Forward and inverse resistivity modelling on complex three dimensional structures using the finite element method
Modélisation directe et inverse en prospection électrique sur les structures 3D complexes par la méthode des éléments finis
Résumé
This work presents the adaptation and the use of the CESAR-LCPC finite element code for the forward and inverse modelling of 3D resistivity data. These codes are better suited for imaging structures with complex geometries.
The forward modelling code uses an electrode-independent mesh that allows to place the electrodes at their exact locations and to use a coarse mesh at the same time. In this approach, the choice for the mesh size is solely governed by the need for accurate results. It is also possible to calculate apparent resistivities, without the use of the geometrical factor (that can be evaluated only for simple structures). To calculate apparent resistivity values, a normalisation approach is used that gives significantly better results than the use of the geometrical factor and allows the modelling of any kind of complex 3D structure. As a singularity removal technique cannot be used on complex 3D models, a minimum of 5 to 6 nodes between two current transmitting electrodes should be considered to guarantee the quality of the results. Synthetic results are presented to illustrate the efficiency of the forward modelling technique.
An inversion code was also presented for the processing of resistivity tomographies on complex 3D structures using any electrode arrangement. This algorithm is well suited for the processing of large data sets with a lot of unknown model parameters. The inversion code uses an original strategy to avoid the explicit calculation of a sensitivity matrix. The adjoint-state of the potential field is used to minimize an objective function for the electrical inverse problem. Then, a steepest descent formulation can be used for the first iteration. Further iterations are carried out using a conjugate gradient approach to improve the convergence. As can be seen on synthetic data, a satisfactory reconstruction of the models can be achieved with a minor computational cost. This kind of inverse problem would have been very difficult to solve using a more traditional Gauss-Newton approach. Strategies are nevertheless needed to improve the stabilisation of the inverse process and to include a priori information in the problem.
Finally, a ROI (Region Of Investigation) index method is used to assess whether features in the model are caused by the data or are artefacts of the inversion process. This method carries out two inversions of the same data set using different values of the reference resistivity model. The two inversions reproduce the same resistivity values in areas where the data contain information about the resistivity of the subsurface whereas the final result depends on the reference resistivity in areas where the data do not constrain the model.
Ce travail a pour objectif la mise au point d'un ensemble d'outils de modélisation directe et inverse en utilisant le code d'éléments finis CESAR-LCPC. Ces outils sont adaptés aux données électriques collectées sur des structures 3D à géométrie complexe. Pour le problème direct, un programme utilitaire servant d'interface avec le solveur CESAR a été créé afin de modéliser des séquences de mesures électriques (tomographies). Afin de pouvoir inverser un nombre important de données sur des modèles de grandes dimensions, une fonction objectif est minimisée en utilisant la technique de l'état adjoint. Cette approche est originale car elle vise le calcul direct de la variation à apporter aux paramètres du modèle, sans évaluation explicite de la matrice de sensibilité. Des données synthétiques ont été utilisées pour valider cet algorithme d'inversion. La fiabilité des modèles inversés est testée en utilisant une méthode de calcul de l'indice ROI (Region of Investigation).
Origine : Fichiers produits par l'(les) auteur(s)
Loading...