Keywords

1 Introduction

Boundary-value problems (BVPs) and their solutions represent an important tool for describing and modelling potential fields such as the Earth’s gravitational field. Solutions to spherical geodetic BVPs lead to spherical harmonic series or surface convolution integrals with Green’s kernel functions. New BVPs have recently been formulated reflecting the development of new sensors. BVPs have also been developed for observables measured by kinematic sensors on moving platforms, i.e., airplanes and satellites. Solutions to BVPs for higher-order gradients of the gravitational potential as boundary conditions are represented by multiple integral transforms. For example, solutions to gravimetric BVP are represented by two integral transforms (Grafarend 2001, Eqs. (199) and (142)):

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & T^{(V)}(r,\varOmega)\\ & \quad ={}-\frac{R}{4\pi}\int_{\varOmega'}\Bigg\{\sum_{n=3}^{N_{max}}\frac{2n+1}{n+1}\left(\frac{R}{r}\right)^{n+2}P_{n,0}\left(\cos\psi\right)\Bigg\}\\ & \quad \quad \times T_{z}(R,\varOmega')\mathrm{d}\varOmega'\,, \end{aligned} \end{array} \end{aligned} $$
(1)
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & T^{(H)}(r,\varOmega)\\ & \quad ={}\frac{R}{4\pi}\int_{\varOmega'}\Bigg\{\sum_{n=3}^{N_{max}}\frac{2n+1}{n(n+1)}\left(\frac{R}{r}\right)^{n+2}P_{n,1}\left(\cos\psi\right)\Bigg\}\\ & \quad \quad \times\left[-T_{x}(R,\varOmega')\cos\alpha'+T_{y}(R,\varOmega')\sin\alpha'\right]\mathrm{d}\varOmega'\,, \end{aligned} \end{array} \end{aligned} $$
(2)

to the gradiometric BVP by three integral transforms (Martinec 2003, Eq. (19)):

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & T^{(VV)}(r,\varOmega)\\ & \; {=}\,{}\frac{R^2}{4\pi}\int_{\varOmega'}\Bigg\{\sum_{n=3}^{N_{max}}\frac{2n+1}{(n+1)(n+2)}\left(\frac{R}{r}\right)^{n+3}\!\!\!P_{n,0}\left(\cos\psi\right)\Bigg\}\;\;\\ & \;\quad \times T_{zz}(R,\varOmega')\mathrm{d}\varOmega'\,, \end{aligned} \end{array} \end{aligned} $$
(3)
(4)
(5)

and to the gravitational curvature BVP by four integral transforms (Šprlák and Novák 2016, Eqs. (49)–(52)):

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & T^{(VVV)}(r,\varOmega)\\ & \;{=}\,{}-\frac{R^3}{4\pi}\int_{\varOmega'}\\ & \;\;{\times}\;\Bigg\{\sum_{n=3}^{N_{max}}\frac{2n+1}{(n+1)(n+2)(n+3)}\left(\frac{R}{r}\right)^{n+4}\!\!\!\!\!P_{n,0}\left(\cos\psi\right)\Bigg\}\quad \\ & \;\;{\times}\; \;T_{zzz}(R,\varOmega')\mathrm{d}\varOmega'\,, \end{aligned} \end{array} \end{aligned} $$
(6)
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & T^{(VVH)}(r,\varOmega)\\ & \;{=}\,{}\frac{R^3}{4\pi}\int_{\varOmega'}\\ & \;{\times}\,\Bigg\{\sum_{n=3}^{N_{max}}\frac{2n+1}{n(n+1)(n+2)(n+3)}\left(\frac{R}{r}\right)^{n+4}\!\!\!P_{n,1}\left(\cos\psi\right)\Bigg\}\\ & \;{\times}\,\left[T_{xzz}(R,\varOmega')\cos\alpha'-T_{yzz}(R,\varOmega')\sin\alpha'\right]\mathrm{d}\varOmega'\,, \end{aligned} \end{array} \end{aligned} $$
(7)
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & T^{(VHH)}(r,\varOmega)\\ & \quad ={}-\frac{R^3}{4\pi}\int_{\varOmega'}\\ & \quad \times\Bigg\{\sum_{n=3}^{N_{max}}\frac{2n+1}{(n-1)n(n+1)(n+2)(n+3)}\left(\frac{R}{r}\right)^{n+4}\\ & \qquad \quad \times P_{n,2}\left(\cos\psi\right)\Bigg\}\\ & \quad \times\left[\left(T_{xxz}(R,\varOmega')-T_{yyz}(R,\varOmega')\right)\cos2\alpha'\right.\\ & \qquad \left.-2T_{xyz}(R,\varOmega')\sin2\alpha'\right]\mathrm{d}\varOmega'\,, \end{aligned} \end{array} \end{aligned} $$
(8)
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \hspace{-3pt}\begin{aligned} & T^{(HHH)}(r,\varOmega)\\ & \quad \!\!\!={}-\frac{R^3}{4\pi}\int_{\varOmega'}\\ & \quad \!\!\times\Bigg\{\sum_{n=3}^{N_{max}}\frac{2n+1}{(n-2)(n-1)n(n+1)(n+2)(n+3)}\\ & \qquad \quad \times\left(\frac{R}{r}\right)^{n+4} P_{n,3}\left(\cos\psi\right)\Bigg\}\\ & \quad \!\!\times\left[\left(T_{xxx}(R,\varOmega')-3T_{xyy}(R,\varOmega')\right)\cos3\alpha'\right.\\ & \quad \quad \!\!\left.+\left(T_{yyy}(R,\varOmega')-3T_{xxy}(R,\varOmega')\right)\cos3\alpha'\right]\mathrm{d}\varOmega'\,. \end{aligned} \end{array} \end{aligned} $$
(9)

