Comparison of different inversion strategies for electrical impedance tomography (EIT) measurements

Abstract

SUMMARY Electrical impedance tomography (EIT) is a promising method to image the frequency-dependent complex electrical conductivity distribution of the subsurface in the mHz to kHz frequency range. In contrast to the well-developed electrical resistivity tomography (ERT) method, the inversion approach for EIT data is less established. Different inversion strategies have been proposed, but the implications of the differences between these methods have not been investigated yet. In this study, we aim to compare four different inversion strategies for EIT measurements. The first strategy (CVI) formulates the inverse problem in the complex number domain and is mathematically the most elegant method. The second strategy (RVI) is the established real-valued inversion method, which decouples the inversion of the real and imaginary parts and completely ignores the complex nature. The third strategy (ALT) is very similar to the RVI strategy in case of small phase angles, but it considers the complex coupling in the forward operator and alternately updates the real and imaginary parts of the model in the case of large phase angles. The fourth and final strategy (CVI+) was newly formulated in this study. It fully considers the complex nature of EIT measurements but separates the treatment of the real and imaginary part in terms of the data weighting and regularization. The different inversion strategies were tested with two synthetic models. The first model has a small phase contrast and the second model has a large phase contrast. In the case of a small phase contrast, the CVI strategy was able to resolve the distribution of electrical conductivity amplitude, but the inversion result for the phase angle was less reliable. The other three strategies presented similar results and the models were well resolved within the expected data misfit. In the case of a model with large phase contrast, only the newly formulated CVI + strategy was able to produce reliable results. It was found that the extremely large phase angle can have a significant influence on the modelled amplitude of data. The cross-sensitivity (i.e. the imaginary part of the sensitivity) that describes the influence on the real part of data due to a change in the imaginary part of model, or that on the imaginary part of data due to a change in the real part of model, provided unique information during the inversion. It was concluded that the CVI + strategy is theoretically the most comprehensive and correct approach for EIT inversion, but that in the case of small phase angles the RVI strategy has the practical advantage that no complex calculations are required, which substantially reduces the required computational effort.