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This work is dedicated to the wavelet modelling of regional and temporal variations of the Earth's gravitational potential observed by GRACE. In the first part, all required mathematical tools and methods involving spherical wavelets are introduced. Then we apply our method to monthly GRACE gravity fields. A strong seasonal signal can be identified, which is restricted to areas, where large-scale redistributions of continental water mass are expected. This assumption is analyzed and verified by comparing the time series of regionally obtained wavelet coefficients of the gravitational signal originated from hydrology models and the gravitational potential observed by GRACE. The results are in good agreement to previous studies and illustrate that wavelets are an appropriate tool to investigate regional time-variable effects in the gravitational field.
In this paper we introduce a multiscale technique for the analysis of deformation phenomena of the Earth. Classically, the basis functions under use are globally defined and show polynomial character. In consequence, only a global analysis of deformations is possible such that, for example, the water load of an artificial reservoir is hardly to model in that way. Up till now, the alternative to realize a local analysis can only be established by assuming the investigated region to be flat. In what follows we propose a local analysis based on tools (Navier scaling functions and wavelets) taking the (spherical) surface of the Earth into account. Our approach, in particular, enables us to perform a zooming-in procedure. In fact, the concept of Navier wavelets is formulated in such a way that subregions with larger or smaller data density can accordingly be modelled with a higher or lower resolution of the model, respectively.
A multiscale method is introduced using spherical (vector) wavelets for the computation of the earth's magnetic field within source regions of ionospheric and magnetospheric currents. The considerations are essentially based on two geomathematical keystones, namely (i) the Mie representation of solenoidal vector fields in terms of toroidal and poloidal parts and (ii) the Helmholtz decomposition of spherical (tangential) vector fields. Vector wavelets are shown to provide adequate tools for multiscale geomagnetic modelling in form of a multiresolution analysis, thereby completely circumventing the numerical obstacles caused by vector spherical harmonics. The applicability and efficiency of the multiresolution technique is tested with real satellite data.
In this work we introduce a new bandlimited spherical wavelet: The Bernstein wavelet. It possesses a couple of interesting properties. To be specific, we are able to construct bandlimited wavelets free of oscillations. The scaling function of this wavelet is investigated with regard to the spherical uncertainty principle, i.e., its localization in the space domain as well as in the momentum domain is calculated and compared to the well-known Shannon scaling function. Surprisingly, they possess the same localization in space although one is highly oscillating whereas the other one shows no oscillatory behavior. Moreover, the Bernstein scaling function turns out to be the first bandlimited scaling function known to the literature whose uncertainty product tends to the minimal value 1.
Die Bestimmung des Erdgravitationspotentials aus den Meßdaten des Forschungssatelliten CHAMP lässt sich als Operatorgleichung formulieren (SST-Problem). Dieser Ansatz geht davon aus, dass ein geometrischer Orbit des Satelliten CHAMP vorliegt. Mittels numerischer Differentiation unter Einsatz eines geeigneten Denoising Verfahrens kann dann aus dem geometrischen Orbit der Gradient des Potentials längs der Bahn bestimmt werden. Damit sind insbesondere die Radialableitung (und der Flächengradient) auf einem Punktgitter auf der Bahn bekannt. In einem erdfesten System stellt sich dies als eine nahezu vollständige Überdeckung der Erde (bis auf Polar Gaps) mit einem ziemlich dichten Datengitter auf Flughöhe des Satelliten dar. Die Lösung der SST-Operatorgleichung (Bestimmung des Potentials auf der Erdoberfläche aus Kenntnis der Radialableitung auf einem Datengitter auf Flughöhe) ist ein schlecht gestelltes inverses Problem, das mit einer geeigneten Regularisierungstechnik gelöst werden muß. Im vorliegenden Fall wurde eine solche Regularisierung mit Hilfe von nicht-bandlimitierten Regularisierungsskalierungsfunktionen und Regularisierungswavelets umgesetzt. Diese sind stark ortslokalisierend und führen daher auf ein Potentialmodell, welches eine Linearkombination stark ortslokalisierender Funktionen ist. Ein solches Modell kann als Lokalmodell auch aus nur lokalen Daten berechnet werden und bietet daher gegenüber Kugelfunktionsmodellen wie EGM96 erhebliche Vorteile für die moderne Geopotentialbestimmung. Die Diskretisierung und numerische Umsetzung der Berechnung eines solchen Modells erfolgt mit Splines, die hier ebenfalls Linearkombinationen stark ortslokalisierender Funktionen sind. Die großen linearen Gleichungssysteme, die zur Berechnung der glättenden oder interpolierenden Splines gelöst werden müssen, können auf schnelle und effiziente Weise mit dem Schwarzschen alternierenden Algorithmus in Verbindung mit schnellen Summationsverfahren (Fast Multipole Methods) gelöst werden. Eine Kombination des Schwarzschen alternierenden Algorithmus mit solchen schnellen Summationsverfahren ermöglicht eine weitere erhebliche Beschleunigung beim Lösen dieser Gleichungssysteme. Zur Bestimmung von Glättungsparametern (Spline-Smoothing) und Regularisierungsparametern kann die L-Curve Method zum Einsatz kommen.
The purpose of satellite-to-satellite tracking (SST) and/or satellite gravity gradiometry (SGG) is to determine the gravitational field on and outside the Earth's 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 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.
Being interested in (rotation-)invariant pseudodi erential equations of satellite problems corresponding to spherical orbits, we are reasonably led to generating kernels that depend only on the spherical distance, i. e. in the language of modern constructive approximation form spherical radial basis functions. In this paper approximate identities generated by such (rotation-invariant) kernels which are additionally locally supported are investigated in detail from theoretical as well as numerical point of view. So-called spherical di erence wavelets are introduced. The wavelet transforms are evaluated by the use of a numerical integration rule, that is based on Weyl's law of equidistribution. This approximate formula is constructed such that it can cope with millions of (satellite) data. The approximation error is estimated on the orbital sphere. Finally, we apply the developed theory to the problems of satellite-to-satellite tracking (SST) and satellite gravity gradiometry (SGG).
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 of 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.
Based on the well-known results of classical potential theory, viz. the limit and jump relations for layer integrals, a numerically viable and e±cient multiscale method of approximating the disturbing potential from gravity anomalies is established on regular surfaces, i.e., on telluroids of ellipsoidal or even more structured geometric shape. The essential idea is to use scale dependent regularizations of the layer potentials occurring in the integral formulation of the linearized Molodensky problem to introduce scaling functions and wavelets on the telluroid. As an application of our multiscale approach some numerical examples are presented on an ellipsoidal telluroid.