Geomathematics

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Geomathematics is a branch of mathematics . Its task is to build a bridge between mathematical theory and geotechnical application. The special attraction of this subsidiary of mathematics is based on the lively exchange of ideas between the group of applied mathematicians who are more interested in modeling, theoretical foundations, and approximate and numerical problem solving, and the group of geo-engineers who are more familiar with measurement technology, data analysis methods, the implementation of routines and software applications and - physicists .

Geomathematics as a cultural asset

According to the oldest written evidence, mathematics emerged in Sumerian Babylon from the practical tasks of measuring, counting and calculating for cultivating fields and keeping stocks. Mathematics, which dealt with relevant geoscientific issues, experienced its first heyday in antiquity; B. with the calculation of the earth's radius by the Alexandrian Eratosthenes (176–195 BC). From the Arabs about the year 827 AD a degree measurement made northwest of Baghdad is handed down. Further key stages in geomathematical research lead us via the Orient into the Western Middle Ages and modern times. Nicolaus Copernicus (1473-1543), the transition from succeeds geocentric system of Ptolemy to heliocentric system. Johannes Kepler (1571–1630) found the laws of planetary motions. Further milestones from a historical point of view include: B. the establishment of the doctrine of geomagnetism by W. Gilbert (1544-1608), the development of triangulation methods for graticule determinations by Tycho Brahe (1547-1601) and Willibrord van Roijen Snellius (1580-1626), the laws of fall of Galileo Galilei (1564 –1642) and the main features of the propagation of earthquake waves by Christiaan Huygens (1629–1695). The gravitational laws formulated by the Englishman Isaac Newton (1643–1727) make it clear that the force of gravity (also called gravity) decreases with distance from the earth. In the 17th and 18th centuries, France played an essential role with the establishment of the Academy in Paris (1666). Stages of success are the theory of isostatic balancing of the mass distribution in the earth's crust by Pierre Bouguer (1698–1758), the calculation of the earth's figure, especially the flattening of the polar, by PL Maupertuis (1698–1759) and Alexis Claude Clairaut (1713–1765) as well as the development the calculus of spherical functions by Adrienne-Marie Legendre (1752–1833) and Pierre Simon Laplace (1749–1829). The 19th century was largely shaped by the work of Carl Friedrich Gauß (1777–1855). Particularly noteworthy are the calculation of the first Fourier coefficients of the spherical function development of the earth's magnetic field, the hypothesis of electrical currents in the ionosphere and the definition of the level surface of the geoid (the term “geoid”, however, comes from the Gauss student Johann Benedict Listing (1808–1882)). At the end of the 19th century the basic idea of ​​the dynamo theory in geomagnetics was born by B. Stewart (1851–1935), u. v. a. This incomplete list (not even including the last century) shows that, historically, geomathematics is one of the great achievements of mankind.

Geomathematics as a task and goal

Geomathematics from today's perspective is dedicated to the qualitative and quantitative properties of the currently existing or possible structures of our earth system. She is both guarantor and godmother for the concept of science in earth system research . The system earth consists of a number of elements that themselves represent systems. The complexity of the earth system as a whole is determined by interacting physical, biological and chemical processes that transform and transport energy, material and information. It is characterized by natural, social and economic processes that lead to mutual influence. Consequently, simple cause-and-effect thinking is completely unsuitable for understanding. It is necessary to think in dynamic structures and to be aware of multiple, unforeseen and sometimes undesirable effects of interventions. Inherent networks must be recognized and used, self-regulation must be observed. All of these aspects make mathematics indispensable, which must be more than a collection of theories and numerical procedures.

Mathematics devoted to geosciences is essentially nothing other than the organization of the complexity of the earth system. This includes clear thinking to clarify abstract, complex issues, correct simplification of the complicated interactions, an appropriate mathematical system of terms for description and the accuracy in thinking and formulating. Geomathematics is thus becoming the key science of the complex earth system. Wherever there is data and observations, e.g. B. with the various scalar , vectorial and tensorial clusters of satellite data, it becomes mathematical. Statistics are used for B. the noise reduction, constructive approximation of the compression and evaluation, special systems of functions provide geo-relevant graphic and numerical representations - all with mathematical algorithms .

The spectrum of modern geosciences, which is in the focus of geomathematics, is broad, not least because of the increasing diversity of observations. At the same time, the “box” of mathematical tools increases. A special feature is that geomathematics is primarily concerned with those areas of the earth that are inadequate for direct measurements or not accessible (even with remote sensing methods). Inverse methods for mathematical evaluation are then inevitable. These usually mean that a physical field is measured close to the earth's surface or at satellite height, in order to then continue it with mathematical methods into the depth areas of interest ("downward continuation").

Geomathematics as a potential solution

The previous methodology of the applied measurement and evaluation procedures varies greatly depending on the measured variable being investigated (acceleration due to gravity, electric or magnetic field strength, temperature and heat flow, stress-strain behavior, etc.), the observed frequency range and the fundamental "field characteristics" (potential field, Diffusion field or wave field, each depending on the underlying differential equations ). In particular, the differential equation has a great influence on the evaluation process. Therefore - as is usual in the geosciences - the typical mathematical exploration methods are listed according to the relevant field characteristics: Potential methods (potential fields, elliptical differential equations) in gravimetry , geomagnetics, geoelectrics, geothermal energy , ..., diffusion methods (diffusion fields, parabolic differential equations) in magnetotellurics, geoelectromagnetics, ..., wave methods (wave fields, hyperbolic differential equations) in seismology and seismics, georadar, ....

The advantage and the benefit of this mathematical approach consist in the better, faster, cheaper and more reliable problem solving, namely with the means of simulation, visualization and the reduction of data floods.

Geomathematics is closely related to geoinformatics , geoengineering and geophysics. But geomathematics is also fundamentally different from these disciplines. Engineers and physicists need mathematical language as an aid and tool. The content of geomathematics is also the further development of the language itself. The subject of geoinformatics is the design and architecture of processors and computers, databases and program languages ​​etc. in a georeflective environment. For geomathematics, however, computers are not objects of study, but technical aids for solving mathematical problems of geo-reality.

literature

  • Willi Freeden : Geomathematics, what is it anyway? , Annual Report of the German Mathematicians Association, Volume 111, 2009, pp. 125–152 (and the quotations contained therein)