The notation in the previous equations is defined as follows. An Earth-fixed coordinate system is used with the geocentric radius r, spherical latitude \(\varphi \) and longitude \(\lambda \). Moreover, in a point positioned in the 3-D space by a triplet of the spherical coordinates \((r,\varphi ',\lambda ') = (r,\varOmega ')\), a right-handed local Cartesian system is defined with the z-axis aligned with the geocentric radius and pointing outwards, and the x-axis pointing to the geodetic North. The symbol T represents the disturbing potential and components of the first-, second- and third-order gradient tensors are \(T_i, T_{ij}\) and \(T_{ijk}\) (with indexes \(i, j, k\) running over the Cartesian coordinates \(x, y, z\)). The spherical distance \(\psi \) and the backward azimuth \(\alpha '\) are defined between the computation point \((r,\varOmega )\) and the integration point \((R,\varOmega ')\) located at the mean Earth’s sphere with the radius R. \(P_{n,m}\) are the associated Legendre functions of the degree n and order m. The minimum degree was homogenized to the smallest common degree \(n = 3\) recoverable from all gradients. The series in Eqs. (1) to (9) theoretically extends to infinity; however, it will always be truncated at the degree \(N_{max}\) for the satellite data where the gravitational signal is attenuated. Note that expressions in curly brackets represent the integral kernel functions in the spectral forms. The superscripts in Eqs. (1) to (9) stand for the name of the corresponding solution (V —vertical, H—horizontal) and could be expressed by a more general index i. For example, the superscript V HH means the vertical-horizontal-horizontal solution. The solutions of geodetic BVPs defined by Eqs. (1) to (9) represent the direct problem. The components of the first-, second- and third-order gradient tensors \(T_i, T_{ij}\) and \(T_{ijk}\) are located at the mean sphere while the unknown disturbing potential is estimated at the sphere with radius r (\(r>R\)).

