Relationship between Cole-Cole model parameters in permittivity and conductivity formulation

Summary For the analysis of spectral induced polarization (SIP) measurements and for the description of frequency-dependent electrical relaxation responses, so-called Cole-Cole models (CCMs) are widely used. Typically, CCM formulations in terms of complex electrical conductivity or complex electrical resistivity are used in geophysical applications. The differences between these model descriptions, in particular between the respective time constants, and their conversion have been studied. A third variant of the model is formulated in terms of complex permittivity, commonly used in materials science. In general, all these model formulations can be used equivalently for fitting SIP data, which, however, results in differing values for some of the model parameters. For a meaningful comparison of CCM parameters of different samples or measurements, it is necessary that they are based on the same model formulation. In this work, the relationships between the Debye model (DM) and CCM parameters in the formulation for complex permittivity and complex conductivity are studied. A direct analytical conversion is possible for generalized DM formulations, both in single- and multi-term model formulations, resulting in relationships between the respective relaxation time distributions (RTDs). Such a direct conversion for CCM formulations is not possible. We however derived an approximate relationship between log -normal RTD and CCM formulations and respective permittivity and conductivity parameter values. Our study also highlights the significance of using consistent model formulations when experimental data are compared in terms of DM or CCM parameters, as parameters used to predict ice temperature are incorrect if the conductivity time constant is used to predict the temperature from interpolation of a permittivity time constant-temperature relationship.