Helga Nutz

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.

Measuring techniques for the very high cycle fatigue behaviour of high strength steel at ultrasonic frequencies

Abstract To analyse the fatigue behaviour of steels in the very high cycle fatigue regime, an ultrasonic testing facility was developed that allows describing the cyclic deformation behaviour at ultrasonic frequencies by measuring characteristic fatigue data at a sufficiently high frequency. In correlation with scanning electron micrographs, fatigue tests prove that the sensitive measuring techniques indicate fatigue induced microstructural changes by a significant rise in the process parameters more than 106 cycles before final failure occurs. By further analysis of the attenuation behaviour of the ultrasonic resonance system, the logarithmic decrement can be used as a reasonable physical value to indicate changes in the microstructure with a very high sensitivity. This allows an improved understanding of the fatigue mechanisms in the very high cycle fatigue regime.

About the Importance of the Runge–Walsh Concept for Gravitational Field Determination

On the one hand, the Runge–Walsh theorem plays a particular role in physical geodesy, because it allows to guarantee a uniform approximation of the Earth’s gravitational potential within arbitrary accuracy by a harmonic function showing a larger analyticity domain. On the other hand, there are some less transparent manifestations of the Runge–Walsh context in the geodetic literature that must be clarified in more detail. Indeed, some authors make the attempt to apply the Runge–Walsh idea to the gravity potential of a rotating Earth instead of the gravitational potential in non-rotating status. Others doubt about the convergence of series expansions approximating the Earth’s gravitational potential inside the whole outer space of the actual Earth.

Geodetic Observables and Their Mathematical Treatment in Multiscale Framework

For the determination of the Earth’s gravitational field various types of observations are available nowadays, e.g., from terrestrial gravimetry, airborne gravimetry, satellite-to-satellite tracking, satellite gravity gradiometry, etc. The mathematical relation between these observables on the one hand and the gravitational field and the shape of the Earth on the other hand is called the integrated concept of physical geodesy. In this paper, an integrated concept of physical geodesy in terms of harmonic wavelets is presented. Essential tools for approximation are Runge–Walsh type integration formulas relating an integral over an internal sphere to suitable linear combinations of observational functionals, i.e., linear functionals representing the geodetic observables in terms of gravitational quantities on and outside the Earth. A scale discrete version of multiresolution is described for approximating the gravitational potential on and outside the Earth’s surface. Furthermore, an exact fully discrete wavelet approximation is developed for the case of bandlimited wavelets. A method for combined global outer harmonic and local harmonic wavelet modeling is proposed corresponding to realistic Earth’s models.

Time-Space Multiscale AnalysisTime-Space Multiscale Analysis Multiscale analysis and Its Application to GRACE and Hydrology Data

We present two concepts of a multiresolution analysis Siehe Multiscale analysis of temporal and spatial variations of the Earth’s gravitational potential. First we apply a separated wavelet analysis using Euclidean wavelets in the time and spherical wavelets in the space domain and, second, we realize a tensor product wavelet analysis using Legendre and spherical wavelets for the time and space domain, respectively. Based on the results of the multiresolution analysis we compute the correlation coefficients between satellite and hydrology data which help us to develop a filter for extracting an improved hydrology model from the satellite data of GRACE (Gravity Recovery and Climate Experiment). Such an extraction is finally realized based on the results of the tensor product analysis.

Gravimetry and Exploration

In this work we are especially concerned with the “mathematization” of gravimetric exploration and prospecting. We investigate the extractable information of the Earth’s gravitational potential and its observables obtained by gravimetry for gravitational modeling as well as geological interpretation. More explicitly, local gravimetric data sets are exploited to visualize multiscale reconstruction and decorrelation features to be found in geophysically and geologically relevant signature bands.

Innovative Explorationsmethoden in der Gravimetrie und Reflexionsseismik

In dieser Arbeit wird eine neuartige Moglichkeit behandelt, zusatzliche Informationen uber die Risikoabschatzung aus geophysikalischen Messungen in die Bewertung eines geplanten Geothermieprojektes eingehen zu lassen. Dazu wird fur die ublicherweise eingesetzten Techniken (Gravimetrie, Reflexionsseismik) ein innovatives geomathematisches Verfahren entwickelt, um vorhandene Auswertungsmethoden sinnvoll zu erganzen. Das Verfahren, das dem Wesen nach ein seismisches Postprocessing darstellt, basiert auf einer Weiterentwicklung moderner Multiskalenverfahren der Konstruktiven Approximation.

Geomathematical Advances in Satellite Gravity Gradiometry (SGG)

A promising technique of globally establishing the fine structure and the characteristics of the external Earth’s gravitational field is satellite gravity gradiometry (SGG). Satellites such as ESA’s gradiometer satellite GOCE are able to provide sufficiently large data material of homogeneous quality and accuracy. In geodesy, traditionally the external Earth’s gravitational potential and its Hesse matrix are described using orthogonal (Fourier) expansions in terms of (outer) spherical harmonics. Spherical and outer harmonics are introduced for the global modeling of (scalar / tensor) fields. We briefly recapitulate the results interconnecting spherically the potential coefficients with respect to tensor spherical harmonics at Low Earth Orbiter’s (LEO) altitude to the corresponding coefficients with respect to scalar spherical harmonics at the Earth’s surface. The relation between the known tensorial measurements g (i.e., gradiometer data) and the gravitational potential F on the Earth’s surface is expressed by a linear integral equation of the first kind. This operator equation is discussed in the framework of pseudodifferential operators as an invertible mapping between Sobolev spaces under the assumption that the data are not erroneous. In reality, however, the data g are noisy such that the Sobolev reference space for the (noisy) tensorial data g must be embedded in a larger Sobolev space. Under these conditions, we base our inversion process on the fact that the reference Sobolev subspace is dense in the larger Sobolev space and that, e.g., a smoothing spline process or a signal-to-noise procedure in multiscale framework open appropriate perspectives to approximate F (in suitable accuracy) from noisy data g.

Mathematische Lösungspotentiale,Strategien und Dilemmata

Das Kapitel stellt eine integrative Zusammenschau aller wesentlichen in der oberflachennahen Exploration verfugbaren Datensysteme und Methoden aus mathematischer Sicht bereit, um auf diese Weise zur Risikoreduzierung in einem geothermischen Projekt beizutragen. Die in der Exploration auftretenden physikalisch motivierten Grundgleichungen werden beschrieben und klassifiziert. Sie fuhren zu sogenannten „Inversen Problemen“. Dieser Begriff wird in seiner mathematischen Bedeutung erlautert und die den Inversen Problemen innewohnende Eigenschaft der Schlecht-Gestelltheit im Hinblick auf die daraus resultierenden Schwierigkeiten bei der Losung dem Leser verdeutlicht. Auf die Behandlung dieser Art von Gleichungen durch Regularisierung wird ausfuhrlich eingegangen.

On the Solution of the Oblique Derivative Problem by Constructive Runge-Walsh Concepts

The goal of this contribution is to provide the conceptual setup of the Runge-Walsh theorem for the oblique derivative problem of physical geodesy. The Runge-Walsh concept presents constructive approximation capabilities of the Earth’s gravitational potential for geoscientifically realistic geometries. The force of gravity is generally not perpendicular to the actual Earth’s surface such that we have to handle an oblique derivative problem.