2 Spectral Combination

The goal of this study is to apply Eqs. (1) to (9) for downward continuation (DWC), i.e., to estimate values of the disturbing potential at the mean Earth’s sphere with the radius of R from the components of the first-, second- and third-order gradient tensors \(T_i, T_{ij}\) and \(T_{ijk}\) located at the mean orbital sphere. The well-known example of such a problem is an estimation of the spherical harmonic coefficients from the satellite observables. To do so, we change the ratio \((R/r)\), called an attenuation factor, to \((r/R)\) in Eqs. (1) to (9). Further, the arguments in brackets on the left-hand side of Eqs. (1) to (9) from \((r,\varOmega )\) to \((R,\varOmega )\) and the arguments related to the components of the first-, second- and third-order gradient tensors \(T_i, T_{ij}\) and \(T_{ijk}\) on the right-hand side from \((R,\varOmega ')\) to \((r,\varOmega ')\). Note that in the same way we modified Eqs. (1) to (9) but their exact formulas are omitted here and we provide two examples in terms of the simplest V  solution and the most complicated HHH solution modified for the DWC:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & T^{(V)}(R,\varOmega)\\ & \;\,{=}\,{}-\frac{R}{4\pi}\int_{\varOmega'}\Bigg\{\sum_{n=3}^{N_{max}}\frac{2n+1}{n+1}\left(\frac{r}{R}\right)^{n+2}P_{n,0}\left(\cos\psi\right)\Bigg\}\\ & \;\,{\times}\, T_{z}(r,\varOmega')\mathrm{d}\varOmega'\,, \end{aligned} \end{array} \end{aligned} $$
(10)
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & T^{(HHH)}(R,\varOmega)\\ & \;={}-\frac{R^3}{4\pi}\int_{\varOmega'}\\ & \;\times\Bigg\{\sum_{n=3}^{N_{max}}\frac{2n+1}{(n-2)(n-1)n(n+1)(n+2)(n+3)}\left(\frac{r}{R}\right)^{n+4}\\ & \qquad \times P_{n,3}\left(\cos\psi\right)\Bigg\}\\ & \;\times\left[\left(T_{xxx}(r,\varOmega')-3T_{xyy}(r,\varOmega')\right)\cos3\alpha'\right.\\ & \;\quad \left.+\left(T_{yyy}(r,\varOmega')-3T_{xxy}(r,\varOmega')\right)\cos3\alpha'\right]\mathrm{d}\varOmega'\,. \end{aligned} \end{array} \end{aligned} $$
(11)

The geometry of this problem is depicted in Fig. 1. Note that all symbols used in this figure are explained in the text above. The problem presented by modified Eqs. (1) to (9) represents DWC. To solve it, we need to control the signal-to-noise ratio of results. Among many methods, we decided to apply the least-squares spectral combination method developed for combination of then-available gravity data by Sjöberg (1980) and Wenzel (1982). Since then it has been used by many scholars for geoid determination. The very first publication, which discussed the application of the spectral combination method for combining solutions of boundary-value problems (BVPs) of the potential theory, was published by Eshagh (2011). Using the spectral combination method, Eshagh (2012) combined three analytical solutions to the spherical gradiometric BVP and applied the spectral combination method for DWC of second-order gradients of the gravitational potential simulated at the satellite orbit. The spectral combination of solutions to the spherical gravitational curvature BVP for estimation of the gravitational potential was investigated by Pitoňák et al. (2018). The method can be used not only for combination of various data types but it can also continue observables from an observation level down to the irregular Earth’s surface (or elsewhere as long as the harmonicity of the gravitational potential is guaranteed) and transform them to corresponding gravitational field quantity, e.g., Sjöberg and Eshagh (2012), Eshagh (2012) or Pitoňák et al. (2018). Despite DWC being an inverse problem, the method does not need any matrix inversion and the signal-to-noise ratio of results is controlled by spectral weights. However, DWC can still be problematic as one continues gradient data inside a geocentric sphere which encloses completely Earth’s masses (Brillouin’s sphere). In this space, an external form of a spherical harmonic series representing the gravitational potential may not be converging.

Fig. 1
figure 1

Geometry of the downward continuation of satellite data

Full size image

In this study, we apply the spectral combination method to all gradients of the gravitational potential of up to the third order. The method combines nine solutions of the respective BVPs, two for the gravimetric BVP, three for the gradiometric BVP and four for the third-order gravitational curvature BVP. The integral kernel functions in Eqs. (1) to (9) can be modified by adding spectral weights \(a_n\) which respect signal and error degree variances of measured gravitational gradients. Moreover, the various solutions can be combined in the spectral domain which provides a minimum expected global mean-square error of estimated parameters. In order to obtain height anomalies we apply the well-known Bruns’s equation (e.g., Heiskanen and Moritz 1967, Eq. 2–144, p. 85) to Eqs. (1)–(9).

The height anomaly estimator based on a single gradient group has the spectral form:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \zeta^{(i)}(\varOmega) = \frac{1}{\gamma}\sum_{n=3}^{N_{max}}\ a_{i,n}\ b_{i,n}\ T_{i,n}(\varOmega)\ , \end{array} \end{aligned} $$
(12)

which is obtained from spherical harmonics \(T_{i,n}\) derived by spherical analysis of the gradient group \(T_i\). Spectral weights \(a_{i,n}\) are defined as follows (Eshagh 2012; Pitoňák et al. 2020):

