currently, for high accuracy scientific and engineering applications, the precision of AUSGEOID93 is not sufficient, especially in mountainous regions (Featherstone ef al. 1997). This paper presents a new gravimetric geoid of Australia, its determination, data used, strategies and methodologies. The methodologies developed in this paper are also pertinent to other part of the world for a gravimetric geoid determination over a large area.
(6)
2 Methodology Compared to other nations, Australia has a relatively large area and is geologically complex. Australian continent is one of the oldest continents and is abundance of mineral deposit. In addition, both the distribution and accuracy of the gravity and terrain data are heterogeneous @hang, 1997). The marine gravity observations along coast are less accurate, whereas the coastal areas are densely inhabited where demand a high precision geoid. This section of the paper will outline the methodologies and data used, evaluation of the SA derived gravity information and their wmbination with marine gravity and determination of an optimal reference field.
where R and a are the mean radius of the spherical Earth and the semi-major axis of the reference ellipsoid respectively; Ag is the freeair gravity anomaly on the geoid, da is the surface integration element; GM and p are the geocentric gravitational constant and the constant density of the Earth’s mass respectively; S(ul) is the Stokes function; r, 0 and il are the spherical polar coordinates of the computation point; F, (co&) are the fully normalised associated Legendre functions; sFm and S,, are the fully normalised spherical harmonic ccefficients of disturbing potential and Histhe height. One dimensional spherical FFT technique (Haagmans, et
2.1 Computational procedures The precise determination of local geoid undulation by gravimetric means has been a dominant research topic of geodesy in the last ten years. Significant developments have taken place in both the theoretical aspects and the practical improvement of the geoid’s precision. Combining a geopotential model with local gravity data and digital terrain elevation data in geoid computation has become a wmmonly used methodology (e.g. For&erg, 1993; Sideris et al. 1992). The Earth geopotential model is generally of long and medium wavelength in nature. The contribution of local gravity data is generally of medium and short wavelength nature. The local topographic effect is relatively small, typically of centimetre to decimetre level and is of short wavelength in nature. The Stokes integral, by applying the so-called removerestore procedure, can be expressed as: N = NY+NGGM+SNm~ (1) with,
al. 1993) is applied to both the computation of the medium
wmponents of the geoid (Ndg) and terrain reduction (&Vh, and AgDm ) since FFT is more efficient for data handling, particularly for a large area. Simulation tests using OSU91A gravity field model as a “ground-truth” data (Tziavos, 1996) indicate that cm-level geoid can be achieved provided the windowing and edge effects, zero padding and removal of the bias in the residual gravity anomalies are properly considered @rang, 1997). In addition, terrain effects computed by Zhang and Featherstone (1997) are also incorporated into this computation (i.e. eq. 5 and eq. 7). 2.2 The data sets and their validation Geoid computation can be considered as a filter with a multiple input and single output. Therefore, to accurately determine the gravimetric geoid of Australia, the data availability, their validation, refinement and preparation are crucial. Since the quality of the data will directly affect the quality of subsequent geoid determination, any physically and geometrically systematic and gross errors should be removed. This procedure gives a homogeneous data source for subsequent geoid determination.
K. Zhang:
On the Determination
Several data sets are used for the new geoid determination of Australia, gravity observations both in land and ocean areas, SA derived gravity anomalies, spot heights, global geopotential model and geometric geoid from GPS/levelling networks. Validation and evaluation of these data sets are described hereafter. 2.2.1 Gravity data The gravity data set used in this research is the AGSO 1992 gravity database (526,091 land and 111,396 marine gravity observations). The spatial resolution of these data is approximately one observation per 7 km in Tasmania and South Australia and one observation per 11 km elsewhere (Gilliland, 1987). However, the resolution is uneven and increases dramatically where detailed gravity surveys have been conducted in areas of geophysical or geological interest. The approaches used to detect erroneous data were based on the strategies described by Featherstone et al. (1997). The marine and land gravity data distribution is shown in Figure 1. The estimated precision of the observed gravity with respect to the Australian gravity network is +0.2-0.5 mGa1 and the precision of the station elevations is better than f 6-10 m (Mather et al. 1976; Dooley and Barlow 1976). This implies that the accuracy of the free-air anomalies is better than +_2-3 mGa1. The accuracy is slightly worse for marine gravity measurements, as these were inevitably made on an accelerating platform. Mather et al. (1976) estimate that the accuracy is f 2-6 mGal for the AGSO marine gravity measurements and their resolution is about 15-30km per observation on average.
