** 4 Numerical results**

**4.5 Tests with a large-scale real data set in the Auvergne area**

**4.5.2 Gravimetric solutions**

The computation procedure is the same as in the previous numerical tests. The employed point mass search model is SMA. First, the initial depth of the point mass RBFs is derived from the empirical covariance function of the input gravity residuals, resulting in the value of 16 km. Second, the depth limits are determined based on the empirical rule described in Section 4.1.3.5. Accordingly, the shallowest upper depth limit is limited to 2 km, and the deepest lower depth limit is set to 55 km.

Here, the upper and lower depth limits are chosen to be about 0.875 and 3 times of the initial depth,

**Table 4.35:** Statistics of the gravity anomalies [mGal] and height anomalies [m].

Mean STD RMS Min Max

∆g 3.066 20.697 20.922 -127.459 177.841

∆g_{EGM}_{08} 7.788 18.686 20.244 -45.761 125.033

∆g*res1* -4.722 16.276 16.947 -206.735 91.440

∆g* _{RT M}* -6.115 15.327 16.502 -179.018 90.691

∆g*res2* 1.394 9.544 9.645 -49.426 72.054

*ζ* 49.550 1.473 49.571 46.774 52.147

*ζ**EGM08* 49.659 1.466 49.680 47.049 52.455
*ζ** _{res1}* -0.109 0.172 0.203 -0.537 0.272

*ζ*

*0.083 0.197 0.212 -0.247 0.562*

_{RT M}*ζ*

*res2*-0.192 0.185 0.266 -0.605 0.108

**Table 4.36:** Model setup used for the numerical tests in the Auvergne area.

Spectral bandwidths of the RBFs 1 *N*_{min} = 51 (unconstrained solution)

2 *N*min = 0 and*n*^{0} = 40 (constrained solution)
Optimization direction Radial-direction

Initial depth [km] 16

Depth limits [km] 14−50

Number*N** _{ε}* of the
nearest point mass RBFs 10
Total number

*K*of point

10000 mass RBFs

Iteration limit*N**it* for each
new point mass RBF 20

resulting in the values of 14 km and 50 km, respectively. Third, the spectral bandwidths of the RBFs
are chosen. For the unconstrained solutions,*N*_{min} = 51 is selected after various tests. For the
cons-trained solutions, the full point mass RBFs are employed while*n*^{0} is chosen to be 40 (see Eq. (3.11)).

Fourth, as the accuracy of the input gravity residuals is assumed to be about 1−2 mGal, the searching for the point mass RBFs is set to be terminated after reaching the target data misfit of 1 mGal. If the target data misfit can not be achieved with a maximum number of 10000 point mass RBFs, then the search process is stopped. The reason why the number of RBFs is limited to 10000 is that the data fit to the input data is very close to the target data misfit. In this case, the data misfit is 1.136 mGal for 10000 RBFs and 1.044 mGal for 20000 RBFs. Obviously, the former choice is better when considering the numerical complexity. Tikhonov regularization is applied for all computations in the second step of PM-FRE, and the regularization parameter is determined by VCE. The detailed model setup is given in Table 4.36.

On the basis of the above model setup, 10000 point mass RBFs are found and positioned in the defined single layer. The depth histogram and the horizontal distribution of these point mass RBFs are shown in Fig. 4.33. Most of the searched RBFs are located around the upper depth limit while only a small part of them are near the lower depth limit. This is because a large amount of shallow point mass RBFs are needed to represent the high-frequency gravity field signals, while only a small number of deep RBFs are required for approximating the long-wavelength signals. The horizontal distribution of the RBFs shows a high correlation to the topography, or equivalently, to the roughness of the gravity field. In the areas with rough features (i.e., rough topography), the point mass RBFs are dense, whereas the RBFs are sparsely distributed in the areas with smooth features (i.e., flat topography).

With the use of the point mass RBFs with known positions, two solutions are computed in the second step of PM-FRE, i.e., the unconstrained solution and the constrained solution. Both are validated by comparing to the observed values at the control points, and the statistics of the differences are presented in Table 4.37. The RMS of the gravity anomaly errors is about 1.2 mGal for both solutions, where the unconstrained solution is slightly better than the constrained one. From the gravity anomaly errors illustrated in Fig. 4.34, a good agreement between the two solutions can be seen except for the

0 10 20 30 40 50

Percent [%]

10 15 20 25 30 35 40 45 50 55

Depth [km]

−2˚ 0˚ 2˚ 4˚ 6˚ 8˚

42˚

44˚

46˚

48˚

50˚

−2˚ 0˚ 2˚ 4˚ 6˚ 8˚

42˚

44˚

46˚

48˚

50˚

−1000 −500 0 500 1000 1500 2000 2500 3000

Topography [m]

**Figure 4.33:**Depth histogram (left) and horizontal distribution (right) for 10000 searched point mass RBFs.

points located near the center of (3^{◦}E,45^{◦}N), where only a small number of gravity observations are
available (see Fig. 4.32). The largest errors for the two solutions are observed at the same points,
which are located in the Alps area around the point of (5.5^{◦}E,45^{◦}N). The STD of the height anomaly
errors is about 0.035 m for the two solutions, and close to the results presented in Ågren et al. (2009).

