1. Auvergne dataset : testing several geoid computation methods
In 2004, the French Institut Géographique National (IGN), upon the request of the steering commitee of European Gravity and Geoid Project, has prepared a data set to test the geoid computation methods. It consists in a set of around gravity points, three digital terrestrial models (an accurate one, a large-step one and a filtered one) and 75 GPS levelled points to evaluate the quality of the computed geoids [Duquenne, 2006]. The GPS levelled points were measured by the Service de Géodésie et Nivellement of the IGN and are very well distributed over the center area of computation. Their precision is close to 2-3 cm for the ellipsoidal heights.
modification used : WONG GORE
EGM08 (deg 360) L 67
115 92 20
100
360
radius of integration of Stokes anomalies 2°
2,5°
bias (cm)
-11,3
-8.2
-10,5
-8.8
-16,2
-10,4
-12,7
standart deviation (cm)
4,5
4,3
5,0
3,8
4,7
8,2
Pierre Valty, Henri Duquenne, Isabelle Panet Institut Géographique National – LAREG, 6-8 Avenue Blaise Pascal, Champs-Sur-Marne, France
This dataset is located in heart of France, and is centered on the semi-mountainous region of Auvergne, in the Northern part of the Massif Central. But the gravity data also covers plain areas (Bassin Parisien and Bassin Aquitain) and high mountains (French Alps). The height varies from 150 to 1886 m, with large plateaux of nearly 1000 m height and deep valleys. See figure 1.The gravity data have been directly extracted from the database of the Bureau Gravimétrique International. Their geographic repartition is quite homogeneous, except in the most mountainous areas and in some very isolated regions (south of Auvergne). The geographic coverage is 43° to 47° N in latitude and -1°W to 7°E in longitude, with a mean density of 0,59 point/km². These data were converted into the IGSN71 system. Although they have already been checked y the BGI and interpolated by collocation, some points are still obviously wrong. There are points which height is not coherent with the DTM's height and moreover the spatial positioning of a lot of points seems to be inaccurate. The accuracy of gravity values was evaluated to 0,25-0,75 mgals, but actually the consistency between gravity campaigns can be worse than 2 mgals. The noise rms was estimated at 2.3 magls during the computation.
Tab 2 : statistics obtained on geoid models computed with the Wong and Gore modified Stokes’ kernel
B. Stochastic methods
Following the studies realized by [Ellmann, 2004], we thought about modelling geoid heights by KTH (Stockholm Royal Institute of Technology) method with Sjöberg stochastic modifications of Stokes formula. The two main characteristics of this approach are that :
Stokes integration is computed with gravity anomalies from which global filed is not removed.
-Coefficients are computed by least-square modifications, so as to minimise the mean square error of the difference between the estimator of equation (3) and the true geoidal height expressed by formula (4).
An other expression of modelled geoid height can be : [Sjöberg, 2003] (3)
And the “true” geoid height [Heiskanen & Moritz, 1967] (4)
So the expected mean square error m of the difference between (3) and (4) can be written as :
Coefficients are computed by solving by least squares the equation obtained by the derivation (depending of the expression of ).
These ones can be evaluated in function of the degree variances and error degree variances of global field model and of the error degree variance of gravity data is function of the a-priori variance of gravity data C(0). The formula also contains coefficients which depend of radius of Stokes’ integration. For more details see [Ellmann, 2004] and [Sjöberg, 2003].
We used the biased Sjöberg modification [Sjöberg, 1984], as we found difficulties with the inversion of normal matrix in the computation of coefficients with the Sjöberg unbiased method. However, the unbiased method is considered to be better than the biased one, [Ellmann, 2004]. In the biased method, least-squares estimated coefficients are equal to the in equation(3).
Several values for the variance of gravity data and for the degree of modification of Stokes formula were tested. As for the other methods, Dmax is not necessarily equal to L, but, here, we must have (because of the presence of in the second term of (3).
Bias is also always reduced compared to Stokes unmodified method, but Wong Gore appears clearly as the best from this point of view.
Fig 1 1
The aim of this dataset was to compute other models in odrer to compare themselves. It was tested with the 75 GPS levelled points. Thus we decided to compute a set of geoid models with the classic remove-compute-restore method, but with changing some parameters. Afterwards, we compare these solutions with the computed models with modified Stokes kernel. Two modifications has been tested : a deterministic one (Wong-Gore modification) and a stochastic one (Sjöberg biased modification). For these two methods we also tried to use differents sets of parameters (-see part II).
Fig 2 : gravity data
I . Using the classical remove-compute-restore computation method
Following the work of Henri Duquenne [Duquenne, 2006], we first computed some geoid models using the classic remove-compute-restore method and the Gravsoft package [Tscherning & al. , 1992] using the residual terrain anomalies.
The computed height model can be (before terrain restauration) written as [Vincent & Marsh, 1974] :
Where is the gravity anomaly from the global field model and the gravity anomaly corrected from the
terrain effects. Dmax is the maximum degree of the global gravity field.
