** 2 Approximation of the regional gravity field**

**2.7 Spherical harmonics and radial basis functions**

**2.7.4 Remarks on RBFs**

When the low frequencies are removed from the functions, the shape of the functions becomes much narrower in the cases of the depths of 50 km and 100 km, whereas the shape remains nearly the same in the case of the depth of 10 km. This indicates that the RBFs at shallow depths are dominated by short-wavelength contributions, while the ones at deep depths are dominated by long-wavelength contributions. As the basis function with the Poisson kernel has a stronger space localizing than the one with the point mass kernel, the former function is expected to be located at a deeper depth such that it can provide a similar shape to the one of the latter function.

coefficients negligible. Regarding the non-perfect spectral localizing property of the RBFs and the
existence of long-wavelength errors in the residuals, we prefer to choose 0≤*N*_{min} ≤*n*_{max}+ 1 while
the value of*N*_{min} being dependent on the particular applications. After fixing the type and spectral
bandwidths of the RBFs, the choice of optimal spatial bandwidths of the RBFs for regional gravity
field modeling is equivalent to the determination of optimal depths of the RBFs. Often, the RBFs are
located at gridded or scattered points at a constant depth, which is selected by “trial-and-error” or
based on empirically found relations with the data spacing or the gravity anomaly covariance function
(e.g., Dampney, 1969; Hardy and Göpfert, 1975; Heikkinen, 1981; Sünkel, 1981; Vermeer, 1995); all
RBFs have the same spatial bandwidths in this case. Sometimes, several grids at various depths
are used (e.g., Reilly and Herbrechtsmeier, 1978; Heikkinen, 1981; Vermeer, 1983, 1984; Ihde et al.,
1998; Chen, 2006), and sometimes, each RBF is placed below one data point at a depth, which is
proportional to the distance to the nearest neighboring data point (e.g., Cordell, 1992). Additionally,
some advanced strategies involve the generalized cross validation (GCV) technique (e.g., Klees et al.,
2008), the adaptation of the RBFs’ depths to the local signal covariance function (e.g., Marchenko
et al., 2001), and the direct determination of the depths as well as the scaling coefficients by solving
a nonlinear problem (e.g., Barthelmes, 1986; Lehmann, 1993; Claessens et al., 2001).

*2) The choice of the horizontal positions of the RBFs*

The choice of the horizontal positions of the RBFs addresses the issue of how many RBFs are to be used for the modeling. This is important as the number of used RBFs defines the resolution of the gravity field solution. In addition, the chosen horizontal positions should be adapted to the input data to achieve a reasonable solution. There are three possible cases for choosing the horizontal positions:

1 RBFs are placed on one or more grids.

2 RBFs are placed on scattered points.

3 RBFs are placed on both gridded and scattered points.

In the first case, the RBFs are to be distributed as homogeneously as possible on the surface of a sphere inside the Earth. Therefore, a grid is designed, and the RBFs are placed on the nodes of the grid. Four factors are needed for designing a grid: (1) the grid type; (2) the grid extent; (3) the grid spacing; and (4) the grid depth. The horizontal positions of the RBFs are determined by the first three factors, and the last one is related to the spatial bandwidth. Several kinds of grids on the sphere, such as geographical grid, Driscoll-Healy grid, Reuter grid, triangle center grid, triangle vertex grid, and recursive quasi random grid are summarized and compared in Eicker (2008), suggesting that the Reuter grid and the triangle vertex grid are very well suited as nodal point patterns for RBFs. In this context, only the geographical grid will be used due to its simplicity, and due to the fact that the selected research areas for numerical tests are at medium latitudes. For the areas at high latitudes, the geographical grid should be avoided. The grid extent is usually set to be as large as the data area.

If the model area is smaller than the data area, such a choice is satisfactory. However, if the model
area is the same as the data area, a larger extent may be preferred such that the edge effect caused
by the lack of data outside the model area could be reduced. It should be pointed out that the larger
the grid extent than the data area is, the serious the numerical instabilities become. A study about
the choice of the grid extent is given in Naeimi (2013). The grid spacing should be carefully chosen
to avoid under- and over-parameterization. It also has a high correlation with the grid depth (i.e.,
spatial bandwidth). Fig. 2.3 shows two examples of three neighboring non-bandlimited RBFs *B*^{PM}
with a grid spacing of 2^{◦} at different depths. It can be found that if the grid depth is too deep (see
Fig. 2.3b), the neighboring RBFs overlap too much in comparison with the case of a shallow grid
(see Fig. 2.3a). This will lead to numerical instabilities. To overcome this problem, we can make the

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**a)**

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**b)**

**Figure 2.3:**Normalized neighboring non-bandlimited *B*^{PM} with a grid spacing of 2^{◦}; **a)** grid at a depth of
10 km;**b)** grid at a depth of 100 km.

grid depth shallower or make the grid spacing larger such that the overlaps between the functions become smaller. Often, one of the two factors (i.e., grid spacing and grid depth) is fixed at first, and then the other one is determined by experimenting with various choices and comparing obtained solutions. Furthermore, the grid spacing should be related to the signal content of the input data. A dense grid is required for the area with rough structures, and a coarse grid is sufficient for smooth signals. However, the signal content is not fully taken into account when employing only one grid for the modeling as all RBFs have the same spatial bandwidths. A hierarchical arrangement of the RBFs by placing them on two or more grids seems to be more reasonable. The deep grid always has a large spacing and extent and is used for modeling the low-frequency signals. The high-frequency signals shall be represented by the grid situated at a shallow depth, of which the spacing and extent is smaller. When the grids are chosen, the number of RBFs is known too. It is also worth mentioning that the effect of data distribution on the choice of the grid spacing is significant. In principle, the grids are well suited for the case with regular data. For the case with irregular data, a relatively coarser grid is advisable to reduce the danger of over-parameterization.

In the second case, the RBF center locations are selected based on the signal content of the input data. Least-squares collocation can be considered as an approach, in which there is one RBF under each observation at an optimal depth, resulting in the high numerical complexity. In practice, it is not necessary to place the RBFs below every data point. The efficient way is to construct a set of RBFs iteratively by adding one RBF at each time. Cordell (1992) designed an algorithm for using the point masses. At each time, one point mass is chosen under the data point with the largest absolute (residual) value at a depth derived from the distance to the nearest neighboring data point, and then the magnitude of the point mass is computed individually. Barthelmes (1986) proposed a similar point mass approach; however, it requires solving a nonlinear problem with four unknowns per point mass, i.e., the 3D position and the magnitude of the point mass. Marchenko et al. (2001) also developed an approach with the use of radial multipoles. The sequential multipole analysis is applied to determine the horizontal position, the order, and the depth of the multipole at each time. The latter two parameters are fixed by using the covariance function of the observations in the vicinity of the data point corresponding to the multipole. Comparing to the approaches using the RBFs on grids, the approaches using scattered RBFs are more flexible and require fewer RBFs for a good approximation; however, they are also more complicated.

An approach that is a compromise between the first and second case is proposed by Klees and Wittwer (2007) and Klees et al. (2008). A coarse grid of RBFs at a depth, which is determined by

the GCV technique, is used to model the smooth signals over the research area. If the approximation quality is not satisfactory, the local refinement is carried out by placing one RBF below the data point with the largest absolute residual value iteratively. A joint least-squares solution is finally computed when all RBFs are fixed. A good approximation can be obtained by this approach with a relatively small number of RBFs.