Jan Kostelecký

Determination of the geopotential scale factor from satellite altimetry

SummaryThe geopotential scale factor R0=GM/W0 has been determined on the basis of satellite altimetry as R0=(6 363 672·5±0·3) m and/or the geopotential value on the geoid W0=(62 636 256·5±3) m2s−2. It has been stated that R0 and/or W0 is independent of the tidal distortion of surface W=W0 due to the zero frequency tide.РезюмеПо ¶rt;aнным сnуmнuковоŭ aльmuмеmрuu вычuслен мaсщmaбныŭ фaкmореоnоmенцuaлa R0=GM/W0=(6 363 672,5±0,3) m u/uлu знaченuееоnоmенцuaлa нa nоверхносmuеоu¶rt;a W0=(62 636 256,5±3) m2 s−2. Оmмечено, чmо R0 u/uлu W0 не зaвuсum оm nосmоянноŭ чaсmu зонaльноо членa nрuлuвных ¶rt;еформaцuŭ nоверхносmu W=W0.

The Use of Resonant Orbits in Satellite Geodesy: A Review

Dynamic resonance, arising from commensurate (orbital or rotational) periods of satellites or planets with each other, has been a strong force in the development of the solar system. The repetition of conditions over the commensurate periods can result in amplified long-term changes in the positions of the bodies involved. Such resonant phenomena driven by the commensurability between the mean motion of certain artificial Earth satellites and the Earth’s rotation originally contributed to the evaluation and assessment of the Stokes parameters (harmonic geopotential coefficients) that specify the Earth’s gravitational field. The technique constrains linear combinations of the harmonic coefficients that are of relevant resonant order (lumped coefficients). The attraction of the method eventually dwindled, but the very accurate orbits of CHAMP and GRACE have recently led to more general insights for commensurate orbits applied to satellite geodesy involving the best resolution for all coefficients, not just resonant ones. From the GRACE mission, we learnt how to explain and predict temporary decreases in the resolution and accuracy of derived geopotential parameters, due to passages through low-order commensurabilities, which lead to low-density ground-track patterns. For GOCE we suggest how to change a repeat orbit height slightly, to achieve the best feasible recovery of the field parameters derived from on-board gradiometric measurements by direct inversion from the measurements to the harmonic geopotential coefficients, not by the way of lumped coefficients. For orbiters of Mars, we have suggestions which orbits should be avoided. The slow rotation of Venus results in dense ground-tracks and excellent gravitational recovery for almost all orbiters.

MASS DISTRIBUTION OF EARTH LANDFORMS DETERMINED BY ASPECTS OF THE GEOPOTENTIAL AS COMPUTED FROM THE GLOBAL GRAVITY FIELD MODEL EGM 2008

Correlations of large-scale landform patterns with some aspects of the geopotential as computed from the global gravity field model EGM 2008, particularly the radial second derivatives of the disturbing gravitational potential Γ33, the strike angle θS and virtual deformations of the ellipse of deformation, are demonstrated. Selected regions with documentation of aspects from EGM 2008 are the Nepal Himalaya and its neighbouring areas, the collision zone of East-Asian and West-Pacific lithospheric plates, the contact region of north-eastern Africa, south-western Asia and south-eastern Europe, morphotectonic contact between the Bohemian Massif, Eastern Alps and the Western Carpathians in Central Europe and regions of ancient rapid events indicated by relics of large impact craters Vredefort, Chicxulub and Popigai. It is suggested that landform patterns with very conspicuous combinations of significantly high positive or negative values of Γ33 are under the strong influence of rapid and/or intensive geomorphic processes. These geophysical signatures supported by values of the strike angle θS and virtual dilatations or compressions of the ellipse of deformation reflect the regional dynamics of Earth surface evolution as characterised by a very effective integration of tectonic and climate-driven morphogenetic processes.

