Volker Michel

Constructive approximation and numerical methods in geodetic research today – an attempt at a categorization based on an uncertainty principle

Abstract. Current activities and recent progress on constructive approximation and numerical analysis in physical geodesy are reported upon. Two major topics of interest are focused upon, namely trial systems for purposes of global and local approximation and methods for adequate geodetic application. A fundamental tool is an uncertainty principle, which gives appropriate bounds for the quantification of space and momentum localization of trial functions. The essential outcome is a better understanding of constructive approximation in terms of radial basis functions such as splines and wavelets.

A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods

This paper provides an overview of two topics. First, it presents a unified approach to various techniques addressing the non-uniqueness of the solution of the inverse gravimetric problem; alternative, simple proofs of some known results are also given. Second, it summarizes in a concise and self-contained way a particular multiscale regularization technique involving scaling functions and wavelets.

Satellite-to-satellite tracking and satellite gravity gradiometry (Advanced techniques for high-resolution geopotential field determination)

The purpose of satellite-to-satellite tracking (SST) and/or satellite gravity gradiometry (SGG) is to determine the gravitational field on and outside the Earths surface from given gradients of the gravitational potential and/or the gravitational field at satellite altitude. In this paper both satellite techniques are analysed and characterized from a mathematical point of view. Uniqueness results are formulated. The justification is given for approximating the external gravitational field by finite linear combination of certain gradient fields (for example, gradient fields of single-poles or multi-poles) consistent to a given set of SGG and/or SST data. A strategy of modelling the gravitational field from satellite data within a multiscale concept is described; illustrations based on the EGM96 model are given.

Regularized wavelet-based multiresolution recovery of the harmonic mass density distribution from data of the Earth's gravitational field at satellite height

The inverse problem of recovering the Earths density distribution from data of the first or second derivative of the gravitational potential at satellite orbit height is discussed for a ball-shaped Earth. This problem is exponentially ill-posed. In this paper, a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part, the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG (satellite gravity gradiometry) satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earths crust. Moreover, the noise sensitivity of the regularization technique is analysed numerically.

Splines on the three-dimensional ball and their application to seismic body wave tomography

In this paper, we construct spline functions based on a reproducing kernel Hilbert space to interpolate/approximate the velocity field of earthquake waves inside the Earth based on travel-time data for an inhomogeneous grid of sources (hypocenters) and receivers (seismic stations). Theoretical aspects including error estimates and convergence results as well as numerical results are demonstrated.

Sparse regularization of inverse gravimetry—case study: spatial and temporal mass variations in South America

Sparse regularization has recently experienced high popularity in the inverse problems community. In this paper, we show that a sparse regularization technique can also be developed for linear geophysical tomography problems. For this purpose, we adapt a known matching pursuit algorithm. The main theoretical features (existence, stability, and convergence) of the new method are given. We also show further properties of some trial functions which we use. Moreover, the algorithm is applied to a static and a monthly varying gravitational field of South America which yields spatial and temporal variations in the mass distribution. The new approach represents essential progress in comparison to a corresponding wavelet method, which is not flexible enough for the use of heterogeneous data, and a respective spline method, where the resolution cannot exceed approximately 104 basis functions due to experienced numerical problems with the ill-conditioned and dense matrix. The novel sparse regularization technique does not require homogeneous data and is not limited in the number of basis functions due to its iterative algorithm.

Optimally Localized Approximate Identities on the 2-Sphere

We introduce a method to construct approximate identities on the 2-sphere that have an optimal localization. This approach can be used to accelerate the calculations of approximations on the 2-sphere essentially with a comparably small increase of the error. The localization measure in the optimization problem includes a weight function that can be chosen under some constraints. For each choice of weight function, existence and uniqueness of the optimal kernel are proved as well as the generation of an approximate identity in the bandlimited case. Moreover, the optimally localizing approximate identity for a certain weight function is calculated and numerically tested.

Orthogonal Zonal, Tesseral and Sectorial Wavelets on the Sphere for the Analysis of Satellite Data

The spherical harmonics Yn,k}n=0,1,...;k=−n,...,n represent a standard complete orthonormal system in ℒ2(Ω), where Ω is the unit sphere. In view of present and future satellite missions (e.g., for the determination of the Earths gravity field) it is of particular importance to treat the different accuracies and sizes of data in dependence of the index pairs (n,k). It is, e.g., known that the GOCE mission yields essentially less accurate data in the zonal (k=0) case. Therefore, this paper presents new ways of constructing multiresolutions for a Sobolev space of functions on Ω allowing the separate treatment of certain classes of pairs (n,k) and, in particular, the separate treatment of different orders k. Orthogonal bandlimited as well as non-bandlimited detail and scale spaces adapted to certain (geo)scientific problems and to the character of the given data can now be used. Finally, an explicit representation of a non-bandlimited wavelet on Ω yielding an orthogonal decomposition of the function space is calculated for the first time.

Electro-magneto-encephalography for the three-shell model: numerical implementation via splines for distributed current in spherical geometry

The basic inverse problems for the functional imaging techniques of electroencephalography (EEG) and magnetoencephalography (MEG) consist in estimating the neuronal current in the brain from the measurement of the electric potential on the scalp and of the magnetic field outside the head. Here we present a rigorous derivation of the relevant formulae for a three-shell spherical model in the case of independent as well as simultaneous MEG and EEG measurements. Furthermore, we introduce an explicit and stable technique for the numerical implementation of these formulae via splines. Numerical examples are presented using the locations and the normal unit vectors of the real 102 magnetometers and 70 electrodes of the Elekta Neuromag (R) system. These results may have useful implications for the interpretation of the reconstructions obtained via the existing approaches.

Inverting GRACE gravity data for local climate effects

Abstract The Amazon area is the largest water shed on Earth. Thus, it is of great importance to observe the water levels regularly. The satellite mission Gravity Recovery and Climate Experiment (GRACE) allows, since its launch in 2002, a monthly global overview of the water distribution on Earth, in particular floods and droughts. In recent years, the Amazon area has experienced a number of extreme weather situations in late summer (July through October), explicitly a drought in 2005 and one in 2010. Furthermore, one can identify the remains of a flood in spring 2009 in the summer season of 2009 as well, where the names of the seasons refer here to the northern hemisphere, though some events were also located on the southern hemisphere. Here we present corresponding results with respect to a recently introduced localized method called the RFMP (Regularized Functional Matching Pursuit) that can be applied to ill-posed inverse problems. In comparison to the usual processing of GRACE data as well as other data types (i.e., the volumetric soil moisture content given by the NOAA-CIRES Twentieth Century Global Reanalysis Version II and the average layer 1 soil moisture given by the GLDAS Noah Land Surface Model L4), we gain an improved spatial resolution with the novel method. We also observe that it is very difficult to validate inverted GRACE data with hydrological models due to, e.g., discrepancies among these models.