Keywords: scattered data approximation, gridding, irregular sampling, potential fields, gravity, magnetics
Published: PhD Thesis, University of Vienna, May 1998
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ABSTRACT

Geophysical potential-field data are in general observed at scattered sampling points and are contaminated by measuring and preprocessing errors. Analysis and further treatment require a representation of the underlying potential fields on a regular grid.

In the presence of noise, a smooth approximation of the data is more appropriate than an exact interpolation.

Analysis of the smoothness of potential fields indicates an approximation by band-limited functions. The resulting least squares problem can be formulated as a linear system of equations, where the system matrix is of block-Toeplitz type. This special structure allows a computationally attractive and robust solution via the fast Fourier transform and the conjugate gradient algorithm. Based on the work of T.Strohmer, I present a new gridding method that incorporates additional knowledge about the physical properties of potential fields and the statistics of the data.

The new method is compared to five standard algorithms. Synthetic data sets are used to study the influence of the sampling geometry and the sampling error, respectively. Experimental results for real world gravity and magnetic data demonstrate the good performance of the new method for highly irregularly spaced noisy data.

Table of contents:

1. Potential theory in geophysical applications
• Introduction
• Gravity and magnetic fields
• Gravity field
• Magnetic field
• The Fourier transform of the 1/r-function
• Green's Identities
• Boundary value problems
• Laplace's equation and the Fourier transform
• Derivation of the Fourier transform
• Fourier transform of potential field functions
2. Gridding - Review of methods
• Introduction
• Minimum curvature
• The generalized equivalent source technique
• Kriging
• Triangulation based interpolation
• A modified quadratic Shepard's method
3. A new gridding method
• Introduction
• Non-uniform sampling theory
• A discrete model of irregular sampling
• Non-uniform sampling and trigonometric polynomials
• The invertibility of T
• Irregular sampling and block-Toeplitz matrices
• Numerical methods
• Toeplitz and circulant matrices
• The conjugate gradient method
• Non-uniform fast Fourier transform
• Efficient approximation of potential fields
• Model based approximation
• The WCT-method
• Complexity of the WCT-method
• Estimation of the bandwidth
• Remarks on the WCT-method
4. Comparison of gridding methods
• Introduction
• Synthetic data
• The influence of the sampling set
• The influence of noise
• Voronoi and statistical weights
• Numerical aspects
• Real world data
• Gravity data from the Neusiedlersee area
• Magnetic data from the Bad Gleichenberg area
5. Conclusion and future research
6. Appendix
7. References
8. Acknowledgements
9. Curriculum Vitae