Resolution, stability and efficiency of resistivity tomography estimated from a generalized inverse approach

Abstract

Inconsistencies between an object and its image delivered by tomographical methods are inevitable. Loss of information occurs during the survey through incomplete and inaccurate data sampling and may also be introduced during the inverse procedure by smoothness constraints inadequate to the resolving power of the experimental setup. A quantitative appraisal of image quality (spatial resolution and image noise) is therefore not only required for successful interpretation of images but can be used together with measures of efficiency of the experimental design to optimize survey and inverse procedures. This paper introduces a low-contrast inversion scheme for electrical resistivity tomography that supports the reconstructed image with estimates of model resolution, model covariance and data importance. The algorithm uses a truncated pseudo-inverse and a line search approach to determine the maximum number of degrees of freedom necessary to fit the data to a prescribed target misfit. Though computationally expensive, the virtue of the method is that it reduces subjectivity by avoiding any empirically motivated model smoothness constraints. The method can be incorporated into a full non-linear inversion scheme for which a posteriori quality estimates can be calculated. In a numerical 2-D example the algorithm yielded reasonable agreement between object and image even for moderate resistivity contrasts of 10:100:1000. On the other hand, the resolving power of an exemplary four-electrode data set containing classical dipole-dipole and non-conventional configurations was shown to be severely affected by data inaccuracy. Insight into the resolving power as a function of space and data accuracy can be used as a guideline to designing optimized data sets, smoothness constraints and model parametrization.