Spaceborne gravitational field determination by means of locally supported wavelets

In space-borne gravitational field determination, two challenges are inherent. First, the continuation of the data down to the surface of the Earth is an ill-posed problem, requiring therefore regularization techniques. Second huge data sets result requiring efficient numerical methods. In this paper, we show how locally supported wavelets on the sphere can be developed by means of a spherical version of the so-called up function. By construction, the corresponding scaling functions and wavelets are infinitely smooth, so that they can be used for regularization purposes. In particular, we show how the ill-posed pseudo-differential equations coming from satellite missions can be regularized by efficient numerical schemes using locally supported wavelets. These methods seem in particular to be interesting for regional gravity field modelling.     

Spectral and Multiscale Signal-to-Noise Thresholding of Spherical Vector Fields

The basic concepts of spectral and multiscale selective reconstruction of (geophysically relevant) vector fields on the sphere from error-affected data is outlined in detail. The reconstruction mechanism is formulated under the assumption that spectral as well as multiscale approximation is well-representable in terms of only a certain number of expansion coefficients at the various resolution levels. It is shown that spectral denoising by means of orthogonal expansions in terms of vector spherical harmonics reflects global a priori information of the noise (e.g., in form of a covariance tensor field), whereas multiscale signal-to-noise thresholding can be performed under locally dependent noise information within a multiresolution analysis in terms of spherical vector wavelets. An application of the multiscale formalism to Earth's magnetic field determination is presented.     

Classical globally reflected gravity field determination in modern locally oriented multiscale framework

The purpose of this paper is the canonical connection of classical global gravity field determination following the concept of Stokes (Trans Camb Philos Soc 8:672–712, 1849), Bruns (Die Figur der Erde, Publikation Königl. Preussisch. Geodätisches Institut, P. Stankiewicz Buchdruckerei, Berlin, 1878), and Neumann (Vorlesungen über die Theorie des Potentials und der Kugelfunktionen. Teubner, Leipzig, pp 135–154, 1887) on the one hand and modern locally oriented multiscale computation by use of adaptive locally supported wavelets on the other hand. The essential tools are regularization methods of the Green, Neumann, and Stokes integral representations. The multiscale approximation is guaranteed simply as linear difference scheme by use of Green, Neumann, and Stokes wavelets. As an application, gravity anomalies caused by plumes are investigated for the Hawaiian and Iceland areas.     

An integrated wavelet concept of physical geodesy

For the determination of the earth's gravity field many types of observations are nowadays available, including terrestrial gravimetry, airborne gravimetry, satellite-to-satellite tracking, satellite gradio-metry, etc. The mathematical connection between these observables on the one hand and gravity field and shape of the earth on the other is called the integrated concept of physical geodesy. In this paper harmonic wavelets are introduced by which the gravitational part of the gravity field can be approximated progressively better and better, reflecting an increasing flow of observations. An integrated concept of physical geodesy in terms of harmonic wavelets is presented. Essential tools for approximation are integration formulas relating an integral over an internal sphere to suitable linear combinations of observation functionals, i.e. linear functionals representing the geodetic observables. A scale discrete version of multiresolution is described for approximating the gravitational potential outside and on the earth's surface. Furthermore, an exact fully discrete wavelet approximation is developed for the case of band-limited wavelets. A method for combined global outer harmonic and local harmonic wavelet modelling is proposed corresponding to realistic earth's models. As examples, the role of wavelets is discussed for the classical Stokes problem, the oblique derivative problem, satellite-to-satellite tracking, satellite gravity gradiometry and combined satellite-to-satellite tracking and gradiometry. Received: 28 February 1997 / Accepted: 17 November 1997     

Multiscale Gravitational Field Recovery from GPS-Satellite-to-Satellite Tracking

The purpose of GPS-satellite-to-satellite tracking (GPS-SST) is to determine the gravitational potential at the earth's surface from measured ranges (geometrical distances) between a low-flying satellite and the high-flying satellites of the Global Positioning System (GPS). In this paper, GPS-satellite-to-satellite tracking is reformulated as the problem of determining the gravitational potential of the earth from given gradients at satellite altitude. The uniqueness and stability of the solution are investigated. The essential tool is to split the gradient field into a normal part (i.e. the first-order radial derivative) and a tangential part (i.e. the surface gradient). Uniqueness is proved for polar, circular orbits corresponding to both types of data (first radial derivative and/or surface gradient). In both cases gravity recovery based on satellite-to-satellite tracking turns out to be an exponentially ill-posed problem. Regularization in terms of spherical wavelets is proposed as an appropriate solution method, based on the knowledge of the singular system. Finally, the extension of this method is generalized to a nonspherical earth and a non-spherical orbital surface, based on combined terrestrial and satellite data.     

On the local multiscale determination of the Earth’s disturbing potential from discrete deflections of the vertical

As a first approximation, the Earth is a sphere; as a second approximation, it may be considered an ellipsoid of revolution. The deviations of the actual Earth’s gravity field from the ellipsoidal “normal” field are so small that they can be understood to be linear. The splitting of the Earth’s gravity field into a “normal” and a remaining small “disturbing” field considerably simplifies the problem of its determination. Under the assumption of an ellipsoidal Earth model, high observational accuracy is achievable only if the deviation (deflection of the vertical) of the physical plumb line, to which measurements refer, from the ellipsoidal normal is not ignored. Hence, the determination of the disturbing potential from known deflections of the vertical is a central problem of physical geodesy. In this paper, we propose a new, well-promising method for modelling the disturbing potential locally from the deflections of the vertical. Essential tools are integral formulae on the sphere based on Green’s function with respect to the Beltrami operator. The determination of the disturbing potential from deflections of the vertical is formulated as a multiscale procedure involving scale-dependent regularized versions of the surface gradient of the Green function. The modelling process is based on a multiscale framework by use of locally supported surface curl-free vector wavelets.      

On the approximation of external gravitational potential with closed systems of (trial) functions

Summary Let S be the (regular) boundary-surface of an exterior regionE _e in Euclidean space ℜ³ (for instance: sphere, ellipsoid, geoid, earth's surface). Denote by {φ_n} a countable, linearly independent system of trial functions (e.g., solid spherical harmonics or certain singularity functions) which are harmonic in some domain containingE _e ∪ S. It is the purpose of this paper to show that the restrictions {ϕ_n} of the functions {φ_n} onS form a closed system in the spaceC (S), i.e. any functionf, defined and continuous onS, can be approximated uniformly by a linear combination of the functions ϕ_n. Consequences of this result are versions of Runge and Keldysh-Lavrentiev theorems adapted to the chosen system {φ_n} and the mathematical justification of the use of trial functions in numerical (especially: collocational) procedures.     

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

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. Received: 4 May 1999 / Accepted: 21 May 1999     

Spherical Tikhonov regularization wavelets in satellite gravity gradiometry with random noise

 A special class of regularization methods for satellite gravity gradiometry based on Tikhonov spherical regularization wavelets is considered, with particular emphasis on the case of data blurred by random noise. A convergence rate is proved for the regularized solution, and a method is discussed for choosing the regularization level a posteriori from the gradiometer data. Received: 23 March 2000 / Accepted: 20 September 2000