The measurement of the gravity field on a global scale has become possible due

to the evolution of spatial gravimetric measurement techniques. These models contain a large part of the content of the gravimetric signal, which motivated the fact that the method of Withdrawal-Calculation-Restoration uses, instead of complete anomalies, the residual anomalies of the gravitational field compared to the global models. The principle of the Remove-Compute-Restore method, called R-C-R in the rest of this paper, is to combine the information which covers three different wavelength ranges [8]:

· the low frequencies given by global models;

· the high frequencies which come mainly from the topography;

· The medium frequencies accessible by gravimetric measurements.

First, the residual anomalies are kept in the gravity anomalies by removing the anomalies from the global model and the corrections from the terrain. These anomalies are then interpolated to a regular grid. The Stokes integral in the next step calculates the residual altitude anomalies. Finally, the altitude anomalies from the topography and the global model are added to those of Stokes in order to have the heights of the quasi-geoid. This method is illustrated by Figure 5: Let ∆g be the anomalies in the open air at the measurement points.

These anomalies can be written using Equation (1), depending on the disturbing potential:

Δ g = − ∂ T ∂ r − 2 r T (1)

The signal is considered to be composed by three wavelength ranges:

T = T M + T R T + T Re s (2)

T_M: is the long-wavelength part determined by spatial gravimetry (M for model).

T_RT: is the short wavelength part that comes from the topography.

T_Res: is the medium wavelength part.

In the first step we calculate the residual anomalies such as:

Δ g R e s = Δ g − Δ g M − Δ g R T (3)

In the second step, the Stokes integral is applied to the values of interpolated Δ g R e s and we obtain ξ R e s , the residual altitude anomaly value can be written as

ξ R e s = T γ = R 4 π γ ∫ s i g m a S ( ψ ) Δ g R e s (4)

At the last stage, the altitude anomalies can be written as:

ξ = ξ M + ξ R e s + ξ R T (5)

The first step in the R-C-R method consists in calculating the residual anomalies ∆_gRes at the points of the measurements:

Δ g R e s = Δ g − Δ g M − Δ g R T (6)

- ∆_g: the anomaly in the open air;

- ∆_gM: the anomaly of the global model which contains the long wavelength part. This part is obtained by taking the fundamental equation of geodesy and applying it to the decomposition into spherical harmonics of the field:

Δ g M = ( r , θ , λ ) = G M r 2 ∑ n = 2 n max ( n − 1 ) ( a r ) n ∑ m = 0 n P n , m cos θ ( Δ C n , m cos m λ + Δ S n , m sin m λ ) (7)

where: G is the universal gravitational constant, M the mass of the Earth, r is the radius, n and m are the degree and order of the geopotential model, a is a scale factor of the field model, n_max is the maximum degree of development in spherical harmonics, P_nm is the function of Legendre, ∆C_n, , ∆S_n,m are the coefficients of development;

- ∆_gRT: contains the short wavelengths of the disturbing field created by the topography. To calculate this component, its long wavelength effects must be removed from the ground, since they are already taken into account in ∆_gM from the field model. The terrain is filtered in order to keep only the short wavelengths of the relief:

Δ g R T = − ∂ T R T ∂ R − 2 R T R T (8)

With

− ∂ T R T ∂ R = G ∭ R T ρ d τ l 3 (9)

T R T = G ∭ R T ρ d τ l (10)

Before being integrated by the Stokes formula, the residual point anomalies must be transformed into a regular grid by interpolation.

The residual altitude anomalies on the grid points are then calculated from the Stokes formula (Equation (11)) applied to the residual anomalies at these points. It is first considered that the area around the gravity station P is flat and horizontal and that the density between the geoid and the earth’s surface is constant ρ = 2.67 g/cm³. The Bouguer plateau exerts an attraction (2πGρh). All the masses above the geoid are removed (the simple Bouguer reduction) [9]. Simple Bouguer anomalies are calculated by the Equation (11) [10]:

Δ g = g p − ( g o − 0.3086 h + 2 π G ρ h ) (11)

To calculate the terrain correction we perform the direct summation of the contributions of each model element (prism, tesseroides etc.).

As mentioned in the general process of the R-C-R method (Figure 5), the gravimetric anomalies used in the calculation are the free air anomalies, their determination is ensured by a translation of the Somigliana and free air formulas described above which give the value of g₀ (gravity on the ellipsoid) and the value of the anomalies. The following figure (Figure 6) shows the free air anomalies.

“Remove” calculation:

This step is the most important step in the geoid calculation; in fact it requires an optimized manipulation of the different types of data as well as a good configuration of the software.

Calculation of the reduction due to the global model:

This step allows subtracting anomalies due to the global model from anomalies in the free air.

