## 3 Results and Discussion
### Distribution Function of Relaxation Times (DFRT) Analysis
The MATLAB application _\"DFRTtoEIS\"_ tool co-developed by some of the current authors [34] was used for obtaining the reconstructed solutions. The difference between the MAP (_Maximum-a-Posteriori_) solution, which maximises the probability density function of the DFRT, and of the mean solution, is also higher for lower \(\lambda\) values (Figure 7a-c). Another interesting feature is that the analytical _Gerischer_ component presents a good match with the two peaks of the _Bayesian_ regression. In contrast, the last peak, appearing for higher \(\tau\)-values (low-frequency region), has a strong deviation between both procedures. In addition, the numerous peaks of this _Bayesian_ solution seem too unrealistic for attributing physical processes separately. However, with the increase of \(\lambda\), the oscillation decreases and the number of peaks, as well. Hence, the best solution was that given by \(\lambda=0.1\), with a good match between the individual time constants and a decreased CI. Figure 7: DFRTs obtained from the _Bayesian_ regularisation of the EIS data obtained in low humidity (\(p_{\rm H2O}\sim 10^{-5}\) atm) conditions at 450 \({}^{\circ}\)C in O\({}_{2}\) atmosphere (the 99% credibility interval is presented in the shadow area around the _Bayesian_ curves) for different regularisation parameters: (**a**) \(\lambda=10^{-3}\), (**b**) \(\lambda=10^{-2}\), (**c**) \(\lambda=0.1\) (FWHM is the full width at half maximum). The reconstructed impedances are plotted together with the original EIS data in (**d**), and the respective residuals in (**e**). Finally, a last approach to obtain a DFRT was based on a EQM with a linear series of \(R\mid\mid\)Q-circuits [44; 45], herein designated as multi-fit procedure (\(m(\text{RQ})_{\text{fit}}\)), which was obtained using a combination of five \(R\mid\mid\)Q-circuits in the series (Figure 6c,d). This method avoids the use of any regularisation parameter, thus, minimising the chance of having extra peaks (oscillations). As a result, for lower time constants (higher frequencies), only one large peak could be obtained in wet conditions. In contrast, two peaks were detected in this region for the data in low humidity conditions. Interestingly, one of these peaks has a characteristic time constant falling outside of the represented \(\tau\)-axis scale. This could be related to some contribution related to the grain boundary of the BCY10 electrolyte support (e.g., due to the very low humidity in this case) or due to the difficulty in fitting the data with five \(R\mid\mid\)Q contributions. Nonetheless, the important point here is that both the _Bayesian_ regression and the \(m(\text{RQ})_{\text{fit}}\) procedure show quite similar behaviour with a broad main peak for higher time constants (lower frequencies), albeit the latter procedure has a slightly sharper peak, which is clearly related to the \(RC\) type behaviour of the _Gerischer_ dispersion at lower frequencies, which should yield a \(\delta\)-function in \(\tau\)-space (i.e., a vertical spike) [45]. The main idea in this discussion is to clearly demonstrate that the impedance reconstructions of the different methodologies are all very close to the original experimental data and different DFRTs. This implies that for any given EIS spectrum there are many possible DFRTs, depending on the employed methodology. It is, thus, very important to put all information together in order to determine the exact number of meaningful physical processes. In this way, it is reasonable to assume that at least three time constants can be expected in low humidity conditions, which is in agreement with those found for the EQM procedure.