of a New Australian
Geoid
63
2.2.2 Geometric geoid from GPSLevelling Independent geometric geoids from several GPS and spirit levelling networks are used in this study to evaluate the accuracy of the new geoid. These GPS/levelling networks are the Australian National and Fiducial Networks (ANNIAFN) (59 stations, -500km per point on average) and three local networks in Australian Capital Territory (86 points, -3km per point), Victoria (18 points, -120km per point) and Western Australia (21 points, -40km per point). They are also used as part of the evaluation process to determine a best-fitted geopotentail model for Australia. The precision of these networks has been examined through analysing their various error sources including GPS related errors (e.g. ionosphere, troposphere, satellite orbit and multipath), tide correction (e.g. ocean and atmosphere loadings) and Australian height datum errors as demonstrated by Zhang (1997). 2.2.3 Topographic data The Australian landscape is strikingly distinctive, with a variety of landforms ranging from extensive plains and plateaus to great expanses of desert. Australia is the lowest, flattest and smallest continent in the world. The highest mountain in Australia is 2228 m (Mt. Kosciusko) and lowest elevation on land is -15 m (Lake Eyre). There are a total of 5,726,698 records of spot heights to form a digital terrain model @TM) of Australia. On average, the density of the spot height is approximately one point per 1.3 km’ which corresponds to a resolution of 1.lkm (full wavelength). The terrain height data have been
20'
25'
a0
,a5
Fig.1 LMdandmMinegrWitydIltallsed
in the gravimeuic gwid determination of AustraIia (linear projcclion).
64
K. Zhang: On the Determination of a New Australian
formatted and transformed to WGS84 where necessary. Any erroneous records have been corrected or removed. In order to minimise the aliasing effect in the terrain reduction, a grid size of l’xl’, which is about 1.8 km resolution, was chosen to produce the digital terrain model. The minimum curvature spline in tension method (Wessel and Smith, 1995) was used to produce the l’xl’ DTM. This DTM is then used to compute terrain corrections and terrain indirect effect in the remove-restore technique (eq 1 and eq. 3).
2.2.4 Global geopotential model (GGM) The selection and verification of an optimal geopotential model is also studied to give the best local reference field through comparisons with the geometric geoid and the land, marine and SA derived gravity anomalies. More than ten sets of high degree and order GGMs from the USA, Germany, Canada and the International Geoid Service (IGeS), Italy are tested. It is demonstrated that both OSU91A (Rapp et al. 1991) and EGM96 (Lemoine et al. 1997) fit the Australian gravity field very well. However, OSU91A is slightly better fitted with land gravity (Zhang, 1997). Because the geoid undulation in land is of most importance and the land gravity measurements are more accurate than marine and SA gravity anomalies, OSU91A is used as the best fitted GGM for Australia. 2.3 Satellite altimetry derived gravity anomalies and their assessment The sparse coverage of marine gravity measurements offshore Australia restricts the gravimetric geoid determination near the coast @hang, 1998). The coast of Australia, however, is densely populated and highly developed and thus demands the most reliable geoid solution. Therefore, recent satellite altimetry results may be a supplementary data source to improve data coverage and quality, and thus improve the geoid near the coast. A wide data coverage around the coast is also very useful to remove the boundary effects in the geoid determination by the fast Fourier transform @hang, 1997). Sandwell et al. (1995) and Sandwell and Smith (1997) have calculated global gravity anomalies through a combination of satellite altimeter profiles from multiple missions, which include data from GEOSAT, ERS-1, SEASAT and TOPEX/Poseidon (frp sire:baltica.ucsd.edu). The accuracy and resolution of the derived field is approaching the best fields derived from declassified GEOSAT Geodetic Mission data south of 30’S (Sandwell ef al., 1995). In order to avoid a multi-satellite crossover adjustment, sea surface topography profiles were first differentiated, then combined to produce grids of east and north vertical deflection. These were, in turn, converted to
Geoid
gravity anomalies using a planar fast Fourier transform algorithm (Sandwell, 1992). This approach has been shown to be superior to the use of an inverse Stokes’s formula by Olgiati et al. (1995). To study the possibility of combining SA derived gravity anomalies with local gravity anomalies, the resolution and accuracy of the SA gravity anomalies must be evaluated. The accuracy of SA gravity anomalies is, in the most accurate cases, 3-6 mGa1 (Nerem et al. 1995; hbelos and Ttiavos, 1994). However, the accuracy of the Australian marine gravity observations is 2-6 mGal (Gilliland, 1987) and their resolution is about 15-30km per observation on average. Another consideration is the actual resolution of the SA observations around Australia, which is approximately 20-30 km (Sandwell and Smith, 1997) although the nominal grid size is 2’. For these reasons, a block mean comparison method is chosen instead of point to point comparison. The comparison is made between the SA gravity anomalies around Australia and marine gravity anomalies for