Assessment of three numerical solution strategies for gravity field recovery from GOCE satellite gravity gradiometry implemented on a parallel platform

 The recovery of a full set of gravity field parameters from satellite gravity gradiometry (SGG) is a huge numerical and computational task. In practice, parallel computing has to be applied to estimate the more than 90 000 harmonic coefficients parameterizing the Earth's gravity field up to a maximum spherical harmonic degree of 300. Three independent solution strategies (preconditioned conjugate gradient method, semi-analytic approach, and distributed non-approximative adjustment), which are based on different concepts, are assessed and compared both theoretically and on the basis of a realistic-as-possible numerical simulation regarding the accuracy of the results, as well as the computational effort. Special concern is given to the correct treatment of the coloured noise characteristics of the gradiometer. The numerical simulations show that the three methods deliver nearly identical results—even in the case of large data gaps in the observation time series. The newly proposed distributed non-approximative adjustment approach, which is the only one of the three methods that solves the inverse problem in a strict sense, also turns out to be a feasible method for practical applications. Received: 17 December 2001 / Accepted: 17 July 2002 Acknowledgments. We would like to thank Prof. W.-D. Schuh, Institute of Theoretical Geodesy, University of Bonn, for providing us with the serial version of the PCGMA algorithm, which forms the basis for the parallel PCGMA package developed at our institute. This study was partially performed in the course of the GOCE project `From E?tv?s to mGal+', funded by the European Space Agency (ESA) under contract No. 14287/00/NL/DC. Correspondence to: R. Pail     

Formation of surface spherical harmonic normal matrices and application to high-degree geopotential modeling

 The structure of normal matrices occurring in the problem of weighted least-squares spherical harmonic analysis of measurements scattered on a sphere with random noises is investigated. Efficient algorithms for the formation of the normal matrices are derived using fundamental relations inherent to the products of two surface spherical harmonic functions. The whole elements of a normal matrix complete to spherical harmonic degree L are recursively obtained from its first row or first column extended to degree 2L with only O(L ⁴) computational operations. Applications of the algorithms to the formation of surface normal matrices from geoid undulations and surface gravity anomalies are discussed in connection with the high-degree geopotential modeling. Received: 22 March 1999 / Accepted: 23 December 1999     

Efficient gravity field recovery from GOCE gravity gradient observations

 An efficient algorithm is proposed for gravity field recovery from Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite gravity gradient observations. The mathematical model is formulated in the time domain, which allows the inclusion of realistic observational noise models. The algorithm combines the iterative solution of the normal equations, using a Richardson-type iteration scheme, with the fast computation of the right-hand side of the normal equations in each iteration step by a suitable approximation of the design matrix. The convergence of the iteration is investigated, error estimates are provided, and the unbiasedness of the method is proved. It is also shown that the method does not converge to the solution of the normal equations. The performance of the approach for white noise and coloured noise is demonstrated along a simulated GOCE orbit up to spherical harmonic degree and order 180. The results also indicate that the approximation error may be neglected. Received: 30 November 1999 / Accepted: 31 May 2000     

Combination procedure for length-of-day time series according to the noise frequency behavior

 Length-of-day (LOD) estimates from seven Global Positioning System (GPS) and three satellite laser ranging (SLR) analysis centers were combined into an even-spaced time series for a 27-month period (1996–1998). This time series was compared to the multi-technique Earth-orientation-parameter (EOP) combined solution (C04) derived at the Central Bureau of the International Earth Rotation Service (IERS/CB). Due to inhomogeneities in the different series derived from the various techniques (time, length, quality, and spatial resolution), the concept of a combined solution is justified. The noise behavior in LOD for different techniques varies with frequency; the data series were divided into frequency windows after removing both biases and trends. Different weight factors were assigned in each window, discriminating by technique, and produced one-technique combined solutions. Finally, these one-technique combined solutions were combined to obtain the final multi-technique solution. The LOD combined time series obtained by the present method based on the frequency windows combined series (FWCS) is very close to the IERS C04 solution. It could be useful to generate a new LOD reference time series to be used in the study of high-frequency variations of Earth rotation. Received: 28 March 2000 / Accepted: 15 February 2001     

The spherical horizontal and spherical vertical boundary value problem – vertical deflections and geoidal undulations – the completed Meissl diagram

