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Estimation ff the Global Root Mean Square Error of Geoid Height Calculated by Integral Transforms
Belinger, Jiří ; Pitoňák, Martin ; Trnka, Petr ; Novák, Pavel ; Šprlák, Michal
Integral transformations of the gravitational field gradients are defined over the entire solid angle on the surface of the sphere. Despite the indisputable progress in satellite gravimetry and gradiometry, gravity field focused satellite missions allow accurate determination of the gravity field with a spatial resolution of 100 km, i.e. only in its long-wavelength part. However, there is also a need for high-resolution gravity field models at regional, national or continental scales, especially concerning the determination of the quasi-geoid or geoid. On the other hand, potential weakness of ground-based data is the long-wavelength gravity field accuracy and limited availability due to several constraints (e.g. deserts, lakes and large rivers, forests, or lack of goodwill between neighboring countries to share sensitive data). The ideal scenario combines ground and satellite data that complement each other. In this paper, relations defining the estimation of the global root mean square errors of geoid heights using integral transformations will be derived and presented. For practical calculation, knowledge about the accuracy of measured terrestrial data and formal errors of global satellite models of the Earth's gravity field will be utilized.
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A New Software For the Calculation of Far–Zone Effects For Spherical Integral
Trnka, Petr ; Pitoňák, Martin ; Belinger, Jiří ; Novák, Pavel ; Šprlák, Michal
Integral transformations are a useful mathematical apparatus for modelling the gravitational field. They represent the mathematical basis for the formulation of integral estimators of gravity field values, including error propagation. One of the basic assumptions of integral transformations is global data coverage. However, the availability of ground measurements is frequently limited. In practice, the global integral is divided into two complementary regions, namely the near and far zones. Non-negligible systematic effects of data in the far zone require accurate evaluation.
For this purpose, a new software library is being created in the MATLAB environment to calculate far-zone effects in integral transformations for gravitational potential gradients up to the third order.
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GRAIL and LOLA Satellite Data Resolve the Long-Lasting Convergence/Divergence Problem for the Analytical Downward Continuation of the External Spherical Harmonic Expansions
Šprlák, Michal ; Han, Shin-Chan ; Pitoňák, Martin ; Novák, Pavel
Spherical harmonic expansions are routinely used to represent the gravitational potential and its higherorder spatial derivatives in global geodetic, geophysical, and planetary science applications. The convergence domain of external spherical harmonic expansions is the space outside the minimum Brillouin sphere (the smallest sphere containing all masses of the planetary body). Nevertheless, these expansions are commonly employed inside this bounding surface without any corrections. Justification of this procedure has been debated for several decades, but conclusions among scholars are indefinite and even contradictory. In this contribution, we examine the use of external spherical harmonic expansions for the gravitational field modelling inside the minimum Brillouin sphere. We employ the most recent lunar topographic LOLA (Lunar Orbiter Laser Altimeter) products and the measurements of the lunar gravitational field by the GRAIL (Gravity Recovery and Interior Laboratory) satellite mission. We analyse selected 39 http://dx.doi.org/10.13164/seminargnss.2023.38 quantities calculated from the most recent GRAIL-derived gravitational field models and forwardmodelled (topography-inferred) quantities synthesised by internal/external spherical harmonic expansions. The comparison is performed in the spectral domain (in terms of degree variances depending on the spherical harmonic degree) and in the spatial domain (in terms of spatial maps). To our knowledge, GRAIL is the first gravitational sensor ever, which helped to resolve the long-lasting convergence/divergence problem for the analytical downward continuation of the external spherical harmonic expansions, see [1].
