Jan Martin Brockmann

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.

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).

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.

Adjustment of Digital Filters for Decorrelation of GOCE SGG Data

GOCE satellite gravity gradiometry (SGG) data are strongly autocorrelated within the various tensor components. Consideration of these correlations in the least-squares adjustment for gravity field determination can be carried out by digital decorrelation filters. Due to the complexity of the correlation pattern the used decorrelation filters consist of a cascade of individual filters. In this contribution some of the properties of these filters and their application to GOCE SGG data decorrelation will be presented.

Optimal regularization for geopotential model GOCO02S by Monte Carlo methods and multi-scale representation of density anomalies

GOCO02S is a combined satellite-only geopotential model, regularized from degrees 180 to 250 of the expansion into spherical harmonics. To investigate the start of the regularization, the normal equations of GOCO02S have been used to compute additional geopotential models by regularizations beginning at degrees 160, 200, 220 and with no regularization. Three different methods are applied to determine where to start the regularization. The simplest one considers the decrease of the degree variances of the not regularized solution. The second one tests for the same solution the hypothesis that the square root of the degree variance is equal to the value computed by the estimated harmonic coefficients. If the hypothesis has to be rejected for a certain degree, the error degree variance is so large that the estimated harmonic coefficients cannot be trusted anymore so that the regularization has to start at that degree. The third method uses the density anomalies by which the disturbing potential is caused resulting from the geopotential model. The density anomalies are well suited to visualize the effects of the higher degree harmonics. In contrast to the base functions of the harmonic coefficients with global support, the density anomalies are expressed by a B-spline surface with local support. Multi-scale representations were applied and the hypotheses tested that the wavelet coefficients are equal to zero. Accepting the hypotheses means that nonsignificant wavelet coefficients were determined which lead to nonsignificant density anomalies. By comparing these anomalies for different regularizations, the degree where to start the regularization is determined. It turns out that beginning the regularization at degree 180, as was done for GOCO02S, is a correct choice.

Completion of Band-Limited Data Sets on the Sphere

In this study we propose the complementation of satellite-only gravity field models by additional a priori information to obtain a complete model. While the accepted gravity field models are restricted to a sub-domain of the frequency space, the complete models form a complete basis in the entire space, which can be represented in the frequency domain (spherical harmonics) as well as in the space domain (data grids). The additional information is obtained by the smoothness of the potential field. Using this a priori knowledge, a stochastic process on the sphere is established as a background model. The measurements of satellite-only models are assimilated to this background model by a subdivision into the commission, transition and omission sub-domain. Complete models can be used for a rigorous fusion of complementary data sets in a multi-mission approach and guarantee also, as stand-alone gravity-field models, full-rank variance/covariance matrices for all vector-valued, linearly independent functionals.

Systematic Effects in Laser Scanning and Visualization by Confidence Regions

Abstract A new method for dealing with systematic effects in laser scanning and visualizing them by confidence regions is derived. The standard deviations of the systematic effects are obtained by repeatedly measuring three-dimensional coordinates by the laser scanner. In addition, autocovariance and cross-covariance functions are computed by the repeated measurements and give the correlations of the systematic effects. The normal distribution for the measurements and the multivariate uniform distribution for the systematic effects are applied to generate random variates for the measurements and random variates for the measurements plus systematic effects. Monte Carlo estimates of the expectations and the covariance matrix of the measurements with systematic effects are computed. The densities for the confidence ellipsoid for the measurements and the confidence region for the measurements with systematic effects are obtained by relative frequencies. They only depend on the size of the rectangular volume elements for which the densities are determined. The problem of sorting the densities is solved by sorting distances together with the densities. This allows a visualization of the confidence ellipsoid for the measurements and the confidence region for the measurements with systematic effects.

Consistent Combination of Gravity Field, Altimetry and Hydrographic Data

The ocean’s mean dynamic topography as the difference between the mean sea surface and the geoid reflects many characteristics of the general ocean circulation. Consequently, it provides valuable information for evaluating or tuning ocean circulation models. However, the determination of the mean dynamic topography from satellite based gravity field and altimetric observations as well as in-situ data is not straightforward. We developed a rigorous combination method where both instrumental errors and omission errors are accounted for, including the determination of optimal relative weights between the observation groups. This method allows the direct determination of the normal equations of the mean dynamic topography on arbitrary grids. In this paper we focus on the steps for preprocessing the in-situ data. We show results for the North Atlantic Ocean based on a combined GRACE/GOCE gravity field, altimetric sea surface height observations from Jason-1 and Envisat and in-situ observations of salinity, temperature and pressure from Argo floats.

Use of High Performance Computing for the Rigorous Estimation of Very High Degree Spherical Harmonic Gravity Field Models

The estimation of the global Earth’s gravity field parameterized as a finite spherical harmonic series is computationally demanding. The computational effort depends on the one hand on the maximal resolution of the spherical harmonic expansion and on the other hand on the number of observations which might be several millions. All global high-resolution Earth’s gravity field models currently available above degree and order 360 were computed introducing approximations, significantly reducing the numerical complexity. For example, the prerequisites for the orthogonality of the spherical harmonic base functions, leading to a block diagonal system of normal equations, are often introduced artificially by working with equally distributed data along parallels assuming constant accuracy. These methods do not allow for a complex modeling of the observation errors, or the inclusion of redundant observations. Within this contribution, we demonstrate how high-performance computers can be used for very high degree gravity field determination without introducing approximations. In addition, complex modeling of the observation errors is made possible within the algorithm to derive consistent error estimates for the spherical harmonic coefficients. Based on the high performance computing library ScaLAPACK, a gravity field solver was implemented which allows for the estimation of high degree gravity fields (e.g. degree and order 720, resulting in more than 500, 000 unknown parameters) from various data sources with the direct solution method using assembly and solution of full normal equations.