Application of Measurement Models to Impedance Spectroscopy III. Evaluation of Consistency with the Kramers-Kronig Relations

The Kramers-Kronig equations and the current methods used to apply them to electrochemical impedance spectra are reviewed. Measurement models are introduced as 'a tool for identification of the frequency-dependent error structure of impedance data and for evaluating the consistency of the data with the Kramers-Kronig relations. Through the use of a measurement model, experimental data can be checked for consistency with the Kramers-Kronig relations without explicit integration of the Kramers-Kronig relations; therefore, inaccuracies associated with extrapolation of an incomplete frequency spectrum are resolved. The measurement model can be used to determine whether the residual errors in the regression are due to an inadequate model, to failure of data to conform to the Kramers-Kronig assumptions, or to noise. This paper is part of a series intended to present the foundation for the application of measurement models to impedance spectroscopy. The basic premise behind this work is that determination of measurement characteristics is an essential aspect of the interpretation of impedance spectra in terms of physical parameters. The importance of the error structure identification for interpretation of impedance measurements has been recognized for some time (see, e.g., Ref. 1-11), but experimental assessment of the error structure was complicated by the difficulty of obtaining truly replicate impedance measurements. Recently, measurement models have been demonstrated to be useful tools for identification of both stochastic and bias contributions to the error structure of impedance spectra, 11-18 and other groups have begun employing the concept for assessing consistency with the Kramers-Kronig relations. 19,2~ In the first paper of this series, 13 it was shown that a measurement model based on Voigt circuit elements can provide a statistically significant fit to typical electrochemical impedance spectra. In the second paper, 21 a method was demonstrated in which the measurement model is used to identify the stochastic component of the frequency-dependent error structure of impedance data, and a preliminary model for the stochastic component of the error was proposed. In this paper of the series we address the use of the measurement model for identification of the bias component of the error structure. This method is placed in context of the current methods used to assess the consistency of impedance data with the Kramers-Kronig relations.