SSP 1998 Project Summary:

Application of BICG method to large satellite magnetic datasets

Student

Supervisors

Douglas Smith, EPCC

Rob Baxter, EPCC

(a) downward-continue the field to the Earth's surface, where more detail can be discerned
or
(b) model the distribution of magnetization that causes the observed anomalies.

Both methods have a multitude of solutions that fit the data; following standard practise in geophysics, we find the smoothest such model. This gives the problems a similar mathematical formulation, which involves solving a data-by-data system of linear equations. An exact solution is impractical for large satellite datasets, but only a small proportion of the coefficients of the linear equation are numerically significant. Thus, we can apply approximate solution methods such as the biconjugate gradient algorithm. This is an iterative, and therefore time consuming, method for large datasets on a single processor; the main aim of the project, therefore, is to develop a parallelized algorithm.

Previous solutions to the downward-continuation problem have involved either a singular value decomposition of the matrix relating model coefficients to data, retaining only the eigenvectors associated with the largest eigenvalues in the solution, or finding an 'almost optimally smooth' solution, using an expansion of the model at only a subset of the data points (and hence solving a smaller system of equations). Both methods work satisfactorily on continental-sized areas, but cannot provide global solutions. One method of modelling the magnetization of crustal rocks is to divide the crust into blocks, put a hypothetical dipole in the centre of each block and then solve for the dipole moment that best fits the data; averaging the dipole moment over the volume of the block then gives the magnetization. Models with one dipole per satellite datum have been obtained using the biconjugate gradient algorithm. The matrix structure for this 'equivalent dipole' and our optimally smooth model is similar, so we are confident that the same approach will work for our modelling method.

The primary goal of the project is to produce a parallelized algorithm to model either the downward-continued field or the distribution of magnetization in the crust; the same algorithm will solve both problems. Investigation of preconditioning the system to speed up the convergence will be investigated. Further work may include incorporating aeromagnetic data into the solution - these are collected closer to the Earth's surface, over small patches. Matching adjacent surveys flown at different times and with different parameters has been a perennial problem in their interpretation; we propose using the satellite data to achieve this. By using an iterative, parallelized technique, it should be possible to preserve the resolution of the aeromagnetic data. The two types of data have vastly different amplitudes, due to being collected at different heights above the sources; thus, a tricky part of this is getting the relative weights of the two types of data correct, so that neither dominates the solution. The matching process involves 'warping' the longer wavelength content of the surveys, which in turn distorts the source depths inferred from them that are a crucial part of interpretation for mineral and hydrocarbon exploration. The project could also investigate the differences in the source depths obtained from the 'raw' aeromagnetic data, and those that have been integrated with satellite data.

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