 In a comparison of the solution of the spherical horizontal and vertical boundary value problems of physical geodesy it is aimed to construct downward continuation operators for vertical deflections (surface gradient of the incremental gravitational potential) and for gravity disturbances (vertical derivative of the incremental gravitational potential) from points on the Earth's topographic surface or of the three-dimensional (3-D) Euclidean space nearby down to the international reference sphere (IRS). First the horizontal and vertical components of the gravity vector, namely spherical vertical deflections and spherical gravity disturbances, are set up. Second, the horizontal and vertical boundary value problem in spherical gravity and geometry space is considered. The incremental gravity vector is represented in terms of vector spherical harmonics. The solution of horizontal spherical boundary problem in terms of the horizontal vector-valued Green function converts vertical deflections given on the IRS to the incremental gravitational potential external in the 3-D Euclidean space. The horizontal Green functions specialized to evaluation and source points on the IRS coincide with the Stokes kernel for vertical deflections. Third, the vertical spherical boundary value problem is solved in terms of the vertical scalar-valued Green function. Fourth, the operators for upward continuation of vertical deflections given on the IRS to vertical deflections in its external 3-D Euclidean space are constructed. Fifth, the operators for upward continuation of incremental gravity given on the IRS to incremental gravity to the external 3-D Euclidean space are generated. Finally, Meissl-type diagrams for upward continuation and regularized downward continuation of horizontal and vertical gravity data, namely vertical deflection and incremental gravity, are produced. Received: 10 May 2000 / Accepted: 26 February 2001     

Regularization of gravity field estimation from satellite gravity gradients

 The performance of the L-curve criterion and of the generalized cross-validation (GCV) method for the Tikhonov regularization of the ill-conditioned normal equations associated with the determination of the gravity field from satellite gravity gradiometry is investigated. Special attention is devoted to the computation of the corner point of the L-curve, to the numerically efficient computation of the trace term in the GCV target function, and to the choice of the norm of the residuals, which is important for the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) in the presence of colored observation noise. The trace term in the GCV target function is estimated using an unbiased minimum-variance stochastic estimator. The performance analysis is based on a simulation of gravity gradients along a 60-day repeat circular orbit and a gravity field recovery complete up to degree and order 300. Randomized GCV yields the optimal regularization parameter in all the simulations if the colored noise is properly taken into account. Moreover, it seems to be quite robust against the choice of the norm of the residuals. It performs much better than the L-curve criterion, which always yields over-smooth solutions. The numerical costs for randomized GCV are limited provided that a reasonable first guess of the regularization parameter can be found. Received: 17 May 2001 / Accepted: 17 January 2002     

Fast spherical collocation: theory and examples

 It has long been known that a spherical harmonic analysis of gridded (and noisy) data on a sphere (with uniform error for a fixed latitude) gives rise to simple systems of equations. This idea has been generalized for the method of least-squares collocation, when using an isotropic covariance function or reproducing kernel. The data only need to be at the same altitude and of the same kind for each latitude. This permits, for example, the combination of gravity data at the surface of the Earth and data at satellite altitude, when the orbit is circular. Suppose that data are associated with the points of a grid with N values in latitude and M values in longitude. The latitudes do not need to be spaced uniformly. Also suppose that it is required to determine the spherical harmonic coefficients to a maximal degree and order K. Then the method will require that we solve K systems of equations each having a symmetric positive definite matrix of only N × N. Results of simulation studies using the method are described. Received: 18 October 2001 / Accepted: 4 October 2002 Correspondence to: F. Sansò     

Seasonal length-of-day changes and atmospheric angular momentum oscillations in their temporal variability

