Research
During the last decades technological progress has changed completely the
observational methods in all fields of geosciences and -engineering with a trend
to achieve immediate results, thus reducing time and costs. Modern high speed
computers and satellite based techniques are entering more and more disciplines
like geomagnetics, geodesy, geology, meteorology, navigation and many others.
The increasing observational accuracy demands adequate mathematical tools;
mathematics concerned with geoscientific problems, i.e. geomathematics, is
becoming more and more indispensable. Geomathematics offers appropriate means of
assimilating, assessing and reducing the comprehensible form the readily increasing
flow of data from geomagnetic, geochemical, geodetic, geological, and satellite
sources and providing an objective basis for scientific interpretation,
classification, testing of concepts and solution of problems.
Undoubtedly, the stage is set for geomathematics to play a major role in all Earth's sciences.
Presents Research Projects
The Geomathematics Group, Kaiserslautern, is especially concerned with the following research projects:
Special functions of mathematical physics
vectorial harmonic basis systems in SST (Satellite-to-Satellite Tracking),
tensorial harmonic basis systems in SGG (Satellite Gravity Gradiometry),
special systems for the solution of the Cauchy-Navier-Equations of the theory of elasticity (earthquakes, deformation analysis, loading problems (e.g. at reservoirs)),
special systems for the solution of the Maxwell equations (geomagnetic field determination via the CHAMP satellite (2000), refraction analysis in geodetic surveys),
vectorial spherical harmonics and radial basis functions for the solution of the Navier-Stokes equations on the sphere (modelling of ocean flows from measurements of the future satellite missions GRACE (2001) and GOCE (2005), wind field modelling)
Partial differential equations
- Potential theory
geoid and geoptential determination from oblique-derivativ boundary value problems, gravimetry (determination of density and discontinuities in the Earth's interior from gravity data),
inverse problems from satellite applications (determination of the gravitational field from measurements of the CHAMP (2000), GRACE (2001) and GOCE (2005) satellite missions),
time-dependent gravitational field determination (from GOCE data),
pseudo-differential equations in 'Satellite-to-Satellite Tracking' and 'Satellite Gravity Gradiometry'
- Theory of Elasticity
Cauchy-Navier-equations of the elastic field (boundary value problems of elasticity, loading problems at reservoirs, causality to seismic phenomena)
- Electromagnetism
geomagnetic field determination (determination of the magnetic induction, modelling of electric current densities in the iono- and magnetosphere from satellite data, regularization),
refraction (i.e. determination of atmospheric refraction via CCD-camera data, turbulence, fractal structure)
- Flow dynamics
Navier-Stokes equations on the sphere (wind field modelling)
Geostrophic flow of ocean currents (ocean circulation)
Constructive approximation
(scalar, vectorial and tensorial) radial basis functions, uncertainty principles, space-frequency behaviour, multivariate approximation (splines, wavelets and their application to partial differential equations), data analysis,
vectorial spline deformation analysis of the Earth's crust,
Approximation of functions depending on time and location, in particular wave phenomena
Regularization of Inverse Problems
Downward continuation of satellite data (satellite-to-satellite tracking, satellite gradiometry) to the Earth's surface
Modelling of the crustal geomagnetic field
Reconstruction of the mass density inside the earth from gravitational data
Determination of the spatial variation of the velocity of seismic waves from the traveltime between earthquake hypocenter and recording station
Determination of the Earth's structure from the analysis of the eigenoscillations of the earth
Numerical Methods (development and implementation)
spherical harmonic expansion,
domain decomposition methods,
fast multipole methods (FMM),
fast wavelet transform (FWT), tree algorithms (pyramid schemes),
spline interpolation and smoothing, best approximation,
wavelet denoising (multiscale signal-to-noise response),
numerical integration (on georelevant surfaces),
data analysis, data compression
Scientific Computing
multiscale modelling of the Earth's gravitational field (from satellite data, e.g. CHAMP, GRACE and GOCE data),
multiscale modelling of the geomagnetic field and electric current distributions (from MAGSAT and CHAMP data),
multiscale modelling of density variations in the Earth's interior from gravity data (using OSA91a, EGM96a),
spline and multiscale modelling of the windfield in Rheinland-Pfalz (from data of the Deutscher Wetterdienst and the Forstliche Versuchsanstalt des Landes Rheinland-Pfalz, Trippstadt)
multiscale modelling of the eigenoscillations of the earth
