# On the equivalence of spherical splines with least-squares collocation and Stokes’s formula for regional geoid computation

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## Abstract

This work is an investigation of three methods for regional geoid computation: Stokes’s formula, least-squares collocation (LSC), and spherical radial base functions (RBFs) using the spline kernel (SK). It is a first attempt to compare the three methods theoretically and numerically in a unified framework. While Stokes integration and LSC may be regarded as classic methods for regional geoid computation, RBFs may still be regarded as a modern approach. All methods are theoretically equal when applied globally, and we therefore expect them to give comparable results in regional applications. However, it has been shown by de Min (Bull Géod 69:223–232, 1995. doi: 10.1007/BF00806734) that the equivalence of Stokes’s formula and LSC does not hold in regional applications without modifying the cross-covariance function. In order to make all methods comparable in regional applications, the corresponding modification has been introduced also in the SK. Ultimately, we present numerical examples comparing Stokes’s formula, LSC, and SKs in a closed-loop environment using synthetic noise-free data, to verify their equivalence. All agree on the millimeter level.

## Keywords

Regional geoid computation Stokes’s formula Least-squares collocation Spherical radial base functions Spline kernel## 1 Introduction

The global gravity field is typically represented using spherical harmonics (SH). Consequently, in regional gravity modeling, one usually splits the gravity signal into a global long-wavelength part which is modeled using SH, and a regional short-wavelength part, which is modeled using a suitable regional method (Sansò and Sideris 2013).

There exist several methods for approximating Earth’s regional gravity field, of which integral formula solutions to geodetic boundary value problems and least-squares techniques have emerged as common approaches (Nerem et al. 1995). A review and comparison of different regional gravity modeling concepts are given by Tscherning (1981). More recently, Schmidt et al. (2015) investigated different regional gravity modeling methods through an International Association of Geodesy Inter-Commission Committee on Theory Joint Study Group. Considering geoid computation in particular, Stokes’s formula (Stokes 1849) and least-squares collocation (LSC) (Krarup 1969; Moritz and Sünkel 1978; Moritz 1980) are treated in most geodetic text books (Heiskanen and Moritz 1967; Vaníček and Krakiwsky 1986; Hofmann-Wellenhof and Moritz 2006; Torge and Müller 2012).

Radial base functions (RBFs) are limited to a certain spatial region, making them suitable for regional gravity field modeling due to their space-localizing properties. There is a vast amount of RBFs to choose from, as long as they represent harmonic kernel functions. They are versatile in that their approximation characteristics and spatial distribution can be adjusted, making it possible to use them for all kinds of data sets and for combining different types of observations (e.g., Lieb et al. 2016). Regional gravity field modeling with RBFs can be done using numerical integration (e.g., Freeden and Schneider 1998; Schmidt et al. 2002; Liu and Sideris 2003; Roland and Denker 2005) or least-squares estimation approaches (e.g., Schmidt et al. 2007; Lieb et al. 2016). In this work, we use the latter, which is the common geodetic approach, facilitating error analysis and propagation. The mathematical foundation of RBFs, the special RBFs known as spherical wavelets, and their application in multiscale analysis are given by, for example, Freeden et al. (1998), Schmidt (2001), Jekeli (2005), or Schmidt et al. (2007). In later years, we have observed an increased use of RBFs for regional gravity modeling (Roland 2005; Klees et al. 2008; Eicker 2008; Tenzer and Klees 2008; Wittwer 2009; Bentel 2013; Naeimi 2013; Bentel et al. 2013a, b; Eicker et al. 2014; Pock et al. 2012; Bucha et al. 2015, 2016; Naeimi et al. 2015; Farahani et al. 2016; Lieb et al. 2016).

