Wolf-Dieter Schuh

EGM_TIM_RL05: An independent geoid with centimeter accuracy purely based on the GOCE mission

After more than 4.5 years in orbit, the Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission ended with the reentry of the satellite on 11 November 2013. This publication serves as a reference for the fifth gravity field model based on the time-wise approach (EGM_TIM_RL05), a global model only determined from GOCE observations. Due to its independence of any other gravity data, a consistent and homogeneous set of spherical harmonic coefficients up to degree and order 280 (corresponding to spatial resolution of 71.5 km on ground) is provided including a full covariance matrix characterizing the uncertainties of the model. The associated covariance matrix realistically describes the model quality. It is the first model which is purely based on GOCE including all observations collected during the entire mission. The achieved mean global accuracy is 2.4 cm in terms of geoid heights and 0.7 mGal for gravity anomalies at a spatial resolution of 100 km.

The processing of band-limited measurements; filtering techniques in the least squares context and in the presence of data gaps

This paper discusses the treatment of correlated measurements in the least squares context. We focus on the processing of band-limited measurements and on long time series with a constant sampling interval. Time domain as well as frequency domain approaches were discussed to offer different ways to integrate the filtering process into the optimization scheme as good as possible. The focus was on long equispaced data sets, The application of discrete filters in the space domain makes it possible to decorrelate the observations during data acquisition. This opens the way to a sequential adjustment procedure, where the design matrix is treated row-by-row. Huge systems with millions of observations can be solved by direct or iterative strategies, and both approaches benefit from well-tailored filter techniques. Because of the sequential access the computational effort of this giant task can be easily distributed to a cluster of parallel processors and offers, in addition. the possibility to treat data gaps in a straightforward way.

Finite covariance functions

Because of the full covariance matrices and the computer storage limitations the number of measurements which can be handled by the collocation method simultaneously, is limited. This paper presents a method to compute covariance functions with a finite support yielding sparse covariance matrices. The theoretical background is pointed out and, for the one- and two-dimensional case, special functions are developed which can be combined with the usually used covariance functions to get a “finite covariance function”. Simulated examples to demonstrate the behaviour of different solution methods to solve these special, sparse covariance matrices supplement our investigations.

GOCE Data Analysis: From Calibrated Measurements to the Global Earth Gravity Field

The goal of this chapter is to describe an in-situ approach to determine a global Earth gravity model and its variance/covariance information on the basis of calibrated measurements from the GOCE mission. As the main characteristics of this procedure, the GOCE data are processed sequentially on a parallel computer system, iteratively via application of the method of preconditioned conjugate gradient multiple adjustment (PCGMA), and in situ via development of the functionals at the actual location and orientation of the gradiometer. We will further explain the adaption of the unknown stochastic model, determined by estimating decorrelation filters and variance components with respect to the GOCE observation types (i.e. SST, SGG, and regularizing prior information).

GOCE Gravity Field Processing

A concept for an operable software system for the processing of a high-accuracy, high-resolution spherical harmonic model of the Earth’s gravity field from GOCE observables (satellite gravity gradiometry (SGG), satellite-to-satellite tracking in high-low mode (hl-SST)) is presented. The software architecture and the data flow are briefly described, and the main software components and recent developments are presented. Selected numerical simulations are performed to demonstrate the functionality of the software. They are based on a realistic mission scenario. Special emphasis is placed on the impact of the new GOCE mission design, i.e. the gravity gradients defined in the Gradiometer Reference Frame (GRF), which deviates from the actual flight direction (Local Orbit Reference Frame; LORF) by a few degrees, and the resulting modified error budget of the GOCE gradiometer. Additionally, the benefits of a combination of the SGG and hl-SST components are presented and discussed.

GOCE Gravity Field Modeling: Computational Aspects — Free Kite Numbering Scheme

The modelling of the Earth’s gravity field by means of a high-resolving spherical harmonic analysis is a numerically demanding task, especially when realistic (non gridded) data sets are analysed. The free kite numbering scheme, presented in the current article, allows a flexible combination of models. It is focussed, in particular, on the combination of a model containing rotation-symmetrical, high-resolving data with a second model comprising fully correlated data, which allows the determination of the lower degrees. This kite scheme may, depending on the degree of conformance with rotation symmetry, be used both with a direct solver and to improve the convergence rate of an iterative solver.