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Validation of Satellite Gravitational Gradients Grids by Spectral Combination Method and GNSS/Levelling Data Over Norway, Czechia and Slovakia
Pitoňák, Martin ; Šprlák, Michal ; Ophaugh, Vegard ; Omang, Ove C. D. ; Novák, Pavel
The launch of gravity-dedicated satellite missions at the beginning of the new millennium led to an accuracy improvement of global Earth gravity field models (GGMs). One of these missions was the Gravity field and steady-state Ocean Circulation Explorer (GOCE) launched in 2009. As the first ESA's Earth Explorer Mission, the satellite carried a novel instrument, a 3-D gradiometer, which allowed to measure of second-order directional derivatives of the gravitational potential (gravitational gradients) 37 http://dx.doi.org/10.13164/seminargnss.2023.36 with uniform quality and near-global coverage. The main mission goal was to determine the static Earth's gravity field with the ambitious precision of 1-2 cm in terms of geoid heights and 1 mGal in terms of gravity anomalies for a spatial resolution of 100 km (half wavelength at the equator). More than three years of outstanding measurements resulted in three levels of data products (Level 0, Level 1b and Level 2), six releases of GGMs, and several global grids of gravitational gradients. The grids, which represent a step between gravitational gradients measured directly along the GOCE orbit and those represented by GGMs, were used mainly in geophysical applications. In this contribution, we validate the official Level 2 product GRD SPW 2 using height anomalies over two test areas in central and northern Europe (Czechia/Slovakia and Norway). A mathematical model based on the least-squares spectral weighting is employed with corresponding spectral weights estimated to validate gravitational gradient grids. This model continues gravitational gradients from the mean orbital altitude of GOCE down to the irregular Earth's surface (not to a sphere) and transforms them to height anomalies in one computational step. Analytical downward continuation errors of the model are estimated using a closed-loop test. Before comparing, height anomalies estimated from gravitational gradients with their reference values derived from GNSS/levelling over the two test areas, the gravitational gradients and reference data are corrected for all systematic effects, such as the tide system conversion. Moreover, the high-frequency part of the gravitational signal is estimated and subtracted from reference data as it is attenuated in the gravitational gradients measured by GOCE. A relative improvement between the release 6 and release 2 gradient grids reaches 48% in terms of height anomalies in Czechia/Slovakia. The relative improvement in Norway is even more significant and reaches 55%. Release 6 of the official Level 2 product GRD SPW 2 gained absolute accuracy with the standard deviation of 9.1 cm over Czechia/Slovakia and 9.6 cm over Norway.
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Far Zone Effects for Integral Transformations: Theory and Implementation
Trnka, Petr ; Belinger, Jiří ; Šprlák, Michal ; Pitoňák, Martin ; Novák, Pavel
Integral transformations are a useful mathematical apparatus for modelling the gravitational field and require the formulation of integral estimates including error propagation. For classical integral transformations, this issue has already been studied, but the formulation for all available gravitational observables has not been studied yet. The assumption of integral transformations is global data coverage. In practice, however, data availability is limited, so we divide the global integration into the effects of the near and far zones. The computation of distant zones is a non-negligible systematic effect requiring an accurate calculation. The theory is implemented in the form of a precise software. In this paper, we present the basic theory for the evaluation of the far zones. We also investigate properties of integral kernels and truncation error coefficients. In the numerical experiments, we compare calculation of the far zones by numerical integration with truncated spherical harmonic series. One of the outputs of this contribution is a software library for computation of the far zones for integral transformations mutually relating all quantities up to the third derivatives of the gravitational potential.
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Crustal Density and Global Gravitational Field Estimation of the Moon from GRAIL and LOLA Satellite Data
Šprlák, Michal ; Han, Shin-Chan ; Featherstone, Will ; Novák, Pavel ; Pitoňák, Martin
We employ Newton’s integral in the spectral domain to solve two geodetic/geophysical tasks for the Moon, see [1]. Firstly, we determine density distribution within the lunar crust (inverse problem). For this purpose, we exploit GL1500E GRAIL gravitational field model and LOLA topography to estimate: 1) constant, 2) laterally variable, and 3) 3D spatially variable crustal density. Secondly, we calculate lunar gravitational field models inferred by these three crustal compositions (forward problem) up to spherical harmonic degree 2519 (corresponding to a spatial resolution of 2.2 km at the lunar equator). We test the performance of our new models, and of recent and independent forward models, against the official Level 1B and Level 2 GRAIL products. Our high resolution global gravitational field models will be an asset to future lunar lander navigation and geophysical exploration of the Moon.
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Estimation of Litospheric Elastic Thicknes from In-orbitGOCE-based Vertical Gradients and CRUST1.0
Pitoňák, Martin ; Eshagh, Mehdi ; Šprlák, Michal ; Novák, Pavel
The lithospheric strength with respect to the loading is represented by a parameter called elastic thickness (Te) and places with larger value of Te flex less. In this contribution, we use the in-orbit vertical gravitational gradients measured by Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite for determining the elastic thickness over Africa. A forward computational method is developed based on the Vening Meinesz-Moritz (VMM) and flexural theories of isostasy to find a mathematical relation between the second-order vertical derivative of the gravitational potential and mechanical properties of the lithosphere. This method is developed in terms of spherical harmonics. Loading effects of topography and bathymetry, sediments and crystalline masses are calculated from CRUST1.0, in addition to estimates of laterally-variable density of the upper mantle, Young’s modulus and Poisson’s ratio. The second-order vertical derivatives of the gravitational potential are synthesised from the crustal model and different a priori values of elastic thickness to find which one matches the GOCE in-orbit vertical gradient. Our map of Te over Africa shows that the high values of Te are mainly associated with the cratonic areas of Congo, Chad and the Western African basin while the intra-continental hotspots and volcanoes, such as Ahaggar, Tibesti, Darfur, Cameroon volcanic line and Libya are connected by corridors of low Te.
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