Keywords

1 Introduction

The study of the resolution and approximation accuracy of the global gravity field (EGF) models is a vital task as regards fine structure of the gravitational field. Current models in the form of standardized coefficients of geopotential spherical harmonics make it possible to construct detailed high accuracy digital models of gravity fields characteristics (Koneshov et al. 2013; Aka Blush Ulfred 2019; Erol et al. 2020; Abd-Elmotaal et al. 2020). Reliable estimate of EGF models accuracy increases opportunity of their application in geodynamics, geophysics and navigation.

The object of this paper is the study of evolution of resolution and approximation accuracy for global models by their spectral characteristics.

Estimated approximation accuracy criterion for quasigeoid height and gravity anomalies implies the values of 1 cm and 1 mGal respectively, as specified by project GOCE developers.

1.1 Theoretical Part

When modelling the Earth’s gravity field, the attraction potential expansion in a Fourier series is used by the system of spherical harmonics of geocentric coordinates—radius vector r, latitude ϕ, and longitude λ in the form (Hoffman-Wellenhof and Moritz 2005)

$$ {{\begin{array}{ll}\displaystyle V\Big(\varphi, \lambda, r\Big)=\frac{fM}{r}&\Big[1+\sum \limits_{n=2}^{\infty }{\Big(\frac{a{}_e}{r}\Big)}^n\sum \limits_{m=0}^n\Big({\overline{C}}_{nm}\cos m\lambda\\ &+{\overline{S}}_{nm}\sin m\lambda \Big){\overline{P}}_{nm}\Big(\sin \varphi \Big)\Big]\end{array}}} $$
(1)

  • where fM—geocentric gravitational constant;

  • a e—Earth’s equatorial radius;

  • \( {\overline{C}}_{nm} \) и \( {\overline{S}}_{nm} \)—normalized dimensionless harmonic factors of geopotential;

  • \( {\overline{P}}_{nm}\left(\sin \varphi \right) \)—normalized associated Legendre polynomials.

Summing-up in formula (1) is performed to infinity, and the Earth’s gravity field models are limited by the maximum degree N 0. Series (1), limited by maximum N 0 presents spectral expansion of the Earth’s gravity field structure by waves lengths 360∘/2N 0 that corresponds to the spatial angular resolution.

$$ \Delta =\frac{180^{\circ }}{N_0}. $$
(2)

Simulation error of the global Earth’s gravity field V(φ, λ, r) by Fourier series V N (φ, λ, r) in point P(φ, λ, r)∈ω with approximate harmonic coefficients \( \left\{{\overline{C}}_{nm},{\overline{S}}_{nm}\right\} \) is written as

$$ {\varDelta}_N=\rho \left(V,{V}_N\right)+\underset{P\in \omega }{\max}\mid \delta N\mid, $$
(3)

where

$$ \rho \left(V,{V}_N\right)=\underset{P\in \omega }{\max}\mid V(P)-{V}_N(P)\mid; $$
(4)

δ N—simulation error of field V(P) due to the error of harmonic factors {δC nm, δS nm}.

Geopotential approximation dispersion error V(φ, λ, r), taking into account the error of harmonic factors, is estimated by formula (Kanushin et al. 2015)

$$ D\left\{{\Delta}_N\right\}=D-\sum \limits_{n=0}^N\left({D}_n-{d}_n\right) $$
(5)

where D is a dispersion of the initial potential V with unlimited spectrum. The degree dispersion of geopotential is calculated by formula

$$ {D}_n={\left(\frac{a_e}{r}\right)}^{2n}\sum \limits_{m=0}^n\left({\overline{C}}_{nm}^2+{\overline{S}}_{nm}^2\right). $$
(6)

Formula (7) is used to calculate the degree dispersion of errors \( \delta \kern0.1em {\overline{C}}_{nm}^2 \) and \( \delta \kern0.1em {\overline{S}}_{nm}^2 \) of geopotential harmonic factors V

$$ {d}_n={\left(\frac{a_e}{r}\right)}^{2n}\sum \limits_{m=0}^n\left(\delta {\overline{C}}_{nm}^2+\delta {\overline{S}}_{nm}^2\right). $$
(7)