Observation of the system earth from space : CHAMP, GRACE, GOCE and future missions

Part I. 1. Lotse CHAMP-GRACE: An interdisciplinary research project for earth observation from space.- 2. Improvement in GPS orbit determination at GFZ.- 3. Using accelerometer data as observations.- 4. GFZ RL05 - an improved time -series of monthly GRACE gravity field solution.- 5. GRACE gravity modeling using the integrated approach.- 6. Comparison of daily GRACE solutions to GPS station height movements.- 7. Identification and reduction of satellite-induced signals is GRACE accelerometer data.- 8. Reprocessing and application of GPS radio occultation data from CHAMP and GRACE.- Part II. 9. Real data analysis GOCE (REAL GOCE).- 10. GOCE gravity gradients: reprocessed gradients and spherical harmonic analyses.- 11. GOCE gravity gradients : combination with GRACE and satellite altimetry.- 12. Incorporating topographic-isostatic information into GOCE gravity gradient processing.- 13. Global gravity fields from different GOCE orbit products.- 14. Adjustment of digital filters for decorrelation of GOCE SGG data.- 15. Stochastic modeling of GOCE gravitational tensor invariants.- 16. Cross-overs assess quality of GOCE gradients.- 17. Consistency of GOCE geoid information with in situ ocean and atmospheric data, tested by ocean state estimation.- 18. Regional validation and combination of GOCE gravity field models and terrestrial data.- 19. Height system unification based on GOCE gravity field models - benefits and challenges.- 20. EIGEN-6C - a high resolution global gravity combination model including GOCE data.

A stochastic framework for inequality constrained estimation

Quality description is one of the key features of geodetic inference. This is even more true if additional information about the parameters is available that could improve the accuracy of the estimate. However, if such additional information is provided in the form of inequality constraints, most of the standard tools of quality description (variance propagation, confidence ellipses, etc.) cannot be applied, as there is no analytical relationship between parameters and observations. Some analytical methods have been developed for describing the quality of inequality constrained estimates. However, these methods either ignore the probability mass in the infeasible region or the influence of inactive constraints and therefore yield only approximate results. In this article, a frequentist framework for quality description of inequality constrained least-squares estimates is developed, based on the Monte Carlo method. The quality is described in terms of highest probability density regions. Beyond this accuracy estimate, the proposed method allows to determine the influence and contribution of each constraint on each parameter using Lagrange multipliers. Plausibility of the constraints is checked by hypothesis testing and estimating the probability mass in the infeasible region. As more probability mass concentrates in less space, applying the proposed method results in smaller confidence regions compared to the unconstrained ordinary least-squares solution. The method is applied to describe the quality of estimates in the problem of approximating a time series with positive definite functions.

Mean dynamic topography estimates purely based on GOCE gravity field models and altimetry

The quality of mean dynamic topography (MDT) models derived from an altimetric mean sea surface and a gravity field model mainly depends on the spatial resolution and accuracy of the particular gravity field model. We use an integrated approach which allows for estimating the MDT and its (inverse) covariance matrix on a predefined grid which is one of the requirements for ocean data assimilation. The quality and accuracy of the MDT directly reflects the quality and accuracy of the used gravity field model. For the first time, MDT estimates along with its full error covariance matrix based on Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) data can be provided. We demonstrate the progress accomplished with GOCE processing and the valuable contribution of the GOCE gravity field models regarding the estimation of the MDT by showing results based on altimetric observations of Jason-1 and Envisat in combination with different GOCE gravity field models for the North Atlantic.

Fast Variance Component Estimation in GOCE Data Processing

For the processing of GOCE (Gravity Field and steady-state Ocean Circulation Explorer) data the program system pcgma (Preconditioned Conjugate Gradient Multiple Adjustment) was designed as a tailored solution strategy for the determination of the Earth’s gravity field in terms of a spherical harmonic analysis. Within GOCE-HPF (High Level Processing Facility) the pcgma algorithm works with the purpose of a tuning machine in that it is used to optimize the filter design and to determine optimal variance components with respect to the combination of satellite-to-satellite tracking (sst) data, satellite gravity gradiometry (sgg) data and additional prior information about the smoothness of the gravity field (the latter especially with regard to the polar regions). pcgma is based on an extended version of the iterative conjugate gradient (CG) algorithm, which allows for data combination in terms of observation and normal equations. A basic prerequisite for handling the nesting of the two iterative methods (variance component estimation (VCE) and parameter estimation using CG) is an efficient and fast implementation, because the VCE requires a repeated solution of the system. In this paper we will show how the nesting can be organized in an optimal way. We will concentrate on the reduction of CG iteration steps.