Jaroslav Klokočník

Intercomparisons of Earth models by means of lumped coefficients

A new procedure for comparing and testing the Earth gravity field models by the order of their harmonic geopotential coefficients (not by degree as is usual) is explained. The differences in the lumped coefficients computed from different Earth models are presented as a set of 20 figures (for order of 6 ⩽ β ⩽ 15 and the orbital inclination of 30° ⩽ I ⩽ 140°). A statistical description of those dispersions is added, too. Conclusions from the intercomparisons are the following: (1) The differences in the lumped values are higher than expected, especially for lower order (∼β < 10), where the Earth models are believed to be “well determined”. The examples of evident imperfections in some of the models are in Table 3. (2) The truncation errors of the series, which define the lumped coefficients as a function of the individual harmonic coefficients, cannot affect the lumped values more than about 10%, if at least five harmonic coefficients (approx.) are available (from the Earth model in question) to compute the lumped coefficients.

Accuracy of the GEM‐T2 geopotential from Geosat and ERS 1 crossover altimetry

Extensive analyses of altimetrically determined sea height differences at crossovers have been used to assess the accuracy of the GEM-T2 geopotential. The orbits used were determined with GEM-T2 for Geosat in its 17-day Exact Repeat Mission (ERM) in 1986–1989 and ERS 1 in both its 3-day ERM in 1991–1992 and its 35-day ERM in 1992. The data examined are completely independent of the data used in GEM-T2s development though GEM-T2 had considerable use of Doppler tracking information on Geosat. The test of the radial accuracy of the ERS 1 orbit (98.5° inclination) is especially significant because it is not “close” to any other orbit well represented in GEM-T2. The assessment consists of a comparison of observed mean height differences at thousands of distinct geographic locations with error projections from the GEM-T2 covariance matrix which was estimated from other data sources. This first comprehensive, independent test of the purely radial accuracy of an orbit-geopotential model clearly shows that the covariant predictions for GEM-T2 are broadly reliable for this purpose. Thus, the agreement of crossover predictions and observations suggests that the total radial errors for these ERMs, due only to GEM-T2 (but excluding the effects of initial state error) are about 23 cm for Geosat and 115 cm (rms) for ERS 1. However, there is little detailed agreement of measurements and predictions for ERS 1 and only partial agreement in detail for Geosat. Our 30,000 mean crossover discrepancies for Geosat (derived from ERM cycles 1–44) are also shown to reduce substantially the crossover height differences in cycles 45–61, almost exactly as predicted if these are the true GEM-T2 errors for this orbit.

A Test of GEM T2 from Geosat Crossovers Using Latitude Lumped Coefficients

SummaryNew Latitude Lumped Coefficients (LLC) of a geopotential model are defined as representing the principal differences of the radial distance to a satellite due to the model at single-orbit crossovers in an Exact Repeat Mission. In contrast with previously defined orbital lumped coefficients, the LLC here are dependent only on the geopotential order (without degree distinction) and the latitude. We examine discrepancies in altimetrically determined sea surface heights at over 30000 crossover positions of GEOSAT during its ERM, 1986–1989, after removal of many variable media and surface effects (Cheney et al., 1991) as well as initial condition orbit error. The mean of these discrepancies along well represented latitude bands in the southern hemisphere are used to determine the LLC errors for Goddard Earth Model T2, which was the reference for the GEOSAT sea surface heights. GEM T2 was derived from satelliteonly tracking data with good representation of the GEOSAT orbit. Relating the ”measured” LLC discrepancies to projections of commission error from the GEM T2 variance-covariance matrix, we find that — except for order 3 — GEM T2s performance is as expected. This test represents the first spectral calibration of a gravity model with independent, purely radial orbit data.

