Multiscale Potential Theory: With Applications to Geoscience by Willi Freeden, Volker Michel, Birkhauser

By Willi Freeden, Volker Michel, Birkhauser

This self-contained booklet offers a easy starting place for college students, practitioners, and researchers drawn to a number of the different new components of multiscale (geo)potential thought. New mathematical tools are constructed allowing the gravitational strength of a planetary physique to be modeled and analyzed utilizing a continual movement of observations from land or satellite tv for pc units. Harmonic wavelet tools are brought, in addition to speedy computational schemes and numerous numerical attempt examples.

The paintings is split into major elements: half I treats well-posed boundary-value difficulties of capability thought and elasticity; half II examines ill-posed difficulties equivalent to satellite-to-satellite monitoring, satellite tv for pc gravity gradiometry, and gravimetry. either sections exhibit how multiresolution representations yield Runge–Walsh sort options which are either exact in approximation and tractable in computation.

Topic and key features:

* accomplished assurance of issues which, up to now, are just scattered in magazine articles and convention proceedings

* vital purposes and advancements for destiny satellite tv for pc eventualities; new modelling thoughts concerning low-orbiting satellites

* Multiscale methods for various geoscientific difficulties, together with geoidal decision, magnetic box reconstruction, deformation research, and density version modelling

* Multilevel stabilization methods for regularization

* remedy of the genuine Earth’s form in addition to a round Earth model

* smooth equipment of confident approximation

* workouts on the finish of every bankruptcy and an appendix with tricks to their solutions

Models and strategies offered convey how quite a few huge- and small-scale tactics might be addressed through a unmarried geoscientific modelling framework for capability selection. Multiscale capability Theory can be utilized as a textbook for graduate-level classes in geomathematics, utilized arithmetic, and geophysics. The booklet is additionally an up to date reference textual content for geoscientists, utilized mathematicians, and engineers.

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Additional info for Multiscale Potential Theory: With Applications to Geoscience

Sample text

2 Spherical Harmonics Spherical harmonics are the functions most commonly used to represent scalar fields on a spherical surface. They are used extensively in the gravitational and magnetic applications involving Laplace's equation. 3 (without proofs) will occupy us in this subsection. Important results of the theory of spherical harmonics are the addition theorem and the formula of Funk and Hecke. The properties of closure and completeness tell us that any member of a large class of functions may be approximated in terms of spherical harmonics.

2. Spherical Nomenclature where the component functions li are given by li(~) = I(~) . e i , ~ E O. 12(0) denotes the space consisting of all square-integrable vector fields on O. 64) I,g E 12(0); 12(0) is a Hilbert space. The space c(p)(O), 0 ~ P ~ 00, consists of all p-times continuously differentiable vector fields on O. For brevity, we write c(O) = c(O)(O). The space c(O) is complete with respect to the norm 1I/IIc([I) 1 E c(O). 66) where the completion is understood with respect to the 12(0)-topology.

TJ), TJ E 0, is called a ~-zonal function on 0 (or ~-axial radial basis function). Zonal functions are constant on the sets M(~jh) = {TJ E Ol~' TJ = h}, h E [-1, +1]. 3). 61) and IIGIILP[-l,+1] = IIG(c3 ')IILP(n) = (L IG(TJ . 62) = (27f [llIG(t)IP dt) liP. Analogously, we define the inner product in L 2 [-1, 1] by (F, G)L2[_1,1] = 27f [11 F(t)G(t) dtj F, G E L 2 [-1, 1]. Next we give some preliminaries for the study of vector fields defined on the unit sphere O. 2. Spherical Nomenclature where the component functions li are given by li(~) = I(~) .

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