Dependence of approximation error ε of gravity anomalies and quasigeoid heights due to series (1) limit in the form of

$$ \left\{\begin{array}{c}{\varepsilon}_{{\Delta{g}}_N}^2\\ {}{\varepsilon}_{\zeta_N}^2\end{array}\right\}=\left\{\begin{array}{c}{D}_{\Delta g}\\ {}{D}_{\zeta}\end{array}\right\}-\sum \limits_{n=2}^N\left(\left\{\begin{array}{c}{D}_{\Delta {g}_n}\\ {}{D}_{\zeta_n}\end{array}\right\}-\left\{\begin{array}{c}{d}_{\Delta {g}_n}\\ {}{d}_{\zeta_n}\end{array}\right\}\right), $$
(8)

spherical degree dispersions of gravity anomalies and quasigeoid heights expressed by factors \( \varDelta {\overline{C}}_{nm} \) and \( {\overline{S}}_{nm} \)

$$ \left\{\begin{array}{c}{D}_{\Delta {g}_n}\\ {}{D}_{\zeta_n}\end{array}\right\}=\left\{\begin{array}{c}{\gamma}^2{\left(n-1\right)}^2\\ {}{R}^2{\left(\frac{a_e}{r}\right)}^{2n}\end{array}\right\}\sum \limits_{m=0}^n\left(\Delta {\overline{C}}_{nm}^2+{\overline{S}}_{nm}^2\right), $$
(9)

where \( \varDelta {\overline{C}}_{nm}={\overline{C}}_{nm}-{\overline{C}}_{nm}^0 \)—the difference between coefficients of normalized spherical functions of real and normal gravity fields;

\( {\overline{C}}_{{nm}}^0 \)—coefficients of normal geopotential are referred to ellipsoid WGS- 84.

The degree dispersion of the errors of gravity anomalies and quasigeoid heights coefficients is written as

$$ \left\{\begin{array}{c}{d}_{\Delta {g}_n}\\ {}{d}_{\zeta_n}\end{array}\right\}=\left\{\begin{array}{c}{\gamma}^2{\left(n-1\right)}^2\\ {}{R}^2{\left(\frac{a_e}{r}\right)}^{2n}\end{array}\right\}\sum \limits_{m=0}^n\left(\delta {\overline{C}}_{nm}^2+\delta {\overline{S}}_{nm}^2\right). $$
(10)

Experimental. The study of evolution of the resolution and accuracy for 70 models under test for global Earth’s gravity field was conducted by the comparison results of geopotential harmonic coefficients degree dispersions and their errors as well as the computed (by them) degree dispersions of gravity anomalies and quasigeoid heights.

For satellite models under test their resolution and accuracy results according to their spectral characteristics are presented in the time range of 2008–2015 (Fig. 1). At the present moment the maximum expansion degree of the satellite models has achieved N0 = 300. The given expansion degree is limitary for satellite models built by current space gravimetry missions.

Fig. 1
figure 1

Resolution capabilities for spectral characteristics obtained according to low-level models, when approximation errors of 1 cm and 1 mGal are achieved (period 2008–2015)

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The evolution of the spatial resolution of the models built by the last decade satellite data shows that the gravity anomalies models resolution averages to 100 km, and for quasigeoid heights 200 km.

The results of global models resolution obtained by the combined data with error approximation 1 cm and 1 mGal are presented for the period of 1996–2019 (Fig. 2). The degree dispersions for the combined models under test practically coincide to the 240th degree. This testifies to the fact that the low-frequency harmonic segment of these geopotential models has been thoroughly studied.

Fig. 2
figure 2

The results of calculating the resolution capabilities from the spectral characteristics obtained from the combined data, with approximation error of 1 cm and 1 mGal (period 1996–2019)

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The data on the evolution of resolution capability for the gravity anomalies and quasigeoid heights models obtained by the combined data shows that the resolution capability was changing insignificantly during the last 15 years, i.e. about 180 km for quasigeoid and about 70 km for gravity anomalies.

The results of spectral estimates of global geopotential ultrahigh-degree models show that their degree dispersions practically coincide over all the frequency range under study. This means that the same initial data were mostly used for constructing these models. Significant improvements for the resolution of gravity field ultrahigh-degree models (Fig. 3) built in 2008–2019 have not been observed. For the gravity anomalies models the resolution is about 45 km and that for quasigeoid height—about 170 km.

Fig. 3
figure 3

The results of calculating the resolution capabilities by the geopotential ultrahigh-degree models characteristics with approximation error of 1 cm and 1 mGal (period 2008–2019)

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Analysis of the obtained research results as regards evolution of gravity field models resolution capability (for combined ultrahigh-degree models made in 2008–2019) revealed that it has not practically changed. Resolution of satellite models (in the period of 2008–2019) changed for the gravity anomalies model from 200 km to 100 km; for the quasigeoid model it changed insignificantly, i.e. from 230 km to 200 km.