Evaluation of JGM 2 geopotential errors from Geosat, TOPEX/Poseidon and ERS-1 crossover altimetry

Abstract World-ocean distribution of the crossover altimetry data from Geosat, TOPEX/Poseidon (T/P) and the ERS 1 missions have provided strong independent evidence that NASAs/CSRs JGM 2 geopotential model (70 × 70 in spherical harmonics) yields accurate radial ephemerides for these satellites. In testing the sea height crossover differences found from altimetry and JGM 2 orbits for these satellites, we have used the sea height differences themselves (of ascending minus descending passes averaged at each location over many exact repeat cycles) and the Lumped Latitude Coefficients (LLC) derived from them. For Geosat we find the geopotential-induced LLC errors (exclusive of non-gravitational and initial state discrepancies) mostly below 6 cm, for TOPEX the corresponding errors are usually below 2 cm, and for ERS 1 (35-day cycle) they are generally below 5 cm. In addition, we have found that these observations agree well overall with predictions of accuracy derived from the JGM 2 variance-covariance matrix; the corresponding projected LLC errors for Geosat, T/P, and ERS 1 are usually between 1 and 4 cm, 1 – 2 cm, and 1 – 4 cm, respectively (they depend on the filtering of long-periodic perturbations and on the order of the LLC). This agreement is especially impressive for ERS 1 since no data of any kind from this mission was used in forming JGM 2. The observed crossover differences for Geosat, T/P and ERS 1 are 8, 3, and 11 cm ( rms ), respectively. These observations also agree well with prediction of accuracy derived from the JGM 2 variance-covariance matrix; the corresponding projected crossover errors for Geosat and T/P are 8 cm and 2.3 cm, respectively. The precision of our mean difference observations is about 3 cm for Geosat (approx. 24,000 observations), 1.5 cm for T/P (approx. 6,000 observations) and 5 cm for ERS 1 (approx. 44,000 observations). Thus, these “global” independent data should provide a valuable new source for improving geopotential models. Our results show the need for further correction of the low order JGM 2 geopotential as well as certain resonant orders for all 3 satellites.

Navigation of satellite measurements without ground control points

Abstract The minimum of information on satellite dynamics (a part of celestial mechanics) which is needed to navigate operational polar orbiting meteorological satellites either for direct or inverse referencing without ground control points, is presented. This software is tailored to the orbit information available. Its validation by ground control points is also included. The accuracy achieved by the improved software in the determination of the time of satellite culmination over a pixel (or pixel line) and in the off-nadir angle (or pixel number) is comparable to the pixel size (1–5 km) of the AVHRR onboard NOAA-N satellites, if the standard orbital elements are not older than about two weeks.

Dual-satellite crossover altimetry for ERS-1/TOPEX

Abstract The advantage of use of the dual-satellite crossovers for the pair “ERS-1 & TOPEX” over the single-satellite crossovers for ERS-1 alone is numerically demonstrated: the dual-satellite crossover error is lower than the single-satellite crossover error [the errors are mapped from the calibrated geopotential variance-covariance matrices of GRIM4S2 and GEM T2].

Orbital rates of earth satellites at resonances to test the accuracy of earth gravity field models

Differences among the Earth gravity field models, which were (in Klokočník and Pospíšilová, 1981) expressed as dispersions of the relevant lumped geopotential coefficients, are here transformed to the differences in variations of orbital quantities.Theoretical formulae, the Lagrange (planetary) equations, describing the orbital rates near resonances due to the geopotential, are derived in a simple and unified form. They are then applied to estimate the orbital uncertainty as a function of Earth models differences. The first set of the Earth models (set I) consists of 11 models from the decade 1970–1980, of greatly varying quality; the set II contains several recent models; we present a test (for the 13th-to 15th-order) based on standard deviations of the lumped values of GEM 10B, which were estimated by means of independent resonant data (in Klokočník, 1982).Maxima of the differences in the variations of the elements for the set I reach 8×10−4 deg day−1, 10–12 m day−1 or 200 m day−1 inI, a, orL0=ω+M0+Ω, respectively, for close and polar orbits (∼15 revs day−1); the values are not higher than 10−4 deg day−1, 1–2 m day−1 or 20 m day−1 inI, a,L0 for higher orbits (∼6–7 revs day−1). For the set II, calibrated by resonant data, the maximum inaccuracy (±3σ) is about 3×10−4 deg day−1, ≤6 m day−1 or ≤100 m day−1 forI, a, andL0 at 15 revs day−1, and is not larger than∼1×10−4 deg day−1, 2 m day−1 or 25 m day−1 for 13 revs day−1.

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.