Recently, effective atmospheric-angular-momentum (AAM) functions as calculated from National Centers for Environmental Prediction (NCEP) (formerly National Meteorological Center, NMC) and National Center for Atmospheric Research (NCAR) Reanalyses have become available for the years 1958 to 1998. Concerning the wind terms, the top level in the atmosphere used here is 10 hPa. Compared with earlier NMC model versions, which incorporate wind fields up to 100 hPa since 1976 and up to 50 hPa since 1981, the reanalyses have produced improved data series over a longer period than before. The axial AAM component χ₃ is associated with changes in length of day (LOD). Motivated by better quality and continuity of the series AAM (NCEP) Reanalysis, the problem of the seasonal imbalances in the solid Earth–atmosphere axial angular momentum budget is re-examined. To assess better the estimates of the annual and semiannual oscillations in LOD and AAM and of the residual oscillations derived as difference series between LOD and AAM, the series of LOD data from three analysis centers [International Earth Rotation Service (IERS), GeoForschungZentrum Potsdam (GFZ) and Jet Propulsion Laboratory Pasadena (JPL)] and of AAM data in terms of χ₃(W), χ₃(P) and χ₃(P+IB) from four meteorological centers [NCEP, Japan Meteorological Agency (JMA), European Centre for Medium-Range Weather Forecasts (ECMWF) and the UK Meteorological Office (UKMO)] are used in this study. The main analysis steps were removing gaps, filtering out the seasonal oscillations, calculating optimal estimates of the parameters of the oscillations and calculating the difference series between the LOD and AAM systems as well as the residuals in the axial angular momentum budget in the LOD–AAM systems. The results derived as difference series between the different LOD, AAM and LOD–AAM systems show to what extent the variations reflect systematic differences and significant signals, respectively, which is important for future activities in this field. Received: 2 February 1999 / Accepted: 30 November 1999     

The ERS-2 altimetric bias and gravity field enhancement using dual crossovers between ERS and TOPEX/Poseidon

 Dual satellite crossovers (DXO) between the two European Remote Sensing satellites ERS-1 and ERS-2 and TOPEX/Poseidon are used to (1) refine the Earth's gravity field and (2) extend the study of the ERS-2 altimetric range stability to cover the first four years of its operation. The enhanced gravity field model, AGM-98, is validated by several methodologies and will be shown to provide, in particular, low geographically correlated orbital error for ERS-2. For the ERS-2 altimetric range study, TOPEX/Poseidon is first calibrated through comparison against in situ tide gauge data. A time series of the ERS-2 altimeter bias has been recovered along with other geophysical correction terms using tables for bias jumps in the range measurements at the single point target response (SPTR) events. On utilising the original version of the SPTR tables the overall bias drift is seen to be 2.6±1.0 mm/yr with an RMS of fit of 12.2 mm but with discontinuities at the centimetre level at the SPTR events. On utilising the recently released revised tables, SPTR2000, the drift is better defined at 2.4±0.6 mm/yr with the RMS of fit reduced to 3.7 mm. Investigations identify the sea-state bias as a source of error with corrections affecting the overall drift by close to 1.2 mm/yr. Received: 25 May 2000 / Accepted: 24 January 2001     

A formula for computing the gravity disturbance from the second radial derivative of the disturbing potential

 A formula for computing the gravity disturbance and gravity anomaly from the second radial derivative of the disturbing potential is derived in detail using the basic differential equation with spherical approximation in physical geodesy and the modified Poisson integral formula. The derived integral in the space domain, expressed by a spherical geometric quantity, is then converted to a convolution form in the local planar rectangular coordinate system tangent to the geoid at the computing point, and the corresponding spectral formulae of 1-D FFT and 2-D FFT are presented for numerical computation. Received: 27 December 2000 / Accepted: 3 September 2001     

Improvement of edge effect of the wavelet time–frequency spectrum: application to the length-of-day series

 The edge effect in the wavelet time–frequency spectrum of a time series is treated. The time series is first extended on both ends by applying a non-linear model, namely the leap-step time series analysis (LSTSA) model, prior to the wavelet transform. The results of a series of simulation experiments and an application to the observed length-of-day (LOD) series demonstrate that the edge effect is effectively reduced this way. Thus, the application of the LSTSA model improves the wavelet time–frequency spectrum, especially enhancing the ability to detect the low-frequency signals. Received: 15 September 1998 / Accepted: 4 October 1999     

Gravity/magnetic potential of uneven shell topography

 A fast spherical harmonic approach enables the computation of gravitational or magnetic potential created by a non-uniform shell of material bounded by uneven topographies. The resulting field can be evaluated outside or inside the sphere, assuming that density of the shell varies with latitude, longitude, and radial distance. To simplify, the density (or magnetization) source inside the sphere is assumed to be the product of a surface function and a power series expansion of the radial distance. This formalism is applied to compute the gravity signal of a steady, dry atmosphere. It provides geoid/gravity maps at sea level as well as satellite altitude. Results of this application agree closely with those of earlier studies, where the atmosphere contribution to the Earth's gravity field was determined using more time-consuming methods. Received: 14 August 2000 / Accepted: 19 March 2001     

