304	RWTH Publication No: 47106        2009        IGPM304.pdf
TITLE	Three-Dimensional Geoelectric Modelling with Optimal Work/Accuracy Rate Using an Adaptive Wavelet Algorithm
AUTHORS	Alain Plattner, Hans-Rudolf Maurer, Jürgen Vorloeper, Wolfgang Dahmen
ABSTRACT	Despite the ever-increasing power of modern computers, realistic modelling of complex three-dimensional Earth models is still a challenging task and requires substantial computing resources. The overwhelming majority of current geophysical modelling approaches includes either finite difference or non-adaptive finite element algorithms, and variants thereof. These numerical methods usually require the subsurface to be discretised with a fine mesh to accurately capture the behaviour of the physical fields. However, this may result in excessive memory consumption and computing times. A common feature of most of these algorithms is that the modelled data discretisations are independent of the model complexity, which may be wasteful when there are only minor to moderate spatial variations in the subsurface parameters. Recent developments in the theory of adaptive numerical solvers have the potential to overcome this problem. Here, we consider an adaptive wavelet-based approach that is applicable to a large range of problems, also including nonlinear problems. To the best of our knowledge such algorithms have not yet been applied in geophysics. Adaptive wavelet algorithms offer several attractive features: (i) for a given subsurface model, they allow the forward modelling domain to be discretised with a quasi minimal number of degrees of freedom, (ii) sparsity of the associated system matrices is guaranteed, which makes the algorithm memory efficient, and (iii) the modelling accuracy scales linearly with computing time. We have implemented the adaptive wavelet algorithm for solving three-dimensional geoelectric problems. To test its performance, numerical experiments were conducted with a series of conductivity models exhibiting varying degrees of structural complexity. Results were compared with a non-adaptive finite element algorithm, which incorporates an unstructured mesh to best fit subsurface boundaries. Such algorithms represent the current state-of-the-art in geoelectric modelling. An analysis of the numerical accuracy as a function of the number of degrees of freedom revealed that the adaptive wavelet algorithm outperforms the finite element solver for simple and moderately complex models, whereas the results become comparable for models with high spatial variability of electrical conductivities. The linear dependence of the modelling error and the computing time proved to be model- independent. This feature will allow very efficient computations using large-scale models as soon as our experimental code is optimised in terms of its implementation.
KEYWORDS	Numerical solutions, wavelet transform, numerical approximations and analysis, non-linear differential equations, electrical properties
DOI	10.1111/j.1365-246X.2010.04677.x
PUBLICATION	Geophysical journal international 182(2), 741-752 (2010)