different grid sizes, namely 6/x6’, 1O’xlO’and 3O’x30’. The block mean values for each grid are determined by using a moving weighted average strategy (Zhang, 1998). In addition, the SA results are expected to be unreliable near coastal areas because of loss of altimeter lock. Therefore, the comparison is performed about 20 km away Comparisons indicate that the from the coastline. corresponding standard deviation errors of the SA derived mean gravity anomalies are 5.89 mGal for 6’x6’, 6.34 mGa1 for lO’x10’ and 4.20 mGal for 3O’x30’respectively. The strategies used to combine the SA gravity anomalies with marine gravity anomalies are: . in the areas where both gravity anomalies exist, the weight for SA gravity anomalies was determined by their accuracy and the number of observations; . in the areas w&e marine gravity anomalies do not exist, SA gravity anomalies were used; and . in Papua New Guinea Strait area, SA gravity data were not used. In order to test the improvement of the new geoid by using SA data, the new geoids both with and without inclusion of SA data are compared at the 35 AFN/ANN coastal stations (see Table 1). Table 1 Grid dXem~ces between the new gaid (N-J and AFN/ANN geoid (NO=)at 35 coastal stationsboth with and without the SA data (units ill-)
Geoid differences N,, -NGPS N,, with SA)-NGPS
Max 0.85 0.70
Min -1.37 -1.22
Mea
RM
STD
-0.15 -0.06
0.50 0.41
0.48 0.40
From Table 1, it is concluded that SA gravity anomalies can not only improve the coverage and resolution of the
K. Zhans: On the Determination
of a New Australian Geoid
-10”
m 90 80 70 60 50
-20’
40 30 -25”
20 12
-30”
4 -4 -12
-35”
-20 -30
-40’
-40 -50 -45”
gravity
field, it can also improve the quality of the gravity
field and thus geoid. The SA data give an 8 cm improvement over the new geoid. 3 Evaluation
of the new geoid
From Table 2, it is concluded that the relative precision of the new geoid is higher than 5cm for mean baseline length of 4-40km, 18 cm for mean baseline length of 120km. The absolute accuracy of the new geoid is higher than 33cm. This precision is comparable with the precision specrfication
A gravtmetnc geoid of Australia using the techniques and data described above is computed and show11 on Figure 2 using GMT software package (Wessel and Smith, 1995). Comparisons of the new geoid with GPYlevelling &cold are listed in Table 2. In addition to the national-wide GPSllevelling network (AFN/ANN), three regronal or local GPYlevelling networks in Australian Capita1 Territory (ACT), Victoria (Vie) and Western Australia (WA) are used for thus comparison. Note that the errors rn the GPS/levelling results are not taken into account in these compartsons.
-60
of third
order
levelling
(12,/a
) of
Australia.
4 Summaries This research produces the most accurate gravimetrrc geoid which is up-to-date for Australia. The 1992 release of the Australian gravity database is used to include the most-upto date land and marine gravity. The optimal geopotential model, OSU91A for the test areas, has been adopted to give the best reference gravity field. The refined gravity reduction procedures are employed for the new geoid. The terrain effects are evaluated using a detailed DTM and combined into the computation of the new geoid. Sandwell’s satellite altimetry-derived gravity anomalies have been assessed and combined with the Australian gravity database to improve the quantity and coverage of the gravity field offshore Australia. It is demonstrated that a proper inclusion of SA gravity data can give -1Ocm root mean squared improvement of the gravimetric geoid.
66
K. Zhang: On the Determination of a New Australian
The new geoid may provide an alternative and costeffective method to geodetic levelling and for many scientific applications, such as geodetic positioning, geodynamics and geophysical exploration. Relative precision of the new geoid is cm-level, with absolute precision higher than existing geoid and resolution of a few kilometres. Satellite altimetry data collection, analysis, assessment and implementation of marine geoid together with the sea surface topography information and marine and continental gravity observations are important for further refinement of the Australian geoid. Satellite altimetryderived gravity anomalies need properly assessed in terms of error features and accuracy so as to be optimally combined with local gravity data. This is of importance for FIT computation and thus is expected to further improve the accuracy and precision of the geoid along the coastline of Australia. This is particularly useful for Australia, since the coastline is densely populated relative to the interior. Finally, the methodologies developed in this paper are pertinent to other part of the world for a gravimetric geoid determination over a large area, particularly for maritime countries. Acbrowledgement~. The Authorwould
Iike to express his gratitude to Curtin University of Technology for awarding him with two prestigious researchscholarships,namely Ovaseas PostgraduateResearch Scholarship and Curtin UniversityPostgraduateResearch Scholarshipand to Associate professorWE Fcatherso(ne,School of Spatial Sciences. Cutin Universityof Technology for his assistancefor pat of this research. Comments from ProfessorAH D&on, The Universityof N&tin&am and from ProfessorLN. Tziavos. AristotleUniversityof Tbessalonikiarehighly appreciated.
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