Kramers-Kronig relations measurement model

The measurement model method for distinguishing between bias and stochastic errors is based on using a generalized model as a filter for nonreplicacy of impedance data. The measurement model is composed of a superposition of line-shapes that can be arbitrarily chosen subject to the constraint that the model satisfies the Kramers-Kronig relations. The model presented in Figure 21.8, composed of Voigt elements in series with a solution resistance, i.e.. 

The optical constants n and k are not independent if k varies strongly, so must n. Either n must be measured by some other method or a theory of optical constants that couples them together properly must be used, such as the oscillator model (9.25) or the Kramers-Kronig relations [(2.49), (2.50)]. 

Bias errors are systematic errors that do not have a mean value of zero and that cannot be attributed to an inadequate descriptive model of the system. Bias errors can arise from instrument artifacts, parts of the measured system that are not part of the system under investigation, and nonstationary behavior of the system. Some types of bias errors lead the data to be inconsistent with the Kramers-Kronig relations. In those cases, bias errors can be identified by checking the impedance data for inconsistencies with the Kramers-Kronig relations. As some bias errors are themselves consistent with the Kramers-Kronig relations, the Kramers-Kronig relations cannot be viewed as providing a definitive tool for identification of bias errors. 

The use of measurement models to identify consistency with the Kramers-Kronig relations is equivalent to the use of Kramers-Kronig transformable circuit analogues. An important advantage of the measurement model approach is that it identifies a small set of model structures that are capable of representing a large... 

It should be emphasized that the approach presented in this section is part of an overall assessment of measurement errors. The measurement model is used as a filter for lack of replicacy to obtain a quantitative value for the standard deviation of the measurement as a fvmction of frequency. The mean error identified in this way is equal to zero thus, the standard deviation of the measurement does not incorporate the bias errors. In contrast, the standard deviation of repeated impedance measurements t)q)ically includes a significant contribution from bias errors because perfectly replicate measurements can rarely be made for electrochemical systems. Since the line-shapes of the measurement model satisfy the Kramers-Kronig relations, the Kramers-Kronig relations then can be used as a statistical observer to assess the bias error in the measurement. 

The Kramers-Kronig relations have been applied to electrochemical systems by direct integration of the equations, by experimental observation of stability and linearity, by regression of specific electrical circuit models, and by regression of generalized measurement models. 

Remember 22.3 An insufficient experimental frequency range makes direct integration of the Kramers-Kronig relations problematic. Regression-based approaches, such as use of a measurement model, are pr erred. 

Example 23.2 demonstrates the utility of the error analysis for determining consistency with the Kramers-Kronig relations. In this case, the low-frequency inductive loops were foimd to be consistent with the Kramers-Kronig relations at frequencies as low as 0.001 Hz so long as the system had reached a steady-state operation. The mathematical models that were proposed to account for the low-frequency features were based on plausible physical and chemical hypotheses. Nevertheless, the models are ambiguous and require additional measurements and observations to identify the most appropriate for the system under study. 

P. Agarwal, M. E. Orazem, and L. H. Garcia-Rubio, "Measurement Models for Electrochemical Impedance Spectroscopy 3. Evaluation of Consistency with the Kramers-Kronig Relations," Journal of The Electrochemical Society, 142 (1995)4159-4168. 

This part provides a conceptual understanding of stochastic, bias, and fitting errors m frequency-domain measurements. A major advantage of frequency-domain measurements is that real and imaginary parts of the response must be internally consistent. The expression of this consistency takes different forms that are known collectively as the Kramers-Kronig relations. The Kramers-Kronig relations and their application to spectroscopy measurements are described. Measurement models, used to assess the error structure, are described and compared with process models used to extract physical properties. 

Fig. 5. a - theoretically found spectral dependences of optical constants of porous model medium, due to the presence of "ideal" silver particles (1 = 8 nm) and "real" particles (1 2 nm) y and An are, respectively, absorption constant and refractive index, b -dependence of phase incursion (AnT/X), due to the presence of silver particles in the medium, on silver coverage, C, of the studied sample experimental measurements with the use of developed holography film plates PFG-02 (dots) calculations by measured attenuation spectra with the use of dispersion Kramers-Kronig relations (crosses). 

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