Analyzing the spectral estimator changes from one geopotentional model to another a clear view may be formed of the resolution capability of certain harmonics or groups of harmonics of these models, and of the uncertainty state of each parameter under consideration due to the model determination features.

The comparison of model data with land point data on gravity anomalies and quasigeoid heights on the territory of Novosibirsk region is presented.

On the territory under study 17 second-order gravity base network stations were chosen with gravity determination accuracy being ± 0.05 mGal. The territory under investigation includes 208 stations where geometric leveling (first to fourth order) was conducted and the values of normal heights were obtained. At the same points, satellite coordinates were determined for developing active base stations networks. As a result of satellite network adjustment geodetic heights were obtained, with mean square errors ranging from 1.5 cm to 3.1 cm, 1.8 cm, on average (Karpik et al. 2010). On the territory under consideration gravity anomalies variations and elevation differences are insignificant.

The scheme of differences between the quasigeoid heights obtained from model EIGEN-6C4 and those of geoimetric leveling and GNSS measurements is given in Fig. 4. Applying model EIGEN-6C4 made it possible to achieve better results on the territory under consideration, with maximum difference being 22 cm and mean square error about 8 cm.

Fig. 4
figure 4

Scheme of differences between the heights of the quasigeoid obtained from the EIGEN-6C4 model and those obtained from geometric leveling and GNSS measurements, m

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The scheme of differences between the gravity anomalies obtained by EIGEN-6C4 model and those based on the ground data is shown in Fig. 5.

Fig. 5
figure 5

Scheme of differences between the EIGEN-6C4 model reconstructed and ground values of gravity anomalies obtained for the Novosibirsk region in mGal

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Maximum difference amounted to 8mGal, with mean square error being about 4 mGal.

The comparison of model and ground data for the gravity and quasigeoid heights anomalies on the territory of Novosibirsk region (for 9 geopotential models) is given in Table 1.

Table 1 Comparisons of ground and model values of gravitation anomalies and quasigeoid heights
Full size table

2 Conclusions. Interpretation

Comparison of degree dispersions of geopotential harmonic coefficients and their errors is presented for 70 current global models built by the results of measurements of space gravimetric missions CHAMP, GRACE and GOCE.

The models under study revealed: the dependences of gravity anomalies and quasigeoid heights approximation errors on the degree of geopotential expansion into Fourier transform; ratio errors of geopotential harmonic coefficients determination; coefficients dispersions and those of gravity anomalies coefficient errors.

Satellite models of the Earth’s gravity field built by only space gravimetry missions are not sufficient for obtaining quasigeoid heights with accuracy of 1–2 cm and spatial resolution less than 100 km. Satellite models are to be used in combination with land and topographic data to reestablish the global quasigeoid.

Models resolution at the satellite altitude level satisfies the requirements on the models accuracy stated by the developers. In case of necessity for recomputation of the gravity field characteristics for the Earth’s surface the specified resolution capability is reduced.

The compared values of the model and ground data on the Novosibirsk region areas demonstrated that the least values of differences were obtained on the global geopotential models based on the GOCE mission data (Kanushin et al. 2015). The global models accuracy values for the territory of Novosibirsk region do not come up to 1 mGal and 1 cm, being in the range of 4–10 mGal and 7–20 cm, respectfully.

The obtained accuracy values of the gravity anomalies and quasigeoid heights in current global geopotential models under study may be considered as maximum for the newly constructed models by the data of GOCE project.

Space gravity mission GOCE made it possible to essentially improve harmonic coefficients for degrees 100–250.

Harmonic coefficients of geopotential obtained for current EGF models are in conformity with each other within mean square errors.

Further research of the global models is required using the results of space gravity mission GOCE as the most successful mission for investigating EGF.

Accuracy characteristics of the models under study approximate the specified ones in process of their construction, but fall short of them as concerns the required resolution capability.

Current global models of gravity field create new opportunities and essentially extend the range of challenges in studying the Earth’s gravity field and its figure.

Comparing the evolution of resolution capability and approximation accuracy as regards current global models of the Earth’s gravity field we may come to the conclusion that at the current stage of technological and methodical level, EGF model resolution capability and accuracy improvement has reached its maximum. For further improvement of EGF approximation accuracy new technological approaches and methodological principles should be developed. This implies that a new concept of the Earth’s gravity field study should be developed.