Multiscale Potential Theory : With Applications to Geoscience

During the last few decades, the subject of potential theory has not been overly popular in the mathematics community. Neglected in favor of more abstract theories, it has been taught primarily where instructors have ac­ tively engaged in research in this field. This situation has resulted in a scarcity of English language books of standard shape, size, and quality covering potential theory. The current book attempts to fill that gap in the literature. Since the rapid development of high-speed computers, the remarkable progress in highly advanced electronic measurement concepts, and, most of all, the significant impact of satellite technology, the flame of interest in potential theory has burned much brighter. The realization that more and more details of potential functions are adequately visualized by "zooming­ in" procedures of modern approximation theory has added powerful fuel to the flame. It seems as if, all of a sudden, harmonic kernel functions such as splines and/or wavelets provide the impetus to offer appropriate means of assimilating and assessing the readily increasing flow of potential data, reducing it to comprehensible form, and providing an objective basis for scientific interpretation, classification, testing of concepts, and solutions of problems involving the Laplace operator.

This self-contained book provides a basic foundation for students, practitioners, and researchers interested in some of the diverse new areas of multiscale (geo)potential theory. New mathematical methods are developed enabling the gravitational potential of a planetary body to be modeled and analyzed using a continuous flow of observations from land or satellite devices. Harmonic wavelet methods are introduced, as well as fast computational schemes and various numerical test examples.

The work is divided into two main parts: Part I treats well-posed boundary-value problems of potential theory and elasticity; Part II examines ill-posed problems such as satellite-to-satellite tracking, satellite gravity gradiometry, and gravimetry. Both sections demonstrate how multiresolution representations yield Runge–Walsh type solutions that are both accurate in approximation and tractable in computation.

Topic and key features:

* Comprehensive coverage of topics which, thus far, are only scattered in journal articles and conference proceedings

* Important applications and developments for future satellite scenarios; new modelling techniques involving low-orbiting satellites

* Multiscale approaches for numerous geoscientific problems, including geoidal determination, magnetic field reconstruction, deformation analysis, and density variation modelling

* Multilevel stabilization procedures for regularization

* Treatment of the real Earth’s shape as well as a spherical Earth model

* Modern methods of constructive approximation

* Exercises at the end of each chapter and an appendix with hints to their solutions

Models and methods presented show how various large- and small-scale processes may be addressed by a single geoscientific modelling framework for potential determination. Multiscale Potential Theory may be used as a textbook for graduate-level courses ingeomathematics, applied mathematics, and geophysics. The book is also an up-to-date reference text for geoscientists, applied mathematicians, and engineers.