Equivalence of relaxation time distribution in spectral induced polarization

SUMMARY Decomposing spectral responses in induced polarization on the basis of elementary Debye relaxation kernels with a distribution of time constants (Relaxation Time Distribution) is a powerful tool for analysing observations in this low-frequency electromagnetic method. Notably, it enables the estimation of the sizes of polarization sites, particularly in the presence of metallic particles, as well as facilitating environmental studies. These decompositions generalize a plethora of historical models, some of which can be considered equivalent to each other in the sense of mathematical equivalence classes. Here, we explicit several types of these equivalence relations, which we recall in their definition in relation to a common property, the elements of a given class belong to a given set. For example, we present a class of models that fall under the same differential equation, meaning this is the class of models that belong to the set of distributions that verify the differential equation. We also exhibit another class of models where we can pass from one to the other by an elementary calculation. Among all the possibilities, a particular class often interests us in IP: RTD classes such as spectra are practically indistinguishable as they are so close according to a defined criterion. In this particular case, we study here the equivalence (or non-equivalence) of certain classical models. We confirm that two models play major roles: the lognormal distribution (because it is the most natural) and the Cole–Cole distribution, which is empirical but also often used for its simplicity (and the associated RTD is analytical, unlike that of the lognormal which requires numerical evaluations). It turns out that these two distributions are equivalent in terms of their quasi-equal spectra, a fact known since Cole and Cole, but whose scope is extended here by an in-depth study of the objective function which separates them in the least squares sense.