Steady-state Ocean Circulation Explorer Mission Research Articles

Replacement of the part of gravity model and of remaining dynamic models by empirical accelerations in the fit processes of a low-Earth satellite orbit and factors limiting their accuracy

This work contains the results of calculations performed to prove the ability of estimated orbital parameters for the replacement of dynamic models in the orbit determination of a sample low-Earth-orbiting satellite. The obtained solutions include two cases of the absorption of dynamic models. In the first case, the contribution of dynamic models, apart from the gravity field, was absorbed, i.e., the satellite motion was described by the gravity field and estimated parameters. In the second case, the contributions of all dynamic models, including the gravity field, were absorbed. For the gravity field model, the absorption concerned its selected parts. In this case, the satellite motion was modeled only by the gravity model truncated to different degrees and orders and an appropriate set of orbital parameters. In both aforementioned cases, the initial conditions were also improved. Cartesian coordinates of the Gravity Field and Steady-State Ocean Circulation Explorer Mission satellite along selected reference arcs of the official reduced-dynamic orbit served as pseudo-observations in this study. The orbital parameters, also known as empirical accelerations, were determined using the least-squares method by a dedicated orbital package. The results were presented and compared in the form of the root-mean-square (RMS) values of the differences between the estimated orbits and the reference orbits, as well as the corresponding values of the obtained empirical accelerations for selected variants of solutions. The obtained accuracy of the process of the fit of the satellite orbit expressed by the corresponding RMS values, reached a millimeter level. For selected typical solutions, the distribution of residuals and power spectra are presented with an indication of characteristic errors: random and systematic periodic components. Key factors influencing the obtained fit accuracies of estimated orbits are given. Contributions of these factors in the error budget of fits of estimated orbits are also presented. Additionally, in the fit process, the performance of selected gravity models coming from different years is compared to assess the impact of gravity field errors on the results of aforementioned process.

Piecewise acceleration orbital modeling: a GOCE satellite case study

In this work, the Precise Science Orbit (PSO) of the Gravity Field and Steady-State Ocean Circulation Explorer Mission (GOCE) satellite, acquired via the European Space Agency, served as the reference orbit for the testing various variants of the GOCE satellite orbit modeling. The GOCE satellite positions along the reduced-dynamic PSO orbit were treated as pseudo-observations in the satellite orbit improvement process in the least squares sense. This process was realized using dedicated extended Toruń Orbit Processor software which enabled determining corrections to the orbital initial conditions and the set of parameters, necessary to determine additional empirical accelerations of the satellite in the radial-track, along-track and cross-track directions. These piecewise accelerations were determined in successive equal intervals (pieces) into which the processed orbital arcs were divided. For modeling the accelerations, polynomials of different degrees were used. The obtained RMS differences between the improved orbits and the reference PSO orbit were determined for various orbital arc lengths up to 1-day. The best RMS of the fits for the 1-day arcs was in the range from 2.0 to 3.2 mm with significantly worst results for along-track direction. Based on the set of solutions determined, the number of orbital parameters for adopted accuracy thresholds and the upper limit of their number, that can be estimated depending on the length of the orbital arc, under given numerical conditions were obtained. The optimal form of the polynomial modeling of the estimated accelerations also depends partly on the length of the processed orbital arc. For shorter arcs (45 min and less), the second- third- and fourth-order polynomial gives the best results, while for longer arcs (90, 180, 360, 720 and 1440 min), zero- and first-degree polynomials are the most effective. A very promising solution with the RMS of the fit of 0.1 mm for a 1-min arc using the fourth-order polynomial was obtained in a perspective of the future use of the short arc approach to the fit of 1-day arcs. Additionally, solution variants with non-equal numbers of orbital pieces for the radial-along-and cross-track directions were obtained. In the case of the solution for an arc length of 1-day, the distribution of residuals and their statistics in the aforementioned radial-, along-and cross-track directions are presented.

Estimation of gravitational curvature through a deterministic approach and spectral combination of space-borne second-order gravitational potential derivatives