In this work, we aim to show that regional geoid computation with RBFs is equivalent to Stokes’s formula and LSC, in theory and in practice. Theoretical and numerical comparisons of Stokes’s formula and LSC were done by de Min (1995), while a theoretical comparison of LSC and SKs was discussed by Eicker (2008), both of which we review and present in a unified framework. We show the theoretical equivalence of Stokes’s formula, LSC, and RBFs in the global case, as well as the breakdown of the equivalence of Stokes’s formula and LSC in regional applications. We introduce the remedial modification of the cross-covariance function of LSC also in the SKs, such that all methods are equal also in regional applications. Ultimately, we present numerical examples comparing the methods in a closed-loop environment, demonstrating their equivalence in practice.

Section 2 introduces the different modeling approaches, while their theoretical equivalence in the global case is shown in Sect. 3. The breakdown of their equivalence in regional applications is shown in Sect. 4, and the remedial modifications of LSC and SKs to restore their equivalence to Stokes’s formula are applied. Numerical examples comparing the methods are given in Sect. 5, while Sect. 6 summarizes our results.

## 2 Modeling approaches

### 2.1 Stokes’s formula

*N*may be obtained from block mean gravity anomalies \(\varDelta \bar{g}\) by the integral formula of Stokes (1849). It globally integrates the gravity anomalies over the whole sphere \(\sigma \), using the Stokes function

*S*as integration kernel (or weight),

*R*is the spherical Earth radius, \(\gamma \) is normal gravity evaluated on the surface of the reference ellipsoid, and \(\psi _{Pq}\) is the spherical distance between computation point

*P*and data point

*q*. Equation (1) is a spherical convolution of the \(\varDelta \bar{g}_q\) function with the kernel \(S(\psi _{Pq})\) and can be solved exactly by either numerical integration or by a one-dimensional fast Fourier transform (1D-FFT) (Haagmans et al. 1993), where the FFT is performed along parallels only. We have used the FFT approach, implemented according to

### 2.2 Least-squares collocation

LSC is an optimal estimation method in the statistical sense, allowing the estimation of arbitrary gravity field quantities from inhomogeneously distributed point observations (Moritz 1980). It takes advantage of the knowledge of the signal covariance and tries to minimize the prediction error.

*N*and \(\varDelta g\) between computation point

*P*and observations

*q*, and \(\mathbf {C}_{qq}^{gg}\) is the auto-covariance matrix between all combinations of observations.

*T*to be the basic covariance function, from which all covariances are computed by covariance propagation. The covariance function can be written as

### 2.3 Spherical splines

RBFs are isotropic functions which store most of their energy in a limited spherical cap around their origin, i.e., they have a distinct space-localizing ability and are therefore said to have quasi-local support (Freeden et al. 1998).

*k*, and \(\hat{d}_k\) are point-specific gravity field parameters in the form of dimensionless coefficients.

*P*and the origin of the SK at grid point

*k*, and \(\lambda _n\) are the spectral eigenvalues including dimensioning of the considered functional (e.g., \(\lambda _n\) is set to \(\lambda _n^N\) in case of geoid computation). The Legendre coefficients are given by

*n*and order

*m*, as defined by Schmidt (2001). \(\bar{P}_{nm}(\cos {\theta })\) are the fully normalized associated Legendre functions.

We note that the Legendre coefficients of the spherical splines differ slightly from the ones of the covariance function, and this is due to norm convergence issues (Eicker 2008).

We determine the regularization parameter \(\alpha \) by the L-curve method proposed by Hansen and O’Leary (1993), as it has proven to be a suitable method for noise-free data (Bentel 2013). Regardless of the regularization method, an initial guess of the regularization parameter, \(\alpha _0\), must be made. Here, we make an initial guess based on the condition number of the normal matrix and the maximum SH degree \(N_\mathrm {max}\), i.e., \(\alpha _0=8||\mathbf {N}||/N_\mathrm {max}^3\), where \(||\mathbf {N}||\) is the norm of the normal matrix \(\mathbf {N}=\mathbf {A}^T\mathbf {A}\) (Naeimi 2013).

In case of signal representation using SKs, \(\mathbf {R}\) contains scalar products of the SKs located at different grid points (Eicker 2008). If the SKs would be orthogonal, \(\mathbf {R}\) would become a diagonal matrix which could be represented by a scaled identity matrix. If the scaling factor is combined with \(\alpha \), we can set \(\mathbf {R}=\mathbf {I}\).

For bandlimited signals, as we employ them in discrete matrix computations, the SKs are not strictly orthogonal, and therefore, \(\mathbf {R}\) is not a diagonal matrix (Eicker 2008). For non-bandlimited signals, orthogonality can theoretically be achieved, but at the cost of infinite energy along the main diagonal of \(\mathbf {R}\). Eicker (2008) discussed different modifications which restrict the elements along the main diagonal to finite values. Among them is the modification of the a priori signal covariance function, which is used to define the smoothness of the solution. This modification could possibly lead to our SK solution becoming inconsistent with LSC, because implicitly, two different covariance functions are used. The solutions of both methods would not represent the same gravity field.

Eicker (2008) compared different modified and unmodified solutions using either a fully occupied \(\mathbf {R}\), or the approximation \(\mathbf {R}=\mathbf {I}\), and found that the considerations regarding infinite energy and non-orthogonality are of rather theoretical significance. The best solutions were in fact achieved using the approximation \(\mathbf {R}=\mathbf {I}\). Even though the numerical experiments of Eicker (2008) are related to downward continuation of satellite gravity data of relatively low spectral resolution, our own numerical examples (Sect. 5) indicate that this approximation also holds for higher spectral resolutions.

## 3 Global equivalence

In the global case, Eqs. (1), (3), and (7) are equivalent gravity field representations. Furthermore, all three are equal to SH, which is our point of departure.

*T*on Earth’s surface is a harmonic function satisfying Laplace’s equation. Its solution may be formulated as a spherical harmonic expansion,

*T*, subject to the boundary condition as given by the well-known spherical fundamental equation of physical geodesy,

*T*and \(\varDelta g\) as

*i*gravity anomalies \(\varDelta g_i\), we can predict gravity anomalies \(\varDelta g_q\) in any point

*q*on Earth, by means of LSC. Then, we can rewrite Eq. (1) as

*i*do not depend on the integration element,

We now follow the interpretation of Eicker (2008) to show that the same prediction as in Eq. (3) can be done in RBF representation using SKs.

*K*equidistant points

*k*. In the limit, the sum will become the integral over the unit sphere, supposing an infinitely dense distribution of SKs. We get

Finally, it may be shown that Tikhonov regularization and LSC are formally equivalent if we consider observation noise \(\varepsilon \), giving \(\bar{\mathbf {C}}_{qq}^{gg}=\mathbf {C}_{qq}^{gg}+\mathbf {C}_{\varepsilon \varepsilon }^{gg}\). Therefore, \(\mathbf {C}_{\varepsilon \varepsilon }^{gg}=\sigma ^2\mathbf {I}=\mathbf {R}\) is interpreted as the regularization matrix. Thus, LSC is equal to Tikhonov regularization with \(\alpha =1\), and LSC may be considered a particular form of regularization with a statistical rationale for determining the regularization matrix. The interested reader is referred to Rummel et al. (1979) or Bouman (1998) for more on this topic.

## 4 Regional applications

As seen in Sect. 3, all methods are equivalent in the global case, and they can, in principle, be applied globally. This, however, is not practical due to the requirement of globally distributed high-resolution gravity data. In addition, global integration using Stokes’s integral, the LSC formula applied to large data sets, as well as the global analysis of spline coefficients all require huge computational efforts.

Terrestrial gravity data of high resolution are not available globally, while GGMs have global coverage, but lack high resolution. Consequently, in practical regional geoid computation, both data sources are combined. Therefore, the long-wavelength part of the gravity signal is determined from a GGM and removed from the terrestrial data. In turn, regional geoid computation is applied to the residual gravity data in a limited area only. The modeling results are residual geoid heights, and the long-wavelength part of the GGM is restored to obtain the final geoid. This procedure is commonly referred to as the *remove–restore* technique (Denker 2013).

Considering the two-step interpretation of LSC in Sect. 3, we see that if Eq. (3) is applied to residual data in the inner zone only, we also include the implicitly extrapolated gravity signal outside the inner zone. Thus, as demonstrated theoretically and numerically by de Min (1995), LSC and Stokes’s formula are not equivalent in regional applications.

## 5 Numerical examples

Test regions

East Frisia | Alpine region | |
---|---|---|

Data area | \(52^\circ \le \varphi \le 55^\circ \) | \(46^\circ \le \varphi \le 49^\circ \) |

\(5^\circ \le \lambda \le 10^\circ \) | \(7.5^\circ \le \lambda \le 13.5^\circ \) | |

Target area | \(53^\circ \le \varphi \le 54^\circ \) | \(47^\circ \le \varphi \le 48^\circ \) |

\(6.5^\circ \le \lambda \le 8.5^\circ \) | \(9^\circ \le \lambda \le 12^\circ \) |

Spline representation

Resolution (arcmin) | East Frisia | Alpine region | ||
---|---|---|---|---|

5 | 2.5 | 5 | 2.5 | |

No. of observations | 2257 | 8833 | 2701 | 10,585 |

No. of SKs | 1842 | 1842 | 2464 | 2464 |

\(\mathrm {cond}(\mathbf {N})\) | \(1.2\times 10^{19}\) | \(9.3\times 10^{18}\) | \(5.7\times 10^{19}\) | \(7.1\times 10^{19}\) |

\(\alpha _0\) | 7115 | \(2.8\times 10^{4}\) | 6872 | \(2.7\times 10^{4}\) |

\(\alpha \) | 72 | 284 | 6837 | \(1.1\times 10^{4}\) |

\(\mathrm {cond}(\mathbf {N}+\alpha \mathbf {I})\) | \(6.0\times 10^7\) | \(6.0\times 10^7\) | \(6.0\times 10^5\) | \(1.5\times 10^{6}\) |

We have considered two regions, namely the North Sea coastal region of East Frisia and the mountainous Alpine region, with, respectively, smooth and moderately rough topography. For practical computational reasons, the input and output grid resolutions (directly related to the number of observations) were set to 5 arcmin (corresponding to the maximum resolution of EGM2008), and the radius of the spherical integration cap was set to \(\psi _0=1^\circ \). This cap gives theoretical omission errors of approximately 2 and 6 cm for East Frisia and the Alpine region, respectively (approximated by computing the contribution to *N* from the area outside the inner zone using EGM2008, with \(251 \le n \le 2190\)). However, the omission error is not relevant in our comparison as it is equal for all methods. Around the target areas, we consider enlarged data areas such that each computation point in the target area is surrounded by a \(1^\circ \) spherical cap containing data, see Table 1.

Geoid heights by Stokes’s formula were computed using Eq. (33), implemented according to Eq. (2) using the closed formula for computing the Stokes function (where the function values were set to zero outside the inner zone). Geoid heights by LSC were computed using Eqs. (36) and (37). Equation (37) was developed to degree 2190, corresponding to the maximum resolution of the observations (5 arcmin). Considering the SKs, dimensionless spline coefficients were estimated using Eq. (16) with Eq. (14). Details regarding the stability of the linear system are shown in Table 2. Subsequently, the spline coefficients were used to compute geoid heights using Eq. (11) with Eq. (39). Similar to LSC, the SKs were developed to degree 2190.

*w*of the RBF grid area were determined by the empirical formula of Bentel (2013), where \(w\approx 4\cdot 180^\circ /(N_\mathrm {max}+1)\). The number of RBFs will typically be slightly smaller than the number of observations, but approximately equal. In the following, when the equality of the number of RBFs and the number of observations is discussed, it is this approximate equality that is meant.

Results from the closed-loop simulation, 5 arcmin resolution

East Frisia | Alpine region | |||||||
---|---|---|---|---|---|---|---|---|

Max | Min | Mean | RMS | Max | Min | Mean | RMS | |

Data area | ||||||||

\(\varDelta g_\mathrm{SHS}\) | 11.617 | \(-\)6.881 | 0.316 | 3.242 | 46.704 | \(-\)63.217 | \(-\)0.677 | 12.762 |

\(N_\mathrm{SHS}\) | 17.474 | \(-\)10.942 | 0.513 | 5.200 | 66.092 | \(-\)93.867 | \(-\)1.013 | 18.765 |

\(N_\mathrm{SHS}-N_\mathrm{Stokes}\) | 14.165 | \(-\)5.426 | 1.204 | 3.410 | 28.229 | \(-\)35.504 | \(-\)2.477 | 7.218 |

\(N_\mathrm{SHS}-N_\mathrm{LSC}\) | 1.880 | \(-\)2.594 | 0.006 | 0.644 | 7.548 | \(-\)7.952 | \(-\)0.007 | 1.354 |

\(N_\mathrm{SHS}-N_\mathrm{Splines}\) | 2.118 | \(-\)3.256 | \(-\)0.024 | 0.795 | 6.518 | \(-\)6.345 | \(-\)0.011 | 1.434 |

Target area | ||||||||

\(N_\mathrm{SHS}\) | 7.297 | \(-\)9.076 | 1.870 | 4.241 | 40.613 | \(-\)35.447 | \(-\)1.669 | 13.678 |

\(N_\mathrm{SHS}-N_\mathrm{Stokes}\) | 0.419 | \(-\)0.638 | 0.082 | 0.173 | 1.087 | \(-\)1.598 | \(-\)0.087 | 0.588 |

\(N_\mathrm{SHS}-N_\mathrm{LSC}\) | 0.003 | \(-\)0.020 | \(-\)0.001 | 0.002 | 0.212 | \(-\)0.208 | 0.000 | 0.065 |

\(N_\mathrm{SHS}-N_\mathrm{Splines}\) | 0.010 | \(-\)0.044 | \(-\)0.003 | 0.007 | 0.162 | 0.017 | 0.107 | 0.113 |

The results of the geoid computations using modified formulas are shown in Table 3 and Figs. 4 and 5. Table 3 shows that all methods agree on the mm level in the target area, with smaller errors in East Frisia than in the Alpine region. Stokes’s formula gives larger errors outside the target area than LSC and the SKs. In the target areas, LSC and SKs show maximum RMS differences of 0.7 and 1.1 mm (both in the Alpine region), respectively, while RMS differences of 1.7 mm (East Frisia) and 5.9 mm (Alpine region) are found using Stokes’s formula. Of all three methods, LSC gives the smallest error.

Results from the closed-loop simulation, 2.5 arcmin resolution

East Frisia | Alpine region | |||||||
---|---|---|---|---|---|---|---|---|

Max | Min | Mean | RMS | Max | Min | Mean | RMS | |

Data area | ||||||||

\(\varDelta g_\mathrm{SHS}\) | 11.794 | \(-\)6.894 | 0.296 | 3.219 | 46.736 | \(-\)63.347 | 0.634 | 12.723 |

\(N_\mathrm{SHS}\) | 17.536 | \(-\)10.942 | 0.481 | 5.158 | 66.515 | \(-\)94.385 | \(-\)0.955 | 18.729 |

\(N_\mathrm{SHS}-N_\mathrm{Stokes}\) | 14.422 | \(-\)5.894 | 1.032 | 3.273 | 30.577 | \(-\)37.650 | \(-\)2.107 | 7.056 |

\(N_\mathrm{SHS}-N_\mathrm{LSC}\) | 3.717 | \(-\)3.266 | 0.031 | 0.996 | 12.432 | \(-\)10.639 | \(-\)0.051 | 2.098 |

\(N_\mathrm{SHS}-N_\mathrm{Splines}\) | 2.317 | \(-\)3.385 | \(-\)0.022 | 0.786 | 4.377 | \(-\)5.683 | \(-\)0.019 | 1.257 |

Target area | ||||||||

\(N_\mathrm{SHS}\) | 7.333 | \(-\)9.076 | 2.097 | 4.247 | 41.130 | \(-\)35.447 | \(-\)1.936 | 13.620 |

\(N_\mathrm{SHS}-N_\mathrm{Stokes}\) | 0.429 | \(-\)0.512 | 0.037 | 0.088 | 0.529 | \(-\)0.663 | \(-\)0.041 | 0.242 |

\(N_\mathrm{SHS}-N_\mathrm{LSC}\) | 0.057 | \(-\)0.103 | \(-\)0.001 | 0.010 | 0.059 | \(-\)0.054 | 0.000 | 0.015 |

\(N_\mathrm{SHS}-N_\mathrm{Splines}\) | \(-\)0.002 | \(-\)0.040 | \(-\)0.004 | 0.006 | 0.110 | 0.024 | 0.079 | 0.082 |

Finally, we explore how the number of SKs affects the RMS differences (and thus the equivalence to the other methods). In Sect. 3, LSC and SKs were found to be theoretically equal in the continuous case, respectively, in case of an infinitely dense distribution of SKs. In the following, we want to explore how dense the point grid needs to be in practical applications. We repeat the computations for the Alpine region, still using gravity anomalies with 5 arcmin signal resolution given on the 5 arcmin grid, but varying the number of SKs. The regularization parameter was kept constant to make the computations consistent, securing that the only variable in the test is the number of SKs. Again, the RMS differences, or modeling errors, are with respect to the truncated SH validation geoid. If the error is small for a certain number of SKs, we can conclude that the SKs are also similar to LSC, thus providing a measure of how dense the SKs need to be placed in practice. Because we already know that the different methods agree on the mm level, we have chosen 1 mm as the modeling error threshold.

Figure 6 shows the modeling error as a function of the number of SKs. The vertical dotted lines denote the number of observations corresponding to the 5 arcmin and 2.5 arcmin grid resolutions, while the horizontal dotted line denotes the 1-mm modeling error threshold with respect to SHS. We observe that the error decreases rapidly with increasing number of SKs until it reaches the 1-mm threshold using roughly 3000 SKs. This corresponds to the number of grid points of the 5 arcmin input grid and, in our case, represents both the number of observations (input grid resolution) and the signal resolution (spectral content). To test which of the two is decisive, we computed solutions using a denser 2.5 arcmin grid while keeping the signal resolution constant at 5 arcmin (leading to a slightly oversampled signal). Based on this input data grid, we performed two modeling runs, where the number of SKs corresponds to either (1) the signal resolution (corresponding to 5 arcmin, see Table 2) or (2) the number of observations (9738 SKs, corresponding to 2.5 arcmin). Both cases give practically equal results (RMS differences between SK geoid solutions of 1 mm and \(5\times 10^{-2}\) mm in the data and target zones, respectively), suggesting that the SKs converge toward LSC depending on the signal resolution rather than the number of observations. Using more observations increases redundancy and helps to filter out observation noise, but it is not necessary to use an equally increased number of SKs to adequately represent the gravity field.

## 6 Summary

We have reviewed the theoretical equivalence of Stokes’s formula, LSC, and SKs in the global case, as well as in regional applications, where Stokes integration is restricted to a spherical cap around the computation point, and no data outside this cap is considered. If LSC is not applied globally, its result will be different from Stokes’s formula, because an unwanted extrapolation outside the cap takes place. If the cross-covariance function is modified appropriately, LSC and Stokes’s formula are again equal. This has already been shown by de Min (1995). As SKs are equivalent to LSC, they have to be modified correspondingly to give equal results as Stokes’s formula.

With a few numerical examples, we have shown that the methods are equal also in practice. Two regions were considered, East Frisia and the Alpine region, with small and large gravity field variations, respectively. At the 5 arcmin resolution, all methods agree within \(2\times 10^{-2}\) mm to 5.9 mm in the target areas, where the largest RMS differences are due to the discretization of Stokes’s formula. At the 2.5 arcmin resolution, all methods agree within \(6\times 10^{-2}\) mm to 2.4 mm. In general, the remaining discrepancies can be expected due to the varying numerical implementations. For example, Novák et al. (2001) found remaining differences on the mm level between the theoretically equivalent numerical integration and 1D-FFT evaluations of Stokes’s formula at the 5 arcmin resolution, which they attribute to the numerical accuracy of their implemented algorithms.

From a theoretical point of view, LSC should give the best results because the covariance function is the only kernel which has the minimum variance property (Moritz 1980). Indeed, this is confirmed in our numerical examples, where LSC generally gives the smallest error. SKs perform very similar to LSC. From the theoretical point of view, both methods should be identical in the continuous case, i.e., when placing the SKs on an infinitely dense point grid. Our numerical examples show that this limit case does not have to be reached in practice. Rather, it suffices to choose the SK grid spacing according to the signal resolution. This is directly related to the ability of the SKs to achieve an adequate representation of the gravity field. Using less SKs than associated with the signal resolution leads to an inadequate gravity field representation, i.e., the modeling error increases with decreasing number of SKs. Using more SKs than associated with the signal resolution leads to overparametrization. This generally increases the numerical effort (inversion of an unnecessarily large matrix) and may lead to instabilities of the normal equation system. Thus, the SKs are principally independent of the number and distribution of observations. When measuring the numerical effort of SKs and LSC in terms of the size of the matrix that needs to be inverted (normal equation matrix in case of the SKs and the auto-covariance matrix of the observations in case of LSC), the effort needed for LSC depends on the number of observations, while the effort needed for the SKs depends on the signal resolution.

In practice, of course, signal and grid resolutions are interrelated. If the distance between adjacent observation points decreases, more small-scale features of the signal can be modeled, which in turn requires a denser grid of SKs. The limit is reached at the point where there are no significant signal variations between the observations. At this point, the number of observations is sufficiently high to represent the full spectral content of the signal. Then, the numerical effort for both SKs and LSC is similar. If the number of observations is higher (which could help filtering out observational noise), the numerical effort increases for LSC, but not for SKs. Furthermore, in reality, the distribution of observations is typically heterogeneous, and placing the SKs on a regular grid will never give optimal results (Wittwer 2009). One possibility to overcome this problem would be to determine the grid spacing for the SKs depending on the resolution necessary to reach a certain accuracy requirement for the final modeling results (e.g., a 1-cm geoid). Another possibility could be to choose a scattered placement of the SKs, where more SKs are used for parts of the data set where the signal is rough, and less in parts where the signal is smoother. Different data-adaptive approaches have been used to generate such a scattered distribution, see, for example, Klees et al. (2008).

To complete the discussion on spacing between and distribution of the SKs, we also stress that the spacing and distribution are not the only variables in the RBF approach, but also the bandwidth (i.e., the maximum degree to which the SKs are developed) needs to be adapted according to the signal resolution (Wittwer 2009). This was not explicitly discussed in our numerical examples, because the signal resolution and bandwidth were always set to \(N_\mathrm {max}=2190\).

Finally, we point out that our aim has not been to decide which regional geoid computation method is the best, as there are advantages and drawbacks to all depending on the data situation. Furthermore, our numerical examples do not represent the most efficient implementations of each method, as they rather aim to compare the outcome of the different methods. Thus, we have not compared the numerical implementations in terms of computation times and limitations. In addition, this work does not present new theory, but is a first attempt to compare the three methods both theoretically and numerically in a unified framework. We have demonstrated that the three methods give equal results in applications, for which a modification of LSC and SKs was necessary. We stress that this modification is not a general necessity when applying the LSC and RBF approaches. However, de Min (1995) points out a few advantages of modifying the covariance function. For example, the modification makes the LSC results less dependent on the validity of the covariance function, as its validity is of most importance for the extrapolated data, which is not considered anymore. Although the modification is not necessary for the application of LSC and RBFs in general, it is critical in direct comparison with Stokes’s formula.

## Notes

### Acknowledgements

The authors would like to thank R. Rummel and M. Šprlák for helpful comments on an early version of the manuscript. It was further improved by constructive comments from the Editor-in-Chief and three anonymous reviewers, which are gratefully acknowledged. Figures were drafted using the M_Map package, with coastlines and political boundaries from the National Oceanic and Atmospheric Administration and Natural Earth, respectively.

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