HEM 3D-inversion

Our problem: How to generate 3D inversion models for HEM data without CPU explosions?

The sub-project SP3 aims at the development of methods for the three-dimensional inversion of helicopter electromagnetic (HEM) data. For this purpose, numerical algorithms have to be developed and implemented. These numerical algorithms utilize a recently developed fast and accurate 3-D forward operator for the reconstruction of the spatial distribution of the main petrophysical parameter, the electrical conductivity.

It is clearly not advisable to invert a complete HEM dataset. Instead, a properly chosen subset of the data has to be defined. To this end, an indicator has to be defined which provides those parts of the data set that cannot be interpreted using conventional 1-D inversion methods. Sub-project SP2 selects parts of the full dataset, which indicate strong lateral variations in the models obtained by a local 1-D interpretation. The reduced partial datasets, which are essentially smaller than the full dataset, will be inverted in a subsequent step by SP3. The obtained local 3-D model will then be integrated into the quasi-1-D models of SP2.

Finally, all obtained 3-D models have to be scrutinized with respect to hydrogeological and sedimentological aspects by SP5.

The 3D inverse problem of HEM data interpretation is underdetermined and ill-posed. Moreover, HEM datasets are spatially incomplete and noisy. State-of-the-art inversion methods are predominantly based on regularized Gauß-Newton methods. The penalty functionals involved that have to be minimized typically comprise linear combinations of a data residual, and a model norm.

The major objective of this research is to decide which variants of known 3-D inversion methods are particularly suited for the interpretation of HEM data.

3D inversion structure Source: Technische Universität Bergakademie Freiberg

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Our solution: Linear equations systems with large, sparse, and structured coefficient matrices!

The solution of the discrete forward problem of electromagnetics leads to linear equations systems with large, sparse, and structured coefficient matrices. Algebraically, the movement of the EM transmitter towed by the helicopter is expressed by multiple right-hand sides of the linear equation systems. The number of column vectors of the right-hand side block is equivalent to the number of discrete transmitter positions. We note that the coefficient matrix remains unchanged. There exist several numerical techniques to solve that kind of equation systems. Among the most important are iterative solvers, which seem attractive due to their low memory requirement when 3-D large scale problems are considered. Recently, multigrid methods become an interesting alternative to the generally slowly convergent iterative methods. We plan to solve systems with multiple right-hand sides with Krylov methods and their block-related variants.

For the solution of the linearized sub-problem, regularized Gauß-Newton methods require the construction of the sensitivity matrix of partial derivatives of the model response with respect to small model parameter perturbations. However, in the case of 3-D inversion, this matrix can only be computed numerically. The most important methods for the construction of the sensitivities are known as adjoint-field and adjoint-equation methods. Both approaches will be investigated with respect to efficiency and applicability.

Finally, we will study variants of regularized Krylov methods. Of great scientific importance will be the choice of the regularisation parameter as well as the selection of efficient numerical methods for the solution of the normal equation system at each iteration of the inverse process.