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} a_{i,n} = \frac{c_n\ t_n} {c_n\ t^2_n + \sigma_{i,n}^2\ b_{i,n}^2}\ , \end{array} \end{aligned} $$
(13)

with the signal degree variances of the height anomaly \(c_n\) and the error degree variance of the particular gradient group \(\sigma _{i,n}^2\). Numerical coefficients for the order of the gradient \(\ell \) are:

$$\displaystyle \begin{aligned} \begin{array}{rcl} b_{i,n} = R^{\ell}\ \frac{(n - j)!} {(n + \ell)!}\ ,\ \ell = \{1,2,3\}\ , \end{array} \end{aligned} $$

and the respective attenuation factors \(t_n\) are defined as follows:

$$\displaystyle \begin{aligned} \begin{array}{rcl} t_{n} = \left(\frac{R}{r}\right)^{n+1+\ell}\ ,\ \ell = \{1,2,3\}\ . \end{array} \end{aligned} $$

The index j represents the order of the gradient in the horizontal coordinates x and y (number of repetitions of the index H, j = {0,1,2,3}). Note that in the practical computation we used modified integral transforms. Two examples in terms of the simplest V  and the most complicated HHH solutions are:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & \zeta^{(V)}(R,\varOmega)\\ & \;={}-\frac{R}{4\pi\gamma}\int_{\varOmega'}\Bigg\{\sum_{n=3}^{N_{max}}\frac{\left(2n+1\right)a_{n}^{V}}{n+1}P_{n,0}\left(\cos\psi\right)\Bigg\}\\ & \;\quad \times T_{z}(r,\varOmega')\mathrm{d}\varOmega'\,, \end{aligned} \end{array} \end{aligned} $$
(14)
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & \zeta^{(HHH)}(R,\varOmega)\\ & \;={}-\frac{R^3}{4\pi\gamma}\int_{\varOmega'}\\ & \;\times\Bigg\{\sum_{n=3}^{N_{max}}\frac{\left(2n+1\right)a_{n}^{HHH}}{(n-2)(n-1)n(n+1)(n+2)(n+3)}\\ & \;\quad \quad \times P_{n,3}\left(\cos\psi\right)\Bigg\}\\ & \;\times\left[\left(T_{xxx}(r,\varOmega')-3T_{xyy}(r,\varOmega')\right)\cos3\alpha'\right.\\ & \;\quad \left.+\left(T_{yyy}(r,\varOmega')-3T_{xxy}(r,\varOmega')\right)\cos3\alpha'\right]\mathrm{d}\varOmega'\,. \end{aligned} \end{array} \end{aligned} $$
(15)

Two or more gradient groups can be then combined in the spectral domain. The combined solution based on two gradient groups reads:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \zeta^{(i,j)}(\varOmega) & =&\displaystyle \frac{1}{\gamma}\sum_{n=3}^{N_{max}} \left[\ a^{(i,j)}_n\ b_{i,n}\ T_{i,n}(r,\varOmega)\right.\\ & &\displaystyle \left.+\; a_n^{(j,i)}\ b_{j,n}\ T_{j,n}(r,\varOmega)\ \right]\ ,\quad \end{array} \end{aligned} $$
(16)

with the respective spectral weights defined as follows (Eshagh 2012):

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} a_n^{(i,j)} = \frac{\overline{\sigma}_{j,n}^2}{t_n\ (\overline{\sigma}_{i,n}^2 + \overline{\sigma}_{j,n}^2)}\ ,\ a_n^{(j,i)} = \frac{\overline{\sigma}_{i,n}^2}{t_n\ (\overline{\sigma}_{i,n}^1 + \overline{\sigma}_{j,n}^2)}\ ,\quad \end{array} \end{aligned} $$
(17)

with

$$\displaystyle \begin{aligned} \begin{array}{rcl} \overline{\sigma}_{i,n}^2 = b_{i,n}^2\ \sigma^2_{i,n}\ ,\ \overline{\sigma}_{j,n}^2 = b_{j,n}^2\ \sigma^2_{j,n}\ . \end{array} \end{aligned} $$