Reduction of crossover errors in the earth gravity model (EGM) 96

Using the established procedure, the calibrated covariance matrix of harmonic geopo‐tential coefficients of the new (Earth Gravity Model) EGM96 (to 70 × 70) is projected to single‐and dual‐satellite crossover errors, and their spectral latitude lumped coefficient constituents. These results are compared with previous gravity solutions, such as JGM 2 and JGM 3, to assess the strengths and weaknesses of the new solution. This analysis quantifies the level of improvement over previous solutions, as well as suggests areas where further refinements are required to achieve subdecimeter accuracy over a wide range of satellite missions.

Accuracy assessments of recent earth gravity models using crossover altimetry

SummaryThe calibrated variance-covariance matrices of the harmonic geopotential coefficients of the recent combined model JGM 2 has been tested and verified by independent crossover altimetry from TOPEX/Poseidon and ERS 1 using the Latitude Lumped Coefficients in the southern oceans area. Although orbits are not yet available for these missions with other recent models for which error matrices have been released, by comparison with JGM 2 results and field differences we also confirm that the error matrices for the satellite model GRIM 4S4p and the combined data model JGM 3 are also generally valid. Projections of these matrices for a variety of inclinations show that many unused orbits of even moderate altitude (≈ 800 km) will still yield trajectory crossover errors at a level of many tens of centimeters even with the latest orbitgeopotential models.

Collocations and thirtieth order resonant harmonics

Abstract The method of collocations (LSC) has been compared with traditional least-squares adjustments (LSA) for determining the values of the individual harmonic coefficients in the expansion for the Earth gravitational potential from the lumped geopotential coefficients of order 30, accumulated from previous analyses of satellite orbits near the 15th-order resonance. The computations are based on the data from King-Hele and Walker (1982b), where the 30th-order harmonic coefficients were determined from the lumped values by means of the usual least-squares method. We take into account the correlation coefficients among the lumped coefficients; we do not omit higher degree harmonics but on the contrary, they are statistically estimated as the signal in LSC. Four groups of runs have been performed: from LSA (similar to that in King- Hele and Walker, 1982b) to as general LSC as possible. The resulting harmonic coefficients are compared mutually, with the resonant solution by King-Hele and Walker (1982b), with our older trials (Kostelecký and Klokocnik, 1979) and with recent comprehensive Earth models (GEM 10 B(C), ‘Rapp 77’ and GRIM 3). The comparison by harmonic coefficients is in Tables 4 and 5 and on Fig. 1, that via lumped coefficients for arbitrary inclination is in Figs. 2–7. The first few pairs of the 30th-order harmonic coefficients, at least C30, 30and S30, 30, are now well determined. King-Hele and Walker (1982b) used better data than we had in our previous solution (Kostelecký and Klokocnik, 1979), so that the LSC does not play so important a role in the determination as played in the older solution. Although our evaluations serve as an example where more complicated LSC is not necessary, LSC ought to be preferred in a situation where LSA does not achieve optimal utilization of the data base.

A support for the existence of paleolakes and paleorivers buried under Saharan sand by means of “gravitational signal” from EIGEN 6C4

The goal of this study is to demonstrate that and how the recent gravitational and topographic data support the findings made by geologists and others as for the existence of the paleolakes and paleoriver systems, now buried under the sands of Sahara. It is always important and useful to have such an independent analysis supporting certain results, and this paper is such a case. We make use of the gravity disturbances (or anomalies), the Marussi tensor of the second derivatives of the disturbing geopotential, the gravitational invariants and their certain ratio, the strike angle and the virtual deformations. The geopotential is represented by the global combined (from satellite and terrestrial data) high-resolution gravity field model EIGEN 6C4 (till degree and order 2160 in spherical harmonic expansion). The topography is derived from the ASTER GDEM and ETOPO 1 models (both are used). With all these data, we confirm the existence of huge paleolakes or paleoriver systems under the Saharan sands known or anticipated in an independent way by geologists for the lakes MegaChad, Fazzan and Chotts; for Tamanrasset river valley; and Kufrah Basin, presumptive previous flow of the Nile River. Moreover, we suggest a part of the Grand Egyptian Sand Sea as another “candidate” for a paleolake and hence for a follow-up survey.