A mean difference of −0.182 m for the unconstrained solution and −0.165 m for the constrained solution are observed, indicating the inconsistency between the gravimetric and GPS/leveling-derived height anomalies. Due to the biases, the height anomaly errors at the GPS/leveling points with the removal of the mean errors are shown in Fig. 4.35 so that some useful information is visible. It clearly shows that most height anomaly errors are in the range between−0.1 m and 0.1 m if the biases are

**Table 4.37:** Statistics of the differences between the predicted and observed values at the control points, i.e.,
1145 observed gravity anomalies [mGal] and 75 GPS/leveling-derived height anomalies [m], for
unconstrained (first two lines) and constrained solutions (second two lines).

Mean STD RMS Min Max

∆g 0.048 1.190 1.191 -6.625 17.412
*ζ* -0.182 0.034 0.185 -0.255 -0.069

∆g 0.078 1.239 1.241 -6.802 17.589
*ζ* -0.165 0.035 0.169 -0.247 -0.070

0˚ 1˚ 2˚ 3˚ 4˚ 5˚ 6˚

44˚

45˚

46˚

47˚

48˚

**a)**

−20 −16 −12 −8 −4 0 4 8 12 16 20

∆g [mGal]

0˚ 1˚ 2˚ 3˚ 4˚ 5˚ 6˚

44˚

45˚

46˚

47˚

48˚

**b)**

−20 −16 −12 −8 −4 0 4 8 12 16 20

∆g [mGal]

**Figure 4.34:**Gravity anomaly errors at 1145 control points associated with**a)**the unconstrained solution and
**b)**the constrained solution.

1˚30' 2˚00' 2˚30' 3˚00' 3˚30' 4˚00' 4˚30' 45˚00'

45˚30' 46˚00' 46˚30' 47˚00'

**a)**

−0.10 −0.05 0.00 0.05 0.10

ζ [m]

1˚30' 2˚00' 2˚30' 3˚00' 3˚30' 4˚00' 4˚30' 45˚00'

45˚30' 46˚00' 46˚30' 47˚00'

**b)**

−0.10 −0.05 0.00 0.05 0.10

ζ [m]

**Figure 4.35:**Height anomaly errors at 75 GPS/leveling points associated with**a)**the unconstrained solution
with the removal of the mean of−0.182 m and**b)** the constrained solution with the removal of
the mean of−0.165 m.

removed. The largest errors for both solutions are at the same points, located in the southeast of the region covered by the GPS/leveling points. Comparing the error distribution for both gravity and height anomalies (Figs 4.34 and 4.35) to the input data locations in Fig. 4.32, large errors can be seen in the area with sparse observations, indicating the importance of using homogeneous input data to compute high-quality regional gravity field models.

Based on the estimated point mass RBFs, a 1^{0}×1^{0} grid of gravity anomalies and height anomalies
on the Earth’s surface associated with the two solutions are computed and shown in Fig. 4.36, so
are the differences between them. The differences between the two gravity anomaly models are very
small with values in the range between −2 mGal and 2 mGal in most areas. Differences with values
larger than 8 mGal can be observed in two places due to the lack of gravity observations (see also
Fig. 4.32). It is obvious that one of the two solutions provides worse predictions in the gaps of the
input data points. According to the comparisons at the control points, the gravity anomaly model

0˚ 1˚ 2˚ 3˚ 4˚ 5˚ 6˚

44˚

45˚

46˚

47˚

48˚ **a)**

−100 −80 −60 −40 −20 0 20 40 60 80 100

∆g [mGal]

0˚ 1˚ 2˚ 3˚ 4˚ 5˚ 6˚

44˚

45˚

46˚

47˚

48˚ **b)**

−100 −80 −60 −40 −20 0 20 40 60 80 100

∆g [mGal]

0˚ 1˚ 2˚ 3˚ 4˚ 5˚ 6˚

44˚

45˚

46˚

47˚

48˚

**c)**

−10 −8 −6 −4 −2 0 2 4 6 8 10

∆g [mGal]

0˚ 1˚ 2˚ 3˚ 4˚ 5˚ 6˚

44˚

45˚

46˚

47˚

48˚

48

49

50

51 52

**d)**

45 46 47 48 49 50 51 52 53 54 55 ζ [m]

0˚ 1˚ 2˚ 3˚ 4˚ 5˚ 6˚

44˚

45˚

46˚

47˚

48˚

48

49

50

51 52

**e)**

45 46 47 48 49 50 51 52 53 54 55 ζ [m]

0˚ 1˚ 2˚ 3˚ 4˚ 5˚ 6˚

44˚

45˚

46˚

47˚

48˚

**f)**

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 ζ [m]

**Figure 4.36:**Gravimetric gravity anomaly models with a resolution of 1^{0}×1^{0}for**a)**the unconstrained solution;

**b)**the constrained solution; **c)**the differences between**a)** and**b);****d),****e), and****f)** the same, but
for the gravimetric quasigeoid models.

associated with the unconstrained solution is regarded as the better one (Fig. 4.36a). As expected, significant height anomaly differences are found in the same two places in Fig. 4.36f. Besides the two small-scale height anomaly differences with large values, large-scale differences reaching up to±30 cm are observed in the border area from Fig. 4.36f. It might be caused by the different computation schemes, i.e., the reduced RBFs without constraints and the full RBFs with constraints.