On the adequacy of identified Cole–Cole models

Abstract

The Cole–Cole model has been widely used to interpret electrical geophysical data. Normally an iterative computer program is used to invert the frequency domain complex impedance data and simple error estimation is obtained from the squared difference of the measured (field) and calculated values over the full frequency range. Recently a new direct inversion algorithm was proposed for the ‘optimal’ estimation of the Cole–Cole parameters, which differs from existing inversion algorithms in that the estimated parameters are direct solutions of a set of equations without the need for an initial guess for initialisation. This paper first briefly investigates the advantages and disadvantages of the new algorithm compared to the standard Levenberg–Marquardt “ridge regression” algorithm. Then, and more importantly, we address the adequacy of the models resulting from both the “ridge regression” and the new algorithm, using two different statistical tests and we give objective statistical criteria for acceptance or rejection of the estimated models. The first is the standard χ 2 technique. The second is a parameter-accuracy based test that uses a joint multi-normal distribution. Numerical results that illustrate the performance of both testing methods are given. The main goals of this paper are (i) to provide the source code for the new ‘‘direct inversion’’ algorithm in Matlab and (ii) to introduce and demonstrate two methods to determine the reliability of a set of data before data processing, i.e., to consider the adequacy of the resulting Cole–Cole model.