Approximate continuation of harmonic functions in geodesy : a weighted least-squares approach based on splines with extension to the multiscale adaptive case / Gabriela Jager. 2010
Inhalt
- Introduction
- Harmonic Functions
- Preliminaries
- Some Basic Notation
- Harmonic Functions in Natural Sciences
- Spherical Harmonics
- Orthogonal Expansions: Laplace Series
- The Earth's Gravitational Field
- Preliminaries
- Fundamental Relationships
- Gravity Acceleration and Gravity Potential
- Differential and Integral Formulas for the Gravity Potential
- Geometry of the Gravity Field
- Classical Gravity Field Model: The Spherical Harmonics Representation
- Determination of the Model
- Coefficients Significance
- Convergence Issues
- Deficiencies of the Spherical Harmonic Representation
- Boundary Problems in Geodesy
- New Approach: Least Squares with Regularization
- General Setup, Justification
- Construction of the Solution
- Concrete Choices
- Computing the Solution
- Alternative Formulations for Comparison Purposes
- Two Dimensional Illustration of the Method
- From the Boundary Data to the System of Equations
- Variation of the Weight Parameter vs. Boundary Data Configurations
- Variation of the Basis Cardinality vs. Strongly Incomplete Boundary Data
- Numerical Results
- Algorithm
- Numerical Setup
- Experiments with Synthetic Harmonic Functions
- Test Data
- Variation of the Boundary Data Set
- Comparison to Finite Differences
- Variation of the Weight Parameter
- Experiments with Earth's Potential Field Data
- Constraints of the Least-Squares-Approach for Potential Field Datasets
- Estimating the Weight Parameter
- Working with Iterative Solvers
- Adaptivity: Experiments with the Hierarchical Cubic B-spline Basis
- Prerequisites
- Background on Multivariate and Multiscale Constructions
- Multiscale Constructions
- General Adaptive Set-up
- Alternative Refinement Strategies
- Conclusions
- Separable Solution of the Laplace Equation
- Notation
- Bibliography