Národní úložiště šedé literatury Nalezeno 5 záznamů.  Hledání trvalo 0.01 vteřin. 
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].
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.
Vliv vzdálených zón pro integrální transformace: teorie a implementace
Trnka, Petr ; Belinger, Jiří ; Šprlák, Michal ; Pitoňák, Martin ; Novák, Pavel
Integrální transformace jsou užitečný matematický aparát pro modelování gravitačního pole a vyžadují formulaci integrálních odhadů včetně chybových charakteristik. Pro klasické integrální transformace byla již tato problematika prozkoumána, ale zatím nebyla studována formulace vzájemně vztahující všechny dostupné gravitační pozorovatelné veličiny. Předpokladem je globální pokrytí daty a globální integrace. Dostupnost dat může být omezená, proto globální integraci rozdělujeme na vliv blízkých a vzdálených zón. Výpočet vzdálených zón je nezanedbatelný systematický efekt, vyžadující přesný výpočet. Potřebná teorie a její implementace se realizují v podobě přesného softwarového nástroje. V tomto příspěvku představujeme základní teorii vlivu vzdálených zón. Dále studujeme vlastnosti integrálních jader a Moloděnského koeficientů. V numerických experimentech porovnáme výpočet vzdálených zón numerickou integrací s omezenou sumací ve formě sférických harmonických řad. Jedním z výstupů tohoto příspěvku je i softwarová knihovna na výpočet vlivu vzdálených zón pro integrální transformace až po třetí derivace gravitačního potenciálu.
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.
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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7 Pitoňák, Michal
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