1. Vertical datum unification on Iberia and Macaronesian islands with a local gravimetric geoid. First results
J. Catalão(1), M. Sevilla(2) 1) University of Lisbon, Faculty of Sciences, IDL, LATTEX, Lisbon,
2) Instituto de Astronomía y Geodesia, Universidad Complutense, Madrid,
ABSTRACT
IBERIA and Macaronesian islands
Geoid validation on the Atlantic
The vertical datum unification between two or more vertical datums separated by sea is one of the present day challenges of geodesy. In this paper, the first results of vertical datums unification between Iberia and Macaronesian islands (Azores, Madeira and Canary archipelagos) are presented. Such a vertical datums can be connected indirectly by the combination of precise geocentric coordinates of points on all datums determined by GNSS, their orthometric height (or potential) in the respective datum and the geoid height. For the later, ICAGM07 gravimetric geoid model was used. This geoid model incorporates updated terrestrial and shipborne free-air gravity anomalies, high resolution digital terrain models and the most recent geopotential model GGM02C.
The geoid boundary value problem approach was used assuming that orthometric heights and gravity anomalies are refered to different height systems. The solution equation has k+1 parameters, where k is the number of local vertical datums considered. The first k parameters are the potential offsets relative to the global vertical datum and the last parameter is the difference in potential between the global vertical datum and the reference ellipsoid potential. The unknown off-sets between various datums were estimated in a least-squares adjustment with respect to a pre-defined W0= m2s-2 using 747 GPS/levelling points with geoid height given by ICAGM07 geoid model. The estimated offsets, in meters, were: Lanzarote, 0.27 Madeira, Faial, Pico, S. Miguel, Terceira, Portugal mainland and Spain mainland. The relative offset between Portugal and Spain (within Iberia) is close to the 17(16) cm obtained from the EUVN, revealing a very good precision of this geoid model for the vertical data unification.
A synthetic geoid model was constructed as the difference of the mean sea surface height (MSSH), given by CLS01 model, and the mean dynamic topography (MDT) given by MDT_Rio05 (produced by CLS Space Oceanography Division). This synthetic model was compared with ICAGM07 geoid model.
The highest residuals are close to the shoreline and inter island which can be explained by satellite altimetry degradation close to land areas and consequently reflected on the mean sea surface.
It can be seen a periodic longitudinal effect with a period of 5 degrees on the residuals and also a circular effect over Canary archipelago.
Both seem to be related with the geopotential model.
Geoid Model – ICAGM07
The concept of remove-restore used on the geoid computation implies that the low-frequencies and the high-frequencies gravity signal from the global geopotential model and topography, respectively, are removed before the application of Stokes integral and restored afterwards as geoid undulation and indirect effect. The final geoid undulation is given by:
where NGM is the geoid undulation derived from GGM02C geopotential model and Nind is the indirect effect as a result of using Helmert gravity anomalies on Stokes’ formula. The computation of the high resolution geoid model was based on the spherical FFT evaluation of Stokes integral. A Wong-Gore kernel modification has been used for spherical harmonic degrees less than 240 to limit the long-wavelength errors.
Introduction
This investigation is part of ICA project, a joint project between University of Lisbon and University Complutense of Madrid funded by GRICES/FCT and CSIC and has the main objective of derive a high precision gravity and geoid model on the North-East Atlantic – Iberia-Canary-Azores that will be useful for: a) vertical datum unification between Portugal, Spain and Azores, Madeira and Canary islands, b) conversion between orthometric and ellipsoidal vertical systems. b) estimation of ocean circulation.
Vertical datum unification
Each GPS/leveling mark represents one equation, and 747 equations were computed. For each equation the weight was determined as the inverse of the variance of the sum (h-H-N). A standard deviation of 2 cm was attributed to the ellipsoidal height (h ), a standard deviation of 3 cm for the orthometric heights on Iberia (Portugal and Spain) and 6 cm for the other areas and a standard deviation ranging from 2.2 cm to 13.3 cm for the geoid (dependent on the geoid comparison results). This last value is highly dependent on the vertical network accuracy.
Over centuries vertical datuns have been defined by long term averages of tide gauges registration, the mean sea level. However, the mean sea level, determined this way, does not coincide with an equipotential surface (the geoid). The difference between the mean sea level and the geoid is the sea surface topography (SST). His way, the knowledge of sea surface topography would permit the unification of vertical datuns separated by sea. But unfortunately the SST is not known on coastal areas with enough accuracy. Moreover mean sea surface changes with time and vertical tectonic movements may affect records and induce false sea level changes.
Rummel and Teunissen (1988) presented a strategy for vertical datum unification between two or more vertical datuns not physically connected, by means of a combination of precise ellipsoidal heights (determined with GPS observations), the potential in the respective height datum and geoid height differences between both datuns.
Geoid height
U0= m2s-2 ;
W0= m2s-2 (Bursa et al.,)
The problem to be solved is clearly illustrated in the next figure. Two vertical systems A and B located in different continents (or islands) must be connected.
All potential differences of system A refer to bench mark A (WA) and system B to bench mark B (WB). Furthermore, we have a global equipotentail surface W0 of the actual gravity filed, the geoid.
Geoid validation on land
The problem solved by Rummel and Teunissen (1988) was the geodetic boundary value problem and its representation in the context of the determination of the potential difference CAB between the fundamental bench marks (and the assigned values WA and WB are known). The solution is:
The geoid model, computed with GGM02C geopotential model, was compared with GPS-leveling data on Iberia (Portugal and Spain), and on 7 different vertical datums used on Azores, Madeira and Canary archipelagos. Residuals of the geoid and the “geometric” geoid determined from the elipsoidal-orthometric pair of measurements (h-H) were computed for each vertical datum and presented in the table below. For each vertical datum, the first line represents the original comparison and the second line the result after a 4 parameter adjustment.
The unknown off-sets between various datums were estimated in a least-squares adjustment with respect to a pre-defined W0= m2s-2 using the 747 GPS/levelling points with geoid height given by ICAGM07 geoid model. The estimated offsets, in meters, were: Lanzarote, 0.27 Madeira, Faial, Pico, S. Miguel, Terceira, Portugal mainland and Spain mainland. The relative offset between Portugal and Spain (within Iberia) is close to the 16 cm obtained from the EUVN, revealing a very good precision of this geoid model for the vertical data unification
4-parameter residuals
where
hP is the ellipsoidal height given by GPS observables.
HiP is the orthometric height on system I
NiP is the geoid height
Ci,0 is the potential difference between the regional datum i and the global vertical datum
W0 = W0 – U0
S() is the Stokes’ function
 normal gravity
REFERENCES
Bursa, M. et al., (2004) A global vertical reference frame based on four regional vertical datums. Studia Geophys.Geod. 48,
Rummel and Teunissen (1988). Height datum definition, height datum connection and the role of the geodetic boundary value problem. Bull. Géod., 62,
From the comparison with GPS-leveling data we see that geoid accuracy is not homogeneous (over land areas) ranging from 6 cm on Faial to 25 cm on S. Miguel. However, on Azores, such results may be interpreted as the result of considerable horizontal and vertical surface deformation over time reaching on extreme volcanic events deformations of meters. Because of this, orthometric and ellipsoidal heights surveys over time, or separated decades, may contain local vertical deformations and may give only a first approximation to the gravimetric geoid accuracy. The same reasoning can be applied to Lanzarote.