For the first time in the International Terrestrial Reference Frame (ITRF) history, the ITRF2014 is generated with an enhanced modeling of nonlinear station motions, including seasonal (annual and semiannual) signals of station positions and postseismic deformation for sites that were subject to major earthquakes. Using the full observation history of the four space geodetic techniques (very long baseline interferometry (VLBI), satellite laser ranging (SLR), Global Navigation Satellite Systems (GNSS), and Doppler orbitography and radiopositioning integrated by satellite (DORIS)), the corresponding international services provided reprocessed time series (weekly from SLR and DORIS, daily from GNSS, and 24 h session-wise from VLBI) of station positions and daily Earth Orientation Parameters. ITRF2014 is demonstrated to be superior to past ITRF releases, as it precisely models the actual station trajectories leading to a more robust secular frame and site velocities. The ITRF2014 long-term origin coincides with the Earth system center of mass as sensed by SLR observations collected on the two LAGEOS satellites over the time span between 1993.0 and 2015.0. The estimated accuracy of the ITRF2014 origin, as reflected by the level of agreement with the ITRF2008 (both origins are defined by SLR), is at the level of less than 3 mm at epoch 2010.0 and less than 0.2 mm/yr in time evolution. The ITRF2014 scale is defined by the arithmetic average of the implicit scales of SLR and VLBI solutions as obtained by the stacking of their respective time series. The resulting scale and scale rate differences between the two solutions are 1.37 (±0.10) ppb at epoch 2010.0 and 0.02 (±0.02) ppb/yr. While the postseismic deformation models were estimated using GNSS/GPS data, the resulting parametric models at earthquake colocation sites were applied to the station position time series of the three other techniques, showing a very high level of consistency which enforces more the link between techniques within the ITRF2014 frame. The users should be aware that the postseismic deformation models are part of the ITRF2014 products, unlike the annual and semiannual signals, which were estimated internally with the only purpose of enhancing the velocity field estimation of the secular frame.

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1002/2016JB013098

ITRF2014: A new release of the International Terrestrial Reference Frame modeling nonlinear station motions

Preface Bibliography 1. Interpolation. Introduction The Taylor expansion The finite Taylor series with the remainder term Interpolation by polynomials The remainder of Lagrangian interpolation formula Equidistant interpolation Local and global interpolation Interpolation by central differences Interpolation around the midpoint of the range The Laguerre polynomials Binomial expansions The decisive integral transform Binomial expansions of the hypergeometric type Recurrence relations The Laplace transform The Stirling expansion Operations with the Stirling functions An integral transform of the Fourier type Recurrence relations associated with the Stirling series Interpolation of the Fourier transform The general integral transform associated with the Stirling series Interpolation of the Bessel functions 2. Harmonic Analysis. Introduction The Fourier series for differentiable functions The remainder of the finite Fourier expansion Functions of higher differentiability An alternative method of estimation The Gibbs oscillations of the finite Fourier series The method of the Green's function Non-differentiable functions Dirac's delta function Smoothing of the Gibbs oscillations by Fejer's method The remainder of the arithmetic mean method Differentiation of the Fourier series The method of the sigma factors Local smoothing by integration Smoothing of the Gibbs oscillations by the sigma method Expansion of the delta function The triangular pulse Extension of the class of expandable functions Asymptotic relations for the sigma factors The method of trigonometric interpolation Error bounds for the trigonometric interpolation method Relation between equidistant trigonometric and polynomial interpolations The Fourier series in the curve fitting 3. Matrix Calculus. Introduction Rectangular matrices The basic rules of matrix calculus Principal axis transformation of a symmetric matrix Decomposition of a symmetric matrix Self-adjoint systems Arbitrary n x m systems Solvability of the general n x m system The fundamental decomposition theorem The natural inverse of a matrix General analysis of linear systems Error analysis of linear systems Classification of linear systems Solution of incomplete systems Over-determined systems The method of orthogonalisation The use of over-determined systems The method of successive orthogonalisation The bilinear identity Minimum property of the smallest eigenvalue 4. The Function Space. Introduction The viewpoint of pure and applied mathematics The language of geometry Metrical spaces of infinitely many dimensions The function as a vector The differential operator as a matrix The length of a vector The scalar product of two vectors The closeness of the algebraic approximation The adjoint operator The bilinear identity The extended Green's identity The adjoint boundary conditions Incomplete systems Over-determined systems Compatibility under inhomogeneous boundary conditions Green's identity in the realm of partial differential operators The fundamental field operations of vector analysis Solution of incomplete systems 5. The Green's Function. Introduction The role of the adjoint equation The role of Green's identity The delta function -- The existence of the Green's function Inhomogeneous boundary conditions The Green's vector Self-adjoint systems The calculus of variations The canonical equations of Hamilton The Hamiltonisation of partial operators The reciprocity theorem Self-adjoint problems Symmetry of the Green's function Reciprocity of the Green's vector The superposition principle of linear operators The Green's function in the realm of ordinary differential operators The change of boundary conditions The remainder of the Taylor series The remainder of the Lagrangian interpolation formula

Linear Differential Operators. By C. Lanczos. Pp. 580. 80s. 1961. (Van Nostrand, London)