Gravity Disturbances, Marussi Tensor, Invariants and Other Functions of the Geopotential Represented by EGM 2008

Gravity disturbances, the Marussi tensor, invariants of the gravity field, their certain ratio and other functions of the geopotential (including newly defined “virtual deformations”) are computed based on the harmonic coefficients of the global gravitational field model EGM 2008. Regional examples of correlations of large-scale morphotectonic and landform patterns with some aspects of the geopotential as computed from the EGM 2008 are presented. It is suggested that morphotectonic and landform patterns with very conspicuous combinations of significantly high positive or negative values of Γ33 are under the strong influence of rapid and/or intensive geomorphic processes. These geophysical signatures supported by values of the strike angle θS and virtual dilatations or compressions of the ellipse of deformation reflect the regional dynamics of Earth surface evolution as characterised by a very effective integration of tectonic and climate-driven morphogenetic processes. Key wordsEarth Gravitational Model 2008; gravity disturbances; the Marussi tensor; invariants of the gravity field; virtual deformations of the ellipse of deformation; landform patterns; geodynamics.

Geosat and ERS-1 Datum Offsets Relative to Topex/Poseidon Estimated Simultaneously with Geopotential Corrections from Multi-Satellite Crossover Altimetry

Crossover residuals, averaged over a sufficient long time period carry the signature of geopotentially based orbit errors. In addition they may exhibit geographical pattern which are inconsistent with orbit dynamics. Tracking system offsets and different orientations, for example, map to the so called “forbidden” harmonics: the degree 1 and the degree 2, order 1 coefficients. Long-term averaged dual satellite crossover residuals among Geosat, ERS-1 and Topex/Poseidon, all with JGM3 based orbits, have been used to solve simultaneously for i) corrections to the gravity field harmonics and ii) parameters which account for a coordinate frame offset relative to Topex/Poseidon. The parameters are estimated by a least squares adjustment. The paper investigates in particular the obstacle of large correlation between the shift parameters and the low degree order one harmonic coefficients. Results of the inversion indicate significant shifts between the coordinate frames of the altimeter missions involved. They must be taken into account, if secular sea level changes are to be investigated.

Tidal friction. Secular variations of the inclination and eccentricity of the Moon's orbit

РезюмеПересмamрuвaюmся вычuсленuя вековых uзмененuŭ нaклонa u эксценmрuцumеma лунноŭ орбumы, вызвaнных nрuлuвнымu сuлaмu. Кроме mоо nрове¶rt;енa оценкa влuянuя сжamuя Землu; эmо влuянuе сосmaвляеm оm 10−5 ¶rt;о 10−6 знaченuя ¶rt;ля шaрообрaзноŭ Землu u сле¶rt;овamельно nренебрежuмо. Резульmamы хорошо солaсуюmся с вычuсленuямu Кaулa [7], в оmлuчuе оm резульmamов МaкДонaль¶rt;a [8],¶rt;е рaзнuцa нa nоря¶rt;ок в uзмененuu эксценmрuцumеma очевu¶rt;но вызвaнa оnечamкоŭ. Нaконец ¶rt;aно решенuе ¶rt;ля рaзных знaченuŭ фaкmорa ¶rt;uссunaцuu.SummaryThe computation of the secular variation of the inclination and eccentricity of the Moons orbit due to tidal forces is revised. Also the effect of the Earths ellipticity, which amounts to 10−6 to 10−5 of the value for the spherical earth, is estimated and found to be negligible. The results agree well with the computations of Kaula [7], as opposed to MacDonalds results [8] which display an order-of-magnitude difference in the variation of the eccentricity, apparently due to a printing error. Finally, solutions are presented for various values of the dissipation factor.