SHORT NOTE: Extending simplified high-degree synthesis methods to second latitudinal derivatives of geopotential

 Simplified techniques for high-degree spherical harmonic synthesis are extended to include gravitational potential second derivatives with respect to latitude. Received: 23 July 2001 / Accepted: 12 April 2002 Acknowledgement. The authors would like to thank Christian Tscherning for recommending Laplace's equation as an accuracy test. Our use of Legendre's differential equation, as the most direct means for extending our simplified synthesis methods to second-order derivatives, was a direct result of this suggestion. Correspondence to: S. A. Holmes     

Somigliana–Pizzetti gravity: the international gravity formula accurate to the sub-nanoGal level

 The Somigliana–Pizzetti gravity field (the International gravity formula), namely the gravity field of the level ellipsoid (the International Reference Ellipsoid), is derived to the sub-nanoGal accuracy level in order to fulfil the demands of modern gravimetry (absolute gravimeters, super conducting gravimeters, atomic gravimeters). Equations (53), (54) and (59) summarise Somigliana–Pizzetti gravity Γ(φ,u) as a function of Jacobi spheroidal latitude φ and height u to the order ?(10⁻¹⁰ Gal), and Γ(B,H) as a function of Gauss (surface normal) ellipsoidal latitude B and height H to the order ?(10⁻¹⁰ Gal) as determined by GPS (`global problem solver'). Within the test area of the state of Baden-Württemberg, Somigliana–Pizzetti gravity disturbances of an average of 25.452 mGal were produced. Computer programs for an operational application of the new international gravity formula with (L,B,H) or (λ,φ,u) coordinate inputs to a sub-nanoGal level of accuracy are available on the Internet. Received: 23 June 2000 / Accepted: 2 January 2001     

Two different methods of geoidal height determinations using a spherical harmonic representation of the geopotential, topographic corrections and the height anomaly–geoidal height difference

 It is suggested that a spherical harmonic representation of the geoidal heights using global Earth gravity models (EGM) might be accurate enough for many applications, although we know that some short-wavelength signals are missing in a potential coefficient model. A `direct' method of geoidal height determination from a global Earth gravity model coefficient alone and an `indirect' approach of geoidal height determination through height anomaly computed from a global gravity model are investigated. In both methods, suitable correction terms are applied. The results of computations in two test areas show that the direct and indirect approaches of geoid height determination yield good agreement with the classical gravimetric geoidal heights which are determined from Stokes' formula. Surprisingly, the results of the indirect method of geoidal height determination yield better agreement with the global positioning system (GPS)-levelling derived geoid heights, which are used to demonstrate such improvements, than the results of gravimetric geoid heights at to the same GPS stations. It has been demonstrated that the application of correction terms in both methods improves the agreement of geoidal heights at GPS-levelling stations. It is also found that the correction terms in the direct method of geoidal height determination are mostly similar to the correction terms used for the indirect determination of geoidal heights from height anomalies. Received: 26 July 2001 / Accepted: 21 February 2002     

A note on Earth's rotation angular velocity in the general-relativity framework

 The value of the Earth's rotational angular velocity determined from observations is given in the GCRS (geocentric celestial reference system) defined by Resolution B1 of the 2000 IAU General Assembly. The same quantity derived from dynamical theories of the Earth's rotation, such as SMART97, is referred to the␣DGRSC (dynamically non-rotating ecliptical reference system). The relativistic theory of reference systems (RSs) enables unambiguous general-relativity relations between these quantities to be derived. Received: 9 November 2000 / Accepted: 9 July 2001     

GPS-based precise orbit determination of the very low Earth-orbiting gravity mission GOCE

 A prerequisite for the success of future gravity missions like the European Gravity field and steady-state Ocean Circulation Explorer (GOCE) is a precise orbit determination (POD). A detailed simulation study has been carried out to assess the achievable orbit accuracy based on satellite-to-satellite tracking (SST) by the US global positioning system (GPS) and in conjunction the implications for gravity field determination. An orbit accuracy at the few centimeter level seems possible, sufficient to support the GOCE gravity mission and in particular its gravity gradiometer. Received: 21 January 2000 / Accepted: 4 July 2000     