And is the Stokes function. In this simple case, the Stokes function is unmodified, so (see part II)
With the expression :
The reference geoid was computed using the GL05C global field model [Förste & al. , 2008] up to degree 360.
Main parameters were :
Integration of terrain corrections in a radius of 100 km for the low-step DTM and 20 km for the accurate DTM.
- integration of Stokes anomalies in a radius of 2° and a 1D-FFT computation.
To check the quality of the computed model, we compare it with GPS levelled points. To do this, we estimate and remove a linear trend function from the latitude and the longitude (5 parameters).
We decided to change some parameters, specially the radius of terrain effects computation and of Stokes integration, but also the global field model (we tested EGM2008 [Pavlis & al., 2008] up to degree 360). We also tried to use a very low noise variance of the covariance function for the interpolation of residual gravity anomalies, in order to have a better coherence between the grid and the values of residual gravity anomalies measured on the points. Main changes are summarized in the table below :
Results with Sjöberg biased modification does not really improve the results in the whole area. But when the boundary areas are removed (i.e the 0.5° on eastern and western sides), the standard deviation is improved compared to the two other methods for and improved compared to classic remove-compute-restore method for We can notice than boundary effects are smaller with GL05C.
Atmospheric [Sjöberg,1998] and ellipsoidal corrections [Sjöberg, 2002] do not clearly improve the precision of models, but they permit to reduce the bias.
Table below summarizes the main results.
* center area geoid is the computed geoid from where two 0,5 degrees stripes are removed at the east and west of the grid
** except L and Dmax which are chosen as optimal when L and Dmax are optimal for Sjöberg biased method also. L = 115 when L (for Sjöberg biased method)=90 and L=92 when L(for Sjöberg method) = 72
These effects are maybe due to the integration of anomalies which still contain the global field model effects. In the eastern part of the area, the GGM up to degree 360 is too much far from the reality of gravity field and leads to have these kind of errors. The 0.5° area at the east side of the target area corresponds exactly to the area where data coming from the Alps are integrated in Stokes formula.
modfication used : Sjöberg biased
EGM08(360)
GL05C L 16 67 90 72
Dmax
radius of integration for Stokes anomalies 2°
2,5°
variance of gravity data C(0) (mgal²) 4 1
atmospheric and ellipsoidal corrections no
yes
bias (cm)
-13,6
-11,5
-14,6
-18,3
-12,0
-11,2
-10,0
standart deviation (cm)
7,4
5,5
6,6
4,7
6,7
5,6
4,2
4,3
bias for the center area geoid*
-15,3
-13,0
-14,7
-20,3
-15,9
-11,8
-10,6
standart deviation for thecenter area geoid*
5,9
3,6
5,7
3,2
5,2
3,5
bias for the center area geoid for Wong-Gore modification (with the same parameters **)
-11,4
-7,9
-11,3
-9,1
standart deviation for thecenter area geoid for Wong-Gore modification (with the same parameters)
4,4
3,8
4,9
bias for the center area geoid forclassic remove-and-restaure method (with the same parameters)
-19,8
-21,8
-17,5
-17,4
standart deviation for thecenter area geoid for classic remove-and-restaure method (with the same parameters)
6,0
3,9
global field
GL05C
EGM08 (deg 360)
EGM08(deg360)
EGM08 (deg360)
radius of inetgration of refined terrain anomalies
20 km
radius of integration of Stokes anomalies 2°
2,5°
noise variance (for interpolation of residual gravity anomalies)
2,3
0,2
bias (cm)
-17,3
-17,5
-19,7
-18,0
-21,8
standart deviation (cm)
5,7
3,9
5,9
4,9
3,8
Tab 3 : statistics obtained on geoid models computed with the KTH method and Sjöberg biased modification and Ccmparison with, other models
Tab 1 : statistics obtained on geoid models computed with the classical remove-compute-restore technic and unmodified Stokes kernel1
Figure 5 shows the discrepancies between the models computed with Sjöberg biased models and with the classic remove-compute-restore method (with the same parameters). The boundary effects (east and west) corresponds to GPS levelled points where the difference of residuals between the two models is the highest, specially on the eastern side, where the rough anomalies of the Alps are integrated by Stokes formula.
By comparing the two maps below, we can assert that the residuals of GPS levelled points on the eastern boundary are better with the unmodified Stokes kernel, that is why we can maybe suppose that there are boundary effects in this area.
Fig 5 : discrepancies between geoids computed with Sjöberg biased modification and unmodified Stokes’ kernel (part I) , and discrepancies between residuals on GPS levelled points obtained with these two methods
Fig 4 : Residuals (in m) obtained on GPS levelled points with the geoid computed with GL05C, Stokes radius set to 2° and refined terrain integration radius set to 20 km.
Fig 3 : geoid model using GL05C global field and a 2° Stokes integration (in meters)
II . Using the modifications of Stokes kernel
Formula (1), in part I is the generalization of Stokes scheme by Vanicek and Sjoberg (1991). In the classic remove-compute-restore method, the Stokes function keeps unmodified. But from a more general point of view, can be expressed as :
Where the coefficients are the modification parameters of Stokes function.
