InSAR datum connection using GNSS-augmented radar transponders

Deformation estimates from Interferometric Synthetic Aperture Radar (InSAR) are relative: they form a ‘free’ network referred to an arbitrary datum, e.g. by assuming a reference point in the image to be stable. However, some applications require ‘absolute’ InSAR estimates, i.e. expressed in a well-defined terrestrial reference frame, e.g. to compare InSAR results with those of other techniques. We propose a methodology based on collocated InSAR and Global Navigation Satellite System (GNSS) measurements, achieved by rigidly attaching phase-stable millimetre-precision compact active radar transponders to GNSS antennas. We demonstrate this concept through a simulated example and practical case studies in the Netherlands.     

‘DEOS_CHAMP-01C_70’: a model of the Earth’s gravity field computed from accelerations of the CHAMP satellite

Performance of a recently proposed technique for gravity field modeling has been assessed with data from the CHAMP satellite. The modeling technique is a variant of the acceleration approach. It makes use of the satellite accelerations that are derived from the kinematic orbit with the 3-point numerical differentiation scheme. A 322-day data set with 30-s sampling has been used. Based on this, a new gravity field model – DEOS_CHAMP-01C_70 - is derived. The model is complete up to degree and order 70. The geoid height difference between the DEOS_CHAMP-01C_70 and EIGEN-GRACE01S models is 14 cm. This is less than for two other recently published models EIGEN-CHAMP03Sp and ITG-CHAMP01E. Furthermore, we analyze the sensitivity of the model to some empirically determined parameters (regularization parameter and the parameter that controls the frequency-dependent data weighting). We also show that inaccuracies related to non-gravitational accelerations, which are measured by the on-board accelerometer, have a minor influence on the computed gravity field model.     

Calculation of strongly singular and hypersingular surface integrals

Efficient numerical computation of integrals defined on closed surfaces in ℝ³ with non-integrable point singularities that arise in physical geodesy is discussed. The method is based on the use of polar coordinates and the definition of integrals with non-integrable point singularities as Hadamard finite part integrals. First the behavior of singular integrals under smooth parameter transformations is studied, and then it is shown how they can be reduced to absolutely integrable functions over domains in ℝ². The correction terms that usually arise if the substitution rule is formally applied, in contrast to absolutely integrable functions, are calculated. It is shown how to compute the regularized integrals efficiently, and, numerical efforts for various orders of singularity are compared. Finally, efficient numerical integration methods are discussed for integrals of functions that are defined as singular integrals, a task that typically arises in Galerkin boundary element methods. Received: 15 April 1997 / Accepted: 7 May 1998     

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     

The choice of the spherical radial basis functions in local gravity field modeling

The choice of the optimal spherical radial basis function (SRBF) in local gravity field modelling from terrestrial gravity data is investigated. Various types of SRBFs are considered: the point-mass kernel, radial multipoles, Poisson wavelets, and the Poisson kernel. The analytical expressions for the Poisson kernel, the point-mass kernel and the radial multipoles are well known, while for the Poisson wavelet new closed analytical expressions are derived for arbitrary orders using recursions. The performance of each SRBF in local gravity field modelling is analyzed using real data. A penalized least-squares technique is applied to estimate the gravity field parameters. As follows from the analysis, almost the same accuracy of gravity field modelling can be achieved for different types of the SRBFs, provided that the depth of the SRBFs is chosen properly. Generalized cross validation is shown to be a suitable technique for the choice of the depth. As a good alternative to generalized cross validation, we propose the minimization of the RMS differences between predicted and observed values at a set of control points. The optimal regularization parameter is determined using variance component estimation techniques. The relation between the depth and the correlation length of the SRBFs is established. It is shown that the optimal depth depends on the type of the SRBF. However, the gravity field solution does not change significantly if the depth is changed by several km. The size of the data area (which is always larger than the target area) depends on the type of the SRBF. The point-mass kernel requires the largest data area.