Local geoid determination combining gravity disturbances and GPS/levelling: a case study in the Lake Nasser area, Aswan, Egypt

 The use of GPS for height control in an area with existing levelling data requires the determination of a local geoid and the bias between the local levelling datum and the one implicitly defined when computing the local geoid. If only scarse gravity data are available, the heights of new data may be collected rapidly by determining the ellipsoidal height by GPS and not using orthometric heights. Hence the geoid determination has to be based on gravity disturbances contingently combined with gravity anomalies. Furthermore, existing GPS/levelling data may also be used in the geoid determination if a suitable general gravity field modelling method (such as least-squares collocation, LSC) is applied. A comparison has been made in the Aswan Dam area between geoids determined using fast Fourier transform (FFT) with gravity disturbances exclusively and LSC using only the gravity disturbances and the disturbances combined with GPS/levelling data. The EGM96 spherical harmonic model was in all cases used in a remove–restore mode. A total of 198 gravity disturbances spaced approximately 3 km apart were used, as well as 35 GPS/levelling points in the vicinity and on the Aswan Dam. No data on the Nasser Lake were available. This gave difficulties when using FFT, which requires the use of gridded data. When using exclusively the gravity disturbances, the agreement between the GPS/levelling data were 0.71 ± 0.17 m for FFT and 0.63 ± 0.15 for LSC. When combining gravity disturbances and GPS/levelling, the LSC error estimate was ±0.10 m. In the latter case two bias parameters had to be introduced to account for a possible levelling datum difference between the levelling on the dam and that on the adjacent roads. Received: 14 August 2000 / Accepted: 28 February 2001     

National height datum,the Gauss–Listing geoid level value w0 and its time variation 0 (Baltic Sea Level Project: epochs 1990.8, 1993.8, 1997.4)

 A methodology for precise determination of the fundamental geodetic parameter w ₀, the potential value of the Gauss–Listing geoid, as well as its time derivative ₀, is presented. The method is based on: (1) ellipsoidal harmonic expansion of the external gravitational field of the Earth to degree/order 360/360 (130 321 coefficients; http://www.uni-stuttgard.de/gi/research/ index.html projects) with respect to the International Reference Ellipsoid WGD2000, at the GPS positioned stations; and (2) ellipsoidal free-air gravity reduction of degree/order 360/360, based on orthometric heights of the GPS-positioned stations. The method has been numerically tested for the data of three GPS campaigns of the Baltic Sea Level project (epochs 1990.8,1993.4 and 1997.4). New w ₀ and ₀ values (w ₀=62 636 855.75 ± 0.21 m²/s², ₀=−0.0099±0.00079 m²/s² per year, w ₀/&γmacr;=6 379 781.502 m,₀/&γmacr;=1.0 mm/year, and &γmacr;= −9.81802523 m²/s²) for the test region (Baltic Sea) were obtained. As by-products of the main study, the following were also determined: (1) the high-resolution sea surface topography map for the Baltic Sea; (2) the most accurate regional geoid amongst four different regional Gauss–Listing geoids currently proposed for the Baltic Sea; and (3) the difference between the national height datums of countries around the Baltic Sea. Received: 14 August 2000 / Accepted: 19 June 2001     

Tectonic plate motion and co-seismic steps surveyed by DORIS field beacons: the New Hebrides experiment

 The New Hebrides experiment consisted of setting up a pair of DORIS beacons in remote tropical islands in the southwestern Pacific, between 1993 and 1997. Because of orbitography requirements on TOPEX/Poséidon, the beacons were only transmitting to SPOT satellites. Root-mean-square (RMS) scatters at the centimeter level on the latitude and vertical components were achieved, but 2-cm RMS scatters affected the longitude component. Nevertheless, results of relative velocity (123 mm/year N250°) are very consistent with those obtained using the global positioning system (GPS) (126 mm/yr N246°). The co-seismic step (12 mm N60°) related to the Walpole event (M _W = 7.7) is consistent with that derived from GPS (10 mm N30°) or from the centroid moment tensor (CMT) of the quake (12 mm N000°). Received: 19 November 1999 / Accepted: 17 May 2000