P-orthogonal matrices; Generalized polar decomposition; Padé approximants; Matrix square root; Matrix logarithm

Reconstruction of electrical capacitance tomography (ECT) images is performed using simulated gas–oil distributions. An inverse problem has to be solved to find the permittivity coefficient, using measurements of the capacitances. The least squares optimal solution is sought using a Gauss–Newton method, with a sufficient descent condition and a backtracking for the steplength. The Tikhonov regularisation method is used, to control the measurement error propagation due to the ill-posednes of the inverse problem. It is shown that the reconstruction is very sensitive to the Tikhonov regularisation parameter and the L-curve method to find its value is used. When the optimal regularisation parameter is used, convergence is attained to points where no further precision in the permittivity parameter is possible. Simulation examples using typical two-phase flow regimes are presented, and the approximated images as well as the range of values for the regularization parameter for different regimes are shown.