There are two main kinds of modification of Stokes formula :deterministic methods and stochastic methods [Ellmann, 2004]. The aim of the deterministic ones is only to reduce the effect of the high-frequency contribution of the neglected integration area (when ), whereas the stochastic ones also take into account the errors of the global model field and of the gravity data which affect the computed geoid height. These ones try to minimise the influence of these errors (see part II.B)
A. deterministic methods
The only deterministic method we tested on Auvergne dataset is Wong and Gore method [Wong and Gore, 1969]. In equation
(2), parameters can be expressed as : Stokes integration is also realized on residual gravity anomalies.
An important parameter to set for Stokes modified kernels methods is the higher degree of modification L. A good choice is to choose L as
for ( is the value of Stokes integration radius) [Ellmann, 2004].
Parameter Dmax is totally independent from L. Better results were obtained here using Dmax=360, but other values (specially Dmax=L) have been tested. Main results are shown in table 2. Bias of the computed geoid is clearly better than with the unmodified Stokes kernel, specially for and precision is slightly better for
Fig 6: residuals on GPS levelled points obtained with Sjöberg biased modification (with EGM08 to degree 360, Stokes integration at 2.5°)
Fig 7: residuals on GPS levelled points obtained with unmodified Stokes kernel (with EGM08 to degree 360, Stokes integration at 2.5°)
The results show that the modifications of Stokes kernel give better results (for bias and accuracy), even though the roughness of gravity field in French mountains can maybe affect locally the quality of the geoid computed. Nevertheless, testing stochastic modifiactions, like Sjöberg biased one, was a good way to undestand what are the ways to investigate so as to improve the quality of geoid modelling in France. Trying to compute a model using unbiased Sjöberg modifications will be the next step of our study. Other set of GPS levelled points, much more dense, has also been tested to evalute the quality of the computed geoids here, and they have mostly confirmed the results obtained with « only » 75 points.
These studies were realized with the aim of the new French geoid computation in The other part of preprocessing will concern the improvement of spatial (2D) positioning of the gravity points. To do this, we are going to use the database of communication axes and of elevation models of the IGN.
Pavlis, N.K., Holmes, S.A., Kenyon, S.C., Factor, J.K.An Earth Gravitational Model to Degree 2160: EGM2008. presented at Session G3: "GRACE Science Applications", EGU Vienna.
Sjöberg, LE (1984) Least-squares modification of Stokes and Vening-Meinesz formulas by accounting for errors of truncation and potential coefficients errors. Mansucripta Geodetica, 9:
Sjöberg, LE (1998) The atmospheric geoid and gravity corrections. In Proc. of the 2nd Continental Workshop on the Geoid in Europe. Report of the Finnish Geodetic Institute 98:4
Sjöberg, LE. (2002) The ellipsoidal correction to Stokes’ formula. In “Gravity and Geoid 2002”, proc of the 3rd Meeting of the ICGC, Ziti editions.
Sjöberg, LE (2003). A general model of modifying Stokes ‘ formula and its least-squares solution Journal of Geodesy,
Tscherning CC, Forsberg R, Knudsen (1992) P. The GRAVSOFT package for geoid determination. Proceedings First Continental Workshop on the Geoid in Europe. Prague.
Vanicek P, Sjöberg LE. (1991) Reformulation of Stokes’ theory for higher than Second-degree reference field and modification of integration kernels. Journal of Geophysical Research, 96(B4),
Vincent S., Marsh J. Gravimetric global geoid. In Proc. Of Int. Symp. on the use of artificial satellites for geodesy and geodynamics, National Institute of Technology, Athens, Greece
Wong L, Gore R. (1969) Accuracy of geoid heights from the modified Stokes kernels. Geophysical Journal Royal Astronomy Society, 18, 81:91.
Duquenne, H, (2006) A data set to test geoid computation methods, in Harita Dergisi, Proceedings of the 1st International Symposium ot the International Gravity Field Service "Gravity Field of the Earth", Harita Dergisi, International Gravity Field Service Meeting, Istanbul, Turkey, p
Ellmann, A. (2004) The geoid for the Baltic countries determined by least-squares modification of Stokes formula. Doctoral Dissertation in Geodesy, Royal Institute of Technology, Stockholm, Sweden.
Förste, C., Flechtner, F., Schmidt, R., Stubenvoll, R., Rothacher, M., Kusche, J., Neumayer, K.H., Biancale, R., Lemoine, J.-M., Barthelmes, F., Bruinsma, S., König, R., Meyer, U. EIGEN-GL05C - A new global combined high-resolution GRACE-based gravity field model of the GFZ-GRGS cooperation; Geophysical Research Abstracts, Vol. 10, EGU2008-A-03426,
Heiskanen WA, Moritz H. (1967) Physical Geodesy. WH Freeman and Company . San Francisco .