: The general principles of the use of known density anomalies for gravity field modelling are reviewed with special emphasis on local applications and utilization of high degree and order spherical harmonic reference fields The natural extension to include also unknown density anomalies will be studied within the framework of geophysical inversion methods, and the prospects for hybrid gravity field modelling/inversion methods will be outlined A very simple case of such methods is the determination of representative topographic densities through collocation with parameters

/pdf/a-study-of-terrain-reductions-density-anomalies-and-166i7zwhoe.pdf

A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modelling

GipsyX/RTGx, a new tool set for space geodetic operations and research

SUMMARY Truncated singular value decomposition (TSVD) techniques have been widely used in inversion. The method of truncation determines the quality of a truncated SVD solution, but truncation has often been done arbitrarily. The ¢rst workable criterion for truncation was based on F-statistical testing, but has only rarely been used in geophysical inversion. Recently, the L-curve approach was proposed for the same purpose, and soon found many applications in interdisciplinary inverse problems. Up to the present, very little has been known of the solution quality using these two workable criteria. We have thus proposed in this paper a new quality-based method for truncation. Six TSVD estimators have been investigated for comparison in regularizing discrete unstably ill-posed problems, based on the statistically frequently used F-statistic, the L-curve and our new quality-based mean squared error (MSE) criteria. The three F-statistic-based TSVD estimators tested can indeed marginally improve the least squares (LS) solution to the ill-posed downward continuation problem in the sense of long-term averaging, depending on pre-selected signi¢cance levels.The simulations have shown that estimators of this type can hardly guarantee the improvement of the condition number of the linearized unstably ill-posed system. In other words, F-statistic criterion can frequently lead to incorrect discardings of components.TheTSVD estimatorby means ofan L-curve is thebest in (over)stabilizing ill-posed problems, but results in an over-discarding of components. It is extremely poor in terms of biases and mean squared error (MSE) roots of the solution. The qualitybased TSVD estimator with the basic ridge estimate of x as its initial value has outperformed the other twoTSVD techniques tested in terms of solution stability, bias and MSE. The simulations have clearly shown signi¢cant quality advantages of the new method. The simulations have also indicated that the new method is able to achieve a mean accuracy of 5 mgal for gravity anomalies from satellite gradiometry, if the few largest biases are left out of the computation.

/pdf/truncated-svd-methods-for-discrete-linear-ill-posed-problems-4xmeisbvh8.pdf

Truncated SVD methods for discrete linear ill-posed problems

A new realization of the International Terrestrial System was computed at the ITRS Combination Centre at DGFI as a contribution to ITRF2008. The solution is labelled DTRF2008. In the same way as in the DGFI computation for ITRF2005 it is based on either normal equation systems or estimated parameters derived from VLBI, SLR, GPS and DORIS observations by weekly or session-wise processing. The parameter space of the ITRS realization comprises station positions and velocities and daily resolved Earth Orientation Parameters (EOP), whereby for the first time also nutation parameters are included. The advantage of starting from time series of input data is that the temporal behaviour of geophysical parameters can be investigated to decide whether the parameters can contribute to the datum realization of the ITRF. In the same way, a standardized analysis of station position time series can be performed to detect and remove discontinuities. The advantage of including EOP in the ITRS realization is twofold: (1) the combination of the coordinates of the terrestrial pole—estimated from all contributing techniques—links the technique networks in two components of the orientation, leading to an improvement of consistency of the Terrestrial Reference Frame (TRF) and (2) in their capacity as parameters common to all techniques, the terrestrial pole coordinates enhance the selection of local ties as they provide a measure for the consistency of the combined frame. The computation strategy of DGFI is based on the combination of normal equation systems while at the ITRS Combination Centre at IGN solutions are combined. The two independent ITRS realizations provide the possibility to assess the accuracy of ITRF by comparison of the two frames. The accuracy evaluation was done separately for the datum parameters (origin, orientation and scale) and the network geometry. The accuracy of the datum parameters, assessed from the comparison of DTRF2008 and ITRF2008, is between 2–5 mm and 0.1–0.8 mm/year depending on the technique. The network geometry (station positions and velocities) agrees within 3.2 mm and 1.0 mm/year. A comparison of DTRF2008 and ITRF2005 provides similar results for the datum parameters, but there are larger differences for the network geometry. The internal accuracy of DTRF2008—that means the level of conservation of datum information and network geometry within the combination—was derived from comparisons with the technique-only multi-year solutions. From this an internal accuracy of 0.32 mm for the VLBI up to 3.3 mm for the DORIS part of the network is found. The internal accuracy of velocities ranges from 0.05 mm/year for VLBI to 0.83 mm/year for DORIS. The internal consistency of DTRF2008 for orientation can be derived from the analysis of the terrestrial pole coordinates. It is estimated at 1.5–2.5 mm for the GPS, VLBI and SLR parts of the network. The consistency of these three and the DORIS network part is within 6.5 mm.

The 2008 DGFI realization of the ITRS: DTRF2008

ITRS Combination Center at DGFI: a terrestrial reference frame realization 2003

The new DGFI-TUM realization of the ITRS: DTRF2014 (data)

ITRS Combination Centre at DGFI - A terrestrial reference frame realization 2003

Refined approaches for terrestrial reference frame computations

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