Calculation of terrain corrections at gravimetric stations:

Terrain correction (TC) makes it possible to calculate the effects of the terrain on the potential and its derivatives. We detail here the parameters in the case of

application of the residual ground. The effects of the terrain are represented by straight prisms taking into account the local curvature of the Earth. The computation of the terrain corrections is performed by numerical integration using the formulas of the gravitational attraction of the right prisms to represent the terrain in the near area and approximate formulas in the far area.

Terrain corrections are obtained by a simple summation of the contribution of these elements. This allows us to calculate residual anomalies at gravity stations which will be interpolated over the entire study area in order to transform them into residual ripples through the Stokes formula.

The following figure (Figure 7) shows the residual anomalies:

Integration:

The Stokes module makes it possible to calculate residual altitude anomalies from the grid of residual gravity anomalies by a discretized convolution with subsampling in the vicinity of zero.

It is important here to point out that the choice of the integration radius is very important; in fact it directly influences the precision of the geoid.

The choice here was based on a publication by Z. Ismail and O. Jamet [11] which introduces the choice of the radius of integration according to the nature of the terrain (mountainous, semi-mountainous or plain) and the used global geopotential model (Figure 8).

Calculation of the terrain effect:

The terrain effect is calculated using the RTM method; the principle is to calculate this effect from the two DEMs, filtered DEM and reference DEM, over the entire study area.

The result of this calculation is a grid of corrections (Figure 9) to restore for the determination of the final geoid.

The radii R1 and R2 are fixed at 0 and 999 km to optimize the corrections.

Calculation of the undulations of the global model:

These undulations are obtained from the file of coefficients in spherical harmonics by using the GEOEGM module of GravSoft software. Figure 10 shows the undulations of the global model.

Restoration

Restoration is the last step in the R-C-R method. It consists in restoring the anomalies of altitudes calculated by combining the three grids, that of the anomalies of residual altitudes, the corrections of the ground and the anomalies of

altitudes due to the global model.

This step is used to determine the provisional geoid which will later be adjusted to leveling marks. The following figure illustrates the provisional geoid (Figure 11).

The main factors that determine the accuracy of a geoid model are mainly:

· The resolution of the digital terrain model, the higher it is, the more the terrain corrections are punctual and precise.

· The accuracy of gravimetric measurements.

· The density of the gravimetric points and their distributions on the study area: an optimized density and a homogeneous distribution lead to a precise interpolation.

For our case study two factors can have an influence on the precision of the calculated model:

· The absence of a high resolution digital terrain model (terrain resolution 30 m × 30 m).

· The non-homogeneous distribution of gravity points:

- Points spaced 1.5 km apart, others 20 km apart.

- Areas almost devoid of points (Tunis area).

To assess the accuracy of the calculated quasi-geoid, a comparison with other models calculated in different countries makes it possible to rank in a precision scale, for this, a research made by Z. Ismail within the framework of a PhD thesis [12] allowed to know the order of precision of certain models calculated in different countries the table below (Table 1) summarizes the results of this research of which I checked the bibliographical references and we selected the models having conditions similar to our case study.

Verification of the accuracy of the model is ensured by calculating the undulation of the geoid in leveling marks with undulations considered to be known (surveyed by GPS) distributed almost throughout the study area (Figure 12).

The following table (Table 2) summarizes the difference between the values of the undulations.

The bias is the average of the deviations from the model to the data. It identifies a gap between the model and the data.

b i a i s = 1 n ∑ i = 1 n e i (11)

With:

n: the number of control points, e_i: deviation from the benchmarks. We have a bias of −0.0005 cm, that we consider almost negligible.

The Root Mean Square RMS provides an indication of the accuracy of the model. It is defined by:

R M S = 1 n ∑ i = 1 n e i 2 (12)

The RMS is about of 0.031 m (3.1 cm)

The maximum deviation is defined by: Maximal deviation = Max |ei| = 0.069 m.

In terms of justification of the value of the standard deviation (3.1 cm) we can note that:

o The area of the nearest leveling mark FF13 is almost devoid of gravity point (distance of almost 20 Km), which has produced a wide interpolation residual anomalies.

Gravimetric points are not compensated for a block compensation so as to homogenize their precision (points of different order).

o The altimetric network has not yet undergone block compensation, in fact the calculation and compensation are done by tracking and not by mesh.

o The ellipsoidal heights of the leveling marks are observed in RTK mode which produced an error of the order of 2 cm on the undulation at these points.

So, according to the statistics of the deviations, we can assimilate the accuracy of the provisional geoid to: 3.1 cm. Figure 13 the deviation between exact and calculated undulations, this deviation is superimposed with the spatial distribution of the control network.

After having assessed the accuracy of the provisional quasi-geoid, an adaptation of this surface to the leveling marks is performed.

The following table (Table 3) shows the exact undulations of the leveling marks as well as the calculated undulations after adjusting the provisional geoid to the leveling marks.

Figure 14 and Figure 15 illustrate the adjusted geoid and the undulations of the geoid after adjustments.