SUMMARY Second- and third-order gravitational potential derivatives can be employed for the determination of the medium- and high-frequency parts of the Earth's gravity field. Due to the Gravity field and steady-state Ocean Circulation Explorer mission, second-order derivatives (SOD) in particular, express currently observed functionals of high accuracy and global coverage. Third-order derivatives (TOD), or gravitational curvature data, provide significant gravity field information when applied regionally. The absence of directly observed TOD data underlines the importance of investigating the relationship between SOD and TOD. This paper discusses the combination of simulated SOD in order to obtain TOD at satellite altitude by applying the spectral combination method. For the determination of TOD integral equations are developed that utilize SOD data at satellite altitude, thus extending the well-known Meissl spectral scheme. The performance of the derived mathematical models is investigated numerically for the test area of Himalayas and the Tibet region. Two different TOD computational strategies are examined. First, we define a deterministic approach that recovers TOD data from noise-free simulated SOD data. Results show that retrieved TOD data at satellite level reach an agreement of the level of 1 × 10−17 m−1s−2 when compared with the true TOD data. Secondly, we propose a new mathematical model based on the spectral combination of integral relations and noisy SOD data with Gaussian noise for recovering TOD. Integral estimators of biased and unbiased types are examined in the cases of SOD data at satellite altitude. The used vertical SOD components show differences between the recovered and true vertical TOD components in the order of 1 × 10−17 m−1s−2 in magnitude, proving the vertical–vertical component of SOD as the best for validating purposes.

A geothermal application for GOCE satellite gravity data: modelling the crustal heat production and lithospheric temperature field in Central Europe

SUMMARYSince the completion of the Gravity field and steady-state Ocean Circulation Explorer mission (GOCE), global gravity models of uniform quality and coverage are available. We investigate their potential of being useful tools for estimating the thermal structure of the continental lithosphere, through simulation and real-data test in Central-Eastern Europe across the Trans-European Suture Zone. Heat flow, measured near the Earth surface, is the result of the superposition of a complex set of contributions, one of them being the heat production occurring in the crust. The crust is enriched in radioactive elements respect to the underlying mantle and crustal thickness is an essential parameter in isolating the thermal contribution of the crust. Obtaining reliable estimates of crustal thickness through inversion of GOCE-derived gravity models has already proven feasible, especially when weak constraints from other observables are introduced. We test a way to integrate this in a geothermal framework, building a 3-D, steady state, solid Earth conductive heat transport model, from the lithosphere–asthenosphere boundary to the surface. This thermal model is coupled with a crust-mantle boundary depth resulting from inverse modelling, after correcting the gravity model for the effects of topography, far-field isostatic roots and sediments. We employ a mixed space- and spectral-domain based forward modelling strategy to ensure full spectral coherency between the limited spectral content of the gravity model and the reductions. Deviations from a direct crustal thickness to crustal heat production relationship are accommodated using a subsequent substitution scheme, constrained by surface heat flow measurements, where available. The result is a 3-D model of the lithosphere characterised in temperature, radiogenic heat and thermal conductivity. It provides added information respect to the lithospheric structure and sparse heat flow measurements alone, revealing a satisfactory coherence with the geological features in the area and their controlling effect on the conductive heat transport.

Local recovery of lithospheric stress tensor from GOCE gravitational tensor

The sublithospheric stress due to mantle convection can be computed from gravity data and propagated through the lithosphere by solving the boundary-value problem of elasticity for the Earth's lithosphere. In this case, a full tensor of stress can be computed at any point inside this elastic layer. Here, we present mathematical foundations for recovering such a tensor from gravitational tensor measured at satellite altitudes. The mathematical relations will be much simpler in this way than the case of using gravity data as no derivative of spherical harmonics (SHs) or Legendre polynomials is involved in the expressions. Here, new relations between the SH coefficients of the stress and gravitational tensor elements are presented. Thereafter, integral equations are established from them to recover the elements of stress tensor from those of the gravitational tensor. The integrals have no closed-form kernels, but they are easy to invert and their spatial truncation errors are reducible. The integral equations are used to invert the real data of the gravity field and steady-state ocean circulation explorer mission (GOCE), in 2009 November, over the South American plate and its surroundings to recover the stress tensor at a depth of 35 km. The recovered stress fields are in good agreement with the tectonic and geological features of the area.

GRAVITY ANOMALY ASSESSMENT USING GGMS AND AIRBORNE GRAVITY DATA TOWARDS BATHYMETRY ESTIMATION

Abstract. The Earth’s potential information is important for exploration of the Earth’s gravity field. The techniques of measuring the Earth’s gravity using the terrestrial and ship borne technique are time consuming and have limitation on the vast area. With the space-based measuring technique, these limitations can be overcome. The satellite gravity missions such as Challenging Mini-satellite Payload (CHAMP), Gravity Recovery and Climate Experiment (GRACE), and Gravity-Field and Steady-State Ocean Circulation Explorer Mission (GOCE) has introduced a better way in providing the information on the Earth’s gravity field. From these satellite gravity missions, the Global Geopotential Models (GGMs) has been produced from the spherical harmonics coefficient data type. The information of the gravity anomaly can be used to predict the bathymetry because the gravity anomaly and bathymetry have relationships between each other. There are many GGMs that have been published and each of the models gives a different value of the Earth’s gravity field information. Therefore, this study is conducted to assess the most reliable GGM for the Malaysian Seas. This study covered the area of the marine area on the South China Sea at Sabah extent. Seven GGMs have been selected from the three satellite gravity missions. The gravity anomalies derived from the GGMs are compared with the airborne gravity anomaly, in order to figure out the correlation (R2) and the root mean square error (RMSE) of the data. From these assessments, the most suitable GGMs for the study area is GOCE model, GO_CONS_GCF_2_TIMR4 with the R2 and RMSE value of 0.7899 and 9.886 mGal, respectively. This selected model will be used in the estimating the bathymetry for Malaysian Seas in future.