In contrary to a single group estimator in Eq. (12), which is based on the signal degree variances \(c_n\) of estimated parameters, the two- and more-group estimators are based only on the error degree variances of input data. We present their unbiased forms herein. Combining two gradient groups in Eq. (16) yields already 36 different combined solutions. One example for the combination of the V  and H gradient groups in the integral form is:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & \zeta^{(V,H)}(R,\varOmega)\\ & \;={}-\frac{R}{4\pi\gamma}\int_{\varOmega'}\Bigg\{\sum_{n=3}^{N_{max}}\frac{\left(2n+1\right)a_{n}^{V,H}}{(n+1)}P_{n,0}\left(\cos\psi\right)\Bigg\}\\ & \;\quad \times T_{z}(r,\varOmega')\mathrm{d}\varOmega'\\ & \;\quad +\frac{R}{4\pi\gamma}\int_{\varOmega'}\Bigg\{\sum_{n=3}^{N_{max}}\frac{\left(2n+1\right)a_{n}^{H,V}}{n(n+1)}P_{n,1}\left(\cos\psi\right)\Bigg\}\\ & \;\quad \times\left[-T_{x}(r,\varOmega')\cos\alpha'+T_{y}(r,\varOmega')\sin\alpha'\right]\mathrm{d}\varOmega'\,, \end{aligned} \end{array} \end{aligned} $$
(18)

Similarly, the solution based on three gradient groups reads:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} & &\displaystyle \zeta^{(i,j,k)}(\varOmega)\\ & &\displaystyle \quad ={}\frac{1}{\gamma}\sum_{n=3}^{N_{max}}\ \left[\ a^{(i,j,k)}_n\ b^{(i)}_n\ T_{i,n}(r,\varOmega) + a_n^{(j,i,k)}\ b_n^{(j)}\ \right.\quad \\ & &\displaystyle \qquad \left.\times T_{j,n}(r,\varOmega) + a_n^{(k,i,j)}\ b_n^{(k)}\ T_{k,n}(r,\varOmega)\ \right]\ , \end{array} \end{aligned} $$
(19)

with the spectral weights defined as follows (Eshagh 2012):

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} a_n^{(i,j,k)} & =&\displaystyle \frac{\overline{\sigma}_{j,n}^2\ \overline{\sigma}_{k,n}^2}{t_n\ D_n}\ ,\ a_n^{(j,i,k)} = \frac{\overline{\sigma}_{i,n}^2\ \overline{\sigma}_{k,n}^2}{t_n\ D_n}\ ,\ \\ a_n^{(k,i,j)} & =&\displaystyle \frac{\overline{\sigma}_{i,n}^2\ \overline{\sigma}_{j,n}^2}{t_n\ D_n}\ , \end{array} \end{aligned} $$

and

$$\displaystyle \begin{aligned} \begin{array}{rcl} D_n = \overline{\sigma}_{i,n}^2\ \overline{\sigma}_{j,n}^2 + \overline{\sigma}_{i,n}^2\ \overline{\sigma}_{k,n}^2 + \overline{\sigma}_{j,n}^2\ \overline{\sigma}_{k,n}^2\ . \end{array} \end{aligned} $$

An integral form of the combination of the V , V V  and V V V  gradient groups is:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \begin{aligned} & \zeta^{(V,VV,VVV)}(R,\varOmega)\\ & \; ={}-\frac{R}{4\pi\gamma}\int_{\varOmega'}\Bigg\{\sum_{n=3}^{N_{max}}\frac{\left(2n+1\right)a_{n}^{V,VV,VVV}}{(n+1)}P_{n,0}\left(\cos\psi\right)\Bigg\}\\ & \;\quad \;\,\times T_{z}(r,\varOmega')\mathrm{d}\varOmega'\\ & \;+\frac{R^2}{4\pi\gamma}\int_{\varOmega'}\Bigg\{\sum_{n=3}^{N_{max}}\frac{\left(2n+1\right)a_{n}^{VV,V,VVV}}{(n+1)(n+2)}P_{n,0}\left(\cos\psi\right)\Bigg\}\\ & \;\times T_{zz}(R,\varOmega')\mathrm{d}\varOmega'-\frac{R^3}{4\pi\gamma}\int_{\varOmega'}\\ & \;\times\Bigg\{\sum_{n=3}^{N_{max}}\frac{\left(2n+1\right)a_{n}^{VVV,V,VV}}{(n+1)(n+2)(n+3)}P_{n,0}\left(\cos\psi\right)\Bigg\}\\ & \;\times T_{zzz}(R,\varOmega')\mathrm{d}\varOmega'\,. \end{aligned} \end{array} \end{aligned} $$
(20)

There are 84 solutions based on the three-group estimator in Eq. (19). Its scheme is then presented in Fig. 2. Each integral transform is represented by one line. Here we select 5 possible solutions out of 84. The four- to nine-group estimators can be derived analogously. The four- and five-group estimators provide the maximum number of combined solutions, 126 solutions each. For more groups, the number of combined solutions decreases again with the nine-group estimator providing a single solution. Generally, the number of the combined solutions is given by the factorial coefficient.

