3-D electrical resistivity tomography using adaptive wavelet parameter grids

We present a novel adaptive model parametrization strategy for the 3-D electrical resistivity tomography problem and demonstrate its capabilities with a series of numerical examples. In contrast to traditional parametrization schemes, which are based on fixed disjoint blocks, we discretize the subsurface in terms of Haar wavelets and adaptively adjust the parametrization as the iterative inversion proceeds. This results in a favourable balance of cell sizes and parameter reliability, that is, in regions where the data constrain the subsurface properties well, our parametrization strategy leads to a fine grid, whereas poorly resolved areas are represented only by a few large blocks. This is documented with eigenvalue analyses and by computing model resolution matrices. During the initial iteration steps, only a few model parameters are involved, which reduces the risk that the regularization dominates the inversion. The algorithm also automatically accounts for non-linear effects caused by pronounced conductivity contrasts. Inside conductive features a finer grid is generated than inside more resistive structures. The automated parameter adaptation is computationally efficient, because the coarsening and refinement subroutines have a nearly linear numerical complexity with respect to the number of model parameters. Because our approach is not tightly coupled to electrical resistivity tomography, it should be straightforward to adapt it to other data type

Three-dimensional geoelectric modelling with optimal work/accuracy rate using an adaptive wavelet algorithm

Despite the ever-increasing power of modern computers, realistic modelling of complex 3-D 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 discretized 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 discretizations 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. In comparison with earlier applications of adaptive solvers to geophysical problems we employ here a new adaptive scheme whose core ingredients arose from a rigorous analysis of the overall asymptotically optimal computational complexity, including in particular, an optimal work/accuracy rate. Our adaptive wavelet algorithm offers several attractive features: (i) for a given subsurface model, it allows the forward modelling domain to be discretized 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 3-D 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-fitting 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 optimized in terms of its implementatio

3-D electrical resistivity tomography using adaptive wavelet parameter grids

ISSN:0956-540XISSN:1365-246

Three-dimensional geoelectric modelling with optimal work/accuracy rate using an adaptive wavelet algorithm

ISSN:1365-246XISSN:0956-540

Adaptive wavelet methods basic concepts and applications to the Stokes problem

SIGLEAvailable from TIB Hannover: RN 8680(215) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Multilevel preconditioners for discretizations of elliptic boundary value problems - experiences with different software packages

SIGLEAvailable from TIB Hannover: RN 8680(194) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

The multilevel library software tools for multiscale methods and wavelets. Version 2.1, documentation

SIGLEAvailable from TIB Hannover: RN 8680(205) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekVolkswagen Stiftung, Hannover (Germany); European Commission (CEC), Brussels (Belgium)DEGerman

The multilevel library software tools for multiscale methods and wavelets. Version 2.1, documentation

SIGLEAvailable from TIB Hannover: RN 8680(205) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekVolkswagen Stiftung, Hannover (Germany); European Commission (CEC), Brussels (Belgium)DEGerman