Goce flight dynamics operations from an orbital perspective

The Gravity Field and Steady-State Ocean Circulation Explorer mission GOCE was successfully launched on the 17 th of March 2009 from the Plesetsk Cosmodrome on a Rockot launch vehicle and is since controlled from ESA's European Space Operations Centre (ESOC) in Darmstadt, Germany. The spacecraft is the first Earth Explorer Core Mission of the ESA Living Planet Program and its main objective is to obtain detailed measurements of the Earth's gravitational field to an unprecedented accuracy. The purpose of this paper is to report on the experiences of operating such an unique spacecraft from a flight dynamics point of view with an emphasis on the orbital aspects of the mission during the different phases of the mission (LEOP, commissioning, routine phases, and contingencies).

GNSS-R concept extended by a fine orbit tuning

Small changes in semimajor axis of the orbits selected for the GNSS-R [R as Reflectometry] satellites, so-called fine orbit tuning, known from the ESA’s Gravity and steady-state Ocean Circulation Explorer mission, can dramatically increase the number of nadir and off-nadir reflecting points and, in turn, can enhance the capability of the concept of bistatic altimetry (GNSS Reflectometry) without additional costs. The application of our suggestion is feasible for a satellite which will be equipped by thrusters for the orbit keeping. During the mission lifetime several orbit tunings are feasible, just to transfer from one to another orbit. Then we can study short-periodic or longer-periodic features, according to scientific goals defined for the mission. The shortest cycles (few days), corresponding to the required revisit time (defined by ESA), may be subcycles of much longer cycles (repeat periods).

Mission design, operation and exploitation of the gravity field and steady-state ocean circulation explorer mission

The European Space Agency’s Gravity field and steady-state ocean circulation explorer mission (GOCE) was launched on 17 March 2009. As the first of the Earth Explorer family of satellites within the Agency’s Living Planet Programme, it is aiming at a better understanding of the Earth system. The mission objective of GOCE is the determination of the Earth’s gravity field and geoid with high accuracy and maximum spatial resolution. The geoid, combined with the de facto mean ocean surface derived from twenty-odd years of satellite radar altimetry, yields the global dynamic ocean topography. It serves ocean circulation and ocean transport studies and sea level research. GOCE geoid heights allow the conversion of global positioning system (GPS) heights to high precision heights above sea level. Gravity anomalies and also gravity gradients from GOCE are used for gravity-to-density inversion and in particular for studies of the Earth’s lithosphere and upper mantle. GOCE is the first-ever satellite to carry a gravitational gradiometer, and in order to achieve its challenging mission objectives the satellite embarks a number of world-first technologies. In essence the spacecraft together with its sensors can be regarded as a spaceborne gravimeter. In this work, we describe the mission and the way it is operated and exploited in order to make available the best-possible measurements of the Earth gravity field. The main lessons learned from the first 19 months in orbit are also provided, in as far as they affect the quality of the science data products and therefore are of specific interest for GOCE data users.

Investigations on the polar gap problem in ESA's gravity field and steady-state ocean circulation explorer mission (GOCE)

Due to the sun-synchronous GOCE orbit two polar regions with a radius of about 700 km cannot be observed. In this contribution effects of these data gaps are investigated using a regional recovery strategy. It consists essentially of a variant of the time-wise approach, where the geopotential is represented by a system of space-localizing base functions. The ability of the three diagonal tensor components of the gradiometer instrument “to look into” the gaps is investigated. Our investigations show that 1°×1° mean anomalies can be recovered at reduced accuracy. Referred to a simulated gravity field model complete up to degree 360 the rms-values increase with geographical latitude from 3 mGal up to 10–12 mGal. It is not possible to overcome these problems solely by introducing additional GPS data, but if terrestrial or airborne gravity information is available, also datum defects can be successfully fixed. This work on the polar gap problem was done in the framework of the ESA contract on GOCE entitled “From Eötvös to Milligal”.