Development of numerical algorithms

Gravity field modelling comprises the processing of heterogeneous types of data and the estimation of gravity field and nuisance parameters using least-squares techniques. Efficient algorithms are a pre-requisite not only for least-squares parameter estimation, but also for outlier detection algorithms, noise modelling, error propagation, and validation. The Physical and Space Geodesy group develops methods, algorithms and software to solve a number of essential problems in gravity field modelling from a large number of terrestrial, airborne and space-borne data:

Optimal data weighting of observation groups and outlier detection.  The variance component estimation (VCE) technique is a powerful technique for optimal data weighting. The application to huge amount of satellite, airborne and ground-based data requires Monte Carlo techniques for stochastic trace estimation. Statistical methods allow the detection of outliers in data sets. Optimal data weighting and outlier detection are mandatory for high-accuracy gravity field modelling.

Frequency-dependent data weighting. If data noise is coloured (frequency-dependent), which is typical for airborne and space-borne sensors, and the number of data is large, the least-squares adjustment becomes problematic. The problem can be solved using ARMA (Auto-regressive Moving Average) filtering in combination with the Pre-Conditioned Conjugate Gradient (PCCG) method. It is rigorous and does not introduce any edge effects when applied to data with gaps. The method has been applied successfully to the processing of satellite and airborne gravity data.

Numerically efficient error propagation. Explicit error propagation in the context of large-scale problems is an extremely time-consuming operation. Approximate error propagation techniques, e.g. the Gibbs sampler, in combination with an optimal parallelization of the algorithms offer a solution to this problem.

Fast spherical harmonic synthesis and co-synthesis. Among the most time-consuming operations when computing spherical harmonic coefficients by least-squares adjustment are the application of the design matrix and its transpose to a vector (synthesis and co-synthesis). A combination of Fast Fourier Techniques and 3D spline interpolation offers an extremely fast and efficient solution. Similar approaches can also be applied to radial basis functions and wavelet representations of gravitational fields.