Maximum Entropy Multi-resolution EM Tomography by Adaptive Subdivision

Li-He Zou, Zhengrong Wang, and Louis E. Roemer
Department of Electrical Engineering
Louisiana Tech Univ, LA 71272, USA


Audio band electromagnetic (EM) waves have a great potential for success in bore hole to bore hole or surface to bore hole tomography for geophysical exploration or environmental tests. Low resolution is generally the major limitation in the EM tomography. If a high resolution grid is used in the model of the transmission medium, the inversion problem in reconstruction turns out to be severely ill-posed. Many artifacts with random patterns will show in the resultant image of the reconstruction if a least square error criterion is applied. The maximum entropy constraint can certainly reduce the artifacts. However, the conflict of high resolution and fewer artifacts still exists. This paper proposes an adaptive procedure which produces a tomography image with different resolution in different subdivisions according to the details the subdivision may possess. This procedure can reduce unnecessary resolution in those areas where no more interesting details are shown while showing high resolution in other areas where interesting details may occur. Thus, the artifacts can be reduced to a minimum. This is a recursive approach. At the first recursive stage, a coarse grid is generated in the model of the medium. A maximum entropy reconstruction image on the grid can be obtained. Since the number of unknowns on the coarse grid is low in comparison with the data size acquired from the test, the result must be robust with high reliability. However, the resolution in this stage is low, to refine the resolution the process comes to the next recursive stage. In this stage, the solution obtained in the previous stage is now applied as initial values for the present reconstruction process on a refined grid. In this grid refinement procedure, each cell splits in each dimension making four subdivisions for a 2-D cell or eight subdivisions for a 3-D cell. The maximum entropy reconstruction algorithm is applied again on this refined grid. A testing procedure is set to monitor the change of the cost function for optimization and to determine whether further refinement is needed for a cell. If the test finds no more resolution in this cell is necessary, the cell will be dropped from the list of cells for further refining. The refining process will continue on the rest of cells in the list and the reconstruction algorithm will start again until the list is empty or a satisfactory reconstruction image is obtained. In the final image, high resolution is shown only in areas with detailed features and low resolution in those with fewer details. Computer simulations on the proposed method compared with least square error method and a conventional maximum entropy method show that the proposed method can produce higher resolution images with significantly reduced artifacts. This technique has been found particularly useful if some prior information about the tested structure is available or special attention should be directed to some region in the tested area. In these cases, a spatial weighting function may be imposed on the cost function. All experimental results are encouraging and show great potential in practical application
MaxEnt 94 Abstracts /