Fig. 2
figure 2

Scheme of the three-group estimator

Full size image

3 Numerical Experiments

The spectral combination method was tested using synthetic disturbing gradients derived from the global geopotential model GO_CONS_GCF_2_TIM_R6e (Zingerle et al. 2019) up to the maximum degree \(N_{max} = 250\). The Geodetic Reference System 1980 was used as the normal field (Moritz 2000). In total, 19 disturbing gradients of up to the third order were synthesized at the equiangular coordinate grid with the resolution of 0.2 arc-deg. The global grid was located at the mean satellite orbit with the geocentric radius 6,633,850 m. The error variances were calculated from the formal errors of the applied global geopotential model.

The height anomalies were computed in the area of Himalayas limited by \(\varphi \in [24.75^{\circ },\ 45.25^{\circ }]\) and \(\lambda \in [69.75^{\circ },\ 105.25^{\circ }]\) from global grids of the gradient groups by the spectral combination method at the Brillouin sphere of radius 6,383,850 m, thus safely outside solid Earth’s masses. Brillouin’s sphere was used instead of the real Earth surface since the solutions of corresponding BVPs are based on the external spherical harmonic series of the gravitational potential, and its first- , second- and third-order gradients. To combine the solutions based on groups of the disturbing gradients in the spectral domain, respective spectral weights must be computed first. Figure 3 shows the required signal and error spectra (signal and error degree variances) corresponding to the first-, second- and third-order gradients, respectively. Note that we calculated signal and error spectra from the spherical harmonic coefficients of GO_CONS_GCF_2_TIM_R6e and their uncertainties.

Fig. 3
figure 3

Signal and error degree variances of the first-order gradients (a), the second-order gradients (b) and the third-order gradients (c)

Full size image

Based on the estimated spectral weights, the combined solutions were computed. As the synthetic gradient data have been used, the closed-loop tests could be performed. We divided values of the disturbing potential by normal gravity generated by the mean sphere of radius 6,383,850 m in order to obtain height anomalies. Thus, the numerical results include basic statistics of the differences between values computed by the spectral combination method and their counterparts synthesized directly from the global geopotential model in the bandwidth 3-250. In total, there are 511 combined solutions; thus, only few selected examples can be presented in this study. As it is clear from the statistics for the one-group estimator, see Table 1, the worst fit with respect to the true values was obtained from the first-order vertical gradient \(T_z\), while the best fit was achieved for the second-order gradient group V H and the third-order gradient groups V V H and V HH. From the statistics obtained using the two-group estimator, see Table 2, one can conclude that the more accurate group (based on results from the one-group estimator) improves the less accurate solutions from the one-group estimator. The same pattern can be observed for the three-group estimator as well as for the rest of the gradient groups, see Tables 3 and 4, respectively.

Table 1 Statistics of the differences between estimated values of the height anomaly by the one-group estimators and their reference counterparts synthesized from the global geopotential model (in metres). The indexes \(1, \dots , 9\) stand for the V , H, V V , V H, HH, V V V , V V H, V HH and HHH solutions, respectively
Full size table
Table 2 Statistics of the differences between values of the height anomaly estimated by the selected two-group estimators and their reference counterparts synthesized from the global geopotential model (in metres)
Full size table
Table 3 Statistics of the differences between values of the height anomaly estimated by the selected three-group estimators and their reference counterparts synthesized from the global geopotential model (in metres)
Full size table
Table 4 Statistics of the differences between values of the height anomaly estimated by the selected multi-group estimators and their reference counterparts synthesized from the global geopotential model (in metres)
Full size table

4 Conclusion

The spectral combination method was applied for the estimation of the height anomaly from gradients of the gravitational potential up to the third order as potentially observed by satellites. The method has been applied to synthetic gradient data synthesized at the global equiangular grids from the state-of-the-art global geopotential model which allowed for applying the closed-loop test. Obtained numerical results revealed some interesting properties of the spectral combination method applied to satellite gradients of the geopotential, namely: (i) The best fit was obtained from the mixed vertical and horizontal second- and third-order gradient groups and their respective combination; (ii) Horizontal gradients of the disturbing gravitational potential in the local coordinate frame influence results more than vertical gradients; and (iii) The combination of more than six groups